Extremist vs. centrist decision behavior: quasi-convex utility functions for interactive multi-objective linear programming problems

Abstract

This paper presents the fundamental theory and algorithms for identifying the most preferred alternative for a decision maker (DM) having a non-centrist (or extremist) preferential behavior. The DM is requested to respond to a set of questions in the form of paired comparison of alternatives. The approach is different than other methods that consider the centrist preferential behavior. In this paper, an interactive approach is presented to solve the multiple objective linear programming (MOLP) problem. The DM's underlying preferential function is represented by a quasi-convex value (utility) function, which is to be maximized. The method presented in this paper solves MOLP problems with quasi-convex value (utility) functions by using paired comparison of alternatives in the objective space. From the mathematical point of view, maximizing a quasi-convex (or a convex) function over a convex set is considered a difficult problem to solve, while solutions for quasi-concave (or concave) functions are currently available. We prove that our proposed approach converges to the most preferred alternative. We demonstrate that the most preferred alternative is an extreme point of the MOLP problem, and we develop an interactive method that guarantees obtaining the global most preferred alternative for the MOLP problem. This method requires only a finite number of pivoting operations using a simplex-based method, and it asks only a limited number of paired comparison questions of alternatives in the objective space. We develop a branch and bound algorithm that extends a tree of solutions at each iteration until the MOLP problem is solved. At each iteration, the decision maker has to identify the most preferred alternatives from a given subset of efficient alternatives that are adjacent extreme points to the current basis. Through the branch and bound algorithm, without asking many questions from the decision maker, all branches of the tree are implicitly enumerated until the most preferred alternative is obtained. An example is provided to show the details of the algorithm. Some computational experiments are also presented. Scope and purpose This paper presents the fundamental theory, algorithm, and examples for identifying the most preferred alternative (solution) for a decision maker (DM) having a non-centrist (or extremist) preferential behavior for Multiple Objective Linear Programming (MOLP) problems. The DM is requested to respond to a set of questions in the form of paired comparison of alternatives. Although widely applied, Linear Programming is limited to a single objective function. In many real world situations, DMs are faced with multiple objective problems in that several competing and conflicting objectives have to be considered. For these problems, there exist many alternatives that are feasible and acceptable. However, the DM is interested in finding “the most preferred alternative”. In the past three decades, many methods have been developed for solving MOLP problems. One class of these methods is called “interactive”, in which the DM responds to a set of questions interactively so that his/her most preferred alternative can be obtained. In most of these methods, the value (utility) function (that presents the DM's preference) is assumed to be linear or additive, concave, pseudo-concave, or quasi-concave. However, for MOLP problems, there has not been any effort to recognize and solve the quasi-convex utility functions, which are among the most difficult class of problems to solve. The quasi-convex class of utility functions represents an extremist preferential behavior, while the other aforementioned methods (such as quasi-concave) represent a conservative behavioral preference. It is shown that the method converges to the optimal (the most preferred) alternative. The approach is computationally feasible for moderately sized problems.