co

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Article history:Received 27 May 2009

a b s t r a c t

tered by companies such as Mattel, Inc., a toy maker (see, e.g.,Barnes-Schuster et al., 2002). Motivated by this observation, weaddress in this paper the issue of whether an option mechanismis a viable alternative to achieving an efcient supply chain. Theoption mechanism is characterized by two parameters, namelythe option price o and the exercise price e. The option price, in es-

in supply chains. A typical example is the numerous operatingmodes based on cooperative relations that have been widelyadopted in the practice of SCM, such as vendor managed inventory(VMI), holding cost subsidies, and consignment. These operatingmodes have signicantly improved the overall performance of sup-ply chains. Such industry environment motivates us to address theproblem under study by way of a cooperative game approach.Again, much of the literature on supply chain coordination has fo-cused on developing incentive contract forms, and did not take into

* Corresponding author.

European Journal of Operational Research 207 (2010) 668675

Contents lists availab

O

.eE-mail address: lgtcheng@polyu.edu.hk (T.C.E. Cheng).1. Introduction

Manufacturerretailer supply chains commonly adopt awholesale price mechanism in practice. This mechanism, however,has often led manufacturers and retailers to situations of conictsof interest, resulting in an inefcient supply chain. For instance,due to uncertain market demand and aversion to incurring inven-tory costs, retailers prefer to order exibly from manufacturers soas to accommodate uctuating market demand. On the other hand,in order to hedge against the risks of over- and under-production,manufacturers prefer retailers to place full orders as early as possi-ble. This conict between the retailer and the manufacturer canlead to actions that are not system-wide optimal, thereby resultingin an inefcient supply chain. Such problems have been encoun-

sence, is an allowance paid by the retailer to the manufacturer forreserving one unit of the production capacity. The exercise price isto be paid by the retailer to the manufacturer for one unit of theproduct purchased by exercising the option. Option mechanismshave attracted substantial attention not only in the context ofnancial market, pioneered by Black and Scholes (1973), but alsoin the area of supply chain management (SCM) (please refer toSection 2). In fact, option mechanisms have been extensivelyapplied in various industries such as fashion apparel (Eppen andIyer, 1997), toys (Barnes-Schuster et al., 2002), and electronics(Billington, 2002; Carbone, 2001).

Relationships between manufacturers and retailers in supplychains have dramatically changed in recent years. One such changeis evidenced by the fact that partnerships have become prevalentAccepted 13 May 2010Available online 20 May 2010

Keywords:Supply chain managementCooperative gameOption contractNegotiating powerChannel coordination0377-2217/$ - see front matter 2010 Elsevier B.V. Adoi:10.1016/j.ejor.2010.05.017Manufacturerretailer supply chains commonly adopt a wholesale price mechanism. This mechanism,however, has often led manufacturers and retailers to situations of conicts of interest. For example,due to uncertain market demand, retailers prefer to order exibly from manufacturers so as to avoidincurring inventory costs and to be able to respond exibly to market changes. Manufacturers, on theother hand, prefer retailers to place full orders as early as possible so that they can hedge against the risksof over- and under-production. Such conicts between retailers and manufacturers can result in an inef-cient supply chain. Motivated by this problem, we take a cooperative game approach in this paper toconsider the coordination issue in a manufacturerretailer supply chain using option contracts. Usingthe wholesale price mechanism as a benchmark, we develop an option contract model. Our study dem-onstrates that, compared with the benchmark based on the wholesale price mechanism, option contractscan coordinate the supply chain and achieve Pareto-improvement. We also discuss scenarios in whichoption contracts are selected according to individual supply chain members risk preferences and nego-tiating powers.

2010 Elsevier B.V. All rights reserved.Production, Manufacturing and Logistics

Coordination of supply chains by optiongame theory approach

Yingxue Zhao a,b, Shouyang Wang a, T.C.E. Cheng c,*, Xa Institute of Systems Science, Academy of Mathematics and Systems Science, Chinese Acb School of International Trade and Economics, University of International Business andcDepartment of Logistics and Maritime Studies, The Hong Kong Polytechnic University, HdDepartment of Applied Mathematics, The Hong Kong Polytechnic University, Hung Home School of Business, Adelphi University, Garden City, NY 11530, USAf School of Business, Renmin University of China, Beijing 100872, PR China

European Journal of

journal homepage: wwwll rights reserved.ntracts: A cooperative

oqi Yang d, Zhimin Huang e,f

y of Sciences, Beijing 100190, PR Chinaomics, Beijing 100029, PR Chinag Hom, Kowloon, Hong Kong, PR Chinaowloon, Hong Kong, PR China

le at ScienceDirect

perational Research

lsevier .com/locate /e jor

leave each party to be at least as well off as that party would bewithout these contracts. Motivated by this, we propose a notion

peracalled the core of a contract set, which is a subset of the contractset consisting of all the contracts satisfying the coordinationrequirement. Again, noting that the wholesale price mechanismis prevalently used in practice, we take the prot level under thewholesale price mechanism as a benchmark and derive the coreof option contracts by taking a cooperative game approach. Wedemonstrate that option contracts can coordinate the supply chainand achieve Pareto-improvement. Hence, the manufacturer andthe retailer will both have strong incentives to adopt the optionmechanism rather than the wholesale price mechanism.

Even though every option contract in the core, compared withthe wholesale price mechanism, can coordinate the supply chainwith Pareto-improvement, the prot allocations between the retai-ler and the manufacturer differ for different option contracts in thecore. As a result, the retailer and manufacturer will negotiate theoption contract from the core to implement. Obviously, the out-come depends heavily on individual members negotiating powers,as well as their risk preferences. By taking these factors into con-sideration, we identify the option contract that the retailer andmanufacturer both agree to implement according to Nashs bar-gaining model and Eliashbergs model, and explore analyticallythe effects of these factors on the ultimate outcome. The remainderof this paper is organized as follows: In Section 2 we review the re-lated literature. In Section 3 we introduce the model. In Section 4we analyze the model. In Section 5 we examine supply chain coor-dination with option contracts and discuss the related characteris-tics. We explore analytically in Section 6 the selection of optioncontracts and the sharing of the extra prot gained from coordina-tion, taking into consideration the effects of members risk prefer-ences and negotiating powers. We conclude the paper and suggesttopics for future research in Section 7. We give all the proofs in thesupplementary Appendix, which appears as an online electroniccompanion of the paper.

2. Literature review

This paper can be regarded as a study of supply chain contracts.To highlight our contributions, we review only the literature that isrepresentative and particularly relevant to our study.

First, this paper is closely related to the literature on the use ofoptions in SCM. Research on this aspect mainly focused on opera-tions exibility and economics efciency derived from options.Barnes-Schuster et al. (2002) studied how options provide mana-gerial exibility in response to uncertain market changes andhow to achieve channel coordination by options based on a modelof two periods with correlated demand. Eppen and Iyer (1997) con-sidered a backup agreement, which is essentially an option con-account issues concerning implementation of these contract forms.However, prot allocations differ under different coordinating con-tracts. The ultimate implementation outcome of a coordinatingcontract form inevitably depends on supply chain members indi-vidual risk preferences and negotiating powers. Therefore, giventhat an incentive contract form has been derived, two importantquestions remain: (i) What coordinating contracts are feasible?(ii) How do supply chain members individual risk preferencesand negotiating powers impact the ultimate outcome?

Supply chain coordination is an important issue in SCM. Variouscontract types have been designed to align supply chain membersincentives to drive the optimal action in the supply chain. How-ever, it is clear that feasible coordinating contracts are those that

Y. Zhao et al. / European Journal of Otract. They showed that backup agreements have substantialeffects on expected prot and committed quantity. Wu et al.(2002) explored the optimal option contract bidding and contract-ing strategies using a model with one seller and one or more buy-ers. Milner and Rosenblatt (2002) and Wang and Tsao (2006)explored from the buyers perspective the bidirectional option bywhich the option buyer can adjust freely the initial order quantityboth upwards and downwards. Cheng et al. (2003) considered anoption model where the option buyer, in addition to a committedorder quantity, has the right of acquiring an additional quantity ofthe supply, if necessary, by purchasing options from the supplier.More studies on the use of options can be found in Cachon andLariviere (2001), Spinler et al. (2003) and Wang and Liu (2007).Other research incorporating the quality of demand informationupdating can be found in Brown and Lee (2003), who analyticallyexamined the impacts of demand signal quality on order decisionswith an options-futures contract.

Second, this paper is closely related to the literature on supplychain coordination with contracts and supply chain contract nego-tiation. We review only a few representative studies in these areas.Pasternack (1985) showed that coordination between a buyer anda supplier can be achieved with buy-back contracts. Cachon andLariviere (2005) demonstrated that revenue sharing can coordinatea two-echelon supply chain with a price-setting newsvendor. Tay-lor (2002) considered sales-rebate contracts with sales effort ef-fects. He showed that when demand is affected by the retailerssales effort, a target rebate and returns contract can achieve chan-nel coordination. The other alternative contracts that have beenshown to achieve coordination in supply chains include quantitydiscounts (see, e.g., Weng, 1995) and quantityexibility contracts(see, e.g., Tsay, 1999). Cachon (2003) gave an excellent survey onsupply chain coordination with contracts. Note that numerous con-tracts can be utilized to achieve supply chain coordination and thesubsequent prot allocations differ. Hence, an issue that needsresolving is to identify the contract that will be accepted by allthe supply chain members and will subsequently be implemented.This issue has attracted some researchers attention. Kohli and Park(1989) considered simply the issues concerning the selection ofquantity discount contracts using Nashs bargaining model, Ka-laiSmorodinskys model, and Eliashbergs model. Sobel and Turcic(2008) developed a general model for supply chain contractnegotiation with risk aversion using Nashs bargaining model. Nag-arajan and Soic (2008) gave a survey on using cooperativebargaining models to allocate prot between supply chainmembers.

This paper focuses on the economics efciency of options andwe deploy a cooperative game approach to address the relevantproblems. Compared with the literature on option models, the crit-ical difference of this paper does not lie in the developed optionmodel but lies in the analytical exploration of the issues concern-ing implementation of option contracts in the supply chain, takinginto account supply chain members risk preferences and negotiat-ing powers. In fact, to more clearly explore these issues, we devel-op in this paper a relatively simple option model, as compared withprevious ones. Compared with the literature on supply chain coor-dination with contracts and supply chain contract negotiation, ourpaper is different in the following ways. First, our paper focuses onemploying the option mechanism to achieve supply chain coordi-nation. Second, more importantly, we analytically derive the re-sults, most of which are in closed-form, on the implementationof the coordinating option contract, including the coordinating op-tion contract that will be accepted by all the supply chain mem-bers, the corresponding prot allocations between them, and theeffects of their risk references and negotiating powers. This wasnot the case in the existing literature such as Sobel and Turcic(2008) and Nagarajan and Soic (2008). Third, different from the

tional Research 207 (2010) 668675 669existing literature such as Sobel and Turcic (2008) and Nagarajanand Soic (2008), our paper incorporates the use of Eliashbergsmodel, the supply chain members risk preferences, and their

risk preferences. In addition, our research approach can easily be

To focus on the implementation issue of the option contract, as

erareviewed in Section 2, we consider a relatively simple option mod-el. The primary purpose of our model is to explore the implemen-tation issue of the coordinating option contract form, taking intoaccount the effects of the supply chain members risk preferencesand negotiating powers.

4. Basic option contract model

Since the wholesale price mechanism is commonly used inmanufacturerretailer supply chains in practice, we use the whole-sale price mechanism as the benchmark against which we willextended to other types of supply chain contract popular in SCMresearch. Of course, we also discuss the limitations of our study.Despite the acknowledged limitations, we believe our paper hasmade a good contribution to the literature on the implementationaspect of supply chain contracts.

3. Model description

We consider a two-echelon supply chain consisting of a singlemanufacturer and a single retailer where the manufacturers prod-uct is sold via the retailer to end consumers. Market demand forthe product is a stochastic variable X, which follows a strictlyincreasing distribution function F(x) with xP 0. We assume theoption mechanism is employed to facilitate the production andprocurement in the supply chain, which is characterized by twoparameters, namely the option price o and the exercise price e.The option price is essentially an allowance paid by the retailerto the manufacturer for reserving one unit of the production capac-ity at the beginning of the production season. The exercise price isto be paid by the retailer to the manufacturer for one unit of theproduct purchased by exercising the option after the market de-mand is realized. In addition, considering the current industryenvironment in which many retailers are as powerful as, or evenmore powerful than, their manufacturers, we assume the manufac-turer uses the make-to-order policy for production. The event se-quence of our model can then be described as follows: At thebeginning of the production season, the retailer reserves a quantityof the production capacity beforehand from the manufacturer, sayQ, at a unit price of o. Using the make-to-order production policy,the manufacturer produces Q units of the product by following thequantity reserved by the retailer. During the selling season, accord-ing to the realized market demand for the product, the retailer pur-chases a quantity of the product up to Q units from themanufacturer at the exercise price e to satisfy the demand, andany unsatised demand is lost with no penalty cost. We assumedemand information is symmetric between the manufacturer andthe retailer. We let the marginal production cost of the manufac-turer be c and the salvage value per unit of unsold product be vfor both the manufacturer and the retailer. Let p be the retailers re-tail price. We focus on the reasonable and non-trivial case wherep > c > v, 0 6 o < c v, and e > v. In fact, assuming o < c v avoidsthe unreasonable case where the manufacturer is risk-free for itsproduction while assuming e > v avoids the trivial case where theretailer always exercises all the options purchased by it.negotiation powers (by a line aggregation rule) into the analysis ofthe issues concerning implementation of the option contract. Wethink doing so is important because the implementation outcomeof supply chain contracts clearly depends on supply chain mem-bers negotiation powers on the negotiation table, as well as their

670 Y. Zhao et al. / European Journal of Opcompare the option mechanism developed in this study. Withthe wholesale price mechanism, the event sequence is identicalto the option model except for the option mechanism, whichshould be replaced by the wholesale price mechanism. Then, theretailers expected prot function under the wholesale price mech-anism is given by

EPwrQwr E pminfQwr ; xg wQwr vmaxfQwr x;0g ; 1where Qwr is the retailers order quantity and w is the wholesaleprice. Obviously, in order to avoid the unreasonable cases, we re-quire p > w > c. The rst term of (1) is the retailers sales revenue,the second term is the order cost, and the third term is the salvagevalue. Hence, with the wholesale price mechanism, the retailersproblem is to maximize the expected prot function (1) with re-spect to Qwr, which yields the following proposition.

Proposition 1. With the wholesale price mechanism, the retailer willearn an expected prot of:

pwr pwQwr p vZ Qwr0

Fxdx; 2

and the manufacturer will earn an expected prot of pwm w cQwr, where Qwr F1 pwpv

.

If the manufacturer does not use the make-to-order produc-tion policy but plans its production for its own interest, then itsoptimal production quantity is to maximize the following expectedprot function with respect to the production quantity Qwm:

EPwmQwm EwminfQwm; xg cQwm vmaxfQwm x;0g:3

Similar to the proof of Proposition 1, we derive the manufac-turers optimal production quantity as Qwm F1 wcwv

. All these

results obtained with the wholesale price mechanism will be takenas benchmarks against which we will compare the option mecha-nism developed in this paper.

In what follows, we consider the option model. With the optioncontract mechanism, say (o, e), the retailers expected prot func-tion is given by

EPorQor E p eminfQor; xg oQor ; 4where Qor is the retailers reserved quantity under the option con-tract mechanism. The rst term of (4) is the retailers sales protand the second term is the allowance payout for the reserved capac-ity. Similarly, when the manufacturer plans its production quantityfor its own interest instead of using the make-to-order productionpolicy, its expected prot function is given by

EPomQom E oQom eminfQom; xg vmaxfQom x;0g cQom :5

Hence, with the option mechanism, the retailers problem is tomaximize the expected prot function (4) and the manufacturersis to maximize (5), which lead to the following proposition.

Proposition 2

(i) Given (o, e), the manufacturers optimal production quantity is

Qom F1 eocev

and the retailers optimal reserved quantity is

Qor F1 poepe

.

(ii) Given o + e, EPor(Qor) is decreasing in o or increasing in e,whereas EPom(Qom) is increasing in o or decreasing in e.

(iii) Only if the option contract satises e p pvcv o o < c vwill the retailers optimal reserved quantity be just consistentwith the manufacturers optimal production quantity.

tional Research 207 (2010) 668675By the option mechanism, we know that o + e is the unit pricefor the quantity of the product purchased by the retailer. FromProposition 2, we see that, given o + e, the higher the retailer is

highera low

tem-wide optimal prot. Based on this idea, we propose a notioncalledtract srequir

In wtakingsee frparamas tha

perathe converse is preferred by the manufacturer. These indeed coin-cide with the intuition from practice. Furthermore, in terms of theoption contract that makes the retailers optimal reserved quantityjust consistent with the manufacturers optimal production quan-tity, the exercise price is a negative linear function of the optionprice.

Comparing with the wholesale price mechanism, we have thefollowing proposition.

Proposition 3

(i) Qwr < Qor iff o cvwe

wv .

Proposition 3 shows that compared with the wholesale pricemechanism, the option contract mechanism with an option pricethat is lower than pewvpv will induce the retailer to reserve more,whereas an option price that is higher than cvwewv will push themanufacturer to produce more.

5. Supply chain coordination with option contracts

To derive the system-wide optimal expected prot for the sup-ply chain, we take the supply chain as a centralized entity. LetEPs(Qs) denote the expected prot of the centralized entity whenthe production quantity is Qs. With some algebra, we have:

EPsQs E pminfQs; xg vmaxfQs x; 0g cQs

p cQs p vZ Qs0

Fxdx: 6

Similar to the proof of Proposition 2, we can show the strict con-cavity of EPs(Qs) with respect to Qs. Therefore, by the rst-orderoptimality condition, we nd that the rst-best production quan-tity for the supply chain is Qs F1 pcpv

, and accordingly the sys-

tem-wide optimal expected prot, denoted by pc, is

pc EPsQs p cQs p vZ Qs0

Fxdx: 7

It can be shown that Qs > Qwr; Qs > Qwm (see the Supplemen-tary Appendix). We discuss below supply chain coordination withthe option contract. Obviously, an option contract that providesthe retailer with an incentive to reserve as much asQs F1 pcpv

will make the supply chain system achieve the

system-wide optimal expected prot. This perspective yields thefollowing proposition.

Proposition 4

(i) The system-wide optimal expected prot of the supply chaincan be achieved under any option contract (o, e) in the followingset M:

M o;e : o kcv;e 1 kp kv ; where k 2 0;1f g:8

Compared with the wholesale price mechanism, any option contract inM can increase the expected prot of the supply chain system byexercise price later. In addition, the retailer prefers to payer option price and a higher exercise price later. However,willing to pay for the option price, the higher is the optimal pro-duction quantity of the manufacturer. Besides, the retailer is proneto reserve more if it is allowed to pay a lower option price and aY. Zhao et al. / European Journal of ODp p cQs Qwr p vZ QsQwr

Fxdx: 9strictly increases in k, only the option contracts in M with k satis-fying 1 > k > pwrpc will make the retailer strictly better off than thatunder the wholesale price mechanism. Any other option contractinMwill make the retailer worse off and subsequently is unaccept-

pwrable twith afacturthe core of a contract set and dene it as a subset of the con-et that consists of all the contracts fullling the coordinationement.hat follows, we derive the core of the option contract setM,the wholesale price mechanism as the benchmark. First, weom (10) that, with the option contract associated with theeter k pwrpc in the set M, the retailer will just earn as mucht it will under the wholesale price mechanism. Since kpc(ii) Under any option contract (o, e) inM, we have Qor Qom Qs.Furthermore, with the option contract associated with k, themaximum expected prot received by the retailer, denoted bypor(k), is given by

pork kpc; 10and the maximum expected prot received by the manufacturer,denoted by pom(k), is given by

pomk 1 kpc; 11where pc, given by (7), is the system-wide optimal expectedprot of the supply chain.

The relationship between o and e can be derived from (8) as

e p p vc v o; 12

which has the following implications: (i) the exercise price nega-tively correlates with the option price, and (ii) an increase in the op-tion price by an unit will induce a decrease in the exercise price bypvcv > 1. This coincides with the intuition in practice that the earlierone pays for the product, the lower the price is. From Qor Qom Qs, we know that, for any option contract inM, the manufac-turers production quantity in our model is just the optimal one,even under the make-to-order production policy, which greatlyenhances the robustness of the option contract in implementation.In addition, Proposition 4 implies that the supply chain system prof-it can be allocated arbitrarily by the option contracts inM using dif-ferent ks 2[0,1). In fact, we see from (10) that k is the fractionalsplit by the retailer of the optimal joint prot of the supply chainsystem with the option contract associated with k in M. An optioncontract inM with a larger k will make the retailer share more prof-it, and the converse holds for the manufacturer. In addition, it isworth noting that cvwewv < o 0), and a risk-neu-tral manufacturer with the utility function Um(Dpm) = Dpm. It isclear that Ur(Dpr) is strictly concave, increasing, twice differential,and Ur(0) = 0. By Proposition 7, we know that Nashs bargainingmodel predicts that the risk-averse retailer will obtain a smallershare of the extra prot than the risk-neutral manufacturer, andthe option contract that the retailer and manufacturer both agreeto implement is given by

o; e ~kc v; 1 ~kp ~kv; 23where ~k, from the proof of Proposition 7, is the unique solution of(24):

expakpc kap2c expapwr apcpc pwm 1 expapwr 0:24

6.2. The case of Eliashbergs model involving negotiating power

It is worth pointing out that Nashs bargaining model does nottake individual members negotiating powers into account whilepredicting the outcome, which is a severe deciency of the modelbecause the selection of a contract clearly depends on supply chainmembers negotiating powers. In order to overcome this de-ciency, an alternative way is to apply the approach introduced byEliashberg (1986). Eliashbergs model predicts an option contractthat maximizes the group utility function reecting the joint pref-erences of the supply chain members. By Eliashbergs model, wecan incorporate supply chain members negotiating powers intothe ultimate implementation outcome, with the use of aggregation

674 Y. Zhao et al. / European Journal of Op(i) If r2r1 6a

b expaDp, then the optimal solution k* = kmax, the corre-

sponding option contract is given by (o; e kmaxc v,(1 kmax)p + kmaxv), and the allocation of the extra protDp between the retailer and the manufacturer is given by(Dpr(kmax), Dpm(kmax)) = (Dp,0).

(ii) If r2r1 Pa expbDp

b , then k* = kmin, the corresponding option con-

tract is given by (o; e kmin (c v), (1 kmin)p + kminv), andthe allocation of the extra prot Dp is given by (Dpr(kmin),Dpm(kmin)) = (0,Dp).

(iii) If r2r1 2 ab expaDp ;a expbDp

b

, then

k aa b kmin

ba b kmax

ln br2ar1pca b ; 26

the corresponding option contract is given by (o; e k (c v),(1 k*)p + k*v), where k* is given by (26), and the allocation of theextra prot Dp is given by

Dprk ba bDpln br2ar1a b ; Dpmk

aa bDp

ln br2ar1a b : 27

Hence, based on Eliashbergs model, for such a supply chain, theretailer will obtain a share bab from the extra prot Dp and themanufacturer will obtain a share aab. A compensation fee between

the retailer and the manufacturer isln

br2ar1

ab , which represents a fee

paid by the retailer to the manufacturer if r2r1 Pab; otherwise, from

the manufacturer to the retailer. Clearly, we see from (27) that theproportions shared by the retailer and the manufacturer do not de-pend on their relative power measurements r1 and r2 but only de-pend on their risk aversion measurements a and b. The more risk-averse a member is, the less share it will obtain from the extraprot Dp. As for the compensation fee, we observe that whenr2r1P ab, an increase in r2 or a decrease in r1 means an increasing

compensation fee from the retailer to the manufacturer. Particu-larly, when r2 increases or r1 decreases to the point wherer2r1P a expbDpb , the manufacturer will receive a compensation fee

from the retailer that is just equal to babDp. In other words, withan increase in the relative power of the manufacturer with respectto the retailer, it will receive a higher compensation fee from theretailer. When the manufacturers relative power is high enoughwith respect to the retailer (e.g., r2r1 >

a expbDpb ), the result of coordi-

nation will be that the manufacturer captures all the extra prot,whereas the retailer receives nothing from coordination. A similaranalysis is applicable to the case where r2r1