A sequential allocation model with random opportunities

Abstract

The present work debits with an investment model where the investment op­portunities are random. VVe have initially A' units of a resource (capital or fertilizer or energy, etc) available for investment. At certain times // — 0. 1...., Ar — ] an opportunity to invest will occur with probability p. As soon as an opportunity arises, we must decide how much of our available resource to invest. If we invest «, then we obtain the profit n(u). the amount a then becomes unavailable for further investment. If no investment opportunity arises, we obtain c (if c > 0) or must pay — c (if c < 0), e.g. as the management cost per time unit. The problem is to decide how much to invest at each opportunity so as to maximize the total expected profit over the N time periods. N is called the horizon. This model has been investigated (with c = 0, = 1 and Vo = 0) by Derman/Lieberman/Ross (1975), cited as D/L/R. The resource may be measured in discrete units or in real numbers. In another model (called the "con tin nous-time case" by D/L/R.) the investment opportunities arise at random times, namely according to a renewal process. VVe now give a, survey on the contents of tin; present work. To introduce the reader in this lield of mathematics we present in detail in the first chapter, what is dynamic programming and its intuitive background. Also we talk about, the advantages of the use of (personal) computer in dynamic programming. The use of PC’s played also an important role in the present work. We wrote numerous programs in 1'UK.BO-PASCAL VERSION 6.0 (cf. Appendix C) for illustration and control of theoretical results and for finding conjectures about the structure of the solut ion. In the second chapter we collect some auxiliary results from mathematics and dynamic program ini ng, without proofs. 'The purpose of this exposition is to be helpful in the main proofs. Mainly the proofs use the induction principle, combined with the value