Stability in possibilistic quadratic programming ∗
Elio Canestrelli
canestre@vega.unive.it
Silvio Giove
sgiove@vega.unive.it
Robert Fullér
rfuller@abo.fi
Abstract
We show that possibilistic quadratic programs with crisp decision variables and 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.
1
Preliminaries
Sensitivity analysis in fuzzy linear programming was first considered by Hamacher et al [13], where a
functional relationship between changes of parameters of the right-hand side and those of the optimal
value of the primal objective function was derived for almost all conceivable cases.
Fullér [5] showed that the solution to fuzzy linear programs with symmetrical triangular fuzzy numbers
is stable with respect to small changes of centres of fuzzy numbers.
Fedrizzi and Fullér [7] proved 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 (in contrast to classical linear programming, where a small error of measurement may produce
a large variation in the optimal value of the objective function).
In this paper we prove that possibilistic quadratic programs with crisp decision variables and continuous fuzzy number coefficients are well-posed, i.e. small changes of the membership function of the
coefficients may cause only a small deviation in the possibility distribution of the objective function.
Throughout this paper fuzzy sets are denoted by capital letters. A fuzzy number A is a fuzzy set of
the real line with a normal, upper semi-continuous and fuzzy convex membership function of bounded
support. Let A be a fuzzy number. Then for any θ ≥ 0 we define, ω(A, θ), the modulus of continuity of
A as
ω(A, θ) = sup |A(u) − A(v)|.
|u−v|≤θ
It is easy to see that if 0 ≤ θ ≤ θ0 then
ω(A, θ) ≤ ω(A, θ0 ).
The following statement holds [8]:
lim ω(A, θ) = 0
θ→0
(1)
Let [A]α = [a1 (α), a2 (α)] denote the α-level set of fuzzy number A. If A and B are fuzzy numbers and
t ∈ R then A + B, A − B, At are defined by Zadeh’s extension principle in the usual way.
∗
The final version of this paper appeared in: E. Canestrelli, S. Giove and R. Fullér, Stability in possibilistic quadratic
programming, Fuzzy Sets and Systems, 82(1996) 51-56. doi: 10.1016/0165-0114(95)00267-7
1
Let A and B be fuzzy numbers and
[A]α = [a1 (α), a2 (α)], [B]α = [b1 (α), b2 (α)]
for all α ∈ [0, 1]. We metricize the set of fuzzy numbers by the metric [9]
d(A, B) = sup max{|a1 (α) − b1 (α)|, |a2 (α) − b2 (α)|}.
α∈[0,1]
According to Zadeh’s possiblity theory [12] we have
Poss[A ∗ B] = sup min{A(x), B(y)} = sup(A − B)(t)
x∗y
(2)
t∗0
where ∗ stands for <, ≤, =, ≥ or >.
We will need the following lemmas [7]
Lemma 1 Let A, B, C and D be fuzzy numbers and let t ∈ R. Then
d(At, Bt) = |t|d(A, B), d(A ± C, B ± D) ≤ d(A, B) + d(C, D)
Lemma 2 Let x1 , x2 be real numbers such that |x1 | + |x2 | > 0 and let A1 and A2 be fuzzy numbers.
Then
θ
ω(A1 x1 + A2 x2 , θ) ≤ Ω
,
|x1 | + |x2 |
where
Ω(θ) = max{ω(A1 , θ), ω(A2 , θ)}
for θ ≥ 0.
Lemma 2 can be easily extended to any finite linear combinations of fuzzy numbers. Namely, let
x1 , . . . xn be real numbers such that |x1 | + · · · + |x2 | > 0 and let A1 , . . . , An be fuzzy numbers. Then
θ
ω(A1 x1 + · · · + An xn , θ) ≤ Ω
,
|x1 | + · · · + |xn |
where
Ω(θ) = max{ω(A1 , θ), . . . , ω(An , θ)}
for θ ≥ 0.
Lemma 3 Let δ ≥ 0 and let A, B be fuzzy numbers. If d(A, B) ≤ δ, then
sup |A(t) − B(t)| ≤ max{ω(A, δ), ω(B, δ)}.
t∈R
Lemma 3 shows that if all α-level sets of two fuzzy numbers are close to each other then there can be
only a small difference between their membership degrees.
A possibilistic quadratic program is
maximize
Z := x0 Cx + D0 x
subject to A0i x ≤ Bi , 1 ≤ i ≤ m, x ≥ 0
2
(3)
where C = (Ckj ) is a matrix of fuzzy numbers, Ai = (Aij ) and D = (Dj ) are vectors of fuzzy numbers
and Bi is a fuzzy number.
We will assume that all fuzzy numbers are non-interactive [2]. We define, Poss[Z = z], the possibility
distribution of the objective function Z. Following Buckley [2, 3] and Luhandjula [11] first specify the
possibility that x satisfies the i-th constraint. Let
Π(ai , bi ) = min{Ai1 (ai1 ), . . . , Ain (ain ), Bi (bi )}
where ai = (ai1 , . . . , ain ), which is the joint possibility distribution of Ai , 1 ≤ j ≤ n and Bi . Then
Poss[x ∈ Fi ] = sup{Π(ai , bi )|a0i x ≤ bi }
ai ,bi
which is the possibility that x is feasible with respect to th i-th constraint. Therefore, for x ≥ 0,
Poss[x ∈ F] = min Poss[x ∈ Fi ]
i=1,...,m
We next construct P oss[Z = z|x] which is the conditional possibility that Z equals z given x. The joint
possibility distribution of C and D is
Π(c, d) = min{Ckj (ckj ), D(dj )}
k,j
where c = (ckj ) is a crisp matrix and d = (dj ) a crisp vector. Therefore,
Poss[Z = z|x] = sup{Π(c, d)|x0 cx + d0 x = z}.
c,d
Finally, applying Bellman and Zadeh’s method of fuzzy decision making [1], the possibility distribution
of the objective function is defined as
Poss[Z = z] = sup min{Poss[Z = z|x], Poss[x ∈ F]}.
x≥0
2
Stability in possibilistic quadratic programming
An important question is the impact of the perturbations of the fuzzy parameters on the solution. We will
assume that there is a collection of fuzzy parameters Aδ , B δ , C δ and Dδ are available with the property
δ
max d(Aij , Aδij ) ≤ δ, max d(Ckj , Ckj
) ≤ δ, max d(Bi , Biδ ) ≤ δ, max d(Dj , Djδ ) ≤ δ
i,j
i
k,j
j
(4)
Then we have to solve the following perturbed problem:
x0 C δ x + (Dδ )0 x
maximize
subject to
(Aδi )0 x
≤
Biδ ,
(5)
1 ≤ i ≤ m, x ≥ 0
Let us denote by Poss[x ∈ Fiδ ] that x is feasible with respect to the i-th constraint in (5). Then the
possibility distribution of the objective function Z δ 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 (4)) of the possibility distribution of the objective function of the possibilistic quadratic programs (3) and (5).
3
δ , D , D δ be continuous
Theorem 2.1 Let δ > 0 be a real number and let Aij , Aδij , Bi , Biδ , Ckj , Ckj
j
j
fuzzy numbers. If (4) hold then
sup |Poss[Z δ = z] − Poss[Z = z]| ≤ Ω(δ)
(6)
z∈R
where
δ
Ω(δ) = max{ω(Aij , δ), ω(Aδij , δ), ω(Ckj , δ), ω(Ckj
, δ), ω(Bi , δ), ω(Biδ , δ), ω(Dj , δ), ω(Djδ , δ)}.
i,j,k
Proof It is sufficient to show that
|Poss[x ∈ Fi ] − Poss[x ∈ Fiδ ]| ≤ Ω(δ)
(7)
|Poss[Z = z|x] − Poss[Z δ = z|x]| ≤ Ω(δ)
(8)
for each z ∈ R, x ≥ 0 and 1 ≤ i ≤ m, because (6) follows from (7) and (8).
Let’s show (7) first. Let x ∈ Rn and 1 ≤ i ≤ m be arbitrarily fixed. From (2) it follows that
n
X
Poss[x ∈ Fi ] = sup(
Aij xj − Bi )(t)
t∗0
j=1
n
X
Poss[x ∈ Fiδ ] = sup(
Aδij xj − Biδ )(t)
t∗0
j=1
Applying Lemma 1 and (4) we have
X
X
n
n
n
X
δ
δ
d
Aij xj − Bi ,
Aij xj − Bi ≤
|xj |d(Aij , Aδij ) + d(Bi , Biδ ) ≤ δ(|x|1 + 1)
j=1
j=1
j=1
where |x|1 = |x1 | + · · · + |xn |. By Lemma 2 we get
max{ω
X
n
Aij xj − Bi , δ , ω
j=1
X
n
Aδij xj
−
Biδ , δ
j=1
} ≤ Ω(
δ
).
|x|1 + 1
Finally, applying Lemma 3 we get
|Poss[x ∈ Fi ] − Poss[x ∈ Fiδ ]| =
X
X
n
n
δ
δ
sup
Aij xj − Bi (t) − sup
Aij xj − Bi (t) ≤
t∗0
t∗0
j=1
j=1
n
X
n
X
δ
δ
sup
Aij xj − Bi (t) −
Aij xj − Bi (t) ≤
t∗0
j=1
j=1
n
X
n
X
δ
δ
sup
Aij xj − Bi (t) −
Aij xj − Bi (t) ≤
t∈R
j=1
j=1
Ω
δ(|x|1 + 1)
|x|1 + 1
4
= Ω(δ).
Let’s prove now (8). Let x ∈ Rn be arbitrarily fixed.
Applying Lemma 1 and (4) we have
d(x0 C δ x + (Dδ )0 x, x0 Cx + D0 x) =
X
X
X
X
δ
δ
d
xk Ckj xj +
Dj xj ,
xk Ckj xj +
Dj xj , ≤
j
k,j
X
δ
|xk xj |d(Ckj , Ckj
)+
j
k,j
X
X
|xj |d(Dj , Djδ ) ≤ δ(
|xk xj | + |x|1 )
j
k,j
k,j
By Lemma 2 we get
max{ω
X
xk Ckj xj +
X
X
X
δ
δ
Dj xj , δ , ω
xk Ckj xj +
Dj xj , δ } ≤
j
k,j
k,j
j
δ
)
k,j |xk xj | + |x|1
Ω( P
Finally, applying Lemma 3 we get
|Poss[Z = z|x] − Poss[Z δ = z|x]| =
| x0 Cx + D0 x (z) − x0 C δ x + (Dδ )0 x (z)| ≤
X
X
X
X
δ
δ
sup
xk Ckj xj +
Dj xj (t) −
xk Ckj xj +
Dj xj (t)
t∈R
j
k,j
k,j
j
P
δ( k,j |xk xj | + |x|1 )
P
= Ω(δ).
≤Ω
k,j |xk xj | + |x|1
Remark 2.1 From (1) and (6) 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 (4).
As an immediate consequence of this theorem we obtain the following result (see [5, 6]: If the fuzzy
numbers in (4) 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 g-fuzzy number coefficients (see [10]).
Sensitivity analysis in quadratic programs with fuzzy variables and coefficients (introduced by Canestrelli
and Giove [4]) is the topic of future research.
5
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