Determining Optimal Reorder Intervals in Capacitated Production-Distribution Systems

• Published in: Management science Vol. 34; no. 8; pp. 938 - 958
• Main Authors: Jackson, Peter L, Maxwell, William L, Muckstadt, John A
• Format: Journal Article
• Language: English
• Published: Linthicum, MD INFORMS 01.08.1988; Institute of Management Sciences; Institute for Operations Research and the Management Sciences
• Series: Management Science
• Subjects: Algorithms; Applied sciences; Capacity utilization; Carrying costs; Distribution; Exact sciences and technology; Iterative solutions; Lagrange multipliers; lot-sizing; Mathematical intervals; Mathematical vectors; Musical intervals; nonlinear programming applications; Operational research and scientific management; Operational research. Management science; Optimal solutions; Production management; production scheduling; Production systems; Scheduling, sequencing; Non linear programming; Integrated planning; Algorithm; Distribution system; Production control; Production
• ISSN: 0025-1909; 1526-5501
• DOI: 10.1287/mnsc.34.8.938

The problem of determining consistent and realistic reorder intervals in complex production-distribution environments was formulated as a large scale, nonlinear, integer programming problem by Maxwell and Muckstadt (Maxwell, W. L., J. A. Muckstadt. 1985. Establishing consistent and realistic reorder intervals in production-distribution systems. Oper. Res. 33 (6, November–December) 1316–1341.). They show how the special structure of the problem permits its solution by a standard network flow algorithm. In this paper, we review the Maxwell-Muckstadt model, provide necessary and sufficient conditions that characterize the solution, and show that the optimal partition of nodes in the production-distribution network is invariant to an arbitrary scaling of the set-up and holding cost parameters. We consider two capacitated versions of the model: one with a single constrained work center, and the other with multiple constrained work centers. For single constraint problems, the invariance corollary provides a simple closed-form solution. For the multiple work center problem, the invariance corollary is exploited in the development of a Lagrange multiplier method of solution. The technique is illustrated by means of a small example problem and a problem taken from a real industrial setting.