A Packaging Problem

A number of packaging problems involve optimization of costs subject to constraints imposed by the geometry of the alternatives. One such problem arises in those consumer goods warehousing systems which pack a large number of similar products, differing in size, into individual boxes for warehousing and subsequent shipment to the ultimate consumer. Box cardboard costs and warehouse space costs are minimized if boxes exactly fit each product size; on the other hand, box inventory, handling, and purchase costs are minimized if only one box size is used for all product sizes. The problem is to select the optimum number and sizes of boxes which minimize the total system costs. An integer programming formulation of the problem is given, and the results obtained from the application of a heuristic procedure to a specific problem are described.