Solution of a Purchase-Storage Programme
E. M. L. Beale, G. Morton, A. H. Land · Journal of the Operational Research Society · 1958
THE problem discussed in this paper was suggested to one of the authors by Dr. J. Bethel, Forestry Department, North Carolina State College. It can be briefly described as follows. A certain commodity is needed in specified amounts in each of a number of time periods. In each of these periods the commodity may be obtained either by purchase, or through storage from earlier periods. At any one time, the commodity may be bought from one or more of several sources, at prices which may vary from one source to another. In each time period, the maximum amount available from each source, the capacity of the storage facilities, and the cost per unit of commodity stored over that period, are known. The cost of storage is presumed to include handling and interest charges as well as rent of the warehouse and losses during storage. The amounts available from each source, their costs, the storage capacity and its cost, can all vary from one period to the next. The time scale is discrete, in that it is assumed that all purchases for a given period are made at the same point in time, the storage costs being measured from that instant to the next purchasing point. It is desired to arrange the programme of purchase and storage in such a way as to minimize total cost and to supply the required amounts of the commodity during the period under consideration, while not exceeding purchase and storage facilities. This problem can easily be put into linear programming form and, because of its simple nature, a remarkably straightforward and rapid method of numerical solution is possible, which does not need elaborate equipment. The method is also applicable in the more general case where the cost function of the amounts bought and stored is convex rather than being piecewise linear. Formal proofs and a numerical example are given below. In the case where storage capacity is unlimited the solution reduces to that given by Johnson.2