Optimisation of a fuzzy non linear function

Fuzzy numbers can be introduced in order to model imprecise situations involving real numbers, and one of the first problem one meets working with these is to decide what type of order to use on the fuzzy number set. In fact this set does not have a natural total ranking. Different methods for ranking fuzzy numbers has been described. Most of these are defined by a function which maps each fuzzy number into an ordered set and transfer the order of one set to the other. We consider that fuzzy numbers may be thought as intervals whose boundaries are blurred, and the difficulty in ranking them arise from the problems created in ranking real intervals. As overlapping is the main difficulty, the problem is overcome when the supports of the fuzzy numbers are disjoint. In this case all the methods give the same solution. By contrast, the decision is not evident when the set intersect. In this case, different methods give different solutions for the same problem. This problem happens even in the classification of real intervals when they are partially overlapped. The problem we study in this paper is optimising a non-linear function of fuzzy variables with values in the fuzzy number set. At the beginning, we had to start with a definition of ranking fuzzy numbers in order to being able to speak about maximum or minimum of a fuzzy valued function. In two papers ([6] e [7]), Canestrelli and Giove faced an analogous problem. These authors decided to use a definition of “linked variables” to approach the problem of ranking fuzzy numbers. In this paper we use a particular real valued ranking function , called average value (AV), generated by two different ranking functions (evaluation functions) on real intervals. The choice has been due to the fact that the AV is defined as dependent on several parameters, allowing flexibility in the final result. Both evaluations functions on real intervals contain a parameter: in the first case it is a real number, in the second it is a function, we call degree of risk, which takes into account of a risk-propension or aversion of the decision maker. The two AV we use are the mean values of the evaluation functions on the α–cuts of the fuzzy numbers obtained by a particular Stieltjes measure generated by a function s(x)= x with r>0. We used r=2 because this choice gives more weight to the high values of α . In the fourth chapter we produce a necessary and sufficient condition for the existence of a solution of the optimising problem and the interesting result is that it is possible to treat the fuzzy optimisation problem without having any information about the minimum and the maximum of the function. This result should give the opportunity to build an algorithm to reach the solution easier. An interesting future development of the study is the use of AV defined not with additive measures, but with fuzzy measure, using the Choquet’s integral.