Introduction

Abstract

Decision making is the process of selecting a possible course of action from all the available alternatives. In almost all such problems the multiplicity of criteria for judging the alternatives is pervasive. That is, for many such problems, the decision maker (DM) wants to attain more than one objective or goal in selecting the course of action while satisfying the constraints dictated by environment, processes, and resources. Another characteristic of these problems is that the objectives are apparently non-commensurable. Mathematically, these problems can be represented as:

$$\left. {_{subject\,to:\,\,\,\,\,g_j \left( {\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{x} } \right)\,\, \leqslant \,\,0,\,\,\,\,\,\,\,\,\,j = 1,\,\,...,\,\,m}^{Max\left[ {\user2{f}_1 \left( {\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{x} } \right),\,\user2{f}_2 \left( {\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{x} } \right),\,...,\,f_k \left( {\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{x} } \right)} \right]} } \right\}$$

(1.1)

where x is an n dimensional decision variable vector. The problem consists of n decision variables, m constraints and k objectives. Any or all of the functions may be nonlinear. In literature this problem is often referred to as a vector maximum problem (VMP).