Optimal Preventive Maintenance of a Production-Inventory System When the Action of “Idling” Is Permissible

Abstract

In this paper we consider a manufacturing system in which an input generating installation (I) supplies a buffer (B) with a raw material, and a production unit (PU) pulls the raw material from the buffer with constant rate d > 0. The capacity of the buffer is equal to K units of raw material. The input rate P is assumed to be a discrete random variable whose possible values belong to the set $\{d,d + 1,\ldots,d + K - x\}$ where $x \in \{ 0,\ldots,K\}$ is the content of the buffer. The installation deteriorates as time evolves and the problem of its preventive maintenance is considered. There are three possible decisions when the installation is at operative condition: (i) the action of allowing the installation to operate, (ii) the action of leaving the installation idle, and (iii) the action of initiating a preventive maintenance of the installation. The objective is to find a policy (i.e., a rule for choosing actions) that minimizes the expected long-run average cost per unit time. The cost structure includes operating costs of the installation, maintenance costs of the installation, storage costs, and costs due to the lost production when a maintenance is performed on the installation and the buffer is empty. Using the dynamic programming equations that correspond to the problem and some results from the theory of Markov decision processes we prove that the average-cost optimal policy initiates a preventive maintenance of the installation if and only if, for some fixed buffer content x, the degree of deterioration of the installation is greater or equal to a critical level i ^∗(x) that depends on x. The optimal policy and the minimum average cost can be computed numerically using the value iteration algorithm. For fixed buffer content x, extensive numerical results provide strong evidence that there exists another critical level $\tilde{i}(x) \leq {i}^{{\ast}}(x)$ such that the average-cost optimal policy allows the installation to operate if its degree of deterioration is smaller than $\tilde{i}(x)$ and leaves the installation idle if its degree of deterioration is greater or equal to $\tilde{i}(x)$ and smaller than i ^∗(x). A proof of this conjecture seems to be difficult.