Optimization under fuzzy if-then rules ∗ Christer Carlsson christer.carlsson@abo.fi Robert Fuller rfuller@abo.fi Abstract The aim of this paper is to introduce a novel statement of fuzzy mathe- matical programming problems and to provide a method for findig a fair so- lution to these problems. 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 con- sequence part consists of a linguistic value of the objective function. We suggest the use of Tsukamoto’s 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. Keywords: Linguistic variable, Tsukamoto’s fuzzy reasoning, fuzzy optimiza- tion, soft constraints 1 Introduction When Bellman and Zadeh [1], and a few years later Zimmermann [14], introduced fuzzy sets into optimization problems, they cleared the way for a new family of methods to deal with problems which had been inaccessible to and unsolvable with standard mathematical programming techniques. There is also the underlying issue of solving optimization problems in a way which gives us relevant and useful optimal solutions. In our earlier works on inter- dependence in multiple criteria decision problems [2, 3, 4, 5] we found out that the standard mcdm models have some limitations and shortcomings, which motivated work on extensions and enhancements of the mcdm models to include interdepen- dence. In order to find solutions to decision problems with multiple, interdependent * The final version of this paper appeared in: Fuzzy Sets and Systems, 119(2001) 111-120. 1