Programming of Economic Lot Sizes

TLDR

The study addresses a machine‑shop planning problem, differing from prior work by incorporating indivisible setup costs. The paper aims to minimize overtime labor while meeting delivery schedules and capacity limits in a multi‑item machine‑shop planning problem. The authors propose a linear‑programming approximation where each activity’s input coefficients encode setup costs at unit or zero levels. The LP solution yields binary activity levels except for at most as many items as capacity constraints, implying accurate approximations when many items are produced relative to constraints.

Abstract

This paper studies the planning problem faced by a machine shop required to produce many different items so as to meet a rigid delivery schedule, remain within capacity limitations, and at the same time minimize the use of premium-cost overtime labor. It differs from alternative approaches to this well-known problem by allowing for setup cost indivisibilities. As an approximation, the following linear programming model is suggested: Let an activity be defined as a sequence of the inputs required to satisfy the delivery requirements for a single item over time. The input coefficients for each such activity may then be constructed so as to allow for all setup costs incurred when the activity is operated at the level of unity or at zero. It is then shown that in any solution to this problem, all activity levels will turn out to be either unity or zero, except for those related to a group of items which, in number, must be equal to or less than the original number of capacity constraints. This result means that the linear programming solution should provide a good approximation whenever the number of items being manufactured is large in comparison with the number of capacity constraints.