Research ArticleDifferential Game Analyses of Logistics Service Supply ChainCoordination by Cost Sharing Contract

Haifeng Zhao Bin Lin Wanqing Mao and Yang Ye

School of Economics and Management Tongji University Shanghai 200092 China

Correspondence should be addressed to Haifeng Zhao hfzhaotongjieducn

Received 28 January 2014 Accepted 12 March 2014 Published 13 April 2014

Academic Editor X Zhang

Copyright copy 2014 Haifeng Zhao 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

Cooperation of all the members in a supply chain plays an important role in logistics service The service integrator can encouragecooperation from service suppliers by sharing their cost during the service which we assume can increase the sales by accumulatingthe reputation of the supply chain A differential game model is established with the logistics service supply chain that consists ofone service integrator and one supplier And we derive the optimal solutions of the Nash equilibrium without cost sharing contractand the Stackelberg equilibrium with the integrator as the leader who partially shares the cost of the efforts of the supplier Theresults make the benefits of the cost sharing contract in increasing the profits of both players as well as the whole supply chainexplicit which means that the cost sharing contract is an effective coordination mechanism in the long-term relationship of themembers in a logistics service supply chain

1 Introduction

A logistics service supply chain abbreviated as LSSC is atypical service supply chain which consists of functionalservice providers service integrators and customers [1] Afunctional logistics service provider (FLSP) is a traditionallogistics enterprise that can provide standard service such astransportation or warehousing A logistics service integrator(LSI) can integrate various functional services in differentregions to satisfy different needs of customers who may bea manufacture or other enterprise LSI is the focal companyin LSSC which integrates and controls all types of resourcesby the use of the information technology to meet the needsof customers and enhance the performance while decreasingthe total cost of serviceMeanwhile LSI and FLSP are separateand independent economic entities which results in theconflict and competition in revenue or the risk during thecooperation in service As a result the key issue is to developmechanisms that can incentivize the members to coordinatetheir activities so as to enhance the performance of the wholesupply chain

The coordination of supply chain is an important area ofsupply chain management and a lot of strategies have been

proposed so far [2] Among them a common mechanismis to develop a set of properly designed contracts [3] Sincethe concept of supply chain contract was first introducedby Pasternack [4] the design of contracts has received aconsiderable interest A large number of contracts have beenproposed including the wholesale price contract [5ndash10] thequantity flexibility contract [11ndash16] the buy-back contract[17ndash22] the revenue sharing contract [23ndash30] the optioncontract [31ndash35] and the cost sharing contract [36ndash38]

Nevertheless most of the previous studies have focusedon contracts with respect to the coordination of traditionalsupply chain and few studies focus on service supply chaincontracts which will be very important given the rapiddevelopment of service industry To quote a few Wei and Hu[39] introduce a wholesale price contract into service supplychain and study the ordering pricing and allocation strategyLiu et al [40] analyze the fairest revenue sharing coefficientin the coordination of a two-stage logistics service supplychain under the stochastic demand condition and expandthe result to a three-stage supply chain Cui and Liu [41]propose a coordination mechanism by the use of the optioncontract and analyze the allocation of extra profits betweenthe integrator and the provider Lu [42] makes use of the

Hindawi Publishing CorporationJournal of Applied MathematicsVolume 2014 Article ID 842409 10 pageshttpdxdoiorg1011552014842409

2 Journal of Applied Mathematics

cost sharing contract to coordinate the service supply chainunder the assumption that the market demand depends onthe efforts made by the service provider and the integrator

On the other hand the supply chain coordination by theuse of contracts is based on one-time transaction betweenmembers But members in a supply chain are often long-term and stable partners in reality That means that timebecomes a critical factor in the game of members becauseall the members may make different decisions by taking thefuture into consideration So we propose a differential gamestrategy which has been widely applied in themanagement ofinvestment dual-oligopolymarket and cooperative advertis-ing [43ndash46]

In this study we consider time to be a critical factorin the coordination of LSSC The rest of this paper isorganized as follows Section 2 presents the differentialmodelof LSSC under the assumption that reputation contributesto the increase of sales In Sections 3 and 4 we come upwith the optimal decisions of both parties in supply chainunder the Nash (without cost sharing contract) feedbackequilibrium and Stackelberg (with cost sharing contract)feedback equilibrium Section 5 compares the results of twodifferent equilibrium strategies and analyzes the coordinatingrole of the cost sharing contract Section 6 concludes

2 Model

The logistics service supply chain in themodel consists of oneservice integrator and one professional service supplier bothof which need to endeavor together to satisfy the customerCorporate reputation as one of themost important intangibleassets is the comprehensive reflection of the behavior ofcorporate in the past [47] The results of empirical researchof service industry indicate that reputation is conducive tothe creation and maintaining of customer trust resulting inthe increase in successive purchase by loyal customers [48]Hence the demand of the LSSC can be reflected by thefunction of the reputation of LSSC and the accumulation ofLSSC reputation depends on the efforts contributed by theLSI and FLSP in their respective service

It is assumed that the demand of logistics service 119878(119905)

adopts the following specification

119878(119905) = 120579(119905) 119877(119905) minus 120574(119905) 1198772(119905) (1)

where 120579(119905) and 120574(119905) are positive parameters capturing theeffect of the reputation on the demand and 120579(119905) ≫ 120574(119905) gt 0

The growth and accumulation of the reputation 119877(119905) ofLSSC depend on the efforts of supply members The serviceintegrator controls his efforts 119864

119894(119905) while the service supplier

controls his efforts 119864119901(119905) during the cooperation in service

The efforts of both parties during the service contributeto the accumulation of the reputation of LSSC 119877(119905) whichevolves according to the Nerlove and Arrow model [49] thatis

1198771015840(119905) = 120582

119894(119905) 119864119894(119905) + 120582

119901(119905) 119864119901(119905) minus 120575(119905) 119877(119905)

119877(0) = 1198770ge 0

(2)

where 120582119894(119905) gt 0 and 120582

119901(119905) gt 0 reflecting the efficiency

of the efforts of the integrator and the supplier respectivelySince the reputation decays by time [50] 120575(119905) gt 0 in factis the constant decay rate of LSSC reputation caused byenvironment disturbance

Let119862119894(119905) and119862

119901(119905) be the cost of both parties to maintain

the reputation which are convex and increasing indicatingincreasing marginal costs of the efforts and assuming to bequadratic

119862119894(119905) =

120583119894(119905)

21198642

119894(119905) 119862

119901(119905) =

120583119901(119905)

21198642

119901(119905) (3)

where 120583119894(119905) and 120583

119901(119905) are the positive coefficient

Let 119863(119905) be the rate of cost that the integrator will sharewith the supplier which ranges from zero (the integrator willnot share the cost) to one (the integrator pays the full cost)Let 120588 denote the discount rate of both parties 120587

119894(119905) and 120587

119901(119905)

represent the profit margin of the integrator and the supplierrespectively Consider

119869119894= int

infin

0

119890minus120588119905

120587119894(119905) [120579(119905) 119877(119905) minus 120574(119905) 119877

2(119905)] minus

120583119894(119905)

21198642

119894(119905)

minus120583119901(119905)

2119863(119905) 119864

2

119901(119905) 119889119905

119869119901= int

infin

0

119890minus120588119905

120587119901(119905) [120579(119905) 119877(119905) minus 120574(119905) 119877

2(119905)]

minus120583119901(119905)

2[1 minus 119863(119905)] 119864

2

119901(119905) 119889119905

(4)

To recapitulate (1) to (4) define a differential game withtwo players three control variables119864

119894(119905)119864119901(119905) and119863(119905) and

one state variable119877(119905)The controls are constrained by119864119894(119905) ge

0 119864119904(119905) ge 0 0 le 119863(119905) le 1 The state constraint 119877(119905) ge 0 is

automatically satisfiedThe optimal decisions of both parties are decided by a

feedback game so that they are all functions of reputation andtime Let 119868(119877(119905) 119905) and 119875(119877(119905) 119905) be the decisions of LSI andFLSP respectively To simplify the model it is assumed thatall the parameters in the model 120583

119894(119905) 120583119901(119905) 120582119894(119905) 120582119901(119905) 120575(119905)

120579(119905) 120574(119905) 120587119894(119905) and 120587

119901(119905) are constants and have nothing to

do with time which means the players are in the same gamein different time horizons Thus the differential games canbe changed to static games [51] The decisions of both partiescan be defined as 119868(119877(119905)) and119875(119877(119905)) In the followingmodel119877(119905) 119864

119894(119905) 119864119901(119905) and 119863(119905) are simplified as 119877 119864

119894 119864119901 and 119863

3 Nash Equilibrium withoutCost Sharing Contract

In this section we try to analyze the decision strategiesof players under the circumstance that LSI does not sharethe cost of FLSP (119863 = 0) We assume that each memberdetermines his optimal decisions independently at the sametime which means the optimal decisions of both players arethe solutions of Nash equilibrium

Journal of Applied Mathematics 3

Theorem 1 If the feasible solutions of the following constraintequations (119901

1 119901

2 119901

3 119902

1 119902

2 119902

3) exist the optimal decisions of

both parties can be derived without the cost sharing contract inLSSC Consider

21205822

1198941199012

1

120583119894

+

41205822

11990111990111199021

120583119901

minus (2120575 + 120588) 1199011minus 120587119894120574 = 0

21205822

11989411990111199012

120583119894

+

21205822

119901(11990111199022+ 11990121199021)

120583119901

minus (120575 + 120588) 1199012+ 120587119894120579 = 0

1205822

1198941199012

2

2120583119894

+

1205822

11990111990121199022

120583119901

minus 1205881199013= 0

41205822

11989411990111199021

120583119894

+

21205822

1199011199022

1

120583119901

minus (2120575 + 120588) 1199021minus 120587119901120574 = 0

21205822

119894(11990111199022+ 11990121199021)

120583119894

+

21205822

11990111990211199022

120583119901

minus (120575 + 120588) 1199022+ 120587119901120579 = 0

1205822

11989411990121199022

120583119894

+

1205822

1199011199022

2

2120583119901

minus 1205881199023= 0

(5)

The optimal efforts of LSI and FLSP under Nash equilib-rium are

119864119894=

120582119894(2119901

1119877+ 119901

2)

120583119894

119864119901=

120582119901(2119902

1119877+ 119902

2)

120583119901

(6)

where 119877= (1198770+ 119904119903)119890

119903119905minus 119904119903 119903 = 2120582

2

1198941199011120583119894+ 21205822

1199011199021120583119901minus 120575

and 119904 = 1205822

1198941199012120583119894+ 1205822

1199011199022120583119901

Proof We apply a standard sufficient condition for a sta-tionary Markov perfect Nash equilibrium and wish to findbounded and continuously differentiable functions 119881

119899(119877)

119899 isin 119894 119901 which satisfy for all 119877 ge 0 the Hamilton-Jacobi-Bellman (HJB) equations [52]

120588119881119894(119877) = Max

119864119894ge0

120587119894(120579119877 minus 120574119877

2) minus

120583119894

21198642

119894

+ 1198811015840

119894(119877) [120582

119894119864119894+ 120582119901119864119901minus 120575119877]

(7)

120588119881119901(119877) = Max

119864119901ge0

120587119901(120579119877 minus 120574119877

2) minus

120583119901

21198642

119901

+1198811015840

119901(119877) [120582

119894119864119894+ 120582119901119864119901minus 120575119877]

(8)

Equations (7) and (8) are both concave in 119864119894and 119864

119901

yielding the unique effort level of both parties

119864119894=

1205821198941198811015840

119894(119877)

120583119894

119864119901=

1205821199011198811015840

119901(119877)

120583119901

(9)

Substitute 119864119894and 119864

119901into (7) to obtain

120588119881119894(119877) = [120587

119894(120579 minus 120574119877) minus 120575119881

1015840

119894(119877)] 119877

+[1205821198941198811015840

119894(119877)]2

2120583119894

+

1205822

1199011198811015840

119894(119877)1198811015840

119901(119877)

120583119901

120588119881119901(119877) = [120587

119901(120579 minus 120574119877) minus 120575119881

1015840

119901(119877)] 119877

+

1205822

1198941198811015840

119894(119877)1198811015840

119901(119877)

120583119894

+

[1205821199011198811015840

119901(119877)]2

2120583119901

(10)

We conjecture that the solutions to (10) will be quadratic

119881119894(119877) = 119901

11198772+ 1199012119877 + 1199013

119881119901(119877) = 119902

11198772+ 1199022119877 + 1199023

(11)

in which 1199011 1199012 1199013and 1199021 1199022 1199023are constants We substitute

119881119894(119877)119881

119901(119877) and their derivatives from (11) into (10) to obtain

120588(11990111198772+ 1199012119877 + 1199013) = [(120587

119894120579 minus 120575119901

2) 119877 minus (120587

119894120574 + 2120575119901

1) 1198772]

+1205822

119894(21199011119877 + 1199012)2

2120583119894

+

1205822

119901(21199011119877 + 1199012) (21199021119877 + 1199022)

120583119901

(12)

120588(11990211198772+ 1199022119877 + 1199023) = [(120587

119901120579 minus 120575119902

2) 119877 minus (120587

119901120574 + 2120575119902

1) 1198772]

+1205822

119894(21199011119877 + 1199012) (21199021119877 + 1199022)

120583119894

+

1205822

119901(21199021119877 + 1199022)2

2120583119901

(13)

Equations (12) and (13) both satisfy for all 119877 ge 0 Thuswe can find out that a set of values for (119901

1 119901

2 119901

3 119902

1 119902

2 119902

3)

4 Journal of Applied Mathematics

which can result in the maximization of the profits is thesolution for the following equations

1205881199011=

21205822

1198941199012

1

120583119894

+

41205822

11990111990111199021

120583119901

minus (120587119894120574 + 2120575119901

1)

1205881199012=

21205822

11989411990111199012

120583119894

+

21205822

119901(11990111199022+ 11990121199021)

120583119901

+ (120587119894120579 minus 120575119901

2)

1205881199013=

1205822

1198941199012

2

2120583119894

+

1205822

11990111990121199022

120583119901

1205881199021=

41205822

11989411990111199021

120583119894

+

21205822

1199011199022

1

120583119901

minus (120587119901120574 + 2120575119902

1)

1205881199022=

21205822

119894(11990111199022+ 11990121199021)

120583119894

+

21205822

11990111990211199022

120583119901

+ (120587119901120579 minus 120575119902

2)

1205881199023=

1205822

11989411990121199022

120583119894

+

1205822

1199011199022

2

2120583119901

(14)

Then the optimal profits of LSI and FLSP can be repre-sented as

119881119894(119877) = 119901

11198772+ 119901

2119877 + 119901

3

119881119901(119877) = 119902

11198772+ 119902

2119877 + 119902

3

(15)

Substituting the derivatives of 119881119894(119877) 119881

119901(119877) obtained

from (15) into (9) the optimal effort level of both parties canbe represented as

119864119894=

120582119894(2119901

1119877 + 119901

2)

120583119894

119864119901=

120582119901(2119902

1119877 + 119902

2)

120583119901

(16)

Substitute the results into (2) to obtain

1198771015840= 119903119877 + 119904 (17)

in which 119903 = 21205822

1198941199011120583119894+21205822

1199011199021120583119901minus120575 119904 = 120582

2

1198941199012120583119894+1205822

1199011199022120583119901

The general solution to (17) is

119877 = 119908119890119896119905

+ 1199081 (18)

By substituting 119877 and its derivative for 119905 (19) can beobtained

1199081= minus

119904

119903 (19)

Hence the general solution can be represented as

119877 = 119908119890119903119905

minus119904

119903 (20)

in which 119908 is a constant Given 119877|119905=0

= 1198770ge 0 a particular

solution can be obtained

119877= (1198770+

119904

119903) 119890119903119905

minus119904

119903 (21)

4 Stackelberg Equilibrium withCost Sharing Contract

In this section we apply a cost sharing contract to realizethe coordination of LSSC in a Stackelberg game with the LSIas the leader of the game and the provider as the followerFirstly the integrator decides the optimal efforts 119864

119894and

the sharing rate 119863 Then the functional service providerdetermines his optimal efforts after observing the strategy ofthe integrator The optimal decisions of both parties are theresults of Stackelberg equilibrium

Theorem 2 If the feasible solutions to the following constraintequations (119891

lowast

1 119891lowast

2 119891lowast

3 119892lowast

1 119892lowast

2 119892lowast

3) exist the optimal decisions

of both parties can be derived after adopting the cost sharingcontract in LSSC

21205822

1198941198912

1

120583119894

+

1205822

119901(21198911+ 1198921)2

2120583119901

minus (2120575 + 120588) 1198911minus 120587119894120574 = 0

21205822

11989411989111198912

120583119894

+

1205822

119901(21198911+ 1198921) (21198912+ 1198922)

2120583119901

minus (120575 + 120588) 1198912+ 120587119894120579 = 0

1205822

1198941198912

2

2120583119894

+

1205822

119901(21198912+ 1198922)2

8120583119901

minus 1205881198913= 0

21205822

11989411989111198921

120583119894

+

1205822

1199011198921(21198911+ 1198921)

120583119901

minus (2120575 + 120588) 1198921minus 120587119901120574 = 0

1205822

119894(11989111198922+ 11989121198921)

120583119894

+

1205822

119901(11989111198922+ 11989121198921+ 11989211198922)

120583119901

minus (120575 + 120588) 1198922+ 120587119901120579 = 0

1205822

11989411989121198922

2120583119894

+

1205822

1199011198922(21198912+ 1198922)

4120583119901

minus 1205881198923= 0

(22)

The optimal decisions of LSI and FLS in the Stackelbergequilibrium with LSI as the leader can be represented as

119864lowast

119894=

120582119894(2119891lowast

1119877lowast+ 119891lowast

2)

120583119894

119864lowast

119901=

120582119901[(4119891lowast

1+ 2119892lowast

1) 119877lowast+ (2119891

lowast

2+ 119892lowast

2)]

2120583119901

119863lowast=

(4119891lowast

1minus 2119892lowast

1) 119877lowast+ (2119891

lowast

2minus 119892lowast

2)

(4119891lowast

1+ 2119892lowast

1) 119877lowast + (2119891

lowast

2+ 119892lowast

2)

(23)

in which 119877lowast= (1198770+ 119889119896)119890

119896119905minus 119889119896 119896 = 2120582

2

119894119891lowast

1120583119894+ 1205822

119901(2119891lowast

1+

119892lowast

1)120583119901minus 120575 and 119889 = 120582

2

119894119891lowast

2120583119894+ 1205822

119901(2119891lowast

2+ 119892lowast

2)2120583119901

Journal of Applied Mathematics 5

Proof Given that the integrator announced decisions 119864119894and

119863 the provider faces a maximization problem where theprofit function must satisfy the HJB equation

120588119881119901(119877) = Max

119864119901ge0

120587119901(120579119877 minus 120574119877

2) minus

120583119901

2(1 minus 119863) 119864

2

119901

+ 1198811015840

119901(119877)[120582

119894119864119894+ 120582119901119864119901minus 120575119877]

(24)

Maximization on the right side of (24) yields

119864119901=

1205821199011198811015840

119901(119877)

120583119901(1 minus 119863)

(25)

The integrator as the leader of game can forecast thedecision of the provider and then the leader HJB equationis

120588119881119894(119877) = Max

119864119894ge0

120587119894(120579119877 minus 120574119877

2) minus

120583119894

21198642

119894minus

120583119901

21198631198642

119901

+ 1198811015840

119894(119877)[120582

119894119864119894+ 120582119901119864119901minus 120575119877]

(26)

Insert (25) into (26) and obtain the efforts and sharingrate as

119864119894=

1205821198941198811015840

119894(119877)

120583119894

(27)

119863 =

21198811015840

119894(119877) minus 119881

1015840

119901(119877)

21198811015840

119894(119877) + 1198811015840

119901(119877)

(28)

Insert (25) (27) and (28) into (24) and (26) to obtain

120588119881119894(119877) = [120587

119894(120579119877 minus 120574119877

2) minus 120575119881

1015840

119894(119877)] 119877 +

[1205821198941198811015840

119894(119877)]2

2120583119894

+

1205822

119901[21198811015840

119894(119877) + 119881

1015840

119901(119877)]2

8120583119901

120588119881119901(119877) = [120587

119901(120579119877 minus 120574119877

2) minus 120575119881

1015840

119901(119877)] 119877

+

1205822

1198941198811015840

119894(119877)1198811015840

119901(119877)

2120583119894

+

1205822

1199011198811015840

119901(119877) [2119881

1015840

119894(119877) + 119881

1015840

119901(119877)]

4120583119901

(29)

We conjecture that the solutions of (29) will be quadratic

119881119894(119877) = 119891

11198772+ 1198912119877 + 1198913 119881

119901(119877) = 119892

11198772+ 1198922119877 + 1198923

(30)

in which11989111198912 and119891

3and 1198921 1198922 and 119892

3are all constantsWe

substitute 119881119894(119877) 119881

119901(119877) and their derivatives from (30) into

(29) to obtain

120588(11989111198772+ 1198912119877 + 1198913)

= [(120587119894120579 minus 120575119891

2) 119877 minus (120587

119894120574 + 2120575119891

1) 1198772] +

1205822

119894(21198911119877 + 1198912)2

2120583119894

+

1205822

119901[2 (2119891

1119877 + 1198912) + (2119892

1119877 + 1198922)]2

8120583119901

(31)

120588(11989211198772+ 1198922119877 + 1198923)

= [(120587119901120579 minus 120575119892

2) 119877 minus (120587

119901120574 + 2120575119892

1) 1198772]

+1205822

119894(21198911119877 + 1198912) (21198921119877 + 1198922)

2120583119894

+

1205822

119901(21198921119877 + 1198922) [2 (2119891

1119877 + 1198912) + (2119892

1119877 + 1198922)]

4120583119901

(32)

Equations (31) and (32) both satisfy for all119877 ge 0Thus wecan find out that a set of values for (119891

lowast

1 119891lowast

2 119891lowast

3 119892lowast

1 119892lowast

2 119892lowast

3)

which can result in the maximization of the profits is thesolution for the following equations

1205881198911=

21205822

1198941198912

1

120583119894

+

1205822

119901(21198911+ 1198921)2

2120583119901

minus (120587119894120574 + 2120575119891

1)

1205881198912=

21205822

11989411989111198912

120583119894

+

1205822

119901(21198911+ 1198921) (21198912+ 1198922)

2120583119901

+ (120587119894120579 minus 120575119891

2)

1205881198913=

1205822

1198941198912

2

2120583119894

+

1205822

119901(21198912+ 1198922)2

8120583119901

1205881198921=

21205822

11989411989111198921

120583119894

+

1205822

1199011198921(21198911+ 1198921)

120583119901

minus (120587119901120574 + 2120575119892

1)

1205881198922=

1205822

119894(11989111198922+ 11989121198921)

120583119894

+

1205822

119901(11989111198922+ 11989121198921+ 11989211198922)

120583119901

+ (120587119901120579 minus 120575119892

2)

1205881198923=

1205822

11989411989121198922

2120583119894

+

1205822

1199011198922(21198912+ 1198922)

4120583119901

(33)

The optimal profits of LSI and FLSP can be represented as

119881lowast

119894(119877) = 119891

lowast

11198772+ 119891lowast

2119877 + 119891

lowast

3 (34)

119881lowast

119901(119877) = 119892

lowast

11198772+ 119892lowast

2119877 + 119892lowast

3 (35)

6 Journal of Applied Mathematics

Substituting the derivative of 119881lowast119894(119877) from (34) into (27)

the optimal efforts of LSI can be represented as

119864lowast

119894=

120582119894(2119891lowast

1119877 + 119891

lowast

2)

120583119894

(36)

Substituting the derivatives of119881lowast119894(119877) and119881

lowast

119901(119877) from (34)

and (35) into (28) the optimal sharing rate can be representedas

119863lowast=

(4119891lowast

1minus 2119892lowast

1) 119877 + (2119891

lowast

2minus 119892lowast

2)

(4119891lowast

1+ 2119892lowast

1) 119877 + (2119891

lowast

2+ 119892lowast

2) (37)

Also we can obtain the optimal efforts of FLSP bysubstituting119881

lowast

119901(119877)rsquos derivative and119863

lowast from (35) and (37) into(25)

119864lowast

119901=

120582119901[(4119891lowast

1+ 2119892lowast

1) 119877 + (2119891

lowast

2+ 119892lowast

2)]

2120583119901

(38)

Substitute (36) and (38) into differential game equation(2) to obtain

1198771015840= 119896119877 + 119889 (39)

in which 119896 = 21205822

119894119891lowast

1120583119894+1205822

119901(2119891lowast

1+119892lowast

1)120583119901minus120575 119889 = 120582

2

119894119891lowast

2120583119894+

1205822

119901(2119891lowast

2+ 119892lowast

2)2120583119901

Same as the proof ofTheorem 2 the general solutions canrepresented as

119877 = 119888119890119896119905

minus119889

119896 (40)

in which 119888 is a constant Given 119877|119905=0

= 1198770ge 0 the particular

solution is

119877lowast= (1198770+

119889

119896) 119890119896119905

minus119889

119896 (41)

5 Numerical Analysis

Consider a case with the following set of parameters

120583119894= 12 120583

119901= 1 120582

119894= 06

120582119901= 05 120575 = 001 120579 = 05

120574 = 005 120587119894= 06 120587

119901= 04

120588 = 09 1198770= 05

(42)

Then we can get the results of (5) and (22)

(1199011 119901

2 119901

3 119902

1 119902

2 119902

3)

= (minus0031269 0315981 0034968

minus 0020664 0208800 0028047)

(119891lowast

1 119891lowast

2 119891lowast

3 119892lowast

1 119892lowast

2 119892lowast

3)

= (minus0031046 0313701 0040789

minus 0020847 0210668 0023275)

119903 =21205822

1198941199011

120583119894

+

21205822

1199011199021

120583119901

minus 120575 = minus0039093

119904 =1205822

1198941199012

120583119894

+

1205822

1199011199022

120583119901

= 0146994

119896 =21205822

119894119891lowast

1

120583119894

+

1205822

119901(2119891lowast

1+ 119892lowast

1)

120583119901

minus 120575 = minus0049362

119889 =1205822

119894119891lowast

2

120583119894

+

1205822

119901(2119891lowast

2+ 119892lowast

2)

2120583119901

= 0198869

(43)

As a result we can obtain the optimal solutions of thedifferential game

(1) Nash Equilibrium without Cost Sharing Contract Theoptimal efforts of LSI and FLSP are

119864119894=

120582119894(2119901

1119877+ 119901

2)

120583119894

= 01020119890minus004119905

+ 00404

119864119901=

120582119901(2119902

1119877+ 119902

2)

120583119901

= 00673119890minus004119905

+ 00267

(44)

Under the circumstances the accumulation of the reputationof LSSC is

119877(119905) = (119877

0+

119904

119903) 119890119903119905

minus119904

119903= minus326119890

minus004119905+ 376 (45)

And the profits of both parties and the whole supply chain are

119881119894(119877) = 119901

11198772+ 119901

2119877 + 119901

3

= minus03323119890minus008119905

minus 02635119890minus004119905

+ 07810

119881119901(119877) = 119902

11198772+ 119902

2119877 + 119902

3

= minus02196119890minus008119905

minus 01741119890minus004119905

+ 05210

119881LSSC = 119881

119894+ 119881

119901

= minus05519119890minus008119905

minus 04376119890minus004119905

+ 13020

(46)

Journal of Applied Mathematics 7

0 50 100 150 200 250

02

015

01

005

0

t

E(t)

Ei(t)

Ep(t)

(a)

0 50 100 150 200 250

02

015

01

005

0

t

E(t)

Elowast

i(t)

Elowast

p(t)

(b)

0 50 100 150 200 250

02

015

01

005

0

t

E(t)

Elowast

i(t)

Ei(t)

(c)

0 50 100 150 200 250

02

015

01

005

0

t

E(t)

Elowast

p(t)

Ep(t)

(d)

Figure 1 Difference of optimal efforts between Nash equilibrium and Stackelberg equilibrium

(2) Stackelberg Equilibrium with Cost Sharing Contract Theoptimal efforts of LSI and FLSP are

119864lowast

119894=

120582119894(2119891lowast

1119877lowast+ 119891lowast

2)

120583119894

= 01096119890minus005119905

+ 00318

119864lowast

119901=

120582119901[(4119891lowast

1+ 2119892lowast

1) 119877lowast+ (2119891

lowast

2+ 119892lowast

2)]

2120583119901

= 01463119890minus005119905

+ 00424

(47)

The optimal sharing rate of LSI is

119863lowast=

(4119891lowast

1minus 2119892lowast

1) 119877lowast+ (2119891

lowast

2minus 119892lowast

2)

(4119891lowast

1+ 2119892lowast

1) 119877lowast + (2119891

lowast

2+ 119892lowast

2)

=02911119890

minus005119905+ 00844

05853119890minus005119905 + 01698

(48)

Under the circumstances the accumulation of the reputationof LSSC is

119877lowast(119905) = (119877

0+

119889

119896) 119890119896119905

minus119889

119896= minus353119890

minus005119905+ 403 (49)

And the profits of both parties and the whole supply chain are

119881lowast

119894(119877) = 119891

lowast

11198772+ 119891lowast

2119877 + 119891

lowast

3

= minus03866119890minus010119905

minus 02242119890minus005119905

+ 08007

119881lowast

119901(119877) = 119892

lowast

11198772+ 119892lowast

2119877 + 119892lowast

3

= minus02596119890minus010119905

minus 01507119890minus005119905

+ 05336

119881lowast

LSSC = 119881lowast

119894+ 119881lowast

119901

= minus06462119890minus010119905

minus 03749119890minus005119905

+ 13343

(50)

In the long-term relationship among members in LSSCLSI and FLSP are faced with dynamic games which meansthat the optimal strategies of LSI and FLSP will change withtime As a result both players will adjust their decisions tomaximize their profits in an infinite-time horizon Howeverthe optimal decisionswill become stable at last suggesting theautonomy of the LSSC

Figure 1 illustrates the difference between efforts of bothparties obtained from Nash equilibrium and Stackelbergequilibrium The efforts will decline with time until theyreach an appropriate levelThis is because themarginal utilityof reputation for revenue is diminishing while the cost to

8 Journal of Applied Mathematics

0 50 100 150 200 250

04973

04972

04972

04971

04971

0497

t

D(t)

Figure 2 The optimal cost sharing rate of the LSI

45

4

35

3

25

2

15

1

05

R(t)

0 50 100 150 200 250

t

R(t)

Rlowast(t)

Figure 3 Change of the reputation after adopting cost sharingcontract

maintain reputation remains the same By comparing theresults from the two equilibria we can see that efforts ofFLSP enhance remarkably under the Stackelberg equilibriumwhich means that the cost sharing contract can help LSI toencourage FLSP to endeavor in service

The optimal sharing rate of the LSI can be observed inFigure 2 declining until it reaches an appropriate level whichis about 05 Thus LSI should share about half of the cost ofFLSP to realize the coordination of the supply chain

Figure 3 illustrates the whole process of the buildingreputation The reputation of LSSC will increase because ofaccumulation But the rate of growth will decrease with thedecline of the efforts of both parties resulting in a stablelevel at last The stable state of reputation gets higher in theStackelberg equilibriumwith the cost sharing contract whichsuggests that the cost sharing contract contributes to thegrowth of reputation Same as reputation the profits of LSSCas well as members can be enhanced by the use of the costsharing contract as seen in Figure 4

To sum up the long-term relationship among the mem-bers of the supply chain contributes to the stability of theservice supply chain In the long-term relationship betweenLSI and FLSP the cost sharing contract is helpful in increasingthe reputation and profits of the supply chain by encouraginga higher level of the efforts by FLSP which implies that costsharing contract is an effective coordination mechanism

14

12

1

08

06

04

02

0

V(t)

Vlowast

i(t)

Vlowast

p(t)

0 50 100 150 200 250

t

Vp(t)

Vi(t)

VLSSC (t) V

lowast

LSSC (t)

Figure 4 Change of profits of LSSC and themembers after adoptingcost sharing contract

6 Conclusions

Given the long-term relationship amongmembers in a supplychain we take time and reputation into consideration asthe main factors in a dynamic differential model in LSSCconsisting of one LSI and one FLSP The results indicate theautonomy of the logistics service system and the effectivenessof cost sharing contract in the coordination of the supplychain

However we simplify the parameters the expression formof whichmay be various in reality resulting in different formsof solutions in the model to obtain the optimal decisions ofboth parties which is an interesting subject for future workMeanwhile further studies in the coordination of the supplychain that consists of multi-integrators and multiprovidersand the empirical analysis of the mathematical models willplay an important role in the practice of supply chainmanagement

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 NSFC Major Program(Grant no 7109040471090400) and NSFC General Program(Grant nos 71272045 and 71101107)

References

[1] K L Choy C L Li S C K So H Lau S K Kwok and DW KLeung ldquoManaging uncertainty in logistics service supply chainrdquoInternational Journal of Risk Assessment and Management vol7 no 1 pp 19ndash25 2007

Journal of Applied Mathematics 9

[2] X H Li and Q N Wang ldquoCoordination mechanisms of supplychain systemsrdquo European Journal of Operational Research vol179 no 1 pp 1ndash16 2007

[3] G P Cachon ldquoSupply chain coordination with contractsrdquo inSupply Chain Management Design Coordination and Opera-tion A G deKok and S C Graves Eds pp 229ndash340 ElsevierAmsterdam The Netherlands 2003

[4] B A Pasternack ldquoOptimal pricing and return policies forperishable commoditiesrdquo Marketing Science vol 27 no 1 pp166ndash176 2008

[5] R Anupindi and Y Bassok ldquoCentralization of stocks retailersvs manufacturerrdquo Management Science vol 45 no 2 pp 178ndash191 1999

[6] F Chen A Federgruen and Y-S Zheng ldquoCoordination mech-anisms for a distribution system with one supplier and multipleretailersrdquoManagement Science vol 47 no 5 pp 693ndash708 2001

[7] L Dong andN Rudi Supply Chain Interaction under Transship-ments Washington University 2001

[8] M A Lariviere and E L Porteus ldquoSelling to the newsvendoran analysis of price-only contractsrdquoManufacturing and ServiceOperations Management vol 3 no 4 pp 293ndash305 2001

[9] D Ni K W Li and X Tang ldquoSocial responsibility allocationin two-echelon supply chains insights from wholesale pricecontractsrdquo European Journal of Operational Research vol 207no 3 pp 1269ndash1279 2010

[10] X H Wang X Y Wang and Y S Su ldquoWholesale-pricecontract of supply chain with information gatheringrdquo AppliedMathematical Modelling vol 37 no 6 pp 3848ndash3860 2013

[11] Y Bassok and R Anupindi ldquoAnalysis of contracts with totalminimum commitmentrdquo IIE Transactions vol 29 no 5 pp373ndash381 1997

[12] A A Tsay and W S Lovejoy ldquoQuantity flexibility contractsand supply chain performancerdquo Manufacturing and ServiceOperations Management vol 1 no 2 pp 118ndash149 1999

[13] E L Plambeck and T A Taylor ldquoSell the plant The impact ofcontract manufacturing on innovation capacity and profitabil-ityrdquoManagement Science vol 51 no 1 pp 133ndash150 2005

[14] O Yazlali and F Erhun ldquoRelating the multiple supply problemto quantity flexibility contractsrdquo Operations Research Lettersvol 35 no 6 pp 767ndash772 2007

[15] Z T Lian and A Deshmukh ldquoAnalysis of supply contracts withquantity flexibilityrdquo European Journal of Operational Researchvol 196 no 2 pp 526ndash533 2009

[16] S Karakaya and I S Bakal ldquoJoint quantity flexibility formultiple products in a decentralized supply chainrdquo Computersamp Industrial Engineering vol 64 no 2 pp 696ndash707 2013

[17] V Padmanabhan and I P L Png ldquoReturns policies makemoney by making goodrdquo SloanManagement Review pp 65ndash721995

[18] T Taylor ldquoChannel coordination under price protectionmidlife returns and end-of-life returns in dynamic marketsrdquoManagement Science vol 47 no 9 pp 1220ndash1234 2001

[19] Z Yao S Leung and K K Lai ldquoAnalysis of the impact of price-sensitivity factors on the returns policy in coordinating supplychainrdquo European Journal of Operational Research vol 187 no 1pp 275ndash282 2008

[20] Y L Wang and P Zipkin ldquoAgentsrsquo incentives under buy-backcontracts in a two-stage supply chainrdquo International Journal ofProduction Economics vol 120 no 2 pp 525ndash539 2009

[21] J Hou A Z Zeng and L Zhao ldquoCoordination with a backupsupplier through buy-back contract under supply disruptionrdquo

Transportation Research E Logistics and Transportation Reviewvol 46 no 6 pp 881ndash895 2010

[22] H C Xiong B T Chen and J X Xie ldquoA composite contractbased on buy back and quantity flexibility contractsrdquo EuropeanJournal of Operational Research vol 210 no 3 pp 559ndash567 2011

[23] J Dana Jr and K E Spier ldquoRevenue sharing and vertical controlin the video rental industryrdquo Journal of Industrial Economicsvol 49 no 3 pp 223ndash245 2001

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

[25] Z Yao S C H Leung and K K Lai ldquoManufacturerrsquos revenue-sharing contract and retail competitionrdquo European Journal ofOperational Research vol 186 no 2 pp 637ndash651 2008

[26] 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

[27] 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

[28] J M Chen H L Cheng and M C Chien ldquoOn channelcoordination through revenue-sharing contracts with priceand shelf-space dependent demandrdquo Applied MathematicalModelling vol 35 no 10 pp 4886ndash4901 2011

[29] D P D Omkar ldquoSupply chain coordination using revenue-dependent revenue sharing contractsrdquoOmega vol 41 no 4 pp780ndash796 2013

[30] G Kannan andN PMaria ldquoReverse supply chain coordinationby revenue sharing contract a case for the personal computersindustryrdquo European Journal of Operational Research vol 233no 2 pp 326ndash336 2014

[31] D J Wu and P R Kleindorfer ldquoCompetitive options supplycontracting and electronic marketsrdquoManagement Science vol51 no 3 pp 452ndash468 2005

[32] XWang and L Liu ldquoCoordination in a retailer-led supply chainthrough option contractrdquo International Journal of ProductionEconomics vol 110 no 1-2 pp 115ndash127 2007

[33] Y X Zhao S Y Wang T C E Cheng X Q Yang and ZM Huang ldquoCoordination of supply chains by option contractsa cooperative game theory approachrdquo European Journal ofOperational Research vol 207 no 2 pp 668ndash675 2010

[34] L Liang X HWang and J G Gao ldquoAn option contract pricingmodel of relief material supply chainrdquoOmega vol 40 no 5 pp594ndash600 2012

[35] Y X Zhao L J Ma G Xie and T C E Cheng ldquoCoordinationof supply chains with bidirectional option contractsrdquo EuropeanJournal of Operational Research vol 229 no 2 pp 375ndash3812013

[36] M M Leng and M Parlar ldquoGame-theoretic analyses of decen-tralized assembly supply chains Non-cooperative equilibria vscoordination with cost-sharing contractsrdquo European Journal ofOperational Research vol 204 no 1 pp 96ndash104 2010

[37] M Kunter ldquoCoordination via cost and revenue sharing inmanufacturer-retailer channelsrdquo European Journal of Opera-tional Research vol 216 no 2 pp 477ndash486 2012

[38] Y C Tsao and G J Sheen ldquoEffects of promotion cost sharingpolicy with the sales learning curve on supply chain coordi-nationrdquo Computers and Operations Research vol 39 no 8 pp1872ndash1878 2012

10 Journal of Applied Mathematics

[39] Y H Wei and Q Y Hu ldquoOrdering pricing and allocationin a service supply chainrdquo International Journal of ProductionEconomics vol 144 no 2 pp 590ndash598 2013

[40] W H Liu X C Xu and A Kouhpaenejad ldquoDeterministicapproach to the fairest revenue-sharing coefficient in logisticsservice supply chain under the stochastic demand conditionrdquoComputers amp Industrial Engineering vol 66 no 1 pp 41ndash522013

[41] A P Cui and W Liu ldquoStudy on capability coordination inlogistics service supply chain with options contractrdquo ChineseJournal of Management Science vol 17 no 2 pp 59ndash65 2009(Chinese)

[42] Q H Lu ldquoCoordinate with cost sharing strategy for servicesupply chainrdquo Control and Decision vol 26 no 11 pp 1649ndash1653 2011 (Chinese)

[43] A Prasad and S P Sethi ldquoCompetitive advertising underuncertainty a stochastic differential game approachrdquo Journal ofOptimization Theory and Applications vol 123 no 1 pp 163ndash185 2004

[44] S Joslashrgensen S Taboubi and G Zaccour ldquoRetail promotionswith negative brand image effects is cooperation possiblerdquoEuropean Journal of Operational Research vol 150 no 2 pp395ndash405 2003

[45] J Navas and J Marın-Solano ldquoInteractions between govern-ment and firms a differential game approachrdquo Annals ofOperations Research vol 158 no 1 pp 47ndash61 2008

[46] S Joslashrgensen S Taboubi and G Zaccour ldquoCooperative adver-tising in a marketing channelrdquo Journal of Optimization Theoryand Applications vol 110 no 1 pp 145ndash158 2001

[47] C J Fombrun and V Rindova ldquoWhorsquos tops and who decidesThe social construction of corporate reputationsrdquo WorkingPaper Stern School of Business New York University 1996

[48] N Nha and L Gaston ldquoCorporate image and corporate repu-tation in customers Retention decisions in servicesrdquo Journal ofRetailing and Consumer Services no 8 pp 227ndash2361 2001

[49] M Nerlove and K J Arrow ldquoOptimal advertising policy underdynamic conditionsrdquo Economica vol 39 pp 129ndash142 1962

[50] K Saxton ldquoWhere do reputations come fromrdquo CorporateReputation Review vol 1 no 4 pp 393ndash399 1998

[51] M G Nagler ldquoAn exploratory analysis of the determinants ofcooperative advertising participation ratesrdquo Marketing Lettersvol 17 no 2 pp 91ndash102 2006

[52] E Dockner N Jorgensen L Van et al Differential Gamesin Economics and Management Science Cambridge UniversityPress Cambridge UK 2000

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

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Mathematical PhysicsAdvances in

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OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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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 Journal of Applied Mathematics

cost sharing contract to coordinate the service supply chainunder the assumption that the market demand depends onthe efforts made by the service provider and the integrator

On the other hand the supply chain coordination by theuse of contracts is based on one-time transaction betweenmembers But members in a supply chain are often long-term and stable partners in reality That means that timebecomes a critical factor in the game of members becauseall the members may make different decisions by taking thefuture into consideration So we propose a differential gamestrategy which has been widely applied in themanagement ofinvestment dual-oligopolymarket and cooperative advertis-ing [43ndash46]

In this study we consider time to be a critical factorin the coordination of LSSC The rest of this paper isorganized as follows Section 2 presents the differentialmodelof LSSC under the assumption that reputation contributesto the increase of sales In Sections 3 and 4 we come upwith the optimal decisions of both parties in supply chainunder the Nash (without cost sharing contract) feedbackequilibrium and Stackelberg (with cost sharing contract)feedback equilibrium Section 5 compares the results of twodifferent equilibrium strategies and analyzes the coordinatingrole of the cost sharing contract Section 6 concludes

2 Model

The logistics service supply chain in themodel consists of oneservice integrator and one professional service supplier bothof which need to endeavor together to satisfy the customerCorporate reputation as one of themost important intangibleassets is the comprehensive reflection of the behavior ofcorporate in the past [47] The results of empirical researchof service industry indicate that reputation is conducive tothe creation and maintaining of customer trust resulting inthe increase in successive purchase by loyal customers [48]Hence the demand of the LSSC can be reflected by thefunction of the reputation of LSSC and the accumulation ofLSSC reputation depends on the efforts contributed by theLSI and FLSP in their respective service

It is assumed that the demand of logistics service 119878(119905)

adopts the following specification

119878(119905) = 120579(119905) 119877(119905) minus 120574(119905) 1198772(119905) (1)

where 120579(119905) and 120574(119905) are positive parameters capturing theeffect of the reputation on the demand and 120579(119905) ≫ 120574(119905) gt 0

The growth and accumulation of the reputation 119877(119905) ofLSSC depend on the efforts of supply members The serviceintegrator controls his efforts 119864

119894(119905) while the service supplier

controls his efforts 119864119901(119905) during the cooperation in service

The efforts of both parties during the service contributeto the accumulation of the reputation of LSSC 119877(119905) whichevolves according to the Nerlove and Arrow model [49] thatis

1198771015840(119905) = 120582

119894(119905) 119864119894(119905) + 120582

119901(119905) 119864119901(119905) minus 120575(119905) 119877(119905)

119877(0) = 1198770ge 0

(2)

where 120582119894(119905) gt 0 and 120582

119901(119905) gt 0 reflecting the efficiency

of the efforts of the integrator and the supplier respectivelySince the reputation decays by time [50] 120575(119905) gt 0 in factis the constant decay rate of LSSC reputation caused byenvironment disturbance

Let119862119894(119905) and119862

119901(119905) be the cost of both parties to maintain

the reputation which are convex and increasing indicatingincreasing marginal costs of the efforts and assuming to bequadratic

119862119894(119905) =

120583119894(119905)

21198642

119894(119905) 119862

119901(119905) =

120583119901(119905)

21198642

119901(119905) (3)

where 120583119894(119905) and 120583

119901(119905) are the positive coefficient

Let 119863(119905) be the rate of cost that the integrator will sharewith the supplier which ranges from zero (the integrator willnot share the cost) to one (the integrator pays the full cost)Let 120588 denote the discount rate of both parties 120587

119894(119905) and 120587

119901(119905)

represent the profit margin of the integrator and the supplierrespectively Consider

119869119894= int

infin

0

119890minus120588119905

120587119894(119905) [120579(119905) 119877(119905) minus 120574(119905) 119877

2(119905)] minus

120583119894(119905)

21198642

119894(119905)

minus120583119901(119905)

2119863(119905) 119864

2

119901(119905) 119889119905

119869119901= int

infin

0

119890minus120588119905

120587119901(119905) [120579(119905) 119877(119905) minus 120574(119905) 119877

2(119905)]

minus120583119901(119905)

2[1 minus 119863(119905)] 119864

2

119901(119905) 119889119905

(4)

To recapitulate (1) to (4) define a differential game withtwo players three control variables119864

119894(119905)119864119901(119905) and119863(119905) and

one state variable119877(119905)The controls are constrained by119864119894(119905) ge

0 119864119904(119905) ge 0 0 le 119863(119905) le 1 The state constraint 119877(119905) ge 0 is

automatically satisfiedThe optimal decisions of both parties are decided by a

feedback game so that they are all functions of reputation andtime Let 119868(119877(119905) 119905) and 119875(119877(119905) 119905) be the decisions of LSI andFLSP respectively To simplify the model it is assumed thatall the parameters in the model 120583

119894(119905) 120583119901(119905) 120582119894(119905) 120582119901(119905) 120575(119905)

120579(119905) 120574(119905) 120587119894(119905) and 120587

119901(119905) are constants and have nothing to

do with time which means the players are in the same gamein different time horizons Thus the differential games canbe changed to static games [51] The decisions of both partiescan be defined as 119868(119877(119905)) and119875(119877(119905)) In the followingmodel119877(119905) 119864

119894(119905) 119864119901(119905) and 119863(119905) are simplified as 119877 119864

119894 119864119901 and 119863

3 Nash Equilibrium withoutCost Sharing Contract

In this section we try to analyze the decision strategiesof players under the circumstance that LSI does not sharethe cost of FLSP (119863 = 0) We assume that each memberdetermines his optimal decisions independently at the sametime which means the optimal decisions of both players arethe solutions of Nash equilibrium

Journal of Applied Mathematics 3

Theorem 1 If the feasible solutions of the following constraintequations (119901

1 119901

2 119901

3 119902

1 119902

2 119902

3) exist the optimal decisions of

both parties can be derived without the cost sharing contract inLSSC Consider

21205822

1198941199012

1

120583119894

+

41205822

11990111990111199021

120583119901

minus (2120575 + 120588) 1199011minus 120587119894120574 = 0

21205822

11989411990111199012

120583119894

+

21205822

119901(11990111199022+ 11990121199021)

120583119901

minus (120575 + 120588) 1199012+ 120587119894120579 = 0

1205822

1198941199012

2

2120583119894

+

1205822

11990111990121199022

120583119901

minus 1205881199013= 0

41205822

11989411990111199021

120583119894

+

21205822

1199011199022

1

120583119901

minus (2120575 + 120588) 1199021minus 120587119901120574 = 0

21205822

119894(11990111199022+ 11990121199021)

120583119894

+

21205822

11990111990211199022

120583119901

minus (120575 + 120588) 1199022+ 120587119901120579 = 0

1205822

11989411990121199022

120583119894

+

1205822

1199011199022

2

2120583119901

minus 1205881199023= 0

(5)

The optimal efforts of LSI and FLSP under Nash equilib-rium are

119864119894=

120582119894(2119901

1119877+ 119901

2)

120583119894

119864119901=

120582119901(2119902

1119877+ 119902

2)

120583119901

(6)

where 119877= (1198770+ 119904119903)119890

119903119905minus 119904119903 119903 = 2120582

2

1198941199011120583119894+ 21205822

1199011199021120583119901minus 120575

and 119904 = 1205822

1198941199012120583119894+ 1205822

1199011199022120583119901

Proof We apply a standard sufficient condition for a sta-tionary Markov perfect Nash equilibrium and wish to findbounded and continuously differentiable functions 119881

119899(119877)

119899 isin 119894 119901 which satisfy for all 119877 ge 0 the Hamilton-Jacobi-Bellman (HJB) equations [52]

120588119881119894(119877) = Max

119864119894ge0

120587119894(120579119877 minus 120574119877

2) minus

120583119894

21198642

119894

+ 1198811015840

119894(119877) [120582

119894119864119894+ 120582119901119864119901minus 120575119877]

(7)

120588119881119901(119877) = Max

119864119901ge0

120587119901(120579119877 minus 120574119877

2) minus

120583119901

21198642

119901

+1198811015840

119901(119877) [120582

119894119864119894+ 120582119901119864119901minus 120575119877]

(8)

Equations (7) and (8) are both concave in 119864119894and 119864

119901

yielding the unique effort level of both parties

119864119894=

1205821198941198811015840

119894(119877)

120583119894

119864119901=

1205821199011198811015840

119901(119877)

120583119901

(9)

Substitute 119864119894and 119864

119901into (7) to obtain

120588119881119894(119877) = [120587

119894(120579 minus 120574119877) minus 120575119881

1015840

119894(119877)] 119877

+[1205821198941198811015840

119894(119877)]2

2120583119894

+

1205822

1199011198811015840

119894(119877)1198811015840

119901(119877)

120583119901

120588119881119901(119877) = [120587

119901(120579 minus 120574119877) minus 120575119881

1015840

119901(119877)] 119877

+

1205822

1198941198811015840

119894(119877)1198811015840

119901(119877)

120583119894

+

[1205821199011198811015840

119901(119877)]2

2120583119901

(10)

We conjecture that the solutions to (10) will be quadratic

119881119894(119877) = 119901

11198772+ 1199012119877 + 1199013

119881119901(119877) = 119902

11198772+ 1199022119877 + 1199023

(11)

in which 1199011 1199012 1199013and 1199021 1199022 1199023are constants We substitute

119881119894(119877)119881

119901(119877) and their derivatives from (11) into (10) to obtain

120588(11990111198772+ 1199012119877 + 1199013) = [(120587

119894120579 minus 120575119901

2) 119877 minus (120587

119894120574 + 2120575119901

1) 1198772]

+1205822

119894(21199011119877 + 1199012)2

2120583119894

+

1205822

119901(21199011119877 + 1199012) (21199021119877 + 1199022)

120583119901

(12)

120588(11990211198772+ 1199022119877 + 1199023) = [(120587

119901120579 minus 120575119902

2) 119877 minus (120587

119901120574 + 2120575119902

1) 1198772]

+1205822

119894(21199011119877 + 1199012) (21199021119877 + 1199022)

120583119894

+

1205822

119901(21199021119877 + 1199022)2

2120583119901

(13)

Equations (12) and (13) both satisfy for all 119877 ge 0 Thuswe can find out that a set of values for (119901

1 119901

2 119901

3 119902

1 119902

2 119902

3)

4 Journal of Applied Mathematics

which can result in the maximization of the profits is thesolution for the following equations

1205881199011=

21205822

1198941199012

1

120583119894

+

41205822

11990111990111199021

120583119901

minus (120587119894120574 + 2120575119901

1)

1205881199012=

21205822

11989411990111199012

120583119894

+

21205822

119901(11990111199022+ 11990121199021)

120583119901

+ (120587119894120579 minus 120575119901

2)

1205881199013=

1205822

1198941199012

2

2120583119894

+

1205822

11990111990121199022

120583119901

1205881199021=

41205822

11989411990111199021

120583119894

+

21205822

1199011199022

1

120583119901

minus (120587119901120574 + 2120575119902

1)

1205881199022=

21205822

119894(11990111199022+ 11990121199021)

120583119894

+

21205822

11990111990211199022

120583119901

+ (120587119901120579 minus 120575119902

2)

1205881199023=

1205822

11989411990121199022

120583119894

+

1205822

1199011199022

2

2120583119901

(14)

Then the optimal profits of LSI and FLSP can be repre-sented as

119881119894(119877) = 119901

11198772+ 119901

2119877 + 119901

3

119881119901(119877) = 119902

11198772+ 119902

2119877 + 119902

3

(15)

Substituting the derivatives of 119881119894(119877) 119881

119901(119877) obtained

from (15) into (9) the optimal effort level of both parties canbe represented as

119864119894=

120582119894(2119901

1119877 + 119901

2)

120583119894

119864119901=

120582119901(2119902

1119877 + 119902

2)

120583119901

(16)

Substitute the results into (2) to obtain

1198771015840= 119903119877 + 119904 (17)

in which 119903 = 21205822

1198941199011120583119894+21205822

1199011199021120583119901minus120575 119904 = 120582

2

1198941199012120583119894+1205822

1199011199022120583119901

The general solution to (17) is

119877 = 119908119890119896119905

+ 1199081 (18)

By substituting 119877 and its derivative for 119905 (19) can beobtained

1199081= minus

119904

119903 (19)

Hence the general solution can be represented as

119877 = 119908119890119903119905

minus119904

119903 (20)

in which 119908 is a constant Given 119877|119905=0

= 1198770ge 0 a particular

solution can be obtained

119877= (1198770+

119904

119903) 119890119903119905

minus119904

119903 (21)

4 Stackelberg Equilibrium withCost Sharing Contract

In this section we apply a cost sharing contract to realizethe coordination of LSSC in a Stackelberg game with the LSIas the leader of the game and the provider as the followerFirstly the integrator decides the optimal efforts 119864

119894and

the sharing rate 119863 Then the functional service providerdetermines his optimal efforts after observing the strategy ofthe integrator The optimal decisions of both parties are theresults of Stackelberg equilibrium

Theorem 2 If the feasible solutions to the following constraintequations (119891

lowast

1 119891lowast

2 119891lowast

3 119892lowast

1 119892lowast

2 119892lowast

3) exist the optimal decisions

of both parties can be derived after adopting the cost sharingcontract in LSSC

21205822

1198941198912

1

120583119894

+

1205822

119901(21198911+ 1198921)2

2120583119901

minus (2120575 + 120588) 1198911minus 120587119894120574 = 0

21205822

11989411989111198912

120583119894

+

1205822

119901(21198911+ 1198921) (21198912+ 1198922)

2120583119901

minus (120575 + 120588) 1198912+ 120587119894120579 = 0

1205822

1198941198912

2

2120583119894

+

1205822

119901(21198912+ 1198922)2

8120583119901

minus 1205881198913= 0

21205822

11989411989111198921

120583119894

+

1205822

1199011198921(21198911+ 1198921)

120583119901

minus (2120575 + 120588) 1198921minus 120587119901120574 = 0

1205822

119894(11989111198922+ 11989121198921)

120583119894

+

1205822

119901(11989111198922+ 11989121198921+ 11989211198922)

120583119901

minus (120575 + 120588) 1198922+ 120587119901120579 = 0

1205822

11989411989121198922

2120583119894

+

1205822

1199011198922(21198912+ 1198922)

4120583119901

minus 1205881198923= 0

(22)

The optimal decisions of LSI and FLS in the Stackelbergequilibrium with LSI as the leader can be represented as

119864lowast

119894=

120582119894(2119891lowast

1119877lowast+ 119891lowast

2)

120583119894

119864lowast

119901=

120582119901[(4119891lowast

1+ 2119892lowast

1) 119877lowast+ (2119891

lowast

2+ 119892lowast

2)]

2120583119901

119863lowast=

(4119891lowast

1minus 2119892lowast

1) 119877lowast+ (2119891

lowast

2minus 119892lowast

2)

(4119891lowast

1+ 2119892lowast

1) 119877lowast + (2119891

lowast

2+ 119892lowast

2)

(23)

in which 119877lowast= (1198770+ 119889119896)119890

119896119905minus 119889119896 119896 = 2120582

2

119894119891lowast

1120583119894+ 1205822

119901(2119891lowast

1+

119892lowast

1)120583119901minus 120575 and 119889 = 120582

2

119894119891lowast

2120583119894+ 1205822

119901(2119891lowast

2+ 119892lowast

2)2120583119901

Journal of Applied Mathematics 5

Proof Given that the integrator announced decisions 119864119894and

119863 the provider faces a maximization problem where theprofit function must satisfy the HJB equation

120588119881119901(119877) = Max

119864119901ge0

120587119901(120579119877 minus 120574119877

2) minus

120583119901

2(1 minus 119863) 119864

2

119901

+ 1198811015840

119901(119877)[120582

119894119864119894+ 120582119901119864119901minus 120575119877]

(24)

Maximization on the right side of (24) yields

119864119901=

1205821199011198811015840

119901(119877)

120583119901(1 minus 119863)

(25)

The integrator as the leader of game can forecast thedecision of the provider and then the leader HJB equationis

120588119881119894(119877) = Max

119864119894ge0

120587119894(120579119877 minus 120574119877

2) minus

120583119894

21198642

119894minus

120583119901

21198631198642

119901

+ 1198811015840

119894(119877)[120582

119894119864119894+ 120582119901119864119901minus 120575119877]

(26)

Insert (25) into (26) and obtain the efforts and sharingrate as

119864119894=

1205821198941198811015840

119894(119877)

120583119894

(27)

119863 =

21198811015840

119894(119877) minus 119881

1015840

119901(119877)

21198811015840

119894(119877) + 1198811015840

119901(119877)

(28)

Insert (25) (27) and (28) into (24) and (26) to obtain

120588119881119894(119877) = [120587

119894(120579119877 minus 120574119877

2) minus 120575119881

1015840

119894(119877)] 119877 +

[1205821198941198811015840

119894(119877)]2

2120583119894

+

1205822

119901[21198811015840

119894(119877) + 119881

1015840

119901(119877)]2

8120583119901

120588119881119901(119877) = [120587

119901(120579119877 minus 120574119877

2) minus 120575119881

1015840

119901(119877)] 119877

+

1205822

1198941198811015840

119894(119877)1198811015840

119901(119877)

2120583119894

+

1205822

1199011198811015840

119901(119877) [2119881

1015840

119894(119877) + 119881

1015840

119901(119877)]

4120583119901

(29)

We conjecture that the solutions of (29) will be quadratic

119881119894(119877) = 119891

11198772+ 1198912119877 + 1198913 119881

119901(119877) = 119892

11198772+ 1198922119877 + 1198923

(30)

in which11989111198912 and119891

3and 1198921 1198922 and 119892

3are all constantsWe

substitute 119881119894(119877) 119881

119901(119877) and their derivatives from (30) into

(29) to obtain

120588(11989111198772+ 1198912119877 + 1198913)

= [(120587119894120579 minus 120575119891

2) 119877 minus (120587

119894120574 + 2120575119891

1) 1198772] +

1205822

119894(21198911119877 + 1198912)2

2120583119894

+

1205822

119901[2 (2119891

1119877 + 1198912) + (2119892

1119877 + 1198922)]2

8120583119901

(31)

120588(11989211198772+ 1198922119877 + 1198923)

= [(120587119901120579 minus 120575119892

2) 119877 minus (120587

119901120574 + 2120575119892

1) 1198772]

+1205822

119894(21198911119877 + 1198912) (21198921119877 + 1198922)

2120583119894

+

1205822

119901(21198921119877 + 1198922) [2 (2119891

1119877 + 1198912) + (2119892

1119877 + 1198922)]

4120583119901

(32)

Equations (31) and (32) both satisfy for all119877 ge 0Thus wecan find out that a set of values for (119891

lowast

1 119891lowast

2 119891lowast

3 119892lowast

1 119892lowast

2 119892lowast

3)

which can result in the maximization of the profits is thesolution for the following equations

1205881198911=

21205822

1198941198912

1

120583119894

+

1205822

119901(21198911+ 1198921)2

2120583119901

minus (120587119894120574 + 2120575119891

1)

1205881198912=

21205822

11989411989111198912

120583119894

+

1205822

119901(21198911+ 1198921) (21198912+ 1198922)

2120583119901

+ (120587119894120579 minus 120575119891

2)

1205881198913=

1205822

1198941198912

2

2120583119894

+

1205822

119901(21198912+ 1198922)2

8120583119901

1205881198921=

21205822

11989411989111198921

120583119894

+

1205822

1199011198921(21198911+ 1198921)

120583119901

minus (120587119901120574 + 2120575119892

1)

1205881198922=

1205822

119894(11989111198922+ 11989121198921)

120583119894

+

1205822

119901(11989111198922+ 11989121198921+ 11989211198922)

120583119901

+ (120587119901120579 minus 120575119892

2)

1205881198923=

1205822

11989411989121198922

2120583119894

+

1205822

1199011198922(21198912+ 1198922)

4120583119901

(33)

The optimal profits of LSI and FLSP can be represented as

119881lowast

119894(119877) = 119891

lowast

11198772+ 119891lowast

2119877 + 119891

lowast

3 (34)

119881lowast

119901(119877) = 119892

lowast

11198772+ 119892lowast

2119877 + 119892lowast

3 (35)

6 Journal of Applied Mathematics

Substituting the derivative of 119881lowast119894(119877) from (34) into (27)

the optimal efforts of LSI can be represented as

119864lowast

119894=

120582119894(2119891lowast

1119877 + 119891

lowast

2)

120583119894

(36)

Substituting the derivatives of119881lowast119894(119877) and119881

lowast

119901(119877) from (34)

and (35) into (28) the optimal sharing rate can be representedas

119863lowast=

(4119891lowast

1minus 2119892lowast

1) 119877 + (2119891

lowast

2minus 119892lowast

2)

(4119891lowast

1+ 2119892lowast

1) 119877 + (2119891

lowast

2+ 119892lowast

2) (37)

Also we can obtain the optimal efforts of FLSP bysubstituting119881

lowast

119901(119877)rsquos derivative and119863

lowast from (35) and (37) into(25)

119864lowast

119901=

120582119901[(4119891lowast

1+ 2119892lowast

1) 119877 + (2119891

lowast

2+ 119892lowast

2)]

2120583119901

(38)

Substitute (36) and (38) into differential game equation(2) to obtain

1198771015840= 119896119877 + 119889 (39)

in which 119896 = 21205822

119894119891lowast

1120583119894+1205822

119901(2119891lowast

1+119892lowast

1)120583119901minus120575 119889 = 120582

2

119894119891lowast

2120583119894+

1205822

119901(2119891lowast

2+ 119892lowast

2)2120583119901

Same as the proof ofTheorem 2 the general solutions canrepresented as

119877 = 119888119890119896119905

minus119889

119896 (40)

in which 119888 is a constant Given 119877|119905=0

= 1198770ge 0 the particular

solution is

119877lowast= (1198770+

119889

119896) 119890119896119905

minus119889

119896 (41)

5 Numerical Analysis

Consider a case with the following set of parameters

120583119894= 12 120583

119901= 1 120582

119894= 06

120582119901= 05 120575 = 001 120579 = 05

120574 = 005 120587119894= 06 120587

119901= 04

120588 = 09 1198770= 05

(42)

Then we can get the results of (5) and (22)

(1199011 119901

2 119901

3 119902

1 119902

2 119902

3)

= (minus0031269 0315981 0034968

minus 0020664 0208800 0028047)

(119891lowast

1 119891lowast

2 119891lowast

3 119892lowast

1 119892lowast

2 119892lowast

3)

= (minus0031046 0313701 0040789

minus 0020847 0210668 0023275)

119903 =21205822

1198941199011

120583119894

+

21205822

1199011199021

120583119901

minus 120575 = minus0039093

119904 =1205822

1198941199012

120583119894

+

1205822

1199011199022

120583119901

= 0146994

119896 =21205822

119894119891lowast

1

120583119894

+

1205822

119901(2119891lowast

1+ 119892lowast

1)

120583119901

minus 120575 = minus0049362

119889 =1205822

119894119891lowast

2

120583119894

+

1205822

119901(2119891lowast

2+ 119892lowast

2)

2120583119901

= 0198869

(43)

As a result we can obtain the optimal solutions of thedifferential game

(1) Nash Equilibrium without Cost Sharing Contract Theoptimal efforts of LSI and FLSP are

119864119894=

120582119894(2119901

1119877+ 119901

2)

120583119894

= 01020119890minus004119905

+ 00404

119864119901=

120582119901(2119902

1119877+ 119902

2)

120583119901

= 00673119890minus004119905

+ 00267

(44)

Under the circumstances the accumulation of the reputationof LSSC is

119877(119905) = (119877

0+

119904

119903) 119890119903119905

minus119904

119903= minus326119890

minus004119905+ 376 (45)

And the profits of both parties and the whole supply chain are

119881119894(119877) = 119901

11198772+ 119901

2119877 + 119901

3

= minus03323119890minus008119905

minus 02635119890minus004119905

+ 07810

119881119901(119877) = 119902

11198772+ 119902

2119877 + 119902

3

= minus02196119890minus008119905

minus 01741119890minus004119905

+ 05210

119881LSSC = 119881

119894+ 119881

119901

= minus05519119890minus008119905

minus 04376119890minus004119905

+ 13020

(46)

Journal of Applied Mathematics 7

0 50 100 150 200 250

02

015

01

005

0

t

E(t)

Ei(t)

Ep(t)

(a)

0 50 100 150 200 250

02

015

01

005

0

t

E(t)

Elowast

i(t)

Elowast

p(t)

(b)

0 50 100 150 200 250

02

015

01

005

0

t

E(t)

Elowast

i(t)

Ei(t)

(c)

0 50 100 150 200 250

02

015

01

005

0

t

E(t)

Elowast

p(t)

Ep(t)

(d)

Figure 1 Difference of optimal efforts between Nash equilibrium and Stackelberg equilibrium

(2) Stackelberg Equilibrium with Cost Sharing Contract Theoptimal efforts of LSI and FLSP are

119864lowast

119894=

120582119894(2119891lowast

1119877lowast+ 119891lowast

2)

120583119894

= 01096119890minus005119905

+ 00318

119864lowast

119901=

120582119901[(4119891lowast

1+ 2119892lowast

1) 119877lowast+ (2119891

lowast

2+ 119892lowast

2)]

2120583119901

= 01463119890minus005119905

+ 00424

(47)

The optimal sharing rate of LSI is

119863lowast=

(4119891lowast

1minus 2119892lowast

1) 119877lowast+ (2119891

lowast

2minus 119892lowast

2)

(4119891lowast

1+ 2119892lowast

1) 119877lowast + (2119891

lowast

2+ 119892lowast

2)

=02911119890

minus005119905+ 00844

05853119890minus005119905 + 01698

(48)

Under the circumstances the accumulation of the reputationof LSSC is

119877lowast(119905) = (119877

0+

119889

119896) 119890119896119905

minus119889

119896= minus353119890

minus005119905+ 403 (49)

And the profits of both parties and the whole supply chain are

119881lowast

119894(119877) = 119891

lowast

11198772+ 119891lowast

2119877 + 119891

lowast

3

= minus03866119890minus010119905

minus 02242119890minus005119905

+ 08007

119881lowast

119901(119877) = 119892

lowast

11198772+ 119892lowast

2119877 + 119892lowast

3

= minus02596119890minus010119905

minus 01507119890minus005119905

+ 05336

119881lowast

LSSC = 119881lowast

119894+ 119881lowast

119901

= minus06462119890minus010119905

minus 03749119890minus005119905

+ 13343

(50)

In the long-term relationship among members in LSSCLSI and FLSP are faced with dynamic games which meansthat the optimal strategies of LSI and FLSP will change withtime As a result both players will adjust their decisions tomaximize their profits in an infinite-time horizon Howeverthe optimal decisionswill become stable at last suggesting theautonomy of the LSSC

Figure 1 illustrates the difference between efforts of bothparties obtained from Nash equilibrium and Stackelbergequilibrium The efforts will decline with time until theyreach an appropriate levelThis is because themarginal utilityof reputation for revenue is diminishing while the cost to

8 Journal of Applied Mathematics

0 50 100 150 200 250

04973

04972

04972

04971

04971

0497

t

D(t)

Figure 2 The optimal cost sharing rate of the LSI

45

4

35

3

25

2

15

1

05

R(t)

0 50 100 150 200 250

t

R(t)

Rlowast(t)

Figure 3 Change of the reputation after adopting cost sharingcontract

maintain reputation remains the same By comparing theresults from the two equilibria we can see that efforts ofFLSP enhance remarkably under the Stackelberg equilibriumwhich means that the cost sharing contract can help LSI toencourage FLSP to endeavor in service

The optimal sharing rate of the LSI can be observed inFigure 2 declining until it reaches an appropriate level whichis about 05 Thus LSI should share about half of the cost ofFLSP to realize the coordination of the supply chain

Figure 3 illustrates the whole process of the buildingreputation The reputation of LSSC will increase because ofaccumulation But the rate of growth will decrease with thedecline of the efforts of both parties resulting in a stablelevel at last The stable state of reputation gets higher in theStackelberg equilibriumwith the cost sharing contract whichsuggests that the cost sharing contract contributes to thegrowth of reputation Same as reputation the profits of LSSCas well as members can be enhanced by the use of the costsharing contract as seen in Figure 4

To sum up the long-term relationship among the mem-bers of the supply chain contributes to the stability of theservice supply chain In the long-term relationship betweenLSI and FLSP the cost sharing contract is helpful in increasingthe reputation and profits of the supply chain by encouraginga higher level of the efforts by FLSP which implies that costsharing contract is an effective coordination mechanism

14

12

1

08

06

04

02

0

V(t)

Vlowast

i(t)

Vlowast

p(t)

0 50 100 150 200 250

t

Vp(t)

Vi(t)

VLSSC (t) V

lowast

LSSC (t)

Figure 4 Change of profits of LSSC and themembers after adoptingcost sharing contract

6 Conclusions

Given the long-term relationship amongmembers in a supplychain we take time and reputation into consideration asthe main factors in a dynamic differential model in LSSCconsisting of one LSI and one FLSP The results indicate theautonomy of the logistics service system and the effectivenessof cost sharing contract in the coordination of the supplychain

However we simplify the parameters the expression formof whichmay be various in reality resulting in different formsof solutions in the model to obtain the optimal decisions ofboth parties which is an interesting subject for future workMeanwhile further studies in the coordination of the supplychain that consists of multi-integrators and multiprovidersand the empirical analysis of the mathematical models willplay an important role in the practice of supply chainmanagement

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 NSFC Major Program(Grant no 7109040471090400) and NSFC General Program(Grant nos 71272045 and 71101107)

References

[1] K L Choy C L Li S C K So H Lau S K Kwok and DW KLeung ldquoManaging uncertainty in logistics service supply chainrdquoInternational Journal of Risk Assessment and Management vol7 no 1 pp 19ndash25 2007

Journal of Applied Mathematics 9

[2] X H Li and Q N Wang ldquoCoordination mechanisms of supplychain systemsrdquo European Journal of Operational Research vol179 no 1 pp 1ndash16 2007

[3] G P Cachon ldquoSupply chain coordination with contractsrdquo inSupply Chain Management Design Coordination and Opera-tion A G deKok and S C Graves Eds pp 229ndash340 ElsevierAmsterdam The Netherlands 2003

[4] B A Pasternack ldquoOptimal pricing and return policies forperishable commoditiesrdquo Marketing Science vol 27 no 1 pp166ndash176 2008

[5] R Anupindi and Y Bassok ldquoCentralization of stocks retailersvs manufacturerrdquo Management Science vol 45 no 2 pp 178ndash191 1999

[6] F Chen A Federgruen and Y-S Zheng ldquoCoordination mech-anisms for a distribution system with one supplier and multipleretailersrdquoManagement Science vol 47 no 5 pp 693ndash708 2001

[7] L Dong andN Rudi Supply Chain Interaction under Transship-ments Washington University 2001

[8] M A Lariviere and E L Porteus ldquoSelling to the newsvendoran analysis of price-only contractsrdquoManufacturing and ServiceOperations Management vol 3 no 4 pp 293ndash305 2001

[9] D Ni K W Li and X Tang ldquoSocial responsibility allocationin two-echelon supply chains insights from wholesale pricecontractsrdquo European Journal of Operational Research vol 207no 3 pp 1269ndash1279 2010

[10] X H Wang X Y Wang and Y S Su ldquoWholesale-pricecontract of supply chain with information gatheringrdquo AppliedMathematical Modelling vol 37 no 6 pp 3848ndash3860 2013

[11] Y Bassok and R Anupindi ldquoAnalysis of contracts with totalminimum commitmentrdquo IIE Transactions vol 29 no 5 pp373ndash381 1997

[12] A A Tsay and W S Lovejoy ldquoQuantity flexibility contractsand supply chain performancerdquo Manufacturing and ServiceOperations Management vol 1 no 2 pp 118ndash149 1999

[13] E L Plambeck and T A Taylor ldquoSell the plant The impact ofcontract manufacturing on innovation capacity and profitabil-ityrdquoManagement Science vol 51 no 1 pp 133ndash150 2005

[14] O Yazlali and F Erhun ldquoRelating the multiple supply problemto quantity flexibility contractsrdquo Operations Research Lettersvol 35 no 6 pp 767ndash772 2007

[15] Z T Lian and A Deshmukh ldquoAnalysis of supply contracts withquantity flexibilityrdquo European Journal of Operational Researchvol 196 no 2 pp 526ndash533 2009

[16] S Karakaya and I S Bakal ldquoJoint quantity flexibility formultiple products in a decentralized supply chainrdquo Computersamp Industrial Engineering vol 64 no 2 pp 696ndash707 2013

[17] V Padmanabhan and I P L Png ldquoReturns policies makemoney by making goodrdquo SloanManagement Review pp 65ndash721995

[18] T Taylor ldquoChannel coordination under price protectionmidlife returns and end-of-life returns in dynamic marketsrdquoManagement Science vol 47 no 9 pp 1220ndash1234 2001

[19] Z Yao S Leung and K K Lai ldquoAnalysis of the impact of price-sensitivity factors on the returns policy in coordinating supplychainrdquo European Journal of Operational Research vol 187 no 1pp 275ndash282 2008

[20] Y L Wang and P Zipkin ldquoAgentsrsquo incentives under buy-backcontracts in a two-stage supply chainrdquo International Journal ofProduction Economics vol 120 no 2 pp 525ndash539 2009

[21] J Hou A Z Zeng and L Zhao ldquoCoordination with a backupsupplier through buy-back contract under supply disruptionrdquo

Transportation Research E Logistics and Transportation Reviewvol 46 no 6 pp 881ndash895 2010

[22] H C Xiong B T Chen and J X Xie ldquoA composite contractbased on buy back and quantity flexibility contractsrdquo EuropeanJournal of Operational Research vol 210 no 3 pp 559ndash567 2011

[23] J Dana Jr and K E Spier ldquoRevenue sharing and vertical controlin the video rental industryrdquo Journal of Industrial Economicsvol 49 no 3 pp 223ndash245 2001

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

[25] Z Yao S C H Leung and K K Lai ldquoManufacturerrsquos revenue-sharing contract and retail competitionrdquo European Journal ofOperational Research vol 186 no 2 pp 637ndash651 2008

[26] 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

[27] 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

[28] J M Chen H L Cheng and M C Chien ldquoOn channelcoordination through revenue-sharing contracts with priceand shelf-space dependent demandrdquo Applied MathematicalModelling vol 35 no 10 pp 4886ndash4901 2011

[29] D P D Omkar ldquoSupply chain coordination using revenue-dependent revenue sharing contractsrdquoOmega vol 41 no 4 pp780ndash796 2013

[30] G Kannan andN PMaria ldquoReverse supply chain coordinationby revenue sharing contract a case for the personal computersindustryrdquo European Journal of Operational Research vol 233no 2 pp 326ndash336 2014

[31] D J Wu and P R Kleindorfer ldquoCompetitive options supplycontracting and electronic marketsrdquoManagement Science vol51 no 3 pp 452ndash468 2005

[32] XWang and L Liu ldquoCoordination in a retailer-led supply chainthrough option contractrdquo International Journal of ProductionEconomics vol 110 no 1-2 pp 115ndash127 2007

[33] Y X Zhao S Y Wang T C E Cheng X Q Yang and ZM Huang ldquoCoordination of supply chains by option contractsa cooperative game theory approachrdquo European Journal ofOperational Research vol 207 no 2 pp 668ndash675 2010

[34] L Liang X HWang and J G Gao ldquoAn option contract pricingmodel of relief material supply chainrdquoOmega vol 40 no 5 pp594ndash600 2012

[35] Y X Zhao L J Ma G Xie and T C E Cheng ldquoCoordinationof supply chains with bidirectional option contractsrdquo EuropeanJournal of Operational Research vol 229 no 2 pp 375ndash3812013

[36] M M Leng and M Parlar ldquoGame-theoretic analyses of decen-tralized assembly supply chains Non-cooperative equilibria vscoordination with cost-sharing contractsrdquo European Journal ofOperational Research vol 204 no 1 pp 96ndash104 2010

[37] M Kunter ldquoCoordination via cost and revenue sharing inmanufacturer-retailer channelsrdquo European Journal of Opera-tional Research vol 216 no 2 pp 477ndash486 2012

[38] Y C Tsao and G J Sheen ldquoEffects of promotion cost sharingpolicy with the sales learning curve on supply chain coordi-nationrdquo Computers and Operations Research vol 39 no 8 pp1872ndash1878 2012

10 Journal of Applied Mathematics

[39] Y H Wei and Q Y Hu ldquoOrdering pricing and allocationin a service supply chainrdquo International Journal of ProductionEconomics vol 144 no 2 pp 590ndash598 2013

[40] W H Liu X C Xu and A Kouhpaenejad ldquoDeterministicapproach to the fairest revenue-sharing coefficient in logisticsservice supply chain under the stochastic demand conditionrdquoComputers amp Industrial Engineering vol 66 no 1 pp 41ndash522013

[41] A P Cui and W Liu ldquoStudy on capability coordination inlogistics service supply chain with options contractrdquo ChineseJournal of Management Science vol 17 no 2 pp 59ndash65 2009(Chinese)

[42] Q H Lu ldquoCoordinate with cost sharing strategy for servicesupply chainrdquo Control and Decision vol 26 no 11 pp 1649ndash1653 2011 (Chinese)

[43] A Prasad and S P Sethi ldquoCompetitive advertising underuncertainty a stochastic differential game approachrdquo Journal ofOptimization Theory and Applications vol 123 no 1 pp 163ndash185 2004

[44] S Joslashrgensen S Taboubi and G Zaccour ldquoRetail promotionswith negative brand image effects is cooperation possiblerdquoEuropean Journal of Operational Research vol 150 no 2 pp395ndash405 2003

[45] J Navas and J Marın-Solano ldquoInteractions between govern-ment and firms a differential game approachrdquo Annals ofOperations Research vol 158 no 1 pp 47ndash61 2008

[46] S Joslashrgensen S Taboubi and G Zaccour ldquoCooperative adver-tising in a marketing channelrdquo Journal of Optimization Theoryand Applications vol 110 no 1 pp 145ndash158 2001

[47] C J Fombrun and V Rindova ldquoWhorsquos tops and who decidesThe social construction of corporate reputationsrdquo WorkingPaper Stern School of Business New York University 1996

[48] N Nha and L Gaston ldquoCorporate image and corporate repu-tation in customers Retention decisions in servicesrdquo Journal ofRetailing and Consumer Services no 8 pp 227ndash2361 2001

[49] M Nerlove and K J Arrow ldquoOptimal advertising policy underdynamic conditionsrdquo Economica vol 39 pp 129ndash142 1962

[50] K Saxton ldquoWhere do reputations come fromrdquo CorporateReputation Review vol 1 no 4 pp 393ndash399 1998

[51] M G Nagler ldquoAn exploratory analysis of the determinants ofcooperative advertising participation ratesrdquo Marketing Lettersvol 17 no 2 pp 91ndash102 2006

[52] E Dockner N Jorgensen L Van et al Differential Gamesin Economics and Management Science Cambridge UniversityPress Cambridge UK 2000

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

Journal of Applied Mathematics 3

Theorem 1 If the feasible solutions of the following constraintequations (119901

1 119901

2 119901

3 119902

1 119902

2 119902

3) exist the optimal decisions of

both parties can be derived without the cost sharing contract inLSSC Consider

21205822

1198941199012

1

120583119894

+

41205822

11990111990111199021

120583119901

minus (2120575 + 120588) 1199011minus 120587119894120574 = 0

21205822

11989411990111199012

120583119894

+

21205822

119901(11990111199022+ 11990121199021)

120583119901

minus (120575 + 120588) 1199012+ 120587119894120579 = 0

1205822

1198941199012

2

2120583119894

+

1205822

11990111990121199022

120583119901

minus 1205881199013= 0

41205822

11989411990111199021

120583119894

+

21205822

1199011199022

1

120583119901

minus (2120575 + 120588) 1199021minus 120587119901120574 = 0

21205822

119894(11990111199022+ 11990121199021)

120583119894

+

21205822

11990111990211199022

120583119901

minus (120575 + 120588) 1199022+ 120587119901120579 = 0

1205822

11989411990121199022

120583119894

+

1205822

1199011199022

2

2120583119901

minus 1205881199023= 0

(5)

The optimal efforts of LSI and FLSP under Nash equilib-rium are

119864119894=

120582119894(2119901

1119877+ 119901

2)

120583119894

119864119901=

120582119901(2119902

1119877+ 119902

2)

120583119901

(6)

where 119877= (1198770+ 119904119903)119890

119903119905minus 119904119903 119903 = 2120582

2

1198941199011120583119894+ 21205822

1199011199021120583119901minus 120575

and 119904 = 1205822

1198941199012120583119894+ 1205822

1199011199022120583119901

Proof We apply a standard sufficient condition for a sta-tionary Markov perfect Nash equilibrium and wish to findbounded and continuously differentiable functions 119881

119899(119877)

119899 isin 119894 119901 which satisfy for all 119877 ge 0 the Hamilton-Jacobi-Bellman (HJB) equations [52]

120588119881119894(119877) = Max

119864119894ge0

120587119894(120579119877 minus 120574119877

2) minus

120583119894

21198642

119894

+ 1198811015840

119894(119877) [120582

119894119864119894+ 120582119901119864119901minus 120575119877]

(7)

120588119881119901(119877) = Max

119864119901ge0

120587119901(120579119877 minus 120574119877

2) minus

120583119901

21198642

119901

+1198811015840

119901(119877) [120582

119894119864119894+ 120582119901119864119901minus 120575119877]

(8)

Equations (7) and (8) are both concave in 119864119894and 119864

119901

yielding the unique effort level of both parties

119864119894=

1205821198941198811015840

119894(119877)

120583119894

119864119901=

1205821199011198811015840

119901(119877)

120583119901

(9)

Substitute 119864119894and 119864

119901into (7) to obtain

120588119881119894(119877) = [120587

119894(120579 minus 120574119877) minus 120575119881

1015840

119894(119877)] 119877

+[1205821198941198811015840

119894(119877)]2

2120583119894

+

1205822

1199011198811015840

119894(119877)1198811015840

119901(119877)

120583119901

120588119881119901(119877) = [120587

119901(120579 minus 120574119877) minus 120575119881

1015840

119901(119877)] 119877

+

1205822

1198941198811015840

119894(119877)1198811015840

119901(119877)

120583119894

+

[1205821199011198811015840

119901(119877)]2

2120583119901

(10)

We conjecture that the solutions to (10) will be quadratic

119881119894(119877) = 119901

11198772+ 1199012119877 + 1199013

119881119901(119877) = 119902

11198772+ 1199022119877 + 1199023

(11)

in which 1199011 1199012 1199013and 1199021 1199022 1199023are constants We substitute

119881119894(119877)119881

119901(119877) and their derivatives from (11) into (10) to obtain

120588(11990111198772+ 1199012119877 + 1199013) = [(120587

119894120579 minus 120575119901

2) 119877 minus (120587

119894120574 + 2120575119901

1) 1198772]

+1205822

119894(21199011119877 + 1199012)2

2120583119894

+

1205822

119901(21199011119877 + 1199012) (21199021119877 + 1199022)

120583119901

(12)

120588(11990211198772+ 1199022119877 + 1199023) = [(120587

119901120579 minus 120575119902

2) 119877 minus (120587

119901120574 + 2120575119902

1) 1198772]

+1205822

119894(21199011119877 + 1199012) (21199021119877 + 1199022)

120583119894

+

1205822

119901(21199021119877 + 1199022)2

2120583119901

(13)

Equations (12) and (13) both satisfy for all 119877 ge 0 Thuswe can find out that a set of values for (119901

1 119901

2 119901

3 119902

1 119902

2 119902

3)

4 Journal of Applied Mathematics

which can result in the maximization of the profits is thesolution for the following equations

1205881199011=

21205822

1198941199012

1

120583119894

+

41205822

11990111990111199021

120583119901

minus (120587119894120574 + 2120575119901

1)

1205881199012=

21205822

11989411990111199012

120583119894

+

21205822

119901(11990111199022+ 11990121199021)

120583119901

+ (120587119894120579 minus 120575119901

2)

1205881199013=

1205822

1198941199012

2

2120583119894

+

1205822

11990111990121199022

120583119901

1205881199021=

41205822

11989411990111199021

120583119894

+

21205822

1199011199022

1

120583119901

minus (120587119901120574 + 2120575119902

1)

1205881199022=

21205822

119894(11990111199022+ 11990121199021)

120583119894

+

21205822

11990111990211199022

120583119901

+ (120587119901120579 minus 120575119902

2)

1205881199023=

1205822

11989411990121199022

120583119894

+

1205822

1199011199022

2

2120583119901

(14)

Then the optimal profits of LSI and FLSP can be repre-sented as

119881119894(119877) = 119901

11198772+ 119901

2119877 + 119901

3

119881119901(119877) = 119902

11198772+ 119902

2119877 + 119902

3

(15)

Substituting the derivatives of 119881119894(119877) 119881

119901(119877) obtained

from (15) into (9) the optimal effort level of both parties canbe represented as

119864119894=

120582119894(2119901

1119877 + 119901

2)

120583119894

119864119901=

120582119901(2119902

1119877 + 119902

2)

120583119901

(16)

Substitute the results into (2) to obtain

1198771015840= 119903119877 + 119904 (17)

in which 119903 = 21205822

1198941199011120583119894+21205822

1199011199021120583119901minus120575 119904 = 120582

2

1198941199012120583119894+1205822

1199011199022120583119901

The general solution to (17) is

119877 = 119908119890119896119905

+ 1199081 (18)

By substituting 119877 and its derivative for 119905 (19) can beobtained

1199081= minus

119904

119903 (19)

Hence the general solution can be represented as

119877 = 119908119890119903119905

minus119904

119903 (20)

in which 119908 is a constant Given 119877|119905=0

= 1198770ge 0 a particular

solution can be obtained

119877= (1198770+

119904

119903) 119890119903119905

minus119904

119903 (21)

4 Stackelberg Equilibrium withCost Sharing Contract

In this section we apply a cost sharing contract to realizethe coordination of LSSC in a Stackelberg game with the LSIas the leader of the game and the provider as the followerFirstly the integrator decides the optimal efforts 119864

119894and

the sharing rate 119863 Then the functional service providerdetermines his optimal efforts after observing the strategy ofthe integrator The optimal decisions of both parties are theresults of Stackelberg equilibrium

Theorem 2 If the feasible solutions to the following constraintequations (119891

lowast

1 119891lowast

2 119891lowast

3 119892lowast

1 119892lowast

2 119892lowast

3) exist the optimal decisions

of both parties can be derived after adopting the cost sharingcontract in LSSC

21205822

1198941198912

1

120583119894

+

1205822

119901(21198911+ 1198921)2

2120583119901

minus (2120575 + 120588) 1198911minus 120587119894120574 = 0

21205822

11989411989111198912

120583119894

+

1205822

119901(21198911+ 1198921) (21198912+ 1198922)

2120583119901

minus (120575 + 120588) 1198912+ 120587119894120579 = 0

1205822

1198941198912

2

2120583119894

+

1205822

119901(21198912+ 1198922)2

8120583119901

minus 1205881198913= 0

21205822

11989411989111198921

120583119894

+

1205822

1199011198921(21198911+ 1198921)

120583119901

minus (2120575 + 120588) 1198921minus 120587119901120574 = 0

1205822

119894(11989111198922+ 11989121198921)

120583119894

+

1205822

119901(11989111198922+ 11989121198921+ 11989211198922)

120583119901

minus (120575 + 120588) 1198922+ 120587119901120579 = 0

1205822

11989411989121198922

2120583119894

+

1205822

1199011198922(21198912+ 1198922)

4120583119901

minus 1205881198923= 0

(22)

The optimal decisions of LSI and FLS in the Stackelbergequilibrium with LSI as the leader can be represented as

119864lowast

119894=

120582119894(2119891lowast

1119877lowast+ 119891lowast

2)

120583119894

119864lowast

119901=

120582119901[(4119891lowast

1+ 2119892lowast

1) 119877lowast+ (2119891

lowast

2+ 119892lowast

2)]

2120583119901

119863lowast=

(4119891lowast

1minus 2119892lowast

1) 119877lowast+ (2119891

lowast

2minus 119892lowast

2)

(4119891lowast

1+ 2119892lowast

1) 119877lowast + (2119891

lowast

2+ 119892lowast

2)

(23)

in which 119877lowast= (1198770+ 119889119896)119890

119896119905minus 119889119896 119896 = 2120582

2

119894119891lowast

1120583119894+ 1205822

119901(2119891lowast

1+

119892lowast

1)120583119901minus 120575 and 119889 = 120582

2

119894119891lowast

2120583119894+ 1205822

119901(2119891lowast

2+ 119892lowast

2)2120583119901

Journal of Applied Mathematics 5

Proof Given that the integrator announced decisions 119864119894and

119863 the provider faces a maximization problem where theprofit function must satisfy the HJB equation

120588119881119901(119877) = Max

119864119901ge0

120587119901(120579119877 minus 120574119877

2) minus

120583119901

2(1 minus 119863) 119864

2

119901

+ 1198811015840

119901(119877)[120582

119894119864119894+ 120582119901119864119901minus 120575119877]

(24)

Maximization on the right side of (24) yields

119864119901=

1205821199011198811015840

119901(119877)

120583119901(1 minus 119863)

(25)

The integrator as the leader of game can forecast thedecision of the provider and then the leader HJB equationis

120588119881119894(119877) = Max

119864119894ge0

120587119894(120579119877 minus 120574119877

2) minus

120583119894

21198642

119894minus

120583119901

21198631198642

119901

+ 1198811015840

119894(119877)[120582

119894119864119894+ 120582119901119864119901minus 120575119877]

(26)

Insert (25) into (26) and obtain the efforts and sharingrate as

119864119894=

1205821198941198811015840

119894(119877)

120583119894

(27)

119863 =

21198811015840

119894(119877) minus 119881

1015840

119901(119877)

21198811015840

119894(119877) + 1198811015840

119901(119877)

(28)

Insert (25) (27) and (28) into (24) and (26) to obtain

120588119881119894(119877) = [120587

119894(120579119877 minus 120574119877

2) minus 120575119881

1015840

119894(119877)] 119877 +

[1205821198941198811015840

119894(119877)]2

2120583119894

+

1205822

119901[21198811015840

119894(119877) + 119881

1015840

119901(119877)]2

8120583119901

120588119881119901(119877) = [120587

119901(120579119877 minus 120574119877

2) minus 120575119881

1015840

119901(119877)] 119877

+

1205822

1198941198811015840

119894(119877)1198811015840

119901(119877)

2120583119894

+

1205822

1199011198811015840

119901(119877) [2119881

1015840

119894(119877) + 119881

1015840

119901(119877)]

4120583119901

(29)

We conjecture that the solutions of (29) will be quadratic

119881119894(119877) = 119891

11198772+ 1198912119877 + 1198913 119881

119901(119877) = 119892

11198772+ 1198922119877 + 1198923

(30)

in which11989111198912 and119891

3and 1198921 1198922 and 119892

3are all constantsWe

substitute 119881119894(119877) 119881

119901(119877) and their derivatives from (30) into

(29) to obtain

120588(11989111198772+ 1198912119877 + 1198913)

= [(120587119894120579 minus 120575119891

2) 119877 minus (120587

119894120574 + 2120575119891

1) 1198772] +

1205822

119894(21198911119877 + 1198912)2

2120583119894

+

1205822

119901[2 (2119891

1119877 + 1198912) + (2119892

1119877 + 1198922)]2

8120583119901

(31)

120588(11989211198772+ 1198922119877 + 1198923)

= [(120587119901120579 minus 120575119892

2) 119877 minus (120587

119901120574 + 2120575119892

1) 1198772]

+1205822

119894(21198911119877 + 1198912) (21198921119877 + 1198922)

2120583119894

+

1205822

119901(21198921119877 + 1198922) [2 (2119891

1119877 + 1198912) + (2119892

1119877 + 1198922)]

4120583119901

(32)

Equations (31) and (32) both satisfy for all119877 ge 0Thus wecan find out that a set of values for (119891

lowast

1 119891lowast

2 119891lowast

3 119892lowast

1 119892lowast

2 119892lowast

3)

which can result in the maximization of the profits is thesolution for the following equations

1205881198911=

21205822

1198941198912

1

120583119894

+

1205822

119901(21198911+ 1198921)2

2120583119901

minus (120587119894120574 + 2120575119891

1)

1205881198912=

21205822

11989411989111198912

120583119894

+

1205822

119901(21198911+ 1198921) (21198912+ 1198922)

2120583119901

+ (120587119894120579 minus 120575119891

2)

1205881198913=

1205822

1198941198912

2

2120583119894

+

1205822

119901(21198912+ 1198922)2

8120583119901

1205881198921=

21205822

11989411989111198921

120583119894

+

1205822

1199011198921(21198911+ 1198921)

120583119901

minus (120587119901120574 + 2120575119892

1)

1205881198922=

1205822

119894(11989111198922+ 11989121198921)

120583119894

+

1205822

119901(11989111198922+ 11989121198921+ 11989211198922)

120583119901

+ (120587119901120579 minus 120575119892

2)

1205881198923=

1205822

11989411989121198922

2120583119894

+

1205822

1199011198922(21198912+ 1198922)

4120583119901

(33)

The optimal profits of LSI and FLSP can be represented as

119881lowast

119894(119877) = 119891

lowast

11198772+ 119891lowast

2119877 + 119891

lowast

3 (34)

119881lowast

119901(119877) = 119892

lowast

11198772+ 119892lowast

2119877 + 119892lowast

3 (35)

6 Journal of Applied Mathematics

Substituting the derivative of 119881lowast119894(119877) from (34) into (27)

the optimal efforts of LSI can be represented as

119864lowast

119894=

120582119894(2119891lowast

1119877 + 119891

lowast

2)

120583119894

(36)

Substituting the derivatives of119881lowast119894(119877) and119881

lowast

119901(119877) from (34)

and (35) into (28) the optimal sharing rate can be representedas

119863lowast=

(4119891lowast

1minus 2119892lowast

1) 119877 + (2119891

lowast

2minus 119892lowast

2)

(4119891lowast

1+ 2119892lowast

1) 119877 + (2119891

lowast

2+ 119892lowast

2) (37)

Also we can obtain the optimal efforts of FLSP bysubstituting119881

lowast

119901(119877)rsquos derivative and119863

lowast from (35) and (37) into(25)

119864lowast

119901=

120582119901[(4119891lowast

1+ 2119892lowast

1) 119877 + (2119891

lowast

2+ 119892lowast

2)]

2120583119901

(38)

Substitute (36) and (38) into differential game equation(2) to obtain

1198771015840= 119896119877 + 119889 (39)

in which 119896 = 21205822

119894119891lowast

1120583119894+1205822

119901(2119891lowast

1+119892lowast

1)120583119901minus120575 119889 = 120582

2

119894119891lowast

2120583119894+

1205822

119901(2119891lowast

2+ 119892lowast

2)2120583119901

Same as the proof ofTheorem 2 the general solutions canrepresented as

119877 = 119888119890119896119905

minus119889

119896 (40)

in which 119888 is a constant Given 119877|119905=0

= 1198770ge 0 the particular

solution is

119877lowast= (1198770+

119889

119896) 119890119896119905

minus119889

119896 (41)

5 Numerical Analysis

Consider a case with the following set of parameters

120583119894= 12 120583

119901= 1 120582

119894= 06

120582119901= 05 120575 = 001 120579 = 05

120574 = 005 120587119894= 06 120587

119901= 04

120588 = 09 1198770= 05

(42)

Then we can get the results of (5) and (22)

(1199011 119901

2 119901

3 119902

1 119902

2 119902

3)

= (minus0031269 0315981 0034968

minus 0020664 0208800 0028047)

(119891lowast

1 119891lowast

2 119891lowast

3 119892lowast

1 119892lowast

2 119892lowast

3)

= (minus0031046 0313701 0040789

minus 0020847 0210668 0023275)

119903 =21205822

1198941199011

120583119894

+

21205822

1199011199021

120583119901

minus 120575 = minus0039093

119904 =1205822

1198941199012

120583119894

+

1205822

1199011199022

120583119901

= 0146994

119896 =21205822

119894119891lowast

1

120583119894

+

1205822

119901(2119891lowast

1+ 119892lowast

1)

120583119901

minus 120575 = minus0049362

119889 =1205822

119894119891lowast

2

120583119894

+

1205822

119901(2119891lowast

2+ 119892lowast

2)

2120583119901

= 0198869

(43)

As a result we can obtain the optimal solutions of thedifferential game

(1) Nash Equilibrium without Cost Sharing Contract Theoptimal efforts of LSI and FLSP are

119864119894=

120582119894(2119901

1119877+ 119901

2)

120583119894

= 01020119890minus004119905

+ 00404

119864119901=

120582119901(2119902

1119877+ 119902

2)

120583119901

= 00673119890minus004119905

+ 00267

(44)

Under the circumstances the accumulation of the reputationof LSSC is

119877(119905) = (119877

0+

119904

119903) 119890119903119905

minus119904

119903= minus326119890

minus004119905+ 376 (45)

And the profits of both parties and the whole supply chain are

119881119894(119877) = 119901

11198772+ 119901

2119877 + 119901

3

= minus03323119890minus008119905

minus 02635119890minus004119905

+ 07810

119881119901(119877) = 119902

11198772+ 119902

2119877 + 119902

3

= minus02196119890minus008119905

minus 01741119890minus004119905

+ 05210

119881LSSC = 119881

119894+ 119881

119901

= minus05519119890minus008119905

minus 04376119890minus004119905

+ 13020

(46)

Journal of Applied Mathematics 7

0 50 100 150 200 250

02

015

01

005

0

t

E(t)

Ei(t)

Ep(t)

(a)

0 50 100 150 200 250

02

015

01

005

0

t

E(t)

Elowast

i(t)

Elowast

p(t)

(b)

0 50 100 150 200 250

02

015

01

005

0

t

E(t)

Elowast

i(t)

Ei(t)

(c)

0 50 100 150 200 250

02

015

01

005

0

t

E(t)

Elowast

p(t)

Ep(t)

(d)

Figure 1 Difference of optimal efforts between Nash equilibrium and Stackelberg equilibrium

(2) Stackelberg Equilibrium with Cost Sharing Contract Theoptimal efforts of LSI and FLSP are

119864lowast

119894=

120582119894(2119891lowast

1119877lowast+ 119891lowast

2)

120583119894

= 01096119890minus005119905

+ 00318

119864lowast

119901=

120582119901[(4119891lowast

1+ 2119892lowast

1) 119877lowast+ (2119891

lowast

2+ 119892lowast

2)]

2120583119901

= 01463119890minus005119905

+ 00424

(47)

The optimal sharing rate of LSI is

119863lowast=

(4119891lowast

1minus 2119892lowast

1) 119877lowast+ (2119891

lowast

2minus 119892lowast

2)

(4119891lowast

1+ 2119892lowast

1) 119877lowast + (2119891

lowast

2+ 119892lowast

2)

=02911119890

minus005119905+ 00844

05853119890minus005119905 + 01698

(48)

Under the circumstances the accumulation of the reputationof LSSC is

119877lowast(119905) = (119877

0+

119889

119896) 119890119896119905

minus119889

119896= minus353119890

minus005119905+ 403 (49)

And the profits of both parties and the whole supply chain are

119881lowast

119894(119877) = 119891

lowast

11198772+ 119891lowast

2119877 + 119891

lowast

3

= minus03866119890minus010119905

minus 02242119890minus005119905

+ 08007

119881lowast

119901(119877) = 119892

lowast

11198772+ 119892lowast

2119877 + 119892lowast

3

= minus02596119890minus010119905

minus 01507119890minus005119905

+ 05336

119881lowast

LSSC = 119881lowast

119894+ 119881lowast

119901

= minus06462119890minus010119905

minus 03749119890minus005119905

+ 13343

(50)

In the long-term relationship among members in LSSCLSI and FLSP are faced with dynamic games which meansthat the optimal strategies of LSI and FLSP will change withtime As a result both players will adjust their decisions tomaximize their profits in an infinite-time horizon Howeverthe optimal decisionswill become stable at last suggesting theautonomy of the LSSC

Figure 1 illustrates the difference between efforts of bothparties obtained from Nash equilibrium and Stackelbergequilibrium The efforts will decline with time until theyreach an appropriate levelThis is because themarginal utilityof reputation for revenue is diminishing while the cost to

8 Journal of Applied Mathematics

0 50 100 150 200 250

04973

04972

04972

04971

04971

0497

t

D(t)

Figure 2 The optimal cost sharing rate of the LSI

45

4

35

3

25

2

15

1

05

R(t)

0 50 100 150 200 250

t

R(t)

Rlowast(t)

Figure 3 Change of the reputation after adopting cost sharingcontract

maintain reputation remains the same By comparing theresults from the two equilibria we can see that efforts ofFLSP enhance remarkably under the Stackelberg equilibriumwhich means that the cost sharing contract can help LSI toencourage FLSP to endeavor in service

The optimal sharing rate of the LSI can be observed inFigure 2 declining until it reaches an appropriate level whichis about 05 Thus LSI should share about half of the cost ofFLSP to realize the coordination of the supply chain

Figure 3 illustrates the whole process of the buildingreputation The reputation of LSSC will increase because ofaccumulation But the rate of growth will decrease with thedecline of the efforts of both parties resulting in a stablelevel at last The stable state of reputation gets higher in theStackelberg equilibriumwith the cost sharing contract whichsuggests that the cost sharing contract contributes to thegrowth of reputation Same as reputation the profits of LSSCas well as members can be enhanced by the use of the costsharing contract as seen in Figure 4

To sum up the long-term relationship among the mem-bers of the supply chain contributes to the stability of theservice supply chain In the long-term relationship betweenLSI and FLSP the cost sharing contract is helpful in increasingthe reputation and profits of the supply chain by encouraginga higher level of the efforts by FLSP which implies that costsharing contract is an effective coordination mechanism

14

12

1

08

06

04

02

0

V(t)

Vlowast

i(t)

Vlowast

p(t)

0 50 100 150 200 250

t

Vp(t)

Vi(t)

VLSSC (t) V

lowast

LSSC (t)

Figure 4 Change of profits of LSSC and themembers after adoptingcost sharing contract

6 Conclusions

Given the long-term relationship amongmembers in a supplychain we take time and reputation into consideration asthe main factors in a dynamic differential model in LSSCconsisting of one LSI and one FLSP The results indicate theautonomy of the logistics service system and the effectivenessof cost sharing contract in the coordination of the supplychain

However we simplify the parameters the expression formof whichmay be various in reality resulting in different formsof solutions in the model to obtain the optimal decisions ofboth parties which is an interesting subject for future workMeanwhile further studies in the coordination of the supplychain that consists of multi-integrators and multiprovidersand the empirical analysis of the mathematical models willplay an important role in the practice of supply chainmanagement

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 NSFC Major Program(Grant no 7109040471090400) and NSFC General Program(Grant nos 71272045 and 71101107)

References

[1] K L Choy C L Li S C K So H Lau S K Kwok and DW KLeung ldquoManaging uncertainty in logistics service supply chainrdquoInternational Journal of Risk Assessment and Management vol7 no 1 pp 19ndash25 2007

Journal of Applied Mathematics 9

[2] X H Li and Q N Wang ldquoCoordination mechanisms of supplychain systemsrdquo European Journal of Operational Research vol179 no 1 pp 1ndash16 2007

[3] G P Cachon ldquoSupply chain coordination with contractsrdquo inSupply Chain Management Design Coordination and Opera-tion A G deKok and S C Graves Eds pp 229ndash340 ElsevierAmsterdam The Netherlands 2003

[4] B A Pasternack ldquoOptimal pricing and return policies forperishable commoditiesrdquo Marketing Science vol 27 no 1 pp166ndash176 2008

[5] R Anupindi and Y Bassok ldquoCentralization of stocks retailersvs manufacturerrdquo Management Science vol 45 no 2 pp 178ndash191 1999

[6] F Chen A Federgruen and Y-S Zheng ldquoCoordination mech-anisms for a distribution system with one supplier and multipleretailersrdquoManagement Science vol 47 no 5 pp 693ndash708 2001

[7] L Dong andN Rudi Supply Chain Interaction under Transship-ments Washington University 2001

[8] M A Lariviere and E L Porteus ldquoSelling to the newsvendoran analysis of price-only contractsrdquoManufacturing and ServiceOperations Management vol 3 no 4 pp 293ndash305 2001

[9] D Ni K W Li and X Tang ldquoSocial responsibility allocationin two-echelon supply chains insights from wholesale pricecontractsrdquo European Journal of Operational Research vol 207no 3 pp 1269ndash1279 2010

[10] X H Wang X Y Wang and Y S Su ldquoWholesale-pricecontract of supply chain with information gatheringrdquo AppliedMathematical Modelling vol 37 no 6 pp 3848ndash3860 2013

[11] Y Bassok and R Anupindi ldquoAnalysis of contracts with totalminimum commitmentrdquo IIE Transactions vol 29 no 5 pp373ndash381 1997

[12] A A Tsay and W S Lovejoy ldquoQuantity flexibility contractsand supply chain performancerdquo Manufacturing and ServiceOperations Management vol 1 no 2 pp 118ndash149 1999

[13] E L Plambeck and T A Taylor ldquoSell the plant The impact ofcontract manufacturing on innovation capacity and profitabil-ityrdquoManagement Science vol 51 no 1 pp 133ndash150 2005

[14] O Yazlali and F Erhun ldquoRelating the multiple supply problemto quantity flexibility contractsrdquo Operations Research Lettersvol 35 no 6 pp 767ndash772 2007

[15] Z T Lian and A Deshmukh ldquoAnalysis of supply contracts withquantity flexibilityrdquo European Journal of Operational Researchvol 196 no 2 pp 526ndash533 2009

[16] S Karakaya and I S Bakal ldquoJoint quantity flexibility formultiple products in a decentralized supply chainrdquo Computersamp Industrial Engineering vol 64 no 2 pp 696ndash707 2013

[17] V Padmanabhan and I P L Png ldquoReturns policies makemoney by making goodrdquo SloanManagement Review pp 65ndash721995

[18] T Taylor ldquoChannel coordination under price protectionmidlife returns and end-of-life returns in dynamic marketsrdquoManagement Science vol 47 no 9 pp 1220ndash1234 2001

[19] Z Yao S Leung and K K Lai ldquoAnalysis of the impact of price-sensitivity factors on the returns policy in coordinating supplychainrdquo European Journal of Operational Research vol 187 no 1pp 275ndash282 2008

[20] Y L Wang and P Zipkin ldquoAgentsrsquo incentives under buy-backcontracts in a two-stage supply chainrdquo International Journal ofProduction Economics vol 120 no 2 pp 525ndash539 2009

[21] J Hou A Z Zeng and L Zhao ldquoCoordination with a backupsupplier through buy-back contract under supply disruptionrdquo

Transportation Research E Logistics and Transportation Reviewvol 46 no 6 pp 881ndash895 2010

[22] H C Xiong B T Chen and J X Xie ldquoA composite contractbased on buy back and quantity flexibility contractsrdquo EuropeanJournal of Operational Research vol 210 no 3 pp 559ndash567 2011

[23] J Dana Jr and K E Spier ldquoRevenue sharing and vertical controlin the video rental industryrdquo Journal of Industrial Economicsvol 49 no 3 pp 223ndash245 2001

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

[25] Z Yao S C H Leung and K K Lai ldquoManufacturerrsquos revenue-sharing contract and retail competitionrdquo European Journal ofOperational Research vol 186 no 2 pp 637ndash651 2008

[26] 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

[27] 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

[28] J M Chen H L Cheng and M C Chien ldquoOn channelcoordination through revenue-sharing contracts with priceand shelf-space dependent demandrdquo Applied MathematicalModelling vol 35 no 10 pp 4886ndash4901 2011

[29] D P D Omkar ldquoSupply chain coordination using revenue-dependent revenue sharing contractsrdquoOmega vol 41 no 4 pp780ndash796 2013

[30] G Kannan andN PMaria ldquoReverse supply chain coordinationby revenue sharing contract a case for the personal computersindustryrdquo European Journal of Operational Research vol 233no 2 pp 326ndash336 2014

[31] D J Wu and P R Kleindorfer ldquoCompetitive options supplycontracting and electronic marketsrdquoManagement Science vol51 no 3 pp 452ndash468 2005

[32] XWang and L Liu ldquoCoordination in a retailer-led supply chainthrough option contractrdquo International Journal of ProductionEconomics vol 110 no 1-2 pp 115ndash127 2007

[33] Y X Zhao S Y Wang T C E Cheng X Q Yang and ZM Huang ldquoCoordination of supply chains by option contractsa cooperative game theory approachrdquo European Journal ofOperational Research vol 207 no 2 pp 668ndash675 2010

[34] L Liang X HWang and J G Gao ldquoAn option contract pricingmodel of relief material supply chainrdquoOmega vol 40 no 5 pp594ndash600 2012

[35] Y X Zhao L J Ma G Xie and T C E Cheng ldquoCoordinationof supply chains with bidirectional option contractsrdquo EuropeanJournal of Operational Research vol 229 no 2 pp 375ndash3812013

[36] M M Leng and M Parlar ldquoGame-theoretic analyses of decen-tralized assembly supply chains Non-cooperative equilibria vscoordination with cost-sharing contractsrdquo European Journal ofOperational Research vol 204 no 1 pp 96ndash104 2010

[37] M Kunter ldquoCoordination via cost and revenue sharing inmanufacturer-retailer channelsrdquo European Journal of Opera-tional Research vol 216 no 2 pp 477ndash486 2012

[38] Y C Tsao and G J Sheen ldquoEffects of promotion cost sharingpolicy with the sales learning curve on supply chain coordi-nationrdquo Computers and Operations Research vol 39 no 8 pp1872ndash1878 2012

10 Journal of Applied Mathematics

[39] Y H Wei and Q Y Hu ldquoOrdering pricing and allocationin a service supply chainrdquo International Journal of ProductionEconomics vol 144 no 2 pp 590ndash598 2013

[40] W H Liu X C Xu and A Kouhpaenejad ldquoDeterministicapproach to the fairest revenue-sharing coefficient in logisticsservice supply chain under the stochastic demand conditionrdquoComputers amp Industrial Engineering vol 66 no 1 pp 41ndash522013

[41] A P Cui and W Liu ldquoStudy on capability coordination inlogistics service supply chain with options contractrdquo ChineseJournal of Management Science vol 17 no 2 pp 59ndash65 2009(Chinese)

[42] Q H Lu ldquoCoordinate with cost sharing strategy for servicesupply chainrdquo Control and Decision vol 26 no 11 pp 1649ndash1653 2011 (Chinese)

[43] A Prasad and S P Sethi ldquoCompetitive advertising underuncertainty a stochastic differential game approachrdquo Journal ofOptimization Theory and Applications vol 123 no 1 pp 163ndash185 2004

[44] S Joslashrgensen S Taboubi and G Zaccour ldquoRetail promotionswith negative brand image effects is cooperation possiblerdquoEuropean Journal of Operational Research vol 150 no 2 pp395ndash405 2003

[45] J Navas and J Marın-Solano ldquoInteractions between govern-ment and firms a differential game approachrdquo Annals ofOperations Research vol 158 no 1 pp 47ndash61 2008

[46] S Joslashrgensen S Taboubi and G Zaccour ldquoCooperative adver-tising in a marketing channelrdquo Journal of Optimization Theoryand Applications vol 110 no 1 pp 145ndash158 2001

[47] C J Fombrun and V Rindova ldquoWhorsquos tops and who decidesThe social construction of corporate reputationsrdquo WorkingPaper Stern School of Business New York University 1996

[48] N Nha and L Gaston ldquoCorporate image and corporate repu-tation in customers Retention decisions in servicesrdquo Journal ofRetailing and Consumer Services no 8 pp 227ndash2361 2001

[49] M Nerlove and K J Arrow ldquoOptimal advertising policy underdynamic conditionsrdquo Economica vol 39 pp 129ndash142 1962

[50] K Saxton ldquoWhere do reputations come fromrdquo CorporateReputation Review vol 1 no 4 pp 393ndash399 1998

[51] M G Nagler ldquoAn exploratory analysis of the determinants ofcooperative advertising participation ratesrdquo Marketing Lettersvol 17 no 2 pp 91ndash102 2006

[52] E Dockner N Jorgensen L Van et al Differential Gamesin Economics and Management Science Cambridge UniversityPress Cambridge UK 2000

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Stochastic AnalysisInternational Journal of

4 Journal of Applied Mathematics

which can result in the maximization of the profits is thesolution for the following equations

1205881199011=

21205822

1198941199012

1

120583119894

+

41205822

11990111990111199021

120583119901

minus (120587119894120574 + 2120575119901

1)

1205881199012=

21205822

11989411990111199012

120583119894

+

21205822

119901(11990111199022+ 11990121199021)

120583119901

+ (120587119894120579 minus 120575119901

2)

1205881199013=

1205822

1198941199012

2

2120583119894

+

1205822

11990111990121199022

120583119901

1205881199021=

41205822

11989411990111199021

120583119894

+

21205822

1199011199022

1

120583119901

minus (120587119901120574 + 2120575119902

1)

1205881199022=

21205822

119894(11990111199022+ 11990121199021)

120583119894

+

21205822

11990111990211199022

120583119901

+ (120587119901120579 minus 120575119902

2)

1205881199023=

1205822

11989411990121199022

120583119894

+

1205822

1199011199022

2

2120583119901

(14)

Then the optimal profits of LSI and FLSP can be repre-sented as

119881119894(119877) = 119901

11198772+ 119901

2119877 + 119901

3

119881119901(119877) = 119902

11198772+ 119902

2119877 + 119902

3

(15)

Substituting the derivatives of 119881119894(119877) 119881

119901(119877) obtained

from (15) into (9) the optimal effort level of both parties canbe represented as

119864119894=

120582119894(2119901

1119877 + 119901

2)

120583119894

119864119901=

120582119901(2119902

1119877 + 119902

2)

120583119901

(16)

Substitute the results into (2) to obtain

1198771015840= 119903119877 + 119904 (17)

in which 119903 = 21205822

1198941199011120583119894+21205822

1199011199021120583119901minus120575 119904 = 120582

2

1198941199012120583119894+1205822

1199011199022120583119901

The general solution to (17) is

119877 = 119908119890119896119905

+ 1199081 (18)

By substituting 119877 and its derivative for 119905 (19) can beobtained

1199081= minus

119904

119903 (19)

Hence the general solution can be represented as

119877 = 119908119890119903119905

minus119904

119903 (20)

in which 119908 is a constant Given 119877|119905=0

= 1198770ge 0 a particular

solution can be obtained

119877= (1198770+

119904

119903) 119890119903119905

minus119904

119903 (21)

4 Stackelberg Equilibrium withCost Sharing Contract

In this section we apply a cost sharing contract to realizethe coordination of LSSC in a Stackelberg game with the LSIas the leader of the game and the provider as the followerFirstly the integrator decides the optimal efforts 119864

119894and

the sharing rate 119863 Then the functional service providerdetermines his optimal efforts after observing the strategy ofthe integrator The optimal decisions of both parties are theresults of Stackelberg equilibrium

Theorem 2 If the feasible solutions to the following constraintequations (119891

lowast

1 119891lowast

2 119891lowast

3 119892lowast

1 119892lowast

2 119892lowast

3) exist the optimal decisions

of both parties can be derived after adopting the cost sharingcontract in LSSC

21205822

1198941198912

1

120583119894

+

1205822

119901(21198911+ 1198921)2

2120583119901

minus (2120575 + 120588) 1198911minus 120587119894120574 = 0

21205822

11989411989111198912

120583119894

+

1205822

119901(21198911+ 1198921) (21198912+ 1198922)

2120583119901

minus (120575 + 120588) 1198912+ 120587119894120579 = 0

1205822

1198941198912

2

2120583119894

+

1205822

119901(21198912+ 1198922)2

8120583119901

minus 1205881198913= 0

21205822

11989411989111198921

120583119894

+

1205822

1199011198921(21198911+ 1198921)

120583119901

minus (2120575 + 120588) 1198921minus 120587119901120574 = 0

1205822

119894(11989111198922+ 11989121198921)

120583119894

+

1205822

119901(11989111198922+ 11989121198921+ 11989211198922)

120583119901

minus (120575 + 120588) 1198922+ 120587119901120579 = 0

1205822

11989411989121198922

2120583119894

+

1205822

1199011198922(21198912+ 1198922)

4120583119901

minus 1205881198923= 0

(22)

The optimal decisions of LSI and FLS in the Stackelbergequilibrium with LSI as the leader can be represented as

119864lowast

119894=

120582119894(2119891lowast

1119877lowast+ 119891lowast

2)

120583119894

119864lowast

119901=

120582119901[(4119891lowast

1+ 2119892lowast

1) 119877lowast+ (2119891

lowast

2+ 119892lowast

2)]

2120583119901

119863lowast=

(4119891lowast

1minus 2119892lowast

1) 119877lowast+ (2119891

lowast

2minus 119892lowast

2)

(4119891lowast

1+ 2119892lowast

1) 119877lowast + (2119891

lowast

2+ 119892lowast

2)

(23)

in which 119877lowast= (1198770+ 119889119896)119890

119896119905minus 119889119896 119896 = 2120582

2

119894119891lowast

1120583119894+ 1205822

119901(2119891lowast

1+

119892lowast

1)120583119901minus 120575 and 119889 = 120582

2

119894119891lowast

2120583119894+ 1205822

119901(2119891lowast

2+ 119892lowast

2)2120583119901

Journal of Applied Mathematics 5

Proof Given that the integrator announced decisions 119864119894and

119863 the provider faces a maximization problem where theprofit function must satisfy the HJB equation

120588119881119901(119877) = Max

119864119901ge0

120587119901(120579119877 minus 120574119877

2) minus

120583119901

2(1 minus 119863) 119864

2

119901

+ 1198811015840

119901(119877)[120582

119894119864119894+ 120582119901119864119901minus 120575119877]

(24)

Maximization on the right side of (24) yields

119864119901=

1205821199011198811015840

119901(119877)

120583119901(1 minus 119863)

(25)

The integrator as the leader of game can forecast thedecision of the provider and then the leader HJB equationis

120588119881119894(119877) = Max

119864119894ge0

120587119894(120579119877 minus 120574119877

2) minus

120583119894

21198642

119894minus

120583119901

21198631198642

119901

+ 1198811015840

119894(119877)[120582

119894119864119894+ 120582119901119864119901minus 120575119877]

(26)

Insert (25) into (26) and obtain the efforts and sharingrate as

119864119894=

1205821198941198811015840

119894(119877)

120583119894

(27)

119863 =

21198811015840

119894(119877) minus 119881

1015840

119901(119877)

21198811015840

119894(119877) + 1198811015840

119901(119877)

(28)

Insert (25) (27) and (28) into (24) and (26) to obtain

120588119881119894(119877) = [120587

119894(120579119877 minus 120574119877

2) minus 120575119881

1015840

119894(119877)] 119877 +

[1205821198941198811015840

119894(119877)]2

2120583119894

+

1205822

119901[21198811015840

119894(119877) + 119881

1015840

119901(119877)]2

8120583119901

120588119881119901(119877) = [120587

119901(120579119877 minus 120574119877

2) minus 120575119881

1015840

119901(119877)] 119877

+

1205822

1198941198811015840

119894(119877)1198811015840

119901(119877)

2120583119894

+

1205822

1199011198811015840

119901(119877) [2119881

1015840

119894(119877) + 119881

1015840

119901(119877)]

4120583119901

(29)

We conjecture that the solutions of (29) will be quadratic

119881119894(119877) = 119891

11198772+ 1198912119877 + 1198913 119881

119901(119877) = 119892

11198772+ 1198922119877 + 1198923

(30)

in which11989111198912 and119891

3and 1198921 1198922 and 119892

3are all constantsWe

substitute 119881119894(119877) 119881

119901(119877) and their derivatives from (30) into

(29) to obtain

120588(11989111198772+ 1198912119877 + 1198913)

= [(120587119894120579 minus 120575119891

2) 119877 minus (120587

119894120574 + 2120575119891

1) 1198772] +

1205822

119894(21198911119877 + 1198912)2

2120583119894

+

1205822

119901[2 (2119891

1119877 + 1198912) + (2119892

1119877 + 1198922)]2

8120583119901

(31)

120588(11989211198772+ 1198922119877 + 1198923)

= [(120587119901120579 minus 120575119892

2) 119877 minus (120587

119901120574 + 2120575119892

1) 1198772]

+1205822

119894(21198911119877 + 1198912) (21198921119877 + 1198922)

2120583119894

+

1205822

119901(21198921119877 + 1198922) [2 (2119891

1119877 + 1198912) + (2119892

1119877 + 1198922)]

4120583119901

(32)

Equations (31) and (32) both satisfy for all119877 ge 0Thus wecan find out that a set of values for (119891

lowast

1 119891lowast

2 119891lowast

3 119892lowast

1 119892lowast

2 119892lowast

3)

which can result in the maximization of the profits is thesolution for the following equations

1205881198911=

21205822

1198941198912

1

120583119894

+

1205822

119901(21198911+ 1198921)2

2120583119901

minus (120587119894120574 + 2120575119891

1)

1205881198912=

21205822

11989411989111198912

120583119894

+

1205822

119901(21198911+ 1198921) (21198912+ 1198922)

2120583119901

+ (120587119894120579 minus 120575119891

2)

1205881198913=

1205822

1198941198912

2

2120583119894

+

1205822

119901(21198912+ 1198922)2

8120583119901

1205881198921=

21205822

11989411989111198921

120583119894

+

1205822

1199011198921(21198911+ 1198921)

120583119901

minus (120587119901120574 + 2120575119892

1)

1205881198922=

1205822

119894(11989111198922+ 11989121198921)

120583119894

+

1205822

119901(11989111198922+ 11989121198921+ 11989211198922)

120583119901

+ (120587119901120579 minus 120575119892

2)

1205881198923=

1205822

11989411989121198922

2120583119894

+

1205822

1199011198922(21198912+ 1198922)

4120583119901

(33)

The optimal profits of LSI and FLSP can be represented as

119881lowast

119894(119877) = 119891

lowast

11198772+ 119891lowast

2119877 + 119891

lowast

3 (34)

119881lowast

119901(119877) = 119892

lowast

11198772+ 119892lowast

2119877 + 119892lowast

3 (35)

6 Journal of Applied Mathematics

Substituting the derivative of 119881lowast119894(119877) from (34) into (27)

the optimal efforts of LSI can be represented as

119864lowast

119894=

120582119894(2119891lowast

1119877 + 119891

lowast

2)

120583119894

(36)

Substituting the derivatives of119881lowast119894(119877) and119881

lowast

119901(119877) from (34)

and (35) into (28) the optimal sharing rate can be representedas

119863lowast=

(4119891lowast

1minus 2119892lowast

1) 119877 + (2119891

lowast

2minus 119892lowast

2)

(4119891lowast

1+ 2119892lowast

1) 119877 + (2119891

lowast

2+ 119892lowast

2) (37)

Also we can obtain the optimal efforts of FLSP bysubstituting119881

lowast

119901(119877)rsquos derivative and119863

lowast from (35) and (37) into(25)

119864lowast

119901=

120582119901[(4119891lowast

1+ 2119892lowast

1) 119877 + (2119891

lowast

2+ 119892lowast

2)]

2120583119901

(38)

Substitute (36) and (38) into differential game equation(2) to obtain

1198771015840= 119896119877 + 119889 (39)

in which 119896 = 21205822

119894119891lowast

1120583119894+1205822

119901(2119891lowast

1+119892lowast

1)120583119901minus120575 119889 = 120582

2

119894119891lowast

2120583119894+

1205822

119901(2119891lowast

2+ 119892lowast

2)2120583119901

Same as the proof ofTheorem 2 the general solutions canrepresented as

119877 = 119888119890119896119905

minus119889

119896 (40)

in which 119888 is a constant Given 119877|119905=0

= 1198770ge 0 the particular

solution is

119877lowast= (1198770+

119889

119896) 119890119896119905

minus119889

119896 (41)

5 Numerical Analysis

Consider a case with the following set of parameters

120583119894= 12 120583

119901= 1 120582

119894= 06

120582119901= 05 120575 = 001 120579 = 05

120574 = 005 120587119894= 06 120587

119901= 04

120588 = 09 1198770= 05

(42)

Then we can get the results of (5) and (22)

(1199011 119901

2 119901

3 119902

1 119902

2 119902

3)

= (minus0031269 0315981 0034968

minus 0020664 0208800 0028047)

(119891lowast

1 119891lowast

2 119891lowast

3 119892lowast

1 119892lowast

2 119892lowast

3)

= (minus0031046 0313701 0040789

minus 0020847 0210668 0023275)

119903 =21205822

1198941199011

120583119894

+

21205822

1199011199021

120583119901

minus 120575 = minus0039093

119904 =1205822

1198941199012

120583119894

+

1205822

1199011199022

120583119901

= 0146994

119896 =21205822

119894119891lowast

1

120583119894

+

1205822

119901(2119891lowast

1+ 119892lowast

1)

120583119901

minus 120575 = minus0049362

119889 =1205822

119894119891lowast

2

120583119894

+

1205822

119901(2119891lowast

2+ 119892lowast

2)

2120583119901

= 0198869

(43)

As a result we can obtain the optimal solutions of thedifferential game

(1) Nash Equilibrium without Cost Sharing Contract Theoptimal efforts of LSI and FLSP are

119864119894=

120582119894(2119901

1119877+ 119901

2)

120583119894

= 01020119890minus004119905

+ 00404

119864119901=

120582119901(2119902

1119877+ 119902

2)

120583119901

= 00673119890minus004119905

+ 00267

(44)

Under the circumstances the accumulation of the reputationof LSSC is

119877(119905) = (119877

0+

119904

119903) 119890119903119905

minus119904

119903= minus326119890

minus004119905+ 376 (45)

And the profits of both parties and the whole supply chain are

119881119894(119877) = 119901

11198772+ 119901

2119877 + 119901

3

= minus03323119890minus008119905

minus 02635119890minus004119905

+ 07810

119881119901(119877) = 119902

11198772+ 119902

2119877 + 119902

3

= minus02196119890minus008119905

minus 01741119890minus004119905

+ 05210

119881LSSC = 119881

119894+ 119881

119901

= minus05519119890minus008119905

minus 04376119890minus004119905

+ 13020

(46)

Journal of Applied Mathematics 7

0 50 100 150 200 250

02

015

01

005

0

t

E(t)

Ei(t)

Ep(t)

(a)

0 50 100 150 200 250

02

015

01

005

0

t

E(t)

Elowast

i(t)

Elowast

p(t)

(b)

0 50 100 150 200 250

02

015

01

005

0

t

E(t)

Elowast

i(t)

Ei(t)

(c)

0 50 100 150 200 250

02

015

01

005

0

t

E(t)

Elowast

p(t)

Ep(t)

(d)

Figure 1 Difference of optimal efforts between Nash equilibrium and Stackelberg equilibrium

(2) Stackelberg Equilibrium with Cost Sharing Contract Theoptimal efforts of LSI and FLSP are

119864lowast

119894=

120582119894(2119891lowast

1119877lowast+ 119891lowast

2)

120583119894

= 01096119890minus005119905

+ 00318

119864lowast

119901=

120582119901[(4119891lowast

1+ 2119892lowast

1) 119877lowast+ (2119891

lowast

2+ 119892lowast

2)]

2120583119901

= 01463119890minus005119905

+ 00424

(47)

The optimal sharing rate of LSI is

119863lowast=

(4119891lowast

1minus 2119892lowast

1) 119877lowast+ (2119891

lowast

2minus 119892lowast

2)

(4119891lowast

1+ 2119892lowast

1) 119877lowast + (2119891

lowast

2+ 119892lowast

2)

=02911119890

minus005119905+ 00844

05853119890minus005119905 + 01698

(48)

Under the circumstances the accumulation of the reputationof LSSC is

119877lowast(119905) = (119877

0+

119889

119896) 119890119896119905

minus119889

119896= minus353119890

minus005119905+ 403 (49)

And the profits of both parties and the whole supply chain are

119881lowast

119894(119877) = 119891

lowast

11198772+ 119891lowast

2119877 + 119891

lowast

3

= minus03866119890minus010119905

minus 02242119890minus005119905

+ 08007

119881lowast

119901(119877) = 119892

lowast

11198772+ 119892lowast

2119877 + 119892lowast

3

= minus02596119890minus010119905

minus 01507119890minus005119905

+ 05336

119881lowast

LSSC = 119881lowast

119894+ 119881lowast

119901

= minus06462119890minus010119905

minus 03749119890minus005119905

+ 13343

(50)

In the long-term relationship among members in LSSCLSI and FLSP are faced with dynamic games which meansthat the optimal strategies of LSI and FLSP will change withtime As a result both players will adjust their decisions tomaximize their profits in an infinite-time horizon Howeverthe optimal decisionswill become stable at last suggesting theautonomy of the LSSC

Figure 1 illustrates the difference between efforts of bothparties obtained from Nash equilibrium and Stackelbergequilibrium The efforts will decline with time until theyreach an appropriate levelThis is because themarginal utilityof reputation for revenue is diminishing while the cost to

8 Journal of Applied Mathematics

0 50 100 150 200 250

04973

04972

04972

04971

04971

0497

t

D(t)

Figure 2 The optimal cost sharing rate of the LSI

45

4

35

3

25

2

15

1

05

R(t)

0 50 100 150 200 250

t

R(t)

Rlowast(t)

Figure 3 Change of the reputation after adopting cost sharingcontract

maintain reputation remains the same By comparing theresults from the two equilibria we can see that efforts ofFLSP enhance remarkably under the Stackelberg equilibriumwhich means that the cost sharing contract can help LSI toencourage FLSP to endeavor in service

The optimal sharing rate of the LSI can be observed inFigure 2 declining until it reaches an appropriate level whichis about 05 Thus LSI should share about half of the cost ofFLSP to realize the coordination of the supply chain

Figure 3 illustrates the whole process of the buildingreputation The reputation of LSSC will increase because ofaccumulation But the rate of growth will decrease with thedecline of the efforts of both parties resulting in a stablelevel at last The stable state of reputation gets higher in theStackelberg equilibriumwith the cost sharing contract whichsuggests that the cost sharing contract contributes to thegrowth of reputation Same as reputation the profits of LSSCas well as members can be enhanced by the use of the costsharing contract as seen in Figure 4

To sum up the long-term relationship among the mem-bers of the supply chain contributes to the stability of theservice supply chain In the long-term relationship betweenLSI and FLSP the cost sharing contract is helpful in increasingthe reputation and profits of the supply chain by encouraginga higher level of the efforts by FLSP which implies that costsharing contract is an effective coordination mechanism

14

12

1

08

06

04

02

0

V(t)

Vlowast

i(t)

Vlowast

p(t)

0 50 100 150 200 250

t

Vp(t)

Vi(t)

VLSSC (t) V

lowast

LSSC (t)

Figure 4 Change of profits of LSSC and themembers after adoptingcost sharing contract

6 Conclusions

Given the long-term relationship amongmembers in a supplychain we take time and reputation into consideration asthe main factors in a dynamic differential model in LSSCconsisting of one LSI and one FLSP The results indicate theautonomy of the logistics service system and the effectivenessof cost sharing contract in the coordination of the supplychain

However we simplify the parameters the expression formof whichmay be various in reality resulting in different formsof solutions in the model to obtain the optimal decisions ofboth parties which is an interesting subject for future workMeanwhile further studies in the coordination of the supplychain that consists of multi-integrators and multiprovidersand the empirical analysis of the mathematical models willplay an important role in the practice of supply chainmanagement

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 NSFC Major Program(Grant no 7109040471090400) and NSFC General Program(Grant nos 71272045 and 71101107)

References

[1] K L Choy C L Li S C K So H Lau S K Kwok and DW KLeung ldquoManaging uncertainty in logistics service supply chainrdquoInternational Journal of Risk Assessment and Management vol7 no 1 pp 19ndash25 2007

Journal of Applied Mathematics 9

[2] X H Li and Q N Wang ldquoCoordination mechanisms of supplychain systemsrdquo European Journal of Operational Research vol179 no 1 pp 1ndash16 2007

[3] G P Cachon ldquoSupply chain coordination with contractsrdquo inSupply Chain Management Design Coordination and Opera-tion A G deKok and S C Graves Eds pp 229ndash340 ElsevierAmsterdam The Netherlands 2003

[4] B A Pasternack ldquoOptimal pricing and return policies forperishable commoditiesrdquo Marketing Science vol 27 no 1 pp166ndash176 2008

[5] R Anupindi and Y Bassok ldquoCentralization of stocks retailersvs manufacturerrdquo Management Science vol 45 no 2 pp 178ndash191 1999

[6] F Chen A Federgruen and Y-S Zheng ldquoCoordination mech-anisms for a distribution system with one supplier and multipleretailersrdquoManagement Science vol 47 no 5 pp 693ndash708 2001

[7] L Dong andN Rudi Supply Chain Interaction under Transship-ments Washington University 2001

[8] M A Lariviere and E L Porteus ldquoSelling to the newsvendoran analysis of price-only contractsrdquoManufacturing and ServiceOperations Management vol 3 no 4 pp 293ndash305 2001

[9] D Ni K W Li and X Tang ldquoSocial responsibility allocationin two-echelon supply chains insights from wholesale pricecontractsrdquo European Journal of Operational Research vol 207no 3 pp 1269ndash1279 2010

[10] X H Wang X Y Wang and Y S Su ldquoWholesale-pricecontract of supply chain with information gatheringrdquo AppliedMathematical Modelling vol 37 no 6 pp 3848ndash3860 2013

[11] Y Bassok and R Anupindi ldquoAnalysis of contracts with totalminimum commitmentrdquo IIE Transactions vol 29 no 5 pp373ndash381 1997

[12] A A Tsay and W S Lovejoy ldquoQuantity flexibility contractsand supply chain performancerdquo Manufacturing and ServiceOperations Management vol 1 no 2 pp 118ndash149 1999

[13] E L Plambeck and T A Taylor ldquoSell the plant The impact ofcontract manufacturing on innovation capacity and profitabil-ityrdquoManagement Science vol 51 no 1 pp 133ndash150 2005

[14] O Yazlali and F Erhun ldquoRelating the multiple supply problemto quantity flexibility contractsrdquo Operations Research Lettersvol 35 no 6 pp 767ndash772 2007

[15] Z T Lian and A Deshmukh ldquoAnalysis of supply contracts withquantity flexibilityrdquo European Journal of Operational Researchvol 196 no 2 pp 526ndash533 2009

[16] S Karakaya and I S Bakal ldquoJoint quantity flexibility formultiple products in a decentralized supply chainrdquo Computersamp Industrial Engineering vol 64 no 2 pp 696ndash707 2013

[17] V Padmanabhan and I P L Png ldquoReturns policies makemoney by making goodrdquo SloanManagement Review pp 65ndash721995

[18] T Taylor ldquoChannel coordination under price protectionmidlife returns and end-of-life returns in dynamic marketsrdquoManagement Science vol 47 no 9 pp 1220ndash1234 2001

[19] Z Yao S Leung and K K Lai ldquoAnalysis of the impact of price-sensitivity factors on the returns policy in coordinating supplychainrdquo European Journal of Operational Research vol 187 no 1pp 275ndash282 2008

[20] Y L Wang and P Zipkin ldquoAgentsrsquo incentives under buy-backcontracts in a two-stage supply chainrdquo International Journal ofProduction Economics vol 120 no 2 pp 525ndash539 2009

[21] J Hou A Z Zeng and L Zhao ldquoCoordination with a backupsupplier through buy-back contract under supply disruptionrdquo

Transportation Research E Logistics and Transportation Reviewvol 46 no 6 pp 881ndash895 2010

[22] H C Xiong B T Chen and J X Xie ldquoA composite contractbased on buy back and quantity flexibility contractsrdquo EuropeanJournal of Operational Research vol 210 no 3 pp 559ndash567 2011

[23] J Dana Jr and K E Spier ldquoRevenue sharing and vertical controlin the video rental industryrdquo Journal of Industrial Economicsvol 49 no 3 pp 223ndash245 2001

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

[25] Z Yao S C H Leung and K K Lai ldquoManufacturerrsquos revenue-sharing contract and retail competitionrdquo European Journal ofOperational Research vol 186 no 2 pp 637ndash651 2008

[26] 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

[27] 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

[28] J M Chen H L Cheng and M C Chien ldquoOn channelcoordination through revenue-sharing contracts with priceand shelf-space dependent demandrdquo Applied MathematicalModelling vol 35 no 10 pp 4886ndash4901 2011

[29] D P D Omkar ldquoSupply chain coordination using revenue-dependent revenue sharing contractsrdquoOmega vol 41 no 4 pp780ndash796 2013

[30] G Kannan andN PMaria ldquoReverse supply chain coordinationby revenue sharing contract a case for the personal computersindustryrdquo European Journal of Operational Research vol 233no 2 pp 326ndash336 2014

[31] D J Wu and P R Kleindorfer ldquoCompetitive options supplycontracting and electronic marketsrdquoManagement Science vol51 no 3 pp 452ndash468 2005

[32] XWang and L Liu ldquoCoordination in a retailer-led supply chainthrough option contractrdquo International Journal of ProductionEconomics vol 110 no 1-2 pp 115ndash127 2007

[33] Y X Zhao S Y Wang T C E Cheng X Q Yang and ZM Huang ldquoCoordination of supply chains by option contractsa cooperative game theory approachrdquo European Journal ofOperational Research vol 207 no 2 pp 668ndash675 2010

[34] L Liang X HWang and J G Gao ldquoAn option contract pricingmodel of relief material supply chainrdquoOmega vol 40 no 5 pp594ndash600 2012

[35] Y X Zhao L J Ma G Xie and T C E Cheng ldquoCoordinationof supply chains with bidirectional option contractsrdquo EuropeanJournal of Operational Research vol 229 no 2 pp 375ndash3812013

[36] M M Leng and M Parlar ldquoGame-theoretic analyses of decen-tralized assembly supply chains Non-cooperative equilibria vscoordination with cost-sharing contractsrdquo European Journal ofOperational Research vol 204 no 1 pp 96ndash104 2010

[37] M Kunter ldquoCoordination via cost and revenue sharing inmanufacturer-retailer channelsrdquo European Journal of Opera-tional Research vol 216 no 2 pp 477ndash486 2012

[38] Y C Tsao and G J Sheen ldquoEffects of promotion cost sharingpolicy with the sales learning curve on supply chain coordi-nationrdquo Computers and Operations Research vol 39 no 8 pp1872ndash1878 2012

10 Journal of Applied Mathematics

[39] Y H Wei and Q Y Hu ldquoOrdering pricing and allocationin a service supply chainrdquo International Journal of ProductionEconomics vol 144 no 2 pp 590ndash598 2013

[40] W H Liu X C Xu and A Kouhpaenejad ldquoDeterministicapproach to the fairest revenue-sharing coefficient in logisticsservice supply chain under the stochastic demand conditionrdquoComputers amp Industrial Engineering vol 66 no 1 pp 41ndash522013

[41] A P Cui and W Liu ldquoStudy on capability coordination inlogistics service supply chain with options contractrdquo ChineseJournal of Management Science vol 17 no 2 pp 59ndash65 2009(Chinese)

[42] Q H Lu ldquoCoordinate with cost sharing strategy for servicesupply chainrdquo Control and Decision vol 26 no 11 pp 1649ndash1653 2011 (Chinese)

[43] A Prasad and S P Sethi ldquoCompetitive advertising underuncertainty a stochastic differential game approachrdquo Journal ofOptimization Theory and Applications vol 123 no 1 pp 163ndash185 2004

[44] S Joslashrgensen S Taboubi and G Zaccour ldquoRetail promotionswith negative brand image effects is cooperation possiblerdquoEuropean Journal of Operational Research vol 150 no 2 pp395ndash405 2003

[45] J Navas and J Marın-Solano ldquoInteractions between govern-ment and firms a differential game approachrdquo Annals ofOperations Research vol 158 no 1 pp 47ndash61 2008

[46] S Joslashrgensen S Taboubi and G Zaccour ldquoCooperative adver-tising in a marketing channelrdquo Journal of Optimization Theoryand Applications vol 110 no 1 pp 145ndash158 2001

[47] C J Fombrun and V Rindova ldquoWhorsquos tops and who decidesThe social construction of corporate reputationsrdquo WorkingPaper Stern School of Business New York University 1996

[48] N Nha and L Gaston ldquoCorporate image and corporate repu-tation in customers Retention decisions in servicesrdquo Journal ofRetailing and Consumer Services no 8 pp 227ndash2361 2001

[49] M Nerlove and K J Arrow ldquoOptimal advertising policy underdynamic conditionsrdquo Economica vol 39 pp 129ndash142 1962

[50] K Saxton ldquoWhere do reputations come fromrdquo CorporateReputation Review vol 1 no 4 pp 393ndash399 1998

[51] M G Nagler ldquoAn exploratory analysis of the determinants ofcooperative advertising participation ratesrdquo Marketing Lettersvol 17 no 2 pp 91ndash102 2006

[52] E Dockner N Jorgensen L Van et al Differential Gamesin Economics and Management Science Cambridge UniversityPress Cambridge UK 2000

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

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Operations ResearchAdvances in

Journal of

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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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

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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

Journal of Applied Mathematics 5

Proof Given that the integrator announced decisions 119864119894and

119863 the provider faces a maximization problem where theprofit function must satisfy the HJB equation

120588119881119901(119877) = Max

119864119901ge0

120587119901(120579119877 minus 120574119877

2) minus

120583119901

2(1 minus 119863) 119864

2

119901

+ 1198811015840

119901(119877)[120582

119894119864119894+ 120582119901119864119901minus 120575119877]

(24)

Maximization on the right side of (24) yields

119864119901=

1205821199011198811015840

119901(119877)

120583119901(1 minus 119863)

(25)

The integrator as the leader of game can forecast thedecision of the provider and then the leader HJB equationis

120588119881119894(119877) = Max

119864119894ge0

120587119894(120579119877 minus 120574119877

2) minus

120583119894

21198642

119894minus

120583119901

21198631198642

119901

+ 1198811015840

119894(119877)[120582

119894119864119894+ 120582119901119864119901minus 120575119877]

(26)

Insert (25) into (26) and obtain the efforts and sharingrate as

119864119894=

1205821198941198811015840

119894(119877)

120583119894

(27)

119863 =

21198811015840

119894(119877) minus 119881

1015840

119901(119877)

21198811015840

119894(119877) + 1198811015840

119901(119877)

(28)

Insert (25) (27) and (28) into (24) and (26) to obtain

120588119881119894(119877) = [120587

119894(120579119877 minus 120574119877

2) minus 120575119881

1015840

119894(119877)] 119877 +

[1205821198941198811015840

119894(119877)]2

2120583119894

+

1205822

119901[21198811015840

119894(119877) + 119881

1015840

119901(119877)]2

8120583119901

120588119881119901(119877) = [120587

119901(120579119877 minus 120574119877

2) minus 120575119881

1015840

119901(119877)] 119877

+

1205822

1198941198811015840

119894(119877)1198811015840

119901(119877)

2120583119894

+

1205822

1199011198811015840

119901(119877) [2119881

1015840

119894(119877) + 119881

1015840

119901(119877)]

4120583119901

(29)

We conjecture that the solutions of (29) will be quadratic

119881119894(119877) = 119891

11198772+ 1198912119877 + 1198913 119881

119901(119877) = 119892

11198772+ 1198922119877 + 1198923

(30)

in which11989111198912 and119891

3and 1198921 1198922 and 119892

3are all constantsWe

substitute 119881119894(119877) 119881

119901(119877) and their derivatives from (30) into

(29) to obtain

120588(11989111198772+ 1198912119877 + 1198913)

= [(120587119894120579 minus 120575119891

2) 119877 minus (120587

119894120574 + 2120575119891

1) 1198772] +

1205822

119894(21198911119877 + 1198912)2

2120583119894

+

1205822

119901[2 (2119891

1119877 + 1198912) + (2119892

1119877 + 1198922)]2

8120583119901

(31)

120588(11989211198772+ 1198922119877 + 1198923)

= [(120587119901120579 minus 120575119892

2) 119877 minus (120587

119901120574 + 2120575119892

1) 1198772]

+1205822

119894(21198911119877 + 1198912) (21198921119877 + 1198922)

2120583119894

+

1205822

119901(21198921119877 + 1198922) [2 (2119891

1119877 + 1198912) + (2119892

1119877 + 1198922)]

4120583119901

(32)

Equations (31) and (32) both satisfy for all119877 ge 0Thus wecan find out that a set of values for (119891

lowast

1 119891lowast

2 119891lowast

3 119892lowast

1 119892lowast

2 119892lowast

3)

which can result in the maximization of the profits is thesolution for the following equations

1205881198911=

21205822

1198941198912

1

120583119894

+

1205822

119901(21198911+ 1198921)2

2120583119901

minus (120587119894120574 + 2120575119891

1)

1205881198912=

21205822

11989411989111198912

120583119894

+

1205822

119901(21198911+ 1198921) (21198912+ 1198922)

2120583119901

+ (120587119894120579 minus 120575119891

2)

1205881198913=

1205822

1198941198912

2

2120583119894

+

1205822

119901(21198912+ 1198922)2

8120583119901

1205881198921=

21205822

11989411989111198921

120583119894

+

1205822

1199011198921(21198911+ 1198921)

120583119901

minus (120587119901120574 + 2120575119892

1)

1205881198922=

1205822

119894(11989111198922+ 11989121198921)

120583119894

+

1205822

119901(11989111198922+ 11989121198921+ 11989211198922)

120583119901

+ (120587119901120579 minus 120575119892

2)

1205881198923=

1205822

11989411989121198922

2120583119894

+

1205822

1199011198922(21198912+ 1198922)

4120583119901

(33)

The optimal profits of LSI and FLSP can be represented as

119881lowast

119894(119877) = 119891

lowast

11198772+ 119891lowast

2119877 + 119891

lowast

3 (34)

119881lowast

119901(119877) = 119892

lowast

11198772+ 119892lowast

2119877 + 119892lowast

3 (35)

6 Journal of Applied Mathematics

Substituting the derivative of 119881lowast119894(119877) from (34) into (27)

the optimal efforts of LSI can be represented as

119864lowast

119894=

120582119894(2119891lowast

1119877 + 119891

lowast

2)

120583119894

(36)

Substituting the derivatives of119881lowast119894(119877) and119881

lowast

119901(119877) from (34)

and (35) into (28) the optimal sharing rate can be representedas

119863lowast=

(4119891lowast

1minus 2119892lowast

1) 119877 + (2119891

lowast

2minus 119892lowast

2)

(4119891lowast

1+ 2119892lowast

1) 119877 + (2119891

lowast

2+ 119892lowast

2) (37)

Also we can obtain the optimal efforts of FLSP bysubstituting119881

lowast

119901(119877)rsquos derivative and119863

lowast from (35) and (37) into(25)

119864lowast

119901=

120582119901[(4119891lowast

1+ 2119892lowast

1) 119877 + (2119891

lowast

2+ 119892lowast

2)]

2120583119901

(38)

Substitute (36) and (38) into differential game equation(2) to obtain

1198771015840= 119896119877 + 119889 (39)

in which 119896 = 21205822

119894119891lowast

1120583119894+1205822

119901(2119891lowast

1+119892lowast

1)120583119901minus120575 119889 = 120582

2

119894119891lowast

2120583119894+

1205822

119901(2119891lowast

2+ 119892lowast

2)2120583119901

Same as the proof ofTheorem 2 the general solutions canrepresented as

119877 = 119888119890119896119905

minus119889

119896 (40)

in which 119888 is a constant Given 119877|119905=0

= 1198770ge 0 the particular

solution is

119877lowast= (1198770+

119889

119896) 119890119896119905

minus119889

119896 (41)

5 Numerical Analysis

Consider a case with the following set of parameters

120583119894= 12 120583

119901= 1 120582

119894= 06

120582119901= 05 120575 = 001 120579 = 05

120574 = 005 120587119894= 06 120587

119901= 04

120588 = 09 1198770= 05

(42)

Then we can get the results of (5) and (22)

(1199011 119901

2 119901

3 119902

1 119902

2 119902

3)

= (minus0031269 0315981 0034968

minus 0020664 0208800 0028047)

(119891lowast

1 119891lowast

2 119891lowast

3 119892lowast

1 119892lowast

2 119892lowast

3)

= (minus0031046 0313701 0040789

minus 0020847 0210668 0023275)

119903 =21205822

1198941199011

120583119894

+

21205822

1199011199021

120583119901

minus 120575 = minus0039093

119904 =1205822

1198941199012

120583119894

+

1205822

1199011199022

120583119901

= 0146994

119896 =21205822

119894119891lowast

1

120583119894

+

1205822

119901(2119891lowast

1+ 119892lowast

1)

120583119901

minus 120575 = minus0049362

119889 =1205822

119894119891lowast

2

120583119894

+

1205822

119901(2119891lowast

2+ 119892lowast

2)

2120583119901

= 0198869

(43)

As a result we can obtain the optimal solutions of thedifferential game

(1) Nash Equilibrium without Cost Sharing Contract Theoptimal efforts of LSI and FLSP are

119864119894=

120582119894(2119901

1119877+ 119901

2)

120583119894

= 01020119890minus004119905

+ 00404

119864119901=

120582119901(2119902

1119877+ 119902

2)

120583119901

= 00673119890minus004119905

+ 00267

(44)

Under the circumstances the accumulation of the reputationof LSSC is

119877(119905) = (119877

0+

119904

119903) 119890119903119905

minus119904

119903= minus326119890

minus004119905+ 376 (45)

And the profits of both parties and the whole supply chain are

119881119894(119877) = 119901

11198772+ 119901

2119877 + 119901

3

= minus03323119890minus008119905

minus 02635119890minus004119905

+ 07810

119881119901(119877) = 119902

11198772+ 119902

2119877 + 119902

3

= minus02196119890minus008119905

minus 01741119890minus004119905

+ 05210

119881LSSC = 119881

119894+ 119881

119901

= minus05519119890minus008119905

minus 04376119890minus004119905

+ 13020

(46)

Journal of Applied Mathematics 7

0 50 100 150 200 250

02

015

01

005

0

t

E(t)

Ei(t)

Ep(t)

(a)

0 50 100 150 200 250

02

015

01

005

0

t

E(t)

Elowast

i(t)

Elowast

p(t)

(b)

0 50 100 150 200 250

02

015

01

005

0

t

E(t)

Elowast

i(t)

Ei(t)

(c)

0 50 100 150 200 250

02

015

01

005

0

t

E(t)

Elowast

p(t)

Ep(t)

(d)

Figure 1 Difference of optimal efforts between Nash equilibrium and Stackelberg equilibrium

(2) Stackelberg Equilibrium with Cost Sharing Contract Theoptimal efforts of LSI and FLSP are

119864lowast

119894=

120582119894(2119891lowast

1119877lowast+ 119891lowast

2)

120583119894

= 01096119890minus005119905

+ 00318

119864lowast

119901=

120582119901[(4119891lowast

1+ 2119892lowast

1) 119877lowast+ (2119891

lowast

2+ 119892lowast

2)]

2120583119901

= 01463119890minus005119905

+ 00424

(47)

The optimal sharing rate of LSI is

119863lowast=

(4119891lowast

1minus 2119892lowast

1) 119877lowast+ (2119891

lowast

2minus 119892lowast

2)

(4119891lowast

1+ 2119892lowast

1) 119877lowast + (2119891

lowast

2+ 119892lowast

2)

=02911119890

minus005119905+ 00844

05853119890minus005119905 + 01698

(48)

Under the circumstances the accumulation of the reputationof LSSC is

119877lowast(119905) = (119877

0+

119889

119896) 119890119896119905

minus119889

119896= minus353119890

minus005119905+ 403 (49)

And the profits of both parties and the whole supply chain are

119881lowast

119894(119877) = 119891

lowast

11198772+ 119891lowast

2119877 + 119891

lowast

3

= minus03866119890minus010119905

minus 02242119890minus005119905

+ 08007

119881lowast

119901(119877) = 119892

lowast

11198772+ 119892lowast

2119877 + 119892lowast

3

= minus02596119890minus010119905

minus 01507119890minus005119905

+ 05336

119881lowast

LSSC = 119881lowast

119894+ 119881lowast

119901

= minus06462119890minus010119905

minus 03749119890minus005119905

+ 13343

(50)

In the long-term relationship among members in LSSCLSI and FLSP are faced with dynamic games which meansthat the optimal strategies of LSI and FLSP will change withtime As a result both players will adjust their decisions tomaximize their profits in an infinite-time horizon Howeverthe optimal decisionswill become stable at last suggesting theautonomy of the LSSC

Figure 1 illustrates the difference between efforts of bothparties obtained from Nash equilibrium and Stackelbergequilibrium The efforts will decline with time until theyreach an appropriate levelThis is because themarginal utilityof reputation for revenue is diminishing while the cost to

8 Journal of Applied Mathematics

0 50 100 150 200 250

04973

04972

04972

04971

04971

0497

t

D(t)

Figure 2 The optimal cost sharing rate of the LSI

45

4

35

3

25

2

15

1

05

R(t)

0 50 100 150 200 250

t

R(t)

Rlowast(t)

Figure 3 Change of the reputation after adopting cost sharingcontract

maintain reputation remains the same By comparing theresults from the two equilibria we can see that efforts ofFLSP enhance remarkably under the Stackelberg equilibriumwhich means that the cost sharing contract can help LSI toencourage FLSP to endeavor in service

The optimal sharing rate of the LSI can be observed inFigure 2 declining until it reaches an appropriate level whichis about 05 Thus LSI should share about half of the cost ofFLSP to realize the coordination of the supply chain

Figure 3 illustrates the whole process of the buildingreputation The reputation of LSSC will increase because ofaccumulation But the rate of growth will decrease with thedecline of the efforts of both parties resulting in a stablelevel at last The stable state of reputation gets higher in theStackelberg equilibriumwith the cost sharing contract whichsuggests that the cost sharing contract contributes to thegrowth of reputation Same as reputation the profits of LSSCas well as members can be enhanced by the use of the costsharing contract as seen in Figure 4

To sum up the long-term relationship among the mem-bers of the supply chain contributes to the stability of theservice supply chain In the long-term relationship betweenLSI and FLSP the cost sharing contract is helpful in increasingthe reputation and profits of the supply chain by encouraginga higher level of the efforts by FLSP which implies that costsharing contract is an effective coordination mechanism

14

12

1

08

06

04

02

0

V(t)

Vlowast

i(t)

Vlowast

p(t)

0 50 100 150 200 250

t

Vp(t)

Vi(t)

VLSSC (t) V

lowast

LSSC (t)

Figure 4 Change of profits of LSSC and themembers after adoptingcost sharing contract

6 Conclusions

Given the long-term relationship amongmembers in a supplychain we take time and reputation into consideration asthe main factors in a dynamic differential model in LSSCconsisting of one LSI and one FLSP The results indicate theautonomy of the logistics service system and the effectivenessof cost sharing contract in the coordination of the supplychain

However we simplify the parameters the expression formof whichmay be various in reality resulting in different formsof solutions in the model to obtain the optimal decisions ofboth parties which is an interesting subject for future workMeanwhile further studies in the coordination of the supplychain that consists of multi-integrators and multiprovidersand the empirical analysis of the mathematical models willplay an important role in the practice of supply chainmanagement

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 NSFC Major Program(Grant no 7109040471090400) and NSFC General Program(Grant nos 71272045 and 71101107)

References

[1] K L Choy C L Li S C K So H Lau S K Kwok and DW KLeung ldquoManaging uncertainty in logistics service supply chainrdquoInternational Journal of Risk Assessment and Management vol7 no 1 pp 19ndash25 2007

Journal of Applied Mathematics 9

[2] X H Li and Q N Wang ldquoCoordination mechanisms of supplychain systemsrdquo European Journal of Operational Research vol179 no 1 pp 1ndash16 2007

[3] G P Cachon ldquoSupply chain coordination with contractsrdquo inSupply Chain Management Design Coordination and Opera-tion A G deKok and S C Graves Eds pp 229ndash340 ElsevierAmsterdam The Netherlands 2003

[4] B A Pasternack ldquoOptimal pricing and return policies forperishable commoditiesrdquo Marketing Science vol 27 no 1 pp166ndash176 2008

[5] R Anupindi and Y Bassok ldquoCentralization of stocks retailersvs manufacturerrdquo Management Science vol 45 no 2 pp 178ndash191 1999

[6] F Chen A Federgruen and Y-S Zheng ldquoCoordination mech-anisms for a distribution system with one supplier and multipleretailersrdquoManagement Science vol 47 no 5 pp 693ndash708 2001

[7] L Dong andN Rudi Supply Chain Interaction under Transship-ments Washington University 2001

[8] M A Lariviere and E L Porteus ldquoSelling to the newsvendoran analysis of price-only contractsrdquoManufacturing and ServiceOperations Management vol 3 no 4 pp 293ndash305 2001

[9] D Ni K W Li and X Tang ldquoSocial responsibility allocationin two-echelon supply chains insights from wholesale pricecontractsrdquo European Journal of Operational Research vol 207no 3 pp 1269ndash1279 2010

[10] X H Wang X Y Wang and Y S Su ldquoWholesale-pricecontract of supply chain with information gatheringrdquo AppliedMathematical Modelling vol 37 no 6 pp 3848ndash3860 2013

[11] Y Bassok and R Anupindi ldquoAnalysis of contracts with totalminimum commitmentrdquo IIE Transactions vol 29 no 5 pp373ndash381 1997

[12] A A Tsay and W S Lovejoy ldquoQuantity flexibility contractsand supply chain performancerdquo Manufacturing and ServiceOperations Management vol 1 no 2 pp 118ndash149 1999

[13] E L Plambeck and T A Taylor ldquoSell the plant The impact ofcontract manufacturing on innovation capacity and profitabil-ityrdquoManagement Science vol 51 no 1 pp 133ndash150 2005

[14] O Yazlali and F Erhun ldquoRelating the multiple supply problemto quantity flexibility contractsrdquo Operations Research Lettersvol 35 no 6 pp 767ndash772 2007

[15] Z T Lian and A Deshmukh ldquoAnalysis of supply contracts withquantity flexibilityrdquo European Journal of Operational Researchvol 196 no 2 pp 526ndash533 2009

[16] S Karakaya and I S Bakal ldquoJoint quantity flexibility formultiple products in a decentralized supply chainrdquo Computersamp Industrial Engineering vol 64 no 2 pp 696ndash707 2013

[17] V Padmanabhan and I P L Png ldquoReturns policies makemoney by making goodrdquo SloanManagement Review pp 65ndash721995

[18] T Taylor ldquoChannel coordination under price protectionmidlife returns and end-of-life returns in dynamic marketsrdquoManagement Science vol 47 no 9 pp 1220ndash1234 2001

[19] Z Yao S Leung and K K Lai ldquoAnalysis of the impact of price-sensitivity factors on the returns policy in coordinating supplychainrdquo European Journal of Operational Research vol 187 no 1pp 275ndash282 2008

[20] Y L Wang and P Zipkin ldquoAgentsrsquo incentives under buy-backcontracts in a two-stage supply chainrdquo International Journal ofProduction Economics vol 120 no 2 pp 525ndash539 2009

[21] J Hou A Z Zeng and L Zhao ldquoCoordination with a backupsupplier through buy-back contract under supply disruptionrdquo

Transportation Research E Logistics and Transportation Reviewvol 46 no 6 pp 881ndash895 2010

[22] H C Xiong B T Chen and J X Xie ldquoA composite contractbased on buy back and quantity flexibility contractsrdquo EuropeanJournal of Operational Research vol 210 no 3 pp 559ndash567 2011

[23] J Dana Jr and K E Spier ldquoRevenue sharing and vertical controlin the video rental industryrdquo Journal of Industrial Economicsvol 49 no 3 pp 223ndash245 2001

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

[25] Z Yao S C H Leung and K K Lai ldquoManufacturerrsquos revenue-sharing contract and retail competitionrdquo European Journal ofOperational Research vol 186 no 2 pp 637ndash651 2008

[26] 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

[27] 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

[28] J M Chen H L Cheng and M C Chien ldquoOn channelcoordination through revenue-sharing contracts with priceand shelf-space dependent demandrdquo Applied MathematicalModelling vol 35 no 10 pp 4886ndash4901 2011

[29] D P D Omkar ldquoSupply chain coordination using revenue-dependent revenue sharing contractsrdquoOmega vol 41 no 4 pp780ndash796 2013

[30] G Kannan andN PMaria ldquoReverse supply chain coordinationby revenue sharing contract a case for the personal computersindustryrdquo European Journal of Operational Research vol 233no 2 pp 326ndash336 2014

[31] D J Wu and P R Kleindorfer ldquoCompetitive options supplycontracting and electronic marketsrdquoManagement Science vol51 no 3 pp 452ndash468 2005

[32] XWang and L Liu ldquoCoordination in a retailer-led supply chainthrough option contractrdquo International Journal of ProductionEconomics vol 110 no 1-2 pp 115ndash127 2007

[33] Y X Zhao S Y Wang T C E Cheng X Q Yang and ZM Huang ldquoCoordination of supply chains by option contractsa cooperative game theory approachrdquo European Journal ofOperational Research vol 207 no 2 pp 668ndash675 2010

[34] L Liang X HWang and J G Gao ldquoAn option contract pricingmodel of relief material supply chainrdquoOmega vol 40 no 5 pp594ndash600 2012

[35] Y X Zhao L J Ma G Xie and T C E Cheng ldquoCoordinationof supply chains with bidirectional option contractsrdquo EuropeanJournal of Operational Research vol 229 no 2 pp 375ndash3812013

[36] M M Leng and M Parlar ldquoGame-theoretic analyses of decen-tralized assembly supply chains Non-cooperative equilibria vscoordination with cost-sharing contractsrdquo European Journal ofOperational Research vol 204 no 1 pp 96ndash104 2010

[37] M Kunter ldquoCoordination via cost and revenue sharing inmanufacturer-retailer channelsrdquo European Journal of Opera-tional Research vol 216 no 2 pp 477ndash486 2012

[38] Y C Tsao and G J Sheen ldquoEffects of promotion cost sharingpolicy with the sales learning curve on supply chain coordi-nationrdquo Computers and Operations Research vol 39 no 8 pp1872ndash1878 2012

10 Journal of Applied Mathematics

[39] Y H Wei and Q Y Hu ldquoOrdering pricing and allocationin a service supply chainrdquo International Journal of ProductionEconomics vol 144 no 2 pp 590ndash598 2013

[40] W H Liu X C Xu and A Kouhpaenejad ldquoDeterministicapproach to the fairest revenue-sharing coefficient in logisticsservice supply chain under the stochastic demand conditionrdquoComputers amp Industrial Engineering vol 66 no 1 pp 41ndash522013

[41] A P Cui and W Liu ldquoStudy on capability coordination inlogistics service supply chain with options contractrdquo ChineseJournal of Management Science vol 17 no 2 pp 59ndash65 2009(Chinese)

[42] Q H Lu ldquoCoordinate with cost sharing strategy for servicesupply chainrdquo Control and Decision vol 26 no 11 pp 1649ndash1653 2011 (Chinese)

[43] A Prasad and S P Sethi ldquoCompetitive advertising underuncertainty a stochastic differential game approachrdquo Journal ofOptimization Theory and Applications vol 123 no 1 pp 163ndash185 2004

[44] S Joslashrgensen S Taboubi and G Zaccour ldquoRetail promotionswith negative brand image effects is cooperation possiblerdquoEuropean Journal of Operational Research vol 150 no 2 pp395ndash405 2003

[45] J Navas and J Marın-Solano ldquoInteractions between govern-ment and firms a differential game approachrdquo Annals ofOperations Research vol 158 no 1 pp 47ndash61 2008

[46] S Joslashrgensen S Taboubi and G Zaccour ldquoCooperative adver-tising in a marketing channelrdquo Journal of Optimization Theoryand Applications vol 110 no 1 pp 145ndash158 2001

[47] C J Fombrun and V Rindova ldquoWhorsquos tops and who decidesThe social construction of corporate reputationsrdquo WorkingPaper Stern School of Business New York University 1996

[48] N Nha and L Gaston ldquoCorporate image and corporate repu-tation in customers Retention decisions in servicesrdquo Journal ofRetailing and Consumer Services no 8 pp 227ndash2361 2001

[49] M Nerlove and K J Arrow ldquoOptimal advertising policy underdynamic conditionsrdquo Economica vol 39 pp 129ndash142 1962

[50] K Saxton ldquoWhere do reputations come fromrdquo CorporateReputation Review vol 1 no 4 pp 393ndash399 1998

[51] M G Nagler ldquoAn exploratory analysis of the determinants ofcooperative advertising participation ratesrdquo Marketing Lettersvol 17 no 2 pp 91ndash102 2006

[52] E Dockner N Jorgensen L Van et al Differential Gamesin Economics and Management Science Cambridge UniversityPress Cambridge UK 2000

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

6 Journal of Applied Mathematics

Substituting the derivative of 119881lowast119894(119877) from (34) into (27)

the optimal efforts of LSI can be represented as

119864lowast

119894=

120582119894(2119891lowast

1119877 + 119891

lowast

2)

120583119894

(36)

Substituting the derivatives of119881lowast119894(119877) and119881

lowast

119901(119877) from (34)

and (35) into (28) the optimal sharing rate can be representedas

119863lowast=

(4119891lowast

1minus 2119892lowast

1) 119877 + (2119891

lowast

2minus 119892lowast

2)

(4119891lowast

1+ 2119892lowast

1) 119877 + (2119891

lowast

2+ 119892lowast

2) (37)

Also we can obtain the optimal efforts of FLSP bysubstituting119881

lowast

119901(119877)rsquos derivative and119863

lowast from (35) and (37) into(25)

119864lowast

119901=

120582119901[(4119891lowast

1+ 2119892lowast

1) 119877 + (2119891

lowast

2+ 119892lowast

2)]

2120583119901

(38)

Substitute (36) and (38) into differential game equation(2) to obtain

1198771015840= 119896119877 + 119889 (39)

in which 119896 = 21205822

119894119891lowast

1120583119894+1205822

119901(2119891lowast

1+119892lowast

1)120583119901minus120575 119889 = 120582

2

119894119891lowast

2120583119894+

1205822

119901(2119891lowast

2+ 119892lowast

2)2120583119901

Same as the proof ofTheorem 2 the general solutions canrepresented as

119877 = 119888119890119896119905

minus119889

119896 (40)

in which 119888 is a constant Given 119877|119905=0

= 1198770ge 0 the particular

solution is

119877lowast= (1198770+

119889

119896) 119890119896119905

minus119889

119896 (41)

5 Numerical Analysis

Consider a case with the following set of parameters

120583119894= 12 120583

119901= 1 120582

119894= 06

120582119901= 05 120575 = 001 120579 = 05

120574 = 005 120587119894= 06 120587

119901= 04

120588 = 09 1198770= 05

(42)

Then we can get the results of (5) and (22)

(1199011 119901

2 119901

3 119902

1 119902

2 119902

3)

= (minus0031269 0315981 0034968

minus 0020664 0208800 0028047)

(119891lowast

1 119891lowast

2 119891lowast

3 119892lowast

1 119892lowast

2 119892lowast

3)

= (minus0031046 0313701 0040789

minus 0020847 0210668 0023275)

119903 =21205822

1198941199011

120583119894

+

21205822

1199011199021

120583119901

minus 120575 = minus0039093

119904 =1205822

1198941199012

120583119894

+

1205822

1199011199022

120583119901

= 0146994

119896 =21205822

119894119891lowast

1

120583119894

+

1205822

119901(2119891lowast

1+ 119892lowast

1)

120583119901

minus 120575 = minus0049362

119889 =1205822

119894119891lowast

2

120583119894

+

1205822

119901(2119891lowast

2+ 119892lowast

2)

2120583119901

= 0198869

(43)

As a result we can obtain the optimal solutions of thedifferential game

(1) Nash Equilibrium without Cost Sharing Contract Theoptimal efforts of LSI and FLSP are

119864119894=

120582119894(2119901

1119877+ 119901

2)

120583119894

= 01020119890minus004119905

+ 00404

119864119901=

120582119901(2119902

1119877+ 119902

2)

120583119901

= 00673119890minus004119905

+ 00267

(44)

Under the circumstances the accumulation of the reputationof LSSC is

119877(119905) = (119877

0+

119904

119903) 119890119903119905

minus119904

119903= minus326119890

minus004119905+ 376 (45)

And the profits of both parties and the whole supply chain are

119881119894(119877) = 119901

11198772+ 119901

2119877 + 119901

3

= minus03323119890minus008119905

minus 02635119890minus004119905

+ 07810

119881119901(119877) = 119902

11198772+ 119902

2119877 + 119902

3

= minus02196119890minus008119905

minus 01741119890minus004119905

+ 05210

119881LSSC = 119881

119894+ 119881

119901

= minus05519119890minus008119905

minus 04376119890minus004119905

+ 13020

(46)

Journal of Applied Mathematics 7

0 50 100 150 200 250

02

015

01

005

0

t

E(t)

Ei(t)

Ep(t)

(a)

0 50 100 150 200 250

02

015

01

005

0

t

E(t)

Elowast

i(t)

Elowast

p(t)

(b)

0 50 100 150 200 250

02

015

01

005

0

t

E(t)

Elowast

i(t)

Ei(t)

(c)

0 50 100 150 200 250

02

015

01

005

0

t

E(t)

Elowast

p(t)

Ep(t)

(d)

Figure 1 Difference of optimal efforts between Nash equilibrium and Stackelberg equilibrium

(2) Stackelberg Equilibrium with Cost Sharing Contract Theoptimal efforts of LSI and FLSP are

119864lowast

119894=

120582119894(2119891lowast

1119877lowast+ 119891lowast

2)

120583119894

= 01096119890minus005119905

+ 00318

119864lowast

119901=

120582119901[(4119891lowast

1+ 2119892lowast

1) 119877lowast+ (2119891

lowast

2+ 119892lowast

2)]

2120583119901

= 01463119890minus005119905

+ 00424

(47)

The optimal sharing rate of LSI is

119863lowast=

(4119891lowast

1minus 2119892lowast

1) 119877lowast+ (2119891

lowast

2minus 119892lowast

2)

(4119891lowast

1+ 2119892lowast

1) 119877lowast + (2119891

lowast

2+ 119892lowast

2)

=02911119890

minus005119905+ 00844

05853119890minus005119905 + 01698

(48)

Under the circumstances the accumulation of the reputationof LSSC is

119877lowast(119905) = (119877

0+

119889

119896) 119890119896119905

minus119889

119896= minus353119890

minus005119905+ 403 (49)

And the profits of both parties and the whole supply chain are

119881lowast

119894(119877) = 119891

lowast

11198772+ 119891lowast

2119877 + 119891

lowast

3

= minus03866119890minus010119905

minus 02242119890minus005119905

+ 08007

119881lowast

119901(119877) = 119892

lowast

11198772+ 119892lowast

2119877 + 119892lowast

3

= minus02596119890minus010119905

minus 01507119890minus005119905

+ 05336

119881lowast

LSSC = 119881lowast

119894+ 119881lowast

119901

= minus06462119890minus010119905

minus 03749119890minus005119905

+ 13343

(50)

In the long-term relationship among members in LSSCLSI and FLSP are faced with dynamic games which meansthat the optimal strategies of LSI and FLSP will change withtime As a result both players will adjust their decisions tomaximize their profits in an infinite-time horizon Howeverthe optimal decisionswill become stable at last suggesting theautonomy of the LSSC

Figure 1 illustrates the difference between efforts of bothparties obtained from Nash equilibrium and Stackelbergequilibrium The efforts will decline with time until theyreach an appropriate levelThis is because themarginal utilityof reputation for revenue is diminishing while the cost to

8 Journal of Applied Mathematics

0 50 100 150 200 250

04973

04972

04972

04971

04971

0497

t

D(t)

Figure 2 The optimal cost sharing rate of the LSI

45

4

35

3

25

2

15

1

05

R(t)

0 50 100 150 200 250

t

R(t)

Rlowast(t)

Figure 3 Change of the reputation after adopting cost sharingcontract

maintain reputation remains the same By comparing theresults from the two equilibria we can see that efforts ofFLSP enhance remarkably under the Stackelberg equilibriumwhich means that the cost sharing contract can help LSI toencourage FLSP to endeavor in service

The optimal sharing rate of the LSI can be observed inFigure 2 declining until it reaches an appropriate level whichis about 05 Thus LSI should share about half of the cost ofFLSP to realize the coordination of the supply chain

Figure 3 illustrates the whole process of the buildingreputation The reputation of LSSC will increase because ofaccumulation But the rate of growth will decrease with thedecline of the efforts of both parties resulting in a stablelevel at last The stable state of reputation gets higher in theStackelberg equilibriumwith the cost sharing contract whichsuggests that the cost sharing contract contributes to thegrowth of reputation Same as reputation the profits of LSSCas well as members can be enhanced by the use of the costsharing contract as seen in Figure 4

To sum up the long-term relationship among the mem-bers of the supply chain contributes to the stability of theservice supply chain In the long-term relationship betweenLSI and FLSP the cost sharing contract is helpful in increasingthe reputation and profits of the supply chain by encouraginga higher level of the efforts by FLSP which implies that costsharing contract is an effective coordination mechanism

14

12

1

08

06

04

02

0

V(t)

Vlowast

i(t)

Vlowast

p(t)

0 50 100 150 200 250

t

Vp(t)

Vi(t)

VLSSC (t) V

lowast

LSSC (t)

Figure 4 Change of profits of LSSC and themembers after adoptingcost sharing contract

6 Conclusions

Given the long-term relationship amongmembers in a supplychain we take time and reputation into consideration asthe main factors in a dynamic differential model in LSSCconsisting of one LSI and one FLSP The results indicate theautonomy of the logistics service system and the effectivenessof cost sharing contract in the coordination of the supplychain

However we simplify the parameters the expression formof whichmay be various in reality resulting in different formsof solutions in the model to obtain the optimal decisions ofboth parties which is an interesting subject for future workMeanwhile further studies in the coordination of the supplychain that consists of multi-integrators and multiprovidersand the empirical analysis of the mathematical models willplay an important role in the practice of supply chainmanagement

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 NSFC Major Program(Grant no 7109040471090400) and NSFC General Program(Grant nos 71272045 and 71101107)

References

[1] K L Choy C L Li S C K So H Lau S K Kwok and DW KLeung ldquoManaging uncertainty in logistics service supply chainrdquoInternational Journal of Risk Assessment and Management vol7 no 1 pp 19ndash25 2007

Journal of Applied Mathematics 9

[2] X H Li and Q N Wang ldquoCoordination mechanisms of supplychain systemsrdquo European Journal of Operational Research vol179 no 1 pp 1ndash16 2007

[3] G P Cachon ldquoSupply chain coordination with contractsrdquo inSupply Chain Management Design Coordination and Opera-tion A G deKok and S C Graves Eds pp 229ndash340 ElsevierAmsterdam The Netherlands 2003

[4] B A Pasternack ldquoOptimal pricing and return policies forperishable commoditiesrdquo Marketing Science vol 27 no 1 pp166ndash176 2008

[5] R Anupindi and Y Bassok ldquoCentralization of stocks retailersvs manufacturerrdquo Management Science vol 45 no 2 pp 178ndash191 1999

[6] F Chen A Federgruen and Y-S Zheng ldquoCoordination mech-anisms for a distribution system with one supplier and multipleretailersrdquoManagement Science vol 47 no 5 pp 693ndash708 2001

[7] L Dong andN Rudi Supply Chain Interaction under Transship-ments Washington University 2001

[8] M A Lariviere and E L Porteus ldquoSelling to the newsvendoran analysis of price-only contractsrdquoManufacturing and ServiceOperations Management vol 3 no 4 pp 293ndash305 2001

[9] D Ni K W Li and X Tang ldquoSocial responsibility allocationin two-echelon supply chains insights from wholesale pricecontractsrdquo European Journal of Operational Research vol 207no 3 pp 1269ndash1279 2010

[10] X H Wang X Y Wang and Y S Su ldquoWholesale-pricecontract of supply chain with information gatheringrdquo AppliedMathematical Modelling vol 37 no 6 pp 3848ndash3860 2013

[11] Y Bassok and R Anupindi ldquoAnalysis of contracts with totalminimum commitmentrdquo IIE Transactions vol 29 no 5 pp373ndash381 1997

[12] A A Tsay and W S Lovejoy ldquoQuantity flexibility contractsand supply chain performancerdquo Manufacturing and ServiceOperations Management vol 1 no 2 pp 118ndash149 1999

[13] E L Plambeck and T A Taylor ldquoSell the plant The impact ofcontract manufacturing on innovation capacity and profitabil-ityrdquoManagement Science vol 51 no 1 pp 133ndash150 2005

[14] O Yazlali and F Erhun ldquoRelating the multiple supply problemto quantity flexibility contractsrdquo Operations Research Lettersvol 35 no 6 pp 767ndash772 2007

[15] Z T Lian and A Deshmukh ldquoAnalysis of supply contracts withquantity flexibilityrdquo European Journal of Operational Researchvol 196 no 2 pp 526ndash533 2009

[16] S Karakaya and I S Bakal ldquoJoint quantity flexibility formultiple products in a decentralized supply chainrdquo Computersamp Industrial Engineering vol 64 no 2 pp 696ndash707 2013

[17] V Padmanabhan and I P L Png ldquoReturns policies makemoney by making goodrdquo SloanManagement Review pp 65ndash721995

[18] T Taylor ldquoChannel coordination under price protectionmidlife returns and end-of-life returns in dynamic marketsrdquoManagement Science vol 47 no 9 pp 1220ndash1234 2001

[19] Z Yao S Leung and K K Lai ldquoAnalysis of the impact of price-sensitivity factors on the returns policy in coordinating supplychainrdquo European Journal of Operational Research vol 187 no 1pp 275ndash282 2008

[20] Y L Wang and P Zipkin ldquoAgentsrsquo incentives under buy-backcontracts in a two-stage supply chainrdquo International Journal ofProduction Economics vol 120 no 2 pp 525ndash539 2009

[21] J Hou A Z Zeng and L Zhao ldquoCoordination with a backupsupplier through buy-back contract under supply disruptionrdquo

Transportation Research E Logistics and Transportation Reviewvol 46 no 6 pp 881ndash895 2010

[22] H C Xiong B T Chen and J X Xie ldquoA composite contractbased on buy back and quantity flexibility contractsrdquo EuropeanJournal of Operational Research vol 210 no 3 pp 559ndash567 2011

[23] J Dana Jr and K E Spier ldquoRevenue sharing and vertical controlin the video rental industryrdquo Journal of Industrial Economicsvol 49 no 3 pp 223ndash245 2001

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

[25] Z Yao S C H Leung and K K Lai ldquoManufacturerrsquos revenue-sharing contract and retail competitionrdquo European Journal ofOperational Research vol 186 no 2 pp 637ndash651 2008

[26] 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

[27] 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

[28] J M Chen H L Cheng and M C Chien ldquoOn channelcoordination through revenue-sharing contracts with priceand shelf-space dependent demandrdquo Applied MathematicalModelling vol 35 no 10 pp 4886ndash4901 2011

[29] D P D Omkar ldquoSupply chain coordination using revenue-dependent revenue sharing contractsrdquoOmega vol 41 no 4 pp780ndash796 2013

[30] G Kannan andN PMaria ldquoReverse supply chain coordinationby revenue sharing contract a case for the personal computersindustryrdquo European Journal of Operational Research vol 233no 2 pp 326ndash336 2014

[31] D J Wu and P R Kleindorfer ldquoCompetitive options supplycontracting and electronic marketsrdquoManagement Science vol51 no 3 pp 452ndash468 2005

[32] XWang and L Liu ldquoCoordination in a retailer-led supply chainthrough option contractrdquo International Journal of ProductionEconomics vol 110 no 1-2 pp 115ndash127 2007

[33] Y X Zhao S Y Wang T C E Cheng X Q Yang and ZM Huang ldquoCoordination of supply chains by option contractsa cooperative game theory approachrdquo European Journal ofOperational Research vol 207 no 2 pp 668ndash675 2010

[34] L Liang X HWang and J G Gao ldquoAn option contract pricingmodel of relief material supply chainrdquoOmega vol 40 no 5 pp594ndash600 2012

[35] Y X Zhao L J Ma G Xie and T C E Cheng ldquoCoordinationof supply chains with bidirectional option contractsrdquo EuropeanJournal of Operational Research vol 229 no 2 pp 375ndash3812013

[36] M M Leng and M Parlar ldquoGame-theoretic analyses of decen-tralized assembly supply chains Non-cooperative equilibria vscoordination with cost-sharing contractsrdquo European Journal ofOperational Research vol 204 no 1 pp 96ndash104 2010

[37] M Kunter ldquoCoordination via cost and revenue sharing inmanufacturer-retailer channelsrdquo European Journal of Opera-tional Research vol 216 no 2 pp 477ndash486 2012

[38] Y C Tsao and G J Sheen ldquoEffects of promotion cost sharingpolicy with the sales learning curve on supply chain coordi-nationrdquo Computers and Operations Research vol 39 no 8 pp1872ndash1878 2012

10 Journal of Applied Mathematics

[39] Y H Wei and Q Y Hu ldquoOrdering pricing and allocationin a service supply chainrdquo International Journal of ProductionEconomics vol 144 no 2 pp 590ndash598 2013

[40] W H Liu X C Xu and A Kouhpaenejad ldquoDeterministicapproach to the fairest revenue-sharing coefficient in logisticsservice supply chain under the stochastic demand conditionrdquoComputers amp Industrial Engineering vol 66 no 1 pp 41ndash522013

[41] A P Cui and W Liu ldquoStudy on capability coordination inlogistics service supply chain with options contractrdquo ChineseJournal of Management Science vol 17 no 2 pp 59ndash65 2009(Chinese)

[42] Q H Lu ldquoCoordinate with cost sharing strategy for servicesupply chainrdquo Control and Decision vol 26 no 11 pp 1649ndash1653 2011 (Chinese)

[43] A Prasad and S P Sethi ldquoCompetitive advertising underuncertainty a stochastic differential game approachrdquo Journal ofOptimization Theory and Applications vol 123 no 1 pp 163ndash185 2004

[44] S Joslashrgensen S Taboubi and G Zaccour ldquoRetail promotionswith negative brand image effects is cooperation possiblerdquoEuropean Journal of Operational Research vol 150 no 2 pp395ndash405 2003

[45] J Navas and J Marın-Solano ldquoInteractions between govern-ment and firms a differential game approachrdquo Annals ofOperations Research vol 158 no 1 pp 47ndash61 2008

[46] S Joslashrgensen S Taboubi and G Zaccour ldquoCooperative adver-tising in a marketing channelrdquo Journal of Optimization Theoryand Applications vol 110 no 1 pp 145ndash158 2001

[47] C J Fombrun and V Rindova ldquoWhorsquos tops and who decidesThe social construction of corporate reputationsrdquo WorkingPaper Stern School of Business New York University 1996

[48] N Nha and L Gaston ldquoCorporate image and corporate repu-tation in customers Retention decisions in servicesrdquo Journal ofRetailing and Consumer Services no 8 pp 227ndash2361 2001

[49] M Nerlove and K J Arrow ldquoOptimal advertising policy underdynamic conditionsrdquo Economica vol 39 pp 129ndash142 1962

[50] K Saxton ldquoWhere do reputations come fromrdquo CorporateReputation Review vol 1 no 4 pp 393ndash399 1998

[51] M G Nagler ldquoAn exploratory analysis of the determinants ofcooperative advertising participation ratesrdquo Marketing Lettersvol 17 no 2 pp 91ndash102 2006

[52] E Dockner N Jorgensen L Van et al Differential Gamesin Economics and Management Science Cambridge UniversityPress Cambridge UK 2000

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

Journal of Applied Mathematics 7

0 50 100 150 200 250

02

015

01

005

0

t

E(t)

Ei(t)

Ep(t)

(a)

0 50 100 150 200 250

02

015

01

005

0

t

E(t)

Elowast

i(t)

Elowast

p(t)

(b)

0 50 100 150 200 250

02

015

01

005

0

t

E(t)

Elowast

i(t)

Ei(t)

(c)

0 50 100 150 200 250

02

015

01

005

0

t

E(t)

Elowast

p(t)

Ep(t)

(d)

Figure 1 Difference of optimal efforts between Nash equilibrium and Stackelberg equilibrium

(2) Stackelberg Equilibrium with Cost Sharing Contract Theoptimal efforts of LSI and FLSP are

119864lowast

119894=

120582119894(2119891lowast

1119877lowast+ 119891lowast

2)

120583119894

= 01096119890minus005119905

+ 00318

119864lowast

119901=

120582119901[(4119891lowast

1+ 2119892lowast

1) 119877lowast+ (2119891

lowast

2+ 119892lowast

2)]

2120583119901

= 01463119890minus005119905

+ 00424

(47)

The optimal sharing rate of LSI is

119863lowast=

(4119891lowast

1minus 2119892lowast

1) 119877lowast+ (2119891

lowast

2minus 119892lowast

2)

(4119891lowast

1+ 2119892lowast

1) 119877lowast + (2119891

lowast

2+ 119892lowast

2)

=02911119890

minus005119905+ 00844

05853119890minus005119905 + 01698

(48)

Under the circumstances the accumulation of the reputationof LSSC is

119877lowast(119905) = (119877

0+

119889

119896) 119890119896119905

minus119889

119896= minus353119890

minus005119905+ 403 (49)

And the profits of both parties and the whole supply chain are

119881lowast

119894(119877) = 119891

lowast

11198772+ 119891lowast

2119877 + 119891

lowast

3

= minus03866119890minus010119905

minus 02242119890minus005119905

+ 08007

119881lowast

119901(119877) = 119892

lowast

11198772+ 119892lowast

2119877 + 119892lowast

3

= minus02596119890minus010119905

minus 01507119890minus005119905

+ 05336

119881lowast

LSSC = 119881lowast

119894+ 119881lowast

119901

= minus06462119890minus010119905

minus 03749119890minus005119905

+ 13343

(50)

In the long-term relationship among members in LSSCLSI and FLSP are faced with dynamic games which meansthat the optimal strategies of LSI and FLSP will change withtime As a result both players will adjust their decisions tomaximize their profits in an infinite-time horizon Howeverthe optimal decisionswill become stable at last suggesting theautonomy of the LSSC

Figure 1 illustrates the difference between efforts of bothparties obtained from Nash equilibrium and Stackelbergequilibrium The efforts will decline with time until theyreach an appropriate levelThis is because themarginal utilityof reputation for revenue is diminishing while the cost to

8 Journal of Applied Mathematics

0 50 100 150 200 250

04973

04972

04972

04971

04971

0497

t

D(t)

Figure 2 The optimal cost sharing rate of the LSI

45

4

35

3

25

2

15

1

05

R(t)

0 50 100 150 200 250

t

R(t)

Rlowast(t)

Figure 3 Change of the reputation after adopting cost sharingcontract

maintain reputation remains the same By comparing theresults from the two equilibria we can see that efforts ofFLSP enhance remarkably under the Stackelberg equilibriumwhich means that the cost sharing contract can help LSI toencourage FLSP to endeavor in service

The optimal sharing rate of the LSI can be observed inFigure 2 declining until it reaches an appropriate level whichis about 05 Thus LSI should share about half of the cost ofFLSP to realize the coordination of the supply chain

Figure 3 illustrates the whole process of the buildingreputation The reputation of LSSC will increase because ofaccumulation But the rate of growth will decrease with thedecline of the efforts of both parties resulting in a stablelevel at last The stable state of reputation gets higher in theStackelberg equilibriumwith the cost sharing contract whichsuggests that the cost sharing contract contributes to thegrowth of reputation Same as reputation the profits of LSSCas well as members can be enhanced by the use of the costsharing contract as seen in Figure 4

To sum up the long-term relationship among the mem-bers of the supply chain contributes to the stability of theservice supply chain In the long-term relationship betweenLSI and FLSP the cost sharing contract is helpful in increasingthe reputation and profits of the supply chain by encouraginga higher level of the efforts by FLSP which implies that costsharing contract is an effective coordination mechanism

14

12

1

08

06

04

02

0

V(t)

Vlowast

i(t)

Vlowast

p(t)

0 50 100 150 200 250

t

Vp(t)

Vi(t)

VLSSC (t) V

lowast

LSSC (t)

Figure 4 Change of profits of LSSC and themembers after adoptingcost sharing contract

6 Conclusions

Given the long-term relationship amongmembers in a supplychain we take time and reputation into consideration asthe main factors in a dynamic differential model in LSSCconsisting of one LSI and one FLSP The results indicate theautonomy of the logistics service system and the effectivenessof cost sharing contract in the coordination of the supplychain

However we simplify the parameters the expression formof whichmay be various in reality resulting in different formsof solutions in the model to obtain the optimal decisions ofboth parties which is an interesting subject for future workMeanwhile further studies in the coordination of the supplychain that consists of multi-integrators and multiprovidersand the empirical analysis of the mathematical models willplay an important role in the practice of supply chainmanagement

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 NSFC Major Program(Grant no 7109040471090400) and NSFC General Program(Grant nos 71272045 and 71101107)

References

[1] K L Choy C L Li S C K So H Lau S K Kwok and DW KLeung ldquoManaging uncertainty in logistics service supply chainrdquoInternational Journal of Risk Assessment and Management vol7 no 1 pp 19ndash25 2007

Journal of Applied Mathematics 9

[2] X H Li and Q N Wang ldquoCoordination mechanisms of supplychain systemsrdquo European Journal of Operational Research vol179 no 1 pp 1ndash16 2007

[3] G P Cachon ldquoSupply chain coordination with contractsrdquo inSupply Chain Management Design Coordination and Opera-tion A G deKok and S C Graves Eds pp 229ndash340 ElsevierAmsterdam The Netherlands 2003

[4] B A Pasternack ldquoOptimal pricing and return policies forperishable commoditiesrdquo Marketing Science vol 27 no 1 pp166ndash176 2008

[5] R Anupindi and Y Bassok ldquoCentralization of stocks retailersvs manufacturerrdquo Management Science vol 45 no 2 pp 178ndash191 1999

[6] F Chen A Federgruen and Y-S Zheng ldquoCoordination mech-anisms for a distribution system with one supplier and multipleretailersrdquoManagement Science vol 47 no 5 pp 693ndash708 2001

[7] L Dong andN Rudi Supply Chain Interaction under Transship-ments Washington University 2001

[8] M A Lariviere and E L Porteus ldquoSelling to the newsvendoran analysis of price-only contractsrdquoManufacturing and ServiceOperations Management vol 3 no 4 pp 293ndash305 2001

[9] D Ni K W Li and X Tang ldquoSocial responsibility allocationin two-echelon supply chains insights from wholesale pricecontractsrdquo European Journal of Operational Research vol 207no 3 pp 1269ndash1279 2010

[10] X H Wang X Y Wang and Y S Su ldquoWholesale-pricecontract of supply chain with information gatheringrdquo AppliedMathematical Modelling vol 37 no 6 pp 3848ndash3860 2013

[11] Y Bassok and R Anupindi ldquoAnalysis of contracts with totalminimum commitmentrdquo IIE Transactions vol 29 no 5 pp373ndash381 1997

[12] A A Tsay and W S Lovejoy ldquoQuantity flexibility contractsand supply chain performancerdquo Manufacturing and ServiceOperations Management vol 1 no 2 pp 118ndash149 1999

[13] E L Plambeck and T A Taylor ldquoSell the plant The impact ofcontract manufacturing on innovation capacity and profitabil-ityrdquoManagement Science vol 51 no 1 pp 133ndash150 2005

[14] O Yazlali and F Erhun ldquoRelating the multiple supply problemto quantity flexibility contractsrdquo Operations Research Lettersvol 35 no 6 pp 767ndash772 2007

[15] Z T Lian and A Deshmukh ldquoAnalysis of supply contracts withquantity flexibilityrdquo European Journal of Operational Researchvol 196 no 2 pp 526ndash533 2009

[16] S Karakaya and I S Bakal ldquoJoint quantity flexibility formultiple products in a decentralized supply chainrdquo Computersamp Industrial Engineering vol 64 no 2 pp 696ndash707 2013

[17] V Padmanabhan and I P L Png ldquoReturns policies makemoney by making goodrdquo SloanManagement Review pp 65ndash721995

[18] T Taylor ldquoChannel coordination under price protectionmidlife returns and end-of-life returns in dynamic marketsrdquoManagement Science vol 47 no 9 pp 1220ndash1234 2001

[19] Z Yao S Leung and K K Lai ldquoAnalysis of the impact of price-sensitivity factors on the returns policy in coordinating supplychainrdquo European Journal of Operational Research vol 187 no 1pp 275ndash282 2008

[20] Y L Wang and P Zipkin ldquoAgentsrsquo incentives under buy-backcontracts in a two-stage supply chainrdquo International Journal ofProduction Economics vol 120 no 2 pp 525ndash539 2009

[21] J Hou A Z Zeng and L Zhao ldquoCoordination with a backupsupplier through buy-back contract under supply disruptionrdquo

Transportation Research E Logistics and Transportation Reviewvol 46 no 6 pp 881ndash895 2010

[22] H C Xiong B T Chen and J X Xie ldquoA composite contractbased on buy back and quantity flexibility contractsrdquo EuropeanJournal of Operational Research vol 210 no 3 pp 559ndash567 2011

[23] J Dana Jr and K E Spier ldquoRevenue sharing and vertical controlin the video rental industryrdquo Journal of Industrial Economicsvol 49 no 3 pp 223ndash245 2001

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

[25] Z Yao S C H Leung and K K Lai ldquoManufacturerrsquos revenue-sharing contract and retail competitionrdquo European Journal ofOperational Research vol 186 no 2 pp 637ndash651 2008

[26] 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

[27] 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

[28] J M Chen H L Cheng and M C Chien ldquoOn channelcoordination through revenue-sharing contracts with priceand shelf-space dependent demandrdquo Applied MathematicalModelling vol 35 no 10 pp 4886ndash4901 2011

[29] D P D Omkar ldquoSupply chain coordination using revenue-dependent revenue sharing contractsrdquoOmega vol 41 no 4 pp780ndash796 2013

[30] G Kannan andN PMaria ldquoReverse supply chain coordinationby revenue sharing contract a case for the personal computersindustryrdquo European Journal of Operational Research vol 233no 2 pp 326ndash336 2014

[31] D J Wu and P R Kleindorfer ldquoCompetitive options supplycontracting and electronic marketsrdquoManagement Science vol51 no 3 pp 452ndash468 2005

[32] XWang and L Liu ldquoCoordination in a retailer-led supply chainthrough option contractrdquo International Journal of ProductionEconomics vol 110 no 1-2 pp 115ndash127 2007

[33] Y X Zhao S Y Wang T C E Cheng X Q Yang and ZM Huang ldquoCoordination of supply chains by option contractsa cooperative game theory approachrdquo European Journal ofOperational Research vol 207 no 2 pp 668ndash675 2010

[34] L Liang X HWang and J G Gao ldquoAn option contract pricingmodel of relief material supply chainrdquoOmega vol 40 no 5 pp594ndash600 2012

[35] Y X Zhao L J Ma G Xie and T C E Cheng ldquoCoordinationof supply chains with bidirectional option contractsrdquo EuropeanJournal of Operational Research vol 229 no 2 pp 375ndash3812013

[36] M M Leng and M Parlar ldquoGame-theoretic analyses of decen-tralized assembly supply chains Non-cooperative equilibria vscoordination with cost-sharing contractsrdquo European Journal ofOperational Research vol 204 no 1 pp 96ndash104 2010

[37] M Kunter ldquoCoordination via cost and revenue sharing inmanufacturer-retailer channelsrdquo European Journal of Opera-tional Research vol 216 no 2 pp 477ndash486 2012

[38] Y C Tsao and G J Sheen ldquoEffects of promotion cost sharingpolicy with the sales learning curve on supply chain coordi-nationrdquo Computers and Operations Research vol 39 no 8 pp1872ndash1878 2012

10 Journal of Applied Mathematics

[39] Y H Wei and Q Y Hu ldquoOrdering pricing and allocationin a service supply chainrdquo International Journal of ProductionEconomics vol 144 no 2 pp 590ndash598 2013

[40] W H Liu X C Xu and A Kouhpaenejad ldquoDeterministicapproach to the fairest revenue-sharing coefficient in logisticsservice supply chain under the stochastic demand conditionrdquoComputers amp Industrial Engineering vol 66 no 1 pp 41ndash522013

[41] A P Cui and W Liu ldquoStudy on capability coordination inlogistics service supply chain with options contractrdquo ChineseJournal of Management Science vol 17 no 2 pp 59ndash65 2009(Chinese)

[42] Q H Lu ldquoCoordinate with cost sharing strategy for servicesupply chainrdquo Control and Decision vol 26 no 11 pp 1649ndash1653 2011 (Chinese)

[43] A Prasad and S P Sethi ldquoCompetitive advertising underuncertainty a stochastic differential game approachrdquo Journal ofOptimization Theory and Applications vol 123 no 1 pp 163ndash185 2004

[44] S Joslashrgensen S Taboubi and G Zaccour ldquoRetail promotionswith negative brand image effects is cooperation possiblerdquoEuropean Journal of Operational Research vol 150 no 2 pp395ndash405 2003

[45] J Navas and J Marın-Solano ldquoInteractions between govern-ment and firms a differential game approachrdquo Annals ofOperations Research vol 158 no 1 pp 47ndash61 2008

[46] S Joslashrgensen S Taboubi and G Zaccour ldquoCooperative adver-tising in a marketing channelrdquo Journal of Optimization Theoryand Applications vol 110 no 1 pp 145ndash158 2001

[47] C J Fombrun and V Rindova ldquoWhorsquos tops and who decidesThe social construction of corporate reputationsrdquo WorkingPaper Stern School of Business New York University 1996

[48] N Nha and L Gaston ldquoCorporate image and corporate repu-tation in customers Retention decisions in servicesrdquo Journal ofRetailing and Consumer Services no 8 pp 227ndash2361 2001

[49] M Nerlove and K J Arrow ldquoOptimal advertising policy underdynamic conditionsrdquo Economica vol 39 pp 129ndash142 1962

[50] K Saxton ldquoWhere do reputations come fromrdquo CorporateReputation Review vol 1 no 4 pp 393ndash399 1998

[51] M G Nagler ldquoAn exploratory analysis of the determinants ofcooperative advertising participation ratesrdquo Marketing Lettersvol 17 no 2 pp 91ndash102 2006

[52] E Dockner N Jorgensen L Van et al Differential Gamesin Economics and Management Science Cambridge UniversityPress Cambridge UK 2000

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 Journal of Applied Mathematics

0 50 100 150 200 250

04973

04972

04972

04971

04971

0497

t

D(t)

Figure 2 The optimal cost sharing rate of the LSI

45

4

35

3

25

2

15

1

05

R(t)

0 50 100 150 200 250

t

R(t)

Rlowast(t)

Figure 3 Change of the reputation after adopting cost sharingcontract

maintain reputation remains the same By comparing theresults from the two equilibria we can see that efforts ofFLSP enhance remarkably under the Stackelberg equilibriumwhich means that the cost sharing contract can help LSI toencourage FLSP to endeavor in service

The optimal sharing rate of the LSI can be observed inFigure 2 declining until it reaches an appropriate level whichis about 05 Thus LSI should share about half of the cost ofFLSP to realize the coordination of the supply chain

Figure 3 illustrates the whole process of the buildingreputation The reputation of LSSC will increase because ofaccumulation But the rate of growth will decrease with thedecline of the efforts of both parties resulting in a stablelevel at last The stable state of reputation gets higher in theStackelberg equilibriumwith the cost sharing contract whichsuggests that the cost sharing contract contributes to thegrowth of reputation Same as reputation the profits of LSSCas well as members can be enhanced by the use of the costsharing contract as seen in Figure 4

To sum up the long-term relationship among the mem-bers of the supply chain contributes to the stability of theservice supply chain In the long-term relationship betweenLSI and FLSP the cost sharing contract is helpful in increasingthe reputation and profits of the supply chain by encouraginga higher level of the efforts by FLSP which implies that costsharing contract is an effective coordination mechanism

14

12

1

08

06

04

02

0

V(t)

Vlowast

i(t)

Vlowast

p(t)

0 50 100 150 200 250

t

Vp(t)

Vi(t)

VLSSC (t) V

lowast

LSSC (t)

Figure 4 Change of profits of LSSC and themembers after adoptingcost sharing contract

6 Conclusions

Given the long-term relationship amongmembers in a supplychain we take time and reputation into consideration asthe main factors in a dynamic differential model in LSSCconsisting of one LSI and one FLSP The results indicate theautonomy of the logistics service system and the effectivenessof cost sharing contract in the coordination of the supplychain

However we simplify the parameters the expression formof whichmay be various in reality resulting in different formsof solutions in the model to obtain the optimal decisions ofboth parties which is an interesting subject for future workMeanwhile further studies in the coordination of the supplychain that consists of multi-integrators and multiprovidersand the empirical analysis of the mathematical models willplay an important role in the practice of supply chainmanagement

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 NSFC Major Program(Grant no 7109040471090400) and NSFC General Program(Grant nos 71272045 and 71101107)

References

[1] K L Choy C L Li S C K So H Lau S K Kwok and DW KLeung ldquoManaging uncertainty in logistics service supply chainrdquoInternational Journal of Risk Assessment and Management vol7 no 1 pp 19ndash25 2007

Journal of Applied Mathematics 9

[2] X H Li and Q N Wang ldquoCoordination mechanisms of supplychain systemsrdquo European Journal of Operational Research vol179 no 1 pp 1ndash16 2007

[3] G P Cachon ldquoSupply chain coordination with contractsrdquo inSupply Chain Management Design Coordination and Opera-tion A G deKok and S C Graves Eds pp 229ndash340 ElsevierAmsterdam The Netherlands 2003

[4] B A Pasternack ldquoOptimal pricing and return policies forperishable commoditiesrdquo Marketing Science vol 27 no 1 pp166ndash176 2008

[5] R Anupindi and Y Bassok ldquoCentralization of stocks retailersvs manufacturerrdquo Management Science vol 45 no 2 pp 178ndash191 1999

[6] F Chen A Federgruen and Y-S Zheng ldquoCoordination mech-anisms for a distribution system with one supplier and multipleretailersrdquoManagement Science vol 47 no 5 pp 693ndash708 2001

[7] L Dong andN Rudi Supply Chain Interaction under Transship-ments Washington University 2001

[8] M A Lariviere and E L Porteus ldquoSelling to the newsvendoran analysis of price-only contractsrdquoManufacturing and ServiceOperations Management vol 3 no 4 pp 293ndash305 2001

[9] D Ni K W Li and X Tang ldquoSocial responsibility allocationin two-echelon supply chains insights from wholesale pricecontractsrdquo European Journal of Operational Research vol 207no 3 pp 1269ndash1279 2010

[10] X H Wang X Y Wang and Y S Su ldquoWholesale-pricecontract of supply chain with information gatheringrdquo AppliedMathematical Modelling vol 37 no 6 pp 3848ndash3860 2013

[11] Y Bassok and R Anupindi ldquoAnalysis of contracts with totalminimum commitmentrdquo IIE Transactions vol 29 no 5 pp373ndash381 1997

[12] A A Tsay and W S Lovejoy ldquoQuantity flexibility contractsand supply chain performancerdquo Manufacturing and ServiceOperations Management vol 1 no 2 pp 118ndash149 1999

[13] E L Plambeck and T A Taylor ldquoSell the plant The impact ofcontract manufacturing on innovation capacity and profitabil-ityrdquoManagement Science vol 51 no 1 pp 133ndash150 2005

[14] O Yazlali and F Erhun ldquoRelating the multiple supply problemto quantity flexibility contractsrdquo Operations Research Lettersvol 35 no 6 pp 767ndash772 2007

[15] Z T Lian and A Deshmukh ldquoAnalysis of supply contracts withquantity flexibilityrdquo European Journal of Operational Researchvol 196 no 2 pp 526ndash533 2009

[16] S Karakaya and I S Bakal ldquoJoint quantity flexibility formultiple products in a decentralized supply chainrdquo Computersamp Industrial Engineering vol 64 no 2 pp 696ndash707 2013

[17] V Padmanabhan and I P L Png ldquoReturns policies makemoney by making goodrdquo SloanManagement Review pp 65ndash721995

[18] T Taylor ldquoChannel coordination under price protectionmidlife returns and end-of-life returns in dynamic marketsrdquoManagement Science vol 47 no 9 pp 1220ndash1234 2001

[19] Z Yao S Leung and K K Lai ldquoAnalysis of the impact of price-sensitivity factors on the returns policy in coordinating supplychainrdquo European Journal of Operational Research vol 187 no 1pp 275ndash282 2008

[20] Y L Wang and P Zipkin ldquoAgentsrsquo incentives under buy-backcontracts in a two-stage supply chainrdquo International Journal ofProduction Economics vol 120 no 2 pp 525ndash539 2009

[21] J Hou A Z Zeng and L Zhao ldquoCoordination with a backupsupplier through buy-back contract under supply disruptionrdquo

Transportation Research E Logistics and Transportation Reviewvol 46 no 6 pp 881ndash895 2010

[22] H C Xiong B T Chen and J X Xie ldquoA composite contractbased on buy back and quantity flexibility contractsrdquo EuropeanJournal of Operational Research vol 210 no 3 pp 559ndash567 2011

[23] J Dana Jr and K E Spier ldquoRevenue sharing and vertical controlin the video rental industryrdquo Journal of Industrial Economicsvol 49 no 3 pp 223ndash245 2001

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

[25] Z Yao S C H Leung and K K Lai ldquoManufacturerrsquos revenue-sharing contract and retail competitionrdquo European Journal ofOperational Research vol 186 no 2 pp 637ndash651 2008

[26] 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

[27] 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

[28] J M Chen H L Cheng and M C Chien ldquoOn channelcoordination through revenue-sharing contracts with priceand shelf-space dependent demandrdquo Applied MathematicalModelling vol 35 no 10 pp 4886ndash4901 2011

[29] D P D Omkar ldquoSupply chain coordination using revenue-dependent revenue sharing contractsrdquoOmega vol 41 no 4 pp780ndash796 2013

[30] G Kannan andN PMaria ldquoReverse supply chain coordinationby revenue sharing contract a case for the personal computersindustryrdquo European Journal of Operational Research vol 233no 2 pp 326ndash336 2014

[31] D J Wu and P R Kleindorfer ldquoCompetitive options supplycontracting and electronic marketsrdquoManagement Science vol51 no 3 pp 452ndash468 2005

[32] XWang and L Liu ldquoCoordination in a retailer-led supply chainthrough option contractrdquo International Journal of ProductionEconomics vol 110 no 1-2 pp 115ndash127 2007

[33] Y X Zhao S Y Wang T C E Cheng X Q Yang and ZM Huang ldquoCoordination of supply chains by option contractsa cooperative game theory approachrdquo European Journal ofOperational Research vol 207 no 2 pp 668ndash675 2010

[34] L Liang X HWang and J G Gao ldquoAn option contract pricingmodel of relief material supply chainrdquoOmega vol 40 no 5 pp594ndash600 2012

[35] Y X Zhao L J Ma G Xie and T C E Cheng ldquoCoordinationof supply chains with bidirectional option contractsrdquo EuropeanJournal of Operational Research vol 229 no 2 pp 375ndash3812013

[36] M M Leng and M Parlar ldquoGame-theoretic analyses of decen-tralized assembly supply chains Non-cooperative equilibria vscoordination with cost-sharing contractsrdquo European Journal ofOperational Research vol 204 no 1 pp 96ndash104 2010

[37] M Kunter ldquoCoordination via cost and revenue sharing inmanufacturer-retailer channelsrdquo European Journal of Opera-tional Research vol 216 no 2 pp 477ndash486 2012

[38] Y C Tsao and G J Sheen ldquoEffects of promotion cost sharingpolicy with the sales learning curve on supply chain coordi-nationrdquo Computers and Operations Research vol 39 no 8 pp1872ndash1878 2012

10 Journal of Applied Mathematics

[39] Y H Wei and Q Y Hu ldquoOrdering pricing and allocationin a service supply chainrdquo International Journal of ProductionEconomics vol 144 no 2 pp 590ndash598 2013

[40] W H Liu X C Xu and A Kouhpaenejad ldquoDeterministicapproach to the fairest revenue-sharing coefficient in logisticsservice supply chain under the stochastic demand conditionrdquoComputers amp Industrial Engineering vol 66 no 1 pp 41ndash522013

[41] A P Cui and W Liu ldquoStudy on capability coordination inlogistics service supply chain with options contractrdquo ChineseJournal of Management Science vol 17 no 2 pp 59ndash65 2009(Chinese)

[42] Q H Lu ldquoCoordinate with cost sharing strategy for servicesupply chainrdquo Control and Decision vol 26 no 11 pp 1649ndash1653 2011 (Chinese)

[43] A Prasad and S P Sethi ldquoCompetitive advertising underuncertainty a stochastic differential game approachrdquo Journal ofOptimization Theory and Applications vol 123 no 1 pp 163ndash185 2004

[44] S Joslashrgensen S Taboubi and G Zaccour ldquoRetail promotionswith negative brand image effects is cooperation possiblerdquoEuropean Journal of Operational Research vol 150 no 2 pp395ndash405 2003

[45] J Navas and J Marın-Solano ldquoInteractions between govern-ment and firms a differential game approachrdquo Annals ofOperations Research vol 158 no 1 pp 47ndash61 2008

[46] S Joslashrgensen S Taboubi and G Zaccour ldquoCooperative adver-tising in a marketing channelrdquo Journal of Optimization Theoryand Applications vol 110 no 1 pp 145ndash158 2001

[47] C J Fombrun and V Rindova ldquoWhorsquos tops and who decidesThe social construction of corporate reputationsrdquo WorkingPaper Stern School of Business New York University 1996

[48] N Nha and L Gaston ldquoCorporate image and corporate repu-tation in customers Retention decisions in servicesrdquo Journal ofRetailing and Consumer Services no 8 pp 227ndash2361 2001

[49] M Nerlove and K J Arrow ldquoOptimal advertising policy underdynamic conditionsrdquo Economica vol 39 pp 129ndash142 1962

[50] K Saxton ldquoWhere do reputations come fromrdquo CorporateReputation Review vol 1 no 4 pp 393ndash399 1998

[51] M G Nagler ldquoAn exploratory analysis of the determinants ofcooperative advertising participation ratesrdquo Marketing Lettersvol 17 no 2 pp 91ndash102 2006

[52] E Dockner N Jorgensen L Van et al Differential Gamesin Economics and Management Science Cambridge UniversityPress Cambridge UK 2000

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

Journal of Applied Mathematics 9

[2] X H Li and Q N Wang ldquoCoordination mechanisms of supplychain systemsrdquo European Journal of Operational Research vol179 no 1 pp 1ndash16 2007

[3] G P Cachon ldquoSupply chain coordination with contractsrdquo inSupply Chain Management Design Coordination and Opera-tion A G deKok and S C Graves Eds pp 229ndash340 ElsevierAmsterdam The Netherlands 2003

[4] B A Pasternack ldquoOptimal pricing and return policies forperishable commoditiesrdquo Marketing Science vol 27 no 1 pp166ndash176 2008

[5] R Anupindi and Y Bassok ldquoCentralization of stocks retailersvs manufacturerrdquo Management Science vol 45 no 2 pp 178ndash191 1999

[6] F Chen A Federgruen and Y-S Zheng ldquoCoordination mech-anisms for a distribution system with one supplier and multipleretailersrdquoManagement Science vol 47 no 5 pp 693ndash708 2001

[7] L Dong andN Rudi Supply Chain Interaction under Transship-ments Washington University 2001

[8] M A Lariviere and E L Porteus ldquoSelling to the newsvendoran analysis of price-only contractsrdquoManufacturing and ServiceOperations Management vol 3 no 4 pp 293ndash305 2001

[9] D Ni K W Li and X Tang ldquoSocial responsibility allocationin two-echelon supply chains insights from wholesale pricecontractsrdquo European Journal of Operational Research vol 207no 3 pp 1269ndash1279 2010

[10] X H Wang X Y Wang and Y S Su ldquoWholesale-pricecontract of supply chain with information gatheringrdquo AppliedMathematical Modelling vol 37 no 6 pp 3848ndash3860 2013

[11] Y Bassok and R Anupindi ldquoAnalysis of contracts with totalminimum commitmentrdquo IIE Transactions vol 29 no 5 pp373ndash381 1997

[12] A A Tsay and W S Lovejoy ldquoQuantity flexibility contractsand supply chain performancerdquo Manufacturing and ServiceOperations Management vol 1 no 2 pp 118ndash149 1999

[13] E L Plambeck and T A Taylor ldquoSell the plant The impact ofcontract manufacturing on innovation capacity and profitabil-ityrdquoManagement Science vol 51 no 1 pp 133ndash150 2005

[14] O Yazlali and F Erhun ldquoRelating the multiple supply problemto quantity flexibility contractsrdquo Operations Research Lettersvol 35 no 6 pp 767ndash772 2007

[15] Z T Lian and A Deshmukh ldquoAnalysis of supply contracts withquantity flexibilityrdquo European Journal of Operational Researchvol 196 no 2 pp 526ndash533 2009

[16] S Karakaya and I S Bakal ldquoJoint quantity flexibility formultiple products in a decentralized supply chainrdquo Computersamp Industrial Engineering vol 64 no 2 pp 696ndash707 2013

[17] V Padmanabhan and I P L Png ldquoReturns policies makemoney by making goodrdquo SloanManagement Review pp 65ndash721995

[18] T Taylor ldquoChannel coordination under price protectionmidlife returns and end-of-life returns in dynamic marketsrdquoManagement Science vol 47 no 9 pp 1220ndash1234 2001

[19] Z Yao S Leung and K K Lai ldquoAnalysis of the impact of price-sensitivity factors on the returns policy in coordinating supplychainrdquo European Journal of Operational Research vol 187 no 1pp 275ndash282 2008

[20] Y L Wang and P Zipkin ldquoAgentsrsquo incentives under buy-backcontracts in a two-stage supply chainrdquo International Journal ofProduction Economics vol 120 no 2 pp 525ndash539 2009

[21] J Hou A Z Zeng and L Zhao ldquoCoordination with a backupsupplier through buy-back contract under supply disruptionrdquo

Transportation Research E Logistics and Transportation Reviewvol 46 no 6 pp 881ndash895 2010

[22] H C Xiong B T Chen and J X Xie ldquoA composite contractbased on buy back and quantity flexibility contractsrdquo EuropeanJournal of Operational Research vol 210 no 3 pp 559ndash567 2011

[23] J Dana Jr and K E Spier ldquoRevenue sharing and vertical controlin the video rental industryrdquo Journal of Industrial Economicsvol 49 no 3 pp 223ndash245 2001

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

[25] Z Yao S C H Leung and K K Lai ldquoManufacturerrsquos revenue-sharing contract and retail competitionrdquo European Journal ofOperational Research vol 186 no 2 pp 637ndash651 2008

[26] 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

[27] 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

[28] J M Chen H L Cheng and M C Chien ldquoOn channelcoordination through revenue-sharing contracts with priceand shelf-space dependent demandrdquo Applied MathematicalModelling vol 35 no 10 pp 4886ndash4901 2011

[29] D P D Omkar ldquoSupply chain coordination using revenue-dependent revenue sharing contractsrdquoOmega vol 41 no 4 pp780ndash796 2013

[30] G Kannan andN PMaria ldquoReverse supply chain coordinationby revenue sharing contract a case for the personal computersindustryrdquo European Journal of Operational Research vol 233no 2 pp 326ndash336 2014

[31] D J Wu and P R Kleindorfer ldquoCompetitive options supplycontracting and electronic marketsrdquoManagement Science vol51 no 3 pp 452ndash468 2005

[32] XWang and L Liu ldquoCoordination in a retailer-led supply chainthrough option contractrdquo International Journal of ProductionEconomics vol 110 no 1-2 pp 115ndash127 2007

[33] Y X Zhao S Y Wang T C E Cheng X Q Yang and ZM Huang ldquoCoordination of supply chains by option contractsa cooperative game theory approachrdquo European Journal ofOperational Research vol 207 no 2 pp 668ndash675 2010

[34] L Liang X HWang and J G Gao ldquoAn option contract pricingmodel of relief material supply chainrdquoOmega vol 40 no 5 pp594ndash600 2012

[35] Y X Zhao L J Ma G Xie and T C E Cheng ldquoCoordinationof supply chains with bidirectional option contractsrdquo EuropeanJournal of Operational Research vol 229 no 2 pp 375ndash3812013

[36] M M Leng and M Parlar ldquoGame-theoretic analyses of decen-tralized assembly supply chains Non-cooperative equilibria vscoordination with cost-sharing contractsrdquo European Journal ofOperational Research vol 204 no 1 pp 96ndash104 2010

[37] M Kunter ldquoCoordination via cost and revenue sharing inmanufacturer-retailer channelsrdquo European Journal of Opera-tional Research vol 216 no 2 pp 477ndash486 2012

[38] Y C Tsao and G J Sheen ldquoEffects of promotion cost sharingpolicy with the sales learning curve on supply chain coordi-nationrdquo Computers and Operations Research vol 39 no 8 pp1872ndash1878 2012

10 Journal of Applied Mathematics

[39] Y H Wei and Q Y Hu ldquoOrdering pricing and allocationin a service supply chainrdquo International Journal of ProductionEconomics vol 144 no 2 pp 590ndash598 2013

[40] W H Liu X C Xu and A Kouhpaenejad ldquoDeterministicapproach to the fairest revenue-sharing coefficient in logisticsservice supply chain under the stochastic demand conditionrdquoComputers amp Industrial Engineering vol 66 no 1 pp 41ndash522013

[41] A P Cui and W Liu ldquoStudy on capability coordination inlogistics service supply chain with options contractrdquo ChineseJournal of Management Science vol 17 no 2 pp 59ndash65 2009(Chinese)

[42] Q H Lu ldquoCoordinate with cost sharing strategy for servicesupply chainrdquo Control and Decision vol 26 no 11 pp 1649ndash1653 2011 (Chinese)

[43] A Prasad and S P Sethi ldquoCompetitive advertising underuncertainty a stochastic differential game approachrdquo Journal ofOptimization Theory and Applications vol 123 no 1 pp 163ndash185 2004

[44] S Joslashrgensen S Taboubi and G Zaccour ldquoRetail promotionswith negative brand image effects is cooperation possiblerdquoEuropean Journal of Operational Research vol 150 no 2 pp395ndash405 2003

[45] J Navas and J Marın-Solano ldquoInteractions between govern-ment and firms a differential game approachrdquo Annals ofOperations Research vol 158 no 1 pp 47ndash61 2008

[46] S Joslashrgensen S Taboubi and G Zaccour ldquoCooperative adver-tising in a marketing channelrdquo Journal of Optimization Theoryand Applications vol 110 no 1 pp 145ndash158 2001

[47] C J Fombrun and V Rindova ldquoWhorsquos tops and who decidesThe social construction of corporate reputationsrdquo WorkingPaper Stern School of Business New York University 1996

[48] N Nha and L Gaston ldquoCorporate image and corporate repu-tation in customers Retention decisions in servicesrdquo Journal ofRetailing and Consumer Services no 8 pp 227ndash2361 2001

[49] M Nerlove and K J Arrow ldquoOptimal advertising policy underdynamic conditionsrdquo Economica vol 39 pp 129ndash142 1962

[50] K Saxton ldquoWhere do reputations come fromrdquo CorporateReputation Review vol 1 no 4 pp 393ndash399 1998

[51] M G Nagler ldquoAn exploratory analysis of the determinants ofcooperative advertising participation ratesrdquo Marketing Lettersvol 17 no 2 pp 91ndash102 2006

[52] E Dockner N Jorgensen L Van et al Differential Gamesin Economics and Management Science Cambridge UniversityPress Cambridge UK 2000

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 Journal of Applied Mathematics

[39] Y H Wei and Q Y Hu ldquoOrdering pricing and allocationin a service supply chainrdquo International Journal of ProductionEconomics vol 144 no 2 pp 590ndash598 2013

[40] W H Liu X C Xu and A Kouhpaenejad ldquoDeterministicapproach to the fairest revenue-sharing coefficient in logisticsservice supply chain under the stochastic demand conditionrdquoComputers amp Industrial Engineering vol 66 no 1 pp 41ndash522013

[41] A P Cui and W Liu ldquoStudy on capability coordination inlogistics service supply chain with options contractrdquo ChineseJournal of Management Science vol 17 no 2 pp 59ndash65 2009(Chinese)

[42] Q H Lu ldquoCoordinate with cost sharing strategy for servicesupply chainrdquo Control and Decision vol 26 no 11 pp 1649ndash1653 2011 (Chinese)

[43] A Prasad and S P Sethi ldquoCompetitive advertising underuncertainty a stochastic differential game approachrdquo Journal ofOptimization Theory and Applications vol 123 no 1 pp 163ndash185 2004

[44] S Joslashrgensen S Taboubi and G Zaccour ldquoRetail promotionswith negative brand image effects is cooperation possiblerdquoEuropean Journal of Operational Research vol 150 no 2 pp395ndash405 2003

[45] J Navas and J Marın-Solano ldquoInteractions between govern-ment and firms a differential game approachrdquo Annals ofOperations Research vol 158 no 1 pp 47ndash61 2008

[46] S Joslashrgensen S Taboubi and G Zaccour ldquoCooperative adver-tising in a marketing channelrdquo Journal of Optimization Theoryand Applications vol 110 no 1 pp 145ndash158 2001

[47] C J Fombrun and V Rindova ldquoWhorsquos tops and who decidesThe social construction of corporate reputationsrdquo WorkingPaper Stern School of Business New York University 1996

[48] N Nha and L Gaston ldquoCorporate image and corporate repu-tation in customers Retention decisions in servicesrdquo Journal ofRetailing and Consumer Services no 8 pp 227ndash2361 2001

[49] M Nerlove and K J Arrow ldquoOptimal advertising policy underdynamic conditionsrdquo Economica vol 39 pp 129ndash142 1962

[50] K Saxton ldquoWhere do reputations come fromrdquo CorporateReputation Review vol 1 no 4 pp 393ndash399 1998

[51] M G Nagler ldquoAn exploratory analysis of the determinants ofcooperative advertising participation ratesrdquo Marketing Lettersvol 17 no 2 pp 91ndash102 2006

[52] E Dockner N Jorgensen L Van et al Differential Gamesin Economics and Management Science Cambridge UniversityPress Cambridge UK 2000

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