Stability in multiobjective possibilistic
linear programs ∗
Robert Fullér
rfuller@abo.fi
Mario Fedrizzi
fedrizzi@cs.unitn.it
Abstract
This paper continues the authors’ research in stability analysis in possibilistic programming in that it extends the results in [7] to possibilistic linear
programs with multiple objective functions. Namely, we show that multiobjective possibilistic linear programs with continuous fuzzy number coefficients are
well-posed, i.e. small changes in the membership function of the coefficients
may cause only a small deviation in the possibility distribution of the objective
function.
Keywords: Multiobjective possibilistic linear programs (MPLP), possibility theory,
fuzzy number, stability
1
Introduction
Stability and sensitivity analysis becomes more and more attractive also in the area
of multiple objective mathematical programming (for excellent surveys see e.g. Gal
[10] and Rios Insua [18]). Publications on this topic usually investigate the impact
of parameter changes (in the righthand side or/and the objective functions or/and
the ’A-matrix’ or/and the domination structure) on the solution in various models of
vectormaximization problems, e.g. linear or non-linear, deterministic or stochastic,
static or dynamic (see e.g. [5, 17, 19]).
There are only few paper dealing with stability or sensitivity analysis in fuzzy mathematical programming: Sensitivity analysis in LP problems with crisp coefficients and
soft constraints was first considered in [11], where a functional relationship between
changes of parameters of the righthand side and those of the optimal value of the
primal objective function was derived for almost all conceivable cases.
The stability of the fuzzy solution in LP problems with fuzzy coefficients of symmetric triangular form was shown in [9]. In this paper, generalizing the authors’ earlier
result [7], we show that the possibility distribution of the objectives of an MPLP
with continuous fuzzy number coefficients is stable under small changes in the membership function of the parameters. In this section we recall Buckley’s [2, 3, 4] and
∗
The final version of this paper appeared in: R. Fullér and M. Fedrizzi, Stability in multiobjective
possibilistic linear programs, European Journal of Operational Research, 74(1994) 179-187.
1
Luhandjula’s [16] solution concept for MPLP and set up the notation needed for our
main result in the next section.
A fuzzy quantity ã : IR → [0, 1] is a fuzzy set of the real line IR. The support of a
fuzzy quantity ã (denoted by suppã) is the crisp set given by {t|ã(t) > 0}. A fuzzy
number ã is a fuzzy quantity with a normalized, upper semicontinuous, fuzzy convex
and bounded supported membership function.
A symmetric triangular fuzzy number ã denoted by (a, α) is defined as

|a − t|

1−
if |a − t| ≤ α,
ã(t) =
α

0
otherwise,
where a ∈ IR is the centre and 2α > 0 is the spread of ã. The fuzzy numbers will represent the continuous possibility distributions for fuzzy parameters. A multiobjective
possibilistic linear programming is
maximize Z = (c̃11 x1 + · · · + c̃1n xn , . . . , c̃k1 x1 + · · · + c̃kn xn )
subject to
ãi1 x1 + · · · ãin xn ∗ b̃i , i = 1, . . . , m, x ≥ 0,
(1)
where ãij , b̃i , and c̃lj are fuzzy numbers, x = (x1 , . . . , xn ) is a vector of (non-fuzzy)
decision variables, the operations addition and multiplication by a real number of
fuzzy numbers are defined by Zadeh’s extension principle [20], and ∗ denotes <, ≤,
=, ≥ or > for each i, i = 1, . . . , m. Even though ∗ may vary from row to row in the
constraints, we will rewrite the MPLP (1) as
maximize Z = (c̃1 x, . . . , c̃k x)
subject to
Ãx ∗ b̃, x ≥ 0,
(2)
where ã = [ãij ] is an m × n matrix of fuzzy numbers and b̃ = (b̃1 , ..., b̃m ) is a vector of
fuzzy numbers. The fuzzy numbers are the possibility distributions associated with
the fuzzy variables and hence place a restriction on the possible values the variable
may assume [20, 21]. For example, Poss[ãij = t] = ãij (t).
We will assume that all fuzzy numbers ãij , b̃i , c̃j are non-interactive [20].
Following Buckley [2-4], 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
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
c̃lj , j = 1, . . . , n, is
π(cl ) = min{c̃l1 (cl1 ), . . . , c̃ln (cln )}
2
where cl = (cl1 , . . . , cln ), l = 1, . . . , k. Therefore,
Poss[Z = z|x] = Poss[c̃1 x = z1 , . . . , c̃k x = zk ] = min Poss[c̃l x = zl ] =
1≤l≤k
min
sup {π(cl )|cl1 x1 + · · · + cln xn = zl }.
1≤l≤k cl1 ,...,clk
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
Let ã be a fuzzy number. Then for any θ ≥ 0 we define ω(ã, θ), the modulus of
continuity of ã as [12]
ω(ã, θ) = max |ã(u) − ã(v)|.
|u−v|≤θ
An α-level set of a fuzzy numer ã is a non-fuzzy set denoted by [ã]α and is defined by
{t ∈ IR | ã(t) ≥ α} if α > 0
α
[ã] =
cl(suppã)
if α = 0
where cl(suppã) denotes the closure of the support of ã.
Let ã and b̃ be fuzzy numbers and
[ã]α = [a1 (α), a2 (α)], [b̃]α = [b1 (α), b2 (α)].
We metricize the set of fuzzy numbers by the metric [13]
D(ã, b̃) = sup max{|ai (α) − bi (α)|},
α∈[0,1] i=1,2
where d denotes the classical Hausdorff metric in the family of compact subsets of
IR2 , i.e.
d([ã]α , [b̃]α ) = max{| a1 (α) − b1 (α) |, | a2 (α) − b2 (α) |}
The proof of the next two lemmas is very simple.
Lemma 1.1 Let n ∈ N and let xi , yi ∈ [0, 1], i = 1, . . . , n. Then
| min xi − min yi |≤ max | xi − yi |
i=1,...,n
i=1,...,n
i=1,...,n
Lemma 1.2 Let X 6= ∅ be a set and let f , g be fuzzy sets of X. Then
| sup f (x) − sup g(x) |≤ sup | f (x) − g(x) |
x∈X
x∈X
x∈X
The following lemma shows that if all the α-level sets of two continuous fuzzy numbers are close to each other, then there can be only a small deviation between their
membership grades.
Lemma 1.3 [7] 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∈IR
Remark 1.1 It should be noted that if ã or b̃ are discontinuous fuzzy numbers, then
there can be a big deviation between their membership grades even if D(ã, b̃) is arbitrarily small.
3
2
Stability in multiobjective possibilistic linear programs
An important question [6, 7, 9, 22] 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(c̃lj , c̃δlj ) ≤ δ.
l,j
(3)
Then we have to solve the following problem:
max/min Z δ = (c̃δ1 x, . . . c̃δk x)
(4)
subject to Ãδ x ∗ b̃δ , x ≥ 0.
Let us denote by Poss[x ∈ Fiδ ] the Possibility that x is feasible with respect to the
i-th constraint in (4). Then the Possibility distribution of the objective function Z δ
in (4) is defined as follows:
Poss[Z δ = z] = sup(min{Poss[Z δ = z | x], Poss[x ∈ F δ ]}).
x≥0
The following theorem shows a stability property (with respect to perturbations (3))
of the possibility dostribution of the objective function of MPLP (1) and (4).
Theorem 2.1 Let δ ≥ 0 be a real number and let ãij , b̃i , ãδij , c̃lj , c̃δlj be continuous
fuzzy numbers. If (3) hold, then
sup | Poss[Z δ = z] − Poss[Z = z] |≤ ω(δ)
(5)
z∈IR
where
ω(δ) = max{ω(ãij , δ), ω(ãδij , δ), ω(b̃i , δ), ω(bδi , δ), ω(c̃lj , δ), ω(c̃δlj , δ)}.
i,j,l
i.e. ω(δ) denotes the maximum of modulus of continuity of all fuzzy coefficients at
the point δ.
Proof. It is sufficient to show that
| Poss[Z = z] − Poss[Z δ = z] |≤ ω(δ), for any (z1 , . . . , zk ) ∈ IRk ,
(6)
Applying Lemmas 1.1-1.3 we have . . . (for full proof see EJOR, 74(1994) 179187) . . . which proves the theorem.
Remark 2.1 From limδ→0 ω(δ) = 0 and (5) it follows that
sup | Poss[Z δ = z] − Poss[Z = z] |→ 0 as δ → 0
z∈IR
which means the stability of the possibility distribution of the objective function with
respect to perturbations (3).
4
Remark 2.2 As an immediate consequence of this theorem we obtain the following
result (see [9]: If the fuzzy numbers in (1) and (4) satisfy the Lipschitz condition with
constant L > 0, then
sup | Poss[Z δ = z] − Poss[Z = z] |≤ Lδ
z∈IR
Furthermore, similar estimations can be obtained in the case of symmetrical trapezoidal fuzzy number parameters (see [15]) and in the case of symmetrical triangular
fuzzy number parameters (see [8, 14]).
Remark 2.3 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.
3
Illustration
As an example, consider the following biobjective possibilistic linear program
max/min (c̃x, c̃x)
(7)
subject to ãx ≤ b̃, x ≥ 0.
where ã = (1, 1), b̃ = (2, 1) and c̃ = (3, 1) are fuzzy numbers of symmetric triangular
form. Here x is one-dimensional (n = 1) and there is only one constraint (m = 1).
We find

if x ≤ 2,
 1
Pos[x ∈ F] =
 3
if x > 2.
x+1
and
Pos[Z = (z1 , z2 )|x] = min{Pos[c̃x = z1 ], Pos[c̃x = z2 ]}
where


zi


4
−
if zi /x ∈ [3, 4],


x

zi
Pos[c̃x = zi ] =

− 2 if zi /x ∈ [2, 3],


x


 0
otherwise,
for i = 1, 2 and x 6= 0, and
(
Pos[Z = (z1 , z2 )|0] = Pos[0 × c̃ = z] =
1 if z = 0,
0 otherwise.
Both possibilities are nonlinear functions of x, however the calculation of Pos[Z =
(z1 , z2 )] is easily performed and we obtain

if z ∈ M1 ,

 θ1
min{θ1 , θ2 , θ3 } if z ∈ M2 ,
Pos[Z = (z1 , z2 )] =


0
otherwise,
5
Figure 1: Pos[c̃x = z2 ] for z2 = 8.
where
M1 = {z ∈ IR2 | |z1 − z2 | ≤ min{z1 , z2 }, z1 + z2 ≤ 12},
M2 = {z ∈ IR2 | |z1 − z2 | ≤ min{z1 , z2 }, z1 + z2 > 12},
and
θi =
24
p
.
zi + 7 + zi2 + 14zi + 1
for i = 1, 2 and
4 min{z1 , z2 } − 2 max{z1 , z2 }
.
z1 + z2
Consider now a perturbed biobjective problem with two different objectives (derived
from (7) by a simple δ-shifting of the centres of ã and c̃):
θ3 =
max/min (c̃x, c̃δ x)
(8)
subject to ãδ x ≤ b̃, x ≥ 0.
where ã = (1 + δ, 1), b̃ = (2, 1), c̃ = (3, 1), c̃δ = (3 − δ, 1) and δ ≥ 0 is the error of
measurement. Then

2


1
if x ≤
,

1
+
δ
δ
Pos[x ∈ F ] =


 3 − δx if x > 2 .
x+1
1+δ
and
Pos[Z δ = (z1 , z2 )|x] = min{Pos[c̃x = z1 ], Pos[c̃δ x = z2 ]}
where


z1


4−


x

z1
Pos[c̃x = z1 ] =

−2


x


 0


 4 − δ − z2



x

δ
z2
Pos[c̃ x = z2 ] =

−2+δ


x


 0
6
if z1 /x ∈ [3, 4],
if z1 i/x ∈ [2, 3],
otherwise,
if z2 /x ∈ [3 − δ, 4 − δ],
if z2 /x ∈ [2 − δ, 3 − δ],
otherwise,
x 6= 0, and
(
Pos[Z δ = (z1 , z2 )|0] = Pos[0 × c̃ = z] =
1 if z = 0,
0 otherwise.
So,

if z ∈ M1 (δ),

 θ1 (δ)
min{θ1 (δ), θ2 (δ), θ3 (δ)} if z ∈ M2 (δ),
Pos[Z δ = (z1 , z2 )] =


0
otherwise,
where
(
)
2(6
−
δ)
z ∈ IR2 | |z1 − z2 | ≤ (1 − 0.5δ) min{z1 , z2 }, z1 + z2 ≤
,
1+δ
M1 (δ) =
(
M2 (δ) =
)
2(6
−
δ)
z ∈ IR2 | |z1 − z2 | ≤ (1 − 0.5δ) min{z1 , z2 }, z1 + z2 >
,
1+δ
p
24 + δ 7 − z1 − z12 + 14z1 + 1 + 4z1 δ
p
θ1 (δ) =
z1 + 7 + z12 + 14z1 + 1 + 4z1 δ + 2δ
p
2
24 − δ δ + z2 − 1 + (1 − δ − z2 ) + 16z2
p
θ2 (δ) =
z2 + 7 + (1 − δ − z2 )2 + 16z2 + δ
and
θ3 (δ) =
(4 − δ) min{z1 , z2 } − 2 max{z1 , z2 }
.
z1 + z2
It is easy to check that
sup |Pos[x ∈ F] − Pos[x ∈ F δ ]| ≤ δ,
x≥0
sup |Pos[Z = z|x] − Pos[Z δ = z|x]| ≤ δ, ∀x ≥ 0,
z
sup |Pos[Z = z] − Pos[Z δ = z]| ≤ δ.
z
On the other hand, from the definition of metric D the modulus of continuity and
Theorem 2.1 it follows that
D(ã, ãδ ) = δ,
D(c̃, c̃δ ) = δ,
D(c̃, c̃) = 0,
D(b̃, b̃) = 0,
ω(δ) = δ,
and, therefore,
sup |Pos[Z = z] − Pos[Z δ = z]| ≤ δ.
z
7
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