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Fixed-charge transportation on a path: optimization, LP formulations and separation

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  • VAN VYVE, Mathieu

    (Université catholique de Louvain, CORE and Louvain School of Management, B-1348 Louvain-la-Neuve, Belgium)

Abstract

The fixed-charge transportation problem is an interesting problem in its own right. This paper further motivates its study by showing that it is both a special case and a strong relaxation of the big-bucket multi-item lot-sizing problem. We then provide a polyhedral analysis of the polynomially solvable special case in which the associated bipartite graph is a path. We give a O(n^3)-time optimization algorithm and two O(n^2)-size linear programming extended formulation. We describe a new class of inequalities that we call "path-modular" inequalities. We give two distinct proofs of their validity. The first one is direct and crucially relies on sub- and super- modularity of an associated set function. The second proof is by showing that the projection of one of the extended linear programming formulations onto the original variable space yields exactly the polyhedron described by the path- modular inequalities. Thus we also show that these inequalities suffice to describe the convex hull of the set of feasible solutions. We finally report on computational experiments comparing extended LP formulation, valid inequalities separation and a standard MIP solver.

Suggested Citation

  • VAN VYVE, Mathieu, 2010. "Fixed-charge transportation on a path: optimization, LP formulations and separation," LIDAM Discussion Papers CORE 2010068, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    • Handle: RePEc:cor:louvco:2010068
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    File URL: https://sites.uclouvain.be/core/publications/coredp/coredp2010.html
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    References listed on IDEAS

    as
    1. CONFORTI, Michele & DI SUMMA, Marco & EISENBRAND, Friedrich & WOLSEY, Laurence A., 2009. "Network formulations of mixed-integer programs," LIDAM Reprints CORE 2146, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    2. DI SUMMA, Marco & WOLSEY, Laurence, 2010. "Mixing sets linked by bidirected paths," LIDAM Discussion Papers CORE 2010063, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    3. Michele Conforti & Marco Di Summa & Friedrich Eisenbrand & Laurence A. Wolsey, 2009. "Network Formulations of Mixed-Integer Programs," Mathematics of Operations Research, INFORMS, vol. 34(1), pages 194-209, February.
    Full references (including those not matched with items on IDEAS)

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    More about this item

    Keywords

    mixed-integer programming; lot-sizing; transportation; convex hull; extended formulation;
    All these keywords.

    JEL classification:

    • Q25 - Agricultural and Natural Resource Economics; Environmental and Ecological Economics - - Renewable Resources and Conservation - - - Water
    • C11 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods and Methodology: General - - - Bayesian Analysis: General
    • C35 - Mathematical and Quantitative Methods - - Multiple or Simultaneous Equation Models; Multiple Variables - - - Discrete Regression and Qualitative Choice Models; Discrete Regressors; Proportions

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