Determining the optimal production–maintenance policy with inspection errors: using a Markov chain

Abstract

Lee and Park examine the effects of an imperfect production process on the optimal production-inventory policy. They consider an imperfect production process which can go out of control after an exponentially distributed production time and then produce some proportion of defective items. The defective item cost is equivalent to the reworking cost and the warranty cost. Some maintenance-inspection mechanisms are introduced to monitor and restore the process to enhance process reliability and thereby lower the number of defective items produced. However, in many situations, a deteriorating production system possesses an increasing failure rate. In general, such a process will not be as good as new after maintenance but will be younger than its real age. Besides, Lee and Park did not investigate the possibility of imperfect inspection based on consideration of type I and type II errors. In this paper, we consider the effects of general time to shift distributions, two types of process inspection errors and general repair policy on the optimal production/inspection/maintenance policy. A mathematical model representing the expected average cost is developed using a Markov chain to jointly determine the production cycle, process inspection intervals, and maintenance level. An optimal production/inspection/maintenance policy is determined by minimizing the expected average cost. A numerical example is given to illustrate the use of the method. Based on this proposed Markov structure model, some process-related assumptions such as maintenance that can cause the system to go out of control and that the random results of general repairs can also be easily relaxed, and they are described in the section of model extensions. An extended production–maintenance model for a deteriorating production system is established. Scope and purpose There are numerous production situations which may cause a production facility to shift from an “in-control” state to an “out-of-control” state due to aging of equipment. Once the process shifts to an “out-of-control” state, an increased number of defective items will be produced until a maintenance–inspection is performed to return the process to an “in-control” state. To reduce the risk that a shift to an out of control state will occur, lot sizing, periodic inspection and maintenance of the process are needed. These tasks also enhance the economical operation of the system. In this study, we investigate a production/inspection/maintenance model under the assumptions that the production process is subject to random deterioration, and that the process can be monitored through inspections, so that the lot size, inspections and maintenance policy are subject to control. Uncertainty probabilities associated with inspections are also considered, i.e., inspection is assumed to be imperfect. Our objective is to determine the optimal production/inspection/maintenance policy under the effect of process inspection errors. A mathematical model which presents the expected average cost is developed. An optimal production/inspection/maintenance policy is found by minimizing the expected average cost. Since this problem arises in many practical situations, the results may be helpful to production managers in developing approaches to control production systems economically.