# Converging preferred regions in multi-objective combinatorial optimization problems

2011
Lokman, Banu
Finding the true nondominated points is typically hard for Multi-objective Combinatorial Optimization (MOCO) problems. Furthermore, it is not practical to generate all of them since the number of nondominated points may grow exponentially as the problem size increases. In this thesis, we develop an exact algorithm to find all nondominated points in a specified region. We combine this exact algorithm with a heuristic algorithm that approximates the possible locations of the nondominated points. Interacting with a decision maker (DM), the heuristic algorithm first approximately identifies the region that is of interest to the DM. Then, the exact algorithm is employed to generate all true nondominated points in this region. We conduct experiments on Multi-objective Assignment Problems (MOAP), Multi-objective Knapsack Problems (MOKP) and Multi-objective Shortest Path (MOSP) Problems; and the algorithms work well. Finding the worst possible value for each criterion among the set of efficient solutions has important uses in multi-criteria problems since the proper scaling of each criterion is required by many approaches. Such points are called nadir points. v It is not straightforward to find the nadir points, especially for large problems with more than two criteria. We develop an exact algorithm to find the nadir values for multi-objective integer programming problems. We also find bounds with performance guarantees. We demonstrate that our algorithms work well in our experiments on MOAP, MOKP and MOSP problems. Assuming that the DM's preferences are consistent with a quasiconcave value function, we develop an interactive exact algorithm to solve MIP problems. Based on the convex cones derived from pairwise comparisons of the DM, we generate constraints to prevent points in the implied inferior regions. We guarantee finding the most preferred point and our computational experiments on MOAP, MOKP and MOSP problems show that a reasonable number of pairwise comparisons are required.

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Citation Formats
B. Lokman, “Converging preferred regions in multi-objective combinatorial optimization problems,” Ph.D. - Doctoral Program, Middle East Technical University, 2011. 