Solving multi-attribute decision making problems with incomplete weights: A dual approach

This paper proposes a method of ranking discrete alternatives when attribute weights are incompletely known. There are a variety of situations in which it is reasonable to consider incomplete attribute weights and several techniques have been developed to solve such multi-attribute decision making problems. Most frequently, a linear programming (LP) problem subject to a set of incomplete attribute weights is solved to identify dominance relations between alternatives. In this paper, we explore a dual problem to find a closed-form solution and determine the extreme points of a set of (strictly) ranked attribute weights. A simple investigation of the dual optimal solution often leads to a preferred alternative and permits to find the optimal attribute weights that can be applied to the primal, based on the primal-dual relationship. Furthermore, we extend the approach to several examples of incomplete attribute weights and to linear partial information expressed as linear inequalities that satisfy some predefined conditions. Finally, we present a case study to demonstrate how the dual approach can be used to establish dominance between alternatives, when preference orders are specified for a subset of alternatives.