Integrating Equipment Investment Strategy with Maintenance Operations under Uncertain Failures

Abstract

This paper studies the issue of coordinating equipment maintenance operations with capital investment strategy in the presence of random equipment failures. This problem represents an important extension of the celebrated Kamien and Schwartz (KS) paper published in Management Science. The traditional KS approach is to formulate the problem as a deterministic optimal control problem with the probability of machine failure as the state variable. Consequently, a deterministic policy is derived. As a major departure from the KS approach, we explicitly model the underlying stochastic process of machine failures. Our analysis of the stochastic dynamic programming model offers new insights into the problem. Under a long planning horizon with a limited replacement opportunity, each individual machine serves as a revenue generator and contributes a significant amount to the profit of the system. On the other hand, when the replacement budget is quite generous in a relatively short planning horizon, adding one extra machine only helps as a backup for unexpected failures of the machines purchased before it. An interesting result derived from this comparison is that a deterministic policy turns out to be optimal for the former, while a state-contingent policy must be applied to the latter. In other words, the deterministic KS approach does not work when a chain of machine replacement is considered. We further discuss the implications of the discount rate, productivity deterioration, learning, decision delay, and technology advancement on the optimal policy.

Keywords: Maintenance and replacement, optimal control, impulse control, equipment investment; stopping time; variational inequality, reliability, hazard rate, the Kamien-Schwartz model, dynamic programming, chain of machines model

JEL Classification: M1, M2, C61, D24