Optimal prices of a job shop with a single work station: a discrete time model

Abstract

The topic is the optimal control of input to a discrete-time FIFO job shop with a single work station. The input is jobs with known processing times. The control is exercised by charging a price for each completed job. The objective is either profit maximization or welfare maximization. The Markov decision processes that maximize the two objectives are studied simultaneously. The decision points are the equally spaced epochs in which customers make requests for service of jobs. At each decision point the state is the shop's work backlog and the action prescribes the price as a function of the processing time. Optimal prices are specified in terms of opportunity costs. The opportunity cost of a job is the expected future loss of earnings caused by having the job submitted to the shop. Results for the cases with and without discounting are established simultaneously. Mild assumptions ensure that the opportunity cost is (i) increasing as a function of the work backlog and (ii) increasing and convex as a function of the processing time. The opportunity costs are discounted sums of difference values. These may be computed by policy-iteration, by value-iteration or by a new fixed-point algorithm.