Supply chain coordination in vendor-managed inventory systems with stockout-cost sharing under limited storage capacity

Accepted Manuscript

Supply Chain Coordination in Vendor-Managed Inventory Systems with Stockout-Cost Sharing under Limited Storage Capacity Jun-Yeon Lee , Richard K. Cho , Seung-Kuk Paik PII: DOI: Reference:

S0377-2217(15)00634-7 10.1016/j.ejor.2015.06.080 EOR 13089

To appear in:

European Journal of Operational Research

Received date: Revised date: Accepted date:

12 June 2014 13 May 2015 30 June 2015

Please cite this article as: Jun-Yeon Lee , Richard K. Cho , Seung-Kuk Paik , Supply Chain Coordination in Vendor-Managed Inventory Systems with Stockout-Cost Sharing under Limited Storage Capacity, European Journal of Operational Research (2015), doi: 10.1016/j.ejor.2015.06.080

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Highlights of the manuscript Examine VMI systems with stockout-cost sharing under limited storage capacity.



Compare the VMI systems with an integrated supplier-customer system.



Provide a condition on which VMI can coordinate the supply chain.



Use EOQ model with shortages allowed under limited storage capacity.

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ABSTRACT

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Supply Chain Coordination in Vendor-Managed Inventory Systems with Stockout-Cost Sharing under Limited Storage Capacity

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We examine vendor-managed inventory (VMI) systems with stockout-cost sharing between a supplier and a customer using an EOQ model with shortages allowed under limited storage capacity, in which a stockout penalty is charged to the supplier when stockouts occur at the customer. In the VMI systems the customer and the supplier minimize their own costs in designing a VMI contract and making replenishment decisions, respectively. We compare the

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VMI systems with an integrated supplier-customer system where the supply chain total cost is minimized. We show that VMI with stockout-cost sharing and the integrated supplier-customer

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system result in the same replenishment decisions and system performance if and only if the supplier‟s reservation cost is equal to the minimum supply chain total cost of the integrated system. On the other hand, we also show how VMI along with fixed transfer payments as well as

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stockout-cost sharing can lead to the supply chain coordination regardless of the supplier‟s

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reservation cost. We also provide several interesting computational results.

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1. INTRODUCTION Vendor-managed inventory (VMI) is a well-known and widely-used supply chain practice between a supplier and a customer (e.g., supplier-manufacturer or distributor-retailer), in which the supplier manages the inventory at the customer and decides when and how much to replenish. VMI started as a pilot program in the retail industry between Wal-Mart and P&G

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(Proctor & Gamble) in the 1980s and has been adopted by many supply chains such as Campbell Soup Company, Barilla SpA, Intel, and Shell Chemical (Bookbinder et al., 2010). VMI has been also studied by many researchers, most of whom investigate the benefits of VMI in various settings, study the problem of designing VMI contracts, or examine various operational issues/decisions in implementing VMI (Guan and Zhao, 2010).

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On the other hand, several recent papers have examined integrated supplier-customer systems in the context of VMI (e.g., Battini et al., 2010; Bertazzi et al., 2005; Braglia and Zavanella, 2003; Persona et al., 2005; Zhang et al., 2007), in which the supplier minimizes the supply chain total cost, rather than his own cost, in making replenishment decisions for the supply chain. As a result, the integrated supplier-customer systems considered in these papers are

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essentially equivalent to the centralized systems with a single decision-maker who bears all the supply chain costs. As pointed out by Darwish and Goyal (2011), although this approach leads to

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the optimal system performance, it may not be in the best interests of the supplier or the customer. The primary purpose of this paper is to examine how VMI may result in different replenishment decisions and system performances when the supplier and the customer minimize

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their own costs, compared with the integrated supplier-customer system in which the supplier minimizes the supply chain total cost. We also explore under what contractual agreements VMI

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can lead to the same replenishment decisions and system performance as in the integrated supplier-customer system.

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The secondary purpose of this paper is to examine a VMI contract with stockout-cost sharing under limited storage capacity, in which a stockout penalty is charged to the supplier when stockouts occur at the customer. Since Fry et al. (2001) examined a (z, Z)-type VMI contract, which specifies minimum and maximum inventory levels and the corresponding underand over-stocking penalties, many researchers have studied the (z, Z)-type VMI contract and its variants. For example, Shah and Goh (2006) examine the (z, Z)-type VMI contract in a deterministic setting, Darwish and Odah (2010) examine a VMI contract that specifies a

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maximum inventory level and a penalty for overstocking, and Lee and Cho (2014) examine a VMI contract that specifies fixed and proportional stockout penalties. In this paper we seek to characterize the optimal VMI contract with stockout-cost sharing under limited storage capacity and examine how the optimal contract is affected by the storage limit. We examine two VMI systems between a supplier and a customer: VMI with stockout-

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cost sharing and VMI with fixed transfer payments and stockout-cost sharing. In our VMI systems, the customer designs and offers a VMI contract to the supplier. The supplier can accept or reject the contract and, if he accepts it, manages the inventory at the customer and makes replenishment decisions. The customer and the supplier minimize their own costs in designing a VMI contract and making replenishment decisions, respectively. In particular, the supplier has a

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reservation cost such that he accepts the contract as long as his minimum cost under the contract is less than or equal to his reservation cost. The supplier‟s reservation cost may be determined by his negotiating power, or, if the supplier and the customer are currently operating under a traditional system mode, it may be his cost in the current system.

In VMI with stockout-cost sharing, the VMI contract specifies a stockout penalty per unit

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backordered per unit time, which is paid by the supplier to the customer whenever stockouts occur at the customer. The VMI contract also specifies that the inventory at the customer is

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owned by the supplier until it is used by the customer (i.e., consignment stock). VMI with fixed transfer payments and stockout-cost sharing is the same as VMI with stockout-cost sharing, except that the VMI contract also specifies a per-period fixed transfer payment between the two

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firms.

We use a deterministic (Q, r) inventory model (also known as EOQ model with shortages

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allowed) under limited storage capacity to examine and compare four business scenarios: two VMI systems described in the above, an integrated supplier-customer system (or integrated

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system) where the supplier minimizes the supply chain total cost, and a traditional system where the customer manages her own inventory. We show that VMI with stockout-cost sharing and the integrated system result in the same replenishment decisions and system performance if and only if the supplier‟s reservation cost is equal to the minimum supply chain total cost of the integrated system. This result implies that VMI and the integrated system may lead to different replenishment decisions and system performances, while providing a condition under which VMI can coordinate the supply chain without fixed transfer payments. On the other hand, we

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also show how VMI with fixed transfer payments and stockout-cost sharing can be designed to achieve supply chain coordination regardless of the supplier‟s reservation cost. We also provide several interesting computational results. In particular, our results suggest that VMI with stockout-cost sharing performs very well when the supplier‟s reservation cost is close to the minimum supply chain total cost of the integrated system, but that it may perform significantly

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worse than the integrated system, especially when the supplier‟s reservation cost is small or the storage limit is small. 2. LITERATURE REVIEW

There exists a substantial amount of literature on VMI. As mentioned in the above, most of the papers investigate the benefits of VMI compared with the traditional system, study the

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problem of designing VMI contracts, or examine various operational issues/decisions in implementing VMI (Guan and Zhao, 2010). In this section we review some of the papers that are closely related to our paper.

Fry et al. (2001) examine a (z, Z)-type VMI contract, which specifies a minimum inventory level (z), a maximum inventory level (Z), and penalties, b- and b+, for under- and over-

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stocking, respectively. In their model, b-, b+, and

are set through mutual agreement

between the supplier and the customer. Given the values of b-, b+, and Q, the customer chooses

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Z and then the supplier makes production and replenishment decisions. They characterize the optimal behavior of the supplier and the customer, and provide guidelines for choosing the

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values of b-, b+, and Q to minimize the supply chain total cost. They suggest that VMI can perform significantly better than the traditional system in many settings, due to better coordination of production and delivery, but can perform worse in others. Note that our VMI

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models can be viewed as a special case of the (z, Z)-type VMI contract, where the minimum inventory level is zero (i.e., z = 0) with a stockout penalty and the maximum inventory level is

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imposed through the storage capacity. Nagarajan and Rajagopalan (2008) consider a business scenario in which both a supplier

and a retailer incur stockout costs when stockouts occur at the retailer. They examine a holding cost subsidy-type VMI contract where the retailer charges the supplier a holding cost based on average inventory at the retailer. Bichescu and Fry (2009) examine the effect of channel power on VMI performance in the (Q, r) inventory system with a VMI agreement in which the supplier chooses order quantity Q and the retailer chooses reorder point r. In their model the backorder-

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penalty costs are split between the supplier and the retailer, but how to split them is not a decision variable but a given parameter. Lee and Cho (2013) examine the problem of designing a VMI contract with consignment stock and stockout-cost sharing in a (Q, r) inventory system between a supplier and a retailer, in which the contract specifies proportional and fixed stockout penalties paid by the supplier to the

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retailer when stockouts occur at the customer. Our VMI models are similar to theirs in that stockout costs are shared between the supplier and the customer. However, our VMI models extend theirs by incorporating limited storage capacity and fixed transfer payments. More importantly, the purpose of our paper is different from theirs: We examine how VMI and the integrated supplier-customer system can be different and under what conditions or contractual

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agreements VMI can coordinate the supply chain.

Lee and Ren (2011) examine the benefits of VMI in a global environment in which the supplier and the retailer face exchange rate uncertainty. In their VMI model, the retailer charges the supplier a stockout penalty, which is equal to her own backorder penalty cost, at the end of each period for the backorders that occurred during the period. They characterize the supplier‟s

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optimal policy and provide computational results on the benefits of VMI to the supplier, the retailer, and the supply chain and the impact of exchange rate uncertainty on the benefits of VMI.

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Their model is similar to ours in that the VMI contract specifies a stockout penalty, but they do not examine the problem of contract design. Darwish and Goyal (2011) consider a VMI contract between a supplier and a buyer,

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which specifies a maximum inventory level and an over-stocking penalty. The supplier produces the product at a finite production rate and delivers a production lot in a number of equal-sized

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shipments. Under the VMI contract the supplier determines production and shipment lot sizes to minimize his total cost including the inventory-holding and ordering costs of himself and the

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buyer and the over-stocking penalty costs. They provide an algorithm to solve the supplier‟s problem. But they do not examine the buyer‟s contract design problem. Recently, several papers have studied supply chain coordination with VMI. Bernstein et

al. (2006) consider a supply chain with a supplier and multiple retailers. They identify a condition called EOA (Echelon Operational Autonomy), which may arise in VMI partnerships where the supplier minimizes the supply chain total cost, and show that perfect coordination can be achieved via simple pricing schemes under the EOA condition. Wong et al. (2009) consider a

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supply chain with a supplier and multiple retailers in a single-period stochastic setting to examine how a sales rebate contract can coordinate the supply chain under VMI with consignment stock. In their model the VMI partnership facilitates the implementation of the sales rebate contract because the supplier has access to the actual sales data. Chen et al. (2010) consider a supply chain consisting of a wholesaler and a retail chain and examine VMI scenarios

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where the two firms negotiate contract parameters (such as revenue-sharing and side payment) beforehand and then the wholesaler determines the retail price and replenishment policy. They show through a numerical study that the supply chain can be coordinated by a revenue-sharing scheme when the wholesaler maximizes the channel profit and by a two-part tariff contract with a revenue-sharing percentage and a lump-sum side payment when the wholesaler maximizes his

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own profits. Choi (2011) develops VMI models between a manufacturer and a retailer with or without RFID (Radio Frequency Identification), where the RFID technology enables the supplier to monitor and replenish the inventory at the retailer on a continuous basis. He shows how a return policy together with the expense sharing on the cost of RFID can coordinate the supply chain. Wang et al. (2004) examines a VMI contract with consignment stock and revenue sharing

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in a single-period model with uncertain and price-elastic demand, in which the customer chooses a revenue share and the supplier decides on the retailer price and delivery quantity. They show

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that the contract cannot coordinate the supply chain, although the channel profit may be close to that of the centralized supply chain in some situations. On the other hand, Li and Hua (2008) propose a cooperative game model for the supply chain of Wang et al. (2004) and show that the

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supply chain can be coordinated if the revenue share, retail price, and delivery quantity are determined by the cooperative model. Chen et al. (2011) consider a model similar to Wang et al.

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(2004) but with a price-dependent revenue share, and show that the contract with a pricedecreasing revenue share performs worse than one with a fixed or price-increasing revenue share.

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Our paper is similar to the above papers in that we study supply chain coordination with VMI, but we use a different form of VMI contract: stockout-cost sharing and/or fixed transfer payments under storage limit. Our paper also suggests that the supplier‟s reservation cost may play an important role in supply chain coordination with VMI. The use of fixed transfer payments is not uncommon in the operations and supply chain management literature. For example, Gerchak et al. (2006) proposes a per-cycle subsidy between the studio and the rental chain in the video rental industry. Corbett et al. (2004) examines various

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types of supply contracts, some of which involve per-period fixed payments. In particular, Cachon (2001) study a supply chain inventory game in a two-echelon supply chain consisting of a supplier and multiple retailers. He shows that the supply chain can be coordinated by transferring inventory control to the supplier and using fixed transfer payments. His result is similar to ours, but in his model the supplier chooses fixed transfer payments as well as

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inventory policies for the supply chain, while in our model the retailer (customer) chooses fixed transfer payments and stockout penalty and the supplier makes replenishment decisions for the retailer under the contract. In fact, it is well-known in the economics literature that the first-best can be achieved in the agency relationship between two risk-neutral parties by using a contract with fixed payments (e.g., Mas-Colell et al., 1995). In this paper, we show that a VMI contract

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can coordinate the supply chain without fixed transfer payments when the supplier‟s reservation cost is equal to the minimum supply chain total cost.

In most of the papers in the above, the supplier and the customer minimize their own costs or maximize their own profits. On the other hand, several recent papers have examined integrated supplier-customer systems in the context of VMI, in which the supplier minimizes the

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supply chain total cost. For example, Braglia and Zavanella (2003) examine a single-vendor, single-buyer VMI model in which the vendor should maintain the inventory level at the buyer

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between a minimum s and a maximum S. The vendor determines the number of shipments per production cycle and the size of shipment to minimize the total system cost. Persona et al. (2005) examine a VMI model similar to the one in Braglia and Zavanella (2003) but with a finite time

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horizon. Battini et al. (2010) extend Persona et al. (2005) to a stochastic single-vendor, multiple buyers model and numerically analyze the problem. Bertazzi et al. (2005) examine a production-

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distribution system, in which a supplier produces and distributes several items to a set of retailers under VMI, and propose solution procedures. In their model the supplier minimizes the total

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system cost in determining the production policy, the retailers‟ replenishment policies, and the transportation policy. Zhang et al. (2007) consider a supply chain consisting of a vendor and multiple buyers, in which the vendor purchases and processes raw materials and then delivers finished items to the buyers. The vendor makes replenishment decisions for the buyers and investment decisions for ordering-cost reduction to minimize the total cost of the system. The rest of this paper is organized as follows: In Section 3, we present the base model. In Section 4 and 5 we analyze the traditional and the integrated system, respectively. In Section 6

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and 7 we present and analyze our VMI systems and compare the results with those of the integrated system. Section 8 provides computational results, and finally Section 9 concludes the paper. 3. BASE MODEL A continuous-review (Q, r) inventory system over an infinite horizon is used to manage

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the inventory of a single item at the customer (e.g., retailer) under limited storage capacity M, who faces deterministic demand for the item at the rate of  units per unit time. In this system, a replenishment order of

is placed whenever the inventory position falls down to reorder point r.

It is well known that the long-run average cost per unit time of this policy is given by: ̃(

̃

| ̃ ̃ ̃)

)

̃(

)

(1)

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(

where ̃ ̃ ̃ , and L are fixed ordering cost, inventory-holding cost per unit per unit time, backorder-penalty cost per unit backordered per unit time, and fixed replenishment leadtime, respectively (see, e.g., Zipkin, 2000). Note that the cost model above is also known as EOQ (Economic Order Quantity) model with shortages allowed.

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It can be shown that the cost function in (1) is strictly convex in

and

(Zipkin, 2000).

Let ( ̂ ̂ ) be the optimal solution to the following problem:

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| ̃ ̃ ̃)

(

Minimize

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Subject to

(2)

where

represents the maximum inventory level given Q and r, which must be ( ̂ ̂ | ̃ ̃ ̃).

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smaller than or equal to the storage capacity, M. Let ̂ √ ̃( ̃

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Lemma 1. (a) If √ ̃( ̃

(b) If ̂

̃( ̂

̃̃

, then ̂ ̃)

̃̃

, then ̂

̃)

√

√

̃ ( ̃ ̃) ̃

̃ ( ̃ ̃) , ̃̃

̃

̂

̃ ̃

̂

, ̂

̂ , and ̂

̃̃ ̃ ̃

̂.

, and

).

Proof. See the Appendix for all the proofs of lemmas and propositions, unless stated otherwise. Lemma 1 provides the formulas of the optimal solution in EOQ model with shortages allowed under limited storage capacity. In particular, it says that when the storage limit, M, is

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small, the storage constraint is binding and the optimal solution is affected by M. It is interesting that, when M is small, the minimum average cost is equal to the maximum level of backorders, ̂

, multiplied by the backorder-penalty cost, ̃. The supplier uses a lot-for-lot production (or purchasing) policy, and does not hold any

inventory. For example, the supplier could be a manufacturer, who uses a lot-for-lot production

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policy and delivers the order as soon as the production is completed. Or, the supplier could be a wholesaler who uses cross-docking and holds no inventory at its warehouse. The replenishment leadtime is constant, which is L.

Q

order quantity

r

reorder point fixed cost of the supplier fixed ordering cost of the customer sum of

(i.e.,

)

h

inventory-holding cost per unit per unit time for inventory at the customer



backorder-penalty cost per unit backordered per unit time for the customer

L

replenishment leadtime



demand rate per unit time at the customer

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K

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and

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The following notations will be used in our model:

The superscripts TS, IS, V1, and V2 will be used for traditional system, integrated system, VMI

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system with stockout-cost sharing, and VMI system with fixed transfer payments and stockoutcost sharing, respectively. The subscripts C, S and SC will be used for the customer, the supplier,

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and the supply chain, respectively.

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4. TRADITIONAL SYSTEM In the traditional system (TS), the customer manages her own inventory and places

replenishment orders with the supplier, who in turn satisfies the customer orders. In this system the customer chooses Q and r to minimize her average cost: (

)

(

|

)

(3)

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, where the cost function TC is given by (1). Let (

subject to

(

customer‟s optimal replenishment policy and

) be the

) her minimum average cost

in the traditional system.

(

√ √

)

)

,

, and

(4)

, then

(

)

(

√

)

,

, and

(

)

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If

, then

(

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√

From Lemma 1, if

(5)

, under the customer‟s optimal policy is

The supplier‟s average cost,

(6)

The minimum average supply chain total cost in the traditional system is

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5. INTEGRATED SYSTEM

.

In the integrated system (IS), the supplier makes all the inventory-related decisions for

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the supply chain and minimizes the supply chain total cost. In this system the supplier chooses Q and r to minimize the average supply chain total cost: )

subject to

|

)

(7)

, where the cost function TC is given by (1) and

) be the optimal replenishment policy and

CE

(

(

PT

(

(

. Let

) the minimum average

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supply chain total cost in the integrated system. From Lemma 1, if √

If

√

(

(

)

,

√

(

, then

)

, and

, then

)

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(8)

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(

√

)

,

, and

(

)

(9)

The supplier‟s and the customer‟s average costs in the integrated system are

and

, respectively.

√

assume

√

√

, then

(

)

for all . To avoid this trivial case, we shall

in the rest of our analysis.

6. VMI WITH STOCKOUT-COST SHARING

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Note that if

In this section we present and analyze the VMI system with stockout-cost sharing. In this , to the supplier, under which

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system the customer designs and offers a VMI contract * +,

the supplier manages the inventory at the customer and makes replenishment decisions, but pays the customer a stockout penalty at the rate of $p per unit backordered per unit time when stockouts occur at the customer. The VMI contract also specifies the use of consignment stock, and that the supplier is responsible for all the inventory-holding costs for the inventory at the

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customer, which is $h per unit per unit time. The supplier incurs the fixed ordering costs at the customer,

, as well as his own fixed cost,

. The supplier can choose to accept or reject the

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proposed contract. The customer and the supplier minimize their own costs in choosing the contract parameter value, p, and making replenishment decisions, respectively. We assume that the supplier has a reservation cost 𝑈 > 0 such that he accepts the contract as long as his average

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cost is less than or equal to 𝑈. 6.1. Supplier’s Problem

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If the supplier accepts the VMI contract * +, he will choose Q and r to minimize his own average cost:

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(

subject to

| )

(

|

)

(10)

, where the cost function TC is given by (1). Note that he incurs

stockout penalty costs p, as well as fixed costs (

( ) (

and inventory-holding costs . Let

( )) be the optimal replenishment policy for the supplier and ( )

( )| ) his minimum average cost under the VMI contract * +.

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( )

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Note from Lemma 1 that the optimal replenishment policy for the supplier depends on √

whether

, which is equivalent to

(

. Let us define

)

as (11)

(

√ ( ) {

(12)

)

( )

{

(13)

( )

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( )

)

(

√

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Then, from Lemma 1, the optimal replenishment policy for the supplier is

( ) decreases and

Note from expressions (12) and (13) that as p increases,

( ) increases.

This is expected because as p increases, the supplier will raise the reorder point to reduce his penalty costs for stockouts but then decrease the lot size to reduce his inventory-holding costs.

( )

( )

{

Lemma 2. (a) (b)

)

(d)

√

(

and

( )

√

(

( )

)

)

( ) and

( ) are strictly increasing and concave in

(

.

)

( ) is continuous and strictly increasing in

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(c)

(

√

)

( ) as

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( )

( ) and

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Let us define

( )

(14)

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(

√

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The corresponding minimum average cost of the supplier is

(15) .

.

.

Lemma 2 characterizes the supplier‟s minimum cost function,

( ), given by (14).

Based on Lemma 2, the following proposition provides the condition on which the supplier accepts the VMI contract * +. Proposition 1.

( )

𝑈 if and only if

̅, where

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𝑈 ̅

𝑈

(16)

𝑈

{

Proposition 1 says that there exists a critical value, ̅, such that the supplier accepts the

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contract if and only if the stockout penalty, p, is less than or equal to that critical value. In ⁄

particular, it says that if his reservation cost, U, is smaller than ⁄

value, ̅ , is finite; otherwise, if 𝑈

⁄ , then the critical

⁄ , the supplier accepts the contract for any

value of stockout penalty. 6.2. Customer’s Problem

and

( )

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If the customer chooses p for the stockout penalty, then the supplier will choose

( ) given by (12) and (13) for the replenishment policy and, hence, the customer‟s

average cost will be ( )

(

( )

( )|

(

)

M

PT

where (

)(

( ))

AC Note that

(

)(

)

) (

( ))

(

√

( )

(

)

,

√ ( )

( )

) )

( )

(

(17)

(

( )

CE (

( )

and receives stockout penalty p from the

( ) ( )

{

( )

)

ED

( )

( )

( )

Note that the customer incurs backorder-penalty costs supplier. From (12) and (13),

)(

(

( )

(

)

for

)

(

) (

)

√

( ))

(

(

)

, and

(

)

)

(19) ( )

customer has to solve the following problem to choose stockout penalty p:

14

(18)

for

. Now the

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( ) ( )

Subject to

𝑈. ( )

The participation constraint,

𝑈 assures that the supplier will accept the contract.

Note from Proposition 1 that the supplier will accept the contract if and only if the stockout

( ) is unimodal in

Lemma 3. (a) (b)

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penalty, , is smaller than or equal to the critical value, ̅ , given by (16). .

( ) is minimized at )

√(

̂

(20)

(c)

( ) is strictly decreasing in

(d)

(

(

.

).

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)

for

( ) is continuous and strictly decreasing in

(e)

for

Lemma 3 characterizes the customer‟s cost function,

.

( ) , given by (17). In

particular, Lemma 3(e) implies that the value of stockout penalty minimizing the customer‟s cost

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function is greater than her backorder-penalty cost, . The following proposition characterizes

sharing. Proposition 2. (a) If 𝑈 (b) If 𝑈

, then

(c) If 𝑈

, then ̅

, then ̅

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CE AC

{

{

* ( )+

, in the VMI system with stockout-cost

̅.

and

.

* ̂

where

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the customer‟s optimal choice of stockout penalty,

{

and

. More specifically,

̅+

{ ̅

𝑈

(

* ̂

+) ̅

( )}}

( )}

𝑈 𝑈

.

Proposition 2 not only characterizes the optimal stockout penalty but also provides a solution procedure: If 𝑈

, then

̅

, which is the case corresponding to 𝑈 and 15

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𝑈 in Figure 1. On the other hand, if 𝑈 𝑈

Figure 1). For example, if

, the solution depends on * ̂

, then

and

(see

̅ +, where ̂ and ̅ are given by

(20) and (16), respectively. Proposition 2 also leads to the following important results. Proposition 3. In the VMI system with stockout-cost sharing,

(b)

(

) (

)

and

(

(

)

)

if and only if

.

if and only if

.

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(a)

is 𝑈

(c) The sufficient and necessary condition for

.

Proposition 3(a) and (b) say that in the VMI system with stockout-cost sharing the supply chain is coordinated (i.e., the replenishment decisions and system performance are the same as in

equal to her backorder-penalty cost,

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the integrated system) if and only if the customer‟s optimal choice of stockout penalty,

, is

. But Proposition 3(c), which comes directly from

if and only if the supplier‟s reservation cost, 𝑈, is exactly the

Proposition 2, says that

same as the minimum average supply chain total cost in the integrated system,

, which is

given by (8) and (9). Hence, Proposition 3 implies that the supply chain can be coordinated if

M

and only if the supplier is willing to incur all of the minimum supply chain total cost of the integrated system (i.e., 𝑈

). If the supplier‟s reservation cost, 𝑈, is different from

,

ED

then the customer‟s optimal choice of stockout penalty will not be equal to her backorder-penalty cost and thus the supply chain will be not coordinated.

PT

7. VMI WITH FIXED TRANSFER PAYMENTS AND STOCKOUT-COST SHARING In this section we present the VMI system with fixed transfer payments and stockout-cost sharing and show how it can coordinate the supply chain. This system is the same as the VMI

CE

system with stockout-cost sharing presented in section 6, except that the VMI contract consists of two parameters:

and s, where s is a fixed transfer payment per unit time that the

AC

supplier pays to the customer. If the supplier accepts the VMI contract *

+, he will choose Q and r to minimize his

average cost:

( Since

(

| |

)

(

|

) is the same as

)

( (

| )

(21)

| ) except for the constant term s, the supplier‟s

optimal replenishment policy is the same as in section 6. That is, if we define

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(

(

)

(

*

+, then (

)) as the supplier‟s optimal replenishment policy under the VMI contract

)

( ) and

( ) and

where

(

)

( )

(22)

( ) are given by (12) and (13), respectively. And, the corresponding

(

)

( )

(23)

( ) is given by (14).

where

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minimum average cost of the supplier is

Given the supplier‟s optimal replenishment policy, the customer‟s average cost will be: (

)

(

(

)

(

( ) is given by (17).

where

)

(24)

AN US

( )

)|

Subject to

(

)

(

)

𝑈.

M

Now the customer has to solve the following problem to choose p and s:

) (

and )

𝑈 and

(

)

(

PT

(b)

(

+ whose parameter values are:

𝑈

and (a)

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Proposition 4. Consider a VMI contract *

(25) .

)

𝑈.

CE

Proof. (a) The proof is straightforward from (22) and Proposition 3 (a). (b) From Lemma 2 (d),

( )

. As noted after (19),

( ) = 0. Now, the proof

AC

comes easily from (23) and (24). Proposition 4(a) says that the contract *

+ induces the supplier to choose the same

replenishment policy as in the integrated system. Proposition 4(b) says that the contract *

+

assures that the supplier accepts the contract (i.e., it satisfies the participation constraint) and achieves the same system performance as in the integrated system. Now, if the use of the VMI contract given by (25) is in the customer‟s best interest, then it tells us that VMI with fixed transfer payments and stockout-cost sharing can lead to the same

17

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replenishment policy and system performance as in the integrated system. But the following proposition shows that the contract *

+ given by (25) is indeed an optimal contract for the

customer. Proposition 5. The VMI contract *

+ given by (25) is an optimal contract for the customer in

the VMI system with fixed transfer payments and stockout-cost sharing.

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8. COMPUTATIONAL STUDY

In our computational study we used the following parameter values for the base case: λ = 100 units per week, L = 1 week, KS = $400 per order, KC = $60 per order, h = $1 per unit per week, π = $10 per unit backordered per week, U = $300 per week, and M = 280 units. We also conducted a sensitivity study by varying the values of important parameters around the base case. ⁄ in our analysis (see Section 5). We followed this assumption

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√

Note that we assume

in choosing parameter values.

Table 2 provides our computational results for the traditional system, integrated system, and VMI with stockout-cost sharing (which will be referred to as V1 system in this section).

M

Figure 2, 3, and 4 shows the impact of the supplier‟s reservation cost, storage limit, and the supplier‟s fixed cost, respectively, on the minimum supply chain total costs in the three systems and the customer‟s optimal stockout penalty in the V1 system.

ED

Note that Table 2 does not include VMI with fixed transfer payments and stockout-cost sharing, because it results in the same replenishment decisions and system performance as in the

PT

integrated system (see Proposition 4 and 5). For example, in the base case, the optimal contract for the customer in this system is

= 10

and

𝑈

= 300

and

CE

Under this contract, the optimal replenishment decisions for the supplier are

289.34 = 10.66. = 309

= 71. As a result, the customer‟s and the supplier‟s costs will be

AC

𝑈 = 289.34

𝑈 = 300, respectively, and the supply chain total cost

300 = 10.66 and

will be

In Table 2,

10.66 + 300 = 289.34, which is equal to ,

, and

are defined as

and

18

.

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,

, represent the cost reduction (%) from VMI with stockout-cost sharing

compared with the traditional system for the customer, the supplier, and the supply chain. represents the cost increase (%) in the V1 system compared with the integrated system.

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Supply chain coordination. Note from the sensitivity results with respect to the supplier‟s reservation cost, U, in Table 2 that the V1 system (i.e., VMI with stockout-cost sharing) has the same optimal Q and r and supply chain total cost as in the integrated system

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when U = 289.34, which is the minimum supply chain total cost in the integrated system. In other words, as expected, the V1 system coordinates the supply chain when 𝑈

. But, we

also observed that the supply chain cost in the V1 system can be significantly larger than in the integrated system when 𝑈

was 20.96% when U = 260, while it was no larger than 1.29% when

M

. For example,

, while the cost increase may not be significant when 𝑈

U > 289.34.

ED

We also observed from Table 2 that the cost increase (%) in the V1 system compared with the integrated system can be significant when the storage capacity is small, the supplier‟s fixed cost is large, or when the demand rate is large. For example, = 520, and 17.51% when

PT

100, 12.63% when

was 47.37% when M =

= 130. Note that in our model, the benefits of

VMI to the supplier arise due to the economies of scale in production/delivery. However, a small

CE

storage capacity limits the supplier‟s flexibility in choosing delivery frequency and hence reduces the benefits of VMI to the supplier. As a result, the customer will have to charge a small

AC

stockout penalty to assure the supplier‟s participation (i.e., ̅ is small), which will make the supplier‟s optimal replenishment decisions in the V1 system deviate from the optimal replenishment decisions in the integrated system. Similarly, when the supplier‟s fixed cost or the demand rate is large, the customer will have to charge a small stockout penalty, resulting in different replenishment decisions in the two systems. From the point of view of supply chain coordination, our computational results can be summarized as follows: The cost increase in the V1 system compared with the integrated system

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is small as long as the supplier‟s reservation cost is close to the minimum supply chain total cost of the integrated system. However, the V1 system may perform significantly worse than the integrated system, especially when the supplier‟s reservation cost is small, the storage capacity is small, the supplier‟s fixed cost is large, or the demand rate is large. It should be noted that in these cases, it would be more desirable to implement fixed transfer payments to coordinate the

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supply chain. Benefits of VMI. We observed from Table 2 that the benefits of VMI compared with the traditional system can be significant. For example, in the base case, the costs of the customer and the supplier were reduced by 103.58% and 14.75%, respectively, in the V1 system compared with the traditional system, while the supply chain total cost was reduced by 35.25%. However,

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it was observed that the V1 system can perform worse than the traditional system when the storage capacity, M, is small. For example, the supply chain total cost increased by 11.99% in the V1 system compared with the traditional system when M = 120. This is because, as explained in the above, the benefits of VMI in our model arise due to the flexibility of the supplier in choosing delivery frequency. But a small storage capacity limits this flexibility and hence results ,

M

in less benefits of VMI. On the other hand, we observed that as the supplier‟s fixed cost,

increases, the cost reduction (%) in the V1 system compared with the traditional system

ED

increased for the supplier but decreased for the customer, while it increased and then decreased for the supply chain. This is because when chain increases as

is small, the overall benefits of VMI for the supply

increases, due to economies of scale. But, when ̅ decreases as

PT

participation constraint is binding and

is large, the

increases, which negatively

affects the system performance through the supplier‟s replenishment decisions. It was observed

CE

that as the customer‟s backorder-penalty cost, , increases, the cost reduction (%) from VMI increased for the supplier but decreased for the customer, while it increased for the supply chain.

AC

Impact of storage capacity. With regard to the impact of limited storage capacity on the customer‟s optimal choice of stockout penalty, we observed that when the storage capacity, M, is small, the optimal stockout penalty increases as the storage capacity increases. On the other hand, when the storage capacity is large, the optimal stockout penalty slightly decreases and then remains the same as the storage capacity increases. This result may be interpreted as follows: When the storage capacity is small, the participation constraint in the customer‟s problem is ̅) for the reasons explained in the above, but

binding at the optimal stockout penalty (i.e.,

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̅ increases as the storage capacity increases. On the

it becomes less tight and, as a result,

other hand, when the storage capacity is large, the participation constraint in the customer‟s problem becomes no longer binding, while the storage constraint in the supplier‟s problem can be still binding at the optimal solution. For example, in our computational results, the participation constraint was binding for

240, while the storage constraint was binding for

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280. When the storage capacity is in the range where the storage constraint is binding, as the storage capacity increases, the supplier will choose a larger order quantity, which means the customer will receive stockout penalties from the supplier less frequently. Hence the customer may want to induce more stockouts by decreasing the stockout penalty. When the storage capacity is in the range where the storage constraint is not binding, the supplier‟s choice of

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replenishment policy is not affected by the storage capacity, so the optimal stockout penalty remains the same.

In our model the storage limit is assumed to be a given parameter. But if the customer were able to choose the storage limit and specify it in the VMI contract, Table 2 suggests that it might be optimal for the customer to choose a storage limit for which the storage limit constraint

, in the V1 system and that

M

is binding. In other words, we can see from Table 2 that M = 240 minimizes the customer‟s cost, 240, which means the storage constraint is

ED

binding. On the other hand, we can also see from Table 2 that, as expected, the supply chain total cost is minimized when the storage limit constraint is non-binding. For example, Table 2 says that

300 or larger minimizes the supply chain total cost,

, in the V1 system and that

PT

296, which is smaller than the storage limit.

CE

9. CONCLUSION

We have studied and compared four business scenarios: traditional system, integrated system, VMI with stockout-cost sharing, and VMI with fixed transfer payments and stockout-

AC

cost sharing by using an EOQ model with shortages allowed under limited storage capacity. We have shown that VMI with stockout-cost sharing and the integrated system result in

the same replenishment decisions and system performance if and only if the supplier‟s reservation cost is equal to the minimum supply chain total cost of the integrated system. This result is important for two reasons. Firstly, note that, as mentioned in Section 2, it is well known in the economics literature that the first-best can be achieved in the agency relationship between two risk-neutral parties by using a contract with fixed payments. But, our result suggests that

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VMI may be able to coordinate the supply chain even without fixed transfer payments if the supplier is willing to incur all of the minimum supply chain total cost of the integrated system. Secondly, the above result also says that VMI and the integrated system may result in different replenishment decisions and system performances without the aforementioned condition. This suggests that VMI and the integrated system should be distinguished in both academia and

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practice. In particular, since the design of VMI contract affects the supplier‟s replenishment decisions and system performance, the VMI contract should be carefully designed when implementing VMI.

We have also shown how VMI along with fixed transfer payments as well as stockoutcost sharing can coordinate the supply chain regardless of the supplier‟s reservation cost. In fact,

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our computational results suggest that VMI with stockout-cost sharing may perform significantly worse than the integrated system in some situations. In this case, fixed transfer payments might be introduced to coordinate the supply chain.

Although our paper provides interesting results on supply chain coordination with VMI, it has been examined in a deterministic setting using a simple form of VMI contract. It would be

M

interesting if the results could be extended to a stochastic model and/or a more general form of VMI contract. Also, note that in our model both the supplier and the customer are assumed to be

ED

risk-neutral. Another interesting extension of our paper might be to consider a supplier-customer relationship where the supplier and the customer have different degrees of risk-averseness and

AC

CE

PT

examine how it will impact on the VMI contract design and the system performance.

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REFERENCES Battini, D., Gunasekaran, A., Faccio, M., Persona, A., & Sgarbossa, F. (2010). Consignment stock inventory model in an integrated supply chain. International Journal of Production Research, 48(2), 477–500. Bernstein, F., Chen, F., & Federgruen, A. (2006). Coordinating supply chains with simple pricing

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schemes: The role of vendor-managed inventories. Management Science, 52(10), 14831492.

Bertazzi, L., Paletta, G., & Speranza, M.G. (2005). Minimizing the total cost in an integrated vendor-managed inventory system. Journal of Heuristics, 11(5-6), 393–419.

Bichescu, B. & Fry, M. (2009). Vendor-managed inventory and the effect of channel power. OR

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Spectrum, 31(1), 195-228.

Bookbinder, J. H., Gumus, M., & Jewkes, E. M. (2010). Calculating the benefits of vendor managed inventory in a manufacturing-retailer system. International Journal of Production Research, 48(19), 5549-5571.

Braglia, M. & Zavanella, L. (2003). Modelling an industrial strategy for inventory management

Research, 41(16), 3793-3808.

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in supply chains: the „Consignment Stock‟ case. International Journal of Production

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Cachon, G.P. (2001). Stock wars: Inventory competition in a two-echelon supply chain with multiple retailers. Operations Research, 49(5), 658-674. Chen, J.-M., Lin, I-C., & Cheng, H.-L. (2010). Channel coordination under consignment and

PT

vendor-managed inventory in a distribution system. Transportation Research Part E, 46(6), 831-843.

CE

Chen, J.-M., Cheng, H.-L., & Lin, I.-C. (2011). On channel coordination under price-dependent revenue-sharing: can eBay‟s fee structure coordinate the channel? Journal of the

AC

Operational Research Society, 62(11), 1992-2001. Choi, T.-M. (2011). Coordination and risk analysis of VMI supply chains with RFID technology. IEEE Transactions on Industrial Informatics, 7(3), 497-504.

Corbett, C.J., Zhou, D., & Tang, C.S. (2004). Designing supply contracts: contract type and information asymmetry. Management Science, 50(4), 550-559.

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Darwish, M.A. & Goyal, S.K. (2011). Vendor-managed inventory model for single-vendor single-buyer Supply Chain. International Journal of Logistics Systems and Management, 8(3), 313-329. Darwish, M.A. & Odah, O.M. (2010). Vendor managed inventory model for single-vendor multi-retailer supply chains. European Journal of Operational Research, 204(3), 473-484.

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Fry, M., Kapuscinski, R., & Olsen, T. (2001). Coordinating production and delivery under a (z, Z)-type vendor-managed inventory contract. Manufacturing & Service Operations Management, 3(2), 151-173.

Gerchak, Y., Cho, R.K., & Ray, S. (2006). Coordination of quantity and shelf-retention timing in the video movie rental industry. IIE Transactions, 38(7), 525–536.

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Guan, R. & Zhao, X. (2010). On contracts for VMI program with continuous review (r, Q) policy. European Journal of Operational Research, 207(2), 656-667.

Lee, J.-Y. & Cho, R.K. (2013). Contracting for vendor-managed inventory with consignment stock and stockout-cost sharing. International Journal of Production Economics, 151(May), 158-173.

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Lee, J.-Y. & Ren, L. (2011). Vendor-managed inventory in a global environment with exchange rate uncertainty. International Journal of Production Economics, 130(Apr), 169-174.

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Li, S. & Hua, Z. (2008). A note on channel performance under consignment contract with revenue sharing. European Journal of Operational Research, 184(2), 793-796. Mas-Colell, A., Whinston, M.D., & Green, J.R. (1995). Microeconomic Theory. Oxford

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University Press, Inc., New York.

Nagarajan, M. & Rajagopalan, S. (2008). Contracting under vendor managed inventory systems

CE

using holding cost subsidies. Production and Operations Management, 17(2), 200-210. Persona, A., Grassi, A., & Catena, M. (2005). Consignment stock of inventories in the presence

AC

of obsolescence. International Journal of Production Research, 43(23), 4969–4988. Shah, J. & Goh, M. (2006). Setting operating policies for supply hubs. International Journal of Production Economics, 100(2), 239-252.

Wang, Y., Jiang L., & Shen, Z.-J. (2004). Channel performance under consignment contract with revenue sharing. Management Science, 50(1), 34-47.

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Wong, W.K., Qi, J., & Leung, S.Y.S. (2009). Coordinating supply chains with sales rebate contracts and vendor managed inventory. International Journal of Production Economics, 120(1), 151-161. Zhang, T., Liang, L., Yu, Y., & Yu, Y. (2007). An integrated vendor-managed inventory model for a two-echelon system with order cost reduction. International Journal of Production

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Economics, 109(1-2), 241-253.

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CE

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M

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Zipkin, P. (2000). Foundations of Inventory Management. McGraw-Hill, New York.

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APPENDIX. PROOFS OF RESULTS IN SECTION 3, 6 and 7 | ̃ ̃ ̃) is strictly convex in

(

Proof of Lemma 1. (a) Note that

and . Let ̆ and ̆ be

the optimal solution to the unconstrained problem. It is well known that ̃

, ̆

̃̃

̃ ̃

, then it must be that ̂ ̆

The condition, ̆ ̃

̆

̆, ̂

̆

̆

√ ̃( ̃

or √ ̃( ̃

̃̃ ̃)

̃̃

( ̂ ̂ | ̃ ̃ ̃)

̃̃ ̃ ̃

| ̃ ̃ ̃) is strictly convex in

(

, then since

̃)

̆.

̆ , and ̂

, is equivalent to ̃ ̃ ̃

̆

̃ ̃

(b) If ̆

̆

̃̃ ̃ ̃

( ̆ ̆ | ̃ ̃ ̃)

̆ , and

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If ̆

̃ ( ̃ ̃)

√

CR IP T

̆

̂.

and ,

the constraint must be binding at the optimal solution. So the optimal solution can be obtained by | ̃ ̃ ̃) subject to

(

( )

By setting

( )

̃

)( , ̂

̃ ( ̃ ̃) ̂

So,

√

̃

(̃

̂

.

̃ ̃

̃

̂

√

(̃

̃)

)

̂ ̃̂

̃ ̂

, then (

̃ ( ̃ ̃)

̃̂

̃( ̂

̃̂

)√

( ) )

).

√

√

√

( )

> 0 and

√

( ) is strictly increasing and concave in ( )

)

√

̃

√ (

)

̂)

̃(

̂

̃ ( ̃ ̃)

Proof of Lemma 2. (a) ( )

(

̃̂

PT

CE ̃̂

AC

Hence, ̂

̃

̃

̂

or ̂

̃

̃ ( ̃ ̃) ̂

)

and ̂

̃

̃

̃ ( ̃ ̃)

If we let

(

(

̃ ( ̃ ̃)

√

( ̂ ̂ | ̃ ̃ ̃)

̂

̃

)

)

̃(

M

( ̃

̃

̃

| ̃ ̃ ̃)

(

( )

.

ED

minimizing

(

√

(

)

(

)

< 0 for

.

)

√ ( )

26

,

, where .

√

.

ACCEPTED MANUSCRIPT

where ( )

(

( )

)

( )

. Note that

for

and

. ( )

( )

√ ( )

( )

But, it is easy to show that is strictly increasing in

(

)

( )

( ) ( ) (

√ ( )

( ))

(

)

( )√ ( )

( )√ ( )

( ) is strictly concave in

(b) Since

√

or )

√ (

√ (

)

(

)

)

(

)

(

, which means that

( )

)

(

) )

√

(

. To show

)

,

)

)

(

)

(

(

)

)

.

ED

(

)

for

M

(

. (

,

(

But,

which means that

( )

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which means that

(

( )

.

( )

( )

√ ( )

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( )

(c) This result comes from Lemma 2(a) and (b). (d) Note from (14) that if (

( )

) or

CE

⁄( ( )

). Since

PT

( )

AC assuming ( ) If 𝑈

⁄( (

√

𝑈 or √ √

𝑈 or or

𝑈 is equivalent to , then

( )

(

⁄(

⁄(

))

( ) and that if

) from (11),

means that

. ( )

, then from Lemma 2, 𝑈

or (

𝑈 ) 𝑈

(see Section 5),

𝑈 is equivalent to

𝑈 . But, since we are (

)

𝑈

and hence

. 𝑈 is equivalent to

27

,

) ). Also, note from Proposition 3 (a) that

( )

. Hence, from (8) and (9),

Proof of Proposition 1. If 𝑈 ( )

( )

,

( )

𝑈 or √

(

)

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(

)

(

𝑈

𝑈) 𝑈

(

or

)

(𝑈

𝑈 , which is equivalent to

or 𝑈

equivalent to

if 𝑈

. This means that if

𝑈

( )

𝑈 for any

completes the proof. (

)

√

( )

( )

.

)

(

√

)

(

)

𝑈)

when 𝑈

AN US

√

where

(

( )

Proof of Lemma 3. (a)

⁄

. Then, (

or 𝑈

. This means that

𝑈 is

by assumption,

. Now suppose that

( )

, then

√

. Note that since

𝑈 for any

) or

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𝑈 or √

√ (

. This

)

, where ( )

(

)

. It is easy to

show that there is one negative and one positive p satisfying θ(p) = 0. Let ̂ be the positive

)

√(

̂

M

solution of θ(p) = 0, that is, .

ED

As θ(p) is negative for 0 < p < ̂ and positive for p > ̂ , so is unimodal in p > 0.

( ) is unimodal for p > 0 with

( ̂ )

( ) is

, where ̂ is given in the

( ) is minimized at ̂ .

above. So, (

( )

)

(

CE

(c)

PT

(b) From Lemma 3(a),

( ). Hence,

AC

where ( )

( )

( )

( )

(

√

( )

( ))

(

( ( )

( )

(

)

(

(

)

)

(

( )

( )

)

. (

)

)

( ) ( )

( )

)

(

(

{

( )

( ))

28

(

( ) (

) ( )

)

( ) ( ( )

}

)

( )

(

( )) ) ( )

)

.

ACCEPTED MANUSCRIPT

(

√

(

)

(

)

(

)

Since ( ) (

(d)

(

)

(

)

(

( )( ( )

( )

( )

and

√

)

(

)

)

(

for all )

(

)

)

√

(

(

) (

)

√

Also, note that

(

(

)

)

)

(

)

(

√

((

√

(

PT

(

√(

(e) Note that ̂

)

√

√

√

(

√

(

)

(

)

) )

(

(

ED

(

,

(

(

)

)

√

)

)

(

)

)

) (

) (

) √(

. Then,

)

(

(

√

(

)

(

√

< ̂ (see the proof of Lemma 3(a)). This means that

CE

)

√

)

)

AC

for

√

̅

. But, since

be that

)

)

(

)

) and that

( )

for 0 < p

( ) is strictly decreasing in

for

( ) is continuous and

.

Proof of Proposition 2. (a) Note from Lemma 2 (c) and (d) that strictly increasing in p > 0 and

) (

̂ . This together with Lemma 3(c) and (d) implies that

strictly decreasing in

.

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(

)

.

)

(

)

M

√

for all

(

)

( Note that since

)

( )

,

, where

√ √

)( ( )

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( ( ) )

( )

( ) is continuous and

. So, it is easy to see that if 𝑈

( ) is strictly decreasing in p for

, then

from Lemma 3 (e), it must

̅.

(b) Similarly, from Lemma 2 (c) and (d) and Lemma 3 (e), it is easy to see that if 𝑈

29

, then

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̅

.

(c) Now, suppose that 𝑈

. Then, from Lemma 2 (c) and (d), ̅

that

( )

for

if

𝑈

, then, from Lemma 2 (b) and (c), ̅ * ̂

,

for

( )

{

for

( )

, and

Proof of Proposition 3. (a) If

( )

, it is easy to see that

( )

( ) is strictly increasing in (

and

)

( ) ( (

(

(

(

|

)

(

, then

( )

)

)

(

( )|

)

and

(

)

)

AC (b)

( )

( 𝑈

and (14). Hence, (c)

. Hence,

)

(

and

)

( ) is strictly (

, then

)

) ( (

(

( )

( )|

)

from Proposition 3(a), )| )

)

(

|

)

from Proposition 3(a). Since

and ,

(

(

)

(

)|

)

(

|

)

. From Proposition (a) and (c), if 𝑈

, then

. This completes the proof.

Proof of Proposition 4. (a) (22). But

)

and

(c) From Proposition 2 (b), if 𝑈 , then

̅+ or a

( )

( )

CE

(

( )| )

)

(

) is strictly convex in

(

. Hence, if

( )|

(

PT

If

)

( )

ED

(

, then since

for

. Note that

M

( ) (

If

* ̂

is either

.

(b) Note that ( )

𝑈 , then, from

AN US

from (8), (9), (12), and (13). Now, suppose that decreasing and

( ) for

̅ , which completes the proof. Finally, if

( )}. ̅

( )

. But, since

. In this case, it is easy to see that

( ) subject to

𝑈,

. More specifically,

̅+ from Lemma 3 (a) and (b). If

Lemma 2 (b) and (c), ̅ minimizer of

. Therefore,

CR IP T

̅

( )

, and

. Also, note from (17)

( 𝑈

and )

( 𝑈 ( ) ( )

( 𝑈 )

)

( 𝑈

)

( ) from

from Proposition 3(a). 𝑈

) ( )

( ) and

from (23). But, 𝑈.

𝑈

from (24). But

30

( )

from (8), (9),

ACCEPTED MANUSCRIPT

( )

(

( )

( 𝑈

Hence,

( )|

)

(

)

)(

( )

)

( )

𝑈.

Lemma A1. In the VMI system with fixed transfer payments and stockout-cost sharing, if a contract * ̃ ̃ + is optimal for the customer, then ( ̃ ̃)

( ̃ ̃) ̃

( ̃)

𝑈. ( ̃ ̃)

𝑈 . From (19) and (20),

̃ . Let

( ̃ ̃)

𝑈

( ̃)

𝑈

. Then, (̃ (̃

)

)

( ̃)

( ̃)

( ̃) ( ̃)

( ̃) ̃

̃ and

CR IP T

Proof: Suppose

( ̃ ̃)

̃

𝑈 and

( ̃)

̃

. Now, let

( ̃ ̃ ),

̃

so * ̃ ̃ + cannot be an optimal contract. This completes the proof. (

) and

( ̆ ̆) (̆

𝑈 and (

Since

)

( ̆ ̆)

) ( ̆ ̆) (

( ̆ ̆)

( ̆ ̆)

PT

(

Hence,

( ̆ ̆)

CE

𝑈

ED

But, ( ̆ ̆)

(

( ̆ ̆ ) , such that

𝑈

).

)

( ̆ ̆ )| ̆ ̆ )

( ̆ ̆ )| (

𝑈

𝑈

( ̆)

( ̆ ̆)

( ̆ ̆)

(

( ̆ ̆)

( ̆ ̆ )|

(

( ̆ ̆)

( ̆ ̆ )|

(

( ̆ ̆)

( ̆ ̆ )|

(

contradiction, because

is the minimum value of

( ̆ ̆ )|

31

) (

.

( ̆ ̆ )|

(A1) ̆)

̆

̆

( ̆ ̆)

(A1) and (A2) mean that

AC

̆

𝑈 by Proposition 4(c),

( ̆ ̆) ( ̆ ̆)

( ̆ ̆)

𝑈, Lemma A1 implies that we can

+ , where

M

)

( ̆ ̆)

𝑈. Note that if

easily construct another contract * ̆ (̆

AN US

Proof of Proposition 5. Suppose that there exists a contract * ̆ ̆ + such that

|

̆)

̆ ̆)

)

̆ (A2)

. But, this gives a ).

ACCEPTED MANUSCRIPT

U5

(p) U2

(p)

hM

(p)

(p)

U4 U1

U3 (p)

CR IP T

hM 0

0

(p)

(p) ̅ (𝑈 )

̅ (𝑈 )

̅ (𝑈 )

̂

̅ (𝑈 )

(b)

AN US

(a)

̅ (𝑈 )

(p)

400

15

ED

300

20

200 100 0 260

270

280

10 5 0

290

300

Optimal Stockout Penalty

25

M

500

PT

Supply Chain Total Costs

Figure 1. Graphical explanation of the solution procedure for optimal stockout penalty,

310

CE

Supplier's Reservation Cost, U

AC

Figure 2. Impact of supplier‟s reservation cost on system performance and optimal stockout penalty,

32

ACCEPTED MANUSCRIPT

25 20

500 400

15

300

10

200 5

100 0

0 100

150

200

250

300

AN US

Storage Limit, M

CR IP T

600

Optimal Stockout Penalty

Supply Chain Total Costs

700

Figure 3. Impact of storage limit on system performance and optimal stockout penalty,

500

15

ED

400

20

300 200 100 0

370

400

5 0

430

460

490

520

Supplier's Fixed Cost,

AC

CE

340

10

Optimal Stockout Penalty

M

25

PT

Supply Chain Total Costs

600

Figure 4. Impact of supplier‟s fixed cost on system performance and optimal stockout penalty,

33

ACCEPTED MANUSCRIPT

Table 1. Summary of the four systems

Contract Parameters

Decisions; Costs

Customer Supplier

VMI with Fixed Transfer Payments and Stockout-Cost Sharing

Stockout penalty (p)

Stockout penalty (p), fixed transfer payment (s)

Q, r; , h,

p;

No decision;

Q, r;

-p

, h,

p, s; , -p, -s Q, r;

, h, p

Supplier (or central planner) minimizes supply chain total cost

, h, p, s

Each supply chain partner minimizes its own cost

AN US

Objective Function

Q, r;

VMI with Stockout-Cost Sharing

CR IP T

Integrated System

Traditional System

AC

CE

PT

ED

M

* “-p” and “-s” mean that the customer receives penalties.

34

Table 2. Computational results Traditional System

Comparisons (%)

104.45

348.16

452.60

309

71

289.34

21.05

294

86

-3.74

296.81

293.08

103.58

14.75

35.25

1.29

90 90 90 90 90 90 90 90 90 90 90 90 90 90 90 90 90 90 90 90 90 90 90 83 87 91 92 80 99 109 118

104.45 104.45 104.45 104.45 104.45 104.45 104.54 104.45 104.45 104.45 104.45 104.45 104.45 104.45 104.45 104.45 104.45 104.45 104.45 104.45 104.45 104.45 104.45 101.42 103.28 105.25 105.83 99.09 109.55 114.42 119.09

348.16 348.16 348.16 348.16 348.16 348.16 362.14 348.16 348.16 348.16 348.16 348.16 348.16 348.16 348.16 348.16 348.16 295.93 322.04 374.27 400.38 426.49 452.60 338.06 344.27 350.82 352.77 330.29 365.15 381.39 396.96

452.60 452.60 452.60 452.60 452.60 452.60 466.68 452.60 452.60 452.60 452.60 452.60 452.60 452.60 452.60 452.60 452.60 400.38 426.49 478.71 504.83 530.94 557.05 439.48 447.55 456.07 458.60 429.38 474.69 495.80 516.05

309 309 309 309 309 309 142 158 175 193 212 231 250 269 289 318 318 297 308 310 311 312 313 327 316 304 301 302 310 312 313

71 71 71 71 71 71 58 62 65 67 68 69 70 71 71 71 71 73 72 70 69 68 67 53 64 76 79 63 80 88 97

289.34 289.34 289.34 289.34 289.34 289.34 421.27 382.40 353.85 332.87 317.55 306.51 298.80 293.70 290.68 289.20 289.20 269.68 279.61 299.03 308.70 318.33 327.94 280.82 286.03 291.63 293.31 274.36 304.19 318.97 333.69

2.77 3.82 5.76 10.00 16.74 21.05 2.14 2.62 3.26 4.17 5.49 7.50 10.71 16.07 21.13 20.74 20.74 20.74 21.01 10.71 6.25 4.41 3.41 13.02 17.04 25.05 29.06 20.99 7.76 4.33 3.00

354 341 329 309 298 294 240 235 232 232 235 240 248 259 274 311 311 290 294 308 328 348 368 302 297 292 290 293 319 349 380

6 29 51 71 82 86 -40 -25 8 28 45 60 72 81 86 86 86 87 86 72 52 32 12 78 83 88 90 77 71 51 30

89.97 45.40 15.21 0.00 -3.52 -3.74 320.83 206.86 122.93 65.17 28.73 8.33 -1.13 -4.09 -4.06 -3.53 -3.53 -3.29 -3.49 -0.91 13.17 37.13 69.35 -5.86 -4.56 -3.16 -2.74 -3.36 5.26 39.03 92.11

260.00 270.00 280.00 289.34 295.00 296.81 300.00 300.00 300.00 300.00 300.00 300.00 300.00 300.00 298.80 296.26 296.26 276.26 286.59 300.00 300.00 300.00 300.00 292.55 295.16 297.96 298.80 281.13 300.00 300.00 300.00

349.97 315.40 295.21 289.34 291.48 293.08 620.83 506.86 422.93 365.17 328.73 308.33 298.87 295.91 294.74 292.73 292.73 272.97 283.10 299.09 313.17 337.13 369.35 286.69 290.59 294.80 296.06 277.77 305.26 339.03 392.11

13.86 56.53 85.44 100.00 103.37 103.58 -206.91 -98.05 -17.70 37.60 72.49 92.02 101.08 103.92 103.89 103.38 103.38 103.15 103.34 100.87 87.39 64.45 33.60 105.78 104.42 103.00 102.59 103.39 95.20 65.88 22.66

25.32 22.45 19.58 16.89 15.27 14.75 17.16 13.83 13.83 13.83 13.83 13.83 13.83 13.83 14.18 14.91 14.91 6.65 11.01 19.84 25.07 29.66 33.72 13.46 14.27 15.07 15.30 14.88 17.84 21.34 24.43

22.68 30.31 34.78 36.07 35.60 35.25 -33.03 -11.99 6.56 19.32 27.37 31.88 33.97 34.62 34.88 35.32 35.32 31.82 33.62 37.52 37.96 36.50 33.70 34.77 35.07 35.36 35.44 35.31 35.69 31.62 24.02

20.96 9.01 2.03 0.00 0.74 1.29 47.37 32.54 19.52 9.70 3.52 0.59 0.02 0.75 1.40 1.22 1.22 1.22 1.25 0.02 1.45 5.90 12.63 2.09 1.60 1.09 0.94 1.24 0.35 6.29 17.51

M

AN US

90

115 115 115 115 115 115 110 115 115 115 115 115 115 115 115 115 115 115 115 115 115 115 115 118 116 114 113 109 120 126 131

AC

CE

M

VMI with Stockout-Cost Sharing

115

ED

U

260 270 280 289.34 295 310 100 120 140 160 180 200 220 240 260 300 302 340 370 430 460 490 520 6 8 12 14 90 110 120 130

PT

Base Case

Integrated System

CR IP T

ACCEPTED MANUSCRIPT

35