Suppose we are given a mathematical programming problem in which the functional relationship between the decision variables and the objective function is not completely known. Our knowledge-base consists of a block of fuzzy

if-then rules, where the antecedent part of the rules contains some linguistic values of the decision variables, and the consequence part is a linear combination of the crisp values of the decision variables. We suggest the use of Takagi and Sugeno fuzzy reasoning method to determine the crisp functional relationship between the objective function and the decision variables, and solve the resulting (usually nonlinear) programming problem to find a fair optimal

solution to the original fuzzy problem

Carlsson, Christer

Fullér, Róbert

Giove, Silvio

English

Repository of the Academy's Library

Optimization under fuzzy rule constraintsChrister CarlssonIAMSRA˚bo Akademi Universitychrister.carlsson@abo.ﬁRobert Fulle´r∗Department of OREo¨tvo¨s Lora´nd Universityrfuller@cs.elte.huSilvio GioveDepartment of Applied MathematicsUniversity of Venicesgiove@unive.itAbstractSuppose we are given a mathematical programming problem in which thefunctional relationship between the decision variables and the objective func-tion is not completely known. Our knowledge-base consists of a block of fuzzyif-then rules, where the antecedent part of the rules contains some linguisticvalues of the decision variables, and the consequence part is a linear combina-tion of the crisp values of the decision variables. We suggest the use of Takagiand Sugeno fuzzy reasoning method to determine the crisp functional rela-tionship between the objective function and the decision variables, and solvethe resulting (usually nonlinear) programming problem to ﬁnd a fair optimalsolution to the original fuzzy problem.Keywords: fuzzy number, constrained optimization, fuzzy reasoning∗Also at Institute for Advanced Management Systems Research, A˚bo Akademi University. Par-tially supported by the Hungarian Research Fund OTKA T 019455.11 IntroductionFuzzy optimization problems can be stated and solved in many diﬀerent ways [2, 3,14, 21, 22]. Usually the authors consider optimization problems of the formmax/min f(x); subject to x ∈ X,where f or/and X are deﬁnd by fuzzy terms. Then they are searching for a crisp x∗which (in a certain) sense maximizes f on X. For example, fuzzy linear programming(FLP) problems can be stated as [9, 13, 15, 17]max/min f(x) = c˜x; subject to A˜x <∼ b˜, (1)where the fuzzy terms are denoted by tilde. Fulle´r and Zimmermann [12] interpretedFLP problems (1) with fuzzy coeﬃcients and fuzzy inequality relations as multiplefuzzy reasoning schemes, where the antecedents of the scheme correspond to theconstraints of the FLP problem and the fact of the scheme is the objective of theFLP problem. Their solution process consists of two steps: ﬁrst, for every decisionvariable x ∈ Rn, compute the (fuzzy) value of the objective function, MAX(x), viasup-min convolution of the antecedents/constraints and the fact/objective. Thenan (optimal) solution to the FLP problem is any point which produces a maximalelement of the set {MAX(x) | x ∈ Rn}.Unlike in (1) the fuzzy value of the objective function f(x) may not be knownfor any x ∈ Rn. More often than not we are only able to describe the causal linkbetween x and f(x) linguistically using fuzzy if-then rules.In [8] we have considered constrained fuzzy optimization problems of the formmax/min f(x); subject to {1(x), . . . ,m(x) | x ∈ X ⊂ Rn}, (2)withi(x) : if x1 is Ai1 and . . . and xn is Ain then f(x) is Ci,where Aij and Ci are fuzzy numbers; and we have suggested the use of Tsukamoto’sfuzzy reasoning method [19] to determine the crisp values of f .In this paper we suppose that our knowledge base contains fuzzy if-then rules ofthe formi(x) : if x1 is Ai1 and . . . and xn is Ain then f(x) = ai1x1 + · · ·+ ainxn + bi (3)where Aij is a fuzzy number, and aij and bi are real numbers. Then we determinethe crisp value of f at u ∈ Rn by the Takagi and Sugeno fuzzy reasoning method,and obtain an optimal solution to (2) by solving the resulting (usually nonlinear)optimization problemmax/min f(u), subject to u ∈ X.We illustrate the proposed method by several examples.22 Constrained Optimization under Fuzzy If-thenRulesA linguistic variable [20] can be regarded either as a variable whose value is a fuzzynumber or as a variable whose values are deﬁned in linguistic terms. A fuzzy setA in X is called a fuzzy point if there exists a u ∈ X such that A(t) = 1 if t = uand A(t) = 0 otherwise. We will use the notation A = u¯. Fuzzy points are usedto represent crisp values of linguistic variables. If x is a linguistic variable in theuniverse of discourse X and u ∈ X then we simple write ”x = u” or ”x is u¯” toindicate that u is a crisp value of x.To ﬁnd a fair solution to the fuzzy optimization problemmax/min f(x); subject to {1(x), . . . ,m(x) | x ∈ X}, (4)with fuzzy if-then rules of form (3) we ﬁrst determine the crisp value of the objectivefunction f at u ∈ Rn, denoted also by f(u), by the compositional rule of inferencef(u) := (x is u¯) ◦ {1(x), · · · ,m(x)}using the Takagi and Sugeno fuzzy reasoning method asf(u) :=α1z1(u) + · · ·+ αmzm(u)α1 + · · ·+ αm .where the ﬁring levels of the rules are computed byαi =n∏j=1Aij(uj), (5)and the individual rule outputs, denoted by zi, are derived from the relationshipszi(u) =n∑j=1aijuj + bi.To determine the ﬁring level of the rules, we suggest the use of the productt-norm (to have a smooth output function). In this manner our constrained opti-mization problem (4) turns into the following crisp (usually nonlinear) mathematicalprogrammimg problemmax/min f(u); subject to u ∈ X.If X is a fuzzy set with membership function µX (e.g. given by soft constraintsas in [21]) then following Bellman and Zadeh [1] we deﬁne the fuzzy solution toproblem (4) asD = µX ∩ µf , (6)3where µf is an appropriate transformation of the values (computed by the Takagiand Sugeno reasoning method) of f to the unit interval [10], and an optimal solutionto (4) is deﬁned to be as any maximizing element of D.Example 1. Consider the optimization problemmax f(x); subject to {1(x),2(x) | x ∈ X = [0, 1]}, (7)where1(x) : if x is small then f(x) = x2(x) : if x is big then f(x) = 1− x.If small(x) = 1−x and big(x) = x, and u is an input to the rule base then the ﬁringlevels of the rules are computed byα1 = 1− u, α2 = u.Then we getf(u) = (1− u)u + u(1− u) = 2(1− u)u.Thus our original fuzzy problem turns into the following crisp nonlinear mathemat-ical programming problemmax 2u(1− u); subject to u ∈ [0, 1].which has the optimal solution u∗ = 1/2.If the membership functions in the rules are deﬁned bysmall(x) =11 + e−(1/2−x), big(x) =11 + e(1/2−x)and u is an input to the rule base then the ﬁring levels of the rules are computed byα1 =11 + e−(1/2−u), α2 =11 + e(1/2−u).Then we getf(u) =u1 + e−(1/2−u)+1− u1 + e(1/2−u).Thus our original fuzzy problem turns into the following crisp nonlinear mathemat-ical programming problemmax(u1 + e−(1/2−u)+1− u1 + e(1/2−u)); subject to u ∈ [0, 1].which has the optimal solution u∗ = 1/2.If we use nonsymmetric membership functions in the rules, for examplesmall(x) =11 + e−10(3/4−x), big(x) =11 + e0.1(1/3−x)4and u is an input to the rule base then the ﬁring levels of the rules are computed byα1 =11 + e−10(3/4−u), α2 =11 + e0.1(1/3−u).Then we getf(u) =u1 + e−10(3/4−u)+1− u1 + e0.1(1/3−u)11 + e−10(3/4−u)+11 + e0.1(1/3−u).Thus our original fuzzy problem turns into the following crisp nonlinear mathemat-ical programming problemmaxu1 + e−10(3/4−u)+1− u1 + e0.1(1/3−u)11 + e−10(3/4−u)+11 + e0.1(1/3−u) ; subject to u ∈ [0, 1].which has the optimal solution u∗ = 0.65 and f(u∗) = 0.52..Even though Example 1 is probably the simplest one, it clearly shows the com-plexity of the problem of optimization under fuzzy if-then rules. Namely, the onlyway to increase f(u) is to decrease the feasibility of u.Example 2. Consider the optimization problemmin f(x); subject to {x1 + x2 = 1/2, 0 ≤ x1, x2 ≤ 1}, (8)where1(x) : if x1 is small and x2 is small then f(x) = x1 + x2,2(x) : if x1 is small and x2 is big then f(x) = −x1 + x2.Let u = (u1, u2) be an input to the fuzzy system. Then the ﬁring levels of the rulesareα1 = (1− u1)(1− u2), α2 = (1− u1)u2,It is clear that if u1 = 1 then no rule applies because α1 = α2 = 0. So we can excludethe value u1 = 1 from the set of feasible solutions. The individual rule outputs arecomputed byz1 = u1 + u2, z2 = −u1 + u2.and, therefore, the overall system output, interpreted as the crisp value of f at u isf(u) =(1− u1)(1− u2)(u1 + u2) + (1− u1)u2(−u1 + u2)(1− u1)(1− u2) + (1− u1)u2 =u1 + u2 − 2u1u2.Thus our original fuzzy problem turns into the following crisp nonlinear mathemat-ical programming problemmin (u1 + u2 − 2u1u2); subject to {u1 + u2 = 1/2, 0 ≤ u1 < 1, 0 ≤ u2 ≤ 1}.5which has the optimal solution u∗1 = u∗2 = 1/4 and its optimal value is f(u∗) = 3/8.Even though the individual rule outputs are linear functions of u1 and u2, thecomputed input/output function f(u) = u1 + u2 − 2u1u2 is a nonlinear one.Example 3. Consider the problemmaxXf (9)where X is a fuzzy susbset of the unit interval with membership functionµX(u) = 1− (1/2− u)2for u ∈ [0, 1] and the fuzzy rules are1(x) : if x is small then f(x) = 1− x,2(x) : if x is big then f(x) = x.Let u ∈ [0, 1] be an input to the fuzzy system {1(x),2(x)}. Then the ﬁring levelsof the rules are α1 = 1−u, α2 = u. The individual rule outputs are z1 = (1−u)(1−u),z2 = u2 and, therefore, the overall system output isf(u) = (1− u)2 + u2 = 2u2 + 2u + 1.Then according to (6) our original fuzzy problem (9) turns into the followingcrisp biobjective mathematical programming problemmax min{2u2 + 2u + 1, 1− (1/2− u)2}; subject to u ∈ [0, 1],which has the optimal value of 0.8333 and two optimal solutions {0.09, 0.91}.The rules represent our knowledge-base for the fuzzy optimization problem. Thefuzzy partitions for lingusitic variables will not ususally satisfy ε-completeness, nor-mality and convexity. In many cases we have only a few (and contradictory) rules.Therefore, we can not make any preselection procedure to remove the rules which donot play any role in the optimization problem. All rules should be considered whenwe derive the crisp values of the objective function. We have chosen the Takagiand Sugeno fuzzy reasoning scheme, because the individual rule outputs are crispfunctions, and therefore, the functional relationship between the input vector u andthe system output f(u) can be easily identiﬁed.3 SummaryWe have addressed mathematical programming problems in which the functionalrelationship between the decision variables and the objective function is known lin-guistically. We suggested the use of the Takagi and Sugeno fuzzy reasoning methodto determine the crisp functional relationship between the objective function and thedecision variables, and solve the resulting (usually nonlinear) programming problemto ﬁnd a fair optimal solution to the original fuzzy problem. We can reﬁne the fuzzyrule base by introducing new lingusitic variables modeling the linguistic dependen-cies between the variables and the objectives [4, 5, 6, 11]. These will be the subjectsof our future research.6References[1] R. E. Bellman and L.A.Zadeh, Decision-making in a fuzzy environment, Man-agement Sciences, Ser. B 17(1970) 141-164.[2] E. Canestrelli and S. Giove, Optimizing a quadratic function with fuzzy linearcoeﬃcients, Control and Cybernetics, 20(1991) 25-36.[3] E. Canestrelli and S. Giove, Bidimensional approach to fuzzy linear goal pro-gramming, in: M. Delgado, J. Kacprzyk, J.L. Verdegay and M.A. Vila eds.,Fuzzy Optimization (Physical Verlag, Heildelberg, 1994) 234-245.[4] C. Carlsson and R. Fulle´r, Interdependence in fuzzy multiple objective pro-gramming, Fuzzy Sets and Systems, 65(1994) 19-29.[5] C. Carlsson and R. Fulle´r, Fuzzy if-then rules for modeling interdependenciesin FMOP problems, in: Proceedings of EUFIT’94 Conference, September 20-23, 1994 Aachen, Germany, Verlag der Augustinus Buchhandlung, Aachen,1994 1504-1508.[6] C. Carlsson and R. Fulle´r, Multiple Criteria Decision Making: The Case forInterdependence, Computers & Operations Research, 22(1995) 251-260.[7] C. Carlsson and R. Fulle´r, Optimization with linguistic values, TUCS Tech-nical Reports, Turku Centre for Computer Science, No. 157/1998. [ISBN 952-12-0138-X, ISSN 1239-1891].[8] C. Carlsson and R. Fulle´r, Optimization under fuzzy if-then rules, Fuzzy Setsand Systems, 2000, (to appear).[9] M.Delgado,J.L.Verdegay and M.A.Vila, A possibilistic approach for multiob-jective programming problems. Eﬃciency of solutions, in: R.SUlowinski andJ.Teghem eds., Stochastic versus Fuzzy Approaches to Multiobjective Mathe-matical Programming under Uncertainty, Kluwer Academic Publisher, Dor-drecht, 1990 229-248.[10] D. Dubois and H. Prade, Decision making under fuzzy constraints and fuzzycriteria – mathematical programming vs. rule-based system approach, in:M.Delgado, J.Kacprzyk, J.L.Verdegay and M.A.Vila eds., Fuzzy Optimiza-tion: Recent Advances, Studies Fuzziness, Vol. 2, Physica-Verlag, Heidelberg,1994 21-32.[11] R.Felix, Relationships between goals in multiple attribute decision making,Fuzzy sets and Systems, 67(1994) 47-52.[12] R.Fulle´r and H.-J.Zimmermann, Fuzzy reasoning for solving fuzzy mathemat-ical programming problems, Fuzzy Sets and Systems 60(1993) 121-133.7[13] F. Herrera, M. Kova´cs, and J. L. Verdegay, Fuzzy linear programming prob-lems with homogeneous linear fuzzy functions, in: Proc. of IPMU’92, Univer-sitat de les Illes Balears, 1992 361-364.[14] M.Inuiguchi, H.Ichihashi and H. Tanaka, Fuzzy Programming: A Survey ofRecent Developments, in: Slowinski and Teghem eds., Stochastic versus FuzzyApproaches to Multiobjective Mathematical Programming under Uncertainty,Kluwer Academic Publishers, Dordrecht 1990, pp 45-68[15] M. Kova´cs, Linear programming with centered fuzzy numbers, Annales Univ.Sci. Budapest, Sectio Computatorica, 12(1991) 159-165.[16] A. Kusiak and J. Wang, Dependency analysis in constraint negotiation, IEEETransactions on Systems, Man, and Cybernetics, 25(1995) 1301- 1313.[17] H. Rommelfanger, Fuzzy linear programming and applications, EuropeanJournal of Operational Research, 92(1996) 512-527.[18] T.Takagi and M.Sugeno, Fuzzy identiﬁcation of systems and its applicationsto modeling and control, IEEE Trans. Syst. Man Cybernet., 1985, 116-132.[19] Y. Tsukamoto, An approach to fuzzy reasoning method, in: M.M. Gupta, R.K.Ragade and R.R. Yager eds., Advances in Fuzzy Set Theory and Applications(North-Holland, New-York, 1979).[20] L.A.Zadeh, The concept of linguistic variable and its applications to approxi-mate reasoning, Parts I,II,III, Information Sciences, 8(1975) 199-251; 8(1975)301-357; 9(1975) 43-80.[21] H.-J. Zimmermann, Description and optimization of fuzzy systems, Internat.J. General Systems 2(1975) 209-215.[22] H.-J.Zimmermann, Fuzzy Mathematical Programming, in: Stuart C. Shapiroed., Encyclopedia of Artiﬁcial Intelligence, John Wiley & Sons, Inc., Vol. 1,1992 521-528.8

Optimization under fuzzy rule constraints

Suppose we are given a mathematical programming problem in which the functional relationship between the decision variables and the objective function is not completely known. Our knowledge-base consists of a block of fuzzy if-then rules, where the antecedent part of the rules contains some linguistic values of the decision variables, and the consequence part is a linear combination of the crisp values of the decision variables. We suggest the use of Takagi and Sugeno fuzzy reasoning method to determine the crisp functional relationship between the objective function and the decision variables, and solve the resulting (usually nonlinear) programming problem to find a fair optimal solution to the original fuzzy problem. 1 Introduction  Fuzzy optimization problems can be stated and solved in many di#erent ways [2, 3, 14, 20, 21]. Usually the authors consider optimization problems of the form max/min f(x); subject to x  #  X,  # Apperaed in: B.De Baets, J. Fodor and L. T. Koczy eds., Proc..

Christer Carlsson

Robert Fullér

Silvio Giove

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Optimization Under Fuzzy Linguistic Rule Constraints

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Optimization under fuzzy rule constraints

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