2022 年 7 巻 3 号 p. 113-124
When system deterioration information is continuously available, it is convenient to use a continuous stochastic process to characterize the deterioration. The gamma process and inverse Gaussian process are usually used to model monotonic deterioration; however, if the system deterioration is measured frequently, the measured amount of deterioration may not be monotonic. A replacement problem is considered here for a system with a non-monotonic amount of deterioration. The deterioration is characterized by a geometric Brownian motion, which can capture deterioration with an increasing rate over time. The system is inspected at equal time intervals, and the deterioration amount can be specified exactly from the inspection results. An optimal replacement policy is derived in accordance with the age and deterioration amount that minimizes the total expected cost over an infinite time horizon. The cost structure includes replacement cost and operating cost, which both increase with age and the amount of deterioration. The optimization problem is formulated as a Markov decision process and provides a set of conditions with which the structural properties of the optimal replacement policy is characterized. The total expected cost is proven to be monotonically non-decreasing in both age and deterioration amount. Moreover, the optimal replacement policy is shown to be a control limit policy in which the optimal control limits do not monotonically increase with either the amount of deterioration or age. A numerical example is given that illustrates the effects of the cost of replacement and the parameters of geometric Brownian motion on the control limit policy.
