Spare parts inventory control considering stochastic growth of an installed base
Introduction
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, 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, 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.
Section snippets
Maintenance demand considering the growth of product installed base
Notations for modeling maintenance demand
- N(t)
the number of new sales by time t, and N(t) + 1 is the installed base
- λ
rate of new sales or the growth rate of the unit
- Wi
the arrival time of the ith new sale
- a
constant failure rate when the product’s failure time distribution is exponential
- F(t)
cumulative distribution function (Cdf) of the product’s lifetime
- F(i)(t)
i-fold convolution of F(t)
- H(t)
the expected number of renewals or failure replacements by time t if a unit is installed at time 0
- S(t)
aggregate
Aggregate maintenance demand under exponentially distributed failures
In many applications, the mean and the variance of S(t) are often used to evaluate the properties of S(t). In this paper, instead of seeking the distribution of S(t), the explicit expressions for E[S(t)] and Var(S(t)) are derived when the product lifetime is exponentially distributed.
The exponential distribution is often used to model product lifetimes, especially for semiconductor devices and electronic equipment. In today’s industry, products usually go through a series of environmental tests
Spare parts inventory control under aggregate maintenance demand
When the maintenance demand S(t) is known, the next step is to determine the type of inventory system and the optimal operating parameters. In this paper, a (Q, r) inventory model is considered with continuous inventory review (see Fig. 3). Higher values of Q and r may reduce the total ordering cost and the chance of stock-out, but the holding cost may increase. On the other hand, lower values of Q and r may reduce the holding cost but increase the total ordering cost and the possibility of
Numerical examples
In this section, two numerical examples are provided to demonstrate the application of the proposed approach. In both examples, it is assumed that the new installations follow the homogeneous Poisson process with the rate of λ = 15 units/month, but the product failure times follow the Weibull and exponential distributions, respectively.
Conclusions
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
Acknowledgement
The authors greatly appreciate two anonymous referees for their valuable and helpful suggestions on the earlier version of the paper.
References (20)
A gamma-normal series truncation approximation for computing the Weibull renewal function
Reliability Engineering and System Safety
(2008)- et al.
Multi-objective spare parts allocation by means of genetic algorithms and Monte Carlo simulations
Reliability Engineering and System Safety
(2005) - et al.
Multiitem service constrained (s, S) policies for spare parts logistics systems
Naval Research Logistics
(1992) - et al.
A characterization of order statistic point process that are mixed Poisson processes and mixed sample processes simultaneously
Journal of Applied Probability
(1985) On the characterization of point processes with order statistic properties
Journal of Applied Probability
(1979)A multi-echelon inventory model for a repairable item with one-to-one replenishment
Management Science
(1985)- et al.
On some common interests among reliability, inventory and queuing
IEEE Transactions on Reliability
(1985) - et al.
An easily implementable hierarchical heuristic for a two-echelon spare parts distribution system
IIE Transactions
(1999) - Jin, T., Liao, H., Xiong, Z., & Sung, C. H. (2006). Computerized repairable inventory management with reliability...
Recoverable inventory systems with time-varying demand
Production and Inventory Management Journal
(1993)