Fuzzy if-then rules for modeling interdependences in FMOP problems Christer Carlsson Institute for Advanced Management Systems Research, Åbo Akademi University, DataCity A 3210, SF-20520 Åbo, Finland Robert Fullér Department of Computer Science, Eötvös Loránd University, Muzeum krt. 6-8, H-1088 Budapest, Hungary Abstract There has been a growing interest and activity in the area of multiple criteria decision making (MCDM), especially in the last 25 years. modeling and optimization methods have been developed in both crisp and fuzzy environments. The overwhelming majority of approaches for finding best compromise solutions to MCDM problems do not make use of the interdependences among the objectives. However, as has been pointed out by [1, 2], in modeling real world problems (especially in management sciences and in group decisions) we often encounter MCDM problems with interdependent objectives. In multiple objective programs (MOP), application functions are established to measure the degree of fulfillment of the decision maker s requirements (achievement of goals, nearness to an ideal point, satisfaction, etc.) about the objective functions (see e.g. [5, 11]) and extensively used in the process of finding good compromise solutions. In [3] we demonstrated that the use of interdependences among objectives of a MOP in the definition of the application functions provides for more correct solutions and faster convergence. In [4], generalizing the principle of application function to fuzzy multiple objective programs (FMOP) with interdependent objectives, we defined a large family of application functions for FMOP in order to provide for a better understanding of the decision problem, and to find effective and more correct solutions. In this paper we define interdependencies among the objectives of FMOP by using fuzzy if-then rules. Keywords: FMOP, interdependencies, fuzzy if-then rules, group decisions 1 Introduction Decision making with interdependent multiple criteria is a surprisingly difficult task. If we have clearly conflicting objectives there normally is no optimal solution which would simultaneously in: Proceedings of EUFIT 94 Conference, September 20-23, 1994 Aachen, Germany, Verlag der Augustinus Buchhandlung, Aachen, 1994 1504-1508. Presently visiting professor at Department of Management and Computer Science, University of Trento, Italy. Partially supported by the Hungarian National Scientific Research Fund OTKA under the contracts T 4281, I/3-2152 and T 7598 1
satisfy all the criteria. On the other hand, if we have pair wisely supportive objectives, such that the attainment of one objective helps us to attain the another objective, then we should exploit this property in order to find effective optimal solutions. Carlsson [2] used the fuzzy Pareto optimal set of nondominated alternatives as a basis for an OWA-type operator [9] for finding a best compromise solution to MCDM problems with interdependent criteria. Felix [8] presented a novel theory for multiple attribute decision making based on fuzzy relations between objectives, in which the interactive structure of objectives is inferred and represented explicitely. In [3] we demonstrated that the use of interdependences among objectives of a MOP in the definition of the application functions provides for more correct solutions and faster convergence. In [4], generalizing the principle of application function to FMOP with interdependent objectives, we defined a large family of application functions for FMOP in order to provide for a better understanding of the decision problem, and to find effective and more correct solutions. The aim of this paper is to provide a new method for defining interdependencies for FMOP via fuzzy if-then rules. 2 Interdependences in MOP In [3] we defined interdependences between the objectives of a MOP as follows: Consider the following problem max x X {f 1(x),..., f k (x)} (1) where f i :R n R are objective functions, x R n is the decision variable, and X is a subset of R n. Definition 2.1 Let f i and f j be two objective functions of (1). We say that (i) f i supports f j on X if f i (x ) f i (x) entails f j (x ) f j (x), for all x, x X; (ii) f i is in conflict with f j on X if f i (x ) f i (x) entails f j (x ) f j (x), for all x, x X; (iii) f i and f j are said to be independent on X, otherwise. Figure 1. A typical example of conflict on R. Figure 2.Supportive functions on R. If the objective functions are differentiable on X then we have
(i) f i supports f j on X iff e f i (x) e f j (x) 0 for all e R n and x X, (ii) f i is in conflict with f j on X iff e f i (x) e f j (x) 0 for all e R n and x X, where e f i (x) denotes the derivative of f i with respect to the direction e R n at x R n. If for a given direction e R n, e f i (x) e f j (x) 0 [ e f i (x) e f j (x) 0] holds for all x X then we say that f i supports f j [f i is in conflict with f j ] with respect to the direction e on X. 3 Interdependencies in FMOP Consider a fuzzy multiple objective program where the values f i (x) are given by fuzzy if-then rules of the form max x X {f 1(x),..., f k (x)} (2) R 1 : R 2 : if x 1 is A 11 and... and x n is A 1n then f 1 (x) is B 11 and... and f k (x) is B 1k if x 1 is A 21 and... and x n is A 2n then f 1 (x) is B 21 and... and f k (x) is B 2k R m : if x 1 is A m1 and... and x n is A mn then f 1 (x) is B m1 and... and f k (x) is B mk where A ij and B ij are fuzzy sets representing our knowledge about the values of the decision variable and the objective functions. Suppose that after normalization we have the following fuzzy partition with 7 primary fuzzy sets (linguistic terms): NB negative big; NM negative medium; NS negative small; ZE zero; PS positive small; PM positive medium; and PB positive big (see Figure 3.). We shall use the natural inequality relation between linguistic terms NB NM NS ZE P S P M P B. (3) It should be noted that by using 7 linguistic terms for the possible values of x 1,..., x n and the objective functions we can have 7 n+k rules, which is a very large number, e.g. even in the simple case, n = 2 and k = 2, there can be 2401 rules. Definition 3.1 Let f i be an objective function of (2) and let the inequality relation between the linguistic terms be defined in the sense of (3). We say that (i) f i is increasing if x x entails that f(x) f(x ); (ii) f i is decreasing if x x entails that f(x) f(x ).
Table 1: Example for an increasing function. x NB NM NS ZE PS PM PB f(x) NM NM ZE ZE ZE ZE PM In the following we define interdependencies among the objective functions of (2). Definition 3.2 Let f i and f j be two objective functions of (2). We say that (i) f i supports f j if f i (x ) f i (x) entails f j (x ) f j (x), for all linguistic terms; (ii) f i is in conflict with f j if f i (x ) f i (x) entails f j (x ) f j (x), for all linguistic terms; (iii) f i and f j are said to be independent, otherwise. It is easy to see that two objective functions support each other if they simultaneously increasing/decreasing and they are in conflict if their behaviour is contradictory. NB NM NS ZE PS PM PB -1 1 Figure 3. A fuzzy partition with seven linguistic terms. Let us consider a five-objective problem, where the rules are given by the following table: Table 2: Look-up table for interdependencies. x f 1 (x) f 2 (x) f 3 (x) f 4 (x) f 5 (x) NB PB NB PB PS ZE NM PM NM NS PS ZE NS PS NS PS ZE ZE ZE ZE ZE ZE ZE NS PS NS PS ZE NS NS PM NM PM NM NS NS PB NB PB PM NM NM It is easy to see that f 1, f 4 and f 5 support each other, f 1 and f 2 are in conflict and f 3 is in no relation with the others. The explicit use of interdependencies in the process of finding a good compromise solution to (2) will be the subject of further research.
A promising area of application of the principle of interdependencies is the consensus reaching modul in group decisions, where the experts knowledge is represented by fuzzy if-then rules ([6, 7]). It is quite natural to suppose that the experts do not modify their ratings for alternatives on each of the criteria independently. On the contrary, if the expert modifies his rating on a criterium then he should modify his ratings on each of related (conflicting, supporting) criteria. References [1] C.Carlsson, On interdependent fuzzy multiple criteria, in: R.Trappl ed.,proceedings of the Eleventh European Meeting on Cybernetics and Systems Research, World Scientific Publisher, London, 1992, Vol.1. 139-146. [2] C.Carlsson, On optimization with interdependent multiple criteria, in: R.Lowen and M.Roubens eds., Fuzzy Logic: State of the Art, Kluwer Academic Publishers, Dordrecht, 1992. [3] C.Carlsson and R.Fullér, Multiple Criteria Decision Making: The Case for Interdependence, Computers & Operations Research (to appear). [4] C.Carlsson and R.Fullér, Interdependence in fuzzy multiple objective programming, submitted to Fuzzy Sets and Systems. [5] M.Delgado,J.L.Verdegay and M.A.Vila, A possibilistic approach for multuobjective programming problems. Efficiency of solutions, in: R.Slowinski and J.Teghem eds., Stochastic versus Fuzzy Approaches to Multiobjective Mathematical Programming under Uncertainty, Kluwer Academic Publisher, Dordrecht, 1990 229-248. [6] M.Fedrizzi and R.Fullér, On stability in group decision support systems under fuzzy production rules, in: R.Trappl ed., Proceedings of the Eleventh European Meeting on Cybernetics and Systems Research, World Scientific Publisher, London, 1992, Vol.1. 471 478. [7] M.Fedrizzi, Fuzzy approach to modeling consensus in group decisions, In: Proceedings of First Workshop on Fuzzy Set Theory and Real Applications, Milano, May 10, 1993, Automazione e strumentazione, Supplement to November 1993 issue, 9 13. [8] R.Felix, Multiple attribute decision making based on fuzzy relationships between objectives, in: Proceedings of the 2nd International Conference on Fuzzy Logic and Neural Networks, Iizuka Japan, July 17-22, 1992 805-808. [9] R.R.Yager, Family of OWA operators, Fuzzy Sets and Systems, 59(1993) 125-148. [10] M.Zeleny, Multiple Criteria Decision Making, McGraw-Hill, New-York, 1982. [11] H.-J.Zimmermann, Fuzzy programming and linear programming with several objective functions, Fuzzy Sets and Systems, 1(1978) 45-55.