Centralization versus Decentralization: Risk Pooling, Risk Diversification, and Supply Uncertainty in a One-Warehouse Multiple-Retailer System

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1 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 Multiple- Retailer (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

2 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. 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. We 2

3 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

4 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

5 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

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7 Using (21) from Lemma 7, Sd E d [C 2 ] = h 2 (S d d) 2 f(d)dd + p 2 At optimality, we have (d S d ) 2 f(d)dd (28) S d = h 2 (Sd 2 2S dd + d 2 )f(d)dd + (p 2 h 2 ) (d S d ) 2 f(d)dd (29) S ( d ) = h 2 (Sd 2 2S dµ + σ 2 + µ 2 ) + 2σ 2 (p 2 h 2 )Φ 2 Sd µ (30) σ E d [(C ) 2 ] = h 2 ((S d )2 2S d µ + σ2 + µ 2 ) + 2σ 2 (p 2 h 2 )Φ 2 (X) (31) In addition, from (3), E d [C ] = h(s d µ) + σ(p + h)φ1 (X). (32) Therefore, V d [C ] = E d [(C ) 2 ] E d [C ] 2 = h 2 ((Sd )2 2Sd µ + σ2 + µ 2 ) + 2σ 2 (p 2 h 2 )Φ 2 (X) [ h(sd µ) + σ(p + h)φ1 (X) ] 2 = h 2 ((Sd )2 2Sd µ + σ2 + µ 2 ) + 2σ 2 (p 2 h 2 )Φ 2 (X) [ h 2 ((Sd )2 2Sd µ + µ2 ) + 2σh(p + h)(sd µ)φ1 (X) + σ 2 (p + h) 2 Φ 1 (X) 2] = h 2 σ 2 + 2σ 2 (p 2 h 2 )Φ 2 (X) 2σh(p + h)(sd µ)φ1 (X) σ 2 (p + h) 2 Φ 1 (X) 2 = σ 2 [ h 2 + 2(p 2 h 2 )Φ 2 (X) 2h(p + h)xφ 1 (X) (p + h) 2 Φ 1 (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 V d [C D] = nv d [C ]. (34) Applying (33), V d [C C] = σ 2 CY = nσ 2 Y = nv d [C ] = V d [C D]. (35) as desired. 25

8 A.4 Proof of Proposition 6, Section 5.2 Using (21) from Lemma 7, we get ( Sb ) E b [C 2 ] = π i 1 h 2 (S b id) 2 f i (id)dd + p 2 (id S b ) 2 f i (id)dd = = ( π i 1 h 2 (Sb 2 2S bid + i 2 d 2 )f i (id)dd + (p 2 h 2 ) V b [C] = E b [C 2 ] (E b [C]) 2 = ( S b ( ( )) π i 1 h 2 (Sb 2 2S biµ + iσ 2 + i 2 µ 2 ) + 2iσ 2 (p 2 h 2 )Φ 2 Sb iµ σ i ( ( π i 1 h 2 (Sb 2 2S biµ + iσ 2 + i 2 µ 2 ) + 2iσ 2 (p 2 h 2 )Φ 2 Sb iµ σ i ( π i 1 h(s b iµ) + σ ( )) ) 2 i(p + h)φ 1 Sb iµ σ i S b ) (id S b ) 2 f i (id)dd )) 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, (36) (37) V b [C C ] = π i 1 (h 2 (SCb 2 2S Cbniµ + niσ 2 + n 2 i 2 µ 2 ) + ( ) ) 2niσ 2 (p 2 h 2 )Φ 2 SCb niµ σ ni ( ( π i 1 h(s Cb niµ) + σ ( )) ) 2 ni(p + h)φ 1 SCb niµ σ ni (38) 26

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