Abstract
AbstractTwo-level, multi-item, multi-period-capacitated dynamic lot-sizing problem with inclusions of backorders and setup times, TL_CLSP_BS, a well-known NP-hard problem, is solved using a novel procedure. Lagrangian relaxation of the material balance constraint reduces TL_CLSP_BS to a single-constraint continuous knapsack problem. Reduced problem is solved using bounded variable linear programs (BVLPs). We obtain promising bounds, which provides a better start to the branch-and-bound procedure. Limited empirical investigations are carried out on four problem sizes. In terms of computational time, the developed procedure is efficient than the CPLEX solver of GAMS. Further, while GAMS could not solve the largest problem size considered here, our procedure could solve the same in around one second time. This clearly highlights the efficacy of the developed procedure. The solution technique is applicable to any problem structure, which is reducible to the application of BVLPs.
Highlights
To create a production plan over a finite number of periods, where the demand has to be fulfilled while facing finite capacities is known to be a difficult problem
Computational time and gap: procedure in Table 4, we show the efficacy of the Procedure and Minimum cost network flow (MCF) to solve the same problem sets which are solved in GAMS (Table 3)
Notes: tgmsbc: Computational time taken by GAMS using branch and cut. tgmsbb: Computational time taken by GAMS using branch and bound. tproin: Computational time taken by procedure, without branch and bound. tprobb: Computational time taken by procedure, with branch and bound. tmcfin: Computational time taken by MCF, without branch and bound. tmcfbb: Computational time taken by MCF, with branch and bound
Summary
To create a production plan over a finite number of periods, where the demand has to be fulfilled while facing finite capacities is known to be a difficult problem It is even harder in case of a multilevel product structure, where items are related to each other by successor–predecessor relationship, as per the bill of material. For the dynamic multi-item multi-level-capacitated lot-sizing problem (ML_CLSP_BS) considered here, the following assumptions apply (Maes, McClain, & Van Wassenhove, 1991):. Formulation of multi-level-capacitated lot-sizing problem with considerations of backorders and setup times (ML_CLSP_BS) is given as follows:. Equation 1 gives the objective function intending to minimize the production, setup, inventory, and backorders cost, summed over all ite∑ms and time periods. Equation 2 is relaxed using Lagrangian procedure to solve a two-level version of the problem
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