Stability in Multiobjective Possibilistic
Linear Programs with Weakly
Noninteractive Fuzzy Number
Coefficients ∗
Patrik Eklund
peklund@ra.abo.fi
Mario Fedrizzi
fedrizzi@cs.unitn.it
Robert Fullér
rfuller@abo.fi
Abstract
Stability and sensitivity analysis becomes more and more attractive also
in the area of multiple objective mathematical programming (for excellent
surveys see e.g. [Gal86] and [Rios90]). Publications on this topic usually
investigate the impact of parameter changes (in the righthand side or/and
the objective function 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. [Dev79,
Rari83, Rios91]). There are only few paper dealing with stability or sensitivity
analysis in fuzzy mathematical programming (e.g. [Ham78, Tan86, Ful89,
Fed92, Fef92, Dut92]).
In this paper, generalizing the earlier results [Fed92,Fef92] of the second
and third co-authors’, we show that the possibility distribution of the objectives of an Multiobjective Possibilistic Linear Program (MPLP) under continuous triangular norms is stable under small changes in the membership
function of the continuous fuzzy number parameters.
Keywords: Multiobjective possibilistic linear programming, fuzzy number, possibility theory, stability.
1
Introduction
Sensitivity analysis in FLP problems with crisp coefficients and soft constraints
was first considered in [Ham78], 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 final version of this paper appeared in: M.Fedrizzi, J.Kacprzyk and M.Roubens eds., Interactive Fuzzy Optimization, Lecture Notes in Economics and Math. Systems, Vol. 368, SpringerVerlag, 1991 45-48.
1
In [17] a FLP problem was formulated and the value of information was discussed
via sensitivity analysis. The stability of fuzzy solution in FLP problems with fuzzy
coefficients of symmetric triangular form was shown in [8]. Dutta et al [5] performed sensitivity analysis in fuzzy linear fractional programming problems with
crisp coefficients and soft constraints. In this paper, generalizing the authors’ earlier results [6, 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] and Luhandjula’s [12] 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 supp ã ) 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 ã(t) = 1 −
|a − t|/α if |a − t| ≤ α and ã(t) = 0 otherwise, where a ∈ IR is the center and 2α > 0
is the spread of ã.
The fuzzy numbers will represent the possibility distributions for fuzzy parameters.
A multiobjective possibilistic linear programming is
Z = (c̃11 x1 + · · · + c̃1n xn , . . . , c̃k1 x1 + · · · + c̃kn xn ) → max
(1)
subject to ãi1 x1 + · · · + ãin xn ∗ b̃i , i = 1, . . . , m, x ≥ 0,
(2)
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 [18], 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 (1,2) as
Z = (c̃1 x, . . . , c̃k x) → max
(3)
subject to Ãx ∗ b̃, x ≥ 0,
(4)
where à = [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 [18]. For example, Poss[ãij = t] = ãij (t) is the possibility that
ãij is equal to t.
We will assume that all fuzzy numbers ãij , b̃ij , c̃lj are weakly noninteractive [18].
Weakly-noninteractivity means that there exists a triangular norm T , such that we
can find the joint possibility distribution of all the fuzzy variables by calculating
the T -intersection of their possibility distributions. Following Buckley [2, 3] and
Luhandjula [12], 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
ΠT (ai , bi ) = T ãi1 (ai1 ), . . . , ãin (ain ), b̃i (bi ) ,
2
where ai = (ai1 , . . . , ain ), which is the joint distribution of ãij , j = 1, . . . , n, and b̃i .
Then
Poss[x ∈ Fi ] = sup{ ΠT (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 c̃1j ,
j = 1, . . . , n, is
ΠT (cl ) = T (c̃l1 (cl1 ), . . . , c̃ln (cln )),
where cl = (cl1 , . . . , cln ), l = 1, . . . , k. Therefore,
Poss[Z = (z1 , . . . , zk )|x] = Poss[c̃1 x = z1 , . . . , c̃k x = zk ] =
min Poss[c̃l x = zl ] = min sup {ΠT (cl ) | cl1 x1 + · · · + cln xn = zl }.
1≤l≤k
1≤l≤k cl1 ,...,clk
Finally, applying Bellman and Zadeh’s method of fuzzy decision making [Bel72], 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 continuous fuzzy number, then for any θ ≥ 0 we define ω(θ), the modulus
of continuity of ã by
ω(ã, θ) = max |ã(u) − ã(v)|.
|u−v|≤θ
We metricize F by the metric [Kal87],
D(ã, b̃) = sup d([ã]α , [b̃]α )
α∈[0,1]
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)|},
[ã]α = [a1 (α), a2 (α)], [b̃]α = [b1 (α), b2 (α)].
Recall that the mapping T : [0, 1] × [0, 1] → [0, 1] is a triangular norm [Sch63] iff it is
commutative, associative, non-decreasing in each argument and T (x, 1) = x, ∀x ∈
[0, 1].
The following lemma shows that if all the α-level sets of two continuous fuzzy numbers are close to each other, there can be only a small deviation between their
membership grades.
Lemma 1.1. [Fed92]. Let δ ≥ 0 be a real number and let ã, b̃ be fuzzy intervals. If
D(ã, b̃) ≤ δ
then
supã(t) − ˜(b)(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 [Tan86, Zim87, Ful89, Fed92] is the impact of small perturbations in the membership functions of fuzzy coefficients on the possibility distribution
of the objective function.
Suppose that instead of the exact coefficients ãij , b̃i , c̃lj there is a collection of fuzzy
coefficients ãδij , b̃δi , c̃δlj available with the property
max D(ãij , ãδij ) ≤ δ, max D(b̃i , b̃δi ) ≤ δ, max D(c̃lj , c̃δlj ) ≤ δ
i,j
i
lj
(5)
where δ > 0 denotes the maximal error of measurement. Then we have to solve the
following problem
Z δ = (c̃δ1 x, . . . , c̃δk ) → max
(6)
subject to Ãδ x ∗ b̃δ , x ≥ 0
(7)
The possibility distribution of the objective function, Z δ , is computed as
Poss[Z δ = z] = sup min{Poss[Z δ = z|x], Poss[x ∈ F δ ]}.
x≥0
The next theorem establishes a stability property (with respect to perturbations (5))
of the possibility distribution of the objective function of MPLP (1,2) and (6,7).
Theorem 2.1. Let δ ≥ 0 be a real number and let ãij , b̃i , ãδij , b̃δi , c̃lj and c̃δlj be
continuous fuzzy numbers. If (5) holds, then
sup |Poss[Z δ = z] − Poss[Z = z]| ≤ ω(T, Ω(δ))
(8)
z∈IRk
where
Ω(δ) = max max{ω(ãij , δ), ω(ãδij , δ), ω(b̃i , δ), ω(b̃δi , δ), ω(c̃lj , , δ), ω(c̃δlj , δ)},
i,j,l
i.e., Ω(δ) denotes the maximum of modulus of continuity of all the fuzzy parameters
at the point δ.
Proof. It is sufficient to show that
|Poss[Z δ = z] − Poss[Z = z]| ≤ ω(T, Ω(δ))
for any z = (z1 , . . . , zk ) ∈ IRk . Applying Lemma 1.1 we have
Poss[Z δ = z] − Poss[Z = z] ≤ ω(T, Ω(δ))
for any z = (z1 , . . . , zk ) ∈ IRk . Applying Lemma 1.1 we have
Poss[Z δ = z] − Poss[Z = z] =
δ
δ
sup min{Poss[Z = z|x], Poss[x ∈ F]} − sup min{Poss[Z = z|x], Poss[x ∈ F ]} ≤
x≥0
x≥0
4
δ
δ
sup max Poss[Z = z|x] − Poss[Z = z|x], Poss[x ∈ F] − Poss[x ∈ F ] ≤
x≥0
sup max max {Poss[c̃l x = zl ] − Poss[c̃δl x = zl ]},
1≤l≤k
x≥0
max {Poss[x ∈ Fi ] − Poss[x ∈ Fiδ ]
1≤i≤m
=
sup max max sup {T (c̃l1 (cl1 ), . . . , c̃ln (cln )) | cl1 x + · · · + cln xn = zl }−
1≤l≤k
x≥0
cl1 ,...,cln
sup {T (c̃δl1 (cl1 ), . . . , c̃δln (cln )) | cl1 x1 + · · · + cln xn = zl },
,...,c
cl1
ln
max sup T (ãi1 (ai1 ), . . . , ãin (ain ), b̃i (bi )) | ai1 x1 + · · · + ain xn ∗ bi }−
1≤i≤m ai1 ,...,ain
δ
δ
δ
sup {T (ãi1 (ai1 ), . . . , ãin (ain ), b̃i (bi )) | ai1 x1 + · · · + ain xn ∗ bi } ≤
ai1 ,...,ain
sup max max
sup max{ω(T, |c̃lj (clj ) − c̃δlj (clj )|) | cl1 x1 + · · · + cln xn = zl },
1≤l≤k cl1 ,...,cln
x≥0
max
sup
max {max{ω(T, |ãij (aij ) − ãδij (aij )|),
1≤i≤m ai1 ,...,ain 1≤j≤n
ω(T, |b̃i (bi ) −
b̃δi (bi ))|}
| ai1 x1 + · · · + ain xn ∗ bi } ≤
δ
δ
δ
sup max ω(T, |ãij (aij )−ãij (aij )|), ω(T, |b̃i (bi )−b̃i (bi )|), ω(T, |c̃lj (clj )−c̃lj (clj )|) ≤
aij ,bi ,clj
max max ω(T, ω(ãij , δ)), ω(T, ω(ãδij , δ)), ω(T, ω(b̃i , δ)), ω(T, ω(b̃δi , δ)),
i,j,l
ω(T, ω(c̃lj , δ)),
ω(c̃δlj , δ))
≤ ω(T, Ω(δ)).
Which ends the proof.
Remark 2.1. From limδ→0 Ω(δ) = 0 and (8) it follows that
sup |Poss[Z δ = z] − Poss[Z = z]| → 0 as δ → 0
z
which means the stability of the possibility distribution of the objective function with
respect to perturbations (5).
We illustrate Theorem 2.1 by a very simple example. Consider the following MPLP
(c̃1 x, c̃2 x) → max
subject to ãx ≤ b̃, x ≥ 0,
where c̃i = (ci , γi ), i = 1, 2, ã = (a, α), b̃ = (b, β) and T (u, v) = max{0, u + v − 1},
(Lukasiewicz triangular norm).
5
Suppose now that instead of the exact centers γi , i = 1, 2, α and β we have to work
with their approximations γi0 , i = 1, 2, α0 and β 0 , such that
|α − α0 | ≤ δ, |β − β 0 | ≤ δ, max |γi − γi0 | ≤ δ,
i=1,2
where δ > 0 is the maximal error of measurement.
Then applying Theorem 2.1. we have the following estimation for the distance
between the possibility distributions of the objective functions of the exact and the
perturbed MPLP:
1 1 1 1 1 1 1 1
δ
sup |Poss[Z = z] − Poss[Z = z]| ≤ 2δ max
, , , , , , ,
.
γ1 γ10 γ2 γ20 α α0 β β 0
z
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