This paper revisits the economic manufacturing quantity (EMQ) model by Cheung and Hausman (Nav. Res. Logist. 44 (1997) 257) from the theoretical point of view, and develops a new stochastic model under somewhat different restrictive assumptions. The optimal policies, the order quantity and safety stock, are derived, respectively, so as to minimize the expected cost per unit time in the steady-state.
When dealing with stocked items, the question regarding how much to order is answered by the economic order quantity (EOQ). Similarly, the economic manufacturing quantity (EMQ) model, which is a natural extension of EOQ, provides the proper production lot size by minimizing the cost components involved, that is, the production cost, the inventory holding cost and the shortage cost if the stockout is permitted. When a machine breakdown takes place in the production phase, however, the basic EMQ model loses its usefulness since the interruption by corrective maintenance operations is aborted. From a practical perspective, the manufacturer should design the production lot from the standpoint of safety, and thus effects of machine breakdown and corrective maintenance in economic production lot sizing decisions should be examined exactly in uncertain environment without reliable manufacturing facilities. In this article, we consider an extended EMQ model with stochastic machine breakdown and repair. To motivate such a modeling, consider the just-in-time (JIT) philosophy. The JIT production whose lot size is required to be small or is ideally equivalent to one, has not been able to be achieved in many manufacturing industries all over the world, since fairly high set-up and machine maintenance costs are needed in practice. In particular, one of the major impediments for the successful operation of such a tightly coupled organization may be formed by breakdowns in bottleneck resources.
Numerous research efforts have been undertaken to extend the manufacturing model subject to stochastic machine breakdowns. Lee and Rosenblatt [1] and [2] focused on the imperfections in the production process and equipment, and determined the optimal EMQ policy and/or inspection schedule. Furthermore, Rosenblatt and Lee [3] analyzed a deteriorating system during the production process. Groenevelt et al. [4] and Ibrahim and Kee [5] developed somewhat different stochastic models from the above ones and investigated the effects of machine breakdowns in the optimal lot size and reorder level decisions in the framework of the EMQ model. Also, Dohi and Osaki [6], Dohi et al. [7], Cheung and Hausman [8], Dohi et al. [9],[10] considered similar production planning models. Their models are based on a typical maintenance model called the age replacement model in Barlow and Proschan [11]. More precisely, Dohi et al. [7] developed an extended EMQ model under the assumption that the lifetime for the production machine obeys a common distribution and derived the optimal lot size which minimizes the expected cost. Cheung and Hausman [8] formulated the optimization problem to determine both the optimal lot size and safety stock, provided that the production rate is equivalent to the demand rate in the normal production phase.
It should be noted that, however, Cheung and Hausman [8] modeled the EMQ process under rather questionable assumptions. This implies that the mathematical treatment is incomplete and at the same time that their model and conclusion cannot be justified. The main purpose of this article is to revisit the economic manufacturing quantity (EMQ) model by Cheung and Hausman [8] from the theoretical point of view, and to develop a new stochastic model under somewhat restrictive assumptions. In order to formulate the expected cost precisely, we assume that the lifetime of the production machine obeys an exponential distribution. This assumption may be more restrictive than the original one by Cheung and Hausman [8], but is required for well-behaved stochastic modelling. This paper is organized as follows. Section 2 describes the EMQ model subject to stochastic machine breakdowns, and introduces the results by Cheung and Hausman [8]. In Section 3, we point out their problems and develop the expected cost function in the improved modelling framework. In Section 4, we derive the optimal lot size and the optimal safety stock minimizing the expected cost, respectively. In addition, the optimization algorithm to seek the optimal lot size and safety stock jointly is developed.
In this article, we have revisited the economic manufacturing quantity (EMQ) model by Cheung and Hausman [8] from the theoretical point of view, and have developed a new stochastic model under somewhat different restrictive assumptions. We have derived an optimization algorithm to calculate the optimal preventive maintenance and safety stocks which minimize the expected cost per unit time in the steady-state for the EMQ model with stochastic machine breakdown. Since the lifetime has been assumed to be distributed exponentially, it has been shown that the resulting preventive maintenance schedule should be the switching policy. From this simple structure, we have derived the joint optimization algorithm for the optimal preventive maintenance and safety stocks.
In numerical examples, we have calculated the optimal policy (m∗,s∗) under the assumption that the repair time is according to the Weibull distribution with IHR property. It has been shown that the optimal safety stock and its associated expected cost value monotonically increase as the failure rate becomes larger. This can be intuitively understood in our daily experiments. On the other hand, when the shape parameter of the repair time distribution increases with the same mean repair time, then the optimal safety stock and the minimum expected cost increase and decrease, respectively. Since this implies that IHR property of the repair time distribution tends to be remarkable, it can be seen that the hazard rate property of the repair time distribution plays an important role to design the optimal control of preventive maintenance schedule and safety stocks in an unreliable manufacturing environment.
