Solving Lotsizing Problems on Parallel Identical Machines Using Symmetry Breaking Constraints

24 Pages Posted: 3 Mar 2008

See all articles by Raf Jans

Raf Jans

Erasmus University Rotterdam (EUR) - Rotterdam School of Management (RSM); HEC Montreal

Date Written: September 20, 2006

Abstract

Production planning on multiple parallel machines is an interesting problem, both from a theoretical and practical point of view. The parallel machine lotsizing problem consists of finding the optimal timing and level of production and the best allocation of products to machines. In this paper we look at how to incorporate parallel machines in a Mixed Integer Programming model when using commercial optimization software. More specifically, we look at the issue of symmetry. When multiple identical machines are available, many alternative optimal solutions can be created by renumbering the machines. These alternative solutions lead to difficulties in the branch-and-bound algorithm. We propose new constraints to break this symmetry. We tested our approach on the parallel machine lotsizing problem with setup costs and times, using a network reformulation for this problem. Computational tests indicate that several of the proposed symmetry breaking constraints substantially improve the solution time, except when used for solving the very easy problems. The results highlight the importance of creative modeling in solving Mixed Integer Programming problems.

Keywords: Mixed Integer Programming, Formulations, Symmetry, Lotsizing

Suggested Citation

Jans, Raf, Solving Lotsizing Problems on Parallel Identical Machines Using Symmetry Breaking Constraints (September 20, 2006). ERIM Report Series Reference No. ERS-2006-051-LIS, Available at SSRN: https://ssrn.com/abstract=1101146

Raf Jans (Contact Author)

Erasmus University Rotterdam (EUR) - Rotterdam School of Management (RSM) ( email )

P.O. Box 1738
Room T08-21
3000 DR Rotterdam, 3000 DR
Netherlands

HEC Montreal ( email )

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Montreal, Quebec H2X 2L3
Canada
+1 514 340 6834 (Phone)

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