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).

No abstract

A system (e.g., a motor vehicle) must operate for t units of time. A certain component (e.g., a battery) is essential for its operation and must be replaced each time it fails. There are n types of replacement components. A type i replacement costs C i and has a random life with distribution depending on i. There is no salvage value associated with the particular component in use when the system terminates. The problem is to assign the initial component and subsequent replacements from among the n types so as to minimize the total expected cost of providing an operative component for the t units of time. This paper treats this problem when the life distributions are exponential for each type and when t is fixed or has a truncated exponential distribution. Related problems are also considered.

No abstract

We consider an N-server queue with arbitrary arrivals and service times which are random but with differing rates for different servers. Customers arriving when all servers are occupied do not enter the system. We show that the policy of always assigning an arrival to that free server whose service rate is largest (smallest) stochastically minimises (maximises) the number in the system. We then show that in a particular component-repair context with exponential repair times the policy of repairing failed components with the smallest failure rate stochastically maximises the number of working components.