Order relations among the optimal values under three evaluation criteria and their applications

The three fundamental evaluation criteria, (1) the maximum-production rate criterion, (2′) the minimum-cost one and (3) the maximum-profit-rate one, have been utilized in manufacturing optimization, especially in the optimal machining speed problems of machine tools. It is well known that the optimal machining speed under (3) lies between those under (1) and (2′). The main purpose of this paper is to extend such property to more general OR models (including stochastic ones). To do this, we introduce (2) the maximum-unit-profit criterion, instead of (2′). The (2) is a generalization of (2′). First we prove the existence of optimal vectors under (1), (2) and (3), and then present a fundamental result to derive order relations among the optimal vectors. When the decision vector is one-dimensional, i.e., a scalar, some order relations among the optimal values are given for special cases which are of great importance in practice. As an application, we unify some results which have been already discussed or developed in several references.