Optimal Control of Production Sequences: A Continuous Parameter Analysis

In a production sequence the presence of random variations in the completion times of successive jobs causes deviations from planned schedules. The probability is high that such deviations persist over long periods, so managerial intervention will generally be required to restore schedule integrity. This paper treats two problems in the optimal control of such sequences, using a continuous work formulation in which the uncontrolled schedule deviation is a Wiener process. The first problem concerns long run production in the presence of a piecewise-linear operating cost rate during off-schedule production that can be expedited or halted at extra cost, with the latter action involving a fixed cost. An explicit average cost minimizing policy is developed and justified. The second problem concerns batch production with a piecewise-linear penalty cost for off-schedule batch completion. In this problem the controller can expedite production only. The existence of a transient single critical-number optimal policy is proved, and bounds on the critical numbers are developed. A simple but remarkably accurate formula for the critical number curve is developed using numerical methods.