Paper Template
Weighted Trigonometric and Hyperbolic Fuzzy
Information Measures and Their Applications in
Optimization Principles
Arunodaya Raj Mishra
Dhara Singh Hooda
Divya Jain
Depatrement of Mathematics
Jaypee University of
Engineering and Technology,
Guna, M. P., India
Depatrement of Mathematics
Jaypee University of
Engineering and Technology,
Guna, M. P., India
Depatrement of Mathematics
Jaypee University of
Engineering and Technology,
Guna, M. P., India
arunodaya87@outlook.com
ds_hooda@juet.ac.in
divya.jain@juet.ac.in
ABSTRACT
In the present communication, two new measures of
weighted trigonometric fuzzy information and two new
measures of weighted hyperbolic fuzzy information are
introduced, and their validity and the essential properties are
studied. Fuzzy information measures can be used for the study
of optimization principles when certain partial information is
available. The proposed measures of weighted trigonometric
0, if xi  A and thereis noambiguity

 A  xi   1, if xi  A and thereis noambiguity
0.5, max ambiguity whether x  A or x  A

i
i
The concept of measuring fuzzy uncertainty without
reference to probabilities began in 1972 with the work of [2]
who defined the entropy of A  FS ( X ) using Shannon’s
fuzzy information and weighted hyperbolic fuzzy information
are applied to study the maximum information principle.
Keywords
Fuzzy set, fuzzy information measure, weighted fuzzy
information, membership function and maximum fuzzy
information principle.
entropy as
n
H ( A)     A ( xi ) log  A ( xi )  (1   A ( xi )) log(1   A ( xi ))  .
i 1
This is called fuzzy information measure. The fuzzy
information measure has found wide applications to
Engineering, Fuzzy traffic control, Fuzzy aircraft control,
Introduction
The concept of entropy was developed to measure the
uncertainty of a probability distribution. [9] introduced the
concept of fuzzy sets and developed his own theory to
measure the ambiguity of a fuzzy set. A fuzzy set A is
represented as
A
 xi   A  xi  :  A  xi   0,1 ;  xi  X  ,
where  A ( xi ) represents the degree of member-ship and is
defined as
Computer sciences, management and Decision making, etc.
[5] proposed a rule to assign the numerical values to
probabilities
in
circumstances
where
certain
partial
information is available known as Maximum Entropy
Principle. It has many applications in physical, biological and
social sciences and stock market analysis. [6] have studied the
use of extended MaxEnt in all these applications. They have
also studied the use of measures of entropy as a measure of
uncertainty, equality, spread out, diversity, interactivity,
flexibility, system complexity, disorder, dependence and
similarity.
In the study of optimization principles, reliability theory,
marketing, measurement of risk in portfolio analysis and
quality control various generalized measures of entropy and
cross entropy have wide applications. [3] used [8] measure in
Evidently,
studying the market behavior and found an anomalous result
which was later overcome on using parametric measure of
n
1
H ( A, W )   wi 
sin   A ( xi )  sin  (1   A ( xi ))  0
i 1
entropy. Similarly, various other methods can be explained if
if and only if either 𝜇𝐴 (𝑥𝑖 ) = 0 or 1 − 𝜇𝐴 (𝑥𝑖 ) = 0 for all 𝑖 =
we use the generalized measures of entropy and directed
1, 2, … … , 𝑛.
divergence.
1
It implies H ( A, W )  0 if and only if A is a crisp set.
[3] and [6] also studied the totally probabilistic maximum
entropy principle. However, there are situations where
probabilistic measures of entropy fails to work and thus these
require extending their scope of application by applying
1
(P2) H ( A, W ) is maximum if and only if A is the fuzziest
set i. e.  A ( xi )  0.5 for all i  1, 2, ......., n .
Differentiating (3) with respect to  A ( xi ) , we have
possibility of fuzzy measures.
Corresponding to weighted entropy introduced by [1], [7]
has developed two new measures of weighted fuzzy entropy
as follows:
1
H ( AW
, )
n
  wi  cos  A ( xi )   cos  (1   A ( xi ))  (5)
i 1
 A ( xi )
n
 (1   A ( xi )) 
  ( xi )
H s ( A, W )   wi sin A
 sin
 1 (1)
i 1 
2
2

which vanishes at  A ( xi )  0.5 .
n
 (1   A ( xi )) 
  ( xi )
H c ( A, W )   wi cos A
 cos
 1 (2)
i 1 
2
2

2 1
n 2
 H ( AW
, )

2  i 1  wi sin  A ( xi )  sin  (1   A ( xi )) 
 A ( xi )
and applied these measures to study the principle of maximum
weighted fuzzy entropy.
In the present paper, we introduce two weighted
trigonometric fuzzy information measures and study their
validity in section 2. In section 3, we introduce two weighted
hyperbolic fuzzy information measures and study their
validity. Their applications to maximum weighted fuzzy
information principle are discussed in section 4.
Again differentiating (5) with respect to  A ( xi ) , we have
(6)
which is less than zero at  A ( xi )  0.5 .
1
H ( A, W )
is
maximum
at
 A ( xi )  0.5
for
all
i  1, 2, 3, ..., n .
1
Further from (5), we see that H ( A, W ) is an increasing
function of  A ( xi ) in the region 0   A ( xi )  0.5 and
1
H ( A, W ) is a decreasing function of  A ( xi ) in the region
Weighted Trigonometric Fuzzy Information
0.5   A ( xi )  1 .
Measures
(P3) Let
Corresponding to [4], we define the following measures of
If
weighted trigonometric fuzzy information:
n
1
H ( A, W )   wi 
sin  A ( xi )  sin  (1   A ( xi ))
i 1
  2  A ( xi )  1  
1 n
2
H ( A, W )   wi cos 
 
n i 1 
2

 

(3)

Here, we prove that (3) and (4) are valid measures.
Theorem 1. The weighted trigonometric fuzzy information
measure given by (3) is valid measure.
Proof: To prove that the given measure is a valid measure, we
shall show that (3) satisfies the four properties (P1) to (P4).
1
(P1) H ( A, W )  0 if and only if A is a crisp set.
0   A ( xi )  0.5,  * ( xi )   A ( xi )
A
for
all
i  1, 2, 3, ..., n
and if
(4)
A* be a sharpened version of A , which means that
0.5   A ( xi )  1,  * ( xi )   A ( xi )
A
for all
i  1, 2, 3, ..., n .
1
Since H ( A, W ) is an increasing function of  A ( xi ) in the
region 0   A ( xi )  0.5
1
and H ( A, W ) is a decreasing
function of  A ( xi ) in the region 0.5   A ( xi )  1 .
1
*
1
 * ( xi )   A ( xi )  H ( A , W )  H ( A, W ) in [0, 0.5] (7)
A
1 *
1
 * ( xi )   A ( xi )  H ( A , W )  H ( A, W ) in [0.5, 1] . (8)
A
Hence (7) and (8) together give
3
H ( AW
, )
n
    wi cos h  A ( xi )  cosh  (1   A ( xi ))  (11)
i 1
 A ( xi )
Which vanishes at  A ( xi )  0.5 .
1 *
1
H ( A , W )  H ( A, W ) .
Again differentiating (11) with respect to  A ( xi ) , we have
(P4) From the definition It is evident that
1
c
1
H ( A , W )  H ( A, W )
c
where A is the complement of A obtained by replacing
 A ( xi ) by (1-  A ( xi ) ).
2 3
n 2
 H ( AW
, )

2  i 1  wi sinh  A ( xi )  sinh  (1   A ( xi )) 
 A ( xi )
Which is less than zero at  A ( xi )  0.5 .
1
Hence the measure H ( A, W ) proposed in (3) is a valid
weighted trigonometric fuzzy information measure.
Proceeding on similar lines, it can easily be proved that the
weighted measure proposed in (4) is a valid measure of
weighted trigonometric fuzzy information measure.
3
H ( A, W )
is
maximum
at
 A ( xi )  0.5
for
all
i  1, 2, 3, ..., n .
3
Further from (11) we see that H ( A, W )
is an increasing
Weighted Hyperbolic Fuzzy Information
function of  A ( xi ) in the region 0   A ( xi )  0.5
Measures
3
H ( A, W ) is a decreasing function of  A ( xi ) in the region
In this section, we introduce the following measures of
weighted hyperbolic fuzzy information as
n
3
H ( A, W )    wi
i 1
sinh  A ( xi )  sinh  (1   A ( xi ))


 sinh 


and
0.5   A ( xi )  1 .
*
(P3) Let A be a sharpened version of A , which means that
(9)
n
cos h  A ( xi )  cos h  (1   A ( xi ))
4
H ( A, W )    wi 
 (10)
i 1
  (1  cos h  )

If
0   A ( xi )  0.5,  * ( xi )   A ( xi )
A
for
all
i  1, 2, 3, ..., n
and if
0.5   A ( xi )  1,  * ( xi )   A ( xi )
A
for all
Firstly, we prove that (9) and (10) are valid measures.
i  1, 2, 3, ..., n .
Theorem 2. The weighted hyperbolic fuzzy information
3
Since H ( A, W ) is an increasing function of  A ( xi ) in the
measure given by (9) is valid measure.
Proof: To prove that the given measure is a valid measure, we
shall show that (9) satisfies the four properties (P1) to (P4).
3
(P1) H ( A, W )  0 if and only if A is a crisp set.
region 0   A ( xi )  0.5
3
and H ( A, W ) is a decreasing
function of  A ( xi ) in the region 0.5   A ( xi )  1 .
Evidently,
3 *
3
 * ( xi )   A ( xi )  H ( A , W )  H ( A, W ) in [0, 0.5] (12)
A
n
sinh  A ( xi )  sinh  (1   A ( xi ))
3
H ( A, W )    wi 
0
i 1
  sinh 

3 *
3
 * ( xi )   A ( xi )  H ( A , W )  H ( A, W ) in [0.5, 1] . (13)
A
if and only if either 𝜇𝐴 (𝑥𝑖 ) = 0 or 1 − 𝜇𝐴 (𝑥𝑖 ) = 0 for all 𝑖 =
1, 2, … … , 𝑛.
3
It implies H ( A, W )  0 if and only if A is a crisp set.
3 *
3
H ( A , W )  H ( A, W ) .
(P4) From the definition, It is evident that
3 c
3
H ( A , W )  H ( A, W )
3
(P2) H ( A, W ) is maximum if and only if A is the fuzziest
set i. e.  A ( xi )  0.5 for all
Hence (12) and (13) together give
i.
Differentiating (9) with respect to  A ( xi ) , we have
c
where A is the complement of A obtained by replacing
 A ( xi ) by (1-  A ( xi ) ).
3
Hence the measure H ( A, W ) proposed in (9) is a valid
weighted hyperbolic fuzzy information measure.
Similarly, it can easily be proved that the weighted
measure proposed in (10) is a valid measure of weighted
hyperbolic fuzzy information measure.
1 n
1  
 cos  1
 i 1
 2 wi

 g r ( xi )  K

Thus when 2  0 , we have
Principle of maximum weighted fuzzy
information
In this section, we study the applications of two newly
introduced weighted trigonometric measures and two newly
introduced
weighted
hyperbolic
measures
of
fuzzy
  1 1    g r ( xi ) 

sin  cos 1 2



2 wi
n




1
H max ( A, W )   wi

i 1 

   g (x ) 
  sin 1  1 cos 1 1 2 r i   
2 wi

 
 
information for the study of maximum weighted fuzziness by
or
considering problems.
n
1
H max ( A, W )   2 wi sin 
i 1
Problem 3.1. Maximize
n
1
H ( A, W )   wi 
sin  A ( xi )  sin  (1   A ( xi )) 

i 1 
Subject to the fuzzy constraints
n
  A ( xi )   0 ,
i 1
(15)
(16)
Consider the following Lagrangian
n
L   wi 
sin  A ( xi )  sin  (1   A ( xi ))

i 1 

 
n
n
 1   A ( xi )   0  2   A ( xi ) g r ( xi )  K
i 1
i 1
 A ( xi )
n
1
Hence H max ( A, W )   f ( , wi ) , where
i 1
'
f ( , wi )  2wi cos  ,

(20)
1
Thus (20) shows that H max ( A, W ) is concave.
(17)
Problem 3.2. In this problem, we apply another weighted
measure to study maximum weighted fuzzy information
 0 gives  A ( xi ) 
1

cos
1  1  2 g r ( xi ) 


2 wi
.

From (15) and (16), we get
(18)
1 n
1    2 g r ( xi ) 
 cos  1
 g r ( xi )  K
 i 1
 2 wi

(19)
where 1 and 2 can be obtained from (16) and (19).
When 2  0 , we have
1 n
1  
 cos  1
 i 1
 2 wi

  0

principle. For this, we consider the following problem:
Maximize
  2  A ( xi )  1  
1 n
2
H ( A, W )   wi cos 
 
n i 1 
2

 

1 n
1    2 g r ( xi ) 
 cos  1
  0
 i 1
 2 wi

and
2 wi
''
and f ( , wi )  2wi sin   0 .
Thus
L
1 1  2 g r ( xi )
f ( , wi )  2 wi sin 
n
  A ( xi ) g r ( xi )  K
i 1
and
where   cos
(14)
(21)

Subject to the set of fuzzy constraints are given in (15) and
(16).
Consider the following Lagrangian
L

  2  A ( xi )  1  
n
1 n
 wi cos 
   1   A ( xi )   0

i 1
n i 1
2



 2


 
n
  A ( xi ) g r ( xi )  K
i 1


(22)
L
 A ( xi )
 A ( xi ) 
Consider the following Lagrangian
 0 gives
n
L   wi 
sinh  A ( xi )  sinh  (1   A ( xi ))  sinh  

i 1 
 1  n
2 sin 
2
  wi
1





 
1 n 
1  n
  2 sin 
( 1  2 g r ( xi ))   1   0
2 i 1 
  wi
 
 A ( xi ) 
 
1 n 
1  n
  2 sin 
( 1  2 g r ( xi ))   1 g r ( xi )  K
2 i 1 
  wi
 
Now when 2  0 , we have
where
1 n 
1   n  
 2 sin  1   1   0
2 i 1 
  wi  
   1  
1 n
2
H max ( A, W )   wi cos 
 
n i 1
2



(23)





( 1  2 g r ( xi ))   1
wi     1  
 
cos 
n   2  
 wi     1  
'
f ( , wi )  
 
sin 
2n   2  
f ( , wi )  
4n
 2(1  cosh  )

c
  0



   1  
 
cos 
2
 
 
(24)
2
Thus (24) shows that H max ( A, W ) is concave function.
Problem 3.3. In this problem, we maximize the weighted
hyperbolic fuzzy information measure introduced in (9) under
the set of fuzzy constraints given in (15) and (16).

c
g r ( xi )  K



When 2  0 , we have


 2(1  cosh  ) 

  0
1  cosh   sinh 



 1 


  wi 
2
and
and
2




2
 (1  2 g r ( xi ))


 wi
1 n
 log 
 i 1
 1  cosh   sinh 



  1
  wi
1 n
 log 
 i 1



n
2
Hence H max ( A, W )   f ( , wi ) where
i 1
 wi
 (  2 g r ( xi )) 
c 1

 wi



c
and
Thus when 2  0 , we have
f ( , wi ) 

 (1  2 g r ( xi ))


 wi
1 n
 log 
 i 1
 1  cosh   sinh 


1 n 
1   n  
  2 sin  1   1 g r ( xi )  K
2 i 1 
  wi  
 1  n
2 sin 
2
  wi
1
 (1  2 g r ( xi ))


 wi
log 
 1  cosh   sinh 


Applying the constraints (15) and (16), we have
and
1
(25)
zero, we get
and
where  

Differentiating (25) with respect to  A ( xi ) and equating to
Applying fuzzy constraints given in (15) and (16), we get
''
 
n
n
 1   A ( xi )   0  2   A ( xi ) g r ( xi )  K
i 1
i 1
( 1  2 g r ( xi ))   1

  1
  wi
1 n
 log 
 i 1




 1 


  wi 
2
1  cosh   sinh 
Thus when 2  0 , we get

 2(1  cosh  ) 

g r ( xi )  K



n
sinh   sinh  (1   )
3
H max ( A, W )    wi 

i 1   sinh 

where  
1

 (1  2 g r ( xi ))


 wi
log 
 1  cosh   sinh 


(26)

c




When 2  0 , we get

 1
  wi
1 n
 log 
 i 1



n
3
Hence H max ( A, W )    f ( , wi )
i 1

where f ( , wi )  sinh   sinh  (1   )  sinh 

'
f ( , wi )   cosh   cosh  (1   )



''
2
and f ( , wi )   sinh   sinh  (1   )
3
Hence (26) shows that H max ( A, W ) is concave.
we maximize another weighted
hyperbolic fuzzy information measure introduced in (10)
under the set of fuzzy constraints (15) and (16).
n
L   wi cosh  A ( xi )  cosh  (1   A ( xi ))  (1  cosh  )
i 1

 

(27)
Differentiating (27) with respect to  A ( xi ) and equating to
zero, we get
 (1  2 g r ( xi ))
 A ( xi ) 
Where d

1
log 




 wi
 d
1  cosh   sinh 
 (  2 g r ( xi )) 
 1

 wi









 2(1  cosh  ) 

  0



1  cosh   sinh 

 1
  wi
1 n
 log 
 i 1





 2(1  cosh  ) 

g r ( xi )  K
1  cosh   sinh 



 1 


  wi 
2
Thus when 2  0 , we have
n
4
H max ( A, W )    wi cosh   cosh  (1   )  (1  cosh  )
i 1
where  
1


 (1  2 g r ( xi ))

 d

 wi

log 
 1  cosh   sinh  




n
4
Hence H max ( A, W )    f ( , wi )
i 1

where f ( , wi )  cosh   cosh  (1   )  (1  cosh  )

'
f ( , wi )   sinh   sinh  (1   )




''
2
and f ( , wi )   cosh   cosh  (1   )  0
2
 2(1  cosh  )
Applying the constraints (15) and (16), we get
 (1  2 g r ( xi ))

 d

 wi
1 n
  0
 log 
 i 1
 1  cosh   sinh  




and

2

The corresponding Lagrangian is given by
n
n
 1   A ( xi )   0  2   A ( xi ) g r ( xi )  K
i 1
i 1
 1 


  wi 
and

3
Thus H max ( A, W )  0
Problem 3.4. Here,
 (1  2 g r ( xi ))

 d

 wi
1 n
g r ( xi )  K
 log 
 i 1
 1  cosh   sinh  




4
Thus H max ( A, W )  0
4
which shows that H max ( A, W ) is concave.
Conclusion
The development of new parametric and non-parametric fuzzy
measures of information will definitely reduce uncertainty,
which will help to increase the efficiency of the process. It is
therefore concluded that though many information measures
have been developed, still there is scope for the better
measures to be developed which will find applications in a
variety of fields. In this paper, we have developed two
measures of weighted trigonometric fuzzy information and
two measures of weighted hyperbolic fuzzy information, and
have studied their applications in optimization principle.
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