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Separation, dimension, and facet algorithms for node flow polyhedra

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Abstract

Production planning problems such as an Available to Promise (ATP) model of Chen et al. (2002) can involve material compatibility constraints that specify when components from various suppliers can be feasibly assembled into a final product. In a companion paper to Chen et al. (2002), Ball et al. (2003) showed that in many cases such constraints can be modeled as the set of feasible source-sink flows through an acyclic network. The flow through a node is the sum of the flows on all paths containing it. The number of paths is often exponential in the number of nodes, and so it is more computationally tractable to consider the set of node flows in place of that of path flows. Here nodes represent components and paths represent product configurations. In the context of NP hard Mixed Integer Programming models such as the ATP model, when the description of the set of node flows is too complicated to be explicitly written out, the material compatibility constraints can be handled in a cutting plane framework by using a separation algorithm. Ball et al. characterized the polyhedron of node flows for some special cases. We extend this work in various practical and theoretical directions: we allow arbitrary directed networks, we allow both upper and lower bounds on flows, we characterize which valid inequalities are facets, we give fast algorithms for separation, violation, and dimension, and we put all the pieces together into an algorithm for facet-separation. All algorithms are very efficient, as they are based on max flow and min-cost flow subroutines.

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Authors and Affiliations

  1. Zuse Institute Berlin, Takustr. 7, 14195, Berlin, Germany

    S. Thomas McCormick & Maurice Queyranne

  2. Sauder School of Business, University of British Columbia, 2053 Main Mall, Vancouver, BC, V6T 1Z2, Canada

    Maren Martens

Authors
  1. Maren Martens
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  2. S. Thomas McCormick
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  3. Maurice Queyranne
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Corresponding author

Correspondence to S. Thomas McCormick.

Additional information

Dedicated to Alan J. Hoffman.

Research of all authors was supported in part by research grants from the Natural Sciences and Engineering Research Council of Canada (NSERC).

S. Thomas McCormick, Maurice Queyranne: This research was initiated during a visit of M. Queyranne to the IMA Special Year on Optimization, whose support is gratefully appreciated; some research was done while at Laboratoire Leibniz, Institut IMAG, 38031 Grenoble cedex.

Maren Martens: This research was initiated during a postdoc at the Sauder School of Business, UBC, whose support is gratefully appreciated.

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Martens, M., McCormick, S.T. & Queyranne, M. Separation, dimension, and facet algorithms for node flow polyhedra. Math. Program. 124, 317–348 (2010). https://doi.org/10.1007/s10107-010-0378-2

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  • DOI: https://doi.org/10.1007/s10107-010-0378-2

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