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...after-sales support service consists in providing necessary service engineers and replacement parts to existing, geographically dispersed customers when they experience any problems with purchased product. The service is provided as part of the contract between the customer and the manufacturer, and thus designing and operating a logistics network capable of serving customers in a time-responsive manner is crucial for a successful after-sales service. A critical part of this process is Service Parts Logistics (SPL) which includes activities such as designing a responsive network of part stocking facilities, deciding inventory ordering policies, stocking parts, and dispatching the required parts from facilities to the customers in need. A major challenge in SPL is to provide the service and satisfy a customer's request within the committed time frame. Surveys by Cohen et al. (1997) and Poole (2003) show that 40 to 50% of the profits made by manufacturers comes from parts, maintenance, and servicing, which makes SPL a $21 billion industry.
Locating inventory stocking facilities, allocating customer demands to these facilities and selecting the stock levels to be maintained at these facilities are the main decisions to be made when designing a SPL system. Traditionally, location and allocation decisions (collectively called Logistic Network Design (LND)) are considered to be part of the strategic and long-term decisions that are typically made before any tactical decisions such as inventory levels. The frequent redesign of an existing network is becoming increasingly common due to outsourced warehousing and delivery services becoming popular choices in SPL. For example, companies such as UPS and FedEx provide access to extensive global logistics networks and offer services ranging from the sourcing of parts from vendors to shipments to customer sites. This allows companies using third-party logistics services to expand or shrink their network as needed without much difficulty. Moreover, due to the time-based service level requirements that are a critical part of any SPL system, there is a stronger interaction between "strategic network design" and "tactical" inventory decisions as the service requirements are not only a coverage issue (whether a customer's demand is covered by a nearby facility), but also a function of the part availability at that facility. Thus, we conjecture that considering the effects of network decisions on inventory (and vice versa) in an integrated model becomes critical for the optimization of a SPL system.
We note that the ultimate decision made in our proposed model is still designing the network. We make inventory stocking part of the network design model to make better overall decisions and find a network design that will not only minimize facility and transportation costs but also minimize inventory costs while ultimately capturing service level tradeoffs. The inventory decisions can and should be updated over time as needed without overhauling the network. Motivated by these real challenges in today's SPL systems, we model the integrated network design and inventory stocking problem. We explicitly consider inventory decisions and costs in the LND problem, which is itself already complicated due to explicit time-based service constraints. We also quantify the benefit of considering both network design and inventory decisions in the same model and identify the conditions and problem settings where this benefit is significant.
2. Literature review
We review papers from: (i) the location/allocation/network design literature; (ii) the multi-location service-constrained inventory management literature; and, of course, (iii) the limited number of papers that investigate similar integration issues.
2.1. Facility location/network design problems
Facility location and network design problems with countless variations have been studied extensively in the literature. The relevant papers in this area study service-constrained, stochastic, or reliability-based problems. There are studies that address global uncertainties such as exchange rates, transfer prices, taxes and market prices along with supplier reliability and lead time uncertainty in a single-echelon model (Vidal and Goetschalckx, 2000). These studies define reliability as the probability of all suppliers being on time and the amount of product-shipped being at least a specified target value. Another related area is the study of fixed-charge location/allocation problems with demand/service coverage restrictions. Typical examples here include emergency facility location problems (e.g., Goldberg and Paz (1991)) and similar logistics applications (e.g., Nozick (2001)). Snyder and Daskin (2005) investigate the Uncapacitated Facility Location (UFL) problem with potentially unreliable facilities, where a customer may not be served by a facility due to the "failure" of the facility which occurs with a certain probability. The recent reviews of Snyder (2004) and Daskin et al. (2005) survey more papers in this area in the supply chain management context. Daskin (1995) is a reference text on discrete facility location problems. Magnanti and Wong (1984) review the early literature on the facility location problems, while Drezner (1995) summarizes the overall research effort by 1995.
2.2. Inventory management and service parts logistics
Similar to the location literature, there is a vast amount of work on inventory management. We only review the ones that are most relevant to our work, and refer to Zipkin (2000) for a recent text on the topic. A stream of research related to the inventory-portion of our study explicitly considers service level constraints. We can list Chen and Krass (2001) for a single facility operating reorder point and order-up-to level policy (s, S), and Agrawal and Seshadri (2000) for order-quantity and reorder point policy (Q, r). One example (Song, 1998) investigates a simplified time-based service level for base stock policies for a group of items. For an example of a multi-facility inventory allocation problem, see Rappold and Muckstadt (2000).
The early literature on spare/service parts inventory management in multi-echelon systems includes Sherbrooke (1968, 1986) and Muckstadt (1973). Successful applications in SPL systems include the automotive industry (Cohen et al., 2000), computers and other electronic equipment (Cohen et al., 1988; Cohen et al., 1990; Cohen et al., 1999), and the military (Rustenburg et al. 2001). One of the early works on multi-echelon service parts inventory management is by Muckstadt and Thomas (1980). A limited number of studies in this group consider fill-rate-based service allocation (Cohen et al., 1988, 1989; Cohen et al., 1992). While the book by Sherbrooke (1992) provides an overall review of multi-echelon inventory management from a military perspective with a focus on repairable parts, a recent text by Muckstadt (2005) is now a reference for general SPL research.
2.3. Network design models with inventory considerations
The integration of the facility location and inventory problems is a very recent research area, hence there are only a handful of papers on this topic. As they are most relevant to our study, we review them in more detail. One of the early papers modifies the UFL problem to implicitly consider limited inventory levels (Barahona and Jensen, 1998). Nozick and Turnquist (1998) approximate inventory costs as part of the fixed facility costs assuming a linear relationship between inventory and the number of open facilities, and propose a model that takes service coverage as a constraint. Nozick and Turnquist (2001) extend this model and treat demand coverage as part of the objective function. The paper by Daskin et al. (2002) is probably the first study that explicitly includes inventory costs as part of a simple UFL model. Their model assumes Economic Order Quantity (EOQ)-based ordering and constant fill-rate-based safety stocks across all facilities. The total cost function including the inventory-related terms makes the overall model a nonlinear integer program which is then solved using Lagrangian relaxation. (A parallel and a very similar model is analyzed in Miranda and Garrido (2004).) Shen et al. (2003) develop a column generation-based method to solve the same model, while Ozsen et al. (2004) extend the model to include facility capacities. The other extensions of the same model include a scenario-based stochastic version (Snyder et al., 2003), an approximation algorithm for the version that ignores transportation costs (Teo et al., 2001), and a version with customer-specific service levels (Shen and Daskin, 2005). Another relevant paper is a grid-based location-inventory model (Erlebacher and Meller, 2002), In contrast to these important contributions, our model tries to achieve a system-level service by allocating it to multiple facilities in an optimal manner, hence considering the varying fill rates as explicit decision variables. Moreover, our model is especially applicable to low-demand settings with one-at-a-time ordering, whereas the papers cited here model inventory as an objective-changing cost component with EOQ ordering, making them more applicable to high-demand settings with significant order set-up costs.
3. Problem definition and modeling
We first define the problem setting and introduce the notation, and then introduce the integrated model formulation, dealing with its complexities along the way.
3.1. Problem setting and notation
For a given set of customers and their mean demands, we seek to: (i) locate a set of stocking facilities selected from a set of candidate facility locations; (ii) allocate customer demands to these located (open) facilities; and (iii) determine stock levels to be used at the open facilities. Decision sets (i) and (ii) make up the network design, and (iii) are the inventory stocking decisions. The goal is to make these decisions with minimum possible total facility, transportation, and inventory costs while achieving the target (required) service levels. We make the following assumptions that facilitate the model development:
* We assume that network design involves the stocking facilities that are all in one echelon facing the direct demand from geographically dispersed customers. We assume that these facilities to be located are replenished from a central warehouse with infinite capacity (that is, the central warehouse can replenish the stocking facilities anytime without any delay). The lead times from the central warehouse to all facilities is the same, known and constant.
* Due to the low-demand nature of the motivating SPL problem, we assume that the facilities use continuous review, one-for-one (or base stock, also called (S - 1, S)) replenishment policy. This is typical as demands are low, and lead times are relatively short in SPL systems. For examples, all the models in Sherbrooke (1992) use this policy, even for higher-echelon facilities, where demands from lower-echelon facilities are aggregated.
* We assume that demand for each part at each demand point arrives one-at-a-time according to an independent Poisson process, typical in low-demand settings. This is a common assumption in the SPL literature (see, e.g., Sherbrooke (1992), and Muckstadt (2005)). We assume that we know the mean demand rates obtained from the part failure-rate distributions and the number of parts used at each demand point. Any unsatisfied demand due to a stockout at a facility is backordered.
* We assume that service contracts, demand aggregation across individual customers to form demand points, and service level aggregation to obtain a target service level for the region are done a priori. This translates into a percentage of demand to be satisfied within a certain service time window. For example, a typical aggregate service level may read "70% of total demand for part 1 must be satisfied from facilities that are within 4 hours of the demand points."
* For simplicity in presentation, we assume just one service time window. Extending the model (as will be seen) for multiple windows is straightforward. In the experiments, however, we vary the time window as a control factor to see its effect on the results.
* We assume that we know which customers a facility can serve within the service time window. As this is usually a function of distance and the mode of transportation available to the facility and customer, we assume that this processing of transportation times is performed for each customer and facility pair a priori. We further assume that each customer's part request is satisfied by a single direct shipment from a facility, without any...
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