Optimal Adjustment of Production Facilities in a Stochastic Production Process

Abstract

A piece of equipment requiring occasional but intricate adjustments or maintenance actions to cut back increasing runtime cost is considered. Immediately after a reset of the system has been performed it starts out with a stochastic initial runtime cost rate, superimposed by a deterministically increasing runtime cost rate thereafter. On the other hand interruption of a runtime phase for maintenance actions is time and cost consuming. Each down-time phase is composed of a stochastic recovery interval and a runtime dependent readjustment interval; readjustment cost consist of a stochastically varying part for material and labour as well as the opportunity cost collected during down-time.

In part I the problem of optimally scheduling a sequence of readjustments is embedded within a compound renewal process, the cost structure of which forms a cumulative process. Using renewal-theoretic results, necessary and sufficient conditions for the existence of a unique optimal operating cost rate to trigger a readjustment phase in order to minimize expected total cost per unit time (per unit runtime) are derived in part II.

In part III the case of a linearly structured problem is discussed and the complete analytic solution is presented. The maintenance problem is generalized in part IV for non-linear opportunity cost per unit down-time and run-time dependent maintenance efforts. By simple reformulations it can be shown that this even case is covered by the former analysis essentially.