Abstract
Despite 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 two-period 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.
Highlights
Lot-sizing is an important part of the planning process in many manufacturing environments
We continue this line of research by investigating how efficient a local cuts approach is in the context of multi-item capacitated lot-sizing problems
One of the advantages of having such small problems is that we might obtain the full description of the convex hull using software like cdd (Fukuda 2014), which is currently investigated in a companion paper (Doostmohammadi et al 2016)
Summary
Lot-sizing is an important part of the planning process in many manufacturing environments. The interested reader is referred to Belvaux and Wolsey (2001) for modeling and reformulation issues and to Pochet and Wolsey (2006) for an excellent, thorough review of lot-sizing problems and solution methods used In spite of this extensive research, the mathematical programming community has focused mainly on single-item problems, and results for multi-item problems are rather limited. A more general approach applicable to mixed-integer programming (MIP) problems is first suggested by Boyd (1994), and the more recent work of Chvátal et al (2013) extended the concept of “local cuts” to general MIP problems through a sophisticated methodology including tilting the cuts to increase their effectiveness and addressing some of the issues inherent in the precision of coefficients We continue this line of research by investigating how efficient a local cuts approach is in the context of multi-item capacitated lot-sizing problems. We conclude with a discussion of possible extensions and generalizations
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