Cone Convexity, Cone Extreme Points, and Nondominated Solutions in Decision Problems with Multiobjectives

Abstract

Although there is no universally accepted solution concept for decision problems with multiple noncommensurable objectives, one would agree that a good solution must not be dominated by the other feasible alternatives. Here, we propose a structure of domination over the objective space and explore the geometry of the set of all nondominated solutions. Two methods for locating the set of all nondominated solutions through ordinary mathematical programming are introduced. In order to achieve our main results, we have introduced the new concepts of cone con-vexity and cone extreme point, and we have explored their main properties. Some relevant results on polar cones and polyhedral cones are also derived. Throughout the paper, we also pay attention to an important special case of nondominated solutions, that is, Pareto-optimal solutions. The geometry of the set of all Pareto solu-tions and methods for locating it are also studied. At the end, we provide an example to show how we can locate the set of all non-dominated solutions through a derived decomposition theorem.

The author would like to thank Professors J. Keilson and M. Zeleny for their helpful discussion and comments. Thanks also go to an anonymous reviewer for his helpful comments concerning the author’s previous working paper (Ref. 1). He is especially obliged to Professors M. Freimer and A. Marshall for their careful reading of the first draft and valuable remarks. The author is also very grateful to Professor G. Leitmann and Dr. W. Stadler for their helpful comments.