The Multiobjective Discrete Optimization Problem: A Weighted Min-Max Two-Stage Optimization Approach and a Bicriteria Algorithm

Abstract

We study the multiple objective discrete optimization (MODO) problem and propose two-stage optimization problems as subproblems to be solved to obtain efficient solutions. The mathematical structure of the first level subproblem has similarities to both Tchebycheff type of approaches and a generalization of the lexicographic max-ordering problem that are applicable to multiple objective optimization. We present some results that enable us to develop an algorithm to solve the bicriteria discrete optimization problem for the entire efficient set. We also propose a modification of the algorithm that generates a sample of efficient solutions that satisfies a prespecified quality guarantee. We apply the algorithm to solve the bicriteria knapsack problem. Our computational results on this particular problem demonstrate that our algorithm performs significantly better than an equivalent Tchebycheff counterpart. Moreover, the computational behavior of the sampling version is quite promising.multiple objective optimization, efficient set, min-max optimization, bicriteria knapsack problem

Algorithm robust for the bicriteria discrete optimization problem

“…Although these are not simple parametric programs, the weights in the above robust problems can be interpreted as realization probabilities of the scenarios. In Sayin and Kouvelis (2005), it has been established that the solution obtained from the two stage mathematical programs (P w ) and (Q w ) is efficient for any w > 0, and for any efficient solution x, there exists w > 0 such that x is an optimal solution to the two stage parametric programs (P w ) and (Q w ). The same result holds for the deviation robust counterparts of subproblems (P w ) and (Q w ).…”

“…We formally state below our parametric search algorithm that generates the entire efficient solution set of the bicriteria discrete optimization problem in the objective space (Sayin and Kouvelis, 2005). The parameter search is conducted over w 2 in problem (P w ) and w 1 = 1 − w 2 .…”

“…Algorithm Robust (Sayin and Kouvelis, 2005) Step 0. Solve problems (P w ) and (Q w ) with w 2 = θ to obtain y 1 .…”

“…This implies, obviously, that the relationship between the two different problems can lead to interesting developments for the solution of both. Recently Sayin and Kouvelis (2005) proposed an algorithm for bicriteria discrete optimization based on ideas motivated by robust optimization. Algorithm Robust performs an exhaustive search in the weight space while solving corresponding weighted robust optimization problems.…”

An Integrated Multi-Objective Optimization Model for Solving the Construction Time-Cost Trade-Off Problem

“…time and cost) of the construction method. Additional information on the fundamentals of multi-objective optimization and GA can be found in the related references (Certa et al 2011;Elbehairy et al 2006;Elegbede, Adjallah 2003;Geem 2010;Kandil et al 2010;Konak et al 2006;Moradi et al 2011;Sayin, Kouvelis 2005;Šešok, Belevicius 2008;Šešok et al 2010;Šostak, Vakrinien 2011;Venskus et al 2010). In this study, a software program called 'OptQuest' (an optimization tool that runs with a software program called 'Crystal Ball') was used to introduce the GA.…”