قطعات یدکی کنترل موجودی با توجه به رشد تصادفی از یک پایه نصب شده

Installed base is a measure describing the number of units of a particular system actually in use. To maintain the performance of the installed units, spare parts inventory control is extremely important and becomes very challenging when the installed base changes over time. This problem is often encountered when a manufacturer starts to deliver a new product to customers and agrees to provide spare parts to replace failed units in the future. To cope with the resulting non-stationary stochastic maintenance demand, a spare parts control strategy needs to be carefully developed. The goal is to ensure that timely replacements can be provided to customers while minimizing the overall cost for spare parts inventory control. This paper provides a model for the aggregate maintenance demand generated by a product whose installed base grows according to a homogeneous Poisson process. Under a special case where the product’s failure time follows the exponential distribution, the closed form solutions for the mean and variance of the aggregate maintenance demand are obtained. Based on the model, a dynamic (Q, r) restocking policy is formulated and solved using a multi-resolution approach. Two numerical examples are provided to demonstrate the application of the proposed method in controlling spare parts inventory under a service level constraint. Simulation is utilized to explore the effectiveness of the multi-resolution approach.

Spare parts inventory control is usually implemented to maximize the availability of a fleet of systems that require service throughout their life cycle. Comprehensive reviews on spare parts inventory control can be found in Kennedy et al., 2002, Nahmias, 1981 and Zipkin, 2005. Existing models can be classified into either a single-echelon inventory model (Cohen, Kleindorfer, Lee, & Pyke, 1992) or a multi-echelon model (Thonemann, Brown, & Hausman, 2002). Furthermore, according to various operating parameters, these models can be further classified into a fixed quantity model and a fixed period model. More specifically, two popular inventory control models have been widely used: (1) the lot-size/reorder point (Q, r) model (Hopp, Zhang, & Spearman, 1999), where Q represents the order quantity, and r is the reorder point; and (2) the reorder point/order-up-to-level (S, s) (Cohen, Kleindorfer, Lee, & Pyke, 1992), where S represents the order-up-to-level, and s is the reorder point. To determine the optimal operating parameters, associated maintenance demands need to be characterized first. In the literature, most inventory models ( Grave, 1985, Gross et al., 1985, Jung, 1993, Kupta and Rao, 1996 and Slay et al., 1996) are derived based on one of the following assumptions: (1) maintenance demands follow a homogeneous Poisson process (constant rate); (2) installed base is fixed, which generates stationary maintenance demands. In other words, those assumptions address inventory control problems when the distribution of maintenance demand does not change over time. Such distribution is usually determined by fitting an assumed distribution to historical demand data or by simply utilizing past experience. In practice, those assumptions may not be realistic, especially for a new product, for which the maintenance demands grow rapidly as the field installations increase. In the literature, however, little effort has been given to spare parts inventory control considering stochastic growth of an installed base. Jin, Liao, Xiong, and Sung, (2006) showed that stationary maintenance demand models underestimate the actual maintenance demands in such cases. As a result, it is important to investigate how the product population grows in the field in order to proactively forecast the upcoming maintenance demands for dynamic control of spare parts inventory. This paper addresses a spare parts inventory control problem for a non-repairable product with a stochastically growing installed base. When an installed unit fails it will be replaced by a new one. The closed form solution for the maintenance demand is derived when new sales occur following a homogenous Poisson process and the failure time of an installed unit follows the exponential distribution. Based on the model for the maintenance demand, a (Q, r) inventory control model is formulated, and the operating parameters are optimized using a multi-resolution technique. The remainder of this paper is organized as follows. In Section 2, the maintenance demand is derived for a product with a generally distributed lifetime. Section 3 addresses the special case where the product’s lifetime follows the exponential distribution. Section 4 presents the formulation for the dynamic (Q, r) spare parts inventory problem and the multi-resolution solution technique. In Section 5, two numerical examples are provided to demonstrate the proposed approach in practical use. Simulation is utilized to investigate the effectiveness of the multi-resolution approach. Finally, conclusions are provided in Section 6.

This paper proposes an approach for controlling spare parts inventory considering the stochastic growth of an installed base. The resulting maintenance demand due to new sales has been formulated, and its mean and variance were derived when the product lifetime follows the exponential distribution. A multi-resolution inventory optimization model is proposed to deal with the non-stationary maintenance demand, and a bisectional search algorithm is developed to search the optimal settings of the associated operating parameters. Combined with the simulation approach, this model can be used for products with generic lifetime distributions. It can be extended to more general inventory control systems involving multiple types of parts used for the installed base. The problem will become more complex when multi-echelon inventory systems and multiple products sharing the same spare units are considered. Another interesting technique that can be applied in the multi-resolution approach is to consider unequal lengths of those time intervals. By adaptively controlling the resolution (length of re-order time) according to the maintenance demand, it is expected to obtain a better performance in terms of reducing the total inventory cost and number of stockouts. These topics will be studied in our future research.