Optimal prices of an M/G/1 jobshop

This paper concerns the optimal control of input to a FIFO jobshop with a single workstation. The input is jobs for which the processing and delivery times are observable upon arrival. The control is exercised by charging a price for each completed job. The objective is either profit maximization or welfare maximization. The semi-Markov decision processes that maximize the two objectives are studied simultaneously. 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 by a new approach. It is based on the idea of studying the infinite-horizon model directly and it allows the state space and the decision set to be denumerable. Mild assumptions ensure that the opportunity cost is increasing as a function of the work backlog, and increasing and convex as a function of the processing time.