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Practical Solution of Large Mixed Integer Programming Problems with Umpire

Author

Listed:
  • J. J. H. Forrest

    (Scicon Ltd., London, England)

  • J. P. H. Hirst

    (British Petroleum Co. Ltd., London, England)

  • J. A. Tomlin

    (Presently affiliated with Stanford University)

Abstract

In this paper we discuss some branch and bound methods implemented in the UMPIRE mathematical programming system for solving practical integer programming problems and give details of computational experience with these methods. We have found that the nontrivial integer programming problems we encounter tend to fall into two classes. The first class of problems is characterized by a relatively small number (less than 100) of binary integer variables embedded in a very large and difficult linear program, with a few thousand constraints and complex matrix structure. The second class of problems tends to have many more integer variables (perhaps several hundred), frequently organized into special ordered sets and expressing complex logical relationships, but embedded in a much simpler linear program with 500-1,000 constraints. Problems in the first group in particular do not generally behave well using a simple-minded "penalty" approach for reasons we will make clear and a number of heuristic estimation procedures are used to guide the branch and bound search. The most important devices used are "priorities," the "best projection criterion" and "pseudo-costs." Some or all of these heuristics can also be very useful in tackling the second group of problems, where it is important to reach a good solution quickly to limit the search.

Suggested Citation

  • J. J. H. Forrest & J. P. H. Hirst & J. A. Tomlin, 1974. "Practical Solution of Large Mixed Integer Programming Problems with Umpire," Management Science, INFORMS, vol. 20(5), pages 736-773, January.
    • Handle: RePEc:inm:ormnsc:v:20:y:1974:i:5:p:736-773
    DOI: 10.1287/mnsc.20.5.736
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    Cited by:

    1. Siqian Shen & Murat Kurt & Jue Wang, 2015. "Chance-Constrained Programming Models and Approximations for General Stochastic Bottleneck Spanning Tree Problems," INFORMS Journal on Computing, INFORMS, vol. 27(2), pages 301-316, May.
    2. R. Misener & C. A. Floudas, 2010. "Piecewise-Linear Approximations of Multidimensional Functions," Journal of Optimization Theory and Applications, Springer, vol. 145(1), pages 120-147, April.
    3. Ellis L. Johnson & George L. Nemhauser & Martin W.P. Savelsbergh, 2000. "Progress in Linear Programming-Based Algorithms for Integer Programming: An Exposition," INFORMS Journal on Computing, INFORMS, vol. 12(1), pages 2-23, February.
    4. Pantelis Broukos & Antonios Fragkogios & Nilay Shah, 2022. "A Linearized Mathematical Formulation for Combined Centralized and Distributed Waste Water Treatment Network Design," SN Operations Research Forum, Springer, vol. 3(3), pages 1-29, September.
    5. W. Ogryczak & K. Zorychta, 1996. "Modular Optimizer for Mixed Integer Programming MOMIP Version 2.3," Working Papers wp96106, International Institute for Applied Systems Analysis.
    6. Kumar Abhishek & Sven Leyffer & Jeff Linderoth, 2010. "FilMINT: An Outer Approximation-Based Solver for Convex Mixed-Integer Nonlinear Programs," INFORMS Journal on Computing, INFORMS, vol. 22(4), pages 555-567, November.
    7. Derpich, Ivan & Vera, Jorge R., 2006. "Improving the efficiency of the Branch and Bound algorithm for integer programming based on "flatness" information," European Journal of Operational Research, Elsevier, vol. 174(1), pages 92-101, October.
    8. W. Ogryczak & K. Zorychta, 1994. "Modular Optimizer for Mixed Integer Programming MOMIP Version 2.1," Working Papers wp94035, International Institute for Applied Systems Analysis.

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