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Article Excerpt
1. Introduction
Service Parts Logistics (SPL) is increasingly considered to be an indispensable part of supply chain management (Cohen and Lee, 1990; Cohen et al., 1997; Cohen et al., 1999; Cohen, Cull, Lee and Willen, 2000). The term SPL mainly refers to after-market service (providing necessary repair services, replacement parts, service technicians, training, etc.) to address the problems an existing customer base is experiencing with products (e.g., mainframe computers, network servers, heavy machinery or medical equipment, etc.) in use at customer sites. SPL has become a multi-billion dollar industry which is growing every year, mainly because manufacturers who experience diminishing profit margins in sales try to differentiate themselves from the competition by focusing on providing high-quality post-sales service through high-margin service contracts. SPL offers potential research areas, one of which is in network design. The main challenge in designing and operating an SPL network is to guarantee a highly responsive service to customers in need (usually within hours of the customers' request) so that customers' mission-critical systems are up and running again quickly. This challenge necessitates an integrated approach to logistics network design and inventory stocking in SPL systems, which are traditionally handled separately. In this paper, we develop optimization-based solution techniques for this problem of designing and stocking a SPL network to satisfy part requests coming from geographically dispersed customers.
As a reflection of the service contracts between a manufacturer and its customers, the company's SPL network must be designed and stocked in such a way that a specified percentage of overall customer demands is satisfied within a specified service time window. This requirement is called the "time-based service level constraint," and the required percentage is called the "target" service level. Hence, while designing and stocking an SPL network, we need to make sure that we: (i) select the right set of facilities to be stocked; (ii) make the right allocations of demands to facilities; and (iii) stock the right levels of the part so that the target service level is achieved. The problem presents challenges in modeling as well as in solution methodology. One challenge is to correctly capture the relationship between the variable facility fill rates (which are functions of the stock levels and demand allocations), and the achieved service level. Hence, satisfying the target service level translates into finding individual facility fill rates that collectively achieve the service level. The main methodological challenge is to devise computationally efficient algorithms that find provably near optimal solutions.
To the best of our knowledge, integrated models for network design and inventory stocking with variable fill rates do not exist in the literature. One exception is Candas and Kutanoglu (2007), whose model provides a basic structure for our modeling effort. Their main focus is on demonstrating the benefits of considering inventory decisions and variable fill rates as part of the SPL network design problems. They linearize the non-linear fill rate functions to solve reasonable size problems and compare the integrated approach with the sequential approach of designing the network first and then stocking the facilities. Taking their model as a starting point, we develop a more direct and efficient modeling and solution technique for the integrated network design and inventory stocking problems. On the modeling side, we assume lost sales for stock-out situations instead of backorders as modeled in Candas and Kutanoglu (2007). In the quest for a more effective and specialized solution technique, we introduce a new variable substitution scheme, which leads to a much smaller and tighter linearized model, and a new relaxation scheme, which forms the basis of a heuristic algorithm producing provably near-optimal solutions to the integrated problem. In the next section, we give a comprehensive survey of the relevant literature primarily intended to discuss chronological development of integrating inventory issues with location/allocation and network design models.
2. Literature review
Network design and facility location models are among the most widely studied models in the operations research literature. Magnanti and Wong (1984) review the early literature on the facility location problems and Drezner (1995) gives a survey of applications and methods for facility location. A book by Daskin (1995) is a reference text on discrete facility location problems. The research relevant to the work in this paper are the studies based on the Uncapacitated Facility Location (UFL) model that consider time-based demand/service coverage restrictions. Goldberg and Paz (1991) and Nozick (2001) are two close examples. More recently, Snyder and Daskin (2005) investigate the UFL problem with potentially unreliable facilities and they also discuss facility location in designing a supply chain in Daskin et al. (2005).
Given the vast literature in inventory-related research, we limit our focus to the studies related to multi-location inventory management models that explicitly consider service level constraints for low-demand service parts. Early work in this area includes Sherbrooke (1968. 1986), Muck-stadt (1973) and Muckstadt and Thomas (1980). More relevant studies include Verrijdt and De Kok (1995, 1996), who discuss distribution planning with service level constraints. Ouyang and Wu (1997) propose an inventory model with variable lead times and service level constraints, and Song (1998) investigates a simplified time-based service level with base stock policies. Cohen et al. (1988) study inventory systems with service constraints and priority demand classes. Studies related to successful implementations in service parts logistics systems include Cohen, Cull, Lee and Willen (2000) in automotive, Cohen et al. (1988), Cohen et al.(1990) and Cohen et al. (1999) in computers and other electronics, and Rustenburg et al. (2001) in military logistics. Sherbrooke (1992) provides an overall review of multi-echelon inventory management from a military perspective with a focus on repairable parts. More recently, Muckstadt (2005) has become a new reference on general SPL models and algorithms and Zipkin (2000) is a recent reference on inventory theory.
Integration of network design and inventory stocking has been under investigation for more than 25 years, with an increased attention more recently. The earliest references include Geoffrion (1979), Geoffrion and Roy (1979) and Geoffrion and Powers (1980). Geoffrion (1989) specifically discusses both modeling and solution techniques for integration. Ignoring transportation, Wagner (1974) and Cohen and Lee (1988) study integration of production and inventory systems for multi-facility and multi-warehouse companies. Barahona and Jensen (1998) were perhaps the first to study a problem that combines discrete fixed-charge facility location and inventory more explicitly. Their model has simple inventory decisions (to stock or not to stock certain parts) along with location decisions and their solution technique uses Dantzig-Wolfe decomposition. Nozick and Turnquist (1998) study the impact of integrating inventory into a fixed-charge location model assuming fixed fill rates. Nozick and Turnquist (2001) propose an approximate inventory cost function embedded in the fixed costs term of a facility location model, which is an extension of the fixed-charge UFL problem. They model two versions of the problem, one minimizing the total logistics costs and the other maximizing the service coverage. Nozick (2001) considers service coverage constraints with limited consideration on inventory. More recently, Daskin et al. (2002) and Shen et al. (2003) are studies of a single-part joint location-inventory model. These models consider an Economic Order Quantity (EOQ)-based ordering policy and constant fill rates across all facilities, which make the models more suitable for high-demand items. A very similar model is proposed in Miranda and Garrido (2004). A similar work that studies inventory issues in multi-location EOQ framework is Schwarz (1981). Shen et al. (2003) develop a column-generation-based scheme to solve the same model.
In contrast to the important contributions discussed above, our model tries to achieve a system level service by allocating it to multiple facilities in an optimal manner, i.e., considering the varying fill rates as explicit decision variables. It is well known that it is beneficial to adjust the stock levels and fill rates depending on several properties such as the costs of the parts and their demands (Johnson and Montgomery (1974), Graves et al. (1993), Hopp and Spearman (2000)). In this regard, the studies that consider fill-rate-based service allocation (Cohen et al. (1989), Cohen et al. (1992) and Candas and Kutanoglu (2007)) are closely related to our work.
3. Problem statement and assumptions
Given a set of customers and their demands for a service part, we seek to select a set of stocking facilities, allocate all demands to these facilities and then determine stock levels at the facilities. The objective is to minimize the joint costs of transportation and inventory stocking while guaranteeing to achieve the target time-based service level. We make the following assumptions for the modeling studies.
1. Facilities use a one-for-one (or base stock) replenishment policy for inventory stocking. This is realistic since the SPL systems have distinctly low demand and short replenishment lead times. Therefore, there is no incentive to order in large quantities, especially considering the low replenishment ordering costs due to bundled and outsourced transportation services used by SPL networks. Furthermore, the replenishment facility (say a central warehouse) is well stocked (i.e., does not stock out) so that it can provide the replenishment part with out any additional delay.
2. There is a reasonably small upper bound on the base stock level of each facility. Again, this is due to the low-demand nature of real SPL systems, where stocking three or more units would give practically a 100% fill rate.
3. Individual customer demand follows a Poisson distribution, which is known to be an accurate approximation for low-demand systems. Part of this assumption is that the demand rates are obtained from aggregating independent part failure rates. Also, a facility serves its customers' individual demands in a First-Come First-Serve (FCFS) basis with no prioritization or stock reservation (especially regarding the demand from within or outside the time window of the facility).
4. There is a system-wide target service level which requires the minimum amount of total demand that needs to be satisfied within the time window. As most service contracts in SPL stipulate a collective service level to be achieved across geographically dispersed customer sites, the system-wide service level can be used as a conservative way of satisfying individual service contracts. Furthermore, we assume that we have a single service time window. The term "service level" is then used to refer to this system-wide time-based service level (associated with the time window under consideration).
5. A customer demand can only be assigned to a single facility, and only if the facility keeps a positive stock level. The demand not satisfied directly from a facility due to stock-out is passed to the central warehouse, which directly ships the part to the requesting customer. In a way, from the stocked-out facility's perspective, the excess demand is lost. Hence, the facility fill rates are computed using the lost-sales assumption. The direct shipments from the central warehouse are always outside the service time window for all customers.
6. Facilities must have positive stock in order to be considered open.
The first three assumptions are common in the SPL literature (see, e.g., Sherbrooke (1992) and Muckstadt (2005)).
4. Modeling
Given a set of stocking locations I (indexed by i) and a set of customer demand points J (indexed by j), we define a binary parameter [[delta].sub.ij] [for all] i, j taking value one if the transportation time from facility i to demand point j is less than or equal to the prespecified service time window, zero otherwise. (If facility i can ship an (already available) part to demand pointy within the service time window, then [[delta].sub.ij] = 1, otherwise, [[delta].sub.ij] = (i.e., the service cannot be provided within the window even if the part were available due to distance or transportation modes available between the customer and facility).) Parameter [d.sub.j] is the mean annual demand rate at demand point j, and [alpha] is the required fraction (target service level) of the system-wide demand to be satisfied within the time window. Parameter [tau] is the replenishment lead time (in years), assumed to be the same for all facilities. Cost parameters [f.sub.i], [c.sub.ij] and [h.sub.i] are the annual cost of opening and operating facility i, the unit transportation cost from facility i to demand point j and the inventory holding cost at facility i, respectively. Decision variable [Y.sub.i] (defined for all i [member of] I) is a binary variable indicating if facility i is open/stocked (a value of one) or not (a value of zero). Allocation variable [X.sub.ij] (defined for all i and j) is also a binary variable, taking value one when demand point j is assigned to facility i, and zero otherwise. Non-negative variable [[beta].sub.i] (defined for all i [member of] I) measures the part availability as the fill rate at facility i. Non-negative variable [[lambda].sub.i] (defined for all i [member of] I) determines the mean lead time demand at facility i. Finally, [S.sub.i] (for all i [member of] I), is a non-negative integer variable indicating (base) stock level maintained at facility i, which is smaller than an upper bound [S.sub.max], loosely found as a stock level giving 100% fill rate for a high level of demand that can be assigned to a facility.
Using the notation defined above, and building upon the model developed by Candas and Kutanoglu (2007), we present our basic model (BM) that completely captures the problem definition. (BM) is primarily a modified UFL formulation with integrated inventory aspects (costs, fill rates and service levels) of the problem:
BM: min [summation over (i[member of]I)][f.sub.i][Y.sub.i] + [summation over (i[member of]I)][summation over (j[member of]J)][c.sub.ij][d.sub.j][X.sub.ij] + [summation over (i[member of]I)][h.sub.i][S.sub.i], (1)
subject to
[summation over (i[member of]I)][X.sub.ij] = 1[for all]j [member of] J, (2)
[X.sub.ij][less than or equal to][Y.sub.i][for all] i [member of] I and j [member of] J, (3)
[summation over (i[member of]I)][summation over (j[member of]J)][[delta].sub.ij][d.sub.j][[beta].sub.j][X.sub.ij][greater than or equal to][alpha][summation over (j[member of]J)][d.sub.j], (4)
[[lambda].sub.i] = [tau][summation over (j[member of]J)][d.sub.j][X.sub.ij][for all]i [member of] I, (5)
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (6)
[S.sub.i][greater than or equal to][X.sub.ij] [for all] i [member of] I and J [member of] J, (7)
[X.sub.ij] [member of] {0,1} [for all] i [member of] I and j [member of] J, (8)
[Y.sub.i][member of]{0,1}[for all] i [member of] I, (9)
0[less than or equal to][S.sub.i][less than or equal to][S.sub.max] and integer [for all] i [member of] I. (10)
In (BM), objective function (1) is the total annual costs of operating facilities, transporting parts to customers and stocking parts at facilities. Constraints (2) and (3) are the UFL constraints, stating all demands must be allocated and demand can only be assigned to an open facility. Constraint (4) is the service level constraint which states that the total demand satisfied within the time window aggregated across all customers and facilities must be at least [alpha] fraction of the total demand.
Since the details of the development of this constraint are in Candas and Kutanoglu (2007), here we mention the underlying logic briefly: due to the assumption that the individual demands at a facility, say i, are met in a FCFS fashion, they are satisfied proportional to the facility fill rate, [[beta].sub.i]. The amount of demand satisfied from facility i "within the time window" is [[beta].sub.i] [[SIGMA].sub.j[member of]J] [[delta].sub.ij][d.sub.j][X.sub.ij]. When this is aggregated across all facilities, we obtain the total demand satisfied within the time window, the left-hand side of the constraint. (Constraint (4) is similar in spirit to the ones in Muckstadt (2005), who combines multiple facility (variable) fill rates to obtain an aggregate service level constraint.)
Constraints (5) calculate [[lambda].sub.i] for all i, the means of the lead time demands, which are used to compute the fill rates in...
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