Some properties of possibilistic linear
equality systems with weakly
noninteractive fuzzy numbers ∗
Robert Fullér
rfuller@abo.fi
Abstract
Possibilistic linear equality systems (PLES) are linear equality systems with
fuzzy parameters and crisp variables. We study the problem: For a given PLES
with weakly noninteractive fuzzy number parameters how does the solution
change if the parameters are varied?
Keywords: Triangular norm, weakly noninteractive fuzzy numbers, possiblity, extension principle, triangular fuzzy number
1
Introduction
We consider the following PLES with weakly noninteractive fuzzy number parameters
of symmetric triangular form
ãi x = b̃i , i = 1, . . . , m
(1)
where x = (x1 , . . . , xn ) is a vector of crisp variables, b̃i is a fuzzy number, ãi =
(ãi1 , . . . , ãin ) is a vector of weakly noninteractive fuzzy numbers ãij , the operations
addition and multiplication by a real number of fuzzy numbers are defined via a supt-norm convolution and the equations are understood in possibilistic sense. In this
paper we shall estimate that how does the solution of PLES (1) change if the fuzzy
parameters are varied.
2
Preliminaries
A fuzzy set ã of the real line IR with a normal upper semicontinuous and finite-support
membership function is called fuzzy number. The family of all fuzzy numbers we shall
∗
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, Springer-Verlag,
[ISBN 3-540-54577-8], 1991 45-48. [Zbl.772.15002]
1
denote by F. A fuzzy number ã of the real line IR given by the membership function

|a − t|

1−
if |a − t| ≤ α,
ã(t) =
α

0
otherwise,
is called a symmetrical triangular fuzzy number with center a ∈ IR and spread α > 0
and we shall refer to it by the pair (a, α). A function T : [0, 1]×[0, 1] → [0, 1] is said to
be triangular-norm or t-norm for short (Schweizer and Sklar, 1961) if T is associative,
symmetric, monotone and T (a, 1) = a, ∀a ∈ [0, 1].
The fuzzy numbers ã, b̃ are said to be T -weakly noninteractive (Dubois and Prade,
1981) if there exists a t-norm T such that their joint possibility distribution (ã, b̃) is
defined as
(ã, b̃)(z) = T (ã(z1 ), b̃(z2 )),
for each z = (z1 , z2 ) ∈ IR2 .
If f : IR × IR → IR is a function, ã and b̃ are T -weakly noninteractive fuzzy numbers
then according to Zadeh’s extension principle (Zadeh, 1965) f (ã, b̃) is defined as follows
f (ã, b̃)(z) = sup T (ã(a), b̃(b))
f (a,b)=z
We define a metric D in F by the equation
D(ã, b̃) = sup d([ã]α , [b̃]α ).
α∈[0,1]
where d is the Hausdorff metric in the nonempty compact convex subsets of IR2 and
[ã]α denotes the α-level set of ã. Let t be a T -norm and ã, b̃ be fuzzy numbers. The
truth value of the assertion ”ã is equal to b̃” is Pos(ã = b̃) defined in Zadeh (1978) as
Pos(ã = b̃) = sup T (ã(t), b̃(t)) = (ã − b̃)(0)
t∈IR
The following lemma can be proved directly by using the properties of metric D and
those result obtained by Fullér (1990), and Fullér and Keresztfalvi (1990).
Lemma 2.1 Let T be a t-norm, x, y ∈ IR, ã = (a, α), b̃ = (b, α), c̃ = (c, α) and
d˜ = (d, α) be fuzzy numbers of symmetric triangular form. Then,
˜ ≤ |x||a − c| + |y||b − d|
(i) D(xã + y b̃, xc̃ + y d)
2
˜
(ii) |(xã + y b̃)(t) − (xc̃ + y d)(t)|
≤ × max{|a − c|, |b − d|}, t ∈ IR.
α
3
Some properties of PLES’s
Consider the following PLES’s
ãi x = b̃i , i = 1, . . . , m
(2)
c̃i x = d˜i , i = 1, . . . , m
(3)
2
where ãi = (ãij ), c̃i = (c̃ij ) are vectors of T -weakly noninteractive fuzzy numbers and
b̃i , d˜i are fuzzy numbers.
We denote by µi (x), νi (x) the degree of satisfaction of the i-th equation at the point
∈ IRn in PLES’s (2) and (3) respectively, i.e.
νi (x) = P os(c̃i x = d˜i )
µi (x) = P os(ãi x = b̃i ),
where Pos denotes the T -norm-based possibility. Then the solution of PLES’s (2) and
(3) is defined by Bellman and Zadeh (1972) as
µ(x) = min µi (x),
ν(x) = min νi (x)
i=1,...,m
i=1,...,m
In the next theorem we shall give an estimation for the sensitivity of the solution of
PLES’s to the variation of fuzzy number parameters.
Theorem 3.1 Let T be a t-norm b̃i = (bi , α), d˜i = (di , α) and let ãij = (aij , α) and
c̃ij = (cij , α) be T -weakly noniteractive fuzzy numbers. Then
|µ(x) − ν(x)| ≤
n
max{|aij − cij |, |bi − di |}
α i,j
(4)
The proof of this theorem follows from Lemma 1.
Illustration. Let T (u, v) = Tw (u, v), where Tw denotes the weak t-norm defined by
Tw (u, v) = u if v = 1, Tw (u, v) = v if u = 1 and Tw (u, v) = 0 otherwise.
Then with the notation of Theorem 1 we have
µ(x) = 1 −
|Ax − b|∞
,
α max{|x|∞ , 1}
so
|µ(x) − ν(x)| ≤
ν(x) = 1 −
|Cx − d|∞
α max{|x|∞ , 1}
||A − C|| + |b − d|∞
,
α
where |x|∞ = |x1 | + · · · + |xn |.
Remark 3.1 From (4) it follows that small changes in the membership function of
fuzzy parameters of PLES’s (2) and (3) may cause only a small variation in the
solution, i.e. unlike crisp linear equality systems, PLES’s always belong to the family
of well-posed problems.
Remark 3.2 Similar results have been obtain for PLES’s with noninteractive fuzzy
number parameters by Fullér (1990) and Kovács (1988).
References
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