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An Improved Implicit Enumeration Approach for Integer Programming

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  • A. M. Geoffrion

    (University of California at Los Angeles, Los Angeles, California)

Abstract

This paper synthesizes the Balasian implicit enumeration approach to integer linear programming with the approach typified by Land and Doig and by Roy, Bertier, and Nghiem. The synthesis results from the use of an imbedded linear program to compute surrogate constraints that are as “strong” as possible in a sense slightly different from that originally used by Glover. A simple implicit enumeration algorithm fitted with optional imbedded linear programming machinery was implemented and tested extensively on an IBM 7044. Use of the imbedded linear program greatly reduced solution times in virtually every case, and seemed to render the tested algorithm superior to the five other implicit enumeration algorithms for which comparable published experience was available. The crucial issue of the sensitivity of solution time to the number of integer variables was given special attention. Sequences were run of set-covering, optimal-routing, and knapsack problems with multiple constraints of varying sizes up to 90 variables. The results suggest that use of the imbedded linear program in the prescribed way may mitigate solution-time dependence on the number of variables from an exponential to a low-order polynomial increase. The dependence appeared to be approximately linear for the first two problem classes, with 90-variable problems typically being solved in about 15 seconds; and approximately cubic for the third class, with 80-variable problems typically solved in less than 2 minutes. In the 35-variable range for all three classes, use of the imbedded linear program reduced solution times by a factor of about 100.

Suggested Citation

  • A. M. Geoffrion, 1969. "An Improved Implicit Enumeration Approach for Integer Programming," Operations Research, INFORMS, vol. 17(3), pages 437-454, June.
  • Handle: RePEc:inm:oropre:v:17:y:1969:i:3:p:437-454
    DOI: 10.1287/opre.17.3.437
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    Cited by:

    1. 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.
    2. Balev, Stefan & Yanev, Nicola & Freville, Arnaud & Andonov, Rumen, 2008. "A dynamic programming based reduction procedure for the multidimensional 0-1 knapsack problem," European Journal of Operational Research, Elsevier, vol. 186(1), pages 63-76, April.
    3. Thomas L. Magnanti, 2021. "Optimization: From Its Inception," Management Science, INFORMS, vol. 67(9), pages 5349-5363, September.
    4. Patriksson, Michael, 2008. "A survey on the continuous nonlinear resource allocation problem," European Journal of Operational Research, Elsevier, vol. 185(1), pages 1-46, February.
    5. Jiang, Bo & Tzavellas, Hector, 2023. "Optimal liquidity allocation in an equity network," International Review of Economics & Finance, Elsevier, vol. 85(C), pages 286-294.
    6. 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.
    7. Sanjiv Sarin & Mark H. Karwan & Ronald L. Rardin, 1987. "A new surrogate dual multiplier search procedure," Naval Research Logistics (NRL), John Wiley & Sons, vol. 34(3), pages 431-450, June.
    8. Asano, Makoto & Ohta, Hiroshi, 1999. "Single machine scheduling to meet due times under shutdown constraints," International Journal of Production Economics, Elsevier, vol. 60(1), pages 537-547, April.
    9. Bala Shetty, 1990. "A relaxation/decomposition algorithm for the fixed charged network problem," Naval Research Logistics (NRL), John Wiley & Sons, vol. 37(2), pages 327-340, April.
    10. Manfred Padberg, 2005. "Classical Cuts for Mixed-Integer Programming and Branch-and-Cut," Annals of Operations Research, Springer, vol. 139(1), pages 321-352, October.
    11. Fox, B. L. & Lenstra, J. K. & Rinnooy Kan, A. H. G. & Schrage, L. E., 1977. "Branching From The Largest Upper Bound: Folklore And Facts," Econometric Institute Archives 272158, Erasmus University Rotterdam.
    12. Shuangyan Li & Yijing Liang & Zhenjie Wang & Dezhi Zhang, 2021. "An Optimization Model of a Sustainable City Logistics Network Design Based on Goal Programming," Sustainability, MDPI, vol. 13(13), pages 1-20, July.
    13. Hanif D. Sherali & J. Cole Smith & Antonio A. Trani, 2002. "An Airspace Planning Model for Selecting Flight-plans Under Workload, Safety, and Equity Considerations," Transportation Science, INFORMS, vol. 36(4), pages 378-397, November.
    14. Mohammadi Bidhandi, Hadi & Mohd. Yusuff, Rosnah & Megat Ahmad, Megat Mohamad Hamdan & Abu Bakar, Mohd Rizam, 2009. "Development of a new approach for deterministic supply chain network design," European Journal of Operational Research, Elsevier, vol. 198(1), pages 121-128, October.
    15. Freville, Arnaud, 2004. "The multidimensional 0-1 knapsack problem: An overview," European Journal of Operational Research, Elsevier, vol. 155(1), pages 1-21, May.
    16. Fu Lin & Sven Leyffer & Todd Munson, 2016. "A two-level approach to large mixed-integer programs with application to cogeneration in energy-efficient buildings," Computational Optimization and Applications, Springer, vol. 65(1), pages 1-46, September.
    17. van Dam, Wim & Telgen, Jan, 1978. "Some Computational Experiments With A Primal-Dual Surrogate Simplex Algorithm," Econometric Institute Archives 272174, Erasmus University Rotterdam.
    18. Syam Menon & Linus Schrage, 2002. "Order Allocation for Stock Cutting in the Paper Industry," Operations Research, INFORMS, vol. 50(2), pages 324-332, April.

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