Retailer Transshipment versus Central Depot Allocation for Supply Network Design Claudia Rosales † Department of Supply Chain Management, Michigan State University, 350 North Business Complex East Lansing MI 48824 e-mail:
[email protected] Uday Rao and David Rogers Department of Operations, Business Analytics, and Information Systems, University of Cincinnati, Cincinnati, OH e-mail:
[email protected],
[email protected]
Decision Sciences 44(2), 329 – 356, April, 2013. The definitive version is available at www.blackwell-synergy.com.
ABSTRACT We consider a supply chain structure with shipments from an external warehouse directly to retailers and compare two enhancement options: costly transshipment among retailers after demand has been realized vs. cost-free allocation to the retailers from the development of a centralized depot. Stochastic programming models are developed for both the transshipment and allocation structures. We study the impact of cost parameters and demand coefficient of variation on both system structures. Our results show an increasing convex relationship between average costs and demand coefficient of variation, and furthermore that this increase is more pronounced for the allocation structure. We employ simulation and non-linear search techniques to computationally compare the cost performance of allocation and transshipment structures under a wide range of system parameters such as demand uncertainty and correlation; lead times from the external warehouse to retailers, from warehouse to central depot, and from depot to retailers; and transshipment, holding, and penalty costs. The transshipment approach is found to †
Corresponding author.
outperform allocation for a broad range of parameter inputs including many situations for which transshipment is not an economically sound decision for a single period. The insights provided enable the manager to choose whether to invest in reducing lead times or demand uncertainty and assist in the selection of investments across identical and non-identical retailers. [Submitted: February 8, 2012. Revised: May 31st, 2012; June 14, 2012. Accepted: June 18, 2012.] Key Words: Allocation, Simulation, Supply Chain Management, Transshipment.
INTRODUCTION Supply chain design, comprising the location of originating warehouses, transfer facilities, and retailers, as well as how they are designed to interact, can have a significant impact on a firm’s profitability. We consider a retailer-based firm that seeks to evaluate the advantages and weaknesses of various options for distribution system design. One option is to ship goods directly from a warehouse to each retailer. To enhance this distribution system a firm may choose to develop a centralized depot to receive shipments from the warehouse and then allocate the inventory to the retailers. This depot effectively postpones the inventory allocation point and results in system risk pooling, inventory reduction, and centralized ordering efficiencies. An alternative strategy is to continue to ship goods directly to the retailers and develop a transshipment system: an information and transportation structure in which a retailer with excess inventory may choose the potential recourse action of shipping to another retailer witnessing a shortage, which mitigates both holding and stock-out costs, but incurs additional transshipment costs. Both allocation and transshipment strategies are common responses for minimizing product shortages, and have been applied, for example, in the automotive and retail industries (Mattfeld, 2006; Herer & Mund, 2011), and have been thoroughly addressed in the literature.
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Hopp (2008) suggests that restructuring supply chains by eliminating the depot and transshipping across retailers to create a “virtual distribution center” should often be advantageous. He posited that an effective information system to track inventory and an efficient distribution system for transshipment could result in the virtual setting with transshipment achieving levels of efficiency impossible with the depot structure and no transshipment. Our investigation is the first to scrutinize this claim and evaluate the relative savings of implementing an allocation or transshipment system structure for given lead times, demand uncertainty, and cost structures. To efficiently address model comparisons this study focuses on two retailers, sufficiently capturing key model differences, many of which can be generalized to the multiretailer case. We seek to understand which system structure provides lower inventory costs for a given set of system parameters including mean demand, demand coefficient of variation, demand correlation, inventory and transshipment costs, and delivery lead time. Moreover, we examine how changes in these system parameters differently affect the two system structures. The structure of this study is similar to earlier studies which found issues with delayed product differentiation (postponement) models as first examined by Lee (1996) and Lee and Tang (1997). The allocation decision is analogous to the postponement decision of how many steps in the production process should be standardized. Each additional standardized step corresponds to a longer lead time from the warehouse to the centralized depot. However, since the postponement setting in production results in finished products that are uniquely different, the analogy deteriorates for the more flexible supply-chain situation where retailers may now incorporate transshipments of identical inventories. The transshipment system structure may provide significant cost benefits despite incurring additional transshipment costs and for a wide range of parameters tested, employing
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the transshipment system structure was quite often more beneficial than using the allocation option. We further explore situations for higher transshipment costs that result in no immediate single-period cost benefit for stock transfers yet may still result in lower multi-period costs. It was found that for both system structures, an increase in demand coefficient of variation had a greater effect than an identical proportional increase in delivery lead time. Furthermore, we show that increases in the demand coefficient of variation beyond a threshold value will produce a higher cost increase in the allocation system structure than in the transshipment structure. We provide guiding principles for allocating resources to one or both retailers with the purpose of achieving reductions in demand variability and/or delivery lead times for both the allocation and transshipment system structures.
LITERATURE REVIEW The choice of facility location during the design of a supply chain system can be critically important in determining whether a business will remain competitive and has long been an important topic of interest in the decision sciences.Examples of efforts on specific facility design issues include the classical work of Markland (1973), where mathematical programming and simulation were employed to determine warehouse configurations; the work of Swink and Robinson, Jr. (1997), who considered the effectiveness of managers incorporating decision support systems to assist with facility network design; and the efforts of Cattani, Perdikaki, and Marucheck (2007), where supply chain length for online grocers was considered. What is absent from the literature are more comprehensive approaches that yield insights into the relative merits of different methods, and we initiate filling this gap by considering a direct comparison of the merits of incorporating an allocation strategy from an intermediate depot versus developing a
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transshipment structure between retailers. Below is a summary of the directly relevant previous research efforts regarding the allocation and transshipment strategies. Allocation from a Depot Research in stochastic inventory allocation models consisting of one warehouse serving multiple retailers under periodic review include the work of Eppen and Schrage (1981), Federgruen and Zipkin (1984), Schwarz (1989), Erkip, Hausman, and Nahmias (1990), and Jackson (1988), among others. For an overview the reader can refer to Axsäter (2003). Axsäter, Marklund, and Silver (2002) studied a two-step allocation in which step 1 determines how much of the depot inventory should be shipped to retailers and step 2 determines how this inventory should be allocated among retailers. Axsäter (2007) developed a heuristic allocation rule for a continuous time setting. Gürbüz, Moinzadeh, and Zhou (2007) compared replenishment strategies in a one-warehouse multi-retailer model with fixed order costs, inventory costs, and transportation costs and found that, even though an echelon-based inventory policy dominates an installation-based policy, a hybrid policy that utilizes both echelon and installation information performs best. Caro and Gallien (2010) documented a recent application of the allocation distribution modeling approach at fast-fashion retailer Zara. The improved allocation process they proposed increased sales and reduced the need for transshipments. Marklund (2011) proposed a model for the joint evaluation of inventory replenishment and shipment consolidation effects in a onewarehouse multiple-retailer system with real-time point-of-sales data. Using a first-come-firstserve (FCFS) policy to allocate stock, the author presented a recursive method to evaluate inventory holding and shortage costs and proposed two heuristics to obtain near-optimal shipment intervals. Howard and Marklund (2011) used the results obtained by Marklund (2011)
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to compare the performance of the FCFS allocation policy with the myopic allocation policy. The authors found a modest average improvement with the myopic allocation policy over the FCFS policy when the allocation was based on information available at the moment of shipment, while substantial benefits were possible using the myopic allocation policy when the allocation was based on information at the moment of delivery. Retailer Transshipment Literature on transshipment performed to avoid product stock-outs includes Robinson (1990), who considered a centralized inventory model with non-identical retailers and negligible replenishment and transshipment lead times. He demonstrated optimality of the base-stock policy and presented approaches to determine S, the order-up-to level. Herer, Tzur, and Yücesan (2006) used infinitesimal perturbation analysis to find optimal base-stock values and a network linear programming formulation to find optimal transshipment quantities. Tagaras (1989) showed that under certain conditions a complete pooling policy is optimal for a system of two retailers with negligible replenishment and transshipment lead times. He also analyzed the effect of inventory pooling on service level. Tagaras and Cohen (1992) extended this result by considering non-negligible replenishment lead times and allowing partial inventory pooling, where some of each retailer’s stock is not available for transshipment. They demonstrated that complete inventory pooling typically outperformed partial pooling. Herer and Rashit (1999) considered a two-location single-period model with fixed individual and joint replenishment costs. They showed the optimal policy is not in the form of a reorder quantity, base-stock policy but rather depended on the distribution of initial inventory at each retailer. Tagaras and Vlachos (2002) explored how the appropriate transshipment policy was affected by
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the distribution and variability of demand when both replenishment and transshipment lead times were non-negligible. Wee and Dada (2005) analyzed the option of inventory pooling from either a common warehouse or other retailers. They derived five different transshipment protocols and defined conditions for which each of them was optimal. Lien, Iravani, Smilowitz, and Tzur (2010) analyzed and compared different configurations for transshipment networks. They showed that chain configurations, where retailers may only transship to adjacent retailers, may be more efficient than particular group configurations where retailers may transship to any other retailer within the same group. The research mentioned thus far was focused on analyzing reactive transshipments where transshipment was carried out as a response to a location facing a stockout. In contrast, Tiacci and Saetta (2011) presented a proactive transshipment heuristic for stock balancing in order to prevent future stockouts. The proposed heuristic considered the impact of stock transfer on expected costs rather than on service levels when deciding the amount and timing of transshipments, and was tested using a simulation study. Naseraldin and Herer (2008, 2011) considered incorporating lateral transshipments and the effect they have upon the optimal number and locations of retail outlets. Paterson, Kiesmüller, Teunter, and Glazebrook (2011) provided a recent comprehensive survey for both proactive and reactive transshipments but did not indicate any articles that directly compared transshipment with any other approaches. Bookbinder and Heath (1988) recognized the importance of considering the relative merits of direct shipments from warehouses to retailers versus incorporation of an intermediate depot between them, and included a very limited consideration of such issues in a paper that was centered upon replenishment in the supply chain. No significant results were revealed for
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assessing the best configuration and transshipment issues were not compared. We will now develop specific mathematical models to focus upon a direct comparison of incorporating an allocation structure versus a transshipment configuration.
PROBLEM AND MODEL DESCRIPTION For a single-product, periodic-review, two-retailer inventory system supplied by a perfectly reliable common warehouse with infinite capacity, we model a variety of system structures illustrated in Figure 1: MA, with allocation (A) of inventory from a centralized depot to retailers; MT, with transshipment (T) between retailers; MLB, a flexible lower bounding (LB) system with both allocation and transshipment; and MUB, an inflexible upper bounding (UB) system with neither transshipment nor allocation. MA benefits from costless postponing of the allocation of inventory to specific retailers, but the allocation must be chosen using stochastic demand information prior to actual demand realization. MT benefits from the quick transshipment recourse option made after demand materializes, while incurring unit transshipment costs. MLB provides a lower bound because it benefits from invoking both allocation and transshipment when appropriate; while MUB provides an upper bound because it can invoke neither allocation nor transshipment (smaller feasible region implies equal or higher average costs). ------------------------------------------------------Insert Figure 1 Here ------------------------------------------------------Let w denote the central warehouse, 0 the depot, and 1and 2 the retailers. The demand at retailer j in period t is stochastic, stationary, and denoted by Dj(t) with mean µj, standard deviation σj, and coefficient of variation cvj = (σj / µj). All lead times are a nonnegative integer multiple of the base time period. Let Lwj denote the lead time for replenishment from the warehouse to location j, for j=0,1,2. Let L0j denote the replenishment lead time from the depot to
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retailer j, for j=1,2, which includes processing time at the depot and transportation time to retailers. Unless otherwise stated, in comparing different systems, we assume Lwj=(Lw0 + L0j). The lead time for transshipment between retailers is negligibly small and assumed to be zero because expedited transportation between retailers is executed in a fraction of a time period. This assumption is common in the transshipment literature (e.g., Tagaras & Cohen, 1992; Herer & Rashit, 1999; Herer, Tzur, & Yücesan, 2006; Paterson et al., 2011). Per Tagaras and Vlachos (2002), incorporation of non-negligible transshipment lead time requires additional decisions relating to the timing and quantity of proactive transshipments to rebalance retailer inventories; these additional decisions do not improve the transshipment option. Hence, we view the unit transshipment cost cij as a proxy for transshipment lead time. Firms routinely use third‐party logistics providers for transportation, so they can chose an expensive but faster transportation mode vs. a standard (less expensive) but slower transportation mode. Thus, one could argue that if the distance between retailers is large (which would imply a positive lead time if transshipped via the standard mode), transshipment is only feasible if it is done through a fast transportation mode (to be able to meet demand in that same period), which necessitates a high cij. The shipment cost for replenishments to each retailer is sunk and not considered explicitly; in contrast, the unit transshipment cost from retailer i to j, cij, is significant because quick emergency shipping is relatively expensive. For retailer j, the per-period unit holding cost is hj and the per-period unit shortage cost is pj. As in Robinson (1990), we assume the following for transshipments: (i) hi ≤ (cij + hj) and (ii) pj ≤ (cij + pi). These assumptions preclude situations for which transshipments are made to circumvent only holding costs or penalty costs, and not for satisfying additional demand. Robinson (1990) also assumed cij ≤ (hi + pj) to guarantee no increase in period costs from transshipment but we allow this condition to be violated while
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forcing transshipments to explore the effects of higher cij on MT. We will sometimes drop the retailer’s subscript for parameters such as hi , pj, and demand cvj when we focus on identical retailers. In period t, the key decision variables are Xj(t), the amount to order at location j; Aj(t), the amount of depot stock to allocate to retailer j; and Zij(t), the amount to transship from retailer i to retailer j. The general sequence of events, shown in Figure 2, is (i) the centralized depot receives delivery and then allocates its stock to retailers, (ii) the retailers receive material shipments, (iii) demand is realized, (iv) transshipments are made, (v) demand is satisfied and costs are incurred, and (vi) replenishment orders are placed. In systems with a depot, it is the only location that places replenishment orders and Xj(t)=0, j=1,2; in the absence of a depot, retailers place replenishment orders and X0(t) and Aj(t) default to zero. When retailer transshipment is not permitted, Zij(t) is zero. ------------------------------------------------------Insert Figure 2 Here ------------------------------------------------------We now develop a mathematical model for the general case with both allocation and transshipments (MLB). As stated earlier and illustrated in Figure 2, the model for MLB may be easily modified to obtain the models for MA, MT, (and MUB) by deleting variables corresponding to disallowed actions. We start the model development within period t just after demand Dj(t) occurs (Figure 2). Transshipment quantities, Zij(t), are based on NIj(t), the net inventory just before transshipment, which is given by
NI j (t )= I j (t − 1) + X j (t − Lwj ) + Aj (t − L0 j ) − D j (t ), j= 1, 2
(1)
Based on results from Tagaras and Cohen (1992), a complete pooling policy for transshipments is employed. That is, a transshipment from i to j is only possible when NI i (t) >0
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and
0, Xj(t) = 0 and AIPj(t) = IPj(t−1). After the depot receipt of X0(t – Lw0), this inventory is entirely allocated to the retailers. In
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general, the allocation problem may be written as a linear program minimizing expected inventory costs that accrue in the time periods in which the allocated amounts are delivered. For the two-retailer case, we use a simple proportional allocation policy, with closed-form expressions for the allocations, that seeks to balance inventory positions after allocation. Specifically, we maintain the ratio [IP1 (t − 1) + A1 (t)] [IP2 (t − 1) + A2 (t)] as close to a constant ρ as permitted by constraint
∑ A (t ) =X j
0
(t − Lw0 ), j = 1, 2
(6)
j
For identical retailers, setting ρ =1 yields the well-known and effective myopic allocation policy (Federgruen & Zipkin, 1984). In the case of non-identical retailers we search for the best value of
ρ that provides minimum average cost. When depot inventory is constrained, the allocation mimics the virtual allocation rule from Graves (1996). Solving [ IP1 (t − 1) + A1 (t )] =ρ
[ IP2 (t − 1) + A2 (t )] and Equation (6) simultaneously yields a retailer 2 allocation of ( X 0 (t − Lw0 ) + IP1 (t − 1) − ρ IP2 (t − 1)) / (1 + ρ ) . When this allocation becomes negative, the available inventory of X0(t – Lw0) is insufficient to maintain the ratio of inventory positions after allocation at ρ and the amount allocated to retailer 2 is zero. Thus A2 (= t ) ( X 0 (t − Lw0 ) + IP1 (t − 1) − ρ IP2 (t − 1)) + / (1 + ρ ) .
(7)
The objective is to minimize expected long-run average inventory holding and shortage costs plus transshipment costs at the retailers. The optimal average cost for the supply chain is specified by (8)
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where the subscript m = A, T, LB, or UB, depending on the system structure. The mathematical program that encompasses all system structures consists of Equation (8) subject to Equations (1)–(7); non-negativity constraints for variables Aj(t), Zij(t), and Xj(t); and additional constraints setting variables such as Aj(t) and/or Zij(t) to zero to disable allocation and/or transshipment in system structures where these actions are not permitted. The objective, with
I j (t ) − max(0, − I j (t )) , may be linearized by adding new nonI j (t ) + = max(0, I j (t )) and = negative variables I j (t ) + and I j (t ) − with additional constraints of the form
.
For any system structure, we use a two-stage procedure to solve the problem. In the first stage we search for Sj. In the second stage, the first-stage variable Sj is known and simulation is used to estimate the resulting expected costs with allocations and transshipments determined for each sample path using Equations (2), (6), and (7). Golden section search, one of several regionelimination search techniques that does not require the usage of derivatives (Bazaraa, Sherali, & Shetty, 1993), is implemented for identical retailers and embedded golden section searches are used for for non-identical retailers (Feng & Rao, 2007). Several alternative search techniques such as interval halving or Fibonaci search may likewise have been explored, but the golden section method proved to be quite efficient. For larger systems with more retailers, other search techniques may be preferred including Nelder-Mead search (Rosales, 2011).
ANALYTICAL RESULTS In this section we focus on the special case of identical retailers, with cij≤(hi+pj) and Lwj=(Lw0+L0j), to prove analytical results on how factors such as the transshipment cost, lead times, demand or cost scaling, and demand coefficient of variation affect the average cost of
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different system structures and the comparison between MA and MT. For conciseness, all missing proofs are presented in the Appendix. The questions motivating our work include the following: (i) Which system, MA or MT, dominates over which parameter settings? At what point does transshipment cost cij become large enough that MA becomes the preferred option? (ii) How do average costs CA and CT change with demand coefficient of variation cvj and how does this affect the relative preference for MA or MT? (iii)Which parameter, demand cvj or lead time Lwj, has greater impact on average costs? Let
, for m = A or T, denote the optimal average cost for demand D′ (t ) = ν D(t ) and
costs h′ = χ h , p′ = χ p , and cij′ = χ cij for all i, j, where ν and χ represent positive scaling of demand and costs, respectively. Comparing average cost for this scaled version with the original unscaled
, we note that
/
=
/
, so that any comparison of optimal average costs
for MA and MT is invariant to demand and cost scaling. While this scaling result is intuitive and also applies to demand location translation (i.e., D′ (t ) = D(t ) +∆ for constant ∆), it does not hold for time scaling (e.g., if all lead times are multiplied by κ with Lij′ = (κ Lij ) for all i, j). Rescaling time affects the end-of-period cost accounting; consequently, the average cost objective is not linear in κ . Comparison of CA and CT clearly depends on values of parameters such as lead times, demand uncertainty, and unit transshipment costs. We use
to denote the optimal average
cost as a function of any parameter(s) with all other parameters remaining unchanged. Primary consideration will be given to ζ = cv and η, a scaling factor for the cij. Thus
denotes the
optimal average cost with positively-scaled transshipment costs satisfying
for all i, j,
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and other costs unchanged, in other words, h′′ = h and p′′ = p . The current
and
is non-decreasing in η .
Proposition 1: There exists a threshold transshipment cost multiplier, η , for which If η ≥ 1 then
≥
; if η ≤ 1 then
≤
=
.
.
The threshold η provides a break-even transshipment cost multiplier value for which the decision maker is indifferent, from an average cost perspective, between MA and MT; larger values of η imply transshipment is more favorable than allocation. Next we discuss the impact of retailer demand coefficient of variation cv on average cost, , under constant mean demand µ. D(cv) denotes period demand parameterized by cv and attention is restricted to continuous demand distributions that are stochastically convex (SCX, Shaked & Shanthikumar, 2007, Chapter 8) in cv. By definition, the SCX property implies that E[ϕ(D(cv))] is convex in cv when ϕ(⋅) is convex; we will use a specific convex (and locationinvariant) function ϕ(⋅) that corresponds to
. Common distributions of demand
include the normal, uniform, and gamma. For normal demand, D(cv)=N(µ,µcv) =µ+µcvN(0,1), where N(µ,σ) is normal with mean µ, and standard deviation σ. For uniform demand, , where U(0,1) is uniform between 0 and 1. For gamma 1 1 demand,= D(cv) γ 2 , µ cv ≡ µ cv ⋅ γ 2 ,1 , where γ(α,β) is gamma with shape α and rate β. cv cv
The first result below confirms that these common demand models are SCX in cv. The next result examines convexity of the average cost
in D(cv).
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Lemma 1: For constant mean, µ, D(cv) is SCX in cv when the demand distributions are normal or uniform. When demand is gamma distributed, D(cv) is SCX in cv if we restrict attention to CX functions that are demand location-invariant and scale linearly with the gamma rate parameter. Lemma 2: The
optimization is a convex program.
is convex in the right hand side b,
where the entries of vector b consist of either zero or a demand, D(cv). The above convexity results coupled with stochastic convexity from Shaked and Shanthikumar (2007) yield the following result on how
changes with cv.
Proposition 2: For normal, uniform, or gamma demand with constant mean,
is
increasing and convex (ICX) in cv. We use Proposition 2 to prove a cv threshold result similar in spirit to the transshipment threshold result in Proposition 1. The proof requires an average cost comparison with two other system structures that may be of independent interest, which we call the super retailer (SR) system and the pooled (P) system. In SR, which is a special case of transshipment, one of the retailers acts as a “super retailer”—it is the only location to order from the warehouse and all of the other retailer’s demand is satisfied by transshipment from the super retailer. In P, both retailers are pooled into one and this pooled retailer experiences the total demand. Clearly, the pooled retailer provides a lower bound on average cost, that is,
,
where m corresponds to the A, T, LB, and UB systems from Figure 1. Further, because SR is a special (restricted) case of T, it provides an upper bound on T, that is, . The shape of these average total costs (and their relative magnitudes) as a function of cv is illustrated in Figure 3 and forms the basis of the Appendix proof of the threshold result in Proposition 3.
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------------------------------------------------------Insert Figure 3 Here ------------------------------------------------------Proposition 3: For SCX D(cv), there exists a threshold value, cvθ , such that
≥
when cv ≥ cvθ . Further, cvθ increases with cij and decreases with L0j (with Lwj fixed). The result on how the threshold value cvθ changes with other parameters such as transshipment cost and lead time may be inferred from Figure 3. Clearly, as cij increases, the CT in Figure 3 increases while CA remains unchanged; consequently the corresponding threshold value, cvθ , increases. Similarly, as the depot-to-retailer lead time, L0j, increases with the warehouse-toretailer lead time, Lwj, staying fixed,
increases with L0j while
remains unchanged,
so the corresponding threshold value cvθ decreases. Overall, our analysis supports the intuition that transshipment will dominate allocation for low transshipment costs, high demand uncertainty, low total (warehouse-to-retailer) lead time, and high depot-to-retailer allocation lead time. However, what constitutes low and high values (e.g., magnitude of the thresholds defined by Propositions 1 and 3) is less clear, as is the relative impact on average costs of demand cv and lead times. Hence we conduct computational experiments to better understand the role of cost parameters, demand cv, and lead times on the relative performance of the MA and MT systems.
NUMERICAL RESULTS For our numerical experiments we use simulation (Law & Kelton, 2000) to obtain point estimates of the optimal average costs for the MT, MA, MUB, and MLB models, denoted as CT, CA, CUB, and CLB, respectively. The simulation model was coded in C++, and a golden section search
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was used to find the optimal base-stock level for retailer j, Sj in MT, and to find the optimal order-up-to level for the depot S0, and the optimal target inventory positions for allocation in MA. The demand for retailer j is an independent gamma distribution, γ(αj,βj), with shape αj and scale
βj, satisfying αj, βj > 0. This demand has mean αjβj, variance αjβj2, and coefficient of variation . We use the gamma distribution because it is flexible and maps to several common distributions such as the Erlang and the normal for different values of shape and scale. Further, the gamma distribution permits higher levels of coefficient of variation than the more common normal distribution. Gamma random variates are generated using the modified rejection technique by Ahrens and Dieter (1982); the uniform random variates used by this technique are generated as in Marse and Roberts (1983). For tests involving both positively- and negativelycorrelated demands, the method of Schmeiser and Lal (1982) as coded by Van Damme (2010) was employed to generate demand variates. Identical streams of random numbers for demand realizations are used as a variance reduction technique for comparing simulation output from the different models. The initialization for the simulation is 5,000 time periods; data is collected over the next 200,000 periods using the batch-means method with 40 batches each of 5,000 periods resulting in a 95% confidence interval on average cost with a half-width of 1.05%. Whenever comparisons of the MT and MA models are made the paired t-test confidence interval method is used. For our numerical experiments we use the following parameter values. We begin with a base case of identical retailers set as μ=100, cv=.5, h=1, p=5, cij=4, Lwj=5, and L0j=1, and then vary with a complete factorial design over cv=0.1, 0.5, 1.0, and 2.0; cij=4, 5, 6, and 7; Lwj=5, 6, 8, 10, 15, 20, and 50; and L0j=1, 2, 3, and 4, yielding 448 identical retailer scenarios, denoted as the base set. We classify parameters to be either low or high as follows: (i) low transshipment cost:
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cij ≤(hi + pj), (ii) low demand cv: cv ≤ 0.5, consistent with the classification in Hopp and Spearman (2011), and (iii) low lead time: L0j < 4 and (L0j/Lwj)≤ 0.4. The objective of this numerical study is to provide insights on the relative performance of the MA and MT models. The following are some of the questions we seek to answer: •
Under different parameter values, which system structure, MA or MT, is more likely to provide lower average costs?
•
How do CA and CT behave relative to the upper and lower bounds?
•
How do CA and CT behave with changes in cv, correlation, and Lwj?
•
Which system parameter, cv or Lwj, has a higher impact on CA and CT?
•
Relative to h and p, how large can we expect the threshold transshipment cost (from Propostion 3) to be for CT to exceed CA?
Major observations from the numerical results will be divided into two sections. In the next section we consider the impact upon average cost that results from changes in cv, correlation, lead times, and transshipment costs, cij. In the subsequent section is an examination of preferred strategies for making monetary investments to enhance system performance. Impact of System Structure Observations 1, 2, and 3 are derived from the results obtained from our numerical experiments on the base set defined above. Observation 1: For either MA or MT, supply chain managers will obtain greater average cost reduction by investing resources to reduce demand cv than by investing resources to produce an identical proportional reduction to Lwj. Figure 4 illustrates how CT and CA behave with changes in Lwj and cv. Note that CA and CT increase in a concave fashion with increases in Lwj, while the increase with cv is convex for
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CA and CT, consistent with Proposition 2. As a result of this behavior, an increase to cv produces a higher impact on CA and CT than an identical proportional increase to Lwj. As an example, notice from Figure 4(a) that a fourfold increase in Lwj (from 5 to 20) produced less than a twofold increase for CA and CT, while a fourfold increase in cv (from 0.5 to 2.0) produced a greater than fourfold increase for CA and CT. Additional scenarios were tested with very low values fir cv and Lwj (cv = 0.01, 0.02, 0.03, and Lwj =1, 2, 3), and similar results were obtained. This observation is consistent with basic supply chain theory. For example, safety stock under continuous review with stationary stochastic demand, with standard deviation σ, deterministic lead time L, and safety stock factor z, is given by zσL1/2, so it increases linearly in σ (or cv) but is a concave function of L. The presence of multiple locations with allocation and transshipment prevents simple closed-form safety stock expressions, but our numerical experiments confirm that typically demand cv has a bigger impact on cost than lead time L. ------------------------------------------------------Insert Figure 4 Here ------------------------------------------------------Table 1 provides an aggregate summary of the results from the 448 base case scenarios for each level of the four parameters L0j, Lwj, cij, and cv. Results include the frequency and percent of instances in which CT
