Economic Lot Scheduling Problem - Model Formulation

Model Formulation

The classic ELSP is concerned with scheduling the production of several products on a single machine in order to minimize the total costs incurred (which include setup costs and inventory holding costs).

We assume a known, non-varying demand for the N products (for example, there might be N=3 products and customers require 7 items a day of Product 1, 5 items a day of Product 2 and 2 items a day of Product 3). Customer demand is met from inventory and the inventory is replenished by our production facility.

A single machine is available which can make all the products, but not in a perfectly interchangeable way. Instead the machine needs to be set up to produce one product, incurring a setup cost and/or setup time, after which it will produce this product at a known rate . When it is desired to produce a different product, the machine is stopped and another costly setup is required to begin producing the next product. Let be the setup cost when switching from product i to product j; and let be the setup time.

In addition, an inventory cost is charged based on average inventory level of each item.

(To give a very concrete example, the machine might be a bottling machine and the products could be cases of bottled apple juice, orange juice and milk. The setup corresponds to the process of stopping the machine, cleaning it out and loading the tank of the machine with the desired fluid. This product switching must not be done too often or the setup costs will be large, but equally too long a production run of apple juice would be undesirable because it would lead to a large inventory cost for unsold cases of apple juice and perhaps stock-outs in orange juice and milk. The ELSP seeks the optimal tradeoff between these two extremes.)