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A multi-item production planning model with setup times: algorithms, reformulations, and polyhedral characterizations for a special case

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Abstract.

 We study a special case of a structured mixed integer programming model that arises in production planning. For the most general case of the model, called PI, we have earlier identified families of facet–defining valid inequalities: (l, S) inequalities (introduced for the uncapacitated lot–sizing problem by Barany, Van Roy, and Wolsey), cover inequalities, and reverse cover inequalities. PI is 𝒩𝒫–hard; in this paper we focus on a special case, called PIC. We describe a polynomial algorithm for PIC, and we use this algorithm to derive an extended formulation of polynomial size for PIC. Projecting from this extended formulation onto the original space of variables, we show that (l, S) inequalities, cover inequalities, and reverse cover inequalities suffice to solve the special case PIC by linear programming. We also describe fast combinatorial separation algorithms for cover and reverse cover inequalities for PIC. Finally, we discuss the relationship between our results for PIC and a model studied previously by Goemans.

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Authors and Affiliations

  1. University of Wisconsin, Department of Industrial Engineering, Madison, WI 53706-1572., , , , , , US

    A.J. Miller

  2. Georgia Institute of Technology, School of Industrial and Systems Engineering, Atlanta, GA 30332-0205, , , , , , US

    G.L. Nemhauser & M.W.P. Savelsbergh

Authors
  1. A.J. Miller
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  2. G.L. Nemhauser
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  3. M.W.P. Savelsbergh
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Additional information

Received: December 13, 2000 / Accepted: December 13, 2001 Published online: October 9, 2002

RID="★"

ID="★" Some of the results in this paper have appeared in condensed form in Miller et al. (2001).

Key words. mixed integer programming – polyhedral combinatorics – production planning – capacitated lot–sizing – fixed charge network flow – setup times

This paper presents research results of the Belgian Program on Interuniversity Poles of Attraction initiated by the Belgian State, Prime Minister's Office, Science Policy Programming. The scientific responsibility is assumed by the authors.

This research was also supported by NSF Grant No. DMI-9700285 and by Philips Electronics North America.

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Miller, A., Nemhauser, G. & Savelsbergh, M. A multi-item production planning model with setup times: algorithms, reformulations, and polyhedral characterizations for a special case. Math. Program., Ser B 95, 71–90 (2003). https://doi.org/10.1007/s10107-002-0340-z

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  • DOI: https://doi.org/10.1007/s10107-002-0340-z

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