Optimization with linguistic variables ∗ Christer Carlsson
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Robert Full´er
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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.
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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. Full´ er, 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, 2003 113-121.
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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 µA : X → [0, 1] and µA (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 µA (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 = u ¯. 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
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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 u ¯” 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 ˜ . ˜b, max/min f (x) = c˜x; subject to Ax
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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 ∈ Rn . 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 max/min f (x); subject to {
