Stability in possibilistic linear programming with continuous
fuzzy number parameters ∗
Mario Fedrizzi
fedrizzi@cs.unitn.it
Robert Fullér
rfuller@abo.fi
Abstract
We prove that possibilistic linear programming problems (introduced by Buckley in [2]) are wellposed, i.e. small changes of the membership function of the parameters may cause only a small
deviation in the possibilistic distribution of the objective function.
1
Introduction
We consider certain possibilistic linear programming problems, which have been introduced in [2]. In
contrast to classical linear programming (wherea small error of measurement may produce a large variation in the objective function), we show that the possibility distribution of the objective function of a
possibilistic linear program with continuous fuzzy number parameters is stable under small perturbations
of the parameters. First, in this section, we will briefly review possibilistic linear programming and set
up notations.
A fuzzy number is a fuzzy set ã, ã : R → [0, 1] = I, which is normal, continuous, fuzzy convex and
compactly supported. The fuzzy numbers will represent the continuous possibility distributions for fuzzy
parameters. A possibility linear programming is
max / min Z = x1 c̃1 + · · · + xn c̃n ,
(1)
subject to x1 ãi1 + · · · + xn ãin ∗ b̃i , 1 ≤ i ≤ m, x ≥ 0.
where ãij , b̃i , c̃j are fuzzy numbers,x = (x1 , . . . , xn ) is a vector of (nonfuzzy) decision variables, and ∗
denotes <, ≤, =, ≥ or > for each i.
We will assume that all fuzzy numbers ãij , b̃i , c̃j are non-interactive [11].
Following Buckley [2], we define Poss[Z = z], the possibility distribution of the objective function Z.
We first specify the possibility that x satisfies the i-th constraints. Let
Π(ai , bi ) = min{ãi1 (ai1 ), . . . , ãin (ain ), b̃i (bi },
where ai = (ai1 , . . . , ain ), which is the joint distribution of ãij , j = 1, . . . , n, and b̃i . Then
Poss[x ∈ Fi ] = sup{ Π(ai , bi ) | ai1 x1 + . . . + ain xn ∗ bi },
ai ,bi
∗
The final version of this paper appeared in: M. Fedrizzi and R. Fullér, Stability in possibilistic linear programming with
continuous fuzzy number parameters, Fuzzy Sets and Systems, 47(1992) 187-191. doi: 10.1016/0165-0114(92)90177-6
1
which is the Possibility that x is feasible with respect to the i-th constraint. Therefore, for x ≥ 0,
Poss[x ∈ F] = min Poss[x ∈ Fi ],
1≤i≤m
which is the Possibility that x is feasible. We next construct Poss[Z = z|x] which is the conditional
possibility that Z equals z given x. The joint distribution of the c̃j is
Π(c) = min{c̃1 (c1 ), . . . , c̃n (cn )}
where c = (c1 , . . . , cn ). Therefore,
Poss[Z = z|x] = sup{Π(c)|c1 x1 + · · · + cn xn = z}.
c
Finally, applying Bellman and Zadeh’s method of fuzzy decision making [1], the Possibility distribution
of the objective function is defined as follows
Poss[Z = z] = sup min{Poss[Z = z|x], Poss[x ∈ F]}
x≥0
It should be noted that Buckley [3] showed that the solution to an appropriate linear program gives the
correct z values in Poss[Z = z] = α for each α ∈ I.
Let ã be a fuzzy number. Then for any θ ≥ 0 we define ω(ã, θ), the modulus of continuity of ã as
ω(ã, θ) = max |ã(u) − ã(v)|.
|u−v|≤θ
The following statements hold [8]:
If 0 ≤ θ ≤ θ0 then ω(ã, θ) ≤ ω(ã, θ0 )
(2)
If α > 0, β > 0, then ω(ã, α + β) ≤ ω(ã, α) + ω(ã, β).
(3)
lim ω(ã, θ) = 0
(4)
θ→0
An α-level set of a fuzzy numer ã is a non-fuzzy set denoted by [ã]α and is defined by
(
[ã]α =
{t ∈ R | ã(t) ≥ α} if α > 0
cl(suppã)
if α = 0
where cl(suppã) denotes the closure of the support of ã.
If ã and b̃ are fuzzy numbers and λ ∈ R then ã + b̃, ã − b̃, λã are defined by Zadeh’s extension principle
[12] in the usual way.
Let ã and b̃ be fuzzy numbers and
[ã]α = [a1 (α), a2 (α)],
[b̃]α = [b1 (α), b2 (α)].
It is well known that
[ã + b̃]α = [a1 (α) + b1 (α), a2 (α) + b2 (α)].
(5)
Poss[ã ∗ b̃] = sup min{ã(x), b̃(y)} = sup(ã − b̃)(t)
(6)
It is easy to see that
x∗y
t∗0
where ∗ stands for <, ≤ , =, ≥ or >. We metricize the set of fuzzy numbers by the metric [9]
D(ã, b̃) = sup max{|ai (α) − bi (α)|},
α∈[0,1] i=1,2
2
2
Auxiliary propositions
Lemma 2.1 [9]. Let ã, b̃, c̃ and d˜ be fuzzy numbers. Then
D(λã, λb̃) = |λ|D(ã, b̃), for λ ∈ R,
˜ ≤ D(ã, b̃) + D(c̃, d)
˜
D(ã + c̃, b̃ + d)
˜ ≤ D(ã, b̃) + D(c̃, d)
˜
D(ã − c̃, b̃ − d)
Lemma 2.2 Let λ 6= 0, µ 6= 0 be real numbers and let ã and b̃ be fuzzy numbers. Then
ω(λã, θ) = ω(ã, θ/|λ|),
θ
ω(λã + λb̃, θ) ≤ ω
,
|λ| + |µ|
(7)
(8)
where
ω(θ) := max{ω(ã, θ), ω(b̃, θ)},
for θ ≥ 0.
Proof. From the equation (λã)(t) = ã(t/λ) for t ∈ R we have
ω(λã, θ) = max |(λã)(u) − (λã)(v)| = max |ã(u/λ) − ã(v/λ)| =
|u−v|≤θ
max
|u−v|≤θ
|u/λ−v/λ|≤θ/|λ|
|ω(u/λ) − ω(v/λ)| = ω(ã, θ/|λ|),
which proves (7).
As to (8), let θ > 0 be arbitrary and u, t ∈ R such that |u − t| ≤ θ. Then with the notations c̃ =: λã,
d˜ := µb̃ we need to show that
˜
˜
|(c̃ + d)(u)
− (c̃ + d)(t)|
≤ ω(
θ
).
|λ| + |µ|
We assume without loss of generality that t < u. From (5) it follows that there are real numbers t1 , t2 ,
u1 , u2 with the properties
t = t1 + t2 , u = u1 + u2 , t1 ≤ u1 , t2 ≤ u2
(9)
˜ 2 ) = (c̃ + d)(t),
˜
c̃(t1 ) = d(t
(10)
˜ 2 ) = (c̃ + d)(u).
˜
c̃(u1 ) = d(u
(11)
Since from (7) we have
|c̃(u1 ) − c̃(t1 )| ≤ ω(ã, |u1 − t1 |/|λ|)
and
˜ 2 ) − d(t
˜ 2 )| ≤ ω(b̃, |u2 − t2 |/|µ|),
|d(u
it follows by (??), (9), (10), (11) that
˜
˜
|(c̃ + d)(u)
− (c̃ + d)(t)|
≤
min{ω(ã, |u1 − t1 |/|λ|), ω(b̃, |u2 − t2 |/|µ|)} ≤
3
min{ω(|u1 − t1 |/|λ|), ω(|u2 − t2 |/|µ|)} ≤
ω
|u1 − t1 | + |u2 − t2 |
|λ| + |µ|
=ω
|u − t|
|λ| + |µ|
≤ω
θ
|λ| + |µ|
,
which proves the lemma.
The following lemma can be proved directly by using the definition of α-level sets.
Lemma 2.3 Let ã be a fuzzy number and [ã]α = [a1 (α), a2 (α)]. Then a1 : [0, 1] → R is strictly increasing and
a1 (ã(t)) ≤ t,
for t ∈ cl(suppã),
ã(a1 (α)) = α,
for α ∈ [0, 1],
a1 (ã(t)) ≤ t ≤ a1 (ã(t) + 0),
for a1 (0) ≤ t < a1 (1), where
a1 (ã(t) + 0) = lim a1 (ã(t) + ).
→+0
(12)
Lemma 2.4 Let ã and b̃ be fuzzy numbers.Then
(i) D(ã, b̃) ≥ |a1 (α + 0) − b1 (α + 0)|, for 0 ≤ α < 1,
(ii) ã(a1 (α + 0)) = α, for 0 ≤ α < 1,
(iii) a1 (α) ≤ a1 (α + 0) < a1 (β), for 0 ≤ α < β ≤ 1.
Proof. (i) From the definition of the metric D we have
|a1 (α + 0) − b1 (α + 0)| = lim |a1 (α + ) − lim b1 (α + )| =
→+0
→+0
lim |a1 (α + ) − b1 (α + )| ≤ sup |a1 (γ) − b1 (γ)| ≤ D(ã, b̃).
→+0
γ∈I
(ii) Since ã(a1 (α + )) = α + , for ≤ 1 − α, we have
ã(a1 (α + 0)) = lim A(a1 (α + )) = lim (α + ) = α.
→+0
→+0
(iii) From strictly monotonity of a1 it follows that a1 (α + ) < a1 (β), for < β − α. Therefore,
a1 (α) ≤ a1 (α + 0) = lim a1 (α + ) < a1 (β),
→+0
which completes the proof.
Lemma 2.5 Let δ ≥ 0 and let ã, b̃ be fuzzy numbers. If D(ã, b̃) ≤ δ, then
sup |ã(t) − b̃(t)| ≤ max{ω(ã, δ), ω(b̃, δ)}.
t∈R
4
(13)
Proof. Let t ∈ R be arbitrarily fixed. It will be sufficient to show that
|ã(t) − b̃(t)| ≤ max{ω(ã, δ), ω(b̃, δ)}.
If t ∈ suppã ∪ suppb̃ then we obtain (13) trivially. Suppose that t ∈ suppã ∪ suppb̃. With no loss of
generality we will assume 0 ≤ b̃(t) < ã(t). Then either of the following must occur:
(a)
(b)
(c)
(d)
t ∈ (b1 (0), b1 (1)),
t ≤ b1 (0),
t ∈ (b2 (1), b2 (0))
t ≥ b2 (0).
In this case of (a) from Lemma 2.4 (with α = b̃(t), β = ã(t)) and Lemma 2.3(iii) it follows that
ã(a1 (b̃(t) + 0)) = b̃(t),
t ≥ a1 (ã(t)) ≥ a1 (b̃(t) + 0)
and
D(ã, b̃) ≥ |a1 (b̃(t) + 0) − a1 (b̃(t) + 0))|.
Therefore from continuity of ã we get
|ã(t) − b̃(t)| = |ã(t) − ã(a1 (b̃(t) + 0))| = ω(ã, |t − a1 (b̃(t) + 0)|) =
ω(ã, t − a1 (b̃(t) + 0)) ≤ ω(ã, b1 (b̃(t) + 0) − a1 (b̃(t) + 0)) ≤ ω(ã, δ).
In this case of (b) we have b̃(t) = 0; therefore from Lemma 2.3(i) it follows that
|ã(t) − b̃(t)| = |ã(t) − 0| = |ã(t) − ã(a1 (0))| ≤ ω(ã, |t − a1 (0)|)
≤ ω(ã, |b1 (0) − a1 (0)|) ≤ ω(ã, δ).
A similar reasoning yields in the cases of (c) and (d); instead of properties a1 we use the properties of
a2 .
3
Stability theorem
An important question [4, 7, 13] is the influence of the perturbations of the fuzzy parameters to the
Possibility distribution of the objective function. We will assume that there is a collection of fuzzy
parameters ãδij , b̃δi , c̃δj available with the property
max D(ãij , ãδij ) ≤ δ,
i,j
max D(b̃i , b̃δi ) ≤ δ,
i
max D(c̃j , c̃δj ) ≤ δ.
j
(14)
Then we have to solve the following problem:
max / min Z δ = x1 c̃δ1 + · · · + xn c̃δn
(15)
subject to x1 ãδi1 + · · · + xn ãδin ∗ b̃δi , 1 ≤ i ≤ m, x ≥ 0.
Let us denote by Poss[x ∈ Fiδ ] the Possibility that x is feasible with respect to the i-th constraint in (15).
Then the Possibility distribution of the objective function Z δ in (15) is defined as follows:
Poss[Z δ = z] = sup(min{Poss[Z δ = z | x], Poss[x ∈ F δ ]}).
x≥0
The next theorem shows a stability property (with respect to perturbations (14) of the Possibility dostribution of the objective function of the Possibilistic linear programming problems (1) and (15).
5
Theorem 3.1 Let δ ≥ 0 be a real number and let ãij , b̃i , ãδij , c̃j , c̃δj be continuous fuzzy numbers. If (14)
hold, then
sup | Poss[Z δ = z] − Poss[Z = z] |≤ ω(δ)
(16)
z∈R
where
ω(δ) = max{ω(ãij , δ), ω(ãδij , δ), ω(b̃i , δ), ω(bδi , δ), ω(c̃j , δ), ω(c̃δj , δ)}.
i,j
Proof. It is sufficient to show that
| Poss[Z = z | x] − Poss[Z δ = z | x] |≤ ω(δ), z ∈ R,
(17)
| Poss[x ∈ F] − Poss[x ∈ Fiδ ] |≤ ω(δ),
(18)
for each x ∈ R and 1 ≤ i ≤ m, because (16) follows from (17) and (18). We shall prove only (18),
because the proof of (17) is carried out analogously.
Let x ∈ R and i ∈ {1, . . . , m} arbitrarily fixed. From (6) it follows that
Poss[x ∈ Fi ] = sup(
n
X
t∗0
Poss[x ∈
Fiδ ]
ãij xj − Bi )(t),
j=1
= sup(
t∗0
n
X
ãδij xj − Biδ )(t),
j=1
Applying Lemma 2.1 and (4) and (14) we have
n
n
X
X
D(
ãij xj − b̃i ,
Aδij xj − b̃δi ) ≤
j=1
n
X
j=1
|xj |D(ãij , ãδij ) + D(b̃i , b̃δi ) ≤ δ(|x|1 + 1),
j=1
where
|x|1 = |x1 | + . . . + |xn |.
By Lemma 2.2 we get
X
X
n
n
δ
ãij xj − b̃i δ, δ
≤ω
ãij xj − b̃i , δ , ω
max ω
j=1
j=1
δ
.
|x|1 + 1
Finally, applying Lemma 2.5 we have
| Poss[x ∈ Fi ] − Poss[x ∈ Fiδ ] |=
| sup
X
n
t∗0
sup |
t∗0
X
n
δ
δ
ãij xj − b̃i (t) − sup
ãij xj − b̃i (t) |≤
t∗0
j=1
X
n
j=1
X
n
δ
δ
ãij xj − b̃i (t) −
ãij xj − b̃i (t) |≤
j=1
j=1
6
sup |
t∈R
X
n
X
n
δ
δ
ãij xj − b̃i (t) −
ãij xj − b̃i (t) |≤
j=1
j=1
ω
δ(|x|1 + 1)
|x|1 + 1
= ω(δ),
which proves the theorem.
Remark 3.1 From (4) and (16) it follows that
sup | Poss[Z δ = z] − Poss[Z = z] |→ 0 as δ → 0
z∈R
which means the stability of the possibility distribution of the objective function with respect to perturbations (14).
Remark 3.2 As an immediate consequence of this theorem we obtain the following result (see [5]: If
the fuzzy numbers in (1) and (15) satisfy the Lipschitz condition with constant L > 0, then
sup | Poss[Z δ = z] − Poss[Z = z] |≤ Lδ
z∈R
Furthermore, similar estimations can be obtained in the case of symmetrical trapezoidal fuzzy number
parameters (see [11]) and in the case of symmetrical triangular fuzzy number parameters (see [6, 10]).
Remark 3.3 It is easy to see that in the case of non-continuous fuzzy parameters the possibility distribution of the objective function may be unstable under small changes of the parameters.
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4
Follow ups
The results of this paper have been improved and generalized later in the following works:
A27-c12 Rafik Aliev, Alex Tserkovny, Systemic approach to fuzzy logic formalization for approximate
reasoning, INFORMATION SCIENCES, 181(2011), pp. 1045-1059. 2011
http://dx.doi.org/10.1016/j.ins.2010.11.021
A27-c11 Rasoul Dadashzadeh; S. B. Nimse, ON STABILITY IN MULTIOBJECTIVE LINEAR PROGRAMMING PROBLEMS WITH SYMMETRIC TRAPEZOIDAL FUZZY NUMBERS, ANNALS OF ORADEA UNIVERSITY - MATHEMATICS FASCICOLA, Tom XIV(2007), pp. 514. 2007
http://stiinte.uoradea.ro/en/pdfs/2007/1.pdf
A27-c10 Rybkin VA, Yazenin AV, On the problem of stability in possibilistic optimization, INTERNATIONAL JOURNAL OF GENERAL SYSTEMS, 30 (1), pp. 3-22. 2001
http://dx.doi.org/10.1080/03081070108960695
A27-c9 Xu Jiuping, A kind of fuzzy linear programming problems based on interval-valued fuzzy sets,
Applied Mathematics - A Journal of Chinese Universities, vol. 15, pp. 65-72. 2000
http://dx.doi.org/10.1007/s11766-000-0010-y
A27-c8 S.Jenei, Continuity in approximate reasoning, Annales Univ. Sci. Budapest, Sect. Comp.,
15(1995) 233-242. 1995
Further investigations have been performed for stability of fuzzy linear systems and
fuzzy linear programming problems in [A27], [A34], [A31], [8], [A35].
A27-c7 B.Julien, An extension to possibilistic linear programming, FUZZY SETS AND SYSTEMS,
64(1994) 195-206. 1994
http://dx.doi.org/10.1016/0165-0114(94)90333-6
The approach deals with imprecise parameters treated as fuzzy variables with possibility distributions assigned to them in the form of fuzzy numbers. A fuzzy number is a
fuzzy set which is normal, continouos, fuzzy convex and compactly supported [A27].
8
in proceedings and in edited volumes
A27-c6 V.A.Rybkin and A.V.Yazenin, Regularization and stability of possibilistic linear programming
problems, in: Proceedings of the Sixth European Congress on Intelligent Techniques and Soft
Computing (EUFIT’98), Aachen, September 7-10, 1998, Verlag Mainz, Aachen, Vol. I, 1998 3741. 1998
The criterion of stability based on uniform metric is given in [A27, 6].
A27-c5 T. Tanino, Sensitivity Analysis in MCDM, in: Gal, Tomas; Stewart, Theodor J.; Hanne, Thomas
(Eds.) Multicriteria Decision Making Advances in MCDM Models, Algorithms, Theory and Applications Series: International Series in Operations Research & Management Science , Vol. 21,
Springer, [ISBN 978-0-7923-8534-9], 1999 pp. 7-1 – 7-29. 1999
A27-c4 Alexander V. Yazenin, Vladimir A. Rybkin, Strong and Weak Stability in Possibilistic Linear
Programming, in: Proceedings of the Seventh European Congress on Intelligent Techniques and
Soft Computing (EUFIT’99), Aachen, September 13-16, 1999 Verlag Mainz, Aachen, [ISBN 389653-808-X], 4 pages, (Proceedings on CD-Rom). 1999
in books
A27-c3 Pedro Salinas, El defensor, (Translated by Juan Marichal), Springer, [ISBN 842064532X],
2002.
A27-c2 Y.J.Lai and C.L.Hwang, Fuzzy Multiple Objective Decision Making, Lecture Notes in Economics and Mathematical Systems, No. 404, Springer Verlag, [ISBN: 978-3-540-57595-5], Berlin
1994.
A27-c1 H. Rommelfanger, Fuzzy Decision Support-Systeme, Springer-Verlag, Heidelberg, 1994 [ISBN
3-540-57793-9] (Second Edition).
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