Valid inequalities for proportional lot-sizing and scheduling problem with fictitious microperiods

Abstract

This paper presents new extended formulations of mixed-integer linear programming (mip) models for small bucket problems. Such problems allow for one machine set-up operation at most during each time period. To ensure a high-quality solution despite this restrictive assumption, real periods (macroperiods) are usually split into several short fictitious microperiods with non-zero demand only at the end of the last microperiod of each macroperiod. The proposed model formulations are based on two properties of such a demand pattern. First, within each macroperiod, it is useful to have one start-up for each product at most. Second, the sequence of lots within a macroperiod has only negligible impact on solution quality, especially for short microperiods.The comprehensive experiments have shown that the computational effort may be significantly reduced with the help of additional macroperiod binary set-up variables, which allow for one lot at most for each product and macroperiod, adding an aggregated decision level in the branch and bound search tree, and allowing us to set inventory lower bounds only for macroperiods. Compared to the best-known model formulation, these new valid inequalities reduce the optimization time of instances with ten products by 98% on average and the number of nodes and iterations by nearly 89% and 94%, respectively.Such large savings in the computational effort make the small bucket models more competitive against the large bucket models and encourage the modeling of multi-level systems with the help of small buckets. The time horizon split in many short microperiods enables the modeling of short lead times between levels, which in turn enables short production cycles and low work-in-process.