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GEOMETRIC APPROACH TO MULTIPLE OBJECTIVE OPTIMIZATION WITH APPLICATION TO MULTIPLE CRITERIA DECISION-MAKING (FACES OF FINITE CONE, ALGORITHM FOR MOLP, UTILITY ANALYSIS)
Abstract
A geometric approach to analyze and solve multiple objective linear programming problems is developed. Since the linear map represented by the cost matrix is not a one-to-one map then not every vertex in constraint space maps to a vertex in objective space. In addition, the dimension of the objective space is far much smaller than the dimension of the constraint space. Hence the objective space will have a simpler geometric structure. Taking advantage of this simpler geometric structure, we focus on analyzing the multiple objective optimization problem in the objective space. Some of the existing approaches to analyze and solve multiple objective linear programs in the constraint space are previewed in the first chapter. We conclude the chapter with a motivational discussion and a formulation of the multiple objective linear programming problem in objective space. From the analysis of the multiple objective simplex tableau, we have, locally at any vertex in objective space, the objective set is contained in the cone generated by the columns of the reduced cost matrix. So the problem of finding the nondominated faces in the objective space is reduced to finding the nondominated faces of a finitely generated cone. The latter problem is addressed in the second chapter. The algorithms for constructing the faces of finitely generated cone utilize the cone's face lattice and structure and determine a hyperplane characterization of the face. An algorithm to construct the entire nondominated set as a union of nondominated faces is developed in the third chapter. This algorithm is based on the algorithms in the second chapter. In addition, we gave a numerical analysis of some of the X-based algorithms for comparison with our Y-based algorithm. Application of the multiple objective optimization algorithm to utility analysis in multiple criteria decision making is investigated in the fourth chapter. A technique for measuring the utility is presented. An algorithm for optimizing a given utility function over the nondominated set is developed.
Subject Area
Operations research
Recommended Citation
EL-ABYAD, ABDELWAHAB MOHAMED, "GEOMETRIC APPROACH TO MULTIPLE OBJECTIVE OPTIMIZATION WITH APPLICATION TO MULTIPLE CRITERIA DECISION-MAKING (FACES OF FINITE CONE, ALGORITHM FOR MOLP, UTILITY ANALYSIS)" (1986). ETD collection for University of Nebraska-Lincoln. AAI8704547.
https://digitalcommons.unl.edu/dissertations/AAI8704547