SS symmetry
Article
Some Fuzzy Inequalities for Harmonically s-Convex Fuzzy
Number Valued Functions in the Second Sense Integral
Jorge E. Macías-Díaz 1,2, * , Muhammad Bilal Khan 3, * , Hleil Alrweili 4 and Mohamed S. Soliman 5
1
2
3
4
5
*
Departamento de Matemáticas y Física, Universidad Autónoma de Aguascalientes, Avenida Universidad 940,
Ciudad Universitaria, Aguascalientes 20131, Mexico
Department of Mathematics, School of Digital Technologies, Tallinn University, Narva Rd. 25,
10120 Tallinn, Estonia
Department of Mathematics, COMSATS University Islamabad, Islamabad 44000, Pakistan
Department of Mathematics, Faculty of Art and Science, Northern Border University, Rafha, Saudi Arabia
Department of Electrical Engineering, College of Engineering, Taif University, P.O. Box 11099,
Taif 21944, Saudi Arabia
Correspondence: jemacias@correo.uaa.mx (J.E.M.-D.); bilal42742@gmail.com (M.B.K.)
Abstract: Many fields of mathematics rely on convexity and nonconvexity, especially when studying
optimization issues, where it stands out for a variety of practical aspects. Owing to the behavior of its
definition, the idea of convexity also contributes significantly to the discussion of inequalities. The
concepts of symmetry and convexity are related and we can apply this because of the close link that
has grown between the two in recent years. In this study, harmonic convexity, also known as harmonic
s-convexity for fuzzy number valued functions (F-NV-Fs), is defined in a more thorough manner. In
this paper, we extend harmonically convex F-NV-Fs and demonstrate Hermite–Hadamard (H.H) and
Hermite–Hadamard Fejér (H.H. Fejér) inequalities. The findings presented here are summaries of a
variety of previously published studies.
Citation: Macías-Díaz, J.E.; Khan,
M.B.; Alrweili, H.; Soliman, M.S.
Some Fuzzy Inequalities for
Keywords: harmonically s-convex fuzzy number valued function in the second sense; Hermite–Hadamard
inequality; Hermite–Hadamard Fejér inequality
Harmonically s-Convex Fuzzy
Number Valued Functions in the
Second Sense Integral. Symmetry
2022, 14, 1639. https://doi.org/
10.3390/sym14081639
Academic Editors: José Carlos
R. Alcantud and Calogero Vetro
Received: 4 July 2022
Accepted: 2 August 2022
Published: 9 August 2022
Publisher’s Note: MDPI stays neutral
with regard to jurisdictional claims in
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Copyright: © 2022 by the authors.
Licensee MDPI, Basel, Switzerland.
This article is an open access article
distributed under the terms and
conditions of the Creative Commons
Attribution (CC BY) license (https://
creativecommons.org/licenses/by/
4.0/).
1. Introduction
A variety of scientific fields, including mathematical analysis, optimization, economics,
finance, engineering, management science, and game theory, have greatly benefited from
the active, interesting, and appealing field of convexity theory study. Numerous scholars
study the idea of convex functions, attempting to broaden and generalize its various
manifestations by using cutting-edge concepts and potent methods. Convexity theory
offers a comprehensive framework for developing incredibly effective, fascinating, and
potent numerical tools to approach and resolve a wide range of issues in both pure and
practical sciences. Convexity has been developed, broadened, and extended in several
sectors during recent years. Inequality theory has benefited greatly from the introduction of
convex functions. Numerous studies have shown strong connections between the theories
of inequality and convex functions.
Functional analysis, physics, statistics theory, and optimization theory all benefit from
integral inequality. Only a handful of the applications of inequalities in research [1–8]
include statistical difficulties, probability, and numerical quadrature equations. Convex
analysis and inequalities have developed into an alluring, captivating, and attentiongrabbing topic for researchers as a result of various generalizations, variants, extensions,
wide-ranging perspectives, and applications; the reader can refer to [9–15]. Kadakal
and Iscan recently presented n-polynomial convex functions, which are an extension of
convexity [16].
A well-known particular instance of the harmonic mean is the power mean. In areas
such as statistics, computer science, trigonometry, geometry, probability, finance, and
Symmetry 2022, 14, 1639. https://doi.org/10.3390/sym14081639
https://www.mdpi.com/journal/symmetry
Symmetry 2022, 14, 1639
2 of 18
electric circuit theory, it is frequently employed when average rates are sought. Since it
equalizes the weights of each piece of data, the harmonic mean is the ideal statistic for rates
and ratios. The harmonic convex set is defined by the harmonic mean. Shi [17] was the first
to present the harmonic convex set in 2003. The harmonic and p-harmonic convex functions
were introduced and explored for the first time by Anderson et al. [18] and Noor et al. [19],
respectively. Awan et al. [20] introduced a new class known as n-polynomial harmonically
convex function while maintaining their focus on extensions.
We learned that there is a certain class of function known as the exponential convex
function and that there are many people working on this subject currently [21,22]. This
information was motivated and encouraged by recent actions and research in the field
of convex analysis. Convexity of the exponential type was described by Dragomir [23].
Dragomir’s work was continued by Awan et al. [24], who investigated and examined
an entirely new family of exponentially convex functions. Kadakal and İşcan presented
a fresh idea for exponential type convexity in [25]. The idea of n-polynomial harmonic
exponential type convex functions was recently put out by Geo et al. [26]. In statistical learning,
information sciences, data mining, stochastic optimization, and sequential prediction [27,28],
and the references therein, the benefits and applications of exponential type convexity
are used.
Additionally, symmetry and inequality have a direct relationship with convexity. Because of their close association, whatever one we focus on may be applied to the other,
demonstrating the important relationship between convexity and symmetry, see [29]. The
traditional ideas of convexity have been successfully extended in a number of instances.
Weir and Mond, for instance, developed the category of preinvex functions. The concept of h-convexity, which also encompasses several other types of convex functions, was
first presented by Varosanec [30]. Additionally, Varosanec has discovered a few h-convex
function-related classical inequalities. Noor et al. [31] introduced the class of h-preinvex
functions and pointed out that by considering multiple viable options for the real function
h, other classes of preinvexity and classical convexity may be retrieved. In a similar work,
they also created a number of entirely new Dragomir–Agarwal and Hermite–Hadamard
inequalities. Cristescu et al. [32] were the first to introduce the class of (h1, h2)-convex
functions, which they also investigated and covered some of its key aspects. By combining ideas from interval analysis and convex analysis, Zhao et al. [33] created a class
of interval-valued h-convex functions and created several entirely new iterations of the
Hermite–Hadamard inequality.
In order to construct the HH inequalities for harmonic convex functions, Iscan [34]
first developed the idea of a harmonic convex set. By defining the Harmonic h-convex
functions on the Harmonic convex set and expanding the HH inequalities that Iscan [34]
developed, Mihai [35] advanced the concept of harmonic convex functions.
Keep in mind that fuzzy mappings are fuzzy functions with numeric values. On the
other hand, Nanda and Kar [36] were the first to introduce the idea of convex F-NV-Fs.
To this one step further, Khan et al. [37,38] recently proposed h-convex F-NV-Fs and
(h 1,h 2)-convex F-NV-Fs and obtained some to offer new iterations of HH and fractional type of inequalities by employing fuzzy Riemann–Liouville fractional integrals
and fuzzy Riemannian integrals, respectively. Similarly, using fuzzy order relations and
fuzzy Riemann–Liouville fractional integrals, Sana and Khan et al. [39] developed new
iterations of fuzzy fractional HH inequalities for harmonically convex F-NV-Fs. We direct
the readers to [40–73] and the references therein for further information on generalized
convex functions, fuzzy intervals, and fuzzy integrals.
We introduced a new generalization of harmonically convex functions known as
harmonic s-convex F-NV-Fs in the second sense by fuzzy order relation. This work was
motivated and inspired by ongoing research. We developed new iterations of the HH
inequalities utilizing Riemann–Liouville fractional operators with the help of this class. In
addition, we explored the applicability of our study in rare instances.
It is a partial order
It is a relation.
partial order relation.
We now use mathematical
We now use mathematical
operations to operations
examine some
to examine
of the characteristics
some of the characteristics
of fuzzy
of fuzzy
numbers. Let numbers.
𝔵, 𝔴 ∈ 𝔽 Let
𝔽 then
and𝔵,𝜌𝔴∈∈ℝ,
andwe
𝜌∈
have
ℝ, then we have
[𝔵+𝔴]
Symmetry 2022, 14, 1639
[𝔴] =
= [𝔵] [𝔵+
+ 𝔴]
, [𝔵] + [𝔴] ,
(5)
(5)
[𝔵 × 𝔴] = [𝔵][𝔵 × [𝔴]
𝔴] =
, [𝔵] × [ 𝔴] ,
(6)
3 of(6)
18
(7)
(7)
[𝜌. 𝔵]
= 𝜌. [𝔵][𝜌. 𝔵]
= 𝜌. [𝔵]
2. Preliminaries
Definition 1. Definition
[49] A
1. number
[49]
A fuzzy
valued
number
𝔖valued
:notations
𝐾 ⊂ ℝmap
→ and
𝔽𝔖:is
𝐾
⊂ ℝ →F-NV-F.
𝔽 We
called
is called
For each
F-NV-F.
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begin
by recalling
themap
basic
definitions.
define
intervalFor
as each
𝜑 ∈ (0, 1] whose
𝜑 ∈ (0,
𝜑 1]
-cuts
whose
define𝜑 -cuts
the family
defineofthe
I-V-Fs
family
𝔖 of
: 𝐾 I-V-Fs
⊂ℝ→𝔖
𝒦 : 𝐾are
⊂ℝ
given
→ 𝒦 byare given by
∗ 𝔖
∗ the𝜑end
∗ (𝒾,
∗ (𝒾,
(𝒾)𝔖=
[𝔖
𝔖 (𝒾) = [𝔖∗ (𝒾,
𝔖 𝜑),
𝜑),
for
all
∈
∈
each
𝜑 ∈Xfor
each
(0,point
1] Xthe
X∗ (𝒾,
∈ 𝐾.
R for
: Here,
X∗all
≤ 𝒾for
≤ 𝐾.
X∗Here,
and
, X1]
∈ R },∈where
≤end
X∗ .point real
[𝜑)]
]=
{ 𝒾 𝜑)]
∗, X
∗(0,
∗real
functions 𝔖∗ (.functions
, 𝜑), 𝔖∗ (. ,𝔖
𝜑):
, 𝜑),
→𝔖
ℝ∗ (.are
, 𝜑):
called
𝐾 → lower
ℝ areand
called
upper
lower
functions
and upper
of 𝔖
functions
.
of 𝔖.
∗ (.𝐾
The collection of all closed and bounded intervals of R is denoted and defined as
∗ ] : X , X∗ ∈ R and X ≤ X∗ }. If X ≥ 0, then [X , X∗ ] is called positive
K[49]
X[49]
{[X𝔖∗ :,2.[𝜇,
∗ℝ →
∗ 𝔽 beThen
∗
∗ 𝔖integral
C =Let
anℝF-NV-F.
an F-NV-F.
fuzzy
integral
Then fuzzy
of
over of 𝔖 over
Definition 2.Definition
𝜈] ⊂Let
𝔖:𝔽[𝜇, be
𝜈] ⊂
→
+
+
interval.
The𝔖set
allgiven
positive
is denoted
[𝜇, 𝜈], denoted
[𝜇,
(𝒾)𝑑𝒾
(𝐹𝑅)
by
𝜈], (𝐹𝑅)
denoted
by of
𝔖(𝒾)𝑑𝒾
, is
levelwise
,intervals
is given
by levelwise
byby KC and defined as KC =
{[X∗ , X∗ ] : [X∗ , X∗ ] ∈ KC and X∗ ≥ 0}.
We now examine some of the properties of intervals using arithmetic operations. Let
∗ ], (𝐹𝑅)
∗𝔖
(𝒾)𝑑𝒾
(𝒾)𝑑𝒾
(𝐹𝑅)[X∗ , 𝔖X(𝒾)𝑑𝒾
(𝒾)𝑑𝒾
= ρ(𝐼𝑅)
𝔖 𝔖(𝒾,
= : 𝔖(𝒾,
𝔖(𝒾,
𝜑)𝜑)𝑑𝒾
∈ ℛ([: 𝔖(𝒾,
, ∈ (8)
ℛ([ , ], ) , (8)
, ], )𝜑)
KC𝔖and
∈=R, then
we𝜑)𝑑𝒾
have
[w∗ , =w(𝐼𝑅)
]∈
∗
∗
w∗ ],
[X∗ℛ, ([X∗,the
]],+)collection
[w
for all 𝜑 ∈ (0,for
1],all
where
𝜑 ∈ (0,
ℛ([1],, ],where
denotes
the
of[X
Riemannian
collection
integrable
Riemannian
funcintegrable func∗, w ] =
∗ + w∗ , Xof+
) denotes
(𝐹𝑅)[𝜇, 𝜈]
(𝒾)𝑑𝒾that,
tions of I-V-Fs.
tions𝔖 ofis I-V-Fs.
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𝔖 is 𝐹𝑅over
-integrable
[𝜇, 𝜈] ifover
𝔖(𝒾)𝑑𝒾
if ∗ (𝐹𝑅)
∈ 𝔽 . 𝔖Note
∈
𝔽
.if Note that, if
min{X∗ w∗ , X w∗ , X∗ w∗ , X∗ w∗ },
∗
∗
∗
∗
X ]Lebesgue-integrable,
× [w∗ , then
w ] =𝔖 is a fuzzy
[X∗ ,are
𝜑), Lebesgue-integrable,
𝔖 (𝒾, 𝜑)
then 𝔖
Aumann-integrable
is a fuzzy Aumann-integrable
funcfunc𝔖∗ (𝒾, 𝜑), 𝔖 (𝒾,𝔖𝜑)
∗ (𝒾,are
max {X∗ w∗ , X∗ w∗ , X∗ w∗ , X∗ w∗ }
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tion over [𝜇, 𝜈].
tion
over [𝜇,
𝜈].14, x FOR
[ρX∗ , ρX∗ ] if ρ ≥ 0,
∗
ρ.[X∗ , X ]=
∗ , ρX ] if ρ < 0.
Theorem 2. [49]
Theorem
Let 𝔖: [𝜇,
2. [49]
𝜈] ⊂Let
ℝ→
𝔖:𝔽[𝜇, be
𝜈] a⊂F-NV-F
ℝ → 𝔽 whose
be a[ρX
F-NV-F
𝜑-cuts
whose the
𝜑-cuts
family
define
of I-V-Fs
the family of I-V-Fs
∗ define
∗ (𝒾,
∗ (𝒾,
∗ ]=
Proposition
1.∗ (𝒾,
[51]
Proposition
Let
1.
𝔵,𝒾defined
𝔴
[51]
∈𝔽
Let
1..by
[51]
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∈fuzzy
𝔵,. 𝔴
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and
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are
→∗ ]𝒦
by
are
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𝔖 : [𝜇, 𝜈] ⊂ ℝ𝔖→ :𝒦
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, X
, [w𝔖
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∈[𝔖
K
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the
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𝜑)
both
are
𝜑)
both
𝜑 ∈ (0, 1]. Then
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Then 𝔖 is over
𝐹𝑅-integrable
[𝜇, 𝜈] if and
overonly
[𝜇, if
𝜈] 𝔖if∗ (𝒾,
and𝜑)
𝔖∗and
[𝔵]
[𝔵]are
[𝔵]𝜑 ∈
[𝔴]
𝔵only
≼and
𝔴if if
𝔵≼
only
𝔴and
if𝔵if,and
≼𝔖
𝔴(𝒾,
only
if≤
and
if,
only
if,
for
≤all[𝔴]
≤
(1)
[X∗ , X∗ ] ⊆ [w∗ , w∗ ], if and only if w∗ ≤ X∗ , X∗ ≤ w∗ .
[𝜇, 𝜈], then
[𝜇, 𝜈], then
𝑅-integrable over
𝑅-integrable
[𝜇, 𝜈]. Moreover,
over [𝜇,if𝜈].𝔖Moreover,
is 𝐹𝑅-integrable
is over
𝐹𝑅-integrable
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1. [51]
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Let
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Proposition
𝔴𝔵,∈𝔴𝔽∈. 1.
𝔽
[51]
Proposition
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Let
𝔵,We
𝔴𝔵,∈
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∈.[51]
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∈
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on K
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numbers.
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𝔴Let
∈∈
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∈ 𝜑)𝑑𝒾
𝔽𝜌we
𝜌
ℝ,𝔴
then
∈and
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have
then
𝜌 ∈ ℝ,
wethen
havewe hav
(𝒾,
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(𝑅)Let
(𝐹𝑅) 1 [40].
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= 𝔖
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= 𝜑)𝑑𝒾
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, 𝜑)𝑑𝒾
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and
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([ reversed
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, ],𝔖the
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for all 𝒾, 𝒿 ∈ [𝜇,
for
𝜐],all
𝜉∈
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1], ≥where
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𝔖 (𝒾) = [𝔖
𝔖 ∗ (𝒾,
=𝔖
𝜑),[X
𝔖
for
all
𝜑)]
𝒾
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𝐾.
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Here,
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∈
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Here,
each
for
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each
(0,
1]
𝜑
the
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end
1]
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tions
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tions
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of
-integrable
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𝜈]over
if [𝜇,over
𝜈] (6)
if
𝔖[𝜇,
+ [[𝔴]
,, ×
+𝔖[[𝔴]
,,𝐹𝑅 [𝜇,
[𝔵
[𝔵]
× 𝔴]
× 𝔴]
𝔴]
=
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×
=
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𝔴]
[𝜇,
[𝜇, of
is called a harmonically
is called a harmonically
concave function
on tions
function
𝜐]. of [𝔵
on
𝜐].
∗concave
∗ (.
(.
(.
(.
functions
functions
𝔖
𝔖
,
𝜑),
𝔖
,
𝜑):
,
𝜑),
𝐾
𝔖
→
ℝ
,
𝜑):
are
𝐾
called
→
ℝ
are
lower
called
and
lower
upper
and
functions
upper
functions
of
𝔖
.
of
𝔖
.
∗
∗
∗
∗
∗
(𝒾)𝑑𝒾
(𝒾)𝑑𝒾
(𝒾)𝑑𝒾
(𝒾)𝑑𝒾
(𝒾)𝑑𝒾
(𝐹𝑅)
(𝐹𝑅)𝔖(𝒾)𝑑𝒾
(𝐹𝑅)
(𝐹𝑅)𝔖𝔖𝔖
(𝒾)𝑑𝒾
(𝐹𝑅)
(𝒾)𝑑𝒾
(𝒾)𝑑𝒾
(𝐼𝑅)
(𝐼𝑅)
(𝐼𝑅)
𝔖(𝒾)𝑑𝒾= (𝐼𝑅)
𝔖(𝒾)𝑑𝒾
= (𝐼𝑅)
𝔖
=(𝐹𝑅)
==
=
𝔖(𝒾,
𝔖(𝒾,
𝜑)𝑑𝒾
𝔖
𝜑)𝑑𝒾
𝔖
:𝜑),
𝔖(𝒾,
=:𝔖
=
𝔖(𝒾,
𝜑)
=
=(𝐼𝑅)
∈
𝜑)
𝔖(𝒾,
ℛ∈𝔖𝔖(𝒾,
ℛ
𝜑)𝑑𝒾
𝔖
𝜑)𝑑𝒾
:, 𝔖(𝒾,
=
𝔖(𝒾,
=𝜑)
𝔖(𝒾,
∈
𝜑)
ℛ
∈([fuzzy
ℛ,𝜑)𝑑𝒾
: 𝔖(𝒾,
,: 𝔖
(𝒾,
(𝒾,
(𝒾,
(𝒾,
([
],Lebesgue-integrable,
) :,(8)
([
],𝔖
],Au
)a
,(𝒾)𝑑𝒾
,) is
𝜑)𝔖
𝜑),
are
𝔖[𝔵]
Lebesgue-integrable,
𝜑)
are
𝜑)
Lebesgue-integrable,
𝔖(8)
is 𝔖(𝒾,
then
a𝜑)𝑑𝒾
then
𝔖𝔖
∗ (𝒾, 𝜑), 𝔖
∗[𝜌.
From these definitions, we have
(6)
[𝔵𝔖×
[𝔵],𝜌.
[ 𝔴]
×(𝒾,
𝔴]
𝔴]
×
𝔴]
× ([are
,,) ],then
(7)
[𝜌.
[𝔵]
[𝔵]
𝔵] =[𝔵∗=
𝔵][=
𝜌.
=
tion over [𝜇,
tion
𝜈].over
tion
[𝜇,over
𝜈]. [𝜇, 𝜈].
be anThen
F-NV-F.
fuzzy
Then
integral
fuzzyofintegral
𝔖 overof 𝔖
Definition
Definition
2. [49] Let2. 𝔖[49]
: [𝜇,Let
𝜈] ⊂ϕ𝔖ℝ: [𝜇,
→ 𝜈]
𝔽 ⊂beℝ an
→∗ 𝔽F-NV-F.
[𝜌.
[𝜌.
[𝔵]
[𝔵]
𝔵]
𝔵]
=ϕ
𝜌.
𝜌.],denotes
for for
all all
𝜑 ∈𝜑(0,
∈ 1],
(0, 1],
where
for
where
for
allℛall
𝜑
∈, 𝜑
∈,) ],1],
(0,
1],
where
for
all
where
all
𝜑=
ℛ∈
𝜑
(0,
ℛ,∈
1],
(0,
1],
where
where
denotes
denotes
the
the
collection
collection
denotes
denotes
Riemannian
of
Riemannian
the
the
integrable
integrable
of Riemannian
the
of funcRiemannian
the
collection
funccollection
integrable
ofintegrab
Rieman
of(7)
Riem
fu
([ ℛ
([
],(0,
) for
([
([
],
) of
([ ℛ
([
],= ,)collection
) denotes
X
X
(
ϕ
,
X
(
)]
,ℛ
[
]
[
,))],
,collection
∗
[𝜇, Definition
[𝜇, Definition
(𝐹𝑅)
(𝒾)𝑑𝒾
(𝐹𝑅)
(𝒾)𝑑𝒾
𝜈], denoted
𝜈],1.by
denoted
by
𝔖
𝔖
,
is
given
levelwise
,
is
given
levelwise
by
by
[49]
A
fuzzy
1.
[49]
number
A
fuzzy
valued
number
map
valued
𝔖
:
𝐾
map
⊂
ℝ
→
𝔖
:
𝔽
𝐾
⊂
ℝ
→
𝔽
is
called
F-NV-F.
is
called
For
F-NV-F.
each
(𝐹𝑅)
(𝐹𝑅)
(𝒾)𝑑𝒾
(𝒾)𝑑𝒾
(𝐹𝑅)
(𝒾)𝑑𝒾
(𝐹𝑅)
tions
tions
of I-V-Fs.
of I-V-Fs.
𝔖 tions
is
𝔖 𝐹𝑅
tions
is of
-integrable
𝐹𝑅I-V-Fs.
of
-integrable
I-V-Fs.
𝔖
tions
over
is
𝔖
tions
over
of
𝐹𝑅
is[𝜇,I-V-Fs.
of
-integrable
𝐹𝑅
𝜈]
[𝜇,
I-V-Fs.
-integrable
𝜈]
if[49]
if𝔖Let
is
𝔖over
is
over
-integrable
𝐹𝑅
[𝜇,
𝔖
-integrable
𝜈]
[𝜇,
if
if
over
over
𝔖
[𝜇,
𝔖
[𝜇,
if
𝔖
𝔖Fd(
∈
𝔽:∈
.Let
𝔽𝔽Note
.(𝐹𝑅)
Note
that,
if𝜈]
∈if
𝔽aif∈𝔽
. (𝐹𝑅)
𝔽Note
.𝜑-cuts
Note
tha
Theorem
2.Theorem
Theorem
2.
𝔖𝐹𝑅
:𝔖
[49]
[𝜇,
𝜈]Let
2.
⊂
[49]
𝔖
ℝ𝜈]
→
[𝜇,
𝜈]
𝔖⊂
: [𝜇,
ℝ
→
𝜈]𝜈]
𝔽⊂(𝒾)𝑑𝒾
ℝbe
→
be
athat,
F-NV-F
whose
F-NV-F
be
a (𝒾)𝑑
F-N
who
∈ (0, 1](𝒾,
𝜑whose
∈(𝒾,(0, ∗1]
𝜑 whose
-cuts
define
𝜑(𝒾,
-cuts∗the
define
family
theoffamily
I-V-Fs of𝔖 I-V-Fs
: 𝐾 ⊂ ℝ𝔖→: 𝒦
𝐾 ⊂ are
ℝ →∗given
𝒦 are
by gi∗
∗ (𝒾,𝜑
∗ (𝒾,
(𝒾,where
(𝒾,
(𝒾,
(𝒾,
(𝒾,
𝔖
𝜑),
𝔖∗Definition
𝜑)
𝜑)
are𝔖are
Lebesgue-integrable,
Lebesgue-integrable,
𝜑),
𝔖
𝜑),(𝒾,
𝔖∗∗𝜑)
𝜑)
are
Lebesgue-integrable,
then
Lebesgue-integrable,
𝔖
𝜑),
then
𝔖
𝔖number
𝜑)
is
𝔖
a𝜑)
are
is𝒦
fuzzy
avalued
are
Lebesgue-integrable,
fuzzy
Aumann-integrable
Aumann-integrable
then
𝔖
is
𝔖
ais
fuzzy
then
Aumann-integrable
functhen
Aumann-integrabl
𝔖
funcis
𝔖
a[𝔖
isfuzzy
a(𝒾,
fuzzy
Aum
fu
A
𝔖∗ (𝒾,
𝔖∗𝜑),
𝔖
𝔖are
Definition
[49]
A
fuzzy
1. [49]
number
A
fuzzy
valued
map
𝔖
:then
𝐾
⊂
ℝ
→
𝔖
:𝔖𝔽
𝐾
⊂fuzzy
ℝcalled
→
𝔽𝔖
isaare
F-NV-F.
is
called
For
each
F
(𝒾)
[𝔖
(𝒾)
(𝒾,
(𝒾)
[𝔖
=
𝜑),
=𝔖
𝔖
𝜑)]
=
𝜑),
for
𝔖
:∗𝜑),
[𝜇,
𝜈]
⊂
ℝ
:→
[𝜇,
𝜈]
⊂
are
: [𝜇,
ℝLebesgue-integrable,
given
→
𝜈]map
𝒦
⊂
ℝ
by
are
→
𝒦
given
by
given
by
𝔖
∗ 𝔖
∗1.
∗𝔖
∗ (𝒾,
∗F-NV-F.
∗ (𝒾
∗
(𝒾) =
[𝔖
(𝒾,
(𝒾)𝔖
[𝔖∗(𝒾,
(𝒾,
(𝒾,
𝔖 𝜈].
𝔖over
𝜑),
=
𝔖
𝜑),
for
all
𝜑)]
∈
for
𝐾.
all
Here,
∈≥𝐾.
for
for
𝜑
∈ each
(0,
1]∈𝜑
∈([(0,
end
the
end
real
X𝜈].
( (𝒾)𝑑𝒾
ϕ(𝐼𝑅)
)𝔖
=-cuts
in
f[𝜇,
∈
R|
X
ϕ}Here,
,𝜑)𝑑𝒾 𝔖(𝒾,
( 𝒾of
)(𝒾)𝑑𝒾
{ 𝒾(𝐼𝑅)
(𝒾)𝑑𝒾
(𝐹𝑅)
(𝒾)𝑑𝒾
(𝐹𝑅)
𝔖
=
=
𝔖
=𝔖
𝔖(𝒾,
=each
: 𝔖(𝒾,
𝜑)𝑑𝒾
𝜑)
: the
𝔖(𝒾,
ℛ
𝜑)
,ℛ
,
∗𝜑)]
∗∈
],1]∈
) point
([ (8)
],by
) po
, are
, are
tiontion
over
over
[𝜇, 𝜈].
[𝜇,
over
[𝜇,
[𝜇,
tion
tion
over
over
𝜈].
[𝜇,
𝜈].
𝜑 ∈ (0,tion
1] tion
𝜑whose
(0,
1]𝜑𝜈].
whose
-cuts
define
𝜑
the
define
family
the
family
I-V-Fs
of
𝔖
I-V-Fs
𝔖
:
𝐾
⊂
ℝ
→
:
𝒦
𝐾
⊂
ℝ
→
given
𝒦
gi
∗ (.(. 𝜑 ∈ (0,
∗ (. 1]. 𝜑
(𝒾,
Then
(0,ℝ
𝔖1].
𝜑are
islower
∈Then
𝐹𝑅-integrable
(0,
1].
𝔖 Then
isupper
𝐹𝑅-integrable
𝔖over
is 𝐹𝑅-integrable
[𝜇, 𝜈] over
iffunctions
over
𝜈] ifof
if [𝜇,
and
𝔖𝔖∗𝜈]
if
functions functions
𝔖 (. , 𝜑), 𝔖𝔖
, 𝜑):
, 𝜑),𝐾𝔖→
ℝ
, 𝜑):
are
𝐾∈called
→
called
and
lower
and
functions
upper
ofand
𝔖[𝜇,
.only
. onl𝜑
∗(𝒾,
(𝒾)𝜑),
[𝔖∗ ∗(𝒾,
𝔖 (𝒾) = [𝔖
𝔖 ∗∗(𝒾,
=𝔖
for
all
𝜑)]
for
𝐾.
Here,
for
each
for
𝜑 ∈ each
(0, 1] 𝜑the
∈ (0,
end
1] point
the end
realpo
X∗𝜑)]
(𝜑),
ϕ)𝔖
=∗ (𝒾,
sup
∈
R|all
X
ϕ}Here,
.
( 𝒾 )∈≥𝐾.
{𝒾 ∈
[𝜇, 𝜈], of
𝑅-integrable
𝑅-integrable
over
[𝜇,
𝑅-integrable
𝜈].
over
[𝜇,
over
𝜈].
ifMoreover,
[𝜇,
𝔖
𝜈].
is
𝐹𝑅-integrable
Moreover,
ifaI-V-Fs
𝔖I-V-Fs
iswhose
if𝐹𝑅-integrable
𝔖over
is
𝐹𝑅-inte
all
𝜑 Let
∈for
all
where
∈
ℛ
where
ℛ
the
denotes
collection
the
collection
of
Riemannian
of
Riemannian
integrable
integrable
func∗1],
)∗ (.
([⊂
],⊂
)Let
,Theorem
,2.
Theorem
Theorem
2.for
[49]
2.functions
[49]
Let
Theorem
𝔖:(0,
[𝜇,
Theorem
𝔖
:∗𝜈]
[𝜇,
𝜈]
2.ℝ(0,
⊂
[49]
2.
→
ℝ
𝔽(.
Let
→
𝔽
Let
𝔖
[𝜇,
𝔖
𝜈]
[𝜇,
2.
[49]
𝜈]
ℝ
[49]
ℝ
𝔽
→
Let
𝔖
𝔽
: [𝜇,
𝔖
:𝜈]
[𝜇,
⊂
𝜈]
ℝ
⊂
→
ℝ
𝔽the
→
𝔽
be
a],:𝔖
be
F-NV-F
a:denotes
whose
whose
𝜑-cuts
be
𝜑-cuts
aMoreover,
be
F-NV-F
define
aand
F-NV-F
define
the
whose
family
whose
be
family
𝜑-cuts
aupper
be
of
F-NV-F
𝜑-cuts
of
F-NV-F
define
the
family
𝜑-cuts
𝜑-cuts
I-o
(.𝜑,⊂
(.
functions
𝔖1],
𝔖
𝜑),
𝔖
,Theorem
𝜑):
,([
𝜑),
𝐾
→
ℝ
,F-NV-F
𝜑):
are
𝐾
called
→
ℝ
are
lower
called
lower
upper
and
functions
functions
ofdefine
𝔖the
. whose
offamily
𝔖of
. def
∗[49]
be
an
F-NV-F.
be
an
Then
F-NV-F.
fuzzy
Then
integral
fuzzy
of
integral
𝔖
over
Definition
Definition
2.
[49]
Let
2.
𝔖
[49]
:
[𝜇,
Let
𝜈]
⊂
𝔖
ℝ
:
[𝜇,
→
𝜈]
𝔽
⊂
ℝ
→
𝔽
(𝐹𝑅)
tions of I-V-Fs.
tions of𝔖I-V-Fs.
is 𝐹𝑅 -integrable
𝔖 is 𝐹𝑅 -integrable
over∗ [𝜇,∗ 𝜈]
over
if (𝐹𝑅)
[𝜇, 𝜈] if𝔖∗(𝒾)𝑑𝒾
∈𝔖
𝔽 (𝒾)𝑑𝒾
. Note
∈∗𝔽that,
.∗ Note
if of
tha
∗
(𝒾)
[𝔖
[𝔖
(𝒾,
(𝒾,
(𝒾,
[𝔖
(𝒾,
[𝔖
(𝒾,
(𝒾)
(𝒾,
(𝒾)
(𝒾,
(𝒾,
[𝔖for
(𝒾,all
(𝒾,
=
=
𝜑),
𝜑),
𝔖
𝔖(𝒾)
𝜑)]
=
𝜑)]
for
=∗given
for
all
𝒾𝜑),
∈𝔖
𝔖
[𝜇,
∈
𝜈]
𝔖
[𝜇,
𝜑)]
=
and
𝜈][𝔖
𝜑)]
=and
for
all
for
all
𝒾𝜑),
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𝜈]
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and
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[𝜇,⊂𝜈]ℝ⊂[𝜇,
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[𝜇,
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𝜈]
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∗𝜑),
(𝒾)𝑑𝒾
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𝜈],
denoted
𝜈],
by
denoted
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Definition
Definition
2.
Let
2. 𝔖[49]
: [𝜇,Lebesgue-integrable,
Let
𝜈] ⊂𝔖ℝ: [𝜇,
→ 𝜈]
𝔽 then
⊂be
ℝ an
→
(𝒾,
𝜑), 𝔖𝔖∗ ∗(𝒾,
𝜑)
𝜑),[49]
are
𝔖∗ (𝒾,
Lebesgue-integrable,
𝜑)
are
𝔖 𝔽F-NV-F.
isthen
fuzzy
𝔖Then
isAumann-integrable
a (𝒾)𝑑𝒾
fuzzy
funcf
𝔖∗ (𝒾,
(𝒾,(𝑅)
(𝒾,
(𝑅)Then
(𝐹𝑅)
(𝒾)𝑑𝒾
(𝐹𝑅)
(𝐹𝑅)
(𝒾)𝑑𝒾
=
𝔖
=∗fuzzy
𝜑)𝑑𝒾
=integral
𝔖
,𝔖(𝑅)
𝜑)𝑑𝒾
𝔖𝔖
𝔖abe
𝔖
𝔖Aumann-integrable
∗(𝑅)
∗
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and
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𝜑)
𝜑)
both
𝜑)
both
are
and
are
and
𝔖
𝔖
bot
𝜑)
𝜑 ∈𝜑(0,
∈ 1].
(0, 1].
Then
Then
𝔖 is𝔖
𝜑 𝐹𝑅-integrable
∈is𝜑(0,
𝐹𝑅-integrable
∈ 1].
(0, 1].
Then
Then
over
𝔖𝜑over
is
∈𝔖
[𝜇,
𝜑(0,
𝐹𝑅-integrable
is
∈
𝜈]
[𝜇,
1].
(0,
𝐹𝑅-integrable
if𝜈]1].
and
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Then
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𝜈]
[𝜇,
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𝔖if∗𝜑)
𝔖∗𝜑)
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(𝐹𝑅) , is𝔖given
(𝒾)𝑑𝒾levelwise
tion[𝜇,over
tion
[𝜇,[𝜇,
𝜈].
over
[𝜇,(𝐹𝑅)
𝜈]. by𝔖(𝒾)𝑑𝒾
𝜈], denoted
𝜈], by
denoted
, is ∗given
levelwise
by
by ∗ ∗
[𝜇,
[𝜇,
[𝜇,
[𝜇,∈𝜈],
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𝑅-integrable
𝑅-integrable
overover
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[𝜇, Moreover,
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Moreover,
if
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over
if[𝜇,
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𝑅-integrable
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Moreover,
𝜈].
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Moreover,
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𝜈],
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𝜈],
then
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is
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𝐹𝑅-integrable
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𝐹𝑅-integrable
𝜈],
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𝜈]
th
(8)
(𝒾)𝑑𝒾
(𝒾)𝑑𝒾
(𝐹𝑅)
(𝒾)𝑑𝒾
(𝐹𝑅)
(𝐼𝑅)
(𝐼𝑅)
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𝔖
=
=
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𝔖
𝔖(𝒾,
=if𝜑)𝑑𝒾
𝔖(𝒾,
:𝔖𝔖(𝒾,
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𝜑)
:then
𝔖(𝒾,
ℛ([then
𝜑)
,[𝜇,
ℛ([𝜈],
, ], ∈
, ], )
(𝒾)𝑑𝒾
(𝒾)𝑑𝒾
(𝐼𝑅)
(𝐼𝑅)
(𝐼𝑅)
=
𝔖
=
=
𝔖
Theorem Theorem
2. [49]
𝔖[49]
: [𝜇, 𝜈]
Let⊂𝔖
𝔖
: (𝐼𝑅)
[𝜇,
→ 𝜈]
𝔽 ⊂be
→
𝔽 =
F-NV-F
be
a(𝒾)𝑑𝒾
F-NV-F
𝜑-cuts
whose
define
𝜑-cuts
the ∈family
define
the
of I-V-Fs
family of I-𝔖
(𝒾)𝑑𝒾
(𝐹𝑅)Let2.𝔖
(𝒾)𝑑𝒾
(𝐹𝑅)
=ℝ(𝒾)𝑑𝒾
=
𝔖ℝa(𝐼𝑅)
𝔖 whose
𝔖(𝒾,
=𝜑)𝑑𝒾
𝔖(𝒾,
: 𝔖(𝒾,
𝜑)𝑑𝒾
𝜑)
: 𝔖(𝒾,
ℛ([ 𝜑)
) ,ℛ ([ (8)
, ], ∈
, ], )
∗ (𝒾,
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all
𝜑ℝ
∈for
(0,
all
1],
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∈ (0,
1],
ℛ(𝐹𝑅)
where
ℛ∗(𝒾)𝑑𝒾
denotes
the
denotes
collection
the
collection
of=
Riemannian
Riemannian
integrab
(𝒾,
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([
],given
)(𝒾)
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=
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𝔖
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=,∗)(𝒾,
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, 𝜑),
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𝔖∗𝜑),
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,(𝑅)
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, func𝔖
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(𝒾,
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𝜑)]
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𝜑)]
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for
[𝜇,
all
𝜈]
𝒾∗and
∈
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for
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all(𝑅)
and𝔖∗fo(
: [𝜇,(𝐹𝑅)
𝜈] (𝐹𝑅)
⊂
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𝒦
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ℝ
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given
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are
by
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𝔖 for
𝔖
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∗(𝒾,
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(𝒾)𝑑𝒾
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(𝒾)𝑑𝒾
tions of I-V-Fs.
tions of𝔖I-V-Fs.
is 𝐹𝑅for
-integrable
𝔖 allis 𝜑𝐹𝑅
-integrable
[𝜇,
𝜈]
over
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𝔽ℱℛ
. Note
𝔽that,
.collectio
Note
if dt
(0, 1],
(0,
∈ for
(0,over
all
1]. 𝜑
For
for
∈ all
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1].
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(0,𝜈]
1].
all
1], if𝜑
For
ℱℛ
all
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1], ∈ℱℛ
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4 of 19
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order relation
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Symmetry 2022, 14, x FOR PEER REVIEW
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1639 on 𝔽 by
4 of 18
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𝔵, 𝔴PEER
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14,
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ERxREVIEW
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2022,
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x
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4
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19
4
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4
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19
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(4)
Proposition 1. 𝔵[51]
Let
𝔵,
𝔴
∈
𝔽
.
Then
fuzzy
order
≼ 1],
” is given on 𝔽 (4)
by
≼ 𝔴 if and only if, [𝔵] ≤ [𝔴]
forrelation
all 𝜑 ∈“(0,
4 of 19
1 [51].
relation “ 41.
is given
F0∈by𝔽 . Then fuzzy orde
Proposition
𝔵, 𝔴
0 . Then fuzzy order
14, x FOR PEER REVIEW 𝔵 ≼ 𝔴 if and only Proposition
4” [51]
of
19Let on
[𝔵] ≤ [𝔴]
forLet
all X,
𝜑w
∈∈
(0,F1],
(4)
examine
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of the characteristics of if,
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partial
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Proposition
Proposition
1. [51] Let 1.
Proposition
𝔵,[51]
𝔴 ∈ Let
𝔽 . Then
𝔵, 𝔴
1. ∈
[51]
𝔽 Let
𝔵, 𝔴fuzzy
∈relation
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. Then
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mmetry
ry is
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[X] ϕ ≤ I [w] ϕ for all ϕ ∈ (0, 14𝔵],≼
[𝔵]
[𝔵]
[𝔵]
[𝔴]
[𝔴]
[𝔴]
ers. Let
𝔵, 𝔴 ∈use
𝔽 mathematical
and 𝜌 ∈ 𝔵ℝ,≼then
have
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and
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and
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and
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all
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1],
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1],
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(4)
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1],
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We
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characteristics
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fuzzy
[51]
𝔵, 𝔴, ∈ 𝔽 . Then fuzzy order relation
(5) “of≼19” is given on 𝔽 by
[𝔵] Let
It is a partial order relation.
+ [𝔴]
Proposition
1.∈[51]
LetIt 𝔵,
𝔴
∈ 𝔽 . Then
order relation “ ≼ ” is given on 𝔽 by
iswe
a partial
order fuzzy
relation.
numbers.
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𝔵,
𝔴
∈
𝔽
and
𝜌
ℝ,
then
have
22,
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x It
FOR
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Symmetry
14,
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𝜑 𝔴∈𝔵,use
(0,
1],
(6)
[𝔵] 1.×𝔵[51]
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Proposition
1.𝔵 𝔴]
[51]
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𝔵,
∈
𝔴
𝔽
∈
𝔽
.
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.
Then
fuzzy
fuzzy
order
order
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relation
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“
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of
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the
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numbers.
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𝔵,
𝔴
∈
𝔽
and
𝜌
∈
ℝ,
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we
have
≼1.𝔴[51]
if
and
only
if,
≤
for
all
𝜑
∈
(0,
1],
(4)
(5)
[𝔵+
[𝔵]
[𝔴]
+ ,and ρ, ∈ R, then we have
Let[𝔴]
Fall
[𝔵
[𝔵]X,=w
[∈𝔴]
× 𝔴] ≤=
×for
0 𝜑 ∈ (0, 1],
𝔵 ≼ 𝔴numbers.
if and
if,
(4) (6)
rder
numbers.
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𝔵,“ ≼
𝔴only
∈ Let
𝔽 given
numbers.
𝔵,[𝔵]
𝔴∈
𝔵,
𝔴
∈
𝔽
and
𝜌 𝔽∈𝔽Let
ℝ,
and
then
𝜌
we
∈
ℝ,
have
then
and
we
𝜌
∈
have
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then
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have
(7)
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= 𝜌.relation.
fuzzy
order
relation
”
is
on
by
≼ 𝔵𝔴≼×if𝔴[and
if and
only
if, [𝔵]
if, [𝔵]
≤ [𝔴]
≤ [𝔴]
for
all all
𝜑(6)
∈𝜑(0,
∈ 1],
(0, 1],
(4) (4)
It is a partial order[𝔵relation.
[𝔵+𝔴] = [𝔵]
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𝔴]
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e
X
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Proposition
[51]
1.
[51]
Let
Let
Proposition
𝔵,
𝔴
Proposition
𝔵,
∈
𝔴
𝔽
∈
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1.
[51]
Proposition
1.
[51]
Let
Proposition
Let
𝔵,
𝔴
𝔵,
Proposition
∈
1.
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𝔽
[51]
∈
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1.
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[51]
𝔵,
1.
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𝔴
[51]
∈
𝔵,
𝔽
Let
𝔴
∈
𝔵,
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𝔴
∈
𝔽
.
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.
Then
fuzzy
fuzzy
order
order
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relation
.
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on
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Symmetry
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2022,
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PEER
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,
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𝜑
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1],
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order
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,
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[
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,
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he
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𝔖
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𝔖(𝒾,
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𝜑)
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ℛ
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1.
[51]
1.
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1],
where
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𝐾.
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each
𝜑
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1]
the
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real
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n
[49]
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𝔖
[𝜇,
𝜈]
⊂
ℝ
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𝔽
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called
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functions
of
.
of
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𝐹𝑅
over
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𝜈]
if
𝔖
∈
𝔽
.
Note
that,
if
e
for
𝜑
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1],
where
ℛ
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hen
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∗ ϕ𝜑)𝑑𝒾
∗: 𝔖(𝒾,
[𝜇, 𝜈], denoted
(𝐹𝑅)
(𝒾)𝑑𝒾
,
given
levelwise
by
([
µ, ν], ϕ)
([
],
)
([
],
)
([
],
)
,
,
,
EVIEW
4
of
19
be an
be
F-NV-F.
an
F-NV-F.
Then
Then
fuzzy
integral
integral
of 𝔖
of over
𝔖 over
Definition
Definition
2.partial
2.∗[49]
Let
Let
𝔖
: [𝜇,
𝔖
:partial
𝜈]
[𝜇,
⊂𝜈]
ℝ
⊂order
→
𝔽→
𝔽
µis
µ
µfuzzy
∗[49]
∗ℝ
∗ (.
∗ (.
∗that,
∗ relation.
(𝒾)𝑑𝒾
It𝔖is∗It(𝒾,
aispartial
a(𝐹𝑅)
partial
order
relation.
relation.
isgiven
a
It
partial
is
a
order
order
relation.
It
relation.
a
It
is
a
partial
order
relation.
It
relation.
is
It
a
is
partial
a
partial
order
order
relation.
noted
by
𝔖order
,Itis-integrable
levelwise
by
(𝐹𝑅)
(𝒾)𝑑𝒾
(.
(.
(.
(.
(.
(.
(.
(.
(.
(.
(.
(.
tions
of
I-V-Fs.
𝔖
is
𝐹𝑅
over
[𝜇,
𝜈]
if
𝔖
∈
𝔽
.
Note
if
𝜑),
𝜑)
are
Lebesgue-integrable,
then
𝔖
is
a
fuzzy
Aumann-integrable
funcfunctions
functions
𝔖
𝔖
functions
functions
𝔖
𝔖
functions
functions
𝔖
functions
𝔖
𝔖
,
𝜑),
,
𝜑),
𝔖
𝔖
,
𝜑):
,
𝜑):
𝐾
→
𝐾
ℝ
→
are
ℝ
,
are
𝜑),
called
,
𝜑),
𝔖
called
lower
,
𝜑):
lower
,
𝜑):
𝐾
and
→
,
𝐾
𝜑),
and
ℝ
upper
→
𝔖
are
ℝ
,
upper
𝜑),
are
called
,
functions
𝜑):
𝔖
,
called
𝜑),
functions
𝐾
,
lower
→
𝜑):
𝔖
ℝ
lower
𝐾
of
,
are
→
𝜑):
and
𝔖
of
ℝ
.
called
𝐾
and
𝔖
upper
are
→
.
upper
ℝ
called
lower
are
functions
functions
called
lower
and
upper
of
lower
and
𝔖
of
.
upper
functions
and
𝔖
.
upper
functions
of
functi
𝔖
.
of
[𝔵+
[𝔵+
[𝔵]
[𝔴]
[𝔴]
𝔴]
=→
+
𝔴]
=
, [𝔵]
+⊂
∗ PEER REVIEW
∗ Definition
∗
∗[49]
for
all
𝜑
(0,
1],
where
ℛ
the
co
Symmetry 2022,
14,
FOR
of
19
Definition
[49]
1.levelwise
Alevelwise
fuzzy
A∗ fuzzy
Definition
number
number
valued
Definition
1.
valued
[49]
A
map
fuzzy
𝔖
1.
[49]
𝐾𝔖
:number
A
ℝ⊂fuzzy
→
ℝ
valued
number
𝔽
map
valued
𝐾
map
→
is
called
F-NV-F.
For
F𝔖
e
], ,called
) denotes
, is
(6)
[𝔵
[𝔵
[𝔵
[𝔵]
[ to
[𝔵]
[𝐾
[𝔵]
[([fuzzy
[𝔵]
[: some
×∗some
×
×
𝔴]
=
𝔴]
×map
𝔴]
=∈[𝔵
, :some
𝔴]
×⊂
=
𝔴]
, 𝔽×
×
𝔴]
𝔴]
×𝔖4F-NV-F.
𝔴]
,ℝ
[𝜇,
[𝜇,∗x 𝜈],
(𝐹𝑅)
(𝐹𝑅)
(𝒾)𝑑𝒾
(𝒾)𝑑𝒾
𝜈],
denoted
denoted
by
𝔖
𝔖now
, issome
given
, 1.
is
given
by
by
be
an
F-NV-F.
Then
fuzzy
integral
of by
𝔖use
𝔽
We
We
now
use
use
mathematical
mathematical
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operations
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examine
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examine
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some
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the
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some
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characteristics
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examine
to=examine
characteristics
some
of
of
fuzzy
the
of tf
∗now
ver
[𝜇,
𝜈].
(8)
(𝒾)𝑑𝒾
(𝒾)𝑑𝒾
(𝐼𝑅)
𝔖
=
𝔖
=
𝔖(𝒾,
𝜑)𝑑𝒾
:
𝔖(𝒾,
𝜑)
∈
ℛ
,
(𝒾,
(𝒾,
𝜑),
𝔖
𝜑)
are
Lebesgue-integrable,
then
𝔖
is
a
fuzzy
Aumann-integrable
func𝔖
for
all
𝜑
∈
for
(0,
all
1],
𝜑
where
∈
(0,
for
1],
ℛ
all
where
𝜑
∈
(0,
ℛ
1],
where
ℛ
denotes
the
denotes
collection
the
collection
of
denotes
Riemannian
the
of
Riemannian
collection
integrable
of
integrable
Riemannian
funcfuncintegrable
func([
],
)
, ([whose
for
all
ϕ
∈
0,
1
,
where
R
denotes
the
collection
of
Riemannian
integrable
functions
of
(
]
tions
of
I-V-Fs.
𝔖
is
𝐹𝑅
-integrable
over
[𝜇,
𝜈]
∗
([
],
)
([
],
)
([
],
)
𝜑
∈
𝜑
(0,
∈
1]
(0,
1]
whose
𝜑
-cuts
𝜑
-cuts
𝜑
define
∈
define
(0,
1]
the
𝜑
whose
the
∈
family
(0,
family
1]
𝜑
of
-cuts
whose
I-V-Fs
of
I-V-Fs
define
𝜑
-cuts
𝔖
𝔖
the
define
family
the
of
family
I-V-Fs
of
𝔖
:
𝐾
⊂
:
𝐾
ℝ
⊂
→
ℝ
𝒦
→
𝒦
are
are
given
giv
,
,
,
µ,
ν
]
,
ϕ
)
Symmetry
2022,
14,
x
FOR
PEER
REVIEW
4
of
19
[𝔵
[𝔵
[𝔵]
[
[𝔵]
[
(8)
×
×
𝔴]
=
×
𝔴]
𝔴]𝔖
=, over
×fuzzy
𝔴]of
(𝒾)𝑑𝒾
(𝐹𝑅)
(𝒾)𝑑𝒾
(𝐼𝑅)
𝔖
𝔖
=ℝ,
𝔖(𝒾,
𝜑)𝑑𝒾
:→
𝔖(𝒾,
𝜑)
∈
ℛ
ven
levelwise
by
)𝔖
numbers.
numbers.
Let
Let
𝔵,
𝔴𝔵,
∈𝔴𝔽∈numbers.
𝔽
numbers.
Let
𝔵,
Let
𝔴
∈=:𝔵,
𝔽numbers.
𝔴⊂𝜈]
∈have
𝔽numbers.
Let
Let
𝔴𝔖
∈F-NV-F.
𝔵,𝔽
numbers.
∈∗⊂𝜈]
𝔽numbers.
Let
Let
𝔵,𝜈]
𝔴
𝔵,
∈F-NV-F.
𝔴
𝔽
∈→
𝔽
and
𝜌2.∈[49]
𝜌ℝ,
∈ then
ℝ,
then
we
we
have
and
𝜌2.
and
∈
ℝ,
then
∈
we
then
have
and
we
and
∈
ℝ,
𝜌[𝜇,
then
∈fuzzy
ℝ,
we
then
have
we
and
have
and
𝜌
∈
𝜌ℝ,
∈over
then
ℝ,
then
we
we
have
have
be𝜌𝔵,(𝐹𝑅)
an
be
an
F-NV-F.
Then
fuzzy
be
be
integral
an
integral
F-NV-F.
Then
be
of
Then
an
over
𝔖
fuzzy
be
fuzzy
an
be
integral
F-NV-F.
an
Then
integral
F-NV-F.
of
Then
fuzzy
Then
fuzzy
𝔖integral
over
integ
Definition
2.and
[49]
Let
Definition
𝔖𝐹𝑅
: [𝜇,
Definition
𝔖
𝜈]
ℝ2.
⊂
→
ℝ
[49]
𝔽
Definition
→
[49]
Let
𝔽
Definition
:ℛ
[𝜇,
𝔖
Definition
2.:𝔴
𝜈]
[𝜇,
[49]
2.
ℝ
⊂𝜌have
[49]
→
ℝ
𝔽
𝔖
2.
Let
:R.if
[49]
𝔽
Let
:∈
⊂
[𝜇,
ℝ
𝜈]
𝔖
:([Note
⊂
[𝜇,
𝔽, of
ℝ],𝜈]
→
⊂∈,𝜌.
𝔽[𝜌.
ℝ
→
𝔽
(7)
[𝜌.
[𝜌.
[𝜌.
[𝔵]
[𝔵]
[𝔵]
𝔵]
𝔵]
𝔵]
𝔵]
=
𝜌.
=
𝜌.
=∗of
𝜌. [𝔵]
νNote
-NV-F
whose
𝜑-cuts
define
the
family
of
I-V-Fs
∗Let
∗=
tion
over
𝜈].
(8)
(𝒾)𝑑𝒾
(𝐹𝑅)
(𝒾)𝑑𝒾
(𝐼𝑅)
e[𝜇,
e𝔖an
(𝒾)𝑑𝒾
(𝐹𝑅)
(𝒾)𝑑𝒾
(𝐹𝑅)
(𝒾)𝑑𝒾
𝔖[𝜇,
=Definition
𝔖
=
𝔖(𝒾,
𝜑)𝑑𝒾
:relation
𝔖(𝒾,
∈
,Then
tions
of
tions
of𝔖𝔵,
I-V-Fs.
is
tions
𝐹𝑅
-integrable
𝔖
of
isI-V-Fs.
I-V-Fs.
-integrable
over
𝔖
is
𝜈]
over
-integrable
𝜈]
over
[𝜇,
𝜈]
𝔖
𝔽
𝔽
that,
.𝔖
𝔽
.F-NV-F.
if𝔖
Note
if
(𝒾)
(𝒾)
[𝔖
[𝔖
(𝒾,
(𝒾,
(𝒾)
(𝒾,
(𝒾,
𝔖[𝜇,𝐹𝑅
𝔖
𝔖
=if𝜑)
=[𝜇,
𝜑),
𝔖],given
𝜑)]
𝜑)]
for
for
all
=
all
𝒾𝔖)∗∈
𝒾𝐾.
∈
=
𝐾.
Here,
for
𝜑)]
𝜑),
for
each
for
all
𝜑𝜑)]
∈ϕfor
∈)𝐾.
1]
all
Here,
the
1]( 𝒾the
end
𝐾.end
point
Here,
each
poi
([ [if
)]𝔖if
S
is
FR-integrable
over
µ,
ν(𝒾,
FR
S
d(𝒾,
∈
Fif𝔖
.∗∗Here,
Note
that,
ifeach
S
,(0,
S
,∈
ϕ)for
,𝔖
(∈
)(𝒾)
([𝔖
(∈𝒾𝜑,(0,
(𝒾,
𝜑),
𝜑)
are
Lebesgue-integrable,
then
𝔖
∗that,
∗“(𝒾,
∗𝜑),
∗that,
0[𝔖
Proposition
1.I-V-Fs.
[51]
Let
𝔴
∈
𝔽
.
Then
fuzzy
order
≼
”
is
on
𝔽
by
µ
(8)
[𝜇,
[𝜇,
(𝐹𝑅)
(𝐹𝑅)
[𝜇,
[𝜇,
(𝒾)𝑑𝒾
(𝒾)𝑑𝒾
[𝜇,(𝐼𝑅)
(𝐹𝑅)
[𝜇,
(𝐹𝑅)
(𝒾)𝑑𝒾
(𝐹𝑅)
(𝐹𝑅)
(𝒾)𝑑𝒾
(𝐹𝑅)
(𝒾)𝑑𝒾
(𝒾)𝑑𝒾
(𝒾)𝑑𝒾
(𝐹𝑅)
(𝐹𝑅)
(𝒾)𝑑𝒾
(𝒾)𝑑𝒾
(𝐼𝑅)
𝜈],
𝜈],
denoted
denoted
by
by
𝜈],
𝔖
𝜈],
denoted
𝔖[𝔵]
denoted
by
by
𝜈],
denoted
𝜈],
𝔖[𝜇,
denoted
𝔖
𝜈],
by
by
𝔖
by
𝔖
𝔖∈
, [𝔵]
is+𝜑-cuts
given
,[𝔴]
is
given
levelwise
levelwise
by
by
,denoted
given
,[𝔴]
is𝔖(𝒾,
given
levelwise
,=
is
given
by
,=
is
by
given
levelwise
,, +
is
given
levelwise
levelwise
by
by
𝔖all
𝔖
=
=
𝔖
𝔖the
=
=
𝔖(𝒾,
𝜑)𝑑𝒾
𝜑)𝑑𝒾
:levelwise
𝔖(𝒾,
:(𝒾)𝑑𝒾
𝔖(𝒾,
𝜑)
𝜑)
ℛ
∈
,(8)
∗ (𝒾,
[𝜌.
[𝜌.
[𝔵]
[𝔵]
rem
2. [49]
Let
𝔖
:
[𝜇,
𝜈]
⊂
ℝ
→
𝔽
be
a
F-NV-F
whose
define
family
of
I-V-Fs
𝔵]
=
𝜌.
=
𝜌.
([ ℛ
([
],, [𝔴]
)by
,) ],,𝔵]
∗(𝒾)𝑑𝒾
∗is
∗[𝔵]
∗[𝔵+
1],
where
ℛ
denotes
the
collection
of
Riemannian
integrable
func∗
∗
∗
(𝒾,
𝜑),
𝔖
𝜑)]
for
all
𝒾
∈
[𝜇,
𝜈]
and
for
(5)
(5)
(5)
(5)
(5)
[𝔵+
[𝔵+
[𝔵+
[𝔵+
[𝔵+
[𝔵+
[𝔵+
[𝔴]
[𝔵]
[𝔵]
[𝔴]
[𝔵]
[𝔴]
[𝔵]
[𝔵]
[𝔴]
[𝔴]
([
],
)
,
𝔴]
𝔴]
=
=
+
,
,
𝔴]
=
𝔴]
=
+
+
,
𝔴]
,
𝔴]
+
𝔴]
,
𝔴]
=
=
+
+
,
,
(.
(.
(.
(.
(.
(.
(.
(.
functions
functions
𝔖
𝔖S
functions
functions
𝔖order
𝔖∗lower
, ∗𝔖
𝜑),
,is𝜑),
𝔖
𝔖
, of
𝜑):
, 𝜑):
𝐾𝔖Aumann-integrable
→𝐾
ℝ→
ℝ
called
, 𝜑),
called
𝔖
lower
,func𝜑):
, and
𝜑),
𝐾
𝔖upper
→
ℝupper
, given
𝜑):
functions
→
lower
ofbycalled
𝔖
of.and
𝔖. low
up
(𝒾,
(𝒾,
∗
𝜑)
𝜑),
𝔖
Lebesgue-integrable,
are
𝜑), are
𝔖
Lebesgue-integrable,
𝜑)
are
then
Lebesgue-integrable,
𝔖[51]
isthe
then
aLet
a𝔽
then
fuzzy
is
a are
fuzzy
funcAumann-integrable
funcfunc𝔖∗ (𝒾, 𝜑), 𝔖for
𝔖(𝒾,
𝔖𝜑)
tion
over
[𝜇,
𝜈].
Lebesgue-integrable,
then
aAumann-integrable
Aumann-integrable
function
µ,
νare
. 𝐾called
]functions
∗ fuzzy
∗are
∗ (𝒾,
∗ (𝒾,where
Proposition
1.
𝔵,is𝔴
∈fuzzy
. Then
fuzzy
relation
“over
≼and
” [is
onℝ𝔽are
all
𝜑are
∈ (0,
1],
ℛ
denotes
collection
Riemannian
integrable
],∗ ) [𝔴]
[𝔵]
𝔴
iftion
and
if,Definition
≤
for
all
𝜑define
∈that,
(0,A
1],
(4)
(8)
(𝒾)𝑑𝒾
=→
𝔖(𝒾,
𝜑)𝑑𝒾
:≼
𝔖(𝒾,
𝜑)
∈𝜈]
ℛ
,([ ,𝔖
(𝒾)𝑑𝒾
2.𝒦
Let
𝔖
:𝔵[𝜇,
𝜈]
⊂
ℝ
→
𝔽if
be
a)𝜈].
F-NV-F
whose
𝜑-cuts
the
family
of
I-V-Fs
s.
𝔖and
is
𝐹𝑅
-integrable
over
[𝜇,
𝔖
∈[49]
𝔽
.A
Note
if1.
(𝒾)
[𝔖
(𝒾,Riemannian
∗𝜈].
=only
𝜑),
𝜑)]
for
all
𝒾integrable
∈
[𝜇,
𝜈]
and
for
all
⊂1],
ℝ
given
bydenotes
𝔖
([both
],(𝒾,
∈, 𝜈]
where
ℛ
the
collection
of
funcDefinition
1.
Definition
fuzzy
1.
[49]
number
fuzzy
Definition
[49]
valued
number
A
fuzzy
map
1.
valued
[49]
number
𝔖
:
𝐾
A
⊂
fuzzy
map
ℝ
valued
→
𝔖
number
:
𝔽
𝐾
⊂
map
ℝ
→
valued
𝔖
:
𝔽
𝐾
⊂
map
ℝ
→
𝔽
𝔖
:
𝐾
⊂
ℝ
→
𝔽
is
called
F-NV-F.
is
called
For
F-NV-F.
is
each
called
For
F, (𝐹𝑅)
tion
over
tion
[𝜇,
𝜈].
over
[𝜇,
over
[𝜇,
∗
([
],
)
,
(𝒾,
(𝒾,
𝜑)
and
𝔖
𝜑)
are
]Theorem
if(0,
only
if[49]
𝔖are
∗tions of I-V-Fs. 𝔖 [𝔵
(𝐹𝑅)
is1],
𝐹𝑅
over
𝜈]
if𝔽
𝔖
∈
.𝔴]
Note
Proposition
1.×ℛ[51]
𝔵,
𝔴
.collection
Then
relation
“𝔴]
≼all
is
on[𝔵]
𝔽(6)
(6)
(6)
(6)
[𝔵∈𝔴
[𝔵
[𝔵Riemannian
[𝔵
[([],×
[𝔵]
[𝔵]
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[𝔵]
[that,
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×[𝔵
×-integrable
×
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×
×given
𝔴]
𝔴]
= [𝔵]
=ℛ[𝔵]
𝔴],)[Let
𝔴]
,[𝔵) [𝜇,
, 𝔵𝔴]
=𝔴]
=
×only
𝔴](𝒾)𝑑𝒾
×
,×
𝔴]
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=
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×”,[𝔵
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=
=
×by[ ×
𝔴][ 𝔴]
, (4)
, (6)
for(𝐹𝑅)
for
allthen
all
𝜑 ∈𝔖
𝜑𝔖
(0,
∈(𝒾)
(0,
1],
where
where
denotes
denotes
the
the
collection
of
Riemannian
of
integrable
func[𝔵]
[𝔴]
≼
if=
and
if,
≤
for
𝜑:⊂
∈
(0,
1],
∗(𝒾)𝑑𝒾
([𝜑
],S
,𝔖
e
∗1]
(8)
(8)
(8)
(8)
be
an
be
F-NV-F.
an
F-NV-F.
Then
Then
fuzzy
fuzzy
be
integral
an
integral
F-NV-F.
of
be
𝔖
an
of
T
o𝐾
Definition
Definition
2.
[49]
2.
[49]
Let
Let
𝔖
:
[𝜇,
Definition
𝔖
:
𝜈]
[𝜇,
⊂
𝜈]
ℝ
⊂
→
Definition
ℝ
2.
𝔽
→
[49]
𝔽
Let
2.
𝔖
[49]
:
[𝜇,
𝜈]
Let
⊂
𝔖
ℝ
:
[𝜇,
→
𝔽
𝜈]
⊂
ℝ
→
𝔽
𝜑
∈
(0,
1]
whose
∈
(0,
1]
𝜑
𝜑
-cuts
whose
∈
(0,
define
𝜑
-cuts
whose
𝜑
the
∈
(0,
define
family
𝜑
1]
-cuts
whose
the
of
define
family
I-V-Fs
𝜑
-cuts
the
of
𝔖
family
define
I-V-Fs
of
the
𝔖
I-V-Fs
family
𝔖
of
I-V-Fs
𝔖
:
𝐾
ℝ
→
:
𝒦
𝐾
⊂
are
ℝ
→
given
:
𝒦
𝐾
⊂
are
ℝ
by
→
give
𝒦
(𝒾)𝑑𝒾
(𝒾)𝑑𝒾
(𝒾)𝑑𝒾
(𝒾)𝑑𝒾
(𝒾)𝑑𝒾
(𝒾)𝑑𝒾
(𝒾)𝑑𝒾
(𝐹𝑅)
(𝒾)𝑑𝒾
(𝒾)𝑑𝒾
(𝐹𝑅)
(𝐹𝑅)
(𝒾)𝑑𝒾
(𝒾)𝑑𝒾
(𝐹𝑅)
(𝐹𝑅)
(𝒾)𝑑𝒾
(𝐹𝑅)
(𝒾)𝑑𝒾
(𝒾)𝑑𝒾
(𝐼𝑅)
(𝐼𝑅)
(𝐼𝑅)
(𝐼𝑅)
(𝐼𝑅)
(𝐼𝑅)
(𝐼𝑅)
(𝐹𝑅)
𝔖
𝔖
𝔖
𝔖
𝔖
=
=
𝔖
=
=
=
𝔖(𝒾,
𝔖(𝒾,
𝜑)𝑑𝒾
𝔖
𝜑)𝑑𝒾
𝔖(𝒾,
𝔖
:
=
𝔖(𝒾,
𝜑)
=
∈
𝜑)
=
=
ℛ
∈
𝔖(𝒾,
𝔖
ℛ
=
𝔖(𝒾,
𝜑)𝑑𝒾
𝔖
,
𝜑)𝑑𝒾
=
,
𝔖(𝒾,
𝔖
𝔖(𝒾,
=
𝜑)
𝔖(𝒾,
∈
𝜑)
𝜑)𝑑𝒾
ℛ
=
𝔖(𝒾,
∈
ℛ
:
𝜑)𝑑𝒾
𝔖(𝒾,
𝔖(𝒾,
,
𝜑)
:
𝜑)𝑑𝒾
𝔖(𝒾,
,
∈
ℛ
𝜑)
:
𝔖(𝒾,
∈
𝜑
I-V-Fs.
𝔖
is
𝐹𝑅
-integrable
over
[𝜇,
𝜈]
if
𝔖
∈
𝔽
.
Note
that,
if
𝜑)
are
Lebesgue-integrable,
is
a
fuzzy
Aumann-integrable
func:
µ,
ν
⊂
R
→
F
Theorem
2
[49].
Let
be
a
F-NV-F
whose
ϕ-cuts
define
the
family
of
I-V-Fs
Definition
1.
Definition
[49]
A
fuzzy
1.
number
[49]
A
fuzzy
valued
number
map
𝔖
valued
:
𝐾
map
𝔖
:
𝐾
⊂
ℝ
is
calle
[𝔖
(𝒾,
(𝒾,
[
]
=
𝜑),
𝔖
𝜑)]
for
all
𝒾
∈
[𝜇,
𝜈]
and
for
all
:
[𝜇,
𝜈]
⊂
ℝ
→
𝒦
are
given
by
𝔖
𝔖
([
([
],
)
],
)
([
([
],
)
],
)
([
],:ℛ
Theorem
2.
[49]
Let
𝔖
:
[𝜇,
𝜈]
⊂
ℝ
→
𝔽
be
a
F-NV(𝒾,
(𝒾,
,
,
,
,
,
0
It
is
a
partial
order
relation.
𝜑)
and
𝔖
𝜑)
both
are
0,
1].
Then
𝔖
is
𝐹𝑅-integrable
over
[𝜇,
𝜈]
if
and
only
if
𝔖
∗
∗ (𝒾,REVIEW
Symmetry
14,
x then
FOR
PEER
4 of 19
∗
[𝜇,𝔖
ntegrable
over2022,
𝜈],
(𝒾,
𝜑),
𝔖tions
are
Lebesgue-integrable,
then
𝔖and
is
a[𝜇,
fuzzy
Aumann-integrable
func(𝐹𝑅)
(𝐹𝑅)
(𝒾)𝑑𝒾
(𝒾)𝑑𝒾
∗over
∗𝔵]
∗(𝒾)𝑑𝒾
∗ (𝒾,
∗of
tions
of𝜑)I-V-Fs.
ofintegrable
I-V-Fs.
𝔖𝔖:x𝔵]
is
𝔖
𝐹𝑅
is
-integrable
𝐹𝑅
-integrable
over
[𝜇,
𝜈]
𝜈]
if[𝔵]
if𝜌.
𝔖
𝔖
∈
𝔽
∈all
.𝜑
𝔽,Note
.𝜑)]
Note
that,
that,
if
if
∗[𝔵]
Symmetry
2022,
14,
FOR
PEER
REVIEW
[𝔵]
[𝔴]
otes
collection
Riemannian
func𝔵
≼
𝔴
if
only
if,
≤
for
𝜑
∈
(0,
1],
(4)
∗ (𝒾, the
(7)
(7)
(7)
(7)
(7)
[𝜌.
[𝜌.
[𝜌.
[𝜌.
[𝜌.
[𝜌.
[𝜌.
[𝜌.
[𝔵]
[𝔵]
[𝔵]
[𝔵]
[𝔵]
[𝔵]
[𝜇,
[𝜇,
(𝐹𝑅)
(𝐹𝑅)
[𝜇,
(𝒾)𝑑𝒾
[𝜇,
(𝐹𝑅)
(𝒾)𝑑𝒾
(𝐹𝑅)
(𝒾)𝑑𝒾
(𝒾)
[𝔖
(𝒾,
(𝒾)
[𝔖
(𝒾,
(𝒾,
(𝒾)
[𝔖
(𝒾,
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[𝔖
(𝒾,
(𝒾,
𝔵]
𝔵]
𝔵]
𝔵]
𝔵]
𝔵]
=
𝜌.
=
𝜌.
=
𝜌.
=
=
𝜌.
=
𝜌.
=
𝜌.
=
𝜌.
𝜈],
𝜈],
denoted
denoted
by
by
𝔖
𝜈],
denoted
𝜈],
by
denoted
by
𝔖
𝔖
,
is
given
,
is
given
levelwise
levelwise
by
by
,
is
given
levelwise
,
is
given
by
levelw
𝔖
𝔖
𝔖
=
𝜑),
=
𝔖
𝜑)]
𝜑),
=
for
𝔖
all
𝜑)]
𝜑),
𝒾
∈
𝔖
for
=
𝐾.
all
Here,
𝜑)]
𝒾
𝜑),
∈
for
for
𝐾.
𝔖
all
Here,
each
𝜑)]
𝒾
∈
for
𝐾.
for
∈
Here,
(0,
each
all
1]
𝒾
𝜑
for
the
∈
∈
𝐾.
each
(0,
end
Here,
1]
point
𝜑
the
∈
for
(0,
end
real
each
1]
point
the
𝜑
S
µ,
ν
⊂
R
→
K
are
given
by
S
,
ϕ
,
S
ϕ
for
all
∈
µ,
ν
and
for
=
S
[
]
(
)
(
[
]
)
[
(
.
𝜑
∈
(0,
1]
whose
𝜑
∈
(0,
𝜑
1]
-cuts
whose
define
-cuts
the
family
define
of
the
I-V-Fs
family
𝔖
of
I-V-Fs
:
𝐾
⊂
ℝ
→
∗
ϕ [49]
ϕ the
∗are
Theorem
2.
Theorem
[49]
LetREVIEW
2.𝔖
:Theorem
[𝜇, is
𝜈]
Let
⊂𝔖ℝ:2.
[𝜇,
→
𝔽
𝜈]
ℝover
𝔖
:only
𝔽
[𝜇,
⊂
ℝrelation.
𝔽 and
be
a→
F-NV-F
be
a𝔖
F-NV-F
𝜑-cuts
whose
be adefine
𝜑-cuts
whose
define
the
𝜑-cuts
family
the𝒦family
I-V-Fs
∗ Aumann-integrable
∗ 𝜑)
∗F-NV-F
Cthen
We
use
mathematical
operations
some
of→
the
characteristics
of ∗fuzzy
𝔖
𝜑)
arenow
Lebesgue-integrable,
then
𝔖
is to
fuzzy
funcIt
isa⊂Let
aexamine
partial
order
(𝒾,
𝔖
𝜑) family
both
𝜑
∈ (0,
1].
Then
𝔖
is
𝐹𝑅-integrable
over
if whose
:I-V-Fs
[𝜇,
𝜈]define
⊂ofℝI-V-Fs
→
are of
given
by 𝔖 (𝒾) = [𝔖∗ (𝒾, 𝜑
𝔖of
[𝜇,𝜈]
grable
over
[𝜇,
Moreover,
if [49]
𝔖
𝐹𝑅-integrable
𝜈],
Symmetry
2022,
14,𝜈].
x tion
FOR
PEER
4∗ of 19
∗ (𝒾,
over
[𝜇,
𝜈].
∗ (𝒾,
∗ (𝒾,[𝜇, 𝜈] if and
∗ (. (. then
∗𝔖
∗ (.
∗ (. ∗
∗𝜑):
(𝒾,
(𝒾,
𝔖
𝜑),
𝜑)
𝜑)
are
are
Lebesgue-integrable,
Lebesgue-integrable,
then
is
𝔖
a𝔖
is
fuzzy
a→
fuzzy
Aumann-integrable
Aumann-integrable
funcfunc𝔖(𝒾)𝑑𝒾
𝔖(0,
e
over
𝜈] Let
if (𝐹𝑅)
𝔖
∈
𝔽
.𝔖Note
that,
if
(.
(.
(.
(.
functions
functions
𝔖
functions
𝔖
𝔖
functions
𝔖
,
𝜑),
𝔖
,
𝜑):
,
𝜑),
𝐾
→
ℝ
,
𝜑):
,
are
𝜑),
𝐾
called
𝔖
,
lower
are
,
𝐾
𝜑),
called
→
𝔖
and
ℝ
are
lower
,
upper
𝜑):
called
𝐾
and
→
functions
ℝ
lower
upper
are
called
and
of
functions
upper
.each
lower
of
functions
𝔖integra
.(0,
upper
o
(𝒾)
[𝔖
(𝒾,
(𝒾)
(𝒾,
[𝔖
(𝒾,
(𝒾,
𝔖
=
𝜑),
=
𝜑)]
𝜑),
for
𝔖
all
𝒾
𝜑)]
∈
𝐾.
for
Here,
all
𝒾
for
∈
Here,
𝜑and
∈
for
eac
1]in
∗∈
∗𝜑),
for
for
all
all
𝜑
𝜑
∈
1],
(0,
1],
where
where
for
ℛ
for
all
ℛ
all
𝜑
∈
𝜑
(0,
∈
1],
(0,
for
1],
where
all
for
𝜑
∈
all
ℛ
(0,
for
𝜑
ℛ
1],
∈
all
(0,
where
𝜑
1],
∈
(0,
where
ℛ
1],
where
ℛ
ℛ
denotes
denotes
the
the
collection
collection
of
denotes
Riemannian
of
denotes
Riemannian
the
the
collection
integrable
collection
denotes
integrable
denotes
of
the
Riemannian
funcof
denotes
collection
Riemannian
functhe
collection
the
integrable
of
collection
Riemannian
integrable
of
Riemannian
funcfuncRiemann
∗where
∗𝔖
∗]ℝ
all
ϕ
∈
0,
1
.
Then
is
FR-integrable
over
µ,
ν
if
and
only
if
S
,
ϕ
and
S
,
ϕ𝔖
both
are
S
(
]
[
(
)
(
)𝐾.
∗
∗
∗
∗
numbers.
𝔵,
𝔴
∈
𝔽
and
𝜌
∈
ℝ,
then
we
have
[𝜇,
𝜈].[𝜇,
We
now
use
mathematical
operations
to
examine
some
of
the
characteristics
of
fuzzy
∗
∗
∗
([
([
],
)
],
)
([
([
],
)
],
)
([
],
)
([
],
)
([
],
)
,
,
,
,
,
,
,
(𝒾)
[𝔖
(𝒾,
(𝒾)
[𝔖
(𝒾,
(𝒾,
(𝒾)
(𝒾,
[𝔖
(𝒾,
(𝒾,
is
partial
=
𝜑),
=relation.
𝔖
𝜑), for
𝔖= all𝜑)]
for
[𝜇,𝔖
all
𝜈] and
𝒾𝜑𝜑)]
∈∈[𝜇,
for
for
𝜈]all
all
and
𝒾 for
∈ [𝜇,
all𝜈]
and for all
𝜈] ⊂
[𝜇,
→
𝒦
𝜈] ⊂are
ℝ𝔖→
given
:if𝒦
[𝜇,𝔖𝜈]
by
are
𝔖
given
ℝa→
𝒦
by are
𝔖order
given
by∗𝜈],𝔖𝜑)]
𝔖 : [𝜇,
𝔖
∗ over
∗ 𝒾 ∈𝜑),
[𝜇,
∗ (𝒾,
𝑅-integrable
over
[𝜇,ℝ:𝔖
𝜈].
Moreover,
isIt⊂𝜈].
𝐹𝑅-integrable
then
(0,
1].
Then
𝔖
is
𝐹𝑅-integrable
over
[𝜇,
𝜈]
if
a
∗
∗
(𝒾,
(𝑅)
𝔖
𝜑)𝑑𝒾
,
𝜑)𝑑𝒾
e
tion
tion
over
over
[𝜇,
𝜈].
[𝜇,
∗ then
(.
(.
R-integrable
over
µ,
νover
.𝔖
Moreover,
S
is
FR-integrable
over
µ,
νnumber
[⊂
]family
](.,ℝ
9]ble,
Let
𝔖: [𝜇,𝔖𝜈]
⊂ations
ℝ
→tions
𝔽of
be
aFOR
F-NV-F
whose
𝜑-cuts
define
the
of
I-V-Fs
functions
𝔖F-NV-F.
functions
𝜑),
𝔖
,𝔖
𝜑):
𝐾
,-integrable
𝜑),
→
𝔖
ℝ
are
,then
𝜑):
called
𝐾
→
lower
ℝ
are
and
called
upper
lower
functio
and
u
(𝐹𝑅)
(𝐹𝑅)
(𝒾)𝑑𝒾
(𝒾)𝑑𝒾
(𝐹𝑅)
(𝐹𝑅)
(𝒾)𝑑𝒾
(𝒾)𝑑𝒾
(𝐹𝑅)
(𝐹𝑅)
(𝒾)𝑑𝒾
(𝐹𝑅)
(𝒾)𝑑𝒾
(𝒾)𝑑𝒾
is
fuzzy
Aumann-integrable
funcI-V-Fs.
of
I-V-Fs.
𝔖 1.
is
𝔖
tions
𝐹𝑅
is
tions
-integrable
𝐹𝑅
of
-integrable
I-V-Fs.
of
I-V-Fs.
over
tions
𝔖
is
tions
[𝜇,
of
𝐹𝑅
is
𝜈]
I-V-Fs.
[𝜇,
-integrable
𝐹𝑅
tions
of
𝜈]
if
-integrable
I-V-Fs.
if
𝔖
of
I-V-Fs.
over
𝐹𝑅
is
𝔖
over
-integrable
[𝜇,
𝔖
𝐹𝑅
𝜈]
is
[𝜇,
-integrable
𝜈]
if
if
over
over
[𝜇,
𝔖
𝜈]
𝔖
over
[𝜇,
if
𝜈]
[𝜇,
if
𝜈]
𝔖
if
𝔖
∈
𝔽→
∈fuzzy
.𝐹𝑅
𝔽(.Note
.𝔖
that,
if
if
∈
𝔽
∈: .𝔖(𝒾,
𝔽
Note
.(𝒾)𝑑𝒾
Note
that,
∈
that,
𝔽
ifℛ
.→
if
∈Note
𝔽
.
numbers.
Let
𝔵,1.
𝔴
∈
𝔽
and
𝜌fuzzy
∈F-NV-F.
ℝ,
then
we
have
∗𝔖
∗𝔖
Definition
Definition
1.
1.Then
A(0,
fuzzy
A
Definition
fuzzy
number
valued
[49]
valued
1.
A
[49]
map
fuzzy
Definition
map
A
𝔖
fuzzy
:𝐹𝑅-integrable
number
Definition
𝐾
𝔖
:and
𝐾
ℝ
number
⊂
→
[49]
valued
ℝ
𝔽→
1.
A
𝔽
[49]
valued
fuzzy
map
Definition
A
Definition
number
𝔖
map
:𝜑-cuts
𝐾
⊂
number
𝔖
ℝ
:,1.
valued
𝐾
→
[49]
⊂
1.
𝔽
ℝ
[49]
valued
A
map
𝔽
A
fuzzy
map
:[Note
number
𝐾
⊂
𝔖
:that,
𝐾
→
⊂
valued
𝔽
ℝ
valued
→
𝔽map
map
: of
𝐾𝔖
⊂
:𝔖
𝐾𝔖
→
ℝ
𝔽
is
called
is
called
For
For
each
is
each
called
is
called
F-NV-F.
F-NV-F.
For
is
each
For
called
is
each
called
F-NV-F.
F-NV-F.
For
is
We
now
use
mathematical
operations
to
examine
some
of
the
characteristics
fuzzy
∗=
∗(𝒾,
∗=
(𝒾)𝑑𝒾
(𝒾)𝑑𝒾
(𝒾)𝑑𝒾
(𝐹𝑅)
(𝐹𝑅)
(𝒾)𝑑𝒾
(𝒾)𝑑𝒾
(𝐹𝑅)
(𝒾)𝑑𝒾
(𝐹𝑅)
(𝒾)𝑑𝒾
(𝐼𝑅)
(𝐼𝑅)
(𝐼𝑅)
𝔖
𝔖
𝔖
𝔖
=
𝔖
=
=
𝔖(𝒾,
𝔖(𝒾,
𝜑)𝑑𝒾
𝜑)𝑑𝒾
𝔖
=
:(𝐼𝑅)
𝔖(𝒾,
𝜑)
∈
𝜑)
=
ℛ
∈⊂
𝔖(𝒾,
=
,𝜑)
EER
REVIEW
Symmetry
2022,
14,
Symmetry
PEER
2022,
14,
REVIEW
x
FOR
PEER
REVIEW
4family
of
19
4ℝ
of
([
([
], each
],
)c
, 19
,) For
∗∈
Theorem
2.xThen
[49]
Let
𝔖
:𝔖
[𝜇,
𝜈]
⊂
ℝ
→
𝔽
be
a
F-NV-F
whose
define
the
of
I-V-Fs
(𝒾,
(𝒾,
(𝒾,
(𝒾,
(𝒾,
𝜑)
and
𝔖
𝜑)
𝜑)
and
both
𝔖
are
𝜑)
𝜑)
and
both
𝔖
are
𝜑)
both
are
𝜑 ∈ (0,
1].[49]
𝜑
∈[49]
𝔖
1].
is
𝐹𝑅-integrable
𝜑Definition
∈number
𝔖
(0,is1].
𝐹𝑅-integrable
Then
over
𝔖
[𝜇,
is
𝜈]
over
if
[𝜇,
only
𝜈]
if
and
over
𝔖
only
[𝜇,
𝜈]
if
𝔖
if
and
only
if
𝔖
Proposition
1.
[51]
Let
𝔵,
𝔴
𝔽
.
Then
fuzzy
order
relation
“
≼
”
is
given
on
𝔽
by
(5)
[𝔵+
[𝔵]
[𝔴]
𝔴]
=
+
,
∗
∗
∗
(𝒾,
(𝒾,
(𝑅)
(𝑅)
(𝐹𝑅)
(𝒾)𝑑𝒾
=
𝜑)𝑑𝒾
,
𝔖
𝜑)𝑑𝒾
𝔖
be
an
F-NV-F.
be
an
Then
F-NV-F.
be
fuzzy
an
Then
F-NV-F.
integral
fuzzy
be
Then
an
of
integral
F-NV-F.
𝔖
fuzzy
over
of
Then
inte
𝔖
Definition
Definition
2.
[49]
Let
Definition
2.
𝔖
[49]
:
[𝜇,
𝜈]
Let
⊂
2.
Definition
𝔖
ℝ
[49]
:
→
[𝜇,
𝔽
𝜈]
Let
⊂
𝔖
ℝ
2.
:
[𝜇,
→
[49]
𝔽
𝜈]
Let
⊂
ℝ
𝔖
→
:
[𝜇,
𝔽
𝜈]
⊂
ℝ
→
𝔽
𝑅-integrable
over
[𝜇,
𝜈].
Moreover,
if
𝔖
is
𝐹𝑅-integ
∗
∗
∗
∗
∗
∗
∗
∗
∗
Proposition
1.
[51]
Let
𝔵,
𝔴
∈
𝔽
.
Then
fuzzy
order
relation
“
≼
”
is
given
on
𝔽
Let
𝔖1]
: [𝜇,by
⊂𝜑∗𝜑),
ℝ
→
𝔽∈(𝒾,
be
a∈ν𝜑)
F-NV-F
whose
the
family
of
I-V-Fs
=
𝔖
𝜑-cuts
-cuts
(𝒾)
[𝔖
(𝒾,
(𝒾,
(𝒾,
(𝒾,
(𝒾,
(𝒾,
(𝒾,
(𝒾,
(𝒾,
(𝒾,
(𝒾,
(𝒾,
(𝒾,
(𝒾,
∈𝜑(0,
∈ 1]
(0,
whose
whose
-cuts
𝜑
𝜑
-cuts
define
(0,
1]
(0,
whose
the
1]
the
family
whose
𝜑
family
-cuts
𝜑 for
∈
of
𝜑(0,
I-V-Fs
-cuts
𝜑
of
define
I-V-Fs
(0,
define
1]
𝔖
the
𝔖𝐾
family
𝜑
the
-cuts
𝜑
family
∈
of
𝜑
-cuts
define
(0,
∈𝜑)
1]
(0,
of
define
1]
whose
the
I-V-Fs
𝔖
whose
family
𝔖
family
of→
-cuts
define
define
I-V-Fs
𝔖
the
𝔖
family
I-V-Fs
of→are
𝔖
:and
⊂
:𝜑),
𝐾
⊂
→
𝒦
→
𝒦
are
are
given
given
: the
𝐾𝜑
by
⊂
ℝ
:𝜑
by
𝐾
𝒦
ℝaZof
→
are
𝒦Aumann-integrable
given
are
: the
𝐾family
⊂
given
by
ℝ
: is
𝐾→⊂
by
ℝAumann-integrab
𝒦
given
are(5)
:𝔖
𝐾give
by
⊂
: 𝐾ℝ
𝜑),
𝔖
𝜑)]
all
𝒾𝔴
∈whose
[𝜇,
𝜈]
and
for
all
→2.𝜑
𝒦[49]
are
given
𝔖
𝔖
𝜑),
𝔖
𝜑)
are
are
Lebesgue-integrable,
Lebesgue-integrable,
𝔖
𝜑),
𝔖
𝜑)
𝜑)
are
then
𝜑),
are
Lebesgue-integrable,
then
𝔖
Lebesgue-integrable,
𝔖
𝔖
𝜑)
𝔖
aℝ
is
𝜑),
fuzzy
are
aZI-V-Fs
𝔖
fuzzy
are
Aumann-integrable
𝜑)
Lebesgue-integrable,
Aumann-integrable
then
are
then
𝔖
Lebesgue-integrable,
is
𝔖
a⊂
isI-V-Fs
fuzzy
then
fuzzy
funcfunc𝔖
then
Aumann-integrable
is
a𝔖
then
fuzzy
aof
𝔖𝒦
fuzzy
is
aI-V-Fs
funcfuzzy
Aumann-in
funcAuma
𝔖𝜈]∗ (𝒾,
𝔖
𝔖
𝔖define
𝔖
𝔖
numbers.
Let
𝔵,1]
𝔽
𝜌ℝ∗is
∈
ℝ,
then
we
have
Zdefine
Z∈
∗𝜑
∗(9)
∗𝜑),
∗whose
∗𝜑
ν ∗∈
νLebesgue-integrable,
[𝔵+
[𝔴]
𝔴]
=
+
,νfor
∗be
(𝒾)
[𝔖
(𝒾,
(𝒾,
Proposition
1.Let
[51]
Let
𝔵,
𝔴
∈𝐹𝑅-integrable
𝔽
.[𝜇,
Then
fuzzy
order
relation
” 𝜈]
is
given
on
𝔽𝜈]
by
=
𝔖
𝜑)]
for
all
∈[𝔵]
[𝜇,
𝜈]
all
: [𝜇,
𝜈]
ℝ
→𝒦
are
given
by
𝔖
𝔖
[𝜇,
[𝜇,
[𝜇,
𝑅-integrable
𝑅-integrable
over
[𝜇,
𝜈].
over
𝑅-integrable
Moreover,
[𝜇,
𝜈].
if
Moreover,
over
𝔖
is
[𝜇,
𝐹𝑅-integrable
𝜈].
if𝜈]
𝔖
Moreover,
is𝔴]
over
𝔖𝜑),
isdenoted
𝜈],
𝐹𝑅-integrable
over
then
𝜈],
then
over
𝜈],
then
[𝜇,
(𝐹𝑅)
(𝒾)𝑑𝒾
(𝐹𝑅)
[𝜇,
(𝐹𝑅)
(𝐹𝑅)
(𝒾)𝑑𝒾
𝜈],
denoted
𝜈],
by
denoted
𝜈],
by
𝔖
𝜈],
by
𝔖
denoted
𝔖
by
𝔖
,the
is
given
levelwise
,𝒾𝐾.
is
given
by
levelwise
,and
is⊂
given
by
levelwise
,of
is
given
by
levelwise
by
Theorem
Theorem
2.
[49]
2.
[49]
Let
𝔖
:𝔖
[𝜇,
𝔖
:𝔖
[𝜇,
⊂
𝜈]
ℝ
→
ℝ
𝔽
→
𝔽
be
a(0,
F-NV-F
a(0,
F-NV-F
whose
whose
𝜑-cuts
𝜑-cuts
define
define
the
the
family
family
of
I-V-Fs
I-V-Fs
ϕ
be
an
F-NV-F.
be
Then
an
F-NV-F.
fuzzy
Definition
2.
Definition
[49]
Let
𝔖
2.
:≼
[𝜇,
Let
ℝ
→
𝔖
:point
𝔽
[𝜇,
⊂
ℝ
→
𝔽1]
∗if
(𝒾,
(𝑅)
(𝑅)
(𝐹𝑅)
(𝒾)𝑑𝒾
=
𝔖
𝜑)𝑑𝒾
,⊂
𝔖
𝜑)𝑑𝒾
𝔖
∗(𝒾)𝑑𝒾
∗ (𝒾,
∗⊂
∗=
∗[𝜇,
∗(𝒾,
∗if,
∗“(𝒾)𝑑𝒾
∗[49]
[𝔵]
[𝔴]
𝔵
≼
𝔴
if
and
only
≤
for
all
𝜑
∈
(0,
1],
(4)
(6)
[𝔵
[𝔵]
[
e
∗
×
𝔴]
×
,
∗
(9)
(𝒾)
(𝒾)
[𝔖
(𝒾,
[𝔖
(𝒾,
(𝒾,
(𝒾)
(𝒾)
[𝔖
(𝒾,
[𝔖
(𝒾,
(𝒾,
(𝒾)
(𝒾,
(𝒾)
[𝔖
(𝒾,
[𝔖
(𝒾,
(𝒾,
(𝒾)
(𝒾,
(𝒾)
[𝔖
(𝒾,
[𝔖
(𝒾,
(𝒾,
(𝒾,
∗
tion
tion
over
over
[𝜇,
𝜈].
[𝜇,
𝜈].
tion
tion
over
over
[𝜇,
𝜈].
[𝜇,
tion
𝜈].
over
tion
[𝜇,
over
tion
𝜈].
[𝜇,
over
𝜈].
[𝜇,
𝜈].
𝔖
𝔖
𝔖
𝔖
𝔖
𝔖
𝔖
=
=
𝜑),
𝜑),
𝔖
𝔖
𝜑)]
𝜑)]
for
=
for
all
=
all
𝒾
∈
𝜑),
𝒾
𝐾.
∈
𝔖
𝐾.
Here,
𝜑),
Here,
𝜑)]
for
=
for
𝜑)]
for
each
all
=
each
for
𝜑),
𝒾
𝜑
∈
all
∈
𝔖
𝜑
𝐾.
𝜑),
𝒾
∈
Here,
1]
∈
𝜑)]
𝔖
𝐾.
the
1]
Here,
for
for
𝜑)]
=
end
all
each
=
end
for
for
point
𝒾
∈
all
point
𝜑),
each
𝜑
∈
real
𝜑),
𝔖
𝒾
(0,
Here,
∈
𝜑
real
𝔖
𝐾.
1]
∈
𝜑)]
(0,
Here,
the
for
𝜑)]
1]
for
end
each
the
for
for
all
all
end
𝒾
each
𝜑
∈
∈
𝒾
𝐾.
(0,
point
real
∈
𝜑
𝐾.
Here,
1]
∈
(0,
Here,
real
the
for
end
the
for
each
point
end
each
𝜑
poin
∈
real
𝜑
(0,
∈
FR
S
=
(
R
)
S
,
ϕ
d
,
(
R
)
S
,
ϕ
d
=
IR
S
(9)
d
d
(
)
(
)
(
)
(
)
)
(
(
)
ϕ([where
∗
∗𝔖∗∈
∗𝜑)
∗ ],denotes
forfor
for
all all
all
𝜑
𝜑(0,
1],
(0,𝜈]
1],
where
where
for
all
∈for
all
1], 𝜑
where
∈the
(0,
1],
ℛ
ℛRiemannian
denotes
the
collection
collection
of
denotes
denotes
integrable
collection
integrabl
the co
fu
o
𝜑),if∗ 𝔖𝔖(𝒾,
𝜑)]
𝒾(𝒾,
∈∈
[𝜇,
and
forℛ
→ 𝐹𝑅-integrable
𝒦 are∗ given
by 𝔖[𝜇,(𝒾)
both
are
n⊂𝔖ℝ is
over
𝜈] =ifµ[𝔖
and∗∗(𝒾,
only
([≼ℛ
([
],= ,)𝜑
) (0,
([all
[𝔵]
[𝔴]
, ∗𝔴
, ],of)Riemannian
, ],𝜑)the
(𝒾)𝑑𝒾
𝔵all
if
and
only
if,
≤
for
(0,
1],
∗ (𝒾, 𝜑) and
(5)
∗ (𝒾,
[𝔵+
[𝔵]
+for
, 𝒾𝔴]
∗(𝒾,
∗by
µ [𝜇,
µ𝔴]
µ𝜈]
[𝜇,
[𝜇,
(𝐹𝑅)
(𝒾)𝑑𝒾
(𝐹𝑅)
(𝒾)𝑑𝒾
𝜈],
denoted
𝜈],
denoted
𝔖
by
𝔖
,[𝜇,
is
given
levelwise
,(𝒾)𝑑𝒾
isfunctions
given
by∈
levelwise
by
∗𝔖
∗define
∗ (.are
∗over
∗=
∗(𝒾,
∗[𝔴]
∗∈
𝜑)
and
𝔖
𝜑)
both
are
𝜑
∈𝜑-cuts
(0,
1].
Then
𝔖𝜈]
is
𝐹𝑅-integrable
𝜈]
ifℝ
and
only
if
𝔖
(𝒾)
(𝒾)
[𝔖
(𝒾,
[𝔖
(𝒾,
(𝒾,
(6)
befunctions
a F-NV-F
whose
the
family
of
I-V-Fs
[𝔵
[𝔵]
[
=
𝜑),
𝜑),
𝔖
𝔖
𝜑)]
𝜑)]
for
all
all
𝒾
∈
[𝜇,
∈
and
𝜈]
and
for
for
all
all
:
[𝜇,
:
𝜈]
[𝜇,
⊂
ℝ
⊂
→
ℝ
𝒦
→
𝒦
are
given
given
by
𝔖
by
𝔖
𝔖
×
𝔴]
=
×
,
[𝔵]
[𝔴]
∗
𝔵
≼
𝔴
if
and
only
if,
≤
for
all
𝜑
(0,
1],
(4)
(.
(.
(.
(.
(.
(.
(.
(.
(.
(.
(.
(.
(.
(.
(.
(𝒾
(𝑅)
(𝐹𝑅)
∗
∗
functions
𝔖
𝔖
functions
functions
𝔖
𝔖
functions
functions
𝔖
𝔖
functions
functions
𝔖
𝔖
,
𝜑),
,
𝜑),
𝔖
𝔖
,
𝜑):
,
𝜑):
𝐾
→
𝐾
ℝ
→
are
ℝ
,
are
𝜑),
called
𝔖
called
,
𝜑),
lower
,
𝜑):
𝔖
lower
𝐾
and
,
→
𝜑):
and
ℝ
upper
𝐾
are
,
→
upper
𝜑),
called
functions
𝔖
are
,
𝜑),
functions
,
called
𝜑):
lower
𝔖
𝐾
of
,
→
𝜑):
lower
and
𝔖
of
ℝ
𝐾
.
𝔖
upper
are
→
and
.
,
ℝ
𝜑),
called
upper
are
,
functions
𝜑),
𝔖
called
lower
𝔖
,
𝜑):
functions
,
𝜑):
𝐾
lower
and
of
→
𝐾
𝔖
ℝ
upper
→
.
and
of
are
ℝ
𝔖
are
upper
called
.
functions
called
lower
lower
of
and
𝔖
.
and
of
upper
𝔖
upper
.
fun
=
𝔖
𝔖
∗ (𝒾,
∗
∗
∗𝔵]
∗ 𝔵,
∗∈
∗𝜈
(𝐹𝑅)
(𝒾)𝑑𝒾
(𝐹𝑅)
tions
tions
of∗and
I-V-Fs.
of
I-V-Fs.
is
𝔖
𝐹𝑅
is
tions
-integrable
𝐹𝑅
-integrable
of𝔽∗ I-V-Fs.
over
of
𝔖[𝜇,
I-V-Fs.
is𝜈]
[𝜇,
𝐹𝑅
𝜈]
𝔖
if”relation
is
𝐹𝑅𝔖-integrable
𝔖
over
[𝜇,
𝔽given
∈𝜈]
.𝔽
Note
.ifon
Note
tha
It𝜈]∈
is∗if𝔽
aand
partial
relation.
(7)
(𝒾)𝑑𝒾
(𝐼𝑅)
=
𝔖
=
(𝒾,
Proposition
Let [𝜇,
𝔵, 𝔴
Proposition
1.
Proposition
[51]
Let
𝔴
1.𝔖
∈
[51]
𝔽”both
𝔵, 𝔴
.[𝜌.
Then
fuzzy
order
relation
“𝜑)
≼
.Let
is
Then
given
fuzzy
on
𝔽(9)
order
.tions
Then
by over
relation
fuzzy
order
“if≼-integrable
is(𝐹𝑅)
given
“ over
on
≼(𝒾)𝑑𝒾
” 𝔽∈is[𝜇,
by
𝔽
𝜑)
𝔖
are
.r Then
is 𝐹𝑅-integrable
over
only
if 𝜌.
𝔖
[𝜇,order
[𝜇, 𝜈].𝔖
Moreover,
if 1.
𝔖 [51]
is 𝐹𝑅-integrable
over
𝜈],
then
∗[𝔵]
∗ (𝒾, (𝑅)
∗ (𝒾,
It𝜑)𝑑𝒾
is
a(𝐹𝑅)
partial
order
relation.
∗ℝ
∗(𝒾)𝑑𝒾
for
all
ϕ⊂
∈
0,
1∗](𝒾,
.Theorem
For
all
ϕ
∈
0,
1,∗→
,ℝ𝜑-cuts
F
R
denotes
the
collection
of
all
FR-integrable
(=
]if(𝒾,
(𝒾,
(𝒾,
(𝑅)
(𝑅)
(𝑅)
(𝑅)
(𝑅)
(𝐹𝑅)
(𝒾)𝑑𝒾
(𝐹𝑅)
(𝒾)𝑑𝒾
(𝐹𝑅)
(𝒾)𝑑𝒾
(8)
=
=
𝔖
,
=
𝜑)𝑑𝒾
𝔖
𝔖
𝜑)𝑑𝒾
𝜑)𝑑𝒾
𝔖
,
𝜑)𝑑𝒾
𝔖
𝜑)𝑑𝒾
𝔖
𝔖
𝔖
(6)
[𝔵
[𝔵]
[
[𝜇,
(𝒾)𝑑𝒾
(𝒾)𝑑𝒾
(𝒾)𝑑𝒾
(𝐹𝑅)
(𝒾)𝑑𝒾
(𝒾)𝑑𝒾
(𝐹𝑅)
(𝒾)𝑑𝒾
(𝐹𝑅)
(𝒾)𝑑𝒾
(𝐼𝑅)
(𝐼𝑅)
(𝐼𝑅)
(𝐼𝑅)
×
𝔴]
=
×
𝔴]
,
𝑅-integrable
over
[𝜇,
𝜈].
Moreover,
if(→
𝔖
is
𝐹𝑅-integrable
over
𝜈],
then
Theorem
Theorem
[49]
2.𝒾1].
[49]
Let
Let
𝔖
: Theorem
[𝜇,
:now
𝜈]
Theorem
[𝜇,
𝜈]
ℝ
⊂
2.
ℝ
[49]
𝔽
2.
→
[49]
𝔽
Let
Let
𝔖
Theorem
:
[𝜇,
𝔖
2.
:
𝜈]
[𝜇,
Theorem
[49]
⊂
𝜈]
ℝ
2.
⊂
Let
[49]
𝔽
𝔖
→
2.
:
Let
[𝜇,
𝔽
[49]
𝜈]
𝔖
:
Let
⊂
[𝜇,
ℝ
𝜈]
𝔖
→
:
⊂
[𝜇,
𝔽
ℝ
𝜈]
→
⊂
𝔽
→
𝔽
be
a
be
F-NV-F
a
F-NV-F
whose
whose
𝜑-cuts
be
define
a
be
F-NV-F
define
a
F-NV-F
the
the
family
whose
family
whose
be
of
a
𝜑-cuts
F-NV-F
I-V-Fs
of
be
𝜑-cuts
I-V-Fs
a
define
F-NV-F
be
whose
define
a
the
F-NV-F
whose
𝜑-cuts
the
family
family
whose
𝜑-cuts
define
of
I-V-Fs
of
𝜑-cuts
I-V-Fs
define
the
family
defin
the
) = [𝔖∗ (𝒾, 𝜑), 𝔖∗ (𝒾,
𝔖
𝔖
𝔖
𝔖
𝔖
=
=
𝔖
=
𝔖(𝒾,
=
𝜑)𝑑𝒾
𝔖
=
:
𝔖(𝒾,
𝔖(𝒾,
=
𝜑)𝑑𝒾
𝜑)
𝔖
∈
:
𝔖(𝒾,
𝔖(𝒾,
ℛ
𝜑)𝑑𝒾
𝜑)
=
∈
:
,
𝔖(𝒾,
ℛ
𝔖(𝒾,
𝜑)
𝜑)𝑑𝒾
∈
,
𝜑)]
for
all
∈
[𝜇,
𝜈]
and
for
all
(𝒾,
(𝒾,
(𝒾,
(𝒾,
∗
∗
∗
∗
([
µ,
ν
]
,
ϕ
)
𝜑)
𝜑)
and
and
𝔖
𝔖
𝜑)
𝜑)
both
both
are
are
𝜑
∈2.𝜑
(0,
∈It
(0,
1].
Then
Then
𝔖
is
𝔖
𝐹𝑅-integrable
is
𝐹𝑅-integrable
over
over
[𝜇,
𝜈]
[𝜇,
if
𝜈]
and
and
only
only
if
𝔖
if
𝔖
∗
∗
([
],
)
([
],
)
,
,
We
use
mathematical
operations
to
examine
some
of
the
characteristics
of
fuzzy
∗ 𝔖𝔖∗(𝒾,
(7)
[𝔵]
(𝒾,
(𝒾, 𝜑)
(𝒾,𝜌.
(𝒾,
=
𝔖
𝜑),(𝒾,
𝔖 𝜑)
are are
Lebesgue-integrable,
Lebesgue-integrable,
𝜑),𝔵]
𝜑)
𝜑),
then
are
𝔖 then
Lebesgue-integrable,
𝔖𝜑)is
𝔖are
aisfuzzy
aLebesgue-integrable,
fuzzy
Aumann-integrable
Aumann-integrabl
then 𝔖 is
then
a fu
fu
𝔖∗ (𝒾,
𝔖∗𝜑),
𝔖∗ (𝒾,[𝜌.
is
a partial
order
relation.
(𝒾)𝑑𝒾
(𝐼𝑅)
=
𝔖
∗operations
denotes
ofifLet
all
𝐹𝑅-integrable
F[𝜇,
ble
over
[𝜇,the
𝜈].collection
Moreover,
𝔖: [𝜇,
is≼
𝐹𝑅-integrable
over
𝜈],
then
) Definition
We
now
use
mathematical
to
examine
some
ofF-NV-F.
the
characteristic
[𝔵]
[𝔵]
[𝔵]
[𝔴]
[𝔴]
[𝔴]
𝔵
𝔴
if
and
only
if,
𝔵
≼
𝔴
if
and
𝔵
only
≼
𝔴
if,
if
and
only
if,
≤
for
all
𝜑
∈
(0,
1],
≤
(4)
for
all
≤
𝜑
∈
(0,
for
1],
all
𝜑
∈
(0,
1],
(4)
F-NV-Fs
over
µ,
ν
.
[
]
be
an
be
F-NV-F.
an
F-NV-F.
Then
Then
fuzzy
be
fuzzy
an
integral
be
F-NV-F.
integral
an
F-NV-F.
of
Then
𝔖
of
over
𝔖
be
Then
fuzzy
over
an
be
fuzzy
F-NV-F.
integral
an
F-NV-F.
integral
Then
of
𝔖
Then
of
fuzzy
over
be
𝔖
an
be
fuzzy
over
integral
an
F-NV-F.
integral
of
Then
𝔖
Then
of
over
fuz
𝔖
Definition
2.
[49]
2.
[49]
Let
Definition
𝔖
𝔖
:
Definition
𝜈]
[𝜇,
⊂
𝜈]
ℝ
2.
⊂
→
ℝ
[49]
𝔽
→
2.
Let
𝔽
[49]
Definition
𝔖
Let
:
[𝜇,
Definition
𝜈]
𝔖
:
⊂
[𝜇,
2.
ℝ
𝜈]
[49]
→
⊂
2.
𝔽
ℝ
Let
[49]
→
Definition
𝔖
𝔽
Let
:
[𝜇,
Definition
𝜈]
𝔖
:
⊂
[𝜇,
ℝ
𝜈]
2.
→
⊂
[49]
2.
𝔽
ℝ
[49]
→
Let
𝔽
Let
𝔖
:
[𝜇,
𝔖
:
𝜈]
[𝜇,
⊂
𝜈]
ℝ
⊂
→
ℝ
𝔽
→
𝔽
∗
∗
∗
∗
∗
∗
∗
∗
(𝒾)𝑑𝒾
(𝒾)𝑑𝒾
(𝐹𝑅)
(𝒾)𝑑𝒾
(𝐹𝑅)
(𝒾)𝑑𝒾
(𝐼𝑅)
(𝐼𝑅)
𝔖
𝔖
=
𝔖
=
=
𝔖
𝔖(𝒾,
𝜑)𝑑𝒾
=
:
𝔖(𝒾,
𝔖(𝒾,
𝜑)
(𝒾)
(𝒾)
[𝔖
(𝒾,
[𝔖
(𝒾,
(𝒾,
(𝒾,
(𝒾)
[𝔖
(𝒾,
[𝔖
(𝒾)
(𝒾,
(𝒾,
(𝒾)
[𝔖
(𝒾)
[𝔖
(𝒾,
(𝒾,
(𝒾,
(𝒾,for
(𝒾,
numbers.
Let
𝔵,⊂𝜈]
𝔴
𝔽ℝ
𝜌𝜈].
∈
ℝ,
then
we
have
=
𝜑),
𝜑),
𝔖→
𝔖
𝜑)]
𝜑)]
for
=→
for
all
=
𝒾given
∈are
𝒾(𝒾,
[𝜇,
∈over
𝜑),
𝔖
𝜈]
[𝜇,
𝔖
and
𝜈]
=
𝜑)]
and
for
=
for
for
all
𝜑),
for
all
=
𝔖all
𝒾[𝔖
𝜑),
∈
𝜑)]
𝔖
∈ 𝜈]
[𝜇,
𝜑),
and
𝜈]
𝜑)]
𝔖all
and
for
for
𝒾𝜑)]
∈
for
all
all[𝜇,
for
all
𝒾𝜈]
∈all[𝜇
an
[𝜇,
:(𝒾,
𝜈]
[𝜇,
⊂𝜈]
ℝ⊂
→ℝ𝒦
→
𝒦
are
are
given
:𝔖
[𝜇,
given
:𝜈]
[𝜇,
by
𝔖
by
ℝ∈
⊂
𝔖
→
→and
:𝔖
[𝜇,
𝒦
are
𝜈]
are
given
:𝐹𝑅-integrable
⊂
ℝ
given
𝜈]
by
:⊂
[𝜇,
𝒦
ℝ
𝔖
by
𝜈]
→are
⊂
𝔖
𝒦
ℝ(𝒾)
given
are
𝒦
by
𝔖
given
by
𝔖
by
𝔖(𝒾,
𝔖if :𝜑
𝔖𝔖
𝔖[𝜇,
𝔖𝒦
𝔖
𝔖
(𝒾,
𝜑)
and
𝔖
𝜑)
both
are
r 𝜑[𝜇,
andFor
only
[𝜇,
[𝜇,
tion
tion
over
over
[𝜇,
[𝜇,
tion
over
tion
[𝜇,
𝜈].
[𝜇,
𝜈].
𝑅-integrable
𝑅-integrable
over
over
𝜈].
[𝜇,
Moreover,
𝜈].
Moreover,
if
if=
𝔖
is
𝐹𝑅-integrable
over
over
𝜈],
𝜈],
then
then
(9)
(9)
(9)
We
now
use
mathematical
operations
to
examine
some
of
the
characteristics
of𝒾∗[𝜇,
fuzzy
∗is
∗[𝜇,
∗𝜑),
∗𝜑)]
∗all
(7)
[𝜌.
[𝔵]
𝔵]
=∗all
𝜌.
(0,
∈𝜈]
(0,if𝜈],
1].
all
∈∗fuzzy
ℱℛ
the
collection
of
all
𝐹𝑅-integrable
FDefinition
1. [49]
number
valued
map
𝔖
⊂
ℝ𝔖
𝔽
isis𝜈].
called
F-NV-F.
For
numbers.
Let
𝔵,𝜈],
𝔴
∈𝜈],
𝔽
and
∈by
ℝ,(𝐹𝑅)
then
we
have
([
],
(𝐼𝑅)
=integrable
(𝒾,
(𝑅)
(𝑅)
(𝐹𝑅)
(𝒾)𝑑𝒾
[𝜇,
[𝜇,
(𝐹𝑅)
(𝐹𝑅)
[𝜇,1],
(𝒾)𝑑𝒾
[𝜇,
(𝒾)𝑑𝒾
(𝐹𝑅)
[𝜇,
(𝐹𝑅)
[𝜇,
(𝒾)𝑑𝒾
(𝒾)𝑑𝒾
[𝜇,
(𝐹𝑅)
[𝜇,
(𝒾)𝑑𝒾
(𝐹𝑅)
(𝒾)𝑑𝒾
(𝒾)𝑑𝒾
=
𝔖
,denotes
𝔖∈:∗for
𝜑)𝑑𝒾
𝔖denoted
𝜈],
denoted
by A
by
𝜈],
𝔖
denoted
𝔖
𝜈],
denoted
by
by𝜑
𝜈],
𝔖𝐾
denoted
𝜈],
denoted
by
𝔖
denoted
𝔖denoted
by
𝔖denotes
𝔖
,,𝜑)𝑑𝒾
is
given
,) is
given
levelwise
levelwise
by
by
,→
is∈by
given
,(𝐹𝑅)
given
levelwise
levelwise
by
,(𝒾)𝑑𝒾
is
given
by
,𝜌each
is
given
levelwise
levelwise
by
,, is],by
given
, )isdenotes
given
levelwise
levelwise
by
by ∗ of𝔖Ri
∗ (𝒾,
for
all
(0,
all
1],
𝜑
where
for
(0,
all
1],
ℛ
𝜑
where
∈
(0,
for
1],
all
ℛ
where
𝜑
∈
(0,
ℛ
1],
where
ℛ
denotes
the
denotes
collection
the
collection
of
Riemannian
the
collection
of
Riemannian
integrable
the
of
collection
Riemannian
func∗
([
],
)
([
],
)
([
],
)
([
,
,
,
∗
∗
∗
∗
It
is
a
partial
order
relation.
It
is
a
partial
It
order
is
a
partial
relation.
order
relation.
numbers.
Let
𝔵,
𝔴
∈
𝔽
and
𝜌
∈
ℝ,
then
we
have
(𝒾,
(𝑅)
(𝑅)
(𝒾)𝑑𝒾
=
𝔖
,υ𝒦
𝔖
𝜑)𝑑𝒾
𝔖
+
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and
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both
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and
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𝜑)𝜑)
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1].
(0,
1].
Then
Then
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(0,of31].
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over
Then
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𝜑
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[𝜇,
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1].
∈
and
𝐹𝑅-integrable
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Then
if(𝒾)𝑑𝒾
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1].
only
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Then
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and
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s over
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𝜈],
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sfor
𝜈].
𝜑all
∈ [𝜇,
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1](0,whose
𝜑all
define
the
I-V-Fs
𝔖
:if𝜈]
𝐾
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are
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by
∗𝐹𝑅-integrable
∗is
∗ (𝒾,
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∗ (𝒾,
∗ (𝒾,
∗𝔖
Definition
1.
[49]
A
fuzzy
valued
map
:𝔖is
𝐾𝜑)
⊂
ℝ
→
𝔽
is𝜑)
called
F-NV-F.
For
(5)
[𝔵+
[𝔵]
[𝔴]
(0,
(𝒾)𝑑𝒾
(𝒾)𝑑𝒾
(𝐼𝑅)
𝔴]
+
[34].
A
set
K
=
µ,
⊂
R
=
0,
∞,𝔖
said
be
a(𝐹𝑅)
convex
set,
if,
for
all
𝜑(𝐹𝑅)
∈
1].over
𝜑=then
∈
the
collection
of
all
F[number
]1].
(is
)ℛ
=
𝔖
=
𝔖
=
𝔖
∗∈
([ family
],
) (𝐼𝑅)
,tions
(𝒾,
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(𝑅)1], ℱℛ
(𝒾)𝑑𝒾
𝔖∗Definition
𝜑)𝑑𝒾
,denotes
𝔖(𝐼𝑅)
𝜑)𝑑𝒾
𝔖now
(𝐹𝑅)
(𝒾)𝑑𝒾
(𝒾)𝑑𝒾
(𝐹𝑅)
(𝒾)𝑑𝒾
(𝐹𝑅)
of(𝑅)
I-V-Fs.
tions
of
𝔖
I-V-Fs.
tions
is
𝐹𝑅
of
-integrable
𝔖
I-V-Fs.
is
tions
𝐹𝑅
-integrable
𝔖
of
over
is
I-V-Fs.
𝐹𝑅
[𝜇,
𝜈]
over
𝔖
if1],
is𝔴]
𝐹𝑅
over
-integrable
if
𝔖
[𝜇,
𝜈]
if
𝔖
[𝜇,
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𝔖fuzzy
if
∈𝜑-cuts
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.⊂
Note
∈
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that,
.Riemanni
Note
ifF-NV
∈𝔽
th
for
all
𝜑
∈
(0,
for
1],
all
where
𝜑
∈-integrable
(0,
where
ℛ
denotes
the
collection
the
of
collection
[𝔵+
[𝔵]
[𝔴]
=
+𝔖
,denotes
([F-NV-F
],to
)𝜈]
([
],
)over
We
use
mathematical
operations
We
now
to
examine
use
mathematical
We
some
now
use
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mathematical
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characteristics
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examine
of
fuzzy
some
examine
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characteristics
some
of
the
characteristics
of
o
(9)
,a[𝜇,
,𝜑-cuts
Theorem
Theorem
2.
[49]
2.
[49]
Let
Let
𝔖
:
[𝜇,
𝔖
:
𝜈]
Theorem
[𝜇,
⊂
𝜈]
ℝ
⊂
→
ℝ
𝔽
2.
Theorem
→
[49]
𝔽
Let
2.
𝔖
[49]
:
[𝜇,
𝜈]
Let
⊂
ℝ
:
[𝜇,
→
𝔽
𝜈]
ℝ
→
𝔽
be
a
be
F-NV-F
whose
whose
define
be
define
a
F-NV-F
the
the
family
be
family
whose
a
of
I-V
∗
∗
∗
∞)
is
said
to
be
a
convex
set,
if,
for
all
𝒾,
𝒿
∈
𝜑(𝐹𝑅)
∈
(0,
1]
-cuts
define
the
family
of
: called
𝐾𝜈],
⊂𝐹𝑅-integrable
ℝover
→𝒦
are
given
by theo𝜑
𝔖 (𝒾)
= [𝔖
𝜑),𝑅-integrable
𝔖 (𝒾, 𝜑)]over
for
all𝜈].
∈(𝐹𝑅)
𝐾.
Here,
for
𝜑
∈[𝔵+
(0,
1]
the
point
real
,Moreover,
∈
K,
ξ𝔖if∈
1each
,iswe
have
[[49]
]A
[𝜇,
[𝜇,
[𝜇,
[𝜇,
[𝜇,
(𝒾,
(𝒾,
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(𝒾)𝑑𝒾
(𝒾)𝑑𝒾
over
[𝜇,
[𝜇,𝒾Definition
𝑅-integrable
𝜈].
Moreover,
𝑅-integrable
over
𝔖
if0,whose
is
over
𝔖
𝑅-integrable
[𝜇,
𝐹𝑅-integrable
𝜈].
[𝜇,
𝐹𝑅-integrable
𝑅-integrable
𝜈].
Moreover,
𝑅-integrable
over
[𝜇,
if=
over
𝔖
𝜈].
ifend
is
[𝜇,
𝔖
Moreover,
𝜈],
over
𝜈].
is
𝜈],
then
[𝜇,
Moreover,
then
𝜈].
if𝐾
Moreover,
𝔖
is
over
if
𝐹𝑅-integrable
𝔖
over
isif𝔖
𝜈],
𝐹𝑅-integrable
𝔖(9)
is
then
then
over
𝜈],
then
over
𝜈], [𝜇,
then
𝜈],
=
=
𝔖
𝔖
𝜑)𝑑𝒾
𝜑)𝑑𝒾
,+𝐹𝑅-integrable
, 𝐹𝑅-integrable
𝔖I-V-Fs
𝜑)𝑑𝒾
𝜑)𝑑𝒾
NV-Fs
over
[𝜇,∗ (𝒾,
𝜈].𝑅-integrable
1.
fuzzy
number
valued
map
𝔖
:𝔖
⊂
ℝ
→
𝔽[𝜇,
is
F-NV-F.
For
each
(5)
[𝔵]
[𝔴]
𝔴]
,for
∗over
∗over
∗𝔖
∗Moreover,
∗∈
(0,
all
𝜑
∈
(0,
1].
For
all
𝜑
∈,𝜑)
1],
ℱℛ
den
(𝐹𝑅)
(𝒾)𝑑𝒾
(𝐹𝑅
(6)
[𝔵
[𝔵]
[is
([is
): (8)
tions
of
I-V-Fs.
tions
𝔖
I-V-Fs.
𝐹𝑅
-integrable
𝔖
is
𝐹𝑅
over
-integrable
[𝜇,
𝜈]
if
over
[𝜇,
𝜈]
𝔖
if
∈
, a,],
×
𝔴]
=
×],∗of
𝔴]
,
(8)
(8)
(8)
(8)
(𝒾,
(𝒾,
(𝒾,
(𝒾,
(𝒾,
(𝒾,
(𝒾,
(𝒾,
numbers.
Let
𝔵,∗𝔖
𝔴
∈𝔽
numbers.
Let
numbers.
𝔵,
𝔴
∈
𝔽
Let
𝔵,
𝔴
∈
𝔽
and
𝜌∈
ℝ,
then
we
have
and
𝜌
ℝ,
then
and
we
𝜌
∈
have
ℝ,
then
we
have
(𝒾)𝑑𝒾
(𝒾)𝑑𝒾
(𝒾)𝑑𝒾
(𝒾)𝑑𝒾
(𝒾)𝑑𝒾
(𝐹𝑅)
(𝐹𝑅)
(𝒾)𝑑𝒾
(𝒾)𝑑𝒾
(𝐹𝑅)
(𝐹𝑅)
(𝒾)𝑑𝒾
(𝒾)𝑑𝒾
(𝐹𝑅)
(𝐹𝑅)
(𝒾)𝑑𝒾
(𝒾)𝑑𝒾
(𝐹𝑅)
(𝐹𝑅)
(𝒾)𝑑𝒾
(𝒾)𝑑𝒾
(𝐼𝑅)
(𝐼𝑅)
(𝐼𝑅)
(𝐼𝑅)
(𝐼𝑅)
(𝐼𝑅)
(𝐼𝑅)
(𝐼𝑅)
𝜑),
𝔖
𝜑)
𝜑),
are
𝔖
Lebesgue-integrable,
𝜑)
𝜑),
are
𝔖
Lebesgue-integrable,
𝜑)
𝜑),
are
𝔖
Lebesgue-integrable,
then
𝜑)
𝔖
are
is
then
Lebesgue-integrable,
a
fuzzy
𝔖
is
Aumann-integrable
then
a
fuzzy
𝔖
is
Aumann-integrable
a
then
fuzzy
𝔖
Aumann-in
funcfuzzy
𝔖
𝔖
𝔖
𝔖
𝔖
𝔖
𝔖
𝔖
𝔖
𝔖
=
=
𝔖
𝔖
=
=
=
𝔖(𝒾,
𝔖(𝒾,
=
𝜑)𝑑𝒾
𝔖
𝜑)𝑑𝒾
:
𝔖(𝒾,
:
𝔖
𝔖(𝒾,
𝜑)
=
𝜑)
=
ℛ
∈
𝔖(𝒾,
ℛ
=
𝜑)𝑑𝒾
𝔖(𝒾,
,
:
𝜑)𝑑𝒾
,
𝔖(𝒾,
𝔖
=
𝜑)
:
𝔖(𝒾,
∈
=
=
𝔖(𝒾,
ℛ
𝜑)
=
∈
𝜑)𝑑𝒾
𝔖(𝒾,
ℛ
𝔖
,
:
𝜑)𝑑𝒾
𝔖(𝒾,
𝔖
:
𝔖(𝒾,
=
∈
=
ℛ
𝜑)
𝔖(𝒾,
∈
𝔖(𝒾,
ℛ
𝜑)𝑑𝒾
𝜑)𝑑𝒾
𝔖(𝒾
,
:
∗
∗
∗
∗
(9)
∗
∗
∗
∗
([
([
],
)
)
([
],
)
([
],
)
([
],
)
([
],
)
(0,
[𝜇,
,
,
,
,
,
,
=
∞)
is
said
to
be
a
convex
set,
if,
for
all
𝒾,
𝒿
∈
ition
3.
[34]
A
set
𝐾
=
𝜐]
⊂
ℝ
(𝒾)
[𝔖
(𝒾,
[𝔖
(𝒾,
(𝒾,
(𝒾)
[𝔖
(𝒾)
[𝔖
=
=∗𝔖
𝜑),
𝔖𝒦ℝ
𝔖
𝜑)]
𝜑)]
for
all
=
all
𝒾𝔖∈point
𝒾[𝜇,
∈by
𝜑),
𝜈]
[𝜇,
=real
𝔖
and
𝜈](𝒾,
and
fo𝜑
:𝔖
[𝜇,
:𝜈]
[𝜇,
⊂𝜈]
ℝ⊂(𝒾,
→
ℝ𝜑)]
𝒦
→𝒦
are
are
given
given
[𝜇,
by𝐾.
𝜈]
𝔖
by
⊂
𝔖I-V-Fs
ℝ: (𝒾)
[𝜇,
→
𝒦
𝜈]
⊂
are
ℝ
given
by
are
𝔖given
bygiven
𝔖family
𝔖
(𝒾)
[𝔖
(.For
(𝒾)𝑑𝒾
(𝐼𝑅)
𝔖1].
=𝔖all
𝔖
for
all
𝒾:.𝐹𝑅-integrable
∈
each
𝜑
∈(𝒾,
(0,
1]
the
end
∗ (𝒾,
functions
𝔖𝜑∗ (.∈for
,, (0,
𝜑),
𝔖=𝜑
,∈
𝜑):
𝐾1].
→
ℝ
lower
and
upper
functions
of
𝔖
𝔖all
[𝔵for
[𝔵]
[𝒦
×
=→
×
𝔴]
, for
∗ (𝒾,
∗ (𝒾,
𝜑1],(0,
∈called
(0,
whose
-cuts
define
the
:∗𝜑),
𝐾
⊂
→
are
(𝑅)
(𝑅)
𝔖
𝔖
𝜑)𝑑𝒾
(0,
(0,
for∗ (𝒾,
all𝜑)𝑑𝒾
all
1].
(0,
allfor
𝜑
∈For
𝜑are
all
∈
𝜑ℱℛ
∈ 1]
For
ℱℛ
𝜑(𝒾,
∈𝜑𝜑),
1],
ℱℛ
denotes
denotes
collection
of
all
the
ofofHere,
collection
𝐹𝑅-integrable
F-𝔴]
of[𝜇,
all
𝐹𝑅-integrable
FFNV-Fs
over
𝜈].
∗denotes
(9) (9)
([
], [𝜇,
)tion
([
], the
)tion
([ 𝜑),
],=tion
)collection
, 1],
, (0,
,the
(𝒾)𝑑𝒾
(𝐼𝑅)
=∗𝜈].
𝔖
∈∗ K.
(10)
tion
over
over
[𝜇,
𝜈].
over
[𝜇,
𝜈].
over
[𝜇,
𝜈].
(6)
(𝒾,
(𝒾,
(𝒾,
(𝒾,
[𝔵]
[called
𝔖
are
𝜑),
𝔖
Lebesgue-integrable,
𝜑)
are
Lebesgue-integrable,
then
𝔖of
is𝔖(7)
a. then
fuzzy
𝔖
Auman
is a
𝔖
𝔖
×
𝔴]
×𝜑)
𝔴]
,all
∗[𝔵
[0, 1], we
have
∗
∗
∗
∗
∗
(0,
[𝜇,
=
∞)
is
said
to
be
a
convex
set,
if,
for
all
𝒾,
𝒿
∈
Definition
3.
[34]
A
set
𝐾
=
𝜐]
⊂
ℝ
(.
(.
[𝜌.
[𝔵]
functions
𝔖
,
𝜑),
𝔖
,
𝜑):
𝐾
→
ℝ
are
lower
and
upper
functions
𝔵]
=
𝜌.
(5)isif𝜈]
[𝔵+
[𝔵+
[𝔵+
[𝔴]
[𝔵]
[𝔴]
[𝔵]
[𝔴]
=
+
,𝔖(𝒾)𝑑𝒾
=
+𝔖
𝔴]
=
, ,and
+1]
(𝒾)
[𝔖∗𝜑(𝒾,
(𝒾,
(𝐼𝑅)
=𝔴]
𝜑),
for
all
𝒾,𝐹𝑅-integrable
∈
𝐾.
Here,
for
each
𝜑
∈
(0,
point
(𝒾,
(𝒾,,[𝜇,
(𝒾,
(𝒾,
=[𝜇,
𝔖𝔖 (𝒾)𝑑𝒾
∗ ∈𝔖
ξ(𝑅)
+
−
(∈1𝔴]
)𝔖over
∗ξ
∗𝜑
∗only
∗𝔖
∗(5)
∗if(𝒾
𝜑)
𝜑)
and
𝔖[𝜇,
𝜑)
𝜑)
both
∈[𝔵]
𝜑
(0,
1].
(0,
1].
Then
Then
𝔖
is𝔖
𝐹𝑅-integrable
is
𝜑(𝑅)
(0,
1].
Then
∈over
[𝜇,
(0,
𝔖
𝜈]
1].
[𝜇,
Then
𝐹𝑅-integrable
if𝔖
and
only
𝔖𝔖
is
if𝔖the
𝐹𝑅-integrable
if,,∗𝜑)𝑑𝒾
over
𝔖
𝜈]∗and
over
ifreal
and
only
𝜈]𝜑)𝑑𝒾
a
NV-Fs
over
NV-Fs
𝜈].where
over
NV-Fs
𝜈].
over
[𝜇,
𝜈].
∗(𝒾,
(𝒾,
(𝒾,
(𝒾,
(𝒾,
(𝒾,
(𝒾,
(𝒾,
(𝒾,
(𝒾,
(𝑅)
(𝑅)
(𝑅)
(𝑅)
(𝑅)
(𝑅)
(𝑅)
(𝑅)
(𝑅)
(𝑅)
(𝐹𝑅)
(𝐹𝑅)
(𝒾)𝑑𝒾
(𝒾)𝑑𝒾
(𝐹𝑅)
(𝐹𝑅)
(𝒾)𝑑𝒾
(𝐹𝑅)
(𝒾)𝑑𝒾
(𝐹𝑅)
(𝒾)𝑑𝒾
(𝒾)𝑑𝒾
=
𝔖𝜑)]
𝜑)𝑑𝒾
,(𝐹𝑅)
=
=
𝔖
𝔖
𝜑)𝑑𝒾
𝔖
𝜑)𝑑𝒾
=
𝜑)𝑑𝒾
,the
=([(𝑅)
=
𝜑)𝑑𝒾
𝔖
𝜑)𝑑𝒾
𝔖end
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𝔖
,𝔖
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𝔖
𝔖
𝔖
𝔖
𝔖the
𝔖
[𝜌.
[𝔵]
𝔵]
=
𝜌.(𝑅)
∗=
∗𝜑)𝑑𝒾
∗(𝒾,
∗𝜑)𝑑𝒾
∗ (𝒾,
∗ function
over
[𝜇,
tion
𝜈].
[𝜇,
𝜈].collection
(9)
∈
𝐾.
for
all
all
𝜑 ∈we
𝜑(0,
∈have
1],
(0,[𝜇,
1],
where
forℛ
all
ℛ
∈
all
(0,
𝜑
∈
where
1],
for
where
all
ℛ
for
𝜑=
all
(0,
𝜑
1],
(0,
where
1],
for
where
ℛ
for
all
𝜑
∈
𝜑𝔖
(0,
1],
(0,
1],
where
where
ℛ
ℛ,collection
denotes
the
the
collection
collection
of
denotes
of
Riemannian
denotes
collection
integrable
collection
integrable
denotes
funcdenotes
of
functhe
Riemannian
integrable
denotes
denotes
funcof
Riemannian
the
the
collection
integrable
collection
integrable
of(𝒾,Riema
funcof𝔖Rie
(10)
([for
],⊂
],denotes
) 1],
([
],ℛ
)([
) fuzzy
([the
],ℛ
)([
],over
)Riemannian
([
], ∗,) of
],integrable
)Riemannian
(𝒾)𝑑𝒾
(𝒾)𝑑𝒾
(𝐼𝑅)
(𝐼𝑅)
,𝜑([
,)ℝ
,∈
,∈],Riemannian
,all
,∈of
=
𝔖
𝔖
∗Then
be
an
F-NV-F.
integral
of
𝔖
over
[49]
𝜈]
→
𝔽(0,
[0,
𝐾,
𝜉For
∈for
𝒾𝒿
(.
(.
(7)
[𝜌.
[𝔵]
functions
𝔖
,
𝜑),
𝔖
,
𝜑):
𝐾
→
ℝ
are
called
lower
and
upper
functions
of
𝔖
.
𝔵]
=
𝜌.
− Definition
𝜉)𝒿
(0,
[𝜇,
1].
all1],
𝜑 ∈2.(0,
1], Let
ℱℛ([𝔖,: [𝜇,
denotes
the
collection
of
all
𝐹𝑅-integrable
F=
∞)
Definition
3.
[34]
𝐾,𝔖
=
𝜐]then
⊂𝔖
ℝ
∗ [𝔵]
[𝜇,
[𝜇,
], )
𝑅-integrable
over
over
[𝜇,
𝜈].
[𝜇,
Moreover,
𝜈].
Moreover,
ifa2.
𝔖
𝑅-integrable
if[𝔵]
is
over
𝔖
𝐹𝑅-integrable
is𝔖
𝐹𝑅-integrable
𝜈].
Moreover,
[𝜇,
over
𝜈].
over
if
Moreover,
𝜈],
is(𝒾)𝑑𝒾
𝐹𝑅-integrable
iswhose
𝐹𝑅-integ
over
(6)
(6)
[𝔵
[𝔵
[𝔵
[ 𝔴]
[[𝜇,
[𝔵]
[set
×
×
×
𝔴]
=
×
,(𝒾)𝑑𝒾
𝔴]
=
𝔴]
𝔴]
=
×
𝔴]
Theorem
2.
Theorem
[49]
Let
2.
Theorem
𝔖
:over
[49]
[𝜇,
𝜈]
Let
⊂
2.
𝔖
ℝ
Theorem
[49]
:⊂
→
[𝜇,
𝜈]
Let
⊂
:of
[𝜇,
→
[49]
𝔽
𝜈]
Let
⊂
ℝ
𝔖
→
:[𝜇,
[𝜇,
𝔽,over
𝜈]
⊂
→
𝔽
be
F-NV-F
be
whose
a(𝒾)𝑑𝒾
F-NV-F
𝜑-cuts
be𝔽(9)
aA
F-NV-F
define
𝜑-cuts
the
whose
be
family
athen
define
F-NV-F
of
the
I-V-Fs
define
of
the
I
∈
𝐾.
(𝐹𝑅)
(𝒾)𝑑𝒾
(𝐹𝑅)
(𝐹𝑅)
(𝒾)𝑑𝒾
(𝐹𝑅)
(𝐹𝑅)
(𝒾)𝑑𝒾
(𝐹𝑅)
(𝐹𝑅)
(𝒾)
tions
tions
of
I-V-Fs.
of I-V-Fs.
𝔖all(𝐹𝑅)
is
𝔖𝜑 tions
𝐹𝑅
is (0,
-integrable
𝐹𝑅
tions
of-integrable
I-V-Fs.
of
I-V-Fs.
over
over
is
tions
[𝜇,
𝔖
𝐹𝑅
𝜈]
[𝜇,
is
tions
-integrable
of
𝜈]
if𝑅-integrable
𝐹𝑅
if[49]
-integrable
of
𝔖
𝔖
is𝔖
𝔖
tions
𝐹𝑅
over
[𝜇,
is
tions
𝜈]
-integrable
of
𝐹𝑅
[𝜇,
I-V-Fs.
𝜈]
-integrable
of
if𝔖ℝ
over
𝔖
𝔖
is
𝔖
over
[𝜇,
𝐹𝑅
is
𝜈]
-integrable
𝐹𝑅
𝜈]
ifwhose
𝔖
over
over
𝔖
[𝜇,
𝜈]
[𝜇,
𝜈]
ifof
𝔖
∈
𝔽
∈
.if𝑅-integrable
𝔽𝔽Note
.(10)
Note
that,
that,
if
if
∈
𝔽if
.-integrable
∈Note
.ℝ
Note
that,
if
that,
∈𝜈],
𝔽
ifif.𝜑-cuts
∈Note
𝔽if𝔖
.family
Note
that,
if
th
(0,
for
∈
1].
For
𝜑𝔖
∈
1],
ℱℛ
denotes
the
collection
all
𝐹𝑅-integrable
F(9)
(9)
(9)𝔖𝜑-c
[𝜇, 𝜈],
(𝒾)𝑑𝒾
be
an
F-NV-F.
Then
fuzzy
integral
over
Definition
2.I-V-Fs.
𝔖
:asaid
[𝜇,
𝜈]
ℝ
→
𝔽I-V-Fs.
denoted
by
,(1
isall
given
levelwise
by
([
], I-V-Fs.
)Let
, (𝐹𝑅)
(𝐼𝑅)
𝔖
𝜉𝒾
+
−
𝜉)𝒿
+
(0,
(0,
(0,
[𝜇,
[𝜇,
[𝜇,
[0,
=
∞)
is
=
said
∞)
to
be
is
=
convex
to
∞)
be
set,
a
is
convex
said
if,
for
to
all
set,
be
𝒾,
a
if,
𝒿
convex
for
∈
all
set,
𝒾,
𝒿
if,
∈
for
all
𝒾,
𝒿
∈
Definition
3. (0,
[34]1],A ℱℛ
set
3.𝔖Definition
[34]
𝐾
=
A
set
𝜐]
𝐾
⊂
3.
=
ℝ
[34]
A
𝜐]
set
⊂
ℝ
𝐾
=
𝜐]
⊂
ℝ
𝒾𝒿
𝐾,
𝜉
∈
1],
we
have
𝜈].
Definition
1.
[49]
A
fuzzy
number
valued
map
𝔖
:
𝐾
⊂
ℝ
→
𝔽
is
called
F-NV-F.
For
each
∈
(0, 1].Definition
For(𝒾)𝑑𝒾
all
𝜑
∈
denotes
the
collection
of
all
𝐹𝑅-integrable
FDefinition
4
[34].
The
S
:
µ,
υ
→
R
is
called
a
harmonically
convex
function
on
µ,
υ
if
[
]
[
]
([𝜈].
, ], ∗)
∗
∗
∗
∗
∗
∗
∗
∗
∗
∗
∗
Theorem
2.
[49]
Theorem
Let
𝔖
:
2.
[𝜇,
[49]
𝜈]
⊂
Let
ℝ
→
𝔖
:
𝔽
[𝜇,
𝜈]
⊂
ℝ
→
𝔽
be
a
F-NV-F
whose
be
a
F-NV-F
𝜑-cuts
whose
define
NV-Fs
over
[𝜇,
∈
𝐾.
(𝒾)
[𝔖
(𝒾,
(𝒾)
[𝔖
(𝒾,
(𝒾,
(𝒾)
[𝔖
(𝒾,
[𝔖
(𝒾,
(𝒾,
(𝒾,
(10)
=
𝜑),
=
𝔖
𝜑)]
=
for
𝔖
all
𝜑)]
𝒾(𝒾)
𝜑),
for
𝔖
=
[𝜇,
𝜈]
all𝔖
and
𝒾is
∈
𝜑),
[𝜇,
for
𝔖aall
𝜈]
all
and
𝒾𝜑)]
∈ [f
:For
[𝜇,
⊂
ℝ
: (𝐼𝑅)
→
[𝜇,
𝒦
𝜈]
are
:Aumann-integrable
[𝜇,
→
given
𝒦
𝜈]
by
are
𝔖→
given
𝒦
𝜈]
by
are
⊂
𝔖
given
→
𝒦
by
are
given
by
𝔖
𝔖
𝔖
𝔖
𝔖
Definition
1.
[49]
A
fuzzy
number
valued
map
𝔖
:a(𝒾,
⊂
ℝ∈then
→
𝔽
is
called
F-NV-F
(𝒾, 𝔖
(𝒾,we
(𝒾,we
(𝒾,
(𝒾,
(𝒾,
(𝒾,
(𝒾,
(𝒾,
(𝒾,
(𝒾,
(𝒾,
(𝒾,
(𝒾,
[𝜇,
(𝐹𝑅)
(𝒾)𝑑𝒾
𝜑),
𝜑),(𝒾,
𝔖 𝜑)
are
are
Lebesgue-integrable,
Lebesgue-integrable,
𝜑),
𝔖
𝜑),
𝜑)
𝔖1].
are
𝜑)
then
Lebesgue-integrable,
are
then
𝔖
𝜑),
Lebesgue-integrable,
is
𝔖
a=
𝜑),
is
fuzzy
a𝜑)
𝔖⊂
fuzzy
are
𝜑)
Lebesgue-integrable,
Aumann-integrable
then
are
Lebesgue-integrable,
then
𝜑),
is[𝜇,
𝔖
𝜑),
agiven
𝔖
𝔖
fuzzy
is
𝜑)
aℝ
func𝜑)
are
fuzzy
Aumann-integrable
functhen
are
Lebesgue-integrable,
Aumann-integrable
Lebesgue-integrable,
𝔖
then
is
a𝜑),
𝔖𝔖
fuzzy
is
fuzzy
funcAumann-integrable
funcAumann-integrable
then
𝔖
afor
is
fuzzy
fuzzy
funcAu
𝔖∗ (𝒾,
𝔖𝐾,
𝔖1],
𝔖𝜉∈
𝔖
𝔖
𝔖ℝ
𝜈],
denoted
by
𝔖
,:𝔽
is
levelwise
by
∗=
∗𝐹𝑅-integrable
∗𝐾
∗𝜑)]
(7)
[𝜌.
[𝜌.
[𝜌.
[𝔵]
[𝔵]
[𝔵]
be
an
F-NV-F.
Then
fuzzy
of
𝔖
over
Definition
2.
[49]
Let
𝔖
:ℝ
[𝜇,
ℝ
→
∗all
∗have
∗𝜈]
∗=
∗⊂
∗𝔖
𝔵]
𝔵]
𝜌.
𝜌.
=
𝜌.
(0,
(0,
for
for
all
𝜑
𝜑
∈
1].
(0,
For
all
𝜑
∈
𝜑𝜑
∈
1],
1],
ℱℛ
ℱℛ
denotes
denotes
the
the
collection
collection
of
of
all
𝐹𝑅-integrable
FF[0,
[0,
𝐾,
𝜉𝜑)
∈ have
𝐾,
∈(0,
1],
we
have
(𝒾)𝑑𝒾
(𝒾)𝑑𝒾
(𝒾)𝑑𝒾
(𝒾)𝑑𝒾
(𝒾)𝑑𝒾
(𝐼𝑅)
(𝐼𝑅)
(𝐼𝑅)
(𝐼𝑅)
(𝐼𝑅)
(𝐼𝑅)
([
([
],
],⊂
)map
=
𝔖
𝔖𝜈]
=
𝔖
𝔖
=
=
=
𝔖integral
𝔖
, (𝒾)𝑑𝒾
,) (𝒾)𝑑𝒾
(1
𝜑
∈[𝜇,
(0,
1]
whose
-cuts
define
the
family
of
𝔖
:(7)
𝐾𝔵]
⊂
ℝ
→
𝒦
are
given
𝜉𝒾
+1.
−
𝜉)𝒿
er a[𝜇,
𝜈].∗ 𝜉 ∈ [0, 1],
d
harmonically
convex
function
on
𝜐]
if[49]
Definition
Aall
fuzzy
number
valued
𝔖
:=
𝐾
⊂I-V-Fs
ℝ
→=
𝔽
isall
called
F-NV-F.
For
each
∗ (𝒾,∗∗by (𝒾,
𝒾𝒿 𝔖
!
∗ (𝒾
(𝒾,
(𝒾,
(𝒾
(𝑅)
(𝑅)
(𝑅)
(𝑅)
(𝑅)
(𝑅)
(𝐹𝑅)
(𝐹𝑅)
(𝒾)𝑑𝒾
(𝒾)𝑑𝒾
(𝐹𝑅)
(𝒾)𝑑𝒾
(𝐹𝑅)
(𝒾)𝑑𝒾
=
𝔖
𝔖
𝜑)𝑑𝒾
𝜑)𝑑𝒾
,
,
=
𝔖
𝜑)𝑑𝒾
=
𝔖
𝜑)𝑑𝒾
𝜑)𝑑𝒾
𝔖
,
𝔖
𝔖
𝔖
𝜑
∈
(0,
1]
whose
𝜑
-cuts
define
the
family
of
I-V-Fs
𝔖
:
𝐾
⊂
ℝ
→
𝒦
are
∗
∗
tion
tion
over
over
[𝜇,
𝜈].
[𝜇,
𝜈].
tion
over
tion
[𝜇,
over
𝜈].
[𝜇,
𝜈].
tion
over
tion
[𝜇,
over
𝜈].
[𝜇,
𝜈].
tion
tion
over
over
[𝜇,
𝜈].
𝜈].
(𝒾)
[𝔖
(𝒾,
(𝒾)
(𝒾,
[𝔖
(𝒾,
=
𝜑),
𝔖
=
𝜑)]
𝜑),
for
all
:
[𝜇,
𝜈]
⊂
ℝ
→
:
𝒦
[𝜇,
𝜈]
are
⊂
given
ℝ
→
𝒦
by
𝔖
are
given
by
𝔖
𝔖
𝔖
∗
∗
∗
∗𝔖
[𝜇,
(𝐹𝑅)
(𝒾)𝑑𝒾
[𝜇,
the
∗𝔖(𝒾,
𝜈],
denoted
by
𝔖
,Then
given
by
∗ ∗point
∗𝔖only
(𝒾,
(𝒾,
(𝒾,
(𝒾,
and
𝔖
𝜑)
𝜑)
and
𝜑)
are𝜑)
and
𝜑
∈
(0,
1].
𝜑𝔖
Then
∈
(0,
𝔖1].
Then
𝐹𝑅-integrable
∈
(0,
𝔖1].
𝜑
𝐹𝑅-integrable
∈over
(0,
𝔖
1].
islevelwise
[𝜇,
𝐹𝑅-integrable
Then
𝜈]
over
if(8)
and
𝔖
[𝜇,
is
only
𝜈]
𝐹𝑅-integrable
over
if and
𝔖
only
𝜈]𝜑)end
ifgiven
over
and
𝔖
only
[𝜇,by
𝜈]ifreal
if𝔖and
if 𝔖bo
NV-Fs
NV-Fs
over
over
𝜈].
𝜈].
(0,
[𝜇,the
=
∞)
is[𝜇,1]
said
to
be
a Fconvex
set,
if,is𝜑for
for
all
𝒾,𝒾𝜑)
𝒿is∈𝒾𝒿
∈is
34]
A set
𝐾=
𝜐]
⊂
ℝ𝜐]
∗ (𝒾,
∗both
∗
(𝒾)𝑑𝒾
(𝐹𝑅)
(𝒾)𝑑𝒾
(𝐼𝑅)
(𝒾)
[𝔖
(𝒾,
(𝒾,
collection
of
all
𝐹𝑅-integrable
𝔖
=
𝔖
=
𝜑)𝑑𝒾
:
𝔖(𝒾,
ℛ
,
𝔖
=
𝜑),
𝜑)]
all
𝐾.
Here,
for
each
𝜑
∈
(0,
1]
[𝜇,
[𝜇,
([ , ], 4.
) denotes
([
],
)
𝜑
∈
(0,
whose
𝜑
-cuts
define
the
family
of
I-V-Fs
𝔖
:
𝐾
⊂
ℝ
→
𝒦
are
,
ition
[34]
The
𝔖:
→
ℝ
is
called
a
harmonically
convex
function
on
𝜐]
if
∗
𝒾𝒿
𝒾𝒿
∗
(1
𝜉𝒾
+
−
𝜉)𝒿
(0,
∞)
is said
to
be
a
convex
set,
if,
for
all
𝒾,
𝒿
∈
Definition 3. [34] A set 𝐾 = [𝜇, 𝜐] ⊂ ℝ =
(𝒾)
[𝔖
(𝒾,
(𝒾,
𝔖
=
𝜑),
𝔖
𝜑)]
for
all
𝒾
∈
𝐾.
Here,
for
each
𝜑
∈
(0,
1]
the
end
S
,
(11)
≤
1
−
ξ
S
+
ξS
(
)
(
)
∗ 𝐾.
∗
∈
𝐾.
∈
∈
𝐾.
(10)
(10)
(10)
e
have
(𝒾,
𝜑)
an
𝜑
∈
(0,
1].
Then
𝜑
∈
𝔖
(0,
is
1].
𝐹𝑅-integrable
Then
𝔖
is
𝐹𝑅-integrable
over
[𝜇,
𝜈]
if
and
over
only
[𝜇,
𝜈]
if
𝔖
if
and
only
∗
(8)
(0,
(𝒾)𝑑𝒾
(𝐹𝑅)
(𝒾)𝑑𝒾
(𝐼𝑅)
=
∞)
is
said
to
be
aℱℛ
convex
set,
if,
for
all
𝒿all
∈
n(1
3.−
[34]
A set
=The
𝜐]
(.
(.([
𝔖
=
𝔖
=
𝜑)𝑑𝒾
: over
𝔖(𝒾,
𝜑)
∈𝔽𝐹𝑅-integrable
ℛthen
,each
[𝜇,
[𝜇,
[𝜇,
∗ then
functions
𝔖
,1],
𝜑),
𝔖
,:,𝜑):
𝐾
→
ℝ
are
called
lower
and
upper
functions
of
𝔖
.)isall
+𝐾𝜉𝔖(𝒿),
≤Definition
𝜉)𝔖(𝒾)
𝑅-integrable
𝑅-integrable
[𝜇,
𝜈].
𝑅-integrable
over
Moreover,
[𝜇,
𝜈].
over
𝑅-integrable
if𝒾,
Moreover,
𝔖∗collection
[𝜇,
is
𝜈].
𝐹𝑅-integrable
if
Moreover,
over
𝔖
is
𝐹𝑅-integrable
𝜈].
over
Moreover,
𝔖
is
𝐹𝑅-integrable
𝜈],
if
then
𝔖
𝜈],
over
over
(11)
Definition
1.
A
fuzzy
number
valued
map
1.over
Definition
[49]
𝔖+
𝐾
⊂
fuzzy
𝔽
number
[49]
A
fuzzy
valued
number
map
𝔖
valued
:[𝜇,
𝐾
⊂
ℝ
map
𝔽denotes
𝔖
:],is
𝐾
⊂
ℝ
is
called
F-NV-F.
For
called
F-NV-F.
is
called
For
F-NV-F.
(𝒾)
[𝔖
(𝒾,
(𝒾,
([collection
],
, [𝜇,
(0,
(0,
(0,
(0,
(0,
𝔖1].
=
𝜑),
𝔖
𝜑)]
for
all
𝒾1.
∈𝜑
𝐾.
Here,
for
each
𝜑
∈
(0,
1]
the
end
real
∗all
for
all
for
all
𝜑⊂
∈
𝜑ℝ
(0,
∈
(0,
1].
For
For
all
for
all
𝜑
all
for
∈
𝜑(0,
𝜑
∈harmonically
∈
𝜑(0,
∈1],
1].
(0,
for
ℱℛ
1].
For
all
for
all
𝜑
∈
all
(0,
∈
for
1].
∈
all
(0,
For
𝜑
1].
1],
ℱℛ
(0,
ℱℛ
1].
all
For
1],
all
ℱℛ
𝜑each
1],
∈([if𝔖(𝒾,
1],
ℱℛ
denotes
denotes
the
of
denotes
all
of
denotes
all
𝐹𝑅-integrable
𝐹𝑅-integrable
the
the
collection
Fof
denotes
all
the
𝐹𝑅-integrable
collection
denotes
the
𝐹𝑅-integrable
the
of)𝜈],
collection
all
Fof
𝐹𝑅-inte
F-all𝔖
o𝐹F
ξ𝜑
+
1the
−
ξcollection
((.
)∈
(1
(1
𝜉𝒾
𝜉𝒾
𝜉𝒾
−
𝜉)𝒿
+
𝜉)𝒿
[𝜇,
[𝜇,
[0,
4.
𝔖:
𝜐]
→
ℝ
is
a+
convex
function
on
if
∗Definition
([
],AFor
],ℝ
)→
([
([
],∈
)𝐾
],𝜑
)∈
],→
)([collection
)([F],of→
𝜉[𝜇,
∈[49]
1],
we
,)−
,𝜑):
,𝜐]
,ℱℛ
,→
,point
𝒾𝒿
(.For
functions
,1],
𝜑),
𝔖
,𝜑⊂
→
ℝ
called
lower
and
of
2.[34]
[49]
2.𝐾,[49]
Let
Let
𝔖
: Theorem
[𝜇,
𝔖Definition
:𝜈]
[𝜇,
Theorem
⊂
𝜈]have
ℝ
⊂2.
→ℝ[34]
[49]
𝔽3.
→called
2.
𝔽
Let
[49]
𝔖
Theorem
:aLet
𝜈]
𝔖
Theorem
:𝜉)𝒿
⊂
[𝜇,
2.
ℝ
𝜈]
[49]
→
⊂
𝔽
2.
ℝ
Let
[49]
→
𝔖
𝔽
Theorem
[𝜇,
𝜈]
𝔖
Theorem
:−
⊂
[𝜇,
ℝ𝜈]
2.
→
[49]
𝔽
2.
ℝ
[49]
→
Let
𝔽aaI-V-Fs
Let
𝔖
:are
[𝜇,
𝔖aset,
:𝜈]
[𝜇,
⊂𝜈]
ℝ
⊂
ℝ
𝔽
→all
𝔽
be
aA
be
F-NV-F
F-NV-F
whose
whose
𝜑-cuts
𝜑-cuts
be
define
a𝔖:Let
F-NV-F
define
be(1
athe
F-NV-F
the
family
whose
family
whose
of
𝜑-cuts
be
of
𝜑-cuts
F-NV-F
define
be
F-NV-F
define
the
whose
family
the
whose
𝜑-cuts
family
of
be
𝜑-cuts
define
aof
be
F-NV-F
I-V-Fs
a F-NV-F
define
thefunctions
family
whose
the
whose
family
of𝜑-cuts
I-V-Fs
𝜑-cu
ofd.I
(0,
(0,
[𝜇,
[𝜇,
=
=
∞)
is
said
is
said
to
be
to
aI-V-Fs
be
convex
convex
set,
if,
for
if,
for
all
𝒾,I-V-Fs
𝒿→
𝒾,
∈
𝒿upper
∈
Definition
3.
[34]
A
set
set
𝐾[𝜇,
=
𝐾
=
𝜐]
⊂
𝜐]
ℝ
⊂
ℝ
∗∞)
∗
1],Theorem
weTheorem
have
(1
+
𝜉𝔖(𝒿),
𝔖
≤
−
𝜉)𝔖(𝒾)
for
all
(0,1]NV-Fs
1],whose
where
ℛ
denotes
the
collection
of
Riemannian
integrable
func(11)
𝜑𝜑∈∈(0,
𝜑
-cuts
define
𝜑
the
∈
(0,
family
1]
whose
𝜑
of
∈
I-V-Fs
(0,
𝜑
1]
-cuts
whose
𝔖
define
𝜑
-cuts
the
family
define
of
the
I-V-Fs
family
𝔖
of
I-V-Fs
𝔖
:
𝐾
⊂
ℝ
→
𝒦
are
given
by
:
𝐾
⊂
ℝ
𝒦
:
𝐾
are
⊂
ℝ
given
→
𝒦
by
are
gi
[𝜇,
(8)
𝑅-integrable
over
𝑅-integrable
[𝜇,
𝜈].
Moreover,
over
[𝜇,
𝜈].
if
𝔖
Moreover,
is
𝐹𝑅-integrable
if
𝔖
is
𝐹𝑅-integrable
over
𝜈],
then
ove
(𝒾)𝑑𝒾
(𝐹𝑅)
(𝒾)𝑑𝒾
(𝐼𝑅)
𝒾𝒿
(.
𝔖𝐾
=NV-Fs
𝔖 [𝜇, and
𝔖(𝒾,
𝜑)𝑑𝒾
: 𝔖(𝒾,
𝜑)
functions
𝔖∗ (.
, 𝜑),
𝔖over
ℝ are
called
lower
upper
functions
of(𝒾)𝑑𝒾
𝔖
. ∈ ℛ=
([ 𝜈].
)
NV-Fs
over
over
[𝜇,
[𝜇,
NV-Fs
NV-Fs
over
[𝜇,, 𝜑):
𝜈].
[𝜇,
NV-Fs
𝜈].∗→NV-Fs
over
[𝜇,
over
𝜈]. [𝜇,
over
𝜈].
𝜈]. =
, ], 𝜈].
(𝒾)𝑑𝒾
(𝒾)𝑑𝒾
(𝐼𝑅)
([ (𝐼𝑅)
= (𝐼𝑅)
=
𝔖4. [34]
𝔖
= (𝐼𝑅)
𝔖h
, ], ) , ∗𝔖
𝜉𝒾
−
𝜉)𝒿
∗ (𝒾,𝒾𝒿
∗ (𝒾, (𝒾)
∗ (𝒾,
∗(𝒾)
∗ (𝒾,
[𝜇,
Definition
The
𝔖:
𝜐]
ℝ𝒾𝜑),
called
aall
[0,
[0,
𝐾,
𝜉+
𝐾,
∈
𝜉given
∈set,
1],
we
1],
we
have
have
∈
𝐾.
(10)
∗[𝜇,
∗(𝒾)
∗[𝜇,
(𝒾)
(𝒾)
[𝔖
(𝒾,
[𝔖
(𝒾,
(𝒾,
(𝒾)
[𝔖
[𝔖
(𝒾,
(𝒾)
[𝔖
(𝒾,
[𝔖
(𝒾,
(𝒾,
(𝒾,
(𝒾)
[𝔖
(𝒾,
[𝔖
(𝒾,
(𝒾,
=
=
𝜑),
𝜑),
𝔖
𝔖
𝜑)]
𝜑)]
for
=
for
all
all
𝒾(𝒾,
=
∈
𝜑),
𝒾S
∈
𝔖
𝜈]
[𝜇,
𝜑),
and
𝜈]
𝜑)]
𝔖
and
for
=
for
𝜑)]
for
all
all
=
for
𝒾𝐾.
𝜑),
∈all
[𝜇,
𝔖]by
𝒾∗.of
𝜑),
𝜈]
∈∈
𝜑)]
[𝜇,
𝔖
and
𝜈]
for
𝜑)]
=
and
all
=𝜑
for
∈→
all
𝜑),
[𝜇,
𝔖
𝜈]
∈Sis
[𝜇,
𝔖
and
𝜑)]
𝜈]
for
𝜑)]
and
for
fa
:𝔖[𝜇,
:𝜈]
[𝜇,
⊂
𝜈]
ℝ=
⊂→[𝔖
ℝ
→
𝒦
are
given
:(1
𝜈]
by
:𝒾𝒿
⊂
[𝜇,
𝔖
by
ℝ𝜈]
𝔖
→
⊂
𝒦
ℝ
→
are
:given
[𝜇,
are
𝜈]
by
⊂
[𝜇,
ℝ
𝜈]
→
by
⊂
𝒦
ℝ
𝔖𝜑)]
→
are
𝒦
given
:𝔽
are
[𝜇,
:be
given
by
[𝜇,
⊂
𝜈]
𝔖
ℝ
⊂
by
→
ℝ
𝔖
𝒦
→
𝒦
are
are
given
given
𝔖
by
𝔖
𝔖
𝔖
𝔖
𝔖
𝔖
𝔖
(0,
[𝜇,
=𝔖
∞)
said
toIf𝒦
be
aare
convex
if,
for
all
𝒾,
𝒿∗𝒦
∈
an
F-NV-F.
Then
integral
of
𝔖
over
Definition
[49]
𝔖
:𝔖[𝜇,
𝜈]
⊂
ℝ
→
for
all
𝒾𝔖
∈is(𝒾)
𝜐].
(11)
is
reversed
then,
𝔖
∗𝜉)𝔖(𝒾)
∗([
∗)𝜈]
∗ all
∗fuzzy
∗𝒾
∗then,
(𝒾,
[𝔖
(𝒾,
(𝒾)
(𝒾,
[𝔖
(𝒾,
(𝒾,
for
all
𝜑
∈Let
(0,
1],
where
ℛ
denotes
the
collection
Riemannian
integrable
func𝔖
𝔖
𝜑),
𝔖
𝜑)]
for
all
𝒾(𝒾)
∈
=
𝐾.
𝜑),
for
𝔖
=
each
𝜑
𝜑),
∈
for
(0,
𝔖
all
1]
𝒾
the
𝜑)]
∈
𝐾.
end
for
Here,
all
point
𝒾
for
∈
real
each
Here,
𝜑
for
(0,
each
1]
the
end
∈
(0,
point
1]
the
real
end
po
(𝒾)𝑑𝒾
for
all
,2.
∈
µ,
υ:given
,Definition
ξ+
∈
0,
1
,
where
≥
0
for
all
∈
µ,
υ
If
(11)
is
reversed
is
tions
of
I-V-Fs.
𝔖
is
𝐹𝑅
-integrable
over
[𝜇,
𝜈]
𝔖
∈
𝔽
.
Note
that,
if
[Here,
](𝐹𝑅)
[
]
(
)
[
(1
∗
∗
∗
(1
𝜉𝔖(𝒿),
𝔖
≤
−
],
(11)
,
+
−
𝜉)𝒿
∈
𝐾.
∗ (𝒾,𝜉𝒾
∗if
∗
(10)
𝒾𝒿
(𝒾,
(𝒾,
(𝒾,
(𝒾,
(𝒾, 𝜑)𝑑𝒾
(𝒾,
(𝑅)
(𝑅)
(𝑅)
(𝑅)
(𝑅)
(𝑅)
(𝑅)o
(𝐹𝑅)
(𝐹𝑅)
(𝒾)𝑑𝒾
(𝐹𝑅)
(𝐹𝑅)
(𝒾)𝑑𝒾
=
=
𝔖[𝜇,
𝜑)𝑑𝒾
𝔖
, [𝜇,
𝜑)𝑑𝒾
𝔖𝔖,on
𝜑)𝑑𝒾
𝜑)𝑑𝒾
𝔖if, (𝒾,
𝔖(𝑅)
𝔖
, 𝜑)𝑑
𝔖is(𝒾)𝑑𝒾
𝔖harmonically
be
Then
fuzzy
integral
2.
[49] ∗Let
𝔖
: [𝜇,on
𝜈]
⊂
ℝ
→ifon
𝔽=
[𝜇,[𝜇,
[𝜇,
[𝜇,
Definition
Definition
4.1],
[34]
The4.∗𝜉𝒾
Definition
𝔖:
[34]
The
𝜐]
→𝜉)𝒿
𝔖:
ℝ4.∈[𝜇,
[34]
The
→∈
ℝ[𝜇,𝔖:
𝜐]
→
is𝜐]called
a𝜉𝒾
is
harmonically
called
aℝ
harmonically
convex
called
a(10)
function
convex
function
𝜐]
convex
function
𝜐]an
if∗F-NV-F.
𝜐]
∗(𝒾)𝑑𝒾
∗𝔖
∗=
∗ 𝜑)𝑑𝒾
(1
(1
+
−
+
−
𝜉)𝒿
∗
∗
∗
∗
∗
∗
𝐾.
[𝜇,
[𝜇,
[0,
𝜐].
𝒾𝒿
(𝐹𝑅)
(𝒾)𝑑𝒾
𝒾,𝜑
𝒿
∈
𝜐],
𝜉
∈
where
𝔖(𝒾)
≥
0
for
all
𝒾
𝜐].
If
(11)
is
reversed
then,
𝔖
𝜈],
denoted
by
𝔖
,
is
given
levelwise
by
𝒾𝒿
𝒾𝒿
(.[34]
(.
(.said
(.1].
called
ais
harmonically
concave
function
on
µ,
υ[34]
.convex
(𝒾,
(𝒾,
(𝒾,
(𝒾,
(𝒾,
(𝒾,
(𝒾,
(𝒾,
(𝒾,
(𝒾,
(𝒾,
[(0,
]if
(0,
(0,
(0,
(0,
(0,
(0,
(0,
[𝜇,
[𝜇,
[𝜇,
[𝜇,
[𝜇,
[𝜇,
be
an
F-NV-F.
Then
fuzzy
integral
of
over
Definition
2.
[49]
Let
𝔖
: Definition
[𝜇,
𝜈]
⊂
ℝ
→
𝔽
(𝐹𝑅)
(𝒾)𝑑𝒾
𝔖Definition
functions
𝔖
functions
, 𝜑):
𝐾
→
ℝ
are
,ℝ
𝜑),
and
upper
,𝔖
𝜑):
𝐾
,𝜑
𝜑),
functions
→
𝔖
ℝ
are
,a𝔖
𝜑):
called
of
𝐾
𝔖
→
.collection
lower
ℝ
are
and
called
upper
lower
functions
and
upper
of
𝔖
functions
.ifbe
of
𝔖
𝜑)
and
𝔖
𝜑)
𝜑)
both
both
are
𝜑)
are
and
𝜑)
𝔖
and
𝔖
𝜑)
are
both
and
𝜑)
are
𝔖
and
𝜑)
𝔖
both
𝜑)
are
𝜑(𝒾
𝜑(0,
∈functions
1].
(0,
Then
is𝔖, 𝜑),
𝐹𝑅-integrable
is
𝜑𝔖
∈
𝐹𝑅-integrable
𝜑
1].
∈[34]
Then
over
1].
over
𝔖
Then
[𝜇,
𝜈]
[𝜇,
𝜑
𝐹𝑅-integrable
𝔖
∈
if𝜈]
(0,
and
is
𝜑
if
𝐹𝑅-integrable
1].
and
∈
only
(0,
Then
only
if
1].
over
𝔖∞)
if
𝔖
Then
𝔖
[𝜇,
is
over
𝐹𝑅-integrable
𝔖
[𝜇,
is
∈
if
𝜑A
and
(0,
𝐹𝑅-integrable
𝜈]
∈
only
if
and
1].
Then
over
only
Then
𝔖
[𝜇,
over
if∞)
𝜈]
𝔖
[𝜇,
if
and
𝐹𝑅-integrable
𝜈]
only
if
and
if𝜑)
only
over
𝔖
over
if𝜑)
[𝜇,
𝔖
[𝜇,
and
and
only
only
if𝒾,
𝔖
=
=
∞)
is
said
to
be
to
be
convex
=
aset
=
∞)
set,
is𝔖
set,
said
if,
is
for
said
if,
=
to
for
all
be
to
all
𝒾,
∞)
a⊂
=
be
𝒿convex
𝒾,
∈
a(𝑅)
is
𝒿convex
∞)
said
=
∈(𝒾,
set,
is
to
∞)
said
set,
be
if,
is
for
a.to
if,
convex
said
for
all
ato
all
𝒾,
convex
𝒿be
set,
∈
a𝔖
𝒿ifif,
conve
∈
set,
Definition
3.
3.
A(0,
set
A
Definition
set
𝐾
=
Definition
𝐾
=called
𝜐]
⊂
3.
𝜐]
ℝ
[34]
⊂
3.
[34]
A
set
A
Definition
set
𝐾
=
𝐾
Definition
[34]
=
𝜐]
3.
⊂
𝜐]
[34]
set
ℝ
⊂
3.
A
𝐾
ℝ
=
A
𝐾
𝜐]
set
=
⊂
𝐾𝐹𝑅-integrable
𝜐]
=
⊂
ℝ
𝜐]
ℝ
tions
of(0,
I-V-Fs.
𝔖
is
𝐹𝑅
-integrable
over
[𝜇,
𝜈]
if
𝔖
∈
𝔽 if𝜈]
Note
that,
if(𝒾,
for
all
𝜑
∈
1],
where
ℛ
denotes
of
Riemannian
integrable
∗.for
(𝒾, Then
∗ (.
∗lower
∗3.
∗(.
∗([
∗ℝ
∗both
∗𝜈]
∗𝔖
∗bo
𝜑),
𝔖∗1].
𝜑)
are
Lebesgue-integrable,
then
𝔖
is
a[𝜇,
Aumann-integrable
𝔖∗∈(𝒾,
(1
],𝜑)
)all
,𝜈]
𝜉𝒾(0,
+
−
𝜉)𝒿
(𝐹𝑅)
(𝒾)𝑑𝒾
denoted
by
𝔖
,𝔖
is
given
by
(𝒾,
(𝑅)
(𝑅)
(𝐹𝑅)
(𝐹𝑅)
(𝒾)𝑑𝒾
=𝜑 𝔖
𝔖𝔖
=
𝜑)𝑑𝒾
,)func𝔖∗([(𝒾,
(0,
(0,
(0,
∈all
𝐾.
𝐾.
for
all
for
all
𝜑 fuzzy
∈𝜈],
𝜑(0,
∈
1].
(0,
1].
For
For
𝜑
for
∈𝜑(0,
all
∈the
1],
𝜑
∈∗func1],
ℱℛ
for
(0,([
ℱℛ
all
1].
For
(0,
all
1].
∈For
all
1],
𝜑
ℱℛ
∈ ([
ℱℛ
denotes
the
the
collection
collection
of
of
denotes
all
𝐹𝑅-integrabl
𝐹𝑅-integ
the
de,c−
(10)
(10)
∗ (𝒾,
([
],𝜑
],denotes
) levelwise
],all
],≤
) (1
, (𝒾)𝑑𝒾
,)∈
, 1],
, 𝜑)𝑑𝒾
(9)
𝒾𝒿
𝒾𝒿
𝒾𝒿
[𝜇,
edtion
aall
harmonically
concave
function
on
𝜐].
(1
∗
𝜉𝒾
+
−
𝜉)𝒿
(1
(1
[0,
[0,
[0,
[0,
[0,
[0,
[0,
[𝜇,
[0,
[𝜇,
[𝜇,
(𝐹𝑅)
(𝒾)𝑑𝒾
𝜉𝒾
𝜉𝒾
+
+
−
𝜉)𝒿
−
𝜉)𝒿
𝐾,
𝜉
𝐾,
∈
𝜉
∈
1],
we
1],
we
have
have
𝐾,
𝜉
𝐾,
∈
𝜉
∈
1],
we
1],
𝐾,
we
have
𝜉
have
∈
𝐾,
𝜉
1],
∈
𝐾,
we
𝜉
1],
have
∈
we
1],
have
we
have
for
𝒾,
𝒿
∈
𝜐],
𝜉
∈
1],
where
𝔖(𝒾)
≥
0
for
all
𝒾
∈
𝜐].
If
(11)
is
reversed
then,
𝔖
𝜈],
denoted
by
𝔖
,
is
given
levelwise
by
𝒾𝒿
(𝐹𝑅)
(𝒾)𝑑𝒾
(𝒾,
(𝒾,
over
[𝜇,over
𝜈].over
I-V-Fs.
𝔖
𝐹𝑅
-integrable
over
𝜈]
ifis
𝔖𝜈].
∈
. 𝐹𝑅-integrable
Note
iffunc𝜑),NV-Fs
𝔖
𝜑)
are
Lebesgue-integrable,
then
𝔖
isMoreover,
a[𝜇,
fuzzy
Aumann-integrable
𝔖
[𝜇,
[𝜇,
[𝜇,
[𝜇,
[𝜇,
[𝜇,
[𝜇, 𝜈],
[𝜇, 𝜈t
𝑅-integrable
[𝜇,
[𝜇,
𝑅-integrable
𝜈].
Moreover,
𝑅-integrable
if over
𝔖iftions
is𝔖𝔖
[𝜇,
𝐹𝑅-integrable
over
is∗of
𝜈].
𝑅-integrable
𝐹𝑅-integrable
[𝜇,
Moreover,
𝜈].
𝑅-integrable
Moreover,
over
over
ifisover
𝔖
[𝜇,
isover
if𝜈],
𝜈].
𝐹𝑅-integrable
𝔖[𝜇,
𝜈],
then
[𝜇,
Moreover,
is
𝑅-integrable
𝜈].
𝐹𝑅-integrable
𝑅-integrable
over
𝔖 [𝜇,
is
over
ifover
[𝜇,
𝐹𝑅-integrable
𝔖
𝜈],
𝜈].
[𝜇,
then
Moreover,
𝜈].
𝐹𝑅-integrable
𝜈],
then
over
if 𝔖
ifover
is𝔖
𝜈],𝔽
𝐹𝑅-integrable
is
then
𝜈],
over
over
(1
(1
+
𝜉𝔖(𝒿),
+(1
𝜉𝔖(𝒿),
+
𝜉𝔖(𝒿),
≤
𝔖
−
𝜉)𝔖(𝒾)
≤
−
𝜉)𝔖(𝒾)
−ifover
𝜉)𝔖(𝒾)
(11)
(11)
(11)thenthat,
34]𝑅-integrable
The
𝔖: [𝜇,
ℝ 𝜈].
is Moreover,
called
a𝔖harmonically
convex
function
on
𝜐]then
if≤Moreover,
NV-Fs
over
over
[𝜇,
𝜈].
[𝜇,
𝜈].
NV-Fs
over
NV-Fs
[𝜇,
𝜈].
over
∈ 𝜐]
𝐾. →
(10)
(1 [𝜇,
𝜉𝒾𝜈]+𝔖:
𝜉𝒾
𝜉𝒾Let
−
𝜉)𝒿
+∗ ℝ
−
𝜉)𝒿
+a(1
−
𝜉)𝒿
be
an
F-NV-F.
Then
fuzzy
integral
be⊂
of
an𝔖
𝔖
F-NV-F.
over
be
Then
an
F-NV-F.
fuzzy
integral
Then
fuzzy
of
𝔖(𝒾)𝑑𝒾
integral
over of
2. [49]
Let
𝔖function
: [𝜇,
⊂(1
Definition
ℝ
→𝜐]
𝔽over
2.
Definition
[49]
𝔖
2.
: [𝜇,
[49]
𝜈]
⊂
Let
ℝ→
𝔖
:=
𝔽[𝜇,
𝜈]
ℝ
→
𝔽 on
[𝜇,
[𝜇,
Definition
4. [34]
The
→
is
called
harmonically
convex
function
𝜐]
if
[𝜇,
a harmonically
concave
on
𝜐].
(𝒾)𝑑𝒾
(𝒾)𝑑𝒾
(𝒾)𝑑𝒾
(𝐼𝑅)
(𝐼𝑅)
(𝐼𝑅)
(𝐼𝑅)
=
=
𝔖
=
𝔖
=
𝔖
(1 − Definition
tion
𝜈].
𝒾is+called
𝜉)𝒿
(8)
(𝒾)𝑑𝒾
(𝐹𝑅)
(𝒾)𝑑𝒾
(𝐼𝑅)
[𝜇,
[0,
𝔖
=
𝔖
𝔖(𝒾,
𝜑)𝑑𝒾
:
𝔖(𝒾,
𝜑)
∈
ℛ
,
𝒾𝒿
𝒾𝒿
𝒾𝒿
𝒾𝒿
𝒾𝒿
𝒾𝒿
𝒾𝒿
for
all
𝒾,
𝒿
∈
𝜐],
𝜉
∈
1],
where
𝔖(𝒾)
≥
0
for
(𝒾,
(𝒾,
𝜑), 𝔖
𝜑) are function
Lebesgue-integrable,
then 𝔖 is a fuzzy Aumann-integrable
func([ , ], )
∗
[𝜇, 𝜐]
[𝜇,
n 4. [34][𝜇,
The
𝔖:
ℝ (𝐹𝑅)
is called
a𝔖
harmonically
convex
on
if
𝒾𝒿 →by
(𝒾)𝑑𝒾
(𝐹𝑅)
(𝒾)𝑑𝒾
(𝐼𝑅)
𝔖,𝜐]
=
𝔖by
= (10)
𝔖(𝒾,
𝜑)𝑑𝒾
:𝐾.
𝔖(𝒾, (10)
𝜑) ∈
ℛ([ ,
∈
𝐾.
∈
𝐾.
∈
𝐾.
∈
𝐾.
∈
𝐾.
∈
𝐾.
∈
(𝒾)𝑑𝒾
[𝜇,
[𝜇,
(𝐹𝑅)
(𝒾)𝑑𝒾
(𝐹𝑅)
(𝒾)𝑑𝒾
(10)
(10)
𝜈],
denoted
𝔖
𝜈],
denoted
by
𝜈],
denoted
𝔖
by
𝔖
,
is
given
levelwise
by
is
given
levelwise
,
is
given
levelwise
by
(𝒾)𝑑𝒾
(𝒾)𝑑𝒾
(𝐼𝑅)
(𝐼𝑅)
Theorem
2.
[49]
Let
𝔖
:
[𝜇,
𝜈]
⊂
ℝ
→
𝔽
be
a
F-NV-F
whose
𝜑-cuts
define
the
family
of
I-V-Fs
=
𝔖
=
𝔖
[𝜇,
[0,
[𝜇,
[0,
[𝜇,
[0,
[𝜇,
[𝜇,
[𝜇,
𝒾𝒿
[𝜇,
[𝜇,
[𝜇,
[𝜇,
[𝜇,
for all
∈ all𝜐],
𝒾,Definition
𝜉𝒿 ∈∈
for1],
𝜐],
all
𝜉where
∈[34]
𝒿tion
∈[34]
1],
𝔖(𝒾)
𝜐],
where
𝜉≥[𝜇,
∈𝔖:
0Definition
𝔖(𝒾)
for
1],
all
≥
where
𝒾+
0ℝ
∈
𝔖(𝒾)
all
𝜐].
𝒾(𝑅)
If
≥∈∗harmonically
(11)
0aset
for
𝜐].
is
all
If
(11)
𝒾+
∈
then,
reversed
𝜐].
If
𝔖A
(11)
then,
is
reversed
then,
𝔖
Definition
4.𝒾,
The
The
𝔖:
𝜐]
→
𝜐]
ℝ
is
called
is
called
a𝔖
convex
convex
function
function
on
𝜐]
if
𝜐]
if
is
called
a+
harmonically
concave
function
on
+
𝜉𝔖(𝒿),
𝔖 𝒾, 𝒿 for
≤
−
𝜉)𝔖(𝒾)
over
(11)
(8)
(1
(1
(1
(1
(1
(0,
(0,
(0,
[𝜇,
[𝜇,
∗A
∗:(𝒾,
∗(1
∗𝜐]
𝜉𝒾
𝜉𝒾
𝜉𝒾
𝜉𝒾
𝜉𝒾
𝜉𝒾
+𝜉𝒾
−
𝜉)𝒿
−
𝜉)𝒿
+
−
𝜉)𝒿
𝜉)𝒿
−
+
𝜉)𝒿
−to
+
𝜉)𝒿
−
𝜉)𝒿
=
=
∞)
∞)
is
said
is
said
be
to
a∗be
convex
a[𝜇,
=
convex
set,
∞)
set,
if,
for
said
if,
for
all
∞)
to𝔖
al
𝒾𝜐b
Definition
3.for
[34]
3.
[34]
set
A
𝐾harmonically
Definition
=
𝐾reversed
=,=
𝜐]
⊂
𝜐]
Definition
⊂
3.
ℝ
[34]
set
3.
[34]
𝐾
=
A
set
𝐾
⊂
ℝ
⊂
ℝ
(𝒾)𝑑𝒾
(𝐹𝑅)
(𝒾)𝑑𝒾
(𝐼𝑅)
𝔖
=
𝔖
=
𝔖(𝒾,
𝜑)𝑑𝒾
𝔖(𝒾,
𝜑)
∈
ℛ𝔖
,=
([
],(1
)𝔖
, ,=
(𝒾,
(𝒾,
(𝒾,
(𝒾,
(𝒾,
(𝒾,
(𝒾,
(𝒾,
(𝒾,
(𝒾,
(𝑅)
(𝑅)
(𝑅)
(𝑅)
(𝑅)
(𝑅)
(𝑅)
(𝑅)
(𝑅)
(𝑅)
(𝑅)
(𝑅)
(𝑅)
(𝒾)𝑑𝒾
(𝐹𝑅)
(𝒾)𝑑𝒾
(𝒾)𝑑𝒾
(𝐹𝑅)
(𝐹𝑅)
(𝒾)𝑑𝒾
(𝐹𝑅)
(𝐹𝑅)
(𝒾)𝑑𝒾
=4.
=
𝔖(𝐹𝑅)
𝔖𝜈].
𝜑)𝑑𝒾
,→
=
=𝔖
𝔖
𝜑)𝑑𝒾
𝜑)𝑑𝒾
𝜑)𝑑𝒾
𝔖
𝜑)𝑑𝒾
,ℝ
=−
𝔖
𝔖
𝜑)𝑑𝒾
𝔖
𝜑)𝑑𝒾
𝔖
𝜑)𝑑𝒾
,on
𝜑)𝑑𝒾
=
𝔖
𝜑)𝑑𝒾
𝔖
𝜑)𝑑𝒾
𝜑)𝑑𝒾
, is=(𝑅)
, (0,(𝑅)
𝔖(𝒾)𝑑𝒾
𝔖(1
𝔖
𝔖
𝔖
𝔖
𝔖
𝔖
(1 −(𝐹𝑅)
∗ (𝒾,
∗𝜑)𝑑𝒾
∗(𝒾,
∗(𝒾)𝑑𝒾
∗ (𝒾,
∗(𝒾)𝑑𝒾
∗𝔖
∗𝜑)𝑑𝒾
𝜉𝒾 + (𝐹𝑅)
𝜉)𝒿
+
𝜉𝔖(𝒿),
𝔖
≤, (1
−
𝜉)𝔖(𝒾)
(11)
𝒾𝒿
∗𝜉[0,
Theorem
2.
[49]
Let
𝔖
:
[𝜇,
𝜈]
⊂
ℝ
→
𝔽
be
a
F-NV-F
whose
𝜑-cuts
define
the
family
of
I-V-Fs
[𝜇,
[𝜇,
[𝜇,
is
called
a
is
harmonically
called
a
harmonically
is
concave
called
a
function
harmonically
concave
on
function
concave
𝜐].
on
function
𝜐].
on
𝜐].
(1
for
all
𝜑
∈
(0,
1],
where
ℛ
denotes
the
collection
of
Riemannian
integrable
func[0,
[0,
[0,
𝜉𝒾
+
−
𝜉)𝒿
𝐾,
𝜉
𝐾,
∈
∈
1],
we
1],
we
have
have
𝐾,
𝜉
∈
1],
𝐾,
we
𝜉
∈
have
1],
we
have
(0,
(0,
(0,
(0,
(𝒾)
[𝔖
(𝒾,
(𝒾,
for
all
𝜑
∈
for
(0,
all
1].
𝜑
For
∈
for
(0,
all
all
1].
𝜑
𝜑
∈
For
∈
(0,
all
1],
for
1].
𝜑
all
ℱℛ
∈
For
𝜑
∈
all
1],
(0,
𝜑
ℱℛ
1].
∈
For
1],
all
ℱℛ
𝜑
∈
1],
ℱℛ
denotes
the
denotes
collection
the
denotes
of
collection
all
𝐹𝑅-integrable
the
of
collection
denotes
all
𝐹𝑅-integra
the
Fof
all
colle
𝐹
(1
=
𝜑),
𝔖
𝜑)]
for
all
𝒾
∈
[𝜇,
𝜈]
and
for
all
:
[𝜇,
𝜈]
⊂
ℝ
→
𝒦
are
given
by
𝔖
𝔖
+
𝜉𝔖(𝒿),
𝔖
≤
−
𝜉)𝔖(𝒾)
[𝜇,
([
],
)
(11) ([ ℛ
s called a harmonically convex function on
𝜐]∗ if
([ , ], )
([ , ], )
([ , ], )
, ], )
𝒾𝒿 all
𝒾𝒿 ,𝜑 ∈ (0, 1], where
for
denotes
the
collection
of Riemannian
integr
𝜉𝒾𝔖(𝒾)
+ (1
−0𝜉)𝒿
([+, 𝜉𝔖(𝒿),
],(9)
) (9)
(9)
(9)
(9)
(1
(1
+
𝜉𝔖(𝒿),
𝔖
𝔖
≤
≤
−
𝜉)𝔖(𝒾)
−
𝜉)𝔖(𝒾)
∗
[𝜇,
(11)
(11)
𝜐], 𝜉 ∈ [0, 1], where
≥(𝒾)𝑑𝒾
for
all
𝒾
∈
𝜐].
If
(11)
is
reversed
then,
𝔖
(8)
(8)
(𝒾)𝑑𝒾
(𝒾)𝑑𝒾
(𝒾)𝑑𝒾
(𝐹𝑅)
(𝐹𝑅)
(𝒾)𝑑𝒾
(𝐹𝑅)
(𝒾)𝑑𝒾
(𝐼𝑅)
(𝐼𝑅)
(𝐼𝑅)
𝔖
𝔖
𝔖
=
𝔖
=
𝔖(𝒾,
𝜑)𝑑𝒾
=
:
𝔖(𝒾,
𝜑)
𝔖
∈
ℛ
=
=
,
𝔖
𝔖(𝒾,
𝜑)𝑑𝒾
=
:
𝔖(𝒾,
𝔖(𝒾,
𝜑)
𝜑)𝑑𝒾
∈
ℛ
:
𝔖(𝒾,
𝜑)
,
∈
ℛ
(𝐹𝑅)
(𝒾)𝑑𝒾
Theorem
2.
[49]
Let
𝔖
:
[𝜇,
𝜈]
⊂
ℝ
→
𝔽
be
a
F-NV-F
whose
𝜑-cuts
define
the
family
of
I-V-Fs
tions
of
I-V-Fs.
𝔖
is
𝐹𝑅
-integrable
over
[𝜇,
𝜈]
if
𝔖
∈
𝔽
.
Note
that,
if
(𝒾)
[𝔖
(𝒾,
(𝒾,
NV-Fs
over
[𝜇,
𝜈].
NV-Fs
[𝜇,
over
[𝜇,
NV-Fs
𝜈].
over
[𝜇,
𝜈].
∗ℝ
=
𝜑),
𝔖
𝜑)]
for
all
𝒾function
[𝜇,
for
all
: [𝜇,
⊂
ℝ
→
are
given
by
𝔖
𝔖1],
(0,
([
],of
)a
([convex
],[𝜇,
)convex
([if
],on
for
all
𝜑over
∈𝔖:
(0,
where
ℛ𝜉𝒾
denotes
collection
Riemannian
integrable
funcfor
all
∈
(0,
for
1].
all
For
𝜑
all
∈∗are
(0,
𝜑
1].
∈
For
1],
𝜑
∈called
1],
denotes
denotes
of
,a
, the
, the
([
],(1
([
],function
)(𝐹𝑅)
([𝜈]
],and
)collection
,if
(1
[𝜇,
[𝜇,
[𝜇,
[𝜇,
[𝜇,
[𝜇,
[𝜇,
[𝜇,
[𝜇,
[𝜇,
𝒾𝒿
𝒾𝒿
𝒾𝒿
𝜉𝒾
+
+
𝜉)𝒿
𝜉)𝒿
(𝒾,
(𝒾,
Definition
[34]
4.
[34]
The
The
Definition
𝔖:
Definition
𝜐]
→
𝜐]
ℝ
→
4.NV-Fs
[34]
ℝis
4.
Definition
[34]
The
The
Definition
𝔖:
𝔖:
4.
Definition
𝜐]
[34]
→
𝜐]𝜑
4.the
ℝ
The
→
𝔖:
4.
The
[34]
𝜐]
𝔖:
The
→
ℝ
𝜐]
𝔖:
→
𝜐]
→
ℝ
called
is
called
a)of
harmonically
a𝜈].
harmonically
is
convex
called
is
convex
called
function
harmonically
function
harmonically
is ℝ
on
called
on
isall
𝜐]
called
convex
aℱℛ
harmonically
if
𝜐]
is
convex
a(0,
harmonically
a∈ℱℛ
harmonically
on
on
𝜐]
function
𝜐]𝔽convex
funct
[0,
[𝜇,
𝜑)
𝔖𝜐].
𝜑)
both
𝜑 ∈ (0, 1]. Then
[𝜇,
𝜈]
if𝜈]
and
only
if,𝒦
𝔖
for𝔖Definition
allis𝒾,𝐹𝑅-integrable
𝒿 ∈4.[𝜇,
𝜐],
𝜉∈
1],
where
𝔖(𝒾)
≥over
0−
for
all
𝒾and
∈
If
(11)
is
reversed
then,
𝔖
∗−
tions
I-V-Fs.
𝔖[34]
is
𝐹𝑅
-integrable
over
𝜈]
𝔖𝒾𝒿,(𝒾)𝑑𝒾
∈if
[𝜇,
monically
function
ontions
𝜐].
∈[𝜇,
𝐾.
∈Note
𝐾.𝒾if(𝐼𝑅)
∈ 𝐾.. No
∗(𝐼𝑅)
∗𝔖
[0, 1],
[𝜇,
∈ [𝜇,≤𝜐],(1𝜉concave
where
𝔖(𝒾)
≥
0
for
all
𝒾
∈
𝜐].
If
(11)
is
reversed
then,
𝔖
+
𝜉𝔖(𝒿),
−∈ 𝜉)𝔖(𝒾)
(11)
(𝒾)𝑑𝒾
(𝒾)𝑑𝒾
(𝒾)𝑑𝒾
(𝒾)𝑑𝒾
(𝒾)𝑑𝒾
(𝒾)𝑑𝒾
(𝒾)𝑑𝒾
(𝒾)𝑑𝒾
(𝐼𝑅)
(𝐼𝑅)
(𝐼𝑅)
(𝐼𝑅)
(𝐼𝑅)
(𝐼𝑅)
∗
=
=
𝔖
𝔖
=
=
𝔖
𝔖
=
=
𝔖
=
=
𝔖
𝔖
NV-Fs
over
[𝜇,
NV-Fs
𝜈].
over
[𝜇,
𝜈].
(𝐹𝑅)
(𝒾)𝑑𝒾
(𝒾,
(𝒾,
of
I-V-Fs.
𝔖
is
𝐹𝑅
-integrable
over
[𝜇,
𝜈]
if
𝔖
∈
𝔽
.
that,
if
𝜑),
𝔖
𝜑)
are
Lebesgue-integrable,
then
𝔖
is
a
fuzzy
Aumann-integrable
func𝔖
(𝒾)
[𝔖
(𝒾,
(𝒾,
=
𝜑),
𝔖
𝜑)]
for
all
∈
[𝜇,
𝜈]
and
for
all
:
[𝜇,
𝜈]
⊂
ℝ
→
𝒦
are
given
by
𝔖
𝔖
[𝜇,
is called
a for
harmonically
function
on
𝜐].
𝜑)
and
𝔖+(𝒾,
𝜑)
both
are− 𝜉)
𝜑
1].
Then
𝔖[𝜇,
is
𝐹𝑅-integrable
over
[𝜇,
and
𝔖∗ (𝒾,
(1
(1
(1Aumann-integr
∗ 𝜈]𝜉𝒾if
𝜉𝒾
𝜉𝒾
+𝜉𝒾
+only
−
𝜉)𝒿
−ifthen
𝜉)𝒿
− 𝜉)𝒿
+ (1
𝜉)𝒿
𝑅-integrable over
[𝜇, 𝜈].for
Moreover,
isconcave
𝐹𝑅-integrable
over
𝜈],
then
𝒾𝒿
𝒾𝒿
𝒾𝒿
𝒾𝒿
𝒾𝒿
[𝜇,
[0,
[𝜇,
(𝒾,
(𝒾,
all 𝒾,
all𝒿on
𝒾,
∈ℛif𝒿∗[𝜇,
∈𝔖𝜐],
𝜉𝜐],
∈ all
𝜉∈[0,
∈(0,
1],
1],
where
where
𝔖(𝒾)
0∗=
≥
for
0𝜑)
for
all
all
𝒾denotes
∈
𝒾[𝜇,
∈
𝜐].
𝜐].
If
(11)
If
(11)
is
reversed
then,
then,
𝔖
𝔖
𝜑),
𝔖
are
Lebesgue-integrable,
𝔖∞)
isbe
aof
fuzzy
𝔖all
∗ 𝔖(𝒾)
(0,
(0,
(0,
(0,to
[𝜇,
[𝜇,
[𝜇,
[𝜇,
=
∞)
is−
=
to
be
=
asaid
convex
to
is
set,
asaid
=
convex
if,
for
∞)
all
set,
ais(11)
convex
said
𝒾,if,𝒿𝜉𝔖(𝒿),
for
∈to all
set,
be a
3.𝒾𝒿
[34]
A≤𝜑
set
Definition
3.
[34]
𝐾
3.𝒾𝒿
Definition
[34]
⊂
𝐾
ℝ
=
set
𝜐]
3.
⊂
[34]
ℝ
A
𝜐]
set
⊂is∞)
𝐾
ℝreversed
=is
𝜐]
⊂
ℝ
a harmonically
function
𝜐].
for all concave
𝜑 ∈ (0, 1],
where
(0,
for
1],
where
∈≥
(0,
1],
where
ℛ
the
collection
of
Riemannian
collection
denotes
functhe
of
Riemannian
collection
integrable
Riemannian
funcintegrab
([[𝜇,
],∗ (for
)Definition
([A
],𝜐]
)+
([≤
],𝐾
)=
(1
(1ℛ
(1
(1
(1
(1
[𝜇,
, set
, the
𝜉𝔖(𝒿),
+A
𝜉𝔖(𝒿),
+
+
𝜉𝔖(𝒿),
+
𝜉𝔖(𝒿),
+be𝜉𝔖(𝒿),
+
𝔖denotes
𝔖
≤
𝔖
−
𝜉)𝔖(𝒾)
𝔖
−
𝜉)𝔖(𝒾)
𝔖integrable
𝔖
≤
−
𝜉)𝔖(𝒾)
𝔖said
𝜉)𝔖(𝒾)
≤𝜉𝔖(𝒿),
−
≤
𝜉)𝔖(𝒾)
−
≤
𝜉)𝔖(𝒾)
−
𝜉)𝔖(𝒾)
∗(1
(11)
(11)
(11)
( tion
) ,over
) 𝜈].𝜑 ∈Definition
It𝜈]
is∈ifa=𝔽
partial
order
relation.
(8)
(8)
(𝒾,
(𝒾)𝑑𝒾
(𝒾)𝑑𝒾
(𝐹𝑅)
(𝐹𝑅)
(𝐼𝑅)
Proposition
1.
Let
𝔵,𝔖
𝔴(𝒾)𝑑𝒾
Proposition
1.It
Proposition
[51]
Let
𝔵,𝔖(𝒾,
𝔴
∈[51]
Let
𝔵, fuzzy
𝔴
.ℝThen
relation
“, 1.
≼
”𝔽,is. relation.
Then
given
on
𝔽𝔽 order
. by
Then
fuzzy
order
“ ≼ ” relation
is given “on
≼ ”𝔽 isby
given on 𝔽
𝜑)
and
𝔖 ℛ(𝒾,
𝜑)
both
are
Then
𝐹𝑅-integrable
over
[𝜇,
and
only
iffuzzy
𝔖
𝔖is(𝒾)𝑑𝒾
𝔖
=
𝔖(𝒾,
𝜑)𝑑𝒾
𝔖:order
𝔖(𝒾,
𝜑)
=∈
: 𝔖(𝒾,
𝜑)∈ ∈
ℛ
[𝜇,
[𝜇,
𝜈].𝔖Moreover,
if=all
𝔖(𝐼𝑅)
is[51]
over
se
mathematical
operations
to
examine
some
of
the
of
fuzzy
([
],each
)𝜑)𝑑𝒾
([ , ],relation
) ,
∗characteristics
49]
A
valued
map
: 𝐾𝜈].
⊂
→
𝔽𝜈],
isthen
called
For
tial
order
is
partial
order
[𝔴]
if,
≤𝜌.relation.
for
𝜑for𝐹𝑅-integrable
∈ all
(0,
(4)
[𝜇,
𝑅-integrable
over
[𝜇,
Moreover,
if
𝔖 [𝔵]
is F-NV-F.
𝐹𝑅-integrable
over
𝜈],called
thensome
)=
[𝔖fuzzy
(𝒾,
We
now
use
mathematical
operations
toaexamine
some
ofexamine
the
characteristics
ofcharacteristics
fuzzy
(7)
(7)of the
𝜑),
𝔖∗Definition
𝜑)]
𝒾1],∈a𝔖A
𝜈]
and
all
[𝜌.for
[𝔵]number
𝜌.
𝔵][𝔵]
𝔵]
=
=
𝜌.
1.
[49]
fuzzy
number
valued
map
𝔖
:
𝐾
⊂
ℝ
→
𝔽
is
F-NV-F.
For
each
∗ (𝒾,
We
now
use
mathematical
operations
to
of
fuzzy
It
is
partial
order
relation.
, 𝔴over
𝔽-cuts
and
𝜌 ∈ ℝ,
then
we
[𝜇,
le
[𝜇,
𝜈].fuzzy
Moreover,
if family
𝔖
is𝔵have
over
𝜈],
then
ose
𝜑
define
the
I-V-Fs
𝔖⊂
:ℝ
𝐾∈→
⊂
→
𝒦
are
given
by
now
mathematical
operations
to
examine
some
of
the
characteristics
of
fuzzy
We
use
mathematical
operations
toall
examine
1.∈use
[49]
A
number
valued
map
:and
𝐾
𝔽ℝ
is
called
For
each
[𝔵]
[𝔵]
[𝔴]
[𝔴](4)
≼𝐹𝑅-integrable
𝔴
if
and
only
if,
𝔵F-NV-F.
≼now
𝔴then
if
and
only
≼⊂𝔴
if,
if→
and
if,
for
all
𝜑
∈𝔖
(0,
1],
≤only
for
≤ 𝜑[𝔴]
∈ (0,some
for
1], allof𝜑the
∈ characteristic
(0, 1], (4)
Let
𝔵,of
𝔴
∈
𝔽𝔖
ℝ,
weoperations
have
∗ (𝒾,
(8)
(𝒾)𝑑𝒾
𝜑numbers.
∈all
(0,
1]
whose
𝜑
-cuts
define
the
family
of
I-V-Fs
:𝔵have
𝐾
ℝsome
𝒦integrable
are
given
by
𝔖on
=
𝔖(𝒾,
𝜑)𝑑𝒾
:∈𝔖(𝒾,
𝜑)
∈numbers.
ℛ
, 𝜌mathematical
𝔵,are
𝔴[𝔵]
∈then
𝔽≤
and
𝜌collection
∈
ℝ,
we
(𝒾,
𝜑)
and
𝔖
both
r,∈
[𝜇,
𝜈]
if
and
only
if
𝔖
([
],ℛuse
)Let
, 𝜑)
We
now
to
examine
ofℝ,the
characteristics
of fuzzy
∗
(0,
1],
where
for
ℛ
𝜑
(0,
1],
where
denotes
the
collection
of
denotes
Riemannian
the
integrable
of
funcRiemannian
func∗
([
],
)
([
],
)
,
,
∗
𝔵, 𝔴
anddefine
𝜌
∈
ℝ,
then
we
have
(𝒾,∈𝔖
𝜑),
𝔖
𝜑)]
for
all
𝒾
∈
𝐾.
Here,
for
each
𝜑
∈
(0,
1]
the
end
point
real
numbers.
Let
𝔵,
𝔴
∈
𝔽
and
𝜌
∈
then
we
have
]Let
whose
𝜑𝔽(𝒾)𝑑𝒾
-cuts
the
family
of
I-V-Fs
𝔖
:
𝐾
⊂
ℝ
→
𝒦
are
given
by
(𝒾,
(𝒾,
(𝑅)
(𝑅)
(𝐹𝑅)
=
𝔖
𝜑)𝑑𝒾
,
𝔖
𝜑)𝑑𝒾
∗
(5) 𝜑relation.
[𝔵+
[𝔵]
𝔴]
=
+ [𝔴]
, of
∗ 𝔖
∗ (𝒾,the end point real
(𝒾)
[𝔖
(𝒾,
𝔖
𝜑),
𝜑)]
for
all
𝒾=
∈:(𝒾)𝑑𝒾
𝐾.
for
each
∈(𝒾)𝑑𝒾
1]
ons
to
some
of=
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is
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amap
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It⊂
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partial
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∗characteristics
numbers.
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𝔵,(𝒾)𝑑𝒾
𝔴
∈
𝔽 [𝔵+
and
𝜌of
∈
ℝ,
we
have
(𝑅)
(𝐹𝑅)
∗examine
𝔖
𝜑)𝑑𝒾
,, (𝑅)
ued
map
Definition
𝔖
: is
𝐾
⊂
ℝ
1.-integrable
→
[49]
𝔽
A
fuzzy
number
valued
𝔖
𝐾order
ℝ
→
𝔽
is
called
For
each
is[𝔴]
called
F-NV-F.
For
[𝔵]
(𝐹𝑅)
(𝐹𝑅)
𝔴]
=
I-V-Fs.
𝔖
𝐹𝑅
tions
of
I-V-Fs.
over
𝔖F-NV-F.
is
[𝜇,(𝒾,
𝜈]
𝐹𝑅
-integrable
if𝔖
over
[𝜇,
𝜈]
if+
𝔖
∈athe
𝔽
.(𝒾,
Note
if(0,𝔖+
∈[𝔴]
𝔽𝜑)𝑑𝒾
.each
[𝜇,
∗ (𝒾,
s, 𝜑),
𝐹𝑅-integrable
over
𝜈],
then
(.It
(5)
𝔖
,𝔖
𝜑):
→
ℝ
are
called
lower
and
upper
functions
𝔖
. ∗then
[𝔵+
[𝔵]
𝔴]that,
=
, Note that, if(5)
∗ (𝒾,
[𝔖
(𝒾,
𝜑),
𝜑)]
for
all
𝒾
∈
𝐾.
Here,
for
each
𝜑
∈
(0,
1]
end
point
real
∗
∗
(𝒾,
(𝑅)
(𝑅)
(𝐹𝑅)
(𝒾)𝑑𝒾
=
𝔖
𝜑)𝑑𝒾
,
𝔖
𝜑)𝑑𝒾
𝔖
(5)
[𝔵+
[𝔵]
[𝔴]
𝔴]
=
+
,
(.
functions
𝔖
,
𝜑),
𝔖
,
𝜑):
𝐾
→
ℝ
are
called
lower
and
upper
functions
of
𝔖. 𝔴]some
∗[(. the
[𝔵+
[𝔵]
en
have
notes
the
collection
of
Riemannian
integrable
func= examine
+ [𝔴]
, of the characteristic
We
now
use
mathematical
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We
to
now
examine
use
mathematical
We
some
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of
use
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examine
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to
of the
characteristics
some
of fuzzy
(9)
∗
family
𝜑
∈
of
(0,
I-V-Fs
1]
whose
𝔖
𝜑
-cuts
define
family
of
I-V-Fs
𝔖
:
𝐾
⊂
ℝ
→
𝒦
are
given
by
:
𝐾
⊂
ℝ
→
𝒦
are
given
by
∗ we
∗
(6)
[𝔵
[𝔵]
×
𝔴]
=
×
𝔴]
,
∗
(𝒾,
(𝒾,
(𝒾,
𝔖
𝜑)
are
Lebesgue-integrable,
𝜑),
𝔖
𝜑)
are
then
Lebesgue-integrable,
𝔖
is
a
fuzzy
Aumann-integrable
then
𝔖
is
a
fuzzy
funcAumann-integrable
func𝔖
(9)
(5) 4(6)
s 𝔖∗ (. Symmetry
, 𝜑),
𝔖 (.2022,
, 𝜑):
𝐾 14,
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ℝ∗ are
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and
upper
of ×
𝔖
.[×𝔴]𝔴]
∗ 14,
[𝔵+
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, 𝔴] ,
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2022,
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×functions
𝔴]
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[
×
(𝐹𝑅)
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numbers.
Let
𝔵,
𝔴
∈
𝔽
Let
numbers.
𝔵,
𝔴
𝔽
Let
𝔵,
𝔴
∈
𝔽
and
𝜌
∈
then
we
have
and
𝜌
∈
ℝ,
then
and
we
𝜌
have
∈
ℝ,
then
we
have
over
[𝜇,
𝜈]
if
𝔖
∈
𝔽
.
Note
that,
if
(𝒾)
[𝔖
(𝒾,
(𝒾,
𝔖 [𝔵]𝔖:for
𝐾.
Here,
=
each
𝜑),
𝜑=
1]
𝜑)]
the
for
end
all point
𝐾.real
Here,
each of
𝜑 ∈𝔖(0,over
1] the
(9)
∗ ⊂, over
be
an
F-NV-F.
fuzzyfor
integral
[49]
Let
[𝜇,
𝜈]
ℝ
→∈[𝜌.
𝔽(0,
(6)end point real
[𝔵
[𝔵]
𝔴]
×𝒾 [∈Then
𝔴]
,(5)
(𝒾)𝑑𝒾
(𝐼𝑅)
[𝜇, 𝜈].
tion
[𝜇,
𝜈].
[𝔵 × 𝔴] = [𝔵] × [ 𝔴] ,
[𝔴]
𝔴]
=
(𝒾,
(𝑅)
(7)Then
[𝔵]
𝔖∗ (𝒾,+
𝜑)𝑑𝒾
, (𝑅)
𝜑)𝑑𝒾
𝔵]×∗2.
=𝔖
𝜌.=
be upper
an[𝔵𝔖F-NV-F.
of 𝔖 over
Definition
[49]
Let
𝔖are
: [𝜇,called
𝜈]
⊂ℝ
→=
𝔽[𝜌.(𝐼𝑅)
∗𝔖
(𝒾)𝑑𝒾
(6) (7)
[𝔵]
[𝔖𝔴]
×
𝔴]
=
×fuzzy
(.
(.
functions
𝔖
called
lower
and
upper
,
𝜑),
𝔖
functions
,
𝜑):
𝐾
of
→
𝔖
ℝ
.
lower
and
functions
of
.[𝔵] ,integral
(7)
able,
then
𝔖
is
a
fuzzy
Aumann-integrable
func[𝔵]
𝔵]
=
𝜌.
(𝐹𝑅)
(𝒾)𝑑𝒾
∗
don
by
𝔖
,
is
given
levelwise
by
[𝜌.𝔖𝔵]over
=
𝜌.
an
F-NV-F.
Then
fuzzy
integral
of
2.REVIEW
[49] LetSymmetry
𝔖[𝜇,
: [𝜇, 𝜈]2022,
⊂ ℝ14,
→x[𝜌.
𝔽
[𝔵+
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[𝔴]
[𝔵]
[𝔴]
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19 ,= [𝔵] + [𝔴] ,
4 of (5)
19
𝔴]
=
+
,
𝔴]
=
+
𝔴]
(𝒾)𝑑𝒾
(𝐼𝑅)
=
𝔖
(7)
(𝐹𝑅)
(𝒾)𝑑𝒾
[𝔵]
𝜈],
denoted
by
𝔖
,
is
given
levelwise
by
𝔵]
=
𝜌.
[𝜌. 𝔵] = 𝜌. [𝔵]
(6)
[𝔵]
[
𝔴] =
× 𝔴] ,
(9)
2.
[49]
Let
𝔖
:
[𝜇,
Theorem
𝜈]
⊂
ℝ
→
2.
𝔽
[49]
Let
𝔖
:
[𝜇,
𝜈]
⊂
ℝ
→
𝔽
be
a
F-NV-F
whose
𝜑-cuts
be
define
a
F-NV-F
the
family
whose
of
𝜑-cuts
I-V-Fs
define
the
family
of
I-V-Fs
(𝐹𝑅)
(𝒾)𝑑𝒾
noted
by
𝔖
,
is
given
levelwise
by
(7)
[𝜌.
[𝔵]
𝔵]F- = 𝜌. function S : [µ, (6)
+ is called a harmonically
].
all
∈ (0, 2.
1],Then
ℱℛ([Let
denotes
the
collection
ofF-NV-F.
all
𝐹𝑅-integrable
): [𝜇,
, ],map
Definition
[36].
The
real-valued
υ“𝔴]
]≼→”=
[𝔵
[𝔵 ×
[𝔵××[fuzzy
[𝔵]
[𝔵]
[on
Proposition
1.
[51]
Let
Proposition
𝔴𝔴]
∈ 𝔽For
1.each
[51]
Let
𝔵,order
𝔴
∈of𝔽
Then
fuzzy
relation
.[𝔵]
Then
is
order
given
≼ ” is given(6)
on 𝔽
×
𝔴]
=
×𝔵,],[positive
, . fuzzy
𝔴]of
=
𝔴]
,R
𝔴] 𝔽, “by
49]
A
fuzzy
number
valued
𝔖𝜈]
:For
𝐾
ℝ→→
𝔽
is51],
called
F-NV-F.
beDefinition
an 𝜑
F-NV-F.
integral
of
𝔖𝔽∈
over
be
an
Then
integral
𝔖
over
𝔽 For
⊂⊂
ℝ
(0,
for[49]
all 𝜑fuzzy
∈𝔖
(0,
1].
all
𝜑
ℱℛ
the
collection
all
𝐹𝑅-integrable
F- ×relation
([ , map
) denotes
∗ (𝒾,
∗⊂
(7)
[𝜌.
[𝔵]
𝔵]
=𝒦
𝜌. (𝒾)𝑑𝒾
(𝐼𝑅)
𝔖
Definition
1.
[49]
A
fuzzy
number
valued
𝔖
:
𝐾
ℝ
→
𝔽
is
called
F-NV-F.
For
each
(𝒾)
[𝔖
(𝒾,
(𝒾)
[𝔖
(𝒾,
(𝒾,
=
𝜑),
𝔖
𝜑)]
for
=
all
𝒾
∈
[𝜇,
𝜑),
𝜈]
𝔖
and
𝜑)]
for
for
all
all
𝒾
∈
[𝜇,
𝜈]
and
for
all
⊂
ℝ
→
are
given
:
[𝜇,
by
𝜈]
⊂
𝔖
ℝ
→
𝒦
are
given
by
𝔖
𝔖
(8)
].
(𝒾)𝑑𝒾
(𝒾)𝑑𝒾
(𝐼𝑅)
𝔖
=
𝔖
=
𝔖(𝒾,
𝜑)𝑑𝒾
:
𝔖(𝒾,
𝜑)
∈
ℛ
,
Definition
1.
[49]
A
fuzzy
number
valued
map
𝔖
:
𝐾
⊂
ℝ
→
𝔽
is
called
F-NV-F. For each
(0,
∗denotes
∗second
(0,
1].
For
all
𝜑
∈
1],
ℱℛ
the
collection
of
all
F([ 𝐹𝑅-integrable
], )senseby
s-convex
function
in
the
on
µ,
υ
if
[
]
, given
([
],
)
,
se
𝜑
-cuts
define
the
family
of
I-V-Fs
𝔖
:
𝐾
⊂
ℝ
→
𝒦
are
[𝜇,
(𝐹𝑅)
(𝒾)𝑑𝒾
𝜈],
denoted
by
𝔖
ven
levelwise
by
,
is
given
levelwise
by
(8)
nbe1.a[49]
A fuzzy
number
map
𝔖: 𝐾of⊂
ℝ
→𝔵]𝔽the
is family
F-NV-F.
For
each
F-NV-F
whose
define
the
family
I-V-Fs
NV-Fs
over
[𝜇, 𝜈].
(𝒾)𝑑𝒾
(𝐹𝑅)
(𝐼𝑅)
𝔖(𝒾)𝑑𝒾
=
𝔖
=
𝔖(𝒾,
𝜑)𝑑𝒾
: 𝔖(𝒾,
𝜑)
∈=only
ℛfor
,map
Definition
1.
[49]
A𝔵[𝔵]
fuzzy
number
𝔖
:[𝔵]
𝐾 ⊂for
ℝ→
is
called
],[𝜌.
)𝜑
[𝔵]
[𝔴]
[𝔴]
(7)
(7)on 𝔽
[𝜌.
[𝔵]
[𝔵]
𝔵called
≼
𝔴
if
and
if,
≼
𝔴
if
and
if,
≤𝔴
∈→
(0,
≤
1],
all𝔽“on
𝜑
∈”𝔽(4)
(0,
1], F-NV-F
𝔵]
𝔵]ℝ
=
𝜌.
𝜌.([valued
=
𝜌.
𝜑
∈𝜑-cuts
(0,[51]
1]valued
whose
-cuts
define
of
I-V-Fs
:(8)
⊂[𝜌.
ℝ
→
are
given
∗ (𝒾,
∈𝜑
1]
whose
𝜑 order
define
the
family
of
I-V-Fs
𝔖
:,all
𝐾
⊂
𝒦
are
given
∗ (𝒾,
Let
𝔵,𝜑𝔴𝜈]
∈(0,
Proposition
1.
Proposition
[51]
𝔵,
𝔴
1.
∈map
[51]
Let
𝔵,
∈
. Then
fuzzy
relation
“only
≼
is
Then
given
on
𝔽𝔽𝒦
order
Then
relation
fuzzy
order
“by
≼
” relation
is
given
≼by
isby
given
r𝜑),
[𝜇,
𝜈].
(𝒾)𝑑𝒾
(𝒾)𝑑𝒾
(𝐼𝑅)
𝔖
=
𝔖
=
𝔖(𝒾,
𝜑)𝑑𝒾
:-cuts
𝔖(𝒾,
𝜑)
∈and
ℛ
,𝔖.𝜑
(𝒾,
(𝒾,
Definition
1.
[49]
A
fuzzy
number
valued
:𝐾𝐾
⊂fuzzy
ℝ
→
𝔽.∗by
is
called
F-NV-F.
For
each
𝜑)
and
𝔖
both
are
𝜑)
and
𝔖
𝜑)
both
are
. (𝐹𝑅)
Then
𝔖Proposition
is-cuts
𝐹𝑅-integrable
𝜑∗ ∈
(0,1.
1].
Then
over
[𝜇,
𝔖
is
𝐹𝑅-integrable
if𝔽each
and
only
if𝔖
𝔖
over
[𝜇,
𝜈]
ifLet
if”)𝔽
𝔖
!
([only
],
𝔖
𝜑)]
for
all
𝒾
∈
𝐾.
Here,
for
𝜑
∈
(0,
1]
the
end
point
real
,𝜑)
∗𝐾(𝒾,
∗𝔖
whose
𝜑
define
the
family
of
I-V-Fs
:
⊂
ℝ
→
𝒦
are
given
by
𝜑
∈
(0,
1]
whose
-cuts
define
the
family
of
I-V-Fs
𝔖
:
𝐾
⊂
ℝ
→
𝒦 are
∗
)
[𝔖
(𝒾,
(𝒾,
=
𝜑),
𝔖
𝜑)]
for
all
𝒾
∈
[𝜇,
𝜈]
and
for
all
∗
(𝒾)
[𝔖
(𝒾,
(𝒾,
𝔖
=
𝜑),
𝔖
𝜑)]
for
all
𝒾
∈
𝐾.
Here,
for
each
𝜑
∈
(0,
1]
the
end
point
real
(0,
[𝜇,
∗
=
∞)
is
said
to
be
a
convex
set,
if,
for
all
𝒾,
𝒿
∈
4]
A
set
𝐾
=
𝜐]
⊂
ℝ
∗𝜑
(𝒾)
[𝔖
(𝒾,
(𝒾,
ℛued
of
all(0,
𝐹𝑅-integrable
F𝔖
=
𝜑),
𝔖
𝜑)]
allisfamily
𝒾order
∈ 𝐾.
Here,
for 𝔖each
𝜑 for
∈ (0,
1]𝒾, the
end point
real
map
𝔖∗:(𝒾,
𝐾𝐾([⊂→
ℝ
𝔽 called
is called
F-NV-F.
For
each
1],
ℛ
denotes
the
collection
of
Riemannian
func([ , where
], ∗)(. ,denotes
∈
whose
𝜑
-cuts
define
the
of
I-V-Fs
:all
⊂
→for
∗over
It1]
is
afor
partial
relation.
Itthen
isintegrable
athe
partial
relation.
],Definition
)collection
s 𝐾 if,
,the
s𝒦
∗𝔵
𝜑),
𝔖
𝜑):
ℝ→
are
lower
and
upper
functions
of
𝔖for
.∞)
(0,
[𝜇,
=
said
to
be
a≤convex
𝒿are
∈,[𝔴]
[34]
A
set
𝐾
=
𝜐]
⊂∈𝜈],
ℝ
[𝜇,
[𝜇,
ble
over
[𝜇,
𝜈].
Moreover,
𝑅-integrable
if(𝒾)𝑑𝒾
is
𝐹𝑅-integrable
[𝜇,
𝜈].
iforder
𝔖=
is
𝐹𝑅-integrable
over
𝜈],
then
𝔖
𝜑),
𝔖
𝜑)]
for
all
𝒾over
∈𝔵3.
𝐾.
Here,
each
𝜑
(0,
1]
end
point
real
[𝔵]
[𝔴]
(𝒾)
(𝒾,
(𝒾,
≼
𝔴
if
and
only
𝔵=
≼[𝔖
𝔴
if
and
only
≼
𝔴
if,
and
only
for
all
𝜑 𝜑),
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1], by
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for
𝒾 ).ℝ
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1] 1],
the(4)end
∗Moreover,
Sthe
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ξ[𝔵]
S
−
ξ,set,
S
(12)
for
all
𝜑
∈𝔖
(0,
1],
where
ℛ
denotes
collection
of
Riemannian
integrable
func(, 1],if[𝔵]
)≤
([𝔴]
∗ (𝒾,
(8)
∗([
∗𝔖(𝒾,
([
],(8)
)ℝ
(𝒾)𝑑𝒾
(𝒾)𝑑𝒾
(𝐹𝑅)
(𝐼𝑅)
, if,
𝔖
=
𝔖(𝒾,
𝜑)𝑑𝒾
:
𝔖(𝒾,
𝜑)
∈
=
ℛ
𝔖
,
𝔖(𝒾,
𝜑)𝑑𝒾
:
∈
ℛ
∗≤
(.
(.
functions
𝔖
,
𝜑),
𝔖
,
𝜑):
𝐾
→
are
called
lower
and
upper
functions
of
𝔖
have
],
)
([
)
∗
,
(𝒾,
(𝒾,
𝜑)
and
𝔖
𝜑)
both
are
[𝜇,
𝜈]
if
and
only
if
𝔖
(0,
[𝜇,
∗
=
∞)
is
said
to
be
a
convex
set,
if,
for
all
𝒾,
𝒿
∈
nrs.
3.
[34]
A
set
𝐾
=
𝜐]
⊂
ℝ
(.
(.
family
of
I-V-Fs
𝔖
:
𝐾
⊂
ℝ
→
𝒦
are
given
by
functions
𝔖
,
𝜑),
𝔖
,
𝜑):
𝐾
→
ℝ
are
called
lower
and
upper
functions
of
𝔖
.
(𝐹𝑅)
(𝒾)𝑑𝒾
Definition
1.
[49]
A
fuzzy
number
Definition
valued
map
1.
𝔖
[49]
Definition
:
𝐾
⊂
A
ℝ
fuzzy
→
𝔽
1.
number
[49]
A
fuzzy
valued
number
map
𝔖
valued
:
𝐾
⊂
ℝ
map
→
𝔽
𝔖
:
𝐾
⊂
ℝ
→
𝔽
is
called
F-NV-F.
For
each
is
called
F-NV-F.
is characteristic
called
For each
F-NV-F
𝔖 1],
is where
𝐹𝑅∗ -integrable
[𝜇,
𝜈]the
if[𝔖
𝔖of
∈ for
𝔽 . We
Note
that,
if
∗], [0,
(𝒾)
(𝒾,
(𝒾,Riemannian
now
use
mathematical
use
examine
operations
some
of the
to examine
characteristics
some
of the
fuzzy
=lower
𝜑),
𝜑)]
all
𝒾now
∈operations
Here,
for
each
𝜑ℝ
∈ are
(0, 1]
the
end point
real
∗ 𝔖
ξintegrable
+
−
ξmathematical
∈𝔖(0,
ℛ
denotes
collection
(1(.
)func∗to
∗We
) are
𝐾,
1],𝔖called
we
have
,∈
𝐾([𝜉→
ℝover
and
upper
functions
of
𝔖
.𝐾.
(.
functions
𝔖
,
𝜑),
𝔖
,
𝜑):
𝐾
→
called
lower
and
upper
functions
of 𝔖.
(𝐹𝑅)
(𝒾)𝑑𝒾
∗ (. , 𝜑), 𝔖 (. , 𝜑):
tions
of
I-V-Fs.
𝔖
is
𝐹𝑅
-integrable
over
[𝜇,
𝜈]
if
𝔖
∈
𝔽
.
Note
that,
if
∗ order relation.
∗
It
is
a
partial
order
relation.
It
is
a
partial
order
It
is
a
relation.
partial
1],
we
have
∈𝜑)
𝐾.
Here,
for
each
𝜑
∈𝜈],
1]
theF-NV-F.
end
real
𝜑
1]
whose
𝜑
-cuts
define
the
𝜑, 𝜑),
∈
family
(0,
of
∈and
(0,
-cuts
1]
whose
define
-cuts
family
define
ofthe
I-V-Fs
family
𝔖 :𝐾
: over
𝐾
⊂
ℝ
→
𝒦
are
by
:𝐾
⊂ℝ→𝒦
are⊂ given
ℝ → 𝒦by are
𝒾𝒿
numbers.
Let
𝔵,1]
∈(𝒾)𝑑𝒾
𝔽
numbers.
Let
𝔵,then
𝔴that,
∈𝜑we
𝔽the
𝜌.𝔖
∈
ℝ,
have
and
𝜌 ∈given
ℝ,
then
we 𝔖
have
(.𝔴
[𝜇,
functions
𝔖point
𝔖fuzzy
, 𝜑):
𝐾𝜑∗I-V-Fs
→
ℝ𝜑𝔽of
are
called
lower
upper
functions
ofof
𝔖.I-V-Fs
sI-V-Fs.
𝐹𝑅-integrable
over
then
be
an
Then
integral
𝔖
49]
Let
𝔖
𝜈]𝐹𝑅
⊂14,
ℝ
→
𝔽 (0,
(𝐹𝑅)
𝔖:∈[𝜇,
is(0,
-integrable
over
[𝜇,is𝜈]
𝔖whose
∈
Note
if
are
Lebesgue-integrable,
then
𝔖
a∗ (.if
fuzzy
Aumann-integrable
func∗
∗
Symmetry
2022,
x
FOR
PEER
REVIEW
4
of
19
𝒾𝒿
𝐾.
be
F-NV-F.
Then
integral
offuzzy
𝔖to
over
Definition
[49]
Let
:all
[𝜇,
𝜈]
⊂
→
𝔽use
(10)
(𝒾,
𝜑),
𝔖mathematical
𝜑)
are
𝔖
is
athe
fuzzy
funcfor∞)
all
𝜑We
∈𝔖
(0,
1],
ℛ
otes
the
collection
of
Riemannian
integrable
denotes
functhe
of
integrable
func∗2.
∗ (𝒾,
(𝒾,
(𝒾,
(𝒾,
(𝒾,
(𝑅)
(𝑅)
(𝑅)
(𝑅)
(𝐹𝑅)
(𝐹𝑅)
(𝒾)𝑑𝒾
=
𝜑)𝑑𝒾
,[49]
𝔖
𝜑)𝑑𝒾
𝜑)𝑑𝒾
, for
𝔖)fuzzy
𝔖
(0,
=
is
to
be
awhere
convex
set,
if,
𝒾,
𝒿We
∈
be
Then
integral
of
𝔖1𝜑
Definition
Let
[𝜇,
𝜈]
⊂Riemannian
ℝ
→
𝔽𝜑),
([
],𝔖
)for
now
use
operations
to
now
examine
mathematical
We
some
now
of
use
mathematical
characteristics
operations
to
examine
operations
of
fuzzy
some
of
some
characteristic
ofisfuzzy
∗(𝒾)𝑑𝒾
,∈
∗Lebesgue-integrable,
∗an
(𝒾)
[𝔖
(𝒾,
(𝒾,
[𝔖
(𝒾,
(𝒾)
[𝔖
(𝒾,
called
and
upper
functions
𝔖
.F-NV-F.
𝔖
𝔖
𝔖,:𝜑),
𝜑),
𝔖(𝒾,
𝜑)]
for
all
𝒾2.
𝐾.
==ℝ
Here,
for
𝔖
each
=
𝜑)]
∈
(0,
1]
all
𝔖
the
𝒾Aumann-integrable
∈F-NV-F.
end
𝐾.0𝜑)𝑑𝒾
for
Here,
point
for
𝒾real
∈∈
𝐾.
Here,
∈the
for
(0,
1]
each
the
end
∈Ifthe
(0,
point
1]
the
realend
Symmetry
2022,
14,
x FOR
PEER
for
all
,(𝒾)
∈
µ,REVIEW
υ𝔖𝔖
ξthen
∈
1𝜑
,of
where
S∗ (𝒾,
for
all
υexamine
and
s characteristics
∈
.over
(12)
[collection
]Definition
[0,
]∈
(an
[µ,
]𝜑
[0,
]of
∗2.
(1
(𝐹𝑅)
(𝒾)𝑑𝒾
𝜉𝒾
+
−
𝜉)𝒿
𝐾.
𝔖said
,𝜈]
given
levelwise
by
∗is
∗Aumann-integrable
∗2.
(10)
].by
𝒾𝒿
beof
an
fuzzy
integral
𝔖𝔴]
over
n
[49]
Let
𝔖=
:x[𝜇,
⊂
ℝ
→
𝔽
(𝒾,
𝔖
𝜑)lower
are
Lebesgue-integrable,
then
𝔖
is
a∈ Then
fuzzy
funcbe
an
F-NV-F.
[49]
Let
𝔖𝜑)]
:≥
[𝜇,
𝜈]
⊂all
ℝ
→
𝔽each
(5) integral o
[𝔵+
[𝔵+
[𝔵]
[𝔴]
[𝔵]
=
+
,
𝔴]
=
+ [𝔴] 4, Then
(1
[𝜇,
(𝐹𝑅)
(𝒾)𝑑𝒾
𝜉𝒾
+
−
𝜉)𝒿
Symmetry
2022,
14,
FOR
PEER
REVIEW
of 19 fuzzy
𝜈],
denoted
by
𝔖
,
is
given
levelwise
by
tion
over
[𝜇,
𝜈].
∗
∗
∗
(𝐹𝑅)
(𝒾)𝑑𝒾
(𝐹𝑅)
(𝒾)𝑑𝒾
over tions
[𝜇,numbers.
𝜈] ofif I-V-Fs.
𝔖
𝔖
is∈𝔖𝐹𝑅
over
[𝜇,S
𝜈]
ifnumbers.
𝔖
∈,-integrable
𝔽
. 𝜈],
Note
that,
ifwe
∈(10)
𝔽𝔽
.
Note
that,
if
∈
𝐾.
Let
𝔵,
𝔴
𝔽
numbers.
Let
𝔵,
𝔴
∈
𝔽
Let
𝔵,
𝔴
∈
and
𝜌
∈
ℝ,
then
have
and
𝜌
∈
ℝ,
then
and
we
𝜌
have
∈
ℝ,
then
we
have
[𝜇,
(𝐹𝑅)
(𝒾)𝑑𝒾
denoted
by
𝔖
,
is
given
levelwise
by
(.
(.
(.
(.
(.
(.
reversed
then,
is
called
a
harmonically
s-concave
function
in
the
second
sense
on
µ,
υ
.
The
set
[
]
be
an
F-NV-F.
Then
fuzzy
integral
of
𝔖
over
Definition
2.
[49]
Let
𝔖
:
[𝜇,
𝜈]
⊂
ℝ
→
𝔽
functions
𝔖
functions
𝔖
functions
𝔖
,
𝜑),
𝜑):
𝐾
→
ℝ
are
called
lower
,
𝜑),
and
𝔖
upper
,
𝜑):
𝐾
functions
,
𝜑),
→
ℝ
𝔖
are
,
𝜑):
of
called
𝔖
𝐾
.
→
lower
ℝ
are
and
called
upper
lower
functions
and
upper
of
𝔖
functions
.
of 𝔖.
(9)
(9)
∗
∗ [𝜇, 𝜈], denoted
∗ by (𝐹𝑅)
∗ (𝒾,
(1levelwise
(𝐹𝑅)
(𝒾)𝑑𝒾
+𝜑)𝑑𝒾
− 𝜉)𝒿
noted
𝔖
, is𝔖𝜉𝒾
given
by
[𝜇, 𝜈].by
𝔖(𝒾)𝑑𝒾, is given levelwise
by
(𝑅)
(𝑅)
𝔖
𝜑)𝑑𝒾
, Then
∗ (𝒾,
∗ (𝒾,
be
an
F-NV-F.
fuzzy
integral
of
𝔖
over
𝔽
(𝒾,
ble,
then
𝜑),
𝔖
is
𝔖
a
fuzzy
𝜑)
are
Aumann-integrable
Lebesgue-integrable,
functhen
𝔖
is
a
fuzzy
Aumann-integrable
func𝔖
of
all
harmonically
s-convex
(harmonically
s-concave)
functions
is
denoted
by
[𝜇,
(𝐹𝑅)
(𝒾)𝑑𝒾
𝜈], denoted
by
𝔖the
, is, given
by ×=[ [𝔵]
(6)4 of 19(5)
[𝔵 levelwise
× 𝔴] [𝔵+
×
= [𝔵]
𝔴][𝔵
,of
𝔴]
𝔴] ,,
∗ : [𝜇,Symmetry
𝒾𝒿
9]
Let
𝔖
⊂
ℝℝ→
𝔽is 14,
bex a=
F-NV-F
whose
𝜑-cuts
define
family
of, I-V-Fs
EER
REVIEW
2022,
FOR
REVIEW
4[𝔵+
19 =
[𝔵+
[𝔵]
[𝔴]
[𝔴]
[𝔵] ×+[[𝔴]
𝔴]
=
+F-NV-F
𝔴]
+ (5)
𝔴]
,=[𝔵]
[𝜇,
4]
The
𝔖: [𝜇,𝜈]
→
called
a=PEER
harmonically
convex
function
𝜐] (8)
if
(𝒾)𝑑𝒾
(𝐼𝑅)
(𝐼𝑅)
(𝒾)𝑑𝒾
(𝒾)𝑑𝒾
(𝐼𝑅)
𝔖: [𝜇,
=
𝔖
=𝜐]𝐾.
𝔖(𝒾,
𝜑)𝑑𝒾
:(9)
𝔖(𝒾,
∈Definition
∈
([ 𝔖
],
)𝔖
(10)
Theorem
[49]
𝔖
𝜈]
⊂𝜐]
ℝ
→be
𝔽𝜑)
be
whose
𝜑-cuts
define
the
family
of”[𝜇,
I-V-Fs
1.
[51]
Let
𝔵,aℛ𝔴
∈, on
𝔽(𝒾)𝑑𝒾
. Then
fuzzy
order
relation
“𝔽
≼
is
given
on 𝔽integral
by fuzzy
iven
levelwise
by
an
F-NV-F.
Then
fuzzy
integral
be
of
an
𝔖
F-NV-F.
over
be
Then
an
F-NV-F.
fuzzy
Then
of
𝔖 integral
over
Definition
[49]2.
Let
𝔖Let
: Proposition
[𝜇,
𝜈]
⊂
Definition
→
𝔽
2.
[49]
Let
:
2.
[𝜇,
[49]
𝜈]
⊂
Let
ℝ
→
𝔖
𝔽
:
[𝜇,
𝜈]
⊂
ℝ
→
tion
over
[𝜇,
𝜈].2. 𝔖
[𝜇,
Definition
4. [34]
The
𝔖:ℝ
→
ℝ
is
called
a
harmonically
convex
function
on
𝜐]
if
(8)
(𝒾)𝑑𝒾
(𝐹𝑅)
(𝒾)𝑑𝒾
(𝐼𝑅)
𝔖
=
𝔖
=
𝔖(𝒾,
𝜑)𝑑𝒾
:
𝔖(𝒾,
𝜑)
∈
ℛ
,
∗
(1
𝒾
+
−
𝜉)𝒿
(8)
([ , :],fuzzy
) 𝜑)
(𝒾)𝑑𝒾
(𝒾)𝑑𝒾
(𝐼𝑅)
𝔖all
=𝜈]
𝔖+
=𝔵,
𝔖(𝒾,
𝔖(𝒾,
∈ [𝔵]
ℛ relation
1.
[51]
Let
𝔴𝜌.∈[𝔵]
𝔽 . 𝜑)𝑑𝒾
Then
order
is given on 𝔽
mn→
2.
Let
𝔖𝔖:
: [𝜇,
⊂
𝔽𝔖
be
a 𝜑),
F-NV-F
whose
the
family
of
I-V-Fs
+
(𝒾)
[𝔖called
, ], ) , “ ≼ ”(7)
𝔖 (𝒾,(𝐹𝑅)
𝜑)] 𝜑)𝑑𝒾
for𝜑-cuts
𝒾 define
∈Proposition
[𝜇,
for
all
given
by𝜈]
𝔖
[𝜌.
[𝜇,
[𝜇,
𝔵]
𝔵]
=
4.𝒦[49]
[34]are
The
→ℝ
ℝ→⊂=(𝐹𝑅)
is
a=
harmonically
convex
function
on
𝜐]
ifis
∗ (𝒾,
(8)
(𝒾)𝑑𝒾
(𝐹𝑅)
(𝒾)𝑑𝒾
(𝐼𝑅)
∗ (𝒾,
HSX
µ,
,,[𝜌.
s𝔖
HSV
µ,
υ(𝒾)𝑑𝒾
R=
,𝜌.
s ×𝔽.([[𝔖(𝒾,
𝔖
=𝜐]
𝔖(𝒾,
:(𝒾)
𝔖(𝒾,
𝜑)
ℛand
[Then
],,𝔖
[×
], for
𝒾𝒿
(𝒾)𝑑𝒾
(𝐼𝑅)
(6)
=𝒾(𝒾)𝑑𝒾
𝔖
=
𝜑)𝑑𝒾
: 𝔖(𝒾, 𝜑) ∈(6)
ℛ([ ,
[𝔵
[𝔵
[𝔵
[𝔵]
[ ∗∈
[𝔵]
[and
[𝔵]
×given
×given
𝔴]Let
=
×
𝔴]
,([υ(𝐹𝑅)
𝔴]
=
×
𝔴]
𝔴]
=
,levelwise
𝔴]
,
],R
)by
[𝜇,
(𝒾)𝑑𝒾
[𝜇,
[𝜇,
(𝐹𝑅)
(𝒾)𝑑𝒾
(𝐹𝑅)
[𝔖
(𝒾,
Proposition
1.
[51]
𝔵,
𝔴
∈
𝔽
.
fuzzy
order
relation
“
≼
”
is
given
on
by
𝜈],
denoted
by
𝔖
𝜈],
denoted
by
𝜈],
denoted
𝔖
𝔖
,
is
levelwise
by
,
levelwise
,
is
given
by
by
=
𝜑),
𝜑)]
for
all
∈
[𝜇,
𝜈]
all
:
[𝜇,
𝜈]
ℝ
→
𝒦
are
given
by
𝔖
𝔖
[𝔵]all≤𝔖(𝒾,
[𝔴]𝜑)𝑑𝒾
𝒾𝒿for
𝔵=
≼all
𝔴
if∈and
only
if,for
for: 𝔖(𝒾,
all 𝜑
∈∈(0,ℛ1],
(1
(𝒾)𝑑𝒾
(𝐼𝑅)
+𝔖𝜉𝔖(𝒿),
≤
−[𝔖𝜉)𝔖(𝒾)
∗𝔖
𝔖𝔖ℛ
(11)
(8) (4)
(𝒾)𝑑𝒾
(𝐹𝑅)
(𝒾)𝑑𝒾
(𝐼𝑅)
∗𝒾(𝒾,
𝔖
=
𝜑)
,
(𝒾)
(𝒾,
(𝒾,
([
],
)
,
=
𝜑),
𝜑)]
[𝜇,
𝜈]
and
⊂
ℝ
→
𝒦
are
given
by
𝔖
1],
where
denotes
the
collection
of
Riemannian
integrable
func(𝒾,
𝜑)
and
𝔖
𝜑)
both
are
n
𝔖
is
𝐹𝑅-integrable
over
[𝜇,
𝜈]
if
and
only
if
𝔖
([
],
)(1
∗all
,for
𝔵𝜉𝔖(𝒿),
𝔴ofifall
and
if, [𝔵]func≤ [𝔴] for all 𝜑 ∈ (0, 1],
(0,
(0,
(1
∈be(0,
1]. For all
𝜑
∈
all
𝜑𝜉)𝒿
1],
∈𝒾𝒿𝔖
(0,
1].
𝜑𝔽of
∈𝔖I-V-Fs
ℱℛ
denotes
the
collection
all
𝐹𝑅-integrable
the
collection
F-≼
𝐹𝑅-integrable
F𝜉𝒾2.
+for
−
+
≤denotes
− 𝜉)𝔖(𝒾)
(11)
∗a1],
([𝜈]
],
)ℝ
([ whose
)the
∗only
Theorem
[49]
Let
: (0,
[𝜇,
⊂where
→𝜑
a F-NV-F
whose
𝜑-cuts
define
the
family
F-NV-F
define
the
family
of
I-V-Fs
,For
, ],of𝜈]
all
∈ℱℛ
1],
denotes
collection
of
integrable
(𝒾,
(8)
𝜑)
and
𝔖by
𝜑)
are
∈𝜑)𝑑𝒾
(0,
1].
Then
𝔖
is
𝐹𝑅-integrable
over
[𝜇,
if, 𝜑-cuts
only
if.isThen
𝔖
([ 𝜐]
],+
(𝒾)𝑑𝒾
,fuzzy
(7)
(7)
[𝜌.
[𝜌.
[𝜌.
[𝔵]
[𝔵]
=
𝔖(𝒾,
:𝜑𝔖(𝒾,
∈
ℛ
,be
(1
𝔵])𝔵𝜉𝔖(𝒿),
𝔵]
𝔵]1],“ =
=
𝜌.
=
𝜌.(𝒾,
𝜌.
for
all
∈)ℛ
(0,
1],
where
ℛ
denotes
the
collection
of
Riemannian
func𝜉𝒾
+
−
𝜉)𝒿
[𝔵]
[𝔴]
∗Riemannian
≼
𝔴
if[𝔵]
and
only
if,
≤
for
all
𝜑relation
∈
(0,both
(4)
(1
([𝔽
, on
𝔖𝜑
≤
−
[𝜇,
([
],and
)𝔴
(11)
s. called
a𝔖
harmonically
convex
function
if
Proposition
[51]
Let𝜑)
𝔵,
𝔴
∈if
Proposition
1.
[51]
Let
𝔵,∗∈
∈𝜑)
𝔽1],
.],𝜉)𝔖(𝒾)
Then
order
relation
“(0,
≼
”[if
given
fuzzy
on
𝔽order
≼𝔖
”:integrable
is
given
on
𝔽sense
byon F-NV-F
e
(𝐹𝑅)
(𝒾)𝑑𝒾
𝔖
is
𝐹𝑅
-integrable
over
[𝜇,
𝜈]
𝔖
∈
𝔽
.
Note
that,
It
is
a
partial
order
relation.
∈
(0,
1],
where
ℛ([if𝜉𝒾,1.
denotes
the
collection
of
Riemannian
integrable
func[36].
The
F-NV-F
is
called
convex
F-NV-F
in
theℝsecond
Definition
6
S
:
µ,
υ
→
F
]
er
[𝜇,
𝜈].
NV-Fs
over
[𝜇,
𝜈].
(𝒾,
(𝒾,
for
all
𝜑
where
ℛ
denotes
the
collection
integr
𝜑)
and
𝔖
both
are
].
Then
is
𝐹𝑅-integrable
over
[𝜇,
𝜈]
if
and
only
if
𝔖
Definition
1.
[49]
A
fuzzy
Definition
number
1.
[49]
valued
A
fuzzy
map
number
𝔖
:
𝐾
⊂
ℝ
valued
→
𝔽
map
𝐾
⊂
→of
𝔽Riemannian
is
called
F-NV-F.
For
is
each
called
],
)
∗
∗
[𝜇,
0
(1
Moreover,
𝔖
is
𝐹𝑅-integrable
over
𝜈],
then
+
−
𝜉)𝒿
([ ∈
) . all
, ],for
∗ 𝜑),
(𝐹𝑅)
(𝒾)𝑑𝒾
) =𝜉 [𝔖
(𝒾,
(𝒾,
(𝒾)
[𝔖
(𝒾,
(𝒾,
[𝜇,
tions
of
I-V-Fs.
𝔖
is
𝐹𝑅
-integrable
over
[𝜇,
𝜈]
if
𝔖
𝔽
Note
that,
if
𝜑),
𝔖
𝜑)]
for
all
𝒾
∈
[𝜇,
𝜈]
and
for
=
all
𝔖
𝜑)]
for
all
𝒾
∈
[𝜇,
𝜈]
and
:
[𝜇,
𝜈]
⊂
ℝ
→
𝒦
are
given
by
𝔖
𝔖[0,
𝜐],
∈
1],
where
𝔖(𝒾)
≥
0
for
all
𝒾
∈
𝜐].
If
(11)
is
reversed
then,
𝔖
It
is
a
partial
order
relation.
(8)
(𝒾)𝑑𝒾
(𝒾)𝑑𝒾
(𝐹𝑅)
(𝐹𝑅)
(𝒾)𝑑𝒾
(𝐹𝑅)over
(𝒾)𝑑𝒾
(𝐼𝑅)
(𝐼𝑅)
(𝐼𝑅)
∗ denotes the
𝔖(𝒾)𝑑𝒾
𝔖
𝔖[𝜇,
=
𝔖where
=
𝔖(𝒾,
𝜑)𝑑𝒾
=
: 𝔖(𝒾,
𝜑)
∈
𝔖
ℛ(𝒾)𝑑𝒾
=, of
=
𝜑)𝑑𝒾
: 𝔖(𝒾,
𝔖(𝒾,
𝜑)func𝜑)𝑑𝒾
∈that,
ℛfuzzy
, ∈(8)
ℛ([ ,
(𝐹𝑅)
(𝒾)𝑑𝒾
tions
of
I-V-Fs.
isis
-integrable
over
[𝜇,
𝜈]then
if
𝔖𝔖(𝒾,
∈=𝔽
. Note
if
𝑅-integrable
over
[𝜇,
𝜈].
Moreover,
if∗𝔖𝔖
𝐹𝑅-integrable
𝜈],
([
], some
) , 𝔖
([: 𝔖(𝒾,
for
all
𝜑
∈
(0,if
1],
ℛ([𝐹𝑅
denotes
the
collection
Riemannian
integrable
, ], ) 𝜑)
ℛ
collection
of
all
𝐹𝑅-integrable
F],∈
)𝔽
,
We
now
use
mathematical
operations
to
examine
of
the
characteristics
of
[𝜇,
[0,
[𝜇,
([ are
for
all
𝒾,
𝒿
∈
𝜐],
𝜉
∈
1],
where
𝔖(𝒾)
≥
0
for
all
𝒾
∈
𝜐].
If
(11)
is
reversed
then,
𝔖
, ], )Lebesgue-integrable,
µ,
υ
if
[
]
(𝐹𝑅)
(𝒾)𝑑𝒾
-V-Fs.
𝔖
is
𝐹𝑅
-integrable
over
[𝜇,
𝜈]
𝔖
.
Note
that,
if
𝜑)
then
𝔖
is
a
fuzzy
Aumann-integrable
funcIt
is
a
partial
order
relation.
𝜑
∈
(0,
1]
whose
𝜑
𝜑
-cuts
∈
(0,
define
1]
whose
the
family
𝜑
-cuts
of
define
I-V-Fs
the
𝔖
family
of
I-V-Fs
𝔖
:
𝐾
⊂
ℝ
→
𝒦
are
given
:
𝐾
⊂
by
ℝ
→
𝒦
(𝐹𝑅)
(𝒾)𝑑𝒾
tions
of
I-V-Fs.
𝔖
is∗if,
𝐹𝑅[𝔵]
-integrable
over
[𝜇,
if 1],
𝔖the
∈(4)
𝔽 . are
No
[𝔵]
[𝔴] then
[𝔴] (4)
𝔵 is
≼
𝔴𝜐].
if∗for
and
only
if,
𝔵reversed
≼
and
only
≤(11)
for
allifis𝜑
∈fuzzy
(0,
1],
≤operations
for all
𝜑𝜈]
∈ (0,
∗ (𝒾,
𝔴
examine
[𝜇,
ble
over
[𝜇,
𝜈].
Moreover,
ifis
𝔖
𝐹𝑅-integrable
over
𝜈],
then
Symmetry
2022,
14,
x
FOR
PEER
REVIEW
4of
of
19 characteristi
[𝜇,
onically
concave
function
on
We
now
use
mathematical
to
some
∗
(𝒾,
[𝜇,
[0,
[𝜇,
(1
𝜑),
𝔖
𝜑)
are
Lebesgue-integrable,
𝔖
a
Aumann-integrable
func𝔖
notes
the
collection
of
Riemannian
integrable
func∈
𝜐],
𝜉
∈
1],
where
𝔖(𝒾)
≥
0
all
𝒾
∈
𝜐].
If
is
then,
𝔖
+
𝜉𝔖(𝒿),
≤
−
𝜉)𝔖(𝒾)
(11)
(𝒾,
(𝒾,
(𝒾,
(𝒾,
𝜑)
and
𝔖
𝜑)
both
are
𝜑)
and
𝔖
𝜑)
both
are
r
[𝜇,
𝜈]
𝜑
∈
if
(0,
and
1].
only
Then
if
𝔖
𝔖
𝐹𝑅-integrable
over
[𝜇,
𝜈]
if
and
only
if
𝔖
(𝐹𝑅)
(𝒾)𝑑𝒾
(𝒾,
(𝒾,
tions
I-V-Fs.
𝔖𝜐]
𝐹𝑅
-integrable
[𝜇,
𝜈]
if
𝔖(valued
∈𝒾,
𝔽
.∈
that,
if𝔽
𝜑),
𝔖
𝜑)
are
Lebesgue-integrable,
then
𝔖
is
a)all
fuzzy
Aumann-integrable
func∗𝔖
∗a
∗𝜐].
Symmetry
2022,
14,
xvalued
FOR
PEER
Definition
1. ∗[49]
fuzzy
number
Definition
map
1.
[49]
Definition
:and
𝐾on
⊂
A
ℝ
fuzzy
→
𝔽
1.
number
[49]
A
fuzzy
valued
number
map
:𝒾the
𝐾
⊂
ℝ
map
→
𝔽, Note
𝔖the
:is𝐾called
⊂
ℝfuzzy
→
isto
called
F-NV-F.
For
each
F-NV-F.
isreal
called
Forthe
each
F-NV-F
numbers.
Let
𝔵,is
𝔴
𝔽
𝜌
∈1if,
ℝ,
then
we
have
[𝜇,
(0,
(0,
[𝜇,
[𝜇,
∗ (𝒾,
is∗ called
harmonically
concave
function
∗of
=𝔖
∞)
is=
said
to
be
a∈REVIEW
convex
=
∞)
set,
is
said
for
to
all
be
𝒾,
𝒿(𝒾,
convex
∈
set,
if,
for
all
𝒿 1]
on
3. [34]
A set
𝐾
Definition
=
𝜐] ⊂aA
3.ℝ
[34]
A
set
𝐾𝔖
⊂
ℝ
e
eof
e
∗over
e
.𝜉)𝒿
We
now
use
mathematical
operations
examine
some
characteristics
of
(𝒾)
[𝔖
(𝒾,
(𝒾,
(𝒾)
[𝔖
(𝒾,
𝔖
𝔖
=
𝜑),
𝔖
𝜑)]
=
for
all
𝜑),
𝒾
∈
𝔖
𝐾.
Here,
𝜑)]
for
for
each
∈
𝜑
𝐾.
∈
(0,
Here,
for
end
each
point
𝜑
∈
(0,
1]
end
p
S
−
ξ
4
1
−
ξ
S
+
ξ
S
(13)
+
ξ
(
)
(
)
𝔖
𝜑)
are
Lebesgue-integrable,
then
is
a
fuzzy
Aumann-integrable
func(𝒾,
(𝒾,
∗
∗
𝜑),
𝔖
𝜑)
are
Lebesgue-integrable,
then
𝔖
is
a
fuzzy
Aumann-integr
𝔖
∗
numbers.
Let
𝔵,, 𝔴],where
∈) 𝔽denotes
and
𝜌the
∈ )ℝ,
then wethe
∗ relation.
Symmetry
2022,
14,
x𝜑
FOR
PEER
REVIEW
4 ofofintegrable
(𝑅)
(𝑅)
(𝐹𝑅)
(𝒾)𝑑𝒾
tion
over
[𝜇,
𝜈].
harmonically
concave
function
on
=
𝔖
𝔖
𝜑)𝑑𝒾
(𝐹𝑅)
(𝒾)𝑑𝒾
for
(0,
1],
where
ℛdefine
for
all
∈(𝒾,
(0,
1],
all
where
𝜑
∈𝜈],
(0,
1],
denotes
the
of
Riemannian
integrable
collection
denotes
funcofhave
Riemannian
integr
over
[𝜇,
if
𝔖1],
∈
𝔽(𝒾,
.[𝜇,
Note
that,
if𝜑
∗],(𝒾,
It
a(0,
partial
order
relation.
is
a𝐹𝑅-integrable
partial
order
∗tion
([𝜑)𝑑𝒾
), the
([
([fuzzy
], of
over
[𝜇,
𝜈].
, given
∗ℛ
∗for
∗the
𝜑𝜈]is
∈𝔖all
1]
𝜑
𝜑
family
(0,
1]
whose
ofcollection
𝜑
I-V-Fs
∈
(0,
𝜑ℝ,
-cuts
1]
𝔖
whose
define
𝜑→
-cuts
family
define
the
I-V-Fs
family
𝔖
of
𝔖 19Riemannian
:ℛ
𝐾
⊂
ℝ
𝒦
are
by
: collection
𝐾 I-V-Fs
⊂and
ℝ→
𝒦
:𝐾
are
⊂ given
ℝ →func𝒦of
by
are
[𝜇,
[𝜇,
[0,
(𝒾,
s1],
𝐹𝑅-integrable
𝑅-integrable
over
over
[𝜇,
𝜈],
𝜈].
then
Moreover,
if,𝜐].
𝔖It
is
over
then
we
have
𝐾,
𝜉∈ whose
∈
we
𝜑),
𝔖
𝜑)
are
Lebesgue-integrable,
then
𝔖
is
a
Aumann-integrable
func𝔖-cuts
numbers.
Let
𝔵,∈
𝔴
∈
𝔽
and
𝜌
∈
then
we
have
(𝒾,
(𝒾,
(𝑅)
(𝑅)
(𝐹𝑅)
(𝒾)𝑑𝒾
∗have
=
𝔖
𝜑)𝑑𝒾
,
𝔖
𝜑)𝑑𝒾
𝔖
(.
(.
(.
(.
functions
𝔖
functions
𝔖
,
𝜑),
𝔖
,
𝜑):
𝐾
→
ℝ
are
,
𝜑),
called
𝔖
,
𝜑):
lower
𝐾
→
and
ℝ
are
upper
called
functions
lower
of
𝔖
upper
.
functions
𝔖
.
[𝜇,
𝜈].
∗
tion
over
[𝜇,
𝜈].
∗𝔖
∗[𝔵+
(5)
[𝔴]
𝔴]
= [𝔵]
+
,fuzzy
∗ (𝒾,
[𝜇,
𝒾)=≥(0,
0then
for
all
𝒾∈
𝜐].
If
(11)
is𝐹𝑅
reversed
then,
∗is
∗is
(𝐹𝑅)
(𝒾)𝑑𝒾
(𝐹𝑅)
(𝐹𝑅)
We
now
use
mathematical
operations
now
toHere,
examine
use
mathematical
some
of
the
operations
characteristics
to
examine
of
some
of
the
characteristics
of1]
fuzzy
tions
of
I-V-Fs.
𝔖=
-integrable
tions
of
[𝜇,
𝜈]
𝔖∈
of
I-V-Fs.
𝐹𝑅
-integrable
𝔖
𝔖
-integrable
[𝜇,
𝜈]
if∈∈
over
[𝜇,
𝜈]
𝔖(𝒾)𝑑𝒾
𝔖
.end
Note
that,
ifeach
Note
that,
∈ real
𝔽 if
. No
(𝒾,
(𝑅)
(𝑅)
(𝐹𝑅)
(𝒾)𝑑𝒾
is
said
to
a𝜑),
convex
set,
if,
all
𝒾,
∈
𝔖
𝜑)𝑑𝒾
,𝒿𝐾.
𝜑)𝑑𝒾
𝔖
able,
𝔖
is
fuzzy
Aumann-integrable
(𝒾)
[𝔖
(𝒾,
(𝒾,
(𝒾)
[𝔖
(𝒾)
(𝒾,
𝔖
𝔖,𝒾𝒿
𝔖
𝜑)]
for
all
𝒾func∈over
=
𝜑),
for
𝔖
each
=
∈
for
𝜑),
(0,
1]
all
𝔖S∗is∈
𝒾𝔽)𝐹𝑅
𝜑)]
∈over
𝐾.
Here,
point
all
for
𝒾real
𝐾.
∈
for
(0,
1]
each
the
(0,
point
∗for
[𝔵]
for
all
,We
∈
µ,
υ𝔖
ξif
0,[𝔖
1𝜑
where
≥
0for
for
all
and
sif[𝔴]
∈
.∈
If(𝒾)𝑑𝒾
(13)
over
[𝜇,
𝜈].
[I-V-Fs.
]tions
[(𝒾,
]I-V-Fs
(the
[µ,Here,
]𝜑
[∈0,
].end
𝔴]
=υ
+
,𝔽1𝜑
(9)
∗ (𝒾,
∗,(𝒾,
] LetREVIEW
𝔖∞)
: [𝜇,
𝜈]
⊂=aℝ
→∗be
𝔽
be
aSymmetry
F-NV-F
whose
𝜑-cuts
define
the
family
of𝜑)]
PEER
Symmetry
2022,
14,
xtion
FOR
PEER
2022,
REVIEW
14,
x𝔖
FOR
PEER
REVIEW
4[𝔵+
of
19
4isthe
of 19end
𝒾𝒿
[𝜇,
on on numbers.
𝜐].
(9)
(5)
[𝔵+
[𝔵]
[𝔴]
𝔴]
=
+
,
∗
∗
∗
Theorem
2.
[49]
Let
𝔖
:
[𝜇,
𝜈]
⊂
ℝ
→
𝔽
be
a
F-NV-F
whose
𝜑-cuts
define
the
family
of
I-V-Fs
∗
∗
∗
Let
𝔵,
𝔴
∈
𝔽
numbers.
Let
𝔵,
𝔴
∈
𝔽
and
𝜌
∈
ℝ,
then
we
have
and
𝜌
∈
ℝ,
then
we
have
Proposition
1.
[51]
Let
𝔵,
𝔴
∈
𝔽
.
Then
fuzzy
order
relation
“
≼
”
is
given
on
𝔽
by
e
𝐾.
∈,→
Theorem
Let
[𝜇,
𝜈]
⊂
𝔽
be
aLet
F-NV-F
whose
𝜑-cuts
the
family
of
I-V-Fs
(𝒾,
(𝒾,
(𝒾,
(𝒾,
(10)
(10)
𝜑), 𝔖𝔖(𝒾,
𝜑)
Lebesgue-integrable,
𝜑),
𝔖
then
𝜑)
𝔖
is
are
𝜑),
aℝ
Lebesgue-integrable,
fuzzy
𝔖
𝜑)
Aumann-integrable
func𝔖Then
iscalled
adefine
fuzzy
then
Aumann-integrable
𝔖(concave)
is a upper
fuzzy
Aumann-integr
func𝔖
𝔖
𝔖
[𝔵
[𝔵]
(.
(.
(.
(.Lebesgue-integrable,
then,
is
called
concave
F-NV-F
on
υ∈
The
of
all
convex
F-NV-Fs
isintegral
S
[𝜑):
].[.then
×
𝔴]
=
×
𝔴]
,set
functions
functions
𝔖
functions
𝔖𝐾.
, 𝜑),are
𝔖 (.
, 𝜑):
𝐾 reversed
→
ℝ∈2.
are
called
lower
, ∗𝜑),
and
𝔖
upper
𝜑):
𝐾
functions
,→
𝜑),
→
ℝ
𝔖
are
, 𝔵,
of
called
𝔖
𝐾
→
lower
ℝThen
are
and
upper
lower
functions
and
of
functions
.(6)is given
ofon
𝔖.o𝔽
∗ (𝒾,
∗[49]
be
an
fuzzy
an
integral
of𝜑-cuts
𝔖𝔖fuzzy
over
Definition
2.
[49]
𝔖
[𝜇,
𝜈]
⊂
2.
[49]
𝔽are
:µ,
[𝜇,
𝜈]
∗ (𝒾,
∗of
∗:define
∗ℝ
(9)
Proposition
1.
[51]
Let
𝔴
fuzzy
order
relation
2.𝒦[49]are
𝔖𝜑)𝑑𝒾
: [𝜇,by𝜈]
⊂∗ (.ℝ
a(1
F-NV-F
the
family
I-V-Fs
(𝒾)
(𝒾,
(1
(𝒾,be
(𝒾,𝒾Let
(𝒾,
(𝑅)
(𝑅)
(𝑅)
(𝑅)
(𝐹𝑅)
(𝒾)𝑑𝒾
=
𝜑),
𝜑)]
for𝜑-cuts
all∗𝜉𝒾
∈Definition
[𝜇,
and
for
→
given
𝜉𝒾
−𝔖𝜉)𝒿
+
−:,𝜈]
𝜉)𝒿
Theorem
2.
[49]
Let
𝔖
:𝔖[𝜇,
𝜈]F-NV-F.
⊂𝔽⊂
ℝ. ℝ
→[𝔵→
𝔽×𝔽𝔴]
be be
a=
F-NV-F
the famil
𝔖Let
, 𝔖
𝔖𝔽∗[𝔖
𝜑)𝑑𝒾
= whose
𝔖
𝜑)𝑑𝒾
𝔖
𝜑)𝑑𝒾
𝔖
(𝒾)𝑑𝒾
=→(𝐼𝑅)
𝔖
[𝔵]F-NV-F.
[ 𝔴] Then
×whose
, “ ≼ ”define
∗+
∗ (𝒾,
∗all
∗
(𝒾)
[𝔖
(𝒾,
(𝒾,
Proposition
1.
[51]
Let
𝔵,
𝔴
∈
𝔽
.
Then
fuzzy
order
relation
“
≼
”
is
given
on
𝔽
by
=
𝜑),
𝔖
𝜑)]
for
all
𝒾
∈
[𝜇,
𝜈]
and
for
all
ℝ
→𝔖𝒦[𝜇,
are
given
by
𝔖
𝜈].𝜈] ⊂ Theorem
tion
over
[𝜇,
𝜈].
tion
over
[𝜇,
𝜈].
denoted
by
(𝒾)𝑑𝒾
(𝐼𝑅)
=
𝔖
2.
[49]
Let
𝔖
:
[𝜇,
𝜈]
⊂
ℝ
→
𝔽
be
a
F-NV-F
whose
𝜑-cuts
define
the
family
of
I-V-Fs
∗
𝒾𝒿 tion over𝔖[𝜇,: [𝜇,
[𝔖
(𝒾,
(𝐹𝑅)
(𝒾)𝑑𝒾
(𝐹𝑅)
(𝒾)𝑑𝒾
[𝔵
[𝔵]
[[𝔴]
𝔖[𝔵]
for
∈ [𝜇, 𝜈]
for all
: [𝜇,𝜈],𝜈] denoted
⊂
→ 𝒦by
given
by
𝔖
denoted
𝔖
,(𝒾)
isby
given
by
,𝜑)]
isall[𝔴]
given
levelwise
byand (6)
×
𝔴]
= [𝔵]
× 𝜑),
𝔴]
, (𝒾,
(5)
(5)
𝔵are
≼[𝜇,
𝔴
if∈𝔖
and
only
if,
≤
for
𝜑 ∈, all
(0,𝒾1],
(4)
[𝔵+
[𝔵+
[𝔵]
[𝔴]
𝔴]
=for
+𝜈],
, 𝜈]
𝔴]
=
+
∗ℝ
∗levelwise
∗𝒾
(7)
[𝜌.
[𝔵]
𝔵]
=
𝜌.
(𝒾)
[𝔖
(𝒾,
(𝒾,
=if
𝔖
𝜑)]
[𝜇,
for
all
→
𝒦Definition
are
given𝜑-cuts
by[49]
𝔖[𝜇,Let
(𝒾)[𝜌.
[𝔖
(𝒾,
=
𝔖𝜑),
𝜑),
𝔖∗ (𝒾,
𝜑)]
∈ [𝜇, 𝜈]
:(𝒾,
[𝜇,
𝜈]
⊂
→
𝒦
by
𝔖
∈ 𝐾.
𝜑)
and
𝔖FSX
𝜑)
both
are
n⊂be
𝔖ℝais
𝐹𝑅-integrable
over
𝜈] (𝐼𝑅)
and
only
ifℝ
𝔖
(10)
[𝔵]
[𝔴]
∗family
𝔵are
≼
𝔴
ifbe
and
only
if,
≤
for
all for
𝜑fuzzy
∈
(0,
∗,an
∗ (𝒾,
µ,
υ: 2.
,and
F
FSV
µ,
υ=
FThen
s))
. [𝔵]
beover
an
F-NV-F.
Then
fuzzy
integral
of
an
𝔖𝔖
F-NV-F.
over
be
F-NV-F.
fuzzy
integral
Then
ofall
𝔖𝒾1],
integral
over
𝔖the
:𝔖[𝜇,
𝜈]
⊂(𝒾)𝑑𝒾
Definition
→
𝔽(9)
2.all
Definition
[49]
Let
𝔖
[𝜇,
[49]
𝜈]
ℝ
→𝔖given
𝔽
: ∗[𝜇,
𝜈]
⊂
ℝ(9)
→
𝔽
([
]ℝ
)Let
(𝜑)]
]𝒾,𝔵]
∗([
F-NV-F
of
I-V-Fs
=
𝜌.
0 ,ifs⊂
0[𝜇,
∗
(𝒾,
(𝒾,
𝒾 + (1
− Proposition
𝜉)𝒿 whose
𝜑)
and
𝔖
𝜑)
both
are
𝜑 ∈2.1.
(0,[51]
1].define
Then
is
𝐹𝑅-integrable
[𝜇,
𝜈]
if
and
only
𝔖
(𝒾)
[𝔖
(𝒾,
(𝒾,
=
𝜑),
𝔖
for
all
∈
𝜈]
and
for
all
:
[𝜇,
𝜈]
⊂
ℝ
→
𝒦
are
given
by
𝔖
𝔖
[𝔵]
∗[𝔴]
(𝒾,
(𝒾,
𝔵
≼
𝔴
if
and
only
if,
≤
for
all
𝜑
∈
(0,
1],
(4)
𝜑)
and
𝔖
𝜑)
both
are
𝜑
∈
(0,
1].
Then
𝔖
is
𝐹𝑅-integrable
over
[𝜇,
𝜈]
if
and
only
if
𝔖
∗
Let
𝔵,
𝔴
∈
𝔽
Proposition
1.
Proposition
[51]
Let
𝔵,
𝔴
1.
∈
[51]
𝔽
Let
𝔵,
𝔴
∈
𝔽
.
Then
fuzzy
order
relation
“
≼
”
is
.
Then
given
fuzzy
on
𝔽
order
.
by
Then
relation
fuzzy
order
“
≼
”
relation
is
given
on
“
≼
𝔽
”
is
by
given
on 𝔽
∗
∗
(7)
[𝜌.
[𝔵]
𝔵]
=
𝜌.
It𝜈]
is(𝒾)𝑑𝒾
partial
order
relation.
[𝜇,
[𝜇,
[𝜇,
[𝜇,
(6)
(6)
(0,
[𝔵
[𝔵]𝜑)
[function
[𝔵]
on
4.For
[34]
The
Definition
𝜐]
→isby
ℝ
4.(𝐹𝑅)
[34]
The
𝜐]
→
ℝdenoted
called
aa𝔖:
harmonically
is
convex
called
aby
harmonically
on
𝜐]
function
ifif anddefine
1].
all
𝜑
∈𝔖:
ℱℛ
denotes
collection
of
all
𝐹𝑅-integrable
F×
𝔴]
×
𝔴]
,:(𝒾,
𝔴]
=
𝔴]
, a𝜐]
(𝒾,
and
𝔖
𝜑)
both
are
Then
is𝜑),
over
if⊂
and
only
ifbe
𝔖athen
∗ (𝒾,
([𝐹𝑅-integrable
(𝒾, 𝜑
[𝜇,
[𝜇,
[𝜇,
(𝐹𝑅)
(𝒾)𝑑𝒾
(𝐹𝑅)
[𝜇,
, ],is
𝔖∗fami
𝜑
∈is
(0,
1].
𝔖ℝ
is
over
[𝜇,
𝜈]
only
𝔖family
𝜈],𝐹𝑅-integrable
𝔖
𝜈],
by
𝜈],
denoted
𝔖
by
𝔖
,[𝔵
is
given
levelwise
,convex
isif𝐹𝑅-integrable
given
levelwise
,×
is
given
by
levelwise
by ifthe
r)(𝐼𝑅)
[𝜇,
𝜈].𝔖
Moreover,
if1],
𝔖
over
𝜈],
Theorem
2.𝜑)]
[49]
Let
𝔖1].
:𝒾)[𝜇,
[𝜇,
𝜈]
ℝthe
→
Theorem
𝔽×
2.
[49]
Theorem
Let
𝔖
[𝜇,
2.Then
[49]
𝜈]
⊂
Let
→
𝔖
𝔽
:the
[𝜇,
𝜈]
⊂
ℝof
→
𝔽[on
F-NV-F
whose
𝜑-cuts
define
be
family
a(𝒾)𝑑𝒾
F-NV-F
I-V-Fs
whose
be
F-NV-F
𝜑-cuts
whose
𝜑-cuts
define
ofand
I-V-Fs
the
∗=
∗ (𝒾, 𝜑)
= [𝔖
𝔖denoted
all
∈For
[𝜇,
𝜈]
and
for
all
It
a
partial
order
relation.
(𝒾)𝑑𝒾
(𝒾)𝑑𝒾
(𝐼𝑅)
𝔖
=
𝔖
(0,
for
all
𝜑for
∈
(0,
all
𝜑
∈
1],
ℱℛ
denotes
the
collection
of
all
𝐹𝑅-integrable
∗F∗ (𝒾,
([
],
)
[𝜇,
,
𝑅-integrable
over
[𝜇,
𝜈].
Moreover,
if
𝔖
is
𝐹𝑅-integrable
over
𝜈],
then
(8)
(𝒾,
(𝒾,
(𝒾)𝑑𝒾
(𝒾)𝑑𝒾
(𝐹𝑅)
(𝒾)𝑑𝒾
(𝐹𝑅)
(𝒾)𝑑𝒾
(𝐼𝑅)
(𝐼𝑅)
𝜑)
and
𝔖
𝜑)
both
are
𝜑
∈
(0,
1].
Then
𝔖
is
𝐹𝑅-integrable
over
[𝜇,
𝜈]
if
and
only
if
𝔖
𝔖
𝔖
=
𝔖
=
=
𝔖(𝒾,
𝜑)𝑑𝒾
𝔖
:
𝔖(𝒾,
=
𝜑)
∈
𝔖(𝒾,
ℛ
𝜑)𝑑𝒾
,
:
𝔖(𝒾,
𝜑)
∈
ℛ
[𝜇,
e
𝑅-integrable
over
[𝜇,
𝜈].
Moreover,
if
𝔖
is
𝐹𝑅-integrable
over
𝜈],
then
We
now
use
mathematical
operations
to
examine
some
of
the
characteristics
of
fuzzy
([
],
)
([
∗
,
, ]
𝜈].
S
:
µ,
υ
→
F
µ,
υ
Definition
7
[39].
The
F-NV-F
is
called
harmonically
convex
F-NV-F
on
[
]
[
]
Definition
1.
[49]
A
fuzzy
number
valued
map
𝔖
:
𝐾
⊂
ℝ
→
𝔽
is
called
F-NV-F.
For
each
It
is
a
partial
order
relation.
(0, if1],𝒦
∈ (0,
1]. 𝔖For
all
𝜑⊂∈ ℝ
ℱℛ
denotes
the
collection
of𝒦
all
𝐹𝑅-integrable
F∗𝔵
0=
[𝔵]
[𝔵][𝔖
[𝔵]
[𝔴]
[𝔴]
[𝔴]
𝔵isare
≼
𝔴
and
only
if,
≼
𝔴
if𝜑for
and
only
≼
if,
𝔴
if
and
if,∗=
≤
for
all
∈by
(0,
≤
(4)
for
allexamine
≤𝜑for
(0,
1],
for
all
𝜑
(0,
𝒾𝒿
𝒾𝒿
([
[𝜇,
, ], if
over
𝜈].
Moreover,
𝔖
𝐹𝑅-integrable
over
𝜈],
then
[𝜇,
∗)(𝒾,
𝑅-integrable
over
[𝜇,
𝜈].
Moreover,
ifonly
𝔖
is
𝐹𝑅-integrable
𝜈],
then
(𝒾)
[𝔖⊂
(𝒾,
(𝒾)
(𝒾,
(𝒾)
(𝒾,
[𝔖
(𝒾,
We
now
use
mathematical
to
some
of𝜈]
the
characteristi
(7)
(7)
=
𝜑),
𝜑)]
all
∈
[𝜇,
𝜈]
and
for
𝜑),
all
𝔖
𝜑)]
𝜑),
𝔖⊂∗over
𝒾ℝ∈→
𝜑)]
for
all
𝒾1],
for
∈(4)
all
[𝜇, 𝜈]
: [𝜇,
𝜈]
given
by
𝔖𝔖
:𝜐]
[𝜇,
𝜈]
ℝ𝜌.
→
:𝔖
[𝜇,
𝜈]
are
⊂
given
ℝ
→
𝒦
𝔖𝔵𝒾1],
are
given
𝔖
𝔖
[𝜌.
[𝜌.
[𝔵]
[𝔵]
𝔵]
𝔵]number
=
=by
𝜌.
NV-Fs
over
𝜈].
Definition
1.
[49]
A
fuzzy
valued
map
:∈𝐾all
𝔽
is∈and
called
F-NV-F
∗ (𝒾,
∗operations
∗ (𝒾,𝔖
(𝒾,
𝜑)[𝜇,
and
𝔖
𝜑)
both
are
rslecalled
[𝜇,
𝜈] [𝜇,
and
only
if 𝔖∗→
[𝜇,
aif harmonically
convex
function
on
if
(1
(1
+
𝜉𝔖(𝒿),
+
𝜉𝔖(𝒿),
𝔖
≤
−
𝔖
𝜉)𝔖(𝒾)
≤
−
𝜉)𝔖(𝒾)
(11)
(11)
numbers.
Let
𝔵,
𝔴
∈
𝔽
and
𝜌
∈
ℝ,
then
we
have
if
[𝜇,
𝑅-integrable
[𝜇,
𝜈].
if 𝔖valued
isLet
𝐹𝑅-integrable
over
𝜈],
then
𝜑 ∈now
(0,over
1]use
whose
𝜑𝜉)𝒿
-cuts
define
the
family
of
I-V-Fs
𝔖
:characteristics
𝐾⊂
ℝhave
→ 𝒦 are
given
by
We
mathematical
operations
to
examine
some
the
of
fuzzy
er (𝐹𝑅)
[𝜇, 𝜈]. (𝒾)𝑑𝒾 𝜉𝒾 +(𝑅)
Definition
1.
[49]
A
fuzzy
number
map
𝔖
𝐾
⊂
→∈of
𝔽ℝ,
is𝔖
called
F-NV-F.
For
∗Moreover,
!:∈𝔖
(1 relation.
(1
𝜉𝒾
−
𝜉)𝒿
−
numbers.
𝔴
∈(𝒾)𝑑𝒾
𝔽-cuts
and
then
we
∗(𝒾)𝑑𝒾
(8)
(8)
(𝒾,
(𝑅)
(𝒾)𝑑𝒾
(𝒾)𝑑𝒾
(𝐹𝑅)
(𝒾)𝑑𝒾
(𝐹𝑅)
(𝒾)𝑑𝒾
(𝐹𝑅)
(𝐼𝑅)
(𝐼𝑅)
(𝐼𝑅)
𝜑∈
∈
(0,
1]
whose
𝜑
the
family
of
I-V-Fs
𝔖𝔖
𝐾:func→
𝒦are
are
=[𝜇,
𝔖
𝜑)𝑑𝒾
,ℱℛ
𝔖
𝜑)𝑑𝒾
𝔖
𝔖
𝔖all
=said
𝔖,+
𝔖(𝒾,
𝜑)𝑑𝒾
=
: 𝔵,
𝔖(𝒾,
𝜑)
ℛ
=ℝ,𝜌
,and
𝔖(𝒾,
𝜑)𝑑𝒾
: 𝔖(𝒾,
𝔖(𝒾,
𝜑)
∈each
ℛ∗:∗(𝒾,
𝔖(𝒾,
, ∈
It
is
a(0,
order
It
is
a(0,
partial
order
It
is
aℛ
relation.
partial
order
relation.
(𝒾,
(𝒾,
(𝒾,
(𝒾,
(0,
𝜑)
and
𝔖
𝜑)
are
𝜑)
and
𝜑)
𝜑)
both
and
𝔖ℛ∗ (𝒾,
𝜑𝐾
∈𝜑=
1].
Then
is
𝐹𝑅-integrable
𝜑
over
∈
[𝜇,
𝜈]
Then
if
and
𝜑
𝔖
only
(0,
is
if
𝐹𝑅-integrable
𝔖
Then
𝔖𝐾.
is
over
𝐹𝑅-integrable
𝜈]
over
only
[𝜇,
ifthe
𝜈]
and
only
if𝜑)𝑑𝒾
𝔖Riemannian
([define
], =
)ifboth
([
], ℝ
) 𝜑)
([
∗∈(𝒾,
all
∈partial
(0,
1].
For
𝜑
1],
the
collection
of
all
𝐹𝑅-integrable
denotes
the
collection
of
𝐹𝑅-integrable
F,⊂
,𝜑
ifcollection
1]𝔖=
∗=
∗[𝜇,
for
all
𝜑
∈
(0,
1],
where
for
all
𝜑∗1].
∈all
(0,
1],
where
ℛcollection
denotes
the
denotes
of
Riemannian
integrable
of
integra
∗𝒾,
∗the
], for
) denotes
([
],F)1].
(0,
[𝜇,
, A
=all
∞)
is
to
be
a∈
convex
set,
if,
for
𝒿
∈
34]
set
𝜐]
⊂
ℝ
s([ 𝐹𝑅-integrable
over
𝜈],𝔖𝔖
then
([
],
)
([
],
)
,
,
∗
numbers.
Let
𝔵,(𝒾)𝑑𝒾
𝔴
𝔽
and
𝜌
∈
ℝ,
then
we
have
(𝒾)
[𝔖
(𝒾,
(𝒾,
(𝒾,
(𝒾,
(𝑅)
(𝑅)
(𝐹𝑅)
𝔖
=
𝜑),
𝔖
𝜑)]
for
all
𝒾
∈
Here,
for
each
𝜑
∈
(0,
end
point
real
=
𝔖
𝜑)𝑑𝒾
,
𝔖
𝜑)𝑑𝒾
𝔖
𝜑
∈
(0,
1]
whose
𝜑
-cuts
define
the
family
of
I-V-Fs
𝔖
:
𝐾
⊂
ℝ
→
𝒦
are
given
by
∗
(𝒾,
(𝒾,
(𝑅)
(𝑅)
(𝐹𝑅)
(𝒾)𝑑𝒾
=
𝔖
𝜑)𝑑𝒾
,
𝔖
𝜑)𝑑𝒾
𝔖
∗
(5)
[𝔵+
[𝔵]
[𝔴]
(0,
[𝜇,
𝔴]
=
+
,
=
∞)
is
said
to
be
a
convex
set,
if,
for
all
𝒾,
𝒿
∈
Definition
3.
[34]
A
set
𝐾
=
𝜐]
⊂
ℝ
∗
e
e
e
∗
e
∗
(𝒾)
[𝔖
(𝒾,
(𝒾,
(1
𝔖
=
𝜑),
𝔖
𝜑)]
for
all
𝒾
∈
𝐾.
Here,
for
each
𝜑
∈
(0,
1]
the
end
+
𝜉𝔖(𝒿),
≤
−
𝜉)𝔖(𝒾)
S
4
1
−
ξ
S
,
(14)
+
ξ
S
(
)
(
)
(11)
We
now
use
mathematical
operations
We
to
now
examine
use
mathematical
We
some
now
of
use
the
operations
characteristics
mathematical
to
examine
operations
of
fuzzy
some
to
examine
of
the
characteristics
some
of
the
characteristi
of
fuzzy
(𝒾,
(𝒾,
(𝑅)
(𝑅)
(𝐹𝑅)
(𝒾)𝑑𝒾
[𝜇,
[0,
[𝜇,
[0,
[𝜇,
[𝜇,
=
𝔖
𝜑)𝑑𝒾
,
𝔖
𝜑)𝑑𝒾
𝔖
𝒿on
∈
𝜐],
𝜉
∈
for
1],
all
where
𝒾,
𝒿
∈
𝔖(𝒾)
𝜐],
≥
𝜉
0
∈
for
1],
all
where
𝒾
∈
𝜐].
𝔖(𝒾)
If
≥
(11)
0
is
for
reversed
all
𝒾
∈
then,
𝜐].
If
𝔖
(11)
is
reversed
then,
𝔖
∗
NV-Fs
over
[𝜇,
𝜈].
(𝒾,
(𝒾,
(𝑅)
(𝑅)
(𝐹𝑅)
(𝒾)𝑑𝒾
∗
=
𝔖
𝜑)𝑑𝒾
,
𝔖
𝜑)𝑑𝒾
𝔖
Definition
1.
[49]
A
fuzzy
number
Definition
valued
map
1.
[49]
𝔖
:
𝐾
A
⊂
fuzzy
ℝ
→
number
𝔽
valued
map
𝔖
:
𝐾
⊂
ℝ
→
𝔽
is
called
F-NV-F.
For
each
is
called
F-NV-F.
For
each
∗
e𝜉)𝒿
have
[𝔵]
[𝔴]
(𝐹𝑅)
(𝒾)𝑑𝒾
(𝐹𝑅)
(𝒾)𝑑𝒾
=
+
tions
ofsaid
is
tions
𝐹𝑅
of
-integrable
𝔖
over
𝐹𝑅
[𝜇,
𝜈]
if 𝜑
𝔖
[𝜇,
ifthen
𝔖𝜈],
∈
.𝔖
Note
that,
if ∈ 𝔽 . Not
[𝜇,
[𝜇,
[𝜇,
∗ (𝒾,
over
𝜈].𝔖Moreover,
𝑅-integrable
𝔖I-V-Fs.
isto𝔖,𝐹𝑅-integrable
over
𝑅-integrable
[𝜇,𝐾
over
𝜈].
Moreover,
over
𝜈],
[𝜇,
then
if)is
𝜈].
Moreover,
is -integrable
𝐹𝑅-integrable
ifupper
𝔖(0,
is𝔴]
over
𝐹𝑅-integrable
𝜈],𝔽
over
then
∗𝜈]
(0,
∞)
be
a𝔖
convex
set,
if,
all
𝒿∗𝔖∈lower
3. [34]𝑅-integrable
A set 𝐾𝐾,=𝜉 [𝜇,
𝜐] [𝜇,
⊂ℝ
(.
functions
𝔖
𝜑),
𝔖
, 𝜑):
→
are
called
functions
of
. , real
∗over
(𝒾)
[𝔖isif
=
𝜑),
𝜑)]
for
all
𝒾ℝ(𝑅)
∈for
𝐾.
Here,
for
each
∈[𝔵+
1]
the
end
point
∗ (.
ξI-V-Fs.
+
1(9)
−
ξ𝒾,
((.
∗ (𝒾,
∈ [0,
we=have
(𝒾,
(𝒾,
(𝑅)
(𝐹𝑅)
(𝒾)𝑑𝒾
=
𝔖
𝜑)𝑑𝒾
,([𝐾,and
𝔖by
𝜑)𝑑𝒾
𝔖
(9)
[𝔵+
[𝔴]
𝔴]
=
+
(.
functions
𝔖
𝜑),
𝔖
,and
𝜑):
→
are
called
lower
and
upper
functions
of 𝔖.
∗ℝ,
numbers.
𝔵,
𝔴1],
∈
numbers.
Let
numbers.
𝔵,
𝔴𝜑
∈-cuts
𝔽𝜐].
Let
𝔵,⊂
∈
𝔽denotes
and
𝜌, ∈
ℝ,
then
we
𝜌,𝔴]
∈
then
we
𝜌the
have
∈
ℝ,
then
we
have
for∈ all
𝜑 called
∈Let
(0,
1],
where
ℛdefine
for
𝜑∗ whose
∈have
(0,
for
1],
all
where
𝜑
∈∗and
(0,
ℛ
where
ℛare
denotes
the
collection
of
Riemannian
integrable
denotes
functhe
of
Riemannian
collection
of(5)
integrable
Riemannian
func[𝜇,
(9)
a1],
harmonically
concave
afunction
harmonically
on
concave
𝜐].
function
on
∗𝐾
([[𝜇,
], 𝜑
) the
([1],
],[𝔵]
)→
], ℝ
)collection
,𝔴
,𝔴]
𝜑
(0, is
1]
whose
𝜑∗ 𝔽
-cuts
∈
(0,
family
1]
of
I-V-Fs
𝔖
define
the
family
of
I-V-Fs
𝔖
:
ℝ
𝒦
given
:
𝐾
⊂
ℝ
→
𝒦
are
given
byintegr
∗all
(6)
[𝔵
[𝔵]
[
×
=
×
,
we
have
(𝒾,
(𝒾,
(𝒾,
(𝒾,
𝜑),
𝜑)(. ,are
𝜑) are
Lebesgue-integrable,
then
is a [𝔵
fuzzy
Aumann-integrable
then
𝔖
a fuzzy
funcAumann-integra
𝔖∗ 𝔖
(9)
𝒾𝒿
(.x𝔖
functions
,FOR
𝜑),
𝔖
𝜑):𝔖Lebesgue-integrable,
𝐾
ℝ𝔖
are called
lower
and𝔖upper
functions
of
𝔖×
. is[ 𝔴]
∗ →𝜑),
(𝒾,
(𝒾,
(𝑅)
(𝑅)
𝔖
𝜑)𝑑𝒾
,
𝔖
𝜑)𝑑𝒾
(𝒾)𝑑𝒾
(𝐼𝑅)
=
𝔖
[𝔵]
∗then,
×
𝔴]
=
,
∗
[𝜇,
𝒾)
0
for
all
𝒾
∈
𝜐].
If
(11)
is
reversed
𝔖
Symmetry
2022,
Symmetry
14,
x
FOR
2022,
Symmetry
PEER
14,
x
REVIEW
FOR
2022,
Symmetry
PEER
14,
x
REVIEW
FOR
2022,
PEER
14,
REVIEW
PEER
REVIEW
4
of
19
4
of
19
of
∗
∗
(0,
[𝜇,
=≥(0,
∞)
is
said
to
be
a
convex
set,
if,
for
all
𝒾,
=
𝒿
∈
∞)
is
said
to
be
a
convex
set,
if,
for
all
𝒾,
𝒿
∈
Definition
3.
[34]
A
set
𝐾
=
𝜐]
⊂
ℝ
(𝐹𝑅)
(𝒾)𝑑𝒾
(𝐹𝑅)
(𝒾)𝑑𝒾
(𝐹𝑅)
(𝒾)𝑑𝒾
e
e
e
tions
I-V-Fs.
𝔖 is 𝐹𝑅Definition
-integrable
over
of[𝔖
I-V-Fs.
[𝜇,
𝔖
of
is
I-V-Fs.
𝐹𝑅
-integrable
𝔖
𝔖
𝐹𝑅
over
-integrable
[𝜇,
𝜈]
ifeach
over
[𝜇,
𝜈]
𝔖
if reversed
𝔖
∈the
𝔽
.0,
Note
that,
if fuzzy
∈of𝔽(9)
Note
that,
∈ 𝔽 if
. 4No
𝒾𝒿
∈ for
𝐾.over
(𝒾) of
(𝒾)
(𝒾,
(10)
𝔖
𝔖
= [𝔖
for
all
𝒾 ,[𝜇,
∈
=
𝐾.
𝜑),
for
𝔖
each
for
∈
(0,
1](be
𝒾is
𝐾.
end
Here,
for
∈(14)
(0,+is
1]
the
end
point
alltions
∈
µ,
,tions
ξ+
∈
0,
1⊂
,𝜑
where
for
all
∈
µ,
υ,]=
.𝜑If
isreal
S
<
[Here,
[(𝒾,
](𝒾)𝑑𝒾
[[𝔴]
)∈
(𝐼𝑅)
=υ]𝜈]
𝔖,𝜑)]
∗ (𝒾, 𝜑), 𝔖 (𝒾, 𝜑)]
∗Let
an
F-NV-F.
Then
integral
𝔖. then,
over
2.𝔴]
[49]
𝔖
:if
[𝜇,
𝜈]
ℝ
→
𝔽all
(6)
[𝔵
[(𝑅)
×
𝔴]
=
×𝔖
𝔴]
, point
(𝒾)𝑑𝒾
(𝐼𝑅)
tion
𝜈].
tion
over
[𝜇,
𝜈].
∗[𝔵]
∗[𝔴]
∗S
=
(5)
(5)
[𝔵+
[𝔵+
[𝔵+
[𝔵]
[𝔴]
[𝔵](𝒾,
[𝔵]
=
𝔴]
=
+real
𝔴]
, Then
(1
𝜉𝒾
+
−
𝜉)𝒿
∈
𝐾.
(10)
(𝒾,
(𝒾,
(𝒾,
(𝒾,
(𝒾,
(𝑅)
(𝑅)
(𝑅)
(𝑅)
(𝑅)
(𝐹𝑅)
(𝒾)𝑑𝒾
(𝐹𝑅)
(𝒾)𝑑𝒾
(𝐹𝑅)
(𝒾)𝑑𝒾
𝒾𝒿
(7)
[𝜌.
[𝔵]
=
𝔖
𝜑)𝑑𝒾
,
𝔖
=
𝜑)𝑑𝒾
𝔖
=
𝜑)𝑑𝒾
,
𝔖
𝔖
𝜑)𝑑𝒾
,
𝜑)𝑑𝒾
𝔖
𝜑)𝑑𝒾
𝔖
𝔖
𝔖
𝔵]
=
𝜌.
(𝒾)𝑑𝒾
(𝐼𝑅)
on on𝐾,[𝜇,
𝜐].
be
an
F-NV-F.
fuzzy
integral
Definition
2.
[49]
Let
𝔖
:
[𝜇,
𝜈]
⊂
ℝ
→
𝔽
∗
∗
∗
=
𝔖
∗
∗
∗
∗
∗
(𝒾)𝑑𝒾
(𝐼𝑅)
[0,
=is
𝜉functions
1], 𝔖
we
have
(9)
(1
𝜉𝒾
−
(.
(.
called
harmonically
concave
F-NV-F
on
µ,are
υF-NV-F.
.of 𝔖. lower
[an
]called
(𝒾,
(𝒾,
𝔖(𝒾,
functions
, 𝜑),are
𝔖 (.Definition
, 𝜑):
→
ℝ
are
called
,+
𝜑),
and
𝔖
upper
,Lebesgue-integrable,
𝜑):
𝐾isfunctions
→
functions
of
𝔖. Aumann-integr
Lebesgue-integrable,
𝜑),
then
𝜑)
𝔖
is
are
a𝜉)𝒿
fuzzy
𝔖
𝜑)
Aumann-integrable
Lebesgue-integrable,
then
𝔖and
a𝔵]fuzzy
then
𝔖[𝔵]
is a𝔖 fuzzy
func𝔖∈∗ (𝒾, 𝜑),
𝔖
[𝜌.upper
=𝔖
𝜌.Aumann-integrable
∈𝔖
𝐾.
(10)
[𝜇,𝐾𝜈],
(𝐹𝑅)
(𝒾)𝑑𝒾
∗ (. 𝜑)
∗lower
denoted
by𝔖
𝔖
,(𝒾,
given
levelwise
by func∗ (𝒾,
∗𝜈]
beℝare
Then
fuzzy
integral
of
over
2.
[49]
Let
𝔖
: [𝜇,
⊂𝜑),
ℝ
→[𝜌.
𝔽
(𝒾)𝑑𝒾
(𝐼𝑅)
=
𝔖(𝐹𝑅)
(1 − 𝜉)𝒿
PEER
REVIEW
4[𝔵
of
19
𝜉𝒾) +denotes
(7)
[𝜇,
(𝒾)𝑑𝒾
[𝔵]
𝜈],
denoted
by
𝔖
,
is
given
levelwise
by
𝔵]
=
𝜌.
(6)
(6)
(0,
[𝔵
[𝔵
[𝔵]
[
[𝔵]
[
[𝔵]
[
]. For
all
𝜑
∈
1],
ℱℛ
the
collection
of
all
𝐹𝑅-integrable
F×
×
×
𝔴]
=
×
𝔴]
,
𝔴]
=
×
𝔴]
𝔴]
,
=
×
𝔴]
,
(9)
([
],
,
tion(𝒾)𝑑𝒾
overfor[𝜇,all𝜈].𝜑 ∈ (0,[𝜇,
over
[𝜇, 𝜈].
tion
over
[𝜇,
𝒾𝒿 𝔖
𝒾𝒿
(𝐼𝑅)
Theorem
2.(0,
[49]
Let
𝔖
Theorem
2.ℝ
[49]
→𝜈].the
𝔽Let
[𝜇,
𝜈]
ℝ4→
𝔽19
be𝔖a: F-NV-F
whose
𝜑-cuts
be a4F-NV-F
define
the
whose
family
𝜑-cuts
of I-V-Fs
define the(9)
famil
(𝐹𝑅)
1].𝜈],
For
all
𝜑tion
∈by
1],
ℱℛ
denotes
collection
of⊂ all
𝐹𝑅-integrable
F- 𝐹𝑅-integrable
denoted
,e⊂is
given
levelwise
by
([𝔖:𝐾.
],∈ 𝜈]
)(0,
,[𝜇,
for
all
𝜑Definition
∈
(0,
1].
For
all
𝜑(𝒾)𝑑𝒾
1],
ℱℛ
denotes
the
collection
of
F-second
ER
4,
xREVIEW
FOR
PEER
REVIEW
of
of all
19
([
],
)
∈
𝐾.
∈
,
(10)
(10)
[𝜇,
[𝜇,
34]
The
𝔖:
𝜐]
→
ℝ
is
called
a
harmonically
convex
function
on
𝜐]
if
𝜈].
Definition
8.
The
F-NV-F
is
called
harmonically
s-convex
F-NV-F
in
the
S
:
µ,
υ
→
F
[
]
Definition
1.
[49]
A
fuzzy
number
valued
map
𝔖
:
𝐾
⊂
ℝ
→
𝔽
is
called
F-NV-F.
For
each
be
an
F-NV-F.
Then
fuzzy
integral
be
of
an
𝔖
F-NV-F.
over
Then
fuzzy
integral
of
𝔖
over
Definition
2.
[49]
Let
𝔖
:
[𝜇,
𝜈]
⊂
ℝ
→
𝔽
2.
[49]
Let
𝔖
[𝜇,
𝜈]
⊂
ℝ
→
𝔽
(0, 1], ℱℛ
∈+
(0,(11].
For all 𝜑Definition
∈
denotes
the
collection
of
all
𝐹𝑅-integrable
F0
Proposition
Proposition
1.
[51]
Let
Proposition
1.
𝔵,
[51]
𝔴
∈
Proposition
Let
𝔽
1.
𝔵,
[51]
𝔴
∈
Let
𝔽
1.
𝔵,
[51]
𝔴
∈
Let
𝔽
𝔵,
𝔴
∈
𝔽
.
Then
fuzzy
.
Then
order
fuzzy
relation
.
Then
order
fuzzy
“
.
relation
≼
Then
”
order
is
fuzzy
given
“
relation
≼
”
order
on
is
𝔽
given
relation
“
by
≼
”
on
is
𝔽
given
“
≼
by
”
on
is
given
𝔽
by
on
(0,
for
all
𝜑
∈
(0,
1].
For
all
𝜑
∈
1],
ℱℛ
denotes
the
collection
of
all
𝐹𝑅-in
([
],
)
,
(1
([
],
)
𝜉𝒾
− 𝜉)𝒿
+
−
𝜉)𝒿
,
[𝜇,
[𝜇,
∗ ℝ → 𝔽 is called (7)
(7)
[34]
The
𝜐]
→
ℝ𝔖
is𝔖called
harmonically
convex
function
𝜐]
if𝒾𝜌.
[𝜌.
[𝜌.
[𝜌.
[𝔵]
[𝔵]
𝔵]
𝔵]
𝔵][𝔖
=
𝜌.
=∗𝔖
𝜌.𝜑)𝑑𝒾
=𝔖
(𝒾)𝑑𝒾
(𝒾)𝑑𝒾
(𝒾)𝑑𝒾
(𝐼𝑅)
(𝐼𝑅)
(𝐼𝑅)
=For
=on
NV-Fs over4.for
[𝜇,
𝜈].
Definition
[49]
A
fuzzy
number
valued
map
𝔖
:[𝔵]
𝐾
⊂
F-NV-F
(𝒾)
[𝔖=
(𝒾,
(𝒾,
(𝒾)
(𝒾,
=are
𝜑),
𝔖
𝜑)]
for
all
𝔖
𝜈]
and
for
all
all 𝒾 ∈ [𝜇,
𝜈] a
:𝔖:
[𝜇,
𝜈]on
⊂
ℝ
→
𝒦
are
: given
[𝜇,
⊂
bythe
ℝ1.
𝔖
→
𝒦
by
𝔖
𝔖
(8)
NV-Fs
over
[𝜇,
𝜈].
(𝒾)𝑑𝒾
(𝒾)𝑑𝒾
=a𝜈](𝐼𝑅)
=
𝔖(𝒾,
:=
𝔖(𝒾,
𝜑)
∈∈
ℛ[𝜇,
(0,
∗given
∗→
all
𝜑
∈
(0,
𝜑-cuts
∈𝔖
1],
ℱℛ
the
collection
all
𝐹𝑅-integrable
([ (𝒾,
],𝜑)]
)F-, for
, given
υall
if2.
[µ,
([[𝜇,
],⊂
)ℝ
, 𝔖
𝜑𝜈]
∈(𝒾)𝑑𝒾
(0,
1]
whose
𝜑
define
family
of
I-V-Fs
𝔖I-V-Fs
:𝔖
𝐾of
⊂
ℝF-NV-F
𝒦𝜑),
are
by
[𝜇,
[𝜇,
[𝜇,
(𝐹𝑅)
(𝒾)𝑑𝒾
on
4. [34]
The
→byLet
ℝ (𝐹𝑅)
is
asense
harmonically
convex
function
on
𝜐]
if
𝜈], 𝔖:
denoted
𝔖
𝜈],
denoted
by
𝔖
, 1].
is(𝐹𝑅)
by
,denotes
is
given
levelwise
by
r [𝜇,
𝜈]. [𝜇,
Definition
1.
[49]
A
number
valued
map
𝔖
:
𝐾
⊂
ℝ
→
𝔽
is
called
F-NV-F.
For
each
𝒾𝒿
Theorem
2.𝜐]
[49]
𝔖
:called
[𝜇,
⊂
ℝ
→
Theorem
𝔽given
[49]
Theorem
Let
𝔖
:
[𝜇,
2.
[49]
𝜈]
Let
→
𝔖
𝔽
:
[𝜇,
𝜈]
⊂
ℝ
→
𝔽
befuzzy
a]levelwise
F-NV-F
whose
𝜑-cuts
define
the
be
family
a
F-NV-F
of
whose
be
a
𝜑-cuts
define
whose
the
𝜑-cuts
family
define
of
I-V-Fs
the
fami
NV-Fs
over
𝜈].
(𝒾)𝑑𝒾
(𝐹𝑅)
(𝒾)𝑑𝒾
(𝐼𝑅)
𝔖
=
=
𝔖(𝒾,
𝜑)𝑑𝒾
:
𝔖(𝒾,
𝜑)
∈
ℛ
!
([ ,
∈(𝐼𝑅)
(0,
1]all
whose
𝜑
-cuts
define
the
family
of
𝔖(0,
:(𝒾,
𝐾𝜑⊂∈ℝ(0,
→1],
𝒦 ∗ (𝒾,
are
[𝔵]
[𝔵]
[𝔴][𝔵]
[𝔴]
[𝔴]
𝔵𝒾𝒿
≼𝔴
if𝜑
and
≼
only
𝔴
ifThen
if,
and
𝔵(𝒾)𝑑𝒾
≼
only
𝔴≤
ifif,
and
𝔵[𝔴]
𝔴
only
if
if,
only
if,
for
≤and
all
𝜑
∈for
≤
(0,
all
1],
𝜑
≤
for
(0,I-V-Fs
all
1],
(4)
all
1],
(4)
∗ 𝜑 for
ℛ
denotes
collection
of≤∞)
all
𝐹𝑅-integrable
F
if∈
(1𝔖
∗ (𝒾,
+(𝒾,
𝜉𝔖(𝒿),
−
𝜉)𝔖(𝒾)
(11)
(8)
([ A
) Proposition
NV-Fs
over
[𝜇,
𝜈].
, ],set
(𝒾)𝑑𝒾
= (0,
said
to
be
afuzzy
convex
for
𝒿over
34]
𝐾𝔖
= [𝜇,the
𝜐] ⊂
ℝ[51]
𝔖
=
𝔖
=
𝔖(𝒾,
𝜑)𝑑𝒾
:⊂
𝔖(𝒾,
𝜑)
∈
ℛ
,if∈𝜑)
𝜑)
and
𝔖
𝜑)
areand
𝔖
𝜑
𝜑∈1]
∈𝔽 (𝐹𝑅)
(0,
1].
Then
isset,
𝜑
𝐹𝑅-integrable
∈𝔵if,
(0,
1].
is
[𝜇,
𝐹𝑅-integrable
𝜈]≼[𝔵]
if
and
only
over
if
𝔖
𝜈]
and
only
𝔖
(𝒾)
[𝔖
1.
Let
𝔴is
.𝐾
Then
order
relation
“𝒾,𝒾≼
is
given
on
𝔽𝔖
([ ,end
], (𝒾,
=
𝜑),
𝔖ℝ
𝜑)]
for
all
∈”∈𝔖
𝐾.
Here,
for
each
𝜑
∈∗[𝜇,
(0,
1]
the
point
real
∗both
∗ (𝒾,
∗(𝒾,
∗) (𝒾,
𝜑
∈𝔵,given
(0,
whose
𝜑
-cuts
define
the
family
of
I-V-Fs
:
𝐾
ℝ
→
𝒦
are
given
by
∗(𝒾)
sby
(1
(1
s
∗
𝜉𝒾
+
−
𝜉)𝒿
+
𝜉𝔖(𝒿),
𝔖
≤
−
𝜉)𝔖(𝒾)
(11)
𝒾𝒿
(0,
[𝜇,
=
∞)
is
said
to
be
a
convex
set,
if,
for
all
𝒾,
𝒿
∈
Definition
3.
[34]
A
set
=
𝜐]
⊂
[𝔖
(𝒾,
(𝒾)
[𝔖
(𝒾,
(𝒾)
(𝒾,
[𝔖
(𝒾,
e
e
e
=
𝜑),
𝔖
𝜑)]
for
all
𝒾
∈
[𝜇,
=
𝜈]
and
for
𝜑),
all
𝔖
=
𝜑)]
for
𝜑),
all
𝔖
𝒾
∈
𝜑)]
[𝜇,
𝜈]
for
and
all
𝒾
for
∈
all
[𝜇,
𝜈]
:
[𝜇,
𝜈]
⊂
ℝ
→
𝒦
are
by
𝔖
:
[𝜇,
𝜈]
⊂
ℝ
→
𝒦
:
[𝜇,
𝜈]
are
⊂
given
ℝ
→
by
𝒦
𝔖
are
given
by
𝔖
𝔖
𝔖
𝔖
e
(𝒾)
[𝔖
(𝒾,
(𝒾,
𝔖
=𝜐]
𝜑),
𝔖
𝜑)]
for
all
𝒾a)Riemannian
∈
𝐾.
Here,
for
each
𝜑
1]
the
end
(0,
[𝜇,
4
1number
−
ξ)∗)For
S
, is
(15)
+
ξthe
S
S
(on
(if
∗(0,
∗𝔽set,
=called
∞)
is
said
to
be
convex
if,
for
all
𝒾,
𝒿∈ is
∈(0,called
Definition
3.
[34]
A
set
𝐾
=
⊂
ℝ
[𝜇,
[𝜇,
[𝜇,
Definition
4.
[34]
The
𝔖:
𝜐]
→(1
ℝThen
snhave
called
aDefinition
convex
function
on
is
called
𝜐]
ifbe
a𝜉𝔖(𝒿),
harmonically
convex
function
𝜐]
∗denotes
(0,
(0,
for
all 𝐾𝜑Proposition
∈1.
(0,
1].
For
all
𝜑
∈
1],
for
ℱℛ
all
𝜑
∈
for
1].
all
For
𝜑
all
∈
(0,
𝜑
∈
1].
For
1],
ℱℛ
𝜑
∈
1],
ℱℛ
denotes
the
collection
of
all
𝐹𝑅-integrable
denotes
Fcollection
denotes
of
the
all
collection
𝐹𝑅-integrable
of all
F𝐹𝑅-in
(1
∗1.
1.[51]
[49]
A
fuzzy
number
Definition
valued
map
𝔖
[49]
Definition
:
𝐾
A
⊂
fuzzy
ℝ
→
𝔽
1.
number
[49]
A
fuzzy
valued
map
𝔖
:
valued
𝐾
⊂
ℝ
map
→
𝔖
:
𝐾
⊂
ℝ
→
𝔽
is
F-NV-F.
each
called
F-NV-F.
For
each
F-NV-F
for
all
𝜑
∈
(0,
1],
where
ℛ
the
collection
of
integrable
func𝜉𝒾
+
−
𝜉)𝒿
([
],
)a
([=
],(0,
([
],
)
,
,
,
∗
+
𝔖
≤
−
𝜉)𝔖(𝒾)
([
],
)
(11)
,
Proposition
Let
1.
𝔵,
[51]
𝔴
∈
Let
𝔽
𝔵,
𝔴
∈
𝔽
.
fuzzy
.
Then
order
fuzzy
relation
order
“
relation
≼
”
is
given
“
≼
”
on
is
𝔽
given
by
on
𝔽
by
(0,
[𝜇,
=
∞)
is
said
to
a
convex
set,
if,
for
all
𝒾,
𝒿
∈
3. [34]
Aharmonically
set
=
𝜐]
⊂
ℝ
(.
(.
functions
𝔖
,
𝜑),
𝔖
,
𝜑):
𝐾
→
ℝ
are
called
lower
and
upper
functions
of
𝔖
.
It
is
a
partial
It
is
order
a
partial
relation.
It
is
order
partial
It
relation.
is
a
order
partial
relation.
order
relation.
(0,
[𝜇,
=
∞)
is
said
to
be
a
convex
set,
if,
fo
Definition
3.
[34]
A
set
𝐾
𝜐]
⊂
ℝ
[𝜇,
[𝜇,∗Riemannian
(𝒾)
[𝔖∗ (𝒾,
(𝒾,
𝑅-integrable
[𝜇,
𝜈].
𝑅-integrable
Moreover,
over
if
[𝜇,
isξ(0,
𝜈].
𝐹𝑅-integrable
Moreover,
ifover
𝔖denotes
is
𝐹𝑅-integrable
𝜈],
then
over
𝜈], then
=and
𝜑),
𝔖(𝒾)𝑑𝒾
𝜑)]
for
allfor
𝒾∈
∈𝔖
𝐾.
for
(0,
1]
the
end
point
real
∗ over
ξ𝜑𝜑)𝑑𝒾
+
−
(1𝜑
)where
∗𝔖
for
all
(0,
1],
ℛ
the
collection
of
integ
[0,(1
𝐾,
𝜉𝜉𝒾
∈+
have
(8)
∗each
∗ (𝒾,
−Definition
𝜉)𝒿
([
],𝜑
)∈
[𝔵]
[𝔴]
(𝒾)𝑑𝒾
(𝐹𝑅)
(𝒾)𝑑𝒾
(𝐹𝑅)
(𝒾)𝑑𝒾
(𝐼𝑅)
𝔵𝔖
≼
𝔴
if
only
if,
≤
all
∈𝜑)
1],
(4)
𝔖1],
𝔖
𝔖
=
𝔖(𝒾,
=
:𝔖
𝔖(𝒾,
𝜑)
∈
ℛ
=given
,and
𝔖(𝒾,
𝜑)𝑑𝒾
:of
𝔖(𝒾,
𝜑)
∈𝒾,upper
(.
(.
[0,
functions
,Here,
𝜑),
𝔖
,𝔖
𝜑):
𝐾
ℝ
are
called
and
functions
of
𝔖.𝜑
𝐾,
𝜉=𝜑
∈(𝐼𝑅)
1],
we
have
([
],,→
)of
([:∗are
[0,
[𝜇,
, ],
(𝒾,
(𝒾,
(𝒾,
(𝒾,
(𝒾,
, 𝜐],we
𝜉 ∈have
1],(0,where
≥
0we
for
all
𝒾∈
∈(0,
𝜐].
Ifover
(11)
is
reversed
then,
𝔖
and
𝜑)
are
𝜑)
and
𝔖
𝜑)
𝜑)
both
and
𝔖
𝜑∈
1].
Then
is
𝐹𝑅-integrable
𝜑I-V-Fs.
over
∈
(0,
[𝜇,
1].
𝜈]𝐾
Then
if
𝜑
∈
𝔖𝜐]
only
(0,
1].
if
𝐹𝑅-integrable
𝔖
Then
𝔖
is
over
𝐹𝑅-integrable
[𝜇,
if
over
only
[𝜇,
if𝔖
𝜈]
𝔖lower
if
and
only
if
𝔖
(0,
[𝜇,
∗define
=
∞)
is
to
be
aboth
convex
set,
for
all
𝒿ℛcharacteristics
∈
3.
[34]
A
set
=
⊂
ℝ
NV-Fs
over
[𝜇,𝔖(𝒾)
𝜈].𝔖
NV-Fs
[𝜇,
NV-Fs
𝜈].
over
[𝜇,
𝜈].
1]
whose
𝜑
-cuts
define
the
family
whose
of
𝜑
I-V-Fs
∈
(0,
𝜑is
-cuts
1]
𝔖
whose
𝜑
the
-cuts
family
define
the
I-V-Fs
family
I-V-Fs
𝔖
:∗we
𝐾
⊂
ℝ
→
𝒦said
are
by
:if,
𝐾
→
𝒦
𝐾
⊂
ℝ,characterist
→ (8)
𝒦are
by
are
(𝐹𝑅)
(𝒾)𝑑𝒾
tions
of
𝔖
is
𝐹𝑅
-integrable
over
[𝜇,
𝜈]
if,is𝜈]
𝔖
𝔽⊂
.ℝ
Note
that,
if)given
∗∈
∗1]
𝒾𝒿
for
all
1],
where
ℛand
denotes
the
collection
of
Riemannian
integrable
func1],
We
now
use
We
mathematical
now
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mathematical
now
operations
We
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mathematical
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examine
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the
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some
examine
characteristics
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to
the
examine
some
characteristics
of
of
the
some
fuzzy
of
of
the
fuzzy
of
fuz
[0,
([
],ℝ
) are
𝐾,
𝜉
∈
1],
have
,
𝒾𝒿
[𝜇,
[0,
[𝜇,
(.
(.
for
all
𝒾,
𝒿
∈
𝜐],
𝜉
∈
1],
where
𝔖(𝒾)
≥
0
for
all
𝒾
∈
𝜐].
If
(11)
reversed
then,
𝔖
functions
𝔖
,
𝜑),
𝔖
,
𝜑):
𝐾
→
called
lower
and
upper
functions
of
𝔖
.
(𝐹𝑅)
(𝒾)𝑑𝒾
∗all
tions
of
I-V-Fs.
𝔖
is
𝐹𝑅
-integrable
over
[𝜇,
𝜈]
if
𝔖
∈
𝔽
.
No
[𝔵]
[𝔵]
[𝔴]
[𝔴]
𝔵
≼
𝔴
if
and
𝔵
≼
only
𝔴
if
if,
and
only
if,
≤
for
≤
all
𝜑
∈
for
(0,
all
1],
𝜑
∈
(0,
1],
(4)
(4)
(1
(1
∗
∗
∗
+
𝜉𝔖(𝒿),
+
𝜉𝔖(𝒿),
≤
−
𝜉)𝔖(𝒾)
𝔖
≤
−
𝜉)𝔖(𝒾)
(11)
(11)
(0,
=
∞)
is
said
to
be
a
convex
set,
if,
for
𝒾,
𝒿
∈
[𝜇,
e
e
monically
concave
function
on
𝜐].
𝒾𝒿
∈
𝐾.
𝐾,
𝜉Moreover,
1],
we
∗2.
(𝒾)
[𝔖where
(𝒾,
(𝒾,
[𝔖
(𝒾,
(𝒾)
(𝒾,
[𝔖
(𝒾,
(𝒾,
(10)
𝔖
=1],
𝜑),
𝔖
𝜑)]
for
𝐾.
=
Here,
𝜑),
for
𝔖
each
𝜑)]
𝜑
∈
for
(0,
𝜑),
1]
all
𝔖
the
𝒾Moreover,
end
𝐾.
for
Here,
point
for
𝒾(𝒾)𝑑𝒾
real
∈
𝐾.
𝜑
(0,
for
1]
𝜑
∈
point
1]isthe
realend
[𝜇,
for
all
,(𝒾)
∈
µ,
υLebesgue-integrable,
,∈numbers.
∈
0,
1ℝ
where
0,
for
all
∈each
υ𝔽[𝜇,
and
sthen
∈each
0, 𝔖
1[𝜇,
.ifover
If (0,
(15)
S
[are
]-integrable
[𝔵,=
]𝜌∗,over
(then
)∈𝜑)]
[µ,
] integral
[the
]end
𝒿𝜉)𝒿
∈ [𝜇, 𝜐],𝔖
𝜉is∈
𝔖(𝒾)
≥∈Definition
0[0,
all
𝒾𝒾have
∈∈is𝔖
𝜐].
(11)
reversed
then,
𝔖
𝒾𝒿
[𝜇,
It
a[0,
partial
order
relation.
∗over
∗If
𝑅-integrable
[𝜇,
𝜈].
if
𝑅-integrable
𝔖
𝐹𝑅-integrable
over
𝑅-integrable
over
𝜈].
over
𝜈],
[𝜇,
then
if
𝔖
is<
𝐹𝑅-integrable
ifwe
𝔖
is
over
𝐹𝑅-integrable
over
𝜈],
then
be
an
F-NV-F.
Then
fuzzy
of
[49]
Let
𝔖
[𝜇,
𝜈]
⊂
→
𝔽
(𝐹𝑅)
(𝒾,
(𝒾,
(1
tions
I-V-Fs.
𝐹𝑅
[𝜇,
𝜈]
if
𝔖
∈Here,
. ∈𝜈],
Note
that,
𝜑),
𝔖
𝜑)
𝔖
is
aall
fuzzy
Aumann-integrable
func𝔖
numbers.
numbers.
Let
𝔵,
𝔴
∈
numbers.
Let
𝔽is
𝔵,
𝔴
𝔽[𝜇,
𝔴Moreover,
∈
Let
𝔽
𝔵,
𝔴
∈𝜈].
𝔽
and
𝜌:ξLet
∈is
ℝ,
and
then
we
∈
ℝ,
and
have
then
𝜌
we
∈
and
ℝ,
have
then
𝜌
∈
ℝ,
have
then
we
have
𝜉𝒾
+
−𝔖
𝜉)𝒿
[𝜇,
is∈called
a+
harmonically
concave
function
on
𝜐].
∗of
(1
𝜉𝒾
−
𝜉)𝒿
𝐾.
∗∈
𝒾𝒿
∗(10)
∗ (𝒾,
𝒾𝒿
∈𝔖
𝐾.
be
an
F-NV-F.
Then
fuzzy
integral
Definition
2.
[49]
Let
𝔖
:∗if,
[𝜇,
𝜈]
⊂ℝ⊂
ℝ
→
𝔽
(10)
(𝒾,
(𝒾,
∗ (. 𝐾
∗𝜉)𝒿
∗ (.
𝜑),
𝔖
𝜑)
are
Lebesgue-integrable,
then
𝔖
is
a
fuzzy
Aumann-integr
𝔖
for
all
𝜑
(0,
1],
where
ℛ
for
all
𝜑
∈
(0,
1],
where
ℛ
denotes
the
collection
of
Riemannian
denotes
integrable
the
collection
funcof
Riemannian
integrable
func(𝒾,
(𝒾,
(𝒾,
(𝑅)
(𝑅)
(𝑅)
(𝑅)
(𝐹𝑅)
(𝒾)𝑑𝒾
(𝐹𝑅)
(𝒾)𝑑𝒾
=
𝔖
𝜑)𝑑𝒾
=
,
𝔖
𝔖
𝜑)𝑑𝒾
𝜑)𝑑𝒾
,
𝔖
𝜑)𝑑𝒾
𝔖
𝔖
e
(0,
(0,
(0,
[𝜇,
[𝜇,
=
∞)
is
said
to
be
a
convex
set,
=
for
∞)
all
𝒾,
is
𝒿
said
∈
=
to
be
∞)
a
convex
is
said
to
set,
be
if,
a
convex
for
all
𝒾,
set,
𝒿of
∈if,𝔖fo
Definition
3.
[34]
A
set
=
𝜐]
⊂
Definition
ℝ
3.
[34]
Definition
A
set
𝐾
3.
=
[34]
𝜐]
A
set
⊂
ℝ
𝐾
𝜐]
ℝ
([
],
)
([
],
)
∗
,
,
(1
∗
𝜉𝒾
+
−
(.
(.
(.
(.
S
reversed,
then
is
called
harmonically
s-concave
F-NV-F
in
the
second
sense
on
µ,
υ
.
The
set
of
[
]
[𝜇,
functions
𝔖
functions
𝔖
functions
𝔖
,
𝜑),
𝔖
,
𝜑):
𝐾
→
ℝ
are
called
lower
,
𝜑),
and
𝔖
upper
,
𝜑):
𝐾
functions
,
𝜑),
→
ℝ
𝔖
are
,
of
𝜑):
called
𝐾
.
→
lower
are
and
called
upper
lower
functions
and
upper
of
𝔖
functions
.
.
a harmonically
concave
function
on
𝜐].
∈
𝐾.
It is a partial
isorder
a∗ use
partial
relation.
order
relation.
∗2.
(10)
WeItnow
mathematical
operations
some
of
the
characteristics
(1
[𝜇,
(𝐹𝑅)
(𝒾)𝑑𝒾
∗[𝜇, 𝜈] 𝔖
−
∈
𝜈],𝔖
denoted
byto
is∗𝜉𝒾
given
levelwise
byof fuzzy
tion
over
[𝜇, 𝜈].
𝒾𝒿
be+then
an
F-NV-F.
Then
fuzzy
integral
of𝐾.𝔖 over
Definition
[49]
Let
𝔖examine
:all
⊂[𝜇,
ℝ𝜐].
→ ,𝔽If
(𝒾,
𝜑),
𝜑)
are
Lebesgue-integrable,
𝔖𝜉)𝒿
is
fuzzy
Aumann-integrable
func𝔖
∗ (𝒾,
(1
[𝜇,𝜐],
[0,
𝜉𝒾
+
−
𝜉)𝒿
𝒾) ≥ 0forfor
allall
𝒾,∈𝒿𝒾of
∈∈[𝜇,
𝜐].
𝜉 If
∈𝔖(11)
1],
is𝐹𝑅
reversed
where
𝔖(𝒾)
then,
≥over
0
𝔖[0,
for
𝒾𝔖
∈𝜈],
(11)
is
reversed
then,
𝔖
[𝜇,
(𝐹𝑅)
(𝒾)𝑑𝒾
𝜉𝒾
+ (1
−𝔖
𝜉)𝒿
denoted
by
𝔖a[𝔵]
,[𝔴]
is
given
levelwise
by
(5)
(5)
[𝔵+
[𝔵+
[𝔵+
[𝔵+
[𝔵]
[𝔴]
[𝔵]
[𝔴]
[𝔵]
[𝔴]
tion
over
[𝜇,
𝜈].
𝔴]
=
𝔴]
+
=
,
𝔴]
+
=
,
𝔴]
+
=
,
+
,
(𝐹𝑅)
(𝒾)𝑑𝒾
(𝐹𝑅)
(𝒾)𝑑𝒾
tions
I-V-Fs.
is
-integrable
tions
of
I-V-Fs.
[𝜇,
𝜈]
if
is
𝐹𝑅
-integrable
𝔖
over
[𝜇,
𝜈]
if
∈
𝔽
.
Note
that,
if
∈
𝔽
.
Note
that,
if
∈
𝐾.
[0,
[0,
(10)
𝐾,
𝜉
1],
we
have
𝐾,
𝜉
∈
1],
we
𝐾,
have
𝜉
∈
1],
we
have
all
harmonically
s-convex
(harmonically
s-concave)
F-NV-F
is
denoted
by
We nowLet
use
We
now
use
mathematical
to
tosome
examine
of
the
some
of the characteristics
of fuzzy of fuzzy
𝒾𝒿 numbers.
𝔵,mathematical
𝔴
∈ 𝔽[𝜇,
and
𝜌operations
∈[𝜇,
ℝ,𝜈].
thenoperations
weexamine
have
(9)
∗ (𝒾,
∗ (𝒾, (𝑅)
∗ (𝒾,
(1characteristics
(𝐹𝑅)
(𝒾)𝑑𝒾
𝜉𝒾
+
−(𝐹𝑅)
𝜉)𝒿
𝜈],
denoted
𝔖
, is
given
levelwise
by
tion
over
(𝒾,
(𝒾,
(𝒾,
(𝑅)
(𝑅)
(𝑅)
(𝑅)
(𝑅)
(𝐹𝑅)
(𝒾)𝑑𝒾
(𝐹𝑅)
(𝒾)𝑑𝒾
(𝒾)𝑑𝒾
=by
𝔖
𝜑)𝑑𝒾
, Then
𝔖
=
𝜑)𝑑𝒾
𝔖
=over
𝜑)𝑑𝒾
, be
𝔖
𝔖
𝜑)𝑑𝒾
,𝜑)𝑑𝒾
𝔖
𝜑)𝑑𝒾
𝔖[𝜇,
𝔖
𝔖
[𝜇,
on on
is[𝜇,
called
𝜐].
a𝐾.
harmonically
concave
function
on
𝜐].
∗ℝ
∗ (𝒾,
(10)
∗𝜑)
∗𝔖
∗Aumann-integrable
[𝜇,∈𝜐]
[𝜇,
34]
The
𝔖:
→
is∈called
a𝔖𝔴harmonically
convex
function
on
𝜐]
if
be
an
F-NV-F.
fuzzy
integral
be
of
an
F-NV-F.
Then
an
F-NV-F.
fuzzy
integral
Then
fuzzy
of
𝔖
integral
over
Definition
2.
[49]
Let
:
𝜈]
⊂
ℝ
Definition
→
𝔽
2.
Definition
[49]
Let
𝔖
:
[𝜇,
2.
[49]
𝜈]
⊂
Let
ℝ
→
𝔖
𝔽
:
[𝜇,
𝜈]
⊂
ℝ
→
𝔽
(𝒾,
(𝒾,
𝔖
𝜑)
are
Lebesgue-integrable,
𝜑),
𝔖
then
𝔖
are
is
a
Lebesgue-integrable,
fuzzy
Aumann-integrable
then
𝔖
funcis
a
fuzzy
func𝔖
numbers.
𝔵,
𝔴
Let
𝔽
𝔵,
∈
𝔽
and
𝜌
∈
ℝ,
and
then
𝜌
we
∈
ℝ,
have
then
we
have
∗ (𝒾, 𝜑),Let
∗
𝒾𝒿
𝒾𝒿
𝒾𝒿
𝒾 + (1 numbers.
−𝔖
𝜉)𝒿
(6)
(6)
[𝔵
[𝔵
[𝔵
[𝔵
[𝔵]
[
[𝔵]
[
[𝔵]
[
[𝔵]
[
[𝜇,
×
×
×
×
𝔴]
=
𝔴]
×
=
𝔴]
,
𝔴]
×
=
𝔴]
,
𝔴]
×
=
𝔴]
,
×
𝔴]
,
Theorem
2.
[49]
Let
𝔖
:
[𝜇,
𝜈]
⊂
ℝ
→
𝔽
be
a
F-NV-F
whose
𝜑-cuts
define
the
family
of
I-V-Fs
Definition 4. [34]
The 𝔖: [𝜇,
𝜐]
→
ℝ
is
called
a
harmonically
convex
function
on
𝜐]
if
[𝜇,
[𝜇,
Definition
4.
[34]
The
𝔖:
𝜐]
→
ℝ
is
called
a
harmonically
convex
function
on
𝜐]
if
(8)
(𝒾)𝑑𝒾
(𝒾)𝑑𝒾
(𝐼𝑅)
(𝐼𝑅)
(𝒾)𝑑𝒾
(𝐹𝑅)
(𝒾)𝑑𝒾
(𝐼𝑅)
(5)
=
𝔖
=
𝔖
[𝔵+
[𝔵]
[𝔴]
𝔖
=
𝔖
=
𝔖(𝒾,
𝜑)𝑑𝒾
:
𝔖(𝒾,
𝜑)
∈
ℛ
,
𝔴]
=
+
,
HFSX
µ,
υ
,
F
,
s
HFSV
µ,
υ
,
F
,
s
.
([
]
)
(
([
]
))
∈
𝐾.
∈
𝐾.
∈
𝐾.
([
],
)
(10)
(10)
,
Theorem
2.
[49]
Let
𝔖
:
[𝜇,
𝜈]
⊂
ℝ
→
𝔽
be
a
F-NV-F
whose
𝜑-cuts
define
the
0
0
[𝜇,[𝜇,
[𝜇,
[𝜇,
[𝜇,
(𝐹𝑅)
(𝒾)𝑑𝒾
(𝒾)𝑑𝒾
n 4. [34]tion
The𝜈],
𝔖:denoted
𝜐]
→ by
ℝ (𝐹𝑅)
is called
a harmonically
convex
function
𝜐](𝐹𝑅)
if 𝔖:
𝔖(𝒾)𝑑𝒾
𝜈],+denoted
by𝜈],
denoted
𝔖[34]
by𝔖
𝔖=
,[𝜇,
is 𝜉𝒾
given
levelwise
by
, is
given
levelwise
, is 𝜉𝒾
given
by
levelwise
by 𝜑)𝑑𝒾convex
over
tion
over
[𝜇,
𝜈].
(9)
(9)ℛfami
[𝜇,
Definition
4.on
The
𝜐]
→
ℝ
is
called
a=
harmonically
function
𝒾𝒿𝜈].
(𝒾)𝑑𝒾
(𝐹𝑅)
(𝒾)𝑑𝒾
(𝐼𝑅)
𝔖
𝔖(𝒾,
: 𝔖(𝒾,
𝜑)
∈
∗ (𝒾,
(1
(1all
𝜉𝒾
𝜉)𝒿
+
−
𝜉)𝒿
+
−(5)
𝜉)𝒿
([ , o
2.
Let
𝔖
:−
[𝜇,
𝜈]
⊂
ℝ[𝔵]
be
a
F-NV-F
whose
𝜑-cuts
define
the
family
of
I-V-Fs
(𝒾)
[𝔖
(𝒾,
=
𝜑),
𝔖
𝜑)]
for
𝒾
∈
[𝜇,
𝜈]
and
for
all
: [𝜇,
𝜈][49]
⊂
ℝ
→=(1
𝒦
are
given
𝔖
𝔖 𝜉)𝔖(𝒾)
(5)
[𝔵+
[𝔵]
[𝔴]
[𝔴]
𝒾𝒿
𝔴]
𝔴]
+
=
, →by𝔽
+
,
(1 −
[𝜇,
[𝜇,
+
𝜉𝔖(𝒿),
𝔖
≤Theorem
Definition
4.[𝔵+
[34]
The
𝔖:
𝜐]
→
ℝ
is
called
a
harmonically
convex
function
on
𝜐]
if
∗
(11)
(8)
(7)
(7)
[𝜌.
[𝜌.
[𝜌.
[𝜌.
[𝔵]
[𝔵]
[𝔵]
[𝔵]
(𝒾)𝑑𝒾
(𝐹𝑅)
(𝒾)𝑑𝒾
(𝐼𝑅)
∗
𝔵]
𝔵]
𝔵]
𝔵]
=
𝜌.
=
𝜌.
=
𝜌.
=
𝜌.
𝔖
=
𝔖
=
𝔖(𝒾,
𝜑)𝑑𝒾
:
𝔖(𝒾,
𝜑)
∈
ℛ
,
𝒾𝒿
([
],
)
,
(𝒾)
[𝔖
(𝒾,
(𝒾,
=
𝜑),
𝔖
𝜑)]
for
all
𝒾
∈
[𝜇,
𝜈]
:
[𝜇,
𝜈]
⊂
ℝ
→
𝒦
are
given
by
𝔖
𝔖
[𝔵 ×
𝔴]
= [𝔵]
×
𝜉𝒾 + (1 − convex
𝜉)𝒿
𝔖
≤𝔴](1 ,− 𝜉)𝔖(𝒾) +≤𝜉𝔖(𝒿),
∗ 𝔖 (𝒾)𝑑𝒾
(11)
𝒾𝒿
(1
(𝒾)𝑑𝒾
(𝐼𝑅)
(𝐼𝑅)
(𝐼𝑅)
+for
𝜉𝔖(𝒿),
𝔖ℛ[by
−
𝜉)𝔖(𝒾)
∗ (𝒾, 𝔖𝒾𝒿
=𝜉𝒾
𝔖𝜉)𝒿(𝒾)𝑑𝒾
=collection
=(6)
(11)
[𝜇,
∗and
s called a harmonically
function
on
𝜐]
if1].
(𝒾)
[𝔖
(𝒾,
(1
=
𝜑),
𝔖
𝜑)]
all
𝒾
∈
[𝜇,
𝜈]
for
all
:
[𝜇,
𝜈]
⊂
ℝ
→
𝒦
are
given
𝔖
𝔖
for
all
𝜑
∈
(0,
1],
where
denotes
the
of
Riemannian
integrable
func+
−
(𝒾,
(𝒾,
(1
𝜑)
and
𝔖
𝜑)
both
are
𝜑
∈
(0,
1].
Then
𝔖
is
𝐹𝑅-integrable
over
[𝜇,
𝜈]
if
and
only
if
𝔖
𝜉𝔖(𝒿),
𝔖
≤
−
𝜉)𝔖(𝒾)
([
],
)
∗
(11)
,
(0,
(0,
(1
(1
for
all
𝜑
∈
(0,
For
for
all
all
𝜑
∈
𝜑
∈
(0,
1],
1].
ℱℛ
For
all
𝜑
∈
1],
ℱℛ
denotes
the
collection
of
denotes
all
𝐹𝑅-integrable
the
collection
Fof
all
𝐹𝑅-int
𝜉𝒾
+
−
𝜉)𝒿
+
𝜉𝔖(𝒿),
𝔖
≤
−
𝜉)𝔖(𝒾)
∗
𝒾𝒿
([where
)befamily
([
) (6)collection
Theorem (𝐹𝑅)
2. [49]
Let
𝔖𝜉)𝒿
: [𝜇, 𝜈]=⊂(𝐼𝑅)
ℝ[𝔵Theorem
→×𝔽𝔴]
2.
Let
𝔖
[𝜇,
𝜈]
⊂: 1],
ℝ
→
be
F-NV-F
whose
𝜑-cuts
define
a set,
F-NV-F
of
I-V-Fs
whose
define
the
family
of I-V-Fs
,𝔽],the
, ],𝜑-cuts
(6)
[𝔵
[𝔵]
[ ]𝔴]
[𝔵]
[(0,
×[49]
=aLet
𝔴]
×for
=
×
𝔴]
,𝔖𝔖
all
∈
ℛ
denotes
the
of
integ
eifcalled
(𝒾,
(8)
and
𝔖
𝜑υ
∈
(0,
1].
Then
is
𝐹𝑅-integrable
over
[𝜇,
𝜈]
if
and
only
if𝜑)𝑑𝒾
𝔖
+ (1
−
([𝜐]
],+
)ℝ
(𝒾)𝑑𝒾
(𝒾)𝑑𝒾
(𝒾)𝑑𝒾
(𝒾)𝑑𝒾
(𝐹𝑅)
(𝒾)𝑑𝒾
(𝐹𝑅)
(𝒾)𝑑𝒾
(𝐼𝑅)
(𝐼𝑅)
,],→
𝔖
𝔖
𝔖is[𝜌.
=
𝔖(𝒾,
𝜑)𝑑𝒾
𝔖(𝒾,
𝜑)
∈
𝔖
ℛ[𝜇,
=
, let
𝔖(𝒾,
𝜑)𝑑𝒾
:→
𝔖(𝒾,
𝔖(𝒾,
∈
ℛRiemannian
:F-NV-F,
𝜑)
, function
∈(8)
ℛ∗([(𝒾,,
(1
3.
be
a,: 𝜑
harmonically
convex
and
be
S
: [µ,
υ∈a=
FNote
[4.
]∗convex
(R)
𝜉𝒾
−𝔖
𝜉)𝒿
∗a
(1
([=
)a[𝜇,
([convex
],𝜑)) [𝜇,
[𝜇,
, on
,𝔖(𝒾,
+
𝜉𝔖(𝒿),
𝔖
≤
−
𝜉)𝔖(𝒾)
[𝜇,
[𝜇,=
C𝜑)
𝜐], 𝜉 ∈ [0,Definition
1], where𝜉𝒾
≥
0for
for
all
𝒾Theorem
∈(0,
𝜐].
If
(11)
is
reversed
then,
𝔖
(11)
4.𝔖(𝒾)
[34]
The
𝔖:
𝜐]
→
ℝ
Definition
[34]
Definition
The
𝔖:
4.
[34]
𝜐]
→
The
ℝ
𝔖:
called
aµ,
harmonically
convex
function
is
called
harmonically
𝜐]
is
harmonically
function
on
𝜐]
if
(7)
[𝔵]
𝔵]
=
𝜌.
(𝐹𝑅)
(𝒾)𝑑𝒾
tions
of
I-V-Fs.
𝔖
is
𝐹𝑅
-integrable
over
[𝜇,
𝜈]
if
𝔽
. 𝜑)
that,
if
all
𝜑∈
∈[0,
1],
where
ℛnumber
denotes
the
collection
of
Riemannian
integrable
funcNV-Fs
over
[𝜇,
𝜈].
NV-Fs
over
[𝜇,
𝜈].
(𝒾,
(𝒾,
𝜑)
and
𝔖
both
are
𝜑
∈
(0,
1].
Then
𝔖
is
𝐹𝑅-integrable
over
[𝜇,
𝜈]
if
and
only
if
𝔖
([
],
∗)(1
∗𝔖
,
[𝜇,
𝑅-integrable
over
[𝜇,
𝜈].
Moreover,
if
𝔖
is
𝐹𝑅-integrable
over
𝜈],
then
[𝜇,
[𝜇,
+
𝜉𝒾
+
−
𝜉)𝒿
for
all
𝒾,
𝒿
∈
𝜐],
𝜉
1],
where
𝔖(𝒾)
≥
0
for
all
𝒾
∈
𝜐].
If
(11)
is
reversed
then,
𝔖
∗
Definition
Definition
1.
[49]
A
Definition
fuzzy
1.
[49]
A
Definition
fuzzy
1.
[49]
valued
number
A
1.
fuzzy
[49]
map
valued
number
A
𝔖
fuzzy
:
𝐾
map
⊂
ℝ
valued
number
→
𝔖
:
𝔽
𝐾
⊂
map
valued
ℝ
→
𝔖
:
𝔽
𝐾
map
⊂
ℝ
𝔖
→
:
𝐾
𝔽
⊂
ℝ
→
𝔽
is
called
F-NV-F.
is
called
For
F-NV-F.
is
each
called
For
F-NV-F.
is
called
each
For
F-NVea
(𝐹𝑅)
(𝒾)𝑑𝒾
(𝒾)
[𝔖
(𝒾,
(𝒾,
(𝒾)
[𝔖
(𝒾,
(𝒾,
[0,
[𝜇,
tions
of
I-V-Fs.
𝔖
-integrable
[𝜇,
𝜈]
if 𝒾over
𝔽 . No
=⊂
𝜑),
𝔖where
𝜑)]
for
all
𝒾ϕisfor
∈
[𝜇,
=allυ
𝜈]
and
𝜑),
for
𝔖K
all
𝜑)]
∈ by
[𝜇,
𝜈]𝔖
and
for∈all
[𝜇,
𝜈] ⊂
ℝ 𝜉𝔖(𝒿),
→ 𝒦 onare
given
by
[𝜇,(11)
𝜈]
ℝ
→
𝒦 family
are
given
by
𝔖
𝔖concave
𝔖
for
all
𝒾,1],
𝒿𝔖for
∈∗:[𝜌.
𝜐],
𝜉=
∈
1],
𝔖(𝒾)
≥
0𝜈].
𝒾∗⊂
∈R
𝜐].
If
(11)
is
reversed
then,
𝔖 then
whose
ϕ-cuts
define
the
of(0,
S
: Moreover,
µ,
→
⊂
K
areallgiven
[𝐹𝑅
+
−:all
𝜉)𝔖(𝒾)
∗𝜌.
Cfor
[𝜇,
[𝜇,
monically
𝜐].
𝑅-integrable
over
[𝜇,
if𝔖
𝔖
isover
𝐹𝑅-integrable
𝜈],
(7)
[𝜌.
[𝔵]
𝔵]
𝔵]
=
𝜌.all
(0,
(0,
CAumann-integrable
for
𝜑
∈function
(0,
1]. For
all
∈(𝒾,
ℱℛ
𝜑
∈𝐹𝑅
(0,
for
1].
all
For
𝜑
∈[𝔵]
𝜑I-V-Fs
∈1].
For
1],
ℱℛ
𝜑
∈, ]where
ℱℛ
denotes
the
collection
of
all
𝐹𝑅-integrable
denotes
Fthe
collection
denotes
ofthat,
the
all
collection
𝐹𝑅-integrable
of is
allreverse
F𝐹𝑅-in
[0,
[𝜇,
∈ [𝜇, ≤
𝜐],(1
𝜉∈
1],is
where
𝔖(𝒾)
≥𝜑
0∗of
for
all
𝒾(0,
∈all
(11)
is
reversed
then,
𝔖
[𝜇,
[0,
[𝜇,
([
],are
)If
([𝔖
],(0,
)a1],
([
)(7)
for
all
𝒾,
𝒿
∈
𝜐],
𝜉
∈
1],
𝔖(𝒾)
≥
for
all
𝒾
∈
𝜐].
If
(11)
,𝜐].
,∈],0𝔽
(𝐹𝑅)
(𝒾)𝑑𝒾
𝒾𝒿
𝒾𝒿
𝒾𝒿
(𝒾,
tions
I-V-Fs.
𝔖
is
-integrable
over
[𝜇,
𝜈]
if
.
Note
if
𝜑),
𝔖
𝜑)
Lebesgue-integrable,
then
is
fuzzy
func𝔖
𝜉)𝒿
[𝜇,
called
a
harmonically
concave
function
on
𝜐].
𝜑
∈
(0,
1]
𝜑
whose
∈
1]
𝜑
𝜑
whose
-cuts
∈
(0,
define
1]
𝜑
-cuts
whose
∈
(0,
the
1]
define
family
𝜑
whose
-cuts
the
of
define
𝜑
family
-cuts
I-V-Fs
the
define
of
𝔖
family
I-V-Fs
the
of
𝔖
family
I-V-Fs
of
𝔖
I-V-Fs
𝔖
:
𝐾
⊂
ℝ
→
:
𝒦
𝐾
⊂
are
ℝ
→
given
:
𝒦
𝐾
⊂
are
ℝ
by
→
:
given
𝐾
𝒦
⊂
ℝ
are
by
→
𝒦
given
are
∗
[𝜇,
𝑅-integrable
𝜈].
if𝐹𝑅-integrable
is
𝐹𝑅-integrable
over
𝜈],
then
[𝜇,
∗ℛ
∗integrable
is𝔖ℛ𝒾,
called
harmonically
concave
function
on
𝜐].
(𝒾,
(𝒾,
[𝜇,
(1
(1
𝔖
are
Lebesgue-integrable,
then
𝔖said
is
ato
fuzzy
Aumann-integr
𝔖
for
𝜑 ∈Then
(0,
1],
for
all
𝜑[0,
∈
for
1],
all
where
𝜑
∈+
(0,
ℛ
1],
where
denotes
the
collection
of
Riemannian
denotes
integrable
the
collection
funcdenotes
of
the
Riemannian
collection
of
Riemannian
funcintegr
all
𝒿, ∈
𝜐],(0,
𝜉[𝜇,
∈
1],
where
𝔖(𝒾)
≥
0𝜑)
for
all
𝒾[𝜇,
∈
𝜐].
If
(11)
reversed
then,
𝔖
+
𝜉𝔖(𝒿),
+
𝜉𝔖(𝒿),
≤Moreover,
−
𝜉)𝔖(𝒾)
𝔖
𝔖
≤
−
𝜉)𝔖(𝒾)
≤
−
𝜉)𝔖(𝒾)
(11)
(11)
(𝒾,
(𝒾,
(𝒾,
(𝒾,
𝔖
𝜑)
both
are
𝜑)
and
𝔖
𝜑)
both
are
𝜑
∈ all
(0,concave
1].
𝔖where
isfor
𝐹𝑅-integrable
𝜑a)over
∈
over
1].
[𝜇,
𝜈]
Then
if
and
𝔖∗called
only
is
if𝔖
𝔖
over
and
only
ifis
𝔖
([
],∗ [𝜇,
([𝜑)
)and
([
],(1
)⊂to
, ],A
, if
[𝜇,
NV-Fs
over
[𝜇,[49]
𝜈].
NV-Fs
[𝜇,
NV-Fs
𝜈].
over
𝜈].
∗𝜉𝔖(𝒿),
∗[𝜇,
harmonically
function
on
𝜐].
(0,
(0,
[𝜇,
[𝜇,
is
a→
harmonically
concave
function
on
𝜐].
∞)
is
said
=a𝐾.
convex
∞)
is
set,
if,
for
be
all
apoint
convex
𝒾,
𝒿(0,
∈
set,
if,end
for
Definition
3.
[34]
A(0,
set
Definition
=𝜑),
𝜐]
3.
[34]
ℝ
set
𝐾
=
𝜐]
ℝ
∗a𝜈]
∗over
∗⊂
∗𝔖
tion
over
[𝜇,
(1
(1
(1
𝜉𝒾
𝜉𝒾
𝜉𝒾
+
−=𝜈].
𝜉)𝒿
+𝒾=,𝔖
−
𝜉)𝒿
+
−𝒾be𝜉)𝒿
Definition
1.
A
fuzzy
number
valued
map
𝔖
:for
𝐾∗𝐾(𝒾,
⊂
ℝ
𝔽[𝜇,
is
called
F-NV-F.
For
each
(𝒾,
(𝒾,
𝜑),
𝔖
𝜑)
are
Lebesgue-integrable,
then
is
fuzzy
Aumann-integrable
func𝔖
(𝒾)
[𝔖
(𝒾,
(𝒾)
[𝔖
(𝒾,
(𝒾,
(𝒾)
[𝔖
(𝒾)
(𝒾,
[𝔖
(𝒾,
(𝒾,
𝔖
𝔖
=
𝜑),
𝔖
𝜑)]
𝜑),
=
𝔖
all
𝜑)]
𝜑),
=
𝒾
∈
𝔖
for
𝐾.
all
Here,
𝜑),
𝜑)]
∈
for
for
𝐾.
all
𝜑)]
Here,
each
𝒾
for
∈
𝜑
for
𝐾.
all
∈
Here,
(0,
∈
1]
𝜑
for
the
∈
Here,
each
(0,
end
1]
for
point
𝜑
the
∈
each
(0,
end
real
1]
𝜑
the
∈
end
real
1]
the
point
re
=
S
S
ϕ
,
S
,
ϕ
,
∀
∈
µ,
υ
.
(16)
(
)
[
(
)
(
)]
[
]
∗
∗
∗
ϕ
∗
∗
∗
∗
[𝜇,
𝒾) ≥ 0 for
all
𝒾
∈
𝜐].
If
(11)
is
reversed
then,
𝔖
[𝜇,
(𝒾,
(𝒾,
(𝑅)
(𝐹𝑅)
(𝒾)𝑑𝒾
tion
over
[𝜇,
𝜈].
is called
a𝜉harmonically
concave
function
on
=
𝔖
𝜑)𝑑𝒾
𝔖[𝜇,
𝜑)𝑑𝒾
𝔖
(𝐹𝑅)
(𝒾)𝑑𝒾
tions of I-V-Fs.
𝔖 𝜈].
𝐹𝑅 -integrable
tions
over
of
I-V-Fs.
[𝜇,
𝜈]
tions
if∗𝔖
of
isMoreover,
I-V-Fs.
𝐹𝑅
-integrable
𝔖
𝔖 𝔖𝜐].
is
[𝜇, ,that,
𝜈](𝑅)
ifover
over
[𝜇,𝜈],
𝔖
𝜈](𝒾)𝑑𝒾
if (𝐹𝑅)
𝔖(𝒾)𝑑𝒾
∈∗is
𝔽𝐹𝑅
. -integrable
Note
if(𝐹𝑅)
∈ 𝔽 . Note
that,
∈ 𝔽 if. No
∗over
∗ (𝒾,
[𝜇,
[0,
[0,
𝑅-integrable
over A[𝜇,
Moreover,
if
𝑅-integrable
𝔖
is
𝐹𝑅-integrable
over
[𝜇,
𝜈].
over
𝜈],
then
if
𝐹𝑅-integrable
then
𝐾,
∈
1],
we
have
𝐾,
𝜉
∈
1],
we
have
∗
∗
(𝒾,
(𝑅)
(𝑅)
(𝐹𝑅)
(𝒾)𝑑𝒾
Definition
Definition
1.
[49]
fuzzy
1.
[49]
number
A
fuzzy
valued
number
map
valued
𝔖
:
𝐾
⊂
map
ℝ
→
𝔖
:
𝔽
𝐾
⊂
ℝ
→
𝔽
is
called
F-NV-F.
is
called
For
F-NV-F.
each
For
each
=
𝔖
𝜑)𝑑𝒾
,
𝔖
𝜑)𝑑𝒾
𝔖
𝜑
∈
(0,
1]
whose
𝜑
-cuts
define
the
family
of
I-V-Fs
𝔖
:
𝐾
⊂
ℝ
→
𝒦
are
given
by
tion
over
[𝜇,
𝜈].
(.
(.
(.
(.
(.
(.
(.
∗
functions
functions
𝔖
functions
𝔖
functions
𝔖
,
𝜑),
𝔖
,
𝜑):
,
𝜑),
𝐾
𝔖
→
ℝ
,
𝜑):
,
are
𝜑),
𝐾
𝔖
called
→
ℝ
,
𝜑):
𝜑),
are
lower
𝐾
𝔖
called
→
and
,
ℝ
𝜑):
are
lower
upper
𝐾
→
called
ℝ
and
functions
are
lower
upper
called
and
functions
of
lower
𝔖
upper
.
and
of
functions
upper
𝔖
.
functions
of
𝔖
.
of
𝔖
∗
∗
∗
∗
on on [𝜇,
𝜐].
∗=
[𝜇,[34]
[0,
[𝜇,[34]
[0,
[𝜇,
[𝜇,
[𝜇,
[𝜇,
for
all𝜑),
𝒾, 𝒿𝔖∈∗ (𝒾,
𝜐], are
𝜉 ∈∗set
1],= where
for
𝔖(𝒾)
all
𝒾,(0,
𝒿𝔖
0then
∈
for
for
all
𝜐],(𝒾,
all
𝜉A
𝒾 𝜑),
∈
𝒾,to
𝒿 be
∈∗1],
𝜐].
where
If
𝜐],(11)
𝜉A
∈set
is[0,
𝔖(𝒾)
reversed
1],
where
for
then,
all
𝔖(𝒾)
𝒾∈
∈≥to
0be
𝜐].
for
If
all
(11)
𝒾 ∈ais
reversed
𝜐].
If (11)
then,
is
reverse
∗∞)
(0,
(0,
[𝜇,
[𝜇,
[𝜇,
(𝒾,
(𝒾,
(𝑅)
(𝑅)
(𝐹𝑅)
(𝒾)𝑑𝒾
=≥
is
said
a[34]
convex
set,
if,
=≥
for
∞)
all
𝒾,
is
∞)
convex
is
said
to
set,
if,
a convex
for
all
𝒾,set,
𝒿𝔖
∈if, fo
Definition
𝐾define
𝜐] ⊂
Definition
ℝ
3.family
Definition
𝐾
3.
=
𝜐]
ℝ
𝐾
𝜐]
⊂
ℝ𝒿said
=
𝔖
,0
𝔖
𝜑)𝑑𝒾
𝔖
(𝒾,
(𝒾,
(𝒾,
Lebesgue-integrable,
𝜑)
𝔖
is
aset
Lebesgue-integrable,
fuzzy
Aumann-integrable
are
Lebesgue-integrable,
then
𝔖
funcis𝔖
fuzzy
Aumann-integrable
𝔖family
is
fuzzy
𝔖
𝔖
∗⊂
∗(0,
∗ 𝜈]
e𝔖→
(9)
𝜑∈
∈∗3.(0,
1]
𝜑A
-cuts
whose
𝜑 for
-cuts
define
the
of
of
𝔖1ℝ
I-V-Fs
:→
𝐾
⊂
ℝ
𝐾
⊂𝜑)𝑑𝒾
are
ℝ
→
given
𝒦
are
by
byathe
Theorem
2.∗𝒾family
[49]
Let
𝔖
:ϕ[𝜇,
⊂
𝔽
be
a∈:𝒦
F-NV-F
whose
𝜑-cuts
ofbe
I-V-Fs
(𝒾)1]
[𝔖
(𝒾,𝜑)
(𝒾,
𝔖
=𝜑whose
𝜑),
𝔖
𝜑)]
all
∈∈ 𝜑),
𝐾.
Here,
for
𝜑(𝒾,
∈𝜑)
(0,
1]
the
end
point
forthe
all𝔖
υharmonically
, I-V-Fs
∈
0,each
Then
S
HFSX
µ,
υfunction
Freal
s)a,define
ifthen
and
only
if, for
all
∈Aumann-integr
,
([
],𝜐].
[µ,
][𝜇,
[are
]a.𝔖
[0, 1]func𝒾𝒿
𝒾𝒿
0 , given
[𝜇,
[𝜇,
is
called
a
harmonically
concave
function
is
called
on
a
𝜐].
is
called
concave
harmonically
function
concave
on
on
𝜐].
Theorem
2.
[49]
Let
𝔖
:
[𝜇,
𝜈]
⊂
ℝ
→
𝔽
be
a
F-NV-F
whose
𝜑-cuts
define
the
fami
[0,
[0,
[0,
𝐾,
𝜉∈
1],
we
have
𝐾, 𝜉 ∈
we
𝐾,have
𝜉 ∈over1],
we
have
over
[𝜇,
𝜈].
[𝜇,
𝜈].
[𝜇,
∗ (𝒾,
∗ (𝒾,
∈Then
𝐾.⊂end
∈∗ofintegral
𝐾.𝔖fuzzy
(10)
∗ (1],
+𝜑
∗ (.
∗𝔽
(𝒾)
[𝔖
(𝒾)
𝔖 tion
𝔖∗ (𝒾,
=
=
𝔖[𝔖
𝜑)]
𝜑),
𝔖
all
𝒾tion
∈for
𝐾.
all
Here,
∈
for
𝐾.
Here,
each
𝜑for
∈
(0,
each
1]
∈
(0,
end
point
real
point
real
(.
S
,[49]
ϕℝ
,over
S
,lower
ϕ
∈
HSX
υ𝜈].
,[49]
R
,Let
s𝔽
([49]
)(𝑅)
)tion
([
]functions
)∗.(𝒾,
(9)
functions
𝔖
, 𝜑),
𝔖
, 𝜑):
𝐾
→
ℝ
are
called
and
of
𝔖
.the
F-NV-F.
be
an
F-NV-F.
be
fuzzy
an
Then
integral
be
fuzzy
an
Then
F-NV-F.
over
Then
ofintegral
𝔖
fuzzy
overof
integral
𝔖 ov
Definition
Definition
2.
Let
Definition
2.
𝔖
[49]
:are
[𝜇,
Definition
Let
𝜈]
⊂
2.𝔖
ℝ
[49]
[𝜇,
→
𝔽
𝜈]
Let
⊂
𝔖
ℝ
:∗an
[𝜇,
→
𝜈]
⊂
𝔖
:1]
ℝ[𝜇,
→
𝜈]
ℝ
→
𝔽, F-NV-F.
∗𝜑)]
Theorem
2.
Let
𝔖
: 𝒾[𝜇,
⊂
ℝ
→
𝔽
be2.
abe
F-NV-F
whose
𝜑-cuts
define
the
family
of𝜑)𝑑𝒾
I-V-Fs
∗ (𝒾,
(𝒾)
[𝔖
(1
(1
∗𝜑),
(𝒾,
(𝒾,
(𝒾,𝜑)]
(𝒾,
(𝑅)
(𝑅)
(𝑅)
(𝐹𝑅)
(𝒾)𝑑𝒾
(𝐹𝑅)
(𝒾)𝑑𝒾
=
𝜑),
𝔖
for
all
𝒾𝜉)𝒿
∈𝔖[𝜇,
𝜈]
and
for
all
: [𝜇,
𝜈]
⊂
→
𝒦
given
by
𝔖
𝔖for
𝜉𝒾
𝜉𝒾
+the
−
𝜉)𝒿
+
−
=
𝔖
𝜑)𝑑𝒾
,:upper
𝔖
=
𝜑)𝑑𝒾
𝔖
𝜑)𝑑𝒾
𝔖
𝔖µ,
(𝒾)𝑑𝒾
(𝐼𝑅)
=
𝔖
∗𝜈]
∗(𝒾,
∗
(𝒾)(𝐼𝑅)
[𝔖∗ (𝒾,
(𝒾, 𝜑)] for all 𝒾 ∈ [𝜇, 𝜈]
∗ (. (.
∗ (.
=𝒾𝒿
𝜑),
𝔖by
:𝜈],
[𝜇,
⊂(𝒾)𝑑𝒾
ℝ →upper
𝒦
are
given
byof𝔖,𝔖
(𝒾)𝑑𝒾
𝒾𝒿
functions functions
𝔖∗ (. , 𝜑), 𝔖𝔖
, 𝜑):
, 𝜑),
𝐾𝔖
→
ℝ
, 𝜑):
are
𝐾by
called
→ (𝐹𝑅)
ℝ
lower
are
called
and
lower
upper
and
functions
of
functions
𝔖levelwise
[𝜇,
[𝜇,
[𝜇,
(𝒾)𝑑𝒾
[𝜇,
(𝐹𝑅)
(𝐹𝑅)
(𝐹𝑅)
(𝒾)𝑑𝒾
(𝒾)𝑑𝒾
𝜈],
denoted
𝜈],
denoted
𝜈],given
by
denoted
𝔖𝔖
denoted
by
𝔖
𝔖
, 𝜈]
is
given
levelwise
,by
is(𝒾,
given
by
,∗.(𝒾,
is𝔖𝒾𝒿
given
by
levelwise
is.=
given
levelwise
by
∗𝔖
Proof.
The
proof
is
similar
to
the
proof
of
Theorem
2.13,
see
[39].
𝔖𝜈]𝔖∈
∗and
(𝒾)
[𝔖
=
𝜑),
𝔖
𝜑)]
for
all
𝒾
∈
[𝜇,
for both
allfamily
:
[𝜇,
𝜈]
⊂
ℝ
→
𝒦
are
by
𝔖
(𝒾)𝑑𝒾
(𝐼𝑅)
=
𝔖
(𝒾,
(𝒾,
∈
𝐾.
∈
𝐾.
𝐾.
𝜑)
and
𝜑)
aredefine
𝜑
∈
(0,
1].
Then
𝔖
is
𝐹𝑅-integrable
over
[𝜇,
𝜈]
if
and
only
if
𝔖
(10)
(10)fami
∗
(9)
(9)
∗
Theorem
[49]
LetLet
𝔖: [𝜇,
𝜈] ⊂
Theorem
𝔽
2.
[49]
Theorem
: Then
[𝜇,
2.Then
𝜈]
[49]
⊂
Let
ℝ→
𝔽the
: 𝜉𝒾
[𝜇,be
𝜈]a(1
⊂
𝔽𝜉𝒾
abeF-NV-F
𝜑-cuts
define
family
F-NV-F
whose
be
a 𝜈]
F-NV-F
𝜑-cuts
whose
of and
I-V-Fs
the
F-NV-F.
fuzzy
integral
of
𝔖→I-V-Fs
over
Definition2. 2.
[49]
𝔖: [𝜇,
𝜈] ℝ
⊂→
ℝ
→𝜉𝒾𝔽be
(1
+ (1
−an
𝜉)𝒿
+
−ℝof𝜉)𝒿
+[𝜇,
− ∗if
𝜉)𝒿
𝔖∗ (𝒾,
𝜑 Let
∈whose
(0,𝔖1].
𝔖
is𝔖
𝐹𝑅-integrable
over
anddefine
only the
if 𝜑-cuts
𝔖∗ (𝒾, 𝜑)
[𝜇,
[𝜇,
[𝜇,
(0,
Definition
4.
[34]
The
Definition
𝔖:
𝜐]
→
4.
ℝ
[34]
The
𝔖:
𝜐]
→
ℝ
is
called
a
harmonically
is
called
convex
a
harmonically
function
on
convex
𝜐]
if
function
o
for
all
𝜑
∈
(0,
1].
For
all
𝜑
∈
1],
ℱℛ
denotes
the
collection
of
all
𝐹𝑅-integrable
F(𝒾,
(𝒾,
𝜑)
and
𝔖
𝜑)
both
are
𝜑
∈
(0,
1].
Then
𝔖
is
𝐹𝑅-integrable
over
[𝜇,
𝜈]
if
and
only
if
𝔖
∗ (𝒾,
∗ (𝒾,
([
], )(𝒾)
[𝜇,
,integral
𝑅-integrable
over
[𝜇,
if⊂given
𝔖
isby
𝐹𝑅-integrable
over
𝜈],
then
∗over
be
an
F-NV-F.
be
an
Then
fuzzy
Then
of[𝔖
integral
𝔖
of
𝔖
over
Definition
2.
[49]
Let
2.are
𝔖[49]
: [𝜇,
𝜈]
Let
⊂by
𝔖
ℝ: 𝔖
[𝜇,
→
𝜈]
⊂
ℝ
→𝜈].
𝔽 Moreover,
(𝒾)
[𝔖
(𝒾)
[𝔖
(𝒾,for
[𝜇, 𝜈],
(𝒾)𝑑𝒾
=𝜈]
𝜑),
𝔖
𝜑)]
for
all
𝒾are
∈fuzzy
[𝜇,
and
𝜑),
all
𝔖
𝜑),all
𝔖∗over
𝒾(𝒾,∈𝜑)]
[𝜇,
𝜈]
forand
all
for
𝒾all
∈ all
[𝜇,
𝜈]
: [𝜇,denoted
𝜈]Definition
⊂ℝ
→by
𝒦 (𝐹𝑅)
given
: [𝜇,
⊂
ℝ
→
𝒦
:all
[𝜇,
are
𝜈]F-NV-F.
ℝ
→
𝒦
𝔖
given
by
𝔖for
𝔖
𝔖
𝔖
,𝔖is𝔽=
given
levelwise
by
(𝒾)𝑑𝒾
(𝒾)𝑑𝒾
(𝐼𝑅)
(𝐼𝑅)
𝔖
=all
(0,
for
𝜑
∈
(0,
1].
For
𝜑=𝜈]
∈
1],
ℱℛ
the
collection
of(8)
𝐹𝑅-in
∗ (𝒾,
∗ (𝒾,
([=
)∗ denotes
[𝜇,
, ],𝜑)]
𝑅-integrable
over
[𝜇,
𝜈].
Moreover,
if
𝔖
is
𝐹𝑅-integrable
𝜈],
then
(8)
(𝒾)𝑑𝒾
(𝒾)𝑑𝒾
(𝒾)𝑑𝒾
(𝒾)𝑑𝒾
(𝐹𝑅)
(𝒾)𝑑𝒾
(𝐹𝑅)
(𝒾)𝑑𝒾
(𝐹𝑅)
(𝐹𝑅)
(𝒾)𝑑𝒾
(𝒾)𝑑𝒾
(𝐼𝑅)
(𝐼𝑅)
(𝐼𝑅)
(𝐼𝑅)
𝔖
𝔖
𝔖
𝔖
=
𝔖
=
=
𝔖
=
𝔖(𝒾,
=
𝜑)𝑑𝒾
𝔖
:
𝔖(𝒾,
𝔖(𝒾,
𝔖
=
𝜑)𝑑𝒾
𝜑)
∈
:
𝔖(𝒾,
𝔖(𝒾,
ℛ
=
𝜑)𝑑𝒾
𝜑)
∈
𝔖(𝒾,
:
,
𝔖(𝒾,
ℛ
𝜑)𝑑𝒾
𝜑)
∈
:
𝔖(𝒾,
,
ℛ
𝜑)
∈
ℛ
,
NV-Fs
over
[𝜇,For
𝜈].
([ , ], )
([ ,F], )
([ , ], )
([
forby
all
𝜑 ∈, (0,
1].
all
𝜑 NV-Fs
∈ (0,
ℱℛ[𝜇,
the collection
of all 𝐹𝑅-integrable
𝒾𝒿
𝒾𝒿
([
], ) denotes
[𝜇,both
,𝜈].
over
𝜈].
Moreover,
if1],
𝔖
𝐹𝑅-integrable
𝜈],
then
[𝜇,𝜑𝜈],∈ denoted
[𝜇, 𝜈],
(𝐹𝑅)
(𝒾)𝑑𝒾
(𝐹𝑅)
(𝒾)𝑑𝒾
denoted
𝔖
is
given
levelwise
, is
given
by
levelwise
by
∗over
∗ (𝒾,
(𝒾,
and
𝔖
𝜑)
𝜑)only
and
𝜑)
𝜑) [𝜇,
both
and
𝔖if∗ (𝒾, 𝜑
(0, 1].
Then
𝔖The
is𝑅-integrable
𝐹𝑅-integrable
over
∈[𝜇,
[𝜇,
1].
𝜈]
Then
and
𝜑∈
𝔖
only
(0,
is over
1].
𝐹𝑅-integrable
if𝔖:is
𝔖Then
𝔖
is
over
𝐹𝑅-integrable
[𝜇,
𝜈]
and
over
only
[𝜇,if≤
𝜈]𝔖
and
if+𝔖𝔖𝜉𝔖(𝒿),
[𝜇,
[𝜇,
Definition
4.by
[34]
𝔖:𝔖[𝜇, 𝜐] →
ℝ𝜑
Definition
4.
[34]
Definition
The
4.
[34]
𝜐]𝜑)
→
The
ℝfunction
𝔖:
→if
ℝ
is(0,
called
aif harmonically
convex
is
called
on
a[𝜇,
harmonically
𝜐]+are
is
if
called
a∗if(𝒾,
convex
harmonically
function
convex
on
function
𝜐]are
(1
(1
∗ (𝒾,
∗ (𝒾,
𝜉𝔖(𝒿),
𝔖
≤
𝔖
−𝜐]
𝜉)𝔖(𝒾)
−
𝜉)𝔖(𝒾)
(11)
NV-Fs over [𝜇, 𝜈].
∗ (𝒾,
(1
(1(𝑅)
𝜉𝒾collection
𝜉𝒾 +
+∈
−
𝜉)𝒿
− (8)
𝜉)𝒿
(𝒾,
(𝑅)
(𝐹𝑅)
(𝒾)𝑑𝒾
=
𝜑)𝑑𝒾
, if
𝔖collection
𝜑)𝑑𝒾
𝔖[𝜇,
(0,
∗([is
for
all 𝜑 ∈(𝐹𝑅)
(0,
1]. [𝜇,
For
allMoreover,
𝜑Definition
∈=
1],
for
ℱℛ
all
1].
all
𝜑where
1],
ℱℛ
denotes
the
of
all
𝐹𝑅-integrable
Fthe
of
all
(𝒾)𝑑𝒾
(𝐼𝑅)
∗ (𝒾, F- integ
𝔖(𝒾)𝑑𝒾
𝔖
=
𝔖(𝒾,
𝜑)𝑑𝒾
:ℝ𝔖(𝒾,
𝜑)
∈Moreover,
ℛ
([𝜑
], (0,
)1],
],
(0,
[𝜇,
[𝜇,
[𝜇,𝒾,
,∈
=
is
said
to),𝒾𝒿
be
a(𝑅)
convex
set,
if,
for
all
𝒿𝔖
∈integrable
3.
[34]
A
set
𝐾
=
𝜐]
⊂
𝑅-integrable
over
𝜈].
if all
𝑅-integrable
𝔖
iswhere
𝐹𝑅-integrable
over
over
Moreover,
over
[𝜇,
then
ifwhere
𝜈].
𝔖𝔖
𝐹𝑅-integrable
𝔖
is
over
𝐹𝑅-integrable
𝜈],
then
over
𝜈],
then
([
],,denotes
,)collection
for
all
𝜑(0,
∈for
(0,
1],
𝜑
∈
for
(0,
all
ℛ𝜑([𝑅-integrable
where
for
∈For
(0,
all
1],
𝜑
ℛ
∈[𝜇,
1],
ℛ
ℛ
denotes
denotes
collection
denotes
of
the
denotes
collection
of
Riemannian
the
integrable
collection
of𝜑)𝑑𝒾
Riemannian
integrable
funcof
Riemannian
funcfun
𝒾𝒿
𝒾𝒿
(𝑅)
(𝐹𝑅)
(𝒾)𝑑𝒾
=
𝔖
,to𝐹𝑅-integrable
𝜑)𝑑𝒾
𝔖
)𝜈].
([
],𝜈],
)the
([
],
),the
([
],)Riemannian
, ],
,(0,
, ∞)
∗ (𝒾,
(0,
[𝜇,
=
∞)
is
said
be
a
convex
set,
Definition
3.
[34]
A
set
𝐾
=
𝜐]
⊂
ℝ
∗
(1
(1
(1
+
𝜉𝔖(𝒿),
+
𝜉𝔖(𝒿),
+
𝜉𝔖(𝒿),
𝔖
≤
−
𝜉)𝔖(𝒾)
𝔖
𝔖
≤
−
𝜉)𝔖(𝒾)
≤
−
𝜉)𝔖(𝒾)
(11)
(11)if, fo
(𝒾,
(𝒾,
(𝑅)
(𝑅)
(𝐹𝑅)
(𝒾)𝑑𝒾
(8)
(8)
[𝜇,
[0,
[𝜇,
[0,
[𝜇,
[𝜇,
=
𝔖
𝜑)𝑑𝒾
,
𝔖
𝜑)𝑑𝒾
𝔖
for
all
𝒾,
𝒿
∈
𝜐],
𝜉
∈
for
all
1],
𝒾,
where
𝒿
∈
𝔖(𝒾)
𝜐],
𝜉
∈
≥
0
1],
for
where
all
𝒾
∈
𝔖(𝒾)
𝜐].
≥
If
0
(11)
for
is
all
reversed
𝒾
∈
𝜐].
then,
If
(11)
𝔖
is
reverse
(𝒾)𝑑𝒾
(𝒾)𝑑𝒾
(𝐹𝑅)[𝜇, 𝜈].
(𝒾)𝑑𝒾
(𝐹𝑅)
(𝒾)𝑑𝒾
(𝐼𝑅)
(𝐼𝑅)
NV-Fs over
NV-Fs
over
[𝜇,
𝜈].
𝔖
𝔖
=
𝔖
=
=
𝔖
𝔖(𝒾,
=
𝜑)𝑑𝒾
:
𝔖(𝒾,
𝔖(𝒾,
𝜑)𝑑𝒾
𝜑)
∈
:
𝔖(𝒾,
ℛ
𝜑)
∈
,
ℛ
,
[0,
∗
𝐾,of𝜉𝜉𝒾I-V-Fs.
∈tions
1],
we
have
([
],
)
([
],
)
,
,
(9)
(𝐹𝑅)
(𝒾)𝑑𝒾
(𝐹𝑅)
(𝒾)𝑑𝒾
(𝐹𝑅)
(𝐹𝑅)
(𝒾)𝑑𝒾
(𝒾)𝑑𝒾
(0,
[𝜇,
tions
of
𝔖
I-V-Fs.
tions
is
𝐹𝑅
of
-integrable
𝔖
tions
I-V-Fs.
is
𝐹𝑅
of
-integrable
𝔖
I-V-Fs.
over
is
𝐹𝑅
[𝜇,
𝔖
-integrable
𝜈]
over
is
if
𝐹𝑅
[𝜇,
-integrable
𝜈]
over
if
𝔖
[𝜇,
over
𝜈]
if
𝔖
[𝜇,
𝜈]
if
𝔖
𝔖
∈
𝔽
.
Note
∈
𝔽
that,
.
Note
if
∈
𝔽
that,
.
Note
if
∈
𝔽
that,
.
N
=
∞)
is
said
to
be
a
convex
set,
if,
for
all
𝒾,
𝒿
∈
Definition
3.
[34]
A
set
𝐾
=
𝜐]
⊂
ℝ
+ (1 − 𝜉)𝒿
+ (1 − 𝜉)𝒿 𝜉𝒾 + (1 − 𝜉)𝒿
𝐾,called
𝜉 ∈ [0,
1],
we𝜉𝒾have
[𝜇,
[𝜇,
is
called
a
harmonically
is
concave
a
harmonically
function
on
concave
𝜐].
function
on
𝜐].
for all 𝜑 ∈ (0, 1], where
ℛ
denotes
the
collection
of
Riemannian
integrable
func∗
∗
∗
∗
[0,𝔖
([
], (𝒾,
)(𝒾,
,𝔖
𝜉𝔖∈𝜑),
1],
we
have
∗ (𝒾,
∗ (𝒾,
∗ (𝒾,
(9) Aumann-integ
𝒾𝒿
(𝒾,
(𝒾,
(𝒾,
(𝒾,
𝜑)
𝜑),
are
𝔖𝔖
𝜑)
𝜑),
are
𝜑),
𝜑)
𝔖
are
𝜑)
then
are
𝔖
Lebesgue-integrable,
is
then
a≥where
fuzzy
𝔖
is=
Aumann-integrable
then
a𝔖
𝔖
is
then
Aumann-integrable
a∗ (𝒾,
fuzzy
𝔖𝜑)𝑑𝒾
Aumann-integrable
ais
fuzzy
func𝔖𝐾,
𝔖
(𝒾,
(𝑅)
(𝑅)
(𝑅)
(𝑅)
(𝑅)
(𝐹𝑅)
(𝒾)𝑑𝒾
(𝐹𝑅)
(𝒾)𝑑𝒾
=
𝔖
𝜑)𝑑𝒾
, [0,
𝔖
=
𝜑)𝑑𝒾
𝔖
𝜑)𝑑𝒾
, [𝜇,
𝔖
𝜑)𝑑𝒾
,func𝔖
𝔖
𝔖
(𝒾)𝑑𝒾
(𝐼𝑅)
=Lebesgue-integrable,
𝔖
∗ (𝒾,
∗Lebesgue-integrable,
∗(𝐹𝑅)
∗ (𝒾,
∗is
[𝜇,
[0,
[𝜇,
[𝜇,
[𝜇,
[0,
[𝜇,
for
𝒾,∈for
𝒿(0,
∈3.
𝜐],
𝜉 (0,
∈
where
𝔖(𝒾)
𝒾,
𝒿denotes
03.
∈
for
for
all
𝜐],
all
𝜉Lebesgue-integrable,
𝒾set
∈𝒾,
𝒿Riemannian
∈
1],
𝜐].
where
If
𝜐],
(11)
𝜉⊂
∈
is
𝔖(𝒾)
reversed
1],
0∞)
for
then,
all
𝔖(𝒾)
𝒾fuzzy
∈≥
0(𝑅)
𝜐].
If
all
(11)
𝒾is∈set,
reversed
𝜐].for
Ifall
(11)
then,
reverse
𝔖 fun
(0,
(0,
[𝜇,
[𝜇,
=≥
∞)
is
said
to
be
a(𝒾)𝑑𝒾
convex
set,
=
if,
for
𝒾,
said
𝒿=∈
to
be
afor
convex
if,
𝒾,is
𝒿𝜑)𝑑𝒾
∈
[34]
A
set
𝐾1],
=-integrable
𝜐]∗Definition
⊂for
ℝ,all
[34]
A
𝐾
=
𝜐]
ℝ
𝒾𝒿
∈
𝐾.all
(10)
forDefinition
allall
𝜑of
all
1],
𝜑
where
∈
1],
ℛ
where
ℛ
denotes
the
collection
the
collection
of
of
Riemannian
integrable
integrable
funcfunc(𝒾)𝑑𝒾
(𝐼𝑅)
𝔖
(𝐹𝑅)
(𝒾)𝑑𝒾
tions
I-V-Fs.
𝔖
is
𝐹𝑅
over
[𝜇,
𝜈]
if
𝔖
∈
𝔽
.
Note
that,
if
([
],
)
([
],
)
,
tionconcave
over tion
[𝜇,function
𝜈].
over
tion
[𝜇,a𝜈].
over
tion
[𝜇, 𝜈].
over
[𝜇,
𝜈].
(1 (𝒾)𝑑𝒾
𝜉𝒾𝒾𝒿+function
− concave
𝜉)𝒿 on [𝜇,
[𝜇, 𝜐].
[𝜇,𝜉)𝒿
(𝐼𝑅)
is called
a1],
harmonically
called
harmonically
is
called
a=concave
harmonically
function
𝜐]. 𝜉𝒾
on
𝜐]. ∈ 𝐾.
𝐾,
𝜉 of
∈ [0,
we𝔖have
𝐾,is
𝜉over
∈ [0, on
1],
we
have
(9)
(1 −
(𝐹𝑅)
tions
I-V-Fs.
tions
is 𝐹𝑅
-integrable
𝔖 is 𝐹𝑅
-integrable
[𝜇,
𝜈]over
if𝔖 (𝐹𝑅)
[𝜇,a𝜈]fuzzy
if𝔖(𝒾)𝑑𝒾
∈𝔖𝔽𝔖(𝒾)𝑑𝒾
. ∈Note
that,
. Note
if(9)+that,
if
𝐾. ∈ 𝔽
(10)
𝜑),
𝔖∗ (𝒾,of
𝜑) I-V-Fs.
are
Lebesgue-integrable,
then
is
Aumann-integrable
func𝔖∗ (𝒾,
(1
𝜉𝒾
+
−
𝜉)𝒿
(0,
for
all
𝜑
∈
(0,
1].
For
all
𝜑
∈
1],
ℱℛ
denotes
the
collection
of
all
𝐹𝑅-integrable
F([ , ], )
∗ (𝒾,
𝒾𝒿
(𝒾, 𝜑),
(𝒾,𝜑)
𝔖𝔖
𝜑),are
𝔖∗ (𝒾,
Lebesgue-integrable,
𝜑) are Lebesgue-integrable,
then
𝔖
is2.𝔖
then
a[49]
fuzzy
𝔖
is
Aumann-integrable
a1].
fuzzy
Aumann-integrable
func𝔖∗tion
over
𝜈].
(𝒾)𝑑𝒾
(𝒾)𝑑𝒾
(𝒾)𝑑𝒾
(𝐼𝑅)
=
𝔖
=
𝔖
for
all
𝜑𝔽
∈ 2.
(0,
For
all
𝜑
∈𝔽(0,
1],
ℱℛ
denotes
the
collection
of(10)
allthe
∗[𝜇,
Theorem
Theorem
2. [49] =
Let(𝐼𝑅)
2.
Theorem
𝔖𝒾𝒿
[49]
: The
[𝜇,𝔖
𝜈]
Let
Theorem
⊂
ℝ
: [𝜇,
→
𝜈]
Let
⊂
𝔖
[49]
ℝ
[𝜇,
→
Let
𝔽
𝜈] ⊂
𝔖
:ℝ
[𝜇,
→
𝜈]
⊂𝜑-cuts
ℝ
𝔽
be
a:is
F-NV-F
be
a(𝐼𝑅)
F-NV-F
be
whose
a→
define
be, funca],the
whose
F-NV-F
define
𝜑-cuts
whose
of
the
I-V-Fs
family
define
𝜑-cuts
of
the
define
family
of𝐹𝑅-in
I-Vfam
([𝜑-cuts
) family
∈[𝜇,
𝐾.
∈F-NV-F
𝐾.
(10)
[𝜇,
Definition
4.
[34]
𝔖:
𝜐] ℱℛ
→
ℝ
called
awhose
harmonically
convex
function
on
ifI-V-Fs
NV-Fs
over1].[𝜇,
𝜈].
(0,
for
all
𝜑
∈
(0,
For
all
𝜑
∈
1],
denotes
the
all 𝐹𝑅-integrable
F- 𝜐]
([
],
)
,
(1 𝜐]
𝜉𝒾 + (1 − 𝜉)𝒿Definition 4. [34] The𝜉𝒾𝔖:
+ [𝜇,
−collection
𝜉)𝒿
tion over [𝜇,
tion𝜈].over [𝜇, 𝜈].
→ ℝ isof
called
a harmonically
convex function
It is a partial order relation.
[𝔴] for all
𝔵 We
≼ 𝔴now
if and
if, [𝔵] ≤ operations
∈ (0, 1],some of the characte
(4)
useonly
mathematical
to 𝜑
examine
It is a partial order relation.
It is
a It
partial
is a partial
order order
relation.
relation.
We now
use
mathematical
operations
to examine some of the characteristics of fuzzy
numbers.
𝔵,examine
and
𝜌
ℝ,
then
we
Proposition
Proposition
1. [51]
Let
Proposition
1.𝔵,[51]
𝔴 ∈ Let
𝔽Let.toThen
𝔵,1.
Proposition
𝔴𝔴
[51]
∈∈fuzzy
𝔽𝔽
Let
Proposition
𝔵,
𝔴
1. fuzzy
[51]
∈of∈
𝔽
Let
1.𝔵,[51]
𝔴
∈ Let
𝔵,
𝔴
∈relation
. Then
order
relation
. Then
order
“fuzzy
relation
≼
”𝔽have
is
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and
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𝔴]
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, , 6 of 18
Symmetry 2022, 14, 1639
(6)
[𝔵[𝔵+
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[[𝔴]
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𝔴]
=
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[𝔵 × 𝔴] = [𝔵] × [ 𝔴] , [𝔵
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=
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[𝔵]𝔵]=×[𝔵]
[ 𝔴]
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, [ 𝔴] ,
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𝔽numbers.
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∈
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Let
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𝔵,
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[𝔵 ×[𝜌.
𝔴]𝔵] ==[𝔵]𝜌. [𝔵]
× [ 𝔴] ,
(7)
[𝜌. 𝔵] = 𝜌. [𝔵]
[𝜌. 𝔵] [𝜌.
[𝔵]= 𝜌. [𝔵]
𝔵]
=
𝜌.
e :[𝔵+
(5)
[𝔵+
[𝔵+
[𝔵+ [𝔴] =, [𝔵+
[𝔵]
[𝔴]
[𝔵]
[𝔵] + [𝔴]
[𝔵]
𝔴]
𝔴]
= valued
, by
𝔴]
+ [𝔴]
= ,[𝔵]
= (7)
, FExample 1. We consider the
F-NV-Fs 1.S
2]fuzzy
→=𝔵]
F
[0,A
(R)
[𝜌.
[𝔵]
=+𝜌.defined
Cnumber
Definition
[49]
map
𝔖: 𝐾+𝔴]
⊂ ℝ → 𝔽 𝔴]
is called
Definition 1. [49] A fuzzy number
valued
map
𝔖
:
𝐾
⊂
ℝ
→
𝔽
is
called
F-NV-F.
For
each

𝜑 ∈ (0,
1] whose
𝜑 -cuts
define
the
:×
𝐾 𝔴]
⊂
𝒦
[𝔵 ×𝔖𝔴]
[𝔵,i×family
[𝔵]
[ 𝔴]= of
[𝔵]
[each
[(6)
×
××𝔴]
𝔴]
=
𝔴]
×
𝔴]F-NV-F.
, [𝔵I-V-Fs
𝔴] 𝔖
=,[𝔵[𝔵]
×ℝ
𝔴]
=→[𝔵]
,
h [is
Definition 1. [49] A fuzzy
number
valued
map
⊂=[𝔵
ℝ[𝔵]
→
𝔽×
called
For
√
∂A: 𝐾
Definition
Definition
1. [49]
1.A
[49]
fuzzy
fuzzy
number
valued
valued
map𝔖𝔖
map
𝐾⊂
⊂𝔖ℝ
ℝ
:𝐾
→⊂𝒦
𝔽ℝ →
𝔽 given
isare
called
is called
F-NV-F.
𝜑 ∈ (0, 1] whose
𝜑 -cuts
define
the
family
of I-V-Fs
:: 𝐾
→
by F
∗number

√
(𝒾)
[𝔖
(𝒾,
(𝒾,
𝔖
=
𝜑),
𝔖
𝜑)]
for
all
𝒾
∈
𝐾.
Here,
for
each
𝜑
∈
(0,
1]
the
∂
∈
0,

Definition
[49] A∗ fuzzy
number
map
𝐾ℝ
⊂→
ℝ𝒦
→ 𝔽 are
is called
F-NV-F.
For each
∗ of valued
𝜑 ∈ (0, 1] whose
𝜑 -cuts1.∈define
the
I-V-Fs
𝔖the
:𝔖
𝐾:family
⊂
given
by
(0,𝜑𝔖
1]
∈ (𝒾,
(0,
whose
1]family
whose
𝜑 -cuts
𝜑
-cuts
define
define
the
of𝜑𝔵]I-V-Fs
I-V-Fs
𝔖
𝔖
:𝐾
⊂=
ℝ
:𝐾
→
⊂[𝔵]
𝒦𝔵]
ℝ(7)
→are
[𝜌.[𝔵]
[𝜌.
[𝜌.
[𝜌.
[𝔵]
[𝔵]
𝔖 (𝒾) =∗ [𝔖𝜑
𝜑)]
for
all [𝜌.
𝒾∈
𝐾. =
Here,
∈ of
(0,
1]
the
point
real
𝔵]for
𝔵]end
𝜌.
=each
= 𝜌.
𝜌.
=𝒦𝜌.g
i𝜌.family
∗𝔵]
∗ (𝒾, 𝜑),
√
√
(.∗, 𝜑),
(.family
functions
𝔖=
𝔖
, 𝜑):𝐾
→∈ℝ
are
called
lower
and
upper
functions
e∗-cuts
𝜑𝜑),
∈ (0,
whose
𝜑S
define
the
of
I-V-Fs
𝔖
:
𝐾
⊂
ℝ
→
𝒦
are
given
by o
(𝒾, 𝜑)]
∗Here,
𝔖 (𝒾) = [𝔖∗ (𝒾,
𝔖 1]
for
all
𝒾
∈
𝐾.
for
each
𝜑
(0,
1]
the
end
point
real
∂
)
(
)(
∗
2
−
∂
√
(𝒾)
(𝒾)
[𝔖
(𝒾,
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(𝒾,
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(𝒾,
𝔖
𝔖
=
=
𝜑),
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𝜑),
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𝔖
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𝜑)]
all
for
𝒾
∈
all
𝐾.
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Here,
∈
𝐾.
Here,
for
each
for
𝜑
each
∈
(0,
𝜑
1]
∈
(0,
the
1]
end
thep
(.
(.
∂
∈
,
2
functions
𝔖
,
𝜑),
𝔖
,
𝜑):
𝐾
→
ℝ
are
called
lower
and
upper
functions
of
𝔖
.
∗
∗ 
∗ (.[𝔖 ∗(𝒾, 𝜑), 𝔖∗ (𝒾,
(𝒾)𝔖=
𝜑)]
for
all
𝐾. upper
Here, functions
for each of
𝜑 ∈𝔖(0,
the end point real

2 𝒾∗∈
functions 𝔖∗ (.𝔖, 𝜑),
, 𝜑):
ℝ 𝔖are
called
and
. 1]
∗lower
∗ 𝐾 →

(.
(.
(.
(.
functions
functions
𝔖
,
𝜑),
𝔖
,
𝜑),
,
𝜑):
𝔖
𝐾
,
→
𝜑):
ℝ
𝐾
are
→
ℝ
called
are
called
lower
lower
and
upper
and
upper
functions
functions
of
𝔖
.→
o
∗ 1.Definition
∗ (.∗number
be
an
Then
fuzzy
Definition
2.
[49]
Let
𝔖Definition
:valued
[𝜇,
𝜈]
ℝfuzzy
→
𝔽:upper
0are
otherwise,
Definition
Definition
1. 𝔖
[49]
A𝜑),
fuzzy
Definition
1.𝔖[49]
A fuzzy
valued
[49]
number
A fuzzy
map
1.
𝔖
[49]
number
: 𝐾⊂A
⊂
map
ℝ
1.
valued
→
𝔖
[49]
𝔽𝐾number
A
⊂ismap
fuzzy
ℝ
→F-NV-F.
𝔖
valued
𝔽:number
𝐾
ℝ
map
→
valued
𝔽𝔖
: each
𝐾
map
ℝinte
𝔖
called
F-NV-F.
is⊂called
For
F-NV-F.
is ⊂
called
For
(.
functions
,
,
𝜑):
𝐾
→
ℝ
called
lower
and
functions
of
𝔖
.
∗
Definition 2. [49] Let 𝔖: [𝜇, 𝜈] ⊂ ℝ → 𝔽 be an F-NV-F. Then fuzzy integral of 𝔖 over
(𝐹𝑅)
(𝒾)𝑑𝒾
(0, 1]Let
𝜑whose
∈𝔖(0,
𝜑[𝜇,
whose
-cuts
𝜑𝜈],
∈ (0,
define
𝜑 -cuts
whose
𝜑
the
∈define
(0,
family
𝜑1]
-cuts
𝜑
the
whose
∈
of
define
(0,
family
I-V-Fs
1]𝜑, is
-cuts
whose
the
of
𝔖 integral
I-V-Fs
family
the
I-V-Fs
define
the
𝔖given
of
family
I-V-Fs
𝔖𝒦
: define
𝐾𝜑
⊂-cuts
ℝ𝔖of
→
:𝒦
𝐾 family
⊂
are
ℝ→
𝒦: 𝐾
⊂
are
by
ℝ give
→of
by
𝔖
given
levelwise
by
be
an
F-NV-F.
Then
of
𝔖
over
Definition𝜑2.∈ [49]
: [𝜇,1]𝜈]
⊂
ℝ
→denoted
𝔽1]
be𝔽ian
be
F-NV-F.
F-NV-F.
Then
Then
fuzzy
fuzzy
integral
integ
o
Definition
2.
Let
[49]
: given
[𝜇,h∗ 𝜈]
𝔖√
:⊂
[𝜇,
ℝ𝜈]→⊂𝔽fuzzy
ℝ∗ √
→
[𝜇, 𝜈], denoted
(𝐹𝑅)2. [49]
byDefinition
𝔖∗(𝒾)𝑑𝒾
,𝔖
isLet
levelwise
by
∗ (𝒾,
∗an
∗point
(𝒾)
[𝔖
(𝒾,
(𝒾)
[𝔖
(𝒾,
(𝒾)
[𝔖
(𝒾,
(𝒾,
(𝒾)
(𝒾,
[𝔖
(𝒾,
(𝒾)
[𝔖
(𝒾,
(𝒾,
(𝒾,
𝔖
𝔖
𝔖
𝔖
𝔖
=
𝜑),
=
𝔖
𝜑)]
𝜑),
=
𝔖
for
all
𝜑)]
𝒾
𝜑),
∈
for
𝐾.
𝔖
=
all
Here,
𝜑)]
𝒾
∈
𝜑),
for
𝐾.
for
=
𝔖
Here,
each
all
𝒾
𝜑)]
∈
𝜑),
𝜑
for
𝐾.
∈
for
𝔖
each
(0,
Here,
all
1]
𝜑)]
𝜑
the
𝒾
for
∈
∈
for
(0,
𝐾.
end
each
all
1]
Here,
the
𝒾
𝜑
∈
∈
for
end
𝐾.
(0,
real
Here,
each
1]
point
th
=
ϕ
Then,
for
each
ϕ
∈
0,
1
,
we
have
S
,
2
−
ϕ
.
Since
S
,
ϕ
,
S
,
ϕ
)
[
]
(
(
)
(
)
(
)
Then fuzzy
integral of 4𝔖ofover
Definition
2.
[49]
𝔖
:PEER
[𝜇, 𝜈]
ϕ⊂ ℝ → 𝔽bybe
∗
[𝜇, 𝜈],
(𝐹𝑅)
(𝒾)𝑑𝒾
∗
∗Let
∗levelwise
∗ an F-NV-F.
∗
denoted
by
𝔖
, FOR
isdenoted
given
Symmetry 2022,Symmetry
14, x FOR2022,
PEER
Symmetry
14,
REVIEW
x FOR
2022,
PEER
14,
Symmetry
xREVIEW
FOR[𝜇,
PEER
2022,
REVIEW
14,
x
REVIEW
19
(𝐹𝑅)
(𝐹𝑅)
(𝒾)𝑑𝒾
(𝒾)𝑑𝒾
𝜈], [𝜇,
denoted
𝜈],
by
by
𝔖
𝔖
,
is
given
,
is
given
levelwise
levelwise
by
by
∗ (.(.
∗ (. (.
∗ (.
∗ (. (.
(.Hence
functions
functions
functions
𝔖
functions
𝔖𝔖
, 𝜑),
→
ℝ,∗𝜑):
are
,ϕ
𝐾called
→
𝔖[0,
ℝ functions
,are
lower
𝜑):
called
,𝐾𝜑),
→
and
𝔖ℝ
upper
are
, 𝜑):
, 𝜑),
called
and
𝐾functions
𝔖→∗([(.upper
ℝ
lower
, 𝜑):
are𝐾of
functions
called
and
→,𝔖ℝ
lower
of
called
functions
𝔖
and
. low
upp
+ ,, 𝜑),
[𝜇,
(𝒾)𝑑𝒾
by
𝔖
,𝜑),
given
by
e𝔖lower
∗ (.
∗(𝐹𝑅)
∗∈
∈ HSX
,𝔖R
s) with
s, 𝜑):
=
1,𝐾𝔖
for
each
1𝔖
F
s.upper
([µ,𝜈],υ]denoted
]∗.levelwise
). are
(𝒾)𝑑𝒾
(𝐹𝑅)
(𝒾)𝑑𝒾
𝔖is∈
= (𝐼𝑅)S
𝔖HFSX
=µ, υ], 𝔖(𝒾,
: 𝔖(𝒾, 𝜑) ∈
0 𝜑)𝑑𝒾
(𝐹𝑅) 𝔖(𝒾)𝑑𝒾 = (𝐼𝑅) 𝔖 (𝒾)𝑑𝒾 =
𝔖(𝒾, 𝜑)𝑑𝒾 : 𝔖(𝒾, 𝜑) ∈ ℛ([ , ], ) , (8)
(8)
(𝒾)𝑑𝒾
(𝐹𝑅)
𝔖(𝒾)𝑑𝒾
= (𝐼𝑅)
𝔖 Let
=
𝔖(𝒾,
𝔖(𝒾,
𝜑)
∈ 𝜈]
ℛ
,𝔽𝜈]
([⊂
],:integral
)𝜑)𝑑𝒾
,Then
be(𝐼𝑅)
an
F-NV-F.
an
F-NV-F.
fuzzy
be
an
F-NV-F.
fuzzy
of
Then
an
integral
over
fuzzy
of
Th
Definition
Definition
2.𝔖
[49]
: [𝜇,
𝜈]
⊂
2.
𝔖ℝ
Definition
: [49]
[𝜇,
→𝔖𝔽
𝜈]
Let
⊂
ℝ
Definition
𝔖
2.
→
:𝜑)𝑑𝒾
[𝜇,
[49]
𝔽
𝜈]
⊂
Let
ℝ
2.𝔖Then
𝔖
→[49]
: (𝒾)𝑑𝒾
𝔽
[𝜇,
Let
𝔖
ℝ
[𝜇,
→
⊂
ℝ
→𝔖
𝔽
(𝒾)𝑑𝒾
(𝐹𝑅)
(𝐹𝑅)
(𝒾)𝑑𝒾
(𝒾)𝑑𝒾
(𝐼𝑅)
𝔖
=
=
𝔖:be
=
=
𝔖(𝒾,
𝔖(𝒾,
:be
𝜑)𝑑𝒾
𝔖(𝒾,
𝜑)
: F-NV-F.
𝔖(𝒾,
∈ be
ℛ
𝜑)
∈int
ℛ
([an
],F
,𝔖
Remark
2. If sDefinition
=2.(𝐹𝑅)
1,[49]
thenLet
Definition
8
reduces
to
the
Definition
7.
for
all
𝜑
∈
1],
where
ℛ
the
collection
Riemannian
i
(8)
(𝒾)𝑑𝒾
(𝒾)𝑑𝒾
(𝐼𝑅)
𝔖
=(0,
𝔖
=
𝔖(𝒾,
𝜑)𝑑𝒾
:”𝔖(𝒾,
𝜑)(𝒾)𝑑𝒾
∈relation
ℛ”of
, ”on
([ 𝔴
)𝔽 denotes
,1.],∈[51]
([is
],order
)≼
, given
Proposition
Proposition
1.
[51]
Let
Proposition
𝔵,
1.
𝔴
[51]
∈
𝔽
Let
1.
Proposition
𝔵,
[51]
𝔴
∈
Let
𝔽
𝔵,
Let
𝔵,
𝔴
∈
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.
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fuzzy
.
order
Then
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relation
.
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≼
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given
Then
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is
𝔽
gi
Symmetry 2022, 14, x FOR[𝜇,
PEER
REVIEW
4
∗
[𝜇,
(𝐹𝑅)
[𝜇,
(𝒾)𝑑𝒾
(𝐹𝑅)
[𝜇,
(𝒾)𝑑𝒾
(𝐹𝑅)
[𝜇,
(𝒾)𝑑𝒾
(𝐹𝑅)
(𝒾)𝑑𝒾
(𝐹𝑅)
𝜈],
denoted
𝜈],
by
denoted
𝜈],
by
𝔖
denoted
𝔖
by
𝜈],
denoted
𝜈],
𝔖
by
denoted
by
𝔖
𝔖
,
is
given
levelwise
,
is
given
by
levelwise
,
is
given
by
levelwise
,
is
given
by
levelwise
,
by
levelw
for
all
𝜑
∈
(0,
1],
where
ℛ
denotes
the
collection
of
Riemannian
integrable
funcIf SPEER
ϕ) = S (tions
υ, ϕ) with
([ϕ , =
], )1, then harmonically s-convex F-NV-F in the second
∗ ( µ, REVIEW
Symmetry 2022, 14, x FOR
(𝐹𝑅) 𝔖(𝒾)𝑑𝒾 ∈ 𝔽
of I-V-Fs.
𝔖 is 𝐹𝑅 -integrable
overintegrable
[𝜇, 𝜈] if funcfor all 𝜑 ∈PEER
(0, 1],
wherefor
ℛ([all,for
denotes
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ofdenotes
Riemannian
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o1
all
(0,
𝜑1],
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1],
where
ℛif,
denotes
the
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the
collection
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Riemannian
Riemannian
integra
in
(𝐹𝑅)
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tions
of
I-V-Fs.
𝔖],𝜑𝔵harmonically
is
𝐹𝑅
over
𝔖
∈
𝔽
. of
if4 fo
sense reduces
to the
classical
s-convex
in
second
sense,
see
[36].
([ and
],ℛ
)[𝜇,
([≤
],
)ifthe
,function
, 𝜈]
[𝔵]
[𝔵]
[𝔵]
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[𝔴]
≼
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and
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only
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if
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if,
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if,
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and
only
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all
𝜑
∈
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for
≤
1],
all
𝜑
∈Note
for
(0,all
1],
≤that,
𝜑[𝔴]
∈
(4)(0,
∗
for all
𝜑 ∈𝐹𝑅
(0,-integrable
ℛ([(𝒾,, 𝜑)
denotes
the collection
func(𝒾, 𝜑),over
(𝐹𝑅)
(𝒾)𝑑𝒾 ∈ of
𝔖
are
Lebesgue-integrable,
𝔖 that,
is(𝐹𝑅)
a integrable
fuzzy
Aumann-i
𝔖where
tions of I-V-Fs.
is
[𝜇,
if=
𝔖over
𝔽𝜈]Riemannian
.then
Note
if
], )𝜈]
∗1],
∗ϕ
∗=
(𝐹𝑅)
(𝒾)𝑑𝒾
(𝒾)𝑑𝒾
tions
tions
of
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of
I-V-Fs.
𝔖
is
𝔖
𝐹𝑅
is
-integrable
𝐹𝑅
-integrable
over
[𝜇,
[𝜇,
if
𝜈]
if
𝔖
𝔖
∈
𝔽
.
∈
Not
𝔽
If𝔖S∗∗(𝒾,
µ,𝜑),
ϕ
S
υ,
with
ϕ
=
1
and
s
1,
then
harmonically
s-convex
F-NV-F
in
the
(𝔖
)
(
)
(8)
𝔖 (𝒾, 𝜑)
are Lebesgue-integrable,
then
is 𝜑)𝑑𝒾
a 𝔖fuzzy
Aumann-integrable
(𝒾)𝑑𝒾
(𝒾)𝑑𝒾
(𝒾)𝑑𝒾
(𝐹𝑅)
(𝒾)𝑑𝒾
(𝐹𝑅)
(𝒾)𝑑𝒾
(𝐹𝑅)
(𝒾)𝑑𝒾
(𝐹𝑅)
(𝒾)𝑑𝒾
(𝐹𝑅)
(𝒾)𝑑𝒾
(𝐼𝑅)
(𝐼𝑅)
(𝐼𝑅)
(𝐼𝑅)
(𝐼𝑅)
𝔖
𝔖𝔖(𝒾,
𝔖
=
𝔖
=𝔖
=𝔖[𝜇,
=(𝒾)𝑑𝒾
=
:𝔖(𝒾,
𝔖(𝒾,
𝜑)𝑑𝒾
=
𝜑) ∈𝔖
=:∈𝔖(𝒾,
ℛ
𝔖(𝒾,
𝜑)𝑑𝒾
=)∈𝔖, ℛ
: (𝒾)𝑑𝒾
𝔖(𝒾,
𝔖(𝒾,
=
∈,
],Note
([func)𝜑)
, ],𝜑)
(𝐹𝑅)
tion
[𝜇,
𝜈].
tions
of
I-V-Fs.
𝔖
is
-integrable
over
𝜈]
if =
𝔖(𝒾)𝑑𝒾
𝔽([ .,𝜑)
that,
if
Itsense
is a are
partial
Itorder
is 𝔖
a partial
relation.
It∗over
is𝐹𝑅
order
a harmonically
partial
order
It
is ais
relation.
partial
order
relation.
𝔖∗ (𝒾,
𝜑)
Lebesgue-integrable,
then
𝔖
a fuzzy
Aumann-integrable
𝔖∗ (𝒾, 𝜑),
∗ (𝒾,relation.
second
reduces
to
the
classical
function,
see [34].
Proposition
1.
[51]
Let
𝔵,𝜑)
𝔴Lebesgue-integrable,
∈
𝔽 Lebesgue-integrable,
.convex
Then
fuzzy
order
relation
“fuzzy
≼isfunc”aisfuzzy
givenAumann-in
on 𝔽 by
(𝒾,
(𝒾,
(𝒾,
𝜑),
𝔖
𝜑),
𝜑)
𝔖
are
are
then
𝔖
then
is
a
𝔖
Aumann-integra
𝔖
𝔖
tion over
[𝜇,
𝜈].
∗
∗
∗ (𝒾,
Proposition
1.use
[51]
Let
𝔵,use
𝔴s∈mathematical
𝔽0to
.operations
Then
fuzzy
relation
“of≼the
” some
ischaracteristics
onchar
𝔽so
use
mathematical
We
now
operations
We
now
examine
use
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some
toaorder
examine
ofs-convex
the
tocharacteristics
some
operations
examine
togiven
examine
ofoffuzzy
the
∗ (We
𝜑),
are
Lebesgue-integrable,
then
𝔖 mathematical
is
fuzzy
Aumann-integrable
func𝔖∗∗(𝒾,
tion over [𝜇,If𝜈].
S
µ,Proposition
ϕnow
S𝜑)
υ,
ϕnow
with
ϕmathematical
1 ∈and
=
then
harmonically
in
the
(We
)𝔖=
)[𝜇,
tion
over
tion
over
𝜈].
[𝜇,
𝜈].
1.
[51]
Let
𝔵,=
𝔴
𝔽and
. ],𝜑
Then
fuzzy
order
relation
“collection
≼
”F-NV-F
is
given
on
𝔽 the
byco
all
𝜑
∈
for
(0,
all
1],
𝜑
where
∈
(0,
for
1],
all
ℛ
where
𝜑
∈
(0,
for
1],
ℛ
all
where
∈
for
(0,
ℛ
all
1],
𝜑
where
∈
(0,
1],
ℛ
where
ℛ
denotes
the
denotes
collection
the
denotes
collection
of
Riemannian
the
of
denotes
Riemannian
integrable
the
of
denotes
collection
Riemannian
integrable
funcof4
[𝔵]
[𝔴]
([
],
)
([
)
([
],
)
([
],
)
([
],
)
,
,
,
,
,
Symmetry 2022, 14, x FORfor
PEER
REVIEW
𝔵
≼
𝔴
if
only
if,
≤
for
all
𝜑
∈
(0,
1],
numbers.
Let
numbers.
𝔵,
𝔴 Theorem
∈classical
𝔽numbers.
Let
𝔵,harmonical
𝔴
∈∈Let
𝔽ℝ,Let
numbers.
𝔵, 𝔴
∈
𝔽∈ have
Let
𝔵,𝜌𝔴
𝔽
and
then
and
𝜌[𝜇,
ℝ,
and
then
∈∈
have
then
and
we
𝜌 have
∈ ℝ,whose
then we
have
2.𝜌 [49]
:we
𝜈]
⊂
ℝ
→we
𝔽ℝ,
be
a[𝔴]
F-NV-F
define the
tion
over
[𝜇,
𝜈].the
sense
reduces
to
P-convex
function,
see
[36].
[𝔵]
Symmetry 2022, second
14, x FOR
PEER
REVIEW
𝔵ℝ
≼→
𝔴𝔽
if𝔖is
and
only
if,I-V-Fs.
≤
for
all
𝜑
∈𝜑-cuts
(0,
1],
Theorem
2. [49]
Let
𝔖
: [𝜇,
𝜈]is
⊂ 𝐹𝑅
be
a[𝜇,
F-NV-F
whose
𝜑-cuts
define
the
family
of
I-V-Fs
(𝐹𝑅)
(𝒾)𝑑𝒾
(𝐹𝑅)
(𝒾)𝑑𝒾
(𝐹𝑅)
(𝒾)𝑑𝒾
(𝐹𝑅)
tions
of
I-V-Fs.
tions
of
𝔖
I-V-Fs.
is
tions
𝐹𝑅
-integrable
𝔖
of
I-V-Fs.
tions
-integrable
over
𝔖
of
I-V-Fs.
tions
𝐹𝑅
𝜈]
-integrable
over
of
if
𝔖
[𝜇,
is
𝜈]
𝐹𝑅
over
-integrable
if
𝔖
is
[𝜇,
𝐹𝑅
𝜈]
-integrable
𝔖
if
over
[𝜇,
𝜈]
over
𝔖
if
[𝜇,
𝜈]𝔽
∈
𝔽
.
Note
∈
𝔽
that,
.
Note
if
∈
th
∗(0,
[𝔵]
[𝔴]=
Symmetry 2022,
14, x FOR 2.
PEER
≼aℝ
𝔴F-NV-F
if𝒦
andare
only
if,𝜑-cuts
≤
for
all𝜑),
𝜑
∈I-V-Fs
1],
Theorem
[49]REVIEW
Let It𝔖is
: [𝜇,
𝜈] ⊂Theorem
be
whose
define
the
family
of
(𝒾)
[𝔖
(𝒾,
(𝒾,
𝔖
𝜑)]
for
all+(5)
𝒾[𝔴]
∈4 [o
:→
[𝜇,𝔽[49]
𝜈]𝔵relation.
⊂
→
given
by
𝔖
𝔖ℝ order
a partial
∗whose
[𝔵+
[𝔵+
[𝔵+
[𝔵+
[𝔵]
[𝔴]
[𝔵]
[𝔴]
[𝔵]
[𝔴]
[𝔵]
𝔴]
=
+
𝔴]
=
,
𝔴]
+
=
,
+
𝔴]
,
=
∗
Theorem
2.
2.
Let
[49]
𝔖
:
Let
[𝜇,
𝜈]
𝔖
:
⊂
[𝜇,
ℝ
𝜈]
→
⊂
𝔽
ℝ
→
𝔽
be
a
F-NV-F
be
a
F-NV-F
whose
𝜑-cuts
𝜑-cuts
define
define
the
family
the
∗
∗
∗
∗
∗
(𝒾)
(𝒾,
(𝒾,
=
𝜑),
𝔖(𝒾,
all 𝔖𝒾 Aumann-integrable
∈
[𝜇,
𝜈] and
for
: [𝜇,𝔖𝜈]
ℝis
→aare
𝒦
given
by𝔖𝜑)
It𝜑)
partial
relation.
(𝒾,
(𝒾,
(𝒾,
(𝒾,
(𝒾,
(𝒾,𝔽
𝜑),
𝜑),
𝔖
Lebesgue-integrable,
𝜑)
𝜑),
are
𝔖Lebesgue-integrable,
are
𝜑),
𝔖
Lebesgue-integrable,
𝔖
𝜑),
are
is
𝔖
then
awhose
Lebesgue-integrable,
fuzzy
𝜑)
𝔖𝜑)]
is
are
Aumann-integrable
afor
then
Lebesgue-integrable,
fuzzy
isthe
a family
then
fuzzy
𝔖
funcAumann
isthen
aallof
fuzf
𝔖∗ 𝔖
𝔖(𝒾,
𝔖
𝔖[𝔖
∗𝜑)
∗⊂
∗are
∗𝔖
∗ (𝒾,
∗then
Theorem
[49]
Let
𝔖
: order
[𝜇,
𝜈]
⊂
→
be
a(𝒾,
F-NV-F
𝜑-cuts
define
of𝜑)
I-V-Fs
We
now
mathematical
operations
to
examine
some
of the
characteristics
(𝒾)
[𝔖ℝ
(𝒾,
=
𝜑),
𝔖
𝜑)]
for
all
𝒾 (𝒾,
∈
[𝜇,
𝜈]
and
for
all
→ 𝒦It is
are
given
by
𝔖
𝔖 : [𝜇, 𝜈] ⊂ ℝ
a2.𝔖
partial
order
relation.
and
𝔖
𝜑
∈:use
(0,
1].
Then
𝔖given
isare
𝐹𝑅-integrable
over
[𝜇,
𝜈]
ifsome
only
if∗𝜑)]
𝔖
∗and
∗ (𝒾,
∗ (𝒾,
∗ (𝒾,
(𝒾)
(𝒾)
[𝔖
[𝔖
(𝒾,
(𝒾,
=
=
𝜑),
𝔖
𝜑),
𝜑)]
𝔖
for
all
for
𝒾
∈
all
[𝜇,
𝒾
𝜈]
∈
[𝜇
a
:
[𝜇,
𝜈]
⊂
[𝜇,
ℝ
𝜈]
→
⊂
𝒦
ℝ
→
are
𝒦
given
by
𝔖
by
𝔖
𝔖
We
now
use
mathematical
operations
to
examine
of
the
characteristics
3. Fuzzy
Hermite–Hadamard
Inequalities
tion
over
tion
[𝜇,
𝜈].
over
[𝜇,
tion
𝜈].
over
[𝜇,
tion
𝜈].
over
tion
[𝜇,
𝜈].
over
[𝜇,
𝜈].
(6)
[𝔵
[𝔵
[𝔵
[𝔵
[𝔵]
[
[𝔵]
[
[𝔵]
[
[𝔵]
[
∗
∗
×
×
×
×
𝔴]
=
×
𝔴]
𝔴]
=
,
𝔴]
×
=
𝔴]
,
×
𝔴]
𝔴]
=
,
×
𝔴]
(𝒾,
𝜑)
𝔖
𝜑)and
both
𝜑𝔖∈:(0,
1].
Then
𝔖
is1.𝔵,
𝐹𝑅-integrable
[𝜇,
𝜈]
if(𝒾,
and
only
if 𝜑)]
𝔖some
∗ (𝒾,
Proposition
[51]
Let
𝔵,
𝔴𝔖∈over
𝔽
.ℝ,
Then
fuzzy
order
relation
“and
≼𝒾 ”∈characteristics
is
given
on
𝔽are
by
∗∗(𝒾,
numbers.
Let
𝔴
∈
𝔽
and
𝜌
∈
then
we
have
(𝒾)
[𝔖
=
𝜑),
𝔖
for
all
[𝜇,
𝜈]
for
all
[𝜇,
𝜈]
⊂
ℝ
→
𝒦
are
given
by
We
now
use
mathematical
operations
to
examine
of
the
of
fu
∗
Proposition
1.𝔵,[51]
Let
𝔵,
𝔴𝜈].
∈only
𝔽∈ .ℝ,
Then
fuzzy
order
relation
“over
≼
”
is
given
on
𝔽
(𝒾,
(𝒾, 𝜑) both
𝜑)
and
𝔖
are
𝜑 ∈ (0, 1]. Then
𝔖
is
𝐹𝑅-integrable
over
[𝜇,
𝜈]
if
and
if
𝔖
[𝜇,
𝑅-integrable
over
[𝜇,
Moreover,
if
𝔖
is
𝐹𝑅-integrable
𝜈],
then
∗
numbers.
Let
𝔴
∈
𝔽
and
𝜌
then
we
have
∗
(𝒾,
(𝒾,
𝜑)
and
𝜑)𝔽
𝔖and
𝜑over
∈ (0,
𝜑of
1].
(0,
Then
1]. 𝔖
Then
is𝔴are
𝔖𝔖
is.isThen
𝐹𝑅-integrable
over
[𝜇,
over
𝜈] relation
[𝜇,
ifThe
and
𝜈]
only
if and
ifH.H
only
𝔖[𝔵]
if
𝔖∗their
Two
different
sorts
inequalities
proven
infuzzy
this
section.
first
is
and
∗ (𝒾,
[𝜇,
Proposition
1.∈𝜈].
[51]
𝔵,
∈
order
“=
≼
given
on
by𝜑𝔖
𝑅-integrable
[𝜇,
Moreover,
if𝐹𝑅-integrable
𝐹𝑅-integrable
over
𝜈],
then
∗ (𝒾,
(7)
[𝜌.
[𝜌.
[𝜌.
[𝔵]
[𝔵]
numbers.
Let
𝔵,
𝔴𝐹𝑅-integrable
∈Let
𝔽𝔵 ≼
and
𝜌𝔽
∈[𝜌.
ℝ,𝔵]
then
we
have
𝔵]
𝔵], 𝜑)
𝔵]
=𝔴]
𝜌.
=
𝜌.
𝜌.
= both
𝜌. [𝔵]
[𝔵]
[𝔴]
𝔴
if
and
only
if,
≤
all
𝜑”whose
∈is
(0,
1],
(𝒾,
and
𝔖
𝜑)
are
𝜑∈
(0,
1].[49]
Then
𝔖
is[𝜇,
over
[𝜇,
𝜈]
if⊂
and
only
if𝜈]
𝔖⊂
[𝔵+
[𝔵]
[𝔴]
=F-NV-F
+
∗ for
[𝜇,
𝑅-integrable
over
[𝜇,
𝜈].
Moreover,
if
𝔖
is
𝐹𝑅-integrable
over
𝜈],
then
Theorem
2.
Theorem
Let
2.
𝔖
Theorem
[49]
:
𝜈]
Let
⊂
𝔖
ℝ
2.
:
[𝜇,
→
[49]
Theorem
𝔽
𝜈]
Let
⊂
ℝ
𝔖
→
:
2.
Theorem
[𝜇,
𝔽
[49]
𝜈]
Let
ℝ
2.
𝔖
→
[49]
:
[𝜇,
𝔽
Let
𝔖
ℝ
:
[𝜇,
→
𝔽
𝜈]
⊂
ℝ
→
𝔽
be
a
F-NV-F
be
whose
a
𝜑-cuts
be
whose
a
define
F-NV-F
𝜑-cuts
the
be
family
define
a
F-NV-F
𝜑-cuts
of
the
I-V-Fs
be
family
whose
a
define
F-NVth
I
different
forms,
and
the
second
H.H.
Fejér
inequalities
for
convex
F-NV-Fs
with
F-NV-Fs
[𝔵]
[𝔴]
𝔵
≼
𝔴
if
and
only
if,
≤
for
all
𝜑
∈
(0,
1],
𝑅-integrable
𝑅-integrable
over [𝜇,
over
𝜈]. [𝜇,
Moreover,
𝜈]. Moreover,
if [𝔵+
𝔖 𝔴]
isif 𝐹𝑅-integrable
𝔖=is[𝔵]𝐹𝑅-integrable
over
𝜈], [𝜇,
then
𝜈], then of𝜑
+ [𝔴]over
, [𝜇,
[𝔵]
[𝔴]
𝔵
≼
𝔴
if
and
only
if,
≤
for
all
𝜑
∈
(0,
1],
[𝜇,
∗
𝑅-integrable
over
[𝜇,
𝜈].
Moreover,
if
𝔖
is
𝐹𝑅-integrable
over
𝜈],
then
∗
∗
∗
∗
[𝔵+
[𝔵]
[𝔴]
𝔴]
=
+
,
as the
integrands.
It
is: [𝜇,
a→partial
relation.
(𝒾,
(𝒾,
(𝑅)
(𝑅)
(𝒾)𝑑𝒾
=
𝔖
𝜑)𝑑𝒾
,𝜈]
𝔖
[𝔖
(𝒾,
(𝒾)
[𝔖
(𝒾)
[𝔖
(𝒾,
(𝒾)
(𝒾,
[𝔖
(𝒾)
[𝔖
(𝒾,
(𝒾,
=𝜈]are
𝜑),
=
𝔖
𝜑),
𝔖
for
all
𝜑)]
𝒾 𝜑),
∈
for
[𝜇,
𝔖=
all
𝒾𝜑)]
∈
[𝜇,
𝜑),
for
for
=𝜈]
𝔖
all
alland
𝒾𝜑
∈𝜑
f
𝒦
𝜈] ⊂are
ℝorder
→
:given
[𝜇,
𝒦𝜈]
by
are
⊂ℝ
𝔖
given
→(𝒾)
:(𝐹𝑅)
[𝜇,
𝒦 by
⊂
𝔖
ℝ
: [𝜇,
→
𝒦
𝜈]
by
are
𝔖(𝒾,
ℝ𝜑)]
→
given
by
are
𝔖∗given
by
𝔖and
𝔖 : [𝜇, 𝜈] ⊂
𝔖ℝ
𝔖
𝔖
𝔖𝔖
[𝔵
[=
×
𝔴]
=⊂(𝒾,
×𝒦
𝔴]
∗given
∗[𝔵]
∗,∗(𝒾,
∗ (𝒾,
∗𝜑)𝑑
It
is
a partial
(𝒾,
(𝑅)
(𝐹𝑅)
(𝒾)𝑑𝒾
=
𝔖1.
𝜑)𝑑𝒾
,[𝔵]
𝔖
𝜑)𝑑𝒾
𝔖number
[𝔵
[⊂
Definition 1.
Definition
[49]
A fuzzy
Definition
1.order
[49]
Arelation.
fuzzy
1.valued
Definition
[49]number
A fuzzy
map
valued
number
[49]
: 𝔴]
𝐾to
⊂examine
A
map
ℝfuzzy
valued
→(𝑅)
𝔖
𝔽 :×number
𝐾is
map
ℝ(𝒾,
𝔖the
valued
:𝔽F-NV-F.
𝐾
⊂isℝcalled
map
→For
𝔽∗𝔖F-NV-F.
:is𝐾 calle
⊂ofℝf
called
each
×
=
𝔴]
,→
∗𝔖
We
now
use
mathematical
operations
some
of
characteristics
∗
∗
It
is
a
partial
order
relation.
(𝒾,
(𝑅)
(𝑅)
(𝐹𝑅)
=(0,
𝔖over
𝜑)𝑑𝒾
,∈
𝔖=
𝜑)𝑑𝒾
𝔖e(𝒾)𝑑𝒾
(𝒾,
(𝒾,
(𝒾,
(𝒾,
(𝒾,
∗𝔖
∗𝜑)
[𝔵𝜈]
[𝔵]
[of
𝜑)
𝔖I-V-Fs
and
𝜑)
both
𝔖
are
𝜑)
and
𝜑𝜑
∈∈
(0,(0,
1]. 𝜑Then
∈whose
(0,
𝔖
1].
is𝜑
Then
𝜑𝐹𝑅-integrable
∈
𝔖
1].
is(𝐹𝑅)
𝐹𝑅-integrable
Then
𝜑-cuts
∈
𝔖
(0,
[𝜇,
is1].
𝐹𝑅-integrable
𝜑
Then
over
if𝔴]
and
(0,
𝔖
1].
only
𝜈]
is
Then
𝐹𝑅-integrable
if(𝒾,
over
and
𝔖
𝔖𝔴]
[𝜇,
only
isdefine
𝜈]
ifsome
ifand
𝔖over
only
[𝜇,
𝜈]
if𝔖over
if(𝒾,
[𝜇,
only
𝜈]
a𝔖
×
=
,⊂𝐹𝑅-integrable
∗(𝐹𝑅)
We
now
use
mathematical
operations
to
examine
of
characteristics
∗𝔖
∗and
(𝒾,
(𝑅)
(𝑅)
(𝑅)
(𝑅)
(𝒾)𝑑𝒾
(𝒾)𝑑𝒾
=[𝜇,
𝔖
𝜑)𝑑𝒾
𝔖
,and
𝜑)𝑑𝒾
, the
𝜑)𝑑𝒾
𝔖
𝜑)𝑑𝒾
𝔖
[𝜌.
[𝔵]
𝜑
(0,
1]
-cuts
𝜑𝔵,whose
∈
(0,
1]
𝜑υand
family
1]
-cuts
whose
the
of
define
I-V-Fs
family
𝜑
-cuts
the
I-V-Fs
of
the
family
𝔖
of
I-V-Fs
:family
𝐾
ℝ
𝒦
:𝜑)
𝐾
⊂
ℝ
given
→
:𝒦
𝐾
⊂
by
are
ℝifbo
→
g
𝔵]
=
𝜌.×
∗∗ (𝒾,
∗ (𝒾,
Theorem
4. 1]
Let
,𝜑
ϕ-cuts
define
the
family
ofare
I-V-Fs
S∈now
∈Let
HFSX
µ,
, the
F𝜑0𝔖
sdefine
([(𝒾)𝑑𝒾
]whose
)(𝑅)
∗→
numbers.
𝔴
∈define
𝔽
𝜌,∈operations
∈(0,
ℝ,whose
then
we
have
(9)
We
use
mathematical
to
examine
some
of
the
characteristics
of
fu
[𝜌.
[𝔵]
(𝒾,
(𝒾,
(𝑅)
(𝐹𝑅)
𝔵]
=
𝜌.
=
𝔖
𝜑)𝑑𝒾
,
𝔖
𝜑)𝑑𝒾
𝔖
∗
+ 𝜈].
∗[𝔖
∗𝔽
∗𝑅-integrable
∗∗
numbers.
Let
𝔵,
𝔴
∈
and
𝜌
∈
ℝ,
then
we
have
[𝜇,
[𝜇,
[𝜇,
[
𝑅-integrable
𝑅-integrable
over
[𝜇,
𝑅-integrable
over
Moreover,
[𝜇,
𝜈].
if
Moreover,
over
𝑅-integrable
𝔖
is
[𝜇,
𝐹𝑅-integrable
𝜈].
if
Moreover,
𝔖
over
is
𝐹𝑅-integrable
[𝜇,
𝜈].
over
if
over
𝔖
Moreover,
is
[𝜇,
𝜈],
𝐹𝑅-integrable
over
𝜈].
then
if
Moreover,
𝔖
𝜈],
is
𝐹𝑅-integrable
over
then
if
𝔖
is
𝜈],
𝐹𝑅-integr
then
over
(9)
(𝒾)
[𝔖
(𝒾,
(𝒾)
(𝒾,
(𝒾,
(𝒾)
[𝔖
(𝒾,
(𝒾,
(𝒾)
(𝒾,
[𝔖
(𝒾,
(𝒾,
𝔖
𝔖
𝔖
𝔖
=
𝜑),
𝔖
=
𝜑)]
𝜑),
=
for
𝔖
all
𝜑)]
𝒾
𝜑),
∈
𝐾.
𝔖
for
=
Here,
all
𝜑)]
𝒾
for
∈
for
𝜑),
𝐾.
each
𝔖
all
Here,
𝒾
𝜑)]
𝜑
∈
∈
for
𝐾.
(0,
for
Here,
each
1]
all
the
𝜑
𝒾
for
∈
∈
end
𝐾.
each
(0,
1]
Here,
point
𝜑
the
∈
real
for
(0,
end
1]
eac
p
= then
S ϕ : [µ, υ] ⊂numbers.
R→
are𝔵,given
S∗ϕ (𝜌 )∈ ℝ,
S=[𝔵]
, ϕ)] for
all
∈ [µ, υ],
[S
)=, 𝜌.
((𝐼𝑅)
[𝜌.
∗∗(we
𝔖 (𝒾)𝑑𝒾
𝔵], ϕhave
∗ KCLet
𝔴∗ ∈ 𝔽 byand
[𝔵]
∗ (. (.
∗ (. (. = (𝐼𝑅)
∗[𝔵+
∗ (.+ [𝔴] ,
𝔖(.𝐾,(𝒾)𝑑𝒾
𝔴]
=
(9)
e ∈
(.
(.
functions
𝔖
functions
functions
𝔖
𝔖
functions
𝔖
,
𝜑),
𝔖
,
𝜑):
,
𝜑),
𝐾
→
𝔖
ℝ
,
are
𝜑):
,
𝜑),
called
𝐾
𝔖
→
ℝ
,
lower
𝜑):
are
called
𝜑),
and
→
ℝ
𝔖
upper
are
lower
,
𝜑):
called
functions
𝐾
and
→
lower
ℝ
upper
are
of
and
called
functions
𝔖.upper
lower
of
functio
and
𝔖
.
ϕ ∈ [0,
1]. If S
F
R
,
then
∗
∗
∗
∗
[𝔵+
[𝔵]
[𝔴]
=
([µ, 1.
υ],[49]
ϕ) =A(𝐼𝑅)
(𝒾)𝑑𝒾 valued 𝔴]
Definition
fuzzy 𝔖
number
map
𝔖(𝐼𝑅)
: 𝐾 ⊂+
ℝ → (𝒾)𝑑𝒾
𝔽, is called
F-NV-F.
For
(𝒾)𝑑𝒾
(𝐼𝑅)
=
=
𝔖
𝔖
Definition 1. [49] A fuzzy number
valued
map
𝔖
:
𝐾
⊂
ℝ
→
𝔽
is
called
F-NV-F.
[𝔵+
[𝔵]
[𝔴]
𝔴]
=
+
,
∗
∗
∗
[𝔵=
[𝔵]
[],ℝ
×
𝔴]
=𝔖
×
𝔴]
, (𝑅)
(𝒾,
(𝒾,
(𝒾,
(𝒾
(𝑅)
(𝑅)
(𝑅)
(𝑅)
(𝑅)
(𝑅)
(𝑅)
(𝑅
(𝐹𝑅)
(𝒾)𝑑𝒾
(𝐹𝑅)
(𝒾)𝑑𝒾
(𝐹𝑅)
(𝒾)𝑑𝒾
(𝐹𝑅)
(𝒾)𝑑𝒾
(𝐹𝑅)
(𝒾)𝑑𝒾
(𝒾)𝑑𝒾
(𝐼𝑅)
𝜑 ∈ (0, 1]
whose
𝜑
the
family
of
I-V-Fs
𝔖
:𝐾
⊂
𝒦
are
give
=
𝔖𝔖
,𝔖
=
𝜑)𝑑𝒾
𝔖→
,𝔖
𝔖ℝ(𝒾,
,→
=
𝔖
𝜑)𝑑𝒾
𝜑)𝑑𝒾
𝔖of(𝒾,
𝔖
, ∗𝜑)
𝔖define
𝔖𝜑)𝑑𝒾
=all
𝔖𝜑)𝑑𝒾
(0,
for
all𝔖A
𝜑
∈ -cuts
(0, 1].
For
𝜑
1],
ℱℛ
denotes
the
collection
all
∗∈
∗:(𝒾,
∗ (𝒾,
Definition
[49]
fuzzy
number
valued
map
𝐾𝔖
⊂
𝔽∗=
is𝜑)𝑑𝒾
F-NV-F.
For
e𝐹
([
[𝔵:2.
[be
×
𝔴]
=
×)fuzzy
𝔴]
,ℝ
𝜑
∈[49]
(0,1.
1]
whose
𝜑Let
-cuts
define
the
family
of
I-V-Fs
:called
𝐾𝐹𝑅-integrable
ℝ
→an
𝒦F-NV-F.
are
be⊂
an
F-NV-F.
an
Then
F-NV-F.
an
Then
integral
F-NV-F.
fuzzy
be
ofThen
𝔖
integral
over
fuzzy
Definition
2.
Definition
Let
Definition
𝔖
2.:
[𝜇,
[49]
𝜈](0,
⊂
2.
ℝ𝔖
Definition
→
[49]
:Z[𝜇,
𝔽υ([𝜈]
Let
𝔖
ℝ
[𝜇,
→
[49]
𝔽
𝜈]e⊂be
Let
ℝ[𝔵]
→
𝔖, :e
𝔽
[𝜇,
𝜈]
⊂
→
𝔽⊂
for all 𝜑
∈(𝒾)
(0,
1].
For
all
𝜑
∈
1],
ℱℛ
denotes
the
collection
of𝔖
all
F- og
∗ (𝒾,
],
)
,
e
e
µυ
[𝔖
(𝒾,
𝔖
=
𝜑),
𝔖
𝜑)]
for
all
𝒾
∈
𝐾.
Here,
for
each
𝜑
∈
(0,
1]
the
end
point
2µυ
S
(
µ
)
+
S
(
υ
)
S
)
(
[𝔵
[𝔵]
[
× family
𝔴]
= of
×all
𝔴]
, :𝐾 ⊂ ℝ →
NV-Fs
over
[𝜇,
𝜈].
𝜑 ∈all(0,
whose
𝜑𝜑),
-cuts
define
the
I-V-Fs
𝔖
𝒦1] the
are(9)
given
∗1], ℱℛ
s1]
−∈
1 e(0,
∗ (𝒾,
for all 𝜑 ∈ (0,
1].
𝜑
the
of𝜌.
𝐹𝑅-integrable
[𝔵]
𝔵]
=(𝐹𝑅)
([
) all
,For
[𝔖
(𝒾,
𝔖
=
𝔖],(𝐹𝑅)
𝜑)]
for
all
𝒾ℱℛ
∈
𝐾.by
Here,
for
each
𝜑the
∈ F(0,
end
2for
Sfor
4
4[𝜌.
(17)
[𝜇,
[𝜇,
(𝐹𝑅)
[𝜇,
(𝒾)𝑑𝒾
[𝜇,𝜑
(𝒾)𝑑𝒾
(𝐹𝑅)
(𝒾)𝑑𝒾
(0,
(0,
𝜈],For
denoted
𝜈],
by
denoted
𝜈],
𝔖
by
denoted
by
𝜈],
𝔖
denoted
𝔖
by
𝔖
, denotes
is
given
levelwise
,d[𝜌.
is
given
, )is
given
by
levelwise
is
given
by
levelwise
byp𝐹
all
𝜑
∈
all
𝜑
1].
1].
For
∈
all
𝜑
∈collection
1],
denotes
denotes
the , collection
collection
of
all𝔖of
𝐹𝑅-int
all
∗(0,
NV-Fs
over
[𝜇,(𝒾)
𝜈].
∗(0,
([ 𝔵]
],levelwise
([
], [𝔵]
) (𝒾)𝑑𝒾
,ℱℛ
,𝜌.
21],
∗∈
=
(.
(.
functions
𝔖
,
𝜑),
𝔖
,
𝜑):
𝐾
→
ℝ
are
called
lower
and
upper
functions
of
.
µ
+
υ
υ
−
µ
1
+
s
(𝒾)
[𝔖
(𝒾,
(𝒾,
µ
𝔖
=
𝜑),
𝔖
𝜑)]
for
all
𝒾
∈
𝐾.
Here,
for
each
𝜑
∈
(0,
1]
the
end
point
∗
(0,
for
all
𝜑
∈
(0,
1].
For
all
𝜑
∈
1],
ℱℛ
denotes
the
collection
of
all
𝐹𝑅-integrable
F∗
NV-Fs over [𝜇, 𝜈].
∗
([→, ℝ
],[𝜌.
)are
(.
(.
functions
𝔖
,
𝜑),
𝔖
,
𝜑):
𝐾
called
lower
and
upper
functions
of
𝔖
.𝔖 r
[𝔵]
𝔵]
=
𝜌.
NV-FsNV-Fs
over ∗[𝜇,
over
𝜈].
𝜈]. =A(𝐼𝑅)
(𝒾)𝑑𝒾
(𝒾)𝑑𝒾
(𝒾)𝑑𝒾 set,
(𝐼𝑅)
(𝐼𝑅)
(𝐼𝑅)
(𝐼𝑅)
𝔖
=[𝜇,
𝔖=(𝒾)𝑑𝒾
𝔖is =
𝔖
=
∗ (. [𝜇,
(0,
=
∞)
said
to
be
a
convex
Definition
3.
[34]
set
𝐾
=
𝜐]
⊂
ℝ
(.
functions
𝔖
,
𝜑),
𝔖
,
𝜑):
𝐾
→
ℝ
are
called
lower
and
upper
functions
of
𝔖
.
NV-Fs over 3.[𝜇,
𝜈].∗A set 𝐾 = [𝜇, 𝜐] ⊂ ℝ = (0, ∞) is said to be a convex set, if, for all 𝒾, 𝒿 ∈
[34]
e ∈ HFSV
Definition
1.𝜉F(𝐹𝑅)
[49]
fuzzy
number
valued
map
𝔖F-NV-F.
:=
𝐾𝔖⊂(𝒾)𝑑𝒾
ℝ
→𝔽
is𝔖ℛ
called
F-NV-F.
For
IfDefinition
S
µ,[𝜇,
υ𝔖𝐾,
s[49]
,Let
then
([=
],(𝒾)𝑑𝒾
)A
an
Then
fuzzy
integral
𝔖
Definition
2.
[49]
𝔖
:∞)
[𝜇,
𝜈]
⊂
ℝ
→(𝐹𝑅)
𝔽valued
[0,
(8)
1],
we
have
(𝒾)𝑑𝒾
(𝒾)𝑑𝒾
(𝐹𝑅)
(𝒾)𝑑𝒾
(𝐹𝑅)
(𝒾)𝑑𝒾
(𝒾)𝑑𝒾
(𝐼𝑅)
(𝐼𝑅)
(𝐼𝑅)
(𝐼𝑅)
0∈,1.ℝ
𝔖
𝔖(𝒾)𝑑𝒾
𝔖
=
𝔖number
=
=
𝔖
=be
𝔖(𝒾,
𝜑)𝑑𝒾
:=
𝔖(𝒾,
𝔖(𝒾,
𝜑)
=𝜑)𝑑𝒾
∈
:𝔖(𝒾,
𝔖(𝒾,
𝜑)
=,∈F-NV-F.
: 𝔖(𝒾,
ℛof
(0,
([
)integral
([𝔖(𝒾
, ],𝜑)𝑑𝒾
,𝜑)
=3.
said
to
be
set,
if,
for
all
𝒾, a𝒿to
∈
Definition 3.
[34]
A
set
𝐾
𝜐]
⊂
Definition
A
fuzzy
map
𝔖F-NV-F.
:is𝐾
⊂
ℝ
→
𝔽
is
called
be
an
Then
fuzzy
of],
Definition
2.
[49]
Let
𝔖
:is
[𝜇,
𝜈]
⊂
ℝ⊂
→ℝa𝜐]
𝔽convex
(0,
(0,
[𝜇,
[𝜇,
[0,
=
∞)
=
∞)
said
is
to
said
be
convex
be
a
convex
set,
if,
set,
for
Definition
Definition
3.
[34]
A
[34]
set
A
𝐾
set
=
𝐾
𝜐]
=
⊂
ℝ
𝐾, 𝜉 ∈ Definition
1],
we
have
𝜑
∈
(0,
1]
whose
𝜑
-cuts
define
the
family
of
I-V-Fs
𝔖
:
𝐾
⊂
ℝ
→
𝒦
are
give
[𝜇,
(𝐹𝑅)
(𝒾)𝑑𝒾
1.
[49]
A
fuzzy
number
valued
map
𝔖
:
𝐾
⊂
ℝ
→
𝔽
is
called
F-NV-F.
For
𝜈],
denoted
by
𝔖
,
is
given
levelwise
by
(0,
(0,
(0,
(0,
(0,
forwe
allhave
𝜑Definition
∈for
(0,[𝜇,
all
1].
𝜑
For
∈2.
(0,
for
all
1].
all
𝜑 Let
∈For
𝜑=∈𝔖
all
1],
(0,
for
𝜑
1].
ℱℛ
∈
all
For
𝜑
1],
all
∈
for
(0,
ℱℛ
𝜑
all
1].
∈
𝜑
For
∈
1],
(0,
all
ℱℛ
1].
𝜑
∈
For
all
1],
𝜑
ℱℛ
∈
1],
ℱℛ
denotes
the
denotes
collection
the
of
collection
denotes
all
𝐹𝑅-integrable
the
of
collection
all
denotes
𝐹𝑅-integra
Fthe
of
den
all
co
be
an
F-NV-F.
Then
fuzzy
integral
of
𝔖
oeg
[49]
:
[𝜇,
𝜈]
⊂
ℝ
→
𝔽
(0,
[𝜇,
=
∞)
is
said
to
be
a
convex
set,
if,
for
all
𝒾,
𝒿
∈
Definition
3.
[34]
A
set
𝐾
𝜐]
⊂
ℝ
𝐾, 𝜉 ∈ [0, 1],
([
],
)
([
],
)
([
],
)
([
],
)
([
],
)
,
,
,
,
,
𝜑
∈
(0,
1]
whose
𝜑
-cuts
define
the
family
of
I-V-Fs
𝔖
:
𝐾
⊂
ℝ
→
𝒦
are
𝒾𝒿
Z
(𝐹𝑅)
𝜈],
denoted
by
𝔖υ(𝒾)𝑑𝒾
,)∈is𝐾.
given
levelwise
by 𝜑 ∈ (0, 1] the end point
∗ (𝒾,
[0,𝜉(𝒾,
[0,
𝐾,1]
𝜉=∈[𝔖
𝐾,
1],
∈𝜑),
we
1],
have
we
have
e
e
e
e
µυ
(𝒾)
𝔖
𝔖
𝜑)]
for
all
𝒾
Here,
for
each
2µυ
S
(
µ
)
+
S
(
υ
)
S
(
∈
𝐾.
𝒾𝒿
𝜑
∈
(0,
whose
𝜑
-cuts
define
the
family
of
I-V-Fs
𝔖
:
𝐾
⊂
ℝ
→
𝒦
are
given
∗
swe
−
1all
∗NV-Fs
NV-Fs
NV-Fs
[𝜇,
over
[𝜇,
𝜈].
over
[𝜇,
over
[𝜇,
𝜈].
[𝜇,
𝜈].
(𝐹𝑅)
(𝒾)𝑑𝒾
for
𝜑[0,
∈
(0,
for
1],
where
∈by
for
(0,
all
ℛ
1],
𝜑,where
∈
(0,
for
1],ℛall
where
𝜑],given
∈) d(0,
1],∈
ℛ
denotes
the
denotes
collection
the
denotes
of
Riemannian
ofdenotes
integrable
the
of
Riemanni
collection
funcintegra
𝜈],
denoted
𝔖
,for
is
levelwise
by
ehave
𝐾, all
𝜉over
∈[𝜇,
1],
([<
],
) 𝜈].
([
([
)𝜉𝒾
([ 𝜉)𝒿
], )collection
(𝒾)
[𝔖
(𝒾,
(𝒾,
, NV-Fs
, ],where
,the
𝔖
=𝜑NV-Fs
𝜑),
𝔖
𝜑)]
all
𝒾ℛover
∈
𝐾.
Here,
for
each
𝜑Riemannian
∈ (0, 1]
the
end p
2𝜈].
S
<
(18)
(1
+collection
−
𝐾.
𝒾𝒿
∗ (.
2 called
∗𝔖
(.𝜑),
𝔖
𝜑),
, 𝜑):
𝐾 𝜉𝒾
→
ℝofis
are
lower
and
upper
functions
of
𝔖(𝒾)𝑑𝒾
.(10)
+,∗𝐹𝑅
υ𝔖-integrable
υI-V-Fs.
− µfor
1[𝜇,+𝐹𝑅
sover
𝒾𝒿isfor
𝒾𝒿
(𝒾)tions
[𝔖𝔖of
(𝒾,
(𝒾,
µ
𝔖functions
=
𝜑)]
all
𝒾
∈
𝐾.
Here,
each
𝜑
∈
(0,
1]
the
end
point
∗µ
(1
+
−
𝜉)𝒿
∗
∈
𝐾.
∗
(10)
(𝐹𝑅)
(𝒾)𝑑𝒾
(𝐹𝑅)
(𝒾)𝑑𝒾
(𝐹𝑅)
(𝐹∈
tions of
I-V-Fs.
is
I-V-Fs.
tions
of
𝔖
is
tions
𝐹𝑅
-integrable
over
𝔖
I-V-Fs.
[𝜇,
𝐹𝑅
𝜈]
-integrable
over
if
𝔖
𝜈]
𝔖
-integrable
if
[𝜇,
𝜈]
if
𝔖
over
[𝜇,
𝜈]
𝔖
∈
𝔽
.
Note
∈
that,
𝔽
.
if
Not
functions
𝜑),
𝔖−(.𝜉)𝒿
,=𝜑):
𝐾 → 𝒾𝒿
ℝ𝔖 are
called
upper
(𝒾)𝑑𝒾
(𝐹𝑅)𝔖∗ (.
(𝐼𝑅)
𝔖,(𝒾)𝑑𝒾
= lower
𝔖(𝒾,
𝜑)𝑑𝒾
𝔖(𝒾,functions
𝜑) ∈ ℛ([ of, if
∈and
𝐾.
∈ :𝐾.
], 𝔖). ,
(1
𝜉𝒾
∗+
(𝒾)𝑑𝒾
(𝐹𝑅)
(𝒾)𝑑𝒾
(𝐼𝑅)
𝔖𝜐]
=
𝔖
=and
𝔖(𝒾,
𝜑)𝑑𝒾
:𝜐]
𝔖(𝒾,
𝜑)
∈
ℛ
(.
(.
(1
(1
(0,
(0,
(0,
(0,
(0,
[𝜇,
[𝜇,
[𝜇,
[𝜇,
[𝜇,
functions
𝔖
,
𝜑),
𝔖
,
𝜑):
𝐾
→
ℝ
are
called
lower
upper
functions
of
𝔖
.
𝜉𝒾
𝜉𝒾
+
−
+
𝜉)𝒿
−
𝜉)𝒿
=
∞)
is
=
said
∞)
to
be
is
said
a
=
convex
to
∞)
be
set,
is
a
said
convex
=
if,
for
to
∞)
be
all
set,
a
is
𝒾,
=
if,
convex
said
𝒿
for
∈
to
all
Definition
Definition
3.
[34]
A
set
3.
Definition
[34]
𝐾
=
A
set
3.
𝐾
Definition
⊂
[34]
=
ℝ
A
𝜐]
set
⊂
Definition
𝐾
3.
ℝ
=
[34]
A
𝜐]
set
⊂
3.
ℝ
[34]
𝐾
=
A
set
𝜐]
𝐾
⊂
=
ℝ
⊂
ℝ
([∞)
],b
∈
𝐾.
,
∗
∗
∗
∗
(10)
∗
(𝒾,(𝐹𝑅)
(𝒾,
(𝒾,
(𝒾,𝜉)𝒿
𝔖 (𝒾,
𝜑),
are
𝔖Lebesgue-integrable,
𝜑)
𝜑),4.
are
𝔖𝔖
Lebesgue-integrable,
𝜑)
are
𝜑),
then
Lebesgue-integrable,
𝔖
𝔖
𝜑)
is
are
athen
fuzzy
Lebesgue-integrable,
𝔖 Aumann-integrable
is
then
a fuzzy
𝔖
is
Aumann-integra
a∈integral
fuzzy
then
func𝔖
Auman
is,𝔖
ase
𝔖∗ (𝒾, 𝜑),Definition
𝔖∗𝜑)
𝔖
𝔖(𝐼𝑅)
∗𝔖
∗ (𝒾,
(𝒾)𝑑𝒾
(𝒾)𝑑𝒾
=
𝔖
=
𝔖(𝒾,
𝜑)𝑑𝒾
: 𝔖(𝒾,
𝜑)
ℛ
𝜉𝒾
+ℝ(1
−
[𝜇,
([ , ], of
)func
be1],
an
F-NV-F.
fuzzy
2.we
[49]
Let
:(𝒾,
[𝜇,
𝜈]
⊂
→
𝔽
Definition
[34]
The
𝔖:
𝜐]
→
ℝ
is
called
aThen
harmonically
convex
e 1],
Proof.
Let
S
∈𝐾,HFSX
µ,
υ
,
F
,
s
.
Then,
by
hypothesis,
we
have
([
]
)
[0,
[0,
[0,
[0,
𝐾,tion
𝜉Definition
∈ [0,
we
𝜉
∈
have
1],
𝐾,
𝜉
∈
have
1],
we
𝐾,
𝜉
have
∈
1],
𝐾,
we
𝜉
∈
have
we
have
0
an F-NV-F.
Then
fuzzy
integral
Definition
2.
[49]
: [𝜇,
𝜈] denotes
⊂ℝ
𝔽thebecollection
[𝜇,
[𝜇, 𝜐] if off
4.𝜈].
[34]
𝔖:𝜈].
𝜐]Let
→
is called
harmonically
convex
function
onintegrable
over
tion
[𝜇,
over
[𝜇,ℛℝ𝔖
𝜈].
tion
[𝜇,a→
𝜈].
for
all
𝜑over
∈ The
(0, tion
1],
where
of Riemannian
, ], over
[𝜇,[𝜇,
(𝐹𝑅)
(𝒾)𝑑𝒾
𝜈],
denoted
byis
is)given
levelwise
by Then
be
an
F-NV-F.
fuzzy
of 𝔖 o
Definition
2.
[49]
Let
𝔖(𝐹𝑅)
: [𝜇,a𝔖([𝜈]
⊂
ℝ), , ],→
𝔽
[𝜇,
[𝜇,
Definition 4. [34] The
𝔖:
→∈
ℝ
called
harmonically
convex
function
on of
𝜐]convex
ifintegral
for
all𝜐]Definition
𝜑
1],
where
ℛ
denotes
the
collection
Riemannian
integrab
𝒾𝒿 levelwise
(0,
4.
([(𝒾)𝑑𝒾
[𝜇,
𝜈],
denoted
by
𝔖
,
is
given
by
[𝜇,
[𝜇,
𝒾𝒿
𝒾𝒿
𝒾𝒿
𝒾𝒿
𝒾𝒿 func
Definition
4.
[34]
The
[34]
𝔖:
The
𝜐]
𝔖:
→
ℝ
𝜐]
→
ℝ
is
called
is
called
a
harmonically
a
harmonically
convex
function
o
(𝐹𝑅)
(𝒾)𝑑𝒾
tions
of[34]
I-V-Fs.
𝔖
is𝜐]𝐹𝑅
-integrable
over
[𝜇,
𝜈]µυif
𝔖∈function
∈𝜉𝔖(𝒿),
𝔽
. [𝜇,
Note
th
(1
µυ
+
𝔖
≤
−
𝜉)𝔖(𝒾)
for
alls e4.
𝜑
∈2µυ
(0,The
1],
where
ℛ
denotes
the
collection
of
Riemannian
integrable
fu
𝒾𝒿
([
],
)
,
[𝜇,
∈
𝐾.
∈
𝐾.
𝐾.
∈
𝐾.
[𝜇,
(𝐹𝑅)
(𝒾)𝑑𝒾
Definition
𝔖:
→
ℝ
is
called
a
harmonically
convex
on
𝜐]
if
(10)
𝜈],
denoted
by
𝔖
,
is
given
levelwise
by
e
e
(𝐹𝑅)
(𝒾)𝑑𝒾
e
tions
of
I-V-Fs.
𝔖
is
𝐹𝑅
-integrable
over
[𝜇,
𝜈]
if
𝔖
∈
𝔽
.
Note
(1
2
S
4
S
+
S
.
𝜉𝒾
+
−
𝜉)𝒿
(1
+
𝜉𝔖(𝒿),
𝔖
≤
−
𝜉)𝔖(𝒾)
𝒾𝒿
(1
(1
(1
(1(11)
𝜉𝒾
𝜉𝒾
𝜉𝒾
𝜉𝒾
+
+is
𝜉)𝒿
−
𝜉)𝒿
+the
−that
𝜉)𝒿f
Theorem
[49]
Let
Theorem
: [𝜇,
[49]
𝜈]
⊂𝜉𝒾
ℝ2.
𝔖+
→
[49]
:(1
[𝜇,
Theorem
𝜈]
Let
ℝF-NV-F
: 𝜉)𝒿
[𝜇,
2.
→𝒾𝒿[49]
𝔽
𝜈](1
⊂
ℝ𝜈]
→
𝔖
[𝜇,
𝜈]
⊂
ℝ𝔖→
𝔽+whose
be
be
whose
a−𝜉)𝒿
F-NV-F
whose
a−
define
F-NV-F
𝜑-cuts
the
family
be∈define
a𝔽F-NV-F
𝜑-cuts
I-V-Fs
family
define
whos
∗ (𝒾,
𝒾𝒿
1−
ξ𝜉𝒾
µ
+be
µ𝔖
+
υ𝔖2.
ξµ
1−+
−
ξ⊂)a𝔖
υ−
(𝒾,Theorem
(𝐹𝑅)
(𝒾)𝑑𝒾
(𝔽−
(Let
):𝔽𝜑-cuts
𝜑),I-V-Fs.
are
Lebesgue-integrable,
then
𝔖
aξυ
fuzzy
Aumann-integrable
𝔖∗2.
tions
of
𝔖
is Let
𝐹𝑅
over
[𝜇,
if
.ofNote
𝜉)𝒿
(1
+ 𝜉𝔖(𝒿),
𝔖
≤+-integrable
𝜉)𝔖(𝒾)
(11)
∗𝜑)
(𝒾)𝑑𝒾
(𝐹𝑅)
(𝒾)𝑑𝒾
(𝐼𝑅)
(1
(1
𝔖
=
𝔖
=
𝔖(𝒾,
𝜑)𝑑𝒾
:
𝔖(𝒾,
𝜑)
∈
ℛ
,
(𝒾,
+
𝜉𝔖(𝒿),
+
𝜉𝔖(𝒿),
𝔖
𝔖
≤
−
≤
𝜉)𝔖(𝒾)
−
𝜉)𝔖(𝒾)
𝒾𝒿
𝜑),
𝔖
𝜑)
are
Lebesgue-integrable,
then
𝔖
is
a
fuzzy
Aumann-integrab
𝔖∗ (𝒾, 𝜉𝒾
(1
([
],
)
,
+
−
𝜉)𝒿
∗
∗
∗
[0,
[𝜇,
for
all
𝒾,
𝒿𝔖Lebesgue-integrable,
∈𝒦
𝜐],
∈:𝒦
1],
where
≥
for
𝒾 (𝒾)
∈
𝜐].
If
(11)
is
∗ (𝒾,
(𝒾)
[𝔖
(𝒾,
(𝒾)
[𝔖
(𝒾,
(𝒾,
(𝒾)
[𝔖
(𝒾,
(𝒾,
[𝔖
(𝒾,
(𝒾
=
𝜑),
𝔖
=
𝜑)]
𝜑),
for
=fuzzy
𝔖
all
𝒾𝔖
𝜑)]
∈
𝜑),
[𝜇,
𝔖
𝜈]
=(𝒾,
all
and
𝜑)]
for
𝜑),
[𝜇,
all
𝜈]
𝔖,∗re
𝒾
𝜈]
⊂ 𝜑),
ℝ
→:𝔖[𝜇,
𝒦
𝜈]𝜑)
are
⊂
given
: [𝜇,
→
𝜈]
by
⊂are
𝔖
ℝ𝔖𝜉→
given
[𝜇,
by
𝜈]
are
given
ℝ
→
𝒦
by𝔖(𝒾)
𝔖are
given
by
𝔖 : [𝜇,𝔖
𝔖
𝔖ℝ
(𝒾)𝑑𝒾
(𝐹𝑅)
(𝒾)𝑑𝒾
(𝐼𝑅)
(1
(1
tion
over
[𝜇,
𝜈].
𝔖[𝜇,
=
𝔖
=
𝔖(𝒾,
𝜑)𝑑𝒾
:for
𝔖(𝒾,
𝜑)
∈for
ℛ(11)
𝜉𝒾
𝜉𝒾
+⊂
+
−
𝜉)𝒿
(1−
+
−𝜉)𝒿
𝜉)𝔖(𝒾)
∗𝔖
∗all
∗𝒾
([all
],a
are
then
is
a0𝜉𝔖(𝒿),
Aumann-integrable
fu
[𝜇,
[0,
for all 𝒾,∗ (𝒾,
𝒿for
∈tion
𝜐],
𝜉ϕ ∈
1],
where
𝔖(𝒾)
≥𝔖
0≤
for
all
𝒾 ∈𝔖∗[𝜇,
𝜐].
If
(11)
is
reversed
then,
𝔖
over
[𝜇,
𝜈].
(1
𝜉𝒾
+
−
𝜉)𝒿
(𝒾)𝑑𝒾
(𝐹𝑅)
(𝒾)𝑑𝒾
(𝐼𝑅)
𝔖
=
=
𝔖(𝒾,
𝜑)𝑑𝒾
:
𝔖(𝒾,
𝜑)
∈
ℛ
,
Therefore,
each
∈
0,
1
,
we
have
[
]
([
],
)
[𝜇,
,
is
called
a
harmonically
concave
function
on
𝜐].
∗
∗
[𝜇,
[𝜇,
[𝜇,
[𝜇,
[𝜇,
[𝜇,
[𝜇
[𝜇,
[0,
[𝜇,
4.Then
[34]
The
4.
Definition
𝔖:
[34]
The
𝜐]𝔖
→
𝔖:
4.
ℝ
Definition
[34]
The
ℝ
Definition
𝔖:
4.
[34]
𝜐]
The
→
ℝ4.ais
𝔖:
[34]
The
𝜐]
→
𝔖:
𝜐]
→
ℝ
is𝜐]
a is
harmonically
called
harmonically
called
convex
aℝharmonically
function
convex
is
called
on
function
a𝜈]and
is
harmonical
convex
𝜐]
on
afun
h
for all 𝒾, 𝒿 ∈Definition
𝜉tion
∈Definition
where
𝔖(𝒾)
0
for
all
𝒾is
∈
If𝔖
(11)
reversed
then,
𝔖
[𝜇,
𝜈].
(𝒾,
(𝒾,
(𝒾,
(𝒾,
𝜑)
and
𝜑)
𝜑)
both
𝔖ifare
𝜑)
𝜑
an
𝜑is∈𝜐],
(0,
1].
𝜑1],∈𝜑
(0,
𝔖∈for
is
𝜑
𝐹𝑅-integrable
Then
∈
1].
is
𝐹𝑅-integrable
over
𝜑
∈called
𝔖→
(0,
[𝜇,
1].
𝜈]
𝐹𝑅-integrable
Then
if𝜐].
over
and
only
[𝜇,
𝜈]
ifis
over
𝐹𝑅-integrable
if≥
𝔖
and
[𝜇,
only
𝜈]
if
and
𝔖
only
[𝜇,
if
𝔖
ifcalled
and
only
[𝜇,
[0,
[0,
[𝜇,
[𝜇,
∗0
∗ (𝒾,
∗(11)
[𝜇,
for
all
𝒾,1].
𝒿all
∈
𝒾,concave
𝒿(0,
𝜐],
∈≥
𝜉[𝜇,
∈Then
𝜐],
𝜉1],
∈
where
1],
𝔖(𝒾)
≥
𝔖(𝒾)
0
for
all
for
𝒾
∈all
𝒾over
𝜐].
∈𝔖
If
(11)
𝜐].
If
is
reversed
is
rev
called
a over
harmonically
function
𝜐].the
for
all
(0,
1],
where
ℛ
denotes
collection
of
Riemannian
integrable
([ , ], ) on
all
[𝜇,
0 where
[𝜇,
[0,
[𝜇,
[𝜇,
for all 𝒾,
𝒿s∈
𝜐],
𝜉
1],
where
𝔖(𝒾)
≥
for
all
𝒾
∈
𝜐].
If
(11)
is
reversed
then,
𝔖
is called a harmonically
concave
function
on
𝜐].
Theorem
2.
[49]
Let
𝔖
:
𝜈]
⊂
ℝ
→
𝔽
be
a
F-NV-F
whose
𝜑-cuts
define
the
family
of
Ifor
𝜑
∈
(0,
ℛ
denotes
the
collection
of
Riemannian
integra
([
],
)
,
2µυ
µυ
µυ
𝒾𝒿
𝒾𝒿
𝒾𝒿
𝒾𝒿
[𝜇,
[𝜇,
is
called
is
a1],
harmonically
aS
harmonically
concave
concave
function
function
𝜐].(𝐹𝑅)
𝜐].
[𝜇,
[𝜇,
[𝜇,
𝑅-integrable
𝑅-integrable
over
[𝜇,
𝜈].
Moreover,
over
[𝜇,
𝜈].
if:over
Moreover,
𝔖
𝑅-integrable
is[𝜇,
𝐹𝑅-integrable
𝜈].
if
Moreover,
𝔖
over
is
[𝜇,
𝐹𝑅-integrable
over
ifF-NV-F
𝜈].
𝔖
Moreover,
is
𝜈],
then
over
if𝜑-cuts
𝔖
is𝜈],over
𝐹𝑅-integrable
then
𝜈],family
then
ov
Theorem
2.
[49]
Let
𝔖
[𝜇,
𝜈]
⊂
ℝ
→
𝔽S𝜉)𝔖(𝒾)
be≤
aon
whose
define
(𝒾)𝑑𝒾
tions
of
I-V-Fs.
𝔖
is
over
[𝜇,
𝜈]
ifon
𝔖𝜉)𝔖(𝒾)
∈
𝔽
.the
Note
th
2aall
S
ϕon
+[𝜇,
,𝐹𝑅-integrable
ϕof(1
,𝒾𝒿
,called
ϕ𝑅-integrable
≤
for
∈
(0,
where
ℛ
collection
Riemannian
integrable
fu
∗𝜑
∗ 𝐹𝑅
∗the
([-integrable
)υ ,denotes
, −],ξ-integrable
(1
(1
(1
(1
µ
+
υ
+
𝜉𝔖(𝒿),
+
𝜉𝔖(𝒿),
+
𝜉𝔖(𝒿),
+
𝔖
𝔖
≤
𝔖
−
−
𝔖
𝜉)𝔖(𝒾)
≤
𝔖
−
≤
−
𝜉)𝔖(𝒾)
≤
(11)
ξµ
+(
1
1
−
ξ
µ
+
ξυ
)
(
)
is called
harmonically
concave
function
𝜐].
∗
(𝐹𝑅)
(𝒾)𝑑𝒾
tions
of
I-V-Fs.
𝔖
is
𝐹𝑅
over
[𝜇,
𝜈]
if
𝔖
∈
𝔽
.
Note
2.⊂∗[49]
𝒦 𝔖
(1
be𝜉)𝒿
(1
(1 −for
Theorem
: [𝜇,
𝜈] −
⊂𝜉)𝒿
ℝby+
→𝔖
𝔽 (𝒾)
a𝜉𝒾F-NV-F
whose
define
family
of I-V
[𝔖
(𝒾,−𝜑),
=
𝔖𝜉𝒾(𝒾,
𝒾the
∈ [𝜇,
𝜈] and
f−
ℝ →Let
are
given
𝔖 : [𝜇, 𝜈]
(1
(1
𝜉𝒾
𝜉𝒾
𝜉𝒾all
+
−
+
+𝜑-cuts
𝜉)𝒿
+∈
∗𝜑)]
2µυ
µυ
sS
∗: [𝜇,
∗𝐹𝑅are
∗[𝜇,
(𝐹𝑅)
(𝒾)𝑑𝒾
(𝒾)
(𝒾,
(𝒾,
𝜑),
𝔖
are
Lebesgue-integrable,
then
𝔖
is
a, 𝔖
fuzzy
Aumann-integrable
𝔖∗2(𝒾,
tions
of
I-V-Fs.
is
-integrable
over
𝜈] ∗µυ
if𝜉)𝒿
𝔖𝜑)]
𝔽−𝒾. 𝜉)𝒿
Note
tha
=∗ [𝔖
𝜑),
for
all
∈
[𝜇,
𝜈]
an
𝜈](𝒾,
⊂∗𝜑)
ℝ𝔖→
𝒦
given
by
𝔖
𝔖
,
ϕ
≤
S
,
ϕ
+
S
ϕ
.
∗ξυ
(𝒾,
µ
+
υ𝔖
∗𝜈]
𝜑),
are
Lebesgue-integrable,
then
𝔖
a fuzzy
Aumann-integra
𝔖
ξµ
+(by
1− ξ 𝔖
−and
ξ )µ𝔖+
)υ (𝒾)
[𝔖𝜈]
(𝒾,([0,
(𝒾, is
∗ (𝒾,
=all
𝜑),
𝜑)]
for
𝒾0and
∈for
[𝜇,
and
forr
:∈[𝜇,
[𝜇,
𝜈]
⊂
ℝ
→
𝒦
are
given
(𝒾,
(𝒾,
𝜑)
𝔖
bo
(0,
1].
Then
𝔖𝔖𝜑)
is∈
𝐹𝑅-integrable
[𝜇,
if1[𝜇,
only
if
∗𝔖
∗ (𝒾,
∗[𝜇,
tion
over
[𝜇,
𝜈].
[0,
[𝜇,
[0,
[𝜇,
[0,
[𝜇,
[𝜇,
∗𝜑)
for all 𝒾, 𝔖
𝒿𝔖𝜑
for
∈
all
𝜐],
𝜉𝒿∗(𝐹𝑅)
∈
∈
for
1],
all
𝜐],
𝒾,
𝜉where
𝒿Lebesgue-integrable,
∈
for
1],
𝔖(𝒾)
𝜐],
where
𝜉 (𝑅)
𝒾,
≥
∈𝒿[0,
0for
∈over
𝔖(𝒾)
for
1],
𝜐],
where
𝒾,
≥𝜉then
𝒿𝒾0∗[𝜇,
∈∈
for
𝔖(𝒾)
1],
all
𝜐],
𝜐].
𝜉𝔖
≥
where
𝒾If
∈
0fuzzy
(11)
for
1],
𝜐].
𝔖(𝒾)
reversed
where
Ifall
𝒾𝜑)𝑑𝒾
(11)
≥∈
𝜐].
then,
reversed
≥𝔖(11)
𝒾𝔖
0
∈for
is
the
(𝒾,
(𝒾,
(𝒾,
(𝒾,
(𝒾,
(𝑅)
(𝑅)
(𝑅)
(𝑅)
(𝑅)
(𝑅)
(𝒾)𝑑𝒾
(𝐹𝑅)
(𝒾)𝑑𝒾
(𝐹𝑅)
(𝒾)𝑑𝒾
(𝐹𝑅)
(𝒾)𝑑𝒾
=all
𝔖all
𝜑)𝑑𝒾
,𝔖=
𝔖
𝜑)𝑑𝒾
𝔖
𝔖
,is
𝜑)𝑑𝒾
=
𝔖is
,𝔖(𝒾)
𝔖all
𝜑)𝑑𝒾
𝜑)𝑑𝒾
𝔖
,
𝔖
𝔖
(𝒾,
𝜑),
𝜑)
are
is
a∈
Aumann-integrable
fu
(𝒾,
∗=
∗ and
∗ (𝒾,
∗If
𝜑)
∈𝒾,𝔖
(0,
1].
Then
is[𝜇,
𝐹𝑅-integrable
over
𝜈]
if
only
if∗all
𝔖
∗ (𝒾,𝜑
∗ (𝒾, 𝜑) and
tion
over
[𝜇,
𝜈]. 𝔖
∗
(𝒾,
(𝒾,
𝜑)
and
𝔖
𝜑)
both
∈
(0,
1].
Then
𝔖
is
𝐹𝑅-integrable
over
[𝜇,
𝜈]
if
and
only
if
𝔖
[𝜇,
[𝜇,
[𝜇,
[𝜇,
[𝜇,
[𝜇,
isThen
called 𝜑
a
is
harmonically
called
a
harmonically
is
called
concave
a
harmonically
function
concave
is
called
on
function
a
is
harmonically
concave
called
𝜐].
on
a
function
harmonically
𝜐].
concave
on
function
concave
𝜐].
on
function
𝜐].
on
𝜐
𝑅-integrable
over
[𝜇,
𝜈].
Moreover,
if
𝔖
is
𝐹𝑅-integrable
over
𝜈],
then
∗
tion over [𝜇, 𝜈].
(9)
𝑅-integrable over [𝜇, 𝜈]. Moreover, if 𝔖 is 𝐹𝑅-integrable over [𝜇, 𝜈], then
Theorem
2.
Let
𝔖Moreover,
: [𝜇,
be a RF-NV-F
𝜑-cuts
define the family of I
[49][𝜇,
𝜈] ⊂ ℝ
whose
then
[𝜇,
𝑅-integrable
over
if →
𝔖 𝔽is 𝐹𝑅-integrable
overwhose
𝜈],𝜑-cuts
R
R𝜈].
1
1Let 𝔖: [𝜇,µυ𝜈] ⊂ ℝ → 𝔽 be
1a F-NV-Fµυ
2µυ
Theorem
2.
[49]
define
the
family
s
(𝒾)𝑑𝒾
(𝒾)𝑑𝒾
(𝒾)𝑑𝒾
(𝐼𝑅)
(𝐼𝑅)
(𝐼𝑅)
(𝐼𝑅)∈ [𝜇,
𝔖=
𝔖
=
=
𝔖𝜈](𝒾)𝑑𝒾
2 0Theorem
S
, 2.ϕ
dξ→Let
≤𝒦 0𝔖S
, ϕ(𝒾)
dξ
+
S∗𝜑),
,𝔖ϕ define
dξ,
∗ given
∗ (𝒾,𝒾the
µ+υ𝜈]
: [𝜇,
𝜈]
⊂=
a=
F-NV-F
whose
𝜑-cuts
family
of I-V
0𝔖
ξµ(𝒾)𝑑𝒾
+(
1ℝ
−
ξ→
−∗ξ (𝒾,
ξυ for
[𝔖
)υ𝔖𝔽
(1𝔖
),µ+
𝜑)]
and
f
⊂[49]
ℝ
are
by
𝔖∗ : [𝜇,
(𝑅)
(𝐹𝑅)
∗ (𝒾,
= 𝔖be
𝜑)𝑑𝒾
𝜑)𝑑𝒾
𝔖
are
𝔖all
∗(𝑅)
∗ (𝒾,
R
R
R
∗
(𝒾)
[𝔖
(𝒾,
(𝒾,
=
𝜑),
𝔖
𝜑)]
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ifdenotes
𝔖ℱℛ
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all
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all
1].
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for
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1].
𝜑 all
∈
For
1],
∈) (0,
𝜑
ℱℛ
∈1].([
For
ℱℛ
𝜑([ ∈, only
1],
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all
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𝐹𝑅-integrable
the
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F- of
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],all
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], the
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, 1],
, ], of
𝜑) then
and 𝔖 (𝒾, 𝜑) both
𝜑𝑅-integrable
∈ (0, 1]. Then
is 𝜈].
𝐹𝑅-integrable
𝜈] if and onlyover
if 𝔖[𝜇,
over𝔖[𝜇,
Moreover, ifover
𝔖 is[𝜇,𝐹𝑅-integrable
∗ (𝒾,𝜈],
over
[𝜇,
NV-Fs
𝜈]. overNV-Fs
[𝜇, 𝜈].[𝜇,
over𝜈].[𝜇,
NV-Fs
𝜈]. over
[𝜇, is
𝜈].𝐹𝑅-integrable
(𝐼𝑅)
ItNV-Fs
follows
that
=𝔖
𝔖 (𝒾)𝑑𝒾 over
[𝜇, 𝜈], then
𝑅-integrable
over
Moreover,
if
(𝒾)𝑑𝒾[𝜇, 𝜈], then
= (𝐼𝑅) 𝔖 over
𝑅-integrable over [𝜇, 𝜈]. Moreover, if 𝔖 is 𝐹𝑅-integrable
= (𝐼𝑅) 𝔖 (𝒾)𝑑𝒾
∗if,
(0,
(0,
(0,(𝑅)
(0,
[𝜇,set
[𝜇,
=
is[𝜇,
said
=𝔖
to
be
a=
=convex
said
∞)
to⊂be
set,
isℝ𝔖asaid
convex
=for
to all
be
∞)
set,
a𝒾,is
convex
𝒿if,
said
∈fort
Definition 3.
Definition
[34] A setDefinition
3.𝐾[34]
= [𝜇,
A𝜐]
set
3.⊂(𝒾)𝑑𝒾
Definition
[34]
𝐾ℝ(0,
=A
𝜐]∞)
𝐾
⊂(𝑅)
3.=
ℝ[34]
A
𝜐]
⊂∞)
ℝ
𝐾 is
𝜐]
(𝒾,
(𝒾,
𝜑)𝑑𝒾
𝔖
∗set
∗𝜑)𝑑𝒾
for all 𝜑 ∈ (0, 1]. (𝐹𝑅)
For (𝐹𝑅)
all 𝜑
∈𝔖
1], =ℱℛ=
the, collection
𝐹𝑅-integrab
([ , (𝑅)
], ) denotes
(𝒾,
(𝑅) of𝔖all
(𝒾)𝑑𝒾
𝔖denotes
, collection
𝜑)𝑑𝒾
∗ (𝒾, 𝜑)𝑑𝒾
(0,
(0,
1].
For
all
𝜑
∈
1],
ℱℛ
the
of
all
𝐹𝑅-inte
[0,∈1],
[0,
[0,
∗
𝐾, 𝜉 ∈ [0, 1],for
𝐾,
we𝜉allhave
∈𝜑
𝐾,
we
𝜉
∈
have
1],
we
𝐾,
have
𝜉
∈
1],
we
have
([𝔖, ],(𝒾,) 𝜑)𝑑𝒾 , (𝑅)
(𝐹𝑅)
𝔖 (𝒾, 𝜑)𝑑𝒾
𝔖(𝒾)𝑑𝒾
∗
[𝜇,
forNV-Fs
allNV-Fs
𝜑over
∈ (0,
1].𝜈].
For𝜈].all 𝜑
∈ (0, 1], =ℱℛ(𝑅)
([ , ], ) denotes the collection of all 𝐹𝑅-integrabl
over
[𝜇,
𝒾𝒿
𝒾𝒿
𝒾𝒿
𝒾𝒿
NV-Fs over [𝜇, 𝜈].
∈ 𝐾.
∈ 𝐾.
(10)
= (𝐼𝑅)∈ 𝐾.𝔖 (𝒾)𝑑𝒾∈ 𝐾.
==
is𝔖
said
a convex
set,
Definition 3. [34] A set 𝐾 = [𝜇,
(1 ℝ
(1 −
(1 be
𝜉𝒾 𝜐]
𝜉𝒾(0,
𝜉𝒾
𝜉𝒾 + (1
+⊂
− 𝜉)𝒿
+∞)
𝜉)𝒿
+ to
− 𝜉)𝒿
− if,
𝜉)𝒿for all
(𝒾)𝑑𝒾
(𝐼𝑅)
Definition 3. [34] A set 𝐾 = [𝜇, 𝜐] ⊂ ℝ = (0, ∞) is said to be a convex set, if, for
numbers.
Letuse
𝔵, 𝔴mathematical
∈𝔵𝔽≼ 𝔴
and
𝜌∈
ℝ, then
we
We now
operations
to have
examine some
of the characterist
if and
Let relation.
𝔵, 𝔴 ∈ 𝔽
andonly
𝜌 ∈if,ℝ,[𝔵]
then≤we[𝔴]
havefor all 𝜑 ∈ (0, 1],
It is a numbers.
partial
order
numbers.
Let
𝔵,
𝔴
∈
𝔽
and
𝜌
∈
ℝ,
then
we
have
It is a partial order relation. [𝔵+𝔴] = [𝔵] + [𝔴] ,
noworder
use1.mathematical
to𝔵,
examine
some
of ,the
characteris
It is Proposition
aWe
partial
relation.
[51]
Proposition
Let
𝔵, 𝔴 ∈operations
1.
𝔽 [51]
Let
𝔴 ∈order
. Then
fuzzy
. Then
relation
fuzzy
“ ≼order
” the
is given
relati
[𝔵+
[𝔵]
𝔴]
=𝔽examine
+ [𝔴]
We now
use
mathematical
operations
to
some
of
chara
Proposition
1.
[51]
Proposition
Let
𝔵,
𝔴
∈
1.
𝔽
[51]
Let
𝔵,
𝔴
∈
𝔽
.
Then
fuzzy
order
.
Then
relation
fuzzy
“≼
orde
” is
Symmetry 2022, 14, x FOR PEER numbers.
REVIEW
[𝔵+
[𝔵]
[𝔴]
𝔴]
=
+
,
Let
𝔵,
𝔴
∈
𝔽
and
𝜌
∈
ℝ,
then
we
have
We
now
use
mathematical
operations
to
examine
some
of
the
characterist
[𝔵
[𝔵]
[
×
𝔴]
=
×
𝔴]
,
Symmetry 2022, 14, x FOR PEERProposition
REVIEW
[51]
Proposition
Let
𝔵,
𝔴
∈
1.
𝔽
[51]
Let
𝔵,
𝔴
∈
𝔽
.
Then
fuzzy
order
.
Then
relation
fuzzy
“
≼
order
”
is
given
relatio
numbers. 1.
Let
𝔵, 𝔴
∈
𝔽
and
𝜌
∈
ℝ,
then
we
have
[𝔵]
[𝔵]
[𝔴]
[𝔴]
𝔵
≼
𝔴
if
and
only
𝔵
if,
≼
𝔴
if
and
only
if,
≤
for
all
𝜑
≤
∈
(0,
1],
fo
[𝔵
[𝔵]
[
×
𝔴]
=
×
𝔴]
,
Symmetry 2022, 14, x FOR PEER REVIEW
numbers. Let 𝔵, 𝔴 ∈ 𝔽 and 𝜌 𝔵∈ ≼
ℝ,[𝔵
then
we=only
have
[𝔵] 𝜑≤∈ [(
𝔴[𝔵+
and
≼
if and
onlyfor
if, all
≤
[𝔵]
[[𝔵]
×if𝔴]
×+𝔴
𝔴]
,, [𝔴]
[𝔵]if𝔵𝜌.if,
[𝔴]
𝔴]
=𝔴
[𝜌.
[𝔵]
[𝔵]
[𝔵]
[𝔴]
[𝔴]
𝔵]
=
𝔵
≼
𝔴
if
and
only
𝔵
if,
≼
and
only
if,
≤
for
all
𝜑
≤
∈
(0,
1],
fo
[𝔵+𝔴]
[𝔵]𝜌. [𝔵]
+ [𝔴] , 7 of 18
Symmetry 2022, 14, 1639
It is a partial order
It is
relation.
a partial order relation.
𝔵] =
[𝔵+𝔴]
[𝔵]
[𝔴] ,
=[𝜌.
+=
It is a partial order
It is
relation.
a partial
order
relation.
[𝜌.
[𝔵]
𝔵]
=
𝜌.
[𝔵
[𝔵]
[
×
𝔴]
=
×
𝔴]
,some to
We
noworder
use
mathematical
operations
mathematical
to examine
operations
of
examine
the
so
1.
[51]
Let
𝔵, now
𝔴 ∈order
𝔽use
. Then
fuzzy
order
relation
“≼
” charact
is given
ItProposition
is aProposition
partial
It
relation.
aWe
partial
× 𝔴]
= [𝔵]order
× [operations
𝔴] , some
We now
use
mathematical
use
operations
mathematical
to examine
of”exam
the
1. is
[51]
Let
𝔵,We
𝔴 ∈now
𝔽 relation.
.[𝔵
Then
fuzzy
relation
“ ≼to
is gc
[𝔵
[𝔵]
[operations
×
𝔴]
=
×
𝔴]
, some
numbers.
Let
𝔵,A𝔴numbers.
∈ 𝔽We
Let
𝔵,valued
𝔴mathematical
∈then
𝔽fuzzy
and
𝜌use
∈
ℝ,
we
and
have
𝜌
∈
ℝ,
then
we
have
Symmetry 2022, 14, x FOR PEER REVIEW
4characte
ofF-NV
19som
Definition
1.
[49]
fuzzy
number
map
𝔖
:
𝐾
⊂
ℝ
→
𝔽
is
called
We
now
use
mathematical
now
operations
to
examine
to
of
examine
the
Proposition
1.
[51]
Let
𝔵,
𝔴
∈
𝔽
.
Then
order
relation
“
≼
”
is
given
[𝜌.
[𝔵]
𝔵]
=
𝜌.map
[𝔵]
[𝔴]
𝔵𝔵, ≼
𝔴
and
only
≤we
for
∈is(0,
numbers.
Let A
𝔴
numbers.
∈ 𝔽if number
𝔵,if,
𝔴[𝜌.
∈
𝔽
andLet
𝜌∈
ℝ,
then
and
have
ℝ,
then
have
Symmetry 2022, 14, x FOR PEER REVIEW
41],
of
Definition
1. [49]
fuzzy
valued
𝔖
:𝜌
𝐾 ∈⊂
ℝall
→ 𝔽𝜑 we
called
𝔵]
=⊂ℝ,
𝜌.
[𝔵]
[𝔴]
𝔵≼
𝔴
if
and
only
if,
≤ℝ[𝔵]
all
𝜑𝒦
∈ (0,ar
𝜑 ∈numbers.
(0, 1] whose
-cuts
define
the
family
of
I-V-Fs
𝔖 𝔽:we
𝐾for
ℝ4→of
Definition
1. Let
[49] 𝔵,A𝜑
fuzzy
number
valued
map
𝔖
:
𝐾
→
is⊂have
called
F-NV
𝔴
numbers.
∈
𝔽
Let
𝔵,
𝔴
∈
𝔽
and
𝜌
∈
ℝ,
then
we
and
have
𝜌
∈
then
Symmetry 2022, 14, x FOR PEER REVIEW
19
[𝜌.
[𝔵]
𝔵]
=
𝜌.
[𝔵+
[𝔵+
[𝔵]
[𝔴]
[𝔵]
[𝔴]
𝔴]
=
+
𝔴]
,
=
+
𝜑 ∈ (0,
1] whose
𝜑
-cuts
define
the
family
of
I-V-Fs
𝔖
:
𝐾
⊂
ℝ
→
𝒦
[𝔵]
[𝔴]
𝔵
≼
𝔴
if
and
only
if,
≤
for
all
𝜑
∈
(0,
1],
R υ all
Itsis
a[𝔖
partial
order
[𝔵+
[𝔵+
[𝔵]
[𝔴]
(𝒾,relation.
= [𝔵]end
µυ
=
𝜑),
𝔖∗-cuts
𝜑)]
for
for =
each
𝜑𝔖∈(𝒾)
(0,
1]
whose
𝜑
family
of 𝔴]
I-V-Fs
𝔖 𝜑
:+𝐾∈
⊂(0,𝔴]
ℝ,1]→ the
𝒦
ar
Sthe
ϕ)𝐾. Here,
∗ (𝒾, ∈
2µυ
∗a(𝒾,
∗define
−
1S
It
is
partial
order
relation.
(𝒾)
[𝔖
(𝒾,
(𝒾,
𝔖
=
𝜑),
𝔖
𝜑)]
for
all
𝒾
∈
𝐾.
Here,
for
each
𝜑
∈
(0,
1]
th
,
ϕ
≤
d
2
,
[𝔵+
[𝔴]
[𝔴]
=to𝔖[𝔵]
+
𝔴]
,𝔽, =
+
∗ 1.now
2 [𝔵+𝔴]
Definition
A𝔖∗fuzzy
map
: 𝐾and
⊂
ℝ
→some
is
called
F-NV
∗mathematical
µ[49]
+, order
υ ∗use
υ−number
µ𝐾
µ all
We
examine
of[𝔵]
the
charact
[𝔵
[𝔵]
[𝔵]
[of
×valued
×
𝔴]
×
𝔴]
𝔴]
=
×called
𝔴]𝔖
(.
(.
(𝒾)
(𝒾,
(𝒾,
𝜑),
,𝜑)]
𝜑):
ℝ valued
are
lower
upper
functions
=a [𝔖
for→
𝒾operations
∈[𝔵called
𝐾.
Here,
for
each
𝜑
∈some
(0,
1]
the
end
It isLet
partial
relation.
∗We
Definition
1.𝔖
[49]
A
fuzzy
number
𝔖
:[[𝔵]
𝐾
⊂
ℝ
→
𝔽
is
∗𝔴
∗ (.
Proposition 1.𝔖functions
[51]
𝔵,𝔖
∈𝜑),
𝔽𝔖
. (.
Then
fuzzy
order
relation
“=
≼map
”tolower
isexamine
given
on
𝔽
now
use
mathematical
operations
of
the
cha
[𝔵
[𝔵
[is
[𝔵]
×
×
𝔴]
=
×
𝔴]
𝔴]
,by
=
functions
,
𝜑),
𝔖
,
𝜑):
𝐾
→
ℝ
are
called
and
upper
function
∗
𝜑
∈
(0,
1]
whose
𝜑
-cuts
define
the
family
of
I-V-Fs
𝔖
:
𝐾
⊂
ℝ
→
𝒦
ar
Proposition
1.
[51]
Let
𝔵,
𝔴
∈
𝔽
.
Then
fuzzy
order
relation
“
≼
”
is
given
on
𝔽
by
Definition
1.
[49]
A
fuzzy
number
valued
map
𝔖
:
𝐾
⊂
ℝ
→
𝔽
called
F-NV
∗
numbers.
Let
𝔵,
𝔴
∈
𝔽
and
𝜌
∈
ℝ,
then
we
have
(.
(.
We
now
use
mathematical
operations
to
examine
some
of
the
characte
functions
𝔖
,
𝜑),
𝔖
,
𝜑):
𝐾
→
ℝ
are
called
lower
and
upper
functions
of
[𝔵
[𝔵
[𝔵]
[
[𝔵]
[
×
×
𝔴]
=
×
𝔴]
𝔴]
,
=
×
𝔴]
𝜑
∈
(0,
𝜑
-cuts
define
the
family
of
I-V-Fs
𝔖
:
𝐾
⊂
ℝ
→𝔖4𝒦
∗∈1]𝔽 whose
numbers.
Let
𝔵,
𝔴
∈
𝔽
and
𝜌
∈
ℝ,
then
we
have
Proposition
1.
[51]
Let
𝔵,
𝔴
.
Then
fuzzy
order
relation
“
≼
”
is
given
on
𝔽
by
[𝜌.
[𝜌.
[𝔵]
[𝔵]
∗ (𝒾,
𝔵]
𝔵]
=
𝜌.
=
𝜌.
∗
Symmetry 2022, 14, x FOR PEER REVIEW
R
[𝔵]
[𝔴]
(𝒾)
[𝔖
(𝒾,
µυ
𝔵
≼
𝔴
if
and
only
if,
≤
for
all
𝜑
∈
(0,
1],
(4)
𝔖
=
𝜑),
𝔖
𝜑)]
for
all
𝒾
∈
𝐾.
Here,
for
each
𝜑
∈
(0,
1]
the
end
𝜑 ∈numbers.
(0,s−1]
whose
𝜑
-cuts
the
family
of [𝜌.
I-V-Fs
𝔖Then
:[𝔵]
𝐾⊂
ℝ→
𝒦 = ar
ϕ𝔽) be an
(ℝ,
υ𝜌
2µυ
F-NV-F.
fuzzy
integra
2.=
[49]
Let
𝔖𝔖: [𝜇,
𝜈]
⊂S∈ℝ
→, all
∗define
[𝜌.
𝔵]
=
𝜌.1],
Let
𝔵,
𝔴
∈≤𝔽
and
then
Symmetry 2022, 14, x FOR PEER REVIEWDefinition
(𝒾)
[𝔖
(𝒾,
(𝒾,
[𝔵]
𝔖 1S
for
𝒾→
∈𝔽
𝐾.have
Here,
for
each
𝜑
∈𝔵]fuzzy
(0,
1] 𝜌.in
th
𝔵∗∗≼
and
only
if,
≤
for
𝜑[𝔵]
∈
(0,
(
,∗2.
ϕ
.we
2 Definition
[𝔵+
[𝔵]
[𝔴]
𝔴]
=all
+F-NV-F.
,∈
2⊂[𝔴]
be
an
Then
Let
𝔖𝜑)]
𝜈]
ℝd[𝜌.
∗𝜑),
∗[49]
µ𝔴
+
υ if
υ
−µ𝐾
µ: [𝜇,
[𝜌.
[𝔵]
𝔵]
𝔵]
=
𝜌.
=
𝜌.
(.
(.
Symmetry 2022, 14, x FOR PEER REVIEW
4𝔖
o
(𝒾)
[𝔖
(𝒾,
(𝒾,
functions
𝔖
,
𝜑),
𝔖
,
𝜑):
→
ℝ
are
called
lower
and
upper
functions
of
𝔖
=
𝜑),
𝔖
𝜑)]
for
all
𝒾
∈
𝐾.
Here,
for
each
𝜑
(0,
1]
the
end
[𝔵]
[𝔴]
[𝔵+
[𝔵]
[𝔴]
𝔵
≼
𝔴
if
and
only
if,
≤
for
all
𝜑
∈
(0,
1],
(4)
∗
𝔴]
=
+
,
[𝜇,
(𝐹𝑅)
(𝒾)𝑑𝒾
∗
𝜈],
denoted
by
𝔖
,
is
given
levelwise
by
∗
be called
an F-NV-F.
Then
fuzzy function
integral
Definition
2. [49]
Let
𝔖:𝔖[𝜇,(.𝜈]
⊂ℝ
→
𝔽
It is a partial order
relation.
(.
functions
𝔖
,
𝜑),
,
𝜑):
𝐾
→
ℝ
are
lower
and
upper
∗ ∗ by (𝐹𝑅)
[𝜇, 𝜈],
(𝒾)𝑑𝒾
denoted
𝔖[49]
, is
given
levelwise
by
[𝔵+
[𝔵]
[𝔴]
𝔴]
=
+
,
It
is
a
partial
order
relation.
(.
(.
Definition
1.
[49]
Definition
A
fuzzy
number
1.
A
valued
fuzzy
map
number
𝔖
:
𝐾
valued
⊂
ℝ
→
map
𝔽
𝔖
:
𝐾
⊂
ℝ
is
called
F
functions
𝔖
,
𝜑),
𝔖
,
𝜑):
𝐾
→
ℝ
are
called
lower
and
upper
functions
of
𝔖
[𝔵
[𝔵]
[
×
𝔴]
=
×
𝔴]
,
that is
∗
[𝜇,
(𝐹𝑅)
(𝒾)𝑑𝒾
denoted
by operations
𝔖Definition
, is given
levelwise
by
We now
use𝜈],
mathematical
tofuzzy
examine
some
of
the=characteristics
Definition
1.𝔖
[49]
A
number
1.
[49]
valued
fuzzy
map
number
𝔖:[𝐾𝔴]
valued
⊂fuzzy
→ fuzzy
map
𝔽of fuzz
:
is𝔖ca
It is a partial
relation.
[𝔵
[𝔵]
×A
𝔴]
×
,ℝ of
besome
an
F-NV-F.
Then
Definition
2.
[49]
Let
:. [𝜇,
𝜈]
⊂
ℝexamine
→
𝔽𝜑fuzzy
Weorder
now
use
mathematical
operations
to[49]
of
the
characteristics
𝜑
∈
(0,
1]
whose
𝜑
𝜑
∈
-cuts
(0,
1]
define
whose
the
-cuts
family
define
of
I-V-Fs
the
family
𝔖
of
I-V-Fs
:
𝐾
⊂
ℝ
Proposition
1.
[51]
Let
𝔵,
𝔴
∈
𝔽
Then
fuzzy
order
relation
“
≼
”
is
given
on
𝔽→⊂𝒦
by
Definition
1.
[49]
Definition
A
fuzzy
number
1.
A
valued
map
number
𝔖
:
𝐾
valued
⊂
ℝ
→
map
𝔽
𝔖
:integra
𝐾
ℝ
is
called
F[𝔵
[𝔵]
[
×
𝔴]
=
×
𝔴]
,
be
an
F-NV-F.
Then
fuzzy
in
2.
[49]
Let
𝔖
:
[𝜇,
𝜈]
⊂
ℝ
→
𝔽
numbers.
Let
𝔵,𝔴
∈Definition
𝔽 1.
and
𝜌Let
∈1]
ℝ,
then
we
have
We
now
𝜑
∈
(0,
whose
𝜑
𝜑
∈
-cuts
(0,
1]
define
whose
the
𝜑
-cuts
family
define
of
I-V-Fs
the
family
𝔖
of
:
𝐾
⊂
ℝ
use
mathematical
operations
to
examine
some
of
the
characteristics
of
fuzzy
(𝒾)𝑑𝒾
(𝐹𝑅)
(𝒾)𝑑𝒾
(𝐼𝑅)
[𝜌.
[𝔵]
Proposition
[51]
𝔵,
𝔴
∈
𝔽
.
Then
fuzzy
order
relation
“
≼
”
is
given
on
𝔽
𝔖
=
𝔖
=
𝔖(𝒾,
𝜑)𝑑𝒾
:
𝔖(𝒾,
𝜑)
∈
ℛ
𝔵]
=
𝜌.
Z
Z
([
∗:(𝒾,
∗ ( 𝒾Then
υ=
[𝜇, 𝜑
(𝒾)𝑑𝒾
numbers.
Let
∈ 2µυ
𝔽
and
𝜌(𝐹𝑅)
∈𝔽𝜑
ℝ,
then
we
𝜈],
denoted
by
𝔖
is(𝐼𝑅)
given
levelwise
by
be
an
F-NV-F.
fuzzy
integral
2.[𝔖
[49]
Let
𝔖𝔖
[𝜇,
𝜈]
⊂∗for
ℝυ,have
→
µυ
(𝒾,
(𝒾)
[𝔖
(𝒾,
(𝒾,
𝔖𝔵,1.∈𝔴
𝔖
=
𝜑),
𝔖
=
𝜑)]
𝜑),
all
𝔖𝒾, ∗,is
∈
𝐾.
Here,
for
all
for
each
∈family
𝐾.
𝜑:Here,
∈on
(0,
1]
for
the
eac
S
2µυ Definition
ϕrelation
,𝔖(𝒾,
ϕgiven
(𝜑
)𝜑)]
) 𝜑)𝑑𝒾
(𝒾)𝑑𝒾
(𝐹𝑅)
(𝒾)𝑑𝒾
𝔖
:ℝ
𝔖(𝒾,
𝜑)
[𝜌.
[𝔵]
𝔵]
=
𝜌.
∗𝔽
(0,
1]
∈
(0,
1]
define
whose
the
-cuts
family
define
of
I-V-Fs
the
𝔖
of
I-V-Fs
𝐾∈
⊂
→
𝒦
∗𝔵,
[51]
Let
𝔴
∈𝜑
.-cuts
Then
order
“S
≼
”for
is
𝔽
by
1
∗(𝒾)
∗fuzzy
∗ (𝒾,
(𝐹𝑅)
(𝒾)𝑑𝒾
𝜈],
denoted
by
𝔖=
given
levelwise
by
(𝒾)
[𝔖
(𝒾,
(𝒾)
(𝒾,
[𝔖
(𝒾,
[𝔵]
[𝔴]
𝔖
𝔖
𝜑),
𝔖
=
𝜑)]
for
𝜑),
all
𝔖
𝒾
∈
𝐾.
𝜑)]
Here,
all
for
𝒾
each
𝐾.
𝜑
Here,
∈
(0,
Let , 𝔵,[𝜇,
𝔽[𝜇,
and
∈ϕ=
ℝ,
then
we
have
𝔵𝜌whose
≼
𝔴
if
and
only
if,
≤
for
all
𝜑
∈
(0,
1],
2s−numbers.
S∗ Proposition
ϕ𝔴𝜈],
,∈S
,
≤
d
,
d
.
(𝒾)𝑑𝒾
(𝐹𝑅)
(𝒾)𝑑𝒾
(𝐼𝑅)
𝔖
=
𝔖
=
𝔖(𝒾,
𝜑)𝑑𝒾
:
𝔖(𝒾,
𝜑)
∈
ℛ
∗
∗
I
[𝜌.
[𝔵]
∗
∗
𝔵]
=
𝜌.
([
(5)
[𝔵+
[𝔵]
[𝔴]
2
2
𝔴]
+
,≤
∗if
(𝒾)𝑑𝒾
denoted
by
𝔖
given
levelwise
by
(.is𝜑),
(.called
[𝔵]
[𝔴]
functions
functions
,(𝐹𝑅)
𝜑),
,𝜑)]
𝜑):
𝐾
,all
ℝ
𝜑),
are
𝔖∗∈
,𝜑)]
𝜑):
lower
ℝ all
and
are
upper
functions
𝔵 𝜑),
≼
𝔴
if,
∈ 𝜑(0,
1],
µ+υ
µ=+[𝔖
υ𝔖
υ(.and
−
µ=only
(𝒾)
(𝒾,
(𝒾,
[𝔖
(𝒾,
(𝒾,
µ,→
µ→for
𝔖
𝔖
𝔖𝔖(𝒾)
=
𝐾.
Here,
for
all
for
𝒾each
∈𝜑called
𝐾.
Here,
∈lower
(0, 1]
forand
the
eacou
∗ (.
∗=
∗𝔖
∗𝐾
∗for
[𝔵+
[𝔵]
[𝔴]
𝔴]
,for
(.[𝔴]
(.called
𝔖
𝔖
,only
𝜑),
𝐾𝔖𝒾∗+
→
,ℝ
𝜑),
are
𝔖
, 𝜑):
𝐾lower
→
ℝ
and
are
called
upper
lowe
func
for all
𝜑 ∈functions
(0,
where
ℛfunctions
denotes
the
collection
of
Riemannian
inte(F
𝔵relation.
≼∗1],
𝔴 [49]
if[𝔵+
and
≤
all
𝜑𝐾
∈⊂
(0,
1],
∗ (.
([ 𝔖
],(. [𝔵]
), 𝜑):
,if,
∗
∗
Definition
1.
A
fuzzy
number
valued
map
𝔖
:
ℝ
→
𝔽
is
called
It
is
a
partial
order
(5)
[𝔵]
[𝔴]
𝔴]
=
+
,
forDefinition
all
𝜑𝔖∈
1],
ℛ
denotes
the
collection
of→
Riemannia
(.(0,
(.
(.([number
(.called
functions
functions
𝔖
, 𝜑),
𝔖×where
, 𝜑):
𝐾[𝔵]
,ℝ
𝜑),
𝔖
→
ℝ
and
are
called
upper
lower
functions
Symmetry 2022, 14, x FOR PEER REVIEW It is a partial
)𝔴]
(𝒾)𝑑𝒾
(𝐹𝑅)
(𝒾)𝑑𝒾
(𝐼𝑅)
,×],𝔖
𝔖
=fuzzy
=𝐾lower
𝔖(𝒾,
𝔖(𝒾,
∈and
ℛ([o4u
(6)
[𝔵
[are
∗relation.
∗→
𝔴]
=
,, 𝜑):
1.
A
valued
map
𝔖:𝜑)𝑑𝒾
⊂𝔖:ℝ
𝔽𝜑)
calle
order
(𝐹𝑅)
(𝒾)𝑑𝒾
for
all
∈I-V-Fs.
(0,
1],
where
ℛ
the
of𝐾
Riemannian
integ
tions
of
𝔖
is[49]
𝐹𝑅
-integrable
over
[𝜇,
𝜈]an
if⊂
∈ℝis𝔽→
. 𝒦N
Symmetry 2022, 14, x FOR
PEER REVIEW
Thus,
(𝒾)𝑑𝒾
(𝐹𝑅)
(𝒾)𝑑𝒾
(𝐼𝑅)
𝔖
=denotes
𝔖
=
𝔖(𝒾,
𝜑)𝑑𝒾
: 𝔖(𝒾,
𝜑)
([ :𝔴]
],define
)=
,[𝜇,
(f
[𝔵operations
[𝔵]
×
×
𝔴]
,collection
𝜑order
∈𝜑
(0,
1]
whose
𝜑
-cuts
the
of
I-V-Fs
𝔖𝔽be
: 𝐾an
⊂
F-NV-F.
Then
fuzzy
F-NV-F.
inte
Definition
2.
[49]Definition
Let
𝔖
2.
𝜈]
[49]
⊂
ℝ
Let
→ [𝔽family
𝔖over
:be
[𝜇,
𝜈]𝔖
ℝ⊂
→
𝔽
We
now
use
mathematical
to
examine
some
of
the
characteristics
of
Definition
1.
[49]
A
fuzzy
number
valued
map
:
𝐾
ℝ
→
is
called
It
is
a
partial
relation.
(𝐹𝑅)
(𝒾)𝑑𝒾
tions
of
I-V-Fs.
𝔖
is
𝐹𝑅
-integrable
[𝜇,
𝜈]
if
𝔖
∈F
(𝒾)𝑑𝒾
(𝐹𝑅)
(𝒾)𝑑𝒾
(𝐼𝑅)
𝔖whose
=
𝔖(𝒾,
𝜑)𝑑𝒾
: 𝔖(𝒾,
∈(6)
ℛ
Symmetry 2022, 14, x FOR PEER REVIEW
o
∈
𝜑I-V-Fs.
(0,mathematical
1]𝔖
𝜑=
-cuts
define
the
family
ofan
I-V-Fs
:∈
𝐾
⊂an
ℝ4F-N
→
F-NV-F.
Then
be
fuz
Definition
[49]
Definition
Let
𝔖
:υ[𝜇,
2.
𝜈]
[49]
⊂
ℝ
Let
→
𝔽𝔖
:be
[𝜇,
𝜈]
⊂
ℝ
→𝔖𝔽 𝜑)
We
now
use
operations
to
some
of
the
characteristics
([N
[𝔵is×2.
[𝔵]
[𝔖e𝔴]
Z𝜌.×
𝔴]
=
,examine
∗ (𝒾,
∗[𝜌.
(𝐹𝑅)
(𝒾)𝑑𝒾
tions
of
𝐹𝑅
-integrable
over
[𝜇,
𝜈]
if
𝔖
𝔽
.
(7)
[𝔵]
(𝒾,
𝔵]
=
𝜑),
𝔖
𝜑)
are
Lebesgue-integrable,
then
𝔖
is
a
fuzzy
Aumann-inte
𝔖
µυ
(𝒾)
[𝔖
(𝒾,
(𝒾,
[𝜇,
[𝜇,
(𝐹𝑅)
(𝒾)𝑑𝒾
(𝐹𝑅)
(𝒾)𝑑𝒾
=∈2µυ
𝜑),
𝜑)]
for
all
𝒾given
∈𝔖𝐾.:be
Here,
for
each
∈⊂
(0,
1]
the
numbers.
Let
𝔵,
𝔴
𝔽2.∗ ∗ and
𝜌𝔖operations
∈ ℝ,
then
we
denoted
by
𝜈],
𝔖2.
by
𝔖
,have
isLet
levelwise
, is→
by
given
levelwise
by
S
(→
)family
∗𝜑
∈use
(0,
whose
𝜑Let
-cuts
define
the
of
I-V-Fs
:𝜑
𝐾an
ℝ
→
𝒦
an
F-NV-F.
Then
be
fuzzy
F-NV-F.
integ
Definition
[49]
Definition
𝔖denoted
[𝜇,
𝜈]
[49]
⊂=
ℝ
𝔽
[𝜇,
𝜈]
⊂
ℝ
𝔽𝔖
We
now
mathematical
to
examine
some
of
the
characteristics
of
fu
s𝔖
−
1𝜈],
∗:[𝜌.
e(𝒾,
(𝒾,
(
[𝔵]
for
all
𝜑
∈1]
(0,
1],
where
ℛ
denotes
the
collection
of
Riemannian
inte
𝜑),
𝔖
𝜑)
are
Lebesgue-integrable,
then
𝔖
is
a
fuzzy
Aumann
𝔖
𝔵]
𝜌.
(𝒾)
[𝔖
(𝒾,
(𝒾,
[𝜇,
[𝜇,
(𝐹𝑅)
(𝒾)𝑑𝒾
(𝐹𝑅)
(𝒾)𝑑𝒾
=
𝜑),
𝔖
𝜑)]
for
all
𝒾
∈
𝐾.
Here,
for
each
𝜑
∈
(0,
1]
numbers.
Let
𝔵,
𝔴
∈
𝔽
and
𝜌
∈
ℝ,
then
we
have
𝜈],
denoted
by
𝜈],
denoted
𝔖
by
𝔖
,
is
given
levelwise
,
is
by
given
levelw
FR
d
.
(19)
2
S
4
(
)
([
],
)
,
∗𝔖
∗[𝜇,
∗ (.
2 denotes
∗[𝜌.
for
all
(0,
1],
where
ℛ𝜌.
of
over
𝜈].
(𝒾,
(𝒾,
𝜑),
𝔖=
𝜑)
are
Lebesgue-integrable,
then
𝔖the
iscollection
aisand
fuzzy
𝔖tion
functions
𝔖∈υand
,∗𝜑),
𝔖−
,then
𝜑):
𝐾
ℝ
called
lower
upper
functions
([→
)are
(7)the
[𝔵]
µ[𝔖
+
υ∈
µdenoted
,have
𝔵]
=for
(𝒾,
∗Let
µall
[𝜇,
[𝜇,
(𝐹𝑅)
(𝒾)𝑑𝒾
(𝐹𝑅)
(𝒾)𝑑𝒾
𝔖
𝜑),
𝔖
𝜑)]
𝒾over
∈ 𝐾.
Here,
each
𝜑Aumann-integ
∈ Riemannia
(0,
1]
∗ (.
numbers.
𝔵,
𝔴
∈
𝔽𝜑
𝜌
ℝ,
we
𝜈],
denoted
by
𝜈],
𝔖
by
𝔖
,],→
is[𝔵]
given
levelwise
,(𝐹𝑅)
by
given
levelwise
by
∗ (.
∗ (𝒾,
tion
over
[𝜇,
𝜈].
(𝒾)𝑑𝒾
for
all (𝒾)
𝜑functions
∈I-V-Fs.
(0,
1],
where
ℛ
the
collection
of
Riemannian
tions
of
𝐹𝑅
-integrable
[𝜇,
𝜈]
iffor
𝔖
∈functio
. by
N
[𝔴]
(.
=
+
,
𝔖
𝔖
𝜑):
𝐾
ℝ
are
called
lower
and
upper
Proposition
1.
[51]
Let
𝔵,𝔖
𝔴
∈ ,𝔽𝜑),
.
Then
fuzzy
order
relation
“
≼
”
is
given
on(𝒾)𝑑𝒾
𝔽𝔽integ
([[𝔵+
) denotes
, ],,𝔴]
∗is
∗
(𝐹𝑅)
tion
over
[𝜇,
𝜈].
tions
of
I-V-Fs.
𝔖
is
𝐹𝑅
-integrable
over
[𝜇,
𝜈]
if
𝔖
[𝔵+
[𝔵]
[𝔴]
𝔴]
=
+
,
Proposition
1.
[51]
Let
𝔵,
𝔴
∈
𝔽
.
Then
fuzzy
order
relation
“
≼
”
is
given
on
𝔽∈
(.
(.
functions
𝔖
,
𝜑),
𝔖
,
𝜑):
𝐾
→
ℝ
are
called
lower
and
upper
functions
of
1. tions
[49]
A
fuzzy
number
valued
map
𝔖
:
𝐾
⊂
ℝ
→
𝔽
is
called
F-NV-F.
For
each
∗
∗
(𝒾)𝑑𝒾
(𝒾)𝑑𝒾
(𝐹𝑅)
(𝒾)𝑑𝒾
(𝐹𝑅)
(𝒾)𝑑𝒾
(𝐼𝑅)
(𝐼𝑅)
InDefinition
a similar
way
as
above,
we
have
𝔖
𝔖
=
𝔖
=
=
𝔖(𝒾,
𝔖
𝜑)𝑑𝒾
:
=
𝔖(𝒾,
𝜑)
𝔖(𝒾,
∈
(𝐹𝑅)
(𝒾)𝑑𝒾
of[51]
I-V-Fs.
𝔖
is𝔽Lebesgue-integrable,
𝐹𝑅
-integrable
over
[𝜇,
𝜈], 𝔖
if“isis
𝔖
∈𝔽For
𝔽 by
. eac
N
(𝒾,
𝔖fuzzy
𝜑)
are
then
acalled
Aumann-inte
𝔖1.
[𝔵+
[𝔵]
[𝔴]
𝔴]
=⊂
Proposition
1.𝜑),
Let
𝔵,2.
𝔴
∈
. valued
Then
fuzzy
order
≼
” fuzzy
given
on
[49]
A
map
𝔖
:≤
𝐾
⊂
ℝ
→
𝔽(𝒾)𝑑𝒾
F-NV-F.
∗ (𝒾,
∗number
Symmetry 2022, 14, x FOR PEERDefinition
REVIEW
an
Then
fuzzy
inte
Definition
𝔖
:(𝒾)𝑑𝒾
[𝜇,
𝜈]
ℝ[𝔵]
→+
𝔽relation
(𝒾)𝑑𝒾
(𝐹𝑅)
(𝐹𝑅)
(𝒾)𝑑𝒾
(𝐼𝑅)
(𝐼𝑅)
𝔖
𝔖
=
𝔖
=
𝔖(𝒾,
𝔖define
𝜑)𝑑𝒾
:by
=
𝔖(𝒾
Theorem
[49]
Let
𝔖and
:Let
[𝜇,
𝜈]
⊂
ℝ
→[𝔵]
𝔽=
be
a[𝔴]
F-NV-F
whose
𝜑-cuts
the
fam
[𝔵
[ be
×
𝔴]
×
, all
𝔵𝔖
≼
𝔴[49]
if
only
if,
for
𝜑
∈is
(0,
1],
(𝒾,∗2.
(𝒾,
𝜑),
𝜑)
are
Lebesgue-integrable,
then
𝔖=
is
aare
fuzzy
Aumann
𝔖-cuts
𝜑 ∈PEER
(0,
1]
whose
𝜑
define
the
family
of
I-V-Fs
𝔖
:𝔴]
𝐾
⊂
ℝF-NV-F.
→
𝒦
given
∗Definition
Symmetry 2022, 14, xDefinition
FOR
REVIEW
1.
[49]
A
fuzzy
number
valued
map
𝔖
:
𝐾
⊂
ℝ
→
𝔽
is
called
F-NV-F.
For
each
be
an
F-NV-F.
Then
fuzzy
2.
[49]
Let
𝔖
:
[𝜇,
𝜈]
⊂
ℝ
→
𝔽
Theorem
2.
[49]
Let
𝔖
:
[𝜇,
𝜈]
⊂
ℝ
→
𝔽
be
a
F-NV-F
whose
𝜑-cuts
define
[𝔵
[𝔵]
[
[𝔵]
×
𝔴]
=
[𝔴]
×
𝔴]
,
(𝒾)𝑑𝒾
(𝒾)𝑑𝒾
(𝐹𝑅)
(𝒾)𝑑𝒾
(𝐹𝑅)
(𝒾)𝑑𝒾
(𝐼𝑅)
(𝐼𝑅)
𝔵
≼
𝔴
if
and
only
if,
≤
for
all
𝜑
∈
(0,
1],
𝔖
𝔖
=
𝔖
=
=
𝔖(𝒾,
𝔖
𝜑)𝑑𝒾
:
=
𝔖(𝒾,
𝜑)
𝔖(𝒾,
∈
tion
over
[𝜇,
𝜈].
(𝒾,
(𝒾,
𝜑),
𝔖
𝜑)ZLet
are
Lebesgue-integrable,
then
𝔖: 𝐾
is⊂∗a ℝ
fuzzy
𝔖whose
𝜑 ∈ (0, 1] Theorem
𝜑2.over
-cuts
define
the
family
of
I-V-Fs
𝔖
→ Then
𝒦 Aumann-integ
are
given
bℛ
[𝜇,
(𝐹𝑅)
(𝒾)𝑑𝒾
denoted
by
𝔖𝜈]
,S
given
levelwise
by
∗𝜈],
[49]
:𝒾[𝜇,
𝜈]
ℝ
→
be
a×)[𝔖
F-NV-F
whose
𝜑-cuts
define
fam
Symmetry 2022, 14, x FOR PEER
REVIEW
4𝜈
υall
an
F-NV-F.
fuzzy
integ
Definition
2.
[49]
Let
𝔖
:⊂
[𝜇,
⊂
ℝ
→
𝔽
tion
[𝜇,
𝜈].
e𝔖
e
eis
[𝔵
[𝔵]
[ be
(𝒾)
(𝒾,
e
[𝔵]
×
𝔴]
=𝔽for
𝔴]
,𝜑),
[𝔴]
=
𝔖
𝜑)]
for
all
𝒾 the
∈
[𝜇,
:for
[𝜇,
𝜈]
⊂
ℝ
→
are
given
by
𝔖
𝔖∗𝜑),
𝔵relation.
≼
𝔴𝒦
if
and
only
if,
≤
for
all
𝜑(𝒾,
∈
(0,∗are
1],
(𝒾)
[𝔖
(𝒾,
=
𝔖µυ
𝜑)]
for
∈
𝐾.
Here,
each
∈
(0,
1]
the
end
point
real
S
S
(
µ
)
+
(
υ
(
)
∗𝜑
𝜑𝔖∈ (0,
1]
whose
𝜑
-cuts
define
the
family
of
I-V-Fs
𝔖
:
𝐾
⊂
ℝ
→
𝒦
given
by
∗ (𝒾,
[𝜇,
(𝐹𝑅)
(𝒾)𝑑𝒾
𝜈],
denoted
by
𝔖
,
is
given
levelwise
by
[𝜌.
[𝔵]
all
𝜑
∈
(0,
1],
for
where
all
𝜑
ℛ
∈
(0,
1],
where
ℛ
denotes
the
collection
denotes
of
Riemannian
the
collection
∗
It
is
a
partial
order
𝔵]
=
𝜌.
([ Here,
) for
) 𝜑),
, ], by
(𝒾)
[𝔖
(𝒾,denotes
tion
over
𝜈].
=
𝔖the
𝜑)]
for
all co
𝒾i
𝜈]
𝒦1],
given
𝔖 1],
𝔖∗ 𝔖: [𝜇,
[𝔖
(𝒾,
𝔖 (𝒾) =
𝜑),
𝜑)]
for
all
∈
𝐾.
each
𝜑, ∗∈],(𝒾,
(0,
1]
end
point
re
FR
d 𝒾are
4
. where
(20)of
((𝒾,
) ⊂relation.
∗the
[𝜌.
[𝔵]([𝜑),
for
all
𝜑ℝ
∈→
(0,
for
where
all
𝜑
ℛ
∈
(0,
ℛ
denotes
collection
Riema
the
It
is
a∗𝜑),
partial
order
𝔵]
[𝜇,
(𝐹𝑅)
(𝒾)𝑑𝒾
𝜈],
denoted
by
𝔖
, is
given
levelwise
2 given
([
) 𝜌.
([by
],of) for
∗𝔖
, ],=
,𝜑)]
∗ (𝒾
(𝒾)
[𝔖
(𝒾,
(𝒾,
=
𝔖
all
𝒾
∈
[𝜇,
:
[𝜇,
𝜈]
⊂
ℝ
→
𝒦
are
by
𝔖
𝔖
(.
(.
functions
𝔖
,
,
𝜑):
𝐾
→
ℝ
are
called
lower
and
upper
functions
𝔖
.
υ
−
µ
s
+
1
∗
(𝒾)
[𝔖
(𝒾,
(𝒾,
µ
(𝒾,
𝔖
=
𝜑),
𝔖
𝜑)]
for
all
𝒾
∈
𝐾.
Here,
for
each
𝜑
∈
(0,
1]
the
end
point
real
∗
𝜑)
and
𝔖
𝜑
∈
(0,
1].
Then
𝔖
is
𝐹𝑅-integrable
over
[𝜇,
𝜈]
if
and
only
if
𝔖
We
now
use
mathematical
operations
to
examine
some
of
the
characteristics
of
(𝐹𝑅)
(𝒾)𝑑𝒾
∗relation.
tions
of
tions
isare
of
𝐹𝑅
-integrable
over
𝐹𝑅upper
-integrable
𝜈]
ifdenotes
over
[𝜇,
𝔖
𝜈] the
if∈ (𝐹
𝔽𝜈]
metry
FOR 2022,
PEER14,
REVIEW
x FOR PEER REVIEW
of
19
4, collection
∗ of
[𝜌.
[𝔵]
for
𝜑
∈I-V-Fs.
(0,
1],
for
where
all
𝜑𝐹𝑅-integrable
ℛ
∈I-V-Fs.
(0,,𝔵]→
where
ℛ
denotes
the
of
Riemannian
the
if
It is a∗ partial
order
Theorem
2.
Let
𝔖
:𝔖[𝜇,
⊂
𝔽
beis
a F-NV-F
whose
𝜑-cuts
define
fam
=𝔖
𝜌.
(.[49]
([ℝ
],1],
)4
([[𝜇,
],of
) 19
functions
𝔖We
,1].
𝜑):
𝐾𝔵,
→
ℝ
called
lower
and
𝔖
.collection
(𝒾,
𝜑)
and
𝜑all
∈𝔖
(0,
Then
is𝜈]
over
[𝜇,
𝜈]
iffunctions
and
only
ifcharacteristics
𝔖
∗ (. , 𝜑),
use
mathematical
operations
to
examine
some
of
the
(𝐹𝑅)
(𝒾)𝑑
tions
of
I-V-Fs.
tions
𝔖
is
of
𝐹𝑅
I-V-Fs.
-integrable
𝔖
is
over
𝐹𝑅
-integrable
[𝜇,
𝜈]
if
over
[𝜇,
𝔖
𝜈]
∗𝔖(𝒾,
Proposition
1.
[51]
Let
𝔴
∈
𝔽
.
Then
fuzzy
order
relation
“
≼
”
is
given
on
𝔽
b
Theorem
2.
[49]
Let
𝔖
:
[𝜇,
𝜈]
⊂
ℝ
→
𝔽
be
a
F-NV-F
whose
𝜑-cuts
define
∗now
∗
(𝒾)𝑑𝒾
(𝐹𝑅)
(𝒾)𝑑𝒾
(𝐼𝑅)
𝔖
=
𝔖
=
𝔖(𝒾,
𝜑)𝑑𝒾
:
𝜑)
∈
(.
(.
∗→
∗Then
functionsnumbers.
𝔖We
,Theorem
𝜑),
𝔖
𝐾[51]
ℝ
are
called
and
upper
functions
of
. is
Let
𝔵,, 𝜑):
𝔴
∈
𝔽
and
∈Lebesgue-integrable,
ℝ,
we
have
(𝒾,
(𝒾
𝜑)
and
𝔖
𝜑𝑅-integrable
∈tions
(0,
1].
Then
𝔖[𝜇,
is𝜈].
𝐹𝑅-integrable
over
𝜈]
if(𝒾,
and
only
∗ Proposition
∗over
now
mathematical
operations
to
examine
some
of
the
characteristics
of
∗𝔖
1.𝔵,
Let
𝔵,
𝔴
∈
𝔽valued
.lower
fuzzy
order
relation
“a𝔖
≼fuzzy
”𝜈],
given
𝔽
[𝜇,
(𝒾,
(𝒾,
(𝒾,
(𝒾,
(𝒾)𝑑𝒾
(𝐹𝑅
over
Moreover,
if𝔖
𝔖
is
𝐹𝑅-integrable
then
𝔖
𝜑)
are
𝜑),
𝔖
𝜑)
are
Lebesgue-integrable,
then
𝔖
isifis(𝐹𝑅)
Aumann-i
is
afu
𝔖
of
I-V-Fs.
tions
is
of
𝐹𝑅
I-V-Fs.
-integrable
𝔖
is
𝐹𝑅
-integrable
[𝜇,
𝜈]
if
over
[𝜇,
𝔖
𝜈]
ifon
∈
𝔽
2.
[49]
Let
𝔖
:∗𝜌[𝜇,
𝜈]
⊂then
ℝ
→
𝔽map
be[𝜇,
aover
F-NV-F
whose
𝜑-cuts
define
the
fam
(𝒾)𝑑𝒾
(𝐹𝑅)
(𝒾)𝑑𝒾
(𝐼𝑅)
Definition
1.
[49]
fuzzy
𝔖
:[𝔖
𝐾
ℝLebesgue-integrable,
→
𝔽then
called
F-NV-F.
For
∗use
=
𝔖
=
𝔖(𝒾,
𝜑)𝑑𝒾
:over
𝔖(𝒾,
𝜑
(𝒾)
(𝒾,
=
𝜑),
𝔖
𝜑)]
forthen
all
𝒾𝔖
∈
[𝜇,
𝜈𝔖
:𝔖[𝜇,
𝜈]𝜑),
⊂
ℝ
→
𝒦
are
given
by
∗and
∗we
numbers.
Let
𝔴
∈
𝔽⊂
𝜌
ℝ,
then
have
Combining
(19)
and
(20),
we
have
∗⊂
∗over
be
an
F-NV-F.
Then
fuzzy
integral
of
𝔖
Definition
2. 𝔖
[49]
Let
𝔖
:A
𝜈]
ℝ
→
𝔽.𝔖
𝑅-integrable
over
[𝜇,
𝜈].
ifmap
𝔖
is
𝐹𝑅-integrable
𝜈],
then
(𝒾,
(𝒾,
(𝒾,
(𝒾,
𝜑),
𝔖number
𝜑)
are
Lebesgue-integrable,
𝜑),
𝔖
𝜑)
are
𝔖
is
a:[𝜇,
fuzzy
then
Aum
𝔖
𝔖∈
Proposition
1.
[51]
Let
𝔵,
𝔴
∈
𝔽
Then
fuzzy
order
relation
“
≼
”
is
given
on
𝔽
by
(𝒾)
[𝔖
(𝒾,
(𝒾,
Definition
1.
[49]
A
fuzzy
number
valued
𝔖
:
𝐾
⊂
ℝ
→
𝔽
is
called
F-NV-F.
∗[𝜇,
∗Moreover,
=
𝜑),
𝔖
𝜑)]
for
all
𝒾
:
[𝜇,
𝜈]
⊂
ℝ
→
𝒦
are
given
by
𝔖
𝔖
(𝒾)𝑑𝒾
(𝐹𝑅)
(𝒾)𝑑𝒾
(𝐼𝑅)
𝔖
=
=
𝔖(𝒾,
𝜑)𝑑𝒾
𝔖(𝒾,
𝜑)
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4
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ove
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2.
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over
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if
𝔖
is
𝐹𝑅-integrable
𝜈],
then
𝜑
∈
(0,
1]
whose
𝜑
-cuts
define
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2.
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𝜈]
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where
ℛ
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Z
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𝔴]
=
+
,
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],
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(0,
1].
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𝔖, over
over
𝜈]
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ifthe
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(0,
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eafor
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𝜈],
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tions
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𝔖
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2.
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→
Let
𝔽
𝔖
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ℝ
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𝔖
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𝑅-integrable
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𝜑),
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𝔖
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𝔖
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𝜈],
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1.
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𝔽
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𝔽Auma
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: [𝜇,
𝜈]
2.
⊂
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Let
𝔽(𝒾)
𝔖
:be
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𝜈]
ℝ
→
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F-NV-F
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be
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𝔖
𝔵]
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𝜑),
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𝔴
∈
𝔽,[49]
𝜌:𝔖
∈
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have
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(𝒾)𝑑𝒾
(𝐹𝑅)
(𝐼𝑅)
(𝒾,
(𝒾,
∗⊂
𝔖
=
𝔖
=→
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𝜑)𝑑𝒾
𝔖(𝒾,
𝜑)
∈
,𝒾𝔖
𝜑),
𝔖
𝜑)
Lebesgue-integrable,
then
is
a𝔖ℛ
𝔖
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.given
Then
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“
≼
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=
𝜑),
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𝜑
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𝒦
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are
[𝜇,
𝜈]
given
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ℝ
by
→
𝒦
𝔖
are
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by
𝔖
𝔖
𝔵]
=
𝜌.
∗
∗
numbers.
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𝔵,
𝔴
∈
𝔽
and
𝜌
∈
ℝ,
then
we
have
Hence,
the
required
result.
tion
over
𝜈].
∗ ∗𝔖
(8)
(𝒾)𝑑𝒾
(𝐹𝑅) 𝔖
(𝒾)𝑑𝒾
(𝐼𝑅)
be
an
F-NV-F.
fuzzy
integral
of
Definition
2.
[49]
𝔖
:∈[𝜇,
𝜈]: Then
⊂
ℝ
→
(𝒾,
(𝒾,
=
𝔖
=𝔽
𝔖(𝒾,
𝜑)𝑑𝒾
:∗ (𝒾,
𝔖(𝒾,
𝜑)
∈
ℛ
, 𝔽all
(𝒾,
(𝑅)
(𝑅)
(𝐹𝑅)
𝜑),
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𝜑)
are
Lebesgue-integrable,
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𝜑),
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𝜑),
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𝜈]
ℝLet
→
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→=
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𝔖
𝔵]
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𝜑∗ [51]
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1].
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fuzzy
integral
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2.
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[𝜇,
𝜈]
⊂
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𝔽
𝔵⊂𝔽≼
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and
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𝜑
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𝔖and
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𝔴]
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1].
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𝜑
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is
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𝔖
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⊂
ℝ
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𝔽
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𝔴]
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1],
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Then
fuzzy
.order
Then
fuzzy
“ over
≼([ifrelation
” and
is
given
“1].
≼
on
𝔽over
isthe
given
by
on
𝔽for
by
[𝜇,
(𝐹𝑅)
(𝒾)𝑑𝒾
2022,
x FOR
REVIEW
1.
[49]
Aorder
fuzzy
number
valued
𝔖
: [𝜇,
𝐾𝐹𝑅-integrable
⊂
ℝ
→and
called
F-NV-F.
For
𝜈],
denoted
by
𝔖
,”Then
is
given
levelwise
by
], ∈
)𝜈].
,𝔖
and
𝔖
∈relation
(0,
1].
Then
𝜑
is
(0,
𝐹𝑅-integrable
𝔖
is
𝜈]of
if
over
if.[𝜇,
𝔖
𝜈]
if𝜑)
and
only
[𝔵]
[𝔴]
[𝜇,
𝔵relation.
≼
only
if,𝜈]
all
𝜑𝔽if→
∈only
(0,
1],
Remark
3.for
Ifand
sDefinition
=
from
(17)
we
the
following
inequality,
see
[39]:
𝑅-integrable
[𝜇,
𝑅-integrable
Moreover,
if≤
[𝜇,
𝔖map
𝜈].
is𝐹𝑅-integrable
Moreover,
𝔖is
over
is
𝜈],
then
∗ (𝒾,
Theorem
2.,𝔴
[49]
Let
𝔖
[𝜇,
⊂
ℝmap
→over
𝔽
F-NV-F
whose
𝜑-cuts
define
th
(𝐹𝑅)
(𝒾)𝑑𝒾
[𝔵+
[𝔴]
of
I-V-Fs.
𝔖
isby
over
[𝜇,
𝜈]levelwise
if
𝔖
∈
𝔽
Note
that,
if4ov
It
is
a1,
partial
order
𝔴]
=
+
,Moreover,
numbers.
Letnumbers.
𝔵,2022,
𝔴 ∈ 14,
𝔽 Let
𝔴
𝔽
𝜌for
∈tions
ℝ,
then
we
𝜌
∈
have
ℝ, where
then
we
have
Definition
1.∈𝐹𝑅
[49]
Aacquire
number
valued
𝔖
𝐾
𝔽
is𝐹𝑅-integrable
called
F-NV-F.
all
𝜑
∈
(0,
1],
ℛ
the
collection
of
Riemannian
integrable
func[𝜇,
(𝐹𝑅)
(𝒾)𝑑𝒾
Symmetry
x and
FOR𝔵,PEER
REVIEW
o
𝜈],
denoted
, :is
given
by
[𝜇,
([-integrable
], fuzzy
)𝔖denotes
𝑅-integrable
over
[𝜇,
𝑅-integrable
𝜈].
Moreover,
over
ifbe
[𝜇,
𝔖a:[denotes
𝜈].
is⊂
𝐹𝑅-integrable
if
𝔖.𝒦
over
is
𝐹𝑅-integr
𝜈],
th
[𝔵
[𝔵]
×
𝔴]
=[𝔵]
×
𝔴]
,ℝ(𝒾)𝑑𝒾
(0,
Theorem
2.
[49]
Let
𝔖
:
[𝜇,
𝜈]
⊂
ℝ
→
𝔽
be
a
F-NV-F
whose
𝜑-cuts
defin
for
all
𝜑
(0,
1].
For
all
𝜑
∈
1],
ℱℛ
the
collection
of
all
𝐹𝑅(𝐹𝑅)
tions
of
I-V-Fs.
𝔖
is
𝐹𝑅
-integrable
over
[𝜇,
𝜈]
if
𝔖
∈
𝔽
Note
that,
It
is
a
partial
order
relation.
𝜑
∈
(0,
1]
whose
𝜑
-cuts
define
the
family
of
I-V-Fs
𝔖
:
𝐾
⊂
ℝ
→
are
give
([
],
)
,
Definition
1.
[49]
A fuzzy
number
valued
map
𝔖
:is
𝐾
⊂
ℝ
→
𝔽of
is
called
F-NV-F.
For
eg
[𝔵(0,
[𝔵]
[(𝒾,
×
𝔴]
=ℱℛ
𝔴]
,𝔖𝔽
(𝒾)𝑑𝒾
(𝐼𝑅)
∗characteristics
[𝜇,
∗We
=
𝔖, ×
(0,
𝑅-integrable
over
[𝜇,
𝑅-integrable
𝜈].
Moreover,
over
if
[𝜇,
𝔖
𝜈].
Moreover,
𝐹𝑅-integrable
if
over
is
𝐹𝑅-integrable
𝜈],
then
ove
for
all
𝜑
∈
(0,
1].
For
all
𝜑
∈
1],
denotes
the
collection
of
a
[𝔵]
[𝔵]
[𝔴]
[𝔴]
𝔵 ≼ 𝔴 if and
≼
𝔴
if,
if
and
only
if,
≤
for
all
≤
𝜑
∈
(0,
for
1],
all
𝜑
∈
1],
(4)
(4)
now
use
mathematical
operations
to
examine
some
the
of
fu
𝜑
∈
(0,
1]
whose
𝜑
-cuts
define
the
family
of
I-V-Fs
𝔖
:
𝐾
⊂
ℝ
→
𝒦
are
([
],
)
Theorem
2.
[49]
Let
𝔖
:
[𝜇,
𝜈]
⊂
ℝ
→
𝔽
be
a
F-NV-F
whose
𝜑-cuts
define
the
(𝒾)
[𝔖
(𝒾,
(𝒾,
(𝒾,
(𝐹𝑅)
(𝒾)𝑑𝒾
=
𝜑),
𝔖
𝜑)]
for
all
𝒾
∈
:
[𝜇,
𝜈]
⊂
ℝ
→
𝒦
are
given
by
𝔖
𝔖
Z
𝜑),
𝔖
𝜑)
are
Lebesgue-integrable,
then
𝔖
is
a
fuzzy
Aumann-integrable
func𝔖𝔵only
tions
of
I-V-Fs.
𝔖
is
𝐹𝑅
-integrable
over
[𝜇,
𝜈]
if
𝔖
∈
.
Note
that,
if
It
is
a
partial
order
relation.
∗1].
∗some
∗ [𝔵+𝔴]
(𝒾)𝑑𝒾
𝔖
υall
∗characteristics
∗We
e
eby
e𝔴]
(𝒾)𝑑𝒾
(𝐹𝑅)
(𝒾)𝑑𝒾
(𝐼𝑅)
(0,
NV-Fs
over
[𝜇,
𝜈].+
[𝔵(are
[𝔵]
[(𝐼𝑅)
𝔖
=
=
𝔖(𝒾,
:of
𝔖(𝒾,
𝜑)
∈𝜑)]
ℛ
for
all
𝜑
∈use
(0,
For
all
𝜑
∈operations
1],
ℱℛ
denotes
the
collection
of
𝐹𝑅-i
×𝔖
𝔴]
=
×
,𝜑)𝑑𝒾
e[𝔵]
now
mathematical
to
examine
the
µυ
(5)
(5)
(𝒾)
[𝔖
(𝒾,
(𝒾,
[𝔵+
[𝔵]
[𝔴]
[𝔵]
[𝔴]
𝔖∈𝜑),
=
𝜑),
𝔖
𝜑)]
for
𝒾
∈
𝐾.
Here,
for
each
𝜑
∈
(0,
1]
the
end
=
+
𝔴]
=
,
,
2µυ
S
S
(
µ
)
+
S
(
υ
)
)
([all
],poin
) ,o
([
],=
)(𝒾)
, given
[𝔖
(𝒾,
(𝒾,
,
(𝒾,
(𝒾,
=
𝜑),
𝔖
for
all
:
[𝜇,
𝜈]
⊂
ℝ
→
𝒦
given
𝔖
𝔖
𝔖
𝜑)
are
Lebesgue-integrable,
then
𝔖
is
a
fuzzy
Aumann-integrable
fun
𝔖∗𝜑
(0,
1]
whose
𝜑
-cuts
define
the
family
of
I-V-Fs
𝔖
:
𝐾
⊂
ℝ
→
𝒦
are
∗
[𝜌.
metry
FOR 2022,
PEER14,
REVIEW
x FOR PEER REVIEW
4
of
19
4
of
19
∗
𝔵]
=
𝜌.
∗
∗
(𝒾)𝑑𝒾
(𝐼𝑅)
=
𝔖(𝒾)𝑑𝒾
e 𝜈].
NV-Fs
over
𝜈].
(𝒾)𝑑𝒾
(𝐹𝑅)
(𝒾)𝑑𝒾
(𝐼𝑅)
numbers.
𝔵,
𝔴
∈ Let
𝔽
𝜌)operations
ℝ,
then
have
𝔖
=
𝔖∈
=
𝔖(𝒾,
𝜑)𝑑𝒾
∈(𝒾,
ℛ
∗We
(𝒾)Let
(𝒾,
(𝒾,
𝔖
=
𝜑),
𝔖and
𝜑)]
for
all
𝒾by
𝐾.
Here,
𝜑
∈:(𝑅)
(0,
1]𝜑)
the
end
S
FR
dwe
4examine
(𝒾,
(𝒾,
(𝑅)
(𝑅)
(𝐹𝑅)
(𝒾)𝑑𝒾
(𝐹𝑅)
([𝜇,
([ 𝜑)𝑑
],p
, [𝜇
tion
over
[𝜇,
=
𝔖
𝜑)𝑑𝒾
=only
, 𝔖(𝒾,
𝔖on
𝜑)𝑑𝒾
,
𝔖
𝔖
now
use
mathematical
to
some
of
of
∗4
[𝔖
(𝒾,
(𝒾,
[𝜌.
(𝒾,
∗each
∗𝔖
∗∈
=
𝜑),
𝔖
𝜑)]
all
∈fuz
:[𝔖
[𝜇,
𝜈]
ℝ
𝒦
are
𝔖
𝔖
Proposition
[51]
𝔵,,→
𝔴
∈
𝔽∈→
Then
fuzzy
order
“the
”∗characteristics
is
given
𝔽
by
𝔵]
=
𝜌.for
𝜑),
𝔖over
𝜑)
are
then
𝔖
is
a(𝒾)
fuzzy
Aumann-integrable
func(𝒾,
It is a partialItorder
is a partial
relation.
order𝔖relation.
𝜑)
and
𝜑
∈
1].
Then
𝔖
is
𝐹𝑅-integrable
over
[𝜇,
𝜈]
if.𝜑
and
if
2 then
∗⊂
∗[𝔵]
numbers.
𝔵,𝔖(𝒾)𝑑𝒾
𝔴
𝔽
and
𝜌. ℝ
∈given
ℝ,
we
have
∗ (𝒾,tion
NV-Fs
over
[𝜇,
𝜈].
(.Lebesgue-integrable,
(.
∗for
functions
𝔖
,(0,
𝜑),
𝔖−
𝜑):
𝐾(𝐼𝑅)
are
called
lower
and
upper
functions
of
𝔖
. 𝒾𝜑)
(𝒾)𝑑𝒾
(𝐹𝑅)
(𝒾,
(𝒾a∗
(𝑅)
(𝑅)
(𝐹𝑅)
(𝒾)𝑑𝒾
(𝐹𝑅)
(𝒾)𝑑𝒾
µ[𝔖
+
υ1.
υ
µLet
2relation
𝔖
=
𝔖
=
𝔖(𝒾,
𝜑)𝑑𝒾
:≼
𝔖(𝒾,
𝜑)
ℛ
,𝔖
[𝜇,
𝜈].
=
𝔖
𝜑)𝑑𝒾
=∈𝔖given
,end
𝔖
𝔖
𝔖
(𝒾)
(𝒾,
(𝒾,
µ
𝔖𝔴]
=
𝜑),
𝜑)]
for
all
𝒾
∈
𝐾.
Here,
for
each
∈
(0,
1]
the
∗Let
([(𝑅)
],point
) ∗𝔽
, on
∗≼
∗ (.
Proposition
1.
[51]
𝔵,
𝔴
∈
𝔽
.
Then
fuzzy
order
relation
“
”
is
∗
(𝒾,
𝜑
∈
(0,
1].
Then
𝔖
is
𝐹𝑅-integrable
over
[𝜇,
𝜈]
if
and
only
if
𝔖
[𝜌.
[𝔵]
𝔵]
=
𝜌.
(6)
(6)
[𝔵
[𝔵
[𝔵]
[
[𝔵]
[
∗
×
×
=
×
𝔴]
𝔴]
=
,
×
𝔴]
,
(.
∗ of
functions
𝜑),
𝔖the
,set
𝜑):
𝐾𝜑then
→
ℝ
are
lower
and
upper
functions
𝔖
.,
(0,
numbers.
Letall
𝔵,[51]
𝔴
∈
𝔽
and
𝜌(𝐹𝑅)
∈all.ℝ,
we
for
𝜑𝔖
(0,
1].
For
∈𝔖
1],
ℱℛ
denotes
the
of
all
𝐹𝑅∗to
(𝒾,
(𝒾,
(𝒾,
(𝑅)
(𝑅)
(𝒾)𝑑𝒾
(𝒾)𝑑𝒾
([=
], [𝔴]
)𝔖
,(𝑅)
tion
over
[𝜇,all
𝜈].
=
𝔖
𝜑)𝑑𝒾
,given
𝔖
𝔖
𝜑)𝑑𝒾
𝜑)𝑑
(0,
[𝜇,
∞)
tocollection
be
a𝔖convex
set,
Definition
3.examine
[34]
Aℛ
𝐾
=
𝜐]
⊂have
ℝ
We now use We
mathematical
now use mathematical
operations
to
operations
examine
some
of
characteristics
some
of
the
characteristics
of
fuzzy
of
[𝔵]
∗𝜑)
∗𝜑(.
𝔴]
=called
+
,],ifis∗fuzzy
Proposition
Let
𝔵,, over
𝔴
𝔽
fuzzy
order
relation
“said
≼Riemannian
”=
on
𝔽.then
by
for
𝜑𝔖
∈
(0,
1],
where
denotes
the
collection
integrable
(𝒾,
and
𝔖
𝜑1.
∈
1].
𝔖
is, For
𝐹𝑅-integrable
over
[𝜇,
𝜈]
and
only
ifthe
(0,
all
∈
(0,
1].
all
𝜑
1],
ℱℛ
denotes
ofif,
as
([
],Then
)[𝔵+
[𝜇,
𝜈].
Moreover,
if(𝐹𝑅)
𝔖⊂
is
𝐹𝑅-integrable
over
𝜈],
(.for
∗collection
functions
,(0,
𝜑),
𝔖
𝜑):
𝐾
→
ℝ
are
called
lower
and
upper
functions
of
[𝔵]
[𝔴]
([
)of
,for
𝔵Then
≼
𝔴
if
and
only
if,denotes
≤
all
𝜑
(0,
1],
(0,
[𝜇,
=
∞)
is∈is
said
to[𝜇,
be
a𝔖convex
Definition
3.∈
[34]
A
set
𝐾
=∈𝔴]
𝜐]
ℝ
∗𝑅-integrable
[𝔵+
[𝔵]
[𝔴]
=
+
,
for
all
𝜑
∈
(0,
1],
where
ℛ
the
collection
of
Riemannian
integra
Definition
1.
[49]
A
fuzzy
number
valued
map
𝔖
:
𝐾
⊂
ℝ
→
𝔽
is
called
F-NV-F.
Fo
[𝜇,
(0,
NV-Fs
over
[𝜇,
𝜈].
([
],
)
𝑅-integrable
over
[𝜇,
𝜈].
Moreover,
if
𝔖
is
𝐹𝑅-integrable
over
𝜈],
then
Theorem
2.
[49]
Let
𝔖
:
[𝜇,
𝜈]
⊂
ℝ
→
𝔽
be
a
F-NV-F
whose
𝜑-cuts
define
the
family
of
I-V-Fs
,
for
all
𝜑
∈
(0,
1].
For
all
𝜑
∈
1],
ℱℛ
denotes
the
collection
all
𝐹𝑅-i
[𝔵]
[𝔴]
If
s
≡
0,
then
from
(17)
we
acquire
the
following
inequality,
see
[36]:
[0,
𝔵
≼
𝔴
if
and
only
if,
≤
for
all
𝜑
∈
(0,
1],
([=(7)
], if
)∞)(𝐹𝑅)
𝐾,
𝜉
∈
1],
we
have
, (0,
[𝜇,
numbers. Symmetry
Letnumbers.
𝔵, 𝔴 2022,
∈ 𝔽 14,
Let
𝔵, 𝔴
𝔽 Theorem
and
𝜌 PEER
∈ ℝ,
then
and
we
𝜌
∈
have
ℝ,
then
we
have
is
said
to
be
a
convex
set,
if,
Definition
3.
[34]
A
set
𝐾
=
𝜐]
⊂
ℝ
be
an
F-NV-F.
Then
fuzzy
integral
of
𝔖
Definition
2.
[49]
Let
𝔖
:
[𝜇,
𝜈]
⊂
ℝ
→
𝔽
(7)
[𝜌.
[𝜌.
[𝔵]
[𝔵]
(𝒾)𝑑𝒾
𝔵]
𝔵]
=
𝜌.
=
𝜌.
tions
of
I-V-Fs.
𝔖
is
𝐹𝑅
-integrable
over
[𝜇,
𝜈]
𝔖
∈
𝔽
.
Note
th
[𝔵+
[𝔵]
[𝔴]
x FOR
REVIEW
𝔴]
==
+
,:𝜑-cuts
NV-Fs
over
𝜈].
Definition
A[𝜇,
fuzzy
number
valued
map
𝔖
𝐾,of
⊂Riemannian
ℝ
→over
𝔽 (𝐼𝑅)
is[𝜇,called
F-NV-F
all𝔽
𝜑
∈[49]
(0,relation
1],
where
ℛ
denotes
the
collection
integrable
fuf4
2.
Let𝜉2.1.
:order
[𝜇,
𝜈]
⊂([[𝜇,
ℝ,have
𝔽
be
F-NV-F
whose
define
the
family
of I-V-F
[0,
𝐾,
∈[49]
1],
we
],→only
)Moreover,
𝑅-integrable
𝜈].
if
𝔖be
is
𝐹𝑅-integrable
then
(𝒾)𝑑𝒾
(𝒾)𝑑𝒾
(𝐼𝑅)
[𝔵
[𝔵]
[I-V-Fs
[𝔵]
=
𝔖
=
𝔖𝔽
×
×
𝔵[49]
≼
𝔴
if
and
if,𝔴]
≤
for
all
𝜑
(0,
1],
an
F-NV-F.
Then
fuzzy
integral
of
Definition
Let
𝔖
:given
[𝜇,
𝜈]
⊂
→
𝔽
(𝐹𝑅)
(𝒾)𝑑𝒾
Proposition Proposition
1.Symmetry
[51] Let2022,
𝔵, 1.
𝔴14,[51]
∈x𝔽FOR
Let
𝔵,for
𝔴
∈
. Then
fuzzy
.order
Then
fuzzy
“over
≼
”→
is
“ valued
≼
on
”ℝa𝔽
is
given
by
on
𝔽𝔴]
tions
of
I-V-Fs.
𝔖
isrelation
𝐹𝑅
-integrable
over
[𝜇,
𝜈]
ifby
𝔖
∈
. Note
𝜑
∈
1]
whose
𝜑
-cuts
define
the
family
of
𝔖
:𝒾∈𝐾
⊂
ℝ
→=𝜈],
𝒦
are
giv
PEER
REVIEW
∗[𝔴]
NV-Fs
over
[𝜇,
𝜈].
It
is⊂
aℝ(0,
partial
order
relation.
Definition
1.
[49]
A
fuzzy
number
map
𝔖
:
𝐾
⊂
ℝ
→
𝔽
is
called
F-NV-F.
For
[0,
Theorem
2.
[49]
Let
𝔖
:
[𝜇,
𝜈]
⊂
ℝ
𝔽
be
a
F-NV-F
whose
𝜑-cuts
define
the
family
of
I-V-Fs
𝐾,
𝜉
∈
1],
we
have
(𝒾)
[𝔖
(𝒾,
(𝒾,
=
𝜑),
𝔖
𝜑)]
for
all
∈
[𝜇,
𝜈]
and
for
all
:
[𝜇,
𝜈]
→
𝒦
are
given
by
𝔖
𝔖
(𝒾)𝑑𝒾
(𝐼𝑅)
(𝐼𝑅)
[𝔵
[𝔵]
[
=
𝔖
𝔖
∗
[𝜇,
(𝐹𝑅)
(𝒾)𝑑𝒾
×
𝔴]
=
×
𝔴]
,
𝜈],
denoted
by
,
is
given
levelwise
by
𝒾𝒿
𝜑
∈aI-V-Fs.
(0,
1]
whose
𝜑[𝜇,
-cuts
define
family
I-V-Fs
𝔖
:𝒾𝐾∈∈⊂
𝒦
are
be
an𝜑),
F-NV-F.
Then
fuzzy
integral
𝔖
Definition
2.
[49]
Let
𝔖
:+
𝜈] ⊂
ℝ
→
𝔽∗isthe
(𝒾,
(𝐹𝑅)
(𝒾)𝑑𝒾
𝜑),
𝔖
then
𝔖
is
a𝒾𝒿
fuzzy
Aumann-integrable
𝔖
of
𝔖
is
𝐹𝑅
-integrable
over
[𝜇,
𝜈]
if∗of
𝔖𝜑)𝑑𝒾
𝔽ℝ𝜈]
. →Note
tha
It
is
partial
order
relation.
Symmetry 2022, 14, x FOR PEER REVIEW
4o
∗of
(5)
(5)
∗Lebesgue-integrable,
[𝔵+
[𝔵+
[𝔵]
[𝔴]
[𝔵]
[𝔴]
(𝒾)
[𝔖
(𝒾,
(𝒾,
∗ (𝒾,
𝔴]
=
+
𝔴]
=
,are
,Z𝐾
=
𝔖
𝜑)]
for
all
[𝜇,
and
for
:𝜑
[𝜇,
⊂
ℝ
→
𝒦
are
given
by
𝔖
𝔖 tions
υ=
∗𝜑)
[𝜇,
(𝐹𝑅)
(𝒾)𝑑𝒾
e
𝜈],
denoted
given
levelwise
by
(0,
[𝜇,
(𝒾,
(𝒾,give
(𝑅)
(𝐹𝑅)
∞)
is
said
to
be
a(𝑅)
convex
set,
if,
Definition
3.
[34]
A
set
⊂∗to
ℝ
𝔖
,Aumann-integra
𝔖
𝔖
(𝒾)𝑑𝒾
(𝒾)𝑑𝒾
(𝐼𝑅)
(𝐼𝑅)
[𝔵
[𝔵]
[I-V-Fs
(𝒾)
[𝔖
(𝒾,
(𝒾,
=
𝔖
=(0,
×
𝔴]
=
×
𝔴]
,each
𝔖
=
𝜑),
𝔖µυ
for
all
𝒾,(𝒾)𝑑𝒾
∈
𝐾.
Here,
for
𝜑𝐾the
∈
1]
the𝔖are
end
poin
1𝜈]
2µυ
S
(𝐾
)𝜐]
∈
𝐾.
∗a
(𝒾,
(𝒾,
We
now
use
mathematical
operations
examine
some
of
characteristics
of
fas
𝜑),
𝔖
𝜑)
are
Lebesgue-integrable,
then
𝔖
is
𝔖
∈a[𝔵]
(0,
1]
whose
𝜑by
-cuts
define
the
family
of
𝔖
:fuzzy
⊂
ℝ
→
𝒦
∗Definition
∗=
∗𝜑)𝑑
∗𝜑)]
∗𝜈]
∗→
It
is
partial
order
relation.
[𝜌.
[𝔵]
𝔵]
=
𝜌.
[𝔵]
[𝔴]
[𝔴]
e
e
e
(0,
[𝜇,
𝔵
≼
𝔴
if
and
𝔵
only
≼
𝔴
if,
if
and
only
if,
≤
for
all
≤
𝜑
∈
(0,
for
1],
all
𝜑
∈
(0,
1],
(4)
(4)
(𝒾)
[𝔖
(𝒾,
(𝒾,
(𝒾,
(𝒾
(𝑅)
(𝑅)
(𝐹𝑅)
(𝒾)𝑑𝒾
=
∞)
is
said
to
be
a
convex
3.
[34]
A
set
=
𝜐]
⊂
ℝ
e
=
𝜑),
𝔖
𝜑)]
for
all
𝒾
∈
[𝜇,
and
for
all
:
[𝜇,
𝜈]
⊂
ℝ
𝒦
are
given
by
𝔖
𝔖
=
𝔖
𝜑)𝑑𝒾
,
𝔖
𝔖
(𝒾)
[𝔖
(𝒾,
(𝒾,
∗
[𝜇,
(𝐹𝑅)
(𝒾)𝑑𝒾
(𝒾,
(𝒾,
𝔖
=
𝜑),
𝔖
𝜑)]
for
all
𝒾
∈
𝐾.
Here,
for
each
𝜑
∈
(0,
1]
the
end
𝜈],
denoted
by
,
is
given
levelwise
by
𝒾𝒿
𝜑)
and
𝔖
𝜑)
both
are
𝜑
∈
(0,
1].
Then
𝔖
is
𝐹𝑅-integrable
over
[𝜇,
𝜈]
if
and
only
if
𝔖
4
S
(
µ
)
+
S
(
υ
)
.
S
4
FR
d
(1
(
)
∈
𝐾.
𝜉𝒾
+
−
𝜉)𝒿
tion
over
[𝜇,
𝜈].
∗
∗
We
now
use
mathematical
operations
to
examine
some
of
the
characteristics
∗map
∗
(𝒾,
(𝒾,
∗
Definition 1.Definition
[49] A fuzzy
1. [49]
number
A fuzzy
valued
number
map
𝔖
valued
:
𝐾
⊂
ℝ
→
𝔽
𝔖
:
𝐾
⊂
ℝ
→
𝔽
is
called
F-NV-F.
is
called
For
each
F-NV-F.
For
each
𝜑),
𝔖
𝜑)
are
Lebesgue-integrable,
then
𝔖
is
a
fuzzy
Aumann-integrable
fu
𝔖
∗
[𝜌.
[𝔵]
∗
𝔵]
=
𝜌.
2
[0,
∗
𝐾,
𝜉
∈
1],
we
have
∗
(0,
[𝜇,
=
is∗(0,
said
be
acollection
convex
set,
if,
Definition
[34]
A
set
𝐾
=ℝ
⊂
ℝ
(.µand
functions
𝔖
,is𝜈].
𝜑),
𝔖−
, we
𝜑):
are
called
lower
and
functions
of
𝔖ofboth
.of
(𝒾,
(0,
(1
𝜑)
and
𝜑)(𝒾,
(0,
1].
Then
over
[𝜇,
𝜈]
ifhave
and
only
(𝒾,
(𝑅)
(𝐹𝑅)
(𝒾)𝑑𝒾
for
𝜑3.
(0,
1].
for
For
all
all
𝜑𝔖
𝜑𝒾∈
∈𝜐]
(0,
1].
1],
For
ℱℛ
all
𝜑
∈
1],
ℱℛ
denotes
the
denotes
all
thaf
2tion
µ[𝔖
+
υ𝐾,
υ∈
𝜉𝒾∞)
+
−
𝜉)𝒿
Let
𝔵,(.
𝔴
∈,𝐹𝑅-integrable
𝔽
𝜌𝐾operations
∈
ℝ,
then
we
=
𝔖
𝜑)𝑑𝒾
, 1]
𝔖
𝜑)𝑑𝒾
over
[𝜇,
∈
𝐾.
(𝒾)
(𝒾,
(𝒾,
µ
𝔖
=
𝜑),
𝜑)]
for
all
𝐾.
Here,
for
each
𝜑
∈to
(0,
the
point
∗all
([
],if
)𝔖
([(𝑅)
], (𝒾,
) end
(6)
(6)
∗upper
[𝔵𝜑×∈numbers.
[𝔵
[𝔵]
[𝔖
[𝔵]
, some
,𝔖
××
𝔴]
=
𝔴]
𝔴]
=
×
𝔴]
,→
∗[(.
We
now
mathematical
to
of
the
characteristics
fuz
∗use
[0,
𝜉𝔖
∈
[𝔵]
𝔵]
=
𝜌.order
(.𝔖
functions
,Let
𝜑),
𝔖
,𝔖
𝜑):
→
ℝ
are
called
lower
and
functions
.,
(0,
(0,
for
all
𝜑
∈
(0,
for
all
𝜑
𝜑
∈examine
∈𝒦
(0,
1].
1],
For
ℱℛ
all
𝜑and
∈)≼
ℱℛ
denotes
the
collection
den
numbers.
Let
𝔴
𝔽
and
𝜌𝐾
∈
ℝ,
then
we
have
𝜉𝒾
+
−
𝜉)𝒿
(𝒾,
(𝒾,
∗𝔵,
𝜑)
𝔖
𝜑)
both
are
𝜑𝜑
∈-cuts
(0, tion
1].
Then
𝔖
is
𝐹𝑅-integrable
𝜈]
and
only
if(1
𝔖
([
],by
([
𝜑 is
∈ a(0,partial
1] whose
𝜑order
∈ a(0,partial
𝜑
1]-cuts
whose
define
the
family
define
of
the
I-V-Fs
family
𝔖
of
𝔖[𝜇,
:∈1],
𝐾I-V-Fs
ℝ
→
𝒦
:For
𝐾[𝜌.
are
⊂ifall
ℝ
given
→
by
are
given
,then
, ],of
(𝒾)𝑑𝒾
(𝐹𝑅)
(𝒾)𝑑𝒾
(𝐼𝑅)
over
[𝜇,
𝜈].
[𝜇,
Proposition
1.
[51]
𝔵,⊂
𝔴over
∈have
𝔽1].
. 𝐹𝑅-integrable
Then
fuzzy
relation
“upper
” ∗1],
is
given
on
𝔽)], 𝔖
by
𝔖
=
𝔖
=
𝜑)𝑑𝒾
:
𝔖(𝒾,
𝜑)
∈
ℛ
𝑅-integrable
over
[𝜇,
𝜈].
Moreover,
if
is
over
𝜈],
It
It
is
relation.
order
relation.
∗𝔖(𝒾,
∗
[0,
([
)
,
𝐾,
𝜉
∈
1],
we
have
NV-Fs
over
[𝜇,
𝜈].
NV-Fs
over
[𝜇,
𝜈].
(.
(.
functions
𝔖
,
𝜑),
𝔖
,
𝜑):
𝐾
→
ℝ
are
called
lower
and
upper
functions
of
𝔖
.
𝒾𝒿
(0,
(0,
for
all
𝜑
∈
(0,
1].
for
For
all
all
𝜑
𝜑
∈
∈
(0,
1].
1],
For
ℱℛ
all
𝜑
∈
1],
ℱℛ
denotes
the
collection
denotes
of
all
the
𝐹
numbers.
Let
𝔵,
𝔴
∈
𝔽
and
𝜌
∈
ℝ,
then
we
have
(𝒾)𝑑𝒾
(𝐹𝑅)
(𝒾)𝑑𝒾
(𝐼𝑅)
[𝜇,
Proposition
1.
[51]
Let
𝔵,
𝔴
∈
𝔽
.
Then
fuzzy
order
relation
“
≼
”
is
given
on
𝔽
𝔖
=
𝔖
=
𝔖(𝒾,
𝜑)𝑑𝒾
:
𝔖(𝒾,
𝜑)
∈
ℛ
𝑅-integrable
over
[𝜇,
𝜈].
Moreover,
if
𝔖
is
𝐹𝑅-integrable
over
𝜈],
then
∗
([
],
)
([
],
)
,
,
([
,
∗ (𝒾, Definition
∗ 𝜌.
[𝔵+
[𝔵]
[𝔴]
𝔴]
=be
+
,following
NV-Fs
over
[𝜇,
NV-Fs
over
[𝜇,
Theorem
[49]
Let
𝔖
[𝜇,
⊂
→
𝔽
be
awe
F-NV-F
whose
define
the
family
of, ],𝔖
Ie
[49]
A
fuzzy
number
valued
map
𝔖
:𝜈].
𝐾
⊂
ℝof
→
𝔽
is” called
F-NV-F.
For
𝒾𝒿
(𝒾)𝑑𝒾
(𝐼𝑅)
(7)
(7)
[𝜌.
[𝜌.
[𝔵]
[𝔵]
[𝔖∗ (𝒾,
(𝒾)
[𝔖𝜑)]
=
𝔖
𝔵]=
𝔵]
=
=
𝜌.:𝔵,
𝔖 (𝒾)
𝔖use
= now
𝜑),
𝔖=∗ (𝒾,
𝜑),
for
𝔖
all
Here,
all
𝒾1.for
∈
𝐾.
each
Here,
𝜑
∈
for
(0,
each
1]
the
𝜑
end
∈
(0,
point
1]
the
real
end
point
real
If
S
,∈
ϕ𝐾.
S
,𝜈].
ϕ2.2.
with
ϕof
=
1,
then
from
(17)
achieve
the
inequality,
∈𝜑-cuts
𝐾.
( 2.
)for
)[51]
We
We
mathematical
now
use
mathematical
operations
to
operations
examine
some
to
examine
the
characteristics
some
of
the
characteristics
of
fuzzy
fuzzy
[𝜇,
Definition
4.
[34]
The
𝔖:
𝜐]
→
ℝ
is
called
a𝜈],
harmonically
convex
∗ ( 𝒾𝜑)]
∗ (𝒾,
(𝒾)𝑑𝒾
(𝐹𝑅)
(𝒾)𝑑𝒾
(𝐼𝑅)
an
F-NV-F.
Then
fuzzy
integral
Definition
[49]
Let
𝔖
:𝜈]
[𝜇,
𝜈]ℝ
⊂
ℝ
→
𝔽
[𝜇,
Proposition
1.
Let
𝔴
∈
𝔽
.𝜈].
Then
fuzzy
order
relation
“→
≼
is
given
𝔽],family
by
𝔖
=
𝔖
=
𝔖(𝒾,
𝜑)𝑑𝒾
:𝔖𝜑-cuts
𝔖(𝒾,
𝜑)
∈
ℛon
𝑅-integrable
over
[𝜇,
Moreover,
if
𝔖
is
𝐹𝑅-integrable
over
then
[𝔵+
[𝔵]
[𝔴]
([F-NV-F.
)of
,function
𝔴]
=
+
,
Theorem
[49]
Let
𝔖
:
[𝜇,
𝜈]
⊂
ℝ
→
𝔽
be
a
F-NV-F
whose
define
the
Definition
1.
[49]
A
fuzzy
number
valued
map
𝔖
:
𝐾
⊂
ℝ
𝔽
is
called
F
[𝔵]
[𝔴]
(𝒾)𝑑𝒾
(𝐼𝑅)
𝔵
≼
𝔴
if
and
only
if,
≤
for
all
𝜑
∈
(0,
1],
𝒾𝒿
=
NV-Fs
over
[𝜇,
𝜈].
NV-Fs
over
[𝜇,
𝜈].
(1
∈
𝐾.
𝜉𝒾
+
−
𝜉)𝒿
[𝜇,
Definition
4.
[34]
The
𝔖:
𝜐]
→
ℝ
is
called
a
harmonically
convex
fu
be
an
F-NV-F.
Then
fuzzy
integral
o
Definition
2.upper
[49]
Let
𝔖
:upper
[𝜇,
𝜈]
⊂
ℝ
→
𝔽
∗ (. see
∗→
𝜑𝔖
∈
1]
whose
-cuts
define
the
family
of
I-V-Fs
𝔖
:𝐾.
𝐾
⊂
ℝall
→
𝒦
are
given
[𝔵]
[𝔴]
[𝔵+
[𝔵]
[𝔴]
(.𝔵,, 𝜑),
[36]:
(1
𝔵:𝜈]
≼Definition
𝔴
if
and
only
if,
≤
for
all
𝜑
∈be
(0,
1],
𝔴]
=
+
,(𝒾,
functions Let
𝔖∗numbers.
functions
𝔖
, and
𝜑):
,𝐾𝜑),
,Definition
𝜑):
called
𝐾we
→
ℝ
lower
are
and
lower
functions
and
of
functions
𝔖
.3.
of
𝔖
.[ℝ
𝜉𝒾
+
−
𝜉)𝒿
Theorem
2.
[49]
Let
𝔖
:𝜑3.
[𝜇,
⊂
ℝ
→
𝔽𝜐]
be
a(𝒾,
F-NV-F
whose
define
the
family
ofset
I-V
numbers.
𝔴 ∈𝔖𝔽
Let
𝔴
𝔽∗ (.are
𝜌→∈𝔖ℝ
ℝ,
then
and
𝜌:(0,
∈
have
ℝ,
then
we
have
1.
[49]
Aby
fuzzy
number
valued
map
𝔖
:ℝcollection
𝐾
⊂
𝔽
is⊂
called
For
ea
∈
(𝒾)
[𝔖
(𝒾,
for
all
𝜑denoted
∈
(0,
1],
where
ℛ
the
Riemannian
integrable
=
𝜑),
𝔖
𝜑)]
for
𝒾F-NV-F.
∈a∞)
[𝜇,
𝜈]
and
ft
[𝜇,
𝜈]
⊂
ℝcalled
→
𝒦𝜑
are
given
𝔖
=
𝔖
[𝜇,
[𝜇,
(𝐹𝑅)
(𝒾)𝑑𝒾
∗(𝒾)𝑑𝒾
∗ (.𝔵,
Definition
4.
[34]
The
𝔖:
→
ℝ
is
called
a𝔴]
function
𝜈],
𝔖
,denotes
is
given
levelwise
by
(0,
(0,
[𝜇,
[𝜇,
([
],
)ℝ
=
∞)
is
said
to
=convex
convex
is
said
Definition
[34]
A
set
𝐾
=
[34]
𝜐]
⊂
set
𝐾
=harmonically
𝜐]
ℝ
, by
∗A
[𝔵
[𝔵]
×
𝔴]
=𝜉𝒾
×
,𝜑-cuts
be
an
F-NV-F.
Then
fuzzy
integral
of
𝔖
Definition
2.
[49]
Let
𝔖
[𝜇,
𝜈]
⊂
→
𝔽
∗of
𝜑
∈
(0,
1]
whose
-cuts
define
the
family
of
I-V-Fs
𝔖
:𝜑)𝑑𝒾
𝐾
⊂
ℝ
→
𝒦
are
gi
(𝒾,
(𝑅)
(𝑅)
(𝐹𝑅)
(𝒾)𝑑𝒾
𝒾𝒿
=
𝔖
𝜑)𝑑𝒾
,(𝐼𝑅)
𝔖
𝔖
Z
(1
+
−
𝜉)𝒿
(𝒾)
[𝔖
(𝒾,
(𝒾,
for
all
𝜑
∈
(0,
1],
where
ℛ
denotes
the
collection
of
Riemannian
integra
∗
∗
[𝔵]
[𝔴]
=
𝜑),
𝔖
𝜑)]
for
all
𝒾
∈
[𝜇,
𝜈]
a
:
[𝜇,
𝜈]
⊂
ℝ
→
𝒦
are
given
by
𝔖
𝔖
𝔵
≼
𝔴
if
and
only
if,
≤
for
all
𝜑
∈
(0,
1],
[𝜇,
(𝐹𝑅)
(𝒾)𝑑𝒾
∗
𝜈],
denoted
by
𝔖
,
is
given
levelwise
by
υ
(0,
(0,
[𝜇,
[𝜇,
([
],
)
=
∞)
is
said
to
=
be
a
∞)
conv
i
Definition
3.
[34]
Definition
A
set
𝐾
=
3.
[34]
𝜐]
⊂
A
ℝ
set
𝐾
=
𝜐]
⊂
ℝ
,
∗
[𝔵
[𝔵]
[
×
𝔴]
=
×
𝔴]
,
µυ
(𝒾)
[𝔖
(𝒾,
(𝒾,
𝔖
=
𝜑),
𝔖
𝜑)]
for
all
𝒾
∈
𝐾.
Here,
for
each
𝜑
∈
(0,
1]
the
end
point
𝒾𝒿
(𝒾,
(𝒾,
(𝑅)
(𝑅)
(𝐹𝑅)
(𝒾)𝑑𝒾
2µυ
1
S
)
(
It
is
a
partial
order
relation.
=
𝔖
𝜑)𝑑𝒾
,
𝔖
𝜑)𝑑𝒾
𝔖
(1
∗
+
𝜉𝔖(𝒿),
𝔖
≤
−
𝜉)𝔖(𝒾)
𝜑
∈
(0,
1]
whose
𝜑
-cuts
define
the
family
of
I-V-Fs
𝔖
:
𝐾
⊂
ℝ
→
𝒦
are
given
∗
∗
(𝐹𝑅)
(𝒾)𝑑𝒾
∗
s
−
1
∗
tions
of
I-V-Fs.
𝔖
is
𝐹𝑅
-integrable
over
[𝜇,
𝜈]
if
𝔖
∈
𝔽
.
Note
th
[0,
(0,
𝐾,
𝜉1],
∈
1],
we
have
𝐾,
𝜉],set
∈)𝔖
1],
we
have
all
𝜑
∈𝔽
(0,
1].
For
all
𝜑
∈
1],
ℱℛ
the
collection
of set,
all
(𝒾)
[𝔖
(𝒾,
(𝒾,
for
𝜑is
∈
(0,
where
ℛ
denotes
the
collection
of
Riemannian
integrable
fu
𝜑),
𝔖
𝜑)]
all
𝒾and
𝜈]
for
:all
[𝜇,
𝜈]
⊂
ℝ
→
𝒦
are
given
by
𝔖
Definition 1.Definition
[49] A fuzzy
1. [49]
number
A fuzzy
valued
map
𝔖
valued
:for
𝐾𝔴]
⊂
map
→
𝔖
:[0,
𝐾
⊂
→
𝔽
called
F-NV-F.
is
called
For
F-NV-F.
each
[𝜇,
(𝐹𝑅)
(𝒾)𝑑𝒾
∗denotes
(𝒾,
(𝒾,
𝜈],
denoted
by
𝔖
,[0,
is
given
levelwise
(0,
(0,+
[𝜇,
[𝜇,
𝜑)
𝔖
bo
𝜑number
∈
(0,
1].
Then
𝔖
𝐹𝑅-integrable
over
[𝜇,
𝜈]
if1],
ifS(𝐹𝑅)
𝔖
],For
)𝔖
([
(𝒾)
[𝔖
(𝒾,
(𝒾,
, by
=
isυ(5)
said
to
=
be
a[𝜇,
∞)
convex
is
said
t
Definition
3.
[34]
Definition
A
𝐾
=
3.
[34]
𝜐]
Aeach
ℝ
set
𝜐]
⊂
,ℝ
𝔖
=of
𝜑),
𝔖
𝜑)]
all
𝒾=
∈
𝐾.
Here,
for
each
𝜑. ℝ
∈
(0,
1]
the
end
po
∗⊂
d[𝜌.
≤
2𝔴]
S
≤ℝ
(is
R
)𝐹𝑅
S
(𝐾
µonly
)=
+,∞)
((1
)]
[and
It
a[𝔵+
partial
order
relation.
[𝔵
[𝔵]
[([ℱℛ
𝒾𝒿
(1
𝔴]
=
×
𝔴]
𝜉𝔖(𝒿),
𝔖
≤
−
𝜉)𝔖(𝒾)
𝜉𝒾
−
𝜉)𝒿
∗for
[𝔵]
[𝔴]
[𝔵]
[𝔴]
(𝒾,
(𝒾,
(𝑅)
(𝑅)
(𝐹𝑅)
(𝒾)𝑑𝒾
=⊂
+
=
,mathematical
+
,for
=
,(5)
𝜑)𝑑𝒾
𝔖
∗𝜑)
(𝒾)𝑑𝒾
tions
I-V-Fs.
𝔖
is
-integrable
over
[𝜇,
𝜈]
𝔖
∈
𝔽
.and
Note
[0,
(0,
𝐾,
𝜉∗is
∈
1],
we
have
𝐾,
𝜉𝔖
∈
we
have
for
all
𝜑
∈
(0,
1].
For
all
𝜑
∈𝜑)𝑑𝒾
denotes
the
collection
ofre
2×𝜐]
∗+
∗𝔖
[𝔵]
[𝜇,
𝔵]
=over
𝜌.
(9)
Definition
4.
The
𝔖:
→
ℝ
is
a([if
harmonically
convex
(𝒾,
(𝒾,
(.[34]
𝜑)
and
𝔖
𝜑)
∈[𝔖
(0,
1].
Then
𝔖
is
𝐹𝑅-integrable
over
[𝜇,
𝜈]
ifand
and
only
𝔖
],
) if
be
an
F-NV-F.
be
Then
an
F-NV-F.
fuzzy
integral
Then
fuzzy
of
𝔖
integral
of
𝔖
over
Definition 2.Definition
[49] Let 𝔖2.: [𝜇,
[49]
𝜈] ⊂
Letℝ[𝔵+
𝔖
→
:functions
𝔽(𝒾)
𝜈]
ℝ
→
𝔽
,𝜑
We
now
operations
to
examine
some
of
the
characteristics
of
𝔖
,∗use
𝜑),
,𝐾
𝜑):
𝐾
→
ℝ
are
called
lower
upper
functions
of
𝔖
.function
µof
+
υ(.order
υrelation.
−
µ
s1],
+
1called
(1
𝜉𝒾
+
−
𝜉)𝒿
(1
∗NV-Fs
(𝒾,
(𝒾,
µall
∗+
𝜉𝔖(𝒿),
𝔖
≤
−
𝜉)𝔖(𝒾)
𝔖tions
=of
𝜑),
𝔖
𝜑)]
for
𝒾
∈
𝐾.
Here,
for
each
∈
(0,
1]
the
end
point
∗(𝒾,
It[𝜇,
is∗functions
a𝜑
partial
∗
∗𝔖
over
[𝜇,
𝜈].
∗
(𝒾,
(𝐹𝑅)
(𝒾)𝑑𝒾
[𝜇,
𝜑 ∈ (0, 1] whose
𝜑 ∈ (0,𝜑1]-cuts
whose
define
𝜑 -cuts
the
family
define
the
I-V-Fs
family
𝔖
of
I-V-Fs
𝔖
:
⊂
ℝ
→
𝒦
:
𝐾
are
⊂
ℝ
given
→
𝒦
by
are
given
by
𝜑),
𝜑)
are
Lebesgue-integrable,
then
𝔖
is
a
fuzzy
Aumann-integrable
𝔖
I-V-Fs.
𝔖
is
𝐹𝑅
-integrable
over
[𝜇,
𝜈]
if
𝔖
∈
𝔽
.
Note
tha
[𝜌.
[𝔵]
Definition
4.
[34]
The
𝔖:
𝜐]
→
ℝ
is
called
a
harmonically
convex
fu
𝔵]
=
𝜌.
[0,
[0,
(
(0,
𝐾,
𝜉
∈
1],
we
have
𝐾,
𝜉
∈
1],
we
have
(.𝜑[𝜇,
(. ,[𝜇,
for
all
∈𝔖𝜈].
(0,
1].
For
all→
𝜑ℝ
∈are
1],
ℱℛ
the
collection
of𝔖
all
𝐹
We
now
mathematical
operations
to
some
of
the
characteristic
𝔖
, use
𝜑),
𝔖
𝜑):
𝐾(𝐼𝑅)
called
lower
and
upper
functions
of([𝜑)
.],both
𝒾𝒿
(1
𝜑)
and
𝔖∈ (𝒾,
𝜑𝑅-integrable
∈ (𝒾)𝑑𝒾
(0,
1].
Then
𝐹𝑅-integrable
over
𝜈]
if
and
𝔖
∗NV-Fs
([examine
) ifdenotes
, ],𝒾𝒿
𝜉𝒾
+
−ishave
𝜉)𝒿
(𝒾)𝑑𝒾
(𝐹𝑅)
(𝒾)𝑑𝒾
∗(𝒾,
=
𝔖[𝜇,
=
𝔖(𝒾,
𝜑)𝑑𝒾
:[𝜇,
𝔖(𝒾,
𝜑)
ℛ
over
Moreover,
ifℝ,
𝔖
is
𝐹𝑅-integrable
over
𝜈],
then
∗𝒾(𝒾,
,rever
over
𝜈].
∗is
𝜑),
𝔖
𝜑)
are
then
is
a[𝜇,
fuzzy
Aumann-integra
𝔖
[𝜇,
[𝜇,
[0,
Definition
4.
[34]
The
𝔖:
𝜐]
→
ℝ
called
a𝔖
harmonically
convex
function
[𝜇, 𝜈], denoted
[𝜇, by
(𝐹𝑅)
(𝒾)𝑑𝒾
(𝐹𝑅)
numbers.
Let
𝔵,𝔖
𝔴
∈
and
𝜌ℝ
∈
then
we
for
all
𝒿over
∈
𝜐],
𝜉𝐾Lebesgue-integrable,
∈by
1],
where
𝔖(𝒾)
≥
0only
for
all
𝜐].
If
is(9)
𝜈],
𝔖by
𝔖
,[𝔵isfunctions
levelwise
,∗𝔖
given
by
levelwise
[𝜌.
[𝔵]
∗ (𝒾,
𝔵]
=
𝜌.
𝒾𝒿
𝒾𝒿
(6)
(6)
[𝔵is
[𝔵]
[𝒾,
[𝔵]
∗ (𝒾,
(.
×given
×
=
𝔴]
=
,𝔵,(.
×
𝔴]
,the
We
now
use
mathematical
to
examine
some
of
the
characteristics
of
fu
∈∈
𝐾.
∈
𝐾.)is
𝜑),
𝔖
,𝒿𝔽
𝜑):
→
are
called
lower
and
upper
functions
of(11)
𝔖. If
(𝒾)𝑑𝒾
(𝐹𝑅)
(𝒾)𝑑𝒾
(𝐼𝑅)
[𝜇,
𝔖
=
𝔖
=we
𝔖(𝒾,
𝜑)𝑑𝒾
:∈𝔖(𝒾,
𝜑)
∈
ℛ
𝑅-integrable
[𝜇,
𝜈].
ifℝ,
𝔖
is
𝐹𝑅-integrable
over
𝜈],
then
𝒾𝒿
(𝒾)𝑑𝒾
=𝜉operations
𝔖
∗𝒾
([
,fu
tion
over
[𝜇,
𝜈].
(𝒾)𝔖denoted
[𝔖𝜑)]
𝔖 (𝒾) = [𝔖∗ (𝒾,
𝔖 𝜑),
=∗ (𝒾,
for
all
∈𝔴]
Here,
all
for
𝐾.
each
Here,
𝜑
for
(0,
each
1]
𝜑
end
∈1,
(0,
point
1]
the
real
end
point
real
If𝔖S
,𝔖
ϕ∗𝐾.
=
S
,×∈,𝔴]
ϕ
with
ϕ
=
1[ Moreover,
and
s(𝐼𝑅)
=
then
from
(17)
acquire
the
following
)for
(∗(𝐹𝑅)
)Let
[𝜇,
[0,
[𝜇,
for
all
𝒾,
𝜐],
∈
1],
where
𝔖(𝒾)
≥
0
for
all
𝒾
𝜐].
(11)
numbers.
𝔴
∈
𝔽
and
𝜌
∈
then
we
have
NV-Fs
over
[𝜇,
𝜈].
(𝒾,
(𝒾,
∗ ( 𝒾𝜑)]
𝜑),
𝔖
𝜑)
are
Lebesgue-integrable,
then
𝔖
is
a
fuzzy
Aumann-integrable
∗ (𝒾, 𝜑),
(1
(1
𝜉𝒾
𝜉𝒾
+
−
𝜉)𝒿
+
−
𝜉)𝒿
∈integral
𝐾.∈(11)
𝒾𝒿
𝒾𝒿
𝒾𝒿
(1 𝜐].
+𝜐].𝜉𝔖(𝒿),
𝔖
−
𝜉)𝔖(𝒾)
(𝒾)𝑑𝒾
(𝐼𝑅)
(𝒾)𝑑𝒾
(𝒾)𝑑𝒾
(𝐼𝑅)
[𝜇,
𝔖
𝔖
=
𝔖
=
𝔖(𝒾,
𝜑)𝑑𝒾
𝔖(𝒾,
𝜑)
ℛ
𝑅-integrable
over
[𝜇,
𝜈].
Moreover,
ifℝ,
𝔖where
is
over
𝜈],
then
tion
𝜈].
be
an
F-NV-F.
Then
𝔖
Definition
2.
[49]
Let
𝔖and
:𝜉[34]
[𝜇,
𝜈]
⊂1],
ℝ=
→
𝔽 𝐹𝑅-integrable
[𝜇,
isover
called
a∈
harmonically
concave
function
on
([(1
)𝜉)𝒿
[𝜇,
[0,
[𝜇,
, revers
for
all
𝒾,
𝒿𝔴
𝜐],
∈
𝔖(𝒾)
≥
0𝐾≤
for
all
𝒾𝔽Then
∈−:fuzzy
If
isof
numbers.
Let
𝔵,[𝜇,
∈
𝔽fuzzy
∈
then
we
have
𝜉𝒾
𝜉𝒾
+
𝜉)𝒿
+
(0,
[𝜇,
=
∞)
is
said
to
be
aintegral
convex
se, o
Definition
Aupper
set
𝐾
=
𝜐]−
⊂
ℝ
Definition
1.
[49]
A
number
map
𝔖
:𝔖
ℝ
→
is
called
F-NV-F.
For
(1
𝒾𝒿
(1
+
𝜉𝔖(𝒿),
𝔖
≤
𝜉)𝔖(𝒾)
𝜉𝒾
+
𝜉)𝒿
∈
𝐾.
∈],−𝐾.
(. , 𝜑), 𝔖∗ (. 𝔖
(. ,𝐾𝜑),
see
[34]:
be
an
F-NV-F.
fuzzy
of
Definition
2.
[49]
Let
𝔖
:𝜌
[𝜇,
𝜈]
⊂valued
ℝ
→
𝔽
functions 𝔖∗functions
, 𝜑):
→𝔖ℝ∗ (.are
, 𝜑):
called
𝐾𝔵]
→allℝ
lower
are
called
and
upper
lower
functions
and
of
functions
𝔖
.𝔴]
of
.⊂
[𝜇,
(𝒾)𝑑𝒾
(𝐼𝑅)
is
called
a𝜌.3.harmonically
concave
function
on
𝜐].
[𝔵+
[𝔵]
[𝔴]
=
𝔖
=
+
,𝜉)𝔖(𝒾)
tion
over
[𝜇,
𝜈].
∗inequality,
(7)
(7)
[𝜌.
[𝜌.
[𝔵]
[𝔵]
𝔵]
=
𝜌.
=
(0,
[𝜇,
for
𝜑
∈
(0,
1],
where
ℛ
denotes
the
collection
of
Riemannian
integrable
=
∞)
is
said
to
be
a
convex
Definition
3.
[34]
A
set
𝐾
=
𝜐]
⊂
ℝ
Definition
1.
[49]
A
fuzzy
number
valued
map
𝔖
:
𝐾
⊂
ℝ
→
𝔽
is
called
F-NV-F.
(1
𝜉𝒾
+
−
𝜉)𝒿
(1
(1
(1
∗
+
𝜉𝔖(𝒿),
𝔖
≤
−
𝜉𝒾
𝜉𝒾
+
−
𝜉)𝒿
+
−
𝜉)𝒿
([
],
)
[𝜇,
(𝐹𝑅)
(𝒾)𝑑𝒾
,
𝜈],(0,
denoted
by
𝔖𝜈]
,→
is
given
levelwise
by
[𝔵+
[𝔵]
[𝜇,
𝔴]
=on
+𝜐].
, of
is
called
a∈(0,
harmonically
concave
function
be
an
F-NV-F.
Then
fuzzy
integral
ofintegr
𝔖
Definition
2.
Let
𝔖
::where
[𝜇,
⊂
ℝ
𝔽
(𝒾,
(𝑅)
(𝑅)
(𝐹𝑅)
(𝒾)𝑑𝒾
=
𝔖
𝜑)𝑑𝒾
,[𝔴]
𝔖
𝔖
[0,
𝐾,
𝜉𝔖(𝒾,
1],
we
have
∈
1]
whose
𝜑
-cuts
define
the
of
I-V-Fs
𝐾
⊂
ℝ
→
𝒦
are
given
for
all
𝜑
∈
1],
ℛ
denotes
the
collection
Riemannian
∗ℛ
∗𝜑)𝑑𝒾
(8)
𝜉𝒾
+
−⊂
(0,
[𝜇,
([valued
],𝔽
)family
(𝒾)𝑑𝒾
(𝒾)𝑑𝒾
(𝐹𝑅) 𝔖(𝒾)𝑑𝒾
(𝐹𝑅) = (𝐼𝑅)
(𝒾)𝑑𝒾
(𝐼𝑅)
=
is,𝒾:(8)
said
to
be
athe
convex
set,
Definition
3.
[34]
A[0,
set
𝐾
=
𝜐]
ℝ
1.
[49]
A
fuzzy
number
map
𝔖
:(𝒾,
𝐾
⊂[𝔴]
ℝ𝜑)𝑑𝒾
→
is
called
F-NV-F.
For
egI
(𝐹𝑅)
(𝒾)𝑑𝒾
,ℛ
Theorem
2.
[49]
Let
𝔖
[𝜇,
𝜈]
⊂
ℝ
→
be
F-NV-F
whose
𝜑-cuts
define
family
ofov
𝔖for
𝔖𝜑
=
=
𝔖[49]
𝜑)𝑑𝒾
=
:1],
𝔖(𝒾,
𝔖(𝒾,
𝜑)
𝜑)𝑑𝒾
∈
:,([𝔖(𝒾,
𝜑)
, [𝔵]
∈𝜉)𝒿
,𝔖𝔽
𝜈],
denoted
𝔖
is
levelwise
by
(0,
],(1
)a
([
],
)∞)
allDefinition
𝜑
∈[𝜇,
(0,
1].
For
all
𝜑
∈
ℱℛ
denotes
the
of
all
𝐹𝑅-integrable
F, given
,(𝒾,
(𝒾,
(𝑅)
(𝑅)
(𝐹𝑅)
(𝒾)𝑑𝒾
𝔖
𝔖
𝜑)𝑑𝒾
𝔖
[𝔵+
𝔴]
=
+
,all
[0,
for
all
𝒾,
𝒿[49]
𝜐],
𝜉1],
∈
1],
where
𝔖(𝒾)
≥
0×
for
∈
𝜐].
If
(11)
is
rever
𝜑
([
],[𝜇,
)=
𝐾,
𝜉∈by
∈[𝜇,
we
have
𝜑
∈
(0,
1]
whose
𝜑
-cuts
define
the
family
of
I-V-Fs
𝔖
:[𝜇,
𝐾
⊂
ℝ𝐹𝑅-integrable
→
𝒦
are
,ℝ
(𝐹𝑅)
(𝒾)𝑑𝒾
∗collection
tions
of
I-V-Fs.
𝔖
is
𝐹𝑅
-integrable
over
[𝜇,
𝜈]
if
𝔖
∈
𝔽
.
Note
tfF
Z
for
all
∈
(0,
1],
where
ℛ
denotes
the
collection
of
Riemannian
integrable
Theorem
2.
Let
𝔖
:
[𝜇,
𝜈]
⊂
→
𝔽
be
a
F-NV-F
whose
𝜑-cuts
define
the
family
[𝜇,
[𝔵
[𝔵]
[
∗
Definition
4.
[34]
Definition
The
𝔖:
4.
𝜐]
[34]
→
ℝ
The
𝔖:
𝜐]
→
ℝ
is
called
a
harmonically
is
called
convex
a
harmoni
func
×
𝔴]
=
𝔴]
,
∗
([
],
)
υ
[𝜇,
(𝐹𝑅)
(𝒾)𝑑𝒾
,
(0,
𝜈],
denoted
by
𝔖
,
is
given
levelwise
by
for
all
𝜑
∈
(0,
1].
For
all
𝜑
∈
1],
ℱℛ
denotes
the
collection
of
all
[𝜇,
[0,
[𝜇,
for
all
𝒾,
𝒿
∈
𝜐],
𝜉
∈
1],
where
𝔖(𝒾)
≥
0
for
all
𝒾
∈
𝜐].
If
(11)
is
µυ
(𝒾)
[𝔖
(𝒾,
(𝒾,
([
],
)
𝔖
=
𝜑),
𝔖
𝜑)]
for
all
𝒾
∈
𝐾.
Here,
for
each
𝜑
∈
(0,
1]
the
end
point
(𝒾,
(𝒾,
(𝑅)
(𝑅)
(𝐹𝑅)
(𝒾)𝑑𝒾
,
2µυ
S
S
(
µ
)
+
S
(
υ
)
=
𝔖
𝜑)𝑑𝒾
,
𝔖
𝜑)𝑑𝒾
𝔖
(
)
(𝐹𝑅)
(𝒾)𝑑𝒾
tions
of
I-V-Fs.
𝔖
is
𝐹𝑅
-integrable
over
[𝜇,
𝜈]
if
𝔖
∈
𝔽
.
No
be
an
F-NV-F.
be
Then
an
F-NV-F.
fuzzy
integral
Then
fuzzy
of
𝔖
integral
over
of
𝔖
over
Definition 2.Definition
[49] Let 𝔖2.: [𝜇,
[49]
𝜈] ⊂
Let
ℝ
𝔖
→
:
[𝜇,
𝔽
𝜈]
⊂
ℝ
→
𝔽
[0,
∗
𝐾,
𝜉
∈
1],
we
have
𝜑
∈
(0,
1]
whose
𝜑
-cuts
define
the
family
of
I-V-Fs
:
𝐾
⊂
ℝ
→
𝒦
are
given
∗
∗
[𝜇,
[𝜇,
[𝔵
[𝔵]
[
∗ (𝒾,
Definition
4.
[34]
Definition
The
𝔖:
4.
𝜐]
[34]
→
ℝ
The
𝔖:
𝜐]
→
ℝ
is
called
a
harmonically
is
called
conv
a
ha
×
𝔴]
=
×
𝔴]
,
𝒾𝒿
Theorem
2.
[49]
Let
𝔖
:
[𝜇,
𝜈]
⊂
ℝ
→
𝔽
be
a
F-NV-F
whose
𝜑-cuts
define
the
family
of
I-V
NV-Fs
over
[𝜇,
𝜈].
(𝒾)
[𝔖
(𝒾,
(𝒾,
=
𝜑),
𝔖
𝜑)]
for
all
𝒾
∈
[𝜇,
𝜈]
and
f
:
[𝜇,
𝜈]
⊂
ℝ
→
𝒦
are
given
by
𝔖
𝔖
[𝜇,
∗
is
called
a
harmonically
concave
function
on
𝜐].
[𝜇,
[0,
[𝜇,
(0,
(𝒾)
[𝔖
(𝒾,
for
all
𝒾,
∈
𝜐],
𝜉is( R
∈
1],
𝔖(𝒾)
≥ collection
0
for
all
𝒾 ∈all
𝜐].
If1]𝔽(11)
isend
revers
for
allvalued
𝜑tions
∈
(0,
For
all
𝜑𝔖𝜑),
∈are
1],
ℱℛ
the
of
𝐹𝑅-integrable
F- th
𝔖(𝒾,
=
𝔖Lebesgue-integrable,
𝜑)]
for
all
𝒾 (𝒾)
∈
𝐾.[𝜇,
Here,
for
each
𝜑(𝒾)𝑑𝒾
∈
(0,∈
theNote
po
S1].
≤
)given
ddenotes
≤
. ∗fuzzy
∗each
],where
)is𝔴]
,F-NV-F.
∗𝒿
(𝐹𝑅)
𝜑),
𝔖
then
𝔖
is
a𝔖
Aumann-integrable
𝔖(𝒾)𝑑𝒾
of
I-V-Fs.
is
-integrable
over
𝜈]
if[𝔖(𝒾,
𝔖
∗𝒦
Definition
1.[𝜇,
Definition
[49]
A(𝐹𝑅)
fuzzy
1. [49]
number
A(𝐹𝑅)
fuzzy
number
map
𝔖
valued
:𝜈]
𝐾υ(𝒾,
⊂
ℝ
map
𝔽
:𝐹𝑅
𝐾
⊂
ℝ→([→
𝔽
called
called
For
F-NV-F.
For
each
𝒾𝒿
∈
𝐾.
2[𝜇,
NV-Fs
[𝜇,
𝜈].
∗over
[𝜇,
[𝔵
[𝔵]
[𝔖
(𝒾,
(𝒾,
∗→
[𝜇,
Definition
4.
[34]
Definition
4.
𝜐]
[34]
→
The
𝔖:
𝜐]
→
ℝ
is
a, is
harmonically
is
called
convex
×
=ℝ
𝔴]
=𝒾𝒿
𝜑),
𝜑)]
for
all
𝒾a.([∈harmonic
𝜈]
ar
:[𝔖
ℝ
→
are=
by
𝔖
𝔖
is⊂
called
a𝔖
function
on
𝜐].
∗𝜑)
(.
functions
𝔖
,by
𝜑),
𝔖
,harmonically
𝜑):
𝐾The
ℝ𝔖:
are
called
lower
and
upper
functions
of
𝔖,[𝜇,
.], func
𝒾𝒿
(𝒾)𝑑𝒾
(𝐹𝑅)
(𝒾)𝑑𝒾
(𝐼𝑅)
µ
+
υ𝜑)
−
µ
2called
[𝜇,
(𝒾)𝑑𝒾
∗×
𝔖
𝔖concave
=
𝜑)𝑑𝒾
:
𝔖(𝒾,
𝜑)
∈
ℛ
,
𝜈],
by
𝜈],
denoted
by
𝔖(𝒾)
, is
given
levelwise
,[𝜇,
is
given
levelwise
by
(𝒾,
(𝒾,
µall
𝔖
=
𝜑),
𝔖
𝜑)]
for
𝒾
∈
𝐾.
Here,
for
each
𝜑
∈
(0,
1]
the
end
point
∗ (.
(𝒾,
(𝒾,
[𝜌.
[𝔵]
)
for
all denoted
𝜑 ∈ (0,
for1],
allwhere
𝜑 ∈ (0,ℛ𝔖
1],
where
ℛ
the
collection
denotes
the
of
collection
Riemannian
of
Riemannian
integrable
funcintegrable
func𝜑),
𝔖
are
Lebesgue-integrable,
then
𝔖
a
fuzzy
Aumann-integr
𝔵]
=
𝜌.
𝔖
∗
∗
∗
(1
𝜉𝒾
+
−
𝜉)𝒿
∈
𝐾.
([ , NV-Fs
], ) denotes
([
],
)
∗
,
∗
𝒾𝒿
[𝜇,
is1].
called
a∗𝒦
function
on
𝜐].
(.harmonically
(.
[𝜇,
𝜈].
functions
,𝔖
𝜑),
𝔖
𝜑):
𝐾
ℝare
called
lower
and
upper
functions
of
𝔖, .],bo
(𝒾)
[𝔖
(𝒾,
(𝒾,
𝒾𝒿
𝒾𝒿
=
𝜑),
𝔖
𝜑)]
all
𝒾and
[𝜇,
fo)f
:∈[𝜇,
𝜈]
⊂
→𝔖
are
by
𝔖→: 𝐾
𝔖𝔖
(𝒾)𝑑𝒾
(𝐹𝑅)
(𝒾)𝑑𝒾
(𝐼𝑅)
(1
𝔖
=concave
𝔖
=
𝜑)𝑑𝒾
: 𝔖(𝒾,
∈𝜈]
ℛ
∗ℝ
+∈𝜑)
𝜉𝔖(𝒿),
𝔖
𝔖given
≤
𝜉)𝔖(𝒾)
≤
−and
𝜉)𝔖(𝒾
(𝒾)𝑑𝒾
(𝐼𝑅)
(𝒾,
(𝒾,
=
𝔖by
𝜑)
𝔖(1
𝜑define
(0,
Then
𝔖are
𝐹𝑅-integrable
over
𝜈]
and
only
if −
𝔖
tion
over
[𝜇,
𝜈].
𝜑 ∈ (0, 1] whose
𝜑 ∈ (0,𝜑1]-cuts
whose
define
𝜑 -cuts
theover
family
of
the
I-V-Fs
family
of:Lebesgue-integrable,
I-V-Fs
𝔖𝒦
𝐾given
⊂, ℝ
→
⊂are
ℝ[𝜇,
given
→
are
by
[𝜌.
[𝔵]
∗𝒦
([
𝔵]if
=
𝜌.
∗for
(1
(𝒾,
𝜉𝒾
−
𝜉)𝒿
𝜑),
𝔖
𝜑)
then
𝔖
is
a𝔖(𝒾,
fuzzy
Aumann-integrable
∗𝜑)
∗is
𝐾.
∗ (𝒾,
(1
(1
(1
(1
𝜉𝒾𝔵]
𝜉𝒾
−
𝜉)𝒿
+
−
𝜉)𝒿
𝔖(𝐼𝑅)
𝔖
≤
−
𝜉)𝔖(𝒾)
≤
−
(𝒾)𝑑𝒾
(𝒾,
(.if
(.𝔖𝜈]
=+
𝔖be
𝜑)
𝜑3.
∈[34]
(0,
1].
Then
is𝜐]
𝐹𝑅-integrable
over
𝜈]
if
and
only
if𝔖(𝒾,
𝔖
functions
𝔖
,set
𝜑),
𝔖𝜈].
,[𝜇,
𝜑):
→ℝℝ
are
called
and
upper
functions
of
𝔖,+
.],𝔖
𝒾𝒿
𝒾𝒿
tion
over
[𝜇,
(𝒾)𝑑𝒾
(𝐹𝑅)
(𝒾)𝑑𝒾
(𝐼𝑅)
∞)
is
said
to
a+
convex
set,
if,
all
𝒾,𝜉𝔖(𝒿),
𝒿(𝒾,
𝐾
=
⊂
𝔖
=
𝔖
=[𝜇,
𝔖(𝒾,
𝜑)𝑑𝒾
:∈
∈ for
ℛand
,∈𝜑)
(𝐹𝑅)
(𝒾)𝑑𝒾
(𝐹𝑅)
(𝒾)𝑑𝒾
tions of I-V-Fs.
tions𝔖of∗isI-V-Fs.
𝐹𝑅 -integrable
𝔖 Definition
is 𝐹𝑅
over
-integrable
[𝜇,
𝜈]
over
[𝜇,
𝔖
if𝐾
𝔖Note
∈ 𝔽=
.(0,
that,
∈lower
𝔽𝜉𝒾
.+
if
Note
that,
if
∗𝜑)
∗A
[𝜌.
[𝔵]
([
)
=
𝜌.
∗
∗
(1
𝜉𝒾
+
−
𝜉)𝒿
∗
(1
(1
𝜉𝒾
−
−
𝜉)𝒿
(𝒾)𝔖= (𝒾,
[𝔖𝜑)]
(𝒾,
𝔖 (𝒾) = [𝔖∗ (𝒾,
𝔖 𝜑),
for
all
𝒾𝜑)]
∈Definition
𝐾.
for
Here,
all3.(Then
𝒾[34]
for
∈over
𝐾.
each
Here,
𝜑
∈
for
(0,
1]
the
𝜑
end
∈
point
1]
the
real
end
point
real
(0,
[𝜇,
(1
(1
If𝔖S
,𝜑
ϕ∈
=over
S
,2.
ϕ
with
ϕ
=
1 each
and
s if
=
0,=(0,
then
from
(17)
we
achieve
the+
following
∞)
said
to
be
a−
set,
if,
for
all
𝒾,
𝒿fu
set
𝐾
=
𝜐]
⊂
ℝ
)all
)A
+integral
𝜉𝔖(𝒿),
𝔖
𝔖
≤
𝜉)𝔖(𝒾)
≤
𝜉)𝔖(𝒾
(𝒾)𝑑𝒾
(𝐼𝑅)
(𝒾,
(𝒾,−𝜑)
=
𝔖is
𝜑)
and
𝔖
both
(0,
1].
𝔖
is
𝐹𝑅-integrable
over
[𝜇,
𝜈]
if
and
only
if𝜉)𝒿
𝔖
tion
[𝜇,
𝜈].
[𝜇,
∗ (Definition
∗ (𝒾, 𝜑),
𝑅-integrable
[𝜇,
𝜈].
Moreover,
𝔖
is
𝐹𝑅-integrable
over
𝜈],
then
be
an
F-NV-F.
Then
fuzzy
of
𝔖
[49]
Let
𝔖
:ℛ
[𝜇,
𝜈]
⊂
ℝ
→
𝔽
∗convex
for
𝜑
∈
(0,
1],
where
denotes
the
collection
of
Riemannian
integrable
∗
∗
[0,
𝐾,
𝜉
∈
1],
we
have
(1
(1set,
([
],∈
)[0,
𝜉𝒾
𝜉𝒾
−∈
𝜉)𝒿
+
(0,
[𝜇,
[𝜇,
=
is+
said
to
a⊂
if,
for
𝒿 ∈is
Definition
3.
[34]
A
set
=
𝜐]
⊂
ℝ
𝑅-integrable
over
[𝜇,
Moreover,
if∞)
is
over
𝜈],
then
an
F-NV-F.
Then
fuzzy
integral
of
Definition
2.𝐾
[49]
Let
𝔖
:𝜉ℛ
[𝜇,
𝜈]
⊂
ℝ
→
𝔽
(8)
(8)
are
𝜑),
are
then
𝔖
is
a𝒾,
then
fuzzy
𝔖
Aumann-integrable
is
a,The
fuzzy
Aumann-integrable
func𝔖
𝔖
∗𝔖
(𝒾)𝑑𝒾
(𝒾)𝑑𝒾
(𝐹𝑅)
(𝒾)𝑑𝒾
(𝐹𝑅)
(𝒾)𝑑𝒾
(𝐼𝑅)
(𝐼𝑅)
[𝜇,
[𝜇,
[0,
[𝜇,≥
𝔖
𝔖𝜉[36]:
=
=we
𝔖
𝔖(𝒾,
𝜑)𝑑𝒾
=
::𝜈].
𝔖(𝒾,
𝔖(𝒾,
𝜑)
𝜑)𝑑𝒾
∈
ℛ
:([
𝔖(𝒾,
𝜑)
,𝐹𝑅-integrable
∈
ℛ
, afuncall
𝒿A
∈
𝜐],
for
all
𝒾,
1],
𝒿valued
∈
where
𝜉be
𝔖(𝒾)
1],
0convex
where
for
all
𝔖(𝒾)
𝒾−
𝜐].
0all
for
If 𝒾,
(11)
all
𝒾fun
∈
re
Definition
1.
[49]
fuzzy
number
𝔖
:be
𝐾
→
𝔽
is∈𝜉)𝒿
called
F-NV-F.
For
∗ (𝒾, 𝜑), 𝔖
∗,(𝒾,
[𝜇,
],be
)𝜐],
([
],whose
)ℝ
all
𝜑for
(0,
1],
where
denotes
the
collection
of
Riemannian
integrab
Definition
4.
[34]
𝜐]
→
ℝ
is
called
harmonically
convex
,.𝔖
,.≥
[0,
(.𝜑)
(.Lebesgue-integrable,
(.𝜑)
see
2.∈
[49]
Let
𝔖
[𝜇,
𝜈]
⊂
→
𝔽
amap
F-NV-F
𝜑-cuts
define
family
of
𝐾,
∈Theorem
1],
have
functions
𝔖(𝒾,
𝔖
𝜑),𝔖
𝔖
, (𝒾,
𝜑):
,=
𝐾𝜑),
→
𝔖
ℝ∗ (.Lebesgue-integrable,
are
, 𝜑):
called
𝐾for
→
ℝ
lower
are
called
and
upper
lower
functions
and
upper
of
functions
𝔖
of
𝔖
([ 𝜐],
,ℝ,],𝔖:
(0,
[𝜇,
(𝐹𝑅)
(𝒾)𝑑𝒾
∗functions
∗inequality,
for
all
𝜑I-V-Fs.
∈
(0,
1].
For
all
𝜑
1],
denotes
the
collection
all
𝐹𝑅-integra
𝜈],
denoted
by
𝔖𝜈]
is
given
by
[𝜇,
[0,
[𝜇,
[0,
[𝜇,
for
all
𝒾,
𝒿𝔖
∈
for
𝜉)ℱℛ
∈
all
𝒾,𝔽
𝒿],levelwise
∈
𝜐],
𝜉𝔖
𝔖(𝒾)
≥
1],
0𝔖
where
for
allof
𝔖(𝒾)
𝒾called
∈the
≥
𝜐].
0 famil
for
If
(1
[𝜇,
([
)where
𝑅-integrable
over
[𝜇,
𝜈].
Moreover,
ifℝ
𝔖
is
𝐹𝑅-integrable
over
𝜈],
be
an
F-NV-F.
Then
fuzzy
integral
of
𝔖
oa
Definition
2.
[49]
Let
𝔖
[𝜇,
⊂
→
𝔽
,1],
Definition
1.
[49]
A:ℛ
fuzzy
number
valued
map
:∈called
𝐾
⊂Riemannian
ℝ
→
𝔽then
is
F-NV-F.
[𝜇,
(𝐹𝑅)
(𝒾)𝑑𝒾
Definition
4.
[34]
The
𝔖:
𝜐]
→
ℝ
is
a
harmonically
convex
tions
of
𝔖
is
𝐹𝑅
-integrable
over
[𝜇,
𝜈]
if
∈
𝔽
.
Note
tha
Theorem
2.
[49]
Let
:
[𝜇,
𝜈]
⊂
ℝ
→
be
a
F-NV-F
whose
𝜑-cuts
define
the
all
𝜑
∈
(0,
1],
where
denotes
the
collection
of
integrable
fun
[0,
1],
we
have
(0,
[𝜇,
(𝐹𝑅)
(𝒾)𝑑𝒾
for
all
𝜑
∈
(0,
1].
For
all
𝜑
∈
1],
ℱℛ
denotes
the
collection
of
all
𝐹𝑅-inte
tion over [𝜇,tion
𝜈]. over [𝜇, 𝜈]. 𝐾, 𝜉 ∈ for
([
],
)
𝜈],
denoted
by
𝔖
,
is
given
levelwise
by
,
[𝜇,
[𝜇,
is
called
harmonically
is
called
concave
a∈harmonically
function
concave
on
𝜐].
function
on
𝜐].
𝜑 tions
∈ (0, 1]
whose
𝜑
-cuts
define
the
family
of
I-V-Fs
𝔖
: 𝐾is
⊂
ℝ≥→
𝒦
are
([
],∈
)[0,
[𝜇,
[0,
[𝜇,
[𝜇,
(𝐹𝑅)
(𝒾)𝑑𝒾
for
all
𝒾,
𝒿acalled
∈𝔖
𝜐],
for
𝜉(0,
∈all
𝒾,is
1],
𝒿𝒾𝒿
where
𝜐],
𝜉,𝒾𝒿
𝔖(𝒾)
≥
1],
0𝔖
for
all
𝔖(𝒾)
𝒾for
∈
𝜐].
0
for
If𝔽
(11)
all
is
𝒾give
∈
re
∗where
of
I-V-Fs.
𝔖
is
𝐹𝑅
-integrable
over
[𝜇,
𝜈]
if
𝔖
∈
.
Note
Definition
1.
[49]
A
fuzzy
number
valued
map
𝔖
:
𝐾
⊂
ℝ
→
𝔽
called
F-NV-F.
For
[𝜇,
Definition
4.
[34]
The
𝔖:
𝜐]
→
ℝ
is
called
a
harmonically
convex
funct
Theorem
2.
[49]
Let
:
[𝜇,
𝜈]
⊂
ℝ
→
𝔽
be
a
F-NV-F
whose
𝜑-cuts
define
the
family
of
I[
NV-Fs
over
[𝜇,
𝜈].
(𝒾)
[𝔖
(𝒾,
(𝒾,
=
𝜑),
𝜑)]
all
𝒾
∈
[𝜇,
𝜈]
and
:
[𝜇,
𝜈]
⊂
ℝ
→
𝒦
are
given
by
𝔖
𝔖
[𝜇,
[𝜇,
is
a
harmonically
called
concave
a
harmonically
function
concave
on
𝜐].
function
on
𝜐]
∗
𝒾𝒿
𝜑
∈
(0,
1]
whose
𝜑
-cuts
define
the
family
of
I-V-Fs
𝔖
:
𝐾
⊂
ℝ
→
𝒦
are
[𝜇,
(𝐹𝑅)
(𝒾)𝑑𝒾
∗
for
allof
𝜑I-V-Fs.
∈
(0,
1].
For
all
𝜑-integrable
∈(𝒾)𝑑𝒾
denotes
all
𝐹𝑅-integrabl
𝜈],
denoted
by
𝔖
,υisℱℛ
given
levelwise
bythe
𝐾.
∗ (𝒾,
(10)
Z1],
∗𝔖
([𝔖
)∈𝜈]
, ],
(𝒾,
(𝑅)
(𝑅)
(𝐹𝑅)
∗𝐹𝑅
=
𝔖
𝜑)𝑑𝒾
, collection
𝔖of∈
𝜑)𝑑𝒾
𝔖
(𝒾,
(𝒾,
(𝐹𝑅)
(𝒾)𝑑𝒾
𝜑),
𝔖
𝜑)
are
Lebesgue-integrable,
then
𝔖
is
a
fuzzy
Aumann-integrable
fu
𝔖
tions
𝔖
is
over
[𝜇,
if
𝔽
.
Note
that,
NV-Fs
over
[𝜇,
𝜈].
(𝒾)
[𝔖
(𝒾,
(𝒾,
∗
𝒾𝒿
=
𝜑),
𝔖
𝜑)]
for
all
𝒾
∈
[𝜇,
𝜈]
:
[𝜇,
𝜈]
⊂
ℝ
→
𝒦
are
given
by
𝔖
(1
∗
µυ
(𝒾)
[𝔖
(𝒾,
(𝒾,
+
𝜉𝔖(𝒿),
𝔖
≤
−
𝜉)𝔖(𝒾)
∗
for
all 𝜑 ∈ (0,
1],allwhere
𝜑 ∈𝔖(0,
1],
where
ℛ
denotes
the
collection
denotes
the
of
collection
Riemannian
of
Riemannian
integrable
funcintegrable
func𝔖
=
𝜑),
𝔖
𝜑)]
for
all
𝒾
∈
𝐾.
Here,
for
each
𝜑
∈
(0,
1]
the
end
poin
(1
1
2µυ
S
(
)
∗
𝜉𝒾
+
−
𝜉)𝒿
∈
𝐾.
[𝜇,
[𝜇,
∗
be
an
F-NV-F.
be
Then
an
F-NV-F.
fuzzy
integral
Then
fuzzy
of
𝔖
integral
over
of
𝔖
over
Definition
2.for
Definition
[49]
Let
2.: ℛ
[𝜇,
[49]
⊂
Let
ℝ
𝔖
→
:
[𝜇,
𝔽
𝜈]
⊂
ℝ
→
𝔽
(1
is
called
a
harmonically
is
called
concave
a
harmonically
function
concave
on
𝜐].
function
on
𝜐].
𝒾𝒿
([𝜈]
],
)
([
],
)
𝜑 ∈𝔖(0,
1]
whose
𝜑𝜑),
-cuts
define
theallfamily
of
I-V-Fs
𝔖
⊂
→
are
given
,
, (𝒾,
∗ (𝒾, 𝜑)
(𝒾,
(𝑅)
(𝑅)
(𝐹𝑅)
(𝒾)𝑑𝒾
∗ (𝒾,
=
𝜑)𝑑𝒾
,: 𝐾
𝔖𝒾𝒦
𝜑)𝑑𝒾
∗for
𝜑),
𝔖
are
Lebesgue-integrable,
then
𝔖
is
a𝜑)]
Aumann-integrab
∗𝜈]
∗ (𝒾,
(1
(1
∗𝔖over
(𝒾)
[𝔖
(𝒾,
𝜉𝒾
+𝜉)𝒿
𝜉)𝒿
+
𝜉𝔖(𝒿),
𝔖
≤fuzzy
−
𝜉)𝔖(𝒾)
=
𝔖𝐹𝑅-integrable
for
𝒾−
∈
𝐾.
Here,
each
𝜑∗ℝ
∈𝜑)
(0,
1][𝜇,
the
end
NV-Fs
𝜈].
(1
(𝒾)
[𝔖
(𝒾,
(𝒾,
S
(=
R𝜑)]
)𝔖
d(𝒾)𝑑𝒾
≤
S
(−
µ
)𝔖
+
Sonly
υ𝜑)𝑑𝒾
)I-V-Fs
.if
≤
𝒾𝒿
𝜉𝒾family
+
=
𝜑),
𝔖
all
∈
and
:aover
𝜈]
⊂[𝜇,
ℝ
→∗(𝐹𝑅)
𝒦
are
given
by
𝔖
𝔖
(𝒾,
(𝒾,
𝜑)
and
𝔖
𝜑⊂
∈[𝜇,
(0,
Then
𝔖
is∗𝜈]
[𝜇,
𝜈]
and
𝔖
∗1].
∈
𝐾.
∗=
(10)
(𝐹𝑅)
(𝒾)𝑑𝒾
(𝐼𝑅)
𝔖
𝔖
𝔖(𝒾,
:∗for
𝔖(𝒾,
∈
ℛ
, bfp
tion
[𝜇,
𝜈].
2over
(𝒾,
(𝒾,
(𝑅)
(𝑅)
(𝒾)𝑑𝒾
∗𝔖
=
𝔖
𝜑)𝑑𝒾
,(of
𝔖
𝜑)𝑑𝒾
𝔖
([ 𝔖
)fun
Theorem
2. [49]
Theorem
Let
𝔖I-V-Fs.
:2.[𝜇,
[49]
𝜈]
⊂
Let
𝔖
→
𝔽
𝜈]
ℝ
→
be
F-NV-F
whose
be
a𝜑),
F-NV-F
𝜑-cuts
whose
define
𝜑-cuts
the
define
of
I-V-Fs
the
family
,𝔖
(𝒾,
(𝒾,
𝜑),
𝔖over
𝜑)
are
Lebesgue-integrable,
then
𝔖.if𝜈]
is
aand
fuzzy
Aumann-integrable
𝔖given
(1
(.
(.
𝜉𝒾
+
−
𝜉)𝒿
functions
𝔖
,𝜈].
,𝐾
𝜑):
𝐾
→
ℝ
are
called
upper
functions
of
.],∗𝜑)
(𝐹𝑅)
(𝒾)𝑑𝒾
(𝐹𝑅)
(𝒾)𝑑𝒾
tions
I-V-Fs.
tions
𝔖of
is
𝐹𝑅
-integrable
𝔖ℝ
is
𝐹𝑅
over
-integrable
[𝜇,
𝜈]
if
over
[𝜇,
𝔖
if
∈µ+
𝔽
.𝒾𝔖
Note
that,
∈
𝔽
if
Note
that,
if
2levelwise
µ𝔽∗given
+
υset
−
µ
∗lower
[𝜇, 𝜈], of
[𝜇,
(𝐹𝑅)
(𝒾)𝑑𝒾
(𝐹𝑅)
(𝒾)𝑑𝒾
denoted
by
𝜈],
denoted
𝔖by
𝔖
, :is[𝜇,
,[𝔖
is
by
levelwise
by
(1
∗𝔖
(𝒾)
(𝒾,
(𝒾,
+
𝜉𝔖(𝒿),
𝔖
≤
−
𝜉)𝔖(𝒾)
(𝒾,
(𝒾,
=
𝜑),
𝔖
𝜑)]
for
all
∈
𝐾.
Here,
for
each
𝜑
∈
(0,
1]
the
end
point
∗υ
𝜑)
and
𝜑
𝜑
∈
(0,
1].
Then
𝔖
is
𝐹𝑅-integrable
over
[𝜇,
if
and
only
𝔖
(1
𝜉𝒾
−
𝜉)𝒿
(0,
[𝜇,
∗
(𝒾)𝑑𝒾
(𝐹𝑅)
(𝒾)𝑑𝒾
(𝐼𝑅)
=
∞)
is
said
to
be
a
convex
set,
if,
for
all
Definition
3.
[34]
A
=
𝜐]
⊂
ℝ
𝔖
=
𝔖
=
𝔖(𝒾,
𝜑)𝑑𝒾
:
𝔖(𝒾,
𝜑)
∈
ℛ
tion
[𝜇,
∗
([
],
,
(.
(.
functions
𝔖
,
𝜑),
𝔖
,
𝜑):
𝐾
→
ℝ
are
called
lower
and
upper
functions
of
𝔖
.
∗
(1
𝜉𝒾
+
−
𝜉)𝒿
∗𝔖
(0,
[𝜇,
==
∞)
is0said
be
a∈𝜑)
convex
set,
if,) is
for
3.
[34]
A
set
𝐾
𝜐]
⊂
ℝ
(𝒾,
[𝜇,
[0,
[𝜇,
and
𝔖([If
𝜑)
bor
𝜑𝑅-integrable
∈ Definition
(0,
1].
Then
𝐹𝑅-integrable
over
[𝜇,
𝜈]
if𝔖(𝒾)
and
only
iffor
𝔖to
∗𝔖
∗(𝐼𝑅)
for
all
𝒾,
𝒿is(𝒾,
∈
𝜐],
𝜉→=
∈𝜑)]
1],
where
≥
all
𝒾𝜑)
𝜐].
(11)
[𝜇,
∗ (.
(𝒾)𝑑𝒾
(𝐹𝑅)
(𝒾)𝑑𝒾
over
[𝜇,
𝜈].
Moreover,
ifare
𝔖
is
𝐹𝑅-integrable
over
𝜈],
then
∗ (𝒾,
=
𝔖
𝔖(𝒾,
𝜑)𝑑𝒾
:
𝔖(𝒾,
∈
ℛ
,
tion
over
[𝜇,
𝜈].
∗ (𝒾,
],
,
(𝒾)
[𝔖
(𝒾,
(𝒾)
(𝒾,
[𝔖
(𝒾,
=
𝜑),
𝔖
=
𝜑)]
𝜑),
for
𝔖
all
𝒾
∈
[𝜇,
for
𝜈]
all
and
𝒾
for
∈
[𝜇,
all
𝜈]
and
for
all
: [𝜇,𝜑),
𝜈]𝔖⊂∗ (𝒾,
ℝ𝔖→
:
[𝜇,
𝒦
𝜈]
are
⊂
ℝ
given
→
𝒦
by
are
𝔖
given
by
𝔖
(.
functions
𝔖
,
𝜑),
𝔖
,
𝜑):
𝐾
ℝ
called
lower
and
upper
functions
of
𝔖
.
(𝒾,
[0,
𝜑)
are
𝜑),
𝔖
Lebesgue-integrable,
𝜑)
are
Lebesgue-integrable,
then
𝔖
is
a
then
fuzzy
𝔖
Aumann-integrable
is
a
fuzzy
Aumann-integrable
funcfunc𝔖∗ (𝒾,
𝐾,
𝜉
∈
1],
we
have
∗ 𝒾,
[𝜇,
[𝜇,
∗ [34]
[𝜇,
[𝜇,
Definition
4. ∗[34][0,
𝔖:
𝜐]
is 𝜉⊂
called
a(𝐼𝑅)
harmonically
convex
function
onif,𝜐].
if 𝒾,
∗
for
all
𝒿→
∈
𝜐],
∈ℝ[0,
where
𝔖(𝒾)to
≥be
0over
all
𝒾∈
Ifall
(11)
[𝜇, 𝜈],
[𝜇,
over
[𝜇,
𝜈].
Moreover,
𝔖(0,
is∞)
𝐹𝑅-integrable
then
=1],
is (𝒾)𝑑𝒾
said
afor
convex
set,
for𝜐]
Definition
3.The
A
set
𝐾
=ℝ
𝜐]
=ifbe
𝔖
𝐾,𝑅-integrable
𝜉𝜑∈
1],
we
[𝜇,
Theorem
[49]
Let
𝔖
:harmonically
[𝜇,
𝜈]
ℝ
→
𝔽concave
a F-NV-F
whose
𝜑-cuts
define
the
I-V
for
all
∈for
(0,
1],
where
ℛ
denotes
the
collection
ofThen
Riemannian
integrable
is
called
ahave
function
on
𝜐].
Definition
4.2.
[34]
The
𝔖:
𝜐]
→∈
ℝ
is
called
a(𝐼𝑅)
harmonically
convex
function
on [𝜇,
be
an
F-NV-F.
fuzzy
integral
of
𝔖f
Definition
2.
[49]
Let
𝔖𝜐],
: [𝜇,
⊂
ℝ
→
𝔽𝐹𝑅-integrable
([
],[0,
) 1],
,𝜈]
[𝜇,
[𝜇,
all
𝒾,
𝒿ifLet
∈
𝜉⊂
where
𝔖(𝒾)
≥
0(𝒾)𝑑𝒾
for
all
𝒾𝜐].
∈then
𝜐].
Iffamily
(11)
isof𝜐]
rev
[𝜇,
=
𝔖
𝑅-integrable
over
[𝜇,
𝜈].
Moreover,
if
𝔖
is
over
𝜈],
∗→
∗ (𝒾,
tion
over
[𝜇,
tion
over
[𝜇,
𝜈].
[𝜇,
Theorem
2.
[49]
𝔖
:
[𝜇,
𝜈]
⊂
ℝ
𝔽
be
a
F-NV-F
whose
𝜑-cuts
define
the
family
for
all
𝜑
∈
(0,
1],
where
ℛ
denotes
the
collection
of
Riemannian
integrab
is
called
a
harmonically
concave
function
on
[0,
(𝒾,
(𝒾,
(𝒾,
𝐾,
𝜉
∈
1],
we
have
𝜑)
and
𝔖
𝜑)
𝜑)
and
both
𝔖
are
𝜑)
both
are
𝜑 ∈ (0,
1].(𝐹𝑅)
Then
𝜑𝜈].∈ 𝔖
(0,
is
1].
𝐹𝑅-integrable
Then
𝔖
is
𝐹𝑅-integrable
over
[𝜇,
𝜈]
if
over
and
only
[𝜇,
𝜈]
𝔖
if
and
only
if
𝔖
be
an
F-NV-F.
Then
fuzzy
integral
oo
Definition
2.
[49]
Let
𝔖
:
[𝜇,
𝜈]
⊂
ℝ
→
𝔽
(8)
(8)
([
],
)
(𝒾)𝑑𝒾
(𝒾)𝑑𝒾
(𝒾)𝑑𝒾
(𝐹𝑅)
(𝒾)𝑑𝒾
(𝐼𝑅)
(𝐼𝑅)
[𝜇,
[𝜇,
, ∗ℛ=
𝔖
= 𝔖
𝔖tions
= 4. [34]
=
𝔖
𝔖(𝒾,
𝜑)𝑑𝒾
=
:
𝔖(𝒾,
𝔖(𝒾,
𝜑)
𝜑)𝑑𝒾
∈
:
𝔖(𝒾,
𝜑)
,
∈
ℛ
,
Definition
The
𝔖:
𝜐]
ℝ
is
called
a
harmonically
convex
function
on
𝜐]
if
𝒾𝒿
∗→
([
],
)
([
],
)
𝒾𝒿
,
,
(𝒾)𝑑𝒾
(𝐼𝑅)
𝔖
∗
(𝐹𝑅)
(𝒾)𝑑𝒾
of
I-V-Fs.
𝔖𝔖
is[𝜇,𝔖
𝐹𝑅
-integrable
over
[𝜇,
𝜈]
if
𝔖define
∈
𝔽
. 𝜈]
Note
tha
[𝜇,
(𝐹𝑅)
(𝒾)𝑑𝒾
𝜈],𝜑
denoted
by
𝔖(𝒾)𝑑𝒾
,→
is≤
levelwise
by
[𝜇,
Theorem
2.
[49]
Let
:harmonically
𝜈]
ℝ
→
𝔽concave
begiven
a(𝑅)
F-NV-F
whose
𝜑-cuts
the
family
of I-Vfor
all
∈⊂
(0,
1],
where
ℛ
denotes
the
collection
of
Riemannian
integrable
fu
is
called
a𝔖
function
on
𝜐].
(𝒾)
[𝔖
(𝒾,
(𝒾,
∗𝒾(𝒾,
=
𝜑),
𝔖
𝜑)]
for
all
∈
[𝜇,
and
fo
: [𝜇,
𝜈]
ℝ
→
𝒦
are
given
by
𝔖
𝒾𝒿
be
an
F-NV-F.
Then
fuzzy
integral
of
𝔖
Definition
2.
[49]
Let
:
[𝜇,
ℝ
𝔽
∈
𝐾.
([⊂
], ⊂
)𝔖
,𝜈]
𝒾𝒿
(1
∗
+
𝜉𝔖(𝒿),
−
𝜉)𝔖(𝒾)
(11)
(𝒾,
(𝑅)
(𝐹𝑅)
∗
=
𝔖
𝜑)𝑑𝒾
,
𝔖
𝜑)𝑑𝒾
𝔖
(𝐹𝑅)
(𝒾)𝑑𝒾
tions
of
I-V-Fs.
𝔖
is
𝐹𝑅
-integrable
over
[𝜇,
𝜈]
if
𝔖
∈
𝔽
.
Note
[𝜇,
(𝐹𝑅)
(𝒾)𝑑𝒾
𝜈],∈
denoted
by
𝔖𝜉)𝒿
, (𝒾)
is
given
levelwise
by
[𝔖denotes
(𝒾,
∗𝒾(𝒾,
=
𝜑),
𝔖
𝜑)]
for ofall𝔖all
∈
[𝜇, 𝜈] an
: [𝜇,
𝜈]
⊂
ℝ
→For
𝒦
are
given
by
𝔖𝜉𝒾
𝔖all
(1
[𝜇,
[𝜇, 𝜈],
+
−
𝜉)𝒿
𝐾.
𝑅-integrable over
𝑅-integrable
[𝜇, 𝜈]. Moreover,
over [𝜇, 𝜈].
if 𝔖
Moreover,
isfor𝐹𝑅-integrable
if
𝔖
is 1].
𝐹𝑅-integrable
over
𝜈],
then
over
then
(1
(1
∗∗(𝒾,
𝜉𝒾
+
−
+𝜑)𝑑𝒾
𝜉𝔖(𝒿),
𝔖
≤𝒾𝒿
𝜉)𝔖(𝒾)
(1
(0,
∗ (0,
(𝒾,a∈
(𝑅)
(𝑅)
(𝐹𝑅)
𝜑
all
𝜑𝒾𝒿
∈(𝒾)𝑑𝒾
1],
ℱℛ
the
collection
𝐹𝑅-integra
𝔖
, (𝒾)𝑑𝒾
𝜑)𝑑𝒾
𝔖(𝒾)𝑑𝒾
∗is
([=
],then
)−
, levelwise
∗(𝐹𝑅)
(𝒾,
𝜑),
are
Lebesgue-integrable,
𝔖
fuzzy
Aumann-integrable
𝔖
of
I-V-Fs.
𝔖
is𝜑-cuts
𝐹𝑅
-integrable
over
[𝜇,
𝜈]
if
𝔖(𝒾,
∈∈𝔽[𝜇,
. ∗𝜈]
Note
that
(𝐹𝑅)
𝜈],
denoted
by
𝔖
,−(𝒾)
is
given
by
𝜉𝒾
−
𝜉)𝒿
[𝔖
(𝒾,
(𝒾,
∗𝒾(𝒾,
∗[𝜇,
=(𝑅)
𝜑),
𝔖
for
all
and
for f
:be
𝜈]
⊂
ℝ
→
𝒦
are
given
by
𝔖
𝔖tions
Theorem
[49]
Theorem
Let
[49]
𝜈]
⊂
Let
ℝ𝔖
→:denotes
[𝜇,
𝔽
𝜈]
⊂
ℝ(𝒾,
→𝔖
𝔽𝜑(𝒾,
afor
F-NV-F
whose
be
a𝔖
F-NV-F
whose
define
𝜑-cuts
the
family
define
of
I-V-Fs
the
family
of𝜑)]
I-V-Fs
𝜉𝒾
+
𝜉)𝒿
(𝒾,
∈
𝐾.
(0,
∗𝜑)
𝜑)
and
𝔖
both
𝜑[𝜇,
∈
(0,
1].
Then
𝔖
is
𝐹𝑅-integrable
over
[𝜇,
𝜈]
if(1
and
only
if
𝔖
all
∈
(0,
1].
For
all
𝜑 (1
∈
1],
ℱℛ
denotes
the
collection
of
all
𝐹𝑅-inte
(1
∗([+
+
𝜉𝔖(𝒿),
≤
−
𝜉)𝔖(𝒾)
(𝒾,
(𝑅)
(𝐹𝑅)
(𝒾)𝑑𝒾
],
)
∗
,
=
𝔖
𝜑)𝑑𝒾
,
𝔖
𝜑)𝑑𝒾
𝔖
(𝒾,
∗𝜑)
for all 𝜑 ∈2.(0,
for1],all
where
𝜑𝔖∈:2.[𝜇,
(0,
ℛ
1],
where
ℛ
the
collection
denotes
the
of
collection
Riemannian
of
Riemannian
integrable
funcintegrable
func𝜑),
𝔖
𝜑)
are
Lebesgue-integrable,
then
𝔖
is
a
fuzzy
Aumann-integrab
𝔖
∗ if and only if 𝔖 (𝒾, 𝜑) and 𝔖(11)
([ , ], )
([
],
)
,
∗
(1
𝜉𝒾
+
−
𝜉)𝒿
(𝒾,
NV-Fs
over
[𝜇,
𝜈].
𝜑)
𝜑
∈
(0,
1].
Then
𝔖
is
𝐹𝑅-integrable
over
[𝜇,
𝜈]
(1
𝜉𝒾
+
−
𝜉)𝒿
(0,
∗
for
all
𝜑𝔖∈𝜐],(𝒾,
(0,
1].
For
all
𝜑 ∈ ∗=𝔖(𝒾)
1], ≥ℱℛ
thefuzzy
collection
of 𝜑)
all ∈
𝐹𝑅-integrabl
tion
over
[𝜇,
𝜈].
[𝜇,
[0,
([for
],then
) denotes
, (𝒾)𝑑𝒾
𝒾,
𝒿=
∈𝜑),
𝜉 𝜑)
∈𝔖𝔖
where
0𝜈]
all
𝒾[𝜇,
∈all[𝜇,
If𝔖
(11)
is
reversed
𝔖
(𝒾,
∗[𝜇,
are
Lebesgue-integrable,
𝔖
is 𝜐].
aover
Aumann-integrable
𝔖
(𝐹𝑅)
(𝒾)𝑑𝒾
NV-Fs
over
𝜈].
𝔖
𝔖
=
𝔖(𝒾,
𝜑)𝑑𝒾
:∗𝔖(𝒾,
ℛ𝜑)
∗over
[𝔖
(𝒾,
(𝒾)
(𝒾,
[𝔖
(𝒾,
(𝒾,𝒾(𝐼𝑅)
(𝒾,then,
=
𝜑)]
𝜑),
for
𝔖all
𝜑)]
[𝜇,
for
all
and
𝒾that,
for
∈𝔽and
𝜈]
for
: [𝜇,of𝜈] I-V-Fs.
⊂ ℝ𝔖
→: [𝜇,
𝒦
are
⊂𝐹𝑅
ℝ
given
→𝒦
byfor
are
𝔖all
given
by
𝔖
𝔖
𝜑)
and
𝔖∗𝜑)
𝜑(𝒾)
∈
(0,
1].
Then
is1],
over
[𝜇,
𝜈]
if that,
([ then,
], )fu,
, both
[𝜇,
tion
over
[𝜇,
𝜈].
𝑅-integrable
over
[𝜇,
𝜈].
Moreover,
if∈
𝔖.𝔖
is
𝐹𝑅-integrable
𝜈],
then
(𝒾)𝑑𝒾
(𝒾)𝑑𝒾
[𝜇,
[0,
[𝜇,
tions
tions
𝔖of𝜈]is
I-V-Fs.
-integrable
𝔖
is
𝐹𝑅
-integrable
[𝜇,
𝜈]𝜑),
if
over
[𝜇,
𝜈]
𝔖
if
∈→
𝔽
Note
∈if(𝒾)𝑑𝒾
.harmonically
if𝒾only
Note
if(11)
∗∈
∗𝐹𝑅-integrable
∗ (𝒾,
for
all
𝒾,
𝒿over
𝜐],
𝜉(𝐹𝑅)
∈
1],
where
𝔖(𝒾)
≥
0
for
all
∈ and
𝜐].
Ifall
is
reversed
∗(𝐹𝑅)
∗ (𝒾,
(𝐹𝑅)
(𝒾)𝑑𝒾
(𝐼𝑅)
𝔖
=
𝔖
=
𝔖(𝒾,
𝜑)𝑑𝒾
:
𝔖(𝒾,
∈
ℛ
[𝜇,
[𝜇]
Definition
4.
[34]
The
𝔖:
𝜐]
ℝ
is
called
a
convex
function
on
NV-Fs
[𝜇,
𝜈].
([
(𝒾,
(𝒾,
(𝒾,
(𝑅)
(𝑅)
(𝑅)
(𝑅)
(𝐹𝑅) 𝔖(𝒾)𝑑𝒾
(𝐹𝑅)
(𝒾)𝑑𝒾
[𝜇,
=
𝔖
=
𝜑)𝑑𝒾
,
𝔖
𝜑)𝑑𝒾
𝔖
,
𝜑)𝑑𝒾
𝔖
𝜑)𝑑𝒾
𝔖
𝑅-integrable
over
[𝜇,
𝜈].
Moreover,
if
𝔖
is
𝐹𝑅-integrable
over
𝜈],
then
[𝜇,
(𝒾)𝑑𝒾
(𝐼𝑅)
is
called
a
harmonically
concave
function
on
𝜐].
=
𝔖
tion
∗[𝜇,
∗ The 𝔖:
[𝜇,a 𝜐],
for allis𝒾,
𝒿 ∈over
𝜉 𝜈].
∈ [0,4.1],
where
≥→
0∗𝔖for
all
𝒾𝜐].
∈ [𝜇,
𝜐].
If 𝜑)𝑑𝒾
(11) is
reversed
then,
𝔖 ,,on
[𝜇,
Definition
[34]
𝜐]
ℝ𝐹𝑅-integrable
called
harmonically
convex
function
∗ (𝒾,
(𝒾)𝑑𝒾
(𝐹𝑅)
(𝒾)𝑑𝒾
(𝐼𝑅)
∗Then
𝔖
=𝔖(𝒾)
=
𝔖(𝒾,
𝜑) ∈set,
ℛ
[𝜇,
(𝒾)𝑑𝒾
(𝐼𝑅)
called
harmonically
concave
function
on
=
𝔖
([
], )all
(0,
[𝜇,
[𝜇,
(𝒾,
(𝒾,
(𝒾,
=is
∞)
isawhose
said
to
be
a: 𝔖(𝒾,
convex
if, ,for
Definition
3.
[34]
A
set
𝐾
=
𝜐]
⊂
ℝ
𝑅-integrable
𝜈].
Moreover,
if
𝔖
is
over
𝜈],
then
𝜑)
and
𝔖
𝜑)
𝜑)
and
both
𝔖
are
𝜑)
both
are
𝜑
1].𝔖∗Then
𝜑
(0,
is
1].𝔖
𝐹𝑅-integrable
is Lebesgue-integrable,
𝐹𝑅-integrable
over
[𝜇,
𝜈]
if
over
and
only
[𝜇,
𝜈]
if
𝔖
if
and
only
(𝒾,
(𝒾,𝔖
(𝒾, 𝜑)𝔖are
𝜑),
are
𝜑),
Lebesgue-integrable,
then
𝔖
is
a
then
fuzzy
𝔖
Aumann-integrable
is
a
fuzzy
Aumann-integrable
funcfunc𝔖∗∈(𝒾,(0,
𝔖𝜑)
∗
∗
∗∈
[𝜇,
[𝜇,
Theorem
2.
[49]
Let
𝔖
:
[𝜇,
𝜈]
⊂
ℝ
→
𝔽
be
a
F-NV-F
𝜑-cuts
define
the
family
of
4. (0,
[34]1],
The
𝔖:
→
ℝ
a(0,
harmonically
convex
function
[𝜇,
∞)
is whose
saidofto
be a convex
set,on
if, forI-𝜐
Definition
3.
[34]
A𝔖
set
𝐾([𝜈]
=
𝜐]is
⊂called
ℝ be=
𝒾𝒿
[𝜇,
(𝒾)𝑑𝒾
(𝐼𝑅)
is calledDefinition
afor
harmonically
concave
function
on
𝜐].
=
𝔖
all
𝜑 1],
∈
where
ℛ𝜐]
the
collection
Riemannian
integrable
(9)
(9)
, ],⊂ )ℝdenotes
2.
[49]
Let
:(𝒾)𝑑𝒾
[𝜇,
→
𝔽denotes
a(1
F-NV-F
𝜑-cuts
the
family
∗ (𝒾,define
𝐾,
𝜉Theorem
∈ [0,
we
have
𝒾𝒿
tion over [𝜇,over
tion
𝜈]. [𝜇,
over
𝜈]. [𝜇, 𝜈].
+
𝜉𝔖(𝒿),
𝔖
≤
−
𝜉)𝔖(𝒾)
(0,
[𝜇,
=
∞)
is
said
to
be
a
convex
set,
if,
for
all 𝒾
Definition
3.
[34]
A
set
𝐾
=
𝜐]
⊂
ℝ
for
all
𝜑
∈
(0,
1],
where
ℛ
the
collection
of
Riemannian
integra
(𝒾,
(𝑅)
(𝑅)
(𝐹𝑅)
[𝜇,
[𝜇,
=
𝔖
𝜑)𝑑𝒾
,
𝔖
𝜑)𝑑𝒾
𝔖
𝑅-integrable
𝑅-integrable
𝜈]. [𝜇,
Moreover,
over
if 𝔖
Moreover,
is
𝐹𝑅-integrable
if
𝔖
is
𝐹𝑅-integrable
over
𝜈],
then
over
𝜈],
then
∗ 𝜑-cuts define ∗the family of
([ 𝔖
)be =
, ],(𝒾)
∗ ≤
𝐾,
∈⊂[0,
1],
we
Theorem
2.
[49]
: [𝜇,
𝜈]𝜑⊂∈𝔖
𝔽
a(F-NV-F
whose
[𝔖
(1
(1
𝜑),
all
𝜈] andI-V
fo
: [𝜇,
ℝ(0,
→Let
𝒦𝔖
are
𝔖for
𝜉𝒾
+
−
𝜉)𝒿
+) (𝜉𝔖(𝒿),
(0,
all𝜉𝜈]𝜑
∈
1].
For
all
1],
ℱℛ
denotes
𝒾𝒿
(𝜉)𝔖(𝒾)
)collection
) 𝒾of(∈all[𝜇,)𝐹𝑅-integr
)→
( 𝔖− (𝒾,
) (𝒾,
)the𝜑)]
( for
(have
)given
(ℝby
numbers. Let
𝔴∈
𝔽numbers.
Let
𝔵, 𝔴𝜌if,
∈∈Let
𝔽ℝ,
𝔵,≤𝔴 ∈
Let
𝔵,𝜌𝔴we
and
then
and
we
𝜌 𝔽∈ have
ℝ,
and
then
∈𝜑∈ℝ,
then
and
𝜌 have
∈ ℝ, then we
[𝔵]numbers.
[𝔴]
𝔵numbers.
≼𝔵,𝔴
if and
only
for
all
∈𝔽have
(0,
1],we
(4)have
≼ 𝔴 if and only if, [𝔵] ≤ [𝔴] for all 𝜑 ∈ (0, 1],
(4)
It is a partial order 𝔵relation.
[𝔵+𝔴] = [𝔵]
[𝔵++
[𝔵] 𝔴]+ [𝔴]
[𝔴] ,= [𝔵] +(5)
[𝔴]
𝔴][𝔴]=,[𝔵+
= [𝔵]
, [𝔵+
+ 𝔴]
It is a partial order relation.
We now
use mathematical
operations to examine some of the characteristics of fuzzy
It is a partial
relation.
Weorder
now use
mathematical operations to examine some of the characteristics of fuzzy
[𝔵 × 𝔴] = [𝔵]
[𝔵 ××𝔴]
[ 𝔴]=[𝔵,[𝔵]
[ 𝔴]
× 𝔴]
× [𝔴]
×[=
𝔴][𝔵], [𝔵 ×
𝔴] =
, [𝔵] ×(6)
numbers.
Letuse
𝔵, 𝔴 ∈ 𝔽 and 𝜌operations
∈ ℝ, then we
have
We
now
to examine
numbers.
Letmathematical
𝔵, 𝔴 ∈ 𝔽 and 𝜌 ∈ ℝ, then
we havesome of the characteristics of fuzzy
numbers. Let 𝔵, 𝔴 ∈ 𝔽 and 𝜌 ∈ ℝ, then
we have
(7)
[𝜌. 𝔵] = 𝜌.
[𝜌.[𝔵]
𝔵] , = 𝜌. [𝔵]
𝔵] = 𝜌. [𝔵][𝜌.
(5)𝔵]8 of=18𝜌. [𝔵]
[𝔵+𝔴]
= [𝔵] +[𝜌.[𝔴]
Symmetry 2022, 14, 1639
(5)
[𝔵+𝔴] = [𝔵] + [𝔴] ,
(5)
[𝔵+
𝔴]𝔴] ==[𝔵][𝔵] +×[𝔴]
(6)
[𝔵 ×
[ 𝔴], ,
(6)
[𝔵number
[𝔵]number
[ 𝔴]1.
×A
𝔴]fuzzy
=
×fuzzy
,number
Definition 1.
Definition
[49] A fuzzy
Definition
1.
[49]
1.
valued
Definition
[49]
A
map
valued
𝔖
[49]
:
𝐾
⊂
A
map
ℝ
fuzzy
valued
→
𝔖
𝔽
:
number
𝐾
⊂
map
ℝ
→
𝔖
valued
:
𝔽
𝐾
⊂isℝcalled
map
→For
𝔽𝔖F-NV-F.
:is𝐾 calle
⊂ℝ
is
called
F-NV-F.
each
(6)
[𝔵 × 𝔴]
[𝔵]
[
=
×
𝔴]
,
(7)
[𝜌.
[𝔵]
𝔵]
=
𝜌.
e
𝜑
∈
(0,
1]
whose
𝜑
∈
(0,
𝜑
1]
-cuts
𝜑
whose
∈
(0,
define
1]
𝜑
-cuts
whose
the
𝜑
∈
family
define
(0,
𝜑
1]
-cuts
whose
the
of
define
I-V-Fs
family
𝜑
-cuts
the
𝔖
of
family
I-V-Fs
define
of
𝔖
the
I-V-Fs
family
𝔖
of
I-V-Fs
:
𝐾
⊂
ℝ
→
𝒦
:
𝐾
are
⊂
ℝ
given
→
:
𝒦
𝐾
⊂
by
are
ℝ
→g
, s𝜌.
Theorem 5. Let S ∈ HFSX ([µ,[𝜌.υ𝔵]
], F0=
),[𝔵]whose ϕ-cuts define the family of I-V-Fs
(7)
+
∗
∗
∗
∗
∗
(7)
[𝜌.
[𝔵]
𝔵]
=
𝜌.
(𝒾,K(𝒾)
(𝒾,
(𝒾,
(𝒾)𝜑),
[𝔖
(𝒾,
[𝔖𝜑)]
(𝒾,
𝔖∗𝜑)]
𝜑),
𝔖
= [𝔖
=
for
all
𝜑)]
𝒾𝜑),
𝐾.
𝔖
for
=(𝒾,
Here,
all
𝒾 ,for
∈ϕfor
𝜑),
𝔖
all
Here,
𝜑
∈ϕ∈
for
𝐾.(0,
for
Here,
each
1]all
𝜑
𝒾for
∈
∈end
each
(0,[µ,
1]
Here,
point
for
(0,
end
1]
eac
p
S ϕ : [𝔖
µ, (𝒾)
υ] ⊂=R[𝔖→
by𝔖
S
=
, each
S
for
allthe
∈𝐾.
υ𝜑the
(𝔖
)∈(𝒾)
[S
)𝐾.
( 𝒾 ,𝜑)]
)]
],∈ real
∗∗((𝒾,
∗𝔖
∗ϕ(𝒾,
C are given
∗ (. (.
∗ (. (.
∗ (.
∗ (.
efuzzy
(.
(.
functions
functions
𝔖
𝔖
functions
𝔖
,
𝜑),
𝔖
,
𝜑):
,
𝜑),
𝐾
→
𝔖
ℝ
,
are
𝜑):
,
𝜑),
called
𝐾
𝔖
→
ℝ
,
lower
𝜑):
are
𝐾
,
called
𝜑),
and
→
ℝ
𝔖
upper
are
lower
,
𝜑):
called
functions
𝐾
and
→
lower
ℝ
upper
are
of
and
called
functions
𝔖
.
upper
lower
of
functio
and
𝔖
.
Definition
[49]
number
valued
map
𝔖
:
𝐾
⊂
ℝ
→
𝔽
is
called
F-NV-F.
For
each
ϕ ∈ [1.
0,
1]. IfAS
∈𝔖functions
F
R
,
then
∗
([µ,
υ], ϕ) ∗ valued ∗map 𝔖: 𝐾 ⊂ ℝ∗ → 𝔽 is called F-NV-F. For each
Definition 1. [49] A fuzzy
number
𝜑 ∈ (0, 1] 1.whose
𝜑
-cutsnumber
define valued
the family
of: 𝐾I-V-Fs
𝐾is ⊂
ℝ → F-NV-F.
𝒦 are For
given
by
Definition
[49]
A
fuzzy
map
𝔖
⊂ I-V-Fs
ℝ →𝔖𝔽 :𝔖
each
→ 𝒦 are
𝜑 ∈(0, 1] whose∗ 𝜑 -cuts Zdefine
the
family
of
:called
𝐾⊂be
⊂ℝiℝ
given
by
h
υ
e
be
an
F-NV-F.
an
Then
F-NV-F.
fuzzy
be
an
Then
integral
F-NV-F.
fuzzy
be
ofThen
an
𝔖
integral
F-NV-F.
over
fuzzyo
Definition
2.
Definition
[49]
Let
Definition
𝔖
2.
:
[𝜇,
[49]
𝜈]
Let
⊂
2.
ℝ
𝔖
Definition
→
[49]
:
[𝜇,
𝔽
𝜈]
Let
⊂
𝔖
ℝ
:
2.
[𝜇,
→
[49]
𝔽
𝜈]
Let
→
𝔖
:
𝔽
[𝜇,
𝜈]
⊂
ℝ
→
𝔽
µυ
(𝒾)1]
(𝒾, ∗𝜑)]
= [𝔖
for all
∈ 𝐾. Here,
for1 each
(0,
1]
the1 are
end1 given
point by
real
S( 𝒾 )family
∈2µυ
(0,
whose
-cuts
define
the
ofHere,
I-V-Fs
𝔖eeach
:𝜑𝐾 ∈⊂
ℝ
→
𝒦
∗ (𝒾, 𝜑),𝜑𝔖
s−1 e𝜑𝔖
e
e
(𝒾)
[𝔖
(𝒾,
(𝒾,
𝔖
=
𝜑),
𝔖
𝜑)]
for
all
𝒾
∈
𝐾.
for
𝜑
∈
(0,
1]
the
end
point
real
FR
S
(
µ
)
+
S
(
υ
)
4
S
4 3
4
d
4
3
4
+
,
(21)
(
)
∗
2
1
∗
s
2 𝜈],called
[𝜇,
(𝐹𝑅)
(𝒾)𝑑𝒾
(𝐹𝑅)
[𝜇,given
(𝒾)𝑑𝒾
(𝐹𝑅)
(𝒾)𝑑𝒾
(𝐹𝑅)
(𝒾)𝑑𝒾
by
denoted
𝔖
by
denoted
by
𝔖
by1]
, is
levelwise
,𝜑is∈𝔖
given
by
levelwise
,the
is given
levelwise
, is given
by levelwise by
∗𝔖
(. ,𝜈],
𝜑),υdenoted
, [𝜇,
𝜑):𝜈],
𝐾→
ℝ[𝜇,
are
lower
and
upper
functions
of𝔖𝔖
.
µ(𝒾)
+ υ= [𝔖∗𝔖
−
s𝜈],
+
1denoted
2end
2by
(𝒾,
(𝒾,(.µ𝜑)]
µ
𝔖functions
∈ 𝐾.
each
(0,
functions∗ 𝜑),
𝔖∗ (.𝔖, 𝜑),
𝔖∗ (. ,for
𝜑):all
𝐾 →𝒾 ℝ
areHere,
calledfor
lower
and
upper
functions
ofpoint
𝔖. real
∗
(. , 𝜑), 𝔖 (. , 𝜑): 𝐾 → ℝ are called
functions 𝔖∗where
functions#of 𝔖.
" lower and upper integral of 𝔖 over
be
an eF-NV-F.
Then fuzzy
Definition 2. [49] Let(𝐹𝑅)
𝔖: [𝜇, 𝜈] (𝒾)𝑑𝒾
⊂ (𝐹𝑅)
ℝ → 𝔽 (𝐼𝑅)
e
e
(𝒾)𝑑𝒾
(𝒾)𝑑𝒾
(𝒾)𝑑𝒾
(𝒾)𝑑𝒾
(𝒾)𝑑𝒾
(𝐹𝑅)
(𝒾)𝑑𝒾
(𝐹𝑅)
(𝐼𝑅)
(𝐼𝑅)
(𝐼𝑅)
𝔖 𝜈] ⊂ ℝ
𝔖F-NV-F.
𝔖(𝒾)𝑑𝒾
𝔖S=
𝔖
= 2µυ
𝔖(𝒾,
𝜑)𝑑𝒾
=𝔖 integral
:=
𝔖(𝒾,
𝔖(𝒾,
𝜑)
=𝜑)𝑑𝒾
ℛ:𝔖(𝒾,
𝔖(𝒾,
𝜑)
ℛ(8)
1=
S(µbe
) +an
(υ) =e Then
fuzzy
of∈ 𝔖
𝔖
over
Definition 2. [49] Let 𝔖: [𝜇,
→𝔖𝔽
([
)=,∈: 𝔖(𝒾,
([𝔖(𝒾
],
, ],𝜑)𝑑𝒾
,𝜑)
e
3
=
+
S
,
[𝜇,
(𝐹𝑅)
(𝒾)𝑑𝒾
𝜈], denoted
byLet 𝔖: [𝜇,𝔖𝜈] ⊂1 ℝ, →
is given
levelwise
by Then fuzzy integral
be
an
F-NV-F.
of
𝔖
over
Definition
2.
[49]
𝔽
s+
2
µ+υ
[𝜇, 𝜈], denoted by (𝐹𝑅) 𝔖(𝒾)𝑑𝒾
, is1 given levelwise
by
[𝜇, 𝜈], denoted
(𝒾)𝑑𝒾
byall(𝐹𝑅)
𝔖1],
isfor
given
levelwise
by
for
𝜑 ∈ (0,
for
all where
𝜑, ∈
(0,
all
ℛ
1],
∈ (0,
for
1],ℛall
where
∈) (0,
ℛ([1],, ],where
ℛ([ ,the
denotes
collection
denotes
ofcollection
Riemannian
ofdenotes
Riemannian
integrable
the
of Riemanni
collection
funcintegra
, 𝜑],the
([𝜑,where
],
) denotes
([
)the
], )collection
4µυ
4µυ
s
−
2
e
e
(8)
(𝒾)𝑑𝒾
(𝐹𝑅)
(𝒾)𝑑𝒾
(𝐼𝑅)
e
𝔖
=
𝔖
=
𝔖(𝒾,
𝜑)𝑑𝒾
:
𝔖(𝒾,
𝜑)
∈
ℛ
,
(𝒾)𝑑𝒾
(𝒾)𝑑𝒾
(𝐹𝑅)[𝜇,
(𝐹∈
tions of I-V-Fs.
tions 𝔖of
tions
𝐹𝑅
of𝔖
is tions
𝐹𝑅 -integrable
over
𝔖 ofis
[𝜇,
𝐹𝑅𝜈]-integrable
over
if 𝔖(𝐹𝑅)
is[𝜇,,𝐹𝑅
𝜈]over
𝔖
-integrable
if
[𝜇,
𝜈]𝔽 if𝔖
𝜈]
𝔖(𝒾)𝑑𝒾
.over
Note
∈that,
𝔽
. ifif
Not
3 isI-V-Fs.
=
2 -integrable
SI-V-Fs.
+ I-V-Fs.
S
([
], ) ∈
, (𝐹𝑅)
(𝐹𝑅) 𝔖(𝒾)𝑑𝒾 =2 (𝐼𝑅) 𝔖 (𝒾)𝑑𝒾
=
𝔖(𝒾, 𝜑)𝑑𝒾
µ + 3υ
3µ + :υ𝔖(𝒾, 𝜑) ∈ ℛ([ , ], ) , (8)
∗ (𝒾,
∗=
∗ (𝒾,: 𝔖(𝒾, 𝜑) ∈ ℛ
(8)
(𝒾)𝑑𝒾
(𝐹𝑅)𝔖∗ (𝒾,𝔖𝜑),
(𝒾)𝑑𝒾
𝔖
𝔖(𝒾,
𝜑)𝑑𝒾
,
(𝒾,(𝐼𝑅)
(𝒾,
(𝒾,
(𝒾,
𝔖∗ (𝒾,
𝜑)
𝜑),
are𝔖Lebesgue-integrable,
𝜑)
𝜑),
are
𝔖
Lebesgue-integrable,
𝜑)
are
𝜑),
then
Lebesgue-integrable,
𝔖
𝔖
𝜑)
is
are
a
then
fuzzy
Lebesgue-integrable,
𝔖
Aumann-integrable
is
then
a
fuzzy
𝔖
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a
fuzzy
then
func𝔖
Auman
is
a
𝔖∗=
𝔖
𝔖
([ , ], )
∗
∗
for all and
𝜑 ∈ (0,
1],
where
ℛ
denotes
the
collection
of
Riemannian
integrable
func, ], ) tion
tion
over
[𝜇,tion
𝜈].([over
[𝜇, 𝜈].over [𝜇, 𝜈].
tion
over [𝜇, 𝜈]. ∗
for all
𝜑14,
∈ x(0,
1],PEER
where
ℛ
the
integrable func∗ collection of Riemannian
([ ,3], )=denotes
Symmetry
FOR
REVIEW
3
32 (𝐹𝑅)
= [3Riemannian
[31the
∗ ,[𝜇,
2 ∗ ,𝔖3
2 ]. ∈ 𝔽
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1 over
(𝒾)𝑑𝒾
tions
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𝔖
𝐹𝑅
. Note that,
for
all Symmetry
𝜑
∈I-V-Fs.
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ℛPEER
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func-if
2022,where
14, xisFOR
REVIEW
([-integrable
],
)
, -integrable over [𝜇, 𝜈] if of(𝐹𝑅)
(𝒾)𝑑𝒾
tions
of
I-V-Fs.
𝔖
is
𝐹𝑅
𝔖
∈
𝔽
. Note that, if
Symmetry 2022,∗ 14, x FOR PEER REVIEW
e
(𝐹𝑅)
(𝒾)𝑑𝒾
Theorem
2.
Theorem
[49]
Let
𝔖
2.
Theorem
:
[𝜇,
[49]
𝜈]
Let
⊂
ℝ
2.
𝔖
→
[49]
:
[𝜇,
Theorem
𝔽
𝜈]
Let
⊂
𝔖
ℝ
:
[𝜇,
2.
→
[49]
𝔽
𝜈]
⊂
Let
ℝ
→
𝔖
:
[𝜇,
𝜈]
⊂
ℝ
→
𝔽
be
a
F-NV-F
be
whose
a
F-NV-F
𝜑-cuts
be
whose
a
define
F-NV-F
𝜑-cuts
the
whose
bedefine
a F-NV-F
𝜑-cuts
of the
I-V-Fs
family
define
whos
𝜑),I-V-Fs.
𝔖 (𝒾,
𝜑)
are
Lebesgue-integrable,
then
𝔖
is
a
fuzzy
Aumann-integrable
func𝔖∗ (𝒾,of
tions
𝔖
is
𝐹𝑅
-integrable
over
[𝜇,
𝜈]
if
𝔖
∈
𝔽
.
Note
that,
iffamily
If S
∈ HFSV ([µ, υ], F0 , s), then inequality (21) is reversed.
𝔖∗ (𝒾, 𝜑) are Lebesgue-integrable,
then 𝔖 is a fuzzy Aumann-integrable func𝔖∗ (𝒾, 𝜑),
∗
∗
∗
∗
tion
over
[𝜇,14,
𝜈].x: FOR
(𝒾,
𝜑),
𝔖
are
Lebesgue-integrable,
fuzzy
Aumann-integrable
func𝔖Symmetry
(𝒾)
[𝔖
(𝒾,
(𝒾,
(𝒾,
(𝒾)
[𝔖
(𝒾,
(𝒾)
[𝔖
=a by
𝜑),
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=
𝜑),
for
=𝔖
all
𝒾𝔖
𝜑)]
∈
𝜑),
[𝜇,
for
𝔖𝜈]
=(𝒾,
all
and
𝜑)]
∈for
for
𝜑),
[𝜇,all
all
𝜈]
𝔖∗ (𝒾
a𝒾
⊂ℝ
→
𝒦 𝜈]are
⊂
: [𝜇,
→𝒦
𝜈]then
by⊂are
𝔖
ℝ𝔖→
given
:is
𝒦
[𝜇,
𝜈]
are
⊂
given
ℝ(𝒾)
→𝒦
by [𝔖
𝔖are
given
by
𝔖𝜑)
𝔖REVIEW
𝔖 ℝgiven
2022,
h𝜈]PEER
i: [𝜇,
∗ (𝒾,tion
∗𝔖
∗𝜑)]
∗ (𝒾,
∗𝒾(𝒾,
over
[𝜇,[𝜇,
𝜈].
Symmetry
2022,
14, x FOR
REVIEW
2µυPEER
Proposition
1.
[51]
Let
𝔵,
𝔴
∈
𝔽
.
Then
fuzzy
order
relation
“
≼
”
is
given
o
Proof.
Take
µ,
,
we
have
tion
over
[𝜇,
𝜈].
∗ (𝒾,
∗ (𝒾,
Symmetry 2022, 14,
PEER
REVIEW
Proposition
1.
Let
𝔵,Then
𝔴
∈𝔽
Then
fuzzy
order
relation
≼𝔖”are
is 𝜑
gi
µ
+
υ∈ (0,
𝜑)
and
𝜑)
an
𝜑 x∈FOR
(0, 1].
Then
𝜑
𝔖 1].
is 𝜑
𝐹𝑅-integrable
Then
∈ (0,𝔖1].isThen
𝐹𝑅-integrable
over
𝜑 [51]
∈𝔖𝔵,
(0,
[𝜇,
is
1].
𝜈]
ifover
𝔖
only
[𝜇,.fuzzy
is
𝜈]
ifover
𝐹𝑅-integrable
if𝔖and
[𝜇,only
𝜈]relation
if and
over
𝔖𝔖
only
[𝜇,
if𝜈]”and
𝔖both
if“∗ (𝒾,
and
only
∗ (𝒾,
∗ (𝒾,
Let
𝔴
∈𝐹𝑅-integrable
𝔽𝜑-cuts
.and
Then
“𝜑)
≼𝜑)
is
given
on
Theorem 2. [49] Let 𝔖: [𝜇, 𝜈]Proposition
⊂
ℝ → 𝔽 be1.a[51]
F-NV-F
whose
define
theorder
family
ofI-V-Fs
[𝔵]
[𝔴]
𝔵
≼
𝔴
if
and
only
if,
≤
for
all
𝜑
∈
(0,
1],



Theorem𝑅-integrable
2. [49] Let
𝔖: [𝜇,[𝜇,
𝜈] 𝜈].
⊂𝑅-integrable
ℝ
→ 𝔽[𝜇,be𝜈].
F-NV-F
define
the
family
ofis𝜈],
I-V-Fs
[𝜇,
[𝜇,
[𝜇, 𝜈],
over
Moreover,
over
ifaover
Moreover,
𝔖
𝑅-integrable
is[𝜇,𝐹𝑅-integrable
𝜈].
Moreover,
over
isand
[𝜇,
𝐹𝑅-integrable
over
if𝜈].
𝔖Moreover,
is
𝐹𝑅-integrable
over
if 𝔖
over
𝐹𝑅-integrable
thenall
ov1
[𝔵]then
[𝔴]
𝔵whose
≼if
𝔴𝔖
if𝜑-cuts
only
if,𝜈],
≤
for
𝜑 ∈then
(0,
4µυ
2µυ
2µυ
4µυ𝑅-integrable
µare
µ µ+
µ∈µ[𝜇,
Theorem
:µ[𝜇,
𝜈] ⊂Proposition
ℝby→𝔖𝔽 (𝒾)
be =
a1.F-NV-F
whose
define
family
ofrelation
I-V-Fs
[𝔖
𝔖
𝜑)]
for
all
𝒾the
𝜈][𝔴]
and
for all“ 𝜑
: [𝜇,
ℝ →Let
𝒦𝔖
𝔖
[𝔵]
𝔵𝜑),
≼
𝔴∗𝔴(𝒾,
if∈𝜑-cuts
and
only
if,fuzzy
≤
∈
1], o
∗µ(𝒾,
υ 𝜈]2.⊂[49]
+υgiven
µ
+
υ
+
υ
∗
[51]
Let
𝔵,
𝔽
.
Then
order
≼
” (0,
is given
s e
e
e




It 
is a4
partial
order
relation.
(𝒾)
[𝔖
(𝒾,
(𝒾,
=
𝜑),
𝔖
𝜑)]
for
all
𝒾
∈
[𝜇,
𝜈]
and
for
all
by
𝔖
𝔖 : [𝜇, 𝜈]+⊂ ℝ → 𝒦 are given
e
2S
S
+
S
.
∗[51]∗2µυ
Proposition
1.
Let 𝔵, 𝔴 ∈ 𝔽 . Then fuzzy
order relation “ ≼ ” is gi
2µυ
2µυ
2µυ
It
is a=ξµ
partial
order
relation.
∗𝜈]
(𝒾)
[𝔖
(𝒾,
𝜑),
𝒾−
∈ξ )[𝜇,
for
all
[𝜇,
⊂
ℝ
→
are
given
byis 𝔖
Symmetryξµ
2022,
14,
x FOR
PEER
REVIEW
FOR
PEER
4 ofon
(𝒾, all
(𝒾,
𝜑)
and
both
are“of≼∗ the
(0,
1].
Then
is)µx𝐹𝑅-integrable
over
[𝜇,
𝜈]
ifrelation.
and
if𝔽𝔖.operations
Proposition
1.
[51]
Let
𝔴
∈𝜑)]
Then
relation
” ischaracter
given
∗ (𝒾,
1𝒦𝔖
− ξ14,
+
ξ14,
µ𝔖order
+
ξ and
+𝔖(𝜑
1:∈−
ξ𝜈]
+
−
ξ𝔖
(2022,
(1fuzzy
)Symmetry
(1mathematical
)𝔵,only
∗for
We
now
use
to
examine
some
∗𝜑)
It
aREVIEW
partial
order
∗+
∗ (𝒾,
µ(0,
+xυ FOR
µ+
µ𝔴
+υonly
µ[𝔴]
υ 𝜑)
[𝔵]
Symmetry 2022,
PEER REVIEW
xυ(𝐹𝑅)
FOR
PEER
REVIEW
𝔵
≼
if
and
only
if,
≤
for
all𝔖
𝜑(𝒾,
∈
(0,
1],
(𝒾,
(𝒾,
𝜑)
and
𝔖
both
are
𝜑 ∈14,
1].Symmetry
Then
𝔖2022,
is 𝐹𝑅-integrable
over
[𝜇,
𝜈]
if
and
if
𝔖
(𝒾,
(𝒾,
(𝒾,
(𝒾,
(𝒾,
(𝑅)
(𝑅)
(𝑅)
(𝑅)
(𝑅)
(𝑅)
(𝑅)
(𝐹𝑅)
(𝒾)𝑑𝒾
(𝐹𝑅)
(𝒾)𝑑𝒾
(𝐹𝑅)
(𝒾)𝑑𝒾
=
𝔖
=
𝜑)𝑑𝒾
,
=
𝔖
𝜑)𝑑𝒾
𝔖
𝔖
,
𝜑)𝑑𝒾
=
𝜑)𝑑𝒾
,
𝔖
𝜑)𝑑𝒾
𝜑)𝑑𝒾
𝔖
,
𝔖(𝒾)𝑑𝒾
𝔖
𝔖
𝔖
We now use mathematical
to
examine
some
of
the
∗ operations
∗
∗
∗
∗
∗
[𝔵]
[𝔴]
𝔵if≼𝔖
and
only
≤ some
allcharacteri
𝜑 ∈4 cha
(0,
Symmetry 2022,𝜑14,
FOR
Symmetry
PEER
REVIEW
2022,
x FOR
PEER
REVIEW
of 11
(𝒾,if
(𝒾,
numbers.
∈𝔵𝔽≼only
and
𝜌𝔴
ℝ,
then
we
have
𝜑)
and
𝔖if,
𝜑)
both
arealloffor
∈ x(0,
1].
Then
𝔖[𝜇,
is14,
𝐹𝑅-integrable
over
[𝜇,𝐹𝑅-integrable
𝜈]𝔵, 𝔴
ifmathematical
and
[𝜇,
We
use
operations
to
examine
the
𝑅-integrable
over
𝜈].
Moreover,
if
𝔖now
isLet
over
𝜈],
then
∗∈
[𝔵]
[𝔴]
𝔴
if
and
only
if,
≤
for
𝜑
∈
(0,
1],
Let
𝔵, 𝔴 ∈ 𝔽 over
and [𝜇,
𝜌∈
then we have
Therefore,
we𝐹𝑅-integrable
have
[if0, 𝔖1],is
𝑅-integrable
over [𝜇,for
𝜈].every
Moreover,
𝜈],ℝ,then
It isϕa∈numbers.
partial
order
relation.
(9)
numbers.
𝔵, 𝔴 ∈order
𝔽 and
𝜌[𝜇,
∈ 𝜈],
ℝ, then
we have
a𝐹𝑅-integrable
partial
relation.
𝑅-integrable over [𝜇, 𝜈]. Moreover,
if 𝔖It isLet
over
then
[𝔵+
[𝔵]
𝔴]
= examine
+ [𝔴]
, of the
Proposition
We
now
use
mathematical
operations
to
some
character
It
is
a
partial
order
relation.
4µυ
4µυLet
2µυ
2µυ
[𝔵]
[𝔴]
𝔴]
=≼examine
+
, 𝔖
1. [51]
Proposition
𝔵, 𝔴now
∈ 𝔽 use
1.
[51]
Let
𝔴
∈𝔽
.=Then
fuzzy
order
relation
. [𝔵+
Then
fuzzy
“to
”𝔖
is(𝒾)𝑑𝒾
given
relation
on
𝔽 “the
≼
by” is
(𝒾)𝑑𝒾
(𝒾)𝑑𝒾
(𝒾)𝑑𝒾
(𝐼𝑅)
(𝐼𝑅)
(𝐼𝑅)
∗ (𝒾,
We
mathematical
operations
some
of
𝔖𝔵,υ=
𝔖
=
µ µ+PEER
µ µ=
µ(𝑅)
µorder
υ Proposition
+υ1. [51]
µ+
Symmetry 2022,
14, x (𝐹𝑅)
FOR
(𝑅)
(𝒾)𝑑𝒾
𝔖
𝔖
𝔖REVIEW
[𝔵+
[𝔵]
Proposition
𝔴
∈
𝔽 1.
[51]
Let
𝔵,
𝔴
∈
., Then
fuzzy
order
relation
. Then
fuzzy
“υ=
”order
given
relation
on cha
𝔽“
𝔴]
=𝔽
+ [𝔴]
,≼ (𝐼𝑅)
∗𝔵,
numbers.
Let, use
𝔵, 𝔴
∈(𝒾,
𝔽∗𝜑)𝑑𝒾
and
𝜌µ+,∈
ℝ,
then
we
have
∗𝜑)𝑑𝒾
We
now
mathematical
operations
to
examine
some
of,isthe
characteri
2s S∗2022,
+𝔖
ϕLet
≤
S
,
ϕ
+
S
ϕ
,
∗
Symmetry
14,
x
FOR
REVIEW
2µυPEER
2µυ
2µυ
2µυ
(𝒾,
(𝒾,
(𝑅)
(𝑅)
(𝐹𝑅)
(𝒾)𝑑𝒾
[𝔵
[𝔵]
[
=
𝔖
𝜑)𝑑𝒾
𝔖
𝜑)𝑑𝒾
×
𝔴]
=
×
𝔴]
,
∗𝔴
∈ξµLet
𝔽if,
and
𝜌υ ∗and
∈
ℝ,
then
we
have
1.
[51]
Proposition
𝔴 if
∈Let
𝔽 1.
[51]
∈
𝔽[𝔴]
.𝔵,Then
fuzzy
order
relation
. [𝔵
Then
fuzzy
“ ([𝔵]
≼
”ξorder
is(0,
relation
on
𝔽
“≼
by
”∈is(0
ξµProposition
+(1−ξ ) µ+υ
+(
1≼
−𝔵,ξ𝔴
ξ )µnumbers.
+Let
ξ𝔵µ+≼υ𝔵,𝔴
1𝜑
−[𝔵]
+given
ξ[𝔴]
)𝔴
(1−
)µ
[
µ+
µ1],
+υ𝔴] for
[𝔵]
×
𝔴]
=
×
,
and
only
𝔵
if
only
if,
≤
for
all
∈
≤
all
𝜑
(g
Symmetry 2022, 14,
FOR PEER
REVIEW
(𝒾,
(𝒾,
(𝑅)
(𝑅)
(𝒾)𝑑𝒾
=
𝔖
𝜑)𝑑𝒾
,
𝔖
𝜑)𝑑𝒾
𝔖
x (𝐹𝑅)
numbers.
𝜌 2µυ
∈ ℝ,
then
we
have
∗ ∈𝔴𝔽 if and
(9)
4µυ Let 𝔵,𝔵𝔴≼
2µυ[𝔵]
4µυ
[𝔵]𝔴]
[𝔴]
[𝔴]
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𝔵]
=
𝜌.
[𝔵
[𝔵]
[
×
𝔴]
=
×
𝔴]
,
Proposition
1.
[51]
Let
𝔵,
𝔴
∈
𝔽
.
Then
fuzzy
order
relation
“
≼
”
is
given
(𝒾)𝑑𝒾
(𝐼𝑅)
=
𝔖
noworder
useProposition
mathematical
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operations
useLet
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to∈examine
operations
of=order
to
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ch
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𝔵, 𝔴
𝔽 .to
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”ofthe
isfuzz
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(𝒾)𝑑𝒾
(𝐼𝑅)
=
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now
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We
nowrelation.
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o
Symmetry 2022, 14, x FOR
PEER REVIEW
[𝔵valued
[𝔵]
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×
𝔴]
=
×
𝔴]
,
Definition
1.
[49]
A
fuzzy
number
map
𝔖
:
𝐾
⊂
ℝ
→
𝔽
is
called
F-N
Proposition
1.
[51]
Let
𝔵,
𝔴
∈
𝔽
.
Then
fuzzy
order
relation
“
≼
”
is
given
o
numbers.
Let
𝔵,
𝔴
∈
numbers.
𝔽
Let
𝔵,
𝔴
∈
𝔽
and
𝜌
∈
ℝ,
then
we
and
have
𝜌
∈
ℝ,
then
we
In
consequence,
we
obtain
We
now
use
mathematical
We
now
operations
use
mathematical
to
examine
operations
some
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the
examine
characteristics
some
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of
the
fuzz
ch
[𝔵]
[𝔴]
[𝜌.
[𝔵]
𝔵
≼
𝔴
if
and
only
if,
≤
for
all
𝜑
∈
(0,
1],
𝔵]
=
𝜌.
Symmetry 2022,
14,
x
FOR
PEER
REVIEW
Definition
1.
[49]
A
fuzzy
number
valued
map
𝔖
:
𝐾
⊂
ℝ
→
𝔽
is
calle
(0,
(0,
(0,
(0,
[𝜇,
[𝜇,
[𝜇,
[𝜇,
=
isand
said
=
to
be
is
a∈
=convex
said
∞)
to⊂
be
set,
is
asaid
if,
convex
=for
for
toall
all
be
∞)
set,
a𝜑
𝒾,is
convex
𝒿if,
said
∈
for1t
Definition
3.
Definition
A 1],
set
Definition
3.
𝐾[34]
A],numbers.
𝜐]
3.⊂
Definition
[34]
𝐾ℝ=A
set
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𝐾
⊂𝔴
3.
=
ℝ
[34]
A
𝜐]and
set
⊂∞)
ℝ
𝐾
=
𝜐]
ℝ
Let
𝔵,=([𝔴
𝔽set
𝔵,∞)
∈
𝔽
and
𝜌Let
∈≼
ℝ,
then
we
have
𝜌if,
ℝ, I-V-Fs
then
we
have
(0,
for all 𝜑
∈ (0,
1].
For
allnumbers.
𝜑[34]
∈Definition
ℱℛ
denotes
the
collection
of
all
𝐹𝑅-integrable
F[𝜌.
[𝔵]
[𝔵]
[𝔴]
) A
𝔵]
=
𝜌.
,∈
𝔵
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if
only
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whose
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define
the
family
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𝔖
:
𝐾
⊂
ℝ
→
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Symmetry
2022,
14,
x
FOR
PEER
REVIEW
1.
[49]
fuzzy
number
valued
map
𝔖
:
𝐾
⊂
ℝ
→
𝔽
is
called
F-N
(0,
for all 𝜑 𝐾,
∈ (0,
allLet
𝜑 [0,
∈𝔴
1],
ℱℛ
theand
collection
of
all
𝐹𝑅-integrable
numbers.
𝔵,
∈numbers.
𝔽 ∈(0,
𝔵,
𝔴
∈𝔴]
𝔽1],
and
𝜌Let
∈
we
𝜌 [𝜌.
∈
ℝ,
then
we
],we
)𝔵ℝ,
[𝔵]
𝔵][𝔵]
=
𝜑
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𝜑[0,
-cuts
define
the
family
of
I-V-Fs
: 𝐾 1],
⊂ℝ→
[𝔴]
[0, For
[0,
≼[𝔵+
𝔴
if
and
only
if,all
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∈1].
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𝜉 (0,
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∈
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(
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[𝔴]F=have
+
, 𝔴]
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It (𝒾)
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aℱℛ
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relation.
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∈ (0,
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denotes
the
𝐹𝑅-integrable
FRcollection
(𝒾,µυ
=
𝔖∗-cuts
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for
∈
Here,
for
each
𝜑
1]→ the
([whose
], order
)𝜑),
1]
𝜑
define
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family
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: 𝐾∈⊂(0,
ℝ[𝔴]
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ϕ)𝐾.
S
,∗ (𝒾,
[𝔵+
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[𝔵]
∗ ( 𝒾 ,of
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µ[𝔵+
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𝔴]
=
+
,
𝔴]
=
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∗
s−(0,
2It
over
[𝜇, 𝜈].all 𝜑 ∈ 𝜑𝔖2∈1],
is
a
partial
order
relation.
(𝒾)
[𝔖
(𝒾,
(𝒾,
𝔖
=
𝜑),
𝔖
𝜑)]
for
all
𝒾
∈
𝐾.
Here,
for
each
𝜑
∈
(0,
S
,
ϕ
≤
d
,
∗
∗
µ
2
Definition
1.
[49]
A
fuzzy
number
valued
map
𝔖
:
𝐾
⊂
ℝ
→
𝔽
is
called
F-N
∗ (.υ[𝔵+
(5
[𝔵+
[𝔵]
[𝔴]
[𝔵]
[𝔴]
∗ (𝒾,
µ𝔖
+3υ
𝔴]
, 𝔴]
=
+
,the
We[𝔖now
mathematical
operations
to
examine
some
of
characte
𝒾𝒿
𝒾𝒿
𝒾𝒿
𝒾𝒿≼
NV-Fs over [𝜇, 𝜈].
It
is
a Definition
partial
relation.
(.1.
(𝒾)
functions
,use
𝜑),
𝔖
,𝜑)]
𝜑):
𝐾
ℝ
are
upper
functions
of
𝔖
=
𝜑),
for→
all
𝒾operations
∈+
𝐾.
for
each
𝜑
(0,
the
en
(
[𝔵−
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[𝔵]
[valued
[𝔵]
[ℝ
×
=
×called
𝔴]
,lower
𝔴]
=and
×𝐾.
𝔴]
,”of
Proposition
[51]
Let
𝔵,µ×
𝔴
𝔽=
.𝐾
Then
fuzzy
order
relation
“∈
isthe
given
∗order
1.𝔖use
[49]
A
fuzzy
number
map
𝔖: 𝐾
⊂
→
𝔽1]
is
calle
∗ (𝒾,
∗𝔴]
We
now
mathematical
to
examine
some
char
∈
𝐾.
∈
𝐾.
∈→
∈is
𝐾.
(10)
(.𝔵,∈,[𝔵
functions
𝔖
,-cuts
𝜑),
𝔖and
𝜑):
→
ℝ
are
called
lower
and
upper
functio
[𝔵
[𝔵]
[
[𝔵]
[
(0,
[𝜇,
×
×
𝔴]
=
×
𝔴]
,
𝔴]
=
×
𝔴]
,
=
∞)
is
said
to
be
a
convex
set,
if,
for
all
𝒾,
𝒿
∈
Definition 3. [34] A set 𝐾 = Definition
𝜐](0,
⊂
ℝ
∗ (.
Proposition
1.
[51]
Let
𝔴
∈
𝔽
.
Then
fuzzy
order
relation
“
≼
”
giv
𝜑
∈
1]
whose
𝜑
define
the
family
of
I-V-Fs
𝔖
:
𝐾
⊂
ℝ
→
𝒦
1.
[49]
A
fuzzy
number
valued
map
𝔖
:
𝐾
⊂
ℝ
𝔽
is
called
F-N
∗
numbers.
Let
𝔵,
𝔴
∈
𝔽
𝜌
ℝ,
then
we
have
(1
(1
(1
(1
𝜉𝒾
𝜉𝒾
𝜉𝒾
𝜉𝒾
+
−
𝜉)𝒿
+
−
𝜉)𝒿
+
−
𝜉)𝒿
+
−
𝜉)𝒿
We
now
use
mathematical
operations
to
examine
some
of
the
character
(.
(.
𝔖
,
𝜑),
𝔖
,
𝜑):
𝐾
→
ℝ
are
called
lower
and
upper
functions
of
(0,
[𝜇,
=
∞)
is
said
to
be
a
convex
set,
if,
for
all
𝒾,
𝒿
∈
Definition 3. [34] A set 𝐾functions
=
𝜐]
⊂
ℝ
𝜑
∈
(0,
1]
whose
𝜑
-cuts
define
the
family
of
I-V-Fs
𝔖
:
𝐾
⊂
ℝ
→
∗
(6
[𝔵
[𝔵
[𝔵]
[
[𝔵]
[
×order
𝔴]
=. Then
𝔴][𝔵]
, [𝜌.
𝔴]
=
×
numbers.
LetLet
𝔵, 𝔴
𝔽[𝜌.
and
𝜌∗only
∈×
ℝ,
then
we
have
Proposition
1.
[51]
𝔵,∈×
𝔴
𝔽
fuzzy
relation
“𝔴]
≼ 𝜑”, ∈
is (0,
given
on
2µυ
[𝔵]
[𝔵]
R∈ifbe
𝔵]
=
𝜌.
=𝒾,
𝜌.
[𝔴]
𝐾, 𝜉 ∈ [0,Symmetry
1],
we 2022,
have
14,𝐾
x FOR
REVIEW
𝔵≼
and
≤𝔵]
all
1],
(𝒾,𝔵,
(𝒾,
µυ
𝔖𝜐]
=
𝜑),
𝔖
𝜑)]
for
all
𝒾→, ∈
Here,
for
each
𝜑
(0,
the
𝜑PEER
∈(𝒾)
(0,
1][𝔖
whose
𝜑
the
family
of
I-V-Fs
𝔖
: 𝐾∈
⊂
ℝ1]→ integr
𝒦 e(a
S
ϕ𝔽if,
(0,
(then
)𝐾.
=
∞)
is∗-cuts
said
convex
set,
if,
for
all
𝒿for
∈
Definition
3.Symmetry
[34]
A set
= [𝜇,
⊂
ℝ
4µυ
µ+
υa
∗2.
be
an
F-NV-F.
Then
fuzzy
Definition
[49]
Let
𝔖and
:𝔴to
[𝜇,
𝜈]
⊂
ℝ
∗define
2S
numbers.
Let
𝔴
∈
𝔽𝜑),
𝜌
∈
ℝ,
we
have
[𝜌.
[𝜌.
[𝔵]
[𝔵]
𝔵]
𝔵]
=
𝜌.
=
𝜌.
[𝔵]
[𝔴]
𝐾, 𝜉 ∈ [0,
1], we
have
2022,
14, x FOR
𝔵
≼
𝔴
if
and
only
if,
≤
for
all
𝜑
∈
(0,
1]
(𝒾)
[𝔖
(𝒾,
(𝒾,
𝔖∗ REVIEW
=
𝔖
𝜑)]
for
all
𝒾
∈
𝐾.
Here,
for
each
𝜑
∈
(0,
1]
2s−PEER
,
ϕ
≤
d
.
[𝔵+
[𝔵]
[𝔴]
𝔴]
=
+
,
∗
µ
2
be
an
F-NV-F.
Then
fuzzy
Definition
2.
[49]
Let
𝔖
:
[𝜇,
𝜈]
⊂
ℝ
→
𝔽
∗
∗ (𝒾,
µ𝔖
+(𝒾,
3υ
µ [𝜌.𝐾
(.𝒾𝒿
(.υ𝔖−
(𝒾)
[𝔖
(7
functions
,
𝜑),
𝔖
,
𝜑):
→
ℝ
are
called
lower
and
upper
functions
of
[𝜌.
[𝔵]
[𝔵]
𝔖[𝜇,
=
𝜑),
𝔖
𝜑)]
for
all
𝒾
∈
𝐾.
Here,
for
each
𝜑
∈
(0,
1]
the
en
𝔵]
𝔵]
=
𝜌.
=
𝜌.
[𝔵]
[𝔴]
𝐾, 𝜉 ∈ [0,Symmetry
1], we 2022,
have14, x FOR PEER
[𝔵+
[𝔵]
[𝔴]
REVIEW
𝔵
≼
𝔴
if
and
only
if,
≤
for
all
𝜑
∈
(0,
1],
𝔴]
=
+
,
∗
(𝐹𝑅)
(𝒾)𝑑𝒾
∗
𝜈],
denoted
by
𝔖
,
is
given
levelwise
by
∗
be
an
F-NV-F.
Then
fuzzy
integr
Definition
2.
[49]
Let
:
[𝜇,
𝜈]
⊂
ℝ
→
𝔽
is aDefinition
partial
order
relation.
(.
functions
,[𝜇,
𝜑),
𝔖→(.𝔖:
, a𝜑):
𝐾𝜐]
→→
ℝℝ, is
are
called
and
upper
functio
[𝜇,
[𝜇,
[𝜇,
Definition 4.
Definition
[34]ItThe
𝔖:
4. 𝜈],
[34]
𝜐]
The
→𝔖4.
ℝ
Definition
[34]
The
𝜐]
ℝ
4.[𝜇,
[34]
The
𝜐]levelwise
→lower
ℝ
is
called
harmonically
is
called
a𝔖:
is
harmonically
convex
called
a[𝔴]
function
harmonically
is
convex
called
on
function
a harmoni
convex
𝜐]
if of
∗𝔖:
[𝜇,
(𝐹𝑅)
(𝒾)𝑑𝒾
𝒾𝒿
denoted
by
𝔖
given
by
[𝔵+
[𝔵]
𝔴]
=
+
,
∈
𝐾.
∗
(10)
It
is
a
partial
order
relation.
(.(1
[𝔵
[𝔵]
[called
functions
𝔖+∗𝒾𝒿
,use
𝜑),
𝔖
, valued
𝜑):
→map
ℝoperations
are
called
lower
and
upper
functions
=
×
𝔴]
, (10)
Definition [𝜇,
1. is
[49]
ADefinition
fuzzy
number
1. (.
[49]
A
fuzzy
𝔖given
:×
𝐾[𝔵𝔴]
⊂×valued
ℝ𝔴]
→
𝔽 map
𝔖
:[𝐾𝔴]
⊂
→ characte
𝔽For
isby
F-NV-F.
isof
eac
ca
(𝐹𝑅)
(𝒾)𝑑𝒾
𝜈],
denoted
by
, number
is
levelwise
𝜉𝒾
− 𝜉)𝒿
∈𝔖𝐾𝐾.
We
now
mathematical
examine
some
of𝔖
the
that is
It
a partial
order
relation.
[𝔵]
,ℝfuzzy
Definition
1. [49]
A
Definition
[49]
valued
Aℝfuzzy
map
number
𝔖to
:𝐾
⊂=
valued
ℝ
→𝒾𝒿𝔽×
map
:𝐾
⊂
ℝintegr
→𝔽
issome
called
F-NV-F.
(1
𝜉𝒾
+fuzzy
−∈number
𝜉)𝒿
𝒾𝒿
𝒾𝒿
𝒾𝒿
We
now
use
mathematical
operations
to
examine
of
the
chara
be
an
F-NV-F.
Then
Definition
2.
[49]
Let
𝔖
:1.
[𝜇,
𝜈]
⊂𝔵,
→
𝔽
𝐾.
(10)
Proposition
1.
[51]
Let
𝔴
∈
𝔽
.
Then
fuzzy
order
relation
“
≼
”
𝜑
∈
(0,
1]
whose
𝜑
𝜑
-cuts
∈
(0,
1]
define
whose
the
𝜑
family
-cuts
define
of
I-V-Fs
the
𝔖
family
of
I-V-Fs
𝔖
:
𝐾
⊂
ℝ
→
𝒦
are
given
:
𝐾
⊂
ℝis
b
[𝔵
[𝔵]
[ called
×
=𝜉𝔖(𝒿),
×is
𝔴]
,𝐾 of
numbers.
Let
𝔴
∈[49]
𝔽define
and
𝜌map
∈−𝜑family
ℝ,
then
we
have
Definition
1.
[49]
A𝜉𝒾
Definition
fuzzy
number
1.(𝒾)𝑑𝒾
[49]
valued
A [51]
fuzzy
number
𝔖
:𝔖
𝐾𝔴]
⊂define
valued
ℝ
→
𝔖
:𝜑)𝑑𝒾
→
𝔽−
F-NV-F.
For
is∈𝔖eac
(1
(1
(1
(1
be
an
F-NV-F.
Then
fuzzy
Definition
2.
Let
𝔖
:operations
[𝜇,
𝜈]
⊂
ℝ
→
𝔽examine
+
+
𝜉𝔖(𝒿),
+
𝜉𝔖(𝒿),
𝔖+
𝔖
≤
𝔖
𝜉)𝔖(𝒾)
≤
−
𝔖
𝜉)𝔖(𝒾)
≤map
−
𝜉)𝔖(𝒾)
𝜉)𝔖(𝒾
(1
(11)
−𝔵,-cuts
"use
#
We
now
mathematical
to
some
the
characteri
[𝜌.
[𝔵]
𝔵]
=𝔽𝜉𝒾
𝜌.
(𝐹𝑅)
(𝐼𝑅)
𝔖(0,
=
=
𝔖(𝒾,
:≤
𝔖(𝒾,
𝜑)
ℛ
Proposition
1.
Let
𝔵,2µυ
𝔴
∈(𝒾)𝑑𝒾
𝔽
.I-V-Fs
Then
fuzzy
order
relation
“cal
≼g
𝜑 ∈ (0,
1]
whose
𝜑
𝜑
∈𝜉)𝒿
1]
whose
the
-cuts
of
the
𝔖
family
of
I-V-Fs
:−
𝐾
⊂⊂
ℝℝ
→
𝒦
are
:
2µυ
ZLet
Z
numbers.
𝔵,
𝔴
∈
𝔽
and
𝜌
∈
ℝ,
then
we
have
(1
(1
(1
(1
[𝜇,
(𝐹𝑅)
(𝒾)𝑑𝒾
𝜉𝒾
𝜉𝒾
𝜉𝒾
+
−
𝜉)𝒿
+
−
𝜉)𝒿
+
−
𝜉)𝒿
+
𝜉)𝒿
𝜈],
denoted
by
𝔖
,
is
given
levelwise
by
∗
∗
be
an
F-NV-F.
Then
fuzzy
integr
Definition
2.
[49]
Let
𝔖
:
[𝜇,
𝜈]
⊂
ℝ
→
𝔽
∗
[𝜌.
[𝔵]
𝔵]
=
𝜌.
(𝒾)𝑑𝒾
(𝐹𝑅)
(𝒾)𝑑𝒾
(𝐼𝑅)
=define
=of
𝔖(𝒾,
𝜑)𝑑𝒾
:𝐾
𝔖(𝒾,
𝜑b
µυ
[𝔖whose
(𝒾, 𝜑),[𝜇,
(𝒾,
(𝒾)
[𝔖
(𝒾,∈𝔵,𝜑)]
𝔖-cuts
=
𝔖
𝜑)]
for
all
𝒾(𝐹𝑅)
𝔖
∈Let
𝐾.
Here,
for
each
𝒾have
∈family
𝜑
∈Here,
(0,
1]→
for
the
end
𝜑
point
∈⊂
(0,is
re
1
µ
+whose
υ∗ (𝒾,
µ
+
υfor
4µυ
4µυ
,𝜑
ϕ-cuts
,𝔖
ϕ𝐾.
(𝔖
)𝜌
(𝔖
): 𝐾
Proposition
1.
[51]
𝔴
𝔽I-V-Fs
.S
Then
fuzzy
relation
“given
≼
∗𝜑),
𝜑𝔖The
∈(𝒾)
(0,𝔖:
1][𝜇,
𝜑
∈
(0,
1]
define
the
family
of
the
I-V-Fs
𝔖
⊂
ℝ
𝒦
are
:each
ℝ
∗𝜐]
(𝒾)𝑑𝒾
𝜈],
denoted
by
𝔖
,all
is
given
∗𝔴
∗∈
numbers.
Let
𝔵,=
𝔽S
and
then
we
[𝜇,
→≤
ℝI ∗𝜑
is
called
a∈(𝐹𝑅)
harmonically
convex
function
on
ifbyeach
[𝔵]
(𝒾)𝑑𝒾
(𝐹𝑅)
(𝒾)𝑑𝒾
𝔵(𝐼𝑅)
𝔴
if
and
only
if,
≤𝜐]
for
𝜑
𝔖
=
𝔖
=
𝔖(𝒾,
𝜑)𝑑𝒾
: 𝔖(𝒾,
𝜑)
∈” ∈
ℛ
[𝔖
(𝒾,
(𝒾,
(𝒾)
[𝔖
(𝒾,
(𝒾,
[𝜌.
[𝔵]
𝔖
𝔖
=𝜈],
𝜑),
𝔖
𝜑)]
=
for
all
𝜑),
𝒾 ,ℝ,
𝔖
∈
𝐾.
𝜑)]
Here,
for
for
all
each
𝒾+
∈order
𝐾.
𝜑
∈
Here,
(0,
1]
for
theall
end
𝜑
p((
𝔵]
=
𝜌.
2s−2 S∗ Definition
, ϕ 4., [34]
S∗ [34]
, (𝒾)
ϕ[𝜇,
d≼
dlevelwise
.[𝔴]
[𝔵+
[𝔵]
𝔴]
=
,[𝔴]
∗𝔖
(𝒾)𝑑𝒾
∗ (.
∗ (.
denoted
by
,
is
given
levelwise
by
2
2
[𝜇,
[𝜇,
∗
∗
Definition
The
𝔖:
𝜐]
→
ℝ
is
called
a
harmonically
convex
function
on
𝜐]
if
[𝔵]
[𝔴]
𝔵
≼
𝔴
if
and
only
if,
≤
for
all
𝜑
(.
(.
functions
𝔖
functions
𝔖
,
𝜑),
𝔖
,
𝜑):
𝐾
→
ℝ
,
𝜑),
are
𝔖
called
,
𝜑):
lower
𝐾
→
and
ℝ
are
upper
called
functions
lower
and
of
upper
𝔖
.
funct
µ+
3υ
µ
+
3υ
υ
−
µ
[𝜇,
[0,
[𝜇,
[0,
[𝜇,
[0,
[𝜇,
[𝜇,
[0,
[𝜇,
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for4.all
𝒾,
𝒿
∈
for
𝜐],
all
𝜉
𝒾,
∈
𝒿
for
∈
1],
all
𝜐],
where
𝒾,
𝜉
𝒿
∈
∈
for
𝔖(𝒾)
1],
𝜐],
all
𝜉
where
≥
∈
𝒾,
0
𝒿
∈
for
1],
𝔖(𝒾)
all
where
𝜐],
𝒾
≥
𝜉
∈
∈
0
𝔖(𝒾)
for
𝜐].
1],
all
≥
If
where
𝒾
(11)
0
∈
for
is
𝔖(𝒾)
all
𝜐].
reversed
𝒾
If
≥
∈
(11)
0
then,
for
𝜐].
is
reversed
all
If
(11)
𝔖
𝒾
∈
[𝔵+
[𝔵]
[𝔴]
(𝒾) [𝜇,
[𝔖∗ for
(𝒾, 𝜑),
(𝒾,
(𝒾)
(𝒾,
(𝒾, 𝜑)]
µ
µ
𝔴]
=∈
+ for
, end and
𝔖𝜑is
𝜑)]
=
∈ℛ(.
𝐾.
Here,
for
all
each
𝒾the
𝐾.
𝜑
Here,
(0,called
1]
the
each
𝜑
point
(0,.rea
1]i
∗all𝜑),
∗[𝔖
∗denotes
all𝔖
∈
(0,
1],
where
collection
of
Riemannian
(. harmonically
(.and
[𝜇,
functions
functions
,called
𝜑),
,∗𝜑):
𝐾𝒾𝔖
→𝔖
,≼𝜑),
𝔖
,for
𝜑):
lower
𝐾=
→∈
ℝ
are
upper
functions
lower
of∈𝜑uppe
𝔖
Definition 4. [34] 𝔖
The 𝔖:=
𝜐]∗→
ℝ
afor
convex
function
on
𝜐]
if
[𝔵]
[𝔴]
],𝔴
)called
𝔵([
if
only
if,and
≤
for
all
∈ in
(0
,are
𝒾𝒿
Definition
1.𝔖
[49]
A1],order
fuzzy
number
valued
map
𝔖
:
𝐾
⊂
ℝ
→
𝔽
is
called
F∗ (.
∗ℝ
[𝔵+
[𝔵]
[𝔴]
𝔴]
+
,
It
is
a
partial
relation.
∗
∗
for
all
𝜑
∈
(0,
where
ℛ
denotes
the
collection
of
Riemann
[𝜇,are
[𝜇,
is called
a harmonically
is called
aDefinition
is
harmonically
called
concave
ais≤
harmonically
function
is
concave
called
on
afunction
harmonically
concave
𝜐].
on
concave
𝜐].
on
function
on→
[𝔵
[𝔵]
[𝜐].
([ 𝐾
)and
(𝒾)𝑑𝒾
(𝐼𝑅)
, ],
×function
𝔴]
= called
×
𝔴]
, and
=
𝔖
=
𝔖(𝒾,
𝜑)𝑑𝒾
: upper
𝔖(𝒾,
𝜑)
∈ ℛ
(. ,all
(.(𝐹𝑅)
(.
(.[𝜇,
functions
𝔖∗tions
functions
𝔖
𝜑),
𝔖
, 𝜑):
𝐾
→
ℝ
,𝜉)𝔖(𝒾)
𝜑),
are
𝔖called
, 𝜑):
lower
→
ℝ
upper
functions
lower
of
𝔖[𝜇,
𝒾𝒿
[49]
A
fuzzy
number
valued
map
𝔖
: 𝐾(11)
⊂
ℝ
𝔽. 𝜐].
is∈functi
calle
(1
+
𝜉𝔖(𝒿),
−
∗(𝒾)𝑑𝒾
It
a1.𝔖
partial
order
relation.
(𝐹𝑅)
(𝒾)𝑑𝒾
𝜑
∈I-V-Fs.
(0,
1],
where
ℛ
denotes
the
collection
int
of
is
𝐹𝑅
-integrable
over
[𝜇,
𝜈]
if
𝔽F-N
.
(𝒾)𝑑𝒾
(𝐹𝑅)
(𝒾)𝑑𝒾
(𝐼𝑅)
It follows
that𝔖 for
[𝔵
[𝔵]
[of
𝔖
=
𝔖
=⊂
𝔖(𝒾,
𝜑)𝑑𝒾
:of
𝔖(𝒾,
𝜑
×
𝔴]
=
×
𝔴]
,:𝔖𝐾
([define
],𝔖
)+
(0,
whose
𝜑
-cuts
the
family
of:be
I-V-Fs
𝔖Riemannian
⊂
ℝ
→
𝒦
, be
(1
(1
𝜉𝒾𝜑[49]
+∈𝒾𝒿
−1]
𝜉)𝒿
𝜉𝔖(𝒿),
𝔖
≤
−→
𝜉)𝔖(𝒾)
(11)
Definition
1.
[49]
A
fuzzy
number
valued
map
𝔖
𝐾
ℝF-NV-F.
→
𝔽
is
called
We
now
use
mathematical
operations
to
examine
some
the
an
F-NV-F.
Then
fuzzy
an
integral
Then
of
𝔖
fuzz
ov
Definition
2.
Let
Definition
𝔖
:
[𝜇,
𝜈]
2.
⊂
[49]
ℝ
Let
𝔽
:
[𝜇,
𝜈]
⊂
ℝ
→
𝔽
(𝐹𝑅)
(𝒾)𝑑𝒾
tions
of
I-V-Fs.
𝔖
is
𝐹𝑅
-integrable
over
[𝜇,
𝜈]
if
𝔖
It
is
a
partial
order
relation.
(𝒾)𝑑𝒾
(𝐹𝑅)
(𝒾)𝑑𝒾
(𝐼𝑅)
𝔖
=
𝔖
=
𝔖(𝒾,
𝜑)𝑑𝒾
:
𝔖(𝒾,
𝜑)
∈
ℛ
𝜑
∈
(0,
1]
whose
𝜑
-cuts
define
the
family
of
I-V-Fs
𝔖
:
𝐾
⊂
ℝ
→tc(
2µυ
(1
[𝔵
[𝔵]
[
Z
𝜉𝒾
+
−
𝜉)𝒿
×
𝔴]
=
×
𝔴]
,
(1
We
now
use
mathematical
operations
to
examine
some
of
∗
+
𝜉𝔖(𝒿),
𝔖 tions
≤
−
𝜉)𝔖(𝒾)
∗
be
an
F-NV-F.
Then
be
fuzzy
an
F-NV-F.
integral
The
of
Definition
2.
[49]
Let
Definition
𝔖
:
[𝜇,
𝜈]
2.
⊂
[49]
ℝ
→
Let
𝔽
𝔖
:
[𝜇,
𝜈]
⊂
ℝ
→
𝔽
(11)
e
(𝐹𝑅)
(𝒾)𝑑𝒾
of−(𝐹𝑅)
I-V-Fs.
𝔖are
is-cuts
𝐹𝑅
-integrable
[𝜇,
if
𝔖
∈𝒦
𝔽the
.
[𝜌.
[𝔵]
𝔵]
=𝜈]𝜌.𝔖
µυ
(𝒾)
(𝒾,
(𝒾,
=numbers.
∈
𝐾.
Here,
for
𝜑
∈
(0,of1]
µ+and
υ all
𝔖[𝔖
𝜑)
Lebesgue-integrable,
then
is
aeach
fuzzy
Aumann-int
𝔖𝔖
4µυ
S𝜈]
)family
( 𝒾𝜌over
𝜑
∈(𝒾,
(0,
1]
whose
𝜑⊂𝔖
define
the
I-V-Fs
𝔖
:aby
𝐾
⊂Then
→
∗[49]
Let
𝔵,
∈Lebesgue-integrable,
𝔽for
∈dis
ℝ,
then
we
have
(1
[𝜇,
[𝜇,
(𝒾)𝑑𝒾
(𝐹𝑅)
(𝒾)𝑑𝒾
∗𝜑),
−
2𝜑),
∗𝜑)]
𝜉𝒾
+
𝜉)𝒿
𝜈],[0,denoted
𝜈],
denoted
by
𝔖
,(𝒾,
is
given
levelwise
given
We
use
mathematical
operations
to
examine
some
the
ch
e
an
F-NV-F.
Then
be
fuzzy
an
integral
of
𝔖
fuzzy
ove
2.
Let
Definition
𝔖
:∗ [𝜇,
𝜈]
2.
[49]
ℝ
→
Let
𝔽
𝔖
: 𝔽[𝜇,
⊂
ℝ
→
𝔽
[𝜌.
[𝔵]
[𝜇,
𝔵]ofcollection
=levelwise
𝜌.
for all 𝒾, 𝒿 ∈ [𝜇, 𝜐],Definition
𝜉∈
1],𝜈],
where
𝔖(𝒾)
≥
0𝔖now
all
𝒾𝔴
∈
𝜐].
If
(11)
reversed
then,
(𝒾,
(𝒾,
[𝔖
(𝒾,
for
all
𝜑
∈∗by
(0,
1],
where
ℛ
denotes
the
of
Riemannian
in
𝜑),
𝔖
are
then
𝔖F-NV-F.
is𝔖
fuzzy
Auma
𝔖
𝔖
=
𝜑),
𝔖
𝜑)]
all
𝒾𝜌,by
∈
𝐾.
Here,
for
each
𝜑ℝ
∈by
(0,
1]
4𝜑)
.is
(22)
2sby
S
],given
)for
,be
∗(𝒾)
∗for
numbers.
𝔵,([
𝔴
∈
and
∈
ℝ,
then
we
have
[𝜇,
(𝐹𝑅)
[𝜇,
(𝒾)𝑑𝒾
(𝐹𝑅)
(𝒾)𝑑𝒾
∗Let
denoted
𝜈],
denoted
𝔖
by
𝔖
,
is
levelwise
,
by
is
given
levelwise
2
∗
[0, 𝔖
[𝜇,
for all 𝒾, 𝒿 ∈ [𝜇, 𝜐], 𝜉 ∈
1],
where
𝔖(𝒾)
≥
0
for
all
𝒾
∈
𝜐].
If
(11)
is
reversed
then,
𝔖
(.
(.
for
all
𝜑
∈
(0,
1],
where
ℛ
denotes
the
collection
of
Riemann
tion
over
[𝜇,
𝜈].
functions
𝔖
,
𝜑),
𝔖
,
𝜑):
𝐾
→
ℝ
are
called
lower
and
upper
functions
(𝒾,
(𝒾,
𝜑),
𝔖
𝜑)
are
Lebesgue-integrable,
then
𝔖
is
a
fuzzy
Aumann-inte
[𝜌.
[𝔵]
µ
+
3υ
υ
−
µ
𝔵]
=
𝜌.
([ (𝒾)𝑑𝒾
) 𝐾.
(𝒾)
[𝔖
(𝒾,
µ levelwise
,𝒾
𝔖∗ all
=
𝜑),
𝜑)]
all
∈
Here,
for
each
∈ (𝒾)𝑑𝒾
(0, 1] theinteo
∗𝔖
∗(𝐹𝑅)
∗ (𝒾,
numbers.
Let
𝔵,
∈
𝔽for
and
𝜌],ℝ
∈over
we
[𝜇,
(𝐹𝑅)
[𝜇,
(𝒾)𝑑𝒾
[𝜇,
by
𝜈],
, by
,by
isthen
given
levelwise
by
is all
called
concave
function
on
𝜐].
tion
over
[𝜇,
𝜈].
(.𝔖
(.
(𝐹𝑅)
functions
𝔖
, is
𝜑),
𝔖
→
are
called
lower
and
upper
for
𝜑
∈I-V-Fs.
(0,
1],
where
ℛ
denotes
the
collection
of+𝜑
Riemannian
tions
of
𝐹𝑅
-integrable
[𝜇,𝔴]
𝜈]
ifhave
𝔖 , function
∈𝔽 .
[𝜇,a 𝜐],
[0, denoted
[𝜇,
for
𝒿 a∈ harmonically
𝜉 ∈𝜈],
1],
where
𝔖(𝒾)
≥
0denoted
for
all
𝒾is𝔴
∈given
𝜐].
(11)
isℝ,
reversed
then,
𝔖
[𝔵+
[𝔵]
[𝔴]
([, 𝜑):
)If𝔖
∗𝔖
, ], 𝐾
=
∗
[𝜇,
is𝒾,called
harmonically
concave
function
on
𝜐].
(𝐹𝑅)
tion
over
[𝜇,
𝜈].
tions
of
-integrable
over
𝜈]
if[𝔵]
𝔖,(𝒾)𝑑𝒾
(. [49]
(. 𝔖
[𝔵+
[𝔴]
functions
𝔖
,I-V-Fs.
𝜑), 𝔖
, 𝜑):isnumber
𝐾 𝐹𝑅
→ℝ
are
called
lower
upper
of
𝔴]
Definition
Aisfuzzy
valued
map
𝔖:[𝜇,
𝐾and
⊂=
ℝ
→ 𝔽𝔖+functions
is
called
F-N
∗1.
In a similarconcave
way(𝐹𝑅)
as
above,
we
have
∗
(𝐹𝑅)
(𝒾)𝑑𝒾
[𝜇,
tions
of
I-V-Fs.
𝔖
𝐹𝑅
-integrable
over
[𝜇,
𝜈]
if
∈
𝔽
.(
is called a harmonically
function
on
𝜐].
(𝒾)𝑑𝒾
(𝒾)𝑑𝒾
(𝒾)𝑑𝒾
(𝐹𝑅)
(𝒾)𝑑𝒾
(𝐼𝑅)
(𝐼𝑅)
𝔖 𝔖2.(𝒾,
𝔖
=
𝔖𝜈]
=
=
𝔖
𝜑)𝑑𝒾
: 𝔖(𝒾,
=
∈
𝔖(𝒾,
ℛ
𝜑)𝑑𝒾
𝔖(𝒾,
,the
𝜑),
𝜑)
are
Lebesgue-integrable,
then
𝔖F-NV-F.
is
a𝐾𝜑)
fuzzy
𝔖Definition
1.
[49]
A
fuzzy
number
valued
map
𝔖
:[𝔵]
⊂𝜑-cuts
ℝ
→Aumann-int
([,𝔽
, ],is):called
[𝔵+
[𝔵]
[𝔴]
𝔴]
=
+
∗ (𝒾,Definition
be
an
Then
fuzzy
integ
2.(𝒾)𝑑𝒾
[49]
Let
𝔖
: [𝜇,
𝜈]ℝ
⊂
ℝ
→ be
𝔽𝔖(𝒾,
∗Let
Theorem
[49]
𝔖
:
[𝜇,
⊂
→
𝔽
a
F-NV-F
whose
define
f
(𝒾)𝑑𝒾
(𝒾)𝑑𝒾
(𝐹𝑅)
(𝐹𝑅)
(𝒾)𝑑𝒾
(𝐼𝑅)
(𝐼𝑅)
[𝔵
[
𝔖
𝔖
(𝒾,
(𝒾,
=
𝔖
=
=
𝔖(𝒾,
𝔖
𝜑)𝑑𝒾
:
𝔖(𝒾,
=
𝜑)
∈
𝔖(𝒾,
ℛ
𝜑)𝑑𝒾
×
𝔴]
=
×
𝔴]
,
𝜑),
𝔖
𝜑)
are
Lebesgue-integrable,
then
𝔖
is
a
fuzzy
Auma
𝔖
𝜑
∈
(0,
1]
whose
𝜑
-cuts
define
the
family
of
I-V-Fs
𝔖
:
𝐾
⊂
ℝ
→
𝒦
([defin
, ], i
∗ ∗ 1. [49]
Definition
A
fuzzy
number
valued
map
𝔖
:
𝐾
⊂
ℝ
→
𝔽
is
called
F-N
be
an
F-NV-F.
Then
fuzzy
Definition
2.
[49]
Let
𝔖
:
[𝜇,
𝜈]
⊂
ℝ
→
𝔽
Theorem
2.
[49]
Let
𝔖
:
[𝜇,
𝜈]
⊂
ℝ
→
𝔽
be
a
F-NV-F
whose
𝜑-cuts
[𝔵(𝒾)𝑑𝒾
[𝔵]
[:,𝔴]
×: 𝔴]
=
×Aumann-inte
,integr
tion
over
[𝜇,
(𝒾,𝜈],
(𝒾,
(8𝜑
𝜑
∈
(0,
1]𝜈].
whose
𝜑𝔖
-cuts
define
the
family
of
𝔖
𝐾], ⊂
ℝ
→fa
𝜑),
𝔖
𝜑)
are
Lebesgue-integrable,
then
𝔖
is=∗aI-V-Fs
fuzzy
𝔖
(𝒾)𝑑𝒾
(𝐹𝑅)
(𝒾)𝑑𝒾
(𝐹𝑅)
(𝒾)𝑑𝒾
(𝐼𝑅)
(𝐼𝑅)
𝔖
=
𝔖
=
=
𝔖(𝒾,
𝔖
𝜑)𝑑𝒾
𝔖(𝒾,
𝜑)
∈
𝔖(𝒾,
ℛ
𝜑)𝑑𝒾
:
𝔖(𝒾,
,
Z(𝒾)𝑑𝒾
(𝐹𝑅)
∗[𝜇,
denoted
by
𝔖
,
is
given
levelwise
by
([
)
∗
υ
be
an
F-NV-F.
Then
fuzzy
Definition
2.
[49]
Let
𝔖
:
[𝜇,
𝜈]
⊂
ℝ
→
𝔽
Theorem
2.
[49]
Let
𝔖
:
[𝜇,
𝜈]
⊂
ℝ
→
𝔽
be
a
F-NV-F
whose
𝜑-cuts
define
the
e
tion
over
(𝒾)
[𝔖
(𝒾,µυ
(𝒾,levelwise
(𝒾,𝔖
=
𝜑),
𝔖by
for
𝒾 )family
∈
𝐾.
Here,
for
each
∈⊂,(0,
the
=
𝜑),
𝔖
𝜑)]
for
all1]
𝒾∈
[𝜇,
[𝜇,
𝜈]
⊂
ℝ
→[𝜇,
𝒦𝜑𝜈].
are
given
by𝔖all
𝔖
𝔖𝔖∈:s(0,
4µυ
S((𝒾)
[𝔵[𝔖
[𝔵]
[∗:𝜑
×
=
𝔴]
𝜑
1]
whose
-cuts
define
the
of
I-V-Fs
𝐾
ℝ
→
∗ (𝒾,
[𝜇,
(𝐹𝑅)
(𝒾)𝑑𝒾
∗𝔴]
𝜈],
denoted
,𝔖
is
by
−
2𝔖
∗𝜑)]
efor
[𝔵]
𝔵]
=𝜑),
𝜌.×𝔖
for all 𝜑 ∈ tion
(0,
where
all
ℛ
𝜑⊂
where
ℛ
denotes
the
denotes
of
Riemannian
collection
integrable
funta𝒾
(𝒾)
[𝔖
(𝒾,
(𝒾,
(𝒾)
[𝔖
(𝒾,
(𝒾,
over
𝜈].
=[𝜌.
𝜑)]
for
all
: [𝜇,
𝜈]
ℝ
→
𝒦
are
given
by
𝔖
=
𝜑),
𝔖
𝜑)]
for
all
𝒾) .given
∈
𝐾.
Here,
each
𝜑of
∈ Rieman
(0,𝒦1]
21],
S
(23)
([
],4
)∗1],
([collection
, (0,
, ],d
∗the
∗∈
∗ for
[𝜇,
(𝐹𝑅)
(𝒾)𝑑𝒾
2µυ
𝜈],
denoted
by
𝔖
,
is
given
levelwise
by
2
∗
[𝜌.
[𝔵]
𝔵]
=
𝜌.
for all
𝜑
∈
(0,
1],
where
for
all
ℛ
𝜑
∈
(0,
1],
where
ℛ
denotes
the
collection
denotes
of
Riemannian
the
collection
integrab
of
R
(.
(.
∗
(𝒾)
[𝔖
(𝒾,
(𝒾,
functions
𝔖
,
𝜑),
𝔖
,
𝜑):
𝐾
→
ℝ
are
called
lower
and
upper
functions
of
=
𝜑),
𝔖
𝜑)]
for
all
𝒾
∈
[𝜇,
:
[𝜇,
𝜈]
⊂
ℝ
→
𝒦
are
given
by
𝔖
𝔖
3µ
+
υ
υ
−
µ
([
],
)
([
],
)
,
,
[𝔖∗Then
(𝒾,
=1].
𝜑)]
for
𝒾𝜈]
∈ if
𝐾.
Here,
for
each
1]function
the
∗ 𝜑),
∗ if and
(𝒾,
∗ (.
𝜑)
and
𝔖en
𝜑 1],
∈(𝒾)
(0,
𝔖(.𝔖
is(𝒾,
𝐹𝑅-integrable
over
[𝜇,(𝐹𝑅)
𝜈]
only
if[𝔵]
𝔖𝔽upper
+all
υ[𝜇,
∗∈
(𝒾)𝑑𝒾
(𝐹𝑅)
(𝒾)𝑑
Theorem
[49]
Let
𝔖
:𝔖
[𝜇,
𝜈]
⊂µ𝐹𝑅
ℝ
𝔽,ℝ
be
adenotes
F-NV-F
whose
𝜑-cuts
define
the
f
tions
𝔖
tions
𝐹𝑅
of
over
-integrable
over
𝔖
[𝜇,
𝜈]and
if𝜑
𝔖
∈only
.if(0,
Note
that,
functions
𝔖
𝔖𝔖
,is
𝜑):
𝐾
→
are
called
lower
[𝜌.
𝔵]𝜈]
=and
𝜌.
for
all of
𝜑 ∈I-V-Fs.
(0,
where
all
ℛ
𝜑-integrable
∈I-V-Fs.
where
ℛ→
denotes
the
collection
of
Riemannian
the
collection
integrable
of
Riemann
func
∗ , (0,
(𝒾,
([
],, 𝜑),
)1],
([
], →
) over
𝜑)
an
𝜑for
∈is2.
(0,
1].
Then
is
𝐹𝑅-integrable
[𝜇,
if
𝔖
∗
∗
Theorem
2.
[49]
Let
𝔖
:
[𝜇,
𝜈]
⊂
ℝ
𝔽
be
a
F-NV-F
whose
𝜑-cuts
defin
(𝐹𝑅)
(𝒾)𝑑𝒾
(𝐹𝑅)
tions∗𝜑
of
I-V-Fs.
𝔖
tions
is
𝐹𝑅
of
-integrable
I-V-Fs.
𝔖
over
is
𝐹𝑅
[𝜇,
-integrable
𝜈]
if
over
𝔖
[𝜇,
𝜈]
if
∈
𝔽
.
Note
∗
(𝐹𝑅)
(𝒾)𝑑𝒾
𝔖
=→(𝐼𝑅)
𝔖[𝜇,(𝒾)𝑑𝒾
=
𝔖(𝒾,
𝜑)𝑑𝒾
:𝜑)
𝔖(𝒾,
𝜑) 𝔖
∈ (ℛ
(. , 𝜑),
(.𝐹𝑅-integrable
functions
𝔖
,Moreover,
𝜑):𝜈]𝐾⊂
ℝ→
are
called
lower
and
upper
functions
of
(𝒾,
and
∈
(0,
1].
Then
𝔖
is
over
𝜈]
if
and
only
if
𝔖
∗over
∗𝔖
∗
∗
[𝜇,
𝑅-integrable
[𝜇,
𝜈].
if
𝔖
is
𝐹𝑅-integrable
over
𝜈],
then
Theorem
2.
[49]
Let
𝔖
:
[𝜇,
ℝ
𝔽
be
a
F-NV-F
whose
𝜑-cuts
define
the
fa
(𝒾)𝑑𝒾
(𝐹𝑅)
(𝒾)𝑑𝒾
(𝐼𝑅)
𝔖
=
𝔖
=
𝔖(𝒾,
𝜑)𝑑𝒾
:
𝔖(𝒾,
𝜑)
(𝒾,
(𝒾,
(𝒾,
(𝒾,
(𝐹𝑅)
(𝒾)𝑑𝒾
(𝐹𝑅)
(𝒾)𝑑𝒾
𝔖 and
𝜑)
are
Lebesgue-integrable,
𝜑),
𝔖
𝜑)
Lebesgue-integrable,
then
𝔖if
abe
Aumann-integrable
𝔖
fuzzy
Aum
fun
𝔖∗ 𝜑),
𝔖is
tions
of
I-V-Fs.
𝔖
tions
𝐹𝑅
of→
-integrable
I-V-Fs.
𝔖
over
isMoreover,
𝐹𝑅
[𝜇,
-integrable
𝜈](𝒾)
if𝔽is
𝔖then
[𝜇,
∈𝜑)]
𝔽is
.afor
Note
(𝒾,
(𝒾,
=𝔖[𝔖
𝜑),
𝔖 𝜈]
all
𝒾that,
∈then
[𝜇,
: [𝜇,
𝜈]
ℝ2.
𝒦over
given
by
𝔖ℝ
𝔖Definition
Definition
1.are
[49]
A
fuzzy
number
valued
map
𝔖
:if
𝐾
⊂
ℝ
→ 𝜈],
𝔽𝔖
is
c
∗⊂
∗over
Combining
(22)
have
an
Then
fuzzy
integ
Let
𝔖
:[49]
[𝜇,
𝜈]
⊂
→
∗𝑅-integrable
∗ are
∗over
[𝜇,
[𝜇,
𝜈].
isfuzzy
𝐹𝑅-integrable
(𝒾,we
(𝒾,
(𝒾,
(𝒾)
[𝔖
(𝒾,
𝜑), (23),
𝔖Definition
are
Lebesgue-integrable,
𝜑),
𝔖Let
𝜑)
Lebesgue-integrable,
then
is
aF-NV-F.
then
Aumann-integrab
𝔖
isℝ𝜑)
a→fuzz
𝔖∗ (𝒾,
𝔖[49]
𝜑),
𝔖
𝜑)]
for
:𝜑)
[𝜇,over
𝜈]
⊂
ℝ[49]
→ Moreover,
𝒦
are
given
by
𝔖(𝒾)𝑑𝒾
𝔖
(𝐹𝑅)
(𝒾)𝑑𝒾
𝔖
=
𝔖
=
𝔖(𝒾,
𝜑)𝑑𝒾
:𝐾𝔖(𝒾,
∈all
Definition
1.
A
fuzzy
number
valued
map
𝔖(𝒾,
:Then
⊂
𝔽 ℛi𝒾
∗2.
∗fuzzy
be
an
F-NV-F.
fuzzy
𝔖
:(𝐼𝑅)
[𝜇,are
𝜈]𝔖define
⊂
ℝ
→[𝔖
𝔽𝔖(𝒾,
∗over
[𝜇,
𝑅-integrable
[𝜇,
𝜈].
if
is
𝐹𝑅-integrable
𝜈],
then
∗[𝜇,
∗[𝜇,
tion
over
𝜈].
tion
over
𝜈].
(𝒾)
(𝒾,
𝜑
∈
(0,
1]
whose
𝜑
-cuts
the
family
of
I-V-Fs
𝔖
:
𝐾
⊂
=
𝜑),
𝔖
𝜑)]
for
all
𝒾
∈
[𝜇,
:
[𝜇,
𝜈]
⊂
ℝ
→
𝒦
are
given
by
𝔖
𝔖
(𝒾,
(𝒾,
(𝒾,
(𝒾,
𝜑),
𝔖
𝜑)
are
Lebesgue-integrable,
𝜑),
𝔖
𝜑)
are
Lebesgue-integrable,
then
𝔖
is
a
fuzzy
then
Aumann-integrable
𝔖
is
a
fuzzy
Aum
func
𝔖
𝔖
(𝐹𝑅)
(𝒾)𝑑𝒾
Definition
1.where
[49]
A
fuzzy
number
valued
map
𝔖:ifThen
𝐾of
ℝfuzzy
→
is :cal
∗an
denoted
by
𝔖𝜈]
, over
isdenotes
given
levelwise
by
(𝒾,
𝜑)𝔽and
𝔖𝐾i∗ℝ
𝜑[𝜇,
∈ 𝜈],
(0,
1].
Then
𝔖Let
is
𝐹𝑅-integrable
[𝜇,
𝜈]
if
and
only
𝔖⊂∗ Riemannian
∗
∗ ∈
be
F-NV-F.
integr
Definition
2.𝜑(0,
[49]
𝔖
: [𝜇,𝜈].
⊂
ℝ
for
all
𝜑
ℛ
the
collection
of
tion over
[𝜇,
𝜈].
tion
over
[𝜇,
∈1],
(0,
1]
whose
𝜑
-cuts
define
the
family
I-V-Fs
𝔖𝜑)
)→
,𝔖],(𝒾)𝑑𝒾
Z𝔽
[𝜇,
(𝐹𝑅)
(𝒾,
𝜈],
denoted
by
,
is
given
levelwise
by
∗([(𝒾,
an
𝜑
∈
(0,
1].
Then
𝔖
is
𝐹𝑅-integrable
over
[𝜇,
𝜈]
if
and
only
if
𝔖
υ
e
∗
∗
for
all
𝜑
∈
(0,
1],
where
ℛ
denotes
the
collection
of
Riemannia
µυ
(𝒾)
[𝔖
(𝒾,
𝔖
=
𝜑),
𝔖
𝜑)]
for
all
𝒾
∈
𝐾.
Here,
for
each
𝜑
∈
(0,
4µυ
4µυ
S
(
)
([
],
)
,= (𝑅)
tion overs−[𝜇,
tion
over
[𝜇,
(𝒾,
(𝒾,
(𝑅)
(𝐹𝑅)
(𝒾)𝑑𝒾
𝜑
∈Then
(0,
1]
whose
-cuts
define
the
of
𝔖
:and
𝐾𝜑)𝑑𝒾
⊂
ℝ∗∈(
∗𝜈].
𝔖family
𝜑)𝑑𝒾
,ifI-V-Fs
𝔖𝜑
∗if
(𝒾,
𝜑)
∈𝜈],
(0,denoted
1].
𝔖[𝜇,
is𝜈].
𝐹𝑅-integrable
over
[𝜇,
𝜈]
if
and
only
𝔖[𝜇,
[𝜇,
(𝐹𝑅)
(𝒾)𝑑𝒾
∗d
by
𝔖
,
is
given
levelwise
by
e𝜈].
e
∗𝔖
(𝐹𝑅)
(𝒾)𝑑𝒾
∗
e 1],
tions
of∈
I-V-Fs.
𝔖
is(𝐹𝑅)
𝐹𝑅
-integrable
over
[𝜇,
𝜈]
if
𝔖
∈
𝔽
𝑅-integrable
over
Moreover,
𝔖
is
𝐹𝑅-integrable
over
𝜈],
then
(𝒾)
[𝔖
(𝒾,
(𝒾,
𝔖
=
𝜑),
𝔖
𝜑)]
for
all
𝒾
∈
𝐾.
Here,
for
each
𝜑
2 2𝜑
S
+
S
4
.
(24)
for
all
𝜑
(0,
where
ℛ
denotes
the
collection
of
Riemannian
int
(𝒾,
(𝒾,
(𝑅)
(𝑅)
(𝒾)𝑑𝒾
=
𝔖
𝜑)𝑑𝒾
,
𝔖
],F-NV-F
)𝜈] ⊂ ℝwhose
∗:,(.
∗define
∗([
(𝐹𝑅)
(𝒾)𝑑𝒾
[𝜇,
∗𝜑-cuts
Theorem 2. [49]µLet
Theorem
𝔖(𝒾)
:of
[𝜇,
𝜈][𝔖
⊂
2.
ℝ
[49]
→
𝔽
𝔖
be
a[𝜇,
𝜑-cuts
be
a[𝜇,
F-NV-F
whose
the
family
of
I-V-F
def
tions
I-V-Fs.
𝔖
𝐹𝑅
over
𝜈]
𝔖𝜈],
𝑅-integrable
over
[𝜇,
𝜈].
Moreover,
ifµall→
𝔖 𝒾𝔽
is2𝔖
𝐹𝑅-integrable
over
then
(.
functions
𝔖
,∗υLet
𝜑),
𝔖
, 𝜑):
→
ℝ
are
called
lower
and
upper
func
+𝔖
3υ
3µ
υ-integrable
µ𝐾 (𝑅)
(𝒾,
(𝒾,
=
𝜑),
𝔖
∈
each
𝜑
∈
(0,
1∈f
∗+
(𝒾,
(𝒾,
(𝑅)
(𝐹𝑅)
(𝒾)𝑑𝒾
=
, if
𝔖
𝜑)𝑑𝒾
𝔖is
∗−
∗𝜈].
∗functions
∗𝐾.
Theorem
2.of
[49]
Let
𝔖
:𝔖[𝜇,
⊂
2.
[49]
→
Let
𝔽𝜑)]
𝔖,for
:a[𝜇,
𝜈]
⊂[𝜇,
ℝwhose
→ 𝜑)𝑑𝒾
𝔽Here,
F-NV-F
𝜑-cuts
be
afor
F-NV-F
define
whose
the
family
𝜑-c
[𝜇,
(.be
𝑅-integrable
over
[𝜇,
if
is
𝐹𝑅-integrable
over
𝜈],
then
𝔖Lebesgue-integrable,
, 𝜑),
𝔖
𝜑):
𝐾
→
ℝ
are
called
lower
and
upper
(𝒾)𝑑𝒾
(𝐹𝑅)
(𝒾)𝑑𝒾
(𝐼𝑅)
(𝒾,Theorem
(𝐹𝑅)
(𝒾)𝑑𝒾
𝔖
=
𝔖
=
𝔖(𝒾,
𝜑)𝑑𝒾
:
𝔖(𝒾,
𝜑)
∈
ℛ
𝜑),
𝔖
𝜑)
are
then
𝔖
is
fuzzy
Aumann-in
𝔖∗ (𝒾,
tions
I-V-Fs.
is𝜈]Moreover,
𝐹𝑅
-integrable
over
𝜈]
if
𝔖
∈
𝔽
.
∗ (.ℝ
∗
∗ (.
∗ (𝒾,
∗𝜑)𝑑𝒾
(𝒾)𝑑𝒾
(𝐹𝑅)
(𝒾)𝑑𝒾
(𝐼𝑅)
(𝒾,
=𝜈]
𝔖
= lower
𝔖(𝒾,
:and
𝔖(𝒾,
𝜑)
Theorem
𝔖are
:𝜑),
𝜈]
⊂
2.
ℝ
[49]
Let
𝔽
𝔖
ℝwhose
→
𝔽(𝒾)
be
F-NV-F
𝜑-cuts
be=then
a[𝔖
F-NV-F
the
family
𝜑-cuts
offor
I-V-F
defi
𝔖
𝜑)
are
𝔖 𝜑),
is
fuzzy
Auman
𝔖∗Theorem
(.→
(𝒾)
(𝒾,
(𝒾,
functions
, 𝜑),
𝔖Lebesgue-integrable,
,[𝔖
𝜑):
𝐾⊂𝜑),
→
ℝ𝔖
are
called
and
upper
funct
=:a[𝜇,
𝜑)]
for
all
𝒾 whose
∈a𝔖[𝜇,
𝜈]𝜑)]
for
alla
⊂ [49]
ℝ → Let
𝒦
:[𝜇,
[𝜇,
given
𝜈](𝒾,
⊂𝔖by
→
𝔖
𝒦
are
given
by
𝔖
𝔖 : [𝜇, 𝜈] 2.
𝔖
∗ℝ
∗ (𝒾,
∗define
∗ (𝒾,
∗ (𝒾,
∗ (𝒾,
over
[𝜇,
𝜈].
∗
(𝒾)
[𝔖
(𝒾,
(𝒾)
[𝔖
(𝒾,
=
𝜑),
𝔖
𝜑)]
=
for
all
𝜑),
𝒾
∈
𝔖
[𝜇,
𝜈]
𝜑)]
an
𝜈]
⊂
ℝ
→
𝒦
are
:
[𝜇,
given
𝜈]
⊂
by
ℝ
→
𝔖
𝒦
are
given
by
𝔖
𝔖 : [𝜇,
𝔖
(𝒾)𝑑𝒾
(𝐹𝑅)
(𝒾)𝑑𝒾
(𝐼𝑅)
(𝒾,
𝔖
=
𝔖
=
𝔖(𝒾,
𝜑)𝑑𝒾
:
𝔖(𝒾,
𝜑)
∈
ℛ
𝜑),
𝔖
𝜑)
are
Lebesgue-integrable,
then
𝔖
is
a
fuzzy
Aumann-int
𝔖tion
(𝒾,(𝒾)𝑑𝒾
(𝒾, 𝜑)𝑑𝒾
(𝑅)
∗⊂∗ℝ →
∗ ∗
==
𝔖𝔖
𝜑)𝑑𝒾
𝔖Then
𝔖(𝒾)𝑑𝒾
(
be an, F-NV-F.
fuz
Definition
2. [49]
Let 𝔖: [𝜇,
𝜈](𝑅)
∗
(𝐼𝑅)
∗𝔽
∗all
tion
over
[𝜇,⊂(𝐹𝑅)
𝜈].
∗ (𝒾,
(𝒾)[49]
[𝔖
(𝒾)
[𝔖
(𝒾,
(𝒾,
(𝑅)
(𝑅)
(𝒾)𝑑𝒾
=
𝜑),
𝜑)]
=
for
all
𝜑),
𝒾an
∈
𝔖
[𝜇,
𝜈]
and
for
a
ℝfor
→𝒦
are
[𝜇,
given
𝜈]
by
ℝ
→
𝔖(𝐹𝑅)
𝒦
given
𝔖
𝔖
𝔖
=
𝔖∗𝔖
𝜑)𝑑𝒾
,(𝒾,
𝔖for
𝔖
F-NV-F.
Then
Definition
2.
Let
:by
[𝜇,𝔖
𝜈](𝒾,
⊂
ℝ𝔖
→
𝔽
(𝒾)𝑑𝒾
𝜑)
and
𝔖
both
𝜑)
𝜑 ∈: [𝜇,
(0,𝜈]
1].⊂ Then
𝔖
𝜑
is
∈:[𝜇,
𝐹𝑅-integrable
(0,
1].
Then
over
𝔖
isare
[𝜇,
𝐹𝑅-integrable
𝜈]
if𝔖denotes
and
only
over
if(𝐼𝑅)
[𝜇,
𝜈]
ifbe
and
only
if𝜑)]
𝔖
=
∗ (𝒾,
∗(𝒾,
∗ (𝒾,
∗ (𝒾,
∗𝜑)
𝜑[𝜇,
∈
(0,
1],
where
ℛ
the
collection
of
Riemannian
in
∗𝜑)𝑑𝒾
tion
over
𝜈].
([ over
)=
, ], is
(𝒾,
(𝒾,
(𝑅)
(𝑅)
(𝐹𝑅)
(𝒾)𝑑𝒾
(𝐹𝑅)
(𝒾)𝑑𝒾
𝔖
𝜑)𝑑𝒾
,
𝔖
𝔖
𝜈],
denoted
by
𝔖
,
is
given
levelwise
by
(𝒾,
(𝒾,
𝜑)
and
𝔖
𝜑)
𝜑 ∈ (0,
1].all
Then
𝔖
𝜑
is
∈
𝐹𝑅-integrable
(0,
1].
Then
𝔖
[𝜇,
𝐹𝑅-integrable
𝜈]
if
and
only
over
if
𝔖
[𝜇,
𝜈]
if
and
only
if
𝔖
(𝒾)𝑑𝒾
(𝐼𝑅)
=
𝔖
be
an ∗levelwise
F-NV-F.
Then
fuzzy
2. [49]
Let
𝔖
: [𝜇,
𝜈]) ⊂
ℝ→𝔽
∗ the
∗
forDefinition
all 𝜑[𝜇,∈𝜈],
(0, denoted
1],
where
ℛ(𝐹𝑅)
denotes
collection
Riemannia
∗of
by
is
given
by
(𝒾,
(𝒾,
𝜑)
and
𝔖over
𝜑)
both
𝜑𝑅-integrable
∈ (0, 1]. Then
𝜑
is
(0,
1]. where
Then
is([𝜈].
[𝜇,
𝐹𝑅-integrable
𝜈]([ℝ
if,denotes
only
[𝜇,
𝜈]if
ifthen
and
only
if(𝒾)𝑑𝒾
𝔖
Theorem
2. [49]
Let
:𝔖[𝜇,
[𝜇,
𝜈]
⊂
→],and
𝔽 𝔖over
beover
whose
𝜑-cuts
define
the
[𝜇,
(𝐹𝑅)
over
[𝜇,𝜑
𝑅-integrable
𝜈].
Moreover,
ifis𝔖over
𝐹𝑅-integrable
Moreover,
if(𝒾)𝑑𝒾
𝔖aifF-NV-F
is, 𝔖
𝐹𝑅-integrable
𝜈],
𝜈],
thar.
tions
of
I-V-Fs.
𝔖over
𝐹𝑅
-integrable
[𝜇,
𝔖
∈𝜑)
𝔽
∗𝜈]
∗ (𝒾,
for
all𝔖Theorem
∈∈𝐹𝑅-integrable
(0,
1],
ℛ
collection
of(𝐹𝑅)
Riemannian
int
],[𝜇,
)is⊂
,𝔖
[𝜇,
(𝐹𝑅)
(𝒾)𝑑𝒾
byis
𝔖
given
levelwise
by
2. [49]
Let
𝔖
:if𝐹𝑅
[𝜇,
𝜈]
ℝMoreover,
→([ ,𝔽, is],the
be
a
F-NV-F
whose
𝜑-cuts
define
[𝜇,
[𝜇
(𝒾)𝑑𝒾
(0,
𝑅-integrable
over
[𝜇,denoted
𝑅-integrable
𝜈].
Moreover,
over
𝜈].𝐹𝑅-integrable
if
over
𝔖
is
𝐹𝑅-integrable
𝜈],
then
over
of
I-V-Fs.
𝔖
-integrable
over
[𝜇,
𝜈]
if
𝔖
∈
for alltions
𝜑
∈𝜈],
(0,
1].
For
all
𝜑
∈
1],
ℱℛ
denotes
the
collection
of
all
𝐹𝑅
)
∗[49]
(𝐼𝑅)
for
all
𝜑ℝ∈𝜑)
(0,
1].
For
all
𝜑
denotes
the
collection
of
Theorem
2.Moreover,
Let
:Lebesgue-integrable,
[𝜇,
𝜈]
⊂Moreover,
ℝby∈→(0,
be
a𝔖
F-NV-F
𝜑-cuts
define
[𝜇,
[𝜇,
(𝒾)
[𝔖
(𝒾,
(𝐹𝑅)
([
], (𝒾)𝑑𝒾
) whose
𝑅-integrable tions
over
𝑅-integrable
𝜈].
over
if are
𝔖
[𝜇,given
is
𝜈].
𝐹𝑅-integrable
if ℱℛ
over
is∗𝔖
𝐹𝑅-integrable
𝜈],
then
over
𝜈],𝒾the
=
𝜑),
𝔖∗(𝒾)𝑑𝒾
for
all∈
:[𝜇,
[𝜇,
𝜈]
⊂
𝔖𝔽=1],
𝔖∗ (𝒾,
𝜑),
𝔖
are
then
𝔖
a(𝒾,
fuzzy
Aumann-in
𝔖
,(𝒾,
of
I-V-Fs.
𝔖𝒦
is𝔖
𝐹𝑅
-integrable
over
[𝜇,
𝜈]
if
𝔖(𝒾)𝑑𝒾
𝔽∈the
. [fa
∗𝜑)]
∗→
(𝐼𝑅)
=
𝔖is
(0,
NV-Fs
over
[𝜇,
𝜈].
for
all𝔖
∈
(0,
1].
For
all
𝜑
∈
1],
ℱℛ
denotes
the
collection
of
allall
𝐹𝑅
(𝒾)𝑑𝒾
(𝐹𝑅)
(𝒾)𝑑𝒾
(𝐼𝑅)
𝔖
=
𝔖
= is𝔖
𝔖(𝒾,
: 𝔖(𝒾
(𝒾)
[𝔖
(𝒾,
(𝒾,
=
𝜑),
𝜑)]𝜑)𝑑𝒾
for
𝒾
:
[𝜇,
𝜈]
⊂
ℝ
→
𝒦
are
given
by
𝔖
𝔖𝜑
𝜑),
𝔖
𝜑)
are
Lebesgue-integrable,
then
a(𝒾,
fuzzy
Auman
([
],
)
,
∗ (𝒾, 𝔖
∗
Symmetry 2022, 14, x FOR PEER REVIEW
Symmetry 2022, 14, x FOR PEER
REVIEW
Proposition
1. [51] Let 𝔵, 𝔴 ∈ 𝔽 . Then fuzzy order relation “ ≼ ”
Symmetry 2022, 14, x FOR PEER REVIEW Proposition 1. [51] Let 𝔵, 𝔴 ∈ 𝔽 . Then fuzzy order relation
4 o
is given on 𝔽 by
“ ≼ ” is given on 4𝔽of
Proposition 1. [51] Let 𝔵, 𝔴 ∈ 𝔽 . Then fuzzy order relation “ ≼ ” is given on 𝔽 by
𝔵 ≼ 𝔴 if and only if, [𝔵] ≤ [𝔴] for all 𝜑 ∈ (0, 1],
𝔵 ≼ 𝔴 if and only if, [𝔵] ≤ [𝔴] for all 𝜑 ∈ (0, 1],
[𝔵] ≤
[𝔴]relation
𝔵≼𝔴
for all “𝜑≼∈” (0,
1],
1.
[51]relation.
Let
𝔵, 𝔴if ∈and
𝔽 .only
Thenif,fuzzy
order
is given
on 𝔽 by (
ItProposition
is a partial
order
Proposition
1.
[51]relation.
Let 𝔵, 𝔴 ∈ 𝔽 . Then fuzzy order relation “ ≼ ” is given on 𝔽
It
is
a
partial
order
[51]mathematical
Let 𝔵, 𝔴 ∈ 𝔽 operations
. Then fuzzy
relation
“of≼the
” ischaracteristics
given9 of
on18𝔽 of
byfuz
now1.
use
to order
examine
some
ItProposition
is aWe
partial
order
relation.
Symmetry 2022, 14, 1639
𝔵 ≼mathematical
𝔴 if and onlyoperations
if, [𝔵]
≤ to[𝔴]
for all
𝜑 ∈of
(0,the
1],characteristics
We now use
examine
some
[𝔵]
[𝔴]
𝔵 ≼𝜌 𝔴
if and
if,
𝜑 ∈ (0, 1], of fuz
numbers.
Letuse
𝔵, 𝔴mathematical
∈ 𝔵𝔽≼ and
∈
ℝ,
thenonly
we
We now
operations
to have
examine≤some
offor
theall
characteristics
numbers.
Let relation.
𝔵, 𝔴 𝔴
∈ 𝔽if and
andonly
𝜌 ∈ if,
ℝ, [𝔵]
then≤we[𝔴]
havefor all 𝜑 ∈ (0, 1],
It
is
a
partial
order
numbers.
𝔵, 𝔴 ∈order
𝔽 and
𝜌 ∈ ℝ, then we have
It isLet
a partial
relation.
[𝔵+𝔴] to
= [𝔵]
+ [𝔴]some
, of the characteristics of fu
noworder
use mathematical
operations
examine
It is aWe
partial
relation.
[𝔵+4,𝔴]
= [𝔵]
+ [𝔴]some
, of the characteristics
We
now
use
mathematical
operations
to
examine
Therefore, for every ϕ ∈ [0, 1], by using Theorem
we
have
[𝔵+
[𝔵]
[𝔴]
𝔴] we
= have
+ some
, of the characteristics
numbers.
Letuse
𝔵, 𝔴mathematical
∈ 𝔽 and 𝜌 ∈
ℝ,
then
We now
operations
to
examine
of fuz(
[𝔵
[𝔵]
[
×
𝔴]
=
×
𝔴]
,
numbers.
Let
then
we
have
𝔵, 𝔴 ∈ 𝔽h and 𝜌 ∈ ℝ,
i
[𝔵
× have
𝔴] = [𝔵] 4µυ
× [ 𝔴] ,
numbers.
Let
we
2µυ𝔵, 𝔴 ∈ 𝔽 and 𝜌 ∈ ℝ, then
4µυ
1 × [ 𝔴] ,
(
[𝔵]
× 𝔴]
==+
[𝔵]
4s−1 S∗ µ+υ , ϕ ≤ 4s−1 21s S∗ [𝔵µ[𝔵+
, ϕ
𝔴]
+
,, ϕ ,
sS
∗ [𝔴]
[𝜌.
[𝔵]
+
3υ
2
3µ
+
𝔵]
=
𝜌.
[𝔵+
𝔴]𝔵] = =
+υ [𝔴]i,
h
[𝜌.
[𝔵]𝜌. [𝔵]
2µυ
4µυ𝔴] = [𝔵]
4µυ,
+∗ [𝔴]
1𝜌. [𝔵]
(
[𝜌., 𝔵]
4s−1 S∗ µ+υ , ϕ ≤ 4s−1 21s S∗ [𝔵+
ϕ =
+=[𝔵]
s S× [3µ
[𝔵+×
𝔴]
µ
3υ 𝔴]
+υ , ϕ ,
[𝔵 × 𝔴]2 = [𝔵]
× [ 𝔴] ,
= 32[𝔵∗valued
,× 𝔴] map
= [𝔵] :×𝐾[⊂
𝔴] ,→ 𝔽 is called F-NV-F. For ea
Definition 1. [49] A fuzzy number
[𝔵] 𝔖ℝ
𝔵] = 𝔖
𝜌.
Definition 1. [49] A fuzzy number
map
: 𝐾 ⊂ ℝ → 𝔽 is called F-NV-F.
∗ , [𝜌.valued
[𝜌.
𝔵]
=
=
3
𝜑 ∈ (0, 1] whose
-cuts number
define 2the
family
of
𝔖 𝔽: 𝐾is⊂called
ℝ→𝒦
are given
Definition
1. [49] A𝜑fuzzy
valued
map
𝔖
: I-V-Fs
𝐾 ⊂ 𝜌.
ℝ[𝔵]
→
F-NV-F.
For ea
[𝜌.
[𝔵]
𝔵]
=
𝜌.
𝜑
∈
(0,
1]
whose
𝜑
-cuts
define
the
family
of
I-V-Fs 𝔖 : 𝐾 ⊂ ℝ → 𝒦 4 are
Symmetry 2022, 14, x FOR PEER REVIEW (𝒾)
of 19g
∗ (𝒾,
R
υ
[𝔖
(𝒾,
µυ
𝔖
=
𝜑),
𝔖
𝜑)]
for
all
𝒾
∈
𝐾.
Here,
for
each
𝜑
∈
(0,
1]
the
end
point
𝜑
∈
(0,
1]
whose
𝜑
-cuts
define
the
family
of
I-V-Fs
𝔖
:
𝐾
⊂
ℝ
→
𝒦
are
given
,
ϕ
S
)
(
∗
Symmetry 2022, 14, x FOR PEER REVIEW 𝔖 (𝒾)∗ = [𝔖 (𝒾, 𝜑), 𝔖∗ (𝒾, 𝜑)] for all 𝒾 ∈ 𝐾. Here, for each 𝜑 ∈ (0, 1] the end4 pro
≤ υ−µnumber
d ,map 𝔖: 𝐾 ⊂ ℝ → 𝔽 is called F-NV-F.
∗ A∗ ∗fuzzy
µ
2
Definition
1.(.[49]
For
e
Symmetry 2022, 14, x FOR PEER REVIEW 𝔖
4 of
19 re
(. ,𝜑)]
(𝒾) =
[𝔖𝔖
(𝒾,
functions
, 𝜑),
𝜑):
𝐾
ℝ
are
lower
functions
of
𝔖. F-NV-F.
𝔖𝔖[49]
for→all
𝒾valued
∈ called
𝐾.valued
Here,
forand
each
𝜑∈
(0,
1] the
end
point
∗ 𝜑),
Definition
1.
A
number
map
𝔖:upper
𝐾and
⊂
ℝupper
→𝔽
is called
∗ (𝒾,
∗fuzzy
(.
(.
functions
𝔖
,
𝜑),
𝔖
,
𝜑):
𝐾
→
ℝ
are
called
lower
functions
of
𝔖
.
∗A𝜑∗fuzzy
𝜑 ∈ (0, 1] 𝔖whose
define
the
family
𝔖 𝔽: 𝐾is⊂called
ℝ→𝒦
are
Definition
1.(.[49]
number
valued
mapof
𝔖: I-V-Fs
𝐾 ⊂ ℝupper
→
F-NV-F.
For ea
functions
, 𝜑),whose
𝔖 ∗(.-cuts
, 𝜑):
𝐾
→ ℝdefine
are
called
of →𝔖
. given
𝜑 ∈ (0,
𝜑 -cuts
the lower
familyand
of I-V-Fs functions
𝔖 :𝐾 ⊂ ℝ
𝒦
are
∗ 1]
∗ 𝒾, ∈
R
(𝒾)
[𝔖
(𝒾,
(𝒾,
µυ
𝔖
=
𝜑),
𝔖
𝜑)]
for
all
𝐾.
Here,
for
each
𝜑
∈
(0,
1]
the
end
point
r
𝜑
∈
(0,
1]
whose
𝜑
-cuts
define
the
family
of
I-V-Fs
𝔖
:
𝐾
⊂
ℝ
→
𝒦
are
given
ϕ
S
)
(
υ
∗
be
an
F-NV-F.
Then
fuzzy
integral
of
𝔖 ov
Definition
2.
[49]
Let
𝔖
:
[𝜇,
𝜈]
⊂
ℝ
→
𝔽
∗𝔽
Proposition
1.
[51]
Let
𝔵,
𝔴
∈
.
Then
fuzzy
order
relation
“
≼
”
is
given
on
𝔽
by
(𝒾)
[𝔖
(𝒾,
(𝒾,
𝔖
=
𝔖
for2⊂all
𝒾 ,∈𝔽order
𝐾.beHere,
for each
𝜑is∈given
(0, 1]on
the𝔽 end
p
d→
≤ 𝔵,Let
∗ Let
an
F-NV-F.
Then
fuzzy
integral
of
Definition
2.
𝔖𝜑)]
:µ[𝜇,
𝜈]
ℝ
∗𝜑),
∗[49]
Proposition
1.
[51]
∈
𝔽
.
Then
fuzzy
relation
“
≼
”
by
υ𝔴
−µ𝐾
(.
(.
(𝒾)
[𝔖
(𝒾,
(𝒾,
functions
𝔖
,
𝜑),
𝔖
,
𝜑):
→
ℝ
are
called
lower
and
upper
functions
of
𝔖
.
𝔖
=
𝜑),
𝔖
𝜑)]
for
all
𝒾
∈
𝐾.
Here,
for
each
𝜑
∈
(0,
1]
the
end
point
r
h(𝐹𝑅)
belevelwise
i by“ Then
[𝜇, 𝜈], functions
(𝒾)𝑑𝒾
∗ ∗ [49]
denoted
by
𝔖
is→
∗ 𝜈]
an
F-NV-F.
fuzzy
integral
ofof 𝔖𝔖.ov
Definition
2.
Let
𝔖∈: ϕ𝔖
[𝜇,
⊂ ℝ, fuzzy
𝔽ℝ order
Proposition
[51]
Let
𝔵,
𝔽
Then
relation
≼and
”∈
isupper
given
on 𝔽 by
(.S∗𝔴
,by
,∗𝜑):
→given
called
lower
2µυ
(µ,
)+.(.S
(υ,𝔖ϕ𝐾
) if,
∗𝜑),
1𝔖
[𝜇,1.
(𝒾)𝑑𝒾
[𝔵]
[𝔴]
𝜈],𝔖
denoted
, ∗isare
given
by
𝔵∗𝔖
≼
𝔴
if(𝐹𝑅)
and
only
≤
for
𝜑
(0,
1], functions
(4)
≤ s𝜑),
+
S
, ϕlevelwise
,by all
(.≼, 𝜑):
functions
𝐾
→
ℝ, only
are
called
lower
and
upper
functions
of 𝔖.
+
1
2
µ
+
υ
[𝔵]
[𝔴]
∗ (. ,by
[𝜇,
(𝐹𝑅)
(𝒾)𝑑𝒾
𝔵
𝔴
if
and
if,
≤
for
all
𝜑
∈
(0,
1],
𝜈],
denoted
𝔖
is
given
levelwise
h14, x∗ FOR PEER
i
Symmetry 2022,Symmetry
14, x FOR2022,
PEER
Symmetry
14,REVIEW
x FOR
2022,
PEER
14,Symmetry
xREVIEW
FOR PEER
2022,
REVIEW
REVIEW 4 of
∗
[𝔵]
[𝔴]
𝔵
≼
𝔴
if
and
only
if,
≤
for
all
𝜑
∈
(0,
1],
(4)
be
an
F-NV-F.
Then
fuzzy
integral
of
𝔖
o
Definition
2.
[49]
Let
𝔖
:
[𝜇,
𝜈]
⊂
ℝ
→
𝔽
S
µ,
ϕ
S
υ,
ϕ
2µυ
(
)+ (
)
1
∗
It is a partial
order
relation.
≤order
ϕ4 an
, 19
F-NV-F.
Then
fuzzy
integral
o
Definition
[49] Let=
: [𝜇, 𝜈]+
⊂Sℝ
→µ19
𝔽υ=, be
s+𝔖
12.(𝒾)𝑑𝒾
2 𝔖
+
14,
ERSymmetry
xREVIEW
FOR PEER
2022,
REVIEW
14, x FOR PEER REVIEW
4(𝒾)𝑑𝒾
of
of
4 of𝜑)
19
(𝐹𝑅)
(𝐼𝑅)
𝔖
𝔖(𝒾,
𝜑)𝑑𝒾
:
𝔖(𝒾,
∈
ℛ
,
It
is
a
partial
relation.
([
],
)
,
[𝜇,
(𝐹𝑅)𝔖
(𝒾)𝑑𝒾
𝜈],now
denoted
by Let
𝔖operations
levelwise
an
F-NV-F.
Then
fuzzy: 𝔖(𝒾,
integral
ofℛfuzzy
𝔖 ov
Definition
2. [49]
: [𝜇,
𝜈]
⊂=
ℝ, is
→given
𝔽to be
(𝒾)𝑑𝒾
(𝐹𝑅)
(𝐼𝑅)
𝔖
=by of
𝔖(𝒾,
𝜑) ∈of
We
use
mathematical
some
the 𝜑)𝑑𝒾
characteristics
([ , ],
It is a partial
relation.
[𝜇,order
(𝐹𝑅)
𝜈],
by𝔖(𝒾)𝑑𝒾
𝔖1(𝒾)𝑑𝒾
, isexamine
given
levelwise
by
=3
∗ ,𝔖 (𝒾)𝑑𝒾
(𝐹𝑅)
(𝒾)𝑑𝒾
(𝐼𝑅)
We
nowdenoted
use
operations
to =
examine
some
of: 𝔖(𝒾,
the characteristics
𝔖mathematical
=(𝒾)𝑑𝒾
𝔖(𝒾,
𝜑)𝑑𝒾
𝜑) ∈ ℛ([ , ], ) of
, fu(
[𝜇,
(𝐹𝑅)
𝜈],
denoted
by
𝔖
,
is
given
levelwise
by
numbers.
Letuse
𝔵, 𝔴
∈ 𝔽 and 𝜌operations
∈=
ℝ,3
then
have
∗ , we
We
now
mathematical
to
examine
some
of
the
characteristics
of
fuzzy
1
numbers.
Let
𝔵,
𝔴
∈
𝔽
and
𝜌
∈
ℝ,
then
we
have
for all 𝜑 ∈ Proposition
(0,
Riemannian
integrable
h1. 1],
iof 𝔵,
Proposition
[51]where
LetProposition
𝔵,1.𝔴ℛ
[51]
∈
1.𝔵,denotes
Proposition
[51]
𝔴 fuzzy
∈ Let
𝔽 .the
𝔵,
𝔴collection
1.
∈fuzzy
𝔽
[51]
Let
𝔴relation
𝔽order
Then
order
relation
. Then
order
“fuzzy
≼
”∈ is
given
. Then
“ relation
≼
on
”fuzzy
is𝔽 given
by
“orde
≼fu”o
([ 𝔽
],. Then
, Let
for all
(0,
1],Swhere
ℛ1)([ , we
denotes
the 𝔖(𝒾,
collection
Riemannian
(υ,∈ϕ=
)ℝ,(𝐼𝑅)
(µ,
)+
∗𝜌
∗𝔽
1 𝔴𝜑S
numbers. Let
𝔵,
∈∈
],𝔖) have
(𝒾)𝑑𝒾
(𝐹𝑅)
(𝒾)𝑑𝒾
𝔖ϕand
=
𝜑)𝑑𝒾
: of
𝔖(𝒾,
∈ ℛ([ , integra
,(5)
], )that
metrySymmetry
2022, 14, 2022,
x FOR14,
PEER
x FOR
REVIEW
PEER REVIEW
4Riemannian
of
19 4 𝜑)
of
19
≤
+],”then
S
µ,=
ϕ
+
S
υ,=,(𝐹𝑅)
ϕ))on
,𝔖(𝒾,
(denotes
)≼[𝜇,
[𝔵+
[𝔵]
[𝔴]
𝔴]
+
srelation
(𝒾)𝑑𝒾
∗ (by
∗ (given
for
all
𝜑
∈
(0,
1],
where
ℛ
the
collection
of
integrable
fun
tions
of
I-V-Fs.
𝔖
is
𝐹𝑅
-integrable
over
𝜈]
if
𝔖
∈
𝔽
.
Note
(𝒾)𝑑𝒾
(𝐹𝑅)
(𝒾)𝑑𝒾
(𝐼𝑅)
𝔖
=
𝔖
𝜑)𝑑𝒾
:
𝔖(𝒾,
𝜑)
∈
ℛ
Proposition
Proposition
1.
[51]
Let
1.
𝔵,
Proposition
[51]
𝔴
∈
Let
𝔽
𝔵,
𝔴
1.
∈
𝔽
[51]
Let
𝔵,
𝔴
∈
𝔽
.
Then
fuzzy
.
Then
order
fuzzy
relation
order
.
Then
“
relation
≼
”
fuzzy
is
given
order
“
≼
on
is
𝔽
given
on
“
𝔽
”
is
by
𝔽
by
s
+
1
2
2
([
)
,
([
, ],[∈
Symmetry 2022, 14, x FOR PEER REVIEW
4[𝔴]
of𝜑19
[𝔵+
[𝔵]if[𝔴]
[𝔴]
𝔴]
=over
+𝔖(𝒾,
,i≼
(𝐹𝑅)
(𝒾)𝑑𝒾
[𝔵]
[𝔵]
[𝔵]
[𝔵]
[𝔴]
tions
ofh SI-V-Fs.
𝔖
is
𝐹𝑅
-integrable
[𝜇,
𝜈]
if
𝔖
∈
𝔽
.
Note
𝔵
≼
𝔴
if
and
𝔵
≼
only
𝔴
if,
if
and
𝔵
≼
only
𝔴
if,
and
only
𝔵
if,
𝔴
if
and
only
if,
≤
for
≤
all
𝜑
∈
(0,
for
≤
1],
all
for
(0,
all
1],
≤
𝜑
(𝒾)𝑑𝒾
(𝐹𝑅)
(𝒾)𝑑𝒾
(𝐼𝑅)
∗
∗ (𝔖
=
𝔖
=
𝜑)𝑑𝒾
:
𝔖(𝒾,
𝜑)
∈
ℛ
,
, ], )that,
ϕ) [𝔵+1𝔴] ∗over
ϕis
)+S
∗1(𝒾,
∗if
ymmetry 2022, 14, x FOR PEER REVIEW
4 (𝒾)𝑑𝒾
of 19 ∈ 𝔽 . ([
(𝐹𝑅)
(5)
tions
of≤I-V-Fs.
𝔖µ,are
𝐹𝑅(υ,
-integrable
[𝜇,
𝜈]
𝔖
Note
[𝔵]
[𝔴]
=
+
,
𝔖s+
𝜑)
Lebesgue-integrable,
then
𝔖
is
a
fuzzy
Aumann-integrable
fu
𝔖∗ (𝒾, 𝜑),
µ,
ϕ
+
S
υ,
ϕ
,
+
S
(
)
(
))
(
s
(6)
[𝔵
[𝔵]
[
∗
𝔴]
=
, 𝔖 is
1(0,
2for
2× denotes
[𝔵]for
[𝔵]
[𝔵]all
[𝔴]
[𝔴]
[𝔴]
𝔵 ≼ 𝔴 if and
𝔵 ≼only
𝔴 ifif,
and
only
𝔵≤all
if,
𝔴∗𝜑(𝒾,
if∈
and
only
if,
for
≤𝔖
all
𝜑𝜑)
∈
(0,
1],
≤𝜑,relation.
∈
for
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𝔴
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𝔽
.
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order
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≼
”
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𝔽
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1
1
1
∗
∗
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tions
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𝔖
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-integrable
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𝔖
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𝔽
.
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that,
(𝒾,
(𝒾,
𝜑),
𝔖
𝜑)
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then
𝔖
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a
fuzzy
Aumann-integrable
fu
𝔖
Let
numbers.
𝔵,
𝔴
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𝔽
numbers.
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𝔵,
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∈
Let
𝔽
𝔵,
numbers.
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Let
𝔵,
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∈
𝔽
and
𝜌
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ℝ,
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we
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have
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υ,
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2.
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:≤
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𝜈]Lebesgue-integrable,
⊂≤
ℝ[𝔴]
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𝔽 for
F-NV-F
defineAumann-integra
the family of I-V
s(𝒾,
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2is given
Proposition 1. [51]𝔵Let
𝔵,if𝔵𝔴≼
∈∗𝔴𝔽
.𝜑),
Then
fuzzy
order
relation
“be≼
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𝔖
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and
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𝜑
1],a∈2𝔖
(0,whose
1],
(4)
(4)
∗[𝜇,
Theorem
2.only
[49]
Let
𝔖
: [𝔵]
[𝜇,for
𝜈] ⊂
ℝ
→
be
F-NV-F
whose
𝜑-cuts
define
the
family
tion
over
𝜈].
(𝒾,
𝜑),
𝔖
𝜑)
are
Lebesgue-integrable,
then
is
a
fuzzy
Aumann-integrable
fun
numbers. Let
numbers.
𝔵, 𝔴 ∈ Let
𝔽 and
𝔵,numbers.
𝔴 ∈𝜌 𝔽∈ ℝ,and
Let
𝔴Theorem
∈∗have
𝔽then
then
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we
∈
ℝ,
and
we
𝜌
have
∈
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then
we
have
𝔵1.≼
𝔴
ifALet
and
if, ⊂ by
≤
all
𝜑
∈[𝔴]
(4)
∗→
[49]
fuzzy
number
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𝔖
: 𝐾𝜑),
⊂
ℝ
𝔽𝜑-cuts
is+for
called
2.
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𝔖only
: [𝜇,
𝜈]
ℝvalued
→
𝔽[𝔴]
be
F-NV-F
define
family
I-Vtion
[𝜇,
𝜈].
(𝒾)
[𝔖
(𝒾,
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𝜑)]
all
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fore
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1],
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fuzzy
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𝜈].[49]
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𝜑),
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𝜑)]
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𝜈]
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ℝ
𝒦define
by
𝔖𝜑 (𝒾)
𝔖 and
It is a partial
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order Definition
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∗
𝜑
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𝔖
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𝐾
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1.
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fuzzy
number
valued
map
𝔖
:
𝐾
⊂
ℝ
→
𝔽
is
called
F-NV-F.
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each
∗
(𝒾)
[𝔖
(𝒾,
(𝒾,
=
𝜑),
𝔖
𝜑)]
for
all
𝒾
∈
[𝜇,
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and
for
:
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given
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𝔖
(5)
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[𝔴]
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+
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(𝒾,
(𝒾,
𝜑)
and
𝔖
𝜑)
both
𝜑
∈
(0,
1].
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𝔖
is
𝐹𝑅-integrable
over
[𝜇,
𝜈]
and
only
if
𝔖
𝜑
∈
(0,
1]
whose
𝜑
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𝔖
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→
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given
Z
∗
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[and
[𝔵]
Theorem
2. 𝜑),
[49]
Let
𝔖examine
: [𝜇,is
𝜈]𝐹𝑅-integrable
⊂
ℝ
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𝔖:υfor
[𝜇,
⊂
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1 e𝜑
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of1𝜑)
fuzzy
e
e𝜈],
e𝜑
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[𝔖
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𝜑)
and
𝔖
𝜑)
both
a
𝜑Theorem
∈
(0,
1].
Then
𝔖
is
𝐹𝑅-integrable
over
[𝜇,
if
and
only
𝔖
𝔖
=
𝜑),
𝔖
all
𝒾
𝐾.
for
each
∈
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1]
the
end
point
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d
FR
4
3
4
S
(
µ
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+
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υ
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4
3
4
+
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∗
∗
∗
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2
1
𝑅-integrable
over
[𝜇,
𝜈].
Moreover,
if
𝔖
is
𝐹𝑅-integrable
over
then
2.
[49]
Let
𝔖
:
[𝜇,
𝜈]
⊂
ℝ
→
𝔽
be
a
F-NV-F
whose
𝜑-cuts
define
the
family
of
I-V
∗
(6)
(6)
(6)
[𝔵
[𝔵
[𝔵
[𝔵]
[
[𝔵]
[
[𝔵]
[
numbers.
numbers.
Let
𝔵,
𝔴
Let
𝔽
𝔵,
𝔴
∈
𝔽
and
𝜌
∈
and
ℝ,
then
𝜌
∈
ℝ,
we
then
have
we
have
×
×
×
=
𝔴]
×
𝔴]
=
,
×
𝔴]
𝔴]
,
=
×
𝔴]
,
s
2
(𝒾)
[𝔖
(𝒾,
(𝒾,
∗𝔖
=
𝜑),
𝔖
𝜑)]
for[𝜇,
all
∈
[𝜇,
and
for
𝜈]
→
𝒦
are
given
by
𝔖𝐾.
𝔖𝔵,υ=:𝔴[𝜇,
We numbers.
now use𝔖mathematical
to[𝜇,
examine
some
of
the
characteristics
of
fuzzy
(.operations
(.µ𝜑)]
functions
𝔖
,ℝ
𝜑),
, 𝜑):
𝐾
→
ℝgiven
are
called
lower
and
upper
functions
of[𝔵]
𝔖
. 𝒾𝜈]
∗ 𝐹𝑅-integrable
µ(𝒾)
+
υ⊂
−
s𝜌.+
1 (𝒾,
2𝒾end
2[𝜌.
∗ (𝒾,
[𝔖
(𝒾,
(𝒾,
µall
[𝜌.
[𝜌.
[𝜌.
[𝔵]
[𝔵]
𝜑),
𝔖
for
𝒾
∈
Here,
for
each
𝜑
∈
(0,
1]
the
point
real
∗⊂
𝔵]
𝔵]
𝔵]
𝔵]
=
=
𝜌.
=
𝜌.
=
𝜌.
𝑅-integrable
over
𝜈].
Moreover,
if
𝔖
is
over
𝜈],
then
∗
Let
∈
𝔽
and
𝜌
∈
ℝ,
then
we
have
∗
(𝒾)
[𝔖
=
𝜑),
𝔖
𝜑)]
for
all
∈
[𝜇,
𝜈]
a
:
[𝜇,
𝜈]
ℝ
→
𝒦
are
by
𝔖
𝔖
functions 𝔖
, 𝜑),
𝔖 (.
, 𝜑): 𝐾 →ifℝ𝔖are
lower∗ and
functions
of 𝔖.
∗ (𝒾,upper
∗ (.
[𝜇, 𝜈],
𝑅-integrable
over
[𝜇, 𝜈].
Moreover,
is called
𝐹𝑅-integrable
over
then
(𝒾)
[𝔖
numbers. Let 𝔵, 𝔴functions
𝜌(.⊂
∈, 𝜑),
ℝ,
then
have
=
𝜑),upper
𝔖
for
all of
𝒾 (5)
∈𝔖[𝜇,
𝜈] 𝜑)
andboth
for
:and
[𝜇,
ℝ
→
are
by
(.
and
𝔖
(0,
1].
Then
𝔖
is
𝐹𝑅-integrable
over
𝜈]∗ (𝒾,
ifand
and
only
if𝜑)]
𝔖∗ (𝒾,
𝔖𝜈]
𝔖
, we
𝜑):
𝐾
→+
ℝ[𝔴]
are
lower
functions
. ∗ (𝒾,
(5)𝜑)
(7)
(7)
(7)
[𝔵+
[𝔵+
[𝔵]
[𝔵]
[𝔴]
[𝜌.∈𝔵]𝔽𝔖𝜑 ∈
[𝜌.
[𝔵]
[𝔵]
[𝔵]
𝔴]
=
𝔴]
==
,𝔖
+called
,[𝜇,
𝔵]given
=
𝜌.[𝜌.
=1].
𝜌.∗𝒦
𝜌.
(𝒾,
𝜑)
and
𝔖∗ (𝒾,
𝜑
𝜑 ∗∈𝔵]
(0,
Then
𝔖
is
𝐹𝑅-integrable
over
[𝜇,
𝜈]
if
and
only
if
𝔖
Definition
(5)
[𝔵+
[𝔵]
[𝔴]
𝔴]
=
+
,
∗
∗
be
an
F-NV-F.
Then
fuzzy
integral
of
𝔖
over
2.
[49]
Let
𝔖
:
[𝜇,
𝜈]
⊂
ℝ
→
𝔽
∗
(𝒾,
(𝒾,
(𝑅)
(𝐹𝑅)
(𝒾)𝑑𝒾
=
𝔖an
𝜑)𝑑𝒾
, if (𝑅)
𝔖(5)
𝜑)𝑑𝒾
𝔖: [𝜇,
(𝒾,
𝜑)
and
𝜑)
both
𝜑𝑅-integrable
∈ (0, 1]. Then
𝔖𝔴]
is
𝐹𝑅-integrable
over
[𝜇,
𝜈](𝑅)
if
and
only
𝔖
∗𝔖
∗𝔖
∗: (𝒾,
[𝜇,
[𝔵+
[𝔵]
[𝔴]
over
[𝜇,
𝜈].
Moreover,
if
𝔖
is
𝐹𝑅-integrable
over
𝜈],
then
be
F-NV-F.
Then
fuzzy
integral
of
𝔖
Definition
2.
[49]
Let
𝔖
𝜈]
⊂
ℝ
→
𝔽
=
+
,
Definition
Definition
1.
[49]
A
fuzzy
Definition
1.
[49]
number
A
fuzzy
1.
[49]
valued
Definition
number
A
fuzzy
map
valued
1.
number
:
[49]
𝐾
⊂
map
A
ℝ
valued
fuzzy
→
𝔖
𝔽
𝐾
number
⊂
map
ℝ
→
𝔖
:
𝔽
valued
𝐾
⊂
ℝ
→
map
𝔽
𝔖
is
called
F-NV-F.
is
called
For
F-N
ea
iso:
(𝒾,
(𝒾,
(𝑅)
(𝐹𝑅)
(𝒾)𝑑𝒾
=
𝔖
𝜑)𝑑𝒾
,
𝔖
𝜑)𝑑𝒾
𝔖
[𝔵by
[𝔵𝔖=
[𝔵]
[Moreover,
[𝔵]
[ 𝔴]
×Let
×
𝔴]
𝔴]
×
=
×
,given
,an
∗ (𝒾,𝜈],(6)
[𝜇, 𝜈], denoted
(𝐹𝑅)
(𝒾)𝑑𝒾
𝑅-integrable
over
[𝜇,
𝜈].
if[ levelwise
𝔖 F-NV-F.
is
𝐹𝑅-integrable
over(6)
then
𝔖
,→
is[𝔵]
by∗Then
be
fuzzy
integral
of 𝔖 over
Definition
2. [49]
: [𝜇,
𝜈]
⊂
ℝ𝔴]
𝔽
(𝒾,
(𝑅)
(𝑅)
(𝐹𝑅)
(𝒾)𝑑𝒾
=
𝔖the
𝜑)𝑑𝒾
,the
𝔖[𝜇,
𝜑)𝑑𝒾
𝔖(0,
(6)
[𝔵
×
𝔴]
=
×
𝔴]
,
∗whose
e
e
[𝜇,
(𝐹𝑅)
(𝒾)𝑑𝒾
[𝜇,
𝜈],
denoted
by
𝔖
,
is
given
levelwise
by
𝑅-integrable
over
[𝜇,
𝜈].
Moreover,
if
𝔖
is
𝐹𝑅-integrable
over
𝜈],
then
𝜑
∈
(0,
1]
whose
𝜑
∈
(0,
1]
𝜑
-cuts
𝜑
whose
∈
define
1]
𝜑
-cuts
whose
the
𝜑
define
∈
family
𝜑
(0,
-cuts
1]
of
define
I-V-Fs
family
𝜑
-cuts
𝔖
of
family
I-V-Fs
define
of
𝔖
the
I-V-Fs
family
𝔖
of
:
𝐾
⊂
ℝ
→
𝒦
:
𝐾
⊂
are
ℝ
given
→
:
𝒦
𝐾
⊂
∈⊂
HFSX
υ
,map
F𝔽0 ,×F-NV-F.
sis
and
Q
υF-NV-F.
, whose
ϕ-cuts
Theorem
][𝔵]
([µ,
], F0 , s)For
DefinitionDefinition
1. [49] A fuzzy
1. [49]
Definition
number
A fuzzy
valued
1.
number
[49]6.map
ALet
valued
fuzzy
𝔖S: 𝐾number
map
ℝ→
𝔖
𝐾([µ,
⊂is=
ℝ
→
𝔖
:𝔴]
𝐾
⊂, ℝ
→∈each
𝔽HFSX
called
called
For
F-NV-F.
is called
For
each
(6)each
[)∗𝜌.
×:𝔽valued
𝔴]
[𝜇, 𝜈], denoted by[𝔵+
(𝐹𝑅)
𝔖(𝒾)𝑑𝒾
, is
given
levelwise
by
(7)
(7)
[𝜌.
[𝜌.
[𝔵]
[𝔵]
∗
∗
∗
𝔵]
𝔵]
=
𝜌.
=
∗
(𝒾)
(𝒾,
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(𝒾,
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(𝒾)
[𝔖
(𝒾,
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𝔖∗ℝ
=I-V-Fs
𝜑),
𝔖
𝜑)]
𝜑),
for
all
𝜑)]
𝒾𝔖
𝜑),
∈𝒦(𝑅)
𝐾.
for(𝒾,
Here,
all
𝜑)]
𝒾∗ℝ
for
𝐾.
𝜑),
each
all
Here,
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for
𝐾.
∈given
(0,
Here,
for
each
all
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for
end
∈each
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point
Here,
𝜑the
∈ (re(
∗ (𝒾,
S
, whose
Qdefine
: [µ,
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R[𝔖
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Kof
are
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S
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and
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)]
( 𝒾∈)(7)
(𝔖
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∈ (0, 1]𝜑 -cuts
whose
𝜑 define
∈𝜑
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1]ϕthe
-cuts
define
𝔖
the
I-V-Fs
family
𝔖𝔖
of
I-V-Fs
𝔖
:=
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→
:=
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are
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by
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by
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ϕ (𝜌.
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∗ (𝒾,
∗=
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(𝑅)
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C
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=
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∗ (.
∗=
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(8)
(𝒾,
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(𝑅)
(𝒾)𝑑𝒾
=
𝔖
𝜑)𝑑𝒾
, functions
𝔖are
𝜑)𝑑𝒾
𝔖
(𝒾)𝑑𝒾
(𝐹𝑅)
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∗ (𝒾,
∗ (𝒾, [𝔖 (𝒾,
𝔖
=
𝔖
=
𝔖(𝒾,
𝜑)𝑑𝒾
𝜑)
∈
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,
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e
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e
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],
)
,
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e
functions
𝔖
functions
functions
𝔖
𝔖
functions
𝔖
,
𝜑),
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,
𝜑):
,
𝜑),
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→
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ℝ
,
are
𝜑):
,
𝜑),
called
𝐾
𝔖
→
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,
lower
𝜑):
are
𝐾
called
,
→
and
𝜑),
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𝔖
upper
are
lower
,
called
𝜑):
and
𝐾
→
lower
upper
ℝ
of
and
functions
called
𝔖
.
upper
lowe
fu
(𝒾)𝜑),
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∗
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𝔖
=𝔖
𝜑)]
𝜑),
for
𝔖
=
all
𝜑)]
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∈
for
𝐾.
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all
Here,
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𝒾
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for
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for
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for
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each
𝐾.
1]
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the
∈
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end
for
1]
each
point
the
end
𝜑
real
∈
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point
1]
the
real
end
point
real
,
ϕ
,
Q
,
ϕ
for
all
∈
µ,
υ
,
ϕ
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0,
1
,
respectively.
If
S
×
Q
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(
)
)]
[
]
[
]
∗
∗ 𝔖(𝒾)𝑑𝒾
∗ ==𝔖
∗ (𝒾)𝑑𝒾
∗ ∗
∗
(𝑅)
(𝑅) : 𝔖(𝒾,
(𝐹𝑅)
(𝒾)𝑑𝒾𝔖𝔖∗=(𝒾,
(𝐹𝑅)
𝜑)𝑑𝒾
, 𝜑)𝑑𝒾
𝔖 (𝒾,𝜑)
𝜑)𝑑𝒾
𝔖(𝒾)𝑑𝒾
= (𝐼𝑅)
𝔖(𝒾,
∈ ℛ([ , ], ) ,of
(𝐼𝑅)
∗ (. (. Definition
∗1.
∗number
(8)
(𝒾)𝑑𝒾
(𝐹𝑅)
(𝒾)𝑑𝒾
(𝐼𝑅)
𝔖
=
𝔖
=
𝔖(𝒾,
𝜑)𝑑𝒾
:
𝔖(𝒾,
𝜑)
∈
ℛ
,
Definition
[49]
A
1.
fuzzy
[49]
A
fuzzy
number
valued
valued
map
𝔖
:
map
𝐾
⊂
𝔖
ℝ
:
→
𝐾
⊂
𝔽
ℝ
→
𝔽
is
called
is
F-NV-F.
called
F-NV-F.
For
each
For
each
(.
(.
(.
(.
F
R
,
then
functions functions
𝔖∗ , 𝜑), 𝔖𝔖∗ , 𝜑):
𝔖µ,∗𝐾υcalled
,functions
𝜑),𝐾𝔖→ ℝ, 𝜑):
are
lower
, called
𝜑):and
𝐾→
lower
upper
ℝ are
and
functions
called
upper
lower
of
functions
𝔖and
. upper
of 𝔖.functions of 𝔖.
([ , ], )
([
],→
,𝜑),
ϕ)ℝ𝔖are
Definition
1.Definition
[49]
A∈𝜑fuzzy
number
valued
map
𝔖
𝐾𝔽([⊂
ℝ
→
𝔽
is ⊂
called
F-NV-F.
For
each
(0,
for
all
∈ (0,
For
all
∈],𝜈]
1],
ℱℛ
the
collection
of
all
𝐹𝑅-integrable
], :collection
)→
, be
all
𝜑-cuts
(0,
1],
where
ℛ
denotes
the
of
Riemannian
integrable
funcan
be
an
Then
F-NV-F.
fuzzy
be
Then
integral
F-NV-F.
fuzzy
be
Then
integ
𝔖
an ov
fu
F
Definition
2.𝜑1].
[49]
Let
Definition
2.
𝔖
:𝜑[49]
[𝜇,
Let
⊂
2.
ℝ
𝔖
[49]
→
Definition
:::[𝜇,
𝜈]
Let
⊂
𝔖
ℝ
:denotes
[𝜇,
→
2.F-NV-F.
𝔽
𝜈]
[49]
Let
ℝgiven
→
𝔖
𝔽
: [𝜇,
𝜈]an
⊂
ℝ
→of
𝔽all of
([
)𝜑
, all
𝜑 Definition
∈ (0, 𝜑
1]∈ whose
(0,1.1][49]
whose
𝜑for
-cuts
𝜑
define
define
the
family
the
family
of
I-V-Fs
of
I-V-Fs
𝔖
𝔖
𝐾
⊂
ℝ
𝐾
⊂
𝒦
ℝ
→
are
𝒦
are
by
given
by
(𝒾)𝑑𝒾
(𝐼𝑅)
=
𝔖
(0,
for
all
∈
(0,
1].
For
∈
1],
ℱℛ
denotes
the
collection
𝐹𝑅-inte
A
fuzzy
number
valued
map
𝔖
:
𝐾
⊂
ℝ
→
𝔽
is
called
F-NV-F.
For
each
([
],
)
,
for
all
𝜑
∈
(0,
1],
where
ℛ
denotes
the
collection
of
Riemannian
integrable
fu
([
],
)
,
(𝒾)𝑑𝒾
(𝐼𝑅)
𝜑 ∈ (0, 1] tions
whose
𝜑
-cuts
define
the
family
of
I-V-Fs
𝔖
:
𝐾
⊂
ℝ
→
𝒦
are
given
by
=
𝔖
Z
Z
Z
(0,
NV-Fs
over
[𝜇,
𝜈].
for
all
∈
(0,
1].
For
all
𝜑
∈
1],
ℱℛ
denotes
the
collection
of
all
𝐹𝑅-integrable
∗
∗
([
],
)
,
υ
1
1
[𝜇,
[𝜇,
(𝐹𝑅)
[𝜇,
(𝒾)𝑑𝒾
(𝐹𝑅)
[𝜇,
(𝒾)𝑑𝒾
(𝐹𝑅)
(𝒾)𝑑𝒾
(𝐹𝑅)
(𝒾)𝑑𝒾
e
(𝐹𝑅)
(𝒾)𝑑𝒾
e
𝜈],
denoted
by
denoted
𝜈],
𝔖
denoted
by
𝔖
𝜈],
denoted
𝔖
by
𝔖𝔖by
, isover
given
levelwise
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is
by
levelwise
,point
is end
given
levelwise
, .is Note
given
by that,
levelw
of
I-V-Fs.
𝔖
is
𝐹𝑅
-integrable
[𝜇,
ifgiven
𝔖 ofreal
∈
𝔽
if
eℝ
be
an
F-NV-F.
be
an
Then
F-NV-F.
fuzzy
be
Then
integral
an
fuzzy
F-NV-F.
of
integral
Then
over
fuzzy
of
𝔖
over
integral
over
DefinitionDefinition
2. [49] Let
2.(𝒾)
𝔖
Definition
: (0,
[𝜇,
𝜈]
Let
⊂
𝔖
ℝ𝜑),
:[𝔖
[𝜇,
→
2.𝔖
𝔽
𝜈]
[49]
⊂
Let
ℝ𝔖
→(𝒾,
𝔖
𝔽
:∗𝜑)]
[𝜇,
⊂
→𝒾Here,
𝔽
µυ
(𝒾)
(𝒾,
(𝒾,
(𝒾,
for
all
𝜑
∈for
(0,
1],
where
ℛ
denotes
collection
of
Riemannian
integrable
func𝔖𝜑
𝔖 [𝔖
=
=
𝜑),
𝜑)]
all
𝒾the
∈
all
𝐾.
∈ ([𝐾.
for
for
𝜑
each
∈𝔖
(0,
𝜑
1]𝜈]
the
(0,
end
the
point
real
S𝜈]
×
Q(
(for
)𝜈],
)is
(𝒾)𝑑𝒾
(𝐼𝑅)
=
𝔖
NV-Fs
over
[𝜇,
𝜈].
],Here,
) each
, by
∈[49]
1]
𝜑
-cuts
define
family
of
𝔖
:the
ℝ
→
𝒦1]
given
s𝐾
e
(𝐹𝑅)
eI-V-Fs
of
I-V-Fs.
𝔖 all
-integrable
[𝜇,
𝜈]
if υare
∈ 𝔽real
. Note that
e
(𝒾) =∗ [𝔖
(𝒾,
(𝒾,
𝔖∗ whose
𝜑),
𝜑)]
for
∈
𝐾.
Here,
each
𝜑
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1]
point
FR
d 𝒾𝐹𝑅
4
Λ
υ) for
ξover
· ξ⊂s dξ
+
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µ,
1 (𝒾)𝑑𝒾
−by
ξ )s dξ,
(tions
)∗ 𝔖
(µ,
) theξ send
(𝔖
over
[𝜇,
𝜈].called
∗∗NV-Fs
∗by
∗𝜑):
2
[𝜇, 𝜈], denoted
[𝜇, 𝜈],by
(𝐹𝑅)
[𝜇,
(𝒾)𝑑𝒾
(𝐹𝑅)
(𝒾)𝑑𝒾
(𝐹𝑅)
(𝒾)𝑑𝒾
denoted
by
𝔖
𝜈],
denoted
𝔖
𝔖
,
is
given
levelwise
,
is
given
by
levelwise
,
is
given
by
levelwise
by
(.
(.
(.
(.
(𝒾,
(𝒾,
(𝐹𝑅)
(𝒾)𝑑𝒾
functions
functions
𝔖
,
𝜑),
𝔖
,
𝜑),
,
𝜑):
𝔖
𝐾
→
,
ℝ
𝐾
are
→
ℝ
are
lower
called
and
lower
upper
and
functions
upper
functions
of
𝔖
.
of
𝔖
.
𝜑),
𝔖
𝜑)
are
Lebesgue-integrable,
then
𝔖
is
a
fuzzy
Aumann-integrable
func𝔖
tions
of
I-V-Fs.
𝔖
is
𝐹𝑅
-integrable
over
[𝜇,
𝜈]
if
𝔖
∈
𝔽
.
Note
that,
if
υ
−
µ
[𝔖
µfor
0 𝜑 ∈ (0, 1]denotes
0point
𝔖 (𝒾) = functions
𝜑)]
𝒾𝐾∈→
𝐾.ℝ
Here,
for
end
real
∗ ∗ (𝒾, 𝜑),
∗ 𝔖∗ (𝒾,
∗ (.
(0,𝜐]each
for
all 𝜑),
∈
(0,
1].
For
∈[𝜇,
1],
ℱℛ
the
collection
𝐹𝑅-integrable
([
],upper
)∞) the
(0,
= ,ℱℛ
is𝔖
said
beof
a convex
set,of
if,all
for𝐹𝑅-int
all 𝒾,fu𝒿
3.
A
setall
𝐾𝜑=
ℝ
(.
(𝒾,
(𝒾,
𝔖∗∗Definition
𝔖𝜑
,∗all
𝜑):
are
called
lower
and
functions
𝔖
. of all
𝔖
𝜑)
are
Lebesgue-integrable,
then
is
atofuzzy
Aumann-integrable
𝔖,∗𝜑),
(0,
all
𝜑[34]
∈ (0,
1].
For
all𝐾𝜑
∈⊂
1],
denotes
the
collection
([
], )∞)
∗for
,is
(0,
[𝜇,
=
is
said
to
be
a
convex
set,
if,∈
Definition
3.
[34]
A
set
=
𝜐]
⊂
ℝ
over
[𝜇,
𝜈].
(.
(𝒾,
(𝒾,
functions 𝔖∗ (. , 𝜑),
𝔖
,
𝜑):
𝐾
→
ℝ
are
called
lower
and
upper
functions
of
𝔖
.
𝜑),
𝔖
𝜑)
are
Lebesgue-integrable,
then
𝔖
a
fuzzy
Aumann-integrable
func𝔖tion
(𝒾)𝑑𝒾
(𝒾)𝑑𝒾
(𝒾)𝑑𝒾
(𝐹𝑅)
(𝒾)𝑑𝒾
(𝐹𝑅)
(𝒾)𝑑𝒾
(𝐹𝑅)
(𝒾)𝑑𝒾
(𝐹𝑅)
(𝒾)𝑑𝒾
(𝐼𝑅)
(𝐼𝑅)
(𝐼𝑅)
(𝐼𝑅)
𝔖
𝔖
𝔖
𝔖
=
𝔖
=
=
𝔖
=
𝔖(𝒾,
𝜑)𝑑𝒾
=
𝔖
:
𝔖(𝒾,
=
𝔖(𝒾,
=
𝜑)
𝜑)𝑑𝒾
∈
𝔖
ℛ
:
𝔖(𝒾,
𝔖(𝒾,
,for
:𝔖
ℛ
(0,
NV-Fs
[𝜇,
𝜈].
for
all
𝜑over
∈1],[𝜇,
(0,
1].
Forsetall𝐾𝜑=∈[𝜇, 𝜐]1],⊂ ℱℛ
denotes
the
collection
of all
𝐹𝑅-integrable
∗ Definition
([(𝒾)𝑑𝒾
],𝜑)
) =
, 𝜑)𝑑𝒾
[0,
([
],
)
𝐾,
𝜉
∈
we
have
,
(0,
tion
over
𝜈].
=
∞)
is
said
to
be
a
convex
set,
if,
for
all
𝒾,
𝒿
3.
[34]
A
ℝ
e
e
e
e
e
e
e
e
e
e
NV-Fs
over
[𝜇,
𝜈].
e
e
e
e
e
e
be
an
F-NV-F.
be
an
F-NV-F.
Then
fuzzy
Then
integral
fuzzy
integral
of
𝔖
over
of
𝔖
over
Definition
Definition
2.
[49]
2.
Let
[49]
𝔖
:
Let
[𝜇,
𝜈]
𝔖
⊂
:
[𝜇,
ℝ
𝜈]
→
⊂
𝔽
ℝ
→
𝔽
S(𝜉𝜈].
µ∈
)×[0,
Q(1],
µ)we
+S
(υ)×Q(υ), ∂(µ, υ) =(8)S(µ)×Q(
where Λtion
υ
)
+
S
(
υ
)
×
Q(
µ
)
,
and
(µ, υover
) = [𝜇,
𝐾,
have
(8)
(𝒾)𝑑𝒾
(𝒾)𝑑𝒾
(𝐹𝑅) 𝔖(𝒾)𝑑𝒾
(𝐹𝑅) =
(𝐹𝑅)
(𝒾)𝑑𝒾
(𝐼𝑅)
(𝐼𝑅)
NV-Fs
over
𝜈].
𝔖(𝒾)𝑑𝒾
𝔖
𝔖
= (𝐼𝑅)
=
𝔖[0,
=
𝔖(𝒾,
=𝜑)𝑑𝒾
𝔖
:𝔖(𝒾,
𝔖(𝒾,
𝜑)
=be
: 𝔖(𝒾,
ℛ([an𝔖(𝒾,
, ℛ([ :Then
𝔖(𝒾,
𝜑)
,fuzzy
∈(8)ℛintegral
,
F-NV-F.
of
𝔖
over
Definition
2.
[49]
Let
𝔖[𝜇,
: [𝜇,
𝜈]∗ ⊂
ℝ(𝒾)𝑑𝒾
→𝜑)𝑑𝒾
𝔽∈
], )∈𝜑)𝑑𝒾
],
)
([
],
)
𝐾,
𝜉
∈
1],
we
have
, 𝜑)
,
,
𝒾𝒿
µ,
υ(𝐹𝑅)
=by
Λ𝔖(𝐹𝑅)
µ,2.𝜈]
υ[49]
ϕ(𝒾)𝑑𝒾
, →Λ𝔽
µ,
υ𝜈])an
, ⊂=
ϕlevelwise
and
µ, υ=
= [∞)
∂∗ 𝒾𝒿
µ,said
υ),𝜑-cuts
ϕ
µ, υ)the
, ϕset,
)Theorem
[Definition
),⊂
((
)]
(ℝ
)fuzzy
((
),be∂𝔖∗adefine
((over
)]. if, for
[𝜇,Definition
[𝜇, 𝜈],Λdenoted
(𝒾)𝑑𝒾
𝜈], denoted
𝔖
, is
given
,A
islevelwise
given
by
by
ϕ
∗𝔖
ϕThen
𝔖
[𝜇,
ℝ[𝜇,
→𝜐]
𝔽∂⊂
be
F-NV-F
whose
family
ofofthe
I-V-Fs
be
F-NV-F.
integral
2.(by
[49]
Let
:(([𝜇,
ℝ)Let
(0,
toof
convex
all
𝒾,
3.
[34]
set
𝐾
for
all
𝜑∈
for
1],
all
where
𝜑:[34]
∈
for
(0,
all
1],
ℛset
∈
(0,
1],
for
ℛ𝜐]([awhere
all
𝜑
ℛ
(0,
1],
ℛ
denotes
the
denotes
collection
denotes
of
collection
Riemannian
the
of
Riemannian
integrable
Riem
co
fun
in(
∈, ∞)
[𝜇, 𝜈],
(𝐹𝑅)
(𝒾)𝑑𝒾
denoted
by
𝔖
, is
given
levelwise
by
([𝜑where
],ℝ
)→
) ∈=
([is
],𝐾.where
)the
([be
) denotes
,𝐾
, ],ℝ
, ],acollection
Theorem
2.(0,
[49]
Let
𝔖
: [𝜇,
𝜈]
⊂
𝔽
be
a
F-NV-F
whose
𝜑-cuts
define
the
family
of
I-V
(0,
[𝜇,
is
said
to
convex
set,
if,
for
Definition
3.
A
=
⊂
𝒾𝒿
(1
∈
𝐾.
𝜉𝒾
+
−
𝜉)𝒿
∗
[𝜇,
(𝐹𝑅)
(𝒾)𝑑𝒾
𝜈],
denoted
by
𝔖
,
is
given
levelwise
by
[0,
𝐾,
𝜉
∈
1],
we
have
(0,
[𝜇,
Theorem
2.
[49]
Let
𝔖
:
[𝜇,
𝜈]
⊂
ℝ
→
𝔽
be
a
F-NV-F
whose
𝜑-cuts
define
the
family
of
I-V-Fs
=
∞)
is
said
to
be
a
convex
set,
if,
for
all
𝒾,
Definition
3.
[34]
A
set
𝐾
=
𝜐]
⊂
ℝ
(𝒾)
[𝔖
(𝒾,
(𝒾,
=
𝜑),
𝔖
𝜑)]
for
all
𝒾
∈
[𝜇,
𝜈]
and
for
all
:
[𝜇,
𝜈]
⊂
ℝ
→
𝒦
are
given
by
𝔖
𝔖
(1
𝜉𝒾
+
−
𝜉)𝒿
(𝐹𝑅)
(𝒾)𝑑𝒾
(𝐹𝑅)
(𝒾)𝑑𝒾
(𝐹𝑅)
(𝒾
∈iffunc𝐾.𝔖∗integrable
tions
of
is
I-V-Fs.
tions
𝐹𝑅of
-integrable
of
𝔖collection
I-V-Fs.
is 𝐹𝑅
tions
-integrable
𝔖
over
is
of
[𝜇,
𝐹𝑅
I-V-Fs.
𝜈]
-integrable
over
[𝜇,
is 𝜈]
𝐹𝑅
over
𝔖if-integrable
[𝜇, 𝜈]
[𝜇,
𝔖𝜈]
∈ 𝔽if𝔖. over
Note ∈
that,
𝔽
.𝒿
(1
for all 𝜑 ∈for
(0,all
1], 𝜑where
∈ (0, 1],
for
ℛ([where
all, ],𝜑) ∈denotes
ℛ
(0, 1],
where
ℛI-V-Fs.
collection
the
of
denotes
Riemannian
the
Riemannian
integrable
of
integrable
func∗Riemannian
)the
) 𝔖
, ], tions
,[0,
𝐾,are
∈harmonically
1],
we
have
eof
edenotes
(𝒾)
[𝔖
(𝒾,
𝜑),
𝔖 (𝒾,we
𝜑)]have
for funcall 𝒾 ∈ [𝜇, 𝜈] and for
: [𝜇,
𝜈]𝜉 ([⊂
ℝ], collection
→
𝒦
are
given
by F-NV-Fs,
𝔖 𝜉𝒾
𝔖
(1then
Proof. ([Since
S,
Q
s-convex
for
each,
+=
−∗𝜉)𝒿
∗
[0,
𝐾,
𝜉
∈
1],
we
have
(8)
(8)
∗
∗
∗
∗
∗
(𝒾)𝑑𝒾
(𝐹𝑅)
(𝒾)𝑑𝒾
(𝒾)𝑑𝒾
(𝐼𝑅)
(𝐼𝑅)
𝔖𝔖
𝔖[𝜇,
=
=
𝔖
𝔖
=(𝒾)𝑑𝒾
𝔖(𝒾,
=(𝒾)𝑑𝒾
𝜑)𝑑𝒾
𝔖(𝒾,
:are
𝔖(𝒾,
𝜑)𝑑𝒾
𝜑)
: 𝔖(𝒾,
∈
ℛ
𝜑)
ℛ
, fuzzy
[𝔖
(𝒾,
(𝒾,
(𝐹𝑅)
(𝒾)𝑑𝒾
(𝐹𝑅)
(𝐹𝑅)
(𝒾)𝑑𝒾
=
𝜑),
𝔖𝔖
all
𝒾)then
[𝜇,
and
for
all
:∈𝔖
[𝜇,(0,
𝜈]
⊂
ℝ
→(𝒾,
𝒦
given
by
𝒾𝒿
tions of I-V-Fs.
tions of𝔖 I-V-Fs.
is 𝐹𝑅
tions
-integrable
𝔖(𝐹𝑅)
is
of 𝐹𝑅
I-V-Fs.
-integrable
over
is
𝜈]
𝐹𝑅
over
if
-integrable
[𝜇,
𝜈]are
if𝔖
over
[𝜇,𝔽are
𝔖
𝜈]
if=
∈
. (𝒾)
Note
∈[𝜇,
𝔽
that,
Note
if
that,
∈
𝔽are
if],𝔖
.fuzzy
Note
if,𝔖𝜈]
([ 𝜑)]
)for
([, 𝔖
],Aumann-integrable
,a∈
,𝜑)
(𝒾,
(𝒾,
(𝒾,
(𝒾,
(𝒾,
(𝒾,
(𝒾,
and
𝔖
both
are
𝜑
1].
Then
is
𝐹𝑅-integrable
over
𝜈]
if𝔖
and
only
if
𝜑),
𝔖
𝜑)
𝜑),
are
𝔖
Lebesgue-integrable,
𝜑)
𝜑),
𝔖
Lebesgue-integrable,
𝜑)
Lebesgue-integrable,
then
𝔖
is
then
Lebesgue-integrable,
isthat,
a∈
is
Aumann-in
a fuzzy
then
fun
A𝔖
𝔖
𝔖
𝔖
∗.𝜑),
∗𝔖
∗ (𝒾,
∗ (𝒾,
(8)
∗𝜑)
(𝐹𝑅)
(𝒾)𝑑𝒾
(𝐼𝑅)
𝔖
=
𝔖(𝒾,
𝜑)𝑑𝒾
:𝜑)
𝔖(𝒾,
𝒾𝒿
([ ,𝜑)
], )and
𝔖[𝜇,(𝒾)𝑑𝒾
(𝒾,
(𝒾,on
∈ only
𝐾.𝜑)if∗∈𝔖ℛ
𝔖
𝜑) [𝜇,
both
𝜑∗ ∈ (0,
1]. 4.
Then
𝔖The
is 𝐹𝑅-integrable
over
[𝜇, 𝜈]
ifharmonically
and
Definition
[34]
𝔖:
𝜐]
→
ℝ
is
called
a
function
𝜐](
∗convex
∗
∗
∗
(8)
∗
µυ
(𝒾)𝑑𝒾
(𝐹𝑅)
(𝒾)𝑑𝒾
(𝐼𝑅)
s
=
𝔖𝔖
=
𝔖(𝒾,
:(if
𝔖(𝒾,
𝜑)
∈over
, then
𝒾𝒿
over
[𝜇,
tion
tion
[𝜇,
𝜈].
over
[𝜇,
𝜈].
tion
over
[𝜇,
𝜈].
(1
𝐾.
sS
𝜉𝒾
+
−
𝜉)𝒿
𝜑)
𝜑),are
𝔖 (𝒾,
Lebesgue-integrable,
𝜑)
are𝜑),
Lebesgue-integrable,
𝔖 (𝒾,𝜑𝔖
𝜑)
are
then
Lebesgue-integrable,
𝔖∗over
is
then
a𝔖𝜈].
fuzzy
is
Aumann-integrable
a The
fuzzy
then
Aumann-integrable
𝔖
isis𝜑)𝑑𝒾
a→
fuzzy
funcAumann-integrable
funcfunc𝔖∗ (𝒾, 𝜑), 𝔖𝔖(𝒾,
𝔖∗ (𝒾,
([𝔖[𝜇,
],
)𝜑)
Definition
4.
[34]
𝔖:
𝜐]
ℝ
is
called
aifℛ
harmonically
convex
,∈
(𝒾,
and
𝔖 (𝒾, 𝜑)function
both areon
∈ tion
(0,
1].
Then
isover
𝐹𝑅-integrable
over
[𝜇,
𝜈]
and
only
𝑅-integrable
[𝜇,
𝜈].
Moreover,
if
𝔖
𝐹𝑅-integrable
𝜈],
∗ (𝒾,
S
,
ϕ
≤
ξ
µ,
ϕ
+
1
−
ξ
S
υ,
,
(
)
)
(
)
∗ϕ
∗
∗
(1
𝜉𝒾
+
−
𝜉)𝒿
∈
𝐾.
−
ξ )µ+The
ξυ
(1over
[𝜇,the
Definition
4.
[34]
𝜐] →
ℝ
isiscalled
a harmonically
convex
function on [𝜇, 𝜐](1
[𝜇,
𝑅-integrable
𝜈].𝔖:
if𝒾𝒿
𝐹𝑅-integrable
over
𝜈], then
Moreover,
for
𝜑 ∈over
all
(0, 𝜑
1],∈ where
(0, 1],
where
ℛ , ], ℛ
denotes
the
collection
collection
of𝔖𝜉𝒾
Riemannian
of
Riemannian
integrable
integrable
funcfunc) ([denotes
, ], )[𝜇,
+
−
𝜉)𝒿
tion over tion
[𝜇, 𝜈].over for
[𝜇, all
𝜈].tion
µυ
s over
for all[𝜇,𝜑𝜈].
∈ (0, 1], ([where
ℛ
the
of(1
Riemannian
integrable
func∗
∗ ( µ,
∗ ( υ,[𝜇,
𝒾𝒿((1
𝑅-integrable
over
[𝜇,
𝜈]., ],Moreover,
ifs S𝔖
iscollection
𝐹𝑅-integrable
𝜈],
then
+
𝜉𝔖(𝒿),
𝔖
≤
−
𝜉)𝔖(𝒾)
([
) ϕdenotes
(
S
,
≤
ξ
ϕ
+
1
−
ξ
S
ϕ
.
)
)
)
(𝐹𝑅)
(𝒾)𝑑𝒾
(𝒾)𝑑𝒾
tions
oftions
I-V-Fs.
𝔖 where
is Theorem
𝐹𝑅
𝔖 -integrable
isℛ([𝐹𝑅
[𝜇,
over
𝜈]𝜈]
if[𝜇,
𝜈]
if𝔽Theorem
𝔖
∈𝔽
𝔽
Note
∈
𝔽𝜉)𝔖(𝒾)
Note
ifdefine
if𝔽whose
(1
1[49]
ξ )over
µLet
+ξυ 𝔖
(1
(Theorem
𝜉𝔖(𝒿),
𝔖
≤
𝜉𝒾(𝐹𝑅)
+
−𝜈]
𝜉)𝒿
for all
𝜑 ∈ of
(0, I-V-Fs.
1],
collection
of
Riemannian
integrable
func2.
2.
Theorem
:the
[𝜇,
[49]
Let
⊂
ℝ
2.
𝔖→
[49]
:𝒾𝒿
[𝜇,
Let
⊂
𝔖
: [𝜇,
→𝔖
2.
𝜈]
[49]
⊂.be
ℝ
Let
→
𝔖
𝔽.𝜑-cuts
:that,
[𝜇,
𝜈]
⊂that,
ℝ 𝜑-cuts
→
be
aℝF-NV-F
whose
a−
F-NV-F
beconvex
whose
a+
F-NV-F
the
family
be
define
aon
𝜑-cuts
F-NVof[𝜇,
the
I-V𝜐f
],-integrable
)−denotes
,
[𝜇,
Definition
4.
[34]
The
𝔖:
𝜐]
→
ℝ
is
called
a
harmonically
function
∗
(𝐹𝑅)
(𝒾)𝑑𝒾
tions
of
I-V-Fs.
𝔖
is
𝐹𝑅
-integrable
over
[𝜇,
𝜈]
if
𝔖
∈
𝔽
.
Note
that,
if
(1
𝜉𝒾
+
−
𝜉)𝒿
(1
+
𝜉𝔖(𝒿),
𝔖
≤
−
𝜉)𝔖(𝒾)
(1
(𝒾,
(𝒾,
(𝑅)
(𝑅)
(𝐹𝑅)
(𝒾)𝑑𝒾
=
𝔖
𝜑)𝑑𝒾
,
𝔖
𝜑)𝑑𝒾
𝔖
[𝜇,
∗ (𝒾,
∗ (𝒾,
∗ called
Definition
4.
[34]
The
𝔖:
𝜐]=
→of
ℝ
is
athe
harmonically
convex
function o
∗func∗[𝔖
∗I-V-Fs
(1
(𝒾,
(𝒾,
(𝐹𝑅)
(𝒾)𝑑𝒾
𝜉𝒾
+
−
𝜉)𝒿
𝜑)
𝔖𝔽𝜈]
are
Lebesgue-integrable,
are
Lebesgue-integrable,
𝔖
is
afamily
𝔖
fuzzy
is
athe
Aumann-integrable
fuzzy
func𝔖
of
I-V-Fs.
𝔖
is
𝐹𝑅
-integrable
[𝜇,
𝜈]then
ifby
𝔖
∈
𝔽
Note
that,
if
Theorem 2.
Theorem
[49] Let𝔖2.𝔖
:[49]
[𝜇,𝜑),
Theorem
𝜈]
Let
⊂
𝔖
ℝ𝜑),
: [𝜇,
→
2.
[49]
⊂be∗𝜑)
ℝ
Let
→
: [𝜇,
ℝ
→
𝔽whose
a𝔖
F-NV-F
be𝒿𝜈]
whose
a∈⊂
F-NV-F
𝜑-cuts
be
define
a1],
𝜑-cuts
F-NV-F
the
define
whose
I-V-Fs
𝜑-cuts
family
define
of
I-V-Fs
family
of
(𝒾,
(𝒾,
(𝑅)
(𝐹𝑅)
(𝒾)𝑑𝒾
𝔖
𝜑)𝑑𝒾
𝔖
𝜑)𝑑𝒾
𝔖
∗tions
∗𝔖
(𝒾)
[𝔖
(𝒾,
(𝒾)
(𝒾,
(𝒾,
(𝒾)
[𝔖
(𝒾,
(𝒾,
(𝒾)
(𝒾,
[𝔖
(𝒾,
=
𝜑),
=
𝔖.𝒾𝒦
𝜑)]
𝜑),
=
for
all
𝒾𝜑)]
𝜑),
∈func[𝜇,
for
𝔖∗𝜈]
=
allon
and
𝜑)]
𝒾then,
for
for
[𝜇
𝜑
:𝔖𝔽
[𝜇,
𝜈]
⊂
ℝ
→
: 𝜐],
[𝜇,
𝒦over
are
⊂
ℝ
:given
[𝜇,
→
𝒦
𝜈]
⊂
are
ℝ
𝔖
→
given
𝒦
:(𝑅)
[𝜇,
𝜈]
𝔖
given
ℝ
→
by
𝔖∗,[𝜇,
are
given
𝔖
𝔖
𝔖[0,
𝔖
∗harmonically
[𝜇,
[𝜇,
[𝜇,
Definition
4.
[34]
The
𝔖:
𝜐]
→
ℝ
called
aAumann-integrable
convex
function
𝜐]
for
all
𝒾,
𝜉𝜈]
∈then
where
𝔖(𝒾)
≥by
0fuzzy
for
all
∈
𝜐].
If𝔖
(11)
is
reversed
∗⊂
∗by
∗∈
∗ (𝒾,
Lebesgue-integrable,
then
𝔖is
is
aare
Aumann-integrable
𝔖and
𝒾𝒿
∗∗(𝒾, 𝜑), 𝔖 (𝒾, 𝜑) are
(𝒾,
(𝑅)
(𝑅)
(𝐹𝑅)
(𝒾)𝑑𝒾
=
𝔖
𝜑)𝑑𝒾
,
𝔖
𝜑)𝑑𝒾
𝔖
[𝜇,
[0,
[𝜇,
for
all
𝒾,
𝒿
∈
𝜐],
𝜉
∈
1],
where
𝔖(𝒾)
≥
0
for
all
𝒾
∈
𝜐].
If
(11)
is
reversed
∗
(9)
tion
over
tion
[𝜇,
over
𝜈].
[𝜇,
𝜈].
∗
∗
∗
𝒾𝒿
µυ
(1
s
+
𝜉𝔖(𝒿),
𝔖
≤
−
𝜉)𝔖(𝒾)
(𝒾,
(𝒾,
𝜑),
𝔖
𝜑)
are
Lebesgue-integrable,
then
𝔖
is
aµ,𝔖
fuzzy
func𝔖
∗ (𝒾,
s for
(𝒾)
[𝔖
(𝒾,
(𝒾)
[𝔖
(𝒾,
(𝒾,
(𝒾)
(𝒾,
[𝔖
(𝒾,
(𝒾,
[𝜇,
=
𝜑),
𝔖
𝜑)]
𝜑),
for
𝔖
=
all
𝜑)]
𝒾∗where
∈
[𝜇,
𝜑),
𝜈]
and
𝒾ϕ−
𝜑)]
[𝜇,
for
𝜈]
all
for
𝒾[𝜇,
∈υ,
all
[𝜇,
and
for
all
→𝒦
𝜈] ⊂are
ℝ∗ →
given
: [𝜇,
by
are
𝜈]𝔖⊂
given
ℝ→
by𝒦
𝔖∈
are
given
𝔖
𝔖 : [𝜇, 𝜈] ⊂
𝔖 ℝ: [𝜇,
𝔖𝒦
is
called
concave
function
𝜐].
[0,
[𝜇,
for
𝒾,
𝒿=
𝜐],
𝜉1].
∈
𝔖(𝒾)
≥
0
for
all
𝒾(1
∈
If
(11)
is
then,
(𝒾,
(𝒾,
∗all
∗by
𝜑)
𝜑)
𝜑)and
both
𝔖
𝜑
(0,
1].
Then
𝜑[𝜇,
∈
(0,
𝔖
is
𝐹𝑅-integrable
Then
∈1],
(0,
1].
is
Then
𝐹𝑅-integrable
over
𝜑
𝔖
∈
[𝜇,
(0,
ison
𝐹𝑅-integrable
1].
ifand
over
and
𝜈]
is
if
over
if)𝜈]
𝐹𝑅-integrable
𝔖
and
[𝜇,
only
𝜈]
ifand
and
𝔖reversed
over
only
[𝜇,
if
𝜈]
𝔖
an(∗𝜑
Q
,𝜑
ϕ
≤
ξ𝔖
Q
+
1𝜈]
−
ξThen
Q
ϕ
,𝜐].
(all
)∈
(Aumann-integrable
)all
(𝔖
tion
over
[𝜇,
𝜈].
∗+
∗a∈harmonically
∗only
𝒾𝒿
∗ (𝒾,
∗𝔖
∗if
(1
+
𝜉𝔖(𝒿),
𝔖
≤for
−
𝜉)𝔖(𝒾)
𝜉𝒾
𝜉)𝒿
[𝜇,
+ξυ
(1−ξa)µharmonically
is
called
concave
function
on
𝜐].
𝜉𝒾(𝐼𝑅)
+ (1on
−≤
𝜉)𝒿
(1
(9) (1
tion over [𝜇, 𝜈].
+ 𝜉𝔖(𝒿),
−
𝜉)𝔖(𝒾)
∗function
∗ (𝒾,
∗ (𝒾,
[𝜇,
is𝜈]called
a∗ only
harmonically
concave
𝜐].
(𝒾)𝑑𝒾
=
𝔖
sboth
(𝒾,
(𝒾,
𝜑)
and
𝔖
𝜑)
and
𝔖
and
are
𝔖𝜈].
𝜑)𝐹𝑅-integrable
both
are
𝜑 ∈ (0, 1]. 𝜑Then
∈ (0,𝔖1].isThen
𝐹𝑅-integrable
𝜑𝔖
∈ (0,
is 1].
𝐹𝑅-integrable
over
Then[𝜇,
𝔖
isover
if𝐹𝑅-integrable
and
[𝜇,
𝜈]over
ifµυand
𝔖
over
only
[𝜇,
if𝔖
𝜈]
𝔖ξ[𝜇,
and
only
ifboth
𝔖
s(𝒾,
∗𝜑)
∗𝐹𝑅-integrable
(1
[𝜇,
[𝜇,
𝜉𝒾
+
−
[𝜇,
𝑅-integrable
Moreover,
over
𝜈].
if
Moreover,
𝔖
𝑅-integrable
is
[𝜇,
𝐹𝑅-integrable
𝜈].
Moreover,
𝔖
is
over
ifthe
𝔖
Moreover,
isI-V-Fs
then
ifis
𝔖reversed
𝜈],
isover
then
𝐹𝑅-integr
𝜈],
∗ (𝒾,
∗if
,F-NV-F
ϕ[0,
≤
Q
µ,
ϕ=
+
1∗are
−𝜑)
ξ𝜑)
Q
.[𝜇,
(over
)𝜉)𝒿
(𝜑-cuts
)over
([𝜇,
)family
(𝒾)𝑑𝒾
(𝐼𝑅)
𝔖
[𝜇,
Theorem
Theorem
2. [49] Let
2. [49]
𝔖: 𝑅-integrable
[𝜇,
Let𝜈]all
𝔖⊂
:Q
[𝜇,
ℝ𝒿 𝜈]
→
⊂
𝔽ξ )ℝ
𝔽∈
be1],
a F-NV-F
whose
𝜑-cuts
whose
define
the
define
family
of
I-V-Fs
for
𝒾,
∈(𝑅-integrable
𝜐],
𝜉a𝜈].
where
𝔖(𝒾)
≥
0if
for
all
𝒾υ,
∈ϕ
𝜐].𝜈],
Ifofover
(11)
then
1[𝜇,
−
µbe
+→
ξυ
[𝜇,
[𝜇, 𝜐]. ofIfI-V-Fs
(𝒾)𝑑𝒾
(𝐼𝑅)
for𝔖
all
𝒾,𝜈]𝒿if⊂
∈
𝜐],
where
𝔖(𝒾)
≥then
0 for
all the
𝒾 ∈family
(11) is reversed
Theorem
[49]
Let
: [𝜇,
ℝ𝜈],
→
𝔽𝜉 ∈[𝜇,
be[0,𝜈],
a 1],
F-NV-F
whose
𝜑-cuts
define
=
𝔖 𝜈],
[𝜇,
[𝜇,
𝑅-integrable
𝑅-integrable
over [𝜇, 𝜈].over
Moreover,
𝑅-integrable
[𝜇, 𝜈].
ifMoreover,
𝔖over
is 2.𝐹𝑅-integrable
[𝜇,
if
𝜈].
𝔖
Moreover,
is
𝐹𝑅-integrable
over
𝔖
is
over
then
𝐹𝑅-integrable
then
over
∗ (𝒾,
∗ (𝒾,≥on
[𝜇,
is
called
a∈
harmonically
concave
function
𝜐].
[𝜇,
[𝜇,
for
allare
𝒾,by
𝒿given
𝜐],=
𝜉be[𝔖
∈(𝒾)
1],
𝔖(𝒾)
0define
for
all
𝒾𝒾family
∈
Ifall(11)
2. 𝜈]
[49]
:given
[𝜇,
𝜈]is
⊂
ℝ
→(𝒾)
a[0,
F-NV-F
whose
the[𝜇,
of
(𝒾,
[𝔖where
(𝒾,
=
𝜑),
𝜑),
𝜑)]
𝔖𝜑-cuts
for∗𝜑)]
all
𝒾for∈on
all
[𝜇,
𝜈]
∈and
[𝜇, 𝜐].
𝜈]
forI-V-Fs
and
forisallreversed then,
: [𝜇, 𝜈]
: [𝜇,
ℝ
→
⊂
𝒦Let
ℝ are
→𝔖
𝒦
𝔖
by
𝔖
𝔖 Theorem
𝔖⊂
∗𝔖
called
a𝔽 harmonically
concave
function
𝜐].
forℝall→ 𝜑
(0,
1].
For
all 𝔖∗𝜑 (𝒾)
∈ (0,
1],
ℱℛ
denotes
the
collection
of∗and
all 𝐹𝑅-integrable
F[𝔖
(𝒾,
=(𝐹𝑅)
𝜑),
𝜑)]
for
all
𝒾 (𝑅)
∈(𝒾,
[𝜇,
𝜈]𝔖
for
all
𝒦∈
are
given
by
𝔖 : [𝜇, 𝜈] ⊂
([
], )(𝒾,
, 𝔖
∗
[𝜇,
is
called
a
harmonically
concave
function
on
𝜐].
(𝒾,
(𝒾,
(𝑅)
(𝑅)
(𝑅)
(𝑅)
(𝑅)
(𝑅)
(𝐹𝑅)
(𝒾)𝑑𝒾
(𝒾)𝑑𝒾
(𝐹𝑅)
(𝒾)𝑑𝒾
(𝐹𝑅)
(𝒾)𝑑𝒾
=
𝔖
=
𝜑)𝑑𝒾
,
=
𝔖
𝜑)𝑑𝒾
𝔖
,
𝜑)𝑑𝒾
=𝜑)𝑑𝒾
𝔖,∗ (𝒾,
𝜑)𝑑𝒾
𝔖∗ (𝒾𝔖
𝔖
𝔖
𝔖
𝔖
(0,
∗
for is
all𝐹𝑅-integrable
𝜑 by
∈ (0,
For
all
𝜑𝜈]
∈only
1],
ℱℛ
the
collection
all
𝐹𝑅-integrabl
∗[𝜇,
∗both
∗of(𝒾,
([iffor
], all
) ∗denotes
,𝔖
(𝒾)
(𝒾,
(𝒾,𝔖
=𝜈][𝔖
𝜑),
𝔖
𝜑)]
𝒾
∈
𝜈]
and
for
all
: [𝜇,
⊂ℝ
𝒦
are
given
𝔖 1].
(𝒾,
(𝒾,
(𝒾,
(𝒾,
𝜑)
and
𝜑)
𝔖
and
𝜑)
𝔖
𝜑)
are
both
are
𝜑𝔖
∈ (0,
𝜑
1].𝜈]
∈ Then
(0,
1].→
𝔖Then
is
𝐹𝑅-integrable
𝔖
over
[𝜇,
over
if
[𝜇,
and
if
and
if
only
∗
NV-Fs
over
[𝜇,
𝜈].
∗
∗
(0, [𝜇,
for
all
𝜑𝔖
∈ (𝑅)
(0,
1]. [𝜇,
For
𝜑
∈∗ (𝒾,
1], 𝜈]
ℱℛ
the𝜑)collection
of 𝜑)
all 𝐹𝑅-integrable
F∗if(𝒾,
([and
) denotes
,𝜑)𝑑𝒾
and 𝔖∗ (𝒾,
both are
𝜑 ∈ (0,
1].
is𝔖
𝐹𝑅-integrable
if 𝔖
(𝒾,
(𝑅)Then
(𝑅)
(𝑅)
(𝑅)
(𝐹𝑅) 𝔖(𝒾)𝑑𝒾
(𝐹𝑅)
(𝐹𝑅)
(𝒾)𝑑𝒾
NV-Fs
over
𝜈].all
=
=
𝔖∗ (𝒾,
𝜑)𝑑𝒾
𝔖
, ∗ (𝒾,
=𝜑)𝑑𝒾
𝔖over
, (𝑅)
𝜑)𝑑𝒾
𝔖∗ (𝒾,
𝔖𝜑)𝑑𝒾
,], only
𝔖∗∗(𝒾,
𝜑)𝑑𝒾
𝔖(𝒾)𝑑𝒾
𝜑)𝜈],
andthen
𝔖∗ (𝒾, 𝜑) both are
𝜑 ∈ (0,
1].over
Then[𝜇,𝔖
𝐹𝑅-integrable
[𝜇, is
𝜈] 𝐹𝑅-integrable
if and
only if[𝜇,over
𝔖𝜈],
[𝜇,
𝑅-integrable
𝑅-integrable
over
𝜈].isMoreover,
[𝜇,
𝜈]. Moreover,
is if𝐹𝑅-integrable
𝔖
over
NV-Fs
over
[𝜇,if𝜈].𝔖 over
∗ (𝒾,then
[𝜇,said
𝑅-integrable
over [𝜇, 𝜈].
𝜈], to
then
(0,over
[𝜇,𝐹𝑅-integrable
= (𝐼𝑅)
a convex
set,
if,=for
all 𝒾, 𝔖
𝒿∈
Definition
3. Moreover,
[34]
A set if𝐾 𝔖= is
𝜐] ⊂ over
ℝ =
(9)is
(9)be =
(𝒾)𝑑𝒾
(𝒾)𝑑𝒾
(𝐼𝑅)
𝔖=
𝔖(9)(𝒾)𝑑𝒾
𝑅-integrable over [𝜇,
𝜈].Definition
Moreover,
if3. 𝔖
𝜈],∞)
(0,then
=
∞)
is(𝐼𝑅)
said to𝔖
be(𝐼𝑅)
a convex
set,
if,
for all 𝒾,
[34]isA𝐹𝑅-integrable
set 𝐾 = [𝜇, 𝜐]
⊂ ℝ[𝜇,
[0,
𝐾, 𝜉 ∈
1],
we have
to be a convex set, if, for all 𝒾, 𝒿 ∈
3. [34]
A
set
𝐾 (𝐼𝑅)
= [𝜇, 𝜐]𝔖⊂ (𝒾)𝑑𝒾
ℝ = (0, ∞)∗ is said
(𝒾)𝑑𝒾
(𝒾)𝑑𝒾
(𝐼𝑅)𝐾,
=Definition
𝔖
=∈ (𝐼𝑅)
𝔖(𝑅)
=
[0,
have
(𝒾, 𝜑)𝑑𝒾
(𝒾, 𝜑)𝑑𝒾
(𝑅)
(𝑅) , 𝔖
(𝑅)
(𝐹𝑅)
(𝐹𝑅)
(𝒾)𝑑𝒾
=we
=
𝔖
𝔖 (𝒾,
, 𝜑)𝑑𝒾
𝔖∗ (𝒾, ∗𝜑)𝑑𝒾
𝔖𝜉(𝒾)𝑑𝒾
𝔖1],
(𝑅)
,
𝔖 (𝒾, 𝜑)𝑑𝒾
𝔖(𝒾)𝑑𝒾 =∗ (𝑅) ∗ 𝔖∗ (𝒾, 𝜑)𝑑𝒾
𝐾, 𝜉 ∈ [0, (𝐹𝑅)
1], we have
𝒾𝒿
∗ (𝒾,
(0,
(0,
(0,, 1],
𝜑 ∈ (0,
for 1].
𝜑 ∈for
all
(0,𝔖
all
𝜑
1].
∈𝜑For
∈ 1],
(0,
all 1].
ℱℛ
for
𝜑 ∈For
all
all
1],
𝜑
(0,
ℱℛ
1].
For
all
∈], (0,
1],
denotes
collection
denotes
the
denotes
collection
allℱℛ
𝐹𝑅-integrable
the ofcollectio
allden
𝐹
(𝑅)
(𝐹𝑅)for all
([
)∈
([
],the
) ℱℛ
([𝜑 ,(9)
) of
, ],𝜑
=all For
𝔖
𝔖(𝒾)𝑑𝒾
𝒾𝒿
∈∈𝜑)𝑑𝒾
𝐾.
∗ (𝒾, 𝜑)𝑑𝒾 , (𝑅)
(9) ([ , ], ) (10)
(1collection
𝜉𝒾
+
−over
𝜉)𝒿
∈all𝐾.
𝒾𝒿
(9)
NV-Fs
over
NV-Fs
[𝜇,
𝜈].
over
NV-Fs
[𝜇,
𝜈].
over
[𝜇,
𝜈].
NV-Fs
[𝜇,
𝜈].
(0,𝜑
(0,
(0,
for all 𝜑 ∈for
(0,all
1]. 𝜑For
∈ (0,
all 1].
for
𝜑 ∈For
all
all
1],∈𝜑(0,
ℱℛ
∈ 1].
1],
For
ℱℛ
all
𝜑
∈
1],
ℱℛ
denotes
the
denotes
collection
the
of
collection
all
denotes
𝐹𝑅-integrable
of
the
all
𝐹𝑅-integrable
Fof
F𝐹𝑅-integrable
F([ , ], )
([ , ], )
([ , ], )
𝜉𝒾 + (1 −∈𝜉)𝒿
𝐾.
(10)
(9)
(𝒾)𝑑𝒾
= (𝐼𝑅) = (𝐼𝑅)
𝔖 (𝒾)𝑑𝒾
𝔖 𝜉𝒾
+ (1 − 𝜉)𝒿
NV-Fs overNV-Fs
[𝜇, 𝜈].over [𝜇, 𝜈].
NV-Fs over [𝜇, 𝜈].
= (𝐼𝑅) 𝔖 (𝒾)𝑑𝒾
numbers. Let 𝔵, 𝔴 ∈ 𝔽 and 𝜌 ∈ ℝ, then we have
[𝔵+𝔴]
Symmetry 2022, 14, 1639
[𝔵+𝔴]
= [𝔵] + [𝔴] ,
(5)
[𝔵 × 𝔴] = [𝔵] × [ 𝔴] ,
(6)
[𝜌. 𝔵]
= 𝜌. [𝔵]
(7)
= [𝔵] +
[𝔵 × 𝔴] = [𝔵] ×
[𝜌. 𝔵]
= 𝜌. [
10 of 18
Definition 1. [49] A fuzzy number valued map 𝔖: 𝐾
Symmetry 2022,
Symmetry
14, x FOR
2022,PEER
14, x FOR
REVIEW
PEER REVIEW
Definition 1. [49] A fuzzy number valued
: 𝐾x ⊂
ℝ PEER
→ 𝔽 REVIEW
is called F-NV-F.
For
𝜑 ∈ (0,
1]each
whose 𝜑 -cuts define the family of ISymmetrymap
2022,𝔖14,
FOR
14, x FOR PEER REVIEW
e ( 𝒾 )∈<𝐾.e
𝜑 ∈ (0, 1] whose 𝜑 -cutsSymmetry
define2022,
the
family
of
I-V-Fs
𝔖
:
𝐾
⊂
ℝ
→
𝒦
are
given
by
(𝒾, 𝜑)] for
𝔖 (𝒾) =
S
From the definition of harmonically s-convexity
of [𝔖
F-NV-Fs,
it ∗follows
thatall
0 Here, fo
∗ (𝒾, 𝜑), 𝔖
∗
∗
e
e
𝔖 (𝒾) = [𝔖∗ (𝒾, 𝜑), 𝔖 (𝒾, 𝜑)] for
for each 𝜑 ∈ (0, 1] the end
point real
andall
Q( 𝒾 )∈<𝐾.0, Here,
so
functions
𝔖∗ (. , 𝜑), 𝔖 (. , 𝜑): 𝐾 → ℝ are called lower
Proposition
1. [51]Let1.
𝔴. ∈Let
𝔽 .𝔵,Then
𝔴∈𝔽
fuzzy
. Then
order
fuzzy
relation
order “relation
≼ ” is given
“ ≼ ” oi
functions 𝔖∗ (. , 𝜑), 𝔖∗ (. , 𝜑): 𝐾 → ℝare called lower
andupper
functions
of𝔵,[51]
𝔖
Proposition
Proposition
1. [51] Let 𝔵, 𝔴 ∈ 𝔽 . Then fuzzy order relation “
µυ
µυ
1.
[51] Let
𝔵, 𝔴 ∈ 𝔽 .2.Then
fuzzy
order
relation
“ ≼be
” is
give
an
F-(
Definition
[49]
Let
𝔖
:
[𝜇,
𝜈]
⊂
ℝ
→
𝔽
S∗ (1−ξ )µ+ξυ , ϕ × Q∗Proposition
,
ϕ
(1−ξ )µ+ξυ
[𝔵] for
𝔵 ≼ 𝔴 if and
𝔵≼𝔴
only
if and
if, [𝔵]
only≤if,[𝔴]
≤ [𝔴]
all 𝜑 for
∈ (0,
all1],
𝜑∈
Symmetry 2022,
Symmetry
14, x FOR
2022,PEER
14, x FOR
REVIEW
PEER REVIEW
[𝔵]
[𝔴]
𝔵
≼
𝔴
if
and
only
if,
≤
for
all
ans14,
F-NV-F.
ThenREVIEW
fuzzy
integral
of
𝔖
over
Definition 2. [49] Let 𝔖: [𝜇, 𝜈] ⊂ ℝ Symmetry
→ 𝔽 be2022,
[𝜇,
(𝐹𝑅)
(𝒾)𝑑𝒾
x FOR PEER
𝜈],
denoted
by
𝔖
,
is
given
levelwi
s
𝔵ϕ≼
≤ PEER
ξ S∗REVIEW
ξ sif
Qand
ϕ) +if,
− ξ )≤
Q∗[𝔴]
) +It(1is−
)s Srelation.
) 𝔴
(1 [𝔵]
(υ, ϕfor
) all 𝜑 ∈ (0, 1],
∗ ( υ,
∗ ( µ, only
2022, 14, x FOR
It(µ,
isbyaϕpartial
order
a ξpartial
order
relation.
[𝜇, 𝜈], denoted by (𝐹𝑅) Symmetry
𝔖(𝒾)𝑑𝒾, is given levelwise
s
s
It
is
a
partial
order
relation.
s
s
= S∗ (µ, ϕ)It×
ϕ)[ξorder
· ξ ]use
+S
ϕ) × Q∗operations
ξ )the
(µ, use
(υ,toϕexamine
) (1 −toξ )some
(1 −of
∗ partial
∗ ( υ,operations
We
We
mathematical
now
mathematical
examine
some
character
of the c
isQanow
relation.
We
operations
to examine
(𝒾)𝑑𝒾some
(𝐹𝑅) 𝔖(𝒾)𝑑𝒾
=
𝔖
= ” oo
s now use mathematical
s (𝐼𝑅)
s
s
numbers.
Let
∈Let
𝔵,[51]
𝔴∗∈(Let
and
𝜌𝔽∈ϕ.𝔵,ℝ,
and
then
then
we
Proposition
Proposition
1.𝔵,
1.+
𝔴
∈
. ℝ,
Then
order
order
“relation
≼ ”ofisthe
given
“ chara
≼
i
use
examine
+S∗ (µ, numbers.
ϕ)Q∗ (υ,We
ϕ)ξnow
1𝔴−
ξ𝔽
S
υ,
×operations
Q𝜌𝔽
µ,
ϕhave
1fuzzy
−
ξrelation
· ξ ,some
([51]
)mathematical
)Then
(∈we
)(to
) have
∗fuzzy
numbers.
Let
𝔵,
𝔴
∈
𝔽
and
𝜌
∈
ℝ,
then
we
have
Proposition
1.
[51]
Let
𝔵,
𝔴
∈
𝔽
.
Then
fuzzy
order
relation
“
(8)
(𝐹𝑅) 𝔖(𝒾)𝑑𝒾 = (𝐼𝑅) 𝔖 (𝒾)𝑑𝒾 = 𝔖(𝒾,
𝜑)𝑑𝒾 : 𝔖(𝒾, 𝜑)
∈ ℛ([ , ], ) ,
numbers.
𝔴 ∈Let
𝔽 𝔵,and
then
we have
Proposition
𝔴 ∈𝜌𝔽∈. ℝ,
Then
fuzzy
order relation “ ≼ ” is give
µυ Let1.𝔵,[51]
µυ
∗
∗
[𝔵]
[𝔵]
[𝔴]
[𝔴]
S (1−ξ )µ+ξυ , ϕ × Q (1−ξ )µ+ξυ , ϕ 𝔵 ≼ 𝔴
[𝔵+
[𝔵+
[𝔵]
[𝔴]
[𝔵]
[𝔴]
if
and
𝔵
≼
𝔴
only
if
and
if,
only
if,
≤
for
≤
all
𝜑
for
∈
(0,
all
1],
𝜑
∈
𝔴] where
= 𝔴]+ℛ([
= , ],, ) +denotes
, the coll(
for all 𝜑 ∈𝔵(0,
[𝔵+𝔴]
≼ 1],
𝔴 if and only
if, [𝔵]= [𝔵]
≤ [𝔴]
for, all
+ [𝔴]
[𝔵]
[𝔴]
s -integrable
[𝔵+
𝔵ϕ
≼
if
if,
≤
for
𝜑 ∈[𝜇,
(0,𝜈]1],
𝔴]
+
s S∗ ( µ, ϕ )of
∗ ( υ,
∗funcfor all 𝜑 ∈ (0, 1], where ℛ([ , ], ) denotes≤
theξcollection
integrable
tions
I-V-Fs.
over
+ItRiemannian
ξ sof
Qand
ϕ𝔴]
−=𝔴]
ξ𝐹𝑅
υ,[𝔴]
(1is−
)s Srelation.
) 𝔴
([𝔵µ,×only
) +𝔖
(=
)[𝔵]
([𝔵]
) ,[ all
It is a partial
order
a ξpartial
order
relation.
[𝔵1is
[𝔵]
[=∗𝔴]
×
×Q
, ϕ×
𝔴]
,
It
is
a
partial
order
relation.
[𝔵
[𝔵]
[
×
𝔴]
=
×
𝔴]
,
ymmetry 2022,
Symmetry
14,
x
FOR
Symmetry
2022,
PEER
14,
REVIEW
2022,
x
FOR
14,
PEER
x
FOR
REVIEW
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REVIEW
4
of
19
4
of
19
4
∗
(𝐹𝑅)
tions of I-V-Fs. 𝔖 is 𝐹𝑅 -integrable over=[𝜇,
𝔖(𝒾)𝑑𝒾
.+𝔖Note
that,
ifQ
s
ssome
∗ partial
s∈· ξ𝔽s ]use
∗ ( υ,
∗×
(𝒾,
(𝒾,
𝜑),
𝜑)
Lebesgue-integrable,
then
𝔖c
We
use
We
mathematical
now
operations
operations
toϕexamine
examine
of
character
of of
the19
isQanow
relation.
[𝔵
𝔴]
=(1[𝔵]
∗mathematical
S∗𝜈](µ,ifϕ)It×
ϕ)[ξorder
S
ϕ𝔖) ×
−toξ×
−
ξ, )the
(µ,
(υ,are
)operations
)some
([1𝔴]
We
now
use
mathematical
to
examine
some
o
[𝜌.
[𝜌.
[𝔵]
[𝔵]
𝔵]
𝔵]
=
𝜌.
=
𝜌.
𝔖
is
a
fuzzy
Aumann-integrable
func𝔖∗ (𝒾, 𝜑), 𝔖∗ (𝒾, 𝜑) are Lebesgue-integrable, then
tion
[𝜇,
𝜈].
numbers.
𝔵, 𝔴
∈Let
𝔽 𝔵,and
𝔴 ∈𝜌
𝔽∈ ℝ,
and
then
𝜌 ∈we
have
then
we [𝜌.
have
s𝔵] ssome
[𝔵]the chara
operations
to)(examine
= 𝜌.of
+S∗ (µ, numbers.
ϕ) × Q∗We
υ, now
ϕ)numbers.
ξ s (use
1 −mathematical
ξ )sLet
+ Sover
Q∗ℝ,
µ,
ϕ
1
−
ξ
ξ
.
(Let
(
)
𝔵,∗ (𝔴υ,∈ϕ𝔽) ×and
𝜌
∈
ℝ,
then
we
have
[𝜌. 𝔵] = 𝜌. [𝔵]
tion over [𝜇, 𝜈].
numbers. Let 𝔵, 𝔴 ∈ 𝔽 and 𝜌 ∈ ℝ, then we have
[𝔵+𝔴] =[𝔵+
[𝔵]𝔴]+ [𝔴]
= [𝔵], + [𝔴] ,
Definition
[49]
A
fuzzy
1. [49]
number
A“over
fuzzy
number
map
:by
𝐾
⊂
ℝℝ𝔖
→𝔽
:𝐾
𝔽[𝔵]
⊂isℝ
→
𝔽 ,F-N
is c
PropositionProposition
1. Integrating
[51]Proposition
Let 𝔵,1.𝔴both
[51]
∈ 𝔽1.
Let
𝔵, of
𝔴
Let
∈Definition
𝔽
𝔵,1.
𝔴
∈ 𝔽 inequality
. [51]
Then
fuzzy
. order
Then
.fuzzy
relation
Then
order
fuzzy
≼relation
order
” [0,
isvalued
given
relation
“ Let
≼on
”valued
𝔽“: 𝔖
≼
given
”𝜈]map
is
given
on
by
𝔽+called
by
[𝔵+
[𝔴]
𝔴]
=
Theorem
[49]
𝔖is
[𝜇,
⊂on
→
𝔽
sides
the
above
1],
we
obtain
ymmetry 2022,Symmetry
14, x FOR
Symmetry
2022,
PEER14,
REVIEW
2022,
x FOR
14,PEER
x FOR
REVIEW
PEER REVIEW
4
of
4 be
4ℝof→19
Definition
1. [49]
A2.fuzzy
number
valued
map
:of
𝐾a19
⊂F-NV-F
𝔽
[𝔵+𝔴]
[𝔵]
[𝔴]
=family
+19
, 𝐾𝔖
∈ Definition
(0, 1]
𝜑 whose
∈ (0,
𝜑whose
-cuts
define
𝜑
-cuts
the
define
family
the
of
I-V-Fs
of
𝔖
I-V-Fs
𝔖
:
⊂
ℝ
→
:
𝐾
𝒦
⊂𝜑)ℝ
1.1][49]
Afamily
fuzzy
number
valued
map
𝔖
:
𝐾
⊂
ℝ
→
𝔽
is
called
[𝔵
[𝔵
[𝔵]
[
[𝔵]
[
Theorem 2. [49] Let 𝔖: [𝜇, 𝜈] ⊂ ℝ → 𝔽 be a F-NV-F𝜑whose
𝜑-cuts
define
the
of
I-V-Fs
×
×
𝔴]
=
𝔴]
×
=
𝔴]
,
×
𝔴]
,
(𝒾)
[𝔖
(𝒾,
=
:
[𝜇,
𝜈]
⊂
ℝ
→
𝒦
are
given
by
𝔖
𝔖
[𝔵]
[𝔵]
[𝔴]
[𝔴]
𝜑and
∈[𝔴]
(0,
1]forif,
whose
-cuts
define
the
family
of I-V-Fs
𝔵 only
≼ 𝔴
if,
𝔵if≼and
𝔴 if
only
if,
only
≤
≤
all[𝔵]
𝜑∈
≤𝜑(0,
for
1],
all for
𝜑 ∈all(0,𝜑[𝔵
1],
∈×(0,
(4)
1],= [𝔵]
[ 𝔴]∗ ,𝔖(4):
𝔴]
× (4)
𝔖
𝔵 ≼ 𝔴 if and
∗(𝒾,
∗R(𝒾,
R
[𝔖
(𝒾)
(𝒾,whose
µυ𝔖and
𝔖 1]
𝜑),
= [𝔖
𝜑),
for
𝜑)]
𝒾 ,for
∈ϕ[𝔵
Here,
for
each
for
𝜑 ,𝔖
∈each
(0,
the
∈→
(0,
e𝒦
∈=
(0,
𝜑𝜑)]
-cuts
define
the
family
ofHere,
I-V-Fs
: 𝐾1]
⊂𝜑ℝ
S
×Q
ϕ)𝐾.
∗𝜑
)𝐾.
[𝔵]
υ all
∗ ∗( all
∗ (𝒾, ∈
µυ
×all
𝔴]
=
×𝐾.[ 𝔴]
∗for
∗ (𝒾,
(𝒾,
𝜑),
𝔖(𝒾)
𝜑)]
all
𝒾𝔖
∈
[𝜇,
𝜈]
for
𝔖∗ (𝒾) =µυ[𝔖∗ (𝒾,
𝔖 : [𝜇, 𝜈] ⊂ ℝ → 𝒦 are given by1 S
(𝒾)
[𝔖
(𝒾,
(𝒾,
𝔖
=
𝜑),
𝔖
𝜑)]
for
all
𝒾
∈
Here,
for
each
𝜑
,
ϕ
×
Q
,
ϕ
=
d
[𝜌.
[𝜌.
[𝔵]
[𝔵]
𝔵]
𝔵]
=
𝜌.
=
𝜌.
𝜑
∈
(0,
1].
Then
𝔖
is
𝐹𝑅-integrable
over
[𝜇,
𝜈]
if
and
∗
∗ µ ∗µ
2
∗
∗
υ
−
1is
−ξa)µpartial
+
ξυ
1
−
ξ
µ
+
ξυ
(
)
It is a partialIt0order
is a partial
It(relation.
order
relation.
order
relation.
(. , ∗𝜑),
(. ,ℝ
(𝒾, 𝔖
(𝒾, 𝐾
functions
𝔖
𝔖∗(.(.𝔖,∗,𝜑):
𝜑),
𝔖→
𝜑):are
𝐾 called
→𝒾 ℝ
are
lower
called
and
lower
upper
functions
upper
ofth
[𝜌.
𝔖 (𝒾)functions
=∗[𝔖
𝜑),
𝜑)]
for
∈ 𝐾.
Here,
for
each
𝜑[𝔵]
∈ (0, 1]fun
𝔵]
=and
𝜌.
∗ all
R
1
(.
(.
functions
𝔖
,
𝜑),
𝔖
,
𝜑):
𝐾
→
ℝ
are
called
lower
and
uppe
(𝒾,
(𝒾,
𝜑)
and
𝜑)
both
are
𝜑 ∈ (0, 1]. Then 𝔖 Proposition
is 𝐹𝑅-integrable
over
[𝜇,We
if
and
only
iffuzzy
[𝜌.
s dξif𝔽 𝔖
𝔵]is
=
𝜌.·is
Proposition
1.use
[51]
Proposition
Let
𝔵,𝜈]
1.𝔴
[51]
∈≤
𝔽1.
Let
Let
𝔵,to
𝔴
∈
.S[51]
Then
.Qorder
Then
.,fuzzy
relation
Then
fuzzy
“υ,to
≼the
relation
order
”) →
is
given
relation
“υ,
≼
on
𝔽
“ Moreover,
≼
given
by
on
given
on
by
𝔽𝐹𝑅-integra
byfuzzy
∗×
∗order
𝑅-integrable
over
[𝜇,
isfuzzy
µ,𝔴
ϕ∈𝔖
µ,
ϕ
+
ϕ
×
Q
ϕ”𝜈].
ξ”sfuzzy
ξ[𝔵]
(operations
)𝔽
)𝔖
(of
))
We now
We
mathematical
now
use
now
mathematical
use
mathematical
operations
examine
some
toS
examine
of
examine
some
characteristics
some
the
characteristics
of
the
of
characteristics
of
of
∗ (𝔵,
∗𝔖(∗𝔽
∗are
(.operations
(.∗∗,(𝜑):
functions
𝜑),
𝔖
𝐾
ℝ
called
and upper
R01lower
ymmetry 2022,Symmetry
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Symmetry
2022,
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4 of 19
4 of functions
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[𝔖∗𝔖
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(𝒾)∗[51]
(𝒾,
[𝔖
(𝒾,𝔵,
(𝒾,
𝔖 (𝒾) = Proposition
𝜑), 𝔖
𝔖
= [𝔖
𝜑)]
=
𝜑),
for
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all
𝜑),
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𝒾∈𝔖
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1]
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1.
𝔴
𝔽ϕ(𝒾,
.𝜑),
Then
order
relation
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𝔽(𝒾)𝑑𝒾
by
S
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×
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ϕ
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𝔖each
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∗=
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(5)
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υ−µLet
[𝔵+
[𝔵+
[𝔵+
[𝔵+
[𝔴]
[𝔵]
[𝔵]
[𝔴]
[𝔴]
[𝔴]
[𝔵]
[𝔵]
[𝔴]
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0
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+
𝔴]
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,
+
+
,
𝔴]
=
,
=
+
,
over
[𝜇,
𝜈].
(0,
[𝜇,
=
∞)
Definition
3.
[34]
A
set
𝐾
=
𝜐]
⊂
ℝ
(𝐹𝑅)
(𝒾)𝑑𝒾“ ≼=” R(𝐼𝑅)
𝔖[𝔴]
𝔖on(𝒾)𝑑𝒾
=
𝔖(𝒾, 𝜑)𝑑𝒾 : 𝔖(𝒾, 𝜑)is∈
Proposition
1. [51]functions
Let
∈𝔖
.ℝ
fuzzy
relation
given
by
∗Then
∗[𝔵]
s.𝔽 functions
tion
[𝜇,
𝜈].+
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≼∗ (.𝔴
if𝔵𝔴,≼
and
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only
if
and
if,𝔖only
if,
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≤
for
all
for
𝜑, ∈ϕfunctions
all
(0,is1𝜑
1],
∈
(0,
(4)
(4) 𝔖.
(.
(.
(.over
s(
functions
𝔖functions
𝔖,𝔵,∗𝜑):
𝔖
𝜑),
𝐾
→
,(.𝜑),
,are
𝜑):
called
𝐾
,→
𝜑):
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𝐾[𝔵]
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→
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called
and
are
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called
lower
and
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of1],
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functions
𝔖
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∗ (. , 𝜑),𝔵
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µ,
υ
ξ
1
−
ξ
dξ,
((
)
)
)
∗
(0,
[𝜇,
[0,
[𝔵]
[𝔴]
= 1.
∞)
is
said
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amap
convex
set,
if,
for
all
𝒾,
𝒿ℝ
∈
DefinitionSymmetry
3. [34] A
set
𝐾 x=FOR2022,
𝜐]
⊂
ℝA
𝐾,
𝜉([𝔖
∈map
1],
we
have
𝔵FOR
≼for
𝔴PEER
if
and
only
if,
≤
for
all
𝜑
∈ℝ
(0,
1],
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0𝔴]
all
𝜑be
for
∈2.
(0,
all
1],
𝜑Let
∈
(0,
1],
ℛ
where
ℛ
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the
denotes
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the
collection
of
int
Definition
1.
Definition
[49]
fuzzy
[49]
number
1.
A
[49]
fuzzy
valued
A
number
fuzzy
number
valued
𝔖
:where
𝐾
⊂
valued
map
ℝ
→
𝔽⊂
:𝔖
𝐾
⊂
𝔖
:𝔽×
𝐾
→
𝔽
→
𝔽𝔴]
is
called
F-NV-F.
is
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For
is×
called
F-NV-F.
each
F-NV-F.
For
each
For
each
],
)→
([
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2022,
Symmetry
14,
PEER
Symmetry
14,REVIEW
xDefinition
FOR
2022,
PEER
14,
Symmetry
xREVIEW
2022,
REVIEW
14,
x=
FOR
PEER
REVIEW
of
,×
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(6)
[𝔵
[𝔵
[𝔵
[𝔵
[𝔵
[𝔵]
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[𝔵]
[𝔵]
[
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[
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[𝔵]
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[of
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𝔴]
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,
×
𝔴]
=
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,
=
×
𝔴]
×
, (6)
𝔴]Riema
,4of
Theorem
Theorem
[49]
2.
𝔖
[49]
:
[𝜇,
Let
𝜈]
ℝ
:
[𝜇,
𝜈]
⊂
ℝ
→
𝔽
be
a
F-NV-F
be
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F-NV-F
𝜑-cuts
whose
define
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the
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for[𝔴]
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1],
ℛ
collection
𝔵 ≼ relation.
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all
𝜑Let
∈where
(0,
1],
(4)a the
([⊂, ],
)→denotes
Theorem
[49]
𝔖
:
[𝜇,
𝜈]
ℝ
𝔽
be
F-NV-F
whose
𝜑It is
a∈ partial
It is
a whose
partial
order
relation.
order
𝐾, 𝜉 ∈ [0, 1], we have
(𝐹𝑅)
(𝐹𝑅)
(𝒾)𝑑𝒾
(𝒾)
for
all
𝜑
∈
(0,
1],
where
ℛ
denotes
the
collection
of
Riemannian
tions
of
tions
I-V-Fs.
of
𝔖
I-V-Fs.
is
𝐹𝑅
𝔖
-integrable
is
𝐹𝑅
-integrable
over
[𝜇,
𝜈]
over
if
[𝜇,
𝜈]
if
𝔖
𝔖
∈
𝔽
.t
𝜑
(0,
1]
𝜑
∈
(0,
𝜑
𝜑
1]
∈
-cuts
(0,
whose
1]
define
whose
𝜑
-cuts
the
𝜑
family
define
-cuts
define
the
of
I-V-Fs
family
the
family
𝔖
of
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of
I-V-Fs
𝔖
𝔖
:
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⊂
ℝ
→
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:
𝐾
are
⊂
ℝ
:
given
𝐾
→
⊂
𝒦
ℝ
by
→
are
𝒦
given
are
given
by
by
([
)→
𝒾𝒿
,19],
14,
ER xSymmetry
REVIEW
FOR PEER
2022,
REVIEW
14, x FOR PEER Definition
REVIEW It is2.
4
of
4
of
19
4
of
19
∗
∗
be
an
F-NV-F.
be
an
Then
be
F-NV-F.
fuzzy
an
F-NV-F.
Then
integral
fuzzy
Then
of
𝔖
fuzzy
integral
over
integral
of
𝔖
over
of
𝔖
over
Definition
Definition
Letorder
𝔖
2.:µυ
[𝜇,
[49]
𝜈]
2.
Let
⊂
[49]
ℝ
𝔖
→
:
Let
[𝜇,
𝔽
𝜈]
𝔖
:
⊂
[𝜇,
ℝ
𝜈]
→
⊂
𝔽
ℝ
→
𝔽
a [49]
partial
relation.
Theorem
2.
[49]
Let
𝔖
:
[𝜇,
𝜈]
⊂
ℝ
𝔽
be
a
F-NV-F
whose
𝜑-cuts
define
R-integrable
Roperations
(𝐹𝑅)
(𝒾)
(𝒾,
[𝔖
(𝒾,
(𝒾,
tions
of
I-V-Fs.
𝔖
is
over
𝜈](𝒾,
if(7)
=
𝜑),
=
𝔖
𝜑)]
𝜑),
𝔖𝜌.
for
all
𝜑)]
∈for(7)
[𝜇,a
[𝜇,
𝜈] to
⊂∗ examine
ℝ
: [𝜇,
→
𝜈]
⊂∗ some
are
ℝ
→
given
𝒦some
by=
given
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by
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𝔖
[𝜌.
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[𝔵]
[𝔵]
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𝔵]
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𝔵]
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We
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It isWe
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=
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𝜑)]
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𝜑),
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𝜑)]
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1]
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-integrable
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∗𝔖
We
use
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examine
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characteristics
of
𝜈],
denoted
𝜈],
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𝜈],
denoted
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given
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0
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=
𝜑),
𝔖
𝜑)]
for
all
𝒾𝜑)
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by
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𝜑),𝐹𝑅-integrable
𝔖𝔖
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Lebesgue-integrable,
then
𝔖
a∗ (𝒾,
fuzz
𝔖1].
∈∗ (.
𝐾.
Rcharacteristics
(10)
∗[𝜇,fuzzy
(𝒾,
numbers.
numbers.
Let
𝔵,(.use
𝔴,Let
∈mathematical
𝔽𝔖
𝔵,∗𝔴
∈𝜑):
𝔽,𝜌𝜑),
and
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and
ℝ,
then
𝜌
∈
ℝ,
we
have
we
have
𝜑)
and
𝔖
𝜑
∈
(0,
1].
𝜑,[𝜇,
Then
∈ℝ
(0,
𝔖(𝒾,
isThen
is𝜑)the
𝐹𝑅-integrable
over
[𝜇,
𝜈]
over
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only
𝜈]
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𝔖
only
ifis𝔖
∗ (.then
∗→
1
∗
We
now
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examine
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of
∗
s
(.
(.
(.
s
functions
𝔖
functions
functions
𝔖
𝔖
𝜑),
,
𝐾
→
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ℝ
,
𝜑),
,
are
𝜑):
𝔖
called
𝐾
→
𝜑):
lower
𝐾
are
ℝ
called
and
are
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called
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functions
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upper
and
of
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functions
𝔖
.
functions
of
.
of
𝔖
.
tion
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tion
𝜈].
over
[𝜇,
𝜈].
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(𝒾,
(𝒾,
𝜉𝒾
𝜑),
𝔖
𝜑)
Lebesgue-integrable,
then
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fuzzy
Aumann
∗
numbers.
Let+
𝔵,∗𝔴−∈𝜉)𝒿
𝔽∗ Proposition
and
ℝ,
then
we
have
𝜑
(0,
1].
Then
𝔖
is
𝐹𝑅-integrable
over
[𝜇,
𝜈]
if
and
only
if
𝔖
+
∂
µ,
υ
,
ϕ
ξ
1
−
ξ
dξ,
((
)
)
(
)
∗𝜌 ∈Let
Proposition
1. 𝔖
[51]
Proposition
𝔵,1.𝔴
[51]
∈
𝔽
Let
1.
𝔵,
Proposition
[51]
𝔴
∈
Let
𝔽
𝔵,
𝔴
1.
∈
𝔽
[51]
Let
𝔵,
𝔴
∈
𝔽
.
Then
fuzzy
.
Then
order
fuzzy
relation
.
Then
order
“
fuzzy
≼
relation
”
is
order
.
“
relation
≼
on
”
fuzzy
is
𝔽
given
by
“
orde
≼
”o
∗
0
tion
over
𝜈].
(𝒾,
𝜑)
and
𝜑
(0,
1].
Then
𝔖
is[𝜇,
𝐹𝑅-integrable
over
[𝜇,
𝜈]For
and
only
if
numbers. Let
𝔵, 𝔴 A
∈ fuzzy
𝔽 1.and
𝜌1.
ℝ,valued
then
we
have
∗𝐾
Definition
1.
Definition
[49]
Definition
[49]
number
Definition
A∈𝑅-integrable
[49]
fuzzy
A∈number
fuzzy
1.
Definition
[49]
map
number
A
Definition
valued
𝔖
fuzzy
: [𝔴]
𝐾
1.
⊂
valued
[49]
map
number
ℝ
→
1.
A𝔖
𝔽
[49]
fuzzy
map
:, 𝐾valued
A
⊂
𝔖
number
fuzzy
ℝ
⊂
𝔽F-NV-F.
number
ℝ
→
𝔖
𝔽
𝐾is
⊂valued
ℝcalled
→
𝔖
𝔽
map
:𝐾
⊂called
𝔖𝔖
ℝ
: each
→
⊂𝔽[𝜇,
ℝeach
→
𝔽t
isMoreover,
is
isifmap
F-NV-F.
each
F-NV-F.
is
For
For
F-NV-F
is
ca
[𝜇,
𝑅-integrable
over
[𝜇,
𝜈].
over
Moreover,
[𝜇,
𝜈].
ifcalled
𝔖: 𝐾→
ismap
𝐹𝑅-integrable
if valued
𝔖:called
𝐹𝑅-integrable
over
𝜈],
over
then
𝜈],
tion
over
[𝜇,
𝜈].
(5)
(5)
[𝔵+
[𝔵+
[𝔵]
[𝔵]
[𝔴]
𝔴]
=
𝔴]
+
=
,
+
Proposition
Proposition
1. [51] Let 1.𝔵,[51]
𝔴
Proposition
∈ Let
𝔽 . Then
𝔵, 𝔴 ∈1.fuzzy
𝔽[51]
Let
𝔵,
𝔴
∈ order
𝔽𝔖
. (𝐹𝑅)
Then
order
fuzzy
relation
.𝔖
Then
“=
relation
≼
”fuzzy
is=
given
“𝔖
order
≼=
”𝔖(𝒾,
on[𝔵]
isrelation
𝔽
given
by
on
“𝔖(𝒾,
𝔽4.
” 𝜑)𝑑𝒾
by
is
given
on
𝔽,∈[𝔵]
by
[𝜇
[𝜇,
𝑅-integrable
over
[𝜇,
𝜈].
Moreover,
if𝜑)
𝔖
is
𝐹𝑅-integrable
over
Definition
[34]
The
𝔖:
𝜐]
→≤∈
ℝ
is
called
a≤𝜑
har
(8)
(8)
(𝒾)𝑑𝒾
(𝒾)𝑑𝒾
(𝒾)𝑑𝒾
(𝒾)𝑑𝒾
(𝐹𝑅)
(𝒾)𝑑𝒾
(𝒾)𝑑𝒾
(𝐼𝑅)
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(𝐼𝑅)
=
𝔖
[𝔵+
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[𝔵+
[𝔵]
[𝔴]
[𝔵]
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𝔴]
=
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+
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+
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(𝐹𝑅)
(𝐹𝑅)
(𝒾)𝑑𝒾
(𝒾)𝑑𝒾
denoted
𝜈],
by
denoted
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denoted
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by
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1],
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1],
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∈ (0,
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efor
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Theorem
Theorem
[49]
Let
Theorem
𝔖
2.
:
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[49]
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2.
Let
⊂
[49]
ℝ
𝔖
→
:
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𝜈]
𝔖
:
⊂
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𝜈]
→
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→
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be
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be
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define
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define
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e
(𝒾)
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=
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𝜑)]
for
all
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:
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=
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∈
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:
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tion
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=
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tions
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(𝐹𝑅)
(𝒾)𝑑𝒾
(𝒾)𝑑𝒾
𝜈], denoted
by
by
𝔖
𝔖
,is
is
given
,𝔖be
is:𝐹𝑅-integrable
levelwise
given
levelwise
by
by
be
an
F-NV-F.
Then
fuzzy
integral
of
𝔖
over
Definition
2.
[49]
Let
𝔖
:
[𝜇,
𝜈]
⊂
ℝ
→
𝔽
Theorem
2.denoted
Theorem
[49]
Let
Theorem
𝔖
2.
:
[𝜇,
[49]
𝜈]
2.
Let
⊂
[49]
ℝ
𝔖
→
:
Let
[𝜇,
𝔽
𝜈]
⊂
[𝜇,
ℝ
𝜈]
→
⊂
𝔽
ℝ
→
𝔽
a
F-NV-F
be
whose
a
F-NV-F
be
𝜑-cuts
a
F-NV-F
whose
define
whose
𝜑-cuts
the
family
𝜑-cuts
define
of
the
I-V-Fs
define
of
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of
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[𝜇,
[𝜇,
[𝜇,
𝑅-integrable
𝑅-integrable
over
[𝜇,
𝑅-integrable
𝜈].
Moreover,
over
[𝜇,
𝜈].
if
Moreover,
𝔖
𝜈].
is
Moreover,
if
𝔖
is
if
𝐹𝑅-integrable
𝔖
over
is
𝐹𝑅-integrable
𝜈],
then
over
over
then
𝜈],
[0,
𝐾,
𝜉
∈
1],
we
have
e
e
[𝜇,
(𝐹𝑅)
(𝒾)𝑑𝒾
𝜈],valued
denoted
by
𝔖
,(0,
is
given
levelwise
by
(0,
1]
whose
𝜑ℝ∈all
(0,
1]
𝜑
-cuts
𝜑
whose
∈1],
𝜑
-cuts
whose
𝜑For
define
∈
family
𝜑
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-cuts
1]𝜑the
of
whose
define
I-V-Fs
family
𝜑ℱℛ
-cuts
𝔖
of0 ,family
of
𝔖
the
I-V-Fs
family
of𝐹𝑅
: I-V-Fs
⊂whose
ℝdenotes
→
𝒦
:ϕ-cuts
𝐾the
⊂
areℝ
→all
:𝒦
𝐾
⊂
Theorem
7.𝜑
Let
∈
HFSX
µ,
υ(0,
F1]
,𝔖
sall
,the
Q
∈
HFSX
µ,each
υ]F-NV-F.
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F
s𝐾)],,define
([
]𝜑,define
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([
DefinitionDefinition
1. [49] A fuzzy
1. [49]
Definition
number
A fuzzy
number
1. [49]
map
A
valued
fuzzy
𝔖:S
number
⊂
map
→
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:𝜉𝔽
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valued
⊂
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→
map
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⊂
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→
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is
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For
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each
is
For
called
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each
0(𝒾,
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for
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for
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1].
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1].
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all
1],
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1],
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we
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) denotes
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𝜑),
are
𝔖
Lebesgue-integrable,
𝜑),
𝜑)
are
𝜑)
𝜑),
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are
𝔖
Lebesgue-integrable,
then
𝜑),
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𝜑),
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then
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are
𝜑)
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then
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athe
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Aum
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𝜈],
denoted
by
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given
levelwise
by
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+
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all
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1].
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1],
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and
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1]
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end
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each
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point
Here,
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the
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:𝜈][∗µ,
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⊂
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,
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and
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∈𝜈]
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-cuts
define
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C
NV-Fs
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all
𝜑[𝜇,
(0,
1].
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the
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of al
([ −
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) denotes
tion over
[𝜇,tion
𝜈].(𝐹𝑅)
over
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over
𝜈].
tion
[𝜇, 𝜈].
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𝜈].
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[𝜇,
over
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, 𝜉𝒾
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(1
∈
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+,→
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NV-Fs
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e
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functions
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functions
functions
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functions
𝔖
,
𝜑),
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,
𝜑):
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,
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,
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→
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ℝ
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and
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=
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,
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= 𝔖∗ 𝜑),
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∈ϕ(𝒾)𝑑𝒾
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1]ifϕpoint
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the
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and
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all
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𝜉𝒾
+
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∗ real
∗ and
(8)
(𝒾)𝑑𝒾
(𝐹𝑅)
=
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over
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𝜈].only
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𝜉𝒾
+𝜑)
𝜉)𝒿
∗ (. (.
∗(𝐹𝑅)
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(𝒾)𝑑𝒾
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𝔖
=
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=
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𝜑)𝑑𝒾
:
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𝜑)
∈
ℛ
,
(.
(.
(.
∈
F
R
,
functions functions
𝔖∗ (. , 𝜑), 𝔖𝔖
functions
𝔖
,
𝜑):
,
𝜑),
𝐾
𝔖
→
ℝ
,
𝜑):
are
𝐾
called
→
,
𝜑),
ℝ
𝔖
are
lower
,
called
𝜑):
and
𝐾
lower
→
upper
ℝ
are
and
functions
called
upper
lower
of
functions
𝔖
and
.
upper
of
𝔖
.
functions
of
𝔖
.
([
],
)
,
(0,
(0,
[𝜇,
[𝜇,
=
∞)
=
is
said
∞)
to
is
be
said
a
convex
to
beI-V-Fs
afamil
set,
con
if
Definition
Definition
3.
[34]
A
set
3.
[34]
𝐾
=
A
set
𝜐]
𝐾
⊂
=
ℝ
𝜐]
⊂
ℝ
∗
∗
([
µ,
υ
]
,
ϕ
)
[𝜇,
[𝜇,
[𝜇,
𝑅-integrable
𝑅-integrable
over
[𝜇,
𝑅-integrable
𝜈].
Moreover,
over
[𝜇,
over
if→
Moreover,
[𝜇,
𝔖𝔽 2.
𝜈].
is𝔖be
𝐹𝑅-integrable
Moreover,
if→Theorem
𝔖𝔽𝔖ℝ
is
ifbe
𝐹𝑅-integrable
𝔖
is
𝐹𝑅-integrable
𝜈],
then
over
over
𝜈],
then
𝜈],
then
(9)
(9)
(9)
[𝜇,
2. Theorem
[49]
Let
Theorem
𝔖
2.
: where
[𝜇,
[49]
𝜈]Theorem
2.
Let
⊂
[49]
ℝ𝜈].
𝔖
:Let
[𝜇,
𝜈]
Theorem
[49]
:⊂
[𝜇,
ℝF-NV-F
𝜈]
Let
⊂
2.
: [𝜇,
→
[49]
𝜈]
𝔽
2.
Let
⊂
[49]
ℝ
→
:of
Let
[𝜇,
𝔽A
𝜈]
𝔖
:→
⊂
[𝜇,
ℝ
𝜈]
→
⊂
𝔽Let
ℝ
→
aDefinition
whose
aover
F-NV-F
be
𝜑-cuts
a𝔖𝜈]
F-NV-F
whose
define
be
whose
a𝜑-cuts
the
F-NV-F
family
𝜑-cuts
define
be
whose
of
a𝔽
the
I-V-Fs
define
F-NV-F
be
𝜑-cuts
family
afuncthe
F-NV-F
whose
family
of
define
I-V-Fs
whose
𝜑-cuts
the
of
𝜑-c
def
(0,
[𝜇,
=aan
∞)
is
said
to
be
Definition
3.
[34]
set
𝐾
=
𝜐]
⊂
ℝ
forTheorem
all for
𝜑 ∈all
(0,
𝜑
1],∈ where
(0,
1],
ℛ
denotes
denotes
the
collection
the
collection
of
Riemannian
Riemannian
integrable
integrable
funcbe
an
F-NV-F.
be
an
Then
F-NV-F.
fuzzy
be
Then
integral
F-NV-F.
fuzzy
of
be
Then
integ
𝔖
an
ov
fu
F
Definition
Definition
2.
Let
2.
𝔖
:
[49]
[𝜇,
𝜈]
Let
⊂
2.
ℝ
𝔖
[49]
→
Definition
:
[𝜇,
𝔽
Let
⊂
𝔖
ℝ
:
2.
𝔽
𝜈]
[49]
⊂
ℝ
→
𝔖
𝔽
:
[𝜇,
𝜈]
⊂
ℝ
→
𝔽
[𝜇,
[𝜇,
Definition
4.
[34]
The
4.
[34]
𝔖:
The
𝜐]
→
𝔖:
ℝ
𝜐]
→
ℝ
is
called
a
is
harmonically
called
harmonically
convex
functi
conv
([ , ], ℛ
)Definition
([
],
)
, [49]
2µυ
2µυ
[0,
[0,
𝐾,
𝜉
∈
𝐾,
1],
𝜉
we
∈
have
1],
we
have
(0,ℝ∞)
=→
said to
be
a convex
s
Definition
3. [34]the
A
set
𝐾[34]
= [𝜇,
𝜐]Riemannian
⊂ℝ
2s)−1denotes
for
all→
𝜑
∈𝔖𝜈](0,
1],
where
collection
funce
[𝜇,
e𝔖
Definition
4.
The
𝔖:
𝜐]
isis𝜑)]
called
a∗𝜈]
harmonically
∗[𝔖
∗of
∗ (𝒾,
∗[𝔖
e
([
],
, are
2(𝐼𝑅)
S
×
Q
[0,
𝐾,
𝜉:⊂
∈(𝐼𝑅)
1],
we
have
(𝒾)
[𝔖
(𝒾,
(𝒾)
(𝒾)
(𝒾,
[𝔖
(𝒾,
(𝒾)
(𝒾,
[𝔖
(𝒾,
(𝒾)
(𝒾,
(𝒾)
(𝒾,
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(𝒾,
(𝒾,
(𝒾,
=fuzzy
𝜑),
𝔖
=
𝜑)]
=
𝜑),
for
𝔖
𝜑),
=
𝒾𝔖
𝜑)]
𝔖
∈
[𝜇,
for
𝜈]
𝜑)]
𝜑),
all.and
𝔖
=
for
𝒾integrable
∈
for
all
=
all
𝒾𝜈]
𝜑),
∈for
and
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for
𝒾and
𝜑)]
𝔖
all
∈∗levelw
[𝜇,
for
for𝜈]
𝜑)]
all
al
::all
[𝜇,
𝜈]
⊂
ℝ
:𝔖
[𝜇,
are
:⊂
[𝜇,
ℝ
given
𝜈]
→
⊂
ℝ
by
: -integrable
[𝜇,
→
are
𝔖ℛ
𝜈]
𝒦
given
⊂
ℝ
→
by
given
: the
[𝜇,
𝒦
𝜈]
by
are
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ℝ
𝔖
given
𝜈]
→,(𝒾,
⊂
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by
→
are
𝔖
𝒦
given
are
by
given
𝔖
by
𝔖
𝔖
𝔖
𝔖𝒦
𝔖
𝔖
(𝒾)𝑑𝒾
(𝒾)𝑑𝒾
(𝒾)𝑑𝒾
(𝐼𝑅)
=
𝔖
=
=
𝔖
𝔖all
[𝜇,
[𝜇,
(𝐹𝑅)
[𝜇,
(𝒾)𝑑𝒾
(𝐹𝑅)
(𝒾)𝑑𝒾
(𝐹𝑅)
(𝒾)𝑑𝒾
(𝒾)𝑑𝒾
(𝐹𝑅)
(𝐹𝑅)
(𝒾)𝑑𝒾
(𝒾)𝑑𝒾
∗𝜈],
∗[𝜇,
∗denoted
∗is
∗if
𝜈],
denoted
𝜈],
by
denoted
by
𝔖
denoted
by
𝔖
𝜈],
𝔖[𝜇,
is
given
levelwise
,𝔖
is
given
by
levelwise
,(𝐹𝑅)
is
given
by
levelwise
,∗𝔖
is
given
by
tions
of
tions
I-V-Fs.
is
𝐹𝑅
𝔖
-integrable
isℛ
over
[𝜇,
over
𝜈]
if
[𝜇,
𝜈]
if
𝔖
∈
𝔽
. by
Note
∈𝒾𝒿
𝔽
that,
Note
that,
ifall
[𝜇,
an
F-NV-F.
be
Then
F-NV-F.
Then
be
integral
an
fuzzy
F-NV-F.
of
integral
𝔖
Then
over
of
fuzzy
𝔖
over
integral
𝔖
over
DefinitionDefinition
2. [49] Let
2.𝔖[49]
Definition
[𝜇,
𝜈]
Let
⊂of
𝔖
ℝ:I-V-Fs.
[𝜇,
→
2.𝒦
𝔽𝜈]
[49]
⊂be
ℝLet
→
𝔽
𝔖
: ,[𝜇,
⊂
ℝ
→
𝔽
Definition
4.
[34]
The
𝔖:
𝜐]
→
ℝ
called
aof
harmonically
convex
fu
for
𝜑
∈
(0,
1],
where
denotes
collection
of
Riemannian
integrable
funcµ
+
υ
µ
+
υ
([𝐹𝑅
], 𝜈]
)an
[0,
𝐾,
𝜉
∈
1],
we
have
𝒾𝒿
(𝐹𝑅)
(𝒾)𝑑𝒾
tions of
I-V-Fs. 𝔖𝔖(𝒾)𝑑𝒾
is
𝐹𝑅 =
-integrable
over
[𝜇,,𝜈]
if
𝔖 ,𝒾𝒿
∈
𝔽𝒾𝒿
.𝔖
that,
if
∗ (𝒾, (𝑅)
∗ (𝒾,
∗ (𝒾,
𝒾𝒿Note
(𝒾,
(𝒾,(𝒾,
(𝒾,
(𝑅)
(𝑅)
(𝑅)
(𝐹𝑅)
(𝐹𝑅)
(𝐹𝑅)
(𝒾)𝑑𝒾
(𝒾)𝑑𝒾
𝔖𝜑
𝜑)𝑑𝒾
𝜑)𝑑𝒾
𝔖
𝔖
,Aumann-integrable
𝜑)𝑑𝒾
𝔖𝔖
𝜑)𝑑𝒾
𝜑)𝑑𝒾
𝔖is
𝔖(0,
∗(1
∗ (𝒾,
∗ (𝒾,
(1
∗ (𝒾,
∗given
+
𝜉𝔖(𝒿),
+both
𝔖
𝔖
≤
−over
𝜉)𝔖(𝒾)
≤
−
𝜉)𝔖(𝒾)
∗𝔖
𝒾𝒿
[𝜇, 𝜈], denoted
[𝜇, 𝜈],by
(𝐹𝑅)
[𝜇,
(𝒾)𝑑𝒾
(𝐹𝑅)
(𝒾)𝑑𝒾
(𝐹𝑅)
(𝒾)𝑑𝒾
(𝒾,
(𝒾,
(𝒾,
(𝒾,
denoted
by
𝔖of
𝜈],
denoted
by
, is
,(0,
is 1].
given
by
levelwise
, (𝑅)
is𝜑[𝜇,
given
by
levelwise
by
𝜑)
and
𝔖
𝜑)
𝜑)
and
both
𝜑)
𝔖
and
𝜑)
𝜑)
both
𝜑)
are
𝔖𝜉𝔖(𝒿)
are
(0,
1].
Then
𝜑
∈𝔖
(0,
𝔖
1].
is
∈
𝐹𝑅-integrable
Then
𝜑𝔖
Then
∈𝔖
is
(0,
𝐹𝑅-integrable
𝔖
1].
over
Then
𝐹𝑅-integrable
∈𝜈]
𝔖
if∗=
1].
over
is
and
∈
𝐹𝑅-integrable
(0,
Then
[𝜇,
1].
over
𝜈]𝔖
if(𝑅)
Then
if[𝜇,
is
𝜈]
𝐹𝑅-integrable
𝔖
over
only
if∗(1
is
and
[𝜇,
𝐹𝑅-integrable
if𝜑)𝑑𝒾
only
𝔖
𝜈]
if(𝒾,
over
and
only
[𝜇,
𝜈]𝐾.
if are
if
[𝜇,
𝔖
𝜈]
if𝐾.
and
if𝜉)𝔖(𝒾)
only
𝔖
if 𝜑)
𝔖𝜑
∈
(𝒾,∈𝜑),
(𝒾,
(𝒾,
(𝐹𝑅)
(𝒾)𝑑𝒾
𝜑),
𝜑)
𝔖
are
Lebesgue-integrable,
are
Lebesgue-integrable,
then
𝔖
then
is
a=
𝔖
fuzzy
is
aand
fuzzy
func𝔖∗𝜑
𝔖
tions
I-V-Fs.
𝔖𝜑𝜑)
islevelwise
𝐹𝑅
-integrable
over
[𝜇,
𝜈]
ifonly
𝔖
∈
𝔽𝔖+
.∗if(1
Note
that,
ifand
∗Aumann-integrable
∗ (𝒾,
∗𝜉)𝒿
∗func∗𝔖
(1
𝒾𝒿
+
≤𝔖∈only
−and
𝜉𝒾
+
−
𝜉)𝒿
−
𝒾𝒿
(1
(1
∈ 𝐾.∗
𝜉𝒾
𝜉𝒾
+
−
𝜉)𝒿
+
−
𝜉)𝒿
is
a
fuzzy
Aumann-integrable
func𝔖∗∗(𝒾, 𝜑), 𝔖∗ (𝒾, 𝜑) are Lebesgue-integrable, then 𝔖 𝜉𝒾
(1
𝜉𝒾
+
−
𝜉)𝒿
(1
+
𝜉𝔖(𝒿),
𝔖
≤
−
𝜉)𝔖(𝒾)
(1
+
−
(0,
(0,
tion
𝜈].
[𝜇,
𝜈].
for
alltion
𝜑[𝜇,
∈over
(0,
for1].
all[𝜇,
For
for
𝜑𝜈].
∈all
(0,
𝜑1].
∈𝑅-integrable
(0,
For
1].
1],
ℱℛ
𝜑𝜈].
all
1],
ℱℛ
ℱℛ
denotes
the
the
of𝐹𝑅-integrable
all
collection
the
all
𝐹𝑅-integrable
of
all
𝐹𝑅-integrable
F∈F𝐾.
(9)
(9)
(𝒾,
(𝒾,
[𝜇,
[𝜇,
[𝜇,
[𝜇,
[𝜇
([𝑅-integrable
],𝜑
)∈𝑅-integrable
([
)([collection
],
)𝔖 denotes
, (𝐹𝑅)
,𝔖
,𝔖if
𝜑),
𝔖
𝜑)
are
Lebesgue-integrable,
then
is𝐹𝑅-integrable
a], is
fuzzy
Aumann-integrable
func𝔖∗over
𝑅-integrable
𝑅-integrable
over
𝑅-integrable
Moreover,
over
[𝜇,
over
𝜈].
ifallMoreover,
[𝜇,
𝔖For
is
over
𝐹𝑅-integrable
Moreover,
[𝜇,
if
𝔖
𝜈].
is1],
Moreover,
if(𝒾)𝑑𝒾
over
over
[𝜇,
over
𝐹𝑅-integrable
𝜈].
𝜈],
Moreover,
[𝜇,
then
over
𝜈].
is
Moreover,
over
if𝐹𝑅-integrable
𝜈],
𝔖collection
then
is=ifof
𝐹𝑅-integrable
𝔖
over
then
is
𝐹𝑅-integrable
𝜈],
then
over
over
𝜈],∈
th
(𝒾)𝑑𝒾
(𝒾)𝑑𝒾
(𝒾)𝑑𝒾
(𝒾)𝑑𝒾
(𝐹𝑅)
(𝒾)𝑑𝒾
(𝐹𝑅)
(𝒾)𝑑𝒾
(𝐹𝑅)
(𝒾)𝑑𝒾
(𝐼𝑅)
(𝐼𝑅)
(𝐼𝑅)
(𝐼𝑅)
𝔖∈
𝔖
=𝔖𝔖
𝔖denotes
=
=
𝔖
=
𝔖(𝒾,
𝜑)𝑑𝒾
𝔖𝜈],
:𝜉𝒾
𝔖(𝒾,
=
𝔖(𝒾,
=
𝜑)
𝜑)𝑑𝒾
∈𝜉)𝒿
𝔖
ℛ
:𝔖(𝒾,
𝔖(𝒾,
𝜑)𝑑𝒾
=
, (9)
:F𝔖
ℛ
(1
−
𝜉)𝒿
],𝜑)
, [𝜇,
(1
tion
over
[𝜇, 𝜈].
𝜉𝒾
−
[0,𝜐],
[𝜇,
for
all 𝒾, 𝒿for
∈ [𝜇,
all 𝜐],
𝒾, 𝒿𝜉 ∈∈ [𝜇,
1],𝜉 where
∈ [0,𝜉𝒾
1],+
𝔖(𝒾)
where
≥
0+𝔖(𝒾)
for
all
≥𝜉)𝒿
0𝒾 ∈
for[𝜇,all
𝜐].𝒾 If∈([(11)
𝜐].
is) reve
If (1
NV-Fs
over
[𝜇,
NV-Fs
𝜈].
NV-Fs
over
[𝜇,
over
𝜈].
[𝜇,
𝜈].
[𝜇,
[0,
[𝜇,
for
all
𝒾,
𝒿
∈
𝜐],
𝜉
∈
1],
where
𝔖(𝒾)
≥
0
for
all
𝒾
∈
𝜐
tion over
[𝜇, 𝜈].
(8)
(8)
(8)
(𝒾)𝑑𝒾
(𝒾)𝑑𝒾
(𝐹𝑅) 𝔖(𝒾)𝑑𝒾
(𝐹𝑅)
(𝐹𝑅)
(𝐼𝑅)
(𝐼𝑅)
𝔖(𝒾)𝑑𝒾
𝔖(𝒾)𝑑𝒾
=
𝔖
= (𝐼𝑅)
=𝔖 (𝒾)𝑑𝒾
𝔖(𝒾,
=
=
𝜑)𝑑𝒾
:
𝔖(𝒾,
𝔖
𝔖(𝒾,
𝜑)𝑑𝒾
𝜑)
∈
=
:
𝔖(𝒾,
ℛ
𝜑)
𝔖(𝒾,
∈
𝜑)𝑑𝒾
,
ℛ
:
𝔖(𝒾,
,
𝜑)
∈
ℛ
,
(𝐼𝑅)
(𝐼𝑅)
(𝐼𝑅)
= all
=
=
𝔖
𝔖
) 1],
([
],(𝒾)𝑑𝒾
) 𝔖(𝒾)
([𝜐].
], all
) [𝜇,𝒾𝜐].
, ],
, on
[𝜇,
is
called
aisharmonically
a(𝒾)𝑑𝒾
harmonically
function
concave
on
function
[0,
for
𝒾,called
∈𝔖
𝜐],
∈concave
where
≥
0family
for
∈ [𝜇,
𝜐]. If (11)
is
[𝜇,
Definition
4.𝒿be[34]
The
4.𝜉([[34]
𝔖:
The
𝜐](𝒾)𝑑𝒾
→
𝔖:
ℝ,[𝜇,
𝜐]
→
ℝthe
is
called
ais
harmonically
called
aI-V-Fs
harmonically
convex
functi
conv
Theorem
Theorem
2. [49] Let
2. [49]
𝔖: for
[𝜇,
Let𝜈]
𝔖⊂
:𝜑
[𝜇,
ℝ∈𝜈]
→
⊂
𝔽1],
ℝbe→
𝔽F-NV-F
aDefinition
a[𝜇,
F-NV-F
whose
𝜑-cuts
whose
define
𝜑-cuts
the
define
family
of
I-V-Fs
of
[𝜇,
is
called
a
harmonically
concave
function
on
𝜐].
[𝜇,
Definition
4.
[34]
The
𝔖:
𝜐]
→
ℝ
is
called
a
harmonically
∗
∗
∗
∗
∗
all
(0,
for
all
where
𝜑
∈
for
(0,
all
1],
ℛ
𝜑
where
∈
(0,
1],
for
ℛ
where
all
𝜑
∈
ℛ
(0,
1],
where
ℛ
denotes
the
denotes
collection
the
denotes
of
collection
Riemannian
the
collection
of
denotes
Riemannian
integrable
of
the
Riem
co
fun
in
([ (𝑅)
],F-NV-F
)(𝒾)𝑑𝒾
([
],𝔖
) (𝑅)
([ 𝔖, (𝒾)𝑑𝒾
],
)(𝒾,
([
],of𝜑)𝑑𝒾
) I-V-Fs
,=
, (𝒾,
Theorem
2.set
[49]
Let
𝔖
[𝜇,
𝜈]
⊂
ℝ
→
𝔽(𝒾)𝑑𝒾
be
a, said
whose
𝜑-cuts
define
the
family
(𝒾,
(𝒾,
(𝒾,
(𝒾,
(𝑅)
(𝑅)
(𝑅)
(𝑅)
(𝑅)
(𝑅)
(𝐹𝑅)
(𝒾)𝑑𝒾
(𝐹𝑅)
(𝐹𝑅)
(𝒾)𝑑𝒾
(𝐹𝑅)
(𝐹𝑅)
(𝐹𝑅)
(𝒾)𝑑𝒾
=
𝔖
=
𝜑)𝑑𝒾
=
,to(𝑅)
𝔖
𝔖
,to
𝜑)𝑑𝒾
𝜑)𝑑𝒾
𝔖
,to
𝔖
𝜑)𝑑𝒾
=
𝜑)𝑑𝒾
𝔖
,harmonically
𝔖
𝜑)𝑑𝒾
𝔖
,𝒾,𝜑)𝑑𝒾
𝜑)𝑑𝒾
𝔖fu
𝔖
𝔖
𝔖
𝔖
[𝜇,
is
called
a=
harmonically
concave
function
on
𝜐].
∗[𝜇,
∗∗(𝒾,
∗ (𝒾,
∗=
∗∈
∗ (𝒾,
[𝜇,
Definition
4.(𝒾,
[34]
The
𝔖:
𝜐]
→
ℝ
called
aset,
convex
(0,
(0,
(0,
[𝜇,
[𝜇,
=
∞)
∞)
=
be
is
a 𝜑)𝑑𝒾
convex
said
∞)
is
said
be
set,
ais
if,
convex
be
for
aall
convex
𝒾,
𝒿(𝑅)
if,
for
set,𝔖
all
if,
for
𝒿(𝑅)
all
∈ ,𝒾, 𝒿(𝑅)
∈(
Definition
3.
Definition
[34]
Definition
3.𝐾𝔖
[34]
=
3.
A:ℝ
𝜐]
[34]
set
⊂(𝒾)
𝐾
A
ℝ
=
set
𝐾
𝜐]
⊂𝜑),
ℝis
𝜐]
⊂
ℝ𝜑)]
∗=
𝒾𝒿
𝒾𝒿
Theorem
2.𝜑
[49]
Let
𝔖
:
[𝜇,
𝜈]
⊂
→
𝔽
be
a
F-NV-F
whose
𝜑-cuts
define
the
family
of
I-V-Fs
[𝔖
(𝒾)
(𝒾,
[𝔖
(𝒾,
(𝒾,
(𝒾,
=
=
𝔖
𝜑),
𝔖
for
𝜑)]
all
𝒾
for
∈
all
[𝜇,
𝜈]
𝒾
∈
and
[𝜇,
𝜈]
for
and
all
for
all
: [𝜇,
𝜈]
⊂
:, [𝜇,
ℝ
𝜈]
→
⊂
𝒦
ℝA
are
→
𝒦
given
are
by
given
𝔖
by
𝔖
𝔖 (0,
𝔖([where
(𝐹𝑅)
(𝒾)𝑑𝒾
(𝐹𝑅)
(𝒾)𝑑𝒾
(𝐹𝑅)
(𝒾
tions
of
I-V-Fs.
tions
𝔖
of
is
I-V-Fs.
tions
𝐹𝑅
-integrable
of
𝔖
I-V-Fs.
is
𝐹𝑅
tions
-integrable
𝔖
over
is
of
[𝜇,
𝐹𝑅
I-V-Fs.
𝜈]
-integrable
over
if
𝔖
[𝜇,
is
𝜈]
𝐹𝑅
over
𝔖
if
-integrable
[𝜇,
𝜈]
if
𝔖
over
[𝜇,
𝔖
𝜈]
∈
𝔽
.
Note
∈
that,
𝔽
for all 𝜑 ∈for
(0,all
1], 𝜑where
∈
1],
for
ℛ
all
∈
ℛ
(0,
1],
where
ℛ
denotes
the
denotes
collection
the
collection
of
Riemannian
denotes
of
the
Riemannian
integrable
collection
integrable
of
funcRiemannian
funcintegrable
func∗
∗
∗
],
)
([
],
)
([
],
)
,𝜑 ℝ
, For
𝒾𝒿
(1
+𝐹𝑅-integrable
𝜉𝔖(𝒿),
+ 𝜉𝔖(𝒿)
𝔖
𝜉)𝔖(𝒾)
≤𝐹𝑅-integrable
𝜉)𝔖(𝒾)
(0,
(0,
for
𝜑𝔖
∈1],(0,
for
all
(0,
𝜑
1].
∈1],
(0,
For
1].
1],
all
ℱℛ
𝜑 by
∈
∈denotes
ℱℛ
the
denotes
denotes
the
of all
collection
the
𝐹𝑅-integrable
of−
all
of
Fall−
FF-.
[𝔖,∗ℱℛ
(𝒾,
=1],
𝜑),
𝜑)]
for
all≤collection
𝒾(1∈
[𝜇,
𝜈]
and
:for
[𝜇,1].
𝜈]
⊂
→
𝒦
are
given
([all
],𝜑1],
)(𝒾)
([
], )𝔖
([collection
], )(𝒾,
,𝔖
, 𝔖
[0,
[0,
𝐾, 𝜉all
∈ [0,
𝐾,
we
𝜉all
have
∈For
𝐾,
𝜉∈1],
∈
we
have
we
have
(9)
(9)
(9)
(1 all
𝒾𝒿
+
𝔖
≤for
− 𝜉)𝔖(𝒾)
𝜉𝒾
+ (1∗−𝜉𝒾
𝜉)𝒿
∗+ (1 − ∗𝜉)𝒿
∗
∗
∗∗
numbers.
numbers.
Let
𝔴Let
∈mathematical
𝔽𝔵, 𝔴and
∈ 𝔽𝜌 ∈
and
ℝ, then
𝜌 ∈ ℝ,
wethen
have
we have some of the characteristics of fuzzy
We
now𝔵,use
examine
numbers.
Let 𝔵, 𝔴 ∈ 𝔽 operations
and 𝜌 ∈ ℝ,tothen
we have
numbers. Let 𝔵, 𝔴 ∈ 𝔽 and 𝜌 ∈ [𝔵+
ℝ, then
we
have
(5)
(5)
[𝔵]
[𝔵] ,+ [𝔴] ,
𝔴] [𝔵+
= 𝔴]
+
= [𝔴]
(5)
[𝔵+𝔴] = [𝔵] + [𝔴] ,
(5)
[𝔵+
[𝔵]
[𝔴]
𝔴]
=
+
,
(6)
(6)
[𝔵 × 𝔴] [𝔵 =
× [𝔵]
𝔴] ×
= [[𝔵]
𝔴] ×
, [ 𝔴] ,
(6)
[𝔵 × 𝔴] = [𝔵] × [ 𝔴] ,
(6)
[𝔵 [𝜌.
[𝔵]
[
× 𝔴]
=
×
𝔴]
,
(7)
(7)
[𝜌.
[𝔵]
[𝔵]
𝔵] = 𝜌.
𝔵] = 𝜌.
Symmetry 2022, 14, 1639
11 of(7)
18
[𝜌. 𝔵] = 𝜌. [𝔵]
(7)
[𝜌. 𝔵] = 𝜌. [𝔵]
Definition
Definition
1. [49] A
1. fuzzy
[49] Anumber
fuzzy number
valued valued
map 𝔖:map
𝐾 ⊂𝔖
ℝ: →
𝐾⊂
𝔽 ℝis→called
𝔽 is F-NV-F.
called F-NV-F.
For each
For each
Definition 1. [49] A fuzzy number valued map 𝔖: 𝐾 ⊂ ℝ → 𝔽 is called F-NV-F. For each
𝜑 Definition
∈ (0, 𝜑
1]∈ whose
(0,1.1][49]
whose
𝜑A-cuts
𝜑
define
-cuts
define
the
family
the
family
of
I-V-Fs
of
I-V-Fs
𝔖
𝔖
:
𝐾
⊂
ℝ
:
→
𝐾
⊂
𝒦
ℝ
→
are
𝒦
given
are
by
given
fuzzy number
valued
map
𝔖: 𝐾 ⊂ ℝ
𝔽 is called
each by
𝜑 ∈ (0, 1] ∗ whose
define
the family
ofZ→I-V-Fs
𝔖 : 𝐾 F-NV-F.
⊂ ℝ → 𝒦For
Z𝜑υ-cuts
Z 1are given by
∗
1
e
e
e
µυ
(𝒾)
(𝒾,
[𝔖 𝔖
(𝒾,𝜑
(𝒾,
𝔖 𝜑(𝒾)
𝔖 [𝔖
=
=𝜑),
𝜑),
𝜑)]
𝔖 (𝒾,
for
𝜑)]
all
𝒾the
𝐾.
𝒾Here,
Here,
each
for𝔖
𝜑each
∈:s𝐾
(0,⊂𝜑
1]ℝ∈the
(0,𝒦1]
end
the
point
end
real
point
S(for
Q(
)∈×all
)∈ 𝐾.offor
∈ (0,
1]
-cuts
define
family
→
are
given
by srealreal
eI-V-Fs
e
(𝒾)
𝔖∗ whose
𝜑),
alld 𝒾 +
∈ 𝐾.
Here,
∈+
(0,e
4 =∗ [𝔖
Λ
υ) for
ξ (each
1 − ξ )𝜑s dξ
∂1]
υ) end
ξ s ·point
ξ dξ,
( FR
)∗ 𝔖∗ (𝒾, 𝜑)] for
(µ,
(µ, the
∗∗(𝒾,
∗ (.
2ℝ are lower
(.
(.
(.
functions
functions
𝔖
𝔖
,
𝜑),
𝔖
,
𝜑),
,
𝜑):
𝔖
𝐾
→
,
𝜑):
ℝ
𝐾
are
→
called
called
and
lower
upper
and
functions
upper
functions
of
𝔖
.
of
𝔖
.
υ
−
µ
(𝒾)
[𝔖
(𝒾,
(𝒾,
µ
0
0
𝔖
= functions
𝜑),∗ 𝔖
𝜑)]
for
all
𝒾
∈
𝐾.
Here,
for
each
𝜑
∈
(0,
1]
the
end
point
real
∗∗
∗
𝔖∗ (. , 𝜑), 𝔖 (. , 𝜑): 𝐾 → ℝ are called lower and upper functions of 𝔖.
functions 𝔖∗ (. e, 𝜑), 𝔖∗ (. , 𝜑):
→e ℝ
are ecalled
upper functions
.e e
e
eofυ)𝔖+
e µ), and
e µ
e υand
e (𝐾
e υ)lower
e (µ)×
e Q(
e Q(
e Q(
where
Λ
µ,
υ)𝔖=
S
):×
)+
S
), F-NV-F.
∂Then
=
S
(over
υof
)×
(µ, υ) fuzzy
(
F-NV-F.
be an
Then
integral
fuzzy
integral
of 𝔖S
𝔖 over
Definition
Definition
2. [49] 2.
Let[49]
: Let
[𝜇,
𝜈]µ
𝔖⊂
[𝜇,
ℝQ(
𝜈]
→⊂
𝔽
ℝbe
→(an
𝔽 ×
∗
∗ ((µ, υof
be
an
F-NV-F.
Then
fuzzy
integral
Definition
2.
[49]
Let
𝔖
:
[𝜇,
𝜈]
⊂
ℝ
→
𝔽
Λ
µ,
υ
=
Λ
µ,
υ
,
ϕ
,
Λ
µ,
υ
,
ϕ
and
∂
µ,
υ
=
∂
µ,
υ
,
ϕ
,
∂
, ϕ𝔖)].over
(
)
[
((
)
)
((
)
)]
(
)
[
((
)
)
)
∗
[𝜇,Definition
[𝜇, 𝜈],ϕ denoted
(𝐹𝑅)
(𝒾)𝑑𝒾
(𝒾)𝑑𝒾
𝜈], denoted
by∗ 𝔖(𝐹𝑅)
𝔖
𝔖
, is
, islevelwise
given
levelwise
by ϕ Then
by
be
F-NV-F.
2.by
[49]
Let
: [𝜇,
𝜈] ⊂
ℝgiven
𝔽
[𝜇, 𝜈],
(𝐹𝑅)
denoted
by
𝔖→(𝒾)𝑑𝒾
, is an
given
levelwise byfuzzy integral of 𝔖 over
[𝜇, 𝜈], denoted
(𝐹𝑅) 𝔖(𝒾)𝑑𝒾
by hypothesis,
, is given
by
Proof. By
for each
ϕ ∈ [levelwise
0, 1], we have
(𝐹𝑅) (𝐹𝑅)
𝔖(𝒾)𝑑𝒾𝔖(𝒾)𝑑𝒾
= (𝐼𝑅) = (𝐼𝑅)
𝔖 (𝒾)𝑑𝒾
𝔖 =(𝒾)𝑑𝒾𝔖(𝒾,
= 𝜑)𝑑𝒾
𝔖(𝒾,
: 𝔖(𝒾,
𝜑)𝑑𝒾𝜑): 𝔖(𝒾,
∈ ℛ𝜑)
, , ],(8)) , (8)
([ , ∈], ℛ
(8)
(𝐹𝑅) 𝔖(𝒾)𝑑𝒾 = (𝐼𝑅) 2µυ𝔖 (𝒾)𝑑𝒾
= 2µυ𝔖(𝒾,𝜑)𝑑𝒾 : 𝔖(𝒾,
𝜑))∈([ ℛ
([ , ], ) ,
Q∗ 𝜑)𝑑𝒾
,: ϕ𝔖(𝒾, 𝜑) ∈ ℛ([ , ], ) , (8)
(𝐹𝑅) 𝔖(𝒾)𝑑𝒾 = (𝐼𝑅) 𝔖S(𝒾)𝑑𝒾
∗ µ+υ=, ϕ ×𝔖(𝒾,
µ
+
υ
2µυ of Riemannian
for all for
𝜑 ∈all
(0, 𝜑
1],∈ where
(0, 1], where
ℛ , ], ℛ
denotes
collection
the collection
integrable
integrable
func- func∗ Riemannian
∗the2µυ
) ([denotes
, ], ) S
Qof
for all 𝜑 ∈ (0, 1], ([where
ℛ
denotes
the
collection
+υ , ϕ ×
µ+υ , ϕ of Riemannian integrable func([ , ], ) µ
(𝒾)𝑑𝒾𝔖∈(𝒾)𝑑𝒾
tions
oftions
I-V-Fs.
𝔖 where
is 𝐹𝑅
𝔖 -integrable
isℛ 𝐹𝑅 -integrable
over [𝜇,
over
𝜈]collection
if[𝜇,(𝐹𝑅)
𝜈] if of(𝐹𝑅)
𝔖Riemannian
𝔽 . integrable
Note
∈ 𝔽 . that,
Note
ifthat, if
for all
𝜑 ∈ of
(0, I-V-Fs.
1],
the
func) denotes
tions of
I-V-Fs. 𝔖 ([ is, ],𝐹𝑅
over [𝜇, 𝜈] if (𝐹𝑅) 𝔖(𝒾)𝑑𝒾
 ∈ 𝔽 . Note that, if
 -integrable
∗ (𝒾,
∗ (𝒾,
(𝒾, 𝜑),
(𝒾,
(𝐹𝑅)
(𝒾)𝑑𝒾
𝔖
𝜑),
𝜑)
𝔖
are
𝜑)
Lebesgue-integrable,
are
Lebesgue-integrable,
then
𝔖
then
is
a
𝔖
fuzzy
is
a
Aumann-integrable
fuzzy
Aumann-integrable
func𝔖∗tions
𝔖of
I-V-Fs.
𝔖
is
𝐹𝑅
-integrable
over
[𝜇,
𝜈]
if
𝔖
∈
𝔽
.
Note
that,
iffuncµυ
µυ
∗
S∗ ξµ+(1−ξ )υ , ϕthen
× Q𝔖∗ isξµa+(fuzzy
, Aumann-integrable
ϕ
func𝔖∗∗(𝒾, 𝜑), 𝔖∗ (𝒾, 𝜑) are Lebesgue-integrable,
1
−
ξ
υ
)
2s
tion
𝜈].
[𝜇, are
𝜈]. Lebesgue-integrable,
(𝒾,tion
(𝒾,over
𝜑),[𝜇,
𝔖over
𝜑)
func𝔖∗over
then 𝔖 isa fuzzy
Aumann-integrable

≤2 
tion
[𝜇, 𝜈].
µυ
+S∗ ξµ+(µυ
, ϕ × Q∗ ξµ+(1−ξ )υ , ϕ
tion over [𝜇, 𝜈].
1− ξ ) υ
𝜑-cuts
define
theoffamily
→a𝔽F-NV-F
Theorem
Theorem
2. [49] Let
2. [49]
𝔖: [𝜇,
Let𝜈]𝔖⊂
: [𝜇,
ℝ 𝜈]
→⊂
𝔽 ℝbe
whose
whose
𝜑-cuts
the
define
family
I-V-Fsof I-V-Fs
µυbe a F-NV-F
µυ
Theorem 2. [49] Let 𝔖: [𝜇, 𝜈]
𝔽 be
𝜑-cuts
S∗⊂ (ℝ1−→
, ϕa ∗F-NV-F
× Q∗ ∗ whose
, ϕ define the family of I-V-Fs
ξ
µ
+
ξυ
ξµ
+(
1
−
ξ
υ
)
)
∈,and
be[𝔖
for∗𝜑)]
2. 𝜈]
[49]
:given
[𝜇,are
𝜈]+
⊂
ℝ
𝔽=
a F-NV-F
whose
the𝜈]
of
(𝒾)
(𝒾,
[𝔖 𝔖(𝒾,
(𝒾,
(𝒾,
=𝜑),
𝜑),
𝜑)]
𝔖𝜑-cuts
alldefine
𝒾for∈ all
[𝜇,
𝒾family
[𝜇, 𝜈]
forI-V-Fs
and
all for all
: [𝜇, 𝜈]
: [𝜇,
ℝ
→
⊂
𝒦Let
ℝ are
→𝔖𝒦
by
given
𝔖→(𝒾)
by
𝔖
𝔖 Theorem
𝔖⊂
22s
(𝒾,Q
𝜑),
𝔖 (𝒾,µυ𝜑)] for
all 𝒾 ∈ [𝜇, 𝜈] and for all
by 𝔖∗ µυ(𝒾) =,∗ [𝔖
𝔖 : [𝜇, 𝜈] ⊂ ℝ → 𝒦 are given
∗∗×
,
ϕ
+S(𝒾)
ϕ
∗ ξµ+(
∗
∗ (𝒾, for all
1𝜑)
−𝔖
ξ )all
µ(𝒾,
+ξυ
−(𝒾,
ξ )𝜈]
υ𝜑),
(for
=𝜈][𝔖if1∗[𝜇,
𝜑)]
𝒾𝜑)
∈∗[𝜇,
: [𝜇,
⊂ℝ
𝒦
are
given
by 
𝔖 [𝜇,
(𝒾,
and
𝔖
and
𝜑)𝔖and
both𝜑)
areboth are
𝜑𝔖
∈ (0,
𝜑
1].𝜈]
∈ Then
(0,
1].→
𝔖Then
is 𝐹𝑅-integrable
𝔖 is
𝐹𝑅-integrable
over
and
only
if𝔖
if(𝒾,𝔖
only
if
∗ (𝒾,
over
and
only
𝜈]

and 𝔖∗ (𝒾, 𝜑) both are
𝜑 ∈ (0, 1]. Then 𝔖 is 𝐹𝑅-integrable
over [𝜇, 𝜈]
if and
if∗ 𝔖∗ (𝒾, 𝜑)
µυ
µυ
∗
∗
∗
S
, ϕonly
×Q
, ϕ𝔖 (𝒾, 𝜑) both are
(𝒾,+(
𝜑)
𝜑 ∈ (0,
1].over
Then[𝜇,𝔖
𝐹𝑅-integrable
[𝜇,+(
𝜈]1−𝐹𝑅-integrable
if
if[𝜇,over
𝔖𝜈],
[𝜇,
𝑅-integrable
𝑅-integrable
over
𝜈].isMoreover,
[𝜇,
𝜈]. Moreover,
if 
𝔖 over
is if𝐹𝑅-integrable
𝔖
is
then
∗ξµ
ξ )υand over
1−𝜈],
ξand
)υ then
ξµ

[𝜇,
𝑅-integrable over [𝜇,≤
𝜈].22s
Moreover,
if 𝔖
is 𝐹𝑅-integrable
over
𝜈], then
µυ
µυ
∗ 𝐹𝑅-integrable
∗ 𝜈], then , ϕ
[𝜇,
𝑅-integrable over [𝜇, 𝜈]. Moreover, if +
𝔖Sis
over
,
ϕ
×
Q
ξµ+(1−ξ )υ
ξµ+(1−ξ )υ

 (𝑅)
(𝑅) (𝑅)
(𝑅)
(𝐹𝑅) (𝐹𝑅)
= ∗ = µυ
𝔖
𝔖∗ (𝒾,
, 𝜑)𝑑𝒾
, 𝔖∗µυ(𝒾, 𝜑)𝑑𝒾
𝔖∗ (𝒾, ∗𝜑)𝑑𝒾
𝔖(𝒾)𝑑𝒾𝔖(𝒾)𝑑𝒾
∗ (𝒾, 𝜑)𝑑𝒾
∗
,
ϕ
, ϕ ×
(𝒾, 𝜑)𝑑𝒾
(𝑅)
(𝑅)
(𝐹𝑅) 2s 𝔖S
(𝒾)𝑑𝒾(1−ξ )=µ+ξυ
𝔖∗Q
,
𝔖 (𝒾, 𝜑)𝑑𝒾
1−∗ ξ )υ

ξµ+(𝔖
+2  = (𝑅)
(𝐹𝑅) 𝔖(𝒾)𝑑𝒾
(9)
(9)
µυ𝔖∗ (𝒾, 𝜑)𝑑𝒾 , (𝑅)
µυ(𝒾, 𝜑)𝑑𝒾 ,
∗
∗
+S ξµ+(1−ξ )υ , ϕ × Q (1−ξ )µ+ξυ , ϕ
(9)
(9)
 = (𝐼𝑅)
= (𝐼𝑅)

𝔖 (𝒾)𝑑𝒾
𝔖 (𝒾)𝑑𝒾
µυ
µυ
S∗ ξµ+(1−=ξ )(𝐼𝑅)
, ϕ ×𝔖Q(𝒾)𝑑𝒾
∗ ξµ+(1−ξ )υ , ϕ

𝔖υ (𝒾)𝑑𝒾
≤ 22s  = (𝐼𝑅)
µυ
µυ
,
ϕ
+
S
,
ϕ
×
Q
∗
∗
(0, 𝜑
(0, ([
for all 𝜑
for∈all
(0, 𝜑
1].∈ For
(0, 1].
all For
𝜑 ∈all
1],∈ ℱℛ
1], ℱℛ
the collection
the ξµ
collection
of all 𝐹𝑅-integrable
FF+(of1−all
ξ )υ 𝐹𝑅-integrable
(1−ξdenotes
)µ, +],ξυ) denotes
for all 𝜑 ∈ (0, 1]. For all 𝜑 
∈ (0,, s],1],) ([ℱℛ
 of all 𝐹𝑅-integrable F([ , ], ) denotes
s the collection
NV-Fs
over
NV-Fs
[𝜇,
over
𝜈].
[𝜇,
𝜈].
ϕ) + (1the
− collection
ξ ) S∗ (υ, of
ϕ)all 𝐹𝑅-integrable Ffor all 𝜑 NV-Fs
∈ (0, 1].
For[𝜇,
all𝜈].𝜑 ∈ (0, 1], ℱℛξ([ S
, ],∗ ()µ,denotes
over
× (1 − ξ )s Q∗ (µ, ϕ) + ξ s Q∗ (υ, ϕ) 
NV-Fs over [𝜇, 𝜈].
2s

,
s Sbe

(0,
[𝜇,𝐾𝜐]+
∞)
=
said
to
isϕ)said
be+aξconvex
to
a convex
set,
if, for
set,all
if, 𝒾,
for𝒿 all
∈ 𝒾, 𝒿 ∈
Definition
Definition
3. [34] A
3.set
[34]𝐾A=set
=⊂2[𝜇,
ℝ𝜐]
1ℝ−
ξℝ)sis
S
µ, ∞)
ϕa) convex
+=⊂𝜐]
((0,
((0,
(υ,
∗∞)
∗to
=
is
said
be
set, if, for all 𝒾, 𝒿 ∈
Definition 3. [34] A set 𝐾 = [𝜇,
⊂
s
s
[0, have
𝐾,Definition
𝜉 ∈ [0,
𝐾, 1],
𝜉 ∈we
1], we
have
ξ Q
µ, ϕis)said
+ (1to
−be
ξ ) aQconvex
(0,∗ (∞)
A set
= [𝜇, 𝜐] ⊂ ℝ× =
∗ ( υ, ϕ )set, if, for all 𝒾, 𝒿 ∈
[0, 1],
𝐾, 𝜉3.∈[34]
we 𝐾have


µυ
µυ
𝐾, 𝜉 ∈ [0, 1], we have
𝒾𝒿
𝒾𝒿, ϕ × Q∗
,
ϕ
S∗ ξµ
𝒾𝒿 ∈ 𝐾. ξµ+(1−ξ )υ
+(1−ξ )υ∈ 𝐾.
(10) (10)
𝜉𝒾

≤ 22s 𝜉𝒾 + (1
−+
𝜉)𝒿(1 − 𝜉)𝒿
(10)
𝒾𝒿
∗∈ 𝐾. µυ
(1
𝜉𝒾
+
−
𝜉)𝒿
+S∗ ξµ+(µυ
ϕ
×
Q
,
ϕ
(10)
1−ξ )υ ∈ 𝐾.
ξµ+(1−ξ )υ
(1
𝜉𝒾
+
−
𝜉)𝒿

 s ∗
s ∗
µ,harmonically
ϕ) +
S (υ, convex
ϕ) function
(1 − ξ )convex
[𝜇, 𝜐]𝔖:→[𝜇,
Definition
Definition
4. [34] The
4. [34]
𝔖:The
ℝ 𝜐] is
→called
ℝξ Sis(acalled
a
harmonically
function
on [𝜇, 𝜐]onif[𝜇, 𝜐] if
∗ ( µ, ϕ
×

[𝜇,
Definition 4. [34] The 𝔖:2s
𝜐] →
called
a )harmonically
function on [𝜇, 𝜐] if
− ξ )iss Q
+ ξ s Q∗ (υ, ϕ)convex
(1 ℝ


2ℝ  is𝒾𝒿
Definition 4. [34] The 𝔖: [𝜇, 𝜐] +
→𝒾𝒿
called as harmonically
on [𝜇, 𝜐] if
∗ ( µ, ϕ ) + ξ sconvex
∗ ( υ, ϕfunction
,
+
1
−
ξ
S
S
(
)
)
(1 − 𝜉)𝔖(𝒾)
+ 𝜉𝔖(𝒿),
+∗ 𝜉𝔖(𝒿),
𝔖
𝔖
≤𝒾𝒿(1
−≤
𝜉)𝔖(𝒾)
(11) (11)
s
s
∗
(1ξ −Q𝜉)𝒿
𝜉𝒾 + (1𝔖
𝜉𝒾
−+
𝜉)𝒿
≤ ((1
−ξ𝜉)𝔖(𝒾)
(11)
𝒾𝒿
×
µ, ϕ) +
1−
Q
υ,+ ϕ𝜉𝔖(𝒿),
(
)
(
)
(1
+𝜉𝔖(𝒿),
𝔖
− 𝜉)𝔖(𝒾)
(11)

 𝜉𝒾
+ ≤−(1𝜉)𝒿
µυ ≥all
µυ is
(1 − 𝜉)𝒿
[0,𝜉𝒾
[𝜇,Q𝜐].
for all 𝒾,for
𝒿 ∈all[𝜇,
𝒾, 𝜐],
𝒿 ∈𝜉 [𝜇,
∈ [0,
𝜐], 1],
𝜉 ∈ where
1],+where
𝔖(𝒾)
𝔖(𝒾)
0 for
0, for
𝒾ϕ ∈ all
𝒾∗∈ If[𝜇,(11)
𝜐].
If reversed
(11)
then, 𝔖then, 𝔖
×
, ϕ is reversed
∗ ≥ξµ
+(𝔖(𝒾)
1−ξ )υ ≥ 0 for all
ξµ
+([𝜇,
1−ξ𝜐].
)υ If (11)
for all 𝒾, 𝒿 ∈ [𝜇, 𝜐], 𝜉 =
∈ 2[0,
1],Swhere
𝒾∈
is reversed then, 𝔖
2s 

[𝜇,
[𝜇,
is for
called
acalled
harmonically
a 𝜐],
harmonically
concave
concave
function
function
on 0µυfor
𝜐].
on
𝜐].
µυ is reversed then, 𝔖
[𝜇,
allis𝒾,
𝒿is∈called
∈ [0,
1], where
∈×[𝜇,
a𝜉harmonically
concave
functionall
on
𝜐].
+𝔖(𝒾)
S∗ ≥
, ϕ𝒾 [𝜇,
Q∗𝜐]. ξµIf+((11)
, ϕ
ξµ+(1−ξ )υ
1− ξ ) υ
[𝜇,
is called a harmonically concave function
on
𝜐].
s s
ξ · ξ + (1 − ξ )s (1 − ξ )s ∂∗ ((µ, υ), ϕ)
+22s s
,
+ ξ (1 − ξ )s + (1 − ξ )s ξ s Λ∗ ((µ, υ), ϕ)


µυ
µυ
S∗ ξµ+(1−ξ )υ , ϕ × Q∗ ξµ+(1−ξ )υ , ϕ

= 22s 
µυ
∗
+S∗ ξµ+(µυ
,
ϕ
×
Q
,
ϕ
1− ξ ) υ
ξµ+(1−ξ )υ
s s
s
s
ξ · ξ + (1 − ξ ) (1 − ξ ) ∂∗ ((µ, υ), ϕ)
+22s s
,
+ ξ (1 − ξ )s + (1 − ξ )s ξ s Λ∗ ((µ, υ), ϕ)
Symmetry 2022, 14, 1639
Symmetry
Symmetry
2022, 14,2022,
x FOR
14,PEER
x FORREVIEW
PEER REVIEW
Symmetry 2022, 14, x FOR PEER REVIEW
Symmetry 2022, 14, x FOR PEER REVIEW
We now useWe
mathematical
now
Weuse
now
mathematical
use
operations
mathematical
to
operations
examine
operations
to
some
examine
to
of examine
thesome
charac
o
s
PropositionProposition
1. [51]
Proposition
Let 𝔵,1.𝔴[51]
∈𝔽
1.Let
[51]
𝔵,Let
𝔴 fuzzy
∈ 𝔵,𝔽𝔴. Then
∈order
𝔽 . fuzzy
. Then
Then
relation
fuzzy
order
“ ≼order
relation
” is given
relat
“
numbers. Let
numbers.
𝔵, 𝔴numbers.
∈ 𝔽Letand
𝔵, Let
𝔴𝜌∈∈𝔵,𝔽ℝ,
𝔴 and
∈
𝔽 𝜌we
then
and
∈ ℝ,
have
𝜌then
∈ ℝ,we
then
have
we have
[𝔵] ifonly
[𝔵]for
[𝔵][𝔴]
[𝔴]1],
𝔵 ≼ 𝔴 if and
𝔵≼
only
𝔴 𝔵if
if,
≼and
𝔴
and[𝔴]
if,
only
if,≤all
≤
𝜑≤∈ (0,
for
allf
[𝔵+𝔴] = [𝔵]
[𝔵+𝔴]
[𝔴]=𝔴]
[𝔴] +, [𝔴]
+ [𝔵+
,[𝔵] =+[𝔵]
It is a partial
It order
is a It
partial
relation.
is a partial
order order
relation.
relation.
[𝔵 × 𝔴] to
[𝔵 ×operations
[𝔵]
[ 𝔴]
[𝔵]
[𝔵]
[ 𝔴]
×
=examine
×
𝔴][𝔵
=𝔴]
, to
=×examine
×, [ 𝔴]
We now useWe
mathematical
now
Weuse
now
mathematical
use
operations
mathematical
operations
to
some
examine
of
the
some
charac
o
so
12 of 18
numbers. Let
numbers.
𝔵, 𝔴numbers.
∈ 𝔽Letand
𝔵, Let
𝔴𝜌∈∈𝔵,𝔽ℝ,
𝔴 and
∈
𝔽[𝜌.𝜌we
then
and
∈
ℝ,
have
𝜌
then
∈
ℝ,
we
then
have
we
have
[𝜌.
[𝔵] 𝔵]4 of
𝔵] = 𝜌. [𝜌.
𝔵]𝜌.
=19
4 [𝔵]
of=19𝜌. [𝔵]
4 of 19
4 ,[𝔵]
of 19
[𝔵+𝔴] = [𝔵]
[𝔵+𝔴]
[𝔴]=𝔴]
[𝔴] +, [𝔴]
+ [𝔵+
=+[𝔵]
1. [49]Definition
A fuzzy
1. [49]
number
1.A[49]
fuzzy
valued
A fuzzy
number
map
number
valued
𝔖: 𝐾 valued
⊂map
ℝ →𝔖𝔽
map
: 𝐾is⊂𝔖called
ℝ
: 𝐾→⊂𝔽F
ℝ
Integrating over [0, Definition
1], we haveDefinition
[𝔵 × 𝔴] = [𝔵]
[𝔵 × ×
[ 𝔴]
[𝔵]
[ 𝔴]×, [ 𝔴]
×=𝔴]
𝔴][𝔵
, =×[𝔵]
∈. Then
(0,
whose
𝜑 ∈ (0,
𝜑∈ -cuts
(0,
whose
1] define
whose
𝜑“ ≼
-cuts
the
𝜑
-cuts
define
family
define
the
of𝔽I-V-Fs
family
the
family
𝔖
of
of ℝI-V-Fs
𝔖 𝒦:
: I-V-Fs
𝐾⊂
→
Proposition
Proposition
1. [51]1.
Let
[51]
𝔴 ∈𝜑
𝔵,
𝔽𝔴
∈ 𝔽1]. fuzzy
Then
fuzzy
order
relation
order
relation
”
is
“
≼
given
”
is
given
on
on
by
𝔽
by
𝔵, Let
𝜑1]
R υrelation
Proposition
1. [51]
order
“𝔵]
≼ all
” for
is𝜌.
on
𝔽𝜌.
by
2µυ Let 𝔵, 𝔴 ∈ 𝔽
2µυ. Then fuzzy
∗[𝔖
∗ (𝒾,
1 (𝒾,
2s−1
[𝜌.
[𝜌.
[𝜌.
[𝔵]
[𝔵]
[𝔵]
(𝒾)
(𝒾,
(𝒾,
=𝐾.𝔵]
=(0,
𝜌.
𝔖
𝔖
=∗ [𝔖
𝜑),𝔖=
𝔖(𝒾)
=
𝜑)]
𝜑),
𝔖
𝜑),
all
𝔖
𝒾” ,∗∈
for
𝜑)]
𝒾given
all
for
Here,
∈
𝜑Here,
for
∈
each
1]
for the
ea
𝜑
×. Q
≤
([𝔖
R
)for
ϕ)𝐾.
× Here,
Q=
,∈
ϕ
d𝔵]𝒾each
∗ µ𝔵,
∗ (on
∗ (𝜑)]
∗ (𝒾,
∗ (𝒾,
Proposition 21. [51]SLet
∈ 𝔽(𝒾)
Then
fuzzy
“≼
is
given
𝔽)𝐾.
by
+υ𝔴, ϕ
µ+
υ , ϕ order
υ−∗µrelation
µ S
[𝔵] if,≤[𝔵][𝔴]
[𝔴]
𝔵 ≼ 𝔴 𝔵if≼and
𝔴 ifonly
andif,only
≤
for
all
for
𝜑
∈
all
(0,
𝜑
1],
∈
(0,
1],
(4)
(4)
∗
∗
∗
R
(.
(.
(.
(.
(.
(.
functions
𝔖
functions
functions
𝔖
𝔖
,
𝜑),
𝔖
,
𝜑):
,
𝜑),
𝐾
→
𝔖
,
ℝ
𝜑),
,
are
𝜑):
𝔖
called
𝐾
,
→
𝜑):
ℝ
𝐾
lower
are
→
ℝ
called
and
are
called
upper
lower
lower
functions
and
uppe
and
𝔵 ≼ 𝔴 if and∗ only if, [𝔵]∗ ≤ 1∗[𝔴]
for all
𝜑 ∈ (0, 1],
(4)
s
s (1 −
+Λ[𝔵]
, ϕ) for
ξ )(0,
dξ1],
𝔵 ≼ 𝔴 if and only if,
≤ υ)[𝔴]
𝜑∈
(4)
∗ ((µ,
0 ξall
It is a partial
It is a partial
order order
relation.
relation.
Rnumber
Definition Definition
1. [49]Definition
A fuzzy
1. [49]
As[49]
fuzzy
valued
A fuzzy
number
map
number
valued
𝔖: 𝐾 valued
⊂map
ℝ →𝔖𝔽
map
: 𝐾is⊂𝔖called
ℝ
: 𝐾→⊂𝔽F
ℝ
11.
It is a partial order relation.
s𝔖
an
F-NV-F.
be
Then
be
F-NV-F.
an
fuzzy
F-NV-F
The
int
Definition
Definition
2.∂∈
Definition
Let
2.
𝔖
[𝜇,
𝜈]
2.
Let
[49]
⊂
ℝ
Let
→
:of
[𝜇,
𝔽the
𝜈]
𝔖:be
⊂
[𝜇,
ℝ
𝜈]
→⊂
𝔽ℝ
𝔽an
+
µ,1]
υ-cuts
,whose
ϕ1]
ξ𝜑
·-cuts
ξthe
dξ,
((
)(0,
): [49]
∗(0,
now
Weuse
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to[49]
examine
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some
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characteristics
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fuzzy
It isWe
a partial
order
relation.
0whose
𝜑
∈
(0,
1]
whose
𝜑
𝜑
𝜑
∈
define
the
𝜑
-cuts
define
family
define
the
of
I-V-Fs
family
the
family
𝔖
of
I-V-Fs
of
I-V-Fs
𝔖
:
𝐾
⊂
ℝ
→
𝒦
:
We now use mathematical
operations
to examine
some
of the(𝒾)𝑑𝒾
characteristics of fuzzy
R (𝐹𝑅)
[𝜇,
[𝜇,
[𝜇,
(𝐹𝑅)
(𝒾)𝑑𝒾
(𝐹𝑅)
(𝒾)𝑑𝒾
υ∗ of
𝜈],
denoted
𝜈],
by
denoted
𝜈],
denoted
by
𝔖for
by
𝔖
𝔖∗,for
is
levelwise
is𝒾 given
, is𝒾by
given
levelwise
levelwise
by
by
2µυ
2µυ
numbers.
numbers.
Let
Let
𝔴
𝔵,
𝔽
∈xREVIEW
𝔽2022,
𝜌
∈(𝒾)
ℝ,
𝜌REVIEW
then
∈[𝔖
ℝ,
we
then
have
have
∗[𝔖
1 (𝒾,
Symmetry
2022,
Symmetry
14,2𝔵,2s
xuse
FOR
Symmetry
PEER
14,and
FOR
PEER
14,
x
PEER
REVIEW
∗FOR
∗, the
−2022,
1∈mathematical
∗𝔴
We
now
operations
to
examine
some
characteristics
of
(𝒾,
(𝒾,
𝔖and
𝔖
𝔖we
=
=
𝔖(𝒾)
𝜑)]
𝜑),
𝔖
𝜑),
all
𝜑)]
𝔖
𝒾 ∗given
∈(𝒾,
𝐾.
for
𝜑)]
Here,
all
∈all
for
𝐾.
each
Here,
∈fuzzy
𝐾. 𝜑Here,
for
∈ (0,
each
1]
for the
ea
𝜑
Q
, (𝒾)
ϕ𝜑),
≤
([𝔖
R
)∗ (𝒾,
S
∗∈(𝒾,
numbers.
Let 𝔵,
𝜌µ+
then
have
µ+𝔴
υ ,∈ϕ𝔽 ×and
υ ℝ,
υwe
−∗µ=
µ S ( , ϕ) × Q ( , ϕ)d
∗ (. (. R (. ∗ (.
∗ (.
numbers. Let 𝔵, 𝔴 ∈ 𝔽 andfunctions
𝜌 ∈ ℝ, then
we
have
(.
𝔖
functions
functions
𝔖
𝔖
,
𝜑),
𝔖
,
𝜑):
,
𝜑),
𝐾
→
𝔖
,
ℝ
𝜑),
,
are
𝜑):
𝔖
called
𝐾
,
→
𝜑):
ℝ
𝐾
lower
are
→
ℝ
called
and
are
called
upper
lower
lower
functions
and
uppe
and
∗PEER
∗ 4 of
∗ s ,
(5)
(5)
[𝔵+
[𝔵+
[𝔵]µ,=
[𝔴]
𝔴]+
+, 119
∗ ((
REVIEW
Symmetry 2022, 14,
x FOR
Λ=𝔴]
υ+)[𝔵]
,[𝔴]
ϕ)[𝔵]
ξ+([𝔴]
1 − ξ, )s dξ
(5)
[𝔵+REVIEW
𝔴]
=
0 𝔖(𝐼𝑅)
(𝒾)𝑑𝒾
(𝒾)𝑑𝒾
(𝒾)𝑑𝒾
(𝐹𝑅)
(𝒾)𝑑𝒾
(𝐹𝑅)
(𝐹𝑅)
(𝒾)𝑑𝒾
(𝒾)𝑑𝒾
(𝐼𝑅)
(𝐼𝑅)
𝔖
𝔖
=
𝔖
=
=
=
𝔖
𝔖(𝒾,
𝔖
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1.
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It
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It
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=→
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→
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is
called
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called
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Then
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have
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1]
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define
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family
of
I-V-Fs
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:
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⊂
ℝ
→
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are
given
by
Proposition
1. [51]
1.
[51]
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𝔴𝜑)]
𝔵,𝔖
∈
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.R𝜑)]
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eThen
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1.∈[51]
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→
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are
called
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𝔖
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𝜑),
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𝜑),
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𝜑)
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Itaispartial
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𝔴]
×
,
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and
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Moreover,
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Moreover,
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𝔖over
is [𝔵]
𝐹𝑅-integrable
then, [𝜇
o
numbers.
numbers.
Let Let
𝔵,𝜈],
𝔴𝔵,∈denoted
𝔴𝔽Fejér
∈𝔽
and
and
𝜌 ∈𝑅-integrable
𝜌ℝ,
∈ then
ℝ,
we𝑅-integrable
we
have
have
=
×whose
𝔴]
[𝜇, 𝔵,
(𝒾)𝑑𝒾
𝜈], denoted
by (𝐹𝑅)
𝔖
, is 2.
given
by
Theorem
Theorem
[49] levelwise
Let
Theorem
𝔖2.: [𝜇,
[49]
𝜈]
2.Let
⊂
[49]
ℝ𝔖→
:Let
[𝜇,
𝔽 𝜈]
𝔖:be
⊂
[𝜇,aℝ𝜈]
→⊂𝔽ℝ 𝔴]
→
𝔽a F-NV-F
F-NV-F
be
whose
be𝜑-cuts
a F-NV-F
define
who
𝜑-c
th
numbers. Let
𝔽𝔵] and
∈ ℝ,we
then
wedevelop.
have
is 𝔴
the∈[𝜌.
first
inequality
(7)
[𝜌. 𝔵] = 𝜌. [𝔵
= 𝜌.𝜌 [𝔵]
(5)
[𝔵+will
𝔴]
= [𝔵] +[𝔴] ,
∗[𝔖
∗called
(5)
(5)
[𝔵]
[𝔴]
𝔴]
𝔴]
=
+
+
, →⊂
,Aℝ
Definition
Definition
1.
[49]
ADefinition
fuzzy
1.=
[49]
number
fuzzy
1.
[49]
valued
number
A
fuzzy
map
valued
number
𝔖
:],[𝜌.
𝐾
map
⊂
ℝ
valued
→
𝔖
:,[𝔖
𝔽𝜑),
𝐾[𝔵]
⊂
map
→
𝔖𝜑)]
:𝔽𝐾
is
(8)
(𝒾)
(𝒾)
(𝒾,
(𝒾,
(𝒾)𝑑𝒾
(𝒾)𝑑𝒾
(𝐹𝑅) (𝐹𝑅)
(𝒾)𝑑𝒾
(𝐼𝑅)
(𝐼𝑅)
=
𝔖
=
𝜑)]
for
𝔖ℝ
all
𝜑),
𝒾𝔖
∈∗ (F
:[𝔵+
[𝜇,
𝜈]𝔖⊂
ℝ [𝔵]
→
:=
[𝜇,
𝒦
𝜈]
:[𝔴]
are
⊂
[𝜇,
ℝ
𝜈]
given
𝒦
by
→
are
𝒦
given
are
given
by
𝔖
𝔖=
𝔖
𝔖
𝔖(𝒾)𝑑𝒾𝔖[𝔵+
=𝔖
= [𝔵+
𝔖
=
𝔖(𝒾,
𝜑)𝑑𝒾
𝔖(𝒾,
:𝔖
𝜑)𝑑𝒾
𝔖(𝒾,
𝜑)
: 𝔖(𝒾,
∈[𝔖
ℛ
𝜑)
∈
ℛ
𝔵],(𝒾)
=
([ by
)([
],∗(8)
) 𝜌.
∗ (𝒾,
∗ (𝒾,
, 𝜑),
,(𝒾,
∗(8)
(5)
𝔴]
==[𝔵]
+×[𝔴]
, , (𝒾)𝑑𝒾𝔖e=
(𝐹𝑅) [𝔵𝔖×
(𝒾)𝑑𝒾
(𝐼𝑅)
=
𝔖
𝔖(𝒾,
𝜑)𝑑𝒾
: 𝔖(𝒾,
𝜑)
∈
ℛ
,⊂
(𝒾,
(𝒾,
(𝒾,
(𝒾,
(𝑅)
(𝑅)
(𝑅)
(𝑅)
(𝑅)
(𝐹𝑅)
(𝒾)𝑑𝒾
(𝐹𝑅)
(𝐹𝑅)
(𝒾)𝑑𝒾
(𝒾)𝑑𝒾
=
𝔖
=
𝜑)𝑑𝒾
=
,
𝔖
𝜑)𝑑𝒾
𝔖
𝔖
,
𝜑)𝑑𝒾
𝜑)
𝔖
𝔖
([
],I-V-Fs
)𝐾
,
(6)
[𝔵]
[
𝔴]
𝔴]
∗
∗
∗
𝜑
∈
(0,
1]
𝜑
whose
∈
(0,
1]
𝜑
𝜑
whose
-cuts
∈
(0,
define
1]
𝜑
whose
-cuts
the
define
family
𝜑
-cuts
the
of
define
family
I-V-Fs
the
of
𝔖
family
of
𝔖
I-V
:
ℝ
→
:
𝒦
𝐾
Theorem
8. (Second
fuzzy𝜑
inequality)
Let
∈
HFSX
υ]𝜈]
,∈ Fℛ
,([and
s), ,],[𝜇,
whose
(8)
(𝒾)𝑑𝒾
(𝐹𝑅)
𝔖(𝒾)𝑑𝒾
=H.H.
=
𝔖(𝒾,
𝔖(𝒾,
𝜑)
, if[𝜇,
) (6)
(6)
[𝔵(0,
[𝔵Fejér
[𝔵]
[𝔵]
[1].
×
×𝔖
𝔴]
𝔴]
=
=(0,
×
𝔴]
𝔴]
,S
, 𝜑)𝑑𝒾
𝜑)
and
∈(𝐼𝑅)
1].
Then
𝜑
∈
𝔖
𝜑×1].
∈
is[∗ (0,
𝐹𝑅-integrable
Then
𝔖
Then
is
𝐹𝑅-integrable
𝔖:[49]
over
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𝐹𝑅-integrable
[𝜇,
if0over
only
over
𝜈]
ifϕ-cuts
𝔖and
𝜈]
ifmap
andif
only
𝔖
∗ (𝒾,only
∗
Definition 1. [49] A fuzzy numberdefine
valued
map
𝔖
:
𝐾
⊂
ℝ
→
𝔽
Definition
1.
A
fuzzy
number
valued
𝔖
:
𝐾
is
called
F-NV-F.
For
each
+
∗
∗
∗
(𝒾)
(𝒾,
(𝒾)
[𝔖
(𝒾)
[𝔖
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𝔖(𝒾,
=
𝜑),
=
𝜑)]
𝜑),
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𝔖
for
all𝜑)]
𝜑),
for
all
Here,
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∈ϕfor
for
𝐾.
all
Here,
each
for
𝐾.
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each
(0, 1] 𝜑
for
the
∈
=𝐾.(𝒾,
the
of1],
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S
µ,
⊂𝔖
Kcollection
are
given
byRiemannian
, S
forHere,
[𝔵ℛ×
[𝔵]
[→
)∈𝔖
[S
[𝔵]
]], [𝔖
( 𝒾Riemannian
)integrable
( 𝒾 ,∈ϕ𝜑
)]
𝔴]
=υ
×R
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∗ ( 𝒾 ,(6)
ϕof
∗ (𝒾,
∗S
for all for
𝜑 ∈all
(0,family
𝜑1],
∈ (0,
where
where
denotes
the
collection
integrable
funcfuncC the
(7)
[𝔵]
([ ϕ,[𝜌.
)([ denotes
)𝜌.∗Definition
, =
!
1.
[49]
A∗𝜈].
fuzzy
number
valued
map
𝔖family
: 𝜈],
𝐾 func⊂over
ℝof→[𝜇
𝔽
[𝜇,
𝑅-integrable
[𝜇,
𝑅-integrable
𝜈].
Moreover,
over
[𝜇,
over
if(., [𝜇,
Moreover,
𝜈].
is
Moreover,
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𝔖
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𝔖
over
is
𝐹𝑅-integrable
then
ov
𝜑 ∈ (0, 1] whose 𝜑 -cuts define the for
family
𝔖𝑅-integrable
𝜑
1]
𝜑
the
I: 𝐾 ⊂[𝜌.
→
𝒦
are
given
by
all 𝜑of∈ I-V-Fs
(0, 1], where
ℛℝ
collection
of
integrable
∗[𝔵]
∗Riemannian
([𝔵]
],over
) denotes
(7)
(7)
,𝔖
[𝜌.
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𝔵]=
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(.
(.∗whose
(.-cuts
functions
functions
𝔖
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𝜑),
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𝜑):
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are
𝜑),
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𝔖
called
→
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𝜑):
lower
𝐾define
called
ℝ
are
lower
upper
called
and
functions
upper
a
∗ (. , [𝜇,
(𝐹𝑅)
(𝐹𝑅)
(𝒾)𝑑𝒾
(𝒾)𝑑𝒾
for all
𝜑
∈of(0,I-V-Fs.
1],
ℛfunctions
the
collection
of
Riemannian
integrable
functions
of
tions
I-V-Fs.
𝔖 where
is 𝔖
𝐹𝑅 is
-integrable
𝐹𝑅
over
over
𝜈]
[𝜇,
if∗whose
𝜈]
if𝔖→
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𝔖
∈
.1are
∈Note
𝔽
.(𝐼𝑅)
Note
that,
ifthat,
iflower
([
], ) =denotes
∗𝔽
, -integrable
(7)
[𝔵]
(𝒾)𝑑𝒾
(𝒾)𝑑𝒾
(𝒾)𝑑𝒾
(𝐼𝑅)
(𝐼𝑅)
𝜑
∈
(0,
1]
𝜑
-cuts
define
the
family
of
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𝔖
:
e
𝔵]
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=
𝔖
=
=
𝔖
𝔖
(𝒾)
[𝔖
(𝒾,
(𝒾,
𝔖 (𝒾) = [𝔖∗ (𝒾, 𝜑), 𝔖∗ (𝒾, 𝜑)] for all
𝔖
𝐾.
𝜑
∈
(0,
1]
the
end
point
real
=
𝜑),
𝔖
𝜑)]
for
all
𝒾
∈
𝐾.
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fo
all 𝒾 ∈
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υof
∈for
1]𝔖
. If[𝜌.
and
∇
:
µ,
υ
→
R
,
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S
∈
F
R
=
∇(
≥
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[µ, Here,
], ϕI-V-Fs.
[0, each
[
]
)
(𝐹𝑅)
(𝒾)𝑑𝒾
is 𝐹𝑅 -integrable
over [𝜇, 𝜈] if∗
𝔖
1 + 1 − 1 ∈ 𝔽 . Note that, if
([
µ,
υ
]
,
ϕ
)
∗I-V-Fs.
∗ (𝒾,𝔖 is 𝐹𝑅 -integrable over [𝜇, 𝜈] if (𝐹𝑅) ∗ 𝔖(𝒾)𝑑𝒾 ∈
µ 𝔽
υ . 𝒾Note
tions
of
that,
if
(𝒾,
(𝒾,
(𝒾,
∗
∗
𝜑),
𝔖
𝜑),
𝜑)
𝔖
are
𝜑)
Lebesgue-integrable,
are
Lebesgue-integrable,
then
𝔖
then
is
a
𝔖
fuzzy
is
a
fuzzy
Aumann-integrable
Aumann-integrable
funcfunc𝔖
𝔖
(𝒾)
[𝔖
(𝒾,
(𝒾,
𝔖
=
𝜑),
𝔖
𝜑)]
for
all
∈
𝐾.
Here,
for
each
𝜑
∗
∗
∗
(. , 𝜑): 𝐾 → 1.
(.
(.
∗𝔖
functions 𝔖∗ (. , 𝜑), 𝔖 Definition
functions
𝔖is𝔖
ℝ [49]
are 𝔖
called
lower
and
upper
functions
of
𝔖.→
,→
𝜑),
𝔖⊂𝔖
𝜑):
𝐾(𝑅)
→
ℝan
are
called
A
fuzzy
number
valued
map Definition
𝔖2.
: 𝐾[49]
⊂ (𝐹𝑅)
ℝ
is
called
F-NV-F.
For
∗ lower
∗ℝ
(𝒾,
(𝒾,
be
an
F-NV-F.
be
Then
F-NV-F.
be
an
F-N
int
Definition
2.𝔽ℝ
[49]
:→
[𝜇,
Let
𝜈]
⊂
2.
[49]
:a[𝜇,
𝔽
𝜈]
Let
𝔖
ℝ,:𝔖
[𝜇,
→
𝔽
𝜈]
⊂
ℝ
→
𝔽
𝜑),
𝔖
𝜑)
are
Lebesgue-integrable,
then
𝔖𝔽
fuzzy
Aumann-integrable
func(𝒾,
(𝒾,
(𝒾,
(𝑅)
(𝑅)
(𝑅)
(𝒾)𝑑𝒾
(𝐹𝑅)
(𝒾)𝑑𝒾
(𝒾)𝑑𝒾
=
=∗each
𝜑)𝑑𝒾
=
,(𝑅)
𝔖
𝔖fuzzy
,Then
𝜑)𝑑𝒾
𝜑)𝑑
𝔖
𝔖called
∗1.
∗𝜈].
∗ (𝒾, 𝜑)𝑑𝒾
∗𝔖
Definition
Definition
1.[𝜇,
[49]
[49]
A
A
fuzzy
number
number
valued
valued
map
map
𝔖
: 𝐾Let
𝔖Definition
⊂
:𝐾
⊂
ℝ
𝔽(𝐹𝑅)
→
is
called
is
F-NV-F.
F-NV-F.
For
For
each
each
∗ (.
then
tion
over
tion
over
[𝜇,fuzzy
𝜈].
(𝒾,
(𝒾,
𝜑),
𝔖
𝜑)
are
Lebesgue-integrable,
then
𝔖
is
a𝜑),
fuzzy
Aumann-integrable
func𝔖whose
(.
functions
𝔖
,
𝔖
,
𝜑):
𝐾
→
ℝ
are
called
lower
and
uppe
∗
𝜑
∈
(0,
1]
𝜑
-cuts
define
the
family
of
I-V-Fs
𝔖
:
𝐾
⊂
ℝ
→
𝒦
are
given
by
∗
(0,
(0,
(0,
for
all
𝜑
∈
(0,
for
1].
all
for
For
𝜑
∈
all
all
(0,
𝜑
𝜑
1].
∈
∈
(0,
For
1].
1],
all
For
ℱℛ
𝜑
∈
all
𝜑
1],
∈
ℱℛ
1],
ℱℛ
denotes
the
collection
denotes
denotes
the
of
coll
allt
Definition
1.∈[49]
Ation
fuzzy
map
𝔖[𝜇,
: 𝐾𝜈],
⊂by
ℝ
→
𝔽
called
[𝜇,the
(𝐹𝑅)
[𝜇,
(𝐹𝑅)
(𝐹𝑅)
(𝒾)𝑑𝒾
over
[𝜇,-cuts
𝜈]. valued
𝜈],
denoted
denoted
𝜈],isby
𝔖Z:(𝒾)𝑑𝒾
by
𝔖
𝔖
,ℝ
is
given
levelwise
, is
given
by
,],by
is)([given
levelwise
([
],are
)each
([
,𝒦
,levelwise
, ], ) by
𝜑 ∈𝜑(0,
1]
(0,
1]
whose
whose
𝜑number
-cuts
𝜑
define
the
family
family
I-V-Fs
of
I-V-Fs
𝔖idenoted
𝔖
𝐾
⊂
: 𝐾F-NV-F.
⊂
→
ℝ(𝒾)𝑑𝒾
𝒦
→ For
are
given
given
by
Z νdefine
hforofeach
∗ (𝒾,
tion
over
[𝜇,
𝜈].
1 the
e
be
an
F-NV-F.
Then
fuzzy
integral
of
𝔖
over
be
an
F-N
Definition 2. [49] Let
𝔖
:
[𝜇,
𝜈]
⊂
ℝ
→
𝔽
Definition
2.
[49]
Let
𝔖
:
[𝜇,
𝜈]
⊂
ℝ
→
𝔽
(𝒾)
[𝔖
(𝒾,
𝔖
=
𝜑),
𝔖
𝜑)]
for
all
𝒾
∈
𝐾.
Here,
𝜑
∈
(0,
1]
end
point
real
µυ
µυ
S
)
(
NV-Fs over
NV-Fs
[𝜇,
𝜈].NV-Fs
over
over
𝜈].
[𝜇,
𝜈].
𝜑 ∈ (0,𝔖
1] (𝒾)
-cuts
define
I-V-Fs
𝔖
𝐾[𝜇,
⊂
ℝ
→
𝒦
are
given
by point
∗
s ∈ 1]
∗ (𝒾,∗ (𝒾, the family
eF-NV-F
e:each
e
(𝒾)
[𝔖
(𝒾,
[𝔖
𝔖whose
= Theorem
=∗𝜑
𝜑),
𝜑),
𝔖
𝔖)𝔖𝜑)]
𝜑)]
for
for
all
all
𝒾→)⊂
∈d𝔽𝒾of
𝐾.
∈
𝐾.
Here,
Here,
for
each
𝜑whose
∈Let
𝜑
(0,
the
1] (𝐹𝑅)
the
end
end
point
real
real
4be
S
(µbe
)for
+
S
(2.
ν𝜈],
)[49]
ξ(0,
∇𝔖
FR
∇(
dξ.
(25) levelwis
(Let
∗[49]
∗(𝒾,
be
an
F-NV-F.
The
Definition
:
[𝜇,
𝜈]
⊂
ℝ
→
𝔽
Theorem
2.
2.
[49]
:
Let
[𝜇,
𝜈]
𝔖
:
⊂
[𝜇,
ℝ
𝜈]
ℝ
→
𝔽
a
a
F-NV-F
whose
𝜑-cuts
𝜑-cuts
define
the
define
family
the
family
of
I-V-Fs
of
I-V-Fs
2
[𝜇, 𝜈], denoted by (𝐹𝑅)
(𝒾)𝑑𝒾
[𝜇,
(𝒾)𝑑𝒾
∗
𝔖
denoted
by
𝔖
,
is
given
levelwise
by
,
is
given
(.
(.
, 𝜑),
, 𝜑):
𝐾
→
ℝ Let
called
lower
and
upper
functions
of=
. (1 −
ν𝔖−
µ𝜑)]
ξµ𝔖
+
)υ(𝒾)𝑑𝒾
(𝐼𝑅)
(𝐼𝑅)
(𝐼𝑅)
(𝒾) = [𝔖∗𝔖
(𝒾,
(𝒾,(.
0 the
µall
𝔖ξ=
𝔖 (𝒾)𝑑𝒾
𝔖 (𝒾)𝑑𝒾
𝔖functions
for
𝒾are
∈
𝐾.
for
each
𝜑 a∈and
(0,
1]
end
point
real
∗ 𝜑),
Theorem
2.
[49]
𝔖
: [𝜇,
𝜈]
⊂called
ℝ lower
→
𝔽lower
be
F-NV-F
whose
𝜑-cuts
the=family
of I-V-Fs
(.
(.
functions
functions
𝔖𝔖
, ∗𝜑),
, 𝜑),
𝔖∗Let
𝔖, ∗𝜑):
,: 𝜑):
𝐾
→𝜈]
𝐾ℝ
→
ℝHere,
are
called
and
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upper
functions
of the
𝔖
ofdefine
𝔖
. (𝐼𝑅)
(𝒾)𝑑𝒾
(𝒾)𝑑𝒾
(𝒾)𝑑𝒾
(𝒾)𝑑𝒾
(𝐹𝑅)
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𝔖
𝔖
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=
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∈:
∗ (.𝔖
∗ (𝒾,
∗functions
[𝜇,
𝜈],
denoted
,.⊂
is
given
levelwise
Theorem
[49]
𝔖
[𝜇,are
⊂ are
ℝ
𝔽(𝐹𝑅)
be[𝔖
a ∗F-NV-F
whose
𝜑-cuts
define
family
of
I-V-Fs
∗2.
(𝒾)
(𝒾)
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[𝔖
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=
=
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𝜑),
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𝜑)]
for
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all
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and
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by
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to
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be
se
Definition
Definition
3.
[34]
Definition
A
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3.
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[34]
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3.
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[34]
𝜐]
set
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=𝔖
𝜐]
=
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(.
functions 𝔖𝔖∗:(.[𝜇,
,𝔖
𝜑),
,
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𝐾
→
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are
called
lower
and
upper
of
𝔖
.
∗
𝜑), 𝔖 (𝒾, 𝜑)] for all 𝒾 ∈ [𝜇, 𝜈] and for all
𝔖 : [𝜇, 𝜈] ⊂ ℝ → 𝒦 are given by 𝔖 (𝒾) = [𝔖∗ (𝒾,
e
∗
If: [𝜇,
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∈Let
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υare
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], is
),𝜉by
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Definition
[49]
:Then
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1],
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1],
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𝜑),
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𝜑)]
for
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𝒾∈
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⊂
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𝔖
0→
∗[0,
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(8)
𝜑)
and
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𝔖𝔖
and
𝜑)
𝔖
both
are
both
are
∈2.(0,
𝜑
1].
∈𝜈]
Then
𝔖
𝐹𝑅-integrable
[𝜇,
over
[𝜇,
if
and
𝜈]
and
only
𝔖
ifFor
𝔖
(𝒾)𝑑𝒾
(𝐹𝑅) 𝔖(𝒾)𝑑𝒾
(𝐹𝑅)
(𝒾)𝑑𝒾
(𝐼𝑅)
𝔖
=𝜑Definition
𝔖
=𝒦is
𝔖(𝒾,
:→
𝔖(𝒾,
𝜑)
∈
ℛ
,𝜑
=
=
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(0,
∗1],
for
all
𝜑
(0,
for
1].
all
for
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all
all
(0,
𝜑
1].
∈
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For
all
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1],
denotes
collection
denotes
denotes
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of
all
be
an
be
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an
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Then
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fuzzy
fuzzy
integral
of
𝔖
of
over
𝔖
over
Definition
2.𝜑(0,
[49]
2.1].
[49]
Let
Let
𝔖
:𝔖
[𝜇,
𝔖:𝜈]
[𝜇,𝜑)𝑑𝒾
⊂
𝜈]ℝ𝐹𝑅-integrable
⊂
ℝover
𝔽𝜑
→∈∈
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∗𝜑)
([
],(0,
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, for
([
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)([collection
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for
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1],
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where
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where
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1],
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where
ℛ
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denotes
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the
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Rith
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and
𝔖
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1].
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over
[𝜇,
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and
only
𝔖
([
],
)
([
],
)
([
],
)
,
,
∗
[𝜇,
(𝐹𝑅)
(𝒾)𝑑𝒾
𝜈],
denoted
by
𝔖
,
is
given
levelwise
by
∗
be
an
F-NV-F.
Then
fuzzy
integral
of
𝔖
over
Definition
2.
[49]
Let
𝔖
:
[𝜇,
𝜈]
⊂
ℝ
→
𝔽
(𝒾)𝑑𝒾
(𝐹𝑅)
(𝒾)𝑑𝒾
(𝐼𝑅)
𝔖
=
𝔖
=
𝔖(𝒾,
𝜑)𝑑
(𝒾,
(𝒾,
𝜑)
and
𝔖
𝜑)
both
are
𝜑
∈
(0,
1].
Then
𝔖
is
𝐹𝑅-integrable
over
[𝜇,
𝜈]
if
and
only
if
𝔖
𝒾𝒿
𝒾𝒿
𝒾𝒿
[𝜇, 𝑅-integrable
[𝜇, 𝜈],
(𝐹𝑅)
(𝐹𝑅)
(𝒾)𝑑𝒾
over
NV-Fs
[𝜇,
NV-Fs
over
over
𝜈].
𝜈],
denoted
denoted
by Sby
𝔖(𝒾)𝑑𝒾
, isif given
,𝔖
is
given
levelwise
levelwise
by[𝜇,
by
𝑅-integrable
over
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over
𝜈].
[𝜇,
Moreover,
𝜈].𝔖NV-Fs
Moreover,
is
iftions
𝐹𝑅-integrable
𝔖 𝜈].
isof
𝐹𝑅-integrable
over
over
𝜈],
then
𝜈],
(𝐹𝑅)
tions
of
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𝔖
tions
is𝐹𝑅-integrable
𝐹𝑅
of
-integrable
𝔖
I-V-Fs.
is
-integrable
𝔖over
is then
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[𝜇,
-integrable
𝜈]over
if 𝐾.(𝐹𝑅)
[𝜇, 𝜈]over
if𝔖
[𝜇, 𝜈]
if𝔽
𝔖
Let
be
aover
s-convex
Then,
each
ϕ𝜈].
∈[𝜇,
0,
1∗𝐹𝑅
, [𝜇,
we
have
[[𝜇,
]over
∈
∈(𝒾)𝑑𝒾
𝐾.
∈ ∈𝐾.
[𝜇,
𝑅-integrable
𝜈].F-NV-F.
Moreover,
ifby
𝔖for
isI-V-Fs.
𝜈],
then
[𝜇, 𝜈], ℛ
(𝒾)𝑑𝒾
denotedProof.
by (𝐹𝑅)
, is[𝜇,
given
levelwise
(1
(1 − 𝜉)𝒿
for all 𝜑 ∈ (0, 1], where
for
all
𝜑
∈
(0,
1],
where
ℛ𝜉𝒾
the𝔖collection
of Riemannian
integrable
funcdenotes
the colle
𝜉𝒾
𝜉𝒾
+
− 𝜉)𝒿
+𝜉)𝒿
[𝜇,
([ , 𝑅-integrable
], ) denotesover
([ ,+],(1
) −
[𝜇, 𝜈].Moreover,
if
𝔖
is
𝐹𝑅-integrable
over
𝜈],
then
∗
∗
∗
(𝒾,
(𝒾,
(𝒾,
(𝒾,
(𝒾,
(𝒾,
𝜑)
𝜑),are
𝔖𝔖∈∗Lebesgue-integrable,
𝜑)
𝜑),
are
𝔖where
Lebesgue-integrable,
𝜑) ℛ
are
Lebesgue-integrable,
then
𝔖 is(0,
then
a fuzzy
𝔖
isAumannthen
a fuzzy
𝔖
i
𝔖∗ 𝜑),𝔖𝔖for
∗ [34]
all
(0,
1],
denotes
collection
of
(8)
(0,
(0,
(𝒾)𝑑𝒾
(𝐹𝑅) 𝔖
(𝒾)𝑑𝒾[𝜇, 𝜈]
=
is ⊂
said
= ℝ the
to
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isa∞)
said
convex
istosaid
be
seR
Definition
3.
A𝜑𝜑)𝑑𝒾
set
3.
𝐾[34]
=of[𝜇,
3.A
[34]
𝜐]
set∈⊂A
𝐾ℛ
ℝ∗set
=
𝜐]
⊂
ℝ
𝜐]
µv
= (𝐼𝑅)
𝔖
= Definition
: 𝔖(𝒾,
,[𝜇,
([[𝜇,
],=𝐹𝑅
)∞)
(𝐹𝑅)
tions of I-V-Fs. 𝔖 is 𝐹𝑅 -integrable
over
𝔖,(𝒾)𝑑𝒾
tions
I-V-Fs.
-integrable
over
[𝜇,
𝜈]
∈
𝔽 𝔖(𝒾,
.µvDefinition
Note
that,
if𝜑)
([𝔖
],,is
,𝐾
Sif∗(𝐹𝑅)
ϕ (𝒾)𝑑𝒾
∇
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(𝐹𝑅)𝔖(𝒾)𝑑𝒾
(𝐼𝑅)
(𝐼𝑅)
(𝒾,
(𝒾,
(𝒾,
(𝑅)
(𝑅)
(𝑅)
(𝑅)
(𝐹𝑅)
(𝒾)𝑑𝒾
𝔖(𝒾)𝑑𝒾
=
=
𝔖=
𝔖
=
=
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𝔖(𝒾,
𝜑)𝑑𝒾
𝜑)𝑑𝒾
: 𝔖(𝒾,
: 𝔖(𝒾,
𝜑)
∈
𝜑)
ℛ∈∗([)(𝒾,
ℛ, ∗([𝜑)𝑑𝒾
, ) ,(8) (8)
tion
over
tion
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over
tion
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𝜈].
over
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𝜑)𝑑𝒾
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,
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,
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𝔖
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𝔖ξv(𝒾)𝑑𝒾
],
)
],
,
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1
−
ξ
)
µ
+
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1
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ξ
)
µ
+
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∗
∗
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[0,
[0,
tions
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I-V-Fs.
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-integrable
over [𝜇, 𝜈] if then
𝜉 ∈Aumann-integrable
we
𝜉=
∈have
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1],
have
we
have
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𝐹𝑅
= (𝐼𝑅)
𝔖𝐾,
=1],𝐾,
𝜑)𝑑𝒾
:we
𝔖(𝒾,
𝜑)
ℛ
,𝔖
∗ (𝒾,
(𝒾,
𝔖 is a (𝐹𝑅)
func𝜑),
𝔖
𝜑)
Lebesgue-integrable,
𝔖∗ (𝒾, 𝜑), 𝔖∗ (𝒾, 𝜑) are Lebesgue-integrable,
𝔖s4.
[𝜇,
[𝜇,
],
Definition
Definition
4. 𝔖(𝒾,
[34]Definition
The
[34]
4.
The
[34]
→
ℝ𝔖:
The
𝜐]
𝔖:)𝜑)𝑑𝒾
→
ℝ
𝜐]
→is𝜑)𝑑𝒾
ℝ
called
a(8)
harmonically
called
is called
a harmonically
convex
a harmon
fun
∈(𝒾,
∗𝔖:
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(𝒾,
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(𝐹𝑅)
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= (𝑅)
𝔖
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,𝜐]
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µv
s
∗
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) Riemannian
S∗𝜑)
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)[𝜇,
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,
for all 𝜑 ∈ (0, 1], where ℛ([ , ], ) denotes
integrable
func(𝒾,
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then
𝔖
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a
fuzz
tion over [𝜇, 𝜈].
over
𝜈].
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∗ (𝒾,
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1−⊂
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+
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(9)
for for
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𝜑 ∈𝜑(0,
∈ 1],
(0, 1],
where
where
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Riemannian
of2.𝔖ℝ
Riemannian
integrable
integrable
funcfunc2.the
[49]the
Letcollection
2.
Theorem
𝔖[49]
: [𝜇,of
𝜈]
Let
⊂
[49]
: [𝜇,
→𝒾𝒿𝔽
𝜈]
ℝ
: [𝜇,
→
𝔽
𝜈] ⊂be∈ℝ
→
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F-NV-F
whose
a𝐾.
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be(26)
whose
a F-NV-F
w
𝒾𝒿
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) denotes
,) ],denotes
([ ℛ, ([],Theorem
Theorem
∈
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∈𝜑-cu
𝐾.th
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over
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tions
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if
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Note
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+
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+
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−if+
𝜉)𝔖(𝒾)
≤
−
≤
𝜉)𝔖(𝒾)
−
𝜉)𝔖(
for
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ℛ([-integrable
denotes
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collection
of(𝒾)𝑑𝒾
Riemannian
integrable
funcµv over
µv
(1
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(𝒾)𝑑𝒾
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+
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+
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=
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],
)
,
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(𝐹𝑅)
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(𝒾)𝑑𝒾
, ϕover
tions
tions
of I-V-Fs.
of I-V-Fs.
𝔖 is
𝔖S
𝐹𝑅
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Note
.(1
Note
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(1𝔖
𝜉𝒾
𝜉𝒾
+
−
−+
𝜉)𝒿
−
(𝒾)
(𝒾)
[𝔖
(𝒾)
[𝔖
(𝒾,𝜑)]
(𝒾)𝑑𝒾
(𝐼𝑅)
=
𝜑),
=
𝜑)]
𝜑),
=𝔖
for
𝜑),
𝒾 ∈f𝔖
: [𝜇,
𝜈]∇
⊂
ℝ([𝜇,
:1[𝜇,
→ξ𝜈]
𝜈]
ℝ
→
given
𝒦
𝜈]
⊂
by
are
ℝ𝔖𝔖
→
given
𝒦
are
given
by
𝔖
𝔖
𝔖are
𝔖 𝜉𝒾
(1−-integrable
ξ )𝔖
µ+ξv
−
)𝒦
µ=
+𝜈]
ξv
∗ (𝒾,
∗ 𝜉)𝒿
∗all
∗ (𝒾,
(𝒾,of
(𝐹𝑅)
(𝒾)𝑑𝒾
Theorem 2. [49] Lettions
𝔖:∗[𝜇,
𝜈]
⊂𝔖ℝ
→𝜑)
𝔽𝔖
Theorem
2.
[49]
Let
𝔖
:
[𝜇,
𝜈]
⊂
ℝ
→
𝔽
be
a
F-NV-F
whose
𝜑-cuts
define
the
family
of
I-V-Fs
be
a
F-NV-F
𝜑),I-V-Fs.
are
Lebesgue-integrable,
then
𝔖
is
a
fuzzy
Aumann-integrable
funcis
𝐹𝑅
-integrable
over
[𝜇,
𝜈]
if
𝔖
∈
𝔽
.
Note
that,
if
(𝒾)𝑑𝒾
(𝐼𝑅)
=
𝔖
∗ (𝒾,∗ (𝒾,
Aumann-integrable
(𝒾, 𝔖
𝜑),
𝔖 𝜑) 𝜑)
are are
Lebesgue-integrable,
Lebesgue-integrable,
then
then
𝔖𝒾,
is
𝔖
fuzzy
Aumann-integrable
func𝔖∗ (𝒾,
𝔖
µv
sais
∗𝜑),
∗𝒾,
∗𝒿a
(∈
[𝜇,
[0,
[𝜇,
[𝜇,
[0,
[𝜇,
and
(0,
Then
(0,
1].
𝜑
Then
∈
𝐹𝑅-integrable
(0,
𝔖
1].
is)Then
𝔖
is
[𝜇,
𝐹𝑅-integrable
over
if.𝔖(𝒾)
[𝜇,
only
𝜈]∈
over
if[𝜇,
iffor
and
[𝜇,
only
𝜈]𝒾𝜑)
𝔖is
forξ∈sall
𝒿𝜑
∈
for
all
𝜐],
for
𝜉 is
𝒿∈
∈
𝒾,
1],
∈fuzzy
where
𝜉ϕ
∈𝐹𝑅-integrable
𝜐],
𝜉over
𝔖(𝒾)
1],
∈𝜈][0,
where
0 funcwhere
for
all
≥
𝒾𝔖(𝒾)
0func≥𝔖
𝜐].
0
all
for
If
(11)
∈if
all
𝒾∗𝜐]
ro
Theorem
[49]
Let
𝔖
: [𝜇,
⊂≥1],
ℝ𝜈]
→
𝔽and
be
F-NV-F
𝜑-c
∗ (𝒾,
∗ (𝒾,≤𝜑
S
(1].
µ,
ϕ
)∈
+
(a1𝔖
−
ξall
)2.
S
(𝜐],
v,
∇
∗ (𝒾,
over
[𝜇,
𝜈].
𝜑),
𝔖
𝜑)
are
Lebesgue-integrable,
then
𝔖
is
fuzzy
Aumann-integrable
(𝒾)
[𝔖
(𝒾,
(𝒾)
[𝔖
=
𝜑),
𝔖
𝜑)]
for
all
𝒾
∈
[𝜇,
𝜈]
and
for
all
=whose
𝜑)
are
by
𝔖
:
[𝜇,
𝜈]
⊂
ℝ
→([𝜇,
𝒦
given
byais
𝔖
𝔖 : [𝜇, 𝜈] ⊂ ℝ → 𝒦 𝔖tion
𝔖
1is
−
)all
µare
+
ξv
(0,
(0,
∗ (𝒾, given
for
all
for
𝜑
∈
all
(0,
𝜑
1].
∈
(0,
For
1].
all
For
𝜑
∈
all
𝜑
1],
∈
ℱℛ
1],
ℱℛ
denotes
denotes
the
collection
the
collection
ofξcalled
𝐹𝑅-integrable
ofℝ𝜐]
[𝜇,
Definition
Definition
4.
[34]
Definition
The
4.
𝔖:
[34]
𝜐]
4.
The
[34]
→
ℝ𝔖:
The
𝜐]
𝔖:
→
→is𝐹𝑅-integrable
ℝ
aall
harmonically
called
called
aFharmonically
convex
aFharmon
∗
∗ (𝒾,fun
([ , ], )([ , ], )
tiontion
over
over
[𝜇,for
𝜈].
[𝜇,all
𝜈]. 𝜑 ∈ (0, 1]. For
[𝜇,
[𝜇,
[𝜇,
(0,
∗
is
called
a
harmonically
is
called
is
called
a
harmonically
concave
a
harmonically
function
concave
concave
on
function
𝜐].
function
on
on
𝜐].
𝜐].
all
𝜑
∈
1],
ℱℛ
denotes
the
collection
of
all
𝐹𝑅-integrable
F([
],
)
,
𝑅-integrable
𝑅-integrable
over
𝑅-integrable
Moreover,
over
Moreover,
if 𝔖given
[𝜇,
is 𝜈].
𝐹𝑅-integrable
if
Moreover,
𝔖𝐹𝑅-integrable
is
𝔖∗ (𝒾,
is[𝜇,
𝐹𝑅-integrabl
𝜈],𝔖
over
then
(𝒾)𝐹𝑅-integrable
[𝔖
(𝒾, [𝜇,
=ifover
𝜑)]𝜈
: [𝜇,
𝜈]𝜈].
⊂over
ℝ →[𝜇,
𝒦𝜈].
are
by
𝔖
over NV-Fs
[𝜇,
∗ [𝜇,
NV-Fs
over
[𝜇,
over
𝜈].
𝜈].only
for𝜈].all
𝜑 ∈[𝜇,
(0,𝜈]
1].if[𝜇,and
For
all 𝜑if ∈
1], ℱℛ
the
F-𝜑),
𝜑)𝜑 both
are
𝜑 ∈ (0, 1]. Then 𝔖 tion
is 𝐹𝑅-integrable
over
𝔖(0,
∈ (0,
1].collection
Then
𝔖 ofis all
𝐹𝑅-integrable
[𝜇, 𝜈] if and
([ ,𝔖], (𝒾,
) denotes
∗ (𝒾, 𝜑) and
𝒾𝒿 family
𝒾𝒿
𝒾𝒿 over
NV-Fs
over
[𝜇,ℝ
and Let
𝜈].
𝜑 ∈whose
Theorem 2.
[49]
𝔖
:
[𝜇,
𝜈]
⊂
→
𝔽
be
a
F-NV-F
𝜑-cuts
define
the
of
I-V-Fs
(1
(1
(1
𝜉𝔖(𝒿),
𝔖𝜑-cuts
𝔖 the 𝔖
≤ family
−
𝜉)𝔖(𝒾)
≤
−
≤𝜉)𝔖(𝒾)
− 𝜉)𝔖(
(0, 1].
Then
𝔖𝜑-cuts
is (1
𝐹𝑅-integrable
over
[𝜇,
𝜈] +
if and
only
if+𝔖𝜉
NV-Fs
[𝜇,
µv
Theorem
Theorem
[49]
2.𝐹𝑅-integrable
[49]
Let𝜈].
Let
𝔖: S
[𝜇,
𝔖:𝜈]
[𝜇,
⊂𝜈]µv
ℝ⊂[𝜇,
→ℝ𝜈],
𝔽→ϕ 𝔽
be∇abe
F-NV-F
a F-NV-F
whose
whose
the
family
of𝜉)𝒿
I-V-Fs
of−if
I-V-Fs
(1
(1
𝑅-integrable over [𝜇, 𝜈]. Moreover,
if 𝔖2.over
is
over
then
𝑅-integrable
overdefine
𝔖 is 𝐹𝑅-integra
𝜉𝒾 +define
𝜉𝒾𝜈].
𝜉𝒾−+
− [𝜇,
𝜉)𝒿
+Moreover,
𝜉)𝒿
∗ →
1−
)=
ξµ
+(
1∗=
−
ξ(0,
)is
v𝜑)]
∗ (𝒾,
Theorem
2.⊂[49]
Let
: [34]
[𝜇,given
𝜈]3.
ℝ
𝔽+([𝜇,
be
av ,F-NV-F
whose
𝜑-cuts
define
the
family
of
I-V-Fs
(0,
[𝜇,ℝ∗𝜐]
(𝒾)
[𝔖
(𝒾,=
(𝒾,
∞)
said
∞)
is
to
said
be
a
to
convex
be
a
convex
set,
if,
set,
for
all
if,
for
𝒾,
𝒿
all
∈
𝒾,
𝒿
∈
Definition
3.are
A⊂
[34]
set
Aξµ𝔖=
set
𝐾ξ𝜐]
⊂
⊂
ℝ
𝜑),
𝔖
for
all
𝒾
∈
[𝜇,
𝜈]
and
for
all
ℝ→
𝒦𝔖
by𝐾
𝔖 : [𝜇, 𝜈]Definition
(𝒾,
(𝒾,
(𝒾,
(𝑅)
(𝑅)
(𝑅)
(𝑅)
(𝑅)
(𝐹𝑅)
(𝒾)𝑑𝒾
(𝐹𝑅)
(𝒾)𝑑𝒾
(𝐹𝑅)
(𝒾)𝑑𝒾
=to
𝔖𝔖
𝜑)𝑑𝒾
=
𝔖, ∗for
𝜑)𝑑𝒾
𝔖𝔖,over
𝜑)
𝔖
𝔖
𝔖
∗(0,
∗∞)
[𝜇
𝑅-integrable
over
𝜈].
Moreover,
if=
isand
𝐹𝑅-integrable
∗and
[𝜇,[𝔖
=[0,
isfor
said
be𝒾µv
a∈convex
set,
if,
all 𝒾,
𝒿∗ ∈ 𝜑)
[34]
A
set
𝐾
𝜐]
⊂ ∗ℝ
(𝒾)
[𝔖
(𝒾, 𝜑),
(𝒾,
(𝒾,[𝜇,
=
𝜑),
𝔖
𝔖1],
𝜑)]
for
all
all
𝒾 [0,
∈
[𝜇,
𝜈]
[𝜇,
𝜈]all
for
all
all𝜐].
[𝜇,⊂𝜈]ℝDefinition
⊂→ℝ𝒦
→𝒦
are 3.
are
given
given
byfor
𝔖
byall
𝔖𝒾,=(𝒾)
𝔖 :𝔖[𝜇,:𝜈]
∗ (𝒾,
[𝜇,
[𝜇,
[0,
[𝜇,
∗ (𝒾,
𝒿=ℝ
∈
for
all
𝜐],
for
𝒾,
𝜉∞)
𝒿∈+
all
∈𝔖
𝒾,
𝒿𝜑)]
𝜐],
∈
where
𝜉[𝜇,
∈
𝜐],
𝜉𝔖
𝔖(𝒾)
1],
∈
where
≥
1],
0
where
for
𝔖(𝒾)
𝒾𝔖(𝒾)
0for
for
≥
0
allfor
If𝒾(11)
∈all[𝜇,is
𝒾𝜐]
∈
r
s𝜑),
sfor
[0,
[0,
𝐾,Definition
𝜉Then
∈→𝐾,
𝜉1],
∈are
we
1],
have
we
∗𝜈]
(0,
=
is
said
to
be
a[𝜇,
convex
set,
if,
for≥
all∈
𝒾,
𝒿∈
3.𝐹𝑅-integrable
[34]
Abyhave
set𝔖𝐾(𝒾)
=≤[𝜇,
𝜐]−
⊂
[𝔖
(𝒾,
=
𝔖
𝜑)]
all
𝒾
∈
and
for
all
𝜈]1].
⊂
ℝ
𝒦𝔖
given
𝔖𝜑 :∈[𝜇,(0,
,
1
ξ
)
S
(
µ,
ϕ
)
ξ
S
(
v,
ϕ
))∇
((
(𝒾,
(𝒾,
𝜑)
and
𝜑)
both
are
is
over
[𝜇,
𝜈]
if
and
only
if
∗
∗
∗
∗
𝐾,Then
𝜉 ∈ [0,
1],
we
have overover
∗ (𝒾,
ξµ+(
1−
ξ𝔖
)v∗𝜑)
∗[𝜇,
(𝒾,
(𝒾,
(𝒾,
𝜑)
𝜑)
and
and
𝔖
𝜑)
both
both
are
are
∈𝜑(0,
(0,
Then
is𝔖
𝐹𝑅-integrable
is
𝐹𝑅-integrable
𝜈]
[𝜇,
if
𝜈]
and
if
and
only
only
if
𝔖
if
𝔖
[𝜇,
[𝜇,
[𝜇,
(𝒾,
(𝒾,
(𝒾,
(𝑅)
(𝑅)
(𝑅)
(𝐹𝑅) 𝔖𝜑(𝒾)𝑑𝒾
(𝐹𝑅)
(𝒾)𝑑𝒾
is
called
a
harmonically
is
called
is
called
a
harmonically
concave
a
harmonically
function
concave
concave
on
function
𝜐].
function
on
on
𝜐].
𝜐].
=
𝜑)𝑑𝒾
,
𝔖
𝜑)𝑑𝒾
=
𝔖
𝜑
𝔖
∗
∗
[0,
(27)
𝐾,∈ 𝜉1].
∈ 1].
1], 𝔖
we𝔖
have
∗
∗
µv
𝜑) then
and 𝔖∗ (𝒾,
𝜑) both=are (𝑅) 𝔖 (𝒾, 𝜑)𝑑𝒾 , (𝑅)
𝜑𝑅-integrable
∈ (0, 1]. Then
is 𝜈].
𝐹𝑅-integrable
𝜈]
if∇and 𝒾𝒿
only
if 𝒾𝒿
𝔖[𝜇,
over𝔖[𝜇,
Moreover,
𝔖µvis[𝜇,𝐹𝑅-integrable
over
𝜈],
∗ ifover
∗ (𝒾,
(𝐹𝑅)
(𝒾)𝑑𝒾
𝔖
𝒾𝒿
S
,
ϕ
∈
𝐾.
∈
𝐾.
(10)
(10)
∗
(𝒾)𝑑𝒾 𝔖
(𝒾)𝑑𝒾 𝔖 (𝒾)
(𝐼𝑅) 𝔖
[𝜇,
[𝜇,
= (𝐼𝑅)
= (𝐼𝑅)
𝑅-integrable
𝑅-integrable
overover
[𝜇, 𝜈].
[𝜇, Moreover,
𝜈]. Moreover,
ifξ )𝔖
𝔖𝜉𝒾
𝐹𝑅-integrable
is+𝐹𝑅-integrable
over
over
𝜈],
then=
then
ξµ+(1−
vif is
ξµ
+(
ξ(1
)v −
(9)𝜈],
𝒾𝒿
(1𝜉𝒾
∈ 𝐾.
−1+−𝜉)𝒿
𝜉)𝒿
(10)
[𝜇,(1
𝑅-integrable over [𝜇, 𝜈]. Moreover, if 𝔖 is 𝐹𝑅-integrable
over
𝜈],
then
𝜉𝒾 +
−
𝜉)𝒿
∈
𝐾.
(10)
µv
s
∗
s
∗
(1
𝜉𝒾
+
−
𝜉)𝒿
≤
1
−
ξ
)
S
(
µ,
ϕ
)
+
ξ
S
(
v,
ϕ
))∇
.
((
(𝒾)𝑑𝒾 = (𝑅) 𝔖 (𝒾, 𝜑)𝑑𝒾 , (𝑅) 𝔖∗ (𝒾, 𝜑)𝑑𝒾ξµ+(1−ξ )v
(𝐼𝑅) 𝔖𝔖(𝒾)𝑑𝒾
=(𝐹𝑅)
= (𝐼𝑅) 𝔖 (𝒾
∗𝜑
(0,(0,
(0,
(0,(𝐼𝑅)
for[𝜇,
all =
𝜑 ∈∗for
(0,𝔖
all
1].
𝜑For
∈
for
(0,
all
all
1].
𝜑(𝑅)
𝜑,∈For
∈(𝑅)
all
1],
1].
ℱℛ
∈∗For
all
∈=([
1],
ℱℛ
denotes
denotes
collection
the
of
collec
all
([function
], 𝜑
) ℱℛ
], the
)𝜐]
) denote
(𝒾,
(𝒾,
(𝒾,
(𝐹𝑅)
(𝐹𝑅)
(𝒾)𝑑𝒾
, 1],
, [𝜇,
, ], if
=
𝔖
𝜑)𝑑𝒾
𝔖
𝔖
𝜑)𝑑𝒾
𝜑)𝑑𝒾
𝔖𝔖:
𝔖(𝒾)𝑑𝒾
(𝒾)𝑑𝒾
𝔖([ 𝜐]
∗ (𝒾,
[𝜇,
[𝜇,
Definition
Definition
4. [34]
4.
The
[34]
The
𝜐]
𝔖:→
ℝ𝜐](𝑅)
→
ℝ
is(𝑅)
called
is
called
a ∗𝜑)𝑑𝒾
harmonically
a, harmonically
convex
convex
function
on
on
if
(𝑅)
(𝐹𝑅)
= The
𝔖
𝜑)𝑑𝒾
, is(𝑅)
𝔖over
𝜑)𝑑𝒾
𝔖(𝒾)𝑑𝒾
Definition
4. [34]
𝔖: [𝜇,
𝜐]NV-Fs
→
ℝ𝜈].over
called
a∗ (𝒾,
harmonically
convex(9)
function on [𝜇, 𝜐] if
∗ (𝒾,
NV-Fs
over
[𝜇,
NV-Fs
[𝜇, 𝜈].
[𝜇, 𝜈].
[𝜇,
[34] The 𝔖: [𝜇, 𝜐] →𝒾𝒿ℝ ofis𝒾𝒿
called
a harmonically
on 1],
(9)𝜐]
(9)if([ , ], ) deno
for all 𝜑 ∈ (0, 1]. For all 𝜑 ∈ (0,Definition
1], ℱℛ([ , 4.
for all 𝜑F-∈ (0,convex
1]. Forfunction
all 𝜑 ∈ (0,
ℱℛ
all
𝐹𝑅-integrable
], ) denotes the collection
𝒾𝒿
(9)
(1𝜑−≤
(1 −
+
𝜉𝔖(𝒿),
+𝜑𝜉𝔖(𝒿),
𝔖 = (𝐼𝑅)
𝔖
≤
𝜉)𝔖(𝒾)
𝜉)𝔖(𝒾)
(11)
(11) the coll
(𝒾)𝑑𝒾
𝔖
(0,
for
all
∈
(0,
1].
For
all
∈
1],
ℱℛ
],∞)
) denotes
,⊂
NV-Fs over [𝜇, 𝜈].
NV-Fs
over
(1A
(1
+=
𝜉𝔖(𝒿),
≤=
−set
𝜉)𝔖(𝒾)
(11)
𝜉𝒾
𝜉𝒾𝒾𝒿
+ (1
−=
+𝜉)𝒿
(0,
[𝜇,
is=([said
to
be
is=said
a(0,
convex
∞)
to be
isse
asa
Definition
Definition
3.+−
[34]
A
Definition
set
3. [34]
𝐾
3.
𝜐][𝜇,
[34]
𝐾
⊂ 𝜈].
=
ℝA[𝜇,
set
𝜐](0,𝐾
⊂∞)
=
ℝ [𝜇,
𝜐]
ℝ
(𝒾)𝑑𝒾
(𝒾)𝑑𝒾
(𝐼𝑅)
(𝐼𝑅)
=𝔖
𝔖𝜉)𝒿
𝔖(1
(1
𝜉𝒾
−
𝜉)𝒿
+
𝜉𝔖(𝒿),
𝔖
≤
−
𝜉)𝔖(𝒾)
(11)
NV-Fs
over
[𝜇,
𝜈].
(𝒾)𝑑𝒾
(𝐼𝑅)
= 𝐾,
𝔖
(1
𝜉𝒾
+
−
𝜉)𝒿
[0,
[0,
[0,
𝜉
∈
1],
𝐾,
𝜉
we
∈
have
1],
𝐾,
we
𝜉
∈
have
1],
we
have
[0,𝜉1],
[𝜇,𝒾𝜐].
for all for
𝒾, 𝒿 all
∈ [𝜇,
𝒾, 𝒿𝜐],
∈ 𝜉[𝜇,∈𝜐],
∈ [0,
where
1], where
𝔖(𝒾) ≥𝔖(𝒾)
0 for
≥ all
0 for
𝒾 ∈all
∈ [𝜇,
If (11)
𝜐]. Ifis(11)
reversed
is reversed
then, then,
𝔖
𝔖
(0,
(0, ∞)
[𝜇,
= all
∞)
said
be([[0,
a, convex
set, 𝔖(𝒾)
if,
𝒿 ∈all
Definition 3. [34] A set
Definition
3.
[34]
= F𝜐] ⊂ ℝ =
[𝜇,
[𝜇,A𝜐].setIf 𝐾
for
𝜐],to
𝜉ℱℛ
∈
1],
≥all0 𝒾,for
𝒾∈
(11)
is reversed
then,
𝔖 is
for 𝐾
all=𝜑[𝜇,
∈ 𝜐]
(0,⊂1].ℝ For
all 𝒾,
𝜑𝒿∈∈is(0,
1],
denotes
theforcollection
of all
𝐹𝑅-integrable
], ) where
[𝜇,
[𝜇,
isall
called
is
called
a∈𝒿harmonically
aFor
harmonically
concave
concave
function
function
on
on
𝜐].
𝜐].
[𝜇,
[0,
[𝜇,
𝒾𝒿
𝒾𝒿
𝒾𝒿
(0,
(0,
for
all
𝒾,
∈
𝜐],
𝜉
∈
1],
where
𝔖(𝒾)
≥
0
for
all
𝒾
∈
𝜐].
If
(11)
is
reversed
then,
𝔖
for
for
all
𝜑
∈
𝜑
(0,
1].
(0,
1].
For
all
all
𝜑
∈
𝜑
∈
1],
1],
ℱℛ
ℱℛ
denotes
denotes
the
the
collection
collection
of
all
of
all
𝐹𝑅-integrable
𝐹𝑅-integrable
FF([ , ([
],Definition
) ], )
,
(0,
[𝜇,
=
∞)
is
said
3.
[34]
A
set
𝐾
=
𝜐]
⊂
ℝ
[0,
𝐾, 𝜉 ∈ [0, 1], we have
𝐾,
𝜉
∈
1],
we
have
[𝜇,
is called
harmonically
function
on
𝜐].all 𝐹𝑅-integrable F- ∈ 𝐾.
over
[𝜇,
∈ 𝐾. to be
∈
forNV-Fs
all 𝜑
∈NV-Fs
(0,
1].𝜈].
For
all
𝜑 ∈a (0,
1], concave
ℱℛ([ , ],concave
the[𝜇,
collection
of
) denoteson
is over
called
harmonically
function
𝜐]. we have
NV-Fs
over
[𝜇,a 𝜈].
[𝜇, 𝜈].
𝜉𝒾 + (1 − 𝜉)𝒿
𝜉𝒾 + (1 − 𝜉)𝒿
𝜉𝒾 + (1 − 𝜉)𝒿
𝐾, 𝜉 ∈ [0,
1],
𝒾𝒿
𝒾𝒿
NV-Fs over [𝜇, 𝜈].
∈ 𝜐]
𝐾.⊂ ℝ = (0, ∞) is said to be a convex
∈
(10) set, if, for all 𝒾, 𝒿 ∈ 𝒾𝒿
Definition 3. [34]
A+set
𝐾 𝜉)𝒿
= [𝜇,
(1
(1
𝜉𝒾
𝜉𝒾
−
+
(0, ∞)
[𝜇,⊂𝜐]ℝ⊂ ℝ
= (0,
= ∞)
is said
is said
to be
toabe
convex
a convex
set,set,
if, for
if, for
all all
𝒾, 𝒿 𝒾,
∈ 𝒿 ∈ ∈ 𝐾.− 𝜉)𝒿
Definition
Definition
3. [34]
3. [34]
A set
A set
𝐾 =𝐾[𝜇,
= 𝜐]
Symmetry 2022, 14, 1639
13 of 18
After adding (26) and (27), and integrating over [0, 1], we obtain
R1 µv
µv
,
ϕ
∇
dξ
S
∗ (1−ξ )µ+ξv
0
(1−ξ )µ+ξv
R1 µv
, ϕ ∇ ξµ+(1−ξ )v dξ
+ 0 S∗ ξµ+(µv
1− ξ ) v



µv


ξ s ∇ (1−ξ )µ+ξv




S∗ (µ, ϕ)

µv
s



+(1 − ξ ) ∇ ξµ+(1−ξ )v 

R 1

 dξ,
≤ 0

µv
s

 (1 − ξ ) ∇


(1−ξ )µ+ξv  


 +S∗ (v, ϕ)

µv
s



 +ξ ∇ ξµ+(1−ξ )v
R 1 ∗
µv
µv
S
,
ϕ
∇
dξ
0
(1−ξ )µ+ξv
(1−ξ )µ+ξv
R 1 ∗
µv
+ 0 S ξµ+(µv
,
ϕ
∇
dξ
1− ξ ) v
ξµ+(1−ξ )v



µv


ξ s ∇ (1−ξ )µ+ξv



 ∗
S
(
µ,
ϕ
)


µv
s



+(1 − ξ ) ∇ ξµ+(1−ξ )v 

R 1

 dξ.
≤ 0


µv
s



(1 − ξ ) ∇ (1−ξ )µ+ξv  


∗

 +S (v, ϕ)
µv

 +ξ s ∇


ξµ+(1−ξ )v
R1 s µv
SymmetrySymmetry
2022,
Symmetry
14, x2022,
FOR
2022,
14,
PEER
x14,
FOR
REVIEW
x FOR
PEER
PEER
REVIEW
REVIEW = 2S∗ (µ, ϕ )
4 of 1
0 ξ ∇ (1−ξ )µ+ξv dξ
R1
dξ,
+2S∗ (v, ϕ) 0 ξ s ∇ ξµ+(µv
1− ξ ) v
R
1 s
µv
Proposition
Proposition
Proposition
1. [51]
1. ∗[51]
1.µ,𝔴
[51]
Let
𝔵,. 𝔴
𝔵,∈𝔴𝔽∈
Then
fuzzy
.𝔽
Then
. Then
order
fuzzy
fuzzy
relation
order
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relation
“ ≼relation
” is “given
≼ “” ≼ison
”given
is𝔽given
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on on
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= Let
2S
(𝔵,
ϕ∈) 𝔽Let
0 ξ ∇ (1−ξ )µ+ξv dξ
R
1 if
µvonly
Symmetry 2022,
Symmetry
Symmetry
14, x2022,
FOR2022,
14,
PEER
x14,
FOR
REVIEW
x FOR
PEER
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[𝔵]
[𝔵]
≼ ∗𝔴(v,if𝔵ϕand
ifif,and
only
if, [𝔴]
if,
≤
≤for[𝔴]
≤all[𝔴]
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∈ for
all
(0, 1],
all
𝜑 ∈𝜑(0,
∈ 1],
(0, 1], 4 of(41
dξ.[𝔵]
+𝔵2S
)≼ 𝔵𝔴≼only
ξ s𝔴
∇and
0
ξµ+(1−ξ )v
It is a partial
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relation.
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2022,
Symmetry
14, x2022,
FOR
2022,
14,
PEER
x14,
FOR
REVIEW
x FOR
PEER
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4 is
419
ofon
19on
Proposition
Proposition
Proposition
1.𝔵,[51]
𝔵,
1.𝔵,
𝔴∈[51]
Let
∈𝔽𝜌∈𝔽Let
𝔵,. ℝ,
𝔴and
𝔵,then
∈𝜌𝔴
∈𝜌.ℝ,
𝔽
Then
Then
.ℝ,
order
fuzzy
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relation
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relation
“ ≼relation
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on
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numbers.
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numbers.
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∈Let
𝔽1.
Let
𝔵,[51]
𝔴
𝔴
and
∈𝔽
and
we
∈
then
have
then
we
we
have
have
1
dξ,
= 2[S∗ (µ, ϕ) + S∗ (ν, ϕ)] 0 ξ s ∇ ξµ+(µυ
1−ξ[𝔵]
)υ≤
[𝔵]
[𝔵]
[𝔵]
[𝔴]
𝔵 ≼ 𝔴 if𝔵and
≼ 𝔵𝔴only
≼[𝔵+
if𝔴and
if,
if and
only
if,
≤if, [𝔴]
≤all[𝔴]
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for
all
all
𝜑 ∈𝜑(0,
∈(28)
1],
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R𝔴]
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[𝔵]
[𝔴]
=only
𝔴]
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+
=[𝔴]
=for
, [𝔴]
+
+𝜑for
, (0,
, 1],
1 s [𝔵+
µυ
∗
∗
= 2[S (µ, ϕ) + S (ν, ϕ)] 0 ξ ∇ ξµ+(1−ξ )υ dξ.
is aLet
partial
It
aLet
order
is
aLet
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2022,
Symmetry
14, x2022,
FOR
2022,
14,
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[𝔵fuzzy
[𝔵=×
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[𝔵]
[ [𝔵]
[on
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Proposition
Proposition
1.It[51]
1. It
[51]
𝔵,
1.is𝔴
[51]
∈partial
𝔽
𝔵,partial
𝔵,∈
𝔴𝔽∈
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Then
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∈
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and
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S
,
ϕ
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It
is
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1. [51] Let
1. [51]
𝔵,
1.0𝔴
[51]
Let
∈ 𝔽Let
𝔵,(.1𝔴
𝔵,
Then
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on
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by by
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[𝔵+
[𝔵]
[𝔴]
[𝔴]
−
ξ∈
µ𝔽
+∈
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ξ𝔴]
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1.
[49]
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1.
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Let
𝜈]
Let
𝔖
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1.
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Proposition
1. [51]
1. [51]
Let Let
𝔵, 𝔴𝔵,∈𝔴𝔽∈. 𝔽
Then
. Then
fuzzy
fuzzy
order
order
relation
relation
“ ≼ “” ≼is”given
is given
on on
𝔽 𝔽
by by
Proposition 1. [51] Let𝔵 ≼
𝔵,[𝔵+
𝔴
∈
𝔽
.
Then
fuzzy
order
relation
“
≼
”
is
given
on
𝔽
by
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[𝔴]
(5)
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[𝔴]
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if and
only
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all
𝜑
∈
(0,
1],
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, , = 𝔴] 𝔴]
+= = , + + , ,
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if, [𝔵]
≤ [𝔴]
≤ [𝔴]for for
all all
𝜑 ∈𝜑(0,
∈ 1],
(0, 1],
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Symmetry 2022, 14, 1639
14
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We
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Symmetry
2022,
FOR
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Definition
Definition
Definition
1. [49] A1.fuzzy
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number
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Symmetry 2022, 14, x FORSymmetry
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2022, 14, x FOR PEER REVIEW[𝜌. 𝔵] = 𝜌. [𝔵]
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[51]
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henceProposition
Definition
Definition
1. [49]
1. [49]
A fuzzy
A1.fuzzy
number
number
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𝔖
:Let
𝐾𝔖mathematical
⊂
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𝔵,
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.
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𝜈],
denoted
𝜈],
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denoted
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denoted
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by
1.∈[49]
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ℝ
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𝔽
called
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each
Proposition
1.
[51]
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Proposition
𝔵,
Proposition
𝔴𝔖∈family
𝔽numbers.
1.
[51]
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𝔵,
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∈
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and
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𝔖) 𝜑)]
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functions
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µ
0
𝔖
=
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𝔖
𝜑)]
for
all
𝒾
∈
𝐾.
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for
each
𝜑
∈
(0,
1]
the
end
point
real
∗
numbers. Let 𝔵, 𝔴 ∈
𝔽
numbers.
Let
𝔵,
𝔴
∈
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and
𝜌
∈
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and
𝜌
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∗ (.
∗ (.(𝐹𝑅)
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(𝒾)𝑑𝒾
(𝒾)𝑑𝒾
(𝐹𝑅) functions
(𝐹𝑅)
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𝔖(𝒾)𝑑𝒾
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,(8)
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(.
(. ,(𝐹𝑅)
functions
𝔖
𝔖
,
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,
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𝐾
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and
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functions
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.
.
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It
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Itaispartial
anumbers.
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relation.
functions 𝔖∗ (. , 𝜑), 𝔖∗It(.is, 𝜑):
𝐾 → ℝorder
are called
andorder
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functions
𝔖.× 𝜌
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[𝔵[𝔵]
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:[𝔵]
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𝜈]
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→
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integral
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Definition
[49]
2.
[49]
Let
Let
𝔖
:
[𝜇,
𝔖
:
𝜈]
[𝜇,
⊂
𝜈]
ℝ
⊂
→
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𝔽
→
𝔽
for all 𝜑for
∈ for
all
(0, 1],
all
𝜑 Definition
∈where
𝜑(0,
∈ 1],
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for
where
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where
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∈
for
all
(0,
ℛ
1],
all
𝜑
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where
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(0,
1],
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1],
where
ℛ
where
ℛ
ℛ
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denotes
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denotes
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the
the
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denotes
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the
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integrable
funcintegrable
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𝔴]
=
+
,
Let
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∈
numbers.
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Let
𝔵,
𝔴
∈
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and
𝜌
∈
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and
have
𝜌
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then
we
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([2.
],
)
([
([
],
)
],
)
([
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)
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examine
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ofthe
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characte
[𝜇, 𝜈], denoted
(𝐹𝑅)
(𝒾)𝑑𝒾
by
𝔖𝜈]
,→
is𝔴,given
levelwise
by
[𝜌.examine
[𝜌.
[𝜌.[𝔵]
[𝔵]
𝔵]
𝔵]
𝔵]
=
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=
𝜌.
=
𝜌
be
an
F-NV-F.
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fuzzy
integral
of
𝔖
over
Definition
2.
[49]
Let
𝔖
:
[𝜇,
⊂
ℝ
𝔽
numbers.
Let
𝔵,
∈
𝔽
numbers.
numbers.
Let
Let
𝔵,
𝔴
𝔴
𝔽
∈
𝔽
and
𝜌
∈
ℝ,
then
we
have
and
and
𝜌
∈
𝜌
ℝ,
ℝ,
then
we
we
have
have
(6)
[𝔵
[𝔵
[𝔵]
[
[𝔵]
×
×
𝔴]
=
×
𝔴]
𝔴]
=
[𝜇,
(𝐹𝑅)
(𝐹𝑅)
(𝒾)𝑑𝒾
denoted
denoted
by
by
𝔖𝜈](𝒾)𝑑𝒾
,ifis
given
,(𝐹𝑅)
is
given
levelwise
levelwise
by
by
(𝐹𝑅)
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(𝐹𝑅)
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(𝒾)𝑑𝒾
(𝐹𝑅)
(𝐹𝑅)
tions oftions
I-V-Fs.
tions
of [𝜇,
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of
𝔖 𝜈],
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is 𝜈],
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tions
-integrable
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-integrable
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∈have
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Note
𝔽(𝐹𝑅)
∈
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Note
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∈
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𝔽𝔖
if.(𝒾)𝑑𝒾
Note
∈F-NV-F,
𝔽∈.that,
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. Note
if that,
th
Next,
we
construct
the
first
Fejér
inequality
for
harmonically
s-convex
numbers.
Let
numbers.
𝔽
Let
𝔴
∈
𝔽we𝔖
and
𝜌H.H.
∈
ℝ,
then
and
∈
ℝ,
then
have
[𝔵
[𝔵]
[
[𝜇,
(𝐹𝑅)
(𝒾)𝑑𝒾
(5)
×
𝔴]
=
×
𝔴]
,
[𝔵+
[𝔵+
[𝔵]
[𝔴]
[𝔵]
[𝔴]
𝜈],
denoted
by
𝔖
,
is
given
levelwise
by
𝔴]
=
+
,
𝔴]
=
+
,
∗ (𝒾,(𝒾, ∗ (𝒾,∗which
∗ (𝒾,(𝒾,
∗first
generalizes
the
H.H.
inequality
for
a=
harmonically
convex
function.
(5)
[𝔵+
[𝔵+
[𝔵]
[𝔴]
[𝔵]
[𝔵]
𝔴]
,ais
𝔴]
𝔴]
=Aumann-integrable
=Aumann-integrable
+ [𝔴]
+ [𝔴]
,func(7)
(𝒾, 𝜑)
(𝒾,
(𝒾,
[𝜌.
[𝜌.
[𝔵]
𝜑)𝔖
𝜑),
are
𝔖 𝜑)
Lebesgue-integrable,
are
are
Lebesgue-integrable,
𝜑),
Lebesgue-integrable,
𝔖
𝜑),
are
then
𝔖∗𝜑)
Lebesgue-integrable,
𝔖𝜑)
are
then
is Fejér
are
then
aLebesgue-integrable,
fuzzy
𝔖Lebesgue-integrable,
is
𝔖 ais
Aumann-integrable
fuzzy
a fuzzy
then
Aumann-integrable
Aumann-integrable
𝔖
then
is then
a+fuzzy
𝔖
funcis
𝔖[𝔵+
Aumann-integrable
fuzzy
afuncfuzzy
func𝔖∗ (𝒾, 𝜑),𝔖𝔖
𝔖∗𝜑),
𝔖
𝔖𝔖
𝔖
𝔵]
𝔵], =fun
=∗𝜑),
𝜌.𝜑)
𝜌
∗ (𝒾,
∗ (𝒾,
∗ (𝒾,
[𝔵+
[𝔵]
[𝔴]
[𝔴]
(8)
𝔴]
=
+×number
,×
𝔴]
=
, map
(𝒾)𝑑𝒾 = [𝔵+
(𝐹𝑅) 𝔖(𝒾)𝑑𝒾 = (𝐼𝑅) 𝔖 Definition
𝔖(𝒾,
𝜑)𝑑𝒾
: [49]
𝔖(𝒾,
𝜑)
∈
ℛ𝔴]
, +valued
Definition
1.×
[49]
A=
Definition
fuzzy
1.
A
fuzzy
1.
[49]
valued
fuzzy
map
number
𝔖
: 𝐾𝔵]
⊂, ℝ
valued
→
𝔖:[𝔵]
𝔽𝐾(5)
⊂
map
→
𝔖
isℝcall
([number
],[𝔵]
)[𝔵]
,A=
[𝜌.
(6)
=
𝜌.
[𝔵
[𝔵
[𝔵]
[
[
𝔴]
𝔴]
,
×
𝔴]
tion over
tion
[𝜇,
tion
over
𝜈].over
[𝜇, 𝜈].
[𝜇, 𝜈].
tion
over
tion
[𝜇,
tion
over
𝜈].
over
[𝜇,
𝜈].
[𝜇,
𝜈].
(8)
(𝐹𝑅)
(𝐹𝑅)𝔖(𝒾)𝑑𝒾
𝔖(𝒾)𝑑𝒾= (𝐼𝑅)
= (𝐼𝑅)𝔖 (𝒾)𝑑𝒾
𝔖 (𝒾)𝑑𝒾
= [𝔵=× 𝔖(𝒾,
𝔖(𝒾,
𝜑)𝑑𝒾
𝜑)𝑑𝒾
: 𝔖(𝒾,
:[𝔖(𝒾,
𝜑)[𝔵∈
𝜑)
ℛ𝔴]
∈([×ℛ,𝔴]
,(8)
([
], ,)[𝔵]
],, )[𝔵]
(6)
[𝔵
[𝔵]
[
[
×
𝔴]
=
×
𝔴]
,
=
=
×
×
𝔴]
𝔴]
,
,
e
𝜑
∈
(0,
1]
𝜑
whose
∈
(0,
1]
𝜑
𝜑
whose
-cuts
∈
(0,
define
1]
𝜑
whose
-cuts
the
define
𝜑
family
-cuts
the
of
define
family
I-V-Fs
the
of
𝔖
family
I-V-Fs
of
𝔖
:
𝐾
⊂
ℝ
→
Fejér
inequality).
Let
S∈×∈
HFSX
F0 , ,s), whose (6)
Theorem
9 (First
([,µ, ×υ(8)
][, 𝔴]
(𝐹𝑅)
(𝐼𝑅) fractional
𝔖(𝒾)𝑑𝒾
=fuzzy
𝔖 (𝒾)𝑑𝒾H.H.
= [𝔵
𝔖(𝒾,
𝜑)𝑑𝒾
: 𝔖(𝒾,
([ , =
], )[𝔵]
[𝔵
[𝔵]
[𝜑)
×
𝔴]
=Definition
×[𝔵]
𝔴]
, ℛ[49]
𝔴]
Definition 1. [49] A for
fuzzy
number
valued
map
𝔖
:
𝐾
⊂
ℝ
→
𝔽
1.
A
fuzzy
number
valued
map
𝔖
is
called
F-NV-F.
For
each
(7)
[𝜌.
[𝜌.
[𝔵]
+
∗
∗
∗
𝔵]
𝔵]
=
𝜌.
=
𝜌.
∗
(𝒾)
[𝔖
(𝒾,
[𝔖
(𝒾,
(𝒾,
(𝒾)
[𝔖
(𝒾,
(𝒾,
(𝒾,
𝔖
𝔖𝔽⊂
𝔖Riemannian
=υ𝜑-cuts
𝜑),
𝔖
𝜑)]
𝜑),
=𝜌.
𝔖
for
all
𝜑)]
𝜑),
∈𝔖
for
𝐾.
all
Here,
𝜑)]
𝒾𝔵]=
∈ϕfor
𝐾.
all
Here,
each
∈ϕ𝜑
for
𝐾.∈of
Here,
each
(0,(7)
1]
=define
define
the
family
I-V-Fs
S
µ,
⊂
R(𝒾)
→
are
by
,define
, [𝔵]
S
,I-V-Fs
allLet
𝜑[49]
∈
1],
where
denotes
the
collection
of
integrable
)of
[S
(the
[F-NV-F
][𝜇,
( 𝒾whose
)for
(of𝒾the
)]
∗funcϕ
ϕfamily
∗whose
∗𝔽
∗S
([be
],2.
)[49]
, of
[𝜌.
[𝜌.
[𝜌.
[𝔵]
[𝔵]
𝔵]
𝔵]
=F-NV-F
𝜌.
=
𝜌.
Cthe
2.
[49]
2.
𝔖
2.ϕ-cuts
:(0,
[49]
[𝜇,
Theorem
𝜈]𝜑Let
𝔖⊂
: [𝜇,
ℝ
𝔖Theorem
:→
𝜈]
[𝜇,
2.𝔽
Theorem
⊂ℛ
𝜈]
[49]
ℝ
⊂I-V-Fs
→
Let
→
𝔖
2.
𝔽
:],𝔖
[49]
[𝜇,
Let
𝜈]
: [𝜇,
ℝ
𝔖
:→
𝜈]
𝜈]
ℝbe
⊂
→=
𝔽
→
aℝ
F-NV-F
be
a,)be
F-NV-F
whose
a:)𝔖⊂
whose
define
𝜑-cuts
aℝK
F-NV-F
𝜑-cuts
be
define
family
agiven
be
define
whose
athe
of
the
family
I-V-Fs
𝜑-cuts
whose
I-V-Fs
𝜑-cuts
of
I-V-Fs
𝜑-cuts
define
the
family
of
!family
Definition
[49]
AF-NV-F
fuzzy
number
valued
map
𝔖family
: family
𝐾(7)
⊂ ℝI-V→
for
for
all
all
𝜑Let
∈
(0,
∈family
1],
(0,
1],
where
where
ℛ𝔽
denotes
the
the
collection
collection
of
Riemannian
of
Riemannian
integrable
funcfunc𝜑 ∈ (0, 1]Theorem
whoseTheorem
𝜑Theorem
-cuts
define
the
of
𝜑
∈
(0,
1]
whose
𝜑𝔖
-cuts
define
the
of
𝐾Let
⊂
ℝ
→
𝒦
are
given
by
([ ℛ
],:denotes
, ([
∗1.
∗ (.[𝜌.
∗ (.integrable
[𝜌.
[𝔵]
[𝔵]
𝔵]
𝔵]
=
𝜌.
=
𝜌.
(.
(.
(.
(.
functions
functions
𝔖
functions
𝔖
𝔖
,
𝜑),
𝔖
,
𝜑):
,
𝜑),
𝐾
𝔖
→
ℝ
,
𝜑):
,
are
𝜑),
𝐾
called
→
ℝ
,
𝜑):
are
lower
𝐾
called
→
and
ℝ
are
lower
upper
called
and
functi
low
up
(𝐹𝑅)
(𝒾)𝑑𝒾
tions
of
I-V-Fs.
𝔖
is
𝐹𝑅
-integrable
over
[𝜇,
𝜈]
if
𝔖
∈
𝔽
.
Note
that,
if
∗ ∗ of Riemannian
∗
∗integrable
for
all 𝜑 ∈ (0, 1], where ℛ(𝒾)
denotes
the
collection
∗ (𝒾,(𝒾,
∗(𝒾)
∗(𝒾)𝑑𝒾
∗define
∗ (𝒾,func([ , ], [𝔖
) (𝒾)
∗[𝜇,
1.𝜑)]
𝜑
∈
(0,
1]
whose
𝜑
-cuts
the
family
of
I-V-Fs
𝔖
e
(𝒾,
(𝒾)
[𝔖
(𝒾,
[𝔖
(𝒾,
(𝒾,
[𝔖
(𝒾)
(𝒾,
(𝒾)
[𝔖
(𝒾,
[𝔖
(𝒾,
(𝒾,
(𝒾,
(𝐹𝑅)
(𝐹𝑅)
(𝒾)𝑑𝒾
=
=
𝜑),
=
𝔖
𝜑),
𝜑)]
𝜑),
𝔖
for
𝔖
𝜑)]
all
=
𝜑)]
𝒾
for
∈
[𝜇,
for
all
=
𝜑),
𝜈]
all
𝒾
=
𝔖
∈
and
𝒾
[𝜇,
∈
𝜑),
𝜑)]
for
𝜈]
[𝜇,
𝜑),
𝔖
and
all
𝜈]
for
𝔖
and
𝜑)]
for
all
for
all
𝒾
for
∈
all
[𝜇,
for
all
𝜈]
all
𝒾
∈
and
𝒾
[𝜇,
∈
for
𝜈]
[𝜇,
and
all
𝜈]
and
for
[𝜇,𝜑),
𝜈]
ℝ
𝜈]
[𝜇,
𝒦
⊂𝜈]ℝ
⊂
are
→
ℝ
𝒦
given
→
𝒦
:
are
[𝜇,
by
are
given
𝜈]
𝔖
⊂
given
:
[𝜇,
ℝ
by
:
→
𝜈]
[𝜇,
𝔖
by
𝒦
⊂
𝜈]
𝔖
ℝ
⊂
are
→
ℝ
𝒦
given
→
𝒦
are
by
are
given
𝔖
given
by
𝔖
by
𝔖
𝔖⊂
𝔖
𝔖
𝔖
tions
tions
of
I-V-Fs.
of
I-V-Fs.
is
𝔖
𝐹𝑅
is
-integrable
𝐹𝑅
-integrable
over
over
[𝜇,
𝜈]
[𝜇,
𝜈]
if
if
𝔖
𝔖
∈
𝔽
∈
.
𝔽
Note
Note
that,
that,
if
if
[𝔖∗:(𝒾,
(𝒾,:→
𝔖 (𝒾) = 𝔖
𝔖:𝔖
𝜑)]
for
all
𝒾
∈
𝐾.
Here,
for
each
𝜑
∈
(0,
1]
the
end
point
real
for
all
𝒾
∈
𝐾.
Here,
for
all
∈
µ,
υ
,
ϕ
∈
0,
1
.
If
S
∈
F
R
and
∇
:
µ,
υ
→
R
,
∇
≥
0,
=
∇(
[
]
[
]
[
]
)
∗
∗
∗ ([µ, ν], ϕ)
∗
∗
1𝔖:1𝐾 F-NV-F.
1. [49] A
Definition
fuzzy
number
[49]
Aa fuzzy
map
number
𝔖: 𝐾∗ ∈⊂∗𝔽
ℝ
valued
𝔽 map
⊂ ℝ → 𝔽Foriseach
is∗µ1that,
called
called F
+
(𝒾,of
(𝐹𝑅)
(𝒾)𝑑𝒾
𝜑),I-V-Fs.
𝔖∗ (𝒾, 𝜑)Definition
are
Lebesgue-integrable,
then
𝔖
is
fuzzy
Aumann-integrable
func𝔖
tions
𝔖
is
𝐹𝑅 -integrable
over
[𝜇,
𝜈]1. 𝔖
ifvalued
𝔖fuzzy
.→
Note
ifmap
υ−
∗
∗
∗
∗
(𝒾)
[𝔖
(𝒾,
(𝒾,
=
𝜑),
𝔖
𝜑)]
for
all
𝒾
∈
𝐾.
Here,
for
each
Definition
1.
[49]
A
fuzzy
Definition
Definition
number
1.
valued
[49]
1.
[49]
A
map
A
fuzzy
𝔖
number
:
𝐾
number
⊂
ℝ
valued
→
𝔽
valued
map
𝔖
:
𝐾
𝔖
⊂
:
𝐾
ℝ
⊂
→
ℝ
𝔽
→
𝔽
is
called
F-NV-F.
For
each
is
ca
∗
∗
∗
∗
∗
∗
(.∈,1].
(.𝔖
(.𝔖
∗if
functions𝜑𝔖∈∗ (.
functions
𝔖the
𝜑):
𝐾(𝒾,
→
ℝ𝔖𝜑
are
called
lower
of[49]
𝔖if
.𝜈]
,[49]
𝜑),
𝔖
,and
𝜑):
𝐾
→
ℝ
called
low
(𝒾,
(𝒾,
(𝒾,
𝜑),
𝔖
𝜑),
𝔖
𝜑)
are
are
Lebesgue-integrable,
Lebesgue-integrable,
then
then
𝔖𝔖
is
𝔖
a∗over
is
fuzzy
fuzzy
Aumann-integrable
funcfunc𝔖1].
(𝒾,
(𝒾,
(𝒾,
(𝒾,
(𝒾,
(𝒾,
(𝒾,
(𝒾,
(𝒾,
(𝒾,
(𝒾,
∗2.
and
𝜑)
𝔖𝔖
𝜑)
and
𝜑)
𝔖
both
are
𝜑)
both
both
𝜑)
𝔖𝔽
𝜑)
and
𝜑)
and
𝔖
both
𝔖
𝜑)
are
𝜑)
both
bi
(0,, 𝜑),
1].
𝜑 ∈𝔖Then
𝜑(0,
(0,
is𝔖
𝐹𝑅-integrable
is∈
𝔖
𝐹𝑅-integrable
is𝜑)
1].
𝜑
𝐹𝑅-integrable
∈
over
Then
𝜑
(0,
∈
1].
[𝜇,
(0,
𝔖and
over
𝜈]
1].
Then
isDefinition
over
if𝜑upper
Then
𝐹𝑅-integrable
[𝜇,
and
𝜈]
[𝜇,
only
is𝔖functions
if
𝜈]
𝐹𝑅-integrable
and
isif1.
ifwhose
𝐹𝑅-integrable
𝔖
and
only
over
only
[𝜇,
𝔖
[𝜇,
and
𝜈]
only
if
and
if⊂
ifAumann-integrable
𝔖
and
only
only
if𝜑)
if
𝔖
be
F-NV-F.
be
an
Then
F-NV-F.
be
fuzzy
anFT
Definition
2.
Let
Definition
2.
𝔖a:[49]
:𝐾
[𝜇,
Let
𝜈]
𝔖
ℝ
:𝜑)
[𝜇,
→
𝔽
𝜈]
Let
⊂
𝔖
ℝ
:𝐾an
[𝜇,
→
𝜈]
⊂
ℝ𝔽are
→
𝔽
∗Then
∗Then
∗valued
∗over
∗(𝒾,
∗are
∗are
𝜑
∈(0,
(0,
1]
whose
𝜑
-cuts
∈𝔖Definition
(0,
define
1]
the
𝜑
family
-cuts
of
define
I-V-Fs
family
of
I-V-Fs
𝔖
: map
𝐾
⊂
ℝ
𝒦
are
given
:𝐾
⊂
ℝ
by
→
𝒦
∗ (𝒾,
over
[𝜇,
𝜈].
Definition
1.
[49]
A
fuzzy
number
[49]
Aifa𝜑)
fuzzy
map
number
⊂and
ℝ
valued
𝔖
:→
⊂
ℝ
→
is
called
F-NV-F.
For
is
each
called
∗ (.→𝔖𝔽
then
𝔖
𝜑)
are
Lebesgue-integrable,
then
𝔖
is
fuzzy
Aumann-integrable
func𝔖tion
(.
functions
𝔖
,
𝜑),
𝔖
,
𝜑):
𝐾
→
ℝ
are
called
lower
and
up
𝜑
∈
(0,
1]
whose
𝜑
-cuts
𝜑
∈
𝜑
(0,
define
∈
1]
(0,
1]
whose
the
whose
family
𝜑
-cuts
𝜑
-cuts
of
define
I-V-Fs
define
the
𝔖
the
family
family
of
I-V-Fs
of
I-V-Fs
𝔖
𝔖
:
𝐾
⊂
ℝ
→
𝒦
are
given
:
𝐾
⊂
:
𝐾
by
ℝ
∗ (𝒾, 𝜑),
Zdenoted
Zdenoted
∗ 𝜈],by
tion
tion
over
over
[𝜇,
[𝜇,
𝜈].
∗ (𝒾,(𝒾)
∗denoted
ν[𝜇,
[𝜇,
(𝐹𝑅)
[𝜇,
(𝒾)𝑑𝒾
(𝐹𝑅)
(𝐹𝑅)
(𝒾)𝑑𝒾
e[𝜇,
eeach
𝜈],
𝜈],
𝔖νall
by
𝔖
𝔖
,∈is
given
levelwise
,𝜈],
is𝒦
given
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by
, given
is
by
levelw
(𝒾)
[𝔖
(𝒾,
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𝔖
=
𝜑),
𝔖
𝜑)]
=
for
all
𝜑),
∈
𝔖
𝐾.
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𝜑)]
for
for
𝒾 )family
𝜑
𝐾.
∈(𝒾)𝑑𝒾
Here,
(0,
for
the
each
end
point
𝜑𝐾given
∈⊂(0,
real
1]
the
[𝜇,
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[𝜇,
[𝜇,1]
[𝜇,
S
S
)family
( 𝒾is
(𝔖
𝑅-integrable
𝑅-integrable
𝑅-integrable
over
[𝜇,over
𝜈].
over
Moreover,
[𝜇,
𝑅-integrable
𝜈].
[𝜇,𝜈].
Moreover,
𝜈].1]
if
𝑅-integrable
Moreover,
𝔖[𝔖
𝑅-integrable
over
is
if𝐹𝑅-integrable
[𝜇,
𝔖
if𝜑over
𝜈].
is𝔖
over
𝐹𝑅-integrable
Moreover,
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[𝜇,
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𝜈].
[𝜇,
over
Moreover,
𝜈].
Moreover,
𝔖over
𝜈],
over
then
ifby
𝐹𝑅-integrable
𝔖if𝜈],
is𝔖
𝜈],
then
𝐹𝑅-integrable
isI-V-Fs
then
𝐹𝑅-integrable
over
over
𝜈],
over
then
𝜈],
then
then
𝜑
∈
(0,
whose
𝜑
-cuts
∈
(0,
define
1]
whose
the
𝜑
-cuts
of
define
the
of
I-V-Fs
𝔖
:
𝐾
⊂
ℝ
→
are
:
ℝ
by
→
𝒦
∗ (𝒾,
∗if(𝒾,
s
−
1
∗
∗
∗
tion
over
[𝜇,
𝜈].
e
(𝒾)an
(𝒾)
[𝔖
[𝔖
(𝒾, 𝜑)]
𝔖 fuzzy
𝔖 (𝒾)
= [𝔖
𝔖 (𝒾,
𝜑)]
=
for
=2all
∈
𝐾.
𝜑),
𝔖
Here,
𝔖)over
𝜑)]
for
for
each
all
𝒾Let
∈d 𝒾𝐾.
∈𝔖
∈(0,
Here,
1]Here,
forℝend
for
each
point
𝜑be∈𝜑an
real
(0,
∈F
1
F-NV-F.
integral
𝔖(𝒾,
Definition 2. [49] Let 𝔖: [𝜇, 𝜈] ⊂ ℝ → 𝔽𝔖 be
Definition
2.2 for
[49]
:𝐾.
[𝜇,
𝜈]the
⊂
→(30)
𝔽each
2 ∗ (𝒾,S𝜑),
d∗ 𝒾∗𝜑),
4of
∇(all
( FR
)𝜑
∗ (𝒾,
∗Then
∗(𝒾,
∗𝔖
(.
(.𝜑),
𝔖
functions
𝔖
,ℝ
𝜑),
,+
𝜑):
𝐾
→
,𝒾are
𝜑),
𝔖
called
, 𝜑):
lower
𝐾
and
ℝ
upper
called
functions
lower
and
of be
𝔖
upper
.𝜑an
functions
µ𝜑)]
νafor
(𝒾) 𝔖
[𝔖∗𝜈]
(𝒾,
(𝒾,
[𝔖
(𝒾,
(𝒾,(.
µall
𝔖functions
𝔖
=: [𝜇,
𝔖→
all
∈𝔖(.
𝐾.
Here,
𝜑)]
for
for
each
𝒾are
∈
𝜑
𝐾.
∈are
Here,
(0,
1]functions
for
the
each
end
point
∈and
(0,
real
1] func
theTf
∗ (.
∗ℝ
Theorem
2. [49]
Let
⊂𝜑),
𝔽(𝒾)
be
F-NV-F
whose
𝜑-cuts
define
the
family
of
I-V-Fs
∗=
∗ (.
∗→
∗µ
F-NV-F.
Definition
2.
[49]
Let
𝔖
:
[𝜇,
𝜈]
⊂
ℝ
→
𝔽
(.
(.
(.
(.
[𝜇, 𝜈], denoted by (𝐹𝑅)
(𝒾)𝑑𝒾
[𝜇,
(𝐹𝑅)
(𝒾)𝑑𝒾
functions
𝔖
functions
functions
𝔖
𝔖
,
𝜑),
𝔖
,
𝜑):
𝐾
→
ℝ
are
,
𝜑),
called
,
𝜑),
𝔖
𝔖
,
lower
𝜑):
,
𝜑):
𝐾
→
and
𝐾
ℝ
→
upper
ℝ
are
called
called
lower
lower
of
and
𝔖
.
upper
upper
𝔖
𝜈],
denoted
by
𝔖
,
is
given
levelwise
by
,
is
given
levelw
∗ ∗ ∗whose
Theorem
Theorem
2. [49]
2. [49]
Let Let
𝔖: [𝜇,
𝔖:∗𝜈]
[𝜇,⊂𝜈]
ℝ𝔽→ 𝔽
be abe
F-NV-F
a F-NV-F
whose
𝜑-cuts
𝜑-cuts
define
define
the the
family
family
of I-V-Fs
of I-V-Fs
∗ (.ℝ⊂→
(.(𝒾,
∗(𝑅)
∗(.
∗ (𝒾,
∗and
∗ (𝒾,
∗ (𝒾,
∗.𝔖(𝒾,
functions
𝔖(𝒾)𝑑𝒾
functions
,ℝ
𝜑),
,be
𝜑):
𝐾
→
ℝ
,(𝐹𝑅)
are
𝜑),
𝔖
called
,(𝑅)
𝜑):
lower
𝐾
→
ℝ
are
upper
called
functions
lower
and
of
𝔖
upper
functions
(𝒾)𝑑𝒾
(𝒾)𝑑𝒾
(𝒾)𝑑𝒾
(𝒾)𝑑𝒾
(𝐹𝑅)
(𝒾)𝑑𝒾
(𝐹𝑅)
(𝒾)𝑑𝒾
(𝐼𝑅)
(𝐼𝑅)
(𝐼𝑅)
𝔖
𝔖
𝔖
=
=
𝔖
=
=𝜑)𝑑𝒾
𝔖
:𝔖(𝒾,
𝔖(𝒾,
=𝜑o𝜑
∗ (.
∗∗(𝑅)
Theorem
Let
:𝔖
[𝜇,
𝜈]
⊂([
𝔽
as(𝒾)𝑑𝒾
F-NV-F
whose
𝜑-cuts
define
the
family
of
[𝜇,
(𝒾)
[𝔖
(𝒾,
e𝔖∈are
by
𝔖𝔖
,for
is=
given
levelwise
by
(𝒾,
(𝒾,
(𝒾,
(𝒾,
(𝒾,
(𝑅)
(𝑅)
(𝑅)
(𝑅)
(𝑅)
(𝑅)
(𝑅)
(𝑅)
(𝒾)𝑑𝒾
(𝐹𝑅)
(𝒾)𝑑𝒾
(𝐹𝑅)
(𝐹𝑅)
(𝐹𝑅)
(𝒾)𝑑𝒾
(𝒾)𝑑𝒾
=
𝜑),
𝔖
𝜑)]
for
all
∈∗(𝑅)
[𝜇,
𝜈]
and
all
𝜈]2.⊂[49]
ℝ(𝐹𝑅)
→
𝒦
by
𝔖
𝔖 : [𝜇,(𝐹𝑅)
=given
=→𝔖
=
𝜑)𝑑𝒾
𝔖
,𝔖
𝔖
𝜑)𝑑𝒾
=
𝜑)𝑑𝒾
,𝜈],
𝔖
,denoted
=
𝜑)𝑑𝒾
𝔖
=
𝔖
𝔖
𝜑)𝑑𝒾
𝜑)𝑑𝒾
𝔖(𝐹𝑅)
𝜑)𝑑𝒾
,𝔖
𝜑)𝑑𝒾
𝜑)𝑑𝒾
,(𝒾)𝑑𝒾
𝔖
, I-V-Fs
𝜑)𝑑𝒾
𝔖
𝔖
𝜑)𝑑𝒾
𝜑)𝑑𝒾
𝔖
𝔖
𝔖
𝔖
𝔖
If
S
HFSV
µ,
υ
,
F
,
,
inequality
(30)
is
reversed.
]
)
∗ (𝒾,
∗ (𝒾,
∗then
∗ (𝒾,
∗𝒾(𝒾,
∗
∗
0
be
an
F-NV-F.
be
fuzzy
an
F-NV-F.
integral
Then
ofall𝔖fuzzy
over int
Definition
2. are
[49]given
Let
Definition
2.[𝔖
ℝ
[49]
→ ∗𝜑),
𝔽
Let
𝔖(𝒾,
:𝔖
[𝜇,𝜑)]
𝜈]
⊂forℝfor
→Then
𝔽
(𝒾)
[𝔖
(𝒾, 𝜑),
(𝒾,
=⊂
=
𝔖
𝜑)]
all
all
𝒾
∈
𝒾
[𝜇,
∈
𝜈]
[𝜇,
and
𝜈]
and
for
for
all
[𝜇,⊂𝜈]
ℝ⊂→ℝ𝒦
→𝒦
are
given
by 𝔖
by: [𝜇,
𝔖 𝜈](𝒾)
𝔖 :𝔖[𝜇,:𝜈]
∗ (𝒾,
∗2.
beLet
an
fuzzy
beall
an
be integral
F-NV-F.
an F-NV-F.
of
Then
𝔖Then
over
fuzzf
Definition
2.𝔖[49]
Let
Definition
𝔖𝜈]
Definition
:(𝒾,
[𝜇,
𝜈] 𝔖
⊂only
ℝ
[49]
→
2.
Let
𝔖
: [𝜇,
𝔖
:∈𝜈]
[𝜇,
⊂
𝜈]∗𝜈]
ℝ
⊂Then
→
ℝ𝔽→both
𝔽
(𝒾)
[𝔖
(𝒾,
=[𝜇,
𝜑)]
all
𝒾F-NV-F.
and
for
𝜈]1].
⊂ℝ
→𝒦
are
given
byby
(𝒾,
(𝒾,
𝜑)
and
𝔖
𝜑)
are
𝜑 :∈[𝜇,(0,
Then
𝔖
is(𝒾)𝑑𝒾
𝐹𝑅-integrable
over
if𝜑),
and
if𝔖𝔽[49]
𝔖
∗(𝒾)𝑑𝒾
(9)
(9)
(9) integ
∗)for
(8)
[𝜇,
(𝐹𝑅)
[𝜇,
(𝐹𝑅)
(𝒾)𝑑𝒾
(𝒾)𝑑𝒾
(𝐹𝑅) 𝔖𝔖
(𝒾)𝑑𝒾
(𝐹𝑅)
(𝒾)𝑑𝒾
(𝐼𝑅)
denoted
𝜈],
denoted
by
𝔖
,→
is
given
levelwise
by
,𝜑)
is[𝜇,
given
levelwise
byboth
∗ (𝒾,(9)
∗ (𝒾,
𝔖
𝔖𝜈],
=
𝔖(𝒾,
𝜑)𝑑𝒾
:𝔖
𝔖(𝒾,
𝜑)
∈
ℛ
,all
=
=co
be
an
F-NV-F.
Then
be
fuzzy
an
F-NV-F.
integral
Then
of
𝔖
fuzzy
over
Definition
2.
[49]
Let
Definition
𝔖
:
[𝜇,
𝜈]
⊂
2.
ℝ
[49]
𝔽
Let
:
[𝜇,
𝜈]
⊂
ℝ
→
𝔽
([
],∈
,where
(𝒾,
(𝒾,
𝜑)
and
and
𝔖
𝔖
𝜑)
𝜑)
both
are
are
𝜑=
∈𝜑(𝐼𝑅)
(0,
∈
1].
(0,
1].
Then
Then
𝔖
is
𝔖
𝐹𝑅-integrable
is
𝐹𝑅-integrable
over
over
[𝜇,
𝜈]
[𝜇,
if
𝜈]
and
if
and
only
only
if
𝔖
if
𝔖
for
all
𝜑
∈
for
(0,
all
1],
𝜑
for
(0,
1],
ℛ
𝜑
where
∈
(0,
1],
ℛ
where
ℛ
denotes
the
denotes
collection
the
denotes
collection
of
Riemann
the
o
∗
∗
[𝜇, 𝜈],
[𝜇, 𝜈],
[𝜇,
(𝒾)𝑑𝒾
(𝐹𝑅)
(𝐹𝑅)([𝔖(𝒾)𝑑𝒾
denoted
by (𝐹𝑅)
𝔖𝜈],
denoted
by
, is1by
given
levelwise
, is=
, (𝐼𝑅)
Proof. Since
S is
a s-convex,
then
for
ϕ ∈denoted
have
[0,
], we
],𝔖by
)(𝒾)𝑑𝒾
([given
) levelwise
([levelwise
,(𝒾)𝑑𝒾
,is], given
, ], ) by by
∗ (𝒾,
(𝒾)𝑑𝒾
(𝐹𝑅)
𝔖
𝔖
=
𝔖(𝒾,
𝜑)
(𝒾,
𝜑)
and
𝔖
𝜑)
both
are
𝜑𝑅-integrable
∈ (0, 1]. Then
𝔖
is
𝐹𝑅-integrable
over
[𝜇,
𝜈]
if
and
only
if
𝔖
[𝜇,
over
[𝜇,
𝜈].
Moreover,
if
𝔖
is
𝐹𝑅-integrable
over
𝜈],
then
∗ 𝔖
[𝜇, 𝜈], denoted
(𝐹𝑅)
[𝜇,
(𝒾)𝑑𝒾
(𝐹𝑅)
(𝒾)𝑑𝒾
by
𝜈],
denoted
𝔖
by
,
is
given
levelwise
by
,
is
given
levelwise
by
(𝒾)𝑑𝒾
(𝒾)𝑑𝒾
(𝒾)𝑑𝒾
(𝒾)𝑑𝒾
(𝒾)𝑑𝒾
(𝒾)𝑑𝒾
(𝐼𝑅)
(𝐼𝑅)
(𝐼𝑅)
(𝐼𝑅)
(𝐼𝑅)
(𝐼𝑅)
=
=
𝔖
=
𝔖
𝔖
=
=
=
𝔖
𝔖
(𝐹𝑅)
tions
of𝔖 I-V-Fs.
tions
is over
𝐹𝑅
of
-integrable
𝔖𝜈],
I-V-Fs.
is𝜈],
𝐹𝑅
-integrable
𝔖over
is 𝐹𝑅
[𝜇,-integrable
𝜈]over
if (𝐹𝑅)
[𝜇, 𝜈]over
if𝔖(𝒾)𝑑𝒾
[𝜇, 𝜈
[𝜇,
[𝜇,
𝑅-integrable
𝑅-integrable
overover
[𝜇, 𝜈].
[𝜇,Moreover,
𝜈]. Moreover,
𝔖ifis𝔖of
𝐹𝑅-integrable
is I-V-Fs.
𝐹𝑅-integrable
over
then
then
iftions
for all 𝜑 ∈ (0, 1], where
ℛ([ , over
for
all
𝜑 𝔖(𝒾,
∈ ∗(0,𝔖
1],(𝒾)𝑑𝒾
where
ℛ([∈𝔖(𝒾,
the
collection
integrable
funcdenotes
the
co
2µνif 𝔖 isof𝐹𝑅-integrable
µυ
µυ
[𝜇,
𝑅-integrable
[𝜇, 𝜈]. Moreover,
over
then
1 = (𝐹𝑅)
], ) denotes
], )𝜑)𝑑𝒾
,ℛ
(8)
∗ (𝒾,
∗𝜈],
(𝒾)𝑑𝒾
(𝐹𝑅)
(𝒾)𝑑𝒾
(𝒾)𝑑𝒾
(𝐼𝑅)
(𝐼𝑅)
𝔖
=
=
𝜑)𝑑𝒾
:
𝔖(𝒾,
=
𝜑)
:
,
𝔖(𝒾,
𝜑)
∈
S
≤𝔖Riemannian
S
,
ϕ
+
S
,
ϕ
([
],
)
,
s
(𝒾,
(𝒾,
(𝒾,
(𝒾,
(𝒾,
∗ µ+ν ,𝔖ϕ
∗
∗
𝜑),=𝔖
𝔖
𝜑)
𝜑),
are
𝔖
Lebesgue-integrable,
𝜑)
Lebesgue-integrable,
then
then
a fuzzy
𝔖 is,Auma
then
a𝔖(𝒾
fu
𝔖
2∗
∗ξ )all
∗Lebesgue-integrable,
1𝔖
−
µ𝜑)
+𝜑),
ξυ
+(
1(𝐼𝑅)
−ℛξ𝜑)𝑑𝒾
υ𝔖 (𝒾)𝑑𝒾
((𝐼𝑅)
)([are
(8)
(𝒾)𝑑𝒾
(𝒾)𝑑𝒾
(𝐹𝑅)
(𝐹𝑅)
(𝐹𝑅)
(𝒾)𝑑𝒾
(𝒾)𝑑𝒾
𝔖(𝒾)𝑑𝒾
𝔖
𝔖1],
𝔖are
=∗ where
=ξµ(𝐼𝑅)
=
𝔖(𝒾,
: 𝐹𝑅
=
𝜑)𝔖∈
=isthe
ℛ
𝔖(𝒾,
𝔖(𝒾,
𝜑)𝑑𝒾
:[𝜇,
:o𝔖
for
𝜑
∈
(0,
denotes
([ collection
], ) 𝜑)𝑑𝒾
)𝔖(𝒾,
, over
,is], 𝔖
(𝐹𝑅)
(𝒾)𝑑𝒾
tions of I-V-Fs. 𝔖 is 𝐹𝑅 -integrable(𝐹𝑅)
over [𝜇,
𝜈]
if
𝔖
tions
of
I-V-Fs.
𝔖
-integrable
𝜈]
∈
𝔽
.
Note
that,
if
(𝒾,
(𝒾,
(𝑅)
(𝑅)
(𝒾)𝑑𝒾
=
𝔖
𝜑)𝑑𝒾
,
𝔖
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𝔖
(8)
(0, 𝜑
(0,
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(𝒾)𝑑𝒾
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∗ ∈
for all 𝜑for
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1],
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1],
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1].
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1],
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([
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) 1],
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-integrable
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a 𝜈].
fuzzy
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func𝜑),
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arethe
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then
𝔖∗ (𝒾, 𝜑), 𝔖
𝔖
2µν
µυwhere
µυ
∗ (𝒾,
for
allover
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1],over
where
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1],
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integrable
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func1ℛ𝜑
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=
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, (𝑅)
𝔖
𝔖∈
NV-Fs
NV-Fs
NV-Fs
[𝜇,
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s S
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ν , 1],
2for
ξµ
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υ) denotes
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],∗ξ𝜑
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([
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) ],ξdenotes
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𝜑)
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tion over [𝜇, 𝜈].
tion
over
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𝜈].
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(𝐹𝑅)
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tions
of∈I-V-Fs.
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tions
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of
𝔖over
is
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[𝜇,
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if) ⊂2.
over
𝔖
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if𝔽
𝔖
∈
.ℝ
Note
that,
ifadefin
∈fu𝔽
for
all 𝜑
(0, 1], where
ℛ𝜑
∈, I-V-Fs.
(0,
where
ℛ
denotes
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denotes
Riemannian
the
collection
integrable
of
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funcin
Theorem
Theorem
2.
[49]
Letthe
2.
Theorem
𝔖
[49]
:-integrable
[𝜇,
Let
𝔖ℝ[49]
:of
[𝜇,
→
𝔽
𝜈]
Let
⊂
𝔖
ℝ
:
[𝜇,
→
𝔽
𝜈]
⊂
→
𝔽
be
a
F-NV-F
be
whose
a
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𝜑-cuts
be
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𝜑
([-integrable
], ) 1],
([
],𝜈]
,
(9)
(𝐹𝑅) over
(𝒾)𝑑𝒾
(𝐹𝑅)
(𝐹𝑅)
(𝒾)𝑑
tions of
I-V-Fs.
𝔖
is
tions
𝐹𝑅
tions
-integrable
of
I-V-Fs.
of
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over
𝔖
is
𝔖
[𝜇,
𝐹𝑅
is
𝜈]
-integrable
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if
-integrable
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over
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𝜈]
[𝜇,
𝜈]
if
if
𝔖
𝔖if(
∈
𝔽
.
Note
that,
tion
over
[𝜇,
𝜈].
(𝒾)𝑑𝒾
(𝐼𝑅)
=
𝔖
∗
∗
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said
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to
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be
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set,
if,
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if,
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Definition
Definition
Definition
3. [34] A3.set
[34]
3.𝐾[34]
Definition
A
=
set
A
set
𝜐]
𝐾
Definition
=
⊂
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Definition
ℝ
3.
=
[34]
𝜐]
⊂
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A
ℝ
3.
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ℝ
3.
𝐾
[34]
A
=
set
A
set
𝜐]
𝐾
=
⊂
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ℝ
=
𝜐]
⊂
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ℝ
⊂
ℝ
∗ (𝒾,
∗∈(𝒾,
(𝒾,𝔖of
(𝒾,
(𝐹𝑅)
(𝒾)𝑑𝒾
(𝐹𝑅)
(𝒾)𝑑𝒾
𝜑),I-V-Fs.
𝔖 (𝒾, ∗𝜑)𝔖 are
𝜑),
𝔖
𝜑)ℝ
are
then
𝔖
aℝfuzzy
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then
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fuzzy
Aumannfunc𝔖∗ (𝒾,of
tions
tions
is𝔖∗Lebesgue-integrable,
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-integrable
𝔖
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-integrable
if𝒦
over
𝔖
[𝜇,
𝜈]
if𝔽(𝒾)
𝔖
.𝜑),
Note
that,
if
𝔽𝜑
(𝒾)𝑑𝒾
(𝐼𝑅)
(𝐼𝑅)
=I-V-Fs.
=𝜈]
𝔖𝒦
(𝒾)
[𝔖
[𝔖
(𝒾,
=
=
𝜑),
𝔖
for
⊂
:over
[𝜇,
→is
𝜈]Lebesgue-integrable,
⊂
are
ℝ𝜈]
:Lebesgue-integrable,
[𝜇,
→
given
𝜈]is
⊂by
are
→
given
𝒦
by
are
𝔖∈
given
by
𝔖
𝔖
𝔖
∗𝔖
∗(𝒾)𝑑𝒾
∗ (𝒾,
∗all
:∗[𝜇,
(𝒾,
(𝒾,
(𝒾,(𝒾)𝑑𝒾
(𝒾,
(𝒾,
𝜑),
𝔖∈have
are
Lebesgue-integrable,
𝔖
𝜑),
𝔖
𝜑)
𝜑)
are
then
Lebesgue-integrable,
𝔖
is 𝔖
a Let
fuzzy
Aumann-integrable
then
then
𝔖 [𝔖
is
𝔖∗𝔽(𝒾,
a(𝒾)
is𝜑)]
fuzzy
a=fuzzy
funcAum
A
𝔖[0,
𝔖
𝔖
Theorem𝐾,
2.𝜉[49]
Let
𝔖
:𝜉[0,
[𝜇,
𝜈] ⊂
→𝐾,have
𝔽𝜉 ∈be
Theorem
2. [49]
𝔖: [𝜇,
𝜈] ⊂
ℝ
→
a(𝒾,1],
F-NV-F
whose
𝜑-cuts
define
the
family
ofare
I-V-Fs
be
a F-NV
(𝐼𝑅)
∗𝐾,
∗𝜑),
=
𝔖
[0,
[0,
[0,𝜑)
∈ [0,
𝐾,
1],
𝜉𝐾,
∈we
∈have
1],
we
1],ℝwe
have
𝜉
𝐾,
∈
we
𝜉
1],
we
1],
we
have
have
∗
∗
µυ
µυ
tion
over
[𝜇,
𝜈].
tion
over
[𝜇,
𝜈].
(𝒾, 𝜑)
(𝒾, 𝜑
(𝒾,
𝜑), 𝔖over
are
𝜑),
𝔖(0,
𝜑)
are
then
𝔖
is
a𝔖
fuzzy
Aumann-integrable
then
𝔖over
isit
a[𝜇,
fuzzy
Aumann-in
func𝔖By
𝔖∗Lebesgue-integrable,
∗ (𝒾,multiplying
(31)
by
∇tion
and
integrate
by
ξ over
over
=Lebesgue-integrable,
∇is
(𝒾,
𝜑)
∈
1].
𝜑
Then
∈
(0,
𝔖
1].
𝜑
Then
∈
𝐹𝑅-integrable
(0,
𝔖
1].
is
Then
𝐹𝑅-integrable
over
𝔖
is
[𝜇,
𝐹𝑅-integrable
𝜈]
if
and
only
𝜈]
if
if
and
𝔖
[𝜇,
only
𝜈]
if
Theorem
2.
[49]
Let
:
[𝜇,
𝜈]
⊂
ℝ
→
𝔽
be
a
F-NV-F
whose
𝜑
tion
[𝜇,
𝜈].
tion
over
over
[𝜇,
𝜈].
[𝜇,
𝜈].
∗
∗
1
−
ξ
µ
+
ξυ
ξµ
+(
1
−
ξ
υ
(
)
)
(0,𝒾𝒿
𝜑 ∈by
(0,𝔖
1]. (𝒾)
For
∈
1],tion
ℱℛ
denotes
of
𝐹𝑅-integrable
F- by 𝔖 (𝒾) = [𝔖∗ (𝒾, a𝜑
[𝔖[𝜇,
(𝒾,
=all
𝜑),
𝔖
𝜑)]
𝒾 ∈ [𝜇,the
𝜈]𝒾𝒿collection
and
for
all
areall
given
⊂all
ℝ→
𝒦 are given
𝔖 : [𝜇, 𝜈] ⊂ ℝ → 𝒦 for
𝔖 : [𝜇,
([ , for
], [𝜇,
)all
∗𝜑(𝒾,
𝒾𝒿
𝒾𝒿
𝒾𝒿𝜈]
𝒾𝒿
𝜈].
over
𝜈].
(0,
(0,
for all
for[0,all
𝜑 1tion
∈]𝜑,(0,
∈over
1].
(0,
1].
For
For
all
all
𝜑 ∈𝜑
∈ 1],
1],
ℱℛ
ℱℛ
denotes
denotes
the
the
collection
collection
of
all
of
all
𝐹𝑅-integrable
𝐹𝑅-integrable
F- F- [𝜇,
([
([
],
)
],
)
,
,
we
obtain
∈ 𝐾.
∈[49]
𝐾.
∈over
∈𝒦
𝐾.
∈given
𝐾.
∈[𝜇,
𝐾.F-NV-F
(10)
(10)
(10)
(10)
𝑅-integrable
𝑅-integrable
[𝜇,
𝜈].
𝑅-integrable
over
Moreover,
[𝜇,
𝜈].
over
Moreover,
if
𝔖
is
𝜈].
𝐹𝑅-integrable
if
Moreover,
is =
𝐹𝑅-integrable
ifover
𝐹𝑅-integ
𝜈],∗over
the(1
(𝒾)
[𝔖
(𝒾,
(𝒾,
𝔖
𝜑
:𝐾.
[𝜇,
𝜈]
⊂
ℝ
→⊂
are
by
𝔖𝔖
𝔖
over
[𝜇,
𝜈].
∗ (𝒾,
∗𝔖
(0,
all
𝜑
∈
(0,
1].
For
all
𝜑
∈
1],
ℱℛ
denotes
the
collection
of
all
𝐹𝑅-integrable
FTheorem
2.
[49]
Let
Theorem
𝔖
:
[𝜇,
𝜈]
⊂
2.
ℝ
→
𝔽
Let
𝔖
:
[𝜇,
𝜈]
ℝ
→
𝔽
be
a
F-NV-F
whose
𝜑-cuts
be
a
define
the
whose
family
𝜑-cuts
ofis𝜑),
I-V-Fs
define
([
],
)
(𝒾,
,
(1
(1
(1
(1
(1
(1
𝜑)
and
𝔖
𝜑)
both
are
𝜑 ∈ (0, 1]. Then 𝔖 for
isNV-Fs
𝐹𝑅-integrable
over
[𝜇,
𝜈]
if
and
only
if
𝔖
𝜑
∈
(0,
1].
Then
𝔖
is
𝐹𝑅-integrable
over
[𝜇,
𝜈] ifth
a
𝜉𝒾
𝜉𝒾
𝜉𝒾
𝜉𝒾
𝜉𝒾
𝜉𝒾
+
−
𝜉)𝒿
+
+
−
𝜉)𝒿
−
𝜉)𝒿
+
−
𝜉)𝒿
+
+
−
𝜉)𝒿
−
𝜉)𝒿
∗Theorem
NV-Fs
NV-Fs
overover
[𝜇,Theorem
𝜈].
[𝜇, 𝜈]. 2.[49] Let 𝔖
:
[𝜇,
Theorem
𝜈]
⊂
ℝ
2.
→
[49]
2.
𝔽
[49]
Let
Let
𝔖
:
[𝜇,
𝔖
:
𝜈]
[𝜇,
⊂
𝜈]
ℝ
⊂
→
ℝ
𝔽
→
𝔽
be
a
F-NV-F
whose
𝜑-cuts
be
a
be
define
F-NV-F
a
F-NV-F
the
whose
family
whose
𝜑-cuts
of
I-V-Fs
𝜑-cuts
de
R
NV-Fs over [𝜇, 𝜈].
∗
∗
𝜑
∈
(0,
1].
Then
𝔖
is
𝐹𝑅-integrable
over
[𝜇,
𝜈]
if
and
only
if
1
2µν
µυ
Theorem
Theorem
𝔖𝔖are
: [𝜇,
𝜈]𝜈],
⊂
2.⊂
ℝby
[49]
→𝔖
𝔽Let
𝜈](𝒾,
⊂𝜑),
ℝwhose
be𝔖=
a: [𝜇,
F-NV-F
𝜑-cuts
be [𝜇,
a F-NV-F
define
whose
family
𝜑-cuts
of is
I-V-Fs
define
(𝒾)
[𝔖
(𝒾)
[𝔖
(𝒾,
(𝒾,
𝔖𝔖𝔽(𝒾,
𝜑)]
=
for
𝜑),
𝒾the
∈𝔖[𝜇,
𝜈]
𝜑)]
and
for
allall𝒾the
∈
𝜈]2.⊂[49]
ℝ∗→Let
𝒦over
: [𝜇,
given
𝜈]
ℝ
→
𝒦
are
given
by→
𝑅-integrable over [𝜇, 𝜈].
Moreover,3.if[34]
𝔖𝔖 A
is: [𝜇,
𝐹𝑅-integrable
then
𝑅-integrable
over
𝜈].
Moreover,
𝔖
𝐹𝑅-integ
∗all
∇
dξ
∗ (𝒾,
∗for
=
∞)
isby
said
to
be
aare
convex
set,
if,
for
all
𝒾,=
𝒿(𝒾)𝑑𝒾
∈(𝑅)
Definition
𝐾=
𝜐]
⊂
ℝϕ [𝜇,
µ+→
ν ,𝒦
0(0,
(𝒾)
[𝔖
(𝒾,
(𝒾)
(𝒾)
[𝔖
(𝒾,if[𝜇,
(𝒾,
ξµ
+(
1𝜈]
−ℝ
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𝒾∗𝜑),
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𝜈]
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are
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for
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for
all
all
𝒾,
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∈
𝒿
∈
Definition
Definition
3.
[34]
3.
[34]
A
set
A
set
𝐾
=
𝐾
=
𝜐]
⊂
𝜐]
ℝ
⊂
ℝ
𝑅-integrable
over
[𝜇,
𝜈].
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if
is
𝐹𝑅-integrable
over
∗
∗
∗
∗
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
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=
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𝜑)]
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for
all
𝜑),
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𝜈]
𝜑)]
and
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for
all
all
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:The
[𝜇,
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→
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Definition
Definition
Definition
4. [34]
The
4.1],
[34]
4.
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[34]
Definition
The
𝜐]
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→
Definition
ℝ
𝔖:
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4.ℝ
→
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𝜐]
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→
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ℝ
[34]
4.
𝔖:
[34]
The
The
𝔖:
→
ℝ
𝔖:
→
𝜐]
ℝ
→
ℝ
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is
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harmonically
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harmonically
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harmonically
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function
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function
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function
harmonically
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𝜐]
harmonically
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∈
(0,
1].
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𝜑
∈
𝐹𝑅-integrable
(0,
1].
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over
𝔖
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𝜈]
if
and
only
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and
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have
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said
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𝒾,𝜑)
𝒿𝜈]
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Definition
[34]
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µυ
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+
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𝔖 𝔖
≤ (1[𝜇,
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≤𝑅-integrable
𝔖
𝜉)𝔖(𝒾)
−𝜉𝔖(𝒿),
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𝜉)𝔖(𝒾)
≤
𝜉)𝔖(𝒾)
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,
+
S
ϕ
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𝜉)𝒿
+𝜈].
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− 𝜉)𝒿(9)
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=
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=
, 𝜐].
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, (𝑅)
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1],
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1],
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where
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For
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For
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oc
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harmonically
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([
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𝔖
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Definition 4. [34] The 𝔖: [𝜇, 𝜐] →
ℝ is 𝔖
called
function
𝜐]
(𝒾)𝑑𝒾
=if(𝐼𝑅)
∗convex
1 a∗harmonically
µυ
(9)
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Definition
Definition
4. [34]
4. [34]
TheThe
𝔖: [𝜇,
𝔖:𝜐]
𝜐]ℝ
→ ℝis
called
is called
a
harmonically
a
harmonically
convex
convex
function
function
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on
𝜐]
if
𝜐]
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,
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1
−
ξ
µ
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(
)
)
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4. 1],
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The 𝔖: [𝜇,denotes
𝜐] → ℝ the
called
convex
function
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if
 R a harmonically
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s
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For
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=
𝔖 (11)
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], ) den
µυ
𝒾𝒿+ 𝒾𝒿
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,
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1].
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1],
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𝜉𝒾𝔖+𝒾𝒿𝔖 − 𝜉)𝒿
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𝜉𝔖(𝒿),
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0
ξµ
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1
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υ
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1
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ξ
υ
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)
)
,
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𝜈].
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toc
Definition
Definition
3.
Definition
set
3.
𝐾(𝒾)𝑑𝒾
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3.
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set
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ℝ be
(𝒾)𝑑𝒾
(𝐼𝑅)+A𝜉𝔖(𝒿),
(𝐼𝑅)
= [34]
𝔖 [34]
= [34]
𝔖
(1𝜉)𝒿
𝜉𝒾 +𝜉𝒾(1
+−
−−
𝜉)𝒿
(1
𝔖
≤
𝜉)𝔖(𝒾)
(11)
NV-Fs
over
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𝜈].have
𝜉𝒾
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−
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1],
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1],
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∈ [0,
where
𝔖(𝒾)
≥𝜉𝜑∈0∈∈(0,
for
all
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If∈(0,
(11)
isthe
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𝜑 1],
∈ (0,
1].
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1],
1].ℱℛ
all
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for
all𝜐]
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all𝒿⊂𝒾,
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∈for
𝜐],
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∈
1],
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where
𝔖(𝒾)
𝔖(𝒾)
≥
0≥
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𝔵,
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∈ (0,
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all all
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numbers.
numbers.
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𝔵, 𝔴𝔵,∈𝔴𝔽∈ 𝔽and
∈
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then
ℝ,
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1]
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𝐾ℝ1],
⊂are
→ℝ𝒦
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[𝔵]
[𝔴]
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if
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:⊂
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→
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are
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by
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ov
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2.
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𝜈]
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s the
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partial
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1.
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A
1.
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valued
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valued
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2
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all
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for
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each
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1.
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𝐾
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called
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tions
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and
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numbers.
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numbers.
numbers.
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Let
𝔵,
𝔴
𝔵,
∈
𝔴
𝔽
∈
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and
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for
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µ ∗are
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𝜑),
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𝔖
𝜑),
are
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𝜑)
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(.
Let
𝔵,if,
𝔴
∈
Let
𝔵,
𝔴
∈
𝔽
ℝ,
then
we
and
have
𝜌𝔖𝜌.
∈
ℝ,
then
we
have
functions
𝔖
functions
𝔖
𝔖
,∈
𝜑),
𝔖
,(𝒾)𝑑𝒾
𝜑),
, (.
𝜑):
,Let
𝜑),
𝔖
𝐾
→
𝔖
,⊂[𝔵]
𝜑):
ℝ
, 𝜑):
𝐾
are
→
𝐾
called
ℝ
→
ℝ[𝔵]
lower
are
called
called
and
lower
upper
lower
functions
upper
upper
function
of
func
.
∗(𝐹𝑅)
∗𝔽
∗and
∗(𝒾)𝑑𝒾
∗(.
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∗ℝ
(𝒾)𝑑𝒾
(𝒾)𝑑𝒾
(𝒾)𝑑𝒾
(𝐹𝑅)
(𝒾)𝑑𝒾
(𝐼𝑅)
(𝐼𝑅)
(𝐼𝑅)
𝔖
𝔖
𝔖
=
=
𝔖
=
𝔖
𝔖
𝔖(𝒾,
𝜑)𝑑𝒾
=
𝔖(𝒾,
𝔖(𝒾,
𝜑)𝑑𝒾
𝜑)
𝜑)𝑑𝒾
𝔖(𝒾,
∈
:[𝔵]
ℛAumann-int
𝔖(𝒾,
𝜑)
∈
𝜑)
∈,([𝔖
ℛ
be
an
F-NV-F.
be
an
be
F-NV-F.
an
Then
F-NV-F.
fuzzy
Then
Then
integral
fuzz
in
Definition
Definition
Let
2.
[49]
2.
𝔖
:[49]
[𝜇,
𝜈]
Let
𝔖
:[𝜌.
[𝜇,
ℝ
:→
𝜈]
[𝜇,
𝔽=
⊂
𝜈]
ℝ
⊂
→
ℝ
𝔽=
→
𝔽
(7)
[𝜌.
[𝜌.
[𝜌.
[𝜌.
[𝜌.
[𝔵]
[𝔵]
[𝔵]
[𝔵]
[𝔵+
[𝔵+
[𝔴]
[𝔵]
[𝔴]
𝔵]
𝔵]
𝔵]
𝔵]
𝔵]
𝔵]=:,and
=
=
𝜌.
=
𝜌.
=
𝜌.
𝜌.
=
𝜌.
([
], (5)
)ℛ
𝔴]
=
+
,are
𝔴]
=
+
, fuzzy
,o
(.[49]
(.(𝐹𝑅)
functions
functions
𝔖2.Definition
𝔖
,∗𝜑),
,Let
𝜑),
𝔖
𝔖
,2.
𝜑):
, 𝔵]
𝜑):
𝐾𝜈]
→
𝐾
→
are
are
called
called
lower
lower
and
and
upper
upper
functions
functions
of
of
𝔖
.of
𝔖
.of
∗ (.
be
be
an
an
F-NV-F.
F-NV-F.
Then
Then
fuzzy
fuzzy
integral
integral
𝔖
over
𝔖
ove
Definition
Definition
[49]
2.
Let
𝔖
:
[𝜇,
𝔖
:
[𝜇,
⊂
𝜈]
ℝ
⊂
→
ℝ
𝔽
→
𝔽
(7)
(7)
[𝜌.
[𝜌.
[𝔵]
[𝔵]
𝔵]
=
𝜌.
=
𝜌.
(5)
[𝔵+
[𝔵+
[𝔵+
[𝔵]
[𝔴]
[𝔵]
[𝔵]
[𝔴]
[𝔴]
tion
over
tion
[𝜇,
tion
over
𝜈].
over
[𝜇,
𝜈].
[𝜇,
𝜈].
𝔴]
=
+
,
𝔴]
𝔴]
=
=
+
+
,
,
It is a partial order relation.
From which, we
have
[𝜇, 𝜈],
[𝜇, 𝜈],
[𝜇, 𝜈],
(𝐹𝑅)
(𝐹𝑅)
(𝒾)𝑑𝒾
(𝐹𝑅)
denoted
denoted
bydenoted
by
by
𝔖=
𝔖(𝒾)𝑑𝒾
,𝔖is
given
,, is𝔴]
levelwise
given
, is given
levelwise
by
levelwise
by
by
(5)
[𝔵+
[𝔵+
[𝔵]
[𝔴]
[𝔵]
[𝔴]
𝔴]
+(𝒾)𝑑𝒾
=F-NV-F.
+×F-NV-F.
, ,fuzzy
[𝜇, 𝜈],
[𝜇, 𝜈],
(𝐹𝑅)
(𝐹𝑅)
(𝒾)𝑑𝒾
denoted
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(8)
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(𝒾)𝑑𝒾
(𝒾)𝑑𝒾
(𝒾)𝑑𝒾
(𝐹𝑅)
(𝐹𝑅)
(𝒾)𝑑𝒾
(𝐹𝑅)
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(𝒾)𝑑𝒾
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(𝐼𝑅)
(𝐼𝑅)
(𝐼𝑅)
(𝐼𝑅)
(𝐼𝑅)
Theorem
Theorem
2.
Theorem
[49]
Let
2.
[49]
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2.
:
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Let
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Let
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→
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be
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define
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=
=
𝔖(𝒾,
=
=
𝜑)𝑑𝒾
=
𝔖
𝔖(𝒾,
=
:
𝔖(𝒾,
𝜑)𝑑𝒾
𝔖
𝜑)
=
𝜑)𝑑𝒾
𝔖
:
𝔖(𝒾,
∈
:
ℛ
𝔖(𝒾,
𝜑)
=
∈
𝜑)𝑑𝒾
𝜑)
=
ℛ
𝔖(𝒾,
∈
,
:
ℛ
𝔖(𝒾,
𝜑)𝑑𝒾
𝜑)
be
an
F-NV-F.
Then
fuzzy
be
an
be
integral
F-NV-F.
an
F-NV-F.
of
Then
𝔖
Then
over
fuzz
Definition
2.
[49]
Let
Definition
𝔖
Definition
:
[𝜇,
𝜈]
⊂
2.
ℝ
[49]
→
2.
𝔽
[49]
Let
Let
𝔖
:
[𝜇,
𝔖
:
𝜈]
[𝜇,
⊂
𝜈]
ℝ
⊂
→
ℝ
𝔽
→
𝔽
([
],
)
([
([
],
)
],, ft
,
,
,
[𝜇,
[𝜇,
𝑅-integrable
𝑅-integrable
over
[𝜇,
𝜈].
over
Moreover,
[𝜇,ℝ⊂=
𝜈].
Moreover,
𝜈].
if𝔖(𝒾,
Moreover,
𝔖𝔖(𝒾,
is
if𝐹𝑅-integrable
𝔖
if𝔖(𝒾,
is𝔖
𝐹𝑅-integrable
is
𝐹𝑅-integrable
over
over
𝜈],
over
then
𝜈],
𝜈],
then
(8)family
(𝒾)𝑑𝒾
(𝒾)𝑑𝒾
(𝐹𝑅)
(𝐹𝑅)
(𝒾)𝑑𝒾
(𝐼𝑅)
(𝐼𝑅)
Theorem
Theorem
2. [49]
[49]
2.=
[49]
Let
Let
𝔖
:: [𝜇,
:𝔖
𝜈]
[𝜇,
⊂
𝜈]
→
ℝ=
𝔽
→(𝐹𝑅)
𝔽
be
alevelwise
be
F-NV-F
a𝔖
F-NV-F
whose
whose
𝜑-cuts
define
define
the
family
ofthen
I-V-Fs
of I-V-F
𝔖𝑅-integrable
𝔖(𝒾)𝑑𝒾
=
𝔖over
𝔖(𝒾)𝑑𝒾
𝜑)𝑑𝒾
𝜑)𝑑𝒾
:⊂
:(𝐼𝑅)
𝜑)
𝜑)
∈𝜑-cuts
ℛ∈fuzzy
ℛ, [𝜇,
,(8)
[𝜇,
(𝐹𝑅)
[𝜇,
(𝒾)𝑑𝒾
([levelwise
([
],(𝒾)𝑑𝒾
],, )the
,)integral
𝜈],
by
𝜈],
denoted
by
,[𝜇,
is
given
by
,𝔖(𝒾,
is
given
by
∗ (𝒾,
be
an
F-NV-F.
Then
be
an
F-NV-F.
Then
of
𝔖
fuzzy
over
integ
Definition
2.
Let
Definition
𝔖
[𝜇,
𝜈]
⊂
2.
ℝ
[49]
→
𝔽
Let
𝔖
:
[𝜇,
𝜈]
ℝ
→
𝔽
(𝒾)𝑑𝒾
(𝒾)𝑑𝒾
(𝐼𝑅)
(𝐼𝑅)
=
=
𝔖
=
𝔖
𝔖
𝔖 (𝒾) = [𝔖∗ (𝒾, 𝜑), 𝔖Remark
𝜑)]
for
all
𝒾
∈
𝐾.
Here,
for
each
𝜑
∈
(0,
1]
the
end
point
real
4. Ifdenoted
∇(
=
1,
then
from
(25)
and
(30),
we
acquire
the
inequality
(17).
∗
∗
∗
)
[𝜇,
(𝐹𝑅)
[𝜇,
[𝜇,
(𝒾)𝑑𝒾
(𝐹𝑅)
(𝐹𝑅)
(𝒾)𝑑𝒾
(𝒾)𝑑𝒾
𝜈], denoted
𝜈],
𝔖
𝜈],
denoted
denoted
by
𝔖 =𝔖
𝔖[𝔖
is by
given
levelwise
by
,(𝒾,
is(𝒾)
given
, is
given
levelwise
levelwise
(𝒾)
(𝒾)
[𝔖
(𝒾,
[𝔖
(𝒾,
(𝒾,
(𝒾,
(𝒾, by
=𝜑),
=𝔖
𝜑),
𝜑)]
𝜑),
𝔖 for
𝔖 by
𝜑)]
all
𝜑)]
𝒾for
∈ [𝜇,
for
all 𝜈]
all
𝒾∈
ℝ :→
𝜈]
[𝜇,𝒦
⊂𝜈]
ℝ⊂
are
→ℝ,𝒦
given
→
𝒦
areby
are
given
𝔖given
by
by
𝔖 : [𝜇,by𝜈]
𝔖 ⊂:𝔖[𝜇,
∗𝔖
∗𝜑)]
∗for
∗all
(𝒾)
(𝒾)
[𝔖
(𝒾,
(𝒾,
(𝒾,
(𝒾,𝔽
Theorem
Theorem
Theorem
[49]
Let
2., acquire
[49]
𝔖
2.given
: [49]
[𝜇,
Let𝜈]
Let
𝔖⊂inequality
:=
[𝜇,
ℝ
𝔖∗[𝔖
:→
𝜈]
[𝜇,
𝔽
⊂𝜈]
⊂∗→
𝔽
→
be
aℝ
F-NV-F
be
abe
F-NV-F
whose
a F-NV-F
whose
whose
𝜑-cuts
𝜑-cuts
the
define
fami
def
=
𝜑),
𝔖
𝔖
𝜑)]
for
all
𝒾 by
∈𝜑-cuts
𝒾[𝜇,
∈F-NV-Fs,
𝜈]
[𝜇,define
𝜈]
andand
for
for
all
al
:𝔖→
[𝜇,
𝜈]
[𝜇,are
⊂
𝜈]from
ℝ⊂
→ℝ(25)
𝒦
→
𝒦
are2.denoted
are
given
given
by
by
𝔖
𝔖
(𝐹𝑅)
[𝜇,
𝜈],
denoted
by
𝜈],
𝔖
by
𝔖
is
levelwise
by
,.ℝ
is
given
levelwise
∗𝜑),
Iffor
s𝐾=
1,:ℝ
and
(30),
we
the
for
harmonically
convex
functions 𝔖∗ (. , 𝜑), 𝔖∗ (. ,[𝜇,
𝜑):
called
lower
and
upper
functions
of
Theorem
Theorem
2.
2.
Let
𝔖
: [𝜇,
𝔖
:(𝒾)𝑑𝒾
𝜈]
[𝜇,
⊂
𝜈]
⊂
ℝ1],
𝔽
→∈
𝔽be
athe
be
a(𝒾)𝑑𝒾
F-NV-F
whose
whose
define
the
the
family
family
of
I-V-Fs
of
I-V-F
all
𝜑then
for
∈ for
all
(0,[49]
1],
all
𝜑 [49]
∈
where
𝜑(0,
∈Let
for
1],
(0,
all
1],
where
ℛ
where
for
∈
for
all
(0,
all
𝜑(𝐹𝑅)
ℛ
𝜑
(0,
1],
(0,
1],
where
ℛ
where
ℛ
ℛ,𝜑-cuts
denotes
denotes
collection
the
denotes
collection
Riemannian
of
the
denotes
Riemannian
of
collection
Riemannian
the
integrable
the
collection
of
collection
integrable
Riemanni
funcintegrab
of
Ri
off
([𝜑
], ℝ
)ℛ→
([
([
], where
)∈
],denotes
)F-NV-F
([𝔖
], the
)collection
([of
([
], 𝜑-cuts
) ],denotes
)define
∗
,
,
,
,
,
∗𝔖
∗𝔖
∗𝔖
(𝒾,
(𝒾,
(𝒾,
(𝒾,
for for
all all
𝜑see∈𝜑[39].
(0,
∈ 1],
(0, 1],
where
where
ℛ
denotes
denotes
the
the
collection
collection
of
Riemannian
of
Riemannian
integrable
integrable
funcfunc𝜑)
and
𝜑)
𝔖
𝜑)
and
𝜑𝔖([∈ℛ:,(0,
1].
𝜑
∈
Then
𝜑
(0,
∈
1].
(0,
𝔖
1].
Then
is
Then
𝐹𝑅-integrable
𝔖
is
𝔖
𝐹𝑅-integrable
is
𝐹𝑅-integrable
over
[𝜇,
over
𝜈]
over
if
[𝜇,
and
𝜈]
[𝜇,
only
if
𝜈]
and
if
if
and
only
only
if
if
∗
∗
∗
(𝒾,
(𝒾,
(𝒾,
(𝒾,
(𝒾,
(𝒾,
(𝑅)
(𝑅)
(𝑅)
(𝑅)
(𝑅)
(𝑅)
(𝐹𝑅)
(𝐹𝑅)
(𝒾)𝑑𝒾
(𝐹𝑅)
(𝒾)𝑑𝒾
(𝒾)𝑑𝒾
([
],
)
],
)
∗
∗
,
=
=
𝔖
=
𝜑)𝑑𝒾
𝔖
,
𝔖
𝜑)𝑑𝒾
𝜑)𝑑𝒾
,
𝔖
,
𝜑)𝑑𝒾
𝔖
𝔖
𝜑)𝑑𝒾
𝜑)𝑑𝒾
𝔖
𝔖
𝔖
∗ for
∗𝜑)
∗only
∗if(𝒾)
∗[𝔖
(𝒾)
[𝔖
(𝒾)
(𝒾,
(𝒾,
[𝔖
(𝒾,
(𝒾,
(𝒾,
=
=
𝜑),
=
𝔖
𝜑),
𝜑)]
𝜑),
𝔖
𝜑)]
all
𝜑)]
𝒾(𝐹𝑅)
for
∈∗.(𝒾)𝑑𝒾
[𝜇,
for
all
𝜈]
al
𝒾∈𝜑
[𝜇,
𝜈]
⊂
:(0,
[𝜇,
ℝ
:→
𝜈]
[𝜇,
𝒦
⊂
𝜈]
ℝ
⊂
are
→
ℝ
𝒦
given
→
𝒦
are
by
are
given
𝔖
given
by
𝔖
by
𝔖tions
𝔖
(𝒾,
(𝒾,
(𝒾,
(𝒾,
𝜑)
𝜑)
and
and
𝔖
𝔖
𝜑)
both
both
are
ar
𝜑𝔖∈𝜑:𝔖(0,
∈
1].
(0,
1].
Then
Then
𝔖
is
𝔖
𝐹𝑅-integrable
is
𝐹𝑅-integrable
over
over
[𝜇,
𝜈]
[𝜇,
𝜈]
if
and
if
and
only
if
𝔖
𝔖
(8)
(𝒾)𝑑𝒾
(𝒾)𝑑𝒾
(𝐹𝑅)
(𝒾)𝑑𝒾
(𝐹𝑅)
(𝒾)𝑑𝒾
(𝐼𝑅)
(𝐼𝑅)
∗𝜈]
∗𝔖
=
𝔖
=
=
𝔖(𝒾,
𝔖
𝜑)𝑑𝒾
:
𝔖(𝒾,
=
𝜑)
∈
𝔖(𝒾,
ℛ
𝜑)𝑑𝒾
:
,
𝔖(𝒾,
𝜑)
∗
∗
∗
(0,
(0,
(0,
∗
∗
(𝐹𝑅)
(𝐹𝑅)
(𝒾)𝑑𝒾
(𝐹𝑅)
(𝒾)𝑑𝒾
(𝒾)𝑑𝒾
(𝐹𝑅)
(𝐹𝑅)
for
all
𝜑
for
∈
all
for
(0,
1].
all
𝜑
∈
For
𝜑
∈
1].
all
(0,
1].
𝜑
For
∈
For
all
1],
all
𝜑
∈
ℱℛ
𝜑
∈
1],
1],
ℱℛ
ℱℛ
denotes
denotes
the
denotes
collection
the
the
collection
of
collection
all
𝐹𝑅-integrable
of
all
of
all
𝐹𝑅-in
𝐹𝑅
tions
of
tions
I-V-Fs.
tions
of
I-V-Fs.
of
𝔖
I-V-Fs.
is
𝐹𝑅
𝔖
-integrable
of
is
𝔖
tions
I-V-Fs.
𝐹𝑅
is
tions
-integrable
𝐹𝑅
of
-integrable
I-V-Fs.
of
over
𝔖
I-V-Fs.
is
[𝜇,
𝐹𝑅
over
𝔖
𝜈]
-integrable
over
is
𝔖
if
[𝜇,
𝐹𝑅
is
[𝜇,
-integrable
𝐹𝑅
𝜈]
if
-integrable
over
𝔖
if
[𝜇,
over
𝜈]
𝔖
over
𝔖
if
[𝜇,
𝜈]
[𝜇,
𝜈]
if
𝔖
if
𝔖
∈
𝔽
.
Note
∈
𝔽
∈
.
that,
𝔽
Note
Note
if
tha
∈
([
],
)
,
([
],
)
([
([
],
)
],
)
(𝒾)
(𝒾)
[𝔖
[𝔖
(𝒾,
(𝒾,
(𝒾,
(𝒾,
,
,
,
=𝔖∗ (𝒾)𝑑𝒾
𝜑),
𝔖
𝔖(𝐼𝑅)
𝜑)]
𝜑)]
for
for
all(𝒾)𝑑𝒾
all
𝒾=
∈𝒾[𝜇,
∈ℛ
𝜈]
[𝜇,
𝜈]
and
and
for, :for
all
[𝜇,:𝔖
𝜈]
[𝜇,𝐹𝑅
⊂
𝜈](𝐹𝑅)
ℝ
⊂∗→ℝ-integrable
𝒦
→𝔖
𝒦are
are
given
given
by
𝔖 𝜈]
𝔖
∗𝜑),
(8)
(𝒾)𝑑𝒾
(𝒾)𝑑𝒾
(𝐹𝑅)
(𝐹𝑅)
(𝒾)𝑑𝒾
(𝐼𝑅)
(𝐼𝑅)
(𝐹𝑅)
(𝐹𝑅)
(𝒾)𝑑𝒾
(𝒾)𝑑𝒾
𝔖(𝒾)𝑑𝒾
=by
=
=
𝔖(𝒾,
𝔖
:Note
𝔖(𝒾,
𝜑)
=
𝔖(𝒾,
𝔖(𝒾,
𝔖(𝒾
:a𝔖
tions
of
I-V-Fs.
𝐹𝑅
-integrable
over
over
[𝜇,
𝜈]
[𝜇,
if=𝔖
if=
𝔖=
∈
∈𝜑)𝑑𝒾
.we
𝔽𝔖
.acquire
that,
that,
if
ifover
([
],𝜈],
) 𝜑)𝑑𝒾
, 𝜑)𝑑𝒾
[𝜇,∈
[𝜇,
[𝜇,
𝑅-integrable
𝑅-integrable
𝑅-integrable
over
[𝜇,
over
𝜈].
over
Moreover,
[𝜇,
𝜈].
[𝜇,
Moreover,
𝜈].
ifMoreover,
𝔖
is𝔖
if𝐹𝑅-integrable
𝔖
if 𝔽is
𝔖
𝐹𝑅-integrable
isNote
𝐹𝑅-integrable
over
over
𝜈],
then
𝜈],
then
the
If
S∗𝜈]
ϕ
=
S
µ,
ϕ
with
ϕ
=
1
and
s
1,
from
(25)
and
(30),
the
inequality
be
an
F-NV-F.
Then
fuzzy
integral
of
over
Definition 2.tions
[49]
Letof
𝔖I-V-Fs.
: [𝜇,
⊂is
ℝ) is
→over
𝔽
(𝔖µ,
(
)
∗ (𝒾,
∗
∗
∗
∗
∗
∗
(𝒾,
(𝒾,
(𝒾,
(8)
NV-Fs
NV-Fs
NV-Fs
over
over
[𝜇,
𝜈].
[𝜇,
𝜈].
[𝜇,
[𝜇,
𝜑)
and
𝜑)
𝔖
𝜑)
and
𝜑
𝜑
∈
(0,
1].
𝜑
∈
Then
𝜑
(0,
∈
1].
(0,
𝔖
1].
Then
is
Then
𝐹𝑅-integrable
𝔖
is
𝔖
𝐹𝑅-integrable
is
𝐹𝑅-integrable
over
[𝜇,
over
𝜈]
over
if
[𝜇,
and
𝜈]
[𝜇,
only
if
𝜈]
and
if
if
𝔖
and
only
only
if
𝔖
if
𝔖
(𝒾)𝑑𝒾
(𝒾)𝑑𝒾
(𝐹𝑅)
(𝒾)𝑑𝒾
(𝐹𝑅)
(𝒾)𝑑𝒾
(𝐼𝑅)
(𝐼𝑅)
𝑅-integrable
𝑅-integrable
over
[𝜇,
𝜈].
[𝜇,
𝜈].
Moreover,
Moreover,
if
𝔖
if
is
𝔖
𝐹𝑅-integrable
is
𝐹𝑅-integrable
over
over
𝜈],
𝜈],
then
then
𝔖𝔖
=
𝔖
=
=𝜑)
𝔖(𝒾,
𝜑)𝑑𝒾
:if𝔖(𝒾,
𝜑)fuzzy
∈
𝔖(𝒾,
: ,𝔖(𝒾,
(𝒾,
(𝒾,
(𝒾,
(𝒾,
(𝒾,
(𝒾,
(𝒾,
(𝒾,
(𝒾,
∗, (𝒾,
∗𝔖
∗𝜑)𝑑𝒾
∗𝔖
∗both
𝜑),
𝔖
𝜑)
𝜑),
are
𝔖is
𝜑)
Lebesgue-integrable,
𝜑)
are
𝜑),
are
Lebesgue-integrable,
𝔖
Lebesgue-integrable,
𝜑)
𝔖
𝜑),
are
𝔖
𝜑)
Lebesgue-integrable,
then
are
are
then
is
Lebesgue-integrable,
then
a𝔖
fuzzy
𝔖if is
𝔖
ais
Aumann-integrable
fuzzy
then
a(𝒾,
Aumann-integrable
𝔖ℛand
then
is
Aumann-integrab
then
a𝔖
is
funcais
Auman
fuzzy
a ∈fuz
𝔖
𝔖1].
𝔖
𝔖
𝔖
([
], fuzzy
)𝜑)
∗𝜑
∗ (𝒾,
∗ (𝒾,
∗𝜑),
∗is
∗ (𝒾,
∗𝜑),
(𝒾,
𝜑)
and
𝔖
𝜑)
both
are
arfℛ
∈𝜑
(0,
∈harmonically
1].
Then
Then
𝔖
𝔖𝔖
𝐹𝑅-integrable
𝐹𝑅-integrable
over
over
[𝜇,
𝜈]
[𝜇,
𝜈]
if 𝔖
and
ifLebesgue-integrable,
and
only
only
𝔖
𝔖=
(𝒾,
(𝒾,
∗ (𝒾,
∗𝜑)
𝜑),
𝔖∗a(𝒾,
𝔖
𝜑)
𝜑)
are
are
Lebesgue-integrable,
Lebesgue-integrable,
then
then
𝔖
is
𝔖
aisthe
fuzzy
aℛfuzzy
Aumann-integrable
Aumann-integrable
funcfunc𝔖∗ (𝒾,
𝔖∗𝜑),
for
classical
convex
function.
[𝜇, 𝜈], denoted
(𝐹𝑅)
(𝒾)𝑑𝒾
by
𝔖
,(0,
is(0,
given
levelwise
by
for
all
𝜑
∈
1],
where
for
all
ℛ
𝜑
∈
(0,
1],
where
denotes
collection
denotes
of
Riemannian
the
collection
integrable
of
Riemannian
func([ tion
], [𝜇,
)over
([Moreover,
], is𝔖
, [𝜇,
, =
tionfor
over
tion
[𝜇,
over
𝜈].
over
[𝜇, 𝜈].
[𝜇,
tion
𝜈].
over
over
𝜈].
over
[𝜇,𝜈].
𝜈].
[𝜇,
𝜈].
[𝜇,over
[𝜇,of
[𝜇,
𝑅-integrable
𝑅-integrable
𝑅-integrable
over
𝜈].
over
Moreover,
[𝜇,
Moreover,
𝜈].
ifthe
𝔖
if𝐹𝑅-integrable
𝔖if is
𝔖
𝐹𝑅-integrable
is(𝒾)𝑑𝒾
𝐹𝑅-integrable
over
𝜈],
over
then
𝜈],
𝜈],
then
th
(𝒾)𝑑𝒾
(𝒾)𝑑𝒾
(𝐼𝑅)
(𝐼𝑅)
=
=
𝔖
𝔖
all
𝜑tion
∈over
(0,
1],
where
for
for
all
ℛ
𝜑tion
𝜑(0,
∈[𝜇,
1],
(0,
1],
where
where
ℛ([)(𝐼𝑅)
ℛ
denotes
collection
of
denotes
Riemannian
the
the
collection
collection
integrable
Rieman
of
funcRie
tiontion
over
over
[𝜇, 𝜈].
[𝜇,
𝜈].
([all
)𝔖
([
], ,)over
],denotes
) 𝜈],
, ∈
, over
[𝜇,
[𝜇,
𝑅-integrable
𝑅-integrable
over
[𝜇,
𝜈].
[𝜇,
𝜈].
Moreover,
Moreover,
if],[𝜇,
if𝜐]
is
𝔖
𝐹𝑅-integrable
is
𝐹𝑅-integrable
𝜈],
then
then
∗convex
∗if
∗fo
(0,
(0,
(0,
[𝜇,
[𝜇,
=
∞)
=
is
=
said
∞)
to
∞)
is
be
said
is
a
said
convex
to
be
to
a
be
set,
convex
a
if,
for
set,
all
set,
if,
𝒾,
Definition
Definition
Definition
3.for
[34]
A
3.
set
[34]
3.
𝐾
[34]
A
=
set
A
set
𝐾
=
⊂
𝐾
ℝ
=
𝜐]
⊂
𝜐]
ℝ
⊂
ℝ
(𝐹𝑅)
(𝒾)𝑑𝒾
(𝐹𝑅)
(𝒾)𝑑𝒾
(𝒾,
(𝒾,
(𝒾,
(𝒾,
(𝒾,
(i𝒿
(𝑅)
(𝑅)
(𝑅)
(𝑅)
(𝑅)
(𝑅)
(𝐹𝑅)
(𝐹𝑅)
(𝒾)𝑑𝒾
(𝐹𝑅)
(𝒾)𝑑𝒾
(𝒾)𝑑𝒾
tions
of
I-V-Fs.
𝔖
tions
is
𝐹𝑅
of
-integrable
I-V-Fs.
𝔖
over
is
𝐹𝑅
[𝜇,
-integrable
𝜈]
if
over
𝔖
[𝜇,
𝜈]
if
𝔖
∈
𝔽
.
Note
that,
∈
𝔽
=
=
𝔖
=
𝜑)𝑑𝒾
𝔖
,
𝔖
𝜑)𝑑𝒾
𝜑)𝑑𝒾
,
𝔖
,
𝜑)𝑑𝒾
𝔖
𝔖
𝜑
𝔖
𝔖
𝔖
for
all
𝜑
∈
(0,
1],
where
all
ℛ
𝜑
∈
(0,
1],
where
ℛ
denotes
the
collection
denotes
of
Riemannian
the
collection
integrable
of
Riemannian
funcin
∗
∗
∗
∗
∗
([ (𝒾)𝑑𝒾
], (𝒾)𝑑𝒾
)
([
], )𝜑)𝑑𝒾
,tions
4. Conclusions
(𝒾,
(𝒾,
(𝒾,
(𝒾,𝜈]
(𝑅)is
(𝑅)
(𝐹𝑅)
(𝐹𝑅)
=(𝑅)over
𝔖∗,[𝜇,
𝔖
𝜑)𝑑𝒾
, (𝐹𝑅)
, (𝑅)
𝔖(𝒾)𝑑𝒾
𝔖[𝜇,
𝜑)𝑑𝒾
𝜑)𝑑𝒾
𝔖
𝔖
tions
of
I-V-Fs.
𝔖
istions
𝐹𝑅
-integrable
of
I-V-Fs.
of=I-V-Fs.
𝔖
𝔖
𝐹𝑅
is
-integrable
𝐹𝑅if
-integrable
over
over
[𝜇,
if. (𝐹𝑅)
if (𝐹𝑅)
𝔖(𝒾)𝑑
𝔖if(
∈
𝔽𝜈]
Note
that,
∗𝜈]
[0,
[0,
[0,
𝐾,
𝜉
∈
𝐾,
1],
𝜉
𝐾,
∈
we
𝜉
∈
have
1],
we
1],
we
have
have
∗
∗
(8)
(𝒾)𝑑𝒾
(𝐹𝑅) 𝔖(𝒾)𝑑𝒾tions
(𝐼𝑅)
(𝒾,of
(𝒾,
(𝒾)𝑑𝒾
(𝐹𝑅)
(𝒾)𝑑𝒾
=
𝔖[49]
=
𝔖(𝒾,
𝜑)𝑑𝒾
:are
𝔖(𝒾,
𝜑)
∈
ℛ𝔖
,𝔽⊂
𝜑),I-V-Fs.
𝔖2.(𝒾,
are
𝜑),
𝔖⊂
Lebesgue-integrable,
then
is
fuzzy
Aumann-integrable
then
afamily
Aumannfunc𝔖
𝔖Let
tions
is𝔖[49]
𝐹𝑅
of
-integrable
I-V-Fs.
𝔖
over
is
𝐹𝑅
[𝜇,
-integrable
𝜈]
if
over
𝔖ℝbe
[𝜇,
𝜈]
if𝜑-cuts
𝔖
𝔽 the
.be
Note
that,
ifdefine
Theorem
Theorem
Theorem
2.
𝔖
2.
Theorem
:of
[49]
[𝜇,
Let
𝜈]
Let
Theorem
: (𝒾,
[𝜇,
ℝ
𝔖
2.Theorem
:𝜑)
→
𝜈]
[49]
[𝜇,
𝔽⊂
𝜈]
Let
ℝ
2.
⊂
→
[49]
𝔖
2.
𝔽
:→
[49]
[𝜇,
Let
𝔽
𝜈]
Let
⊂F-NV-F
:(𝐹𝑅)
ℝ
:→
[𝜇,
𝜈]
⊂
→
𝔽
→∈𝔖
𝔽
be
aℝ
F-NV-F
be
a([𝔖be
whose
a𝔖
𝜑-cuts
whose
whose
aℝdefine
F-NV-F
𝜑-cuts
be
ais
define
F-NV-F
whose
a fuzzy
define
F-NV-F
the
𝜑-cuts
of
whose
the
family
I-V-Fs
whose
family
𝜑-cu
of𝔽∗I𝜑
],aF-NV-F
)𝜈]
, [𝜇,
∗Lebesgue-integrable,
∗∈
∗𝜑)
∗ (𝒾,
∗𝔖
(0,
(0,
(0,
We
presented
the
idea
fuzzy
number
valued
functions
in
for
all
for
∈⊂
all
for
(0,
all
𝜑
𝜑
∈𝔽∗be(𝒾,
1].
all
(0,
1].
𝜑
For
For
all
all
𝜑
∈whose
ℱℛ
𝜑𝜑)
∈([𝜑-cuts
1],
ℱℛ
ℱℛ
denotes
denotes
the
denotes
collection
the
the
of𝔖collection
all,is
𝐹𝑅-integrable
offuzzy
offuzzy
𝐹𝑅-in
𝐹𝑅
Theorem
Theorem
2. [49]
2.∗ [49]
Let
Let
𝔖𝜑),
: [𝜇,
𝔖𝔖
:𝜑𝜈]
[𝜇,
𝜈]
ℝ⊂1].
→
ℝ∈
𝔽For
→(0,
abe
F-NV-F
a∈𝔖
F-NV-F
whose
𝜑-cuts
define
the
family
family
I-V-Fs
I-V-Fs
(𝒾,
(𝒾,
(𝒾,
(𝒾,
(𝒾,
(
(𝑅)
(𝑅)
(𝑅)
(𝑅)
(𝑅)
(𝐹𝑅)
(𝐹𝑅)
(𝒾)𝑑𝒾
(𝐹𝑅)
(𝒾)𝑑𝒾
(𝒾,
(𝒾,
(𝒾,
=
=
𝜑)𝑑𝒾
𝔖∗of
,𝔖
𝜑)𝑑𝒾
𝜑)𝑑𝒾
𝜑)𝑑𝒾
𝔖all
𝔖(9
𝜑
𝔖
𝔖
𝜑)
are
Lebesgue-integrable,
𝜑),
𝔖1],
𝜑)
are
are
Lebesgue-integrable,
then
Lebesgue-integrable,
𝔖
is([=
Aumann-integrable
then
then
𝔖𝔖(𝑅)
a∗,is
aall
funcAum
A
𝔖
𝔖
𝔖
],1],
)(𝒾)𝑑𝒾
([
], a
],fuzzy
)s-convex
, harmonically
,define
,)𝔖
∗of
∗the
∗(𝒾,
∗collection
∗ (𝒾,
∗𝜑),
(9)
(𝒾,
(𝒾,
(𝒾,
(𝑅)
(𝑅)
(𝑅)
(𝑅)
(𝐹𝑅)
(𝐹𝑅)
(𝒾)𝑑𝒾
(𝒾)𝑑𝒾
∗ (𝒾,
∗ (𝒾,
=
=
𝔖
𝔖
𝜑)𝑑𝒾
𝜑)𝑑𝒾
,
,
𝔖
𝔖
𝜑)𝑑𝒾
𝜑)𝑑𝒾
𝔖
𝔖
∗
∗
∗
∗
∗
∗
tion
over
[𝜇,
𝜈].
tion
over
[𝜇,
𝜈].
𝒾𝒿
𝒾𝒿
𝒾𝒿
∗
∗
(𝒾,
(𝒾,
𝜑),
𝔖
𝜑)
are
Lebesgue-integrable,
𝜑),
𝔖
𝜑)
are
Lebesgue-integrable,
then
𝔖
is
a
fuzzy
Aumann-integrable
then
𝔖
is
a
fuzzy
Aumann-in
func𝔖study.
𝔖
(𝒾)
[𝔖
(𝒾)
(𝒾,
(𝒾)
[𝔖
(𝒾,
[𝔖
(𝒾,
(𝒾,
(𝒾)
(𝒾,
[𝔖
(𝒾,
(𝒾)
(𝒾,
(𝒾)
[𝔖
(𝒾,
[𝔖
(𝒾,
(𝒾,
(𝒾,
(𝒾,
=
=
𝜑),
=
𝔖
𝜑),
𝜑)]
𝜑),
𝔖
=
for
𝜑)]
all
𝜑)]
𝒾
=
for
𝜑),
∈
[𝜇,
=
for
all
𝔖
𝜈]
all
𝒾
𝜑),
∈
and
𝜑)]
𝒾
[𝜇,
𝜑),
𝔖
∈
for
𝜈]
[𝜇,
for
𝔖
and
all
𝜈]
𝜑)]
all
an
𝜑
𝒾
fo
[𝜇,Some
𝜈]
⊂
:
[𝜇,
ℝ
:
→
𝜈]
[𝜇,
𝒦
⊂
𝜈]
ℝ
⊂
are
→
ℝ
𝒦
given
:
→
[𝜇,
𝒦
are
𝜈]
by
⊂
are
given
:
[𝜇,
ℝ
𝔖
given
:
→
𝜈]
[𝜇,
by
𝒦
⊂
𝜈]
ℝ
𝔖
by
⊂
are
→
ℝ
𝔖
𝒦
given
→
𝒦
are
by
are
given
𝔖
given
by
𝔖
by
𝔖
𝔖
𝔖
𝔖
𝔖
𝔖
∗𝔖 :tion
∗
this
new
fuzzy
Hermite–Hadamard
type
integral
inequalities
are
produced
∗
∗
∗
∗
∗
∗
∗
∗
NV-Fs
over
NV-Fs
NV-Fs
[𝜇,
over
𝜈].
over
[𝜇,
𝜈].
[𝜇,
𝜈].
[𝜇,
𝜈].
over
over
[𝜇,
𝐾.
∈ 𝜈]
𝐾.𝔖
∈ and
𝐾.
(f
(𝒾)𝑑𝒾
(𝒾)𝑑𝒾
(𝐼𝑅)
(𝐼𝑅)
(𝒾)
(𝒾)
[𝔖
[𝔖
(𝒾,[𝜇,
(𝒾,𝜈].
(𝒾,
(𝒾, 𝜑)]
= (𝐼𝑅)
= (𝒾)𝑑𝒾
𝔖 for
=tion
=∗of
𝜑),
𝔖 𝜈].
𝔖(𝐼𝑅)
𝜑)]
for
for
all
all
𝒾∈=
∈func𝒾𝔖
[𝜇,
∈
𝜈]
[𝜇,
and
for
all all
:𝔖[𝜇,where
: 𝜈]
[𝜇,⊂
𝜈]ℝ⊂ℛ→ℝ 𝒦
→ 𝒦over
aredenotes
are
given
given
by
by
𝔖collection
𝔖tion
𝔖 1],
∗𝜑),
for all 𝜑 ∈ (0,
the
Riemannian
integrable
(𝒾)𝑑𝒾
(1 (𝒾)𝑑𝒾
= (𝐼𝑅)
= 𝜉𝒾
𝔖
𝜉𝒾𝜉)𝒿
𝜉𝒾(1+thoroughly
+𝔖
−
+
−(1𝜉)𝒿
− 𝜉)𝒿 deduced.
tion
𝜈].
tion
over
[𝜇,
𝜈].
([over
], )[𝜇,
, new
∗
∗
∗
(9)
(9
using
this
class.
A
number
of
exceptional
instances
are
We
(𝒾,
(𝒾,
(𝒾,
(𝒾,
(𝒾,
and
𝔖
𝜑)
and
𝜑)
both
𝔖only
𝜑)
are𝔖𝜑)
bot
an
𝜑 ∈ (0, 1].
𝜑 ∈Then
𝜑(0,
∈ 1].
(0,
𝔖1].
Then
is 𝜑
Then
𝐹𝑅-integrable
∈𝔖(0,
is𝔖1].
𝜑𝐹𝑅-integrable
is∈Then
𝜑𝐹𝑅-integrable
(0,
∈over
1].
(0,
𝔖1].
Then
[𝜇,
is over
𝜈]
Then
𝐹𝑅-integrable
𝔖
over
if [𝜇,
and
is𝔖𝜈]
[𝜇,
𝐹𝑅-integrable
only
is if𝜈]
𝐹𝑅-integrable
and
ifover
if 𝔖
and
only
[𝜇,
only
if
over
𝜈]𝔖ifover
if∗ (𝒾,
[𝜇,
and
𝔖∗𝜑)
𝜈]
[𝜇,
only
if𝜈]and
ifand
if𝔖
𝔖
and
only
if𝜑)
if
∗𝜑)
∗∗(𝒾,
∗(
(𝒾,
(𝒾,
(𝒾,𝔖
𝜑)
𝜑)
and
and
𝔖
𝔖
𝜑)
𝜑)
both
are
are
𝜑 ∈𝜑(0,
∈ 1].
(0,
Then
Then
𝔖 is
𝔖some
𝐹𝑅-integrable
is [49]
𝐹𝑅-integrable
over
over
[𝜇,
𝜈]
[𝜇,
𝜈]
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2.
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𝔖:𝜈]
[𝜇,
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2.
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[49]
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[49]
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𝒾𝜉𝒾(1
∈+
𝒾[𝜇,
∈
If
(11)
If (11)
is reversed
reversed
𝒾𝒿
(1
(1𝔖(𝒾)
𝜉𝒾
𝜉𝒾
+
+
−
𝜉)𝒿
−≥
𝜉)𝒿
∈
𝐾.
∈=𝜐].
𝐾.
∈on
𝐾.
∈isif,
𝐾.
∈athen,
𝐾.
∈then,
𝐾.∈𝔖 se𝔖
(10)
(0,
(0,
[𝜇,
[𝜇,
=
∞)
is
said
to
be
a
convex
∞)
is
set,
said
to
for
be
all
convex
𝒾,
𝒿
Definition 3. is
[34]
A Definition
set
𝐾
=
3.
𝜐]𝒾𝒿
[34]
⊂harmonically
ℝconcave
A
set
𝐾
=
𝜐]
⊂
ℝ
[𝜇,
[𝜇,
[𝜇,
called
is
a
called
harmonically
is
called
a
harmonically
a
concave
function
concave
function
on
function
𝜐].
on
𝜐].
𝜐].
∈[34]
𝐾.
𝐾.
(10)
(10)
(1
(1+
(1
(1
(1
𝜉𝒾
𝜉𝒾
𝜉𝒾
𝜉𝒾
+∈[34]
𝜉)𝒿
+𝜉𝒾
−
𝜉)𝒿
−𝜐]
𝜉)𝒿
+
−
𝜉)𝒿
+𝜉𝒾
+∞)
−(1
𝜉)𝒿
−
𝜉)𝒿
Acknowledgments:
authors
would
like
to3.
thank
the
Rector,
COMSATS
University
Islamabad,
(0,
(0,
(0,
[𝜇,
[𝜇,
[𝜇,
[𝜇,
[𝜇,
=∈−
∞)
is
said
to
be
aℝ
=≥convex
=
∞)
set,
if,
for
to
be
all
to
a(11)
be
𝒾,If
conv
𝒿a(11
∈
c
Definition
3. [34]
set
Definition
𝐾
Definition
=
𝜐]
⊂
ℝ
set
A
set
𝐾
=𝜐].
𝐾
=
𝜐]
ℝ
⊂
is called
is called
a harmonically
aThe
harmonically
concave
concave
function
function
on
on
𝜐].
[𝜇,
[𝜇,
[0,
[𝜇,
[0,
[0,
[𝜇,
[𝜇,
[𝜇,is
(1
(1
for
all
𝒾,A
for
∈𝜉𝒾
for
all
𝒾,
all
𝜐],
𝒿+𝒾,
∈
𝜉−
𝒿∈
∈
𝜐],
1],
𝜉3.
𝜐],
∈where
𝜉A
1],
1],
𝔖(𝒾)
where
where
≥
𝔖(𝒾)
0⊂
for
𝔖(𝒾)
≥
all
0
𝒾for
0∈
for
all
𝜐].
all
𝒾is∈said
If𝒾is
(11)
∈said
𝜐].
𝜐].
Ifreverse
is
𝜉𝒾
+
𝜉)𝒿
−have
𝜉)𝒿
[0,𝒾,
[0,
𝐾,
𝜉 (0,
∈for
1],
we
have
𝐾,
𝜉𝒿𝐾
∈
1],
we
(0,
(0,
[𝜇,
[𝜇,
[𝜇,
[𝜇,
[0,
[0,
[𝜇,
[𝜇,
=
∞)
is
said
to
be
=
a
convex
∞)
is
set,
said
if,
to
for
be
all
a
convex
𝒾,
𝒿
∈
set,
3.
[34]
A
Definition
set
=
3.
𝜐]
[34]
⊂
ℝ
A
set
𝐾
=
𝜐]
⊂
ℝ
all
all
𝒿
𝒾,
∈
𝒿
∈
𝜐],
𝜉
𝜐],
∈
𝜉
∈
1],
1],
where
where
𝔖(𝒾)
𝔖(𝒾)
≥
0
≥
for
0
for
all
all
𝒾
∈
𝒾
∈
𝜐].
𝜐].
If
(11)
If
(11)
is
reversed
is
reversed
then,
then,
𝔖
𝔖
Pakistan,
for
providing
excellent
research
and
academic
environments.
for all 𝜑 ∈ (0, 1]. ForIslamabad,
allDefinition
𝜑for
∈
1],
ℱℛ
denotes
the
collection
of
all
𝐹𝑅-integrable
F[0,
([ we
)
𝐾, 𝜉 ∈ [0, 1],
𝐾,called
𝜉𝐾,
∈a𝜉[0,
∈a1],
we
1],
we
have
have
, ], have
[𝜇, 𝜐].
[𝜇, 𝜐].
[𝜇, 𝜐].
is
called
isacalled
harmonically
is
harmonically
harmonically
concave
concave
function
concave
function
onfunction
on on
[0,called
[𝜇, 𝜐].
[𝜇, 𝜐].
𝐾, 𝜉is∈called
1],awe
𝐾, 𝜉 ∈ [0,
1],
we have
is
harmonically
ahave
harmonically
concave
concave
function
function
on on
𝒾𝒿
𝒾𝒿
NV-Fs over [𝜇, 𝜈].
[𝜇,
[𝜇,
[𝜇,
[𝜇,
[𝜇,
[𝜇,convex
Definition
Definition
Definition
4. [34] The
4. [34]
Definition
4.
𝔖:[34]
The𝜐]
The
Definition
𝔖:
→Definition
4.
ℝ
𝔖:[34]
𝜐]
𝜐]
The
4.
ℝ→[34]
4.
𝔖:
ℝis
The
𝜐]
The
𝔖:
→a[𝜇,
ℝ
𝔖:
𝜐]
ℝ→ ℝis
is→
called
a[34]
called
harmonically
is∈
called
harmonically
a𝜐]
harmonically
is→
called
convex
acalled
convex
function
harmonically
is𝐾.
called
convex
a function
harmonically
on
a function
harmonical
𝜐]
onif [𝜇,
oncf
𝒾𝒿
𝒾𝒿
𝐾.
(10)
[𝜇,→
[𝜇, 𝜐]
Definition
Definition
4. [34]
4. [34]
TheThe
𝔖: [𝜇,
𝔖: 𝜐]
𝜐] ℝ
→ ℝis called
is called
a harmonically
a𝜉𝒾harmonically
convex
function
function
on∈𝒾𝒿
𝜐]
if∈ if
(1 − 𝜉)𝒿convex
(1 −on
𝜉𝒾 +
+𝒾𝒿
𝜉)𝒿[𝜇,
∈ 𝐾.
𝐾.∈ 𝐾.
(10)
𝒾𝒿
(1
𝜉𝒾𝒾𝒿
+ set,
+𝜉𝒾(1
+∈−(1
𝜉)𝒿
− 𝜉)𝒿
𝒾𝒿 −∈
𝒾𝒿𝒾, 𝒿 𝜉𝒾
𝒾𝒿
𝐾.
𝐾.
(10)
said to 𝔖
be𝒾𝒿
a𝔖convex
if,𝜉)𝒿
for(1
all
∈−𝒾𝒿
Definition 3. [34] A set 𝐾 = [𝜇, 𝜐] ⊂ ℝ = (0, ∞)
𝒾𝒿 is𝒾𝒿
(1
(1
𝜉𝒾
𝜉𝒾−(1
+ (1 −≤𝜉)𝒿
+
𝜉)𝒿
+
𝜉𝔖(𝒿),
+ (1
𝜉𝔖(𝒿),
+−𝜉𝔖(𝒿),
+ 𝜉𝔖
+
𝔖
𝔖−
𝜉)𝔖(𝒾)
≤
𝔖
≤
𝔖
𝜉)𝔖(𝒾)
−
𝜉)𝔖(𝒾)
≤
𝜉)𝔖(𝒾)
≤ (1
≤−(1+
𝜉)𝔖(𝒾)
−𝜉𝔖(𝒿),
𝜉)𝔖(𝒾)
(11)
(1
(1
+
𝜉𝔖(𝒿),
+
𝜉𝔖(𝒿),
𝔖
𝔖
≤
≤
−
𝜉)𝔖(𝒾)
−
𝜉)𝔖(𝒾)
(11)
(11)
(1
(1
(1
(1
(1
(1
𝜉𝒾
𝜉𝒾
𝜉𝒾
𝜉𝒾
𝜉𝒾
𝜉𝒾
+
−
𝜉)𝒿
+
+
−
𝜉)𝒿
−
𝜉)𝒿
+
−
𝜉)𝒿
+
+
−
𝜉)𝒿
−
𝜉)𝒿
𝐾, 𝜉 ∈ [0, 1], we have
(1
(1[𝜇,
𝜉𝒾
+
+𝔖:
−
𝜉)𝒿
−𝜐]
𝜉)𝒿
Definition 4. [34]𝜉𝒾The
Definition
→
4. [34]
ℝ is
The
𝔖: [𝜇,
𝜐] → ℝ is called
called
a harmonically
convex
a harmonically
function on [𝜇,
convex
𝜐] if fun
[𝜇, 𝜐] →4.ℝ[34]
[𝜇,→
Definition 4. [34] TheDefinition
𝔖:Definition
4.is[34]
The
The
𝔖:a[𝜇,
𝔖:
𝜐]
𝜐]ℝ→ ℝis called
called
harmonically
convex
is called
a function
harmonically
a harmonically
on [𝜇,conve
𝜐] co
if
Symmetry 2022, 14, 1639
16 of 18
Conflicts of Interest: The authors declare no conflict of interest.
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