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Dynamic programming algorithms for the bi-objective integer knapsack problem

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  • Rong, Aiying
  • Figueira, José Rui

Abstract

This paper presents two new dynamic programming (DP) algorithms to find the exact Pareto frontier for the bi-objective integer knapsack problem. First, a property of the traditional DP algorithm for the multi-objective integer knapsack problem is identified. The first algorithm is developed by directly using the property. The second algorithm is a hybrid DP approach using the concept of the bound sets. The property is used in conjunction with the bound sets. Next, the numerical experiments showed that a promising partial solution can be sometimes discarded if the solutions of the linear relaxation for the subproblem associated with the partial solution are directly used to estimate an upper bound set. It means that the upper bound set is underestimated. Then, an extended upper bound set is proposed on the basis of the set of linear relaxation solutions. The efficiency of the hybrid algorithm is improved by tightening the proposed upper bound set. The numerical results obtained from different types of bi-objective instances show the effectiveness of the proposed approach.

Suggested Citation

  • Rong, Aiying & Figueira, José Rui, 2014. "Dynamic programming algorithms for the bi-objective integer knapsack problem," European Journal of Operational Research, Elsevier, vol. 236(1), pages 85-99.
    • Handle: RePEc:eee:ejores:v:236:y:2014:i:1:p:85-99
    DOI: 10.1016/j.ejor.2013.11.032
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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. 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.
    4. 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.
    5. José Figueira & Luís Paquete & Marco Simões & Daniel Vanderpooten, 2013. "Algorithmic improvements on dynamic programming for the bi-objective {0,1} knapsack problem," Computational Optimization and Applications, Springer, vol. 56(1), pages 97-111, September.
    6. 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.
    7. 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.
    8. 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.
    9. 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.
    10. Mavrotas, George & Florios, Kostas, 2013. "An improved version of the augmented epsilon-constraint method (AUGMECON2) for finding the exact Pareto set in Multi-Objective Integer Programming problems," MPRA Paper 105034, University Library of Munich, Germany.
    11. Zhang, Weihua & Reimann, Marc, 2014. "A simple augmented ∊-constraint method for multi-objective mathematical integer programming problems," European Journal of Operational Research, Elsevier, vol. 234(1), pages 15-24.
    12. Francis Sourd & Olivier Spanjaard, 2008. "A Multiobjective Branch-and-Bound Framework: Application to the Biobjective Spanning Tree Problem," INFORMS Journal on Computing, INFORMS, vol. 20(3), pages 472-484, August.
    13. 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.
    14. 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.
    15. 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.
    16. 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.
    17. 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.
    18. 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.
    19. 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.
    20. 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.
    21. 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.
    22. 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.
    23. 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.
    24. 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.
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