Replacement policy for a single-component machine with limited spares in a finite time horizon

Abstract

In a maintenance scheduling problem, the optimal policy obtained by using Markov Decision Process (MDP) can be in a control-limit form, which facilitates both computation and implementation of the optimal control. The optimality of control limit policy (CLP) in one dimension has been proven in different models, but few researchers have explored the existence of multidimensional CLP. Besides, most models on maintenance optimization aimed to minimize the long-run expected discounted or average cost, assuming no constraint on the supply of spare parts. However, in industrial systems, since the lead time of spare parts ordering is sometimes fixed and rather long compared to the length of inspection interval, the total number of available spares in a certain time horizon can be limited. Therefore, it is useful to study the replacement policy with limited spares and in a finite time horizon. In this paper, we consider a discrete-time model where the component of a single-unit machine deteriorates according to a Markovian process and needs to be replaced correctively or preventively by consuming a limited number of spare parts. The objective is to find the optimal replacement policy which minimizes the expected total cost in a finite time horizon. MDP is applied to formulate and solve the three-dimensional optimization problem. Structural properties, especially twodimensional control-limit form of the optimal solution, are studied under certain assumptions. Numerical experiments are performed to verify and explore the behaviors and performance of the optimal policy. Finally, we conclude that the optimal solution is a CLP in health condition of the currently working part, but not necessarily in the remained number of spares. However, the numerical experiments show that the optimal two-dimensional CLP has almost the same total cost rate as the true optimal policy, which will help make replacement decisions in real-world maintenance problems. Keywords-replacement