The scheduling of lot sizes in multistage production environments is a fundamental problem in many Material Requirements Planning Systems. Many heuristics have been suggested for this problem with varying degrees of success. Research to date on obtaining optimal solutions has been limited to small problems. This paper presents a new formulation of the lot-sizing problem in multistage assembly systems which leads to an effective optimization algorithm for the problem. The problem is reformulated in terms of "echelon stock" which simplifies its decomposition by a Lagrangean relaxation method. A Branch and Bound algorithm which uses the bounds obtained by the relaxation was developed and tested. Computational results are reported on 120 randomly generated problems involving up to 50 items in 15 stages and up to 18 time periods in the planning horizon.MRP systems, lot sizing in MRP systems, multistage assembly systems
Lot sizing of products that have complex bills of materials plays an important role in the efficient operations of modern manufacturing and assembly processes. In this paper we develop algorithms for optimal lot sizing of products with a complex product structure. We convert the classical formulation of the general structure problem into a simple but expanded assembly structure with additional constraints, and solve the transformed problem by a branch-and-bound based procedure. The algorithm uses a Lagrangean relaxation and subgradient optimization procedure to generate tight lower bounds on the optimal solutions. In computational experiments, a code based on this method was able to solve single end product problems with up to 40 stages in the product structure. The model is extended to handle problems with multi-end items in the product structure, but with less favorable computational results.
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