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A branch and bound method for the solution of multiparametric mixed integer linear programming problems

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  • Richard Oberdieck
  • Martina Wittmann-Hohlbein
  • Efstratios Pistikopoulos

Abstract

In this paper, we present a novel algorithm for the solution of multiparametric mixed integer linear programming (mp-MILP) problems that exhibit uncertain objective function coefficients and uncertain entries in the right-hand side constraint vector. The algorithmic procedure employs a branch and bound strategy that involves the solution of a multiparametric linear programming sub-problem at leaf nodes and appropriate comparison procedures to update the tree. McCormick relaxation procedures are employed to overcome the presence of bilinear terms in the model. The algorithm generates an envelope of parametric profiles, containing the optimal solution of the mp-MILP problem. The parameter space is partitioned into polyhedral convex critical regions. Two examples are presented to illustrate the steps of the proposed algorithm. Copyright Springer Science+Business Media New York 2014

Suggested Citation

  • Richard Oberdieck & Martina Wittmann-Hohlbein & Efstratios Pistikopoulos, 2014. "A branch and bound method for the solution of multiparametric mixed integer linear programming problems," Journal of Global Optimization, Springer, vol. 59(2), pages 527-543, July.
    • Handle: RePEc:spr:jglopt:v:59:y:2014:i:2:p:527-543
    DOI: 10.1007/s10898-014-0143-9
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    References listed on IDEAS

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    1. Vivek Dua & Efstratios Pistikopoulos, 2000. "An Algorithm for the Solution of Multiparametric Mixed Integer Linear Programming Problems," Annals of Operations Research, Springer, vol. 99(1), pages 123-139, December.
    2. Tomas Gal, 1975. "Rim Multiparametric Linear Programming," Management Science, INFORMS, vol. 21(5), pages 567-575, January.
    3. F. Borrelli & A. Bemporad & M. Morari, 2003. "Geometric Algorithm for Multiparametric Linear Programming," Journal of Optimization Theory and Applications, Springer, vol. 118(3), pages 515-540, September.
    4. Tomas Gal & Josef Nedoma, 1972. "Multiparametric Linear Programming," Management Science, INFORMS, vol. 18(7), pages 406-422, March.
    5. GAL, Thomas & NEDOMA, Jozef, 1972. "Multiparametric linear programming," LIDAM Reprints CORE 115, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    6. Mitsos, Alexander & Barton, Paul I., 2009. "Parametric mixed-integer 0-1 linear programming: The general case for a single parameter," European Journal of Operational Research, Elsevier, vol. 194(3), pages 663-686, May.
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    Cited by:

    1. Faraz Salehi & S. Mohammad J. Mirzapour Al-E-Hashem & S. Mohammad Moattar Husseini & S. Hassan Ghodsypour, 2023. "A bi-level multi-follower optimization model for R&D project portfolio: an application to a pharmaceutical holding company," Annals of Operations Research, Springer, vol. 323(1), pages 331-360, April.
    2. Richard Oberdieck & Nikolaos A. Diangelakis & Styliani Avraamidou & Efstratios N. Pistikopoulos, 2017. "On unbounded and binary parameters in multi-parametric programming: applications to mixed-integer bilevel optimization and duality theory," Journal of Global Optimization, Springer, vol. 69(3), pages 587-606, November.
    3. Iosif Pappas & Nikolaos A. Diangelakis & Efstratios N. Pistikopoulos, 2021. "The exact solution of multiparametric quadratically constrained quadratic programming problems," Journal of Global Optimization, Springer, vol. 79(1), pages 59-85, January.
    4. Styliani Avraamidou & Efstratios N. Pistikopoulos, 2019. "Multi-parametric global optimization approach for tri-level mixed-integer linear optimization problems," Journal of Global Optimization, Springer, vol. 74(3), pages 443-465, July.

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