Optimization with linguistic variables

Transcription

1 Optimization with linguistic variables Christer Carlsson Robert Fullér Abstract We consider fuzzy mathematical programming problems (FMP) in which the functional relationship between the decision variables and the objective function is not completely known. Our knowledge-base is supposed to 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 either a linguistic value of the objective function or a linear combination of the crisp values of the decision variables. In this paper we suggest the use of an adequate fuzzy reasoning method to determine the crisp functional relationship between the objective function and the decision variables, and to solve the resulting (usually nonlinear) programming problem to find a fair optimal solution to the original fuzzy problem. Furthermore, we illustrate how the optimal solution may change if we are able to refine the rule base by introducing some non-monotonicity (dependency) rules. 1 Introduction After Bellman and Zadeh [1], and then Zimmermann[14] introduced fuzzy sets into methods for handling optimization problems, they cleared the way for a new family of methods to deal with problems which had been inaccessible to and not solvable with standard mathematical programming (MP) techniques. These problems include a number of limitations and simplifications, which have become apparent, as the use of MP models became widespread and common. The parameters used in MP models need to be valid and accurate, and the estimates used to define them should be built from reliable data sets, which are extensive enough to allow repeated verification and validation. This is not often the case as data may be spotty and incomplete, the data sets are limited and quite often ad hoc, and the validation procedures need to be simplified out of necessity. Nevertheless, MP models are used with the simplifying assumption, that the parameter estimates are good enough for the actual problem - or for the time being (which includes a promise of later adjustments and modifications). The end-result appears to be that optimal solutions found with MP models do not fare too well in the long run, nor do they survive for very long. We need methods to deal with the problems of poor, incomplete data head-on. The final version of this paper appeared in: C. Carlsson and R. Fullér, Optimization with linguistic variables, in: J.L.Verdegay ed., Fuzzy Sets based Heuristics for Optimization, Studies in Fuzziness and Soft Computing. Vol.126, Springer Verlag,

2 The traditional use of one-objective MP models is in many cases an over-simplification as the intentions formulated in decision problems cannot be adequately dealt with without using multiple objectives. It is true that MP models with multiple objectives become complex and difficult to handle and work with, which is why they are avoided. There is even more to multiple objective models: in the standard case we assume that the objectives are independent, which has given us a set of standard tools for finding optimal solutions in cases with multiple objectives. In practical cases, we normally have interdependence among the objectives: some of the objectives may be conflicting, other may be supportive; in cases with greater numbers of objectives, we may have complex combinations of interdependencies [2, 7, 3]. Model builders shy away from these more complex issues as the methods for finding identifiable optimal solutions have been rather few and, in some cases, have not yet been developed. Thus, we need methods for handling interdependence issues in multiple objective MP models. In practical problem solving work, and in handling decision problems in the business world, it has become evident for us that probably a majority of the problem solving and decision processes is not of the type, which can be resolved with the precise MPtype models and methods. These practical situations have actors, who are vague about objectives and constraints in the beginning of a process, and then they become more focused and precise as the process is progressing towards some results. This appears to be more of a search-learning process than a systematic process of analysis progressing in well-defined steps towards a predefined goal. Then, for the early stages of the searchlearning process we need tools, models and methods, which allow for knowledge-rich imprecision but which have an inner core of methodological structure, such that we in later stages of the process can become analytically more precise as objectives and constraints become more focused. Nevertheless, we want to keep the knowledge-rich substance of the models and methods as we progress to more MP-like modelling. This has long been considered a methodological contradiction, but it has become apparent in recent years [5] that this standard objection is not necessarily true. A combination of linguistic variables and fuzzy logic is emerging as a good approach to have knowledgerich imprecision, a systematic and firm methodological structure, and effective and fast analytical MP-algorithms. In this paper, we explore the conditions for optimisation with linguistic variables. Fuzzy sets were introduced by Zadeh[12] as a means of representing and manipulating data that was not precise, but rather fuzzy. Let X be a nonempty set. A fuzzy set A in X is characterized by its membership function µ A : X [0, 1] and µ A (x) is interpreted as the degree of membership of element x in fuzzy set A for each x X. Frequently we will write simply A(x) instead of µ A (x). The family of all fuzzy (sub)sets in X is denoted by F(X). A fuzzy set A in X is called a fuzzy point if there exists a u X such that A(t) = 1 if t = u and A(t) = 0 otherwise. We will use the notation A = ū. The use of fuzzy sets provides a basis for a systematic way for the manipulation of vague and imprecise concepts. In particular, we can employ fuzzy sets to represent 2

3 linguistic variables. A linguistic variable [13] can be regarded either as a variable whose value is a fuzzy number or as a variable whose values are defined in linguistic terms. If x is a linguistic variable in the universe of discourse X and u X then we simple write x = u or x is ū to indicate that u is a crisp value of x. Fuzzy optimization problems can be stated in many different ways [8]. Authors usually consider fuzzy optimization problems of the form max/min f(x); subject to x X, where f or/and X are defind in 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 max/min f(x) = cx; subject to Ãx b, (1) where the fuzzy terms are denoted by tilde. Unlike in (1) the fuzzy value of the objective function f(x) may not be known for any x R n. More often than not we are only able to describe the partial causal link between x and f(x) linguistically using some fuzzy if-then rules. In [6] we have considered constrained fuzzy optimization problems of the form with max/min f(x); subject to {R 1 (x),..., R m (x) x X R n }, (2) R i (x) : if x 1 is A i1 and... and x n is A in then f(x) is C i, (3) where A ij and C i are fuzzy numbers with strictly monotone membership functions; and we have suggested the use of the Tsukamoto fuzzy reasoning method [11] to determine the crisp values of f. In [4] we assumed that the knowledge base is given in the form R i (x) : if x 1 is A i1 and... and x n is A in then f(x) = n j=1 a ijx j + b i (4) where A ij is a fuzzy number, and a ij and b i are real numbers. Crisp values of f have been determined by the Takagi and Sugeno[10] fuzzy reasoning method. In both cases the firing levels of the rules have been computed by the product t-norm [9], T (a, b) = ab, (to have a smooth output function), and then a solution to the original fuzzy problem (2) has been defined as a solution to the resulting deterministic (usually nonlinear) optimization problem max/min f(u), subject to u X. where the crisp value of the objective function f at u R n, denoted also by f(u), has been determined by Zadeh s compositional rule of inference. 2 Optimization with linguistic variables To find a fair solution to the fuzzy optimization problem max/min f(x); subject to {R 1 (x),..., R m (x) x X}, (5) 3

4 with fuzzy if-then rules of form (3) or (4) we first determine the crisp value of the objective function f at u R n, denoted also by f(u), by the compositional rule of inference f(u) := (x is ū) {R 1 (x),, R m (x)}. That is, in case (3) we apply the Tsukamoto fuzzy reasoning method as f(u) := α 1C 1 1 (α 1) + + α m C 1 m (α m ) α α m where the firing levels are computed according to as α i = n A ij (u j ). (6) j=1 Furthermore, in case (4) we apply the Takagi and Sugeno fuzzy reasoning method f(u) := α 1z 1 (u) + + α m z m (u) α α m. where the firing levels of the rules are computed by (6) and the individual rule outputs, denoted by z i, are derived from the relationships z i (u) = n a ij u j + b i. j=1 In this manner our constrained optimization problem (5) turns into the following crisp (usually nonlinear) mathematical programmimg problem max/min f(u); subject to u X. If X is a fuzzy set with membership function µ X (e.g. given by soft constraints as in [14]) then following Bellman and Zadeh[1] we define the fuzzy solution to problem (5) as D = µ X µ f, (7) where µ f is an appropriate transformation of the values of f to the unit interval, and an optimal solution to (5) is defined to be as any maximizing element of D. 3 Examples Example 1 Consider the optimization problem min f(x); subject to {x 1 + x 2 = 1/2, 0 x 1, x 2 1}, (8) 4

5 where R 1 (x) : if x 1 is small and x 2 is small then f(x) = x 1 + x 2, R 2 (x) : if x 1 is small and x 2 is big then f(x) = x 1 + x 2. Let u = (u 1, u 2 ) be an input to the fuzzy system. Then the firing levels of the rules are α 1 = (1 u 1 )(1 u 2 ), α 2 = (1 u 1 )u 2, It is clear that if u 1 = 1 then no rule applies because α 1 = α 2 = 0. So we can exclude the value u 1 = 1 from the set of feasible solutions. The individual rule outputs are computed by z 1 = u 1 + u 2, z 2 = u 1 + u 2. and, therefore, the overall system output, interpreted as the crisp value of f at u is f(u) = (1 u 1)(1 u 2 )(u 1 + u 2 ) + (1 u 1 )u 2 ( u 1 + u 2 ) (1 u 1 )(1 u 2 ) + (1 u 1 )u 2 = u 1 + u 2 2u 1 u 2. Thus our original fuzzy problem turns into the following crisp nonlinear mathematical programming problem min (u 1 + u 2 2u 1 u 2 ) subject to {u 1 + u 2 = 1/2, 0 u 1 < 1, 0 u 2 1}. which 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 u 1 and u 2, the computed input/output function f(u) = u 1 + u 2 2u 1 u 2 is a nonlinear one. Example 2 Consider the problem max f (9) X where X is a fuzzy susbset of the unit interval with membership function for u [0, 1], and the fuzzy rules are µ X (u) = 1 (1/2 u) 2, R 1 (x) : if x is small then f(x) = 1 x, R 2 (x) : if x is big then f(x) = x. Let u [0, 1] be an input to the fuzzy system {R 1 (x), R 2 (x)}. Then the firing levels of the rules are α 1 = 1 u, α 2 = u. The individual rule outputs are z 1 = (1 u)(1 u), z 2 = u 2 and, therefore, the overall system output is f(u) = (1 u) 2 + u 2 = 2u 2 + 2u

6 Then according to (7) our original fuzzy problem (9) turns into the following crisp biobjective mathematical programming problem max min{2u 2 + 2u + 1, 1 (1/2 u) 2 }; subject to u [0, 1], which has the optimal value of and two optimal solutions {0.09, 0.91}. 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 the Takagi and Sugeno and the Tsukamoto fuzzy reasoning scheme, because the individual rule outputs are crisp functions, and therefore, the functional relationship between the input vector u and the system output f(u) can be easily identified. 4 Extensions Assume that besides {R 1,..., R m } we are able to justify some monotonicity properties in the functional link between x and f(x), for example, if x i is very A ij then f(x) is very C i, where very A ij and very C i are new values of linguistic variables x i and C i, respectively. Consider the following very simple optimization problem where max f(x); subject to {R 1 (x), R 2 (x) x X = [0, 1]}, (10) R 1 (x) : if x is small then f(x) is small R 2 (x) : if x is big then f(x) is big Let small(x) = 1 x and big(x) = x, and let u be an input to the rule base R = {R 1, R 2 } then the firing levels of the rules are computed by Then we get α 1 = 1 u, α 2 = u. f(u) = (1 u)u + u u = u. Thus our original fuzzy problem turns into the following trivial crisp problem max u; subject to u [0, 1]. (11) which has the optimal solution u = 1. Assume that besides {R 1, R 2 } we are able to justify the following monotonicity properties between x and f(x), if x is very small then f(x) is very small and if x 6

7 is very big then f(x) is very big. Assume further that these monotonicity rules are implemented by where R 3 (x) : if x is very small then f(x) is very small R 4 (x) : if x is very big then f(x) is very big { 1 2u if 0 u 1/2, (very small)(u) = 0 otherwise, and { 2u 1 if 1/2 u 1, (very big)(u) = 0 otherwise, Then the fuzzy problem max f(x); subject to {R 1 (x), R 2 (x), R 3 (x), R 4 (x) x X = [0, 1]}, where R 1 (x) : if x is small then f(x) is small R 2 (x) : if x is big then f(x) is big R 3 (x) : if x is very small then f(x) is very small R 4 (x) : if x is very big then f(x) is very big turns into the same crisp optimization problem (11), and therefore, the optimal solution remains the same, u = 1. If, however, R 2 does not entail R 4, but we have if x is very big then f(x) is not very big instead, then the optimal solution changes. Really, the problem max f(x); subject to {R 1 (x), R 2 (x), R 3 (x), R 4 (x) x X = [0, 1]}, where R 1 (x) : if x is small then f(x) is small R 2 (x) : if x is big then f(x) is big R 3 (x) : if x is very small then f(x) is very small R 4 (x) : if x is very big then f(x) is not very big has the following crisp objective function (see Figure 2) f(u) = { u if 0 u 1/2, 5u 2u 2 3/2 2u otherwise, and the solution to the resulting crisp optimization problem max f(u); subject to u [0, 1]. is u = and its optimal value is f(u ) =

8 Figure 1: Illustration of Example 2. 5 Summary We have addressed FMP problems where the functional relationship between the decision variables and the objective function is known linguistically. We have suggested the use of an appropriate fuzzy reasoning method to determine the crisp functional relationship between the decision variables and the objective function and solve the resulting (usually nonlinear) programming problem to find a fair optimal solution to the original fuzzy problem. We have shown that the refinement of the intial fuzzy rule base (by introducing some non-monotonicity rules) can result in a substantial change of the solution. References [1] R. E. Bellman and L.A.Zadeh, Decision-making in a fuzzy environment, Management Sciences, Ser. B 17(1970) [2] C. Carlsson and R. Fullér, Interdependence in fuzzy multiple objective programming, Fuzzy Sets and Systems, 65(1994) [3] C. Carlsson and R. Fullér, Multiple Criteria Decision Making: The Case for Interdependence, Computers & Operations Research, 22(1995) [4] C. Carlsson, R. Fullér and S. Giove, Optimization under fuzzy rule constraints, The Belgian Journal of Operations Research, Statistics and Computer Science 38(1998) [5] C. Carlsson and R. Fullér, Multiobjective linguistic optimization, Fuzzy Sets and Systems, 115(2000) [6] C. Carlsson and R. Fullér, Optimization under fuzzy if-then rules, Fuzzy Sets and Systems, 119(2001) [7] R.Felix, Relationships between goals in multiple attribute decision making, Fuzzy sets and Systems, 67(1994) [8] M.Inuiguchi, H.Ichihashi and H. Tanaka, Fuzzy Programming: A Survey of Recent Developments, in: Slowinski and Teghem eds., Stochastic versus Fuzzy 8

9 Approaches to Multiobjective Mathematical Programming under Uncertainty, Kluwer Academic Publishers, Dordrecht [9] B.Schweizer and A.Sklar, Associative functions and abstract semigroups, Publ. Math. Debrecen, 10(1963) [10] T.Takagi and M.Sugeno, Fuzzy identification of systems and its applications to modeling and control, IEEE Trans. Syst. Man Cybernet., 1985, [11] 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). [12] L.A.Zadeh, Fuzzy Sets, Information and Control, 8(1965) [13] L.A.Zadeh, The concept of linguistic variable and its applications to approximate reasoning, Parts I,II,III, Information Sciences, 8(1975) ; 8(1975) ; 9(1975) [14] H.-J. Zimmermann, Description and optimization of fuzzy systems, Internat. J. General Systems 2(1975)