Optimal Allocation for a Class of Finite Horizon Processes

The purpose of this paper is to derive a fundamental frame of reference for the analysis of a class of multistage decision problems that have been separately treated in the literature. It is hoped that exposition of the basic structure of these problems in a general setting will provide a convenient point of departure for future investigations. Specific examples of problems in this class are the capital budgeting problem of Dorfman [Dorfman, R. 1963. Regional allocation of investment: Comment. Quart. J. Econom. (Feb.)], the warehouse problem of Dreyfus [Dreyfus, S. 1957. An analytic solution of the warehouse problem. Management Sci. 4(1, Oct.)], the price speculation model of Arrow and Karlin [Arrow, K. J., S. Karlin. 1958. Price speculation under certainty. Chapter 13 in Arrow, K. J., L. Hurwicz, H. Uzawa. Studies in Linear and Non-linear Programming. Stanford University Press, Stanford, Calif.] and the work on multistage linear programs by Dantzig [Dantzig, G. B. 1959. On the status of multistage linear programming problems. Management Sci. 2(1, Oct.)]. The special results elucidating the structure of optimal solutions to these problems may all be obtained by application of the Linearity Theorem proven in this paper.