Period Decompositions for the Capacitated Lot Sizing Problem with Setup Times

Abstract

W e study the multi-item capacitated lot sizing problem with setup times. Based on two strong reformulations of the problem, we present a transformed reformulation and valid inequalities that speed up column generation and Lagrange relaxation. We demonstrate computationally how both ideas enhance the performance of our algorithm and show theoretically how they are related to dual space reduction techniques. We compare several solution methods and propose a new efficient hybrid scheme that combines column generation and Lagrange relaxation in a novel way. Computational experiments show that the proposed solution method for finding lower bounds is competitive with textbook approaches and state-of-the-art approaches found in the literature. Finally, we design a branch-and-price-based heuristic and report computational results. The heuristic scheme compares favorably or outperforms other approaches.

Local Cuts and Two-Period Convex Hull Closures for Big-Bucket Lot-Sizing Problems

“…We see that the difference of the branch-and-pricebased methods of Pimentel et al (2010) andde Araujo et al (2015) and the strength of the cuts generated by the two-closure procedure is even more profound for this data set. The average gap is just above 1%, more than 40% improvement over Pimentel et al (2010) and 17% improvement over de Araujo et al (2015).…”

“…Although a comparison based on six instances offers limited conclusions, the fact that these instances have been widely used (Miller et al 2000, Jans and Degraeve 2004, Van Vyve and Wolsey 2006, de Araujo et al 2015 allows us to obtain an indication of how the strength of the two-period closure lower bound compares to that of other approaches.…”

“…Next, we report results on a subset of instances utilized from Süral et al (2009 Notes. LR denotes the Lagrange relaxation approach of Süral et al (2009) and PD denotes the period decomposition bound of de Araujo et al (2015). A time limit of 10,800 seconds was imposed.…”

“…The lower bounds of Süral et al (2009) are obtained by solving the Lagrange dual of the facility location formulation using subgradient optimization. We compare the strength of the lower bound obtained by the two-period closure with their approach, and also the period decomposition approach of de Araujo et al (2015). Table 3 summarizes the comparison of these three different methods by presenting the root integrality gaps.…”

“…We excluded the instances for which the gap was simply closed by S inequalities. Table 4 presents the integrality gap obtained by the two-period closure, compared to the gap obtained by Pimentel et al (2010) andde Araujo et al (2015), the two most recent approaches that have considered this data set. We refer the interested reader to the online supplement for detailed results for all instances.…”