Abstract
It is shown that when backorders, setup times and dynamic demand are included in capacitated lot sizing problem, the resulting classical formulation and one of the transportation formulations of the problem (referred to as CLSP_BS) are equivalent. And it is shown that both the formulations are “weak” formulations (as opposed to “strong” formulation). The other transportation version is a strong formulation of CLSP_BS. Extensive computational studies are presented for medium and large sized problems. In case of medium-sized problems, strong formulation produces better LP bounds, and takes lesser number of branch-and-bound (B&B) nodes and less CPU time to solve the problem optimally. However for large-sized problems strong formulation takes more time to solve the problem optimally, defeating the benefit of strength of bounds. This essentially is because of excessive increase in the number of constraints for the large sized problems. Hybrid formulations are proposed where only few most promising strong constraints are added to the weak formulation. Hybrid formulation emerges as the best performer against the strong and weak formulations. This concept of hybrid formulation can efficiently solve a variety of complex real life large-sized problems.
Highlights
Introduction and Literature SurveyResearchers have extensively studied the lot sizing problem in last five decades, but finding an effective and practicable solution to this problem in real time remains to be a challenge faced by the production planners in aHow to cite this paper: Verma, M. and Sharma, R.R.K. (2015) Hybrid Formulation of the Multi-Item Capacitated Dynamic Lot Sizing Problem
This work has attempted to cater to a variant of multi-item multi-period capacitated lot sizing problem (CLSP), which considers dynamic demand, backorders and setup times (CLSP_BS)
It was observed that for large sized instances of CLSP_BS, strong formulation took lesser number of branch-and-bound nodes but more CPU time to optimally solve the problem, compared to the weak formulations. The reason of this behavior is an extreme increase in number of constraints for the strong formulation, due to which solving linear programming (LP) relaxation at each B&B node becomes time intensive
Summary
Researchers have extensively studied the lot sizing problem in last five decades, but finding an effective and practicable solution to this problem in real time remains to be a challenge faced by the production planners in a. Multi-item multi-period capacitated lot sizing problem with dynamic demands, backorders and setup times (CLSP_BS) is considered in this work. Apart from the formulation PC, two PT formulations (PTa and PTb) of CLSP_BS are considered in this work; and the relative strength of these three formulations is investigated This is important because it has an impact on the choice of model formulation and the corresponding solution procedure. Chen & Thizy [3] and Barany et al [13] did mathematical comparison of the Lagrangian and linear relaxations for the classical version of the multi-item CLSP They did not consider the variables of backorders and setup times in their model.
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