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A reduction dynamic programming algorithm for the bi-objective integer knapsack problem

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  • Rong, Aiying
  • Figueira, José Rui

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.

Suggested Citation

  • Rong, Aiying & Figueira, José Rui, 2013. "A reduction dynamic programming algorithm for the bi-objective integer knapsack problem," European Journal of Operational Research, Elsevier, vol. 231(2), pages 299-313.
    • Handle: RePEc:eee:ejores:v:231:y:2013:i:2:p:299-313
    DOI: 10.1016/j.ejor.2013.05.045
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    1. Laumanns, Marco & Thiele, Lothar & Zitzler, Eckart, 2006. "An efficient, adaptive parameter variation scheme for metaheuristics based on the epsilon-constraint method," European Journal of Operational Research, Elsevier, vol. 169(3), pages 932-942, March.
    2. Kwak, Wikil & Shi, Yong & Lee, Heeseok & Lee, Cheng F., 1996. "Capital Budgeting with Multiple Criteria and Multiple Decision Makers," Review of Quantitative Finance and Accounting, Springer, vol. 7(1), pages 97-112, July.
    3. Mavrotas, George & Figueira, José Rui & Florios, Kostas, 2009. "Solving the bi-objective multidimensional knapsack problem exploiting the concept of core," MPRA Paper 105087, University Library of Munich, Germany.
    4. Gomes da Silva, Carlos & Climaco, Joao & Figueira, Jose, 2006. "A scatter search method for bi-criteria {0, 1}-knapsack problems," European Journal of Operational Research, Elsevier, vol. 169(2), pages 373-391, March.
    5. Dhaenens, C. & Lemesre, J. & Talbi, E.G., 2010. "K-PPM: A new exact method to solve multi-objective combinatorial optimization problems," European Journal of Operational Research, Elsevier, vol. 200(1), pages 45-53, January.
    6. Bazgan, Cristina & Hugot, Hadrien & Vanderpooten, Daniel, 2009. "Implementing an efficient fptas for the 0-1 multi-objective knapsack problem," European Journal of Operational Research, Elsevier, vol. 198(1), pages 47-56, October.
    7. Sylva, John & Crema, Alejandro, 2007. "A method for finding well-dispersed subsets of non-dominated vectors for multiple objective mixed integer linear programs," European Journal of Operational Research, Elsevier, vol. 180(3), pages 1011-1027, August.
    8. Bas, Esra, 2011. "An investment plan for preventing child injuries using risk priority number of failure mode and effects analysis methodology and a multi-objective, multi-dimensional mixed 0-1 knapsack model," Reliability Engineering and System Safety, Elsevier, vol. 96(7), pages 748-756.
    9. Andonov, R. & Poirriez, V. & Rajopadhye, S., 2000. "Unbounded knapsack problem: Dynamic programming revisited," European Journal of Operational Research, Elsevier, vol. 123(2), pages 394-407, June.
    10. Aytug, Haldun & SayIn, Serpil, 2009. "Using support vector machines to learn the efficient set in multiple objective discrete optimization," European Journal of Operational Research, Elsevier, vol. 193(2), pages 510-519, March.
    11. Özlen, Melih & Azizoglu, Meral, 2009. "Multi-objective integer programming: A general approach for generating all non-dominated solutions," European Journal of Operational Research, Elsevier, vol. 199(1), pages 25-35, November.
    12. Schweigert, D. & Neumayer, P., 1997. "A reduction algorithm for integer multiple objective linear programs," European Journal of Operational Research, Elsevier, vol. 99(2), pages 459-462, June.
    13. Jenkins, Larry, 2002. "A bicriteria knapsack program for planning remediation of contaminated lightstation sites," European Journal of Operational Research, Elsevier, vol. 140(2), pages 427-433, July.
    14. Silvano Martello & Paolo Toth, 1988. "A New Algorithm for the 0-1 Knapsack Problem," Management Science, INFORMS, vol. 34(5), pages 633-644, May.
    15. George Mavrotas & José Figueira & Alexandros Antoniadis, 2011. "Using the idea of expanded core for the exact solution of bi-objective multi-dimensional knapsack problems," Journal of Global Optimization, Springer, vol. 49(4), pages 589-606, April.
    16. Jones, D. F. & Mirrazavi, S. K. & Tamiz, M., 2002. "Multi-objective meta-heuristics: An overview of the current state-of-the-art," European Journal of Operational Research, Elsevier, vol. 137(1), pages 1-9, February.
    17. Zhang, Cai Wen & Ong, Hoon Liong, 2004. "Solving the biobjective zero-one knapsack problem by an efficient LP-based heuristic," European Journal of Operational Research, Elsevier, vol. 159(3), pages 545-557, December.
    18. Giorgio P. Ingargiola & James F. Korsh, 1977. "A General Algorithm for One-Dimensional Knapsack Problems," Operations Research, INFORMS, vol. 25(5), pages 752-759, October.
    19. Florios, Kostas & Mavrotas, George & Diakoulaki, Danae, 2010. "Solving multiobjective, multiconstraint knapsack problems using mathematical programming and evolutionary algorithms," European Journal of Operational Research, Elsevier, vol. 203(1), pages 14-21, May.
    20. Gomes da Silva, Carlos & Figueira, Jose & Climaco, Joao, 2007. "Integrating partial optimization with scatter search for solving bi-criteria {0, 1}-knapsack problems," European Journal of Operational Research, Elsevier, vol. 177(3), pages 1656-1677, March.
    21. Sylva, John & Crema, Alejandro, 2004. "A method for finding the set of non-dominated vectors for multiple objective integer linear programs," European Journal of Operational Research, Elsevier, vol. 158(1), pages 46-55, October.
    22. Meir J. Rosenblatt & Zilla Sinuany-Stern, 1989. "Generating the Discrete Efficient Frontier to the Capital Budgeting Problem," Operations Research, INFORMS, vol. 37(3), pages 384-394, June.
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    7. Wilbaut, Christophe & Todosijevic, Raca & Hanafi, Saïd & Fréville, Arnaud, 2023. "Heuristic and exact reduction procedures to solve the discounted 0–1 knapsack problem," European Journal of Operational Research, Elsevier, vol. 304(3), pages 901-911.
    8. Cacchiani, Valentina & D’Ambrosio, Claudia, 2017. "A branch-and-bound based heuristic algorithm for convex multi-objective MINLPs," European Journal of Operational Research, Elsevier, vol. 260(3), pages 920-933.

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