A New Representation of Efficient Point Sets and Its Applications in DEA

Abstract

E(M,K)  (Ew(M,K)), the set of Pareto efficient (weak efficient) points of a setMwith respect to a coneKinRn, is expressed as a differencebetweentwo setsMandM+K∖{0}(MandM+intK). Using the new representation, the properties ofE(M,K)are proved more easily than before. WhenMorKis in the form of union, intersection, sum, or difference of two sets or two cones, respectively, the properties ofE(M,K)are considered. Most of the properties are proved by the binary operations of sets, which is a new method in the multiobjective optimization. Then these properties are used to solve some types of multiobjective linear programming problems corresponding to Data Envelopment Analysis (DEA) models. The structures of the DEA efficient solution sets of four most representative DEA models are developed. Further more, the relationships between efficiencies of the four DEA models are deduced.