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

The textbook Dantzig-Wolfe decomposition for the Capacitated Lot Sizing Problem (CLSP), as already proposed by Manne in 1958, has an important structural deficiency. Imposing integrality constraints on the variables in the full blown master will not necessarily give the optimal IP solution as only production plans which satisfy the Wagner-Whitin condition can be selected. It is well known that the optimal solution to a capacitated lot sizing problem will not necessarily have this Wagner-Whitin property. The columns of the traditional decomposition model include both the integer set up and continuous production quantity decisions. Choosing a specific set up schedule implies also taking the associated WagnerWhitin production quantities. We propose the correct Dantzig-Wolfe decomposition reformulation separating the set up and production decisions. This formulation gives the same lower bound as Manne’s reformulation and allows for branch-and-price. We use the Capacitated Lot Sizing Problem with Set Up Times to illustrate our approach. Computational experiments are presented on data sets available from the literature. Column generation is speeded up by a combination of simplex and subgradient optimization for finding the dual prices. The results show that branch-and-price is computationally tractable and competitive with other approaches. Finally, we briefly discuss how this new Dantzig-Wolfe reformulation can be generalized to other mixed integer programming problems, whereas in the literature, branch-and-price algorithms are almost exclusively developed for pure integer programming problems. 5001-6182 Business 5201-5982 Business Science Library of Congress Classification (LCC) QA 1-939 Mathematics M Business Administration and Business Economics M 11 R 4 Production Management Transportation Systems Journal of Economic Literature (JEL) C 61 Programming models 85 A Business General 260 K 240 B Logistics Information Systems Management European Business Schools Library Group (EBSLG) 250 A Mathematics Gemeenschappelijke Onderwerpsontsluiting (GOO) 85.00 Bedrijfskunde, Organisatiekunde: algemeen 85.34 85.20 Logistiek management Bestuurlijke informatie, informatieverzorging Classification GOO 31.80 Toepassingen van de wiskunde Bedrijfskunde / Bedrijfseconomie Bedrijfsprocessen, logistiek, management informatiesystemen