Comments on properties of the minmax solutions in goal programming

This note discusses the properties of solutions generated by the minmax models of goal programming (GP) and compromise programming (CP). GP approaches use a certain target point in the criterion (attribute) space to model decision maker's preferences. When the ideal (utopia) point is used as the target, the minmax GP model coincides with the minmax (Chebyshev) CP model. In a recent review of the current GP state-of-the-art, there have been included suggestions that the two equivalent models ensure Pareto efficiency of solutions and they guarantee a perfectly balanced allocation among the achievement of the individual targets. In this note, it is shown that the models, in general, do not ensure the efficiency of solutions and they do not guarantee the perfect equity among the individual achievements. Moreover, there are given sufficient and necessary conditions clarifying when the discussed properties of minmax solutions do occur.