Approximate resolution of an imprecise goal programming model with nonlinear membership functions

In standard goal programming (GP) it is assumed that the decision maker (DM) is able to determine goal values accurately. This is unrealistic; usually DM expresses his/her aspirations in an imprecise way. The imprecise nature of the DM's judgments has led to an important development of the fuzzy multiobjective approaches. In this paper, we will assume that the DM's goals may be expressed through fuzzy sets and therefore we deal with an imprecise goal programming (IGP). Determining the membership functions that represent the fuzzy goals of the DM use to be a difficult task and it is necessary to fit them in the sense of being robust; hence, some results on sensibility analysis are established. We show that small changes in membership functions produce small differences on the DM's global satisfaction degree and on the efficient frontier. On the other hand, when membership functions are nonlinear, IGP becomes a nonlinear program that may be difficult to solve. This difficulty may be overcome by approximating each nonlinear fuzzy membership function through functions belonging to a class of simpler/easier functions. In this paper, based on the sensibility analysis that we develop, we prove the goodness of subrogated functions. An illustrative example is also provided.