May 27, 2008 - in both a centralized and a decentralized inventory system. We show that using a ... These papers provide a foundation for our analysis, but.
Centralization versus Decentralization: Risk Pooling, Risk Diversification, and Supply Uncertainty in a One-Warehouse Multiple-Retailer System Amanda J. Schmitt Lawrence V. Snyder Dept. of Industrial and Systems Engineering Lehigh University Bethlehem, PA, USA
Zuo-Jun Max Shen Dept. of Industrial Engineering and Operations Research University of California Berkeley, CA, USA
May 27, 2008 ABSTRACT We investigate optimal system design in a One-Warehouse Multiple-Retailer system in which supply is subject to disruptions. We examine the expected costs and cost variances of the system in both a centralized and a decentralized inventory system. We show that using a decentralized inventory design reduces cost variance through the risk-diversification effect, and that when demand is deterministic and supply may be disrupted, a decentralized inventory system is optimal. This is in contrast to the classical result that when supply is deterministic and demand is stochastic, centralization is optimal due to the risk-pooling effect. When both supply may be disrupted and demand is stochastic, we demonstrate that a risk-averse firm should typically choose a decentralized inventory system design.
1
Introduction
As supply chains expand globally, supply risk increases. Classical inventory models have generally focused on demand uncertainty and established best practices to mitigate demand risk. However, supply risk can have very different impacts on the optimal inventory management policies and can even reverse what is known about best practices for system design. In this paper, we focus on the impact of supply uncertainty on the One-Warehouse MultipleRetailer (OWMR) system, and compare two policies: centralization (stocking inventory at the warehouse only) and decentralization (stocking inventory at the retailers only). While most research 1
on the OWMR model allows inventory to be held at both echelons, we allow inventory to be held at only one echelon in order to consider two opposing effects that can occur: risk pooling and risk diversification. The risk pooling effect occurs when inventory is held at a central location, which allows the demand variance at each retailer to be combined, resulting in a lower expected cost [12]. The risk diversification effect occurs when inventory is held at a decentralized set of locations, which allows the impact of each disruption to be reduced, resulting in a lower cost variance [23]. Whereas the risk-pooling effect reduces the expected cost but (as we prove) not the cost variance, the risk-diversification effect reduces the variance of cost but not the expected cost. We prove that the risk diversification effect occurs in systems with supply disruptions. We also consider systems with both supply and demand uncertainty, in which both risk pooling and risk diversification have some impact, and numerically examine the tradeoff between the two. We employ a risk-averse objective to determine which effect dominates the system and drives the choice for optimal inventory system design. Specifically, comparing centralized and decentralized inventory policies, we contribute the following: • The exact relationship between optimal costs and inventory levels when demand is deterministic and supply may be disrupted • The exact relationship between optimal cost variances when: – demand is deterministic and supply may be disrupted – supply is deterministic and demand is stochastic • Formulations of the expected cost and cost variance when supply is disrupted and demand is stochastic • Evidence that decentralization is usually optimal under risk-averse objectives The remainder of the paper is organized as follows. In Section 2 we review the relevant literature. In Section 3 we analyze the risk-diversification effect in the OWMR system with deterministic demand and disrupted supply. We consider stochastic demand in Section 4. In Section 5 we consider both demand uncertainty and disrupted supply and again compare inventory strategies, using a risk-averse objective to choose the optimal inventory design. We summarize our conclusions in Section 6.
2
2
Literature Review
2.1
Single-Echelon Models
The two most commonly considered forms of supply uncertainty are supply disruptions (in which supply is halted entirely for a stochastic amount of time) and yield uncertainty (in which the quantity delivered from the supplier is random). The literature on single-echelon systems with both of these types of supply uncertainty is extensive, and we omit an exhaustive review here. The reader is referred to Yano and Lee [29] for a discussion of the literature on single-echelon systems with yield uncertainty. Supply disruptions have been considered in several settings, including the EOQ model [e.g., 4, 18, 22] and newsboy settings [e.g., 10]. Chopra et al. [8] and Schmitt and Snyder [20] consider systems that have both types of supply uncertainty simultaneously. In this paper we will rely on several results for single-echelon base-stock systems with disruptions developed by Schmitt et al. [21] and Tomlin [25]. These papers provide a foundation for our analysis, but our application to the OWMR model provides new insights on the impact of supply disruptions in complex systems. We consider risk-averse objectives for inventory models, a topic which is gaining momentum in the operations literature. Risk-aversion has been considered in newsboy models to mitigate demand uncertainty. For example, Eeckhoudt et al. [11] show that order quantities decrease with increasing risk-aversion. Van Meighem [28] considers resource diversification in newsboy models with riskaverse objectives, advocating diversifying resource availability to protect against risk. Tomlin and Wang [27] consider a single-period newsboy setting with supply disruptions; they model loss-averse and conditional value-at-risk (CVaR) objectives when deciding between single- and dual-sourcing and between dedicated and flexible resource availability. Chen et al. [7] consider multi-period inventory models with risk-aversion, modeling both replenishment and pricing decisions. Tomlin [25] also considers a multi-period setting and employs a mean–variance approach to consider risk in a two-supplier system in which one supplier is subject to disruptions and the other is perfectly reliable but more expensive. He uses risk-averse objectives to decide between single- and dualsourcing.
2.2
Multi-Echelon Systems
The literature on supply uncertainty in multi-echelon inventory systems is not extensive. The review by Yano and Lee [29] on yield uncertainty includes a section on multiple-facility systems, but most 3
of the literature they discuss focuses on single-period models and/or deterministic demands. One exception is the paper by Bassok and Akella [3], in which the firm orders from an uncertain-yield supplier to ensure that material is available for a constrained production setting; upstream inventory level decisions and downstream production level decisions are integrated in their model. Another is Tang [24], who considers a multi-stage, multi-period model with uncertain yield; his focus is on solving for production rates under the assumption that the fixed inventory levels are always able to satisfy downstream demand. Another body of literature considering multi-echelon systems with uncertain supply is the literature on produce-to-order models; see Grosfed-Nir and Gerchak [14] for a review. This literature tends to focus on a single, known customer order that must be satisfied through multiple uncertainyield production batches. Only recently have researchers begun to address supply uncertainty in multi-echelon systems with multiple periods and/or stochastic demand. The primary focus is on identifying the form of optimal inventory policies. Gurnani et al. [15] consider an assembly system with two raw materials subject to random yield that are used to produce a single end-product. They develop closed-form solutions for the optimal order and production quantities by assuming that the two suppliers will never be simultaneously unavailable. Bollapragada et al. [5, 6] consider a two-echelon serial system and a two-echelon pure assembly system. They assume that the incoming raw material has a stochastic capacity, which creates uncertain replenishment lead times. They suggest that using an internal service-level requirement is a robust way to decompose the model into individual singleechelon systems and that the internal service level does not relate linearly to the required external service level. Hopp and Yin [16] examine a multi-echelon assembly system in which each node may be subject to supply disruptions. They assume deterministic demand and zero lead times and consider both capacity and inventory to protect against disruptions. They show, under certain conditions, that at most one node in each path to the customer will employ capacity or inventory buffers, and that as disruptions become more severe, capacity and inventory buffers shift upstream toward the disrupted node.
2.3
OWMR Systems
In this paper we consider supply uncertainty in a multiple-demand-point, single-warehouse model, which we refer to as a One-Warehouse Multiple-Retailer (OWMR) system, and focus on inventory 4
held at a single level only in order to draw clear conclusions. Eppen [12] considers this system and shows that under demand uncertainty, a centralized inventory strategy provides risk-pooling benefits and reduces expected costs versus a decentralized strategy. The OWMR system with uncertain demand is notoriously difficult to analyze when inventory may be held at both echelons; no closed-form optimal solutions for this case are known. Chu and Shen [9] develop an approximation based on power-of-two policies that guarantees a worst-case error bound of 26%. Gallego et al. [13] introduce three related heuristics for the OWMR system with local control, each involving a different inventory policy at the warehouse, and show that one of them is asymptotically optimal in the number of retailers. They also present a lower bound for the system with centralized control. Lystad and Ferguson [17] find approximate order-up-to levels at the warehouse by decomposing the system into separate 2-stage serial systems and using newsvendor bounds. Agrawal and Seshadri [1] examine a similar setting with multiple retailers and a central distributor in a single-period model. The retailers are risk-averse and the distributor offers a single menu of contracts to all retailers to induce them to order. The paper shows that risk sharing increases the attractiveness of decentralization, allowing for different contract menus for different levels of retailer risk aversion; on the other hand, risk-pooling benefits make centralization attractive. Snyder and Shen [23] use simulation to study multiple complex inventory systems, including the OWMR system with supply uncertainty with inventory at a single level. Their simulation results show that, under supply disruptions, expected costs are equal for centralized and decentralized systems, but the variance of the cost is higher in centralized systems. They call this the riskdiversification effect and suggest that it occurs because a disruption in a centralized system affects every retailer and causes more drastic cost variability. They conclude that risk diversification increases the appeal of inventory decentralization in a system with disruptions. In this paper, we develop a theoretical model for the expected cost and cost variance of the OWMR system subject to supply uncertainty. We analytically prove the presence of risk diversification in this system, discuss its impact, and examine the system under uncertainty in both supply and demand. When demand is deterministic and supply is subject to either disruptions or yield uncertainty, we determine the optimal inventory levels and costs. For those cases, as well as the case in which demand is stochastic and supply is deterministic, we quantify the cost variance. We combine supply disruptions and stochastic demand in a subsequent model and formulate the expected costs and cost variances. We show that under a risk-averse objective function, decentralization is 5
usually the optimal inventory design.
3
OWMR with Supply Disruptions
3.1
Assumptions and Notation
In this section we examine the impact of supply disruptions on the OWMR system. We consider a base-stock inventory system with one central warehouse and multiple retailers, and compare policies for stocking inventory at the warehouse only (centralized system) and at the retailers only (decentralized system). We assume orders are placed every period regardless of any fixed-order costs, which is not unreasonable in retailer settings, so we set order costs equal to zero in our model (we discuss what impact this might have in our conclusions). We assume zero lead times since deterministic lead times would not impact optimal base-stock levels or order quantities, but would simply require that orders be placed exactly the lead time quantity in advance. We also assume deterministic demand, but will relax the latter assumption in Section 4. We assume that in both the centralized and decentralized settings, disruptions occur randomly at the inventory location(s); that is, disruptions occur at the warehouse in the centralized system and at the retailers in the decentralized system. We make this assumption in order to consider the impact of inventory location; if we modeled disruptions as affecting the entire supply chain, then no mitigation through inventory location could be achieved and alternate protection or recovery plans would be required. Relevant examples of disruptions of this nature could include transportation delays, capacity issues at the supplier which cause individual orders to be delayed, and labor disputes or weather issues that interrupt material handling at a single location. We assume disruptions follow a random process governed by the pmf πi and cdf F (i), where i is the number of consecutive periods during which a given stage has been disrupted, as is typical in supply disruption literature [e.g., 8, 16, 25]. Therefore πi is the probability that a given stage has been disrupted for precisely i periods (i ≥ 0) and F (i) is the probability that it has been disrupted for i periods or fewer. Disruptions may be governed by a Markov chain or a more general process. They follow the same process at every location, and they are independent over time and (in the decentralized system) across locations. In both systems, disruptions “pause” the flow of supply to the disrupted stage, but any inventory at that stage may still be used during the disruption; therefore, disruptions affect a stage’s supply-receiving function but not its demand-receiving function. A holding cost of h per unit per period is incurred at the warehouse in the centralized system 6
Notation n d p h i πi F (i) S∗ ∗ SD SC∗ E[C] E[CD ] E[CC ] V [C] V [CD ] V [CC ]
Table 1: Notation Definition number of retailers deterministic demand per period at each retailer penalty cost per period at each retailer holding cost per period at each retailer or the warehouse index representing being in a state with i disrupted periods in a row probability of being in state i cumulative distribution function for the failure states optimal base-stock level for an individual facility optimal base-stock level for each retailer in the decentralized system optimal base-stock level for the warehouse in the centralized system expected cost for an individual facility expected cost for the decentralized system expected cost for the centralized system variance of the cost for an individual facility variance of the cost for the decentralized system variance of the cost for the centralized system
and at the retailers in the decentralized system. Unmet demands are backordered, and a stockout penalty of p is incurred at the retailers (in both systems); there is no stockout penalty for backorders at the warehouse. The parameters, decision variables, and performance measures for the OWMR system are summarized in Table 1. We use an additional subscript s to denote the disrupted-supply model discussed ∗ is defined as the optimal base-stock level for an individual retailer in this section. Note that since SD ∗. in the decentralized system, the total inventory for that system is nSD
3.2
Mean and Variance of Optimal Cost
For a base-stock inventory policy at a single retailer, the costs in a given period depend on the state of the system, defined as the number of consecutive periods for which it has been disrupted. Tomlin [25] derives the expected cost for such a system, given in (1), and shows that the optimal base-stock level is as given in (2): Es [C] =
∞ X
πi [h(Ss − (i + 1)d)+ + p((i + 1)d − Ss )+ ]
(1)
i=0
Ss∗ = jd, where j is the smallest integer such that F (j − 1) ≥
p p+h
(2)
Schmitt et al. [21] discuss these results, demonstrating that the optimal base-stock level is an increasing step function of the newsboy fractile and the disruption probability. As the cost of 7
disruptions increases, the optimal solution increases by discrete jumps to the next whole period’s worth of demand. We make use of these results in our analysis of the OWMR system. The relationships between the inventory levels, expected costs, and cost variances for the centralized and decentralized systems are given in the following theorem. Theorem 1 In the OWMR system subject to supply disruptions, the decentralized, centralized, and single-facility optimal base-stock levels and performance measures are related as follows: ∗ = nS ∗ = nS ∗ 1. SCs s Ds ∗ ] = nE [C ∗ ] 2. Es [CC∗ ] = Es [CD s ∗ ] = n2 V [C ∗ ] 3. Vs [CC∗ ] = nVs [CD s
Proof : See Appendix, Section A.2. Thus the total system inventory and expected costs are equal in the centralized and decentralized systems, but the variance is n times greater for the centralized system, suggesting that, at least for a risk-averse decision maker, the decentralized system is the better design for this system. The intuition behind Theorem 1 is as follows. Because the total inventory is the same in both systems, the total inventory devoted to each retailer is the same in both systems. Therefore each retailer feels the effects of disruptions for the same percentage of periods, on average, in both systems. That is, a given disruption in either system causes a retailer to experience the same number of stock-out periods. We state this formally in the following Corollary. Corollary 2 In the OWMR system subject to supply disruptions, if base-stock levels are set optimally, the distribution of consecutive periods for which a given retailer is stocked out is identical for the centralized and decentralized systems. This causes the expected holding and stock-out costs to be the same. However, disruptions are less frequent but more severe in the centralized system, and therefore that system has a greater cost variance. The relationships presented in Theorem 1 demonstrate the risk-diversification effect in the OWMR system. If demand is deterministic, there is no risk-pooling benefit from centralization (since there is no risk from uncertain demand), thus expected costs for the centralized and decentralized systems are equal. However, the risk from uncertain supply is better mitigated in the 8
decentralized system, as the lower variance for that system demonstrates. Snyder and Shen [23] demonstrate a similar result using simulation, though their model differs slightly from ours in that they assume that inventory at a given stage may not be used when that stage is disrupted.
4
OWMR with Stochastic Demand
We now consider the classical model discussed by Eppen [12]: the OWMR model with deterministic supply and stochastic demand. Following Eppen [12], we assume that demand is normally distributed with mean µ and variance σ 2 at each of the retailers. Therefore, the demand at the 2 = nσ 2 . We employ a subscript warehouse in the centralized system has mean nµ and variance σC
d to denote this model. In Section 4.1, we review the results concerning the optimal base-stock levels and expected cost presented by Eppen [12]. We then evaluate the cost variance at optimality in Section 4.2. To the best of our knowledge, ours is the first study to consider the cost variance at optimality in this system.
4.1
The Risk-Pooling Effect
The expected cost and optimal base-stock level for a single retailer subject to stochastic demand are well known: Ed [C] = h(Sd − µ) + σ(p + h)Φ Sd∗
= µ + σΦ
where Φ1 (x) =
−1
R∞ x
p p+h
1
Sd − µ σ
(3)
(4)
(v −x)ϕ(v)dv is the standard normal loss function. Therefore, for the centralized
system, Ed [CC ] = h(SCd − nµ) + σC (p + h)Φ ∗ SCd
= nµ + σC Φ
−1
p p+h
1
SCd − nµ σC
(5)
(6)
∗ = Since the decentralized system functions like n single-retailer systems, E[CDd ] = nEd [C] and SDd
Sd∗ . We have the following Theorem:
9
Theorem 3 (Eppen, 1979) The optimal expected costs and inventory levels for the centralized and decentralized OWMR systems subject to stochastic demand are as follows: ∗ Ed [CD ]= ∗ nSDd
−
√
∗ SCd
nEd [CC∗ ] = (n −
√
(7) n)σΦ
−1
p p+h
(8)
Proof: Given by [12]. This is the classical risk-pooling effect: by serving the retailers’ demand from a centralized inventory site, we pool the risk from demand uncertainty. As a result, the optimal expected cost √ is n times smaller for the decentralized system, and less inventory is required.
4.2
Cost Variance
The following theorem presents the relationship of cost variance for this system. Theorem 4 At optimality, the cost variance for the centralized and decentralized OWMR systems subject to stochastic demand are equal: ∗ Vd [CC∗ ] = Vd [CD ]
(9)
Proof: See Appendix, Section A.3. The theorem suggests that, under stochastic demand and deterministic supply, centralization is optimal due to the risk-pooling effect, which affects the expected cost only and has no impact on cost variance. In contrast, under supply disruptions and deterministic supply, decentralization is optimal due to the risk-diversification effect, which affects the cost variance only and has no impact on the expected cost. A natural question is, when both demand uncertainty and disruptions are present, which system is optimal, i.e., which prevails—risk pooling or risk diversification? We address this question in the next section.
5
OWMR with Supply Disruptions and Stochastic Demand
In this section we compare the cost means and variances for centralized and decentralized OWMR systems in which each retailer is subject to random demand and the whole system is subject to disrupted supply. We assume that when supply is not disrupted, the yield is deterministic. (We focus on supply disruptions rather than yield uncertainty for ease of exposition since the 10
risk-diversification effect always holds under disruptions but only sometimes holds under yield uncertainty.) We use a subscript b to denote this model since both supply and demand are stochastic. We formulate the expected costs and cost variances in Sections 5.1 and 5.2, respectively. We perform numerical analysis to compare the performance of the two systems in Section 5.3. Finally, in Section 5.4, we introduce a mean–variance objective that allows us to choose optimally between the two systems when risk aversion is accounted for. We summarize our findings in Section 5.5.
5.1
Expected Costs
At a single retailer, a single period’s demand is denoted D and is distributed as N (µ, σ 2 ), while the total demand in i periods is distributed as N (iµ, iσ 2 ). The cost for a single retailer subject to normally distributed demand and disrupted supply is given by Schmitt et al. [21] as: Z Sb Z ∞ ∞ X Eb [C] = πi−1 h (Sb − id)fi (id)dd + p (id − Sb )fi (id)dd =
i=1 ∞ X
−∞
Sb
√
πi−1 h(Sb − iµ) + σ i(p + h)Φ
i=1
1
Sb − iµ √ σ i
(10)
Schmitt et al. [21] prove that Eb [C] is convex and argue that the optimal base-stock level cannot √ appears in the first-order condition for i = 0, 1, . . . , ∞. be found in closed form because Φ S−iµ σ i Using (10), we derive the expected costs for the two systems in the following Proposition. Proposition 5 When demand is normally distributed at the retailers and supply is subject to disruptions, the expected costs for the centralized and decentralized OWMR systems are: ∞ X √ 1 SDb − iµ √ Eb [CD ] = n πi−1 h(SDb − iµ) + σ i(p + h)Φ σ i i=1 ∞ X √ 1 SCb − inµ √ Eb [CC ] = πi−1 h(SCb − inµ) + σ in(p + h)Φ σ in i=1
(11) (12)
Proof: Eb [CD ] is simply equal to nEb [C]. Similarly, Eb [CC ] = nEb [C], but with the pooled demand √ standard deviation equal to σ n and the demand mean equal to nµ. ∗ = S ∗ . For the centralized system, we have confirmed Clearly, for the decentralized system, SDb b ∗ 6= nS ∗ in general, as in the deterministic-demand model in Section 3. However, numerically that SCb b ∗ is neither consistently greater than nor consistently less than nS ∗ when unlike that model, SCb b
demand is stochastic.
11
5.2
Cost Variance
We present the cost variance for arbitrary (not necessarily optimal) base-stock levels in the following Proposition. Proposition 6 When demand is normally distributed at the retailers and supply is subject to disruptions, the cost variances for the decentralized and centralized OWMR systems are: " ∞ ! X S − iµ Db 2 √ Vb [CD ] = n − 2SDb iµ + iσ 2 + i2 µ2 ) + 2iσ 2 (p2 − h2 )Φ2 πi−1 h2 (SDb − σ i i=1 !2 ∞ X √ S − iµ Db √ πi−1 h(SDb − iµ) + σ i(p + h)Φ1 (13) σ i i=1 ! ∞ X S − niµ Cb 2 √ − − 2SCb niµ + niσ 2 + n2 i2 µ2 ) + 2niσ 2 (p2 − h2 )Φ2 Vb [CC ] = πi−1 h2 (SCb σ ni i=1 !2 ∞ X √ S − niµ Cb √ πi−1 h(SCb − niµ) + σ ni(p + h)Φ1 , (14) σ ni i=1 where Φ1 (·) is the standard normal loss function and Φ2 (·) is the standard normal second-order loss function. Proof: See Appendix, Section A.4. ∗ ]. However, we demonstrate We have been unable to prove analytically that Vb [CC∗ ] ≥ Vb [CD
numerically in Section 5.3 that this inequality holds for every instance we tested. Thus, the riskdiversification effect appears to prevail under supply disruptions, even when demand is stochastic.
5.3
Numerical Study
In this section, we perform a numerical study to compare the performance of the two systems when supply disruptions and stochastic demand are present simultaneously. We consider n = 4 retailers. We model disruptions as a two-state, discrete-time Markov process with disruption probability α and recovery probability β. (That is, the probability of being disrupted [not disrupted] next period given that the system is not disrupted [disrupted] in the current period is given by α [β].) The steady-state probabilities for the disruption process are given by Schmitt et al. [21]: π0 =
αβ β ; πi = (1 − β)i−1 , i ≥ 1 α+β α+β
(15)
In Section 5.3.1, we vary the input parameters one by one to determine the impact of each on the performance of the two systems. In Section 5.3.2 we examine the impact of the disruption characteristics on the systems. 12
Table 2: Parameter Levels Variable Values α (failure probability) 0.05, 0.1, 0.2, 0.3, 0.4 β (recovery probability) 0.25, 0.5, 0.75 σ (supply st.dev.) 5, 15, 25 p (penalty cost) 2, 5, 10, 15, 20
Inv. Diff. ∗ − S∗ nSDb Cb Mean Med. Min. Max. 5.3.1
12 14 -78 89
Table 3: Comparison of Performance Measures E[Cost] Diff. % E[Cost] Diff. CVC CVD ∗ ]−E [C ∗ ] Eb [CD b C ∗ ∗ ∗ ) ∗ Eb [CD ] − Eb [CC ] ρ(CCb ) ρ(CDb Eb [C ∗ ] C
13.27 9.10 0.00 53.64
2.5% 0.9% 0.0% 41.9%
2.12 1.95 0.89 4.14
1.03 0.97 0.44 1.97
CV Diff. ∗ ) − ρ(CDb
∗ ) ρ(CCb
1.09 1.01 0.38 2.17
Impact of Parameters
We conducted a full-factorial experiment to determine the effect of the input parameters on the optimal base-stock levels and the cost mean and variance at optimality in the two OWMR systems. We varied σ, keeping µ fixed to 100, and varied p, keeping h fixed to 1. (This can be done with nearly no loss of generality since the model is most sensitive to changes in the ratios σ/µ and p/(p + h) rather than to changes in the individual parameters.) In addition, we varied the disruption parameters α and β. Table 2 lists the values tested for each parameter. The total number of parameter combinations is 5 × 3 × 3 × 5 = 225. For each combination of parameters, we optimized the expected cost functions (11) and (12) numerically to solve for the optimal base-stock levels in both systems. Table 3 lists the mean, median, minimum, and maximum, over the 225 instances, of several performance measures: difference in total inventory (column 2), difference in expected cost (column 3), percent difference in expected cost (column 4), coefficient of variation (CV) of the cost for the centralized and decentralized systems (columns 5 and 6, resp.), and difference in CV (column 7). The CV is given by √ ∗ V [C ] ρ(C ∗ ) = E[C ∗ ] . From Table 3 it is evident that, as suggested in Section 5.1, the decentralized system does not always stock more than the centralized one, despite the risk-pooling effect; in 80 out of 225 cases (36%), the centralized system has a higher total base-stock level. This occurs because of the discrete nature of disruptions. If demand is deterministic, then the optimal base-stock level is always an integer multiple of the demand (see Section 3.2), and demand stochasticity perturbs these
13
$10,000 $8,000
Eb[Cc] SDb[Cc] Eb[Cd] SDb[Cd]
$6,000 $4,000 $2,000 $0.980
0.985
0.990
0.995
1.000
p/(p+h)
Figure 1: Centralized and Decentralized Expected Cost and Cost Standard Deviation vs.
p p+h
discrete base-stock levels only slightly. Since the total standard deviation of demand is smaller in the centralized system, the jumps are less smooth (closer to the deterministic-demand solution) and may jump slightly above the optimal decentralized system levels. However, the optimal expected cost of the centralized system is always less than or equal to that of the decentralized system. Therefore, the expected-cost benefit of the risk-pooling effect still exists when supply disruptions are present, even though the ordering of the magnitude of base-stock levels does not. We found that the difference in expected cost is most sensitive to changes in β and σ: it increases as β increases (disruptions become shorter) or as σ increases (demand is more uncertain). In these cases, the impact of disruptions is smaller compared to that of the demand uncertainty, and so the benefits from risk pooling become more pronounced. The cost variance is always greater for the centralized system, confirming the risk-diversification effect. We found that the difference in cost variance between the two systems generally increases as p, α, and the percent down-time (1 − π0 ) increase, or as β decreases. As the impact of disruptions increases (either because they last longer—α increases or β decreases—or because stockouts cost more), the variance increases more for the centralized system than it does for the decentralized system, because the risk-diversification effect mitigates the impact of disruptions in the decentralized system. Figure 1 plots the mean and standard deviation (SD) of cost for both systems as a function of the newsboy fractile, p/(p + h). In this plot, we fixed α = 0.05, β = 0.8, and σ = 15. The mean and SD for both systems generally increase with the newsboy fractile. However, there is a sharp decrease in the SD for the centralized system at p/(p + h) = 0.99. At this value, the optimal ∗ = 891 to S ∗ = 1154. This large increase in inventory base-stock level increases sharply from SCb Cb
results in a significantly greater protection against disruptions and, as a result, in a reduction in 14
Group I
II
III
IV
V
Table 4: Disruption Profiles % Down-Time α a 0.001 b 1% 0.005 c 0.01 a 0.01 b 5% 0.025 c 0.05 a 0.01 b 10% 0.05 c 0.1 a 0.01 b 15% 0.05 c 0.1 a 0.01 b 20% 0.05 c 0.1
β 0.099 0.495 0.99 0.19 0.475 0.95 0.09 0.45 0.9 0.057 0.283 0.567 0.04 0.2 0.4
cost SD. The decentralized system has a similar, but less pronounced, decrease at the same point. Note also from Figure 1 that the expected costs for the two systems are extremely close, while the SD is substantially larger for the centralized system. We revisit this issue in Section 5.4. 5.3.2
Impact of Disruption Profile
Two pairs of (α, β) values may result in the same fraction of periods disrupted, defined as 1 − π0 = α/(α + β),
(16)
but still have very different characteristics. For example, if α and β are both close to 0, then disruptions are rare but long, while if they are both close to 1, then disruptions are frequent but short. Tomlin and Snyder [26] refer to the frequency/duration characteristics of a disruption as the disruption profile. In this section, we study the effect of the disruption profile on the optimal base-stock levels and performance measures of the two systems. We generated five groups of three (α, β) pairs each. Within each group, the (α, β) pairs have the same percentage down-time (16) but a different disruption profile. The 15 instances are listed in Table 4. The percent down-time increases with the group number (I, II, ...), while the disruption length decreases with the letter within a group (a, b, c). In all instances, we set σ = 15 and p = 10 (the middle value from Table 2) and, as before, set h = 1 and µ = 100. Table 5 presents the optimal base-stock levels and the mean and CV of cost for both systems.
15
Table 5: Base-Stock Levels and Performance ∗ Group SCb Eb [CC∗ ] ρ(CC∗ ) a 441.8 402.57 11.4 I b 441.8 130.17 7.3 c 441.8 89.76 4.0 a 451.5 1,075.26 5.7 II b 451.5 449.73 4.9 c 451.5 239.21 3.5 a 810.9 3,691.54 3.8 III b 764.6 854.80 3.0 c 745.9 411.87 1.2 a 2774.6 5,782.11 2.5 IV b 1205.8 1,791.29 2.7 c 821.7 812.71 2.4 a 4374.9 6,844.85 1.9 V b 2009.5 2,994.01 2.4 c 1214.0 1,402.56 2.2
Measures vs. Disruption Profile ∗ ∗] ∗) nSDb Eb [CD ρ(CD 483.6 452.05 5.1 483.6 179.53 2.6 483.6 139.12 1.2 502.8 1,103.96 2.7 502.8 478.41 2.2 502.8 267.89 1.4 821.7 3,693.17 1.9 729.2 860.72 1.5 691.7 419.30 0.8 2751.4 5,784.26 1.3 1211.5 1,798.20 1.4 843.3 826.61 1.2 4357.5 6,847.41 1.0 2017.9 3,000.02 1.2 1227.8 1,413.12 1.1
The disruption profile has a significant impact on the optimal base-stock levels and performance measures. As the percent down-time increases, expected costs and optimal base-stock levels generally increase for both systems. Moreover, within a group (fixed percent down-time), the optimal base-stock levels and cost mean and variance are significantly greater for rare/long disruptions than for frequent/short ones. For example, instance I(a), which has a recovery probability (β) of less than 0.1, has particularly large CVs: 5.1 for the decentralized system and 11.4 for the centralized system. Disruptions are rare in instance I(a), so the expected cost is small, but when disruptions occur, the cost spikes, increasing the variance significantly. Note also that in groups I and II, disruptions are rare enough that it is optimal to barely protect against them at all—the optimal base-stock levels are almost the same as in the no-disruption case (in which SC∗ = 440.1 ∗ = 480.1). Therefore, disruptions are quite costly when they occur. These results confirm and SD
results by Tomlin [25], Tomlin and Snyder [26], and others that suggest that rare/long disruptions are more difficult to plan for than frequent/short ones. Table 6 compares the performance of the centralized and decentralized systems for these instances. Either system may have a greater total base-stock level (as in Table 3), and there is no clear relationship between the difference in base-stock levels and the disruption profile. However, as the mean disruption duration increases (within a group), the difference in expected cost decreases and the difference in cost variance increases. In other words, when disruptions are longer, the risk-pooling effect is less pronounced and the risk-diversification effect is more pronounced.
16
Table 6: Comparison of Performance Measures vs. Disruption Profile Inv. Diff. E[Cost] Diff. % E[Cost] Diff. CV Diff. ∗ ]−E [C ∗ ] Eb [CD b C ∗ ∗ ∗ ∗ ∗) Group nSDb − SCb Eb [CD ] − Eb [CC ] ρ(CC∗ ) − ρ(CD Eb [C ∗ ] C
I
II
III
IV
V
5.4
a b c a b c a b c a b c a b c
41.8 41.8 41.8 51.5 51.5 51.5 10.8 -35.4 -54.1 -23.3 5.7 21.6 -17.4 8.4 13.8
49.48 49.36 49.36 28.70 28.68 28.68 1.62 5.92 7.43 2.15 6.91 13.89 2.56 6.01 10.57
12.29 % 37.92 % 54.99 % 2.67 % 6.38 % 11.99 % 0.04 % 0.69 % 1.80 % 0.04 % 0.39 % 1.71 % 0.04 % 0.20 % 0.75 %
6.4 4.7 2.8 2.9 2.7 2.1 1.9 1.5 0.5 1.3 1.4 1.3 1.0 1.2 1.1
Optimal Risk-Averse Design
We have shown that for the OWMR system with both supply disruptions and stochastic demand, inventory decentralization reduces the cost variance via the risk-diversification effect while centralization reduces the expected cost via the risk-pooling effect. It is natural to ask which effect is more pronounced, and therefore which system a risk-averse decision maker should prefer. 5.4.1
Mean–Variance Objective
We employ the classical mean–variance approach, minimizing the following objective function: (1 − κ)E[C] + κV [C]
(17)
where κ ∈ [0, 1]. The larger κ is, the more risk-averse the decision maker is. Many other objectives for risk-averse decision making have been proposed in the finance and operations literature. We chose to focus on the mean–variance objective primarily for analytical tractability. Numerical studies using the Conditional Value-at-Risk (CVaR) objective produced similar insights, but a more formal theoretical analysis was precluded by the increased complexity of the objective function [19]. We optimized (17) numerically using a line-search procedure. This function is neither convex nor concave for all values of κ. There are values of κ and Sb such that (17) is convex and values for which it is not convex. By testing the function numerically, we found that for κ ≤ 0.05, the range 17
newsboy fractile
1 0.9 Decentralized Optimal
0.8 0.7 0.6 0.5 0.00
Centralized Optimal
0.01
0.02
0.03
0.04
0.05
kappa
Figure 2: Optimal Inventory System for 4.76% Down-Time we used in our tests below, (17) is convex at nearly every value of Sb , with the exception of Sb ≈ 0. We also found that the function is convex for all S ≥ µ, in the instances we tested, and this range typically includes the optimal base-stock level. Moreover, the second derivative of (17) is positive at the solution found by our optimization procedure for every instance we tested. Therefore, we are confident that our results represent the globally optimal solutions for each instance. 5.4.2
Numerical Study
Our numerical study shows that even for very small values of κ, the decentralized system is almost always optimal under the mean–variance objective. We set µ = 100, σ = 15, α = 0.025, and β = 0.5, resulting in a percent down-time of 4.76%. Figure 2 plots the regions in which each system is optimal for κ ∈ [0, 0.05] and p/(p + h) ∈ [0.5, 1) (keeping h fixed to 1, i.e., p ∈ [1, ∞)). The region in which the centralized system is optimal is barely visible. Figure 3 zooms in on the range κ ∈ [0, 0.001] so that this region is more visible. Clearly, the centralized system is optimal only for very risk-neutral decision makers operating under low service levels: for κ ≥ 0.0008 and p/(p + h) ≥ 0.5, the decentralized system is always optimal. The centralized system can also be optimal for systems with very low disruption risk. For example, Figure 4 plots the optimal systems for 0.12% down-time (α = 0.001 and β = 0.8). However, even in this case, the decentralized system is optimal for all κ ≥ 0.02 and p/(p+h) ≥ 0.75. We next examined the impact of α and β on the optimal design. In Figure 5, we set p/(p + h) = 0.9 and κ = 0.001. The graph shows that the decentralized system is always optimal when the failure probability exceeds 0.7%, regardless of the recovery probability. These figures all employ small values of κ and the newsboy fractile and reasonably small disruption probabilities. They demonstrate that the decentralized system is far more likely to be the 18
newsboy fractile
1 0.9 Decentralized Optimal
0.8 0.7 Centralized Optimal
0.6
0.5 0.0000
0.0002
0.0004
0.0006
0.0008
0.0010
kappa
Figure 3: Optimal Inventory System for 4.76% Down-Time: Narrower κ Range
newsboy fractile
1 0.9
Decentralized Optimal
0.8 0.7
Centralized Optimal
0.6 0.5 0.00
0.01
0.02
0.03
0.04
0.05
kappa
Figure 4: Optimal Inventory System for 0.12% Down-Time
1
beta
0.8
Centralized Optimal
0.6 Decentralized Optimal
0.4 0.2 0 0.000
0.002
0.004
0.006
0.008
0.010
alpha
Figure 5: Optimal Inventory System for Varying α and β Values
19
optimal inventory design for a risk-averse firm. Since the variance is typically several orders of magnitude greater than the expected cost for our model, one could argue that the mean-variance objective is naturally biased toward the lowerrisk strategy, i.e., the decentralized system. Therefore, we also performed similar tests using the standard deviation of the cost in place of the variance, and with κ values roughly equal to the square root of the original values. The results were very similar. For the same parameter values discussed above, we found that when the disruption probability is 4.76%, the decentralized system is optimal for all κ ≥ 0.0075 and p/(p + h) ≥ 0.5, and when the disruption probability is 0.12%, the decentralized system is optimal for all κ ≥ 0.1 and p/(p + h) ≥ 0.75. We also varied α and β as above and found that when κ = 0.001 and p/(p + h) = 0.9, the decentralized system is always optimal when the failure probability exceeds 1.8%, regardless of the recovery probability. A more detailed description of these results is omitted due to space considerations.
5.5
Summary of Numerical Comparisons
For the OWMR system subject to both demand uncertainty and supply disruptions: • The risk-diversification effect exists, resulting in lower cost variance in the decentralized system – The risk-diversification effect is more pronounced when disruptions are longer and more costly • The risk-pooling effect exists, resulting in lower expected cost in the centralized system – The risk-pooling effect does not always result in lower total base-stock levels – The risk-pooling effect is more pronounced when disruptions are less frequent and shorter, or when demand is more variable • Disruption profiles affect the optimal solutions – For the same percent down-time, longer disruptions result in higher optimal base-stock levels, higher expected costs, and higher cost variances – The difference in cost variances between the centralized and decentralized systems is greater for longer disruptions, but the difference in expected cost is greater for shorter disruptions. 20
• For a risk-averse firm, the decentralized system is typically the optimal inventory design – Unless service level (newsboy fractile), κ value, or failure probabilities are very low
6
Conclusions
In this paper, we consider the One-Warehouse Multiple-Retailer (OWMR) system with both supply and demand uncertainty. We investigate the risk-diversification effect, in which the expected cost is the same in the centralized and decentralized systems but the cost variance is smaller in the decentralized system. The intuition behind this effect is that by distributing inventory at multiple sites, the impact of any one disruption is smaller, even though each site is still affected by the same number of disruptions. The firm benefits from not putting all its eggs in one basket; although the same number of eggs may be destroyed, they are not all destroyed at once. We proved that the risk-diversification effect occurs in OWMR systems with supply disruptions and provided conditions under which it occurs in OWMR systems with yield uncertainty. In contrast, the classical risk-pooling effect prevails under demand uncertainty; under this effect, the expected cost is smaller in the centralized system, and we prove that the variance is equal in the two systems. We showed numerically that, when disruptions and demand uncertainty are both present, both effects occur to a certain extent, but that in most cases the risk-diversification effect strongly dominates the risk-pooling effect. Therefore, a decentralized system is optimal when disruptions and demand uncertainty are both present, except when the service level (newsboy fractile) is very low, the firm is very risk neutral, and/or the system is very reliable. Natural extensions to this research can be made by relaxing the assumptions in our models. For example, we assumed zero fixed-ordering costs. An argument could be made that this ignores an additional cost benefit from centralization; synchronizing orders could reduce the total fixed costs. However, as we have shown, risk diversification would suggest that a risk-averse firm should not synchronize orders if individual shipments may be disrupted at the supplier. Another extension could relax our assumption of independently distributed disruptions. Eppen [12] shows that demand correlation affects the magnitude of the risk pooling effect; positively correlated demand decreases the risk pooling benefits of centralization and negatively correlated demand increases them. A similar study could be conducted for supply uncertainty. We would expect that positively correlated supply disruptions would decrease the risk diversification benefits of decentralization, whereas negatively correlated disruptions would increase those benefits. 21
Another obvious extension to our work could address the assumption that inventory can only be held at one echelon of the supply chain, which we made to simplify our analysis and clarify our conclusions. Of course, centralized and decentralized systems are not the only choices available to a firm operating in an OWMR environment; the firm may also choose a hybrid system in which inventory is held at both echelons. Such a system may well provide a desirable balance between the risk-diversification and risk-pooling effects. These systems are significantly harder to analyze (as the literature on the OWMR system attests), but an investigation of these two competing effects in this context is an important avenue for future research.
7
Acknowledgments
This research was supported in part by National Science Foundation grants DGE-9972780, DMI0522725, and DMI-0621433. This support is gratefully acknowledged.
22
A
Appendix
A.1
Loss Functions
Let X be a random variable with pdf f (x) and cdf F (x). The loss function for X is given by: Z
∞
Z
∞
(t − x)f (t)dt =
G(x) = x
(1 − F (t))dt
(18)
x
and the second-order loss function is: 1 H(x) = 2
Z
∞ 2
Z
∞
(t − x) f (t)dt = x
G(t)dt.
(19)
x
Let Φ1 (z) be the standard normal loss function and Φ2 (z) as the standard normal second-order loss function. The following lemma presents properties we use in the proofs below. The proof of the lemma is omitted; it follows from well known results concerning loss functions [e.g., 2, 30]. Lemma 7
1. Let X have cdf F and loss function G. Then for any function θ(x), d G(θ(x)) = −θ0 (x)[1 − F (θ(x))]. dx
(20)
2. If X is normally distributed with mean µ and variance σ 2 and f (x) is its pdf, then Z ∞ S−µ . (x − S)2 f (x)dx = 2σ 2 Φ2 σ S
A.2 A.2.1
(21)
Proof of Theorem 1, Section 3.2 Optimal Base-Stock Levels
∗ = S ∗ since each retailer acts as a single-facility system. Moreover, from (2) we know that SDs s
Ss∗ = jd, where j is the smallest integer such that F (j − 1) ≥
p p+h .
Similarly, in the centralized
∗ = jnd for the same j (since the definition of j does not system, the optimal base-stock level is SCs ∗ = nS ∗ . depend on the demand). Therefore, SCs s
A.2.2
Expected Costs at Optimality
In the decentralized system, the total expected cost is simply the sum of the single-facility costs, so Es [CD ] = nEs [C].
23
In the centralized system, the optimal expected cost at the warehouse, using nSs∗ in place of ∗ in (1), is given by: SCs
Es [CC∗ ] =
∞ X
πi [h(nSs∗ − (i + 1)nd)+ + p((i + 1)nd − nSs∗ )+ ]
i=1 ∞ X
= n
πi [h(Ss∗ − (i + 1)d)+ + p((i + 1)d − Ss∗ )+ ]
i=1 ∗ = nEs [C ∗ ] = Es [CD ]
A.2.3
(22)
Cost Variances at Optimality
For the decentralized system, since each retailer operates independently, the variance of the cost is equal to the sum of the single-retailer variances, or Vs [CD ] = nVs [C]. Now, for a single retailer, ∞ h X 2 2 i ∗ 2 Es [(C ) ] = πi h2 (Ss∗ − (i + 1)d)+ + p2 ((i + 1)d − Ss∗ )+
(23)
i=1 ∗ = nS ∗ , In the centralized system, using SCs s
Es [(CC∗ )2 ] =
∞ X
h 2 2 i πi h2 (nSs∗ − (i + 1)nd)+ + p2 ((i + 1)nd − nSs∗ )+
i=1
= n2
∞ X
h 2 2 i πi h2 (Ss∗ − (i + 1)d)+ + p2 ((i + 1)d − Ss∗ )+
i=1
= n Es [(C ∗ )2 ] 2
(24)
Thus, using (22) and (24), Vs [CC∗ ] = Es [(CC∗ )2 ] − Es [CC∗ ]2 = n2 Es [(C ∗ )2 ] − (nEs [C ∗ ])2 = n2 (Es [(C ∗ )2 ] − Es [C ∗ ]2 ) = n2 Vs [C ∗ ]
(25)
∗ ]. Thus Vs [CC∗ ] = nVs [CD
A.3
Proof of Theorem 4, Section 4.2
At optimality, the centralized and decentralized base-stock levels satisfy ∗ −µ SDd S ∗ − nµ p =Φ = Cd . σ p+h σC
(26)
Let X≡Φ
p p+h
(27) 24
Using (21) from Lemma 7, Z ∞ Z Sd 2 2 2 2 (d − Sd )2 f (d)dd (Sd − d) f (d)dd + p Ed [C ] = h Sd −∞ Z ∞ Z ∞ (Sd2 − 2Sd d + d2 )f (d)dd + (p2 − h2 ) (d − Sd )2 f (d)dd = h2 −∞ S d 2 2 2 2 2 2 2 2 Sd − µ = h (Sd − 2Sd µ + σ + µ ) + 2σ (p − h )Φ σ
(28) (29) (30)
At optimality, we have Ed [(C ∗ )2 ] = h2 ((Sd∗ )2 − 2Sd∗ µ + σ 2 + µ2 ) + 2σ 2 (p2 − h2 )Φ2 (X)
(31)
In addition, from (3), Ed [C ∗ ] = h(Sd∗ − µ) + σ(p + h)Φ1 (X) .
(32)
Therefore, Vd [C ∗ ] = Ed [(C ∗ )2 ] − Ed [C ∗ ]2 = h2 ((Sd∗ )2 − 2Sd∗ µ + σ 2 + µ2 ) + 2σ 2 (p2 − h2 )Φ2 (X) 2 − h(Sd∗ − µ) + σ(p + h)Φ1 (X) = h2 ((Sd∗ )2 − 2Sd∗ µ + σ 2 + µ2 ) + 2σ 2 (p2 − h2 )Φ2 (X) − h2 ((Sd∗ )2 − 2Sd∗ µ + µ2 ) + 2σh(p + h)(Sd∗ − µ)Φ1 (X) + σ 2 (p + h)2 Φ1 (X)2 = h2 σ 2 + 2σ 2 (p2 − h2 )Φ2 (X) − 2σh(p + h)(Sd∗ − µ)Φ1 (X) − σ 2 (p + h)2 Φ1 (X)2 = σ 2 h2 + 2(p2 − h2 )Φ2 (X) − 2h(p + h)XΦ1 (X) − (p + h)2 Φ1 (X)2 (33) Let Y equal the sum of the terms inside the brackets. Note that Y is a function of X, not of σ or µ alone; therefore, Y is the same for an individual retailer or for the warehouse in the centralized system. Since the decentralized system functions as n individual retailers, we have ∗ Vd [CD ] = nVd [C ∗ ].
(34)
Applying (33), 2 ∗ Vd [CC∗ ] = σC Y = nσ 2 Y = nVd [C ∗ ] = Vd [CD ].
as desired. 25
(35)
A.4
Proof of Proposition 6, Section 5.2
Using (21) from Lemma 7, we get Z Sb Z ∞ X 2 2 2 2 (Sb − id) fi (id)dd + p Eb [C ] = πi−1 h = =
i=1 ∞ X i=1 ∞ X
−∞
πi−1 h
2
Z
πi−1 h
2
2
(id − Sb ) fi (id)dd
Sb
∞
−∞
∞
(Sb2
(Sb2
2 2
2
Z
2
∞ 2
(id − Sb ) fi (id)dd
− 2Sb id + i d )fi (id)dd + (p − h ) Sb 2
2 2
2
2
2
− 2Sb iµ + iσ + i µ ) + 2iσ (p − h )Φ
2
i=1
Sb − iµ √ σ i
Vb [C] = Eb [C 2 ] − (Eb [C])2 ∞ X 2 2 2 2 2 2 2 2 2 Sb − iµ √ = πi−1 h (Sb − 2Sb iµ + iσ + i µ ) + 2iσ (p − h )Φ − σ i i=1 !2 ∞ X √ S − iµ b √ πi−1 h(Sb − iµ) + σ i(p + h)Φ1 σ i i=1
(36)
(37)
In the decentralized system, the total variance is just the sum of the individual-retailer variances since each retailer operates independently. In the centralized system, the total demand seen by the warehouse is distributed as N (niµ, niσ 2 ). Therefore, Vb [CC ] =
∞ X
2 − 2SCb niµ + niσ 2 + n2 i2 µ2 ) + πi−1 h2 (SCb
i=1 2
2
2
2niσ (p − h )Φ ∞ X
2
SCb − niµ √ σ ni
!
√
−
πi−1 h(SCb − niµ) + σ ni(p + h)Φ1
i=1
26
SCb − niµ √ σ ni
!2 (38)
References [1] V. Agrawal and S. Seshadri. Risk intermediation in supply chains. IIE Transactions, 32:819– 831, 2000. [2] S. Axsater. Inventory Control. Kluwer Academic Publishers, Boston, MA, first edition, 2000. [3] Y. Bassok and R. Akella. Ordering and production decisions with supply quality and demand uncertainty. Management Science, 37(12):1556–1574, Dec. 1991. [4] E. Berk and A. Arreola-Risa. Note on “Future supply uncertainty in EOQ models”. Naval Research Logistics, 41:129–132, 1994. [5] R. Bollapragada, U.S. Rao, and J. Zhang. Managing two-stage serial inventory systems under demand and supply uncertainty and customer service level requirements. IIE Transactions, 36:73–85, 2004. [6] R. Bollapragada, U.S. Rao, and J. Zhang. Managing inventory and supply performance in assembly systems with random supply capacity and demand. Management Science, 50(12):1729– 1743, Dec. 2004. [7] X. Chen, M. Sim, D. Simchi-Levi, and P. Sun. Risk aversion in inventory management. Operations Research, 55(5):828–842, 2007. [8] S. Chopra, G. Reinhardt, and U. Mohan. The importance of decoupling recurrent and disruption risks in a supply chain. Naval Research Logistics, 54(5):544–555, 2007. [9] L.Y. Chu and Z.J.M. Shen. A power-of-two ordering policy for one warehouse, multi-retailer systems with stochastic demand. Working paper, Marshall School of Business, University of Southern California, Los Angeles, CA, 2006. [10] M. Dada, N. Petruzzi, and L. Schwarz. A newsvendor’s procurement problem when suppliers are unreliable. Manufacturing & Service Operations Management, 9(1):9–32, 2007. [11] L. Eeckhoudt, C. Gollier, and H. Schlesinger. The risk-averse (and prudent) newsboy. Management Science, 41(5):786–794, May 1995. [12] G.D. Eppen. Effects of centralization on expected costs in a multi-location newsboy problem. Management Science, 25(5):498–501, May 1979. ¨ ¨ [13] Guilliermo Gallego, Ozalp Ozer, and Paul Zipkin. Bounds, heuristics, and approximations for distribution systems. Operations Research, 55(3):503–517, 2007. [14] A. Grosfeld-Nir and Y. Gerchak. Multiple lotsizing in production to order with random yields: review of recent advances. Annals of Operations Research, 126:43–69, 2004. [15] H. Gurnani, R. Akella, and J. Lehoczky. Supply management in assembly systems with random yield and random demand. IIE Transactions, 32:701–714, 2000. [16] W.J. Hopp and Z. Yin. Protecting supply chain networks against catastrophic failures. Working paper, Dept. of Industrial Engineering and Management Science, Northwestern University, Evanston, IL, 2006.
27
[17] E. Lystad and M. Ferguson. Simple newsvendor heuristics for two-echelon distrbution networks. Working paper, Georgia Institute of Technology, 2007. [18] M. Parlar and D. Berkin. Future supply uncertainty in EOQ models. Naval Research Logistics, 38:107–121, 1991. [19] A. Schmitt. Strategic inventory management for supply chains subject to supply uncertainty. PhD Dissertation, Industrial and Systems Engineering Department, Lehigh University, Bethlehem, PA, 2007. [20] A.J. Schmitt and L.V. Snyder. Infinite-horizon models for inventory control under yield uncertainty and disruptions. Working paper, P.C. Rossin College of Engineering and Applied Sciences, Lehigh University, Bethlehem, PA, 2007. [21] A.J. Schmitt, L.V. Snyder, and Z.J.M. Shen. Inventory systems with stochastic demand and supply: Properties and approximations. Working paper, P.C. Rossin College of Engineering and Applied Sciences, Lehigh University, Bethlehem, PA, 2008. [22] L.V. Snyder. A tight approximation for a continuous-review inventory model with supplier disruptions. Working paper, P.C. Rossin College of Engineering and Applied Sciences, Lehigh University, Bethlehem, PA, September 2006. [23] L.V. Snyder and Z.J.M. Shen. Supply and demand uncertainty in multi-echelon supply chains. Working paper, P.C. Rossin College of Engineering and Applied Sciences, Lehigh University, Bethlehem, PA, 2006. [24] C.S. Tang. The impact of uncertainty on a production line. Management Science, 36:1518– 1531, Dec. 1990. [25] B. Tomlin. On the value of mitigation and contingency strategies for managing supply chain disruption risks. Management Science, 52(5):639–657, May 2006. [26] B. Tomlin and L.V. Snyder. Inventory management with advanced warning of disruptions. Working Paper, Kenan-Flagler Business School, University of North Carolina, Chapel Hill, NC, 2007. [27] B. Tomlin and Y. Wang. On the value of mix flexibility and dual sourcing in unreliable newsvendor networks. Manufacturing & Service Operations Management, 7(1):37–57, 2005. [28] J.A. Van Meighem. Risk mitigation in newsvendor networks: Resource diversification, flexibility, sharing and hedging. Management Science, 53(8):1269–1288, 2007. [29] C.A. Yano and H.L. Lee. Lot sizing with random yields: A review. Operations Research, 43(2):311–334, March-April 1995. [30] P.H. Zipkin. Foundations of Inventory Management. McGraw-Hill Higher Education, Boston, MA, first edition, 2000.
28
