Robust Goal Programming with Interactive Fuzzy Coefficients

In this paper, goal programming problems with interactive fuzzy coefficients are treated. Two types of targets can be expressed by fuzzy sets in goal programming problems with fuzzy coefficients. One is the ambiguous target whose true value is not known precisely and the other is the target distribution to which the fuzzy set of objective function values is brought close. Corresponding to the difference of targets, we use two kinds of deviations naturally obtained from the extension principle. On the other hand, to treat the interaction among fuzzy coefficients, we introduce oblique fuzzy vectors (OFVs). An OFV can be obtained from the expert knowledge about the behavior of coefficients as well as from the principal component analysis of the stored coefficient data. It is shown that linear functions with OFVs can be obtained easily. The goal programming problems are formulated based on the necessity measure maximization model. It is shown that the reduced programming problems can be solved by a bisection method together with a simplex method. Moreover, it is shown that the constraints of the reduced programming problems have special structures such as a dual block angular structure and a bordered angular structure so that some decomposition methods are applicable.