Fuzzy goals under fuzzy constraints (Q1080773)

The paper deals with a classical problem of decision making in a fuzzy environment, viz. in situations where a goal G as well as constraints \(C_ 1,C_ 2,...,C_ n\) imposed are viewed as fuzzy sets defined in a certain universe of discourse. Discussed are two forms of aggregation of goals and constraints: (1) intersection of G and the \(C_ i's\). The fuzzy decision is hence obtained via \(D=G\cap C_ 1\cap C_ 2\cap...\cap C_ n\) (where \(\cap\) stands for taking the minimum of their membership functions), (2) their convex combination, \(D(x)=a_ 0G(x)+\sum^{n}_{i=1}a_ iC_ i(x)\) with \(a_ 0+a_ 1+...+a_ n=1\), \(x\in X.\) A definition of optimum decision (in the sense of Pareto) and its existence for both formulations given above is discussed in detail. Secondly, some computational hints useful for determining the optimum (resulting from an analysis of the Lagrange saddle point and the formulation of the Lagrange problem) are provided.