On the Computational Testing of Procedures for Interactive Multiple Objective Linear Programming

Abstract

The problem to be solved is the multiple objective linear program (MOLP) $$\begin{array}{*{20}{l}} {\max \{ {c^1}x = {z_1}\} } \\ {\begin{array}{*{20}{c}} \vdots \end{array}} \\ {\max \{ {c^k}x = {z_k}\} } \\ {s.t.\,x \in S} \end{array}$$ where S is the feasible region in decision space. Let \(Z \subset {R^k}\) be the feasible region in criterion space. Criterion vector \(\bar z \in Z\) iff there exists an \(\bar x \in S\) such that \(\bar z = ({c^1}\bar x,...,{c^k}\bar x)\). A point \(\bar x \in S\) is efficient iff there does not exist any x∈S such that \({c^i}x \geqslant {c^i}\bar x\) for all i and \({c^i}x > {c^i}\bar x\) for at least one i. A criterion vector is nondominated iff its inverse image is an efficient point. The set of all efficient points is the efficient set E and the set of all nondominated criterion vectors is the nondominated set N. Methods for computing all efficient extreme points and characterizing the entire nondominated set have been developed by Ecker et al. (1980), Fandel (1972), Gal (1977), Gal and Leberling (1981), Isermann (1977), and others.