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

Abstract

D espite the significant attention they have drawn, big-bucket lot-sizing problems remain notoriously difficult to solve. Previous literature contained results (computational and theoretical) indicating that what makes these problems difficult are the embedded single-machine, single-level, multiperiod submodels. We therefore consider the simplest such submodel, a multi-item, two-period capacitated relaxation. We propose a methodology that can approximate the convex hulls of all such possible relaxations by generating violated valid inequalities. To generate such inequalities, we separate two-period projections of fractional linear programming solutions from the convex hulls of the two-period closure we study. The convex hull representation of the twoperiod closure is generated dynamically using column generation. Contrary to regular column generation, our method is an outer approximation and can therefore be used efficiently in a regular branch-and-bound procedure. We present computational results that illustrate how these two-period models could be effective in solving complicated problems.

Progressive Selection Method for the Coupled Lot-Sizing and Cutting-Stock Problem

“…Parameters (α, β) were respectively set to (4, 2), (3, 2), (2, 1), (3,1), (4,3) and (4, 1) to derive the initial population G y and to solve subproblems. The size of G y was set to 6.…”

“…It is also interesting to apply the two-period convex hull closures ( [1]) and the similar horizon decomposition approach ( [21]) to improve lower bounds for the LS-CS problem.…”

A Horizon Decomposition Approach for the Capacitated Lot-Sizing Problem with Setup Times

“…5 neighborhood search combined with strong formulations and report results on new hard instances. Finally, Akartunal and Miller (2012) give insights on which lot-sizing substructures are computationally challenging, and Akartunali et al (2014) use column generation to generate cuts from a 2-period relaxation of CLST. In the latter work, the authors use several distance functions, by which they are able to generate valid inequalities that cut-off the linear programming relaxation solution, when this solution has a positive distance from a predefined 2-period lot-sizing set.…”

“…Moreover, an automatic procedure that selects the member of a family of decompositions that strikes the best balance between bound quality and CPU time performance is an area that has not been explored yet on its entirety. Finally, an alternative scheme that generates cutting planes from problem substructures, such as the one devised by Akartunali et al (2014) is a promising direction that generates cuts from lower dimensional projections of the initial formulation and yet does not require an inner approximation of the feasible region, which is the case for column generation based approaches.…”

MIP approaches for a lot sizing and scheduling problem on multiple production lines with scarce resources, temporary workstations, and perishable products

“…In addition to the fact that the LSP is N P-hard (James & Almada-Lobo, 2011), the real-world instances of the problem are very challenging from a computational perspective. Moreover, even the pure lot sizing problem with multiple items sharing resources is often computationally challenging (Akartunalı, Fragkos, Miller, & Wu, 2016;Doostmohammadi & Akartunalı, 2018), and sequencing lot sizes necessitates novel approaches (Suerie & Stadtler, 2003) due to further complications, as also further discussed in the recent lot sizing review of Brahimi, Absi, Dauzère-Pérès, and Nordli (2017). Therefore, many researchers proposed heuristic approaches to solve the LSP, and MIP heuristics in particular achieved promising results for problems arising in various practical applications (Sel & Bilgen, 2014;Toledo, da Silva Arantes, Hossomi, França, & Akartunalı, 2015).…”