Research ArticleCoordinating Three-Level Supply Chain by Revenue-SharingContract with Sales Effort Dependent Demand

Qinghua Pang Yuer Chen and Yulu Hu

Business Management School HoHai University Changzhou 213022 China

Correspondence should be addressed to Qinghua Pang pangqh77126com

Received 19 March 2014 Revised 4 July 2014 Accepted 15 July 2014 Published 5 August 2014

Academic Editor Xiaolin Xu

Copyright copy 2014 Qinghua Pang et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

Revenue-sharing contract is a kind of mechanism to improve performance or to achieve perfect coordination of supply chainConsidering a three-level supply chain consisting of a manufacturer a distributor and a retailer who faces a stochastic and saleseffort dependent demand the paper analyzes the impact of sales effort on supply chain coordination and expounds the reasonswhy traditional revenue-sharing contract cannot coordinate supply chain in this condition Given three cases only the retailerbears the sales effort cost only the manufacturer bears the sales effort cost and the retailer bears the sales effort cost with themanufacturer the paper proposes an improved revenue-sharing contract based on quantity discount policy to coordinate the supplychain It illustrates that improved revenue sharing contract can coordinate supply chain by implementing it in one transactionor two transactions of three-level supply chain The model of improved revenue-sharing contract is optimized respectively byaddition form and multiplication form with sales effort dependent demand Formulas are given to determine the optimal contractparameters Finally numerical experiments are given to test the accuracy of the model of improved revenue-sharing contract

1 Introduction

Supply chain is made up of several different decision makerswho pursue different objectives andmay conflict among eachother Then the contractual supply chain actually faces aldquodouble-marginal utilityrdquo each individual of the supply chainmakes the decision respectively in order to achieve optimumprofits But the decision is not identical with the one in thecoordinated supply chain So we need a kind of coordinativemechanism or contract to make the whole profits of supplychain achieve the optimum and guarantee each company getsmore profits than it does in the noncontractual supply chain

Revenue-sharing contract [1] is a kind of contract tocoordinate supply chain The contract can be described bytwo parameters (119908 120593) the supplier charges the retailers a unitwholesale price 119908 lower than the unit marginal cost 119888 inexchange for a percentage (1 minus 120593) of the retailerrsquos revenueThe position119908 lt 119888 guarantees channel coordination whereas120593 determines the distribution of total profits between thesupplier and the retailer In particular 120593 is the supply chainprofit quota gained by the retailer

The revenue-sharing contract initially appeared in theindustry of video-rent and was later successfully extendedto other industries such as franchise model commonly usedin China From an economic point of view of revenue-sharing contract applications in the video rental industry(1998-1999) Mortimer [2] carried out an empirical study andfound that the contract can make different enterprises in thesupply chain increase their profits by 3ndash6 Due to thesuccessful applications of revenue-sharing contract in supplychain practice the study on revenue-sharing contract showsgrowing trends and the literatures [3ndash11] are representativeBased on the literature of Cachon and Lariviere [1] Gian-noccaro and Pontrandolfo [12] proposed a model based onrevenue-sharing contract to coordinate a three-level supplychain On the basis of Giannoccaro and Pontrandolfo [12]the model was further improved [13ndash15]

By reviewing the extensive literature available we can seethatmost of the literatures do not consider the effort activitiesof supply chainmembers that affectmarket demand And it isknown that there exists a wide gap between market demandand the actual situation In reality the effort activity of

Hindawi Publishing CorporationDiscrete Dynamics in Nature and SocietyVolume 2014 Article ID 561081 10 pageshttpdxdoiorg1011552014561081

2 Discrete Dynamics in Nature and Society

RetailerManufacturer Distributor Uncertaindemand

cm

cd

cr

pwdw

m

Qs

Qs

Qd

Qd

Min (x Q)

x

Figure 1 Three-level supply chain

supply chain members as important factors affecting marketdemand [16] is beneficial to the entire supply chain systemIt can increase the market demand to some extent [17ndash19]such as increasing investment in advertising hiring moresales staff to promote products and considering customersrsquopersonality in productmanufacturing However it needs costto do these effort activitiesTherefore if only one supply chainmember bears the cost of effort activities he will choose theeffort level which is best beneficial to him and often thischoice can not make the supply chain coordinate [20]

Cachon [21] and Cachon and Lariviere [1] pointed outthat the revenue-sharing contract can not coordinate supplychain with retailerrsquos sales effort dependent market demandthe kind of situation many scholars have studied He etal [22 23] proposed a mixed contract based on bothrevenue-sharing contract and rebate and penalty contractto coordinate supply chain Hu and Wang [24] proposed anevolution contract based on revenue-sharing contract andstudied this problem in the framework of principal-agentQu and Guo [11] proposed an improved revenue-sharingcontract considering a hybrid model of supply chain to makethe supply chain achieve the coordination of order quantityand effort J Chen and J F Chen [25] studied the applicationof revenue-sharing contract in the virtual enterprise

These contradictions may be resolved when all supplychain members share the effort cost For example Wang andGerchak [26] proposed that a supplier can compensate forthe retailerrsquos the effort cost by way of stock subsides whileconsidering shelf space as an effort variable Krishnan etal [27] proposed a contract to coordinate supply chain bysharing effort cost He [28] and He et al [22] proposed aflexible quantity contract by sharing effort cost Xu et al[20] suggested a return-back contract by sharing effort costHowever in many cases due to the fact that the effort of eachsupply chainmember cannot be observed all kinds of hiddeneffort moral hazard will make the policy of sharing effort costfail to coordinate supply chainTherefore it is not a desirableway to coordinate supply chain by sharing effort cost amongsupply chain members

It should be noticed that the research object of litera-tures mentioned above is a two-level supply chain Throughliteratures searching we have not found the literatures thattake effort into account in three-level supply chainThereforethe paper tries to further study in the following aspectsFirst the research object is extended to three-level supplychain with sales effort dependent demand which meets theactual circs better Second by analyzing the behavior of thedecentralized supply chain under the traditional revenue-sharing contract with sales effort dependent demand we

find that the contract fails to coordinate the supply chainThird by supposing only the retailer or the manufacturerbears the effort cost we propose an improved revenue-sharing contract based on quantity discount policy which cancoordinate the three-level supply chain Because there are twodifferent transaction phases in the three-level supply chaintwo different conditions are discussed implementing theimproved revenue-sharing contract in one transaction phaseand in two transaction phases Further by supposing thedemand and the effort satisfy addition form ormultiplicationform we characterize the optimal solution to the three-level supply chain with two decision variables (sales effortand inventory quantity) and numerical examples are givento illustrate the model and the solution process Finally byconsidering the retailer bears the effort cost together with themanufacturer an improved revenue-sharing contract basedon quantity discount policy is also discussed in this paperwhich can coordinate the three-level supply

The rest of this paper is organized as follows Section 2introduces model assumptions and notations Section 3examines the traditional revenue-sharing contract In Sec-tion 4 we analyze three conditions only the retailer bearsthe sales effort only the manufacturer bears the sales effortand the retailer bears the sales effort with the manufacturerSection 5 provides concluding remarks and describes futureresearch

2 Model Descriptions

The supply chain studied in this paper is made up of onemanufacturer (119898) one distributor (119889) and one retailer (119903)see Figure 1 Upstream member provides a single product todownstream member and the demand is stochastic Beforethe sale season both the distributor and the retailer have onlyone chance to buy products Allmembers are risk neutral andinformation is symmetric among them

Suppose 119901 is sales price of unit product 119888119894is supply chain

memberrsquosmarginal unit costs (119894 = 119898 119889 119903) and 119888 = 119888119903+119888119889+119888119898

V is salvage value for unsold unit product (V lt 119888) 119876 is orderquantity 119908119895

119894is the wholesale price that upstream member

charges downstream member in 119895 condition (119895 = 119888 119889 119894 =

119898 119889) 119890 is the retailerrsquos effort and 119890 ge 0 119892(119890) is the retailerrsquoseffort costwhenhis effort is 119890 supposing119892(0) = 0 and1198921015840(119890) gt0 and 11989210158401015840(119890) gt 0when 119890 gt 0 119909 is the stochastic demand whenretailerrsquos effort is 119890 with probability density function 119891(119909 |

119890) and differentiable cumulated distribution function 119865(119909 |

119890) and 119865(119909 | 119890) is continuously differentiable because thedemand is the increasing function of effort 120597119865(119909 | 119890)120597119890 lt

0 119876 is the order quantity when retailerrsquos effort is 119890 the

Discrete Dynamics in Nature and Society 3

expectation sale quantity is 119878(119876 119890) = 119864min(119876 119909) = 119876 minus

int119876

0119865(119909 | 119890)119889119909 and 120597119878(119876 119890)120597119890 gt 0 the expectation unsold

quantity is 119868(119876 119890) and 119868(119876 119890) = 119876 minus 119878(119876 119890) In particular itis reasonable that 119908119889

119889gt 119908119889119898+ 119888119889 V lt 119888 lt 119901 119908119889

119898+ 119888119903lt 119901

3 Analysis of Revenue-Sharing Contract

Supply chain contracts often apply incentive measures toadjust the relationship among the members to coordinatesupply chain and to make the entire profit of decentralizedsupply chain equal to that of centralized supply chain asmuchas possible The goal of the analysis is to design the revenue-sharing contract so as to achieve channel coordination(maximum profit for the whole supply chain) Besides itaims at analyzing whether and how the contract parametercan be modified so as to more evenly share the profitalong the chain (win-win condition for the chain partners)guaranteeing channel coordination So the maximum profitof centralized supply chain should first be considered asthe goal of revenue-sharing contract to coordinate supplychain The profit function of centralized supply chain can bedescribed as

Π119905 (119876 119890) = 119901119878 (119876 119890) + V119868 (119876 119890) minus 119888119876 minus 119892 (119890)

= (119901 minus V) 119878 (119876 119890) minus (119888 minus V) 119876 minus 119892 (119890) (1)

Supposing 119890lowast is the retailerrsquos optimal effort when givenorder quantity 119876 then 119890lowast should satisfy

120597Π119905(119876 119890lowast)

120597119890= (119901 minus V)

120597119878 (119876 119890lowast)

120597119890minus 1198921015840(119890lowast) = 0 (2)

Supposing 119876lowast is the retailerrsquos optimal order quantitywhen given effort 119890 then 119876lowast should satisfy 120597Π

119905(119876lowast 119890)120597119876 =

(119901minus V)(120597119878(119876lowast 119890)120597119876)minus (119888 minus V) = 0 namely119876lowast should satisfy

119865 (119876lowast| 119890) =

119901 minus 119888

119901 minus V (3)

Therefore if revenue-sharing contract can coordinatesupply chain then formulae (2) and (3) are the necessaryconditions to achieve supply chain coordination

Giannoccaro and Pontrandolfo [12] proposed a modelof revenue-sharing contract to coordinate three-level supplychain without considering supply chain memberrsquos stockoutloss and salvage value of unsold unit product In this paper weanalyze whether the revenue-sharing contract can coordinatethree-level supply chain or not with taking salvage value ofunsold unit product and the retailerrsquos effort into accountSupposing the retailer would keep a quota 120601

2of his revenue

giving the rest (1 minus 1206012) to the distributor and this would be

balanced by a lower price119908119888119889 Similarly the distributor would

keep a quota 1206011of his revenue giving the rest (1 minus 120601

1) to the

manufacturer and thiswould be balanced by a lower price119908119888119898

Also assume only the retailer bears the effort cost Then theexpected profit function of the retailer can be described as

Π119888

119903(119876 119890) = 1206012 [119901119878 (119876 119890) + V119868 (119876 119890)] minus 119888119903119876 minus 119908

119888

119889119876 minus 119892 (119890)

= 1206012(119901 minus V) 119878 (119876 119890) minus (119888119903 + 119908

119888

119889minus 1206012V) 119876 minus 119892 (119890)

(4)

From (4) we get 120597Π119888119903(119876 119890)120597119890 = 120601

2(119901 minus V)(120597119878(119876 119890)120597119890) minus

1198921015840(119890) Compared with (2) it can be seen that the retailerrsquoseffort 119890 in this condition is smaller than the optimaleffort of centralized supply chain namely 120597Π119888

119903(119876 119890)120597119890 lt

120597Π119905(119876 119890)120597119890 That is to say in this condition the revenue-

sharing contract cannot coordinate supply chainThe reasonslie in that the retailer bears all the effort cost of supply chainbut he only get partial profit of the whole supply chain

Similarly if only the distributor or themanufacturer bearsthe sales effort cost the revenue-sharing contract cannotcoordinate the supply chain Then we come to a conclusionthe traditional revenue-sharing contract cannot coordinatethree-level supply chain with sales effort dependent demand

In this paper an improved revenue-sharing contractbased on quantity discount policy is proposed to coordinatethree-level supply chain when demand depends on effort

4 The Improved Revenue-Sharing ContractBased on Quantity Discount Policy

From the discussion above it can be seen that the optimaleffort of the supply chain member under revenue-sharingcontract is not equal to that of the centralized supply chainThe reasons lie in that the supply chain member only getspartial gains but bears all the effort cost 119892(119890) of the wholesupply chain To solve this problem we canmake his gains beequal to a fixed ratio of the gains of the whole supply chainHere we propose an improved revenue-sharing contractbased on quantity discount policy to coordinate supply chainBecause there are two different transaction phases upstreamprocess and downstream process it needs to consider howto implement this improved contract in three-level supplychain

41 Only Retailer Bearing the Effort Cost

411 Only Implement between Retailer and Distributor Thisproblem can be described as follows we implement therevenue-sharing contract based on quantity policy betweenthe retailer and the distributor and the traditional revenue-sharing contract between the distributor and the manufac-turer

Suppose the distributor provides the retailer the revenue-sharing contract (120601

2 119908119888119889(119876)) and

119908119888

119889(119876) = 1206012 (119888 minus V) minus (1 minus 1206012)

119892 (119890lowast)

119876minus 119888119903+ 1206012V (5)

Here 119890lowast is the optimal effort of the supply chainIn this condition the profit function of retailer can be

described asΠ119888

119903(119876 119890)

= 1206012[119901119878 (119876 119890) + V119868 (119876 119890)] minus 119888119903119876 minus 119908

119888

119889(119876)119876 minus 119892 (119890)

= 1206012(119901 minus V) 119878 (119876 119890)minus1206012 (119888 minus V) 119876 + (1 minus 120601

2) 119892 (119890lowast) minus 119892 (119890)

(6)

According to the paper of Giannoccaro and Pontrandolfo[12] without considering demand dependent effort the ratio

4 Discrete Dynamics in Nature and Society

that the retailerrsquos profit shares the profit of the supply chainis 1206012 Here supposing when only the retail bears the effort

cost the ratio that his profit shares the profit of the wholesupply chain is also 120601

2 It is easily known that the effort of

the retail would be equal to the optimal effort of the supplychain namely

Π119888

119903(119876 119890lowast)

= 1206012[119901119878 (119876 119890

lowast) + V119868 (119876 119890lowast)] minus 119888

119903119876 minus 119908

119888

119889(119876)119876 minus 119892 (119890

lowast)

= 1206012(119901 minus V) 119878 (119876 119890lowast) minus 120601

2 (119888 minus V) 119876 minus 1206012119892 (119890lowast)

= 1206012Π119905(119876 119890lowast)

(7)

From (7) it can be seen that the optimal order quantitywould be equal to that of the supply chain Then the profitfunction of the distributor can be described as

Π119888

119889(119876 119890) = 1206011 [(1 minus 1206012) (119901119878 (119876 119890) + V119868 (119876 119890)) + 119908119888

119889(119876)119876]

minus 119888119889119876 minus 119908

119888

119898119876

= 1206011(1 minus 120601

2) (119901 minus V) 119878 (119876 119890)

minus [119888119889+ 119908119888

119898minus 1206011(1 minus 120601

2) V minus 120601

1119908119888

119889(119876)]119876

(8)

Because the distributor is in the middle of the supplychain to maximize his profit he will make his optimal orderquantity equal to the optimal order quantity of retailer inperfect information symmetry condition namely

120597Π119888119889(119876lowast 119890)

120597119876=120597Π119888

119903(119876lowast 119890)

120597119876=120597Π119905(119876lowast 119890)

120597119876 (9)

From (9) it can get

119908119888

119898= (1 + 120601

11206012) 119888 minus 120601

1119888119903minus 119888119889minus [1 minus 120601

1(1 minus 120601

2)] V (10)

Take (5) and (10) into the profit function of the supplychain members and we can get

Π119888

119903(119876 119890) = 1206012Π119905 (119876 119890)

Π119888

119889(119876 119890) = 1206011 (1 minus 1206012)Π119905 (119876 119890)

Π119888

119898(119876 119890) = (1 minus 1206011) (1 minus 1206012)Π119905 (119876 119890)

(11)

From (11) it can be seen that the profit functions ofthe supply chain members are all affine functions of thewhole supply chainrsquos profit function So in this conditionthe revenue-sharing contract can coordinate the three-levelsupply chain

Therefore if we implement the revenue-sharing contractbased on quantity discount policy between the retailer andthe distributor and the traditional revenue-sharing contractbetween the distributor and the manufacturer and the con-tract parameters which satisfy (5) and (10) the improvedrevenue-sharing can coordinate the three-level supply chain

412 Implement among Supply Chain Members This prob-lem can be described as follows we implement both therevenue-sharing contract based on quantity discount policybetween the retailer and the distributor and between thedistributor and the manufacturer

Suppose the distributor provides the retailer the revenue-sharing contract (120601

2 119908119888

119889(119876)) and the manufacturer provides

the distributor the revenue-sharing contract (1206011 119908119888

119898(119876))

Here 119890lowast is the optimal effort of the supply chain 119908119888119889(119876) is

equal to (5) and

119908119888

119898(119876) = 1206011 (119888 minus V) minus 1206011 (1 minus 1206012)

119892 (119890lowast)

119876minus 1206011119888119903+ 1206011V minus 119888119889

(12)

Here we can also suppose that when only the retail bearsthe effort cost the ratio that his profit shares the profit ofthe whole supply chain is also 120601

2 It is easy to know that the

optimal effort and the optimal order quantity of the retailerwould be equal to those of the supply chain

In this condition the profit function of the distributor canbe described asΠ119888

119889(119876 119890) = 1206011 [(1 minus 1206012) (119901119878 (119876 119890) + V119868 (119876 119890)) + 119908119888

119889(119876)119876]

minus 119888119889119876 minus 119908

119888

119898(119876)119876

= 1206011(1 minus 120601

2) (119901 minus V) 119878 (119876 119890)

minus [119888119889+ 119908119888

119898(119876) minus 1206011 (1 minus 1206012) V minus 1206011119908

119888

119889] 119876

= 1206011(1 minus 120601

2) (119901 minus V) 119878 (119876 119890)minus1206011 (1 minus 1206012) (119888 minus V) 119876

(13)

Similarly the optimal order quantity of the distributorshould satisfy

120597Π119888119889(119876lowast 119890)

120597119876=120597Π119888

119903(119876lowast 119890)

120597119876=120597Π119905(119876lowast 119890)

120597119876 (14)

Take (5) and (12) into the profit function of the supplychain member and we can get

Π119888

119903(119876 119890) = 120601

2Π119905 (119876 119890)

Π119888

119889(119876 119890) = 120601

1(1 minus 120601

2) (Π119905 (119876 119890) + 119892 (119890))

Π119888

119898(119876 119890) = (1 minus 120601

1) (1 minus 120601

2)Π119905 (119876 119890) minus 1206011 (1 minus 1206012) 119892 (119890)

(15)

From (15) it can be seen that the profit functions ofthe supply chain members are all affine functions of thewhole supply chainrsquos profit function So in this conditionthe revenue-sharing contract can coordinate the three-levelsupply chain Comparing (11) with (15) it can be seen thatthe retailerrsquos profit keeps unchanged the distributorrsquos profitincreases 120601

1(1 minus 120601

2)119892(119890) while the manufacturerrsquos profit

decreases 1206011(1 minus 120601

2)119892(119890)

Therefore if we implement the revenue-sharing contractbased on quantity discount policy both between the retailerand the distributor and between the distributor and themanufacturer and the contract parameters which satisfy(5) and (12) the improved revenue-sharing contract cancoordinate the three-level supply chain

Discrete Dynamics in Nature and Society 5

413 Model Optimization From the discussion mentionedabove we can come to a conclusion that the traditionalrevenue-sharing contract cannot coordinate three-level sup-ply chain with sales effort dependent demand Howeverthe improved revenue-sharing contract based on quantitydiscount policy can coordinate three-level supply chain Andin this condition the optimal order quantity and optimaleffort of retailer are equal to those of the centralized supplychainThen it needs to determine the optimal order quantityand the optimal effort to maximize the profit of supply chainnamely maxΠ

119905(119876 119890) The following gives the method to

determine the optimal order quantity 119876lowast and the optimaleffort 119890lowast

Suppose the market demand 119883(119890 120585) is the function ofeffort 119890 and random factor 120585 and 120585 is independent of 119890 119891(120585)and 119865(120585) are respectively probability density function anddifferentiable cumulated distribution function of 120585 Usuallywe can use two forms to describe how the effort affectsdemand addition form and multiplication form [29] In thispaper suppose 119883(119890 120585) = 119910(119890) + 120585 Because the influence ofeffort on marker demand is diminishing marginal utility wecan suppose 119910(119890) is the monotone increasing and concavefunction of effort 119890 namely 1199101015840(119890) gt 0 11991010158401015840(119890) le 0

When the market demand satisfies119883(119890 120585) = 119910(119890) + 120585 wecan get

119878 (119876 119890) = 119876 minus int119876

0

119865 (119909 | 119890) 119889119909 = 119876 minus int119876

119910(119890)

119865 (119909 minus 119910 (119890)) 119889119909

= 119876 minus int119876minus119910(119890)

0

119865 (119905) 119889119905

(16)

So the profit function of supply chain can be described as

Π119905 (119876 119890) = (119901 minus V) 119878 (119876 119890) minus (119888 minus V) 119876 minus 119892 (119890)

= (119901 minus 119888)119876 minus (119901 minus V) int119876minus119910(119890)

0

119865 (119905) 119889119905 minus 119892 (119890)

(17)

Given effort 119890 it can be seen that Π119905(119876 119890) is the concave

function of order quantity 119876 So the optimal order quantity119876lowast should satisfy

120597Π119905(119876lowast 119890)

120597119876= (119901 minus 119888) minus (119901 minus V) 119865 (119876lowast minus 119910 (119890)) = 0 (18)

From (18) we can get

119876lowast= 119876lowast(119890) = 119910 (119890) + 119865

minus1(119901 minus 119888

119901 minus V) (19)

Then the profit function of supply chain can be describedasΠ119905(119876lowast 119890)

= (119901 minus 119888)119876lowastminus (119901 minus V) int

119876lowast

minus119910(119890)

0

119865 (119905) 119889119905 minus 119892 (119890)

= (119901 minus 119888) 119910 (119890) + (119901 minus V) int119865minus1

((119901minus119888)(119901minusV))

0

119905119891 (119905) 119889119905 minus 119892 (119890)

(20)

From (20) it can be seen that there is only one variable119890 in Π

119905(119876lowast 119890) Therefore the next work is to determine the

optimal effort 119890lowast to maximize Π119905(119876lowast 119890) From (20) it can

get 120597Π119905(119876lowast 119890)120597119890 = (119901 minus 119888)1199101015840(119890) minus 1198921015840(119890) 1205972Π

119905(119876lowast 119890)1205971198902 =

(119901 minus 119888)11991010158401015840(119890) minus 11989210158401015840(119890) Because 11989210158401015840(119890) gt 0 and 11991010158401015840(119890) le

0 1205972Π119905(119876lowast 119890)1205971198902 lt 0 namely Π

119905(119876lowast 119890) is the concave

function of effort 119890 That is to say there is an optimal effort119890lowast which can make Π

119905(119876lowast 119890) achieve the maximum By

120597Π119905(119876lowast 119890)120597119890 = 0 we get

(119901 minus 119888) 1199101015840(119890lowast) minus 1198921015840(119890lowast) = 0 (21)

Then the optimal effort 119890lowast should satisfy (21) Equation(21) is the first-order differential equation of the optimal effort119890lowast By solving (21) we can get the expression of 119890lowast

414 An Example Suppose a supply chain is made up ofa retailer a distributor and a manufacturer and the supplychain parameters are 119888

119903= 1 119888119889= 2 119888119898= 5 V = 3 119908119889

119889= 18

119908119889119898= 10 119901 = 35 120601

1= 055 and 120601

2= 055

Here assume the relationship between demand and effortsatisfies addition form and 119892(119890) = 119890

22 119910(119890) = 119890 120585 satisfiesuniform distribution in [50 100] It is easy to get119891(120585) = 150119865(120585) = (120585 minus 50)50 and 119865minus1(120585) = 50 + 50120585

(1) The revenue-sharing contract based on quantity dis-count policy

From the discussion above we know that the revenue-sharing contract based on quantity discount policy cancoordinate the supply chain whether it is implemented in oneor two transactions According to the formulas mentionedabove it is easy to get the optimal effort the optimal orderquantity and the profit of the whole supply chain Here 119890lowast =27 119876lowast = 119 and Π

119905(119876lowast 119890lowast) = 22725

(2) The traditional revenue-sharing contract

Similarly under the traditional revenue-sharing contractthe optimal effort and order quantity of the retailer and theprofit of the whole supply chain are respectively 119890119903 = 1485119876119903 = 107 and Π

119905(119876119903 119890119903) = 22027

(3) The decentralized supply chain (the wholesale pricecontract)

Similarly under the traditional revenue-sharing contractthe optimal effort and order quantity of the retailer and theprofit of the whole supply chain are respectively 119890119889 = 16119876119889 = 91 and Π

119905(119876119889 119890119889) = 2137

Table 1 shows the parameters of the supply chain indifferent mode A means that we implement the revenue-sharing contract based on quantity discount policy onlybetween the retailer and the distributor B means that weimplement the revenue-sharing contract based on quantitydiscount policy in two transactions of the supply chain

Table 1 shows that the retailerrsquos effort in the revenue-sharing contract based on quantity discount policy is thehighest in the three modes which can make the supply chain

6 Discrete Dynamics in Nature and Society

Table 1 The parameters of supply chain in different modes with only retailer bearing effort cost

Revenue-sharing contract based onquantity discount policy Traditional

revenue-sharing contract Wholesale price contractA B

Optimal effort 119890 27 27 1485 16Optimal order quantity 119876 119 119 107 91Profit of the whole supply chain Π

119905 22725 22725 22027 2137

maximize its profit the retailerrsquos effort in traditional revenue-sharing contract is the lowest in the threemodes but his orderquantity is still higher than that in wholesale price contractdue to the incentive of the revenue-sharing contract So theprofit of the supply chain in the traditional revenue-sharingcontract is still higher than that in wholesale price contractSo it can be seen that the traditional revenue-sharing contractis better than the wholesale price contract with sales effortdependent demand

42 Only Manufacturer Bearing the Effort Cost

421 Only Implement between Manufacturer and DistributorThis problem can be described as follows we implement therevenue-sharing contract based on quantity policy betweenthe manufacturer and the distributor and the traditionalrevenue-sharing contract between the distributor and theretailer

Supposing the manufacturer provides the distributor therevenue-sharing contract (120601

1 119908119888119898(119876)) and

119908119888

119898(119876) = minus (1 minus 1206011) 119888 + [1 minus (1 minus 1206011) (1 minus 1206012)]

119892 (119890lowast)

119876

+ (1 minus 1206011) 119888119903+ 119888119898

(22)

Here 119890lowast is the optimal effort of the supply chainTaking (22) into the manufacturerrsquos profit function we

can get

Π119888

119898(119876 119890)

= (1 minus 1206011) [(1 minus 120601

2) (119901119878 (119876 119890) + V119868 (119876 119890)) + 119908119888

119889119876]

+ 119908119888

119898(119876)119876 minus 119888

119898119876 minus 119892 (119890)

= (1 minus 1206011) (1 minus 120601

2) (119901 minus V) 119878 (119876 119890)

+ [1 minus (1 minus 1206011) (1 minus 120601

2)] 119892 (119890

lowast)

minus (1 minus 1206011) [119888 minus 119888

119903minus (1 minus 120601

2) V minus 119908119888

119889] 119876 minus 119892 (119890)

(23)

According to the paper of Giannoccaro and Pontrandolfo[12] without considering demand dependent effort the ratiothat the manufacturerrsquos profit shares the profit of the supplychain is (1 minus 120601

1)(1 minus 120601

2) Here supposing when only the

manufacturer bears the effort cost the ratio that his profitshares the profit of the whole supply chain is also (1 minus 120601

1)(1 minus

1206012) It is easily known that the effort of the manufacturer

would be equal to the optimal effort of the supply chainnamely

Π119888

119898(119876 119890)

= (1 minus 1206011) [(1 minus 120601

2) (119901119878 (119876 119890

lowast) + V119868 (119876 119890lowast)) + 119908119888

119889119876]

+ 119908119888

119898(119876)119876 minus 119888

119898119876 minus 119892 (119890

lowast)

= (1 minus 1206011) (1 minus 120601

2)Π119905 (119876 119890)

(24)

From (24) we can get

119908119888

119889= 1206012119888 minus 119888119903 (25)

Taking (25) and (22) into the profit function of the supplychain members we can get

Π119888

119903(119876 119890) = 1206012 [Π119905 (119876 119890) + 119892 (119890)]

Π119888

119889(119876 119890) = 1206011 (1 minus 1206012)Π119905 (119876 119890) minus 1206012119892 (119890)

Π119888

119898(119876 119890) = (1 minus 1206011) (1 minus 1206012)Π119905

(26)

From (26) it can be seen that the profit functions ofthe supply chain members are all affine functions of thewhole supply chainrsquos profit function So in this conditionthe revenue-sharing contract can coordinate the three-levelsupply chain

Therefore if we implement the revenue-sharing contractbased on quantity discount policy between the retailer andthe distributor and the traditional revenue-sharing contractbetween the distributor and the manufacturer and the con-tract parameters which satisfy (22) and (25) the improvedrevenue-sharing can coordinate the three-level supply chain

Compared with the paper of Giannoccaro and Pontran-dolfo [12] it can be seen that the retailerrsquos profit increases1206012119892(119890lowast) while that of the distributor decreases 120601

2119892(119890lowast)

422 Implement among Supply Chain Members This prob-lem can be described as follows we implement both therevenue-sharing contract based on quantity discount policybetween the retailer and the distributor and between thedistributor and the manufacturer

Suppose the distributor provides the retailer the revenue-sharing contract (120601

2 119908119888119889(119876)) and the manufacturer provides

Discrete Dynamics in Nature and Society 7

the distributors the revenue-sharing contract (1206011 119908119888119898(119876))

Here 119890lowast is the optimal effort of the supply chain and

119908119888

119889(119876) = 120601

2119888 minus (1 minus 120601

2)119892 (119890lowast)

119876minus 119888119903

119908119888

119898(119876) = 120601

1119888 +

119892 (119890lowast)

119876minus 1206011119888119903minus 119888119889

(27)

Taking (27) into the profit function of the manufacturerwe can get

Π119888

119898(119876 119890)

= (1 minus 1206011) [(1 minus 120601

2) (119901119878 (119876 119890) + V119868 (119876 119890)) + 119908119888

119889(119876)119876]

+ 119908119888

119898(119876)119876 minus 119888

119898119876 minus 119892 (119890)

= (1 minus 1206011) (1 minus 120601

2) (119901 minus V) 119878 (119876 119890)

minus (1 minus 1206011) (1 minus 120601

2) (119888 minus V) 119876

+ [1 minus (1 minus 1206011) (1 minus 120601

2)] 119892 (119890

lowast) minus 119892 (119890)

(28)

Here we also suppose that the ratio that his profit sharesthe profit of the whole supply chain is also (1 minus 120601

1)(1 minus 120601

2)

It is easy to know that the optimal effort of the manufacturerwould be equal to that of the supply chain In this conditionthe profit function of the manufacturer can be described as

Π119888

119898(119876 119890) = (1 minus 1206011) (1 minus 1206012)Π119905 (119876 119890) (29)

Taking (27) into the profit function of the retailer we canget

Π119888

119903(119876 119890) = 1206012 [119901119878 (119876 119890

lowast) + V119868 (119876 119890lowast)] minus 119888

119903119876 minus 119908

119888

119889(119876)119876

= 1206012(119901 minus V) 119878 (119876 119890lowast)minus120601

2 (119888 minus V) 119876 + (1 minus 1206012) 119892 (119890)

(30)

From (30) it is easy to know that the optimal orderquantity of the retailer is equal to that of the supply chainSo (30) can be described as

Π119888

119903(119876 119890) = 1206012Π119905 (119876 119890) + 119892 (119890) (31)

Taking (27) into the profit function of the distributor itcan get

Π119888

119889(119876 119890)

= 1206011[(1 minus 120601

2) (119901119878 (119876 119890

lowast) + V119868 (119876 119890lowast)) + 119908119888

119889(119876)119876]

minus 119888119889119876 minus 119908

119888

119898(119876)119876 minus 119892 (119890

lowast)

= 1206011(1 minus 120601

2) (119901 minus V) 119878 (119876 119890lowast) minus 120601

1(1 minus 120601

2) (119888 minus V) 119876

minus (1 + 1206011(1 minus 120601

2)) 119892 (119890)

(32)

From (32) it can be seen that the optimal order quantityof the distributor is equal to that of the retailer so (32) canalso be described as

Π119888

119889(119876 119890) = 1206011 (1 minus 1206012)Π119905 (119876 119890) minus 119892 (119890) (33)

From (29) (31) and (33) it can be seen that the profitfunctions of the supply chainmembers are all affine functionsof the whole supply chainrsquos profit function So in this condi-tion the revenue-sharing contract can coordinate the three-level supply chain Therefore if we implement the revenue-sharing contract based on quantity discount policy bothbetween the retailer and the distributor and between thedistributor and themanufacturer and the contract parameterswhich satisfy (27) the improved revenue-sharing can coordi-nate the three-level supply chain

423 Model Optimization Similarly suppose the marketdemand 119883(119890 120585) is the function of effort 119890 and random factor120585 and 120585 is independent of 119890 119891(120585) and 119865(120585) are respectivelyprobability density function and differentiable cumulateddistribution function of 120585 Here suppose 119883(119890 120585) = 119910(119890) sdot 120585As the influence of effort on marker demand is diminishingmarginal utility we can suppose 119910(119890) is the monotoneincreasing and concave function of effort 119890 namely1199101015840(119890) gt 0and 11991010158401015840(119890) le 0

When the market demand satisfies 119883(119890 120585) = 119910(119890) sdot 120585 wecan get

119878 (119876 119890) = 119876 minus int119876

0

119865 (119909 | 119890) 119889119909 = 119876 minus int119876

0

119865(119909

119910 (119890)) 119889119909

= 119876 minus 119910 (119890) int119876119910(119890)

0

119865 (119905) 119889119905

(34)

So the profit function of supply chain can be described as

Π119905 (119876 119890) = (119901 minus V) 119878 (119876 119890) minus (119888 minus V) 119876 minus 119892 (119890)

= (119901 minus 119888)119876 minus (119901 minus V) 119910 (119890) int119876119910(119890)

0

119865 (119905) 119889119905 minus 119892 (119890)

(35)

Given effort 119890 it can be seen that Π119905(119876 119890) is the concave

function of order quantity 119876 So the optimal order quantity119876lowast should satisfy

120597Π119905(119876lowast 119890)

120597119876= (119901 minus 119888) minus (119901 minus V) 119865(

119876lowast

119910 (119890)) = 0 (36)

From (36) we can get

119876lowast= 119876lowast(119890) = 119910 (119890) 119865

minus1(119901 minus 119888

119901 minus V) (37)

Then the profit function of supply chain can be describedas

Π119905(119876lowast 119890) = (119901 minus 119888) 119910 (119890) 119865

minus1(119901 minus 119888

119901 minus V)

minus (119901 minus V) 119910 (119890) int119865minus1

((119901minus119888)(119901minusV))

0

119865 (119905) 119889119905 minus 119892 (119890)

= (119901 minus V) 119910 (119890) int119865minus1

((119901minus119888)(119901minusV))

0

119905119891 (119905) 119889119905 minus 119892 (119890)

(38)

8 Discrete Dynamics in Nature and Society

Table 2 The parameters of supply chain in different modes with only manufacturer bearing effort cost

Revenue-sharing contract based onquantity discount policy Traditional

revenue-sharing contract Wholesale price contractA B

Optimal effort 119890 1962 1962 397 379Optimal order quantity 119876 1674 1674 339 288Profit of the whole supply chain Π

119905 1924628 1924628 700833 654436

According to the supposition Π119905(119876lowast 119890) is the concave

function of effort 119890 So the optimal effort 119890lowast of the supplychain should satisfy

120597Π119905(119876lowast 119890lowast)

120597119890= (119901 minus V) 1199101015840 (119890lowast) int

119865minus1

((119901minus119888)(119901minusV))

0

119905119891 (119905) 119889119905

minus 1198921015840(119890lowast) = 0

(39)

From (39) it can easy to get that the optimal effort 119890lowast ofthe supply chain should satisfy

(119901 minus V) 1199101015840 (119890lowast) int119865minus1

((119901minus119888)(119901minusV))

0

119905119891 (119905) 119889119905 = 1198921015840(119890lowast) (40)

The method to solve the expression of 119890lowast in (40) is thesame as the method in (21)

424 An Example Suppose a supply chain is made up ofa retailer a distributor and a manufacturer and the supplychain parameters are 119888

119903= 1 119888119889= 2 119888119898= 5 V = 3 119908119889

119889= 18

119908119889119898= 10 119901 = 35 120601

1= 1206012= 055 and 119886 = 100

Here assume the relationship between demand and effortsatisfies multiplication form and 119892(119890) = 119886119890

22 119910(119890) = 119890120585 satisfies uniform distribution in [60 90] It is easy to get119891(120585) = 130 119865(120585) = (120585 minus 60)30 and 119865minus1(120585) = 60 + 30120585

(1) The revenue-sharing contract based on quantity dis-count policy

From the discussion above we know that the revenue-sharing contract based on quantity discount policy cancoordinate the supply chain whether it is implemented in oneor two transactions According to the formulas mentionedabove it is easy to get the optimal effort of the manufacturerthe optimal order quantity of the retailer and the profitof the whole supply chain Here 119890lowast = 1962 119876lowast =

1674 Π119905(119876lowast 119890lowast) = 1924628

(2) The traditional revenue-sharing contract

Similarly under the traditional revenue-sharing contractthe optimal effort of the manufacturer and order quantityof the retailer and the profit of the whole supply chain arerespectively 119890119898119888 = 397 119876119903119888 = 339 and Π

119905(119876119903119888 119890119898119888) =

700833

(3) The decentralized supply chain (the wholesale pricecontract)

Similarly under the traditional revenue-sharing contractthe optimal effort of the manufacturer and the order quantityof the retailer and the profit of the whole supply chain arerespectively 119890119898119889 = 379 119876119903119889 = 288 and Π

119905(119876119903119889 119890119889119898) =

654436Table 2 shows the parameters of the supply chain in

different modes A means that we implement the revenue-sharing contract based on quantity discount policy onlybetween the retailer and the distributor B means that weimplement the revenue-sharing contract based on quantitydiscount policy in two transactions of the supply chain

Table 2 shows that the manufacturerrsquos effort in therevenue-sharing contract based on quantity discount policyis the highest in the three modes which can make thesupply chain maximize its profit and the revenue-sharingcontract based on quantity discount policy is better than thetraditional revenue-sharing contract with effort dependentdemand the manufacturerrsquos effort in traditional revenue-sharing contract is higher than that of the supply chain inthe wholesale price contract and retailerrsquos order quantity isstill higher than that in wholesale price contract due to theincentive of the revenue-sharing contract so the profit ofthe supply chain in the traditional revenue-sharing contractis still higher than that in wholesale price contract So itcan be seen that the traditional revenue-sharing contract isbetter than thewholesale price contract with effort dependentdemand

43TheRetailer Bearing the Effort Cost with theManufacturerSuppose the effort cost that the retailer bears is 120572119892(119890) (0 le

120572 le 1) then the manufacturer bears (1 minus 120572)119892(119890) effort costIf 120572 = 0 it means that only the manufacturer bears the effortcost if 120572 = 1 it means that only the retailer bears the effortcost

In this condition we find that the traditional revenue-sharing contract fails to coordinate the supply chain whetherit is implemented in one transaction phase or in two transac-tion phases of the three-level supply chain The reason lies inthat the distributor does not bear the effort cost but he can getpartial profit of thewhole supply chain which is unfair for theretailer and themanufacturer So the effort level of the retailerand the manufacturer will be lower than the effort level of thecentralized supply chain Now the following discusses howto design the improved revenue-sharing implemented in twotransaction phases to coordinate the supply chain

Suppose the distributor provides the retailer the revenue-sharing contract (120601

2 119908119888119889(119876 119890)) and the manufacturer pro-

vides the distributors the revenue-sharing contract (1206011

Discrete Dynamics in Nature and Society 9

119908119888119898(119876 119890)) Then the profit functions of the supply chain

members can be described as

Π119888

119903(119876 119890) = 1206012 (119901 minus V) 119878 (119876 119890)

minus [119888119903+ 119908119888

119889(119876 119890) minus 1206012V] 119876 minus 120572119892 (119890)

Π119888

119889(119876 119890) = 1206011 (1 minus 1206012) (119901 minus V) 119878 (119876 119890)

minus [119888119889+ 119908119888

119898(119876 119890) minus 1206011 (1 minus 1206012) V

minus1206011119908119888

119889(119876 119890)] 119876

Π119888

119898(119876 119890) = (1 minus 1206011) (1 minus 1206012) (119901 minus V) 119878 (119876 119890)

minus [119888119898minus 119908119888

119898(119876 119890) minus (1 minus 1206011) (1 minus 1206012) V

minus (1 minus 1206011) 119908119888

119889(119876 119890)] 119876

minus (1 minus 120572) 119892 (119890)

(41)

If the improved revenue-sharing contract can coordinatethe supply chain the optimal effort level 119890lowast of the retailerand the manufacturer should satisfy (2) Namely the optimalorder (production) quantity 119876

lowast of the retailer and themanufacturer should satisfy (3) Then we can get

120597Π119888119903(119876 119890lowast)

120597119890=120597Π119888

119898(119876 119890lowast)

120597119890= 0

(119901 minus V)120597119878 (119876 119890

lowast)

120597119890minus 1198921015840(119890lowast) = 0

120597Π119888119903(119876lowast 119890)

120597119876=120597Π119888

119898(119876lowast 119890)

120597119876= 0

(119901 minus V)120597119878 (119876lowast 119890)

120597119876minus (119888 minus V) = 0

(42)

From (42) we can get

119908119888

119889(119876 119890) = 1206012 (119888 minus V) minus (1 minus 120572)

119892 (119890)

119876minus 119888119903+ 1206012V

119908119888

119898(119876 119890) = 1206011 (119888 minus V) + 1206011 (1 minus 120572)

119892 (119890)

119876minus 1206011119888119903+ 1206011V minus 119888119889

(43)

Taking (43) into the profit functions of the supply chainmembers (41) we can get

Π119888

119903(119876 119890) = 1206012Π119905 (119876 119890)

Π119888

119889(119876 119890) = 1206011 (1 minus 1206012)Π119905 (119876 119890)

Π119888

119898(119876 119890) = (1 minus 1206011) (1 minus 1206012)Π119905 (119876 119890)

(44)

We find that (44) equals (11) That is to say when theretailer bears the effort with the manufacturer the improvedrevenue-sharing contract can coordinate the three-level sup-ply chain

Therefore if we implement the revenue-sharing contractbased on quantity discount policy among the supply chain

members and the contract parameters which satisfy (43)the improved revenue-sharing can coordinate the three-levelsupply chain

The method to determine the optimal order quantity 119876lowastand the optimal effort 119890lowast can follow the methods given inSections 413 and 423

5 Conclusion

The traditional revenue-sharing contract cannot coordinatethree-level supply chain with sales effort dependent demandThe paper proposes an improved revenue-sharing contractbased on quantity discount policy which can be implementedin one transaction or two transactions of three-level supplychain to coordinate the supply chain The paper also showsthat the profit of the supply chain members depends on thevalue of the contract parameters which is determined by thestatus of the members in supply chain and their bargainingpower It also gave the method to determine the optimalorder quantity and the optimal effort which can provide thedecision makers of supply chain with decision support Butit also should be noticed that this research is carried outin the symmetrical information condition So the next stepis to carry out the research in the asymmetric informationcondition

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This research was supported by the Ministry of Educationof China Grant-in-aid for Humanity and social ScienceResearch (no 10YJC630188) and the Fundamental ResearchFunds for the Central Universities (nos 2012B13914 and2012B13814)

References

[1] G P Cachon and M A Lariviere ldquoSupply chain coordinationwith revenue-sharing contracts strengths and limitationsrdquoManagement Science vol 51 no 1 pp 30ndash44 2005

[2] J H Mortimer The Effects of Revenue-Sharing Contracts onWelfare in Vertically SeparatedMarkets Evidence from the VideoRental Industry University of California at Los Angeles 2000

[3] J D Dana Jr and K E Spier ldquoRevenue sharing and verticalcontrol in the video rental industryrdquo Journal of IndustrialEconomics vol 49 no 3 pp 223ndash245 2001

[4] B Pasternack ldquoUsing revenue-sharing to achieve channel coor-dination for a newsboy type inventory modelrdquo in Supply ChainManagement Models Applications and Research DirectionsJ Geunes P Panos and H Edwin Eds vol 62 pp 271ndash290 Kluwer Academic Publishers DordrechtTheNetherlands2002

[5] Y Wang L Jiang and Z J Shen ldquoChannel performance underconsignment contract with revenue-sharingrdquoManagement Sci-ence vol 50 no 1 pp 34ndash47 2004

10 Discrete Dynamics in Nature and Society

[6] Y Gerchak and Y Wang ldquoRevenue-sharing vs wholesale-price contracts in assembly systems with random demandrdquoProduction andOperationsManagement vol 13 no 1 pp 23ndash332004

[7] C Koulamas ldquoA newsvendor problemwith revenue sharing andchannel coordinationrdquo Decision Sciences vol 37 no 1 pp 91ndash100 2006

[8] S S Chauhan and J Proth ldquoAnalysis of a supply chain partner-ship with revenue sharingrdquo International Journal of ProductionEconomics vol 97 no 1 pp 44ndash51 2005

[9] C T Linh and Y S Hong ldquoChannel coordination through arevenue sharing contract in a two-period newsboy problemrdquoEuropean Journal of Operational Research vol 198 no 3 pp822ndash829 2009

[10] K Pan K K Lai S C H Leung andD Xiao ldquoRevenue-sharingversus wholesale price mechanisms under different channelpower structuresrdquo European Journal of Operational Researchvol 203 no 2 pp 532ndash538 2010

[11] D G Qu and Y J Guo ldquoCoordination of supply chain withhybrid distribution channels when retailerrsquos demand relies onits sales effortrdquo Chinese Journal of Management Science vol 21no 3 pp 19ndash23 2008

[12] I Giannoccaro and P Pontrandolfo ldquoSupply chain coordinationby revenue sharing contractsrdquo International Journal of Produc-tion Economics vol 89 no 2 pp 131ndash139 2004

[13] S F Ji M J Liu W Ding and X Huang ldquoCoordination ofthree-level supply chain and revenue sharing contractrdquo Journalof Northeastern University vol 29 no 11 pp 1652ndash1656 2008

[14] L Hu Z Jiang M Meng and J Qin ldquoCoordination of a three-level supply chain through revenue sharingrdquo Journal of HarbinEngineering University vol 29 no 2 pp 198ndash203 2008

[15] Y L Hou D Q Zhou and B Y Tian ldquoOrdering decisionson supply chain under revenue sharing policiesrdquo IndustrialEngineering Journal vol 12 no 1 pp 24ndash27 2009

[16] Y He Q Wu and L Zhao ldquoQuantity flexibility contract witheffort and price-dependent demandrdquo Systems Engineering andElectronics vol 29 no 12 pp 2056ndash2059 2007

[17] P Desai andK Srinivasan ldquoDemand signaling under unobserv-able effort in franchisingrdquo Management Science vol 20 no 4pp 2608ndash2623 1995

[18] W Chu and P Desai ldquoChannel coordination mechanisms forcustomer satisfactionrdquoMarketing Science vol 21 no 6 pp 475ndash490 1995

[19] T A Taylor ldquoSupply chain coordination under channel rebateswith sales effort effectsrdquoManagement Science vol 48 no 8 pp992ndash1007 2002

[20] Z Xu D L Zhu and W G Zhu ldquoSupply chain coordinationunder buy-back contract with sales effort effectsrdquo System Engi-neering Theory and Practice vol 28 no 4 pp 1ndash11 2008

[21] G P Cachon ldquoSupply chain coordination with contractsrdquoin Handbooks in Operations Research and Management Sci-ence Supply Chain Management North-Holland Elseriver SanDiego Calif USA 2000

[22] Y He D Yang andQWu ldquoRevenue-sharing contract of supplychain with effort dependent demandrdquo Computer IntegratedManufacturing Systems vol 12 no 11 pp 1865ndash1868 2006

[23] Y He X Zhao L Zhao and J He ldquoCoordinating a supply chainwith effort and price dependent stochastic demandrdquo AppliedMathematical Modelling vol 33 no 6 pp 2777ndash2790 2009

[24] B Y Hu and X Y Wang ldquoSupply chain revenue-sharingevolvement-contract with sales effort effectsrdquo Journal of Indus-trial Engineering andEngineeringManagement vol 29 no 2 pp35ndash39 2010

[25] J Chen and J F Chen ldquoStudy on revenue sharing contractin virtual enterprisesrdquo Journal of Systems Science and SystemsEngineering vol 15 no 1 pp 95ndash113 2006

[26] Y Wang and Y Gerchak ldquoSupply Chain Coordination whendemand is shelf-space dpendentrdquo Manufacturing and ServiceOperations Management vol 3 no 1 pp 82ndash87 2001

[27] H Krishnan R Kapuscinski and D A Butz ldquoCoordinatingcontracts for decentralized supply chains with retailer promo-tional effortrdquo Management Science vol 50 no 1 pp 48ndash632004

[28] Y He Supply Chain Contract Models with Stochastic DemandDalian University of Technology Dalian China 2004

[29] L Yao Supply Chain Modeling Pricing Contracts and Coordi-nation Chinese University of Hong Kong Hong Kong 2002

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Discrete Dynamics in Nature and Society

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

2 Discrete Dynamics in Nature and Society

RetailerManufacturer Distributor Uncertaindemand

cm

cd

cr

pwdw

m

Qs

Qs

Qd

Qd

Min (x Q)

x

Figure 1 Three-level supply chain

supply chain members as important factors affecting marketdemand [16] is beneficial to the entire supply chain systemIt can increase the market demand to some extent [17ndash19]such as increasing investment in advertising hiring moresales staff to promote products and considering customersrsquopersonality in productmanufacturing However it needs costto do these effort activitiesTherefore if only one supply chainmember bears the cost of effort activities he will choose theeffort level which is best beneficial to him and often thischoice can not make the supply chain coordinate [20]

Cachon [21] and Cachon and Lariviere [1] pointed outthat the revenue-sharing contract can not coordinate supplychain with retailerrsquos sales effort dependent market demandthe kind of situation many scholars have studied He etal [22 23] proposed a mixed contract based on bothrevenue-sharing contract and rebate and penalty contractto coordinate supply chain Hu and Wang [24] proposed anevolution contract based on revenue-sharing contract andstudied this problem in the framework of principal-agentQu and Guo [11] proposed an improved revenue-sharingcontract considering a hybrid model of supply chain to makethe supply chain achieve the coordination of order quantityand effort J Chen and J F Chen [25] studied the applicationof revenue-sharing contract in the virtual enterprise

These contradictions may be resolved when all supplychain members share the effort cost For example Wang andGerchak [26] proposed that a supplier can compensate forthe retailerrsquos the effort cost by way of stock subsides whileconsidering shelf space as an effort variable Krishnan etal [27] proposed a contract to coordinate supply chain bysharing effort cost He [28] and He et al [22] proposed aflexible quantity contract by sharing effort cost Xu et al[20] suggested a return-back contract by sharing effort costHowever in many cases due to the fact that the effort of eachsupply chainmember cannot be observed all kinds of hiddeneffort moral hazard will make the policy of sharing effort costfail to coordinate supply chainTherefore it is not a desirableway to coordinate supply chain by sharing effort cost amongsupply chain members

It should be noticed that the research object of litera-tures mentioned above is a two-level supply chain Throughliteratures searching we have not found the literatures thattake effort into account in three-level supply chainThereforethe paper tries to further study in the following aspectsFirst the research object is extended to three-level supplychain with sales effort dependent demand which meets theactual circs better Second by analyzing the behavior of thedecentralized supply chain under the traditional revenue-sharing contract with sales effort dependent demand we

find that the contract fails to coordinate the supply chainThird by supposing only the retailer or the manufacturerbears the effort cost we propose an improved revenue-sharing contract based on quantity discount policy which cancoordinate the three-level supply chain Because there are twodifferent transaction phases in the three-level supply chaintwo different conditions are discussed implementing theimproved revenue-sharing contract in one transaction phaseand in two transaction phases Further by supposing thedemand and the effort satisfy addition form ormultiplicationform we characterize the optimal solution to the three-level supply chain with two decision variables (sales effortand inventory quantity) and numerical examples are givento illustrate the model and the solution process Finally byconsidering the retailer bears the effort cost together with themanufacturer an improved revenue-sharing contract basedon quantity discount policy is also discussed in this paperwhich can coordinate the three-level supply

The rest of this paper is organized as follows Section 2introduces model assumptions and notations Section 3examines the traditional revenue-sharing contract In Sec-tion 4 we analyze three conditions only the retailer bearsthe sales effort only the manufacturer bears the sales effortand the retailer bears the sales effort with the manufacturerSection 5 provides concluding remarks and describes futureresearch

2 Model Descriptions

The supply chain studied in this paper is made up of onemanufacturer (119898) one distributor (119889) and one retailer (119903)see Figure 1 Upstream member provides a single product todownstream member and the demand is stochastic Beforethe sale season both the distributor and the retailer have onlyone chance to buy products Allmembers are risk neutral andinformation is symmetric among them

Suppose 119901 is sales price of unit product 119888119894is supply chain

memberrsquosmarginal unit costs (119894 = 119898 119889 119903) and 119888 = 119888119903+119888119889+119888119898

V is salvage value for unsold unit product (V lt 119888) 119876 is orderquantity 119908119895

119894is the wholesale price that upstream member

charges downstream member in 119895 condition (119895 = 119888 119889 119894 =

119898 119889) 119890 is the retailerrsquos effort and 119890 ge 0 119892(119890) is the retailerrsquoseffort costwhenhis effort is 119890 supposing119892(0) = 0 and1198921015840(119890) gt0 and 11989210158401015840(119890) gt 0when 119890 gt 0 119909 is the stochastic demand whenretailerrsquos effort is 119890 with probability density function 119891(119909 |

119890) and differentiable cumulated distribution function 119865(119909 |

119890) and 119865(119909 | 119890) is continuously differentiable because thedemand is the increasing function of effort 120597119865(119909 | 119890)120597119890 lt

0 119876 is the order quantity when retailerrsquos effort is 119890 the

Discrete Dynamics in Nature and Society 3

expectation sale quantity is 119878(119876 119890) = 119864min(119876 119909) = 119876 minus

int119876

0119865(119909 | 119890)119889119909 and 120597119878(119876 119890)120597119890 gt 0 the expectation unsold

quantity is 119868(119876 119890) and 119868(119876 119890) = 119876 minus 119878(119876 119890) In particular itis reasonable that 119908119889

119889gt 119908119889119898+ 119888119889 V lt 119888 lt 119901 119908119889

119898+ 119888119903lt 119901

3 Analysis of Revenue-Sharing Contract

Supply chain contracts often apply incentive measures toadjust the relationship among the members to coordinatesupply chain and to make the entire profit of decentralizedsupply chain equal to that of centralized supply chain asmuchas possible The goal of the analysis is to design the revenue-sharing contract so as to achieve channel coordination(maximum profit for the whole supply chain) Besides itaims at analyzing whether and how the contract parametercan be modified so as to more evenly share the profitalong the chain (win-win condition for the chain partners)guaranteeing channel coordination So the maximum profitof centralized supply chain should first be considered asthe goal of revenue-sharing contract to coordinate supplychain The profit function of centralized supply chain can bedescribed as

Π119905 (119876 119890) = 119901119878 (119876 119890) + V119868 (119876 119890) minus 119888119876 minus 119892 (119890)

= (119901 minus V) 119878 (119876 119890) minus (119888 minus V) 119876 minus 119892 (119890) (1)

Supposing 119890lowast is the retailerrsquos optimal effort when givenorder quantity 119876 then 119890lowast should satisfy

120597Π119905(119876 119890lowast)

120597119890= (119901 minus V)

120597119878 (119876 119890lowast)

120597119890minus 1198921015840(119890lowast) = 0 (2)

Supposing 119876lowast is the retailerrsquos optimal order quantitywhen given effort 119890 then 119876lowast should satisfy 120597Π

119905(119876lowast 119890)120597119876 =

(119901minus V)(120597119878(119876lowast 119890)120597119876)minus (119888 minus V) = 0 namely119876lowast should satisfy

119865 (119876lowast| 119890) =

119901 minus 119888

119901 minus V (3)

Therefore if revenue-sharing contract can coordinatesupply chain then formulae (2) and (3) are the necessaryconditions to achieve supply chain coordination

Giannoccaro and Pontrandolfo [12] proposed a modelof revenue-sharing contract to coordinate three-level supplychain without considering supply chain memberrsquos stockoutloss and salvage value of unsold unit product In this paper weanalyze whether the revenue-sharing contract can coordinatethree-level supply chain or not with taking salvage value ofunsold unit product and the retailerrsquos effort into accountSupposing the retailer would keep a quota 120601

2of his revenue

giving the rest (1 minus 1206012) to the distributor and this would be

balanced by a lower price119908119888119889 Similarly the distributor would

keep a quota 1206011of his revenue giving the rest (1 minus 120601

1) to the

manufacturer and thiswould be balanced by a lower price119908119888119898

Also assume only the retailer bears the effort cost Then theexpected profit function of the retailer can be described as

Π119888

119903(119876 119890) = 1206012 [119901119878 (119876 119890) + V119868 (119876 119890)] minus 119888119903119876 minus 119908

119888

119889119876 minus 119892 (119890)

= 1206012(119901 minus V) 119878 (119876 119890) minus (119888119903 + 119908

119888

119889minus 1206012V) 119876 minus 119892 (119890)

(4)

From (4) we get 120597Π119888119903(119876 119890)120597119890 = 120601

2(119901 minus V)(120597119878(119876 119890)120597119890) minus

1198921015840(119890) Compared with (2) it can be seen that the retailerrsquoseffort 119890 in this condition is smaller than the optimaleffort of centralized supply chain namely 120597Π119888

119903(119876 119890)120597119890 lt

120597Π119905(119876 119890)120597119890 That is to say in this condition the revenue-

sharing contract cannot coordinate supply chainThe reasonslie in that the retailer bears all the effort cost of supply chainbut he only get partial profit of the whole supply chain

Similarly if only the distributor or themanufacturer bearsthe sales effort cost the revenue-sharing contract cannotcoordinate the supply chain Then we come to a conclusionthe traditional revenue-sharing contract cannot coordinatethree-level supply chain with sales effort dependent demand

In this paper an improved revenue-sharing contractbased on quantity discount policy is proposed to coordinatethree-level supply chain when demand depends on effort

4 The Improved Revenue-Sharing ContractBased on Quantity Discount Policy

From the discussion above it can be seen that the optimaleffort of the supply chain member under revenue-sharingcontract is not equal to that of the centralized supply chainThe reasons lie in that the supply chain member only getspartial gains but bears all the effort cost 119892(119890) of the wholesupply chain To solve this problem we canmake his gains beequal to a fixed ratio of the gains of the whole supply chainHere we propose an improved revenue-sharing contractbased on quantity discount policy to coordinate supply chainBecause there are two different transaction phases upstreamprocess and downstream process it needs to consider howto implement this improved contract in three-level supplychain

41 Only Retailer Bearing the Effort Cost

411 Only Implement between Retailer and Distributor Thisproblem can be described as follows we implement therevenue-sharing contract based on quantity policy betweenthe retailer and the distributor and the traditional revenue-sharing contract between the distributor and the manufac-turer

Suppose the distributor provides the retailer the revenue-sharing contract (120601

2 119908119888119889(119876)) and

119908119888

119889(119876) = 1206012 (119888 minus V) minus (1 minus 1206012)

119892 (119890lowast)

119876minus 119888119903+ 1206012V (5)

Here 119890lowast is the optimal effort of the supply chainIn this condition the profit function of retailer can be

described asΠ119888

119903(119876 119890)

= 1206012[119901119878 (119876 119890) + V119868 (119876 119890)] minus 119888119903119876 minus 119908

119888

119889(119876)119876 minus 119892 (119890)

= 1206012(119901 minus V) 119878 (119876 119890)minus1206012 (119888 minus V) 119876 + (1 minus 120601

2) 119892 (119890lowast) minus 119892 (119890)

(6)

According to the paper of Giannoccaro and Pontrandolfo[12] without considering demand dependent effort the ratio

4 Discrete Dynamics in Nature and Society

that the retailerrsquos profit shares the profit of the supply chainis 1206012 Here supposing when only the retail bears the effort

cost the ratio that his profit shares the profit of the wholesupply chain is also 120601

2 It is easily known that the effort of

the retail would be equal to the optimal effort of the supplychain namely

Π119888

119903(119876 119890lowast)

= 1206012[119901119878 (119876 119890

lowast) + V119868 (119876 119890lowast)] minus 119888

119903119876 minus 119908

119888

119889(119876)119876 minus 119892 (119890

lowast)

= 1206012(119901 minus V) 119878 (119876 119890lowast) minus 120601

2 (119888 minus V) 119876 minus 1206012119892 (119890lowast)

= 1206012Π119905(119876 119890lowast)

(7)

From (7) it can be seen that the optimal order quantitywould be equal to that of the supply chain Then the profitfunction of the distributor can be described as

Π119888

119889(119876 119890) = 1206011 [(1 minus 1206012) (119901119878 (119876 119890) + V119868 (119876 119890)) + 119908119888

119889(119876)119876]

minus 119888119889119876 minus 119908

119888

119898119876

= 1206011(1 minus 120601

2) (119901 minus V) 119878 (119876 119890)

minus [119888119889+ 119908119888

119898minus 1206011(1 minus 120601

2) V minus 120601

1119908119888

119889(119876)]119876

(8)

Because the distributor is in the middle of the supplychain to maximize his profit he will make his optimal orderquantity equal to the optimal order quantity of retailer inperfect information symmetry condition namely

120597Π119888119889(119876lowast 119890)

120597119876=120597Π119888

119903(119876lowast 119890)

120597119876=120597Π119905(119876lowast 119890)

120597119876 (9)

From (9) it can get

119908119888

119898= (1 + 120601

11206012) 119888 minus 120601

1119888119903minus 119888119889minus [1 minus 120601

1(1 minus 120601

2)] V (10)

Take (5) and (10) into the profit function of the supplychain members and we can get

Π119888

119903(119876 119890) = 1206012Π119905 (119876 119890)

Π119888

119889(119876 119890) = 1206011 (1 minus 1206012)Π119905 (119876 119890)

Π119888

119898(119876 119890) = (1 minus 1206011) (1 minus 1206012)Π119905 (119876 119890)

(11)

From (11) it can be seen that the profit functions ofthe supply chain members are all affine functions of thewhole supply chainrsquos profit function So in this conditionthe revenue-sharing contract can coordinate the three-levelsupply chain

Therefore if we implement the revenue-sharing contractbased on quantity discount policy between the retailer andthe distributor and the traditional revenue-sharing contractbetween the distributor and the manufacturer and the con-tract parameters which satisfy (5) and (10) the improvedrevenue-sharing can coordinate the three-level supply chain

412 Implement among Supply Chain Members This prob-lem can be described as follows we implement both therevenue-sharing contract based on quantity discount policybetween the retailer and the distributor and between thedistributor and the manufacturer

Suppose the distributor provides the retailer the revenue-sharing contract (120601

2 119908119888

119889(119876)) and the manufacturer provides

the distributor the revenue-sharing contract (1206011 119908119888

119898(119876))

Here 119890lowast is the optimal effort of the supply chain 119908119888119889(119876) is

equal to (5) and

119908119888

119898(119876) = 1206011 (119888 minus V) minus 1206011 (1 minus 1206012)

119892 (119890lowast)

119876minus 1206011119888119903+ 1206011V minus 119888119889

(12)

Here we can also suppose that when only the retail bearsthe effort cost the ratio that his profit shares the profit ofthe whole supply chain is also 120601

2 It is easy to know that the

optimal effort and the optimal order quantity of the retailerwould be equal to those of the supply chain

In this condition the profit function of the distributor canbe described asΠ119888

119889(119876 119890) = 1206011 [(1 minus 1206012) (119901119878 (119876 119890) + V119868 (119876 119890)) + 119908119888

119889(119876)119876]

minus 119888119889119876 minus 119908

119888

119898(119876)119876

= 1206011(1 minus 120601

2) (119901 minus V) 119878 (119876 119890)

minus [119888119889+ 119908119888

119898(119876) minus 1206011 (1 minus 1206012) V minus 1206011119908

119888

119889] 119876

= 1206011(1 minus 120601

2) (119901 minus V) 119878 (119876 119890)minus1206011 (1 minus 1206012) (119888 minus V) 119876

(13)

Similarly the optimal order quantity of the distributorshould satisfy

120597Π119888119889(119876lowast 119890)

120597119876=120597Π119888

119903(119876lowast 119890)

120597119876=120597Π119905(119876lowast 119890)

120597119876 (14)

Take (5) and (12) into the profit function of the supplychain member and we can get

Π119888

119903(119876 119890) = 120601

2Π119905 (119876 119890)

Π119888

119889(119876 119890) = 120601

1(1 minus 120601

2) (Π119905 (119876 119890) + 119892 (119890))

Π119888

119898(119876 119890) = (1 minus 120601

1) (1 minus 120601

2)Π119905 (119876 119890) minus 1206011 (1 minus 1206012) 119892 (119890)

(15)

From (15) it can be seen that the profit functions ofthe supply chain members are all affine functions of thewhole supply chainrsquos profit function So in this conditionthe revenue-sharing contract can coordinate the three-levelsupply chain Comparing (11) with (15) it can be seen thatthe retailerrsquos profit keeps unchanged the distributorrsquos profitincreases 120601

1(1 minus 120601

2)119892(119890) while the manufacturerrsquos profit

decreases 1206011(1 minus 120601

2)119892(119890)

Therefore if we implement the revenue-sharing contractbased on quantity discount policy both between the retailerand the distributor and between the distributor and themanufacturer and the contract parameters which satisfy(5) and (12) the improved revenue-sharing contract cancoordinate the three-level supply chain

Discrete Dynamics in Nature and Society 5

413 Model Optimization From the discussion mentionedabove we can come to a conclusion that the traditionalrevenue-sharing contract cannot coordinate three-level sup-ply chain with sales effort dependent demand Howeverthe improved revenue-sharing contract based on quantitydiscount policy can coordinate three-level supply chain Andin this condition the optimal order quantity and optimaleffort of retailer are equal to those of the centralized supplychainThen it needs to determine the optimal order quantityand the optimal effort to maximize the profit of supply chainnamely maxΠ

119905(119876 119890) The following gives the method to

determine the optimal order quantity 119876lowast and the optimaleffort 119890lowast

Suppose the market demand 119883(119890 120585) is the function ofeffort 119890 and random factor 120585 and 120585 is independent of 119890 119891(120585)and 119865(120585) are respectively probability density function anddifferentiable cumulated distribution function of 120585 Usuallywe can use two forms to describe how the effort affectsdemand addition form and multiplication form [29] In thispaper suppose 119883(119890 120585) = 119910(119890) + 120585 Because the influence ofeffort on marker demand is diminishing marginal utility wecan suppose 119910(119890) is the monotone increasing and concavefunction of effort 119890 namely 1199101015840(119890) gt 0 11991010158401015840(119890) le 0

When the market demand satisfies119883(119890 120585) = 119910(119890) + 120585 wecan get

119878 (119876 119890) = 119876 minus int119876

0

119865 (119909 | 119890) 119889119909 = 119876 minus int119876

119910(119890)

119865 (119909 minus 119910 (119890)) 119889119909

= 119876 minus int119876minus119910(119890)

0

119865 (119905) 119889119905

(16)

So the profit function of supply chain can be described as

Π119905 (119876 119890) = (119901 minus V) 119878 (119876 119890) minus (119888 minus V) 119876 minus 119892 (119890)

= (119901 minus 119888)119876 minus (119901 minus V) int119876minus119910(119890)

0

119865 (119905) 119889119905 minus 119892 (119890)

(17)

Given effort 119890 it can be seen that Π119905(119876 119890) is the concave

function of order quantity 119876 So the optimal order quantity119876lowast should satisfy

120597Π119905(119876lowast 119890)

120597119876= (119901 minus 119888) minus (119901 minus V) 119865 (119876lowast minus 119910 (119890)) = 0 (18)

From (18) we can get

119876lowast= 119876lowast(119890) = 119910 (119890) + 119865

minus1(119901 minus 119888

119901 minus V) (19)

Then the profit function of supply chain can be describedasΠ119905(119876lowast 119890)

= (119901 minus 119888)119876lowastminus (119901 minus V) int

119876lowast

minus119910(119890)

0

119865 (119905) 119889119905 minus 119892 (119890)

= (119901 minus 119888) 119910 (119890) + (119901 minus V) int119865minus1

((119901minus119888)(119901minusV))

0

119905119891 (119905) 119889119905 minus 119892 (119890)

(20)

From (20) it can be seen that there is only one variable119890 in Π

119905(119876lowast 119890) Therefore the next work is to determine the

optimal effort 119890lowast to maximize Π119905(119876lowast 119890) From (20) it can

get 120597Π119905(119876lowast 119890)120597119890 = (119901 minus 119888)1199101015840(119890) minus 1198921015840(119890) 1205972Π

119905(119876lowast 119890)1205971198902 =

(119901 minus 119888)11991010158401015840(119890) minus 11989210158401015840(119890) Because 11989210158401015840(119890) gt 0 and 11991010158401015840(119890) le

0 1205972Π119905(119876lowast 119890)1205971198902 lt 0 namely Π

119905(119876lowast 119890) is the concave

function of effort 119890 That is to say there is an optimal effort119890lowast which can make Π

119905(119876lowast 119890) achieve the maximum By

120597Π119905(119876lowast 119890)120597119890 = 0 we get

(119901 minus 119888) 1199101015840(119890lowast) minus 1198921015840(119890lowast) = 0 (21)

Then the optimal effort 119890lowast should satisfy (21) Equation(21) is the first-order differential equation of the optimal effort119890lowast By solving (21) we can get the expression of 119890lowast

414 An Example Suppose a supply chain is made up ofa retailer a distributor and a manufacturer and the supplychain parameters are 119888

119903= 1 119888119889= 2 119888119898= 5 V = 3 119908119889

119889= 18

119908119889119898= 10 119901 = 35 120601

1= 055 and 120601

2= 055

Here assume the relationship between demand and effortsatisfies addition form and 119892(119890) = 119890

22 119910(119890) = 119890 120585 satisfiesuniform distribution in [50 100] It is easy to get119891(120585) = 150119865(120585) = (120585 minus 50)50 and 119865minus1(120585) = 50 + 50120585

(1) The revenue-sharing contract based on quantity dis-count policy

From the discussion above we know that the revenue-sharing contract based on quantity discount policy cancoordinate the supply chain whether it is implemented in oneor two transactions According to the formulas mentionedabove it is easy to get the optimal effort the optimal orderquantity and the profit of the whole supply chain Here 119890lowast =27 119876lowast = 119 and Π

119905(119876lowast 119890lowast) = 22725

(2) The traditional revenue-sharing contract

Similarly under the traditional revenue-sharing contractthe optimal effort and order quantity of the retailer and theprofit of the whole supply chain are respectively 119890119903 = 1485119876119903 = 107 and Π

119905(119876119903 119890119903) = 22027

(3) The decentralized supply chain (the wholesale pricecontract)

Similarly under the traditional revenue-sharing contractthe optimal effort and order quantity of the retailer and theprofit of the whole supply chain are respectively 119890119889 = 16119876119889 = 91 and Π

119905(119876119889 119890119889) = 2137

Table 1 shows the parameters of the supply chain indifferent mode A means that we implement the revenue-sharing contract based on quantity discount policy onlybetween the retailer and the distributor B means that weimplement the revenue-sharing contract based on quantitydiscount policy in two transactions of the supply chain

Table 1 shows that the retailerrsquos effort in the revenue-sharing contract based on quantity discount policy is thehighest in the three modes which can make the supply chain

6 Discrete Dynamics in Nature and Society

Table 1 The parameters of supply chain in different modes with only retailer bearing effort cost

Revenue-sharing contract based onquantity discount policy Traditional

revenue-sharing contract Wholesale price contractA B

Optimal effort 119890 27 27 1485 16Optimal order quantity 119876 119 119 107 91Profit of the whole supply chain Π

119905 22725 22725 22027 2137

maximize its profit the retailerrsquos effort in traditional revenue-sharing contract is the lowest in the threemodes but his orderquantity is still higher than that in wholesale price contractdue to the incentive of the revenue-sharing contract So theprofit of the supply chain in the traditional revenue-sharingcontract is still higher than that in wholesale price contractSo it can be seen that the traditional revenue-sharing contractis better than the wholesale price contract with sales effortdependent demand

42 Only Manufacturer Bearing the Effort Cost

421 Only Implement between Manufacturer and DistributorThis problem can be described as follows we implement therevenue-sharing contract based on quantity policy betweenthe manufacturer and the distributor and the traditionalrevenue-sharing contract between the distributor and theretailer

Supposing the manufacturer provides the distributor therevenue-sharing contract (120601

1 119908119888119898(119876)) and

119908119888

119898(119876) = minus (1 minus 1206011) 119888 + [1 minus (1 minus 1206011) (1 minus 1206012)]

119892 (119890lowast)

119876

+ (1 minus 1206011) 119888119903+ 119888119898

(22)

Here 119890lowast is the optimal effort of the supply chainTaking (22) into the manufacturerrsquos profit function we

can get

Π119888

119898(119876 119890)

= (1 minus 1206011) [(1 minus 120601

2) (119901119878 (119876 119890) + V119868 (119876 119890)) + 119908119888

119889119876]

+ 119908119888

119898(119876)119876 minus 119888

119898119876 minus 119892 (119890)

= (1 minus 1206011) (1 minus 120601

2) (119901 minus V) 119878 (119876 119890)

+ [1 minus (1 minus 1206011) (1 minus 120601

2)] 119892 (119890

lowast)

minus (1 minus 1206011) [119888 minus 119888

119903minus (1 minus 120601

2) V minus 119908119888

119889] 119876 minus 119892 (119890)

(23)

According to the paper of Giannoccaro and Pontrandolfo[12] without considering demand dependent effort the ratiothat the manufacturerrsquos profit shares the profit of the supplychain is (1 minus 120601

1)(1 minus 120601

2) Here supposing when only the

manufacturer bears the effort cost the ratio that his profitshares the profit of the whole supply chain is also (1 minus 120601

1)(1 minus

1206012) It is easily known that the effort of the manufacturer

would be equal to the optimal effort of the supply chainnamely

Π119888

119898(119876 119890)

= (1 minus 1206011) [(1 minus 120601

2) (119901119878 (119876 119890

lowast) + V119868 (119876 119890lowast)) + 119908119888

119889119876]

+ 119908119888

119898(119876)119876 minus 119888

119898119876 minus 119892 (119890

lowast)

= (1 minus 1206011) (1 minus 120601

2)Π119905 (119876 119890)

(24)

From (24) we can get

119908119888

119889= 1206012119888 minus 119888119903 (25)

Taking (25) and (22) into the profit function of the supplychain members we can get

Π119888

119903(119876 119890) = 1206012 [Π119905 (119876 119890) + 119892 (119890)]

Π119888

119889(119876 119890) = 1206011 (1 minus 1206012)Π119905 (119876 119890) minus 1206012119892 (119890)

Π119888

119898(119876 119890) = (1 minus 1206011) (1 minus 1206012)Π119905

(26)

From (26) it can be seen that the profit functions ofthe supply chain members are all affine functions of thewhole supply chainrsquos profit function So in this conditionthe revenue-sharing contract can coordinate the three-levelsupply chain

Therefore if we implement the revenue-sharing contractbased on quantity discount policy between the retailer andthe distributor and the traditional revenue-sharing contractbetween the distributor and the manufacturer and the con-tract parameters which satisfy (22) and (25) the improvedrevenue-sharing can coordinate the three-level supply chain

Compared with the paper of Giannoccaro and Pontran-dolfo [12] it can be seen that the retailerrsquos profit increases1206012119892(119890lowast) while that of the distributor decreases 120601

2119892(119890lowast)

422 Implement among Supply Chain Members This prob-lem can be described as follows we implement both therevenue-sharing contract based on quantity discount policybetween the retailer and the distributor and between thedistributor and the manufacturer

Suppose the distributor provides the retailer the revenue-sharing contract (120601

2 119908119888119889(119876)) and the manufacturer provides

Discrete Dynamics in Nature and Society 7

the distributors the revenue-sharing contract (1206011 119908119888119898(119876))

Here 119890lowast is the optimal effort of the supply chain and

119908119888

119889(119876) = 120601

2119888 minus (1 minus 120601

2)119892 (119890lowast)

119876minus 119888119903

119908119888

119898(119876) = 120601

1119888 +

119892 (119890lowast)

119876minus 1206011119888119903minus 119888119889

(27)

Taking (27) into the profit function of the manufacturerwe can get

Π119888

119898(119876 119890)

= (1 minus 1206011) [(1 minus 120601

2) (119901119878 (119876 119890) + V119868 (119876 119890)) + 119908119888

119889(119876)119876]

+ 119908119888

119898(119876)119876 minus 119888

119898119876 minus 119892 (119890)

= (1 minus 1206011) (1 minus 120601

2) (119901 minus V) 119878 (119876 119890)

minus (1 minus 1206011) (1 minus 120601

2) (119888 minus V) 119876

+ [1 minus (1 minus 1206011) (1 minus 120601

2)] 119892 (119890

lowast) minus 119892 (119890)

(28)

Here we also suppose that the ratio that his profit sharesthe profit of the whole supply chain is also (1 minus 120601

1)(1 minus 120601

2)

It is easy to know that the optimal effort of the manufacturerwould be equal to that of the supply chain In this conditionthe profit function of the manufacturer can be described as

Π119888

119898(119876 119890) = (1 minus 1206011) (1 minus 1206012)Π119905 (119876 119890) (29)

Taking (27) into the profit function of the retailer we canget

Π119888

119903(119876 119890) = 1206012 [119901119878 (119876 119890

lowast) + V119868 (119876 119890lowast)] minus 119888

119903119876 minus 119908

119888

119889(119876)119876

= 1206012(119901 minus V) 119878 (119876 119890lowast)minus120601

2 (119888 minus V) 119876 + (1 minus 1206012) 119892 (119890)

(30)

From (30) it is easy to know that the optimal orderquantity of the retailer is equal to that of the supply chainSo (30) can be described as

Π119888

119903(119876 119890) = 1206012Π119905 (119876 119890) + 119892 (119890) (31)

Taking (27) into the profit function of the distributor itcan get

Π119888

119889(119876 119890)

= 1206011[(1 minus 120601

2) (119901119878 (119876 119890

lowast) + V119868 (119876 119890lowast)) + 119908119888

119889(119876)119876]

minus 119888119889119876 minus 119908

119888

119898(119876)119876 minus 119892 (119890

lowast)

= 1206011(1 minus 120601

2) (119901 minus V) 119878 (119876 119890lowast) minus 120601

1(1 minus 120601

2) (119888 minus V) 119876

minus (1 + 1206011(1 minus 120601

2)) 119892 (119890)

(32)

From (32) it can be seen that the optimal order quantityof the distributor is equal to that of the retailer so (32) canalso be described as

Π119888

119889(119876 119890) = 1206011 (1 minus 1206012)Π119905 (119876 119890) minus 119892 (119890) (33)

From (29) (31) and (33) it can be seen that the profitfunctions of the supply chainmembers are all affine functionsof the whole supply chainrsquos profit function So in this condi-tion the revenue-sharing contract can coordinate the three-level supply chain Therefore if we implement the revenue-sharing contract based on quantity discount policy bothbetween the retailer and the distributor and between thedistributor and themanufacturer and the contract parameterswhich satisfy (27) the improved revenue-sharing can coordi-nate the three-level supply chain

423 Model Optimization Similarly suppose the marketdemand 119883(119890 120585) is the function of effort 119890 and random factor120585 and 120585 is independent of 119890 119891(120585) and 119865(120585) are respectivelyprobability density function and differentiable cumulateddistribution function of 120585 Here suppose 119883(119890 120585) = 119910(119890) sdot 120585As the influence of effort on marker demand is diminishingmarginal utility we can suppose 119910(119890) is the monotoneincreasing and concave function of effort 119890 namely1199101015840(119890) gt 0and 11991010158401015840(119890) le 0

When the market demand satisfies 119883(119890 120585) = 119910(119890) sdot 120585 wecan get

119878 (119876 119890) = 119876 minus int119876

0

119865 (119909 | 119890) 119889119909 = 119876 minus int119876

0

119865(119909

119910 (119890)) 119889119909

= 119876 minus 119910 (119890) int119876119910(119890)

0

119865 (119905) 119889119905

(34)

So the profit function of supply chain can be described as

Π119905 (119876 119890) = (119901 minus V) 119878 (119876 119890) minus (119888 minus V) 119876 minus 119892 (119890)

= (119901 minus 119888)119876 minus (119901 minus V) 119910 (119890) int119876119910(119890)

0

119865 (119905) 119889119905 minus 119892 (119890)

(35)

Given effort 119890 it can be seen that Π119905(119876 119890) is the concave

function of order quantity 119876 So the optimal order quantity119876lowast should satisfy

120597Π119905(119876lowast 119890)

120597119876= (119901 minus 119888) minus (119901 minus V) 119865(

119876lowast

119910 (119890)) = 0 (36)

From (36) we can get

119876lowast= 119876lowast(119890) = 119910 (119890) 119865

minus1(119901 minus 119888

119901 minus V) (37)

Then the profit function of supply chain can be describedas

Π119905(119876lowast 119890) = (119901 minus 119888) 119910 (119890) 119865

minus1(119901 minus 119888

119901 minus V)

minus (119901 minus V) 119910 (119890) int119865minus1

((119901minus119888)(119901minusV))

0

119865 (119905) 119889119905 minus 119892 (119890)

= (119901 minus V) 119910 (119890) int119865minus1

((119901minus119888)(119901minusV))

0

119905119891 (119905) 119889119905 minus 119892 (119890)

(38)

8 Discrete Dynamics in Nature and Society

Table 2 The parameters of supply chain in different modes with only manufacturer bearing effort cost

Revenue-sharing contract based onquantity discount policy Traditional

revenue-sharing contract Wholesale price contractA B

Optimal effort 119890 1962 1962 397 379Optimal order quantity 119876 1674 1674 339 288Profit of the whole supply chain Π

119905 1924628 1924628 700833 654436

According to the supposition Π119905(119876lowast 119890) is the concave

function of effort 119890 So the optimal effort 119890lowast of the supplychain should satisfy

120597Π119905(119876lowast 119890lowast)

120597119890= (119901 minus V) 1199101015840 (119890lowast) int

119865minus1

((119901minus119888)(119901minusV))

0

119905119891 (119905) 119889119905

minus 1198921015840(119890lowast) = 0

(39)

From (39) it can easy to get that the optimal effort 119890lowast ofthe supply chain should satisfy

(119901 minus V) 1199101015840 (119890lowast) int119865minus1

((119901minus119888)(119901minusV))

0

119905119891 (119905) 119889119905 = 1198921015840(119890lowast) (40)

The method to solve the expression of 119890lowast in (40) is thesame as the method in (21)

424 An Example Suppose a supply chain is made up ofa retailer a distributor and a manufacturer and the supplychain parameters are 119888

119903= 1 119888119889= 2 119888119898= 5 V = 3 119908119889

119889= 18

119908119889119898= 10 119901 = 35 120601

1= 1206012= 055 and 119886 = 100

Here assume the relationship between demand and effortsatisfies multiplication form and 119892(119890) = 119886119890

22 119910(119890) = 119890120585 satisfies uniform distribution in [60 90] It is easy to get119891(120585) = 130 119865(120585) = (120585 minus 60)30 and 119865minus1(120585) = 60 + 30120585

(1) The revenue-sharing contract based on quantity dis-count policy

From the discussion above we know that the revenue-sharing contract based on quantity discount policy cancoordinate the supply chain whether it is implemented in oneor two transactions According to the formulas mentionedabove it is easy to get the optimal effort of the manufacturerthe optimal order quantity of the retailer and the profitof the whole supply chain Here 119890lowast = 1962 119876lowast =

1674 Π119905(119876lowast 119890lowast) = 1924628

(2) The traditional revenue-sharing contract

Similarly under the traditional revenue-sharing contractthe optimal effort of the manufacturer and order quantityof the retailer and the profit of the whole supply chain arerespectively 119890119898119888 = 397 119876119903119888 = 339 and Π

119905(119876119903119888 119890119898119888) =

700833

(3) The decentralized supply chain (the wholesale pricecontract)

Similarly under the traditional revenue-sharing contractthe optimal effort of the manufacturer and the order quantityof the retailer and the profit of the whole supply chain arerespectively 119890119898119889 = 379 119876119903119889 = 288 and Π

119905(119876119903119889 119890119889119898) =

654436Table 2 shows the parameters of the supply chain in

different modes A means that we implement the revenue-sharing contract based on quantity discount policy onlybetween the retailer and the distributor B means that weimplement the revenue-sharing contract based on quantitydiscount policy in two transactions of the supply chain

Table 2 shows that the manufacturerrsquos effort in therevenue-sharing contract based on quantity discount policyis the highest in the three modes which can make thesupply chain maximize its profit and the revenue-sharingcontract based on quantity discount policy is better than thetraditional revenue-sharing contract with effort dependentdemand the manufacturerrsquos effort in traditional revenue-sharing contract is higher than that of the supply chain inthe wholesale price contract and retailerrsquos order quantity isstill higher than that in wholesale price contract due to theincentive of the revenue-sharing contract so the profit ofthe supply chain in the traditional revenue-sharing contractis still higher than that in wholesale price contract So itcan be seen that the traditional revenue-sharing contract isbetter than thewholesale price contract with effort dependentdemand

43TheRetailer Bearing the Effort Cost with theManufacturerSuppose the effort cost that the retailer bears is 120572119892(119890) (0 le

120572 le 1) then the manufacturer bears (1 minus 120572)119892(119890) effort costIf 120572 = 0 it means that only the manufacturer bears the effortcost if 120572 = 1 it means that only the retailer bears the effortcost

In this condition we find that the traditional revenue-sharing contract fails to coordinate the supply chain whetherit is implemented in one transaction phase or in two transac-tion phases of the three-level supply chain The reason lies inthat the distributor does not bear the effort cost but he can getpartial profit of thewhole supply chain which is unfair for theretailer and themanufacturer So the effort level of the retailerand the manufacturer will be lower than the effort level of thecentralized supply chain Now the following discusses howto design the improved revenue-sharing implemented in twotransaction phases to coordinate the supply chain

Suppose the distributor provides the retailer the revenue-sharing contract (120601

2 119908119888119889(119876 119890)) and the manufacturer pro-

vides the distributors the revenue-sharing contract (1206011

Discrete Dynamics in Nature and Society 9

119908119888119898(119876 119890)) Then the profit functions of the supply chain

members can be described as

Π119888

119903(119876 119890) = 1206012 (119901 minus V) 119878 (119876 119890)

minus [119888119903+ 119908119888

119889(119876 119890) minus 1206012V] 119876 minus 120572119892 (119890)

Π119888

119889(119876 119890) = 1206011 (1 minus 1206012) (119901 minus V) 119878 (119876 119890)

minus [119888119889+ 119908119888

119898(119876 119890) minus 1206011 (1 minus 1206012) V

minus1206011119908119888

119889(119876 119890)] 119876

Π119888

119898(119876 119890) = (1 minus 1206011) (1 minus 1206012) (119901 minus V) 119878 (119876 119890)

minus [119888119898minus 119908119888

119898(119876 119890) minus (1 minus 1206011) (1 minus 1206012) V

minus (1 minus 1206011) 119908119888

119889(119876 119890)] 119876

minus (1 minus 120572) 119892 (119890)

(41)

If the improved revenue-sharing contract can coordinatethe supply chain the optimal effort level 119890lowast of the retailerand the manufacturer should satisfy (2) Namely the optimalorder (production) quantity 119876

lowast of the retailer and themanufacturer should satisfy (3) Then we can get

120597Π119888119903(119876 119890lowast)

120597119890=120597Π119888

119898(119876 119890lowast)

120597119890= 0

(119901 minus V)120597119878 (119876 119890

lowast)

120597119890minus 1198921015840(119890lowast) = 0

120597Π119888119903(119876lowast 119890)

120597119876=120597Π119888

119898(119876lowast 119890)

120597119876= 0

(119901 minus V)120597119878 (119876lowast 119890)

120597119876minus (119888 minus V) = 0

(42)

From (42) we can get

119908119888

119889(119876 119890) = 1206012 (119888 minus V) minus (1 minus 120572)

119892 (119890)

119876minus 119888119903+ 1206012V

119908119888

119898(119876 119890) = 1206011 (119888 minus V) + 1206011 (1 minus 120572)

119892 (119890)

119876minus 1206011119888119903+ 1206011V minus 119888119889

(43)

Taking (43) into the profit functions of the supply chainmembers (41) we can get

Π119888

119903(119876 119890) = 1206012Π119905 (119876 119890)

Π119888

119889(119876 119890) = 1206011 (1 minus 1206012)Π119905 (119876 119890)

Π119888

119898(119876 119890) = (1 minus 1206011) (1 minus 1206012)Π119905 (119876 119890)

(44)

We find that (44) equals (11) That is to say when theretailer bears the effort with the manufacturer the improvedrevenue-sharing contract can coordinate the three-level sup-ply chain

Therefore if we implement the revenue-sharing contractbased on quantity discount policy among the supply chain

members and the contract parameters which satisfy (43)the improved revenue-sharing can coordinate the three-levelsupply chain

The method to determine the optimal order quantity 119876lowastand the optimal effort 119890lowast can follow the methods given inSections 413 and 423

5 Conclusion

The traditional revenue-sharing contract cannot coordinatethree-level supply chain with sales effort dependent demandThe paper proposes an improved revenue-sharing contractbased on quantity discount policy which can be implementedin one transaction or two transactions of three-level supplychain to coordinate the supply chain The paper also showsthat the profit of the supply chain members depends on thevalue of the contract parameters which is determined by thestatus of the members in supply chain and their bargainingpower It also gave the method to determine the optimalorder quantity and the optimal effort which can provide thedecision makers of supply chain with decision support Butit also should be noticed that this research is carried outin the symmetrical information condition So the next stepis to carry out the research in the asymmetric informationcondition

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This research was supported by the Ministry of Educationof China Grant-in-aid for Humanity and social ScienceResearch (no 10YJC630188) and the Fundamental ResearchFunds for the Central Universities (nos 2012B13914 and2012B13814)

References

[1] G P Cachon and M A Lariviere ldquoSupply chain coordinationwith revenue-sharing contracts strengths and limitationsrdquoManagement Science vol 51 no 1 pp 30ndash44 2005

[2] J H Mortimer The Effects of Revenue-Sharing Contracts onWelfare in Vertically SeparatedMarkets Evidence from the VideoRental Industry University of California at Los Angeles 2000

[3] J D Dana Jr and K E Spier ldquoRevenue sharing and verticalcontrol in the video rental industryrdquo Journal of IndustrialEconomics vol 49 no 3 pp 223ndash245 2001

[4] B Pasternack ldquoUsing revenue-sharing to achieve channel coor-dination for a newsboy type inventory modelrdquo in Supply ChainManagement Models Applications and Research DirectionsJ Geunes P Panos and H Edwin Eds vol 62 pp 271ndash290 Kluwer Academic Publishers DordrechtTheNetherlands2002

[5] Y Wang L Jiang and Z J Shen ldquoChannel performance underconsignment contract with revenue-sharingrdquoManagement Sci-ence vol 50 no 1 pp 34ndash47 2004

10 Discrete Dynamics in Nature and Society

[6] Y Gerchak and Y Wang ldquoRevenue-sharing vs wholesale-price contracts in assembly systems with random demandrdquoProduction andOperationsManagement vol 13 no 1 pp 23ndash332004

[7] C Koulamas ldquoA newsvendor problemwith revenue sharing andchannel coordinationrdquo Decision Sciences vol 37 no 1 pp 91ndash100 2006

[8] S S Chauhan and J Proth ldquoAnalysis of a supply chain partner-ship with revenue sharingrdquo International Journal of ProductionEconomics vol 97 no 1 pp 44ndash51 2005

[9] C T Linh and Y S Hong ldquoChannel coordination through arevenue sharing contract in a two-period newsboy problemrdquoEuropean Journal of Operational Research vol 198 no 3 pp822ndash829 2009

[10] K Pan K K Lai S C H Leung andD Xiao ldquoRevenue-sharingversus wholesale price mechanisms under different channelpower structuresrdquo European Journal of Operational Researchvol 203 no 2 pp 532ndash538 2010

[11] D G Qu and Y J Guo ldquoCoordination of supply chain withhybrid distribution channels when retailerrsquos demand relies onits sales effortrdquo Chinese Journal of Management Science vol 21no 3 pp 19ndash23 2008

[12] I Giannoccaro and P Pontrandolfo ldquoSupply chain coordinationby revenue sharing contractsrdquo International Journal of Produc-tion Economics vol 89 no 2 pp 131ndash139 2004

[13] S F Ji M J Liu W Ding and X Huang ldquoCoordination ofthree-level supply chain and revenue sharing contractrdquo Journalof Northeastern University vol 29 no 11 pp 1652ndash1656 2008

[14] L Hu Z Jiang M Meng and J Qin ldquoCoordination of a three-level supply chain through revenue sharingrdquo Journal of HarbinEngineering University vol 29 no 2 pp 198ndash203 2008

[15] Y L Hou D Q Zhou and B Y Tian ldquoOrdering decisionson supply chain under revenue sharing policiesrdquo IndustrialEngineering Journal vol 12 no 1 pp 24ndash27 2009

[16] Y He Q Wu and L Zhao ldquoQuantity flexibility contract witheffort and price-dependent demandrdquo Systems Engineering andElectronics vol 29 no 12 pp 2056ndash2059 2007

[17] P Desai andK Srinivasan ldquoDemand signaling under unobserv-able effort in franchisingrdquo Management Science vol 20 no 4pp 2608ndash2623 1995

[18] W Chu and P Desai ldquoChannel coordination mechanisms forcustomer satisfactionrdquoMarketing Science vol 21 no 6 pp 475ndash490 1995

[19] T A Taylor ldquoSupply chain coordination under channel rebateswith sales effort effectsrdquoManagement Science vol 48 no 8 pp992ndash1007 2002

[20] Z Xu D L Zhu and W G Zhu ldquoSupply chain coordinationunder buy-back contract with sales effort effectsrdquo System Engi-neering Theory and Practice vol 28 no 4 pp 1ndash11 2008

[21] G P Cachon ldquoSupply chain coordination with contractsrdquoin Handbooks in Operations Research and Management Sci-ence Supply Chain Management North-Holland Elseriver SanDiego Calif USA 2000

[22] Y He D Yang andQWu ldquoRevenue-sharing contract of supplychain with effort dependent demandrdquo Computer IntegratedManufacturing Systems vol 12 no 11 pp 1865ndash1868 2006

[23] Y He X Zhao L Zhao and J He ldquoCoordinating a supply chainwith effort and price dependent stochastic demandrdquo AppliedMathematical Modelling vol 33 no 6 pp 2777ndash2790 2009

[24] B Y Hu and X Y Wang ldquoSupply chain revenue-sharingevolvement-contract with sales effort effectsrdquo Journal of Indus-trial Engineering andEngineeringManagement vol 29 no 2 pp35ndash39 2010

[25] J Chen and J F Chen ldquoStudy on revenue sharing contractin virtual enterprisesrdquo Journal of Systems Science and SystemsEngineering vol 15 no 1 pp 95ndash113 2006

[26] Y Wang and Y Gerchak ldquoSupply Chain Coordination whendemand is shelf-space dpendentrdquo Manufacturing and ServiceOperations Management vol 3 no 1 pp 82ndash87 2001

[27] H Krishnan R Kapuscinski and D A Butz ldquoCoordinatingcontracts for decentralized supply chains with retailer promo-tional effortrdquo Management Science vol 50 no 1 pp 48ndash632004

[28] Y He Supply Chain Contract Models with Stochastic DemandDalian University of Technology Dalian China 2004

[29] L Yao Supply Chain Modeling Pricing Contracts and Coordi-nation Chinese University of Hong Kong Hong Kong 2002

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Decision SciencesAdvances in

Discrete MathematicsJournal of

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Discrete Dynamics in Nature and Society 3

expectation sale quantity is 119878(119876 119890) = 119864min(119876 119909) = 119876 minus

int119876

0119865(119909 | 119890)119889119909 and 120597119878(119876 119890)120597119890 gt 0 the expectation unsold

quantity is 119868(119876 119890) and 119868(119876 119890) = 119876 minus 119878(119876 119890) In particular itis reasonable that 119908119889

119889gt 119908119889119898+ 119888119889 V lt 119888 lt 119901 119908119889

119898+ 119888119903lt 119901

3 Analysis of Revenue-Sharing Contract

Supply chain contracts often apply incentive measures toadjust the relationship among the members to coordinatesupply chain and to make the entire profit of decentralizedsupply chain equal to that of centralized supply chain asmuchas possible The goal of the analysis is to design the revenue-sharing contract so as to achieve channel coordination(maximum profit for the whole supply chain) Besides itaims at analyzing whether and how the contract parametercan be modified so as to more evenly share the profitalong the chain (win-win condition for the chain partners)guaranteeing channel coordination So the maximum profitof centralized supply chain should first be considered asthe goal of revenue-sharing contract to coordinate supplychain The profit function of centralized supply chain can bedescribed as

Π119905 (119876 119890) = 119901119878 (119876 119890) + V119868 (119876 119890) minus 119888119876 minus 119892 (119890)

= (119901 minus V) 119878 (119876 119890) minus (119888 minus V) 119876 minus 119892 (119890) (1)

Supposing 119890lowast is the retailerrsquos optimal effort when givenorder quantity 119876 then 119890lowast should satisfy

120597Π119905(119876 119890lowast)

120597119890= (119901 minus V)

120597119878 (119876 119890lowast)

120597119890minus 1198921015840(119890lowast) = 0 (2)

Supposing 119876lowast is the retailerrsquos optimal order quantitywhen given effort 119890 then 119876lowast should satisfy 120597Π

119905(119876lowast 119890)120597119876 =

(119901minus V)(120597119878(119876lowast 119890)120597119876)minus (119888 minus V) = 0 namely119876lowast should satisfy

119865 (119876lowast| 119890) =

119901 minus 119888

119901 minus V (3)

Therefore if revenue-sharing contract can coordinatesupply chain then formulae (2) and (3) are the necessaryconditions to achieve supply chain coordination

Giannoccaro and Pontrandolfo [12] proposed a modelof revenue-sharing contract to coordinate three-level supplychain without considering supply chain memberrsquos stockoutloss and salvage value of unsold unit product In this paper weanalyze whether the revenue-sharing contract can coordinatethree-level supply chain or not with taking salvage value ofunsold unit product and the retailerrsquos effort into accountSupposing the retailer would keep a quota 120601

2of his revenue

giving the rest (1 minus 1206012) to the distributor and this would be

balanced by a lower price119908119888119889 Similarly the distributor would

keep a quota 1206011of his revenue giving the rest (1 minus 120601

1) to the

manufacturer and thiswould be balanced by a lower price119908119888119898

Also assume only the retailer bears the effort cost Then theexpected profit function of the retailer can be described as

Π119888

119903(119876 119890) = 1206012 [119901119878 (119876 119890) + V119868 (119876 119890)] minus 119888119903119876 minus 119908

119888

119889119876 minus 119892 (119890)

= 1206012(119901 minus V) 119878 (119876 119890) minus (119888119903 + 119908

119888

119889minus 1206012V) 119876 minus 119892 (119890)

(4)

From (4) we get 120597Π119888119903(119876 119890)120597119890 = 120601

2(119901 minus V)(120597119878(119876 119890)120597119890) minus

1198921015840(119890) Compared with (2) it can be seen that the retailerrsquoseffort 119890 in this condition is smaller than the optimaleffort of centralized supply chain namely 120597Π119888

119903(119876 119890)120597119890 lt

120597Π119905(119876 119890)120597119890 That is to say in this condition the revenue-

sharing contract cannot coordinate supply chainThe reasonslie in that the retailer bears all the effort cost of supply chainbut he only get partial profit of the whole supply chain

Similarly if only the distributor or themanufacturer bearsthe sales effort cost the revenue-sharing contract cannotcoordinate the supply chain Then we come to a conclusionthe traditional revenue-sharing contract cannot coordinatethree-level supply chain with sales effort dependent demand

In this paper an improved revenue-sharing contractbased on quantity discount policy is proposed to coordinatethree-level supply chain when demand depends on effort

4 The Improved Revenue-Sharing ContractBased on Quantity Discount Policy

From the discussion above it can be seen that the optimaleffort of the supply chain member under revenue-sharingcontract is not equal to that of the centralized supply chainThe reasons lie in that the supply chain member only getspartial gains but bears all the effort cost 119892(119890) of the wholesupply chain To solve this problem we canmake his gains beequal to a fixed ratio of the gains of the whole supply chainHere we propose an improved revenue-sharing contractbased on quantity discount policy to coordinate supply chainBecause there are two different transaction phases upstreamprocess and downstream process it needs to consider howto implement this improved contract in three-level supplychain

41 Only Retailer Bearing the Effort Cost

411 Only Implement between Retailer and Distributor Thisproblem can be described as follows we implement therevenue-sharing contract based on quantity policy betweenthe retailer and the distributor and the traditional revenue-sharing contract between the distributor and the manufac-turer

Suppose the distributor provides the retailer the revenue-sharing contract (120601

2 119908119888119889(119876)) and

119908119888

119889(119876) = 1206012 (119888 minus V) minus (1 minus 1206012)

119892 (119890lowast)

119876minus 119888119903+ 1206012V (5)

Here 119890lowast is the optimal effort of the supply chainIn this condition the profit function of retailer can be

described asΠ119888

119903(119876 119890)

= 1206012[119901119878 (119876 119890) + V119868 (119876 119890)] minus 119888119903119876 minus 119908

119888

119889(119876)119876 minus 119892 (119890)

= 1206012(119901 minus V) 119878 (119876 119890)minus1206012 (119888 minus V) 119876 + (1 minus 120601

2) 119892 (119890lowast) minus 119892 (119890)

(6)

According to the paper of Giannoccaro and Pontrandolfo[12] without considering demand dependent effort the ratio

4 Discrete Dynamics in Nature and Society

that the retailerrsquos profit shares the profit of the supply chainis 1206012 Here supposing when only the retail bears the effort

cost the ratio that his profit shares the profit of the wholesupply chain is also 120601

2 It is easily known that the effort of

the retail would be equal to the optimal effort of the supplychain namely

Π119888

119903(119876 119890lowast)

= 1206012[119901119878 (119876 119890

lowast) + V119868 (119876 119890lowast)] minus 119888

119903119876 minus 119908

119888

119889(119876)119876 minus 119892 (119890

lowast)

= 1206012(119901 minus V) 119878 (119876 119890lowast) minus 120601

2 (119888 minus V) 119876 minus 1206012119892 (119890lowast)

= 1206012Π119905(119876 119890lowast)

(7)

From (7) it can be seen that the optimal order quantitywould be equal to that of the supply chain Then the profitfunction of the distributor can be described as

Π119888

119889(119876 119890) = 1206011 [(1 minus 1206012) (119901119878 (119876 119890) + V119868 (119876 119890)) + 119908119888

119889(119876)119876]

minus 119888119889119876 minus 119908

119888

119898119876

= 1206011(1 minus 120601

2) (119901 minus V) 119878 (119876 119890)

minus [119888119889+ 119908119888

119898minus 1206011(1 minus 120601

2) V minus 120601

1119908119888

119889(119876)]119876

(8)

Because the distributor is in the middle of the supplychain to maximize his profit he will make his optimal orderquantity equal to the optimal order quantity of retailer inperfect information symmetry condition namely

120597Π119888119889(119876lowast 119890)

120597119876=120597Π119888

119903(119876lowast 119890)

120597119876=120597Π119905(119876lowast 119890)

120597119876 (9)

From (9) it can get

119908119888

119898= (1 + 120601

11206012) 119888 minus 120601

1119888119903minus 119888119889minus [1 minus 120601

1(1 minus 120601

2)] V (10)

Take (5) and (10) into the profit function of the supplychain members and we can get

Π119888

119903(119876 119890) = 1206012Π119905 (119876 119890)

Π119888

119889(119876 119890) = 1206011 (1 minus 1206012)Π119905 (119876 119890)

Π119888

119898(119876 119890) = (1 minus 1206011) (1 minus 1206012)Π119905 (119876 119890)

(11)

From (11) it can be seen that the profit functions ofthe supply chain members are all affine functions of thewhole supply chainrsquos profit function So in this conditionthe revenue-sharing contract can coordinate the three-levelsupply chain

Therefore if we implement the revenue-sharing contractbased on quantity discount policy between the retailer andthe distributor and the traditional revenue-sharing contractbetween the distributor and the manufacturer and the con-tract parameters which satisfy (5) and (10) the improvedrevenue-sharing can coordinate the three-level supply chain

412 Implement among Supply Chain Members This prob-lem can be described as follows we implement both therevenue-sharing contract based on quantity discount policybetween the retailer and the distributor and between thedistributor and the manufacturer

Suppose the distributor provides the retailer the revenue-sharing contract (120601

2 119908119888

119889(119876)) and the manufacturer provides

the distributor the revenue-sharing contract (1206011 119908119888

119898(119876))

Here 119890lowast is the optimal effort of the supply chain 119908119888119889(119876) is

equal to (5) and

119908119888

119898(119876) = 1206011 (119888 minus V) minus 1206011 (1 minus 1206012)

119892 (119890lowast)

119876minus 1206011119888119903+ 1206011V minus 119888119889

(12)

Here we can also suppose that when only the retail bearsthe effort cost the ratio that his profit shares the profit ofthe whole supply chain is also 120601

2 It is easy to know that the

optimal effort and the optimal order quantity of the retailerwould be equal to those of the supply chain

In this condition the profit function of the distributor canbe described asΠ119888

119889(119876 119890) = 1206011 [(1 minus 1206012) (119901119878 (119876 119890) + V119868 (119876 119890)) + 119908119888

119889(119876)119876]

minus 119888119889119876 minus 119908

119888

119898(119876)119876

= 1206011(1 minus 120601

2) (119901 minus V) 119878 (119876 119890)

minus [119888119889+ 119908119888

119898(119876) minus 1206011 (1 minus 1206012) V minus 1206011119908

119888

119889] 119876

= 1206011(1 minus 120601

2) (119901 minus V) 119878 (119876 119890)minus1206011 (1 minus 1206012) (119888 minus V) 119876

(13)

Similarly the optimal order quantity of the distributorshould satisfy

120597Π119888119889(119876lowast 119890)

120597119876=120597Π119888

119903(119876lowast 119890)

120597119876=120597Π119905(119876lowast 119890)

120597119876 (14)

Take (5) and (12) into the profit function of the supplychain member and we can get

Π119888

119903(119876 119890) = 120601

2Π119905 (119876 119890)

Π119888

119889(119876 119890) = 120601

1(1 minus 120601

2) (Π119905 (119876 119890) + 119892 (119890))

Π119888

119898(119876 119890) = (1 minus 120601

1) (1 minus 120601

2)Π119905 (119876 119890) minus 1206011 (1 minus 1206012) 119892 (119890)

(15)

From (15) it can be seen that the profit functions ofthe supply chain members are all affine functions of thewhole supply chainrsquos profit function So in this conditionthe revenue-sharing contract can coordinate the three-levelsupply chain Comparing (11) with (15) it can be seen thatthe retailerrsquos profit keeps unchanged the distributorrsquos profitincreases 120601

1(1 minus 120601

2)119892(119890) while the manufacturerrsquos profit

decreases 1206011(1 minus 120601

2)119892(119890)

Therefore if we implement the revenue-sharing contractbased on quantity discount policy both between the retailerand the distributor and between the distributor and themanufacturer and the contract parameters which satisfy(5) and (12) the improved revenue-sharing contract cancoordinate the three-level supply chain

Discrete Dynamics in Nature and Society 5

413 Model Optimization From the discussion mentionedabove we can come to a conclusion that the traditionalrevenue-sharing contract cannot coordinate three-level sup-ply chain with sales effort dependent demand Howeverthe improved revenue-sharing contract based on quantitydiscount policy can coordinate three-level supply chain Andin this condition the optimal order quantity and optimaleffort of retailer are equal to those of the centralized supplychainThen it needs to determine the optimal order quantityand the optimal effort to maximize the profit of supply chainnamely maxΠ

119905(119876 119890) The following gives the method to

determine the optimal order quantity 119876lowast and the optimaleffort 119890lowast

Suppose the market demand 119883(119890 120585) is the function ofeffort 119890 and random factor 120585 and 120585 is independent of 119890 119891(120585)and 119865(120585) are respectively probability density function anddifferentiable cumulated distribution function of 120585 Usuallywe can use two forms to describe how the effort affectsdemand addition form and multiplication form [29] In thispaper suppose 119883(119890 120585) = 119910(119890) + 120585 Because the influence ofeffort on marker demand is diminishing marginal utility wecan suppose 119910(119890) is the monotone increasing and concavefunction of effort 119890 namely 1199101015840(119890) gt 0 11991010158401015840(119890) le 0

When the market demand satisfies119883(119890 120585) = 119910(119890) + 120585 wecan get

119878 (119876 119890) = 119876 minus int119876

0

119865 (119909 | 119890) 119889119909 = 119876 minus int119876

119910(119890)

119865 (119909 minus 119910 (119890)) 119889119909

= 119876 minus int119876minus119910(119890)

0

119865 (119905) 119889119905

(16)

So the profit function of supply chain can be described as

Π119905 (119876 119890) = (119901 minus V) 119878 (119876 119890) minus (119888 minus V) 119876 minus 119892 (119890)

= (119901 minus 119888)119876 minus (119901 minus V) int119876minus119910(119890)

0

119865 (119905) 119889119905 minus 119892 (119890)

(17)

Given effort 119890 it can be seen that Π119905(119876 119890) is the concave

function of order quantity 119876 So the optimal order quantity119876lowast should satisfy

120597Π119905(119876lowast 119890)

120597119876= (119901 minus 119888) minus (119901 minus V) 119865 (119876lowast minus 119910 (119890)) = 0 (18)

From (18) we can get

119876lowast= 119876lowast(119890) = 119910 (119890) + 119865

minus1(119901 minus 119888

119901 minus V) (19)

Then the profit function of supply chain can be describedasΠ119905(119876lowast 119890)

= (119901 minus 119888)119876lowastminus (119901 minus V) int

119876lowast

minus119910(119890)

0

119865 (119905) 119889119905 minus 119892 (119890)

= (119901 minus 119888) 119910 (119890) + (119901 minus V) int119865minus1

((119901minus119888)(119901minusV))

0

119905119891 (119905) 119889119905 minus 119892 (119890)

(20)

From (20) it can be seen that there is only one variable119890 in Π

119905(119876lowast 119890) Therefore the next work is to determine the

optimal effort 119890lowast to maximize Π119905(119876lowast 119890) From (20) it can

get 120597Π119905(119876lowast 119890)120597119890 = (119901 minus 119888)1199101015840(119890) minus 1198921015840(119890) 1205972Π

119905(119876lowast 119890)1205971198902 =

(119901 minus 119888)11991010158401015840(119890) minus 11989210158401015840(119890) Because 11989210158401015840(119890) gt 0 and 11991010158401015840(119890) le

0 1205972Π119905(119876lowast 119890)1205971198902 lt 0 namely Π

119905(119876lowast 119890) is the concave

function of effort 119890 That is to say there is an optimal effort119890lowast which can make Π

119905(119876lowast 119890) achieve the maximum By

120597Π119905(119876lowast 119890)120597119890 = 0 we get

(119901 minus 119888) 1199101015840(119890lowast) minus 1198921015840(119890lowast) = 0 (21)

Then the optimal effort 119890lowast should satisfy (21) Equation(21) is the first-order differential equation of the optimal effort119890lowast By solving (21) we can get the expression of 119890lowast

414 An Example Suppose a supply chain is made up ofa retailer a distributor and a manufacturer and the supplychain parameters are 119888

119903= 1 119888119889= 2 119888119898= 5 V = 3 119908119889

119889= 18

119908119889119898= 10 119901 = 35 120601

1= 055 and 120601

2= 055

Here assume the relationship between demand and effortsatisfies addition form and 119892(119890) = 119890

22 119910(119890) = 119890 120585 satisfiesuniform distribution in [50 100] It is easy to get119891(120585) = 150119865(120585) = (120585 minus 50)50 and 119865minus1(120585) = 50 + 50120585

(1) The revenue-sharing contract based on quantity dis-count policy

From the discussion above we know that the revenue-sharing contract based on quantity discount policy cancoordinate the supply chain whether it is implemented in oneor two transactions According to the formulas mentionedabove it is easy to get the optimal effort the optimal orderquantity and the profit of the whole supply chain Here 119890lowast =27 119876lowast = 119 and Π

119905(119876lowast 119890lowast) = 22725

(2) The traditional revenue-sharing contract

Similarly under the traditional revenue-sharing contractthe optimal effort and order quantity of the retailer and theprofit of the whole supply chain are respectively 119890119903 = 1485119876119903 = 107 and Π

119905(119876119903 119890119903) = 22027

(3) The decentralized supply chain (the wholesale pricecontract)

Similarly under the traditional revenue-sharing contractthe optimal effort and order quantity of the retailer and theprofit of the whole supply chain are respectively 119890119889 = 16119876119889 = 91 and Π

119905(119876119889 119890119889) = 2137

Table 1 shows the parameters of the supply chain indifferent mode A means that we implement the revenue-sharing contract based on quantity discount policy onlybetween the retailer and the distributor B means that weimplement the revenue-sharing contract based on quantitydiscount policy in two transactions of the supply chain

Table 1 shows that the retailerrsquos effort in the revenue-sharing contract based on quantity discount policy is thehighest in the three modes which can make the supply chain

6 Discrete Dynamics in Nature and Society

Table 1 The parameters of supply chain in different modes with only retailer bearing effort cost

Revenue-sharing contract based onquantity discount policy Traditional

revenue-sharing contract Wholesale price contractA B

Optimal effort 119890 27 27 1485 16Optimal order quantity 119876 119 119 107 91Profit of the whole supply chain Π

119905 22725 22725 22027 2137

maximize its profit the retailerrsquos effort in traditional revenue-sharing contract is the lowest in the threemodes but his orderquantity is still higher than that in wholesale price contractdue to the incentive of the revenue-sharing contract So theprofit of the supply chain in the traditional revenue-sharingcontract is still higher than that in wholesale price contractSo it can be seen that the traditional revenue-sharing contractis better than the wholesale price contract with sales effortdependent demand

42 Only Manufacturer Bearing the Effort Cost

421 Only Implement between Manufacturer and DistributorThis problem can be described as follows we implement therevenue-sharing contract based on quantity policy betweenthe manufacturer and the distributor and the traditionalrevenue-sharing contract between the distributor and theretailer

Supposing the manufacturer provides the distributor therevenue-sharing contract (120601

1 119908119888119898(119876)) and

119908119888

119898(119876) = minus (1 minus 1206011) 119888 + [1 minus (1 minus 1206011) (1 minus 1206012)]

119892 (119890lowast)

119876

+ (1 minus 1206011) 119888119903+ 119888119898

(22)

Here 119890lowast is the optimal effort of the supply chainTaking (22) into the manufacturerrsquos profit function we

can get

Π119888

119898(119876 119890)

= (1 minus 1206011) [(1 minus 120601

2) (119901119878 (119876 119890) + V119868 (119876 119890)) + 119908119888

119889119876]

+ 119908119888

119898(119876)119876 minus 119888

119898119876 minus 119892 (119890)

= (1 minus 1206011) (1 minus 120601

2) (119901 minus V) 119878 (119876 119890)

+ [1 minus (1 minus 1206011) (1 minus 120601

2)] 119892 (119890

lowast)

minus (1 minus 1206011) [119888 minus 119888

119903minus (1 minus 120601

2) V minus 119908119888

119889] 119876 minus 119892 (119890)

(23)

According to the paper of Giannoccaro and Pontrandolfo[12] without considering demand dependent effort the ratiothat the manufacturerrsquos profit shares the profit of the supplychain is (1 minus 120601

1)(1 minus 120601

2) Here supposing when only the

manufacturer bears the effort cost the ratio that his profitshares the profit of the whole supply chain is also (1 minus 120601

1)(1 minus

1206012) It is easily known that the effort of the manufacturer

would be equal to the optimal effort of the supply chainnamely

Π119888

119898(119876 119890)

= (1 minus 1206011) [(1 minus 120601

2) (119901119878 (119876 119890

lowast) + V119868 (119876 119890lowast)) + 119908119888

119889119876]

+ 119908119888

119898(119876)119876 minus 119888

119898119876 minus 119892 (119890

lowast)

= (1 minus 1206011) (1 minus 120601

2)Π119905 (119876 119890)

(24)

From (24) we can get

119908119888

119889= 1206012119888 minus 119888119903 (25)

Taking (25) and (22) into the profit function of the supplychain members we can get

Π119888

119903(119876 119890) = 1206012 [Π119905 (119876 119890) + 119892 (119890)]

Π119888

119889(119876 119890) = 1206011 (1 minus 1206012)Π119905 (119876 119890) minus 1206012119892 (119890)

Π119888

119898(119876 119890) = (1 minus 1206011) (1 minus 1206012)Π119905

(26)

From (26) it can be seen that the profit functions ofthe supply chain members are all affine functions of thewhole supply chainrsquos profit function So in this conditionthe revenue-sharing contract can coordinate the three-levelsupply chain

Therefore if we implement the revenue-sharing contractbased on quantity discount policy between the retailer andthe distributor and the traditional revenue-sharing contractbetween the distributor and the manufacturer and the con-tract parameters which satisfy (22) and (25) the improvedrevenue-sharing can coordinate the three-level supply chain

Compared with the paper of Giannoccaro and Pontran-dolfo [12] it can be seen that the retailerrsquos profit increases1206012119892(119890lowast) while that of the distributor decreases 120601

2119892(119890lowast)

422 Implement among Supply Chain Members This prob-lem can be described as follows we implement both therevenue-sharing contract based on quantity discount policybetween the retailer and the distributor and between thedistributor and the manufacturer

Suppose the distributor provides the retailer the revenue-sharing contract (120601

2 119908119888119889(119876)) and the manufacturer provides

Discrete Dynamics in Nature and Society 7

the distributors the revenue-sharing contract (1206011 119908119888119898(119876))

Here 119890lowast is the optimal effort of the supply chain and

119908119888

119889(119876) = 120601

2119888 minus (1 minus 120601

2)119892 (119890lowast)

119876minus 119888119903

119908119888

119898(119876) = 120601

1119888 +

119892 (119890lowast)

119876minus 1206011119888119903minus 119888119889

(27)

Taking (27) into the profit function of the manufacturerwe can get

Π119888

119898(119876 119890)

= (1 minus 1206011) [(1 minus 120601

2) (119901119878 (119876 119890) + V119868 (119876 119890)) + 119908119888

119889(119876)119876]

+ 119908119888

119898(119876)119876 minus 119888

119898119876 minus 119892 (119890)

= (1 minus 1206011) (1 minus 120601

2) (119901 minus V) 119878 (119876 119890)

minus (1 minus 1206011) (1 minus 120601

2) (119888 minus V) 119876

+ [1 minus (1 minus 1206011) (1 minus 120601

2)] 119892 (119890

lowast) minus 119892 (119890)

(28)

Here we also suppose that the ratio that his profit sharesthe profit of the whole supply chain is also (1 minus 120601

1)(1 minus 120601

2)

It is easy to know that the optimal effort of the manufacturerwould be equal to that of the supply chain In this conditionthe profit function of the manufacturer can be described as

Π119888

119898(119876 119890) = (1 minus 1206011) (1 minus 1206012)Π119905 (119876 119890) (29)

Taking (27) into the profit function of the retailer we canget

Π119888

119903(119876 119890) = 1206012 [119901119878 (119876 119890

lowast) + V119868 (119876 119890lowast)] minus 119888

119903119876 minus 119908

119888

119889(119876)119876

= 1206012(119901 minus V) 119878 (119876 119890lowast)minus120601

2 (119888 minus V) 119876 + (1 minus 1206012) 119892 (119890)

(30)

From (30) it is easy to know that the optimal orderquantity of the retailer is equal to that of the supply chainSo (30) can be described as

Π119888

119903(119876 119890) = 1206012Π119905 (119876 119890) + 119892 (119890) (31)

Taking (27) into the profit function of the distributor itcan get

Π119888

119889(119876 119890)

= 1206011[(1 minus 120601

2) (119901119878 (119876 119890

lowast) + V119868 (119876 119890lowast)) + 119908119888

119889(119876)119876]

minus 119888119889119876 minus 119908

119888

119898(119876)119876 minus 119892 (119890

lowast)

= 1206011(1 minus 120601

2) (119901 minus V) 119878 (119876 119890lowast) minus 120601

1(1 minus 120601

2) (119888 minus V) 119876

minus (1 + 1206011(1 minus 120601

2)) 119892 (119890)

(32)

From (32) it can be seen that the optimal order quantityof the distributor is equal to that of the retailer so (32) canalso be described as

Π119888

119889(119876 119890) = 1206011 (1 minus 1206012)Π119905 (119876 119890) minus 119892 (119890) (33)

From (29) (31) and (33) it can be seen that the profitfunctions of the supply chainmembers are all affine functionsof the whole supply chainrsquos profit function So in this condi-tion the revenue-sharing contract can coordinate the three-level supply chain Therefore if we implement the revenue-sharing contract based on quantity discount policy bothbetween the retailer and the distributor and between thedistributor and themanufacturer and the contract parameterswhich satisfy (27) the improved revenue-sharing can coordi-nate the three-level supply chain

423 Model Optimization Similarly suppose the marketdemand 119883(119890 120585) is the function of effort 119890 and random factor120585 and 120585 is independent of 119890 119891(120585) and 119865(120585) are respectivelyprobability density function and differentiable cumulateddistribution function of 120585 Here suppose 119883(119890 120585) = 119910(119890) sdot 120585As the influence of effort on marker demand is diminishingmarginal utility we can suppose 119910(119890) is the monotoneincreasing and concave function of effort 119890 namely1199101015840(119890) gt 0and 11991010158401015840(119890) le 0

When the market demand satisfies 119883(119890 120585) = 119910(119890) sdot 120585 wecan get

119878 (119876 119890) = 119876 minus int119876

0

119865 (119909 | 119890) 119889119909 = 119876 minus int119876

0

119865(119909

119910 (119890)) 119889119909

= 119876 minus 119910 (119890) int119876119910(119890)

0

119865 (119905) 119889119905

(34)

So the profit function of supply chain can be described as

Π119905 (119876 119890) = (119901 minus V) 119878 (119876 119890) minus (119888 minus V) 119876 minus 119892 (119890)

= (119901 minus 119888)119876 minus (119901 minus V) 119910 (119890) int119876119910(119890)

0

119865 (119905) 119889119905 minus 119892 (119890)

(35)

Given effort 119890 it can be seen that Π119905(119876 119890) is the concave

function of order quantity 119876 So the optimal order quantity119876lowast should satisfy

120597Π119905(119876lowast 119890)

120597119876= (119901 minus 119888) minus (119901 minus V) 119865(

119876lowast

119910 (119890)) = 0 (36)

From (36) we can get

119876lowast= 119876lowast(119890) = 119910 (119890) 119865

minus1(119901 minus 119888

119901 minus V) (37)

Then the profit function of supply chain can be describedas

Π119905(119876lowast 119890) = (119901 minus 119888) 119910 (119890) 119865

minus1(119901 minus 119888

119901 minus V)

minus (119901 minus V) 119910 (119890) int119865minus1

((119901minus119888)(119901minusV))

0

119865 (119905) 119889119905 minus 119892 (119890)

= (119901 minus V) 119910 (119890) int119865minus1

((119901minus119888)(119901minusV))

0

119905119891 (119905) 119889119905 minus 119892 (119890)

(38)

8 Discrete Dynamics in Nature and Society

Table 2 The parameters of supply chain in different modes with only manufacturer bearing effort cost

Revenue-sharing contract based onquantity discount policy Traditional

revenue-sharing contract Wholesale price contractA B

Optimal effort 119890 1962 1962 397 379Optimal order quantity 119876 1674 1674 339 288Profit of the whole supply chain Π

119905 1924628 1924628 700833 654436

According to the supposition Π119905(119876lowast 119890) is the concave

function of effort 119890 So the optimal effort 119890lowast of the supplychain should satisfy

120597Π119905(119876lowast 119890lowast)

120597119890= (119901 minus V) 1199101015840 (119890lowast) int

119865minus1

((119901minus119888)(119901minusV))

0

119905119891 (119905) 119889119905

minus 1198921015840(119890lowast) = 0

(39)

From (39) it can easy to get that the optimal effort 119890lowast ofthe supply chain should satisfy

(119901 minus V) 1199101015840 (119890lowast) int119865minus1

((119901minus119888)(119901minusV))

0

119905119891 (119905) 119889119905 = 1198921015840(119890lowast) (40)

The method to solve the expression of 119890lowast in (40) is thesame as the method in (21)

424 An Example Suppose a supply chain is made up ofa retailer a distributor and a manufacturer and the supplychain parameters are 119888

119903= 1 119888119889= 2 119888119898= 5 V = 3 119908119889

119889= 18

119908119889119898= 10 119901 = 35 120601

1= 1206012= 055 and 119886 = 100

Here assume the relationship between demand and effortsatisfies multiplication form and 119892(119890) = 119886119890

22 119910(119890) = 119890120585 satisfies uniform distribution in [60 90] It is easy to get119891(120585) = 130 119865(120585) = (120585 minus 60)30 and 119865minus1(120585) = 60 + 30120585

(1) The revenue-sharing contract based on quantity dis-count policy

From the discussion above we know that the revenue-sharing contract based on quantity discount policy cancoordinate the supply chain whether it is implemented in oneor two transactions According to the formulas mentionedabove it is easy to get the optimal effort of the manufacturerthe optimal order quantity of the retailer and the profitof the whole supply chain Here 119890lowast = 1962 119876lowast =

1674 Π119905(119876lowast 119890lowast) = 1924628

(2) The traditional revenue-sharing contract

Similarly under the traditional revenue-sharing contractthe optimal effort of the manufacturer and order quantityof the retailer and the profit of the whole supply chain arerespectively 119890119898119888 = 397 119876119903119888 = 339 and Π

119905(119876119903119888 119890119898119888) =

700833

(3) The decentralized supply chain (the wholesale pricecontract)

Similarly under the traditional revenue-sharing contractthe optimal effort of the manufacturer and the order quantityof the retailer and the profit of the whole supply chain arerespectively 119890119898119889 = 379 119876119903119889 = 288 and Π

119905(119876119903119889 119890119889119898) =

654436Table 2 shows the parameters of the supply chain in

different modes A means that we implement the revenue-sharing contract based on quantity discount policy onlybetween the retailer and the distributor B means that weimplement the revenue-sharing contract based on quantitydiscount policy in two transactions of the supply chain

Table 2 shows that the manufacturerrsquos effort in therevenue-sharing contract based on quantity discount policyis the highest in the three modes which can make thesupply chain maximize its profit and the revenue-sharingcontract based on quantity discount policy is better than thetraditional revenue-sharing contract with effort dependentdemand the manufacturerrsquos effort in traditional revenue-sharing contract is higher than that of the supply chain inthe wholesale price contract and retailerrsquos order quantity isstill higher than that in wholesale price contract due to theincentive of the revenue-sharing contract so the profit ofthe supply chain in the traditional revenue-sharing contractis still higher than that in wholesale price contract So itcan be seen that the traditional revenue-sharing contract isbetter than thewholesale price contract with effort dependentdemand

43TheRetailer Bearing the Effort Cost with theManufacturerSuppose the effort cost that the retailer bears is 120572119892(119890) (0 le

120572 le 1) then the manufacturer bears (1 minus 120572)119892(119890) effort costIf 120572 = 0 it means that only the manufacturer bears the effortcost if 120572 = 1 it means that only the retailer bears the effortcost

In this condition we find that the traditional revenue-sharing contract fails to coordinate the supply chain whetherit is implemented in one transaction phase or in two transac-tion phases of the three-level supply chain The reason lies inthat the distributor does not bear the effort cost but he can getpartial profit of thewhole supply chain which is unfair for theretailer and themanufacturer So the effort level of the retailerand the manufacturer will be lower than the effort level of thecentralized supply chain Now the following discusses howto design the improved revenue-sharing implemented in twotransaction phases to coordinate the supply chain

Suppose the distributor provides the retailer the revenue-sharing contract (120601

2 119908119888119889(119876 119890)) and the manufacturer pro-

vides the distributors the revenue-sharing contract (1206011

Discrete Dynamics in Nature and Society 9

119908119888119898(119876 119890)) Then the profit functions of the supply chain

members can be described as

Π119888

119903(119876 119890) = 1206012 (119901 minus V) 119878 (119876 119890)

minus [119888119903+ 119908119888

119889(119876 119890) minus 1206012V] 119876 minus 120572119892 (119890)

Π119888

119889(119876 119890) = 1206011 (1 minus 1206012) (119901 minus V) 119878 (119876 119890)

minus [119888119889+ 119908119888

119898(119876 119890) minus 1206011 (1 minus 1206012) V

minus1206011119908119888

119889(119876 119890)] 119876

Π119888

119898(119876 119890) = (1 minus 1206011) (1 minus 1206012) (119901 minus V) 119878 (119876 119890)

minus [119888119898minus 119908119888

119898(119876 119890) minus (1 minus 1206011) (1 minus 1206012) V

minus (1 minus 1206011) 119908119888

119889(119876 119890)] 119876

minus (1 minus 120572) 119892 (119890)

(41)

If the improved revenue-sharing contract can coordinatethe supply chain the optimal effort level 119890lowast of the retailerand the manufacturer should satisfy (2) Namely the optimalorder (production) quantity 119876

lowast of the retailer and themanufacturer should satisfy (3) Then we can get

120597Π119888119903(119876 119890lowast)

120597119890=120597Π119888

119898(119876 119890lowast)

120597119890= 0

(119901 minus V)120597119878 (119876 119890

lowast)

120597119890minus 1198921015840(119890lowast) = 0

120597Π119888119903(119876lowast 119890)

120597119876=120597Π119888

119898(119876lowast 119890)

120597119876= 0

(119901 minus V)120597119878 (119876lowast 119890)

120597119876minus (119888 minus V) = 0

(42)

From (42) we can get

119908119888

119889(119876 119890) = 1206012 (119888 minus V) minus (1 minus 120572)

119892 (119890)

119876minus 119888119903+ 1206012V

119908119888

119898(119876 119890) = 1206011 (119888 minus V) + 1206011 (1 minus 120572)

119892 (119890)

119876minus 1206011119888119903+ 1206011V minus 119888119889

(43)

Taking (43) into the profit functions of the supply chainmembers (41) we can get

Π119888

119903(119876 119890) = 1206012Π119905 (119876 119890)

Π119888

119889(119876 119890) = 1206011 (1 minus 1206012)Π119905 (119876 119890)

Π119888

119898(119876 119890) = (1 minus 1206011) (1 minus 1206012)Π119905 (119876 119890)

(44)

We find that (44) equals (11) That is to say when theretailer bears the effort with the manufacturer the improvedrevenue-sharing contract can coordinate the three-level sup-ply chain

Therefore if we implement the revenue-sharing contractbased on quantity discount policy among the supply chain

members and the contract parameters which satisfy (43)the improved revenue-sharing can coordinate the three-levelsupply chain

The method to determine the optimal order quantity 119876lowastand the optimal effort 119890lowast can follow the methods given inSections 413 and 423

5 Conclusion

The traditional revenue-sharing contract cannot coordinatethree-level supply chain with sales effort dependent demandThe paper proposes an improved revenue-sharing contractbased on quantity discount policy which can be implementedin one transaction or two transactions of three-level supplychain to coordinate the supply chain The paper also showsthat the profit of the supply chain members depends on thevalue of the contract parameters which is determined by thestatus of the members in supply chain and their bargainingpower It also gave the method to determine the optimalorder quantity and the optimal effort which can provide thedecision makers of supply chain with decision support Butit also should be noticed that this research is carried outin the symmetrical information condition So the next stepis to carry out the research in the asymmetric informationcondition

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This research was supported by the Ministry of Educationof China Grant-in-aid for Humanity and social ScienceResearch (no 10YJC630188) and the Fundamental ResearchFunds for the Central Universities (nos 2012B13914 and2012B13814)

References

[1] G P Cachon and M A Lariviere ldquoSupply chain coordinationwith revenue-sharing contracts strengths and limitationsrdquoManagement Science vol 51 no 1 pp 30ndash44 2005

[2] J H Mortimer The Effects of Revenue-Sharing Contracts onWelfare in Vertically SeparatedMarkets Evidence from the VideoRental Industry University of California at Los Angeles 2000

[3] J D Dana Jr and K E Spier ldquoRevenue sharing and verticalcontrol in the video rental industryrdquo Journal of IndustrialEconomics vol 49 no 3 pp 223ndash245 2001

[4] B Pasternack ldquoUsing revenue-sharing to achieve channel coor-dination for a newsboy type inventory modelrdquo in Supply ChainManagement Models Applications and Research DirectionsJ Geunes P Panos and H Edwin Eds vol 62 pp 271ndash290 Kluwer Academic Publishers DordrechtTheNetherlands2002

[5] Y Wang L Jiang and Z J Shen ldquoChannel performance underconsignment contract with revenue-sharingrdquoManagement Sci-ence vol 50 no 1 pp 34ndash47 2004

10 Discrete Dynamics in Nature and Society

[6] Y Gerchak and Y Wang ldquoRevenue-sharing vs wholesale-price contracts in assembly systems with random demandrdquoProduction andOperationsManagement vol 13 no 1 pp 23ndash332004

[7] C Koulamas ldquoA newsvendor problemwith revenue sharing andchannel coordinationrdquo Decision Sciences vol 37 no 1 pp 91ndash100 2006

[8] S S Chauhan and J Proth ldquoAnalysis of a supply chain partner-ship with revenue sharingrdquo International Journal of ProductionEconomics vol 97 no 1 pp 44ndash51 2005

[9] C T Linh and Y S Hong ldquoChannel coordination through arevenue sharing contract in a two-period newsboy problemrdquoEuropean Journal of Operational Research vol 198 no 3 pp822ndash829 2009

[10] K Pan K K Lai S C H Leung andD Xiao ldquoRevenue-sharingversus wholesale price mechanisms under different channelpower structuresrdquo European Journal of Operational Researchvol 203 no 2 pp 532ndash538 2010

[11] D G Qu and Y J Guo ldquoCoordination of supply chain withhybrid distribution channels when retailerrsquos demand relies onits sales effortrdquo Chinese Journal of Management Science vol 21no 3 pp 19ndash23 2008

[12] I Giannoccaro and P Pontrandolfo ldquoSupply chain coordinationby revenue sharing contractsrdquo International Journal of Produc-tion Economics vol 89 no 2 pp 131ndash139 2004

[13] S F Ji M J Liu W Ding and X Huang ldquoCoordination ofthree-level supply chain and revenue sharing contractrdquo Journalof Northeastern University vol 29 no 11 pp 1652ndash1656 2008

[14] L Hu Z Jiang M Meng and J Qin ldquoCoordination of a three-level supply chain through revenue sharingrdquo Journal of HarbinEngineering University vol 29 no 2 pp 198ndash203 2008

[15] Y L Hou D Q Zhou and B Y Tian ldquoOrdering decisionson supply chain under revenue sharing policiesrdquo IndustrialEngineering Journal vol 12 no 1 pp 24ndash27 2009

[16] Y He Q Wu and L Zhao ldquoQuantity flexibility contract witheffort and price-dependent demandrdquo Systems Engineering andElectronics vol 29 no 12 pp 2056ndash2059 2007

[17] P Desai andK Srinivasan ldquoDemand signaling under unobserv-able effort in franchisingrdquo Management Science vol 20 no 4pp 2608ndash2623 1995

[18] W Chu and P Desai ldquoChannel coordination mechanisms forcustomer satisfactionrdquoMarketing Science vol 21 no 6 pp 475ndash490 1995

[19] T A Taylor ldquoSupply chain coordination under channel rebateswith sales effort effectsrdquoManagement Science vol 48 no 8 pp992ndash1007 2002

[20] Z Xu D L Zhu and W G Zhu ldquoSupply chain coordinationunder buy-back contract with sales effort effectsrdquo System Engi-neering Theory and Practice vol 28 no 4 pp 1ndash11 2008

[21] G P Cachon ldquoSupply chain coordination with contractsrdquoin Handbooks in Operations Research and Management Sci-ence Supply Chain Management North-Holland Elseriver SanDiego Calif USA 2000

[22] Y He D Yang andQWu ldquoRevenue-sharing contract of supplychain with effort dependent demandrdquo Computer IntegratedManufacturing Systems vol 12 no 11 pp 1865ndash1868 2006

[23] Y He X Zhao L Zhao and J He ldquoCoordinating a supply chainwith effort and price dependent stochastic demandrdquo AppliedMathematical Modelling vol 33 no 6 pp 2777ndash2790 2009

[24] B Y Hu and X Y Wang ldquoSupply chain revenue-sharingevolvement-contract with sales effort effectsrdquo Journal of Indus-trial Engineering andEngineeringManagement vol 29 no 2 pp35ndash39 2010

[25] J Chen and J F Chen ldquoStudy on revenue sharing contractin virtual enterprisesrdquo Journal of Systems Science and SystemsEngineering vol 15 no 1 pp 95ndash113 2006

[26] Y Wang and Y Gerchak ldquoSupply Chain Coordination whendemand is shelf-space dpendentrdquo Manufacturing and ServiceOperations Management vol 3 no 1 pp 82ndash87 2001

[27] H Krishnan R Kapuscinski and D A Butz ldquoCoordinatingcontracts for decentralized supply chains with retailer promo-tional effortrdquo Management Science vol 50 no 1 pp 48ndash632004

[28] Y He Supply Chain Contract Models with Stochastic DemandDalian University of Technology Dalian China 2004

[29] L Yao Supply Chain Modeling Pricing Contracts and Coordi-nation Chinese University of Hong Kong Hong Kong 2002

Submit your manuscripts athttpwwwhindawicom

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Differential EquationsInternational Journal of

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Decision SciencesAdvances in

Discrete MathematicsJournal of

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

4 Discrete Dynamics in Nature and Society

that the retailerrsquos profit shares the profit of the supply chainis 1206012 Here supposing when only the retail bears the effort

cost the ratio that his profit shares the profit of the wholesupply chain is also 120601

2 It is easily known that the effort of

the retail would be equal to the optimal effort of the supplychain namely

Π119888

119903(119876 119890lowast)

= 1206012[119901119878 (119876 119890

lowast) + V119868 (119876 119890lowast)] minus 119888

119903119876 minus 119908

119888

119889(119876)119876 minus 119892 (119890

lowast)

= 1206012(119901 minus V) 119878 (119876 119890lowast) minus 120601

2 (119888 minus V) 119876 minus 1206012119892 (119890lowast)

= 1206012Π119905(119876 119890lowast)

(7)

From (7) it can be seen that the optimal order quantitywould be equal to that of the supply chain Then the profitfunction of the distributor can be described as

Π119888

119889(119876 119890) = 1206011 [(1 minus 1206012) (119901119878 (119876 119890) + V119868 (119876 119890)) + 119908119888

119889(119876)119876]

minus 119888119889119876 minus 119908

119888

119898119876

= 1206011(1 minus 120601

2) (119901 minus V) 119878 (119876 119890)

minus [119888119889+ 119908119888

119898minus 1206011(1 minus 120601

2) V minus 120601

1119908119888

119889(119876)]119876

(8)

Because the distributor is in the middle of the supplychain to maximize his profit he will make his optimal orderquantity equal to the optimal order quantity of retailer inperfect information symmetry condition namely

120597Π119888119889(119876lowast 119890)

120597119876=120597Π119888

119903(119876lowast 119890)

120597119876=120597Π119905(119876lowast 119890)

120597119876 (9)

From (9) it can get

119908119888

119898= (1 + 120601

11206012) 119888 minus 120601

1119888119903minus 119888119889minus [1 minus 120601

1(1 minus 120601

2)] V (10)

Take (5) and (10) into the profit function of the supplychain members and we can get

Π119888

119903(119876 119890) = 1206012Π119905 (119876 119890)

Π119888

119889(119876 119890) = 1206011 (1 minus 1206012)Π119905 (119876 119890)

Π119888

119898(119876 119890) = (1 minus 1206011) (1 minus 1206012)Π119905 (119876 119890)

(11)

From (11) it can be seen that the profit functions ofthe supply chain members are all affine functions of thewhole supply chainrsquos profit function So in this conditionthe revenue-sharing contract can coordinate the three-levelsupply chain

Therefore if we implement the revenue-sharing contractbased on quantity discount policy between the retailer andthe distributor and the traditional revenue-sharing contractbetween the distributor and the manufacturer and the con-tract parameters which satisfy (5) and (10) the improvedrevenue-sharing can coordinate the three-level supply chain

412 Implement among Supply Chain Members This prob-lem can be described as follows we implement both therevenue-sharing contract based on quantity discount policybetween the retailer and the distributor and between thedistributor and the manufacturer

Suppose the distributor provides the retailer the revenue-sharing contract (120601

2 119908119888

119889(119876)) and the manufacturer provides

the distributor the revenue-sharing contract (1206011 119908119888

119898(119876))

Here 119890lowast is the optimal effort of the supply chain 119908119888119889(119876) is

equal to (5) and

119908119888

119898(119876) = 1206011 (119888 minus V) minus 1206011 (1 minus 1206012)

119892 (119890lowast)

119876minus 1206011119888119903+ 1206011V minus 119888119889

(12)

Here we can also suppose that when only the retail bearsthe effort cost the ratio that his profit shares the profit ofthe whole supply chain is also 120601

2 It is easy to know that the

optimal effort and the optimal order quantity of the retailerwould be equal to those of the supply chain

In this condition the profit function of the distributor canbe described asΠ119888

119889(119876 119890) = 1206011 [(1 minus 1206012) (119901119878 (119876 119890) + V119868 (119876 119890)) + 119908119888

119889(119876)119876]

minus 119888119889119876 minus 119908

119888

119898(119876)119876

= 1206011(1 minus 120601

2) (119901 minus V) 119878 (119876 119890)

minus [119888119889+ 119908119888

119898(119876) minus 1206011 (1 minus 1206012) V minus 1206011119908

119888

119889] 119876

= 1206011(1 minus 120601

2) (119901 minus V) 119878 (119876 119890)minus1206011 (1 minus 1206012) (119888 minus V) 119876

(13)

Similarly the optimal order quantity of the distributorshould satisfy

120597Π119888119889(119876lowast 119890)

120597119876=120597Π119888

119903(119876lowast 119890)

120597119876=120597Π119905(119876lowast 119890)

120597119876 (14)

Take (5) and (12) into the profit function of the supplychain member and we can get

Π119888

119903(119876 119890) = 120601

2Π119905 (119876 119890)

Π119888

119889(119876 119890) = 120601

1(1 minus 120601

2) (Π119905 (119876 119890) + 119892 (119890))

Π119888

119898(119876 119890) = (1 minus 120601

1) (1 minus 120601

2)Π119905 (119876 119890) minus 1206011 (1 minus 1206012) 119892 (119890)

(15)

From (15) it can be seen that the profit functions ofthe supply chain members are all affine functions of thewhole supply chainrsquos profit function So in this conditionthe revenue-sharing contract can coordinate the three-levelsupply chain Comparing (11) with (15) it can be seen thatthe retailerrsquos profit keeps unchanged the distributorrsquos profitincreases 120601

1(1 minus 120601

2)119892(119890) while the manufacturerrsquos profit

decreases 1206011(1 minus 120601

2)119892(119890)

Therefore if we implement the revenue-sharing contractbased on quantity discount policy both between the retailerand the distributor and between the distributor and themanufacturer and the contract parameters which satisfy(5) and (12) the improved revenue-sharing contract cancoordinate the three-level supply chain

Discrete Dynamics in Nature and Society 5

413 Model Optimization From the discussion mentionedabove we can come to a conclusion that the traditionalrevenue-sharing contract cannot coordinate three-level sup-ply chain with sales effort dependent demand Howeverthe improved revenue-sharing contract based on quantitydiscount policy can coordinate three-level supply chain Andin this condition the optimal order quantity and optimaleffort of retailer are equal to those of the centralized supplychainThen it needs to determine the optimal order quantityand the optimal effort to maximize the profit of supply chainnamely maxΠ

119905(119876 119890) The following gives the method to

determine the optimal order quantity 119876lowast and the optimaleffort 119890lowast

Suppose the market demand 119883(119890 120585) is the function ofeffort 119890 and random factor 120585 and 120585 is independent of 119890 119891(120585)and 119865(120585) are respectively probability density function anddifferentiable cumulated distribution function of 120585 Usuallywe can use two forms to describe how the effort affectsdemand addition form and multiplication form [29] In thispaper suppose 119883(119890 120585) = 119910(119890) + 120585 Because the influence ofeffort on marker demand is diminishing marginal utility wecan suppose 119910(119890) is the monotone increasing and concavefunction of effort 119890 namely 1199101015840(119890) gt 0 11991010158401015840(119890) le 0

When the market demand satisfies119883(119890 120585) = 119910(119890) + 120585 wecan get

119878 (119876 119890) = 119876 minus int119876

0

119865 (119909 | 119890) 119889119909 = 119876 minus int119876

119910(119890)

119865 (119909 minus 119910 (119890)) 119889119909

= 119876 minus int119876minus119910(119890)

0

119865 (119905) 119889119905

(16)

So the profit function of supply chain can be described as

Π119905 (119876 119890) = (119901 minus V) 119878 (119876 119890) minus (119888 minus V) 119876 minus 119892 (119890)

= (119901 minus 119888)119876 minus (119901 minus V) int119876minus119910(119890)

0

119865 (119905) 119889119905 minus 119892 (119890)

(17)

Given effort 119890 it can be seen that Π119905(119876 119890) is the concave

function of order quantity 119876 So the optimal order quantity119876lowast should satisfy

120597Π119905(119876lowast 119890)

120597119876= (119901 minus 119888) minus (119901 minus V) 119865 (119876lowast minus 119910 (119890)) = 0 (18)

From (18) we can get

119876lowast= 119876lowast(119890) = 119910 (119890) + 119865

minus1(119901 minus 119888

119901 minus V) (19)

Then the profit function of supply chain can be describedasΠ119905(119876lowast 119890)

= (119901 minus 119888)119876lowastminus (119901 minus V) int

119876lowast

minus119910(119890)

0

119865 (119905) 119889119905 minus 119892 (119890)

= (119901 minus 119888) 119910 (119890) + (119901 minus V) int119865minus1

((119901minus119888)(119901minusV))

0

119905119891 (119905) 119889119905 minus 119892 (119890)

(20)

From (20) it can be seen that there is only one variable119890 in Π

119905(119876lowast 119890) Therefore the next work is to determine the

optimal effort 119890lowast to maximize Π119905(119876lowast 119890) From (20) it can

get 120597Π119905(119876lowast 119890)120597119890 = (119901 minus 119888)1199101015840(119890) minus 1198921015840(119890) 1205972Π

119905(119876lowast 119890)1205971198902 =

(119901 minus 119888)11991010158401015840(119890) minus 11989210158401015840(119890) Because 11989210158401015840(119890) gt 0 and 11991010158401015840(119890) le

0 1205972Π119905(119876lowast 119890)1205971198902 lt 0 namely Π

119905(119876lowast 119890) is the concave

function of effort 119890 That is to say there is an optimal effort119890lowast which can make Π

119905(119876lowast 119890) achieve the maximum By

120597Π119905(119876lowast 119890)120597119890 = 0 we get

(119901 minus 119888) 1199101015840(119890lowast) minus 1198921015840(119890lowast) = 0 (21)

Then the optimal effort 119890lowast should satisfy (21) Equation(21) is the first-order differential equation of the optimal effort119890lowast By solving (21) we can get the expression of 119890lowast

414 An Example Suppose a supply chain is made up ofa retailer a distributor and a manufacturer and the supplychain parameters are 119888

119903= 1 119888119889= 2 119888119898= 5 V = 3 119908119889

119889= 18

119908119889119898= 10 119901 = 35 120601

1= 055 and 120601

2= 055

Here assume the relationship between demand and effortsatisfies addition form and 119892(119890) = 119890

22 119910(119890) = 119890 120585 satisfiesuniform distribution in [50 100] It is easy to get119891(120585) = 150119865(120585) = (120585 minus 50)50 and 119865minus1(120585) = 50 + 50120585

(1) The revenue-sharing contract based on quantity dis-count policy

From the discussion above we know that the revenue-sharing contract based on quantity discount policy cancoordinate the supply chain whether it is implemented in oneor two transactions According to the formulas mentionedabove it is easy to get the optimal effort the optimal orderquantity and the profit of the whole supply chain Here 119890lowast =27 119876lowast = 119 and Π

119905(119876lowast 119890lowast) = 22725

(2) The traditional revenue-sharing contract

Similarly under the traditional revenue-sharing contractthe optimal effort and order quantity of the retailer and theprofit of the whole supply chain are respectively 119890119903 = 1485119876119903 = 107 and Π

119905(119876119903 119890119903) = 22027

(3) The decentralized supply chain (the wholesale pricecontract)

Similarly under the traditional revenue-sharing contractthe optimal effort and order quantity of the retailer and theprofit of the whole supply chain are respectively 119890119889 = 16119876119889 = 91 and Π

119905(119876119889 119890119889) = 2137

Table 1 shows the parameters of the supply chain indifferent mode A means that we implement the revenue-sharing contract based on quantity discount policy onlybetween the retailer and the distributor B means that weimplement the revenue-sharing contract based on quantitydiscount policy in two transactions of the supply chain

Table 1 shows that the retailerrsquos effort in the revenue-sharing contract based on quantity discount policy is thehighest in the three modes which can make the supply chain

6 Discrete Dynamics in Nature and Society

Table 1 The parameters of supply chain in different modes with only retailer bearing effort cost

Revenue-sharing contract based onquantity discount policy Traditional

revenue-sharing contract Wholesale price contractA B

Optimal effort 119890 27 27 1485 16Optimal order quantity 119876 119 119 107 91Profit of the whole supply chain Π

119905 22725 22725 22027 2137

maximize its profit the retailerrsquos effort in traditional revenue-sharing contract is the lowest in the threemodes but his orderquantity is still higher than that in wholesale price contractdue to the incentive of the revenue-sharing contract So theprofit of the supply chain in the traditional revenue-sharingcontract is still higher than that in wholesale price contractSo it can be seen that the traditional revenue-sharing contractis better than the wholesale price contract with sales effortdependent demand

42 Only Manufacturer Bearing the Effort Cost

421 Only Implement between Manufacturer and DistributorThis problem can be described as follows we implement therevenue-sharing contract based on quantity policy betweenthe manufacturer and the distributor and the traditionalrevenue-sharing contract between the distributor and theretailer

Supposing the manufacturer provides the distributor therevenue-sharing contract (120601

1 119908119888119898(119876)) and

119908119888

119898(119876) = minus (1 minus 1206011) 119888 + [1 minus (1 minus 1206011) (1 minus 1206012)]

119892 (119890lowast)

119876

+ (1 minus 1206011) 119888119903+ 119888119898

(22)

Here 119890lowast is the optimal effort of the supply chainTaking (22) into the manufacturerrsquos profit function we

can get

Π119888

119898(119876 119890)

= (1 minus 1206011) [(1 minus 120601

2) (119901119878 (119876 119890) + V119868 (119876 119890)) + 119908119888

119889119876]

+ 119908119888

119898(119876)119876 minus 119888

119898119876 minus 119892 (119890)

= (1 minus 1206011) (1 minus 120601

2) (119901 minus V) 119878 (119876 119890)

+ [1 minus (1 minus 1206011) (1 minus 120601

2)] 119892 (119890

lowast)

minus (1 minus 1206011) [119888 minus 119888

119903minus (1 minus 120601

2) V minus 119908119888

119889] 119876 minus 119892 (119890)

(23)

According to the paper of Giannoccaro and Pontrandolfo[12] without considering demand dependent effort the ratiothat the manufacturerrsquos profit shares the profit of the supplychain is (1 minus 120601

1)(1 minus 120601

2) Here supposing when only the

manufacturer bears the effort cost the ratio that his profitshares the profit of the whole supply chain is also (1 minus 120601

1)(1 minus

1206012) It is easily known that the effort of the manufacturer

would be equal to the optimal effort of the supply chainnamely

Π119888

119898(119876 119890)

= (1 minus 1206011) [(1 minus 120601

2) (119901119878 (119876 119890

lowast) + V119868 (119876 119890lowast)) + 119908119888

119889119876]

+ 119908119888

119898(119876)119876 minus 119888

119898119876 minus 119892 (119890

lowast)

= (1 minus 1206011) (1 minus 120601

2)Π119905 (119876 119890)

(24)

From (24) we can get

119908119888

119889= 1206012119888 minus 119888119903 (25)

Taking (25) and (22) into the profit function of the supplychain members we can get

Π119888

119903(119876 119890) = 1206012 [Π119905 (119876 119890) + 119892 (119890)]

Π119888

119889(119876 119890) = 1206011 (1 minus 1206012)Π119905 (119876 119890) minus 1206012119892 (119890)

Π119888

119898(119876 119890) = (1 minus 1206011) (1 minus 1206012)Π119905

(26)

From (26) it can be seen that the profit functions ofthe supply chain members are all affine functions of thewhole supply chainrsquos profit function So in this conditionthe revenue-sharing contract can coordinate the three-levelsupply chain

Therefore if we implement the revenue-sharing contractbased on quantity discount policy between the retailer andthe distributor and the traditional revenue-sharing contractbetween the distributor and the manufacturer and the con-tract parameters which satisfy (22) and (25) the improvedrevenue-sharing can coordinate the three-level supply chain

Compared with the paper of Giannoccaro and Pontran-dolfo [12] it can be seen that the retailerrsquos profit increases1206012119892(119890lowast) while that of the distributor decreases 120601

2119892(119890lowast)

422 Implement among Supply Chain Members This prob-lem can be described as follows we implement both therevenue-sharing contract based on quantity discount policybetween the retailer and the distributor and between thedistributor and the manufacturer

Suppose the distributor provides the retailer the revenue-sharing contract (120601

2 119908119888119889(119876)) and the manufacturer provides

Discrete Dynamics in Nature and Society 7

the distributors the revenue-sharing contract (1206011 119908119888119898(119876))

Here 119890lowast is the optimal effort of the supply chain and

119908119888

119889(119876) = 120601

2119888 minus (1 minus 120601

2)119892 (119890lowast)

119876minus 119888119903

119908119888

119898(119876) = 120601

1119888 +

119892 (119890lowast)

119876minus 1206011119888119903minus 119888119889

(27)

Taking (27) into the profit function of the manufacturerwe can get

Π119888

119898(119876 119890)

= (1 minus 1206011) [(1 minus 120601

2) (119901119878 (119876 119890) + V119868 (119876 119890)) + 119908119888

119889(119876)119876]

+ 119908119888

119898(119876)119876 minus 119888

119898119876 minus 119892 (119890)

= (1 minus 1206011) (1 minus 120601

2) (119901 minus V) 119878 (119876 119890)

minus (1 minus 1206011) (1 minus 120601

2) (119888 minus V) 119876

+ [1 minus (1 minus 1206011) (1 minus 120601

2)] 119892 (119890

lowast) minus 119892 (119890)

(28)

Here we also suppose that the ratio that his profit sharesthe profit of the whole supply chain is also (1 minus 120601

1)(1 minus 120601

2)

It is easy to know that the optimal effort of the manufacturerwould be equal to that of the supply chain In this conditionthe profit function of the manufacturer can be described as

Π119888

119898(119876 119890) = (1 minus 1206011) (1 minus 1206012)Π119905 (119876 119890) (29)

Taking (27) into the profit function of the retailer we canget

Π119888

119903(119876 119890) = 1206012 [119901119878 (119876 119890

lowast) + V119868 (119876 119890lowast)] minus 119888

119903119876 minus 119908

119888

119889(119876)119876

= 1206012(119901 minus V) 119878 (119876 119890lowast)minus120601

2 (119888 minus V) 119876 + (1 minus 1206012) 119892 (119890)

(30)

From (30) it is easy to know that the optimal orderquantity of the retailer is equal to that of the supply chainSo (30) can be described as

Π119888

119903(119876 119890) = 1206012Π119905 (119876 119890) + 119892 (119890) (31)

Taking (27) into the profit function of the distributor itcan get

Π119888

119889(119876 119890)

= 1206011[(1 minus 120601

2) (119901119878 (119876 119890

lowast) + V119868 (119876 119890lowast)) + 119908119888

119889(119876)119876]

minus 119888119889119876 minus 119908

119888

119898(119876)119876 minus 119892 (119890

lowast)

= 1206011(1 minus 120601

2) (119901 minus V) 119878 (119876 119890lowast) minus 120601

1(1 minus 120601

2) (119888 minus V) 119876

minus (1 + 1206011(1 minus 120601

2)) 119892 (119890)

(32)

From (32) it can be seen that the optimal order quantityof the distributor is equal to that of the retailer so (32) canalso be described as

Π119888

119889(119876 119890) = 1206011 (1 minus 1206012)Π119905 (119876 119890) minus 119892 (119890) (33)

From (29) (31) and (33) it can be seen that the profitfunctions of the supply chainmembers are all affine functionsof the whole supply chainrsquos profit function So in this condi-tion the revenue-sharing contract can coordinate the three-level supply chain Therefore if we implement the revenue-sharing contract based on quantity discount policy bothbetween the retailer and the distributor and between thedistributor and themanufacturer and the contract parameterswhich satisfy (27) the improved revenue-sharing can coordi-nate the three-level supply chain

423 Model Optimization Similarly suppose the marketdemand 119883(119890 120585) is the function of effort 119890 and random factor120585 and 120585 is independent of 119890 119891(120585) and 119865(120585) are respectivelyprobability density function and differentiable cumulateddistribution function of 120585 Here suppose 119883(119890 120585) = 119910(119890) sdot 120585As the influence of effort on marker demand is diminishingmarginal utility we can suppose 119910(119890) is the monotoneincreasing and concave function of effort 119890 namely1199101015840(119890) gt 0and 11991010158401015840(119890) le 0

When the market demand satisfies 119883(119890 120585) = 119910(119890) sdot 120585 wecan get

119878 (119876 119890) = 119876 minus int119876

0

119865 (119909 | 119890) 119889119909 = 119876 minus int119876

0

119865(119909

119910 (119890)) 119889119909

= 119876 minus 119910 (119890) int119876119910(119890)

0

119865 (119905) 119889119905

(34)

So the profit function of supply chain can be described as

Π119905 (119876 119890) = (119901 minus V) 119878 (119876 119890) minus (119888 minus V) 119876 minus 119892 (119890)

= (119901 minus 119888)119876 minus (119901 minus V) 119910 (119890) int119876119910(119890)

0

119865 (119905) 119889119905 minus 119892 (119890)

(35)

Given effort 119890 it can be seen that Π119905(119876 119890) is the concave

function of order quantity 119876 So the optimal order quantity119876lowast should satisfy

120597Π119905(119876lowast 119890)

120597119876= (119901 minus 119888) minus (119901 minus V) 119865(

119876lowast

119910 (119890)) = 0 (36)

From (36) we can get

119876lowast= 119876lowast(119890) = 119910 (119890) 119865

minus1(119901 minus 119888

119901 minus V) (37)

Then the profit function of supply chain can be describedas

Π119905(119876lowast 119890) = (119901 minus 119888) 119910 (119890) 119865

minus1(119901 minus 119888

119901 minus V)

minus (119901 minus V) 119910 (119890) int119865minus1

((119901minus119888)(119901minusV))

0

119865 (119905) 119889119905 minus 119892 (119890)

= (119901 minus V) 119910 (119890) int119865minus1

((119901minus119888)(119901minusV))

0

119905119891 (119905) 119889119905 minus 119892 (119890)

(38)

8 Discrete Dynamics in Nature and Society

Table 2 The parameters of supply chain in different modes with only manufacturer bearing effort cost

Revenue-sharing contract based onquantity discount policy Traditional

revenue-sharing contract Wholesale price contractA B

Optimal effort 119890 1962 1962 397 379Optimal order quantity 119876 1674 1674 339 288Profit of the whole supply chain Π

119905 1924628 1924628 700833 654436

According to the supposition Π119905(119876lowast 119890) is the concave

function of effort 119890 So the optimal effort 119890lowast of the supplychain should satisfy

120597Π119905(119876lowast 119890lowast)

120597119890= (119901 minus V) 1199101015840 (119890lowast) int

119865minus1

((119901minus119888)(119901minusV))

0

119905119891 (119905) 119889119905

minus 1198921015840(119890lowast) = 0

(39)

From (39) it can easy to get that the optimal effort 119890lowast ofthe supply chain should satisfy

(119901 minus V) 1199101015840 (119890lowast) int119865minus1

((119901minus119888)(119901minusV))

0

119905119891 (119905) 119889119905 = 1198921015840(119890lowast) (40)

The method to solve the expression of 119890lowast in (40) is thesame as the method in (21)

424 An Example Suppose a supply chain is made up ofa retailer a distributor and a manufacturer and the supplychain parameters are 119888

119903= 1 119888119889= 2 119888119898= 5 V = 3 119908119889

119889= 18

119908119889119898= 10 119901 = 35 120601

1= 1206012= 055 and 119886 = 100

Here assume the relationship between demand and effortsatisfies multiplication form and 119892(119890) = 119886119890

22 119910(119890) = 119890120585 satisfies uniform distribution in [60 90] It is easy to get119891(120585) = 130 119865(120585) = (120585 minus 60)30 and 119865minus1(120585) = 60 + 30120585

(1) The revenue-sharing contract based on quantity dis-count policy

From the discussion above we know that the revenue-sharing contract based on quantity discount policy cancoordinate the supply chain whether it is implemented in oneor two transactions According to the formulas mentionedabove it is easy to get the optimal effort of the manufacturerthe optimal order quantity of the retailer and the profitof the whole supply chain Here 119890lowast = 1962 119876lowast =

1674 Π119905(119876lowast 119890lowast) = 1924628

(2) The traditional revenue-sharing contract

Similarly under the traditional revenue-sharing contractthe optimal effort of the manufacturer and order quantityof the retailer and the profit of the whole supply chain arerespectively 119890119898119888 = 397 119876119903119888 = 339 and Π

119905(119876119903119888 119890119898119888) =

700833

(3) The decentralized supply chain (the wholesale pricecontract)

Similarly under the traditional revenue-sharing contractthe optimal effort of the manufacturer and the order quantityof the retailer and the profit of the whole supply chain arerespectively 119890119898119889 = 379 119876119903119889 = 288 and Π

119905(119876119903119889 119890119889119898) =

654436Table 2 shows the parameters of the supply chain in

different modes A means that we implement the revenue-sharing contract based on quantity discount policy onlybetween the retailer and the distributor B means that weimplement the revenue-sharing contract based on quantitydiscount policy in two transactions of the supply chain

Table 2 shows that the manufacturerrsquos effort in therevenue-sharing contract based on quantity discount policyis the highest in the three modes which can make thesupply chain maximize its profit and the revenue-sharingcontract based on quantity discount policy is better than thetraditional revenue-sharing contract with effort dependentdemand the manufacturerrsquos effort in traditional revenue-sharing contract is higher than that of the supply chain inthe wholesale price contract and retailerrsquos order quantity isstill higher than that in wholesale price contract due to theincentive of the revenue-sharing contract so the profit ofthe supply chain in the traditional revenue-sharing contractis still higher than that in wholesale price contract So itcan be seen that the traditional revenue-sharing contract isbetter than thewholesale price contract with effort dependentdemand

43TheRetailer Bearing the Effort Cost with theManufacturerSuppose the effort cost that the retailer bears is 120572119892(119890) (0 le

120572 le 1) then the manufacturer bears (1 minus 120572)119892(119890) effort costIf 120572 = 0 it means that only the manufacturer bears the effortcost if 120572 = 1 it means that only the retailer bears the effortcost

In this condition we find that the traditional revenue-sharing contract fails to coordinate the supply chain whetherit is implemented in one transaction phase or in two transac-tion phases of the three-level supply chain The reason lies inthat the distributor does not bear the effort cost but he can getpartial profit of thewhole supply chain which is unfair for theretailer and themanufacturer So the effort level of the retailerand the manufacturer will be lower than the effort level of thecentralized supply chain Now the following discusses howto design the improved revenue-sharing implemented in twotransaction phases to coordinate the supply chain

Suppose the distributor provides the retailer the revenue-sharing contract (120601

2 119908119888119889(119876 119890)) and the manufacturer pro-

vides the distributors the revenue-sharing contract (1206011

Discrete Dynamics in Nature and Society 9

119908119888119898(119876 119890)) Then the profit functions of the supply chain

members can be described as

Π119888

119903(119876 119890) = 1206012 (119901 minus V) 119878 (119876 119890)

minus [119888119903+ 119908119888

119889(119876 119890) minus 1206012V] 119876 minus 120572119892 (119890)

Π119888

119889(119876 119890) = 1206011 (1 minus 1206012) (119901 minus V) 119878 (119876 119890)

minus [119888119889+ 119908119888

119898(119876 119890) minus 1206011 (1 minus 1206012) V

minus1206011119908119888

119889(119876 119890)] 119876

Π119888

119898(119876 119890) = (1 minus 1206011) (1 minus 1206012) (119901 minus V) 119878 (119876 119890)

minus [119888119898minus 119908119888

119898(119876 119890) minus (1 minus 1206011) (1 minus 1206012) V

minus (1 minus 1206011) 119908119888

119889(119876 119890)] 119876

minus (1 minus 120572) 119892 (119890)

(41)

If the improved revenue-sharing contract can coordinatethe supply chain the optimal effort level 119890lowast of the retailerand the manufacturer should satisfy (2) Namely the optimalorder (production) quantity 119876

lowast of the retailer and themanufacturer should satisfy (3) Then we can get

120597Π119888119903(119876 119890lowast)

120597119890=120597Π119888

119898(119876 119890lowast)

120597119890= 0

(119901 minus V)120597119878 (119876 119890

lowast)

120597119890minus 1198921015840(119890lowast) = 0

120597Π119888119903(119876lowast 119890)

120597119876=120597Π119888

119898(119876lowast 119890)

120597119876= 0

(119901 minus V)120597119878 (119876lowast 119890)

120597119876minus (119888 minus V) = 0

(42)

From (42) we can get

119908119888

119889(119876 119890) = 1206012 (119888 minus V) minus (1 minus 120572)

119892 (119890)

119876minus 119888119903+ 1206012V

119908119888

119898(119876 119890) = 1206011 (119888 minus V) + 1206011 (1 minus 120572)

119892 (119890)

119876minus 1206011119888119903+ 1206011V minus 119888119889

(43)

Taking (43) into the profit functions of the supply chainmembers (41) we can get

Π119888

119903(119876 119890) = 1206012Π119905 (119876 119890)

Π119888

119889(119876 119890) = 1206011 (1 minus 1206012)Π119905 (119876 119890)

Π119888

119898(119876 119890) = (1 minus 1206011) (1 minus 1206012)Π119905 (119876 119890)

(44)

We find that (44) equals (11) That is to say when theretailer bears the effort with the manufacturer the improvedrevenue-sharing contract can coordinate the three-level sup-ply chain

Therefore if we implement the revenue-sharing contractbased on quantity discount policy among the supply chain

members and the contract parameters which satisfy (43)the improved revenue-sharing can coordinate the three-levelsupply chain

The method to determine the optimal order quantity 119876lowastand the optimal effort 119890lowast can follow the methods given inSections 413 and 423

5 Conclusion

The traditional revenue-sharing contract cannot coordinatethree-level supply chain with sales effort dependent demandThe paper proposes an improved revenue-sharing contractbased on quantity discount policy which can be implementedin one transaction or two transactions of three-level supplychain to coordinate the supply chain The paper also showsthat the profit of the supply chain members depends on thevalue of the contract parameters which is determined by thestatus of the members in supply chain and their bargainingpower It also gave the method to determine the optimalorder quantity and the optimal effort which can provide thedecision makers of supply chain with decision support Butit also should be noticed that this research is carried outin the symmetrical information condition So the next stepis to carry out the research in the asymmetric informationcondition

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This research was supported by the Ministry of Educationof China Grant-in-aid for Humanity and social ScienceResearch (no 10YJC630188) and the Fundamental ResearchFunds for the Central Universities (nos 2012B13914 and2012B13814)

References

[1] G P Cachon and M A Lariviere ldquoSupply chain coordinationwith revenue-sharing contracts strengths and limitationsrdquoManagement Science vol 51 no 1 pp 30ndash44 2005

[2] J H Mortimer The Effects of Revenue-Sharing Contracts onWelfare in Vertically SeparatedMarkets Evidence from the VideoRental Industry University of California at Los Angeles 2000

[3] J D Dana Jr and K E Spier ldquoRevenue sharing and verticalcontrol in the video rental industryrdquo Journal of IndustrialEconomics vol 49 no 3 pp 223ndash245 2001

[4] B Pasternack ldquoUsing revenue-sharing to achieve channel coor-dination for a newsboy type inventory modelrdquo in Supply ChainManagement Models Applications and Research DirectionsJ Geunes P Panos and H Edwin Eds vol 62 pp 271ndash290 Kluwer Academic Publishers DordrechtTheNetherlands2002

[5] Y Wang L Jiang and Z J Shen ldquoChannel performance underconsignment contract with revenue-sharingrdquoManagement Sci-ence vol 50 no 1 pp 34ndash47 2004

10 Discrete Dynamics in Nature and Society

[6] Y Gerchak and Y Wang ldquoRevenue-sharing vs wholesale-price contracts in assembly systems with random demandrdquoProduction andOperationsManagement vol 13 no 1 pp 23ndash332004

[7] C Koulamas ldquoA newsvendor problemwith revenue sharing andchannel coordinationrdquo Decision Sciences vol 37 no 1 pp 91ndash100 2006

[8] S S Chauhan and J Proth ldquoAnalysis of a supply chain partner-ship with revenue sharingrdquo International Journal of ProductionEconomics vol 97 no 1 pp 44ndash51 2005

[9] C T Linh and Y S Hong ldquoChannel coordination through arevenue sharing contract in a two-period newsboy problemrdquoEuropean Journal of Operational Research vol 198 no 3 pp822ndash829 2009

[10] K Pan K K Lai S C H Leung andD Xiao ldquoRevenue-sharingversus wholesale price mechanisms under different channelpower structuresrdquo European Journal of Operational Researchvol 203 no 2 pp 532ndash538 2010

[11] D G Qu and Y J Guo ldquoCoordination of supply chain withhybrid distribution channels when retailerrsquos demand relies onits sales effortrdquo Chinese Journal of Management Science vol 21no 3 pp 19ndash23 2008

[12] I Giannoccaro and P Pontrandolfo ldquoSupply chain coordinationby revenue sharing contractsrdquo International Journal of Produc-tion Economics vol 89 no 2 pp 131ndash139 2004

[13] S F Ji M J Liu W Ding and X Huang ldquoCoordination ofthree-level supply chain and revenue sharing contractrdquo Journalof Northeastern University vol 29 no 11 pp 1652ndash1656 2008

[14] L Hu Z Jiang M Meng and J Qin ldquoCoordination of a three-level supply chain through revenue sharingrdquo Journal of HarbinEngineering University vol 29 no 2 pp 198ndash203 2008

[15] Y L Hou D Q Zhou and B Y Tian ldquoOrdering decisionson supply chain under revenue sharing policiesrdquo IndustrialEngineering Journal vol 12 no 1 pp 24ndash27 2009

[16] Y He Q Wu and L Zhao ldquoQuantity flexibility contract witheffort and price-dependent demandrdquo Systems Engineering andElectronics vol 29 no 12 pp 2056ndash2059 2007

[17] P Desai andK Srinivasan ldquoDemand signaling under unobserv-able effort in franchisingrdquo Management Science vol 20 no 4pp 2608ndash2623 1995

[18] W Chu and P Desai ldquoChannel coordination mechanisms forcustomer satisfactionrdquoMarketing Science vol 21 no 6 pp 475ndash490 1995

[19] T A Taylor ldquoSupply chain coordination under channel rebateswith sales effort effectsrdquoManagement Science vol 48 no 8 pp992ndash1007 2002

[20] Z Xu D L Zhu and W G Zhu ldquoSupply chain coordinationunder buy-back contract with sales effort effectsrdquo System Engi-neering Theory and Practice vol 28 no 4 pp 1ndash11 2008

[21] G P Cachon ldquoSupply chain coordination with contractsrdquoin Handbooks in Operations Research and Management Sci-ence Supply Chain Management North-Holland Elseriver SanDiego Calif USA 2000

[22] Y He D Yang andQWu ldquoRevenue-sharing contract of supplychain with effort dependent demandrdquo Computer IntegratedManufacturing Systems vol 12 no 11 pp 1865ndash1868 2006

[23] Y He X Zhao L Zhao and J He ldquoCoordinating a supply chainwith effort and price dependent stochastic demandrdquo AppliedMathematical Modelling vol 33 no 6 pp 2777ndash2790 2009

[24] B Y Hu and X Y Wang ldquoSupply chain revenue-sharingevolvement-contract with sales effort effectsrdquo Journal of Indus-trial Engineering andEngineeringManagement vol 29 no 2 pp35ndash39 2010

[25] J Chen and J F Chen ldquoStudy on revenue sharing contractin virtual enterprisesrdquo Journal of Systems Science and SystemsEngineering vol 15 no 1 pp 95ndash113 2006

[26] Y Wang and Y Gerchak ldquoSupply Chain Coordination whendemand is shelf-space dpendentrdquo Manufacturing and ServiceOperations Management vol 3 no 1 pp 82ndash87 2001

[27] H Krishnan R Kapuscinski and D A Butz ldquoCoordinatingcontracts for decentralized supply chains with retailer promo-tional effortrdquo Management Science vol 50 no 1 pp 48ndash632004

[28] Y He Supply Chain Contract Models with Stochastic DemandDalian University of Technology Dalian China 2004

[29] L Yao Supply Chain Modeling Pricing Contracts and Coordi-nation Chinese University of Hong Kong Hong Kong 2002

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Discrete Dynamics in Nature and Society 5

413 Model Optimization From the discussion mentionedabove we can come to a conclusion that the traditionalrevenue-sharing contract cannot coordinate three-level sup-ply chain with sales effort dependent demand Howeverthe improved revenue-sharing contract based on quantitydiscount policy can coordinate three-level supply chain Andin this condition the optimal order quantity and optimaleffort of retailer are equal to those of the centralized supplychainThen it needs to determine the optimal order quantityand the optimal effort to maximize the profit of supply chainnamely maxΠ

119905(119876 119890) The following gives the method to

determine the optimal order quantity 119876lowast and the optimaleffort 119890lowast

Suppose the market demand 119883(119890 120585) is the function ofeffort 119890 and random factor 120585 and 120585 is independent of 119890 119891(120585)and 119865(120585) are respectively probability density function anddifferentiable cumulated distribution function of 120585 Usuallywe can use two forms to describe how the effort affectsdemand addition form and multiplication form [29] In thispaper suppose 119883(119890 120585) = 119910(119890) + 120585 Because the influence ofeffort on marker demand is diminishing marginal utility wecan suppose 119910(119890) is the monotone increasing and concavefunction of effort 119890 namely 1199101015840(119890) gt 0 11991010158401015840(119890) le 0

When the market demand satisfies119883(119890 120585) = 119910(119890) + 120585 wecan get

119878 (119876 119890) = 119876 minus int119876

0

119865 (119909 | 119890) 119889119909 = 119876 minus int119876

119910(119890)

119865 (119909 minus 119910 (119890)) 119889119909

= 119876 minus int119876minus119910(119890)

0

119865 (119905) 119889119905

(16)

So the profit function of supply chain can be described as

Π119905 (119876 119890) = (119901 minus V) 119878 (119876 119890) minus (119888 minus V) 119876 minus 119892 (119890)

= (119901 minus 119888)119876 minus (119901 minus V) int119876minus119910(119890)

0

119865 (119905) 119889119905 minus 119892 (119890)

(17)

Given effort 119890 it can be seen that Π119905(119876 119890) is the concave

function of order quantity 119876 So the optimal order quantity119876lowast should satisfy

120597Π119905(119876lowast 119890)

120597119876= (119901 minus 119888) minus (119901 minus V) 119865 (119876lowast minus 119910 (119890)) = 0 (18)

From (18) we can get

119876lowast= 119876lowast(119890) = 119910 (119890) + 119865

minus1(119901 minus 119888

119901 minus V) (19)

Then the profit function of supply chain can be describedasΠ119905(119876lowast 119890)

= (119901 minus 119888)119876lowastminus (119901 minus V) int

119876lowast

minus119910(119890)

0

119865 (119905) 119889119905 minus 119892 (119890)

= (119901 minus 119888) 119910 (119890) + (119901 minus V) int119865minus1

((119901minus119888)(119901minusV))

0

119905119891 (119905) 119889119905 minus 119892 (119890)

(20)

From (20) it can be seen that there is only one variable119890 in Π

119905(119876lowast 119890) Therefore the next work is to determine the

optimal effort 119890lowast to maximize Π119905(119876lowast 119890) From (20) it can

get 120597Π119905(119876lowast 119890)120597119890 = (119901 minus 119888)1199101015840(119890) minus 1198921015840(119890) 1205972Π

119905(119876lowast 119890)1205971198902 =

(119901 minus 119888)11991010158401015840(119890) minus 11989210158401015840(119890) Because 11989210158401015840(119890) gt 0 and 11991010158401015840(119890) le

0 1205972Π119905(119876lowast 119890)1205971198902 lt 0 namely Π

119905(119876lowast 119890) is the concave

function of effort 119890 That is to say there is an optimal effort119890lowast which can make Π

119905(119876lowast 119890) achieve the maximum By

120597Π119905(119876lowast 119890)120597119890 = 0 we get

(119901 minus 119888) 1199101015840(119890lowast) minus 1198921015840(119890lowast) = 0 (21)

Then the optimal effort 119890lowast should satisfy (21) Equation(21) is the first-order differential equation of the optimal effort119890lowast By solving (21) we can get the expression of 119890lowast

414 An Example Suppose a supply chain is made up ofa retailer a distributor and a manufacturer and the supplychain parameters are 119888

119903= 1 119888119889= 2 119888119898= 5 V = 3 119908119889

119889= 18

119908119889119898= 10 119901 = 35 120601

1= 055 and 120601

2= 055

Here assume the relationship between demand and effortsatisfies addition form and 119892(119890) = 119890

22 119910(119890) = 119890 120585 satisfiesuniform distribution in [50 100] It is easy to get119891(120585) = 150119865(120585) = (120585 minus 50)50 and 119865minus1(120585) = 50 + 50120585

(1) The revenue-sharing contract based on quantity dis-count policy

From the discussion above we know that the revenue-sharing contract based on quantity discount policy cancoordinate the supply chain whether it is implemented in oneor two transactions According to the formulas mentionedabove it is easy to get the optimal effort the optimal orderquantity and the profit of the whole supply chain Here 119890lowast =27 119876lowast = 119 and Π

119905(119876lowast 119890lowast) = 22725

(2) The traditional revenue-sharing contract

Similarly under the traditional revenue-sharing contractthe optimal effort and order quantity of the retailer and theprofit of the whole supply chain are respectively 119890119903 = 1485119876119903 = 107 and Π

119905(119876119903 119890119903) = 22027

(3) The decentralized supply chain (the wholesale pricecontract)

Similarly under the traditional revenue-sharing contractthe optimal effort and order quantity of the retailer and theprofit of the whole supply chain are respectively 119890119889 = 16119876119889 = 91 and Π

119905(119876119889 119890119889) = 2137

Table 1 shows the parameters of the supply chain indifferent mode A means that we implement the revenue-sharing contract based on quantity discount policy onlybetween the retailer and the distributor B means that weimplement the revenue-sharing contract based on quantitydiscount policy in two transactions of the supply chain

Table 1 shows that the retailerrsquos effort in the revenue-sharing contract based on quantity discount policy is thehighest in the three modes which can make the supply chain

6 Discrete Dynamics in Nature and Society

Table 1 The parameters of supply chain in different modes with only retailer bearing effort cost

Revenue-sharing contract based onquantity discount policy Traditional

revenue-sharing contract Wholesale price contractA B

Optimal effort 119890 27 27 1485 16Optimal order quantity 119876 119 119 107 91Profit of the whole supply chain Π

119905 22725 22725 22027 2137

maximize its profit the retailerrsquos effort in traditional revenue-sharing contract is the lowest in the threemodes but his orderquantity is still higher than that in wholesale price contractdue to the incentive of the revenue-sharing contract So theprofit of the supply chain in the traditional revenue-sharingcontract is still higher than that in wholesale price contractSo it can be seen that the traditional revenue-sharing contractis better than the wholesale price contract with sales effortdependent demand

42 Only Manufacturer Bearing the Effort Cost

421 Only Implement between Manufacturer and DistributorThis problem can be described as follows we implement therevenue-sharing contract based on quantity policy betweenthe manufacturer and the distributor and the traditionalrevenue-sharing contract between the distributor and theretailer

Supposing the manufacturer provides the distributor therevenue-sharing contract (120601

1 119908119888119898(119876)) and

119908119888

119898(119876) = minus (1 minus 1206011) 119888 + [1 minus (1 minus 1206011) (1 minus 1206012)]

119892 (119890lowast)

119876

+ (1 minus 1206011) 119888119903+ 119888119898

(22)

Here 119890lowast is the optimal effort of the supply chainTaking (22) into the manufacturerrsquos profit function we

can get

Π119888

119898(119876 119890)

= (1 minus 1206011) [(1 minus 120601

2) (119901119878 (119876 119890) + V119868 (119876 119890)) + 119908119888

119889119876]

+ 119908119888

119898(119876)119876 minus 119888

119898119876 minus 119892 (119890)

= (1 minus 1206011) (1 minus 120601

2) (119901 minus V) 119878 (119876 119890)

+ [1 minus (1 minus 1206011) (1 minus 120601

2)] 119892 (119890

lowast)

minus (1 minus 1206011) [119888 minus 119888

119903minus (1 minus 120601

2) V minus 119908119888

119889] 119876 minus 119892 (119890)

(23)

According to the paper of Giannoccaro and Pontrandolfo[12] without considering demand dependent effort the ratiothat the manufacturerrsquos profit shares the profit of the supplychain is (1 minus 120601

1)(1 minus 120601

2) Here supposing when only the

manufacturer bears the effort cost the ratio that his profitshares the profit of the whole supply chain is also (1 minus 120601

1)(1 minus

1206012) It is easily known that the effort of the manufacturer

would be equal to the optimal effort of the supply chainnamely

Π119888

119898(119876 119890)

= (1 minus 1206011) [(1 minus 120601

2) (119901119878 (119876 119890

lowast) + V119868 (119876 119890lowast)) + 119908119888

119889119876]

+ 119908119888

119898(119876)119876 minus 119888

119898119876 minus 119892 (119890

lowast)

= (1 minus 1206011) (1 minus 120601

2)Π119905 (119876 119890)

(24)

From (24) we can get

119908119888

119889= 1206012119888 minus 119888119903 (25)

Taking (25) and (22) into the profit function of the supplychain members we can get

Π119888

119903(119876 119890) = 1206012 [Π119905 (119876 119890) + 119892 (119890)]

Π119888

119889(119876 119890) = 1206011 (1 minus 1206012)Π119905 (119876 119890) minus 1206012119892 (119890)

Π119888

119898(119876 119890) = (1 minus 1206011) (1 minus 1206012)Π119905

(26)

From (26) it can be seen that the profit functions ofthe supply chain members are all affine functions of thewhole supply chainrsquos profit function So in this conditionthe revenue-sharing contract can coordinate the three-levelsupply chain

Therefore if we implement the revenue-sharing contractbased on quantity discount policy between the retailer andthe distributor and the traditional revenue-sharing contractbetween the distributor and the manufacturer and the con-tract parameters which satisfy (22) and (25) the improvedrevenue-sharing can coordinate the three-level supply chain

Compared with the paper of Giannoccaro and Pontran-dolfo [12] it can be seen that the retailerrsquos profit increases1206012119892(119890lowast) while that of the distributor decreases 120601

2119892(119890lowast)

422 Implement among Supply Chain Members This prob-lem can be described as follows we implement both therevenue-sharing contract based on quantity discount policybetween the retailer and the distributor and between thedistributor and the manufacturer

Suppose the distributor provides the retailer the revenue-sharing contract (120601

2 119908119888119889(119876)) and the manufacturer provides

Discrete Dynamics in Nature and Society 7

the distributors the revenue-sharing contract (1206011 119908119888119898(119876))

Here 119890lowast is the optimal effort of the supply chain and

119908119888

119889(119876) = 120601

2119888 minus (1 minus 120601

2)119892 (119890lowast)

119876minus 119888119903

119908119888

119898(119876) = 120601

1119888 +

119892 (119890lowast)

119876minus 1206011119888119903minus 119888119889

(27)

Taking (27) into the profit function of the manufacturerwe can get

Π119888

119898(119876 119890)

= (1 minus 1206011) [(1 minus 120601

2) (119901119878 (119876 119890) + V119868 (119876 119890)) + 119908119888

119889(119876)119876]

+ 119908119888

119898(119876)119876 minus 119888

119898119876 minus 119892 (119890)

= (1 minus 1206011) (1 minus 120601

2) (119901 minus V) 119878 (119876 119890)

minus (1 minus 1206011) (1 minus 120601

2) (119888 minus V) 119876

+ [1 minus (1 minus 1206011) (1 minus 120601

2)] 119892 (119890

lowast) minus 119892 (119890)

(28)

Here we also suppose that the ratio that his profit sharesthe profit of the whole supply chain is also (1 minus 120601

1)(1 minus 120601

2)

It is easy to know that the optimal effort of the manufacturerwould be equal to that of the supply chain In this conditionthe profit function of the manufacturer can be described as

Π119888

119898(119876 119890) = (1 minus 1206011) (1 minus 1206012)Π119905 (119876 119890) (29)

Taking (27) into the profit function of the retailer we canget

Π119888

119903(119876 119890) = 1206012 [119901119878 (119876 119890

lowast) + V119868 (119876 119890lowast)] minus 119888

119903119876 minus 119908

119888

119889(119876)119876

= 1206012(119901 minus V) 119878 (119876 119890lowast)minus120601

2 (119888 minus V) 119876 + (1 minus 1206012) 119892 (119890)

(30)

From (30) it is easy to know that the optimal orderquantity of the retailer is equal to that of the supply chainSo (30) can be described as

Π119888

119903(119876 119890) = 1206012Π119905 (119876 119890) + 119892 (119890) (31)

Taking (27) into the profit function of the distributor itcan get

Π119888

119889(119876 119890)

= 1206011[(1 minus 120601

2) (119901119878 (119876 119890

lowast) + V119868 (119876 119890lowast)) + 119908119888

119889(119876)119876]

minus 119888119889119876 minus 119908

119888

119898(119876)119876 minus 119892 (119890

lowast)

= 1206011(1 minus 120601

2) (119901 minus V) 119878 (119876 119890lowast) minus 120601

1(1 minus 120601

2) (119888 minus V) 119876

minus (1 + 1206011(1 minus 120601

2)) 119892 (119890)

(32)

From (32) it can be seen that the optimal order quantityof the distributor is equal to that of the retailer so (32) canalso be described as

Π119888

119889(119876 119890) = 1206011 (1 minus 1206012)Π119905 (119876 119890) minus 119892 (119890) (33)

From (29) (31) and (33) it can be seen that the profitfunctions of the supply chainmembers are all affine functionsof the whole supply chainrsquos profit function So in this condi-tion the revenue-sharing contract can coordinate the three-level supply chain Therefore if we implement the revenue-sharing contract based on quantity discount policy bothbetween the retailer and the distributor and between thedistributor and themanufacturer and the contract parameterswhich satisfy (27) the improved revenue-sharing can coordi-nate the three-level supply chain

423 Model Optimization Similarly suppose the marketdemand 119883(119890 120585) is the function of effort 119890 and random factor120585 and 120585 is independent of 119890 119891(120585) and 119865(120585) are respectivelyprobability density function and differentiable cumulateddistribution function of 120585 Here suppose 119883(119890 120585) = 119910(119890) sdot 120585As the influence of effort on marker demand is diminishingmarginal utility we can suppose 119910(119890) is the monotoneincreasing and concave function of effort 119890 namely1199101015840(119890) gt 0and 11991010158401015840(119890) le 0

When the market demand satisfies 119883(119890 120585) = 119910(119890) sdot 120585 wecan get

119878 (119876 119890) = 119876 minus int119876

0

119865 (119909 | 119890) 119889119909 = 119876 minus int119876

0

119865(119909

119910 (119890)) 119889119909

= 119876 minus 119910 (119890) int119876119910(119890)

0

119865 (119905) 119889119905

(34)

So the profit function of supply chain can be described as

Π119905 (119876 119890) = (119901 minus V) 119878 (119876 119890) minus (119888 minus V) 119876 minus 119892 (119890)

= (119901 minus 119888)119876 minus (119901 minus V) 119910 (119890) int119876119910(119890)

0

119865 (119905) 119889119905 minus 119892 (119890)

(35)

Given effort 119890 it can be seen that Π119905(119876 119890) is the concave

function of order quantity 119876 So the optimal order quantity119876lowast should satisfy

120597Π119905(119876lowast 119890)

120597119876= (119901 minus 119888) minus (119901 minus V) 119865(

119876lowast

119910 (119890)) = 0 (36)

From (36) we can get

119876lowast= 119876lowast(119890) = 119910 (119890) 119865

minus1(119901 minus 119888

119901 minus V) (37)

Then the profit function of supply chain can be describedas

Π119905(119876lowast 119890) = (119901 minus 119888) 119910 (119890) 119865

minus1(119901 minus 119888

119901 minus V)

minus (119901 minus V) 119910 (119890) int119865minus1

((119901minus119888)(119901minusV))

0

119865 (119905) 119889119905 minus 119892 (119890)

= (119901 minus V) 119910 (119890) int119865minus1

((119901minus119888)(119901minusV))

0

119905119891 (119905) 119889119905 minus 119892 (119890)

(38)

8 Discrete Dynamics in Nature and Society

Table 2 The parameters of supply chain in different modes with only manufacturer bearing effort cost

Revenue-sharing contract based onquantity discount policy Traditional

revenue-sharing contract Wholesale price contractA B

Optimal effort 119890 1962 1962 397 379Optimal order quantity 119876 1674 1674 339 288Profit of the whole supply chain Π

119905 1924628 1924628 700833 654436

According to the supposition Π119905(119876lowast 119890) is the concave

function of effort 119890 So the optimal effort 119890lowast of the supplychain should satisfy

120597Π119905(119876lowast 119890lowast)

120597119890= (119901 minus V) 1199101015840 (119890lowast) int

119865minus1

((119901minus119888)(119901minusV))

0

119905119891 (119905) 119889119905

minus 1198921015840(119890lowast) = 0

(39)

From (39) it can easy to get that the optimal effort 119890lowast ofthe supply chain should satisfy

(119901 minus V) 1199101015840 (119890lowast) int119865minus1

((119901minus119888)(119901minusV))

0

119905119891 (119905) 119889119905 = 1198921015840(119890lowast) (40)

The method to solve the expression of 119890lowast in (40) is thesame as the method in (21)

424 An Example Suppose a supply chain is made up ofa retailer a distributor and a manufacturer and the supplychain parameters are 119888

119903= 1 119888119889= 2 119888119898= 5 V = 3 119908119889

119889= 18

119908119889119898= 10 119901 = 35 120601

1= 1206012= 055 and 119886 = 100

Here assume the relationship between demand and effortsatisfies multiplication form and 119892(119890) = 119886119890

22 119910(119890) = 119890120585 satisfies uniform distribution in [60 90] It is easy to get119891(120585) = 130 119865(120585) = (120585 minus 60)30 and 119865minus1(120585) = 60 + 30120585

(1) The revenue-sharing contract based on quantity dis-count policy

From the discussion above we know that the revenue-sharing contract based on quantity discount policy cancoordinate the supply chain whether it is implemented in oneor two transactions According to the formulas mentionedabove it is easy to get the optimal effort of the manufacturerthe optimal order quantity of the retailer and the profitof the whole supply chain Here 119890lowast = 1962 119876lowast =

1674 Π119905(119876lowast 119890lowast) = 1924628

(2) The traditional revenue-sharing contract

Similarly under the traditional revenue-sharing contractthe optimal effort of the manufacturer and order quantityof the retailer and the profit of the whole supply chain arerespectively 119890119898119888 = 397 119876119903119888 = 339 and Π

119905(119876119903119888 119890119898119888) =

700833

(3) The decentralized supply chain (the wholesale pricecontract)

Similarly under the traditional revenue-sharing contractthe optimal effort of the manufacturer and the order quantityof the retailer and the profit of the whole supply chain arerespectively 119890119898119889 = 379 119876119903119889 = 288 and Π

119905(119876119903119889 119890119889119898) =

654436Table 2 shows the parameters of the supply chain in

different modes A means that we implement the revenue-sharing contract based on quantity discount policy onlybetween the retailer and the distributor B means that weimplement the revenue-sharing contract based on quantitydiscount policy in two transactions of the supply chain

Table 2 shows that the manufacturerrsquos effort in therevenue-sharing contract based on quantity discount policyis the highest in the three modes which can make thesupply chain maximize its profit and the revenue-sharingcontract based on quantity discount policy is better than thetraditional revenue-sharing contract with effort dependentdemand the manufacturerrsquos effort in traditional revenue-sharing contract is higher than that of the supply chain inthe wholesale price contract and retailerrsquos order quantity isstill higher than that in wholesale price contract due to theincentive of the revenue-sharing contract so the profit ofthe supply chain in the traditional revenue-sharing contractis still higher than that in wholesale price contract So itcan be seen that the traditional revenue-sharing contract isbetter than thewholesale price contract with effort dependentdemand

43TheRetailer Bearing the Effort Cost with theManufacturerSuppose the effort cost that the retailer bears is 120572119892(119890) (0 le

120572 le 1) then the manufacturer bears (1 minus 120572)119892(119890) effort costIf 120572 = 0 it means that only the manufacturer bears the effortcost if 120572 = 1 it means that only the retailer bears the effortcost

In this condition we find that the traditional revenue-sharing contract fails to coordinate the supply chain whetherit is implemented in one transaction phase or in two transac-tion phases of the three-level supply chain The reason lies inthat the distributor does not bear the effort cost but he can getpartial profit of thewhole supply chain which is unfair for theretailer and themanufacturer So the effort level of the retailerand the manufacturer will be lower than the effort level of thecentralized supply chain Now the following discusses howto design the improved revenue-sharing implemented in twotransaction phases to coordinate the supply chain

Suppose the distributor provides the retailer the revenue-sharing contract (120601

2 119908119888119889(119876 119890)) and the manufacturer pro-

vides the distributors the revenue-sharing contract (1206011

Discrete Dynamics in Nature and Society 9

119908119888119898(119876 119890)) Then the profit functions of the supply chain

members can be described as

Π119888

119903(119876 119890) = 1206012 (119901 minus V) 119878 (119876 119890)

minus [119888119903+ 119908119888

119889(119876 119890) minus 1206012V] 119876 minus 120572119892 (119890)

Π119888

119889(119876 119890) = 1206011 (1 minus 1206012) (119901 minus V) 119878 (119876 119890)

minus [119888119889+ 119908119888

119898(119876 119890) minus 1206011 (1 minus 1206012) V

minus1206011119908119888

119889(119876 119890)] 119876

Π119888

119898(119876 119890) = (1 minus 1206011) (1 minus 1206012) (119901 minus V) 119878 (119876 119890)

minus [119888119898minus 119908119888

119898(119876 119890) minus (1 minus 1206011) (1 minus 1206012) V

minus (1 minus 1206011) 119908119888

119889(119876 119890)] 119876

minus (1 minus 120572) 119892 (119890)

(41)

If the improved revenue-sharing contract can coordinatethe supply chain the optimal effort level 119890lowast of the retailerand the manufacturer should satisfy (2) Namely the optimalorder (production) quantity 119876

lowast of the retailer and themanufacturer should satisfy (3) Then we can get

120597Π119888119903(119876 119890lowast)

120597119890=120597Π119888

119898(119876 119890lowast)

120597119890= 0

(119901 minus V)120597119878 (119876 119890

lowast)

120597119890minus 1198921015840(119890lowast) = 0

120597Π119888119903(119876lowast 119890)

120597119876=120597Π119888

119898(119876lowast 119890)

120597119876= 0

(119901 minus V)120597119878 (119876lowast 119890)

120597119876minus (119888 minus V) = 0

(42)

From (42) we can get

119908119888

119889(119876 119890) = 1206012 (119888 minus V) minus (1 minus 120572)

119892 (119890)

119876minus 119888119903+ 1206012V

119908119888

119898(119876 119890) = 1206011 (119888 minus V) + 1206011 (1 minus 120572)

119892 (119890)

119876minus 1206011119888119903+ 1206011V minus 119888119889

(43)

Taking (43) into the profit functions of the supply chainmembers (41) we can get

Π119888

119903(119876 119890) = 1206012Π119905 (119876 119890)

Π119888

119889(119876 119890) = 1206011 (1 minus 1206012)Π119905 (119876 119890)

Π119888

119898(119876 119890) = (1 minus 1206011) (1 minus 1206012)Π119905 (119876 119890)

(44)

We find that (44) equals (11) That is to say when theretailer bears the effort with the manufacturer the improvedrevenue-sharing contract can coordinate the three-level sup-ply chain

Therefore if we implement the revenue-sharing contractbased on quantity discount policy among the supply chain

members and the contract parameters which satisfy (43)the improved revenue-sharing can coordinate the three-levelsupply chain

The method to determine the optimal order quantity 119876lowastand the optimal effort 119890lowast can follow the methods given inSections 413 and 423

5 Conclusion

The traditional revenue-sharing contract cannot coordinatethree-level supply chain with sales effort dependent demandThe paper proposes an improved revenue-sharing contractbased on quantity discount policy which can be implementedin one transaction or two transactions of three-level supplychain to coordinate the supply chain The paper also showsthat the profit of the supply chain members depends on thevalue of the contract parameters which is determined by thestatus of the members in supply chain and their bargainingpower It also gave the method to determine the optimalorder quantity and the optimal effort which can provide thedecision makers of supply chain with decision support Butit also should be noticed that this research is carried outin the symmetrical information condition So the next stepis to carry out the research in the asymmetric informationcondition

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This research was supported by the Ministry of Educationof China Grant-in-aid for Humanity and social ScienceResearch (no 10YJC630188) and the Fundamental ResearchFunds for the Central Universities (nos 2012B13914 and2012B13814)

References

[1] G P Cachon and M A Lariviere ldquoSupply chain coordinationwith revenue-sharing contracts strengths and limitationsrdquoManagement Science vol 51 no 1 pp 30ndash44 2005

[2] J H Mortimer The Effects of Revenue-Sharing Contracts onWelfare in Vertically SeparatedMarkets Evidence from the VideoRental Industry University of California at Los Angeles 2000

[3] J D Dana Jr and K E Spier ldquoRevenue sharing and verticalcontrol in the video rental industryrdquo Journal of IndustrialEconomics vol 49 no 3 pp 223ndash245 2001

[4] B Pasternack ldquoUsing revenue-sharing to achieve channel coor-dination for a newsboy type inventory modelrdquo in Supply ChainManagement Models Applications and Research DirectionsJ Geunes P Panos and H Edwin Eds vol 62 pp 271ndash290 Kluwer Academic Publishers DordrechtTheNetherlands2002

[5] Y Wang L Jiang and Z J Shen ldquoChannel performance underconsignment contract with revenue-sharingrdquoManagement Sci-ence vol 50 no 1 pp 34ndash47 2004

10 Discrete Dynamics in Nature and Society

[6] Y Gerchak and Y Wang ldquoRevenue-sharing vs wholesale-price contracts in assembly systems with random demandrdquoProduction andOperationsManagement vol 13 no 1 pp 23ndash332004

[7] C Koulamas ldquoA newsvendor problemwith revenue sharing andchannel coordinationrdquo Decision Sciences vol 37 no 1 pp 91ndash100 2006

[8] S S Chauhan and J Proth ldquoAnalysis of a supply chain partner-ship with revenue sharingrdquo International Journal of ProductionEconomics vol 97 no 1 pp 44ndash51 2005

[9] C T Linh and Y S Hong ldquoChannel coordination through arevenue sharing contract in a two-period newsboy problemrdquoEuropean Journal of Operational Research vol 198 no 3 pp822ndash829 2009

[10] K Pan K K Lai S C H Leung andD Xiao ldquoRevenue-sharingversus wholesale price mechanisms under different channelpower structuresrdquo European Journal of Operational Researchvol 203 no 2 pp 532ndash538 2010

[11] D G Qu and Y J Guo ldquoCoordination of supply chain withhybrid distribution channels when retailerrsquos demand relies onits sales effortrdquo Chinese Journal of Management Science vol 21no 3 pp 19ndash23 2008

[12] I Giannoccaro and P Pontrandolfo ldquoSupply chain coordinationby revenue sharing contractsrdquo International Journal of Produc-tion Economics vol 89 no 2 pp 131ndash139 2004

[13] S F Ji M J Liu W Ding and X Huang ldquoCoordination ofthree-level supply chain and revenue sharing contractrdquo Journalof Northeastern University vol 29 no 11 pp 1652ndash1656 2008

[14] L Hu Z Jiang M Meng and J Qin ldquoCoordination of a three-level supply chain through revenue sharingrdquo Journal of HarbinEngineering University vol 29 no 2 pp 198ndash203 2008

[15] Y L Hou D Q Zhou and B Y Tian ldquoOrdering decisionson supply chain under revenue sharing policiesrdquo IndustrialEngineering Journal vol 12 no 1 pp 24ndash27 2009

[16] Y He Q Wu and L Zhao ldquoQuantity flexibility contract witheffort and price-dependent demandrdquo Systems Engineering andElectronics vol 29 no 12 pp 2056ndash2059 2007

[17] P Desai andK Srinivasan ldquoDemand signaling under unobserv-able effort in franchisingrdquo Management Science vol 20 no 4pp 2608ndash2623 1995

[18] W Chu and P Desai ldquoChannel coordination mechanisms forcustomer satisfactionrdquoMarketing Science vol 21 no 6 pp 475ndash490 1995

[19] T A Taylor ldquoSupply chain coordination under channel rebateswith sales effort effectsrdquoManagement Science vol 48 no 8 pp992ndash1007 2002

[20] Z Xu D L Zhu and W G Zhu ldquoSupply chain coordinationunder buy-back contract with sales effort effectsrdquo System Engi-neering Theory and Practice vol 28 no 4 pp 1ndash11 2008

[21] G P Cachon ldquoSupply chain coordination with contractsrdquoin Handbooks in Operations Research and Management Sci-ence Supply Chain Management North-Holland Elseriver SanDiego Calif USA 2000

[22] Y He D Yang andQWu ldquoRevenue-sharing contract of supplychain with effort dependent demandrdquo Computer IntegratedManufacturing Systems vol 12 no 11 pp 1865ndash1868 2006

[23] Y He X Zhao L Zhao and J He ldquoCoordinating a supply chainwith effort and price dependent stochastic demandrdquo AppliedMathematical Modelling vol 33 no 6 pp 2777ndash2790 2009

[24] B Y Hu and X Y Wang ldquoSupply chain revenue-sharingevolvement-contract with sales effort effectsrdquo Journal of Indus-trial Engineering andEngineeringManagement vol 29 no 2 pp35ndash39 2010

[25] J Chen and J F Chen ldquoStudy on revenue sharing contractin virtual enterprisesrdquo Journal of Systems Science and SystemsEngineering vol 15 no 1 pp 95ndash113 2006

[26] Y Wang and Y Gerchak ldquoSupply Chain Coordination whendemand is shelf-space dpendentrdquo Manufacturing and ServiceOperations Management vol 3 no 1 pp 82ndash87 2001

[27] H Krishnan R Kapuscinski and D A Butz ldquoCoordinatingcontracts for decentralized supply chains with retailer promo-tional effortrdquo Management Science vol 50 no 1 pp 48ndash632004

[28] Y He Supply Chain Contract Models with Stochastic DemandDalian University of Technology Dalian China 2004

[29] L Yao Supply Chain Modeling Pricing Contracts and Coordi-nation Chinese University of Hong Kong Hong Kong 2002

Submit your manuscripts athttpwwwhindawicom

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Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Discrete Dynamics in Nature and Society

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

6 Discrete Dynamics in Nature and Society

Table 1 The parameters of supply chain in different modes with only retailer bearing effort cost

Revenue-sharing contract based onquantity discount policy Traditional

revenue-sharing contract Wholesale price contractA B

Optimal effort 119890 27 27 1485 16Optimal order quantity 119876 119 119 107 91Profit of the whole supply chain Π

119905 22725 22725 22027 2137

maximize its profit the retailerrsquos effort in traditional revenue-sharing contract is the lowest in the threemodes but his orderquantity is still higher than that in wholesale price contractdue to the incentive of the revenue-sharing contract So theprofit of the supply chain in the traditional revenue-sharingcontract is still higher than that in wholesale price contractSo it can be seen that the traditional revenue-sharing contractis better than the wholesale price contract with sales effortdependent demand

42 Only Manufacturer Bearing the Effort Cost

421 Only Implement between Manufacturer and DistributorThis problem can be described as follows we implement therevenue-sharing contract based on quantity policy betweenthe manufacturer and the distributor and the traditionalrevenue-sharing contract between the distributor and theretailer

Supposing the manufacturer provides the distributor therevenue-sharing contract (120601

1 119908119888119898(119876)) and

119908119888

119898(119876) = minus (1 minus 1206011) 119888 + [1 minus (1 minus 1206011) (1 minus 1206012)]

119892 (119890lowast)

119876

+ (1 minus 1206011) 119888119903+ 119888119898

(22)

Here 119890lowast is the optimal effort of the supply chainTaking (22) into the manufacturerrsquos profit function we

can get

Π119888

119898(119876 119890)

= (1 minus 1206011) [(1 minus 120601

2) (119901119878 (119876 119890) + V119868 (119876 119890)) + 119908119888

119889119876]

+ 119908119888

119898(119876)119876 minus 119888

119898119876 minus 119892 (119890)

= (1 minus 1206011) (1 minus 120601

2) (119901 minus V) 119878 (119876 119890)

+ [1 minus (1 minus 1206011) (1 minus 120601

2)] 119892 (119890

lowast)

minus (1 minus 1206011) [119888 minus 119888

119903minus (1 minus 120601

2) V minus 119908119888

119889] 119876 minus 119892 (119890)

(23)

According to the paper of Giannoccaro and Pontrandolfo[12] without considering demand dependent effort the ratiothat the manufacturerrsquos profit shares the profit of the supplychain is (1 minus 120601

1)(1 minus 120601

2) Here supposing when only the

manufacturer bears the effort cost the ratio that his profitshares the profit of the whole supply chain is also (1 minus 120601

1)(1 minus

1206012) It is easily known that the effort of the manufacturer

would be equal to the optimal effort of the supply chainnamely

Π119888

119898(119876 119890)

= (1 minus 1206011) [(1 minus 120601

2) (119901119878 (119876 119890

lowast) + V119868 (119876 119890lowast)) + 119908119888

119889119876]

+ 119908119888

119898(119876)119876 minus 119888

119898119876 minus 119892 (119890

lowast)

= (1 minus 1206011) (1 minus 120601

2)Π119905 (119876 119890)

(24)

From (24) we can get

119908119888

119889= 1206012119888 minus 119888119903 (25)

Taking (25) and (22) into the profit function of the supplychain members we can get

Π119888

119903(119876 119890) = 1206012 [Π119905 (119876 119890) + 119892 (119890)]

Π119888

119889(119876 119890) = 1206011 (1 minus 1206012)Π119905 (119876 119890) minus 1206012119892 (119890)

Π119888

119898(119876 119890) = (1 minus 1206011) (1 minus 1206012)Π119905

(26)

From (26) it can be seen that the profit functions ofthe supply chain members are all affine functions of thewhole supply chainrsquos profit function So in this conditionthe revenue-sharing contract can coordinate the three-levelsupply chain

Therefore if we implement the revenue-sharing contractbased on quantity discount policy between the retailer andthe distributor and the traditional revenue-sharing contractbetween the distributor and the manufacturer and the con-tract parameters which satisfy (22) and (25) the improvedrevenue-sharing can coordinate the three-level supply chain

Compared with the paper of Giannoccaro and Pontran-dolfo [12] it can be seen that the retailerrsquos profit increases1206012119892(119890lowast) while that of the distributor decreases 120601

2119892(119890lowast)

422 Implement among Supply Chain Members This prob-lem can be described as follows we implement both therevenue-sharing contract based on quantity discount policybetween the retailer and the distributor and between thedistributor and the manufacturer

Suppose the distributor provides the retailer the revenue-sharing contract (120601

2 119908119888119889(119876)) and the manufacturer provides

Discrete Dynamics in Nature and Society 7

the distributors the revenue-sharing contract (1206011 119908119888119898(119876))

Here 119890lowast is the optimal effort of the supply chain and

119908119888

119889(119876) = 120601

2119888 minus (1 minus 120601

2)119892 (119890lowast)

119876minus 119888119903

119908119888

119898(119876) = 120601

1119888 +

119892 (119890lowast)

119876minus 1206011119888119903minus 119888119889

(27)

Taking (27) into the profit function of the manufacturerwe can get

Π119888

119898(119876 119890)

= (1 minus 1206011) [(1 minus 120601

2) (119901119878 (119876 119890) + V119868 (119876 119890)) + 119908119888

119889(119876)119876]

+ 119908119888

119898(119876)119876 minus 119888

119898119876 minus 119892 (119890)

= (1 minus 1206011) (1 minus 120601

2) (119901 minus V) 119878 (119876 119890)

minus (1 minus 1206011) (1 minus 120601

2) (119888 minus V) 119876

+ [1 minus (1 minus 1206011) (1 minus 120601

2)] 119892 (119890

lowast) minus 119892 (119890)

(28)

Here we also suppose that the ratio that his profit sharesthe profit of the whole supply chain is also (1 minus 120601

1)(1 minus 120601

2)

It is easy to know that the optimal effort of the manufacturerwould be equal to that of the supply chain In this conditionthe profit function of the manufacturer can be described as

Π119888

119898(119876 119890) = (1 minus 1206011) (1 minus 1206012)Π119905 (119876 119890) (29)

Taking (27) into the profit function of the retailer we canget

Π119888

119903(119876 119890) = 1206012 [119901119878 (119876 119890

lowast) + V119868 (119876 119890lowast)] minus 119888

119903119876 minus 119908

119888

119889(119876)119876

= 1206012(119901 minus V) 119878 (119876 119890lowast)minus120601

2 (119888 minus V) 119876 + (1 minus 1206012) 119892 (119890)

(30)

From (30) it is easy to know that the optimal orderquantity of the retailer is equal to that of the supply chainSo (30) can be described as

Π119888

119903(119876 119890) = 1206012Π119905 (119876 119890) + 119892 (119890) (31)

Taking (27) into the profit function of the distributor itcan get

Π119888

119889(119876 119890)

= 1206011[(1 minus 120601

2) (119901119878 (119876 119890

lowast) + V119868 (119876 119890lowast)) + 119908119888

119889(119876)119876]

minus 119888119889119876 minus 119908

119888

119898(119876)119876 minus 119892 (119890

lowast)

= 1206011(1 minus 120601

2) (119901 minus V) 119878 (119876 119890lowast) minus 120601

1(1 minus 120601

2) (119888 minus V) 119876

minus (1 + 1206011(1 minus 120601

2)) 119892 (119890)

(32)

From (32) it can be seen that the optimal order quantityof the distributor is equal to that of the retailer so (32) canalso be described as

Π119888

119889(119876 119890) = 1206011 (1 minus 1206012)Π119905 (119876 119890) minus 119892 (119890) (33)

From (29) (31) and (33) it can be seen that the profitfunctions of the supply chainmembers are all affine functionsof the whole supply chainrsquos profit function So in this condi-tion the revenue-sharing contract can coordinate the three-level supply chain Therefore if we implement the revenue-sharing contract based on quantity discount policy bothbetween the retailer and the distributor and between thedistributor and themanufacturer and the contract parameterswhich satisfy (27) the improved revenue-sharing can coordi-nate the three-level supply chain

423 Model Optimization Similarly suppose the marketdemand 119883(119890 120585) is the function of effort 119890 and random factor120585 and 120585 is independent of 119890 119891(120585) and 119865(120585) are respectivelyprobability density function and differentiable cumulateddistribution function of 120585 Here suppose 119883(119890 120585) = 119910(119890) sdot 120585As the influence of effort on marker demand is diminishingmarginal utility we can suppose 119910(119890) is the monotoneincreasing and concave function of effort 119890 namely1199101015840(119890) gt 0and 11991010158401015840(119890) le 0

When the market demand satisfies 119883(119890 120585) = 119910(119890) sdot 120585 wecan get

119878 (119876 119890) = 119876 minus int119876

0

119865 (119909 | 119890) 119889119909 = 119876 minus int119876

0

119865(119909

119910 (119890)) 119889119909

= 119876 minus 119910 (119890) int119876119910(119890)

0

119865 (119905) 119889119905

(34)

So the profit function of supply chain can be described as

Π119905 (119876 119890) = (119901 minus V) 119878 (119876 119890) minus (119888 minus V) 119876 minus 119892 (119890)

= (119901 minus 119888)119876 minus (119901 minus V) 119910 (119890) int119876119910(119890)

0

119865 (119905) 119889119905 minus 119892 (119890)

(35)

Given effort 119890 it can be seen that Π119905(119876 119890) is the concave

function of order quantity 119876 So the optimal order quantity119876lowast should satisfy

120597Π119905(119876lowast 119890)

120597119876= (119901 minus 119888) minus (119901 minus V) 119865(

119876lowast

119910 (119890)) = 0 (36)

From (36) we can get

119876lowast= 119876lowast(119890) = 119910 (119890) 119865

minus1(119901 minus 119888

119901 minus V) (37)

Then the profit function of supply chain can be describedas

Π119905(119876lowast 119890) = (119901 minus 119888) 119910 (119890) 119865

minus1(119901 minus 119888

119901 minus V)

minus (119901 minus V) 119910 (119890) int119865minus1

((119901minus119888)(119901minusV))

0

119865 (119905) 119889119905 minus 119892 (119890)

= (119901 minus V) 119910 (119890) int119865minus1

((119901minus119888)(119901minusV))

0

119905119891 (119905) 119889119905 minus 119892 (119890)

(38)

8 Discrete Dynamics in Nature and Society

Table 2 The parameters of supply chain in different modes with only manufacturer bearing effort cost

Revenue-sharing contract based onquantity discount policy Traditional

revenue-sharing contract Wholesale price contractA B

Optimal effort 119890 1962 1962 397 379Optimal order quantity 119876 1674 1674 339 288Profit of the whole supply chain Π

119905 1924628 1924628 700833 654436

According to the supposition Π119905(119876lowast 119890) is the concave

function of effort 119890 So the optimal effort 119890lowast of the supplychain should satisfy

120597Π119905(119876lowast 119890lowast)

120597119890= (119901 minus V) 1199101015840 (119890lowast) int

119865minus1

((119901minus119888)(119901minusV))

0

119905119891 (119905) 119889119905

minus 1198921015840(119890lowast) = 0

(39)

From (39) it can easy to get that the optimal effort 119890lowast ofthe supply chain should satisfy

(119901 minus V) 1199101015840 (119890lowast) int119865minus1

((119901minus119888)(119901minusV))

0

119905119891 (119905) 119889119905 = 1198921015840(119890lowast) (40)

The method to solve the expression of 119890lowast in (40) is thesame as the method in (21)

424 An Example Suppose a supply chain is made up ofa retailer a distributor and a manufacturer and the supplychain parameters are 119888

119903= 1 119888119889= 2 119888119898= 5 V = 3 119908119889

119889= 18

119908119889119898= 10 119901 = 35 120601

1= 1206012= 055 and 119886 = 100

Here assume the relationship between demand and effortsatisfies multiplication form and 119892(119890) = 119886119890

22 119910(119890) = 119890120585 satisfies uniform distribution in [60 90] It is easy to get119891(120585) = 130 119865(120585) = (120585 minus 60)30 and 119865minus1(120585) = 60 + 30120585

(1) The revenue-sharing contract based on quantity dis-count policy

From the discussion above we know that the revenue-sharing contract based on quantity discount policy cancoordinate the supply chain whether it is implemented in oneor two transactions According to the formulas mentionedabove it is easy to get the optimal effort of the manufacturerthe optimal order quantity of the retailer and the profitof the whole supply chain Here 119890lowast = 1962 119876lowast =

1674 Π119905(119876lowast 119890lowast) = 1924628

(2) The traditional revenue-sharing contract

Similarly under the traditional revenue-sharing contractthe optimal effort of the manufacturer and order quantityof the retailer and the profit of the whole supply chain arerespectively 119890119898119888 = 397 119876119903119888 = 339 and Π

119905(119876119903119888 119890119898119888) =

700833

(3) The decentralized supply chain (the wholesale pricecontract)

Similarly under the traditional revenue-sharing contractthe optimal effort of the manufacturer and the order quantityof the retailer and the profit of the whole supply chain arerespectively 119890119898119889 = 379 119876119903119889 = 288 and Π

119905(119876119903119889 119890119889119898) =

654436Table 2 shows the parameters of the supply chain in

different modes A means that we implement the revenue-sharing contract based on quantity discount policy onlybetween the retailer and the distributor B means that weimplement the revenue-sharing contract based on quantitydiscount policy in two transactions of the supply chain

Table 2 shows that the manufacturerrsquos effort in therevenue-sharing contract based on quantity discount policyis the highest in the three modes which can make thesupply chain maximize its profit and the revenue-sharingcontract based on quantity discount policy is better than thetraditional revenue-sharing contract with effort dependentdemand the manufacturerrsquos effort in traditional revenue-sharing contract is higher than that of the supply chain inthe wholesale price contract and retailerrsquos order quantity isstill higher than that in wholesale price contract due to theincentive of the revenue-sharing contract so the profit ofthe supply chain in the traditional revenue-sharing contractis still higher than that in wholesale price contract So itcan be seen that the traditional revenue-sharing contract isbetter than thewholesale price contract with effort dependentdemand

43TheRetailer Bearing the Effort Cost with theManufacturerSuppose the effort cost that the retailer bears is 120572119892(119890) (0 le

120572 le 1) then the manufacturer bears (1 minus 120572)119892(119890) effort costIf 120572 = 0 it means that only the manufacturer bears the effortcost if 120572 = 1 it means that only the retailer bears the effortcost

In this condition we find that the traditional revenue-sharing contract fails to coordinate the supply chain whetherit is implemented in one transaction phase or in two transac-tion phases of the three-level supply chain The reason lies inthat the distributor does not bear the effort cost but he can getpartial profit of thewhole supply chain which is unfair for theretailer and themanufacturer So the effort level of the retailerand the manufacturer will be lower than the effort level of thecentralized supply chain Now the following discusses howto design the improved revenue-sharing implemented in twotransaction phases to coordinate the supply chain

Suppose the distributor provides the retailer the revenue-sharing contract (120601

2 119908119888119889(119876 119890)) and the manufacturer pro-

vides the distributors the revenue-sharing contract (1206011

Discrete Dynamics in Nature and Society 9

119908119888119898(119876 119890)) Then the profit functions of the supply chain

members can be described as

Π119888

119903(119876 119890) = 1206012 (119901 minus V) 119878 (119876 119890)

minus [119888119903+ 119908119888

119889(119876 119890) minus 1206012V] 119876 minus 120572119892 (119890)

Π119888

119889(119876 119890) = 1206011 (1 minus 1206012) (119901 minus V) 119878 (119876 119890)

minus [119888119889+ 119908119888

119898(119876 119890) minus 1206011 (1 minus 1206012) V

minus1206011119908119888

119889(119876 119890)] 119876

Π119888

119898(119876 119890) = (1 minus 1206011) (1 minus 1206012) (119901 minus V) 119878 (119876 119890)

minus [119888119898minus 119908119888

119898(119876 119890) minus (1 minus 1206011) (1 minus 1206012) V

minus (1 minus 1206011) 119908119888

119889(119876 119890)] 119876

minus (1 minus 120572) 119892 (119890)

(41)

If the improved revenue-sharing contract can coordinatethe supply chain the optimal effort level 119890lowast of the retailerand the manufacturer should satisfy (2) Namely the optimalorder (production) quantity 119876

lowast of the retailer and themanufacturer should satisfy (3) Then we can get

120597Π119888119903(119876 119890lowast)

120597119890=120597Π119888

119898(119876 119890lowast)

120597119890= 0

(119901 minus V)120597119878 (119876 119890

lowast)

120597119890minus 1198921015840(119890lowast) = 0

120597Π119888119903(119876lowast 119890)

120597119876=120597Π119888

119898(119876lowast 119890)

120597119876= 0

(119901 minus V)120597119878 (119876lowast 119890)

120597119876minus (119888 minus V) = 0

(42)

From (42) we can get

119908119888

119889(119876 119890) = 1206012 (119888 minus V) minus (1 minus 120572)

119892 (119890)

119876minus 119888119903+ 1206012V

119908119888

119898(119876 119890) = 1206011 (119888 minus V) + 1206011 (1 minus 120572)

119892 (119890)

119876minus 1206011119888119903+ 1206011V minus 119888119889

(43)

Taking (43) into the profit functions of the supply chainmembers (41) we can get

Π119888

119903(119876 119890) = 1206012Π119905 (119876 119890)

Π119888

119889(119876 119890) = 1206011 (1 minus 1206012)Π119905 (119876 119890)

Π119888

119898(119876 119890) = (1 minus 1206011) (1 minus 1206012)Π119905 (119876 119890)

(44)

We find that (44) equals (11) That is to say when theretailer bears the effort with the manufacturer the improvedrevenue-sharing contract can coordinate the three-level sup-ply chain

Therefore if we implement the revenue-sharing contractbased on quantity discount policy among the supply chain

members and the contract parameters which satisfy (43)the improved revenue-sharing can coordinate the three-levelsupply chain

The method to determine the optimal order quantity 119876lowastand the optimal effort 119890lowast can follow the methods given inSections 413 and 423

5 Conclusion

The traditional revenue-sharing contract cannot coordinatethree-level supply chain with sales effort dependent demandThe paper proposes an improved revenue-sharing contractbased on quantity discount policy which can be implementedin one transaction or two transactions of three-level supplychain to coordinate the supply chain The paper also showsthat the profit of the supply chain members depends on thevalue of the contract parameters which is determined by thestatus of the members in supply chain and their bargainingpower It also gave the method to determine the optimalorder quantity and the optimal effort which can provide thedecision makers of supply chain with decision support Butit also should be noticed that this research is carried outin the symmetrical information condition So the next stepis to carry out the research in the asymmetric informationcondition

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This research was supported by the Ministry of Educationof China Grant-in-aid for Humanity and social ScienceResearch (no 10YJC630188) and the Fundamental ResearchFunds for the Central Universities (nos 2012B13914 and2012B13814)

References

[1] G P Cachon and M A Lariviere ldquoSupply chain coordinationwith revenue-sharing contracts strengths and limitationsrdquoManagement Science vol 51 no 1 pp 30ndash44 2005

[2] J H Mortimer The Effects of Revenue-Sharing Contracts onWelfare in Vertically SeparatedMarkets Evidence from the VideoRental Industry University of California at Los Angeles 2000

[3] J D Dana Jr and K E Spier ldquoRevenue sharing and verticalcontrol in the video rental industryrdquo Journal of IndustrialEconomics vol 49 no 3 pp 223ndash245 2001

[4] B Pasternack ldquoUsing revenue-sharing to achieve channel coor-dination for a newsboy type inventory modelrdquo in Supply ChainManagement Models Applications and Research DirectionsJ Geunes P Panos and H Edwin Eds vol 62 pp 271ndash290 Kluwer Academic Publishers DordrechtTheNetherlands2002

[5] Y Wang L Jiang and Z J Shen ldquoChannel performance underconsignment contract with revenue-sharingrdquoManagement Sci-ence vol 50 no 1 pp 34ndash47 2004

10 Discrete Dynamics in Nature and Society

[6] Y Gerchak and Y Wang ldquoRevenue-sharing vs wholesale-price contracts in assembly systems with random demandrdquoProduction andOperationsManagement vol 13 no 1 pp 23ndash332004

[7] C Koulamas ldquoA newsvendor problemwith revenue sharing andchannel coordinationrdquo Decision Sciences vol 37 no 1 pp 91ndash100 2006

[8] S S Chauhan and J Proth ldquoAnalysis of a supply chain partner-ship with revenue sharingrdquo International Journal of ProductionEconomics vol 97 no 1 pp 44ndash51 2005

[9] C T Linh and Y S Hong ldquoChannel coordination through arevenue sharing contract in a two-period newsboy problemrdquoEuropean Journal of Operational Research vol 198 no 3 pp822ndash829 2009

[10] K Pan K K Lai S C H Leung andD Xiao ldquoRevenue-sharingversus wholesale price mechanisms under different channelpower structuresrdquo European Journal of Operational Researchvol 203 no 2 pp 532ndash538 2010

[11] D G Qu and Y J Guo ldquoCoordination of supply chain withhybrid distribution channels when retailerrsquos demand relies onits sales effortrdquo Chinese Journal of Management Science vol 21no 3 pp 19ndash23 2008

[12] I Giannoccaro and P Pontrandolfo ldquoSupply chain coordinationby revenue sharing contractsrdquo International Journal of Produc-tion Economics vol 89 no 2 pp 131ndash139 2004

[13] S F Ji M J Liu W Ding and X Huang ldquoCoordination ofthree-level supply chain and revenue sharing contractrdquo Journalof Northeastern University vol 29 no 11 pp 1652ndash1656 2008

[14] L Hu Z Jiang M Meng and J Qin ldquoCoordination of a three-level supply chain through revenue sharingrdquo Journal of HarbinEngineering University vol 29 no 2 pp 198ndash203 2008

[15] Y L Hou D Q Zhou and B Y Tian ldquoOrdering decisionson supply chain under revenue sharing policiesrdquo IndustrialEngineering Journal vol 12 no 1 pp 24ndash27 2009

[16] Y He Q Wu and L Zhao ldquoQuantity flexibility contract witheffort and price-dependent demandrdquo Systems Engineering andElectronics vol 29 no 12 pp 2056ndash2059 2007

[17] P Desai andK Srinivasan ldquoDemand signaling under unobserv-able effort in franchisingrdquo Management Science vol 20 no 4pp 2608ndash2623 1995

[18] W Chu and P Desai ldquoChannel coordination mechanisms forcustomer satisfactionrdquoMarketing Science vol 21 no 6 pp 475ndash490 1995

[19] T A Taylor ldquoSupply chain coordination under channel rebateswith sales effort effectsrdquoManagement Science vol 48 no 8 pp992ndash1007 2002

[20] Z Xu D L Zhu and W G Zhu ldquoSupply chain coordinationunder buy-back contract with sales effort effectsrdquo System Engi-neering Theory and Practice vol 28 no 4 pp 1ndash11 2008

[21] G P Cachon ldquoSupply chain coordination with contractsrdquoin Handbooks in Operations Research and Management Sci-ence Supply Chain Management North-Holland Elseriver SanDiego Calif USA 2000

[22] Y He D Yang andQWu ldquoRevenue-sharing contract of supplychain with effort dependent demandrdquo Computer IntegratedManufacturing Systems vol 12 no 11 pp 1865ndash1868 2006

[23] Y He X Zhao L Zhao and J He ldquoCoordinating a supply chainwith effort and price dependent stochastic demandrdquo AppliedMathematical Modelling vol 33 no 6 pp 2777ndash2790 2009

[24] B Y Hu and X Y Wang ldquoSupply chain revenue-sharingevolvement-contract with sales effort effectsrdquo Journal of Indus-trial Engineering andEngineeringManagement vol 29 no 2 pp35ndash39 2010

[25] J Chen and J F Chen ldquoStudy on revenue sharing contractin virtual enterprisesrdquo Journal of Systems Science and SystemsEngineering vol 15 no 1 pp 95ndash113 2006

[26] Y Wang and Y Gerchak ldquoSupply Chain Coordination whendemand is shelf-space dpendentrdquo Manufacturing and ServiceOperations Management vol 3 no 1 pp 82ndash87 2001

[27] H Krishnan R Kapuscinski and D A Butz ldquoCoordinatingcontracts for decentralized supply chains with retailer promo-tional effortrdquo Management Science vol 50 no 1 pp 48ndash632004

[28] Y He Supply Chain Contract Models with Stochastic DemandDalian University of Technology Dalian China 2004

[29] L Yao Supply Chain Modeling Pricing Contracts and Coordi-nation Chinese University of Hong Kong Hong Kong 2002

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Discrete Dynamics in Nature and Society 7

the distributors the revenue-sharing contract (1206011 119908119888119898(119876))

Here 119890lowast is the optimal effort of the supply chain and

119908119888

119889(119876) = 120601

2119888 minus (1 minus 120601

2)119892 (119890lowast)

119876minus 119888119903

119908119888

119898(119876) = 120601

1119888 +

119892 (119890lowast)

119876minus 1206011119888119903minus 119888119889

(27)

Taking (27) into the profit function of the manufacturerwe can get

Π119888

119898(119876 119890)

= (1 minus 1206011) [(1 minus 120601

2) (119901119878 (119876 119890) + V119868 (119876 119890)) + 119908119888

119889(119876)119876]

+ 119908119888

119898(119876)119876 minus 119888

119898119876 minus 119892 (119890)

= (1 minus 1206011) (1 minus 120601

2) (119901 minus V) 119878 (119876 119890)

minus (1 minus 1206011) (1 minus 120601

2) (119888 minus V) 119876

+ [1 minus (1 minus 1206011) (1 minus 120601

2)] 119892 (119890

lowast) minus 119892 (119890)

(28)

Here we also suppose that the ratio that his profit sharesthe profit of the whole supply chain is also (1 minus 120601

1)(1 minus 120601

2)

It is easy to know that the optimal effort of the manufacturerwould be equal to that of the supply chain In this conditionthe profit function of the manufacturer can be described as

Π119888

119898(119876 119890) = (1 minus 1206011) (1 minus 1206012)Π119905 (119876 119890) (29)

Taking (27) into the profit function of the retailer we canget

Π119888

119903(119876 119890) = 1206012 [119901119878 (119876 119890

lowast) + V119868 (119876 119890lowast)] minus 119888

119903119876 minus 119908

119888

119889(119876)119876

= 1206012(119901 minus V) 119878 (119876 119890lowast)minus120601

2 (119888 minus V) 119876 + (1 minus 1206012) 119892 (119890)

(30)

From (30) it is easy to know that the optimal orderquantity of the retailer is equal to that of the supply chainSo (30) can be described as

Π119888

119903(119876 119890) = 1206012Π119905 (119876 119890) + 119892 (119890) (31)

Taking (27) into the profit function of the distributor itcan get

Π119888

119889(119876 119890)

= 1206011[(1 minus 120601

2) (119901119878 (119876 119890

lowast) + V119868 (119876 119890lowast)) + 119908119888

119889(119876)119876]

minus 119888119889119876 minus 119908

119888

119898(119876)119876 minus 119892 (119890

lowast)

= 1206011(1 minus 120601

2) (119901 minus V) 119878 (119876 119890lowast) minus 120601

1(1 minus 120601

2) (119888 minus V) 119876

minus (1 + 1206011(1 minus 120601

2)) 119892 (119890)

(32)

From (32) it can be seen that the optimal order quantityof the distributor is equal to that of the retailer so (32) canalso be described as

Π119888

119889(119876 119890) = 1206011 (1 minus 1206012)Π119905 (119876 119890) minus 119892 (119890) (33)

From (29) (31) and (33) it can be seen that the profitfunctions of the supply chainmembers are all affine functionsof the whole supply chainrsquos profit function So in this condi-tion the revenue-sharing contract can coordinate the three-level supply chain Therefore if we implement the revenue-sharing contract based on quantity discount policy bothbetween the retailer and the distributor and between thedistributor and themanufacturer and the contract parameterswhich satisfy (27) the improved revenue-sharing can coordi-nate the three-level supply chain

423 Model Optimization Similarly suppose the marketdemand 119883(119890 120585) is the function of effort 119890 and random factor120585 and 120585 is independent of 119890 119891(120585) and 119865(120585) are respectivelyprobability density function and differentiable cumulateddistribution function of 120585 Here suppose 119883(119890 120585) = 119910(119890) sdot 120585As the influence of effort on marker demand is diminishingmarginal utility we can suppose 119910(119890) is the monotoneincreasing and concave function of effort 119890 namely1199101015840(119890) gt 0and 11991010158401015840(119890) le 0

When the market demand satisfies 119883(119890 120585) = 119910(119890) sdot 120585 wecan get

119878 (119876 119890) = 119876 minus int119876

0

119865 (119909 | 119890) 119889119909 = 119876 minus int119876

0

119865(119909

119910 (119890)) 119889119909

= 119876 minus 119910 (119890) int119876119910(119890)

0

119865 (119905) 119889119905

(34)

So the profit function of supply chain can be described as

Π119905 (119876 119890) = (119901 minus V) 119878 (119876 119890) minus (119888 minus V) 119876 minus 119892 (119890)

= (119901 minus 119888)119876 minus (119901 minus V) 119910 (119890) int119876119910(119890)

0

119865 (119905) 119889119905 minus 119892 (119890)

(35)

Given effort 119890 it can be seen that Π119905(119876 119890) is the concave

function of order quantity 119876 So the optimal order quantity119876lowast should satisfy

120597Π119905(119876lowast 119890)

120597119876= (119901 minus 119888) minus (119901 minus V) 119865(

119876lowast

119910 (119890)) = 0 (36)

From (36) we can get

119876lowast= 119876lowast(119890) = 119910 (119890) 119865

minus1(119901 minus 119888

119901 minus V) (37)

Then the profit function of supply chain can be describedas

Π119905(119876lowast 119890) = (119901 minus 119888) 119910 (119890) 119865

minus1(119901 minus 119888

119901 minus V)

minus (119901 minus V) 119910 (119890) int119865minus1

((119901minus119888)(119901minusV))

0

119865 (119905) 119889119905 minus 119892 (119890)

= (119901 minus V) 119910 (119890) int119865minus1

((119901minus119888)(119901minusV))

0

119905119891 (119905) 119889119905 minus 119892 (119890)

(38)

8 Discrete Dynamics in Nature and Society

Table 2 The parameters of supply chain in different modes with only manufacturer bearing effort cost

Revenue-sharing contract based onquantity discount policy Traditional

revenue-sharing contract Wholesale price contractA B

Optimal effort 119890 1962 1962 397 379Optimal order quantity 119876 1674 1674 339 288Profit of the whole supply chain Π

119905 1924628 1924628 700833 654436

According to the supposition Π119905(119876lowast 119890) is the concave

function of effort 119890 So the optimal effort 119890lowast of the supplychain should satisfy

120597Π119905(119876lowast 119890lowast)

120597119890= (119901 minus V) 1199101015840 (119890lowast) int

119865minus1

((119901minus119888)(119901minusV))

0

119905119891 (119905) 119889119905

minus 1198921015840(119890lowast) = 0

(39)

From (39) it can easy to get that the optimal effort 119890lowast ofthe supply chain should satisfy

(119901 minus V) 1199101015840 (119890lowast) int119865minus1

((119901minus119888)(119901minusV))

0

119905119891 (119905) 119889119905 = 1198921015840(119890lowast) (40)

The method to solve the expression of 119890lowast in (40) is thesame as the method in (21)

424 An Example Suppose a supply chain is made up ofa retailer a distributor and a manufacturer and the supplychain parameters are 119888

119903= 1 119888119889= 2 119888119898= 5 V = 3 119908119889

119889= 18

119908119889119898= 10 119901 = 35 120601

1= 1206012= 055 and 119886 = 100

Here assume the relationship between demand and effortsatisfies multiplication form and 119892(119890) = 119886119890

22 119910(119890) = 119890120585 satisfies uniform distribution in [60 90] It is easy to get119891(120585) = 130 119865(120585) = (120585 minus 60)30 and 119865minus1(120585) = 60 + 30120585

(1) The revenue-sharing contract based on quantity dis-count policy

From the discussion above we know that the revenue-sharing contract based on quantity discount policy cancoordinate the supply chain whether it is implemented in oneor two transactions According to the formulas mentionedabove it is easy to get the optimal effort of the manufacturerthe optimal order quantity of the retailer and the profitof the whole supply chain Here 119890lowast = 1962 119876lowast =

1674 Π119905(119876lowast 119890lowast) = 1924628

(2) The traditional revenue-sharing contract

Similarly under the traditional revenue-sharing contractthe optimal effort of the manufacturer and order quantityof the retailer and the profit of the whole supply chain arerespectively 119890119898119888 = 397 119876119903119888 = 339 and Π

119905(119876119903119888 119890119898119888) =

700833

(3) The decentralized supply chain (the wholesale pricecontract)

Similarly under the traditional revenue-sharing contractthe optimal effort of the manufacturer and the order quantityof the retailer and the profit of the whole supply chain arerespectively 119890119898119889 = 379 119876119903119889 = 288 and Π

119905(119876119903119889 119890119889119898) =

654436Table 2 shows the parameters of the supply chain in

different modes A means that we implement the revenue-sharing contract based on quantity discount policy onlybetween the retailer and the distributor B means that weimplement the revenue-sharing contract based on quantitydiscount policy in two transactions of the supply chain

Table 2 shows that the manufacturerrsquos effort in therevenue-sharing contract based on quantity discount policyis the highest in the three modes which can make thesupply chain maximize its profit and the revenue-sharingcontract based on quantity discount policy is better than thetraditional revenue-sharing contract with effort dependentdemand the manufacturerrsquos effort in traditional revenue-sharing contract is higher than that of the supply chain inthe wholesale price contract and retailerrsquos order quantity isstill higher than that in wholesale price contract due to theincentive of the revenue-sharing contract so the profit ofthe supply chain in the traditional revenue-sharing contractis still higher than that in wholesale price contract So itcan be seen that the traditional revenue-sharing contract isbetter than thewholesale price contract with effort dependentdemand

43TheRetailer Bearing the Effort Cost with theManufacturerSuppose the effort cost that the retailer bears is 120572119892(119890) (0 le

120572 le 1) then the manufacturer bears (1 minus 120572)119892(119890) effort costIf 120572 = 0 it means that only the manufacturer bears the effortcost if 120572 = 1 it means that only the retailer bears the effortcost

In this condition we find that the traditional revenue-sharing contract fails to coordinate the supply chain whetherit is implemented in one transaction phase or in two transac-tion phases of the three-level supply chain The reason lies inthat the distributor does not bear the effort cost but he can getpartial profit of thewhole supply chain which is unfair for theretailer and themanufacturer So the effort level of the retailerand the manufacturer will be lower than the effort level of thecentralized supply chain Now the following discusses howto design the improved revenue-sharing implemented in twotransaction phases to coordinate the supply chain

Suppose the distributor provides the retailer the revenue-sharing contract (120601

2 119908119888119889(119876 119890)) and the manufacturer pro-

vides the distributors the revenue-sharing contract (1206011

Discrete Dynamics in Nature and Society 9

119908119888119898(119876 119890)) Then the profit functions of the supply chain

members can be described as

Π119888

119903(119876 119890) = 1206012 (119901 minus V) 119878 (119876 119890)

minus [119888119903+ 119908119888

119889(119876 119890) minus 1206012V] 119876 minus 120572119892 (119890)

Π119888

119889(119876 119890) = 1206011 (1 minus 1206012) (119901 minus V) 119878 (119876 119890)

minus [119888119889+ 119908119888

119898(119876 119890) minus 1206011 (1 minus 1206012) V

minus1206011119908119888

119889(119876 119890)] 119876

Π119888

119898(119876 119890) = (1 minus 1206011) (1 minus 1206012) (119901 minus V) 119878 (119876 119890)

minus [119888119898minus 119908119888

119898(119876 119890) minus (1 minus 1206011) (1 minus 1206012) V

minus (1 minus 1206011) 119908119888

119889(119876 119890)] 119876

minus (1 minus 120572) 119892 (119890)

(41)

If the improved revenue-sharing contract can coordinatethe supply chain the optimal effort level 119890lowast of the retailerand the manufacturer should satisfy (2) Namely the optimalorder (production) quantity 119876

lowast of the retailer and themanufacturer should satisfy (3) Then we can get

120597Π119888119903(119876 119890lowast)

120597119890=120597Π119888

119898(119876 119890lowast)

120597119890= 0

(119901 minus V)120597119878 (119876 119890

lowast)

120597119890minus 1198921015840(119890lowast) = 0

120597Π119888119903(119876lowast 119890)

120597119876=120597Π119888

119898(119876lowast 119890)

120597119876= 0

(119901 minus V)120597119878 (119876lowast 119890)

120597119876minus (119888 minus V) = 0

(42)

From (42) we can get

119908119888

119889(119876 119890) = 1206012 (119888 minus V) minus (1 minus 120572)

119892 (119890)

119876minus 119888119903+ 1206012V

119908119888

119898(119876 119890) = 1206011 (119888 minus V) + 1206011 (1 minus 120572)

119892 (119890)

119876minus 1206011119888119903+ 1206011V minus 119888119889

(43)

Taking (43) into the profit functions of the supply chainmembers (41) we can get

Π119888

119903(119876 119890) = 1206012Π119905 (119876 119890)

Π119888

119889(119876 119890) = 1206011 (1 minus 1206012)Π119905 (119876 119890)

Π119888

119898(119876 119890) = (1 minus 1206011) (1 minus 1206012)Π119905 (119876 119890)

(44)

We find that (44) equals (11) That is to say when theretailer bears the effort with the manufacturer the improvedrevenue-sharing contract can coordinate the three-level sup-ply chain

Therefore if we implement the revenue-sharing contractbased on quantity discount policy among the supply chain

members and the contract parameters which satisfy (43)the improved revenue-sharing can coordinate the three-levelsupply chain

The method to determine the optimal order quantity 119876lowastand the optimal effort 119890lowast can follow the methods given inSections 413 and 423

5 Conclusion

The traditional revenue-sharing contract cannot coordinatethree-level supply chain with sales effort dependent demandThe paper proposes an improved revenue-sharing contractbased on quantity discount policy which can be implementedin one transaction or two transactions of three-level supplychain to coordinate the supply chain The paper also showsthat the profit of the supply chain members depends on thevalue of the contract parameters which is determined by thestatus of the members in supply chain and their bargainingpower It also gave the method to determine the optimalorder quantity and the optimal effort which can provide thedecision makers of supply chain with decision support Butit also should be noticed that this research is carried outin the symmetrical information condition So the next stepis to carry out the research in the asymmetric informationcondition

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This research was supported by the Ministry of Educationof China Grant-in-aid for Humanity and social ScienceResearch (no 10YJC630188) and the Fundamental ResearchFunds for the Central Universities (nos 2012B13914 and2012B13814)

References

[1] G P Cachon and M A Lariviere ldquoSupply chain coordinationwith revenue-sharing contracts strengths and limitationsrdquoManagement Science vol 51 no 1 pp 30ndash44 2005

[2] J H Mortimer The Effects of Revenue-Sharing Contracts onWelfare in Vertically SeparatedMarkets Evidence from the VideoRental Industry University of California at Los Angeles 2000

[3] J D Dana Jr and K E Spier ldquoRevenue sharing and verticalcontrol in the video rental industryrdquo Journal of IndustrialEconomics vol 49 no 3 pp 223ndash245 2001

[4] B Pasternack ldquoUsing revenue-sharing to achieve channel coor-dination for a newsboy type inventory modelrdquo in Supply ChainManagement Models Applications and Research DirectionsJ Geunes P Panos and H Edwin Eds vol 62 pp 271ndash290 Kluwer Academic Publishers DordrechtTheNetherlands2002

[5] Y Wang L Jiang and Z J Shen ldquoChannel performance underconsignment contract with revenue-sharingrdquoManagement Sci-ence vol 50 no 1 pp 34ndash47 2004

10 Discrete Dynamics in Nature and Society

[6] Y Gerchak and Y Wang ldquoRevenue-sharing vs wholesale-price contracts in assembly systems with random demandrdquoProduction andOperationsManagement vol 13 no 1 pp 23ndash332004

[7] C Koulamas ldquoA newsvendor problemwith revenue sharing andchannel coordinationrdquo Decision Sciences vol 37 no 1 pp 91ndash100 2006

[8] S S Chauhan and J Proth ldquoAnalysis of a supply chain partner-ship with revenue sharingrdquo International Journal of ProductionEconomics vol 97 no 1 pp 44ndash51 2005

[9] C T Linh and Y S Hong ldquoChannel coordination through arevenue sharing contract in a two-period newsboy problemrdquoEuropean Journal of Operational Research vol 198 no 3 pp822ndash829 2009

[10] K Pan K K Lai S C H Leung andD Xiao ldquoRevenue-sharingversus wholesale price mechanisms under different channelpower structuresrdquo European Journal of Operational Researchvol 203 no 2 pp 532ndash538 2010

[11] D G Qu and Y J Guo ldquoCoordination of supply chain withhybrid distribution channels when retailerrsquos demand relies onits sales effortrdquo Chinese Journal of Management Science vol 21no 3 pp 19ndash23 2008

[12] I Giannoccaro and P Pontrandolfo ldquoSupply chain coordinationby revenue sharing contractsrdquo International Journal of Produc-tion Economics vol 89 no 2 pp 131ndash139 2004

[13] S F Ji M J Liu W Ding and X Huang ldquoCoordination ofthree-level supply chain and revenue sharing contractrdquo Journalof Northeastern University vol 29 no 11 pp 1652ndash1656 2008

[14] L Hu Z Jiang M Meng and J Qin ldquoCoordination of a three-level supply chain through revenue sharingrdquo Journal of HarbinEngineering University vol 29 no 2 pp 198ndash203 2008

[15] Y L Hou D Q Zhou and B Y Tian ldquoOrdering decisionson supply chain under revenue sharing policiesrdquo IndustrialEngineering Journal vol 12 no 1 pp 24ndash27 2009

[16] Y He Q Wu and L Zhao ldquoQuantity flexibility contract witheffort and price-dependent demandrdquo Systems Engineering andElectronics vol 29 no 12 pp 2056ndash2059 2007

[17] P Desai andK Srinivasan ldquoDemand signaling under unobserv-able effort in franchisingrdquo Management Science vol 20 no 4pp 2608ndash2623 1995

[18] W Chu and P Desai ldquoChannel coordination mechanisms forcustomer satisfactionrdquoMarketing Science vol 21 no 6 pp 475ndash490 1995

[19] T A Taylor ldquoSupply chain coordination under channel rebateswith sales effort effectsrdquoManagement Science vol 48 no 8 pp992ndash1007 2002

[20] Z Xu D L Zhu and W G Zhu ldquoSupply chain coordinationunder buy-back contract with sales effort effectsrdquo System Engi-neering Theory and Practice vol 28 no 4 pp 1ndash11 2008

[21] G P Cachon ldquoSupply chain coordination with contractsrdquoin Handbooks in Operations Research and Management Sci-ence Supply Chain Management North-Holland Elseriver SanDiego Calif USA 2000

[22] Y He D Yang andQWu ldquoRevenue-sharing contract of supplychain with effort dependent demandrdquo Computer IntegratedManufacturing Systems vol 12 no 11 pp 1865ndash1868 2006

[23] Y He X Zhao L Zhao and J He ldquoCoordinating a supply chainwith effort and price dependent stochastic demandrdquo AppliedMathematical Modelling vol 33 no 6 pp 2777ndash2790 2009

[24] B Y Hu and X Y Wang ldquoSupply chain revenue-sharingevolvement-contract with sales effort effectsrdquo Journal of Indus-trial Engineering andEngineeringManagement vol 29 no 2 pp35ndash39 2010

[25] J Chen and J F Chen ldquoStudy on revenue sharing contractin virtual enterprisesrdquo Journal of Systems Science and SystemsEngineering vol 15 no 1 pp 95ndash113 2006

[26] Y Wang and Y Gerchak ldquoSupply Chain Coordination whendemand is shelf-space dpendentrdquo Manufacturing and ServiceOperations Management vol 3 no 1 pp 82ndash87 2001

[27] H Krishnan R Kapuscinski and D A Butz ldquoCoordinatingcontracts for decentralized supply chains with retailer promo-tional effortrdquo Management Science vol 50 no 1 pp 48ndash632004

[28] Y He Supply Chain Contract Models with Stochastic DemandDalian University of Technology Dalian China 2004

[29] L Yao Supply Chain Modeling Pricing Contracts and Coordi-nation Chinese University of Hong Kong Hong Kong 2002

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

8 Discrete Dynamics in Nature and Society

Table 2 The parameters of supply chain in different modes with only manufacturer bearing effort cost

Revenue-sharing contract based onquantity discount policy Traditional

revenue-sharing contract Wholesale price contractA B

Optimal effort 119890 1962 1962 397 379Optimal order quantity 119876 1674 1674 339 288Profit of the whole supply chain Π

119905 1924628 1924628 700833 654436

According to the supposition Π119905(119876lowast 119890) is the concave

function of effort 119890 So the optimal effort 119890lowast of the supplychain should satisfy

120597Π119905(119876lowast 119890lowast)

120597119890= (119901 minus V) 1199101015840 (119890lowast) int

119865minus1

((119901minus119888)(119901minusV))

0

119905119891 (119905) 119889119905

minus 1198921015840(119890lowast) = 0

(39)

From (39) it can easy to get that the optimal effort 119890lowast ofthe supply chain should satisfy

(119901 minus V) 1199101015840 (119890lowast) int119865minus1

((119901minus119888)(119901minusV))

0

119905119891 (119905) 119889119905 = 1198921015840(119890lowast) (40)

The method to solve the expression of 119890lowast in (40) is thesame as the method in (21)

424 An Example Suppose a supply chain is made up ofa retailer a distributor and a manufacturer and the supplychain parameters are 119888

119903= 1 119888119889= 2 119888119898= 5 V = 3 119908119889

119889= 18

119908119889119898= 10 119901 = 35 120601

1= 1206012= 055 and 119886 = 100

Here assume the relationship between demand and effortsatisfies multiplication form and 119892(119890) = 119886119890

22 119910(119890) = 119890120585 satisfies uniform distribution in [60 90] It is easy to get119891(120585) = 130 119865(120585) = (120585 minus 60)30 and 119865minus1(120585) = 60 + 30120585

(1) The revenue-sharing contract based on quantity dis-count policy

From the discussion above we know that the revenue-sharing contract based on quantity discount policy cancoordinate the supply chain whether it is implemented in oneor two transactions According to the formulas mentionedabove it is easy to get the optimal effort of the manufacturerthe optimal order quantity of the retailer and the profitof the whole supply chain Here 119890lowast = 1962 119876lowast =

1674 Π119905(119876lowast 119890lowast) = 1924628

(2) The traditional revenue-sharing contract

Similarly under the traditional revenue-sharing contractthe optimal effort of the manufacturer and order quantityof the retailer and the profit of the whole supply chain arerespectively 119890119898119888 = 397 119876119903119888 = 339 and Π

119905(119876119903119888 119890119898119888) =

700833

(3) The decentralized supply chain (the wholesale pricecontract)

Similarly under the traditional revenue-sharing contractthe optimal effort of the manufacturer and the order quantityof the retailer and the profit of the whole supply chain arerespectively 119890119898119889 = 379 119876119903119889 = 288 and Π

119905(119876119903119889 119890119889119898) =

654436Table 2 shows the parameters of the supply chain in

different modes A means that we implement the revenue-sharing contract based on quantity discount policy onlybetween the retailer and the distributor B means that weimplement the revenue-sharing contract based on quantitydiscount policy in two transactions of the supply chain

Table 2 shows that the manufacturerrsquos effort in therevenue-sharing contract based on quantity discount policyis the highest in the three modes which can make thesupply chain maximize its profit and the revenue-sharingcontract based on quantity discount policy is better than thetraditional revenue-sharing contract with effort dependentdemand the manufacturerrsquos effort in traditional revenue-sharing contract is higher than that of the supply chain inthe wholesale price contract and retailerrsquos order quantity isstill higher than that in wholesale price contract due to theincentive of the revenue-sharing contract so the profit ofthe supply chain in the traditional revenue-sharing contractis still higher than that in wholesale price contract So itcan be seen that the traditional revenue-sharing contract isbetter than thewholesale price contract with effort dependentdemand

43TheRetailer Bearing the Effort Cost with theManufacturerSuppose the effort cost that the retailer bears is 120572119892(119890) (0 le

120572 le 1) then the manufacturer bears (1 minus 120572)119892(119890) effort costIf 120572 = 0 it means that only the manufacturer bears the effortcost if 120572 = 1 it means that only the retailer bears the effortcost

In this condition we find that the traditional revenue-sharing contract fails to coordinate the supply chain whetherit is implemented in one transaction phase or in two transac-tion phases of the three-level supply chain The reason lies inthat the distributor does not bear the effort cost but he can getpartial profit of thewhole supply chain which is unfair for theretailer and themanufacturer So the effort level of the retailerand the manufacturer will be lower than the effort level of thecentralized supply chain Now the following discusses howto design the improved revenue-sharing implemented in twotransaction phases to coordinate the supply chain

Suppose the distributor provides the retailer the revenue-sharing contract (120601

2 119908119888119889(119876 119890)) and the manufacturer pro-

vides the distributors the revenue-sharing contract (1206011

Discrete Dynamics in Nature and Society 9

119908119888119898(119876 119890)) Then the profit functions of the supply chain

members can be described as

Π119888

119903(119876 119890) = 1206012 (119901 minus V) 119878 (119876 119890)

minus [119888119903+ 119908119888

119889(119876 119890) minus 1206012V] 119876 minus 120572119892 (119890)

Π119888

119889(119876 119890) = 1206011 (1 minus 1206012) (119901 minus V) 119878 (119876 119890)

minus [119888119889+ 119908119888

119898(119876 119890) minus 1206011 (1 minus 1206012) V

minus1206011119908119888

119889(119876 119890)] 119876

Π119888

119898(119876 119890) = (1 minus 1206011) (1 minus 1206012) (119901 minus V) 119878 (119876 119890)

minus [119888119898minus 119908119888

119898(119876 119890) minus (1 minus 1206011) (1 minus 1206012) V

minus (1 minus 1206011) 119908119888

119889(119876 119890)] 119876

minus (1 minus 120572) 119892 (119890)

(41)

If the improved revenue-sharing contract can coordinatethe supply chain the optimal effort level 119890lowast of the retailerand the manufacturer should satisfy (2) Namely the optimalorder (production) quantity 119876

lowast of the retailer and themanufacturer should satisfy (3) Then we can get

120597Π119888119903(119876 119890lowast)

120597119890=120597Π119888

119898(119876 119890lowast)

120597119890= 0

(119901 minus V)120597119878 (119876 119890

lowast)

120597119890minus 1198921015840(119890lowast) = 0

120597Π119888119903(119876lowast 119890)

120597119876=120597Π119888

119898(119876lowast 119890)

120597119876= 0

(119901 minus V)120597119878 (119876lowast 119890)

120597119876minus (119888 minus V) = 0

(42)

From (42) we can get

119908119888

119889(119876 119890) = 1206012 (119888 minus V) minus (1 minus 120572)

119892 (119890)

119876minus 119888119903+ 1206012V

119908119888

119898(119876 119890) = 1206011 (119888 minus V) + 1206011 (1 minus 120572)

119892 (119890)

119876minus 1206011119888119903+ 1206011V minus 119888119889

(43)

Taking (43) into the profit functions of the supply chainmembers (41) we can get

Π119888

119903(119876 119890) = 1206012Π119905 (119876 119890)

Π119888

119889(119876 119890) = 1206011 (1 minus 1206012)Π119905 (119876 119890)

Π119888

119898(119876 119890) = (1 minus 1206011) (1 minus 1206012)Π119905 (119876 119890)

(44)

We find that (44) equals (11) That is to say when theretailer bears the effort with the manufacturer the improvedrevenue-sharing contract can coordinate the three-level sup-ply chain

Therefore if we implement the revenue-sharing contractbased on quantity discount policy among the supply chain

members and the contract parameters which satisfy (43)the improved revenue-sharing can coordinate the three-levelsupply chain

The method to determine the optimal order quantity 119876lowastand the optimal effort 119890lowast can follow the methods given inSections 413 and 423

5 Conclusion

The traditional revenue-sharing contract cannot coordinatethree-level supply chain with sales effort dependent demandThe paper proposes an improved revenue-sharing contractbased on quantity discount policy which can be implementedin one transaction or two transactions of three-level supplychain to coordinate the supply chain The paper also showsthat the profit of the supply chain members depends on thevalue of the contract parameters which is determined by thestatus of the members in supply chain and their bargainingpower It also gave the method to determine the optimalorder quantity and the optimal effort which can provide thedecision makers of supply chain with decision support Butit also should be noticed that this research is carried outin the symmetrical information condition So the next stepis to carry out the research in the asymmetric informationcondition

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This research was supported by the Ministry of Educationof China Grant-in-aid for Humanity and social ScienceResearch (no 10YJC630188) and the Fundamental ResearchFunds for the Central Universities (nos 2012B13914 and2012B13814)

References

[1] G P Cachon and M A Lariviere ldquoSupply chain coordinationwith revenue-sharing contracts strengths and limitationsrdquoManagement Science vol 51 no 1 pp 30ndash44 2005

[2] J H Mortimer The Effects of Revenue-Sharing Contracts onWelfare in Vertically SeparatedMarkets Evidence from the VideoRental Industry University of California at Los Angeles 2000

[3] J D Dana Jr and K E Spier ldquoRevenue sharing and verticalcontrol in the video rental industryrdquo Journal of IndustrialEconomics vol 49 no 3 pp 223ndash245 2001

[4] B Pasternack ldquoUsing revenue-sharing to achieve channel coor-dination for a newsboy type inventory modelrdquo in Supply ChainManagement Models Applications and Research DirectionsJ Geunes P Panos and H Edwin Eds vol 62 pp 271ndash290 Kluwer Academic Publishers DordrechtTheNetherlands2002

[5] Y Wang L Jiang and Z J Shen ldquoChannel performance underconsignment contract with revenue-sharingrdquoManagement Sci-ence vol 50 no 1 pp 34ndash47 2004

10 Discrete Dynamics in Nature and Society

[6] Y Gerchak and Y Wang ldquoRevenue-sharing vs wholesale-price contracts in assembly systems with random demandrdquoProduction andOperationsManagement vol 13 no 1 pp 23ndash332004

[7] C Koulamas ldquoA newsvendor problemwith revenue sharing andchannel coordinationrdquo Decision Sciences vol 37 no 1 pp 91ndash100 2006

[8] S S Chauhan and J Proth ldquoAnalysis of a supply chain partner-ship with revenue sharingrdquo International Journal of ProductionEconomics vol 97 no 1 pp 44ndash51 2005

[9] C T Linh and Y S Hong ldquoChannel coordination through arevenue sharing contract in a two-period newsboy problemrdquoEuropean Journal of Operational Research vol 198 no 3 pp822ndash829 2009

[10] K Pan K K Lai S C H Leung andD Xiao ldquoRevenue-sharingversus wholesale price mechanisms under different channelpower structuresrdquo European Journal of Operational Researchvol 203 no 2 pp 532ndash538 2010

[11] D G Qu and Y J Guo ldquoCoordination of supply chain withhybrid distribution channels when retailerrsquos demand relies onits sales effortrdquo Chinese Journal of Management Science vol 21no 3 pp 19ndash23 2008

[12] I Giannoccaro and P Pontrandolfo ldquoSupply chain coordinationby revenue sharing contractsrdquo International Journal of Produc-tion Economics vol 89 no 2 pp 131ndash139 2004

[13] S F Ji M J Liu W Ding and X Huang ldquoCoordination ofthree-level supply chain and revenue sharing contractrdquo Journalof Northeastern University vol 29 no 11 pp 1652ndash1656 2008

[14] L Hu Z Jiang M Meng and J Qin ldquoCoordination of a three-level supply chain through revenue sharingrdquo Journal of HarbinEngineering University vol 29 no 2 pp 198ndash203 2008

[15] Y L Hou D Q Zhou and B Y Tian ldquoOrdering decisionson supply chain under revenue sharing policiesrdquo IndustrialEngineering Journal vol 12 no 1 pp 24ndash27 2009

[16] Y He Q Wu and L Zhao ldquoQuantity flexibility contract witheffort and price-dependent demandrdquo Systems Engineering andElectronics vol 29 no 12 pp 2056ndash2059 2007

[17] P Desai andK Srinivasan ldquoDemand signaling under unobserv-able effort in franchisingrdquo Management Science vol 20 no 4pp 2608ndash2623 1995

[18] W Chu and P Desai ldquoChannel coordination mechanisms forcustomer satisfactionrdquoMarketing Science vol 21 no 6 pp 475ndash490 1995

[19] T A Taylor ldquoSupply chain coordination under channel rebateswith sales effort effectsrdquoManagement Science vol 48 no 8 pp992ndash1007 2002

[20] Z Xu D L Zhu and W G Zhu ldquoSupply chain coordinationunder buy-back contract with sales effort effectsrdquo System Engi-neering Theory and Practice vol 28 no 4 pp 1ndash11 2008

[21] G P Cachon ldquoSupply chain coordination with contractsrdquoin Handbooks in Operations Research and Management Sci-ence Supply Chain Management North-Holland Elseriver SanDiego Calif USA 2000

[22] Y He D Yang andQWu ldquoRevenue-sharing contract of supplychain with effort dependent demandrdquo Computer IntegratedManufacturing Systems vol 12 no 11 pp 1865ndash1868 2006

[23] Y He X Zhao L Zhao and J He ldquoCoordinating a supply chainwith effort and price dependent stochastic demandrdquo AppliedMathematical Modelling vol 33 no 6 pp 2777ndash2790 2009

[24] B Y Hu and X Y Wang ldquoSupply chain revenue-sharingevolvement-contract with sales effort effectsrdquo Journal of Indus-trial Engineering andEngineeringManagement vol 29 no 2 pp35ndash39 2010

[25] J Chen and J F Chen ldquoStudy on revenue sharing contractin virtual enterprisesrdquo Journal of Systems Science and SystemsEngineering vol 15 no 1 pp 95ndash113 2006

[26] Y Wang and Y Gerchak ldquoSupply Chain Coordination whendemand is shelf-space dpendentrdquo Manufacturing and ServiceOperations Management vol 3 no 1 pp 82ndash87 2001

[27] H Krishnan R Kapuscinski and D A Butz ldquoCoordinatingcontracts for decentralized supply chains with retailer promo-tional effortrdquo Management Science vol 50 no 1 pp 48ndash632004

[28] Y He Supply Chain Contract Models with Stochastic DemandDalian University of Technology Dalian China 2004

[29] L Yao Supply Chain Modeling Pricing Contracts and Coordi-nation Chinese University of Hong Kong Hong Kong 2002

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Discrete Dynamics in Nature and Society 9

119908119888119898(119876 119890)) Then the profit functions of the supply chain

members can be described as

Π119888

119903(119876 119890) = 1206012 (119901 minus V) 119878 (119876 119890)

minus [119888119903+ 119908119888

119889(119876 119890) minus 1206012V] 119876 minus 120572119892 (119890)

Π119888

119889(119876 119890) = 1206011 (1 minus 1206012) (119901 minus V) 119878 (119876 119890)

minus [119888119889+ 119908119888

119898(119876 119890) minus 1206011 (1 minus 1206012) V

minus1206011119908119888

119889(119876 119890)] 119876

Π119888

119898(119876 119890) = (1 minus 1206011) (1 minus 1206012) (119901 minus V) 119878 (119876 119890)

minus [119888119898minus 119908119888

119898(119876 119890) minus (1 minus 1206011) (1 minus 1206012) V

minus (1 minus 1206011) 119908119888

119889(119876 119890)] 119876

minus (1 minus 120572) 119892 (119890)

(41)

If the improved revenue-sharing contract can coordinatethe supply chain the optimal effort level 119890lowast of the retailerand the manufacturer should satisfy (2) Namely the optimalorder (production) quantity 119876

lowast of the retailer and themanufacturer should satisfy (3) Then we can get

120597Π119888119903(119876 119890lowast)

120597119890=120597Π119888

119898(119876 119890lowast)

120597119890= 0

(119901 minus V)120597119878 (119876 119890

lowast)

120597119890minus 1198921015840(119890lowast) = 0

120597Π119888119903(119876lowast 119890)

120597119876=120597Π119888

119898(119876lowast 119890)

120597119876= 0

(119901 minus V)120597119878 (119876lowast 119890)

120597119876minus (119888 minus V) = 0

(42)

From (42) we can get

119908119888

119889(119876 119890) = 1206012 (119888 minus V) minus (1 minus 120572)

119892 (119890)

119876minus 119888119903+ 1206012V

119908119888

119898(119876 119890) = 1206011 (119888 minus V) + 1206011 (1 minus 120572)

119892 (119890)

119876minus 1206011119888119903+ 1206011V minus 119888119889

(43)

Taking (43) into the profit functions of the supply chainmembers (41) we can get

Π119888

119903(119876 119890) = 1206012Π119905 (119876 119890)

Π119888

119889(119876 119890) = 1206011 (1 minus 1206012)Π119905 (119876 119890)

Π119888

119898(119876 119890) = (1 minus 1206011) (1 minus 1206012)Π119905 (119876 119890)

(44)

We find that (44) equals (11) That is to say when theretailer bears the effort with the manufacturer the improvedrevenue-sharing contract can coordinate the three-level sup-ply chain

Therefore if we implement the revenue-sharing contractbased on quantity discount policy among the supply chain

members and the contract parameters which satisfy (43)the improved revenue-sharing can coordinate the three-levelsupply chain

The method to determine the optimal order quantity 119876lowastand the optimal effort 119890lowast can follow the methods given inSections 413 and 423

5 Conclusion

The traditional revenue-sharing contract cannot coordinatethree-level supply chain with sales effort dependent demandThe paper proposes an improved revenue-sharing contractbased on quantity discount policy which can be implementedin one transaction or two transactions of three-level supplychain to coordinate the supply chain The paper also showsthat the profit of the supply chain members depends on thevalue of the contract parameters which is determined by thestatus of the members in supply chain and their bargainingpower It also gave the method to determine the optimalorder quantity and the optimal effort which can provide thedecision makers of supply chain with decision support Butit also should be noticed that this research is carried outin the symmetrical information condition So the next stepis to carry out the research in the asymmetric informationcondition

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This research was supported by the Ministry of Educationof China Grant-in-aid for Humanity and social ScienceResearch (no 10YJC630188) and the Fundamental ResearchFunds for the Central Universities (nos 2012B13914 and2012B13814)

References

[1] G P Cachon and M A Lariviere ldquoSupply chain coordinationwith revenue-sharing contracts strengths and limitationsrdquoManagement Science vol 51 no 1 pp 30ndash44 2005

[2] J H Mortimer The Effects of Revenue-Sharing Contracts onWelfare in Vertically SeparatedMarkets Evidence from the VideoRental Industry University of California at Los Angeles 2000

[3] J D Dana Jr and K E Spier ldquoRevenue sharing and verticalcontrol in the video rental industryrdquo Journal of IndustrialEconomics vol 49 no 3 pp 223ndash245 2001

[4] B Pasternack ldquoUsing revenue-sharing to achieve channel coor-dination for a newsboy type inventory modelrdquo in Supply ChainManagement Models Applications and Research DirectionsJ Geunes P Panos and H Edwin Eds vol 62 pp 271ndash290 Kluwer Academic Publishers DordrechtTheNetherlands2002

[5] Y Wang L Jiang and Z J Shen ldquoChannel performance underconsignment contract with revenue-sharingrdquoManagement Sci-ence vol 50 no 1 pp 34ndash47 2004

10 Discrete Dynamics in Nature and Society

[6] Y Gerchak and Y Wang ldquoRevenue-sharing vs wholesale-price contracts in assembly systems with random demandrdquoProduction andOperationsManagement vol 13 no 1 pp 23ndash332004

[7] C Koulamas ldquoA newsvendor problemwith revenue sharing andchannel coordinationrdquo Decision Sciences vol 37 no 1 pp 91ndash100 2006

[8] S S Chauhan and J Proth ldquoAnalysis of a supply chain partner-ship with revenue sharingrdquo International Journal of ProductionEconomics vol 97 no 1 pp 44ndash51 2005

[9] C T Linh and Y S Hong ldquoChannel coordination through arevenue sharing contract in a two-period newsboy problemrdquoEuropean Journal of Operational Research vol 198 no 3 pp822ndash829 2009

[10] K Pan K K Lai S C H Leung andD Xiao ldquoRevenue-sharingversus wholesale price mechanisms under different channelpower structuresrdquo European Journal of Operational Researchvol 203 no 2 pp 532ndash538 2010

[11] D G Qu and Y J Guo ldquoCoordination of supply chain withhybrid distribution channels when retailerrsquos demand relies onits sales effortrdquo Chinese Journal of Management Science vol 21no 3 pp 19ndash23 2008

[12] I Giannoccaro and P Pontrandolfo ldquoSupply chain coordinationby revenue sharing contractsrdquo International Journal of Produc-tion Economics vol 89 no 2 pp 131ndash139 2004

[13] S F Ji M J Liu W Ding and X Huang ldquoCoordination ofthree-level supply chain and revenue sharing contractrdquo Journalof Northeastern University vol 29 no 11 pp 1652ndash1656 2008

[14] L Hu Z Jiang M Meng and J Qin ldquoCoordination of a three-level supply chain through revenue sharingrdquo Journal of HarbinEngineering University vol 29 no 2 pp 198ndash203 2008

[15] Y L Hou D Q Zhou and B Y Tian ldquoOrdering decisionson supply chain under revenue sharing policiesrdquo IndustrialEngineering Journal vol 12 no 1 pp 24ndash27 2009

[16] Y He Q Wu and L Zhao ldquoQuantity flexibility contract witheffort and price-dependent demandrdquo Systems Engineering andElectronics vol 29 no 12 pp 2056ndash2059 2007

[17] P Desai andK Srinivasan ldquoDemand signaling under unobserv-able effort in franchisingrdquo Management Science vol 20 no 4pp 2608ndash2623 1995

[18] W Chu and P Desai ldquoChannel coordination mechanisms forcustomer satisfactionrdquoMarketing Science vol 21 no 6 pp 475ndash490 1995

[19] T A Taylor ldquoSupply chain coordination under channel rebateswith sales effort effectsrdquoManagement Science vol 48 no 8 pp992ndash1007 2002

[20] Z Xu D L Zhu and W G Zhu ldquoSupply chain coordinationunder buy-back contract with sales effort effectsrdquo System Engi-neering Theory and Practice vol 28 no 4 pp 1ndash11 2008

[21] G P Cachon ldquoSupply chain coordination with contractsrdquoin Handbooks in Operations Research and Management Sci-ence Supply Chain Management North-Holland Elseriver SanDiego Calif USA 2000

[22] Y He D Yang andQWu ldquoRevenue-sharing contract of supplychain with effort dependent demandrdquo Computer IntegratedManufacturing Systems vol 12 no 11 pp 1865ndash1868 2006

[23] Y He X Zhao L Zhao and J He ldquoCoordinating a supply chainwith effort and price dependent stochastic demandrdquo AppliedMathematical Modelling vol 33 no 6 pp 2777ndash2790 2009

[24] B Y Hu and X Y Wang ldquoSupply chain revenue-sharingevolvement-contract with sales effort effectsrdquo Journal of Indus-trial Engineering andEngineeringManagement vol 29 no 2 pp35ndash39 2010

[25] J Chen and J F Chen ldquoStudy on revenue sharing contractin virtual enterprisesrdquo Journal of Systems Science and SystemsEngineering vol 15 no 1 pp 95ndash113 2006

[26] Y Wang and Y Gerchak ldquoSupply Chain Coordination whendemand is shelf-space dpendentrdquo Manufacturing and ServiceOperations Management vol 3 no 1 pp 82ndash87 2001

[27] H Krishnan R Kapuscinski and D A Butz ldquoCoordinatingcontracts for decentralized supply chains with retailer promo-tional effortrdquo Management Science vol 50 no 1 pp 48ndash632004

[28] Y He Supply Chain Contract Models with Stochastic DemandDalian University of Technology Dalian China 2004

[29] L Yao Supply Chain Modeling Pricing Contracts and Coordi-nation Chinese University of Hong Kong Hong Kong 2002

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

10 Discrete Dynamics in Nature and Society

[6] Y Gerchak and Y Wang ldquoRevenue-sharing vs wholesale-price contracts in assembly systems with random demandrdquoProduction andOperationsManagement vol 13 no 1 pp 23ndash332004

[7] C Koulamas ldquoA newsvendor problemwith revenue sharing andchannel coordinationrdquo Decision Sciences vol 37 no 1 pp 91ndash100 2006

[8] S S Chauhan and J Proth ldquoAnalysis of a supply chain partner-ship with revenue sharingrdquo International Journal of ProductionEconomics vol 97 no 1 pp 44ndash51 2005

[9] C T Linh and Y S Hong ldquoChannel coordination through arevenue sharing contract in a two-period newsboy problemrdquoEuropean Journal of Operational Research vol 198 no 3 pp822ndash829 2009

[10] K Pan K K Lai S C H Leung andD Xiao ldquoRevenue-sharingversus wholesale price mechanisms under different channelpower structuresrdquo European Journal of Operational Researchvol 203 no 2 pp 532ndash538 2010

[11] D G Qu and Y J Guo ldquoCoordination of supply chain withhybrid distribution channels when retailerrsquos demand relies onits sales effortrdquo Chinese Journal of Management Science vol 21no 3 pp 19ndash23 2008

[12] I Giannoccaro and P Pontrandolfo ldquoSupply chain coordinationby revenue sharing contractsrdquo International Journal of Produc-tion Economics vol 89 no 2 pp 131ndash139 2004

[13] S F Ji M J Liu W Ding and X Huang ldquoCoordination ofthree-level supply chain and revenue sharing contractrdquo Journalof Northeastern University vol 29 no 11 pp 1652ndash1656 2008

[14] L Hu Z Jiang M Meng and J Qin ldquoCoordination of a three-level supply chain through revenue sharingrdquo Journal of HarbinEngineering University vol 29 no 2 pp 198ndash203 2008

[15] Y L Hou D Q Zhou and B Y Tian ldquoOrdering decisionson supply chain under revenue sharing policiesrdquo IndustrialEngineering Journal vol 12 no 1 pp 24ndash27 2009

[16] Y He Q Wu and L Zhao ldquoQuantity flexibility contract witheffort and price-dependent demandrdquo Systems Engineering andElectronics vol 29 no 12 pp 2056ndash2059 2007

[17] P Desai andK Srinivasan ldquoDemand signaling under unobserv-able effort in franchisingrdquo Management Science vol 20 no 4pp 2608ndash2623 1995

[18] W Chu and P Desai ldquoChannel coordination mechanisms forcustomer satisfactionrdquoMarketing Science vol 21 no 6 pp 475ndash490 1995

[19] T A Taylor ldquoSupply chain coordination under channel rebateswith sales effort effectsrdquoManagement Science vol 48 no 8 pp992ndash1007 2002

[20] Z Xu D L Zhu and W G Zhu ldquoSupply chain coordinationunder buy-back contract with sales effort effectsrdquo System Engi-neering Theory and Practice vol 28 no 4 pp 1ndash11 2008

[21] G P Cachon ldquoSupply chain coordination with contractsrdquoin Handbooks in Operations Research and Management Sci-ence Supply Chain Management North-Holland Elseriver SanDiego Calif USA 2000

[22] Y He D Yang andQWu ldquoRevenue-sharing contract of supplychain with effort dependent demandrdquo Computer IntegratedManufacturing Systems vol 12 no 11 pp 1865ndash1868 2006

[23] Y He X Zhao L Zhao and J He ldquoCoordinating a supply chainwith effort and price dependent stochastic demandrdquo AppliedMathematical Modelling vol 33 no 6 pp 2777ndash2790 2009

[24] B Y Hu and X Y Wang ldquoSupply chain revenue-sharingevolvement-contract with sales effort effectsrdquo Journal of Indus-trial Engineering andEngineeringManagement vol 29 no 2 pp35ndash39 2010

[25] J Chen and J F Chen ldquoStudy on revenue sharing contractin virtual enterprisesrdquo Journal of Systems Science and SystemsEngineering vol 15 no 1 pp 95ndash113 2006

[26] Y Wang and Y Gerchak ldquoSupply Chain Coordination whendemand is shelf-space dpendentrdquo Manufacturing and ServiceOperations Management vol 3 no 1 pp 82ndash87 2001

[27] H Krishnan R Kapuscinski and D A Butz ldquoCoordinatingcontracts for decentralized supply chains with retailer promo-tional effortrdquo Management Science vol 50 no 1 pp 48ndash632004

[28] Y He Supply Chain Contract Models with Stochastic DemandDalian University of Technology Dalian China 2004

[29] L Yao Supply Chain Modeling Pricing Contracts and Coordi-nation Chinese University of Hong Kong Hong Kong 2002

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of