Impact of consignment inventory and vendor-managed inventory for a two-party supply chain

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Int. J. Production Economics 113 (2008) 502–517 www.elsevier.com/locate/ijpe

Impact of consignment inventory and vendor-managed inventory for a two-party supply chain$ Mehmet Gu¨mu¨s-1, Elizabeth M. Jewkes, James H. Bookbinder Department of Management Sciences, University of Waterloo, Waterloo, Ontario, Canada N2L 3G1 Received 16 November 2006; accepted 19 October 2007 Available online 13 February 2008

Abstract Vendor-managed inventory (VMI) and consignment inventory (CI) are supply-chain sourcing practices between a vendor and customer. VMI allows the vendor to initiate orders on behalf of the customer. This presumably benefits the vendor who can then make replenishment decisions according to her own preferences. In CI, as in the usual independentsourcing approach to doing business, the customer has authority over the timing and quantity of replenishments. CI seems to favor the customer because, in addition, he pays for the goods only upon use. Our main aim is to analyze CI in this supply chain under deterministic demand, and provide some general conditions under which CI creates benefits for the vendor, for the customer, and for the two parties together. We also consider similar issues for the combined use of CI and VMI. r 2008 Elsevier B.V. All rights reserved. Keywords: Consignment stock; Direct replenishment; Inventory sourcing; Logistics; Supplier-managed inventory

1. Introduction Planning, sourcing raw materials, making the product, and delivering to customers are typical operational processes for a company within a supply chain. Here we consider a customer who purchases goods from a vendor. The customer’s processes comprise the planning of his requirements, sourcing $ Research partially supported by NSERC and by SSHRC. The authors are grateful to Prakash Abad for extensive discussions of the calculations and results. Corresponding author. Tel.: +1 519 888 4013; fax: +1 519 746 7383. E-mail address: [email protected] (J.H. Bookbinder). 1 Present address: School of Business and Management, American University of Sharjah, United Arab Emirates.

goods from the vendor, and releasing those goods to end-consumers. The vendor, similarly, plans her requirements and sources materials or parts for production, manufactures goods, and releases those goods to the customer. When these two firms are independent and linked in a supply chain as in Fig. 1, decisions on operational processes are, in general, made individually. In the usual sequence of events, the customer first develops his requirements plan and sourcing method based on his own costs. The vendor then reacts to fulfill the customer’s requirements. Hence, replenishment decisions made by the customer do not necessarily consider the choices of his upstream business partner. A common focus of research and supply-chain practice is to seek mechanisms to align the decisions

0925-5273/$ - see front matter r 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.ijpe.2007.10.019

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Goods

Vendor’s Facility

Plan

Source

Make

503

Customer’s Facility

Deliver

Plan

Source

Make

Deliver

The supply chain between the two parties Fig. 1. The supply chain between the vendor and customer: the primary interrelated operations are the customer’s plan and source choices, and the corresponding make and deliver decisions of the vendor.

Purchase Order/Release Order

Goods Receipt

Material Inspection

Transfer of Ownership in Inventory Sourcing

Material Warehousing

Selling

Transfer of Ownership in Consignment Inventory

Fig. 2. The customer’s sourcing activities: transfer of ownership in inventory sourcing and consignment inventory.

of chain members by means of contracts or agreements. Those arrangements aim to increase the overall supply-chain performance. Vendormanaged inventory (VMI), one such agreement, was analyzed by Gu¨mu¨s- et al. (2006) to obtain conditions under which it may lower the costs of each party and of the chain. There are, however, other practices that seem to unbalance the total costs of supply-chain members. In this paper, we will analyze in detail one of those practices, consignment inventory (CI). Our aim is, similarly, to determine conditions whereby consignment stocks create benefits for the customer, the vendor, or for both parties. In CI, goods are owned by the vendor until they are used by the customer. Those goods are stored at the customer’s premises. Although the customer may have authority over the timing and quantity of orders, he pays for the goods only upon use or just afterward. Hence, the customer does not tie up his capital in inventory. In the traditional way of doing business, which we will call ‘‘inventory sourcing’’ (IS) throughout, the customer orders from the vendor based on his total inventory holding costs (both costs of opportunity and physical storage, where opportunity cost refers to the cost of capital), and costs of ordering. IS is generally characterized by a purchasing contract including shipment terms, annual demand specified by the customer, and the price per unit purchased by him. Under this practice, which will be our base case

for analysis, the customer is invoiced by the vendor once the goods arrive at his premises (see Fig. 2). He owns the product at that point. In CI, ownership of goods is transferred to the customer only after they leave his in-house warehouse for sale or manufacturing. If other terms of the purchasing contract stay the same as in IS, one major benefit to the customer is deferral of payment until use. When end-consumer demand is unknown, CI also allows the customer to hedge against uncertainties in production and sales. This will influence his total inventory carrying cost. With the customer’s inventory now off his balance sheet, conventional wisdom holds that the customer gains most from CI. The benefits of CI are less clear for the vendor. One situation that favors CI is where the vendor offers new products that the customer hesitates to buy, or expensive items difficult for the customer to own. In that case, the vendor can use CI as a strategic means to create new sales channels (Piasecki, 2004). This motivation, however, does not explain why a vendor would accept a CI contract when demand is stable and the material purchased is not new. An example of such is seen in the Automation and Drives division of Siemens, where standard parts such as metal springs and nuts can be consigned from suppliers even though the demand during a year can be quite stable. Other scenarios when a vendor might accept a CI contract include a power differential between a strong

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customer and a ‘‘weaker’’ vendor who needs to accommodate the customer’s wishes, or when the vendor at least has sufficient power to negotiate more favorable terms in the CI agreement. There appears to be very little previous work that examines analytically the impact of CI. The focus of the present paper is to establish closed-form results that specify general conditions under which CI is beneficial to one or both parties. To the best of our knowledge, there is no academic work that treats CI in this context. In the literature, CI is mostly taken to be synonymous with VMI or with CI plus VMI (‘‘C&VMI’’). In VMI, replenishment decisions are made by the vendor on behalf of the customer. In CI, even though the vendor is informed about the consumption of goods at the customer’s premises, it is still the customer who finalizes the timing and quantity of orders. We will consider both types of agreements in this paper. The framework we use is similar to those in joint economic lot sizing (JELS) decisions. The JELS literature generally assumes a central decision maker that can optimize the sum of total costs of the vendor plus the customer. The context is very similar in each paper, and the contributions are incremental. Banarjee (1986), the first to analyze the integrated vendor–buyer case, examines a lot-for-lot model in which the vendor V manufactures each shipment as a separate batch. Goyal (1988) extends this work in that he formulates a joint total-relevant-cost model for a single vendor and customer productioninventory system, where the vendor’s lot size is an integer multiple of the customer’s order size. Lu (1995) extended Goyal’s (1988) work by allowing V to supply some quantity to the purchaser before completing the entire lot. Goyal (1995) employed the example provided by Lu for the single vendor and buyer but showed that a different shipment policy could result in a better solution. Hill (1997) considers a single vendor who manufactures a product at a finite rate and in batches, and supplies a sole buyer whose external demand is level and fixed. The vendor incurs a batch setup cost and a fixed delivery cost associated with each shipment. Hill’s policy assumes that successive shipment sizes increase by a factor whose value lies between one and the ratio of manufacturing rate to the product’s demand rate. He concludes that although Goyal’s (1995) policy may perform much better than Lu’s equal-size-shipment policy, his own policy outperforms all.

Similar to the JELS literature, we use a base case (IS) for comparison purposes, contrasting that to other models which assume that the parties in the supply chain still make decisions independently (whether coordinated or not). JELS studies do not discuss how the savings created by central decision making should be divided between parties involved. Benefits achieved are difficult to generalize because, for example, the customer’s ordering cost is not explicit in those models. The CI or C&VMI sourcing models that we consider require a shift of certain costs from one actor to another to reflect changes in decisionmaking responsibility or ownership of inventory. We provide a breakdown of cost parameters so as to identify the impact of such changes on each member. Sucky (2005) extends JELS to a bargaining model, where the vendor offers a side payment to the customer whose costs under JELS go up compared to individual decision making. It is assumed that the vendor, who achieves cost savings under JELS, makes a ‘‘take-it-or-leave-it’’ offer of joint policy with a side payment. The customer may accept the vendor’s offer, or if he is not satisfied with it, can enforce his economic order quantity (EOQ). The bargaining then ends. Sucky assumes that the vendor has full information regarding the customer’s costs. A number of papers have also been written on combined use of CI and VMI. This literature discusses various C&VMI systems that differ in the costs considered, the demand structure, and the nature and number of supply-chain members. Boyaci and Gallego (2002) study a system of a single wholesaler and retailer under deterministic but price-sensitive demand. They analyze the impacts of coordinating pricing and replenishment when decisions are made jointly. They use wholesaler-owned inventory with delayed payment versus CI to extend the models of Crowther (1964) and Monahan (1984). They conclude that pricing and inventory decisions are best made with a coordinated-channel’s profit function. In our paper, we analyze the impacts of CI from the operational point of view. That is, under CI, there is no change in pricing terms from those in the purchasing contract under inventory sourcing. This enables us to focus on operational benefits to both parties. If one party is not satisfied with the outcome, a price change may then become an option, as it would be in industry.

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Valentini and Zavanella (2003) discuss the use of consignment stock by a manufacturer who provides automotive parts and manages the inventory of her customer. While the authors’ main aim was to qualitatively analyze the advantages and disadvantages of this sourcing practice, they compare it numerically with Hill’s (1997) solution, using the same deterministic model. Based on numerical examples only, they conclude that consignment stock outperforms the usual inventory models. Building on the analysis of Valentini and Zavanella, Persona et al. (2005) assume the same characteristics of the agreement and analyze the consequences of product obsolescence, concluding that obsolescence decreases the optimal level of consignment stock. Other articles examine C&VMI in various contexts. For example, Dong and Xu (2002) explore the economics of C&VMI in the short and long terms. Gerchak and Khmelnitsky (2003) provide an interesting example of C&VMI when reported demand cannot be verified. They consider a retailer selling newspapers, and his vendor (a publisher), under VMI and revenue sharing. They analyze the impacts on coordination of the retailer’s sales reports (to the publisher). Although we take both CI and VMI into account in this paper, the problem setting, the approach we use, and our goal are quite distinct from those of Dong and Xu (2002) or of Gerchak and Khmelnitsky (2003). We consider a well-known problem but analyze it under different partnerships, accounting for changes in certain cost parameters. We provide closed-form solutions to see under what conditions a partnership is more favorable than others. 2. Problem definition Suppose a customer purchases an established product (or standard product mix) from a vendor. As in the usual EOQ setting, yearly demand is deterministic with no backordering. Demand is realized at the customer, at a constant rate per unit time. The vendor and customer are independent firms, each with the goal of minimizing their own total cost. Under IS, the customer orders from the vendor based on his total cost of planning (fixed cost per order), sourcing (fixed cost per shipment received), and inventory holding (physical storage and opportunity cost of inventory). The vendor bears production setup costs, costs per shipment released to the

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customer, and inventory holding costs for both work-in-process and finished goods not yet shipped to the customer. The customer buys goods from the vendor based on a purchasing contract that specifies the (minimum) annual quantity, the price per item, and shipment terms. We assume that the price per item as well as shipment terms were negotiated between the two parties based on yearly requirements, and a shipment destination was set by the customer. Our aim is not to optimize these parameters by arranging a new purchasing contract between the two parties. Rather, we will compare different business partnerships to see if any of them creates more benefits when the contract parameters are the same. The customer in IS plans the optimal quantity and timing of his orders, and sources from the vendor according to this plan. The vendor releases shipments based on the customer’s ordering decisions. Upon receipt of the goods, the customer is invoiced by the vendor and owns the product from that point on. Until such items are sold to endconsumers, inventory holding costs are accumulated at the customer. Under CI, goods are owned by the vendor until they are used by the customer, i.e. until sold or employed in the customer’s manufacturing process. The customer pays physical storage costs (e.g. rent, electricity) but does not own the inventory, and hence incurs no capital costs for holding that stock. Those carrying costs accrue to the vendor. The customer still sets the timing and quantity of orders. We will determine under what conditions the consignment of stock can create benefits for the customer, the vendor, and for both. We will also study the use of CI and VMI combined. When CI is coupled with VMI, even though it is the vendor who pays the opportunity cost of goods stored at the customer, the vendor now takes responsibility as well for setting the quantity and timing of shipments released to the customer. This transfer of authority also shifts the decision-making costs to the vendor, but the vendor may benefit from this agreement by decreasing her total inventory holding cost. Table 1 contrasts the three cases we consider. The remainder of this paper is as follows. Section 3 introduces our notation. In Section 4, we develop a model for IS and find the analytical solution for our base case. We then extend the base-case model to incorporate CI (Section 5) and C&VMI (Section 6), and compare those solutions to that of

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Table 1 Comparison of the basic characteristics of IS, CI, and C&VMI

Ordering decision made by Bearer of ordering cost Ownership of stock at customer Bearer of opportunity cost

IS

CI

C&VMI

C C C C

C C V V

V V V V

and customer. Since it is the traditional way of doing business, we will take IS as the base case to contrast with CI and C&VMI. We note that in our analysis and comparisons of different agreements, the terms ‘‘better off’’ and ‘‘worse off’’ will, respectively, mean strictly lower and strictly higher costs for the party in question.

C: the customer, V: the vendor.

4. Inventory sourcing (IS)

the base case. We provide numerical examples in Section 7, while Section 8 contains a summary and conclusions.

In IS, the customer first makes replenishment plans based on his costs Ac and hc, and the end-user demand d. The customer’s decisions concern the frequency and in what quantity to order from the vendor. His optimal order quantity is q1 ¼ EOQ ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi p 2Ac d=h andffi his optimal total cost is pcffiffiffiffiffiffiffiffiffiffiffiffiffiffi TCc1 ¼ 2Ac dhc . The customer passes the replenishment decision to the vendor, who produces at a rate pXd. The vendor, who must satisfy the customer’s orders fully, finds her economic production quantity (Q1) based on her costs of production setup (S), inventory holding (hv), and shipment release (av). To describe system inventory levels, we assume that the vendor switches from other items and begins manufacturing this one when the customer’s inventory level is q1. From that moment, the vendor produces at a rate p during an interval T ¼ kq/p, where k is the number of shipments from vendor to customer during the vendor’s manufacturing cycle. When the vendor is producing, total system inventory increases at a rate pd. After production stops, the vendor supplies goods to the customer from her stock until that is depleted. When the vendor is not producing, the system-wide inventory decreases at a rate d. We denote the time between successive production runs at the vendor by T0 (see Fig. 3). The vendor’s total production quantity in her cycle is Q1 ¼ kq1. All items carried by the vendor are charged holding costs at a rate hv. From Fig. 3, the average total system inventory is q1 þ ðp  dÞQ1 =2p, and the vendor’s mean stock level is q1 =2 þ ðp  dÞQ1 =2p. The vendor’s total cost per period is then

3. Notation Our models for IS, CI, and C&VMI employ the following basic notations. Ac

hc

S av hv p d

Customer’s fixed cost of ordering ($ per order). Ac consists of the cost of issuing an order, ao, and the cost per shipment received. (The latter does not need to be defined separately.) Annual cost to carry one unit in stock at customer’s retail store ($/unit/year). This per-item inventory holding cost is composed of ho, the opportunity cost, and hs, the physical storage cost: hc ¼ ho+hs. Vendor’s fixed cost ($ per setup) incurred at the start of each production cycle. Vendor’s cost per shipment release ($ per shipment to the customer). Annual cost to hold a unit in inventory at vendor’s production site ($/unit/year). Vendor’s annual production rate (units/ year). Annual demand rate at the customer (units/ year).

The vendor is assumed to have sufficient capacity to meet the customer’s demand (i.e. pXd). In IS, each party pays its own costs as defined above. In CI and C&VMI, portions of Ac and/or hc are paid by the vendor on behalf of the customer. In our formulations, the subscripts v and c refer to vendor and customer, respectively. Subscripts 1, 2, and 3, used both for variables and total costs, denote IS, CI, and C&VMI. In the next section, we analyze inventory sourcing, where there is no agreement between vendor

TCv1 ¼ d½S=Q1 þ av =q1  þ hv ½q1 þ ð1  d=pÞQ1=2. (1) The vendor’s total cost is composed of the total production setup and shipment release costs (the first two terms in (1)), and inventory carrying costs (the third and fourth terms). To be able to compare

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Total system inventory …. Customer’s inventory position - - - Vendor’s inventory position System-wide inventory position

p–d

H

d

q1

T

Time

T′

Fig. 3. Inventory positions of the parties (H is the maximal system inventory).

different partnerships analytically, we assume throughout that the number of shipments per cycle is a continuous variable. Therefore, the optimal value of Q1, the production quantity in one cycle, is ffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2Sd=½hv ð1  d=pÞ ¼ EPQ. As in Gu¨mu¨s- et al. (2006), the strict convexity of TCv1 means that the optimal integer value for k is     Min EPQ=q1 ; EPQ=q1 . kint ¼ |ffl{zffl} TCv1 ðk Þ

Based on the optimal values of Q1 and q1, the vendor’s minimal total cost is pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi TCv1 ¼ 2Sdhv ð1  d=pÞ þ dav = 2Ac d=hc pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ hv 2Ac d=hc =2. Letting g ¼ av/Ac and f ¼ hv/hc, and defining pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi C 0 ¼ 2Sdhv ð1  d=pÞ, the vendor’s pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi total cost is now TCv1 ¼ C 0 þ ðg þ fÞ dAc hc =2. The systemwide cost under inventory sourcing (TC c1+TC pffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi v1) is therefore TC1 ¼ C 0 þ ½1 þ ðg þ fÞ=2 2Ac dhc . The preceding model for the base case assumed no agreement between customer and vendor. Benefits of any CI or C&VMI agreement will be relative to total costs under inventory sourcing. 5. Consignment inventory (CI) In the first type of agreement, CI, the customer maintains control over the timing and quantity of orders, and pays Ac every time he places an order. However, he does not incur the opportunity-cost portion of carrying inventory, since the vendor

owns the goods at the customer’s premises until they are used. For i ¼ 1 and 2, let ei (0oeio1) denote the ratios e1 ¼ hs/hc and e2 ¼ ho/hc (where e1+e2 ¼ 1) of portions of the customer’s inventory holding cost per item (hc) under IS. The customer’s total cost in CI is the sum of ordering and physical storage costs: TCc2 ¼ Acd/q2+hsq2/2. Based p upon those costs, his ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffi optimal order size is q2 ¼ 2Ac d=hs ¼ q1 = 1 , which is strictly greater than q1 since e1o1. His optimal cost under CI is then ffiffiffiffiffiffiffiffiffiffiffiffiffiffi total p pffiffiffiffi 2Ac dhs ¼ 1 TCc1 , which is strictly less than TCc1. Therefore, the customer is always better off under CI when compared to IS. The vendor, who bears the opportunity cost of goods stored at the customer, ships less frequently under CI than IS. (Assume for now that when vendor orders on behalf of customer, there is no ‘‘efficiency factor.’’ That is, she pays the same opportunity cost ho as the customer.) Denoting the vendor’s production batch size by Q2, TCv2 ¼

Sd av d 1 þ þ hv ½q2 þ ð1  d=pÞQ2  Q2 q2 2 1 þ ho q2 . 2

Since backordering is not allowed, the vendor’s optimal production batch size is  pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi max q2 ; 2Sd=ðhv ð1  d=pÞÞ . We assume that pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffi S=ðhv ð1  d=pÞÞ4 Ac =hs , and hence her optimal pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi production quantity is Q2 ¼ 2Sd=ðhv ð1  d=pÞÞ ¼ Q1 . Then, TCv2 ¼ C 0 þ ðav d=q2 Þ þ ð1=2Þhv q2 þ ð1=2Þho q2 (where C0 is as defined in IS).

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We can p also write this cost as TCv2 ¼ pffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi C 0 þ ð1= 1 Þ Ac dhcp =2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð1 g þ fffi þ 2 Þ. Recall that TCv1 ¼ C 0 þ ðg þ fÞ Ac dhc =2, from Section 4. The vendor is better off under CI p ifffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi and only if ffi TCv14TCv2. This requires (since Ac dhc =2 and pffiffiffiffi pffiffiffiffi 1 ðg þ fÞ4ð1 g þ f þ 1  1 Þ. Then, 1= 1 40): pffiffiffiffi pffiffiffiffi pffiffiffiffi pffiffiffiffi gð 1  1 Þ4fð1  1 Þ þ ð1  1 Þð1 þ 1 Þ. Bepffiffiffiffi cause 1  1 40, pffiffiffiffi pffiffiffiffi 1 g4f þ 1 þ 1 .

(2)

Proposition 1. A necessary condition that the vendor be better off under CI is g4f+2. pffiffiffiffi Proof. We see in (2) that 1 ðg  1Þ4f þ 1. Since f, e140, (g1) must be positive for the inequality to hold. Therefore, g41 and we re-write (2) as pffiffiffiffi 1 4ðf þ 1Þ=ðg  1Þ. The result follows because e1 pffiffiffiffi (and thus 1 ) are less than one. & Proposition 1 states that the vendor will be better off under a CI agreement if her cost per shipment released exceeds ð1 þ ðhv =hc ÞÞAc . Observe that (2) is more likely to hold when the customer has the higher inventory carrying cost. Consider, for example, an inventory sourcing agreement where the vendor delivers goods to the customer’s premises and pays transportation costs. Some of the vendor’s shipment costs are usually passed on to the customer through an increased price per item. Hence, his inventory holding cost can be higher than the vendor’s. A consignment agreement in such a setting is more likely to create benefits for both parties. What happens if condition (2) does not hold? There are two possible cases: (i) CI achieves system-wide cost savings where the customer is no worse off but the vendor is worse off. In practice, the vendor has recourse: If the vendor has sufficient bargaining power, a better price may be negotiable. Alternatively, without this power, she may simply accept the terms to maintain business with her customer. (ii) The system-wide cost is greater in CI than in IS. Then, it is in neither party’s interest to change the traditional way of doing business. To explore these two possible situations, we formulate the total cost under a CI agreement and compare it with inventory sourcing. The system-

wide cost under CI is 1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi TC2 ¼ C 0 þ pffiffiffiffi 2Ac dhc ð1 g þ f þ 2 Þ 2 1 pffiffiffiffipffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ 1 2Ac dhc . Recall from ISffi that TC1 ¼ C 0 þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffi ½1 þ ð1=2Þðg þ fÞ 2Ac dhc . Therefore, CI leads to system-wide cost savings if TC14TC2, which requires



 pffiffiffiffi 1 1 1 þ ðg þ fÞ 4 pffiffiffiffi ð1 g þ f þ 1  1 Þ þ 1 . 2 2 1 (3) Proposition 2. A necessary condition for system-wide cost savings under CI is fog. pffiffiffiffi Proof. Since 1 40, these inequalities must be satisfied for (3) to hold: pffiffiffiffi 1 ðg þ f þ 2Þ4ð1 g þ f þ 1  1 Þ þ 21 pffiffiffiffi pffiffiffiffi ) 1 ðg þ 1Þ þ 1 ðf þ 1Þ41 ðg þ 1Þ þ ðf þ 1Þ pffiffiffiffi pffiffiffiffi pffiffiffiffi ) 1 ðg þ 1Þð1  1 Þ4ðf þ 1Þð1  1 Þ. pffiffiffiffi Noting that 1 o1, the result follows. & Proposition 2 implies that if the vendor is relatively more efficient in inventory holding costs than shipment release costs, she more likely achieves cost savings under CI. Intuitively, the customer’s replenishment quantities increase under CI versus IS. That increase can benefit the vendor, who prefers fewer shipments if her cost per shipment release is high. This concludes the analytical results for a basic CI agreement, where it is assumed that the vendor pays exactly the same opportunity cost per item, ho, that the customer pays in IS, and that the wholesale price of an item does not change from IS to CI. Table 2 summarizes those findings. In the next two subsections, we will analyze the impacts of the vendor’s efficiency on the opportunity cost of an item, and of cost sharing through a wholesale price adjustment. 5.1. Impacts of the vendor’s efficiency factor Various considerations might create a situation where the vendor and customer may not have the same capital costs of the holding inventory. An organization’s capabilities in financing, and the firm’s relative power in industry, can make tremendous changes in capital costs. Thus, in a CI agreement, the vendor pays b2ho per unit held at

ARTICLE IN PRESS M. Gu¨mu¨-s et al. / Int. J. Production Economics 113 (2008) 502–517 Table 2 Summary of conditions when CI is beneficial for the customer, pffiffiffiffi the vendor, and the whole system; m ¼ ðf þ 1Þ= 1 Necessary and sufficient condition

Benefits under CI compared to IS Customer

Vendor

Supply chain

m1ogom+1 g ¼ m+1

Better off Better off

Better off Better off

g4m+1 g ¼ m1

Better off Better off

Worse off No worse off Better off Worse off

gom1

Better off

Worse off

Better off No worse off Worse off

the customer’s premises. The vendor’s capital cost efficiency compared to that of the customer is b240. Whatever the value of this parameter, the customer’s order quantity and total cost are still pffiffiffiffi pffiffiffiffi q1 = 1 and 1 TCc1 , respectively. However, the total cost of the vendor is now TCv2 ¼ C 0 þ ðav d=q2 Þ þ ð1=2Þhv q2 þ ð1=2Þb2 ho q2 , p which can be pffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi written as TCv2 ¼ C 0 þ ð1= 1 Þ Ac dhc =2ð1 gþ f þ b2 2 Þ. Comparison of TCv2 and TCv1 shows that the vendor is better off if pffiffiffiffi 1 ðg þ fÞ41 g þ f þ b2 ð1  1 Þ pffiffiffiffi pffiffiffiffi pffiffiffiffi ) 1 gð1  1 Þ4fð1  1 Þ pffiffiffiffi pffiffiffiffi þ b2 ð1  1 Þð1 þ 1 Þ. pffiffiffiffi pffiffiffiffi pffiffiffiffi Since 1  1 40, 1 g4f þ b2 ð1 þ 1 Þ. A necessary condition for this inequality to hold is pffiffiffiffi g4b2. We then have 1 4ðf þ b2 Þ=ðy  b2 Þ. Similar to the proof of Proposition 1, we now see that a necessary condition for the vendor to be better off is g4f þ 2b2 . The system-wide costs become rffiffiffiffiffiffiffiffiffiffiffiffi

pffiffiffiffi Ac dhc 1 0 TC2 ¼ C þ pffiffiffiffi ð1 g þ f þ b2 2 Þ þ 2 1 . 1 2 As compared to TC1, we find that system-wide cost savings are achieved when pffiffiffiffi 1 4ðf þ b2 Þ=ðg þ 2  b2 Þ. Again as in the proof of Proposition 2, it is necessary that g þ 24f þ 2b2 . The above analysis shows that system-wide costs, as well as vendor’s costs, improve as b2, the vendor’s cost factor, becomes smaller. A CI agreement is more promising for both parties when the vendor can develop efficiencies in the opportunity cost of capital.

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Even if the vendor is unable to develop these efficiencies, CI can create a situation where there is potential to lower system-wide costs. We call this a ‘‘potentially efficient system,’’ and now examine it in detail. 5.2. Cost sharing Let us define an ‘‘efficient system’’ as one with total-cost improvements with respect to a base case, and where neither party is worse off. We demonstrated that the customer always gains under CI compared to IS. Accordingly, a potentially efficient system in CI is one where, although the vendor is worse off, there are overall system-wide cost savings. A potentially efficient system can be turned into an efficient one by some sort of incentive, offered by the customer to transfer a portion of his benefits to the vendor. When CI is applied, standard industry practice allows the vendor to increase the unit price to share total savings. Without getting into details on cost-sharing research, we briefly explain how this could work. Let c be the original price per item paid by the customer. The vendor suggests a price increment over c in order to make CI beneficial to herself. In the absence of information sharing between parties, the customer may be unsure that a price increase is in his best interest (e.g. when he receives an equal, or even smaller, share of system-wide savings due to CI). Let yhigh be the maximum percentage increase in price acceptable to the customer: pffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi yhigh cd ¼ ð1  1 Þ 2Ac dhc ; which means 100 rffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffi 2Ac hc 100 high ð1  1 Þ ¼ y . c d We now determine the smallest price increment acceptable to the vendor, the value that makes her no worse off than under IS. We assume that inequality (2) does not hold; this is why the vendor is motivated to ask for a price change. Taking b2 ¼ 1 results in rffiffiffiffiffiffiffiffiffiffiffiffi

ylow cd Ac dhc 1 g þ f þ 2 ¼  ðg þ fÞ , pffiffiffiffi 100 1 2 and hence y

low

100 ¼ c

rffiffiffiffiffiffiffiffiffiffi

A c hc  1 g þ f þ  2  ðg þ fÞ . pffiffiffiffi 1 2d

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Note that the maximum price increase the customer will accept, yhigh, is the upper bound on the price increase that would erase his gains under CI. On the other hand, ylow is the lower bound that would compensate the vendor for her increase in costs, but still leave the customer with some benefit. Therefore, when CI creates a potentially efficient system, a wholesale price increment 2 ðylow ; yhigh  will make the vendor willing to accept that agreement rather than IS. The customer favors CI as long as price increments are in the range [ylow, yhigh). Another means of creating possible cost savings for both vendor and customer may be the use of CI and VMI combined. While CI always benefits the customer, VMI has the potential to generate benefits for the vendor. C&VMI is the subject of the next section. 6. Consignment and vendor-managed inventory (C&VMI) In a C&VMI agreement, the vendor owns the goods at the customer’s location until they are sold, but also initiates orders on behalf of the customer. The vendor pays ho per item stored at the customer and ao for every order she places for him. The customer is then exempt from those expenses. In light of these changes, the vendor’s total cost in C&VMI is TCv3 ¼

Sd av d 1 þ þ hv ½q3 þ ð1  d=pÞQ3  Q3 q3 2 ao d 1 þ ho q3 . þ q3 2

(4)

This total cost is also equal to C 0 þ ðav þ ao Þ qd þ 3 IS. Letting previously, we re-write the vendor’s total cost

1 0 2 ðhv þ ho Þq3 where C is as explained in d1 ¼ ao/Ac, and with the ratios defined

TCv3

Ac d 1 ¼ C þ ðg þ d1 Þ þ ðf þ 2 Þhc q3 . q3 2 0

Therefore, the optimal order quantity determined by the vendor on behalf of the customer is sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sffiffiffiffiffiffiffiffiffiffiffiffiffi 2ðg þ d1 ÞAc d g þ d1 ¼ q. q3 ¼ ðf þ 2 Þhc f þ 2 1 Incorporating the p optimal order quantity ffiffiffiffiffiffiffiffiffiffiffiffiffip ffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiin ffi (4), we get TCv3 ¼ C 0 þ g þ d1 f þ 2 2A dhc . Recffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi p call for IS that TCv1 ¼ C 0 þ ðg þ fÞ dAc hc =2. Therefore, the vendor’s cost under C&VMI is

less than her cost of IS if TCv3oTCv1, which reduces to pffiffiffiffiffiffiffiffiffiffiffiffiffipffiffiffiffiffiffiffiffiffiffiffiffiffi g þ f . g þ d1 f þ  2 o 2

(5)

After some algebra, (5) can be written in the form 4½ðg þ d1 Þ2 þ fd1 oðg  fÞ2 .

(6)

The right-hand side of (6) is zero when g ¼ f. Therefore, C&VMI would create benefits for the vendor if she has, compared to the customer, efficiency or inefficiency in either her ordering or inventory holding, but not in both costs. That is, the vendor can make better use of the ordering authority created by C&VMI when she has an advantage or a disadvantage in either her ordering or inventory holding costs. Suppose the vendor’s ordering cost is greater than the customer’s but their inventory holding cost per item is around the same. The vendor can ship larger quantities to decrease her total ordering cost. This holds true if her carrying cost is lower, but she has no clear efficiency in ordering costs relative to the customer. On the other hand, if the vendor’s inventory holding cost is too high, she can replenish the customer frequently in small quantities to achieve savings. In the meantime, the vendor’s costs associated with C&VMI influence the benefits that the agreement can create for her. The vendor’s costs under C&VMI increase linearly in the ratios d1 and e2. Hence, as those parameters get lower, it is more likely that the vendor achieves cost savings, since there is a decrease in the left-hand side of (6). Now, the customer would accept C&VMI if his costs under this agreement were not higher than his costs in IS. The customer’s total cost under C&VMI is ðAc  ao Þd 1 þ hs q3 q3 2 ð1  d1 ÞAc d 1 ¼ þ  1 hc q3 . q3 2

TCc3 ¼

The optimal ordering quantity q3 was determined by the vendor on behalf of the customer. Inserting that optimal quantity in the customer’s cost function yields ! rffiffiffiffiffiffiffiffiffiffiffiffi Ac dhc ð1  d1 Þðf þ 2 Þ þ 1 ðg þ d1 Þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi TCc3 ¼ . 2 ðg þ d1 Þðf þ 2 Þ

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The customer’s total cost in IS is TCc1 ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi 2Ac dhc . C&VMI thus benefits him if pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðg þ d1 Þðf þ 2 Þ. (7) pffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffi Let g þ d1 ¼ m1 and f þ 2 ¼ m2 . Then (7) reduces to ð1  d1 Þ=ðm1 =m2 Þ þ 1 m1 =m2 o2. Note that (1d1) and e1 are both o1. If the vendor’s replenishment quantity q3 exceeds the customer’s quantity q1 under IS (m1/m241), it is more likely for the customer to achieve savings under C&VMI when e1 is low. That is, the customer would not mind large order quantities as long as his physical storage cost per item is low. Similarly, the customer can still benefit when the vendor replenishes him very frequently (m1/m2o1), if his cost per shipment received is not high. In general, the customer is more likely to achieve cost savings under C&VMI because he does not pay the opportunity cost of items in stock nor the cost of placing orders. We can now check whether both parties can be better off under C&VMI. ð1  d1 Þðf þ 2 Þ þ 1 ðg þ d1 Þo2

Proposition 3. For C&VMI to create an efficient system, it is necessary that 2d1 2 42 ð1  gÞþ d1 ð1  fÞ. Proof. Inequalities (5) and (7) together imply that ð1  d1 Þðf þ 2 Þ þ 1 ðg þ d1 Þog þ f, which is a necessary (but not sufficient) condition for both parties to be better off compared to IS. With some algebra, this condition reduces to d1 f þ 2d1 2 þ 2 g42 þ d1 , and then to 2d1 2 4 2 ð1  gÞ þ d1 ð1  fÞ. & We see in the proof of Proposition 3 that this necessary condition (required to achieve an efficient system) holds when g41 and f41, and also when gb1 or fb1. The latter is more likely the case where both parties are better off. This can be explained by our analytical results on C&VMI for the vendor and customer. We observed previously that the vendor can make use of the C&VMI agreement to offset inefficiency in one of her costs. Depending on which cost parameter is high, the vendor can decrease or increase the order quantity to achieve cost savings. That order quantity is also acceptable to the customer, as long as the costs from which he is exempt (cost of placing orders and opportunity cost of inventory) compensate his increased costs result-

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ing from ordering decisions made by the vendor for him. It may be less likely to achieve an efficient system than a potentially efficient system that can be worked out to satisfy both parties. Recall that a system is potentially efficient if there are systemwide cost savings. Proposition 4. C&VMI creates system-wide cost savings relative to IS if 1 þ f m2 o o1; 1 þ g m1

or

1 þ g m1 o o1. 1 þ f m2

The proof of Proposition 4 is provided in Appendix A. Through numerical examples in the next section, we will see when C&VMI can create a potentially efficient system, and highlight as well the analytical results found in the inventory sourcing and CI models. We note in passing that the cost sharing argument discussed for CI in Section 5.2 can also be applied to C&VMI. 7. Numerical examples IS, CI, and C&VMI will now be contrasted numerically when certain parameters are varied. In all those examples, Ac ¼ $100 per order, hc ¼ $1.5 per item stored, d ¼ 1300 items/year, and p ¼ 1600 items/ year. We do not assume any efficiency of the vendor over ho or ao in case of a CI or C&VMI agreement. Figs. 4–6 test the impact of g on different sourcing options. Those examples fix the values e1 ¼ 0.4, d1 ¼ 0.1, and f ¼ 0.8, while g is between (0, 5]. In the next three figures, we vary f over the interval (0, 5], but set g ¼ 1.5, e1 ¼ 0.4, d1 ¼ 0.1. For Figs. 10–12, we change the value of e1 over (0, 1) while g ¼ 1.5, f ¼ 0.8, and d1 ¼ 0.1. The final figure varies d1 between (0, 1) with g ¼ 1.5, f ¼ 0.8, and e1 ¼ 0.4. In line with our analytical results, we observe in Fig. 4 that the customer’s cost saving under CI is fixed, yet the system-wide and the vendor’s savings increase linearly as g increases. CI is beneficial for the vendor when her cost per shipment is at least 3.8 times the customer’s cost per order. System-wide savings are achieved for lower g. For C&VMI, we see in Fig. 5 that cost savings are possible for the vendor when g is very low or very high. When gp0.02, the vendor replenishes the customer frequently to save on inventory holding costs, but such a large number of shipments increases the customer’s and the system-wide total

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CI vs IS

500 400 Cost Savings ($)

300 200 100 0 -100

0.01 0.2

0.5

0.8

1.1

1.4

1.7

2

2.3

2.6

2.9

3.2

3.5

3.8

4.1

4.4

4.7

5

4.7

5

4.7

5

-200 -300 -400

TCv1-TCv2

-500

TCc1-TCc2

TC1-TC2

Gamma ( γ ) Fig. 4. CI versus IS; e1 ¼ 0.4, d1 ¼ 0.1, and f ¼ 0.8; g is between (0, 5].

C&VMI vs IS

600

Cost Savings ($)

400 200 0 -200

0.01 0.2

0.5

0.8

1.1

1.4

1.7

2

2.3

2.6

2.9

3.2

3.5

3.8

4.1

4.4

-400 TCv1-TCv3

-600

TCc1-TCc3

TC1-TC3

Gamma (γ ) Fig. 5. C&VMI versus IS; e1 ¼ 0.4, d1 ¼ 0.1, and f ¼ 0.8; g is between (0, 5].

C&VMI vs CI

1400 1200

TCv2-TCv3

TCc2-TCc3

TC2-TC3

Cost Savings ($)

1000 800 600 400 200 0

0.01 0.2

-200

0.5

0.8

1.1

1.4

1.7

2

2.3

2.6

2.9

3.2

3.5

3.8

4.1

4.4

-400 -600

Gamma (γ ) Fig. 6. C&VMI versus CI; e1 ¼ 0.4, d1 ¼ 0.1, and f ¼ 0.8; g is between (0, 5].

cost compared to IS. The higher values of g, however, enable system-wide cost savings; both parties are better off under C&VMI when gX4.0.

When we compare C&VMI to CI for varying values of g (Fig. 6), we observe that C&VMI almost always generates more system-wide savings,

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CI vs IS

400

Cost Savings ($)

200 0

0.01 0.2

-200

0.5

0.8

1.1

1.4

1.7

2

2.3

2.6

2.9

3.2

3.5

3.8

4.1

4.4

4.7

5

4.4

4.7

5

-400 -600 -800 TCv1-TCv2

-1000

TCc1-TCc2

TC1-TC2

-1200

Phi (ϕ) Fig. 7. CI versus IS; e1 ¼ 0.4, d1 ¼ 0.1, g ¼ 1.5; f is between (0, 5].

C&VMI vs IS

300

Cost Savings ($)

200 100 0 0.01 0.2

0.5

0.8

1.1

1.4

1.7

2

2.3

2.6

2.9

3.2

3.5

3.8

4.1

-100 -200

TCv1-TCv3

TCc1-TCc3

TC1-TC3

-300 Phi (ϕ) Fig. 8. C&VMI versus IS; e1 ¼ 0.4, d1 ¼ 0.1, g ¼ 1.5; f is between (0, 5].

although one party is sometimes worse off. (Compared to CI, the vendor is worse off when 2.3ogo4.7; for go1.6, the customer is worse off.) If an equal split of the benefits can be negotiated, C&VMI is a better option for both actors. We also see in Fig. 6 that as g increases, the vendor and customer each become indifferent between C&VMI and CI. This is logical: Compared to IS, the customer under CI orders larger quantities, and this is what the vendor would do under C&VMI if her shipment costs were high. In Proposition 1, we stated that g4f+2 is a necessary condition for the vendor to be better off under CI. Therefore, the vendor never achieves cost savings in Fig. 7, where g ¼ 1.5 and f varies between (0, 5]; the vendor’s total cost is increasing in f. Under a C&VMI agreement, however, it is possible for all parties to achieve cost savings for high enough f (fX3.7, Fig. 8). As discussed in our formulations, C&VMI becomes an opportunity for

a vendor, one relatively inefficient in inventory holding cost (fbg), to decrease her carrying costs via frequent shipments. When we compare C&VMI to CI for varying f values (Fig. 9), we see that C&VMI becomes a better option for the vendor and the whole system for larger f. This makes sense: The customer under CI increases the order quantity, causing average system stocks to grow. The vendor, on the other hand, prefers more frequent shipments and less inventory when f is high, and she can decide so under C&VMI. In line with analytical results, we see in Fig. 10 that varying e1 changes all cost savings nonlinearly. As 1 approaches one, the system returns to the costs under IS. While the customer’s savings decrease, the vendor’s as well as system-wide savings increase as e1-1. We also observe in this example that no e1 value creates an efficient system; the customer is always better off. The system is

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C&VMI vs CI

1400 1200

TCv2-TCv3

TCc2-TCc3

TC2-TC3

Cost Savings ($)

1000 800 600 400 200 0

0.01 0.2

-200

0.5

0.8

1.1

1.4

1.7

2

2.3

2.6

2.9

3.2

3.5

3.8

4.1

4.4

4.7

5

-400 -600

Phi (ϕ) Fig. 9. C&VMI versus CI; e1 ¼ 0.4, d1 ¼ 0.1, g ¼ 1.5; f is between (0, 5].

CI vs IS

1000 0 Cost Savings ($)

0.01 0.04 0.1 0.16 0.22 0.28 0.34 0.4 0.46 0.52 0.58 0.64 0.7 0.76 0.82 0.88 0.94 0.99

-1000 -2000 -3000 -4000 -5000

TCv1-TCv2

-6000

TCc1-TCc2

TC1-TC2

ε1

Fig. 10. CI versus IS; d1 ¼ 0.1, f ¼ 0.8, g ¼ 1.5; e1 is between (0, 1).

potentially efficient when e1X0.52, but for e1 near zero, the system-wide and vendor’s costs increase enormously. Fig. 11 compares C&VMI to IS. Vendor’s costs and the customer’s savings are each decreasing in e1. An efficient system is never attained. The vendor achieves cost savings only when e1X0.98. (Contrast this to CI in Fig. 10, where the vendor never realizes cost savings.) System-wide costs in Fig. 11 do not change much as e1 varies. We see in Fig. 12 that low values of e1 make a big difference for the vendor’s and system-wide costs under C&VMI compared to CI. However, CI becomes a preferred option for the whole system when e1X0.69, and for the vendor when 0.95Xe1X0.51. The customer favors CI when e1p0.41.

We present only in a single graph the implication of varying d1, since it does not influence costs for IS or CI. Comparing C&VMI to IS in Fig. 13, we see that the customer’s savings and vendor’s costs under C&VMI are increasing in d1. System-wide savings, on the other hand, do not change much, remaining near zero. 8. Summary and conclusions In this paper, we studied a case where a customer and vendor initially consider consignment inventory (CI) for a single item. Comparing it to our base case, which is inventory sourcing, we obtained analytical conditions under which CI creates benefits for one or more parties. In contrast to the general belief that CI is beneficial only for the

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C&VMI vs IS

400

Cost Savings ($)

300 200 100 0

0.01 0.04 0.1 0.16 0.22 0.28 0.34 0.4 0.46 0.52 0.58 0.64 0.7 0.76 0.82 0.88 0.94 0.99

-100 -200 -300

TCv1-TCv3

-400

TCc1-TCc3

TC1-TC3

ε1

Fig. 11. C&VMI versus IS; d1 ¼ 0.1, f ¼ 0.8, g ¼ 1.5; e1 is between (0, 1). C&VMI vs CI

5000

TCv2-TCv3

TCc2-TCc3

TC2-TC3

Cost Savings ($)

4000 3000 2000 1000 0

0.01 0.04 0.1 0.16 0.22 0.28 0.34 0.4 0.46 0.52 0.58 0.64 0.7 0.76 0.82 0.88 0.94 0.99

-1000

ε1

Fig. 12. C&VMI versus CI; d1 ¼ 0.1, f ¼ 0.8, g ¼ 1.5; e1 is between (0, 1). C&VMI vs IS

600

Cost Savings ($)

400

TCv1-TCv3

TCc1-TCc3

TC1-TC3

200 0 0.01 0.04 0.1 0.16 0.22 0.28 0.34 0.4 0.46 0.52 0.58 0.64 0.7 0.76 0.82 0.88 0.94 0.99

-200 -400 -600

δ1

Fig. 13. C&VMI versus IS; e1 ¼ 0.4, f ¼ 0.8, g ¼ 1.5; d1 is between (0, 1).

customer, we showed that it may be favorable for the vendor as well. Depending on the costs of shipment, and who pays for transportation, CI can be beneficial for both parties.

We showed that if the CI agreement results in a potentially efficient system, it can be turned into an efficient one. To achieve that, we found the minimum and maximum amounts by which

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the wholesale price may increase, such that the customer may accept to share his benefits with the vendor. When the system is inefficient under CI, the vendor can offer a C&VMI agreement to realize savings for herself and for the system. We considered the option of CI plus VMI (C&VMI) and extended our analysis to find the optimal costs and potential savings under that agreement. We showed how the vendor can make use of C&VMI to improve her costs in areas where she is inefficient. Although the vendor prefers C&VMI rather than CI, and the customer vice versa, we observed that C&VMI is more likely to generate system-wide cost savings. This paper provided closed-form results for different sourcing options. Outcomes of those choices were shown to depend on cost parameters of the parties involved. We identified conditions under which one option is preferred to another. Our findings can help a vendor or customer decide a priori if CI or C&VMI works for them. Future research may evolve in alternate directions. One can study economies of scale created by a C&VMI agreement when there are multiple customers. Some of them will be offered C&VMI by the vendor, whose goals are to achieve flexibility in production and to reduce operational costs such as shipment expenses. Customers under that agreement should be no worse off than for IS. Secondly, a model can be developed for a customer to choose between vendors for CI when there are several suppliers of certain items. CI is always beneficial for the customer without any change in wholesale price. But various suppliers could enforce a price adjustment when CI is offered. In that case, the customer should carefully select the CI-vendor to maximize his savings. In light of the multiple customers or vendors, computer simulation may be required for either of the above extensions when end-consumer demand is uncertain. In settings where demand is more stable, however, it may be possible to find analytical solutions. Appendix A Proof of Proposition 4: We have 1 pffiffiffiffiffiffiffiffiffiffiffiffi TC1 ¼ TCc1 þ TCv1 ¼ C 0 þ pffiffiffi Ac dhc ð2 þ g þ fÞ 2

and 1 pffiffiffiffiffiffiffiffiffiffiffiffi TC3 ¼ TCc3 þ TCv3 ¼ C 0 þ pffiffiffi Ac dhc 2

m2 m1 þ ð1  2 Þ þ 2m1 m2 .  ð1  d1 Þ m1 m2 Then, TC3oTC1 if ð1  d1 Þm22 þ ð1  2 Þm21 þ 2m21 m22 om1 m2  ð2 þ g þ fÞ. After some algebra, this reduces to ð1 þ gÞm22 þ ð1 þ fÞm21 om1 m2 ½ð1 þ fÞ þ ð1 þ gÞ, and then to ð1 þ fÞm1 ðm1  m2 Þo ð1 þ gÞm2 ðm1  m2 Þ. Thus, if ðm1  m2 Þ40, we have ð1 þ fÞ=ð1 þ gÞo ðm2 =m1 Þo1, but if ðm1  m2 Þo0, then ð1 þ gÞ= ð1 þ fÞoðm1 =m2 Þo1. &

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