Stability in possibilistic linear equality
systems under continuous triangular
norms, ∗
Robert Fullér†
rfuller@abo.fi
Abstract
We consider linear equality systems where all the parameters may be fuzzy
variables specified by their possibility distribution, the operations addition
and multiplication by a real number of fuzzy parameters are defined via a suptriangular norm composition, and the equations are understood in possibilistic
sense. We show that when the triangular norm defining the operations and
equations is continuous, then the possibility distribution of the solution of
these systems depend continuously on the fuzzy parameters.
Keywords: Fuzzy set, α-level set, fuzzy number, Zadeh’s extension principle, generalized Hausdorff metric, possibility, triangular norm, modulus of continuity, stability
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Stability in Possibilistic Linear Equality Systems under Continuous Triangular Norms
This paper continues the author’s research in possibilistic systems in that it extends
the results in [4] to possibilistic equality systems under continuous triangular norms.
In contrast to classical linear equality systems (where a small error of measurement
may produce a large variation in the solution), we show the possibility distribution
of the solution of a possibilistic linear equality system (PLES) with continuous
fuzzy number parameters and continuous triangular norm is stable under small
perturbations of the parameters.
First, we shall briefly review PLES’s and set up the notations. A fuzzy number is a
fuzzy set [11] ã, ã : IR → [0, 1] = I, which is normal continuous, fuzzy convex and
compactly supported. The fuzzy numbers will represent the continuous possibility
distributions for fuzzy parameters. A function T : I ×I → I is said to be a triangular
norm [10] (t-norm for short) iff T is symmetric, associative, non-decreasing in each
argument, and T (x, 1) = x for all x ∈ I. Let ã and b̃ be fuzzy numbers and T be a
∗
in: Proceedings of the Annual Conference of the Operational Research Society of Italy, September 18-20, 1991 Riva del Garda, Italy 130-133.
†
Supported by the German Academic Exchange Service (DAAD)
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t-norm. The grade of possibility [12] of the assertion ”ã is equal to b̃”, denoted by
ΠT (ã = b̃), is defined as
ΠT (ã = b̃) = sup T (ã(t), b̃(t)),
(1)
t∈IR
and the T -sum [12] of ã and b̃, denoted by ã + b̃, is defined as
(ã + b̃)(z) = sup T (ã(x), b̃(y)).
(2)
x+y=z
A possibilistic linear equality system [2] is
ãi1 x1 + . . . + ãin xn = b̃i , i = 1, . . . , m
(3)
where ãij , b̃i are fuzzy sets of IR, x = (x1 , . . . , xn ) is a vector of (non-fuzzy) variables,
the addition of fuzzy numbers is defined by (2), and the equations are defined by
(1).
Following Bellman and Zadeh [1], we define µ(x), the possibility distribution of the
solution by
µ(x) = min µi (x)
i=1,...,m
where µi (x) denotes the possibility that x satisfies the i-th equation at the point
x ∈ IRn in, i.e.
µi (x) = ΠT (ãi1 x1 + . . . + ãin xn = b̃i ).
In many important cases the fuzzy parameters ãij , b̃i of the system (3) are not known
exactly [8,9,13] and we have to work with their approximations ãδij , b̃δi such that
max D(ãij , ãδij ) ≤ δ,
i,j
max D(b̃i , b̃δi ) ≤ δ,
i
(4)
where D denote the generalized Hausdorff metric for fuzzy numbers [6] and δ > 0
is a real number.
Then we get the following system with perturbed fuzzy parameters
ãδi1 x1 + . . . + ãδin xn = b̃δi , i = 1, . . . , m.
(5)
The following theorem establishes a stability property (with respect to perturbations
(4)) of the possibility distribution of the solution of the PLES (3).
Theorem 1.1 Let ãij , ãδij , b̃i , b̃δi be continuous fuzzy numbers and let T be a continuous t-norm. If (4) holds, then
sup | µ(x) − µδ (x) |≤ ω(nδ)
(6)
x∈IRn
where ω is maximum of the modulus of continuity of fuzzy parameters and µ(x),
µδ (x) are the possibility distributions of the solution of systems (3) and (5), respectively.
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The proof of this theorem is principially based: on the relationship [6]
[ãi1 x1 + . . . + ãin xn − b̃i ]α =
(x1 [ãi1 ]α1 + . . . + xn [ãin ]αn − [b̃i ]αn+1 ),
[
T (α1 ,...,αn+1 )≥α
where [ã]α denotes the α-level set of ã; and on the inequality [7]
D
X
n
ãij xj − b̃i ,
j=1
n
X
ãδij xj
−
b̃δi
≤ δ(|x|1 + 1),
j=1
where |x|1 = |x1 | + . . . + |xn |.
Remark 1.1 From limδ→0 ω(δ) = 0 and (6) it follows that
sup | µ(x) − µδ (x) |→ 0 as δ → 0
x∈IRn
which means the stability of the possibility distribution of the solution of PLES (3)
under small changes of the fuzzy parameters.
Other results along this line have appeared in [3,5].
References
[1] R.E.Bellman and L.A.Zadeh, Decision-making in a fuzzy environment, Management Sci. 17, No 4, 1970, B141-B154.
[2] D.Dubois and H.Prade, System of linear fuzzy constraints, Fuzzy Sets and
Systems, 3(1980), 37-48.
[3] M.Fedrizzi and R.Fullér, Stability in Possibilistic Linear Programming Problems with Continuous Fuzzy Number Parameters, Fuzzy Sets and Systems,
(to appear).
[4] R.Fullér, On Stability in Possibilistic Linear Equality Systems with Lipschitzian Fuzzy Numbers,Fuzzy Sets and Systems, 34(1990) 347-353.
[5] R.Fullér and H.J.-Zimmermann, On Zadeh’s Compositional Rule of Inference,
In: Proc. of Fourth IFSA Congress, Brussels, July 7-13, 1991 (to appear).
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Sets and Systems, (to appear).
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301-317.
[8] M.Kovács, F.P.Vasil’jev and R.Fullér. On Stability in Fuzzified Linear Equality
Systems, Vestnik Moscow State University, 1989, Ser. 15, No.1 5-9.
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[9] M.Kovács, Fuzzification of ill-posed linear systems, in: D. Greenspan and
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Methods, North-Holland, Amsterdam, 1988 521-532.
[10] B.Schweizer, A.Sklar, Associative functions and abstract semigroups, Publ.
Math. Debrecen, 10(1963) 69-81.
[11] L.A.Zadeh, Fuzzy Sets, Information and Control, 8(1965) 338-353.
[12] L.A.Zadeh, The concept of linguistic variable and it applications to approximate reasoning, Parts I, II, III, Information Sciencis, 8(1975) 199-251; 8(1975)
301-357; 9(1975) 43-80.
[13] H.-J.Zimmermann, Fuzzy Set Theory - and Its Applications, Kluwer, Nijhoff
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