Computation of ideal and Nadir values and implications for their use in multiple-criteria decision making methods

in this paper we investigate the problem of finding the Nadir point for multicriteria optimization problems (MOP). The Nadir point is characterized by the componentwise maximal values of efficient points for MOP. It can be easily computed in the bicriteria case. However, in general this problem is very difficult. We review some existing methods and heuristics and also discuss some new ones. We propose a general method to compute Nadir values based on theoretical results on Pareto optimal solutions of subproblems with fewer criteria. The general scheme is valid for any number of criteria and practical for the case of three objectives. We also investigate the use of the Nadir point for compromise programming, when the goal is to be as far away as possible from the worst outcomes. We prove some results about (weak) Pareto optimality of the resulting solutions and present examples, which show the limitations of this idea, and therefore limitations of multicritera methods using this type of problems.