A New Dantzig-Wolfe Reformulation and Branch-and-Price Algorithm for the Capacitated Lot Sizing Problem with Set Up Times

37 Pages Posted: 27 May 2003

See all articles by Zeger Degraeve

Zeger Degraeve

London Business School

Raf Jans

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

Date Written: April 2003 3,

Abstract

The textbook Dantzig-Wolfe decomposition for the Capacitated Lot Sizing Problem (CLSP),as already proposed by Manne in 1958, has an important structural deficiency. Imposingintegrality constraints on the variables in the full blown master will not necessarily give theoptimal IP solution as only production plans which satisfy the Wagner-Whitin condition canbe selected. It is well known that the optimal solution to a capacitated lot sizing problem willnot necessarily have this Wagner-Whitin property. The columns of the traditionaldecomposition model include both the integer set up and continuous production quantitydecisions. Choosing a specific set up schedule implies also taking the associated Wagner-Whitin production quantities. We propose the correct Dantzig-Wolfe decompositionreformulation separating the set up and production decisions. This formulation gives the samelower bound as Manne's reformulation and allows for branch-and-price. We use theCapacitated Lot Sizing Problem with Set Up Times to illustrate our approach. Computationalexperiments are presented on data sets available from the literature. Column generation isspeeded up by a combination of simplex and subgradient optimization for finding the dualprices. The results show that branch-and-price is computationally tractable and competitivewith other approaches. Finally, we briefly discuss how this new Dantzig-Wolfe reformulationcan be generalized to other mixed integer programming problems, whereas in the literature,branch-and-price algorithms are almost exclusively developed for pure integer programmingproblems.

Keywords: lot sizing, Dantzig-Wolfe decomposition, branch-and-price, Lagrange relaxation, mixed-integer programming

JEL Classification: M, M11, R4, C61

Suggested Citation

Degraeve, Zeger and Jans, Raf, A New Dantzig-Wolfe Reformulation and Branch-and-Price Algorithm for the Capacitated Lot Sizing Problem with Set Up Times (April 2003 3,). Available at SSRN: https://ssrn.com/abstract=411647

Zeger Degraeve

London Business School ( email )

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United Kingdom

Raf Jans (Contact Author)

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

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Room T08-21
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Netherlands

HEC Montreal ( email )

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

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