A reduction dynamic programming algorithm for the bi-objective integer knapsack problem |
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Authors: | Aiying Rong,José Rui Figueira |
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Affiliation: | 1. Cemapre (Center of Applied Mathematics and Economics), ISEG-Technical University of Lisbon, Rua do Quelhas 6, 1200-781 Lisboa, Portugal;2. CEG-IST, Center for Management Studies, Instituto Superior Técnico, Technical University of Lisbon, Av. Rovisco Pais, 1049-001 Lisboa, Portugal |
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Abstract: | This paper presents a backward state reduction dynamic programming algorithm for generating the exact Pareto frontier for the bi-objective integer knapsack problem. The algorithm is developed addressing a reduced problem built after applying variable fixing techniques based on the core concept. First, an approximate core is obtained by eliminating dominated items. Second, the items included in the approximate core are subject to the reduction of the upper bounds by applying a set of weighted-sum functions associated with the efficient extreme solutions of the linear relaxation of the multi-objective integer knapsack problem. Third, the items are classified according to the values of their upper bounds; items with zero upper bounds can be eliminated. Finally, the remaining items are used to form a mixed network with different upper bounds. The numerical results obtained from different types of bi-objective instances show the effectiveness of the mixed network and associated dynamic programming algorithm. |
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Keywords: | Multi-objective programming Integer knapsack problem Dynamic programming Dominance relation Core concept State reduction |
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