TO BE OPTIMAL IN AN UNRELIABLE MANUFACTURING SYSTEM

Though it may be argued that inventories are a buffer against uncertainty. and that therefore one should strive to maintain a posi- tive inventory whenever there is any uncertainty. we show that there are ranges of values for mean time between failures and mean repair times for which zero-inventory policies are exactly optimal. This provable optimality is initially surprising since it runs counter to the argument made above : instead it reinforces the case for zero- inventory policies which is currently made on the grounds that it enforces a healthy discipline on the entire manufacturing process. In this paper we show that aside from the above mentioned consequence. there are quantifiable ranges of parameter values such as mean time between failures and mean repair times etc., for which such policies are provably optimal. We consider a simple model of a failure prone manufacturing system producing only one commodity. The time between failures is random and modeled as an exponentially distributed random vari- able with mean l/q, while the repair time is exponentially distri- buted with mean l/qo. Though the commodity is in constant demand at a rate d units per unit time. the manufacturing system cannot produce the commodity at all when it is broken down. When functioning, the system can produce at any rate up to a maximum of r unitsltime. The total inventory may at times become negative, which corresponds to a backlog. and while so it is assessed a cost of c- dollars per unit commodity per unitltime. Positive inventories are also assessed a cost at a rate of c+ dollars per unit commodity per unit/time. Let x(t) = total inventory level at time t. and x+ = max (0.x) be its positive part, while x- = max (O.-x) is its negative part. Let I (t be 0 or 1 respectively depending on whether the system is functioning or down, and suppose that u(t) is the production rate at time t . (Clearly u (t = 0 if the manufacturing system is down at time t : while its value is to be chosen from (O , r 1 when the manufacturing system is functioning at time t ). Note that x (t ) = (U (X )-d Ids