Some operations on dynamic programmings with one-dimensional state space

Abstract

From the practical viewpoint, dynamic programming (DP), namely, Bellman’s Principle of Optimality [l] has been applied in engineering, economics and operations research. On the other hand, its theoretical aspect has been analyzed by Mitten [7], Nemhauser [8] and others. Nevertheless, it seems that many research workers have paid their attention to “a” DP itself in the individual case. In this paper we are concerned with a class of dynamic programmings with one-dimensional state space. The n-th feasible action space A,(s,) at state s, is assumed to be independent of s, , namely, A,(s,) = A, for all s, . We focus our attention on the relationship between these DP’s. One-dimensionality of state space enables us to develop an algebraic theory of DP. Such algebraic or automaton-like operations as inverse, reversal, composition, concatenation, maximum and minimum are introduced on these DP’s. Section 2 defines the fundamental operations on the class of all strictly increasing functions from [0, 03) onto [0, cc). The properties concerning these operations suggest results of Section 3. Our main results are Inverse Theorem, Reverse Theorem and Decomposition Theorem: The first is a version of author’s Inverse Theorem [2-6]. The second is new. It has a broard applicability. The third is a refinement of Nemhauser’s decomposition or Mitten’s composition. Another results are interesting algebraic relations between DP’s generated by the above operations (Section 3). Illustrating a simple DP, the last section applies Inverse Theorem and Reverse Theorem.