Multiobjective optimization with linguistic variables

Multiobjective optimization with linguistic variables ∗ Christer Carlsson ˚ Akademi University, IAMSR, Abo ˚ DataCity A 3210, SF-20520 Abo, Finland Robert Full´er † Turku Centre for Computer Science, ˚ DataCity A 4150, SF-20520 Abo, Finland

Abstract Generalizing our earlier results on optimization with linguistic variables [3, 6, 7] we introduce a novel statement of fuzzy multiobjective mathematical programming problems and provide a method for findig a fair solution to these problems. Suppose we are given a multiobjective mathematical programming problem in which the functional relationship between the decision variables and the objective functions 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 consists of a linguistic value of the objective functions. We suggest the use of Tsukamoto’s fuzzy reasoning method to determine the crisp functional relationship between the decision variables and objective functions. We model the anding of the objective functions by a t-norm and solve the resulting (usually nonlinear) programming problem to find a fair optimal solution to the original fuzzy multiobjective problem.

Keywords: Tsukamoto’s fuzzy reasoning, multiobjective fuzzy optimization, ∗

in: Proceedings of the Sixth European Congress on Intelligent Techniques and Soft Computing (EUFIT’98), Aachen, September 7-10, 1998, Verlag Mainz, Aachen, [ISBN 3-89653-500-5], Vol. II, 1998 1038-1042. † On leave from Department of Operations Research, E¨otv¨os Lor´and University, Muzeum k¨orut 6-8, H-1088 Budapest, Hungary

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1

Introduction

Fuzzy multiobjective optimization problems can be stated and solved in many different ways (for good surveys see [5, 9, 11, 12, 15]). Usually the authors consider optimization problems of the form max/min {G1 (x), . . . , GK (x)}; subject to x ∈ X, where Gk , k = 1, . . . , K, or/and X are defind by fuzzy terms. Then they are searching for a crisp x∗ which (in certain) sense maximizes the Gk ’s under the (fuzzy) constraints X. For example, multiobjective fuzzy linear programming (FMLP) problems can be stated as ˜ ≤ ˜b, max/min {˜ c1 x, . . . , c˜K x}; subject to Ax

(1)

where x ∈ IRn is the vector of crisp decision variables, A˜ = (˜ aij ), ˜b = (˜bi ) and c˜j = (˜ cij ) are fuzzy quantities, the inequality relation ≤ is given by a certain fuzzy relation and the (implicite) X is a fuzzy set describing the concept ”x satisfies ˜ ≤ ˜b”. Ax In many important cases (e.g. in strategy formation processes) the values of the objective functions are not known for all x ∈ IRn , and we are able to describe the causal link between x and the Gk ’s linguistically using fuzzy if-then rules. In this paper we consider a new statement of multiobjective fuzzy optimization problems (FMOP), namely max/minX {G1 , . . . , GK }; subject to {1 , . . . , m },

(2)

where x1 , . . . , xn are linguistic variables, and i : if x1 is Ai1 and . . . and xn is Ain then G1 is Ci1 and . . . and GK is CiK , constitutes the only knowledge available about the values of G, and Aij and Cik are fuzzy numbers. In our earlier work [3] we interpreted FMLP problems (1) with fuzzy coefficients and fuzzy inequality relations as multiple fuzzy reasoning schemes, where the antecedents of the scheme correspond to the constraints of the MFLP problem and the facts of the scheme are the objectives of the MFLP problem. Generalizing the fuzzy reasoning approach introduced in [6, 7] we determine the crisp value of the Gj ’s at y ∈ X by Tsukamoto’s fuzzy reasoning method [13], and obtain an optimal solution to (2) by solving the resulting (usually nonlinear) optimization problem max/min t-norm (G1 (y), . . . , GK (y)); subject to y ∈ X. We illustrate the proposed method by a simple example. 2

(3)

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Multiobjective optimization under fuzzy if-then rules

Consider the FMOP problem (2) with continuous Aij representing the linguistic values of xi , and with strictly monotone and continuous Cik , i = 1, . . . , m representing the linguistic values of Gk , k = 1, . . . , K. To find a fair solution to the fuzzy optimization problem (2) we first determine the crisp value of the k-th objective function Gk at y ∈ IRn from the fuzzy rule-base  using Tsukamoto’s fuzzy reasoning method as Gk (y) :=

−1 −1 (α1 ) + · · · + αm Cmk (αm ) α1 C1k α1 + · · · + αm

where αi = t-norm(Ai1 (y1 ), . . . , Ain (yn )) denotes the firing level of the i-th rule, i . To determine the firing level of the rules, we suggest the use of the product t-norm (to have a smooth output function). In this manner the constrained optimization problem (2) turns into the crisp (usually nonlinear) mathematical programmimg problem (3). The same principle is applied to constrained maximization problems. Example 1 Consider the optmization problem max{G1 , G2 }

(4)

subject to y ∈ X = {(y1 , y2 ) ∈ [0, 1] × [0, 1] | y1 + y2 = 3/4} where 1 : if x1 is small and x2 is small then G1 is small and G2 is big, 2 : if x1 is small and x2 is big then G1 is big and G2 is small, and the universe of discourse for the linguistic values of G1 and G2 is also the unit interval [0, 1] (see Figure 1). We will compute the firing levels of the rules by the product t-norm. Let the membership functions in the rule-base  be defined by small(t) = 1 − t,

big(t) = t.

Let y1 , y2 ∈ [0, 1] be an input to the fuzzy system. Then the firing leveles of the rules are α1 = (1 − y1 )(1 − y2 ), α2 = (1 − y1 )y2 , It is clear that if y1 = 1 then no rule applies because α1 = α2 = 0. So we can exclude the value y1 = 1 from the set of feasible solutions. The individual rule outputs are z11 = 1 − (1 − y1 )(1 − y2 ), z21 = (1 − y1 )y2 , 3

x 1 is small

G1 is small

x2 is small

G2 is big

α1 1 x 1 is small

y1

1

z11

1 x2 is big

y2

1

1

G1 is big

product

α2 z21

1

z12

G2 is small

1

z22

Figure 1: Illustration of Example 1. z12 = (1 − y1 )(1 − y2 ),

z22 = 1 − (1 − y1 )y2 ,

and, therefore, the overall system outputs are G1 (y) =

(1 − y1 )(1 − y2 )(1 − (1 − y1 )(1 − y2 )) + (1 − y1 )y2 (1 − y1 )y2 = (1 − y1 )(1 − y2 ) + (1 − y1 )y2 y1 + y2 − 2y1 y2 ,

G2 (y) =

(1 − y1 )(1 − y2 )(1 − y1 )(1 − y2 ) + (1 − y1 )y2 (1 − (1 − y1 )y2 ) = (1 − y1 )(1 − y2 ) + (1 − y1 )y2 1 − (y1 + y2 − 2y1 y2 ).

Modeling the anding of the objective functions by the minimum t-norm our original fuzzy problem (4) turns into the following crisp nonlinear mathematical programming problem max min{y1 + y2 − 2y1 y2 , 1 − (y1 + y2 − 2y1 y2 )}

(5)

subject to y1 + y2 = 3/4, 0 ≤ y1 < 1, 0 ≤ y2 ≤ 1. which has the optimal solutions (1/2, 1/4) and (1/4, 1/2), and its optimal value is (1/2, 1/2). Remark 2.1 We can introduce trade-offs among the objectives function by using an OWA-operator in (5). However, as Yager has pointed out in [14], constrained OWA-aggregations are not easy to solve, because the usually lead to a mixed integer mathematical programming problem of very big dimension. 4

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The rules represent our knowledge-base for the fuzzy optimization problem. The fuzzy partitions for lingusitic variables will not ususally satisfy ε-completeness, normality 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 do not play any role in the optimization problem. All rules should be considered when we derive the crisp values of the objective function. We have chosen Tsukamoto’s fuzzy reasoning scheme, because the individual rule outputs are crisp numbers, and therefore, the functional relationship between the input vector y and the system outputs (G1 (y), . . . , GK (y)) can be relatively easily identified (the only thing we have to do is to perform inversion operations).

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Summary

We have introduced a novel statement of multiobjective fuzzy mathematical programming problems and priveded a method for findig a fair solution to these problems. We addressed multiobjective mathematical programming problems in which the functional relationship between the decision variables and the objective functions are not completely known. Our knowledge-base is assumed to consist only 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 consists of a linguistic value of the objective functions. We suggested the use of Tsukamoto’s fuzzy reasoning method to determine the crisp functional relationship between the the decision variables and objective functions, and solve the resulting (usually nonlinear) programming problem to find a fair optimal solution to the original fuzzy problem. We can refine the fuzzy rule-base by introducing new lingusitic variables modeling the linguistic dependencies between the variables and the objectives [1, 2, 4, 8, 10]. This will be the subject of our future research.

References [1] C.Carlsson and R.Full´er, Interdependence in fuzzy multiple objective programming, Fuzzy Sets and Systems, 65(1994) 19-29. [2] C.Carlsson and R.Full´er, Fuzzy if-then rules for modeling interdependencies in FMOP problems, in: Proceedings of EUFIT’94 Conference, September 20-23, 1994 Aachen, Germany, Verlag der Augustinus Buchhandlung, Aachen, 1994 1504-1508.

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[3] C.Carlsson and R.Full´er, Fuzzy reasoning for solving fuzzy multiple objective linear programs, in: R.Trappl ed., Cybernetics and Systems ’94, Proceedings of the Twelfth European Meeting on Cybernetics and Systems Research, World Scientific Publisher, London, 1994, vol.1, 295-301. [4] C.Carlsson and R.Full´er, Multiple Criteria Decision Making: The Case for Interdependence, Computers & Operations Research, 22(1995) 251-260. [5] C.Carlsson and R.Full´er, Fuzzy multiple criteria decision making: Recent developments, Fuzzy Sets and Systems, 78(1996) 139-153. [6] C.Carlsson and R.Full´er, Optimization with linguistic values, TUCS Technical Reports, Turku Centre for Computer Science, No. 157/1998. [7] C.Carlsson and R.Full´er, Optimization under fuzzy if-then rules, Fuzzy Sets and Systems, (submitted). [8] R.Felix, Relationships between goals in multiple attribute decision-making, Fuzzy Sets and Systems, 67(1994) 47-52. [9] M.Inuiguchi, H.Ichihashi and H. Tanaka, Fuzzy Programming: A Survey of Recent Developments, in: Slowinski and Teghem eds., Stochastic versus Fuzzy Approaches to Multiobjective Mathematical Programming under Uncertainty, Kluwer Academic Publishers, Dordrecht 1990, pp 45-68 [10] A. Kusiak and J. Wang, Dependency analysis in constraint negotiation, IEEE Transactions on Systems, Man, and Cybernetics, 25(1995) 1301- 1313. [11] Y.-J.Lai and C.-L.Hwang, Fuzzy Multiple Objective Decision Making: Methods and Applications, Lecture Notes in Economics and Mathematical Systems, Vol. 404 (Springer-Verlag, New York, 1994). [12] M.K. Luhandjula, Fuzzy optimization: an appraisal, Fuzzy Sets and Systems, 30(1989) 257-282. [13] 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). [14] R.R. Yager, Constrained OWA aggregation, Fuzzy Sets and Systems, 81(1996) 89-101. [15] H.-J.Zimmermann, Methods and applications of fuzzy mathematical programming, in: R.R.Yager and L.A.Zadeh eds., An Introduction to Fuzzy 6

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