We suggest solving fuzzy mathematical programming problems via the

use of multiple fuzzy reasoning techniques. We show that our approach gives

Buckley’s solution [1] to possibilistic mathematical programs when the inequality

relations are understood in possibilistic sense

Fullér, Róbert

English

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Multiple fuzzy reasoning approach tofuzzy mathematical programmingproblems∗Robert Fulle´rrfuller@ra.abo.ﬁAbstractWe suggest solving fuzzy mathematical programming problems via theuse of multiple fuzzy reasoning techniques. We show that our approach givesBuckley’s solution [1] to possibilistic mathematical programs when the in-equality relations are understood in possibilistic sense.Keywords: Compositional rule of inference, multiple fuzzy reasoning, fuzzy mathe-matical programming, possibilistic mathematical programming, fuzzy implication1 PreliminariesYager [4] introduced an approach to making inferences in a knowledge-based systemwhich uses mathematical programming techniques. In this paper we apply multiplefuzzy reasoning (MFR) techniques to solve mathematical programming problemswith fuzzy parameters. We show that our approach yields Buckley’s solution [1] topossibilistic mathematical programs, when the inequality relations are understoodin possibilistic sense. We shall use the following inference rules:The compositional rule of inferenceAntecedent 1: x and y have property WFact: x has property PConsequence: y has property Qwhere P and Q are fuzzy sets, W is a fuzzy relation and Q is obtained by the sup-mincomposition of P and W, i.e.Q(y) = (P ◦W )(y) = supxmin{P (x),W (x, y)}.MFR scheme∗in: Proceedings of the Annual Conference of the Operational Research Society of Italy, Septem-ber 18-20, 1991 Riva del Garda, Italy 126-129.1Antecedent 1: x and y have relation W1. . .Antecedent m: x and y have relation WmFact: x has relation PConsequence: y has relation QwhereQ(y) = (P ◦ mini=1,...,mWi)(y) = supxmin{P (x), mini=1,...,mWi(x, y)}2 The new approachConsider the fuzzy mathematical programming (FMP) problemg(c˜, x) → m˜ax (1)f1(a˜1, x) ≤ b˜1. . .fm(a˜m, x) ≤ b˜mwhere c˜ = (c˜1, . . . , c˜k) and a˜i = (a˜i1, . . . , a˜in) are vectors of fuzzy quantities (i.e.fuzzy sets of the real line IR), b˜i is a fuzzyquantity, x = (x1, . . . , xn) is a vector ofdecision variables, g(c˜, x) and fi(a˜i, x) are deﬁned by Zadeh’s extension principle,and the inequality relation ≤ is deﬁned by a certain fuzzy relation. Now, by us-ing FMR techniques, we shall determine a fuzzy quantity, m˜ax, which satisﬁes theinequalityg(c˜, x) ≤ m˜ax, ∀x ∈ IRn,under the constraints fi(a˜i, x) ≤ b˜i, i = 1, . . . ,m. We consider the inequality relation≤ as a fuzzy relation on IR and for every x ∈ IRn determine m˜axx from the followingMFR schemeAntecedent 1: f1(a˜1, x) ≤ b˜1. . .Antecedent m: fm(a˜m, x) ≤ b˜mFact: g(c˜, x)Consequence: m˜axxwhere, according to the MFR inference rule,m˜axx = g(c˜, x) ◦ mini=1,...,m(fi(a˜i, x) ≤ b˜i).It is clear that m˜axx is a fuzzy set realizing the inequality g(c˜, x) ≤ m˜axx under thepremises fi(a˜i, x) ≤ b˜i, i = 1, . . . ,m. Finally, we deﬁne the solution, m˜ax, of theFMP problem (P1) asm˜ax = supxg(c˜, x) ◦ mini=1,...,m(fi(a˜i, x) ≤ b˜i)It is clear that m˜ax is a fuzzy set realizing the inequality g(c˜, x) ≤ m˜ax, ∀x ∈ X,under the premises fi(a˜i, x) ≤ b˜i, i = 1, . . . ,m.2Remark 2.1 It can be shown that our approach (i) yields Buckley’s solution [1] topossibilistic mathematical programs, when the inequality relations are understood inpossibilistic sense; (ii) under well-chosen inequality relations and objective functioncoincides with those ones suggested by Delgado et al [2], Ramik and Rimanek [3]and Zimmermann [5].3 IllustrationExample 3.1 We illustrate our approach on simple FLP problem, where the in-equality relation ≤ is deﬁned by the Go¨del implication:x ≤ y = x→ y ={1 if x ≤ yy otherwiseThe FLP problem The adequate MFR schemex1a˜1 ≤ b˜1 Antecedent 1: x1a˜1 ≤ b˜1x2a˜1 ≤ b˜2 Antecedent 2: x2a˜1 ≤ b˜2x1a˜1 + x2a˜1 → m˜ax Fact: x1a˜1 + x2a˜1Consequence: m˜ax = b˜1 + b˜2References[1] J.J.Buckley, Possibilistic linear programming with triangular fuzzy numbers,Fuzzy Sets and Systems, (26)1988 135-138.[2] M.Delgado, J.L.Verdegay and M.A. Vila, Optimization models in fuzzy-logic-based decision support systems, Technical Report,No. 90-1-3, Universidad deGranada, 1990.[3] J.Ramik and J.Rimanek, Inequality relation between fuzzy numbers and itsuse in fuzzy optimization, Fuzzy Sets and Systems, 16(1985) 123-138.[4] R.R. Yager, A mathematical programming approach to inference with the ca-pability of implementing default rules, International Journal of Man-MachineStudies, 29(1988) 685-714.[5] H.-J.Zimmermann, Description and optimization of fuzzy systems, Interna-tional Journal of General Systems, 2(1976) 209-215.3

A mathematical programming approach to inference with the capability of implementing default rules,

Description and optimization of fuzzy systems,

Inequality relation between fuzzy numbers and its use in fuzzy optimization, Fuzzy Sets and Systems,

Optimization models in fuzzy-logicbased decision support systems,

Possibilistic linear programming with triangular fuzzy numbers, Fuzzy Sets and Systems,

Multiple fuzzy reasoning approach to fuzzy mathematical programming problems

http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.24.2755

Multiple fuzzy reasoning approach to fuzzy mathematical programming problems

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