Publication: IIE Transactions
Publication Date: 01-MAY-07
Delivery: Immediate Online Access
Author: Sourirajan, Karthik ; Ozsen, Leyla ; Uzsoy, Reha
Article Excerpt
1. Introduction
Most existing distribution network design models focus on the trade-off between the fixed costs of facility location and transportation costs (Daskin, 1995; Geoffrion and Powers, 1995). Operational aspects such as lead times and customer service levels are viewed as tactical decisions, treated by coordinating the flows through the network. This requires considerable simulation and post-processing of the results obtained from these models to make them more effective. Vidal and Goetschalckx (1997) and Erenguc et al. (1999) observe that combining tactical decisions with the traditional strategic objectives in integrated models provides interesting research directions and, specifically, that lead times and service levels need to be considered in strategic network design models.
In this paper, we integrate fixed facility location costs, lead times and service levels into a location-allocation model for distribution system design. We consider a two-stage supply chain in which a single product is replenished at retailers from a production facility. The objective is to locate Distribution Centers (DCs) at certain locations to serve groups of retailers. Each DC has limited capacity, and must hold enough safety stock to guarantee the desired service level at its assigned retailers. The relationships between flows in the network, lead times and service levels are explicitly modeled. The objective is to minimize the sum of the facility location, pipeline inventory and safety stock costs. The model is a variant of the standard fixed charge capacitated facility location models with single sourcing (the same constraints but a nonlinear objective function), as well as of several previous models that have considered lead time or safety stocks separately. Since the model is easily recognized to be NP-hard, we develop a Lagrangian heuristic to obtain near-optimal solutions in modest CPU times.
In Section 2 we review previous related work. In Section 3, we describe the problem and the modeling of lead times and service levels. Section 4 presents the model formulation and the solution procedure. In Section 5, we present the test problem instances used in our computational experiments. Section 6 presents our computational results and analysis. In Section 7, we present some extensions to our model. We conclude the paper with some conclusions and future directions in Section 8.
2. Literature review
Given sets of candidate facility locations and demand points, the objective of the Uncapacitated Facility Location Problem (UFLP) is to locate DCs among the candidate locations to serve the demand points while minimizing the sum of fixed location and transportation costs. The objective and constraints are linear but the problem is NP-hard (Krarup and Pruzan, 1983). Numerous researchers have studied this problem and its variations including Erlenkotter (1978), Daskin (1995), Klose (1998), Holmberg (1999), Melkote and Daskin (2001) and Alp et al. (2003). The Capacitated Facility Location Problem (CFLP) has the same objective as the UFLP, with the addition of capacity constraints that limit the demand that can be served by each candidate location. Cornuejols et al. (1991) and Sridharan (1995) review various heuristics and relaxations for the CFLP. Geoffrion and Graves (1974), Klincewicz and Luss (1986), Van Roy (1986), Sridharan (1993), Daskin (1995), Mazzola and Neebe (1999) and Klose (2000) present models and solution procedures for the standard CFLP and its extensions. However, they ignore lead times and service levels.
One body of previous work constituted by the papers of Berman and Larson (1985), Crainic and Laporte (1997), Owen and Daskin (1998), Jamil et al. (1999) and Eskigun et al. (2005) explicitly considers lead times in network design. Eskigun (2002) models the lead time at a candidate location as a function of the amount of flow sent through that location, considering the effects of both the time required to fill trucks and wait times due to high utilization of servers on lead times at the DCs. He then uses this model to develop an outbound network design model with transportation mode selection whose objective is to minimize the sum of transportation, facility and lead-time-related costs. A Lagrangian heuristic is used to obtain approximate solutions for large instances with reasonable computational requirements. Wang et al. (2002) present a facility location model with stochastic customer demand and immobile servers. Their lead time model is very similar to that of Eskigun (2002). Wang et al. (2004) extend this work to present models from the perspectives of the service provider and the customer, and show how these models reduce to standard facility location models in the absence of explicit lead time modeling. Huang et al. (2005) develop a network design model in which the flow between an origin and a destination experiences congestion at the connection nodes. Congestion is determined by the mean and variance of the service at the connection nodes and they formulate the problem with both fixed and variable service rates. These authors also develop efficient solution procedures that obtain high-quality solutions for large problems with reasonable CPU times. However, safety stocks and service level requirements are not captured in these models.
Graves and Willems (2000) and Jung et al. (2004) along with a number of other authors have considered the problem of locating safety stocks in a given supply chain network. Some recent work considers the effects of safety stock risk-pooling within the framework of traditional discrete location problems that aims to minimize the sum of the location and transportation costs along with the operating costs and safety stock costs at a DC. Daskin et al. (2002) and Shen et al. (2003) present a location-inventory model that incorporates safety stock placement into a location problem for a two-stage network. Their model is an extension of the UFLP in which inventory risk-pooling aspects are captured. Balcik (2003), Miranda and Garrido (2004) and Jia et al. (2005) study extensions to the location-inventory model. Ozsen (2004) presents a capacitated location-inventory model which determines the ordering policy at the DCs along with the amount of safety stock to be maintained such that the maximum possible accumulation at the DCs does not exceed their capacities.
In this paper, we develop an integrated network design model that simultaneously considers the operational aspects of lead time derived from queueing analysis and maintaining enough safety stock to satisfy service level requirements. This enables us to study the effects of resource utilization on the lead times and the safety stock risk-pooling benefits. We describe the problem and the modeling of lead times and service levels in the next section.
3. Problem description and model analysis
We consider a distribution network design problem for a two-stage single-product supply chain. A production facility replenishes retailers located at discrete points within a given region where the demand for the product occurs. We aim to locate DCs, each of which serves a set of retailers, such that the sum of the location and inventory (pipeline and safety stock) costs is minimized. We assume single sourcing, i.e., the entire demand of a given retailer must be met by a single DC. The retailers' demands are assumed to be independent of each other and follow a Poisson process (Daskin et al., 2002; Shen et al., 2003; Ozsen, 2004; Ozsen et al., 2006). This implies that the variance of the demand is equal to the mean. The decisions made by our model regarding the location of DCs and retailer assignments to the DCs have a given lifetime and we assume a time-stationary demand distribution at the retailers over this lifetime. Given the stochastic demand, a DC carries the safety stock for the retailers assigned to it to achieve risk-pooling benefits, and the retailers carry negligible safety stock compared to that at the DC. Montgomery et al. (1998) show that approximating a Poisson process to be Normally distributed is appropriate for sufficiently large demand values. Thus, we assume that the demands at the retailers are Normally distributed when evaluating the safety stock requirement. We also limit the average service rates at the DCs, implying that the total mean demand assigned to the DC must be less than its maximum service rate.
3.1. Modeling the replenishment lead time at the DCs
We assume that the variability of the lead time is negligible compared to its mean. While this assumption is clearly not always realistic, it provides an initial point of departure for solutions to more realistic problems. We assume that the products are sent from the production facility to a DC in full truckloads and thus incur a waiting time at the production facility until enough material is accumulated for a shipment to be made. While sending full truckloads is not necessarily optimal in all situations, we assume that the DCs are far enough from the production facility and that the shipment quantities are high enough (since we group the demands of multiple retailers) to justify it. When shipments arrive at the DC, they dock at an unloading zone, where they wait in a first-in first-out queue to be unloaded and sent to the retailers.
[FIGURE 1 OMITTED]
This process is shown in Fig. 1 and is similar to the operation of a cross-docking facility and thus, the replenishment lead time at a DC has three components:
1. Load make-up time: The time spent in the waiting area of the production facility before the products are sent to the DC. This wait time depends on the demand volume assigned to a DC. As more demand is assigned to a DC, the average load make-up time per unit decreases.
2. Constant DC replenishment time (time/unit): The replenishment lead time between the production facility and the DC due to the physical locations of facilities. We assume that it also includes the time spent due to delays such as material handling, and also general inefficiency and unavoidable processing, such as paperwork.
3. Congestion time: The time spent in the unloading zone. At high utilization of the resources at the unloading zone, shipments have to wait longer in the queue. This implies that congestion increases as the demand assigned to a location approaches its capacity.
We have a common load make-up time component for the demand served by a DC and thus, the total mean demand served by a DC is included in the calculation of expected load make-up time. Since shipments compete for the same resources at the unloading zone, the total mean demand served by...
NOTE: All illustrations and photos have been removed from this article.
More articles from
IIE Transactions
Locating capacitated facilities to maximize captured demand, 01-NOV-07
Erratum, 01-NOV-07
Sequencing with limited flexibility, 01-OCT-07
Looking for additional articles?
Click here to search our database of over 3 million articles.
Looking for more in-depth information on this industry?
Click here to search our complete database of Industry & Market reports by text, subject, publication name or publication date.
About Goliath
Whether you're looking for sales prospects, competitive information, company analysis or best practices in managing your organization, Goliath can help you meet your business needs.
Our extensive business information databases empower business professionals with both the breadth and depth of credible, authoritative information they need to support their business goals. Whether it be strategic planning, sales prospecting, company research or defining management best practices - Goliath is your leading source for accurate information. |
