No abstract

Suppose there are n men available to perform n jobs. The n jobs occur in sequential order with the value of each job being a random variable X. Associated with each man is a probability p. If a "p" man is assigned to an "X = x" job, the (expected) reward is assumed to be given by px. After a man is assigned to a job, he is unavailable for future assignments. The paper is concerned with the optimal assignment of the n men to the n jobs, so as to maximize the total expected reward. The optimal policy is characterized, and a recursive equation is presented for obtaining the necessary constants of this optimal policy. In particular, if p 1 \leqq p 2 \leqq \cdots \leqq p n the optimal choice in the initial stage of an n stage assignment problem is to use p i if x falls into an ith nonoverlapping interval comprising the real line. These intervals depend on n and the CDF of X, but are independent of the p's. The optimal policy is also presented for the generalized assignment problem, i.e., the assignment problem where the (expected) reward if a "p" man is assigned to an "x" job is given by a function r(p, x).

Certain sets of numbers {a in }, i = 0,..., n, n = 1, 2,..., are known characterize an optimal sequential assignment policy. In this paper the limiting behavior as n -> \infty of the a in 's is studied.

No abstract

Several problems in the optimal control of dynamic systems are considered. When observed, a system is classifiable into one of a finite number of states and controlled by making one of a finite number of decisions. The sequence of observed states is a stochastic process dependent upon the sequence of decisions, in that the decisions determine the probability laws that operate on the system. Costs are associated with the sequence of states and decisions. It is shown that, for the problems considered, the optimal rules for controlling the system belong to a subclass of all possible rules and, within this subclass, the optimal rules can be derived by solving linear programming problems.