On stability in fuzzy linear programming problems

On Stability in Fuzzy Linear Programming Problems ∗
Robert Full´ er rfuller@ra.abo.fi, http://www.abo.fi/˜rfuller/robert.html
Abstract This study focuses on the problem of stability (with respect to changes of centres of fuzzy parameters) of the solution in Fuzzy Linear Programming (FLP) problems with symmetrical triangular fuzzy numbers and extended operations and inequalities.
i,j
max |bi − bi (δ )| ≤ δ,
i
(11)
Then we get the following FLP problem:
< (bi (δ ), di ), i = 0, . . . , m. (ai1 (δ ), α)x1 + · · · + (ain (δ ), α)xn ∼
(12)
< (ai1 , α)x1 + · · · + (ain , α)xn ∼ (bi , di ), i = 0, . . . , m.
(6)
(7)
We denote by µi (x) the degree of satisfaction of the i-th restriction at the point x ∈ I Rn in (7), i.e. µi (x) = υ (˜ ai1 x1 + · · · + a ˜in xn , ˜ bi ). Then the solution of the FLP problem (7) is n defined as a fuzzy set on I R whose membership function is given by µ(x) = min µi (x)
3
Stability of the solution for the FLP problem
a0 , x → min; subject to Ax ≤ b.
The conventional model of LP can be stated as (5)
In many real-world problems instead of minimization of the objective function (5) it may be sufficient to determine an x such that a01 x1 + · · · + a0n xn ≤ b0 , a11 x1 + · · · + a1n xn ≤ b1 , . . . am1 x1 + · · · + amn xn ≤ bm , where b0 is a predetermined aspiration level. Assume that all parameters in (6) are fuzzy and are described by symmetrical triangular fuzzy numbers. Then the following FLP problem can be obtained by replacing crisp parameters aij , bi with fuzzy numbers (1) a ˜ij = (aij , α), ˜ bi = (bi , di ) respectively and by < replacing crisp relation ≤ with extended relation ∼.
<˜ The degree of satisfaction of the assertion ”a ˜ is less than ˜ b”, which we write a ˜∼ b, is υ (˜ a, ˜ b) defined in [1] as
υ (˜ a, ˜ b) = sup a ˜(x) ∧ ˜ b(y ).
x∈ I Rn
(13)
In the following theorem we establish a stability of the solution of the FLP problem (7). Theorem 3.1. Let µ(x) and µδ (x) be solution of FLP problems (7) and (12) respectively. Then ||µ − µδ ||C = sup |µ(x) − µδ (x)| ≤ δ
1
Let a ˜ = (a, α) and ˜ b = (b, α ) be two fuzzy numbers of the form (1), λ ∈ R. Then it is easily verified from Zadeh’s extension principle that we have a ˜+˜ b = (a + b, α + α ), a ˜−˜ b = (a − b, α + α ), λa ˜ = (λa, |λ|α). (2)
It will be sufficient to show that |µi (x) − µδ i (x)| ≤ δ because from (16) follows (14). Let x ∈ I Rn and i ∈ {0, . . . , m} be arbitrarily fixed. Consider the following cases: (1) µi (x) = µδ i (x). In this case (16) is trivially obtained. (2) 0 < µi (x) < 1 and 0 < µδ i (x) < 1. In this case from (10), (11) we have |µi (x) − µδ i (x)| = |1 − ai (δ ), x − bi (δ ) ai , x − b i − [1 − ]| = α|x|1 + di α|x|1 + di |bi − bi (δ ) + ai (δ ), x − ai , x | ≤ α|x|1 + di |bi − bi (δ ) + | ai (δ ), −ai , x | δ + |ai (δ ) − ai |∞ |x|1 ≤ ≤ α|x|1 + di α|x|1 + di δ + δ |x|1 1 1 1 1 ≤ δ( + ) ≤ δ + , α|x|1 + di α di α d where ai (δ ) = (ai1 (δ ), . . . , ain (δ )) and |ai (δ ) − ai |∞ = maxj |aij (δ ) − aij |. (3) µi (x) = 1 and 0 < µδ i (x) < 1. In this case from (10) we have ai , x ≤ b i . Hence |µi (x) − µδ i (x)| = |1 − [1 − ai (δ ), x − bi (δ ) ]| = α|x|1 + di (17) 1 1 + , ∀x ∈ I Rn , i = 0, . . . , m, α d (16)
x≤y
(3)
This formula uses once more Zadeh’s extension principle. Let a ˜ = (a, α) and b = (b, α ) be two fuzzy numbers of the form (1). Then it can be easily checked that 1 if a ≤ b, a−b (4) υ (˜ a, ˜ b) = if b < a ≤ b + α + α , 1 − α+α 0 if a > b + α + α .
2
Preliminaries
Definition 2.1. A fuzzy set of the real line given by the membership function |a − t| if |a − t| ≤ α, 1− a ˜(t) = α 0 otherwise,
(1)
i=0,... ,m
(8)
2
and the maximizing solution x∗ of the FLP problem (7) satisfies the equation µ(x). µ(x∗ ) = µ∗ = max n
x∈ I R
(9)
From (2) and (4) it follows that the degree of satisfaction of the i-th restriction at the point x∈I Rn in (7) is the following: 1 if ai , x ≤ bi , ai , x − b i µi (x) = (10) otherwise, 1− α | x | + d ) 1 i 0 if ai , x > bi + α|x|1 + di , where |x|1 = |x1 | + · · · + |xn | and ai , x = ai1 x1 + · · · + ain xn , i = 0, 1, . . . , m. Remark 3.1. In the extremal case α = 0 but di > 0 in (10), we get a linear membership function, i.e. Zimmermann’s principle [4]. Assume that instead of exact centers aij , bi in (7) only their approximations aij (δ ), bi (δ ) are known such that max |aij − aij (δ )| ≤ δ,