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

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

Mehmet Gumus-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 independent-

sourcing 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 theproduct, and delivering to customers are typicaloperational processes for a company within a supplychain. Here we consider a customer who purchasesgoods from a vendor. The customer’s processescomprise the planning of his requirements, sourcing

e front matter r 2008 Elsevier B.V. All rights reserved

e.2007.10.019

artially supported by NSERC and by SSHRC.

re grateful to Prakash Abad for extensive

e calculations and results.

ng author. Tel.: +1 519 888 4013;

7383.

ss: [email protected]

er).

ress: School of Business and Management,

ersity of Sharjah, United Arab Emirates.

goods from the vendor, and releasing those goods toend-consumers. The vendor, similarly, plans herrequirements and sources materials or parts forproduction, manufactures goods, and releases thosegoods to the customer.

When these two firms are independent and linkedin a supply chain as in Fig. 1, decisions onoperational processes are, in general, made indivi-dually. In the usual sequence of events, the customerfirst develops his requirements plan and sourcingmethod based on his own costs. The vendor thenreacts to fulfill the customer’s requirements. Hence,replenishment decisions made by the customer donot necessarily consider the choices of his upstreambusiness partner.

A common focus of research and supply-chainpractice is to seek mechanisms to align the decisions

.

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Make Deliver Plan Source

Vendor’s Facility Customer’s Facility Goods

Make Plan Deliver Source

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.

Material Warehousing

Selling Material Inspection

Goods Receipt

Purchase Order/Release

Order

Transfer of Ownership in Inventory Sourcing

Transfer of Ownership in Consignment Inventory

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

M. Gumus- et al. / Int. J. Production Economics 113 (2008) 502–517 503

of chain members by means of contracts oragreements. Those arrangements aim to increasethe overall supply-chain performance. Vendor-managed inventory (VMI), one such agreement,was analyzed by Gumus- et al. (2006) to obtainconditions under which it may lower the costs ofeach party and of the chain.

There are, however, other practices that seem tounbalance the total costs of supply-chain members.In this paper, we will analyze in detail one of thosepractices, consignment inventory (CI). Our aim is,similarly, to determine conditions whereby consign-ment stocks create benefits for the customer, thevendor, or for both parties.

In CI, goods are owned by the vendor until theyare used by the customer. Those goods are stored atthe customer’s premises. Although the customermay have authority over the timing and quantity oforders, he pays for the goods only upon use or justafterward. Hence, the customer does not tie up hiscapital in inventory.

In the traditional way of doing business, which wewill call ‘‘inventory sourcing’’ (IS) throughout, thecustomer orders from the vendor based on his totalinventory holding costs (both costs of opportunityand physical storage, where opportunity cost refersto the cost of capital), and costs of ordering. IS isgenerally characterized by a purchasing contractincluding shipment terms, annual demand specifiedby the customer, and the price per unit purchased byhim. Under this practice, which will be our base case

for analysis, the customer is invoiced by the vendoronce the goods arrive at his premises (see Fig. 2). Heowns the product at that point.

In CI, ownership of goods is transferred to thecustomer only after they leave his in-house ware-house for sale or manufacturing. If other terms ofthe purchasing contract stay the same as in IS, onemajor benefit to the customer is deferral of paymentuntil use. When end-consumer demand is unknown,CI also allows the customer to hedge againstuncertainties in production and sales. This willinfluence his total inventory carrying cost. With thecustomer’s inventory now off his balance sheet,conventional wisdom holds that the customer gainsmost from CI.

The benefits of CI are less clear for the vendor.One situation that favors CI is where the vendoroffers new products that the customer hesitates tobuy, or expensive items difficult for the customer toown. In that case, the vendor can use CI as astrategic means to create new sales channels(Piasecki, 2004). This motivation, however, doesnot explain why a vendor would accept a CIcontract when demand is stable and the materialpurchased is not new. An example of such is seen inthe Automation and Drives division of Siemens,where standard parts such as metal springs and nutscan be consigned from suppliers even though thedemand during a year can be quite stable. Otherscenarios when a vendor might accept a CI contractinclude a power differential between a strong

ARTICLE IN PRESSM. Gumus- et al. / Int. J. Production Economics 113 (2008) 502–517504

customer and a ‘‘weaker’’ vendor who needs toaccommodate the customer’s wishes, or when thevendor at least has sufficient power to negotiatemore favorable terms in the CI agreement.

There appears to be very little previous work thatexamines analytically the impact of CI. The focus ofthe present paper is to establish closed-form results thatspecify general conditions under which CI is beneficialto 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 besynonymous with VMI or with CI plus VMI(‘‘C&VMI’’). In VMI, replenishment decisions aremade by the vendor on behalf of the customer. InCI, even though the vendor is informed about theconsumption of goods at the customer’s premises, itis still the customer who finalizes the timing andquantity of orders. We will consider both types ofagreements in this paper.

The framework we use is similar to those in jointeconomic lot sizing (JELS) decisions. The JELSliterature generally assumes a central decisionmaker that can optimize the sum of total costs ofthe vendor plus the customer. The context is verysimilar in each paper, and the contributions areincremental.

Banarjee (1986), the first to analyze the integratedvendor–buyer case, examines a lot-for-lot model inwhich the vendor V manufactures each shipment asa separate batch. Goyal (1988) extends this work inthat he formulates a joint total-relevant-cost modelfor a single vendor and customer production-inventory system, where the vendor’s lot size is aninteger multiple of the customer’s order size. Lu(1995) extended Goyal’s (1988) work by allowing V

to supply some quantity to the purchaser beforecompleting the entire lot. Goyal (1995) employedthe example provided by Lu for the single vendorand buyer but showed that a different shipmentpolicy could result in a better solution.

Hill (1997) considers a single vendor whomanufactures a product at a finite rate and inbatches, and supplies a sole buyer whose externaldemand is level and fixed. The vendor incurs a batchsetup cost and a fixed delivery cost associated witheach shipment. Hill’s policy assumes that successiveshipment sizes increase by a factor whose value liesbetween one and the ratio of manufacturing rate tothe product’s demand rate. He concludes thatalthough Goyal’s (1995) policy may perform muchbetter than Lu’s equal-size-shipment policy, his ownpolicy outperforms all.

Similar to the JELS literature, we use a base case(IS) for comparison purposes, contrasting that toother models which assume that the parties in thesupply chain still make decisions independently(whether coordinated or not).

JELS studies do not discuss how the savingscreated by central decision making should bedivided between parties involved. Benefits achievedare difficult to generalize because, for example, thecustomer’s ordering cost is not explicit in thosemodels. The CI or C&VMI sourcing models that weconsider require a shift of certain costs from oneactor to another to reflect changes in decision-making responsibility or ownership of inventory.We provide a breakdown of cost parameters so as toidentify the impact of such changes on eachmember.

Sucky (2005) extends JELS to a bargaining model,where the vendor offers a side payment to thecustomer whose costs under JELS go up comparedto individual decision making. It is assumed that thevendor, who achieves cost savings under JELS,makes a ‘‘take-it-or-leave-it’’ offer of joint policywith a side payment. The customer may accept thevendor’s offer, or if he is not satisfied with it, canenforce his economic order quantity (EOQ). Thebargaining then ends. Sucky assumes that thevendor has full information regarding the custo-mer’s costs.

A number of papers have also been written oncombined use of CI and VMI. This literaturediscusses various C&VMI systems that differ inthe costs considered, the demand structure, and thenature and number of supply-chain members.Boyaci and Gallego (2002) study a system of asingle wholesaler and retailer under deterministicbut price-sensitive demand. They analyze theimpacts of coordinating pricing and replenishmentwhen decisions are made jointly. They use whole-saler-owned inventory with delayed payment versusCI to extend the models of Crowther (1964) andMonahan (1984). They conclude that pricing andinventory decisions are best made with a coordi-nated-channel’s profit function.

In our paper, we analyze the impacts of CI fromthe operational point of view. That is, under CI,there is no change in pricing terms from those in thepurchasing contract under inventory sourcing. Thisenables us to focus on operational benefits to bothparties. If one party is not satisfied with theoutcome, a price change may then become anoption, as it would be in industry.

ARTICLE IN PRESSM. Gumus- et al. / Int. J. Production Economics 113 (2008) 502–517 505

Valentini and Zavanella (2003) discuss the use ofconsignment stock by a manufacturer who providesautomotive parts and manages the inventory of hercustomer. While the authors’ main aim was toqualitatively analyze the advantages and disadvan-tages of this sourcing practice, they compare itnumerically with Hill’s (1997) solution, using thesame deterministic model. Based on numericalexamples only, they conclude that consignmentstock outperforms the usual inventory models.Building on the analysis of Valentini and Zavanella,Persona et al. (2005) assume the same characteristicsof the agreement and analyze the consequences ofproduct obsolescence, concluding that obsolescencedecreases the optimal level of consignment stock.

Other articles examine C&VMI in various con-texts. For example, Dong and Xu (2002) explore theeconomics of C&VMI in the short and long terms.Gerchak and Khmelnitsky (2003) provide an inter-esting example of C&VMI when reported demandcannot be verified. They consider a retailer sellingnewspapers, and his vendor (a publisher), underVMI and revenue sharing. They analyze the impactson coordination of the retailer’s sales reports (to thepublisher).

Although we take both CI and VMI into accountin this paper, the problem setting, the approach weuse, and our goal are quite distinct from those ofDong and Xu (2002) or of Gerchak and Khmel-nitsky (2003). We consider a well-known problembut analyze it under different partnerships, account-ing for changes in certain cost parameters. Weprovide closed-form solutions to see under whatconditions a partnership is more favorable thanothers.

2. Problem definition

Suppose a customer purchases an establishedproduct (or standard product mix) from a vendor.As in the usual EOQ setting, yearly demand isdeterministic with no backordering. Demand isrealized at the customer, at a constant rate per unittime. The vendor and customer are independentfirms, each with the goal of minimizing their owntotal cost.

Under IS, the customer orders from the vendorbased on his total cost of planning (fixed cost perorder), sourcing (fixed cost per shipment received),and inventory holding (physical storage and oppor-tunity cost of inventory). The vendor bears produc-tion setup costs, costs per shipment released to the

customer, and inventory holding costs for bothwork-in-process and finished goods not yet shippedto the customer.

The customer buys goods from the vendor basedon a purchasing contract that specifies the (mini-mum) annual quantity, the price per item, andshipment terms. We assume that the price per itemas well as shipment terms were negotiated betweenthe two parties based on yearly requirements, and ashipment destination was set by the customer. Ouraim is not to optimize these parameters by arran-ging a new purchasing contract between the twoparties. Rather, we will compare different businesspartnerships to see if any of them creates morebenefits when the contract parameters are the same.

The customer in IS plans the optimal quantityand timing of his orders, and sources from thevendor according to this plan. The vendor releasesshipments based on the customer’s ordering deci-sions. Upon receipt of the goods, the customer isinvoiced by the vendor and owns the product fromthat point on. Until such items are sold to end-consumers, inventory holding costs are accumulatedat the customer.

Under CI, goods are owned by the vendor untilthey are used by the customer, i.e. until sold oremployed in the customer’s manufacturing process.The customer pays physical storage costs (e.g. rent,electricity) but does not own the inventory, andhence incurs no capital costs for holding that stock.Those carrying costs accrue to the vendor. Thecustomer still sets the timing and quantity of orders.We will determine under what conditions theconsignment of stock can create benefits for thecustomer, the vendor, and for both.

We will also study the use of CI and VMIcombined. When CI is coupled with VMI, eventhough it is the vendor who pays the opportunitycost of goods stored at the customer, the vendornow takes responsibility as well for setting thequantity and timing of shipments released to thecustomer. This transfer of authority also shiftsthe decision-making costs to the vendor, butthe vendor may benefit from this agreement bydecreasing her total inventory holding cost. Table 1contrasts the three cases we consider.

The remainder of this paper is as follows. Section3 introduces our notation. In Section 4, we developa model for IS and find the analytical solution forour base case. We then extend the base-casemodel 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

IS CI C&VMI

Ordering decision made by C C V

Bearer of ordering cost C C V

Ownership of stock at customer C V V

Bearer of opportunity cost C V V

C: the customer, V: the vendor.

M. Gumus- et al. / Int. J. Production Economics 113 (2008) 502–517506

the base case. We provide numerical examples inSection 7, while Section 8 contains a summary andconclusions.

3. Notation

Our models for IS, CI, and C&VMI employ thefollowing basic notations.

Ac Customer’s fixed cost of ordering ($ perorder). Ac consists of the cost of issuing anorder, ao, and the cost per shipmentreceived. (The latter does not need to bedefined separately.)

hc Annual cost to carry one unit in stock atcustomer’s retail store ($/unit/year). Thisper-item inventory holding cost is com-posed of ho, the opportunity cost, and hs,the physical storage cost: hc ¼ ho+hs.

S Vendor’s fixed cost ($ per setup) incurred atthe start of each production cycle.

av Vendor’s cost per shipment release ($ pershipment to the customer).

hv Annual cost to hold a unit in inventory atvendor’s production site ($/unit/year).

p Vendor’s annual production rate (units/year).

d Annual demand rate at the customer (units/year).

The vendor is assumed to have sufficient capacityto meet the customer’s demand (i.e. pXd). In IS,each party pays its own costs as defined above. InCI and C&VMI, portions of Ac and/or hc are paidby the vendor on behalf of the customer.

In our formulations, the subscripts v and c referto 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 sour-cing, where there is no agreement between vendor

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

4. Inventory sourcing (IS)

In IS, the customer first makes replenishmentplans based on his costs Ac and hc, and the end-userdemand d. The customer’s decisions concern thefrequency and in what quantity to order from thevendor. His optimal order quantity is q1 ¼ EOQ ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

2Acd=hc

pand his optimal total cost is

TCc1 ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2Acdhc

p. The customer passes the replen-

ishment decision to the vendor, who produces at arate pXd. The vendor, who must satisfy thecustomer’s orders fully, finds her economic produc-tion quantity (Q1) based on her costs of productionsetup (S), inventory holding (hv), and shipmentrelease (av).

To describe system inventory levels, we assumethat the vendor switches from other items andbegins manufacturing this one when the customer’sinventory level is q1. From that moment, the vendorproduces at a rate p during an interval T ¼ kq/p,where k is the number of shipments from vendor tocustomer during the vendor’s manufacturing cycle.When the vendor is producing, total systeminventory increases at a rate p�d. After productionstops, the vendor supplies goods to the customerfrom her stock until that is depleted. When thevendor is not producing, the system-wide inventorydecreases at a rate d. We denote the time betweensuccessive production runs at the vendor by T0 (seeFig. 3).

The vendor’s total production quantity in hercycle is Q1 ¼ kq1. All items carried by the vendorare charged holding costs at a rate hv. From Fig. 3,the average total system inventory is q1þ

ðp� dÞQ1=2p, and the vendor’s mean stock level isq1=2þ ðp� dÞQ1=2p. The vendor’s total cost perperiod is then

TCv1 ¼ d½S=Q1 þ av=q1� þ hv½q1 þ ð1� d=pÞQ1�=2.

(1)

The vendor’s total cost is composed of the totalproduction setup and shipment release costs (thefirst two terms in (1)), and inventory carrying costs(the third and fourth terms). To be able to compare

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T Time

Total system inventory

q1

Hp – d

d

T

…. Customer’s inventory position

Vendor’s inventory position

System-wide inventory position

- - -

′

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


different partnerships analytically, we assumethroughout that the number of shipments per cycleis a continuous variable. Therefore, the optimalvalue of Q1, the production quantity in one cycle, isffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

2Sd=½hvð1� d=pÞ�p

¼ EPQ. As in Gumus- et al.(2006), the strict convexity of TCv1 means that theoptimal integer value for k is

kint¼ Min|ffl{zffl}

TCv1ðk�Þ

EPQ=q1

� �; EPQ=q1

� �� .

Based on the optimal values of Q1 and q1, thevendor’s minimal total cost is

TCv1 ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2Sdhvð1� d=pÞ

pþ dav=

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2Acd=hc

pþ hv

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2Acd=hc

p=2.

Letting g ¼ av/Ac and f ¼ hv/hc, and definingC0 ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2Sdhvð1� d=pÞ

p, the vendor’s total cost is

now TCv1 ¼ C0 þ ðgþ fÞffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffidAchc=2

p. The system-

wide cost under inventory sourcing (TCc1+TCv1)is therefore TC1 ¼ C0 þ ½1þ ðgþ fÞ=2�

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2Acdhc

p.

The preceding model for the base case assumedno agreement between customer and vendor.Benefits of any CI or C&VMI agreement will berelative to total costs under inventory sourcing.

5. Consignment inventory (CI)

In the first type of agreement, CI, the customermaintains control over the timing and quantity oforders, and pays Ac every time he places an order.However, he does not incur the opportunity-costportion of carrying inventory, since the vendor

owns the goods at the customer’s premises until theyare used.

For i ¼ 1 and 2, let ei (0oeio1) denote the ratiose1 ¼ hs/hc and e2 ¼ ho/hc (where e1+e2 ¼ 1) ofportions of the customer’s inventory holding costper item (hc) under IS. The customer’s total cost inCI is the sum of ordering and physical storage costs:TCc2 ¼ Acd/q2+hsq2/2. Based upon those costs, hisoptimal order size is q2 ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2Acd=hs

p¼ q1=

ffiffiffiffi�1p

,which is strictly greater than q1 since e1o1. Hisoptimal total cost under CI is thenffiffiffiffiffiffiffiffiffiffiffiffiffiffi2Acdhs

p¼

ffiffiffiffi�1p

TCc1, which is strictly less thanTCc1. Therefore, the customer is always better offunder CI when compared to IS.

The vendor, who bears the opportunity cost ofgoods stored at the customer, ships less frequentlyunder CI than IS. (Assume for now that whenvendor orders on behalf of customer, there is no‘‘efficiency factor.’’ That is, she pays the sameopportunity cost ho as the customer.) Denoting thevendor’s production batch size by Q2,

TCv2 ¼Sd

Q2

þavd

q2

þ1

2hv½q2 þ ð1� d=pÞQ2�

þ1

2hoq2.

Since backordering is not allowed, the vendor’soptimal production batch size is

max q2;ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2Sd=ðhvð1� d=pÞÞ

p� . We assume thatffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

S=ðhvð1� d=pÞÞp

4ffiffiffiffiffiffiffiffiffiffiffiffiAc=hs

p, and hence her optimal

production quantity is Q2 ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2Sd=ðhvð1� d=pÞÞ

p¼

Q1. Then, TCv2 ¼ C0 þ ðavd=q2Þ þ ð1=2Þhvq2 þ

ð1=2Þhoq2 (where C0 is as defined in IS).


We can also write this cost as TCv2 ¼

C0 þ ð1=ffiffiffiffi�1pÞffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiAcdhc=2

pð�1gþ fþ �2Þ. Recall that

TCv1 ¼ C0 þ ðgþ fÞffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiAcdhc=2

p, from Section 4.

The vendor is better off under CI if and only ifTCv14TCv2. This requires (since

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiAcdhc=2

pand

1=ffiffiffiffi�1p

40):ffiffiffiffi�1pðgþ fÞ4ð�1gþ fþ 1� �1Þ. Then,

gðffiffiffiffi�1p� �1Þ4fð1�

ffiffiffiffi�1pÞ þ ð1�

ffiffiffiffi�1pÞð1þ

ffiffiffiffi�1pÞ. Be-

cause 1�ffiffiffiffi�1p

40,ffiffiffiffi�1p

g4fþ 1þffiffiffiffi�1p

. (2)

Proposition 1. A necessary condition that the vendor

be better off under CI is g4f+2.

Proof. We see in (2) thatffiffiffiffi�1pðg� 1Þ4fþ 1. Since

f, e140, (g�1) must be positive for the inequality tohold. Therefore, g41 and we re-write (2) asffiffiffiffi�1p

4ðfþ 1Þ=ðg� 1Þ. The result follows because e1(and thus

ffiffiffiffi�1p

) are less than one. &

Proposition 1 states that the vendor will be betteroff under a CI agreement if her cost per shipmentreleased exceeds ð1þ ðhv=hcÞÞAc. Observe that (2) ismore likely to hold when the customer has thehigher inventory carrying cost. Consider, forexample, an inventory sourcing agreement wherethe vendor delivers goods to the customer’spremises and pays transportation costs. Some ofthe vendor’s shipment costs are usually passed on tothe customer through an increased price per item.Hence, his inventory holding cost can be higherthan the vendor’s. A consignment agreement in sucha setting is more likely to create benefits for bothparties.

What happens if condition (2) does not hold?There are two possible cases:

(i)
CI achieves system-wide cost savings wherethe customer is no worse off but the vendor isworse 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 acceptthe terms to maintain business with hercustomer.
(ii)
The system-wide cost is greater in CI than in IS.Then, it is in neither party’s interest to changethe traditional way of doing business.
To explore these two possible situations, weformulate the total cost under a CI agreement andcompare it with inventory sourcing. The system-

wide cost under CI is

TC2 ¼ C0 þ1

2ffiffiffiffi�1p

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2Acdhc

pð�1gþ fþ �2Þ

þffiffiffiffi�1p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

2Acdhc

p.

Recall from IS that TC1 ¼ C0þ

½1þ ð1=2Þðgþ fÞ�ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2Acdhc

p. Therefore, CI leads to

system-wide cost savings if TC14TC2, whichrequires

1þ1

2ðgþ fÞ

�4

1

2ffiffiffiffi�1p ð�1gþ fþ 1� �1Þ þ

ffiffiffiffi�1p

�.

(3)

Proposition 2. A necessary condition for system-wide

cost savings under CI is fog.

Proof. Sinceffiffiffiffi�1p

40, these inequalities must besatisfied for (3) to hold:ffiffiffiffi�1pðgþ fþ 2Þ4ð�1gþ fþ 1� �1Þ þ 2�1

)ffiffiffiffi�1pðgþ 1Þ þ

ffiffiffiffi�1pðfþ 1Þ4�1ðgþ 1Þ þ ðfþ 1Þ

)ffiffiffiffi�1pðgþ 1Þð1�

ffiffiffiffi�1pÞ4ðfþ 1Þð1�

ffiffiffiffi�1pÞ.

Noting thatffiffiffiffi�1p

o1, the result follows. &

Proposition 2 implies that if the vendor isrelatively more efficient in inventory holding coststhan shipment release costs, she more likely achievescost savings under CI. Intuitively, the customer’sreplenishment quantities increase under CI versusIS. That increase can benefit the vendor, whoprefers fewer shipments if her cost per shipmentrelease is high.

This concludes the analytical results for a basic CIagreement, where it is assumed that the vendor paysexactly the same opportunity cost per item, ho, thatthe customer pays in IS, and that the wholesale priceof an item does not change from IS to CI. Table 2summarizes those findings. In the next two subsec-tions, we will analyze the impacts of the vendor’sefficiency on the opportunity cost of an item, and ofcost sharing through a wholesale price adjustment.

5.1. Impacts of the vendor’s efficiency factor

Various considerations might create a situationwhere the vendor and customer may not have thesame capital costs of the holding inventory. Anorganization’s capabilities in financing, and thefirm’s relative power in industry, can make tremen-dous changes in capital costs. Thus, in a CIagreement, the vendor pays b2ho per unit held at

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Table 2

Summary of conditions when CI is beneficial for the customer,

the vendor, and the whole system; m ¼ ðfþ 1Þ=ffiffiffiffi�1p

Necessary and

sufficient condition

Benefits under CI compared to IS

Customer Vendor Supply

chain

m�1ogom+1 Better off Worse off Better off

g ¼ m+1 Better off No worse

off

Better off

g4m+1 Better off Better off Better off

g ¼ m�1 Better off Worse off No worse

off

gom�1 Better off Worse off Worse off


the customer’s premises. The vendor’s capital costefficiency compared to that of the customer is b240.

Whatever the value of this parameter, thecustomer’s order quantity and total cost are stillq1=

ffiffiffiffi�1p

andffiffiffiffi�1p

TCc1, respectively. However, thetotal cost of the vendor is now TCv2 ¼ C0þ

ðavd=q2Þ þ ð1=2Þhvq2 þ ð1=2Þb2hoq2, which can bewritten as TCv2 ¼ C0 þ ð1=

ffiffiffiffi�1pÞffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiAcdhc=2

pð�1gþ

fþ b2�2Þ.Comparison of TCv2 and TCv1 shows that the

vendor is better off ifffiffiffiffi�1pðgþ fÞ4�1gþ fþ b2ð1� �1Þ

)ffiffiffiffi�1p

gð1�ffiffiffiffi�1pÞ4fð1�

ffiffiffiffi�1pÞ

þ b2ð1�ffiffiffiffi�1pÞð1þ

ffiffiffiffi�1pÞ.

Since 1�ffiffiffiffi�1p

40,ffiffiffiffi�1p

g4fþ b2ð1þffiffiffiffi�1pÞ. A

necessary condition for this inequality to hold isg4b2. We then have

ffiffiffiffi�1p

4ðfþ b2Þ=ðy� b2Þ. Simi-lar to the proof of Proposition 1, we now see that anecessary condition for the vendor to be better off isg4fþ 2b2.

The system-wide costs become

TC2 ¼ C0 þ

ffiffiffiffiffiffiffiffiffiffiffiffiAcdhc

2

r1ffiffiffiffi�1p ð�1gþ fþ b2�2Þ þ 2

ffiffiffiffi�1p

� .

As compared to TC1, we find that system-widecost savings are achieved whenffiffiffiffi�1p

4ðfþ b2Þ=ðgþ 2� b2Þ. Again as in the proofof Proposition 2, it is necessary that gþ 24fþ 2b2.

The above analysis shows that system-wide costs,as well as vendor’s costs, improve as b2, the vendor’scost factor, becomes smaller. A CI agreement ismore promising for both parties when the vendorcan develop efficiencies in the opportunity cost ofcapital.

Even if the vendor is unable to develop theseefficiencies, CI can create a situation where there ispotential to lower system-wide costs. We call this a‘‘potentially efficient system,’’ and now examine it indetail.

5.2. Cost sharing

Let us define an ‘‘efficient system’’ as one withtotal-cost improvements with respect to a base case,and where neither party is worse off. We demon-strated that the customer always gains under CIcompared to IS. Accordingly, a potentially efficientsystem in CI is one where, although the vendor isworse off, there are overall system-wide costsavings.

A potentially efficient system can be turned intoan efficient one by some sort of incentive, offered bythe customer to transfer a portion of his benefits tothe vendor. When CI is applied, standard industrypractice allows the vendor to increase the unit priceto share total savings. Without getting into detailson cost-sharing research, we briefly explain how thiscould work.

Let c be the original price per item paid by thecustomer. The vendor suggests a price incrementover c in order to make CI beneficial to herself. Inthe absence of information sharing between parties,the customer may be unsure that a price increase isin his best interest (e.g. when he receives an equal, oreven smaller, share of system-wide savings due toCI). Let yhigh be the maximum percentage increasein price acceptable to the customer:

yhighcd

100¼ ð1�

ffiffiffiffi�1pÞffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2Acdhc

p; which means

yhigh ¼100

cð1�

ffiffiffiffi�1pÞ

ffiffiffiffiffiffiffiffiffiffiffiffi2Achc

d

r.

We now determine the smallest price incrementacceptable to the vendor, the value that makes herno worse off than under IS. We assume thatinequality (2) does not hold; this is why the vendoris motivated to ask for a price change. Takingb2 ¼ 1 results in

ylowcd

100¼


2

r�1gþ fþ �2ffiffiffiffi

�1p � ðgþ fÞ

� ,

and hence

ylow ¼100

c

ffiffiffiffiffiffiffiffiffiffiAchc

2d

r�1gþ fþ �2ffiffiffiffi

�1p � ðgþ fÞ

� .


Note that the maximum price increase thecustomer will accept, yhigh, is the upper bound onthe price increase that would erase his gains underCI. On the other hand, ylow is the lower bound thatwould compensate the vendor for her increase incosts, but still leave the customer with some benefit.Therefore, when CI creates a potentially efficientsystem, a wholesale price increment 2 ðylow; yhigh�

will make the vendor willing to accept thatagreement rather than IS. The customer favorsCI as long as price increments are in the range[ylow, yhigh).

Another means of creating possible cost savingsfor both vendor and customer may be the use of CIand VMI combined. While CI always benefits thecustomer, VMI has the potential to generatebenefits for the vendor. C&VMI is the subject ofthe next section.

6. Consignment and vendor-managed inventory

(C&VMI)

In a C&VMI agreement, the vendor owns thegoods 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 customerand ao for every order she places for him. Thecustomer is then exempt from those expenses. Inlight of these changes, the vendor’s total cost inC&VMI is

TCv3 ¼Sd

Q3

þavd

q3

þ1

2hv½q3 þ ð1� d=pÞQ3�

þaod

q3

þ1

2hoq3. (4)

This total cost is also equal to C0 þ ðav þ aoÞdq3þ

12ðhv þ hoÞq3 where C0 is as explained in IS. Letting

d1 ¼ ao/Ac, and with the ratios defined previously,we re-write the vendor’s total cost

TCv3 ¼ C0 þ ðgþ d1ÞAcd

q3

þ1

2ðfþ �2Þhcq3.

Therefore, the optimal order quantity determinedby the vendor on behalf of the customer is

q3 ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2ðgþ d1ÞAcd

ðfþ �2Þhc

s¼

ffiffiffiffiffiffiffiffiffiffiffiffiffigþ d1fþ �2

sq1.

Incorporating the optimal order quantity in (4),we get TCv3 ¼ C0 þ

ffiffiffiffiffiffiffiffiffiffiffiffiffigþ d1

p ffiffiffiffiffiffiffiffiffiffiffiffiffifþ �2

p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2Acdhc

p. Re-

call for IS that TCv1 ¼ C0 þ ðgþ fÞffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffidAchc=2

p.

Therefore, the vendor’s cost under C&VMI is

less than her cost of IS if TCv3oTCv1, whichreduces toffiffiffiffiffiffiffiffiffiffiffiffiffi

gþ d1p ffiffiffiffiffiffiffiffiffiffiffiffiffi

fþ �2p

ogþ f2

. (5)

After some algebra, (5) can be written in the form

4½ðgþ d1Þ�2 þ fd1�oðg� fÞ2. (6)

The right-hand side of (6) is zero when g ¼ f.Therefore, C&VMI would create benefits for thevendor if she has, compared to the customer,efficiency or inefficiency in either her ordering orinventory holding, but not in both costs. That is, thevendor can make better use of the orderingauthority created by C&VMI when she has anadvantage or a disadvantage in either her orderingor inventory holding costs.

Suppose the vendor’s ordering cost is greater thanthe customer’s but their inventory holding cost peritem is around the same. The vendor can ship largerquantities to decrease her total ordering cost. Thisholds true if her carrying cost is lower, but she hasno clear efficiency in ordering costs relative to thecustomer. On the other hand, if the vendor’sinventory holding cost is too high, she can replenishthe customer frequently in small quantities toachieve savings.

In the meantime, the vendor’s costs associatedwith C&VMI influence the benefits that the agree-ment can create for her. The vendor’s costs underC&VMI increase linearly in the ratios d1 and e2.Hence, as those parameters get lower, it is morelikely that the vendor achieves cost savings, sincethere is a decrease in the left-hand side of (6).

Now, the customer would accept C&VMI if hiscosts under this agreement were not higher thanhis costs in IS. The customer’s total cost underC&VMI is

TCc3 ¼ðAc � aoÞd

q3

þ1

2hsq3

¼ð1� d1ÞAcd

q3

þ1

2�1hcq3.

The optimal ordering quantity q3 was determinedby the vendor on behalf of the customer. Insertingthat optimal quantity in the customer’s cost func-tion yields

TCc3 ¼


2

rð1� d1Þðfþ �2Þ þ �1ðgþ d1Þffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

ðgþ d1Þðfþ �2Þp

!.


The customer’s total cost in IS is TCc1 ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2Acdhc

p. C&VMI thus benefits him if

ð1� d1Þðfþ �2Þ þ �1ðgþ d1Þo2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðgþ d1Þðfþ �2Þ

p.

(7)

Letffiffiffiffiffiffiffiffiffiffiffiffiffigþ d1

p¼ m1 and

ffiffiffiffiffiffiffiffiffiffiffiffiffifþ �2

p¼ m2. Then (7)

reduces to ð1� d1Þ=ðm1=m2Þ þ �1m1=m2o2.Note that (1�d1) and e1 are both o1. If the

vendor’s replenishment quantity q3 exceeds thecustomer’s quantity q1 under IS (m1/m241), it ismore likely for the customer to achieve savingsunder C&VMI when e1 is low. That is, the customerwould not mind large order quantities as long as hisphysical storage cost per item is low.

Similarly, the customer can still benefit when thevendor replenishes him very frequently (m1/m2o1),if his cost per shipment received is not high. Ingeneral, the customer is more likely to achieve costsavings under C&VMI because he does not pay theopportunity cost of items in stock nor the cost ofplacing orders. We can now check whether both

parties can be better off under C&VMI.

Proposition 3. For C&VMI to create an efficient

system, it is necessary that 2d1�24�2ð1� gÞþd1ð1� fÞ.

Proof. Inequalities (5) and (7) together imply thatð1� d1Þðfþ �2Þ þ �1ðgþ d1Þogþ f, which is a ne-cessary (but not sufficient) condition for bothparties to be better off compared to IS.

With some algebra, this condition reduces tod1fþ 2d1�2 þ �2g4�2 þ d1, and then to 2d1�24�2ð1� gÞ þ d1ð1� fÞ. &

We see in the proof of Proposition 3 that thisnecessary condition (required to achieve an efficientsystem) holds when g41 and f41, and also whengb1 or fb1. The latter is more likely the casewhere both parties are better off. This can beexplained by our analytical results on C&VMI forthe vendor and customer.

We observed previously that the vendor can makeuse of the C&VMI agreement to offset inefficiencyin one of her costs. Depending on which costparameter is high, the vendor can decrease orincrease the order quantity to achieve cost savings.That order quantity is also acceptable to thecustomer, as long as the costs from which he isexempt (cost of placing orders and opportunity costof inventory) compensate his increased costs result-

ing from ordering decisions made by the vendorfor him.

It may be less likely to achieve an efficient systemthan a potentially efficient system that can beworked out to satisfy both parties. Recall that asystem is potentially efficient if there are system-wide cost savings.

Proposition 4. C&VMI creates system-wide cost

savings relative to IS if

1þ f1þ g

om2

m1o1; or

1þ g1þ f

om1

m2o1.

The proof of Proposition 4 is provided inAppendix A. Through numerical examples in thenext section, we will see when C&VMI can create apotentially efficient system, and highlight as well theanalytical results found in the inventory sourcingand CI models. We note in passing that the costsharing argument discussed for CI in Section 5.2 canalso be applied to C&VMI.

7. Numerical examples

IS, CI, and C&VMI will now be contrastednumerically when certain parameters are varied. Inall those examples, Ac ¼ $100 per order, hc ¼ $1.5 peritem stored, d ¼ 1300 items/year, and p ¼ 1600 items/year. We do not assume any efficiency of the vendorover ho or ao in case of a CI or C&VMI agreement.

Figs. 4–6 test the impact of g on different sourcingoptions. Those examples fix the values e1 ¼ 0.4,d1 ¼ 0.1, and f ¼ 0.8, while g is between (0, 5]. Inthe 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) whileg ¼ 1.5, f ¼ 0.8, and d1 ¼ 0.1. The final figurevaries d1 between (0, 1) with g ¼ 1.5, f ¼ 0.8, ande1 ¼ 0.4.

In line with our analytical results, we observe inFig. 4 that the customer’s cost saving under CI isfixed, yet the system-wide and the vendor’s savingsincrease linearly as g increases. CI is beneficial forthe vendor when her cost per shipment is at least 3.8times the customer’s cost per order. System-widesavings are achieved for lower g.

For C&VMI, we see in Fig. 5 that cost savings arepossible for the vendor when g is very low or veryhigh. When gp0.02, the vendor replenishes thecustomer frequently to save on inventory holdingcosts, but such a large number of shipmentsincreases the customer’s and the system-wide total

ARTICLE IN PRESS

CI vs IS

-500-400-300-200-1000

100200300400500

0.01

Cos

t Sav

ings

($)

TCv1-TCv2 TCc1-TCc2 TC1-TC2

Gamma (γ )

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

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

-400

-200

0

200

400

600

0.01

Cos

t Sav

ings

($)


Gamma (γ )

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

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

-600

-400

-200

0

200

400

600

800

1000

1200

1400

0.01

Cos

t Sav

ings

($)


Gamma (γ )

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

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; bothparties are better off under C&VMI when gX4.0.

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

ARTICLE IN PRESS

-1200

-1000

-800

-600

-400

-200

0

200

400

0.01

Cos

t Sav

ings

($)


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

Phi (ϕ)

CI vs IS

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

-200

-100

0

100

200

300

0.01

Cos

t Sav

ings

($)


Phi (ϕ)

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

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. (Com-pared to CI, the vendor is worse off when2.3ogo4.7; 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 alsosee in Fig. 6 that as g increases, the vendor andcustomer each become indifferent between C&VMIand CI. This is logical: Compared to IS, thecustomer under CI orders larger quantities, andthis is what the vendor would do under C&VMI ifher shipment costs were high.

In Proposition 1, we stated that g4f+2 is anecessary condition for the vendor to be better offunder CI. Therefore, the vendor never achieves costsavings 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 ispossible for all parties to achieve cost savings forhigh enough f (fX3.7, Fig. 8). As discussed in ourformulations, C&VMI becomes an opportunity for

a vendor, one relatively inefficient in inventoryholding cost (fbg), to decrease her carrying costsvia frequent shipments.

When we compare C&VMI to CI for varying fvalues (Fig. 9), we see that C&VMI becomes abetter option for the vendor and the whole systemfor larger f. This makes sense: The customer underCI increases the order quantity, causing averagesystem stocks to grow. The vendor, on the otherhand, prefers more frequent shipments and lessinventory when f is high, and she can decide sounder C&VMI.

In line with analytical results, we see in Fig. 10that varying e1 changes all cost savings nonlinearly.As �1 approaches one, the system returns to thecosts under IS. While the customer’s savingsdecrease, the vendor’s as well as system-widesavings increase as e1-1. We also observe in thisexample that no e1 value creates an efficient system;the customer is always better off. The system is

ARTICLE IN PRESS

C&VMI vs CI

-600

-400

-200

0

200

400

600

800

1000

1200

1400

0.01

Phi (ϕ)

Cos

t Sav

ings

($)


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

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

-6000

-5000

-4000

-3000

-2000

-1000

0

1000

0.01

Cos

t Sav

ings

($)


ε1

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

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 nearzero, the system-wide and vendor’s costs increaseenormously.

Fig. 11 compares C&VMI to IS. Vendor’s costsand the customer’s savings are each decreasing in e1.An efficient system is never attained. The vendorachieves cost savings only when e1X0.98. (Contrastthis to CI in Fig. 10, where the vendor never realizescost savings.) System-wide costs in Fig. 11 do notchange much as e1 varies.

We see in Fig. 12 that low values of e1 make a bigdifference for the vendor’s and system-wide costsunder C&VMI compared to CI. However, CIbecomes a preferred option for the whole systemwhen e1X0.69, and for the vendor when0.95Xe1X0.51. The customer favors CI whene1p0.41.

We present only in a single graph the implicationof varying d1, since it does not influence costs for ISor CI. Comparing C&VMI to IS in Fig. 13, we seethat the customer’s savings and vendor’s costs underC&VMI are increasing in d1. System-wide savings,on the other hand, do not change much, remainingnear zero.

8. Summary and conclusions

In this paper, we studied a case where a customerand vendor initially consider consignment inventory(CI) for a single item. Comparing it to our basecase, which is inventory sourcing, we obtainedanalytical conditions under which CI creates bene-fits for one or more parties. In contrast to thegeneral belief that CI is beneficial only for the

ARTICLE IN PRESS

C&VMI vs IS

-400

-300

-200

-100

0

100

200

300

400

0.01

Cos

t Sav

ings

($)


ε1

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

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

0.580.460.4

C&VMI vs CI

-1000

0

1000

2000

3000

4000

5000

0.01

Cos

t Sav

ings

($)


ε1

0.04 0.1 0.16 0.22 0.28 0.34 0.52 0.64 0.7 0.76 0.82 0.88 0.94 0.99

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

-400

-200

0

200

400

600

0.01

Cos

t Sav

ings

($)


δ1

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

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 forthe vendor as well. Depending on the costs ofshipment, and who pays for transportation, CI canbe beneficial for both parties.

We showed that if the CI agreement resultsin a potentially efficient system, it can be turnedinto an efficient one. To achieve that, we foundthe minimum and maximum amounts by which


the wholesale price may increase, such that thecustomer may accept to share his benefits withthe vendor. When the system is inefficient under CI,the vendor can offer a C&VMI agreement to realizesavings for herself and for the system.

We considered the option of CI plus VMI(C&VMI) and extended our analysis to find theoptimal costs and potential savings under thatagreement. We showed how the vendor canmake use of C&VMI to improve her costs inareas where she is inefficient. Although thevendor prefers C&VMI rather than CI,and the customer vice versa, we observed thatC&VMI is more likely to generate system-widecost savings.

This paper provided closed-form results fordifferent sourcing options. Outcomes of thosechoices were shown to depend on cost parametersof the parties involved. We identified conditionsunder which one option is preferred to another. Ourfindings 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 scalecreated by a C&VMI agreement when there aremultiple customers. Some of them will be offeredC&VMI by the vendor, whose goals are to achieveflexibility in production and to reduce operationalcosts such as shipment expenses. Customers underthat agreement should be no worse off than for IS.

Secondly, a model can be developed for acustomer to choose between vendors for CI whenthere are several suppliers of certain items. CI isalways beneficial for the customer without anychange in wholesale price. But various supplierscould enforce a price adjustment when CI is offered.In that case, the customer should carefully select theCI-vendor to maximize his savings.

In light of the multiple customers or vendors,computer simulation may be required for either ofthe above extensions when end-consumer demand isuncertain. In settings where demand is more stable,however, it may be possible to find analyticalsolutions.

Appendix A

Proof of Proposition 4: We have

TC1 ¼ TCc1 þ TCv1 ¼ C0 þ1ffiffiffi2p


pð2þ gþ fÞ

and

TC3 ¼ TCc3 þ TCv3 ¼ C0 þ1ffiffiffi2p


p� ð1� d1Þ

m2

m1þ ð1� �2Þ

m1

m2þ 2m1m2

� .

Then, TC3oTC1 if

ð1� d1Þm22 þ ð1� �2Þm

21 þ 2m2

1m22om1m2

� ð2þ gþ fÞ.

After some algebra, this reduces to ð1þ gÞm22þ

ð1þ fÞm21om1m2½ð1þ fÞ þ ð1þ gÞ�, and then to

ð1þ fÞm1ðm1 �m2Þo ð1þ gÞm2ðm1 �m2Þ. Thus, ifðm1 �m2Þ40, we have ð1þ fÞ=ð1þ gÞoðm2=m1Þo1, but if ðm1 �m2Þo0, then ð1þ gÞ=ð1þ fÞoðm1=m2Þo1. &

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

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