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A Multiphase-Dual Algorithm for the Zero-One Integer Programming Problem

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  • Fred Glover

    (Carnegie Institute of Technology, Pittsburgh, Pennsylvania)

Abstract

Following a line of approach recently applied to the 0-1 integer programming problem with some success by Egon Balas, the algorithm of this paper is based upon an underlying tree-search structure upon which a series of tests is superimposed to exclude large portions of the tree of all possible 0-1 solutions from examination. In our method, the specific design of the enumeration and tests, supplemented by the use of a special type of constraint called a “surrogate constraint,” results in an algorithm that appears to be quite efficient in relation to other algorithms currently available for solving the 0-1 integer programming problem. Early indications of efficiency must, however, be regarded as suggestive rather than conclusive, due to the limited range and size of problems so far examined. Following the analytical development of the method, three example problems are solved in detail with the Multiphase-Dual Algorithm to illustrate various aspects of its application. An extension of the algorithm to the general integer programming problem in bounded variables is briefly sketched in a concluding section.

Suggested Citation

  • Fred Glover, 1965. "A Multiphase-Dual Algorithm for the Zero-One Integer Programming Problem," Operations Research, INFORMS, vol. 13(6), pages 879-919, December.
    • Handle: RePEc:inm:oropre:v:13:y:1965:i:6:p:879-919
    DOI: 10.1287/opre.13.6.879
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    Cited by:

    1. Woiler, Samsão, 1969. "Enumeração implícita aplicada à seleção de investimentos," RAE - Revista de Administração de Empresas, FGV-EAESP Escola de Administração de Empresas de São Paulo (Brazil), vol. 9(4), October.
    2. Satoshi Suzuki & Daishi Kuroiwa, 2012. "Necessary and Sufficient Constraint Qualification for Surrogate Duality," Journal of Optimization Theory and Applications, Springer, vol. 152(2), pages 366-377, February.
    3. Glover, Fred, 2013. "Advanced greedy algorithms and surrogate constraint methods for linear and quadratic knapsack and covering problems," European Journal of Operational Research, Elsevier, vol. 230(2), pages 212-225.
    4. Jiang, Bo & Tzavellas, Hector, 2023. "Optimal liquidity allocation in an equity network," International Review of Economics & Finance, Elsevier, vol. 85(C), pages 286-294.
    5. Audrey Cerqueus & Xavier Gandibleux & Anthony Przybylski & Frédéric Saubion, 2017. "On branching heuristics for the bi-objective 0/1 unidimensional knapsack problem," Journal of Heuristics, Springer, vol. 23(5), pages 285-319, October.
    6. Alidaee, Bahram, 2014. "Zero duality gap in surrogate constraint optimization: A concise review of models," European Journal of Operational Research, Elsevier, vol. 232(2), pages 241-248.
    7. Edirisinghe, Chanaka & Jeong, Jaehwan, 2019. "Indefinite multi-constrained separable quadratic optimization: Large-scale efficient solution," European Journal of Operational Research, Elsevier, vol. 278(1), pages 49-63.
    8. I. Kaliszewski & J. Miroforidis, 2022. "Probing the Pareto front of a large-scale multiobjective problem with a MIP solver," Operational Research, Springer, vol. 22(5), pages 5617-5673, November.
    9. van Dam, Wim & Telgen, Jan, 1978. "Some Computational Experiments With A Primal-Dual Surrogate Simplex Algorithm," Econometric Institute Archives 272174, Erasmus University Rotterdam.
    10. Hasan Pirkul, 1987. "A heuristic solution procedure for the multiconstraint zero‐one knapsack problem," Naval Research Logistics (NRL), John Wiley & Sons, vol. 34(2), pages 161-172, April.
    11. Ablanedo-Rosas, José H. & Rego, César, 2010. "Surrogate constraint normalization for the set covering problem," European Journal of Operational Research, Elsevier, vol. 205(3), pages 540-551, September.
    12. Suzuki, Satoshi & Kuroiwa, Daishi & Lee, Gue Myung, 2013. "Surrogate duality for robust optimization," European Journal of Operational Research, Elsevier, vol. 231(2), pages 257-262.
    13. J. Glover & V. Quan & S. Zolfaghari, 2021. "Some new perspectives for solving 0–1 integer programming problems using Balas method," Computational Management Science, Springer, vol. 18(2), pages 177-193, June.
    14. Amen, Matthias, 2006. "Cost-oriented assembly line balancing: Model formulations, solution difficulty, upper and lower bounds," European Journal of Operational Research, Elsevier, vol. 168(3), pages 747-770, February.
    15. Joseph, Anito & Gass, Saul I. & Bryson, Noel, 1998. "An objective hyperplane search procedure for solving the general all-integer linear programming (ILP) problem," European Journal of Operational Research, Elsevier, vol. 104(3), pages 601-614, February.
    16. Dell'Amico, Mauro & Iori, Manuel & Martello, Silvano & Monaci, Michele, 2006. "Lower bounds and heuristic algorithms for the ki-partitioning problem," European Journal of Operational Research, Elsevier, vol. 171(3), pages 725-742, June.
    17. Satoshi Suzuki & Daishi Kuroiwa, 2020. "Duality Theorems for Convex and Quasiconvex Set Functions," SN Operations Research Forum, Springer, vol. 1(1), pages 1-13, March.
    18. Monique Guignard & Ellis Johnson & Kurt Spielberg, 2005. "Logical Processing for Integer Programming," Annals of Operations Research, Springer, vol. 140(1), pages 263-304, November.
    19. Saïd Hanafi & Christophe Wilbaut, 2011. "Improved convergent heuristics for the 0-1 multidimensional knapsack problem," Annals of Operations Research, Springer, vol. 183(1), pages 125-142, March.
    20. Yuji Nakagawa & Ross J. W. James & César Rego & Chanaka Edirisinghe, 2014. "Entropy-Based Optimization of Nonlinear Separable Discrete Decision Models," Management Science, INFORMS, vol. 60(3), pages 695-707, March.
    21. Marco Antonio Boschetti & Vittorio Maniezzo, 2022. "Matheuristics: using mathematics for heuristic design," 4OR, Springer, vol. 20(2), pages 173-208, June.
    22. Arnaud Fréville & SaÏd Hanafi, 2005. "The Multidimensional 0-1 Knapsack Problem—Bounds and Computational Aspects," Annals of Operations Research, Springer, vol. 139(1), pages 195-227, October.
    23. Yoon, Yourim & Kim, Yong-Hyuk & Moon, Byung-Ro, 2012. "A theoretical and empirical investigation on the Lagrangian capacities of the 0-1 multidimensional knapsack problem," European Journal of Operational Research, Elsevier, vol. 218(2), pages 366-376.
    24. Raymond A. Patterson & Erik Rolland, 2002. "Hybrid Fiber Coaxial Network Design," Operations Research, INFORMS, vol. 50(3), pages 538-551, June.

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