On Supportedness in Multi‐Objective Integer Linear Programming

Abstract

This paper addresses an inconsistency in various definitions of supported non‐dominated points within multi‐objective integer linear optimization problems (MOILPs). MOILP problems are known to contain supported and unsupported non‐dominated points, with the latter typically outnumbering the former. Supported points are, in general, easier to determine, can serve as representations and are used in two‐phase methods to generate the entire non‐dominated point set. Despite their importance, several different characterizations for supported efficient solutions (and supported non‐dominated points) are used in the literature. While these definitions are equivalent for multi‐objective linear optimization problems, they can yield different sets of supported non‐dominated points for MOILP problems. We show by an example that these definitions are not equivalent for MOILP or general multi‐objective optimization problems. Moreover, we analyse the structural and computational properties of the resulting sets of supported non‐dominated points. These considerations motivate us to summarise equivalent definitions and characterizations for supported efficient solutions and to introduce a distinction between supported and weakly supported efficient solutions.