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Article Excerpt
1. Introduction
In electronics, computer, automobile, and many other industry sectors, a manufacturing and supply system usually takes the form of a complex network of suppliers, fabrication/assembly locations, distribution centers, and customer locations, through which materials, components, products, and information flow (Ettl et al. 2000). Throughout the network, there are different sources of uncertainties associated with supplies (availability, quality, and delivery times), processes (transportation times, machine breakdown, and human performance), and demands (arrival times, batch sizes, and types). These uncertainties and other factors affect the performance of a system, including its service level in terms of fill rate or delivery lead time, which in turn affects the bottom line of an enterprise in today's competitive environment. Among other things, inventories can be used to hedge uncertainties and achieve a specific service level. Because inventory placed at different locations usually incurs different costs and results in different service levels for end customers, the efficient allocation and control of inventory assets presents enormous opportunities and, at the same time, poses a great challenge to many companies.
Motivated by this challenge, in this paper we develop an effective approach to deal with complex supply network design problems involving both queueing delay and stocking control at every node in the network. By modeling the interactions of the queueing delay and stocking control in a network setting, we expand the boundary of the system design methodology.
For a serial supply system, we propose a multistage inventory-queue model. By "inventory queue," we refer to a queueing model that incorporates an inventory control mechanism such as the base-stock control. To evaluate the performance of a multistage system, we decompose it into multiple single-stage inventory queues, each with a modified input (raw material arrival process). Our decomposition approach is computationally simple and provides accurate performance estimates. It also enables us to solve an optimization problem that minimizes the total inventory cost subject to a required service level. Our numerical results reveal a number of insights; some of them are notably different from conclusions reached in prior studies. For example, we demonstrate that, depending on the cost structure, it may be better to assign less-variable servers to downstream stations instead of upstream stations, as commonly suggested in the literature (for systems in which objectives other than inventory costs are considered). We also demonstrate that by considering the processing delay and inventory holding costs together, there is a definite benefit in managing work-in-process inventory (WIP) actively throughout a supply chain.
The rest of this paper is organized as follows. The related literature is reviewed in [section]2, followed by model formulation in [section]3, along with some preliminary results. In [section]4, we propose a decomposition method that treats the queue length at each stage as an independent sum of a material queue and material backorders (see definitions in [section]4). Since the material queue and backorders can be readily computed, this decomposition leads to an efficient procedure for network performance evaluation. In [section]5, we first relax the integer requirement on the base-stock level of the last downstream stage so as to utilize the underlying quasi-unimodal property of the cost function. Based on this property, we construct a recursive optimization procedure to compute the optimal solution of the relaxed multistage problem. The optimal solution to the original problem can be recovered from the solution to the relaxed problem. In [section]6, based on extensive numerical experiments, we present results that demonstrate the impact of various parameters and provide managerial insights to the design and control of networks of inventory queues. The concluding [section]7 summarizes the main findings and points out future research opportunities.
2. Literature Review
We are concerned with the performance evaluation and optimization of manufacturing and supply chain systems. In the research literature, queueing-network models are usually used for performance evaluation of multistage discrete manufacturing systems, whereas optimizing inventory control in a network system is commonly associated with multiechelon inventory models. Our problem requires an integration of these two types of models.
Clark and Scarf (1960) consider a multiechelon serial system under periodic review, with constant lead times, unlimited processing capacity, stochastic demands, and a finite decision horizon. This multiechelon inventory optimization problem is decomposed into a set of single-location inventory control problems, and the optimal policy is found to be a modified base-stock policy, i.e., order up to the target echelon base-stock level and ship as much as possible if the entire order cannot be filled. This result has since stimulated significant research efforts in multiechelon periodic-review systems; refer to the details in the survey articles by Graves (1988) and by Federgruen (1993).
The METRIC model of Sherbrooke (1968) has motivated another important stream of research activities in multiechelon systems under continuous review. While the original work on the METRIC model provides an approximate solution, a number of attempts have since been made to obtain the exact solution, e.g., Axsater (1990). Svoronos and Zipkin (1991) study continuous-review hierarchical inventory systems with exogenous stochastic replenishment lead times and a one-for-one replenishment policy. By preserving the order of replenishments, the authors are able to approximate the steady-state system performance and to bring out the important role played by the lead-time variance (in contrast to the METRIC model). Refer to Axsater (1993) for a comprehensive review of multiechelon models under continuous review. Recently, a number of authors have developed models for supply chains based on multiechelon inventory theory. For example, Lee and Billington (1993) use a single-node periodic-review inventory model as a building block to analyze a decentralized supply chain with normally distributed demands and processing lead times. The book edited by Tayur et al. (1999) provides a few more examples of supply chain models.
An extension of the standard periodic-review model is to impose a capacity limit at each stage--the maximum amount of outputs per time unit. Glasserman and Tayur (1994) demonstrate that in a serial system with an echelon base-stock policy, the inventory and backorders are stable if the mean demand per period is less than the capacity at every node. Glasserman (1997) develops bounds and approximations for setting the base-stock levels in the above system. Glasserman and Wang (1998) use a large deviations approach to obtain an asymptotic linear relationship between lead time and inventory as the fill rate approaches 100%.
Buzacott and Shanthikumar (1993) study a multicell system, where each cell has a stocking point and the material flow is controlled by a production authorization card (PAC) mechanism. The focus is on deriving bounds and approximations for key performance measures. Other related studies include those on kanban-controlled production lines, e.g., Glasserman and Yao (1994, 1996).
Closely related to our work are two papers by Lee and Zipkin (1992, 1995), where the authors study tandem and distributed production systems with exponential processing times and inventory control at every stage. By assuming that the effective production lead time is equal to the sum of order delay and sojourn time (at the production facility), they transform the production system into a multiechelon model studied by Svoronos and Zipkin (1991). As such, they are able to use the method from Lee and Zipkin (1992) to obtain system performance measures through approximations involving phase-type distributions. Duri et al. (2000) demonstrate that the approximation method of Lee and Zipkin (1992, 1995) can be extended to systems with general service times with the same phase-type approximation used in Svoronos and Zipkin (1991). The basic structure of the system studied in this paper is similar to that of the systems in Lee and Zipkin (1992, 1995) and Duri et al. (2000). Like those works, we also use a decomposition approach. However, ours is based on a very different idea--decompose the queue at each stage into two components, a backlog queue and a material queue, combined with an effort to characterize the arrival process from the upstream stage (see the sections below for details). Furthermore, we optimize the inventory allocation in the system based on the performance model, whereas prior studies focus entirely on performance evaluation.
Ettl et al. (2000) develop a network of inventory-queue model to analyze complex supply chains. Each stocking location is modeled as an [M.sup.X]/G/[infinity] inventory queue operating under a base-stock control policy. By considering the possible delay caused by stock-out and modifying the lead time accordingly, they derive analytical expressions for performance measures and develop a constrained nonlinear optimization model. Like the METRIC model, the work was motivated by industrial applications and has since enjoyed successful implementation. Our study adopts the inventory-service optimization framework of Ettl et al. (2000). Our main focus, however, is to capture the queueing delay at each stage due to limited production capacity, whereas the infinite-server model in Ettl et al. is uncapacitated.
Zipkin (2000) provides a systematic discussion of inventory models with stochastic lead times. Based on the system structure, the models are divided into three groups: exogenous sequential systems, parallel systems, and limited-capacity systems. Exogenous sequential systems (see, for example, Kaplan 1970 and Zipkin 1986) are essentially standard inventory systems with constant lead times replaced by stochastic lead times. In a parallel system, an infinite-server queue is used to model the supply process. With an unlimited capacity, the order lead times are independent and identically distributed random variables. The category of parallel systems (with unlimited capacity) includes a number of interesting works such as Sherbrooke (1968), Berg and Posner (1990), and Ettl et al. (2000). Our model belongs to the third category, models with limited capacity. While we draw heavily on methodologies and results from queueing theory, our model mainly addresses inventory issues. By treating each node of a supply chain as an inventory queue, we emphasize the connection between inventory theory and queueing theory in supply chain applications.
[FIGURE 1 OMITTED]
3. The Model and Preliminary Analysis
We consider a manufacturing/supply system with m + 1 stages in series (Figure 1), m [greater than or equal to] 1, in which each stage consists of a single production/distribution facility. We model this system as a multistage inventory-queue system with m + 1 nodes, indexed as i = 0, 1,..., m. Each node i in the system consists of two parts, a server with service rate [[mu].sub.i] and service time SCV (squared coefficient of variation) [C.sub.si.sup.2], and an output store for semifinished products. We assume that the setup/order cost in the system is relatively insignificant, and thus a base-stock policy with a base-stock level [R.sub.i] [greater than or equal to] is used to control...
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