A model for solving incompatible fuzzy goal programming: an application to portfolio selection

For many fuzzy goal programming (GP) approaches, in order to build the membership functions of fuzzy aspiration levels, a tolerance threshold for each one of them should be determined. In this paper, we address the case in which the decision maker proposes incompatible thresholds, which could lead to an infeasible problem. We propose an alternative algebraic formulation of the membership functions, which allows us to formulate models capable of providing solutions, although some tolerance thresholds are surpassed. The objective values that do not violate their corresponding threshold are evaluated positively according to the degree of achievement to their fuzzy target, and in turn those who violate the threshold are penalized according to their unwanted deviation with respect to the threshold. Thus, our model jointly uses the fuzzy GP approach and the standard GP approach, which also allows incorporating fuzzy and crisp targets into the same problem. The proposed procedure is applied to socially responsible portfolio selection problems.