Research ArticleThe Model of Distributor Chain Financing Based on Buy BackGuarantee Contract

Jian-xin Chen12 and Jia-yin Chen2

1 Faculty of Applied Mathematics Guangdong University of Technology Guangzhou 510520 China2 School of Business Administration South China University of Technology Guangzhou 510641 China

Correspondence should be addressed to Jian-xin Chen chenjianxin403163com

Received 13 December 2013 Accepted 11 August 2014 Published 20 August 2014

Academic Editor X Zhang

Copyright copy 2014 J-x Chen and J-y Chen This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited

This paper considers the strategy employed by a buy back guarantee contract with a capital-constrained distributor and a coreenterprise The distributor faces a nonnegative random demand and the core enterprise applies buy back guarantee contract inorder to interact with the capital-constrained distributor Mathematical model is built to get the optimal ordering quantity of thedistributor and the optimal wholesale price of the core enterpriseThen sensitivity analysis of optimal ordering quantity is obtainedabout the wholesale price the initial funds and the salvage of the product On that basis the comparison is made between twofinancing modesmdashtrade credit contract and buy back guarantee contract In the end a numerical analysis is illustrated The resultsshow that the different financing modes bring the different expected profits to supply chain system with the different initial fundsfinding that the financingmodes buy back guarantee contract discussed in the paper can create more value for supply chain systemthan trade credit contract

1 Introduction

With the rapid development of science and technology andthe coming of the information age the relation betweensupply chain enterprises is unavoidable more and moreclose in three aspects of information flow cash flow andlogistics Recently supply chain management has been moreand more perfect in the information flow and logisticsHowever the equally important cash flow has not beensynchronized with development and the related academicresearch is also relatively scarce Capital constraints are a realproblem that small and medium-sized enterprises (SMEs)can face when they enter the production and sales chainObtaining sufficient capital to finance these elements oftheir operations is a challenge for SMEs worldwide Capitalconstraints are one of the factors that must be considered inthe production operations and inventory decisions Due tothe lag of the cash flowmanagement research it is difficult tosynchronize the logistics and information flow In traditionalsupply chain research they often supposed the capital ofSMEs was sufficient Caldentey and Haugh pointed out that

capital constraints were the common problem in supply chainmanagement practice Buzacott and Zhang [1] researchedthe strategy of the enterprise under capital constraints andtheir results were published in Management Science in2004 This shows that the research fields were beginningto receive attention Then Xu and Birge [2] assumed thatthe demand was uncertain and the market was not perfectthe joint decision problem between enterprise financing andproduction was discussed They also discussed the mutualinfluence of the enterprisersquos productivity and long-termcapital structure Applying principal agent theory and thenewsvendor model Xu and Brige [3] established the modelof enterprise capital structure management incentives andproduction decision Swinney et al [4] studied the influenceof enterprise bankruptcy risk to the newly established enter-prise investment decisionThey chose the survival probabilityas the objective decision for newly established enterpriseand made the comparison with the mature enterprise whosedecision was the profit maximization Li et al [5] studied adynamic model of coordination in an equity financed firmin which inventory and financial decisions interact in the

Hindawi Publishing CorporationJournal of Applied MathematicsVolume 2014 Article ID 902739 7 pageshttpdxdoiorg1011552014902739

2 Journal of Applied Mathematics

presence of demand uncertainty capital constraints and arisk of default Babich et al [6] identified several economicand business factors thatmight affect the number of supplierssupply risk fixed costs of working with suppliers and accessto financing Xu andZhang [7] allowed amanufacture to offerloans and provide goods to its capital-constrained retailersin a monopoly setting They examined whether the typicalsupply chain contract could achieve channel coordinationDada andHu [8] pointed out that when cost of borrowingwasnot too high the capital-constrained newsvendor borrowedfunds to procure an amount that was less than would beideal The lender charged an interest rate that decreased inthe newsvendorrsquos equity and they derived a nonlinear loanschedule that coordinates the channel Gupta [9] chose thediscount rate of the retailer advance payment as the soledecision variables of the supplier under the assumption ofunchanged wholesale price Lai et al [10] examined theimpact of capital constraints and investigated the supplychain efficiency under each mode Based on a Stackelberggame with the supplier being the leader they showed thatwithout financial constraint the supplier always preferredthe consignment mode taking full inventory risk They alsoshowed that with capital constraints the combination modewas the most efficient mode even if the retailer earned zerointernal capital Kouvelis and Zhao [11] considered a supplychainwith a retailer and a supplier a newsvendor-like retailerhad a single opportunity to order products from a supplierto satisfy future uncertain demand Both the retailer andsupplier were capital constraints and in need of short-termfinancing They used the supplier early payment discountscheme as a decision framework to analyze all decisionsinvolved in optimally structuring the trade credit contractfrom the supplierrsquos perspective Kouvelis and Zhao [12] stud-ied a supply chain of a supplier selling via a wholesale pricecontract to a capital-constrained retailer who faced stochasticdemand They identified the retailerrsquos optimal order quantityas a function of the wholesale price and his total wealth Theanalysis of the supplierrsquos optimal wholesale price problemas a Stackelberg game with the supplier the leader and theretailer the follower led to unique equilibrium solution inwholesale price and ordering quantity with the equilibriumordering quantity smaller than the traditional newsvendorone Chen and Wan [13] examined the impact of financingon the performance of a two-level supply chain consistingof a supplier and a capital-constrained retailer They setup a three-stage Stackelberg game under a wholesale pricecontract with a financial market and demonstrated that ina competitive financial market with symmetric informationon the retailerrsquos initial budget the retailerrsquos operational andfinancial decisions could be decoupled Chen and Wang [14]investigated the impacts of trade credit and limited liabilityon the performance of a two-level supply chain with capitalconstraintsThey found that trade credit contract could createvalue in a supply chain with capital constraints and partlycoordinate the supply chain

The motivation of the paper comes from the operationspractice of Hengda Steel in China a famous companythat affords the goods buy back contract for the capital-constrained distributor Buy back guarantee contract is a kind

of financing service provided by core enterprise through buyback guarantee of the real goods to facilitate downstreamdistributorrsquos purchasing in which core enterprise is thesubject of risk control and the basis is the advance paymentgenerated after downstream distributor and core enterprisesigning a real business contract

Our model differs from the previous work in the fol-lowing aspects First we assume that the retailer is capitalconstraints Capital flow is an important factor in supplychain although the extant literatures typically assume that theretailers or distributors are deep pocket Second there is nomodel to discuss the strategy of the distributor and the coreenterprise under the buy back guarantee contract (buy backguarantee contract will be discussed in detail in Section 2)In this paper with the buy back guarantee contract we modelthe expected profits of the distributor and the core enterpriseFurthermore we discuss the optimal ordering quantity ofthe distributor and the optimal wholesale price of the coreenterprise finding that the buy back guarantee contract couldcreate value for the supply chain with capital constraints

The remainder of the paper is organized as followsSection 2 gives themodel description and assumptionsSection 3 discusses the expected profit models of distributorand the core enterprise obtaining the optimal strategy ofsupply chain Section 4 illustrates a numerical example

2 Model Description and Assumptions

Buy back guarantee contract is a kind of financing serviceprovided by core enterprise through buy back guarantee ofthe real goods to facilitate downstream distributorrsquos purchas-ing Core enterprise is the subject of risk control and the basisis the advance payment generated after downstream distrib-utor and core enterprise signing a real business contract

Hengda Steel for example is a core steel enterprise Ifthere is a steel distributor with 3million self-funds wanting topurchase a batch of steel worth 10 million from the core steelenterprise then it has to apply for fund by chain financing ofbuy back guaranteewith the assistance of core steel enterpriseHengda Steel Company The following are the details

(1) The bank core enterprise and distributor sign thecontract and the core enterprise promises to buyback the unsold steel when the bankrsquos acceptance bill(BA) has reached the appointed date The distributorshould remit money into its appointed account inbank and authorize the bank to deduct the sales forfinancing

(2) The distributor pays no less than 30 of the paymentto core enterprise

(3) Thebank checks the core enterprisersquos line of credit andratifies BA to the distributor

(4) Core enterprise pays the guarantee money of the BAto the bank (generally 30 of the face value) and thecore supplier gives it to core enterprise directly

(5) Core supplier delivers to the distributor after receiv-ing BA and sends the certificate of the steel to thebank

Journal of Applied Mathematics 3

Distributor Core

Warehousingenterprise

Bank

(2) 30 payment

(3) BA

(4) Guarantee money(6) Buy back and refund

(5) Deliver goods

(5) Notice of delivery

(7) Bill payment

enterprise

Figure 1 The process of buy back guarantee contract

(6) Distributor remits the sales to the appointed accountin bank after selling steel to redeem certificates tofulfill selling

(7) If the distributor sells all steel when the BA is due thebank withdraws the capital and releases the BA If thedistributor fails to sell all steel at the end of the sellingperiod the bank releases the BA after core enterprisebuys back the unsold steel

The implemented process is shown in Figure 1 Thedashed box shows that warehousing enterprise is notrequired

We consider a simple supply chain with a core enterpriseand a capital-constrained distributorThe core enterprise pro-duces a single product and the distributor faces a nonnegativerandom demand 120585 with a distribution function (similar tothe model discussed in Cachon (2003) and Lariviere andPorteus (2001)) We formulate the interaction between thedistributor and the core enterprise in the framework of aStackelberg game at the beginning of the selling period thecore enterprise as the Stacklberg leader and the distributor asthe follower sign buy back guarantee contract After the endof the selling period the enterprise buys back the left goodsaccording to the exposure between bank acceptance and thecash deposit Let us assume denotations as follows

120585 market demand is a nonnegative random vari-able with probability density function 119891(119909) and thecumulative distribution function 119865(119909) respectivelySuppose the hazard function ℎ(119909) = 119891(119909)119865(119909)(IFR) and the generalized failure rate (GFR) 119867(119909) =119909119891(119909)119865(119909) where 119865(119909) = 1 minus 119865(119909) is the taildistribution of 119865(119909)

119876 is the ordering quantity of the retailer

119901 is the unit retail price (suppose 119901 = 1)

119908 is the wholesale price (the unit ordering cost)

119861 is the initial capital of distributor

119888 is the unit production cost

119904 is the salvage of the product Without loss ofgenerality assume 119901 = 1 ge 119908 gt 119888 gt 119904

3 Result Concerned with the Paper

When the retailer has no capital constraints the profit of theretailer can be written as follows min119909 119876minus119908119876+119904(119876minus119909)+The profit of the supplier can be written as follows (119908 minus 119888)119876

The optimal decentralized ordering quantity 1198760is

(1 minus 119904) 119865 (1198760) = (119908 minus 119904) (1)

The optimal centralized ordering quantity 119876119862 is

(1 minus 119904) 119865 (119876119862

) = (119888 minus 119904) (2)

We have 1198760lt 119876119862

When the capital-constrained retailer (distributor) has noaccess to the financial market its optimal ordering quantityis

119876NF = min 1198611199081198760 (3)

(see [13])Under the trade credit contract the optimal inventory

level of the retailer 119876119879 is [14]

(1 minus 119904) 119865 (119876119879

) = (119908 minus 119904) 119865 [(119908 minus 119904)

(1 minus 119904)119876119879

minus119861

(1 minus 119904)] (4)

4 Model Calculations

41 The Distributor The expected profit of the capital-constrained distributor can be written as follows

119864Π119877

= int119908119876minus119861

0

[119904 (119876 minus 119909) minus1

119908[(119908119876 minus 119861) minus 119909] minus 119861]

times 119891 (119909) 119889119909

+ int119876

119908119876minus119861

[119909 minus (119908119876 minus 119861) + 119904 (119876 minus 119909) minus 119861] 119891 (119909) 119889119909

+ int+infin

119876

[119876 minus (119908119876 minus 119861) minus 119861] 119891 (119909) 119889119909

(5)

Theorem 1 (i) Under buy back guarantee contract the optimalordering quantity of the distributor is

(1 minus 119904) 119865 (119876lowast

) minus (119908 minus 119904) 119865 (119908119876lowast

minus 119861) = 0 (6)

(ii) Under buy back guarantee contract the optimal orderingquantity of the distributor 119876lowast decreases in 119908 and 119861

Proof (i) It is easy to get 120597119864Π119877120597119876 = (1 minus 119908) int+infin

119876119891(119909)119889119909 minus

(119908 minus 119904) int119876

119908119876minus119861119891(119909)119889119909 If 119876 satisfies 120597119864Π119877120597119876 = 0 then

1205972119864Π1198771205971198762 = minus(1 minus 119904)119865(119876)ℎ(119876) minus 119908ℎ(119908119876 minus 119861) lt 0 So theoptimal ordering quantity is (1minus119904)119865(119876)minus(119908minus119904)119865(119908119876minus119861) = 0

(ii) Let 119866(119876119908) = (1 minus 119904)119865(119876) minus (119908 minus 119904)119865(119908119876 minus 119861) then120597119876120597119908 = minus1minus (119908minus119904)119876ℎ(119908119876minus119861)(119908minus119904)ℎ(119876)minus119908ℎ(119908119876minus119861)

4 Journal of Applied Mathematics

According to the assumption we have119908 isin [119904 1] Suppose1198722(119908) = (119908 minus 119904)119865(119908119876 minus 119861) then1198721015840

2(119908) = 119865(119908119876 minus 119861)[1 minus

(119908 minus 119904)119876ℎ(119908119876 minus 119861)]119865(119908119876 minus 119861) and [1 minus (119908 minus 119904)119876ℎ(119908119876 minus 119861)] both decrease

in 119908 as the IFR property of random demand 120585 Then1198722(119908)

is unimodal in 119908 and achieves the maximum at 1 = (119908 minus119904)119876ℎ(119908119876minus119861) We have119872

2(119904) = 0 and119872

2(1) = (1 minus 119904)119865(119876minus

119861) gt (1 minus 119904)119865(119876)We conclude that for 119908 isin [119904 1] (1 minus 119904)119865(119876) = (119908 minus

119904)119865(119908119876 minus 119861) can only be satisfied on the positive slope of1198722(119908) so 1198721015840

2(119908) gt 0 [1 minus (119908 minus 119904)119876ℎ(119908119876 minus 119861)] gt 0 and

120597119876120597119908 lt 0 It is easy to get 120597119876120597119861 = minusℎ(119908119876 minus 119861)(ℎ(119876) minus119908ℎ(119908119876 minus 119861)) lt 0 This then indicates that Theorem 1 iscorrect

Recall that in the traditional newsvendor model (Hadleyand Whitin 1963) with no capital constraints the optimalordering quantity of the retailer is (1) That is a first-orderoptimality condition that requires the marginal revenue of anextra unit (1 minus 119904)119865(119876

0) to be equal to the cost of this extra

unit (119908 minus 119904)Under buy back guarantee contract the first-order opti-

mality condition satisfies (6) Since (119908minus 119904) ge (119908minus 119904)119865(119908119876lowast minus119861) it follows that a distributor (retailer) has a lower marginalcost in this case and 119876lowast ge 119876

0 It is because that the low

demand reduces the distributorrsquos downside risk Under thetrade credit contract the distributorrsquos (retailerrsquos) optimalordering quantity satisfies (4) since (119908 minus 119904)119865[((119908 minus 119904)(1 minus119904))119876119879 minus 119861(1 minus 119904)] ge (119908 minus 119904)119865(119908119876lowast minus 119861) it follows that 119876lowast ge119876119879 ge 119876

0ge 119876NF In particular 119904 = 0 the optimal ordering

quantity of distributor (retailer) satisfies119865(119876)minus119908119865(119908119876minus119861) =0 It is equal to the optimal ordering quantity under tradecredit contract [14] At this time the incentive effect of buyback guarantee contract and the trade credit contract is thesame Furthermore Theorem 1 also shows that the optimalordering quantity decreases in 119861 the reason why 120597119876120597119861 lt 0is that the distributor with a low budget has little stake atrisk when choosing its ordering quantity in the sense thatits potential losses are bounded by 119861 on the other handthe reason is the incentive effect of the buy back guaranteecontract

Theorem 2 Under buy back guarantee contract the optimalorder quantity of distributor 119876lowast increases in 119904

Proof Let 119866(119876119908) = (1 minus 119904)119865(119876) minus (119908 minus 119904)119865(119908119876 minus 119861)Then

120597119876

120597119904= minus

[119865 (119876) minus 119865 (119908119876 minus 119861)]

(119908 minus 119904) 119865 (119908119876 minus 119861) [ℎ (119876) minus 119908ℎ (119908119876 minus 119861)]gt 0

(7)

The increase of 119904 reduces the random market risk of dis-tributor faces and increases the expected profit of distributorso the distributor will increase the optimal ordering quantityto raise profit

42 The Core Enterprise The expected profit of the coreenterprise can be written as follows

119864Π119872= int119908119876minus119861

0

119876 minus[(119908119876 minus 119861) minus 119909]

119908 (119908 minus 119888) 119891 (119909) 119889119909

+ int+infin

119908119876minus119861

(119908 minus 119888)119876119891 (119909) 119889119909

(8)

Lemma 3 The optimal ordering quantity of distributor 119876satisfies 12059721198761205971199082 lt 0

Proof Let

120597119876

120597119908= minus

1 minus (119908 minus 119904)119876ℎ (119908119876 minus 119861)

(119908 minus 119904) ℎ (119876) minus 119908ℎ (119908119876 minus 119861) (9)

Then

119889119864Π119872

119889119908=120597119864Π119872

120597119876

120597119876

120597119908+120597119864Π119872

120597119908

= (119908 minus 119888) 119865 (119908119876 minus 119861)120597119876

120597119908

+ [119876119865 (119908119876 minus 119861) +119861119888

1199082119865 (119908119876 minus 119861)

+119888

1199082int119908119876minus119861

0

119909 119889119865 (119909)]

(10)

1205972119876

1205971199082= minus(

119865 (119908119876 minus 119861) [1 minus (119908 minus 119904)119876ℎ (119908119876 minus 119861)]

(1 minus 119904) 119891 (119876) minus (119908 minus 119904)119908119891 (119908119876 minus 119861))

1015840

119908

(11)

then

(119865 (119908119876 minus 119861) [1 minus (119908 minus 119904)119876ℎ (119908119876 minus 119861)]

(1 minus 119904) 119891 (119876) minus (119908 minus 119904)119908119891 (119908119876 minus 119861))

1015840

119908

=119865 (119908119876 minus 119861) [1 minus (119908 minus 119904)119876ℎ (119908119876 minus 119861)]

1015840

119908

[(1 minus 119904) 119891 (119876) minus (119908 minus 119904)119908119891 (119908119876 minus 119861)]2

times [(1 minus 119904) 119891 (119876) minus (119908 minus 119904)119908119891 (119908119876 minus 119861)]

minus[(1 minus 119904) 119891 (119876) minus (119908 minus 119904)119908119891 (119908119876 minus 119861)]

1015840

119908

[(1 minus 119904) 119891 (119876) minus (119908 minus 119904)119908119891 (119908119876 minus 119861)]2

times 119865 (119908119876 minus 119861) [1 minus (119908 minus 119904)119876ℎ (119908119876 minus 119861)]

=minus2119876119891 (119908119876 minus 119861) minus (119908 minus 119904)11987621198911015840 (119908119876 minus 119861)

[(1 minus 119904) 119891 (119876) minus (119908 minus 119904)119908119891 (119908119876 minus 119861)]2

times [(1 minus 119904) 119891 (119876) minus (119908 minus 119904)119908119891 (119908119876 minus 119861)]

Journal of Applied Mathematics 5

minus[minus (2119908 minus 119904) 119891 (119908119876 minus 119861) minus (119908 minus 119904)1199081198761198911015840 (119908119876 minus 119861)]

[(1 minus 119904) 119891 (119876) minus (119908 minus 119904)119908119891 (119908119876 minus 119861)]2

times 119865 (119908119876 minus 119861) [1 minus (119908 minus 119904)119876ℎ (119908119876 minus 119861)]

=1

119908119876

minus2119908119876119891 (119908119876 minus 119861) minus (119908 minus 119904)11990811987621198911015840 (119908119876 minus 119861)

[(1 minus 119904) 119891 (119876) minus (119908 minus 119904)119908119891 (119908119876 minus 119861)]2

times [(1 minus 119904)119876119891 (119876) minus (119908 minus 119904)119908119876119891 (119908119876 minus 119861)]

+1

119908119876[ (2119908 minus 119904)119876119891 (119908119876 minus 119861)

+ (119908 minus 119904)1199081198762

1198911015840

(119908119876 minus 119861)]

times ([(1 minus 119904) 119891 (119876) minus (119908 minus 119904)119908119891 (119908119876 minus 119861)]2

)minus1

times [119908119865 (119908119876 minus 119861) minus (119908 minus 119904)119908119876119891 (119908119876 minus 119861)]

=1

119908119876

minus 2119908119876119891 (119908119876 minus 119861) + (119908 minus 119904)11990811987621198911015840 (119908119876 minus 119861)

[(1 minus 119904) 119891 (119876) minus (119908 minus 119904)119908119891 (119908119876 minus 119861)]2

times [(1 minus 119904)119876119891 (119876) minus (119908 minus 119904)119908119876119891 (119908119876 minus 119861)]

+1

119908119876[2119908119876119891 (119908119876 minus 119861) + (119908 minus 119904)119908119876

2

1198911015840

(119908119876 minus 119861)

minus 119904119876119891 (119908119876 minus 119861)]

times ([(1 minus 119904) 119891 (119876) minus (119908 minus 119904)119908119891 (119908119876 minus 119861)]2

)minus1

times [(119908 minus 119904) 119865 (119908119876 minus 119861) minus (119908 minus 119904)119908119876119891 (119908119876 minus 119861)

+119904119865 (119908119876 minus 119861)]

=1

119908119876[(1 minus 119904) 119891 (119876) minus (119908 minus 119904)119908119891 (119908119876 minus 119861)]2times Δ1

+1

119908119876[(1 minus 119904) 119891 (119876) minus (119908 minus 119904)119908119891 (119908119876 minus 119861)]2Δ2

(12)

where

Δ1= minus [2119908119876119891 (119908119876 minus 119861) + (119908 minus 119904)119908119876

2

1198911015840

(119908119876 minus 119861)]

times [(1 minus 119904)119876119891 (119876) minus (119908 minus 119904)119908119876119891 (119908119876 minus 119861)]

+ [2119908119876119891 (119908119876 minus 119861) + (119908 minus 119904)1199081198762

1198911015840

(119908119876 minus 119861)]

times [(119908 minus 119904) 119865 (119908119876 minus 119861) minus (119908 minus 119904)119908119876119891 (119908119876 minus 119861)]

= [2119908119876119891 (119908119876 minus 119861) + (119908 minus 119904)1199081198762

1198911015840

(119908119876 minus 119861)]

times (1 minus 119904) 119865 (119908119876 minus 119861) 1 minus 119876ℎ (119876)

(13)

Since the demand distribution has the property of IFR Δ1gt

0Δ2= minus119904119876119891 (119908119876 minus 119861)

times [119908119865 (119908119876 minus 119861) minus (119908 minus 119904)119908119876119891 (119908119876 minus 119861)]

+ 119904119865 (119908119876 minus 119861)

times [(2119908 minus 119904)119876119891 (119908119876 minus 119861) + (119908 minus 119904)1199081198762

1198911015840

(119908119876 minus 119861)]

= (119908 minus 119904)1199081199041198912

(119908119876 minus 119861) + (119908 minus 119904) 119904119876119891 (119908119876 minus 119861)

times 119865 (119908119876 minus 119861) [1 minus 119908119876ℎ (119908119876 minus 119861)] gt 0

(14)

So 12059721198761205971199082 lt 0

Theorem 4 Under buy back guarantee contract the optimalwholesale price of the core enterprise 119908lowast can be expressed asfollows

(119908 minus 119888) 119865 (119908119876 minus 119861)120597119876

120597119908

+ [119876119865 (119908119876 minus 119861) +119861119888

1199082119865 (119908119876 minus 119861) +

119888

1199082int119908119876minus119861

0

119909 119889119865 (119909)]

= 0

(15)

Proof Consider1198892119864Π

119872

1198891199082= 2119865 (119908119876 minus 119861)

120597119876

120597119908[1 minus (119908 minus 119888)119876ℎ (119908119876 minus 119861)]

minus (119908 minus 119888)119908119891 (119908119876 minus 119861) (120597119876

120597119908)2

+ (119908 minus 119888) 119865 (119908119876 minus 119861)1205972119876

1205971199082

minus (1 minus119888

119908)1198762

119891 (119908119876 minus 119861) minus 2119861119888

1199083119865 (119908119876 minus 119861)

minus 2119888

1199083int119908119876minus119861

0

119909119891 (119909) 119889119909 lt 0

(16)

From Lemma 3 the optimal wholesale price of the coreenterprise can be expressed as follows

(119908 minus 119888) 119865 (119908119876 minus 119861)120597119876

120597119908+ 119876119865 (119908119876 minus 119861) +

119861119888

1199082119865 (119908119876 minus 119861)

+119888

1199082int119908119876minus119861

0

119909 119889119865 (119909) = 0

(17)

5 Numerical Example

Figure 2 gives the relation between the distributorrsquos initialcapital and the optimal ordering quantity We assume that

6 Journal of Applied Mathematics

0 50 100 150 200 250 300 350 400 450 5000

500

1000

1500

2000

B

Q

Retailerrsquos optimal ordering level under buy back

Retailerrsquos optimal ordering level under trade credit contractThe optimal ordering level in traditional newsvendor modelRetailerrsquos optimal ordering level without financing

guarantee contract

Figure 2 Relation between the optimal ordering quantities withdifferent financial modes

Table 1 Optimal ordering quantity of the distributor and the totalprofit of supply chain system

119861

119876

119876NF 1198760

119876119879 119876lowast 119876119862

Π

ΠNF Π0

Π119879 Πlowast Π119862

119861 = 5063 405 935 1406 176630 159 243 404 321

119861 = 100125 405 709 1237 176658 159 236 374 321

119861 = 200250 405 405 866 176689 159 192 294 321

119861 = 400405 405 405 405 1766158 160 192 159 321

119901 = 1 119908 = 08 and 119904 = 04 and 120585simexponential (11000)From Figure 2 we know that the optimal ordering quantity119876lowast is bigger than 119876119879 the optimal ordering quantity of thetrade credit contractThe optimal ordering quantities119876lowast and119876119879 are both bigger than 119876NF the optimal ordering quantityof the retailer without financing service It also indicates thatthe 119876lowast decreases in the initial budget 119861 of the retailer Theseconclusions are all consistent withTheorem 1

Let ΠNF Π0 Π119879 Πlowast and Π119862 be the total expected

profits of supply chain without system finance the traditionalnewsvendormodel traded credit financing buy back guaran-tee financing and the centralized decision without financeWith the change of the initial capital let 119861 = 50 100 200 and400 Table 1 gives the optimal ordering quantity of the retailerand the total profit of supply chain system

6 Conclusions

Working under the assumption of stochastic market demandand a distributor that is capital constraints this paper hasconsidered the optimal ordering quantity of the distributor

and the wholesale price of the core enterprise with buy backguarantee contract in a single stage supply chain systemWe compare the optimal ordering quantities of the differ-ent financing modes trade credit contract the traditionalnewsvendor model and buy back guarantee contract Thestudy has highlighted how the capital-constrained distributorcould be financed well facing different financing modes andthe research results show that the different financing modesbring different expected profits for supply chain systemwith the capital-constrained distributor The results couldbe applied for the distributor and the core enterprise whenthey make decisions in reality The paper has also raised thepotential for further research into finance contact of supplychain where more than one distributor and core enterprisecan also be considered

Conflict of Interests

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

Acknowledgments

This work was financially supported by the Key Project oftheNationalNatural Science Foundation of China (71131003)Project of the National Natural Science Foundation of China(71371075) China Postdoctoral Science Foundation fundedProject (2013M542183)GuangdongProvinceNatural ScienceFund (S2013040014920) of China and Research Fund for theDoctoral ProgramofGuangdongUniversity of Technology ofChina (13ZK0155)

References

[1] J A Buzacott and R Q Zhang ldquoInventory management withasset-based financingrdquo Management Science vol 50 no 9 pp1274ndash1292 2004

[2] X D Xu and J R Birge ldquoJoint production and financing deci-sions modeling and analysisrdquoWorking PaperTheUniversity ofChicago Graduate School of Business 2004

[3] X D Xu and J R Birge ldquoOperational decisions capitalstructure and managerial compensation a news vendor per-spectiverdquo Working Paper The University of Chicago GraduateSchool of Business 2005

[4] R Swinney G Cachon and S Netessine ldquoCapacity investmentby competitive startupsrdquo Working Paper The Wharton SchoolUniversity of Pennsylvania 2005

[5] L Li M Shubik and M Sobel ldquoControl of dividends capitalsubscriptions and physical inventoriesrdquo Working Paper YaleSchool of Management 2005

[6] V Babich G Aydin P Brunet J Keppo and R Saigal ldquoRiskfinancing and optimal number of suppliersrdquo Working PaperIOE University of Michigan Ann Arbor Mich USA 2005

[7] Y Xu and J Zhang ldquoOn the selection of supply chain coordi-nating contracts the role of capital constraintrdquo Working PaperUniversity of Miami 2007

[8] M Dada and Q Hu ldquoFinancing newsvendor inventoryrdquo Oper-ations Research Letters vol 36 no 5 pp 569ndash573 2008

Journal of Applied Mathematics 7

[9] D Gupta ldquoTechnical note financing the newsvendorrdquoWorkingPaper Industrial and Systems Engineering University of Min-nesota 2008

[10] G Lai L G Debo and K Sycara ldquoSharing inventory risk insupply chain the implication of financial constraintrdquo Omegavol 37 no 4 pp 811ndash825 2009

[11] P Kouvelis and H W Zhao ldquoFinancing the NewsvendorSupplier vs Bank and the Structure of Optimal Trade CreditContractsrdquo Working paper Olin Business School WashingtonUniversity September 2009

[12] P Kouvelis and W H Zhao ldquoThe newsvendor problem andprice-only contract when bankruptcy costs existrdquo WorkingPaper Olin Business School Washington University 2010

[13] X F Chen and G Wan ldquoThe effect of financing on a budget-constrained supply chain under wholesale price contractrdquoAsia-Pacific Journal of Operational Research vol 28 no 4 pp 457ndash485 2011

[14] X F Chen and A Wang ldquoTrade credit contract with limitedliability in the supply chain with budget constraintsrdquo Annals ofOperations Research vol 196 pp 153ndash165 2012

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Stochastic AnalysisInternational Journal of

2 Journal of Applied Mathematics

presence of demand uncertainty capital constraints and arisk of default Babich et al [6] identified several economicand business factors thatmight affect the number of supplierssupply risk fixed costs of working with suppliers and accessto financing Xu andZhang [7] allowed amanufacture to offerloans and provide goods to its capital-constrained retailersin a monopoly setting They examined whether the typicalsupply chain contract could achieve channel coordinationDada andHu [8] pointed out that when cost of borrowingwasnot too high the capital-constrained newsvendor borrowedfunds to procure an amount that was less than would beideal The lender charged an interest rate that decreased inthe newsvendorrsquos equity and they derived a nonlinear loanschedule that coordinates the channel Gupta [9] chose thediscount rate of the retailer advance payment as the soledecision variables of the supplier under the assumption ofunchanged wholesale price Lai et al [10] examined theimpact of capital constraints and investigated the supplychain efficiency under each mode Based on a Stackelberggame with the supplier being the leader they showed thatwithout financial constraint the supplier always preferredthe consignment mode taking full inventory risk They alsoshowed that with capital constraints the combination modewas the most efficient mode even if the retailer earned zerointernal capital Kouvelis and Zhao [11] considered a supplychainwith a retailer and a supplier a newsvendor-like retailerhad a single opportunity to order products from a supplierto satisfy future uncertain demand Both the retailer andsupplier were capital constraints and in need of short-termfinancing They used the supplier early payment discountscheme as a decision framework to analyze all decisionsinvolved in optimally structuring the trade credit contractfrom the supplierrsquos perspective Kouvelis and Zhao [12] stud-ied a supply chain of a supplier selling via a wholesale pricecontract to a capital-constrained retailer who faced stochasticdemand They identified the retailerrsquos optimal order quantityas a function of the wholesale price and his total wealth Theanalysis of the supplierrsquos optimal wholesale price problemas a Stackelberg game with the supplier the leader and theretailer the follower led to unique equilibrium solution inwholesale price and ordering quantity with the equilibriumordering quantity smaller than the traditional newsvendorone Chen and Wan [13] examined the impact of financingon the performance of a two-level supply chain consistingof a supplier and a capital-constrained retailer They setup a three-stage Stackelberg game under a wholesale pricecontract with a financial market and demonstrated that ina competitive financial market with symmetric informationon the retailerrsquos initial budget the retailerrsquos operational andfinancial decisions could be decoupled Chen and Wang [14]investigated the impacts of trade credit and limited liabilityon the performance of a two-level supply chain with capitalconstraintsThey found that trade credit contract could createvalue in a supply chain with capital constraints and partlycoordinate the supply chain

The motivation of the paper comes from the operationspractice of Hengda Steel in China a famous companythat affords the goods buy back contract for the capital-constrained distributor Buy back guarantee contract is a kind

of financing service provided by core enterprise through buyback guarantee of the real goods to facilitate downstreamdistributorrsquos purchasing in which core enterprise is thesubject of risk control and the basis is the advance paymentgenerated after downstream distributor and core enterprisesigning a real business contract

Our model differs from the previous work in the fol-lowing aspects First we assume that the retailer is capitalconstraints Capital flow is an important factor in supplychain although the extant literatures typically assume that theretailers or distributors are deep pocket Second there is nomodel to discuss the strategy of the distributor and the coreenterprise under the buy back guarantee contract (buy backguarantee contract will be discussed in detail in Section 2)In this paper with the buy back guarantee contract we modelthe expected profits of the distributor and the core enterpriseFurthermore we discuss the optimal ordering quantity ofthe distributor and the optimal wholesale price of the coreenterprise finding that the buy back guarantee contract couldcreate value for the supply chain with capital constraints

The remainder of the paper is organized as followsSection 2 gives themodel description and assumptionsSection 3 discusses the expected profit models of distributorand the core enterprise obtaining the optimal strategy ofsupply chain Section 4 illustrates a numerical example

2 Model Description and Assumptions

Buy back guarantee contract is a kind of financing serviceprovided by core enterprise through buy back guarantee ofthe real goods to facilitate downstream distributorrsquos purchas-ing Core enterprise is the subject of risk control and the basisis the advance payment generated after downstream distrib-utor and core enterprise signing a real business contract

Hengda Steel for example is a core steel enterprise Ifthere is a steel distributor with 3million self-funds wanting topurchase a batch of steel worth 10 million from the core steelenterprise then it has to apply for fund by chain financing ofbuy back guaranteewith the assistance of core steel enterpriseHengda Steel Company The following are the details

(1) The bank core enterprise and distributor sign thecontract and the core enterprise promises to buyback the unsold steel when the bankrsquos acceptance bill(BA) has reached the appointed date The distributorshould remit money into its appointed account inbank and authorize the bank to deduct the sales forfinancing

(2) The distributor pays no less than 30 of the paymentto core enterprise

(3) Thebank checks the core enterprisersquos line of credit andratifies BA to the distributor

(4) Core enterprise pays the guarantee money of the BAto the bank (generally 30 of the face value) and thecore supplier gives it to core enterprise directly

(5) Core supplier delivers to the distributor after receiv-ing BA and sends the certificate of the steel to thebank

Journal of Applied Mathematics 3

Distributor Core

Warehousingenterprise

Bank

(2) 30 payment

(3) BA

(4) Guarantee money(6) Buy back and refund

(5) Deliver goods

(5) Notice of delivery

(7) Bill payment

enterprise

Figure 1 The process of buy back guarantee contract

(6) Distributor remits the sales to the appointed accountin bank after selling steel to redeem certificates tofulfill selling

(7) If the distributor sells all steel when the BA is due thebank withdraws the capital and releases the BA If thedistributor fails to sell all steel at the end of the sellingperiod the bank releases the BA after core enterprisebuys back the unsold steel

The implemented process is shown in Figure 1 Thedashed box shows that warehousing enterprise is notrequired

We consider a simple supply chain with a core enterpriseand a capital-constrained distributorThe core enterprise pro-duces a single product and the distributor faces a nonnegativerandom demand 120585 with a distribution function (similar tothe model discussed in Cachon (2003) and Lariviere andPorteus (2001)) We formulate the interaction between thedistributor and the core enterprise in the framework of aStackelberg game at the beginning of the selling period thecore enterprise as the Stacklberg leader and the distributor asthe follower sign buy back guarantee contract After the endof the selling period the enterprise buys back the left goodsaccording to the exposure between bank acceptance and thecash deposit Let us assume denotations as follows

120585 market demand is a nonnegative random vari-able with probability density function 119891(119909) and thecumulative distribution function 119865(119909) respectivelySuppose the hazard function ℎ(119909) = 119891(119909)119865(119909)(IFR) and the generalized failure rate (GFR) 119867(119909) =119909119891(119909)119865(119909) where 119865(119909) = 1 minus 119865(119909) is the taildistribution of 119865(119909)

119876 is the ordering quantity of the retailer

119901 is the unit retail price (suppose 119901 = 1)

119908 is the wholesale price (the unit ordering cost)

119861 is the initial capital of distributor

119888 is the unit production cost

119904 is the salvage of the product Without loss ofgenerality assume 119901 = 1 ge 119908 gt 119888 gt 119904

3 Result Concerned with the Paper

When the retailer has no capital constraints the profit of theretailer can be written as follows min119909 119876minus119908119876+119904(119876minus119909)+The profit of the supplier can be written as follows (119908 minus 119888)119876

The optimal decentralized ordering quantity 1198760is

(1 minus 119904) 119865 (1198760) = (119908 minus 119904) (1)

The optimal centralized ordering quantity 119876119862 is

(1 minus 119904) 119865 (119876119862

) = (119888 minus 119904) (2)

We have 1198760lt 119876119862

When the capital-constrained retailer (distributor) has noaccess to the financial market its optimal ordering quantityis

119876NF = min 1198611199081198760 (3)

(see [13])Under the trade credit contract the optimal inventory

level of the retailer 119876119879 is [14]

(1 minus 119904) 119865 (119876119879

) = (119908 minus 119904) 119865 [(119908 minus 119904)

(1 minus 119904)119876119879

minus119861

(1 minus 119904)] (4)

4 Model Calculations

41 The Distributor The expected profit of the capital-constrained distributor can be written as follows

119864Π119877

= int119908119876minus119861

0

[119904 (119876 minus 119909) minus1

119908[(119908119876 minus 119861) minus 119909] minus 119861]

times 119891 (119909) 119889119909

+ int119876

119908119876minus119861

[119909 minus (119908119876 minus 119861) + 119904 (119876 minus 119909) minus 119861] 119891 (119909) 119889119909

+ int+infin

119876

[119876 minus (119908119876 minus 119861) minus 119861] 119891 (119909) 119889119909

(5)

Theorem 1 (i) Under buy back guarantee contract the optimalordering quantity of the distributor is

(1 minus 119904) 119865 (119876lowast

) minus (119908 minus 119904) 119865 (119908119876lowast

minus 119861) = 0 (6)

(ii) Under buy back guarantee contract the optimal orderingquantity of the distributor 119876lowast decreases in 119908 and 119861

Proof (i) It is easy to get 120597119864Π119877120597119876 = (1 minus 119908) int+infin

119876119891(119909)119889119909 minus

(119908 minus 119904) int119876

119908119876minus119861119891(119909)119889119909 If 119876 satisfies 120597119864Π119877120597119876 = 0 then

1205972119864Π1198771205971198762 = minus(1 minus 119904)119865(119876)ℎ(119876) minus 119908ℎ(119908119876 minus 119861) lt 0 So theoptimal ordering quantity is (1minus119904)119865(119876)minus(119908minus119904)119865(119908119876minus119861) = 0

(ii) Let 119866(119876119908) = (1 minus 119904)119865(119876) minus (119908 minus 119904)119865(119908119876 minus 119861) then120597119876120597119908 = minus1minus (119908minus119904)119876ℎ(119908119876minus119861)(119908minus119904)ℎ(119876)minus119908ℎ(119908119876minus119861)

4 Journal of Applied Mathematics

According to the assumption we have119908 isin [119904 1] Suppose1198722(119908) = (119908 minus 119904)119865(119908119876 minus 119861) then1198721015840

2(119908) = 119865(119908119876 minus 119861)[1 minus

(119908 minus 119904)119876ℎ(119908119876 minus 119861)]119865(119908119876 minus 119861) and [1 minus (119908 minus 119904)119876ℎ(119908119876 minus 119861)] both decrease

in 119908 as the IFR property of random demand 120585 Then1198722(119908)

is unimodal in 119908 and achieves the maximum at 1 = (119908 minus119904)119876ℎ(119908119876minus119861) We have119872

2(119904) = 0 and119872

2(1) = (1 minus 119904)119865(119876minus

119861) gt (1 minus 119904)119865(119876)We conclude that for 119908 isin [119904 1] (1 minus 119904)119865(119876) = (119908 minus

119904)119865(119908119876 minus 119861) can only be satisfied on the positive slope of1198722(119908) so 1198721015840

2(119908) gt 0 [1 minus (119908 minus 119904)119876ℎ(119908119876 minus 119861)] gt 0 and

120597119876120597119908 lt 0 It is easy to get 120597119876120597119861 = minusℎ(119908119876 minus 119861)(ℎ(119876) minus119908ℎ(119908119876 minus 119861)) lt 0 This then indicates that Theorem 1 iscorrect

Recall that in the traditional newsvendor model (Hadleyand Whitin 1963) with no capital constraints the optimalordering quantity of the retailer is (1) That is a first-orderoptimality condition that requires the marginal revenue of anextra unit (1 minus 119904)119865(119876

0) to be equal to the cost of this extra

unit (119908 minus 119904)Under buy back guarantee contract the first-order opti-

mality condition satisfies (6) Since (119908minus 119904) ge (119908minus 119904)119865(119908119876lowast minus119861) it follows that a distributor (retailer) has a lower marginalcost in this case and 119876lowast ge 119876

0 It is because that the low

demand reduces the distributorrsquos downside risk Under thetrade credit contract the distributorrsquos (retailerrsquos) optimalordering quantity satisfies (4) since (119908 minus 119904)119865[((119908 minus 119904)(1 minus119904))119876119879 minus 119861(1 minus 119904)] ge (119908 minus 119904)119865(119908119876lowast minus 119861) it follows that 119876lowast ge119876119879 ge 119876

0ge 119876NF In particular 119904 = 0 the optimal ordering

quantity of distributor (retailer) satisfies119865(119876)minus119908119865(119908119876minus119861) =0 It is equal to the optimal ordering quantity under tradecredit contract [14] At this time the incentive effect of buyback guarantee contract and the trade credit contract is thesame Furthermore Theorem 1 also shows that the optimalordering quantity decreases in 119861 the reason why 120597119876120597119861 lt 0is that the distributor with a low budget has little stake atrisk when choosing its ordering quantity in the sense thatits potential losses are bounded by 119861 on the other handthe reason is the incentive effect of the buy back guaranteecontract

Theorem 2 Under buy back guarantee contract the optimalorder quantity of distributor 119876lowast increases in 119904

Proof Let 119866(119876119908) = (1 minus 119904)119865(119876) minus (119908 minus 119904)119865(119908119876 minus 119861)Then

120597119876

120597119904= minus

[119865 (119876) minus 119865 (119908119876 minus 119861)]

(119908 minus 119904) 119865 (119908119876 minus 119861) [ℎ (119876) minus 119908ℎ (119908119876 minus 119861)]gt 0

(7)

The increase of 119904 reduces the random market risk of dis-tributor faces and increases the expected profit of distributorso the distributor will increase the optimal ordering quantityto raise profit

42 The Core Enterprise The expected profit of the coreenterprise can be written as follows

119864Π119872= int119908119876minus119861

0

119876 minus[(119908119876 minus 119861) minus 119909]

119908 (119908 minus 119888) 119891 (119909) 119889119909

+ int+infin

119908119876minus119861

(119908 minus 119888)119876119891 (119909) 119889119909

(8)

Lemma 3 The optimal ordering quantity of distributor 119876satisfies 12059721198761205971199082 lt 0

Proof Let

120597119876

120597119908= minus

1 minus (119908 minus 119904)119876ℎ (119908119876 minus 119861)

(119908 minus 119904) ℎ (119876) minus 119908ℎ (119908119876 minus 119861) (9)

Then

119889119864Π119872

119889119908=120597119864Π119872

120597119876

120597119876

120597119908+120597119864Π119872

120597119908

= (119908 minus 119888) 119865 (119908119876 minus 119861)120597119876

120597119908

+ [119876119865 (119908119876 minus 119861) +119861119888

1199082119865 (119908119876 minus 119861)

+119888

1199082int119908119876minus119861

0

119909 119889119865 (119909)]

(10)

1205972119876

1205971199082= minus(

119865 (119908119876 minus 119861) [1 minus (119908 minus 119904)119876ℎ (119908119876 minus 119861)]

(1 minus 119904) 119891 (119876) minus (119908 minus 119904)119908119891 (119908119876 minus 119861))

1015840

119908

(11)

then

(119865 (119908119876 minus 119861) [1 minus (119908 minus 119904)119876ℎ (119908119876 minus 119861)]

(1 minus 119904) 119891 (119876) minus (119908 minus 119904)119908119891 (119908119876 minus 119861))

1015840

119908

=119865 (119908119876 minus 119861) [1 minus (119908 minus 119904)119876ℎ (119908119876 minus 119861)]

1015840

119908

[(1 minus 119904) 119891 (119876) minus (119908 minus 119904)119908119891 (119908119876 minus 119861)]2

times [(1 minus 119904) 119891 (119876) minus (119908 minus 119904)119908119891 (119908119876 minus 119861)]

minus[(1 minus 119904) 119891 (119876) minus (119908 minus 119904)119908119891 (119908119876 minus 119861)]

1015840

119908

[(1 minus 119904) 119891 (119876) minus (119908 minus 119904)119908119891 (119908119876 minus 119861)]2

times 119865 (119908119876 minus 119861) [1 minus (119908 minus 119904)119876ℎ (119908119876 minus 119861)]

=minus2119876119891 (119908119876 minus 119861) minus (119908 minus 119904)11987621198911015840 (119908119876 minus 119861)

[(1 minus 119904) 119891 (119876) minus (119908 minus 119904)119908119891 (119908119876 minus 119861)]2

times [(1 minus 119904) 119891 (119876) minus (119908 minus 119904)119908119891 (119908119876 minus 119861)]

Journal of Applied Mathematics 5

minus[minus (2119908 minus 119904) 119891 (119908119876 minus 119861) minus (119908 minus 119904)1199081198761198911015840 (119908119876 minus 119861)]

[(1 minus 119904) 119891 (119876) minus (119908 minus 119904)119908119891 (119908119876 minus 119861)]2

times 119865 (119908119876 minus 119861) [1 minus (119908 minus 119904)119876ℎ (119908119876 minus 119861)]

=1

119908119876

minus2119908119876119891 (119908119876 minus 119861) minus (119908 minus 119904)11990811987621198911015840 (119908119876 minus 119861)

[(1 minus 119904) 119891 (119876) minus (119908 minus 119904)119908119891 (119908119876 minus 119861)]2

times [(1 minus 119904)119876119891 (119876) minus (119908 minus 119904)119908119876119891 (119908119876 minus 119861)]

+1

119908119876[ (2119908 minus 119904)119876119891 (119908119876 minus 119861)

+ (119908 minus 119904)1199081198762

1198911015840

(119908119876 minus 119861)]

times ([(1 minus 119904) 119891 (119876) minus (119908 minus 119904)119908119891 (119908119876 minus 119861)]2

)minus1

times [119908119865 (119908119876 minus 119861) minus (119908 minus 119904)119908119876119891 (119908119876 minus 119861)]

=1

119908119876

minus 2119908119876119891 (119908119876 minus 119861) + (119908 minus 119904)11990811987621198911015840 (119908119876 minus 119861)

[(1 minus 119904) 119891 (119876) minus (119908 minus 119904)119908119891 (119908119876 minus 119861)]2

times [(1 minus 119904)119876119891 (119876) minus (119908 minus 119904)119908119876119891 (119908119876 minus 119861)]

+1

119908119876[2119908119876119891 (119908119876 minus 119861) + (119908 minus 119904)119908119876

2

1198911015840

(119908119876 minus 119861)

minus 119904119876119891 (119908119876 minus 119861)]

times ([(1 minus 119904) 119891 (119876) minus (119908 minus 119904)119908119891 (119908119876 minus 119861)]2

)minus1

times [(119908 minus 119904) 119865 (119908119876 minus 119861) minus (119908 minus 119904)119908119876119891 (119908119876 minus 119861)

+119904119865 (119908119876 minus 119861)]

=1

119908119876[(1 minus 119904) 119891 (119876) minus (119908 minus 119904)119908119891 (119908119876 minus 119861)]2times Δ1

+1

119908119876[(1 minus 119904) 119891 (119876) minus (119908 minus 119904)119908119891 (119908119876 minus 119861)]2Δ2

(12)

where

Δ1= minus [2119908119876119891 (119908119876 minus 119861) + (119908 minus 119904)119908119876

2

1198911015840

(119908119876 minus 119861)]

times [(1 minus 119904)119876119891 (119876) minus (119908 minus 119904)119908119876119891 (119908119876 minus 119861)]

+ [2119908119876119891 (119908119876 minus 119861) + (119908 minus 119904)1199081198762

1198911015840

(119908119876 minus 119861)]

times [(119908 minus 119904) 119865 (119908119876 minus 119861) minus (119908 minus 119904)119908119876119891 (119908119876 minus 119861)]

= [2119908119876119891 (119908119876 minus 119861) + (119908 minus 119904)1199081198762

1198911015840

(119908119876 minus 119861)]

times (1 minus 119904) 119865 (119908119876 minus 119861) 1 minus 119876ℎ (119876)

(13)

Since the demand distribution has the property of IFR Δ1gt

0Δ2= minus119904119876119891 (119908119876 minus 119861)

times [119908119865 (119908119876 minus 119861) minus (119908 minus 119904)119908119876119891 (119908119876 minus 119861)]

+ 119904119865 (119908119876 minus 119861)

times [(2119908 minus 119904)119876119891 (119908119876 minus 119861) + (119908 minus 119904)1199081198762

1198911015840

(119908119876 minus 119861)]

= (119908 minus 119904)1199081199041198912

(119908119876 minus 119861) + (119908 minus 119904) 119904119876119891 (119908119876 minus 119861)

times 119865 (119908119876 minus 119861) [1 minus 119908119876ℎ (119908119876 minus 119861)] gt 0

(14)

So 12059721198761205971199082 lt 0

Theorem 4 Under buy back guarantee contract the optimalwholesale price of the core enterprise 119908lowast can be expressed asfollows

(119908 minus 119888) 119865 (119908119876 minus 119861)120597119876

120597119908

+ [119876119865 (119908119876 minus 119861) +119861119888

1199082119865 (119908119876 minus 119861) +

119888

1199082int119908119876minus119861

0

119909 119889119865 (119909)]

= 0

(15)

Proof Consider1198892119864Π

119872

1198891199082= 2119865 (119908119876 minus 119861)

120597119876

120597119908[1 minus (119908 minus 119888)119876ℎ (119908119876 minus 119861)]

minus (119908 minus 119888)119908119891 (119908119876 minus 119861) (120597119876

120597119908)2

+ (119908 minus 119888) 119865 (119908119876 minus 119861)1205972119876

1205971199082

minus (1 minus119888

119908)1198762

119891 (119908119876 minus 119861) minus 2119861119888

1199083119865 (119908119876 minus 119861)

minus 2119888

1199083int119908119876minus119861

0

119909119891 (119909) 119889119909 lt 0

(16)

From Lemma 3 the optimal wholesale price of the coreenterprise can be expressed as follows

(119908 minus 119888) 119865 (119908119876 minus 119861)120597119876

120597119908+ 119876119865 (119908119876 minus 119861) +

119861119888

1199082119865 (119908119876 minus 119861)

+119888

1199082int119908119876minus119861

0

119909 119889119865 (119909) = 0

(17)

5 Numerical Example

Figure 2 gives the relation between the distributorrsquos initialcapital and the optimal ordering quantity We assume that

6 Journal of Applied Mathematics

0 50 100 150 200 250 300 350 400 450 5000

500

1000

1500

2000

B

Q

Retailerrsquos optimal ordering level under buy back

Retailerrsquos optimal ordering level under trade credit contractThe optimal ordering level in traditional newsvendor modelRetailerrsquos optimal ordering level without financing

guarantee contract

Figure 2 Relation between the optimal ordering quantities withdifferent financial modes

Table 1 Optimal ordering quantity of the distributor and the totalprofit of supply chain system

119861

119876

119876NF 1198760

119876119879 119876lowast 119876119862

Π

ΠNF Π0

Π119879 Πlowast Π119862

119861 = 5063 405 935 1406 176630 159 243 404 321

119861 = 100125 405 709 1237 176658 159 236 374 321

119861 = 200250 405 405 866 176689 159 192 294 321

119861 = 400405 405 405 405 1766158 160 192 159 321

119901 = 1 119908 = 08 and 119904 = 04 and 120585simexponential (11000)From Figure 2 we know that the optimal ordering quantity119876lowast is bigger than 119876119879 the optimal ordering quantity of thetrade credit contractThe optimal ordering quantities119876lowast and119876119879 are both bigger than 119876NF the optimal ordering quantityof the retailer without financing service It also indicates thatthe 119876lowast decreases in the initial budget 119861 of the retailer Theseconclusions are all consistent withTheorem 1

Let ΠNF Π0 Π119879 Πlowast and Π119862 be the total expected

profits of supply chain without system finance the traditionalnewsvendormodel traded credit financing buy back guaran-tee financing and the centralized decision without financeWith the change of the initial capital let 119861 = 50 100 200 and400 Table 1 gives the optimal ordering quantity of the retailerand the total profit of supply chain system

6 Conclusions

Working under the assumption of stochastic market demandand a distributor that is capital constraints this paper hasconsidered the optimal ordering quantity of the distributor

and the wholesale price of the core enterprise with buy backguarantee contract in a single stage supply chain systemWe compare the optimal ordering quantities of the differ-ent financing modes trade credit contract the traditionalnewsvendor model and buy back guarantee contract Thestudy has highlighted how the capital-constrained distributorcould be financed well facing different financing modes andthe research results show that the different financing modesbring different expected profits for supply chain systemwith the capital-constrained distributor The results couldbe applied for the distributor and the core enterprise whenthey make decisions in reality The paper has also raised thepotential for further research into finance contact of supplychain where more than one distributor and core enterprisecan also be considered

Conflict of Interests

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

Acknowledgments

This work was financially supported by the Key Project oftheNationalNatural Science Foundation of China (71131003)Project of the National Natural Science Foundation of China(71371075) China Postdoctoral Science Foundation fundedProject (2013M542183)GuangdongProvinceNatural ScienceFund (S2013040014920) of China and Research Fund for theDoctoral ProgramofGuangdongUniversity of Technology ofChina (13ZK0155)

References

[1] J A Buzacott and R Q Zhang ldquoInventory management withasset-based financingrdquo Management Science vol 50 no 9 pp1274ndash1292 2004

[2] X D Xu and J R Birge ldquoJoint production and financing deci-sions modeling and analysisrdquoWorking PaperTheUniversity ofChicago Graduate School of Business 2004

[3] X D Xu and J R Birge ldquoOperational decisions capitalstructure and managerial compensation a news vendor per-spectiverdquo Working Paper The University of Chicago GraduateSchool of Business 2005

[4] R Swinney G Cachon and S Netessine ldquoCapacity investmentby competitive startupsrdquo Working Paper The Wharton SchoolUniversity of Pennsylvania 2005

[5] L Li M Shubik and M Sobel ldquoControl of dividends capitalsubscriptions and physical inventoriesrdquo Working Paper YaleSchool of Management 2005

[6] V Babich G Aydin P Brunet J Keppo and R Saigal ldquoRiskfinancing and optimal number of suppliersrdquo Working PaperIOE University of Michigan Ann Arbor Mich USA 2005

[7] Y Xu and J Zhang ldquoOn the selection of supply chain coordi-nating contracts the role of capital constraintrdquo Working PaperUniversity of Miami 2007

[8] M Dada and Q Hu ldquoFinancing newsvendor inventoryrdquo Oper-ations Research Letters vol 36 no 5 pp 569ndash573 2008

Journal of Applied Mathematics 7

[9] D Gupta ldquoTechnical note financing the newsvendorrdquoWorkingPaper Industrial and Systems Engineering University of Min-nesota 2008

[10] G Lai L G Debo and K Sycara ldquoSharing inventory risk insupply chain the implication of financial constraintrdquo Omegavol 37 no 4 pp 811ndash825 2009

[11] P Kouvelis and H W Zhao ldquoFinancing the NewsvendorSupplier vs Bank and the Structure of Optimal Trade CreditContractsrdquo Working paper Olin Business School WashingtonUniversity September 2009

[12] P Kouvelis and W H Zhao ldquoThe newsvendor problem andprice-only contract when bankruptcy costs existrdquo WorkingPaper Olin Business School Washington University 2010

[13] X F Chen and G Wan ldquoThe effect of financing on a budget-constrained supply chain under wholesale price contractrdquoAsia-Pacific Journal of Operational Research vol 28 no 4 pp 457ndash485 2011

[14] X F Chen and A Wang ldquoTrade credit contract with limitedliability in the supply chain with budget constraintsrdquo Annals ofOperations Research vol 196 pp 153ndash165 2012

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Differential EquationsInternational Journal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Operations ResearchAdvances in

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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

Journal of Applied Mathematics 3

Distributor Core

Warehousingenterprise

Bank

(2) 30 payment

(3) BA

(4) Guarantee money(6) Buy back and refund

(5) Deliver goods

(5) Notice of delivery

(7) Bill payment

enterprise

Figure 1 The process of buy back guarantee contract

(6) Distributor remits the sales to the appointed accountin bank after selling steel to redeem certificates tofulfill selling

(7) If the distributor sells all steel when the BA is due thebank withdraws the capital and releases the BA If thedistributor fails to sell all steel at the end of the sellingperiod the bank releases the BA after core enterprisebuys back the unsold steel

The implemented process is shown in Figure 1 Thedashed box shows that warehousing enterprise is notrequired

We consider a simple supply chain with a core enterpriseand a capital-constrained distributorThe core enterprise pro-duces a single product and the distributor faces a nonnegativerandom demand 120585 with a distribution function (similar tothe model discussed in Cachon (2003) and Lariviere andPorteus (2001)) We formulate the interaction between thedistributor and the core enterprise in the framework of aStackelberg game at the beginning of the selling period thecore enterprise as the Stacklberg leader and the distributor asthe follower sign buy back guarantee contract After the endof the selling period the enterprise buys back the left goodsaccording to the exposure between bank acceptance and thecash deposit Let us assume denotations as follows

120585 market demand is a nonnegative random vari-able with probability density function 119891(119909) and thecumulative distribution function 119865(119909) respectivelySuppose the hazard function ℎ(119909) = 119891(119909)119865(119909)(IFR) and the generalized failure rate (GFR) 119867(119909) =119909119891(119909)119865(119909) where 119865(119909) = 1 minus 119865(119909) is the taildistribution of 119865(119909)

119876 is the ordering quantity of the retailer

119901 is the unit retail price (suppose 119901 = 1)

119908 is the wholesale price (the unit ordering cost)

119861 is the initial capital of distributor

119888 is the unit production cost

119904 is the salvage of the product Without loss ofgenerality assume 119901 = 1 ge 119908 gt 119888 gt 119904

3 Result Concerned with the Paper

When the retailer has no capital constraints the profit of theretailer can be written as follows min119909 119876minus119908119876+119904(119876minus119909)+The profit of the supplier can be written as follows (119908 minus 119888)119876

The optimal decentralized ordering quantity 1198760is

(1 minus 119904) 119865 (1198760) = (119908 minus 119904) (1)

The optimal centralized ordering quantity 119876119862 is

(1 minus 119904) 119865 (119876119862

) = (119888 minus 119904) (2)

We have 1198760lt 119876119862

When the capital-constrained retailer (distributor) has noaccess to the financial market its optimal ordering quantityis

119876NF = min 1198611199081198760 (3)

(see [13])Under the trade credit contract the optimal inventory

level of the retailer 119876119879 is [14]

(1 minus 119904) 119865 (119876119879

) = (119908 minus 119904) 119865 [(119908 minus 119904)

(1 minus 119904)119876119879

minus119861

(1 minus 119904)] (4)

4 Model Calculations

41 The Distributor The expected profit of the capital-constrained distributor can be written as follows

119864Π119877

= int119908119876minus119861

0

[119904 (119876 minus 119909) minus1

119908[(119908119876 minus 119861) minus 119909] minus 119861]

times 119891 (119909) 119889119909

+ int119876

119908119876minus119861

[119909 minus (119908119876 minus 119861) + 119904 (119876 minus 119909) minus 119861] 119891 (119909) 119889119909

+ int+infin

119876

[119876 minus (119908119876 minus 119861) minus 119861] 119891 (119909) 119889119909

(5)

Theorem 1 (i) Under buy back guarantee contract the optimalordering quantity of the distributor is

(1 minus 119904) 119865 (119876lowast

) minus (119908 minus 119904) 119865 (119908119876lowast

minus 119861) = 0 (6)

(ii) Under buy back guarantee contract the optimal orderingquantity of the distributor 119876lowast decreases in 119908 and 119861

Proof (i) It is easy to get 120597119864Π119877120597119876 = (1 minus 119908) int+infin

119876119891(119909)119889119909 minus

(119908 minus 119904) int119876

119908119876minus119861119891(119909)119889119909 If 119876 satisfies 120597119864Π119877120597119876 = 0 then

1205972119864Π1198771205971198762 = minus(1 minus 119904)119865(119876)ℎ(119876) minus 119908ℎ(119908119876 minus 119861) lt 0 So theoptimal ordering quantity is (1minus119904)119865(119876)minus(119908minus119904)119865(119908119876minus119861) = 0

(ii) Let 119866(119876119908) = (1 minus 119904)119865(119876) minus (119908 minus 119904)119865(119908119876 minus 119861) then120597119876120597119908 = minus1minus (119908minus119904)119876ℎ(119908119876minus119861)(119908minus119904)ℎ(119876)minus119908ℎ(119908119876minus119861)

4 Journal of Applied Mathematics

According to the assumption we have119908 isin [119904 1] Suppose1198722(119908) = (119908 minus 119904)119865(119908119876 minus 119861) then1198721015840

2(119908) = 119865(119908119876 minus 119861)[1 minus

(119908 minus 119904)119876ℎ(119908119876 minus 119861)]119865(119908119876 minus 119861) and [1 minus (119908 minus 119904)119876ℎ(119908119876 minus 119861)] both decrease

in 119908 as the IFR property of random demand 120585 Then1198722(119908)

is unimodal in 119908 and achieves the maximum at 1 = (119908 minus119904)119876ℎ(119908119876minus119861) We have119872

2(119904) = 0 and119872

2(1) = (1 minus 119904)119865(119876minus

119861) gt (1 minus 119904)119865(119876)We conclude that for 119908 isin [119904 1] (1 minus 119904)119865(119876) = (119908 minus

119904)119865(119908119876 minus 119861) can only be satisfied on the positive slope of1198722(119908) so 1198721015840

2(119908) gt 0 [1 minus (119908 minus 119904)119876ℎ(119908119876 minus 119861)] gt 0 and

120597119876120597119908 lt 0 It is easy to get 120597119876120597119861 = minusℎ(119908119876 minus 119861)(ℎ(119876) minus119908ℎ(119908119876 minus 119861)) lt 0 This then indicates that Theorem 1 iscorrect

Recall that in the traditional newsvendor model (Hadleyand Whitin 1963) with no capital constraints the optimalordering quantity of the retailer is (1) That is a first-orderoptimality condition that requires the marginal revenue of anextra unit (1 minus 119904)119865(119876

0) to be equal to the cost of this extra

unit (119908 minus 119904)Under buy back guarantee contract the first-order opti-

mality condition satisfies (6) Since (119908minus 119904) ge (119908minus 119904)119865(119908119876lowast minus119861) it follows that a distributor (retailer) has a lower marginalcost in this case and 119876lowast ge 119876

0 It is because that the low

demand reduces the distributorrsquos downside risk Under thetrade credit contract the distributorrsquos (retailerrsquos) optimalordering quantity satisfies (4) since (119908 minus 119904)119865[((119908 minus 119904)(1 minus119904))119876119879 minus 119861(1 minus 119904)] ge (119908 minus 119904)119865(119908119876lowast minus 119861) it follows that 119876lowast ge119876119879 ge 119876

0ge 119876NF In particular 119904 = 0 the optimal ordering

quantity of distributor (retailer) satisfies119865(119876)minus119908119865(119908119876minus119861) =0 It is equal to the optimal ordering quantity under tradecredit contract [14] At this time the incentive effect of buyback guarantee contract and the trade credit contract is thesame Furthermore Theorem 1 also shows that the optimalordering quantity decreases in 119861 the reason why 120597119876120597119861 lt 0is that the distributor with a low budget has little stake atrisk when choosing its ordering quantity in the sense thatits potential losses are bounded by 119861 on the other handthe reason is the incentive effect of the buy back guaranteecontract

Theorem 2 Under buy back guarantee contract the optimalorder quantity of distributor 119876lowast increases in 119904

Proof Let 119866(119876119908) = (1 minus 119904)119865(119876) minus (119908 minus 119904)119865(119908119876 minus 119861)Then

120597119876

120597119904= minus

[119865 (119876) minus 119865 (119908119876 minus 119861)]

(119908 minus 119904) 119865 (119908119876 minus 119861) [ℎ (119876) minus 119908ℎ (119908119876 minus 119861)]gt 0

(7)

The increase of 119904 reduces the random market risk of dis-tributor faces and increases the expected profit of distributorso the distributor will increase the optimal ordering quantityto raise profit

42 The Core Enterprise The expected profit of the coreenterprise can be written as follows

119864Π119872= int119908119876minus119861

0

119876 minus[(119908119876 minus 119861) minus 119909]

119908 (119908 minus 119888) 119891 (119909) 119889119909

+ int+infin

119908119876minus119861

(119908 minus 119888)119876119891 (119909) 119889119909

(8)

Lemma 3 The optimal ordering quantity of distributor 119876satisfies 12059721198761205971199082 lt 0

Proof Let

120597119876

120597119908= minus

1 minus (119908 minus 119904)119876ℎ (119908119876 minus 119861)

(119908 minus 119904) ℎ (119876) minus 119908ℎ (119908119876 minus 119861) (9)

Then

119889119864Π119872

119889119908=120597119864Π119872

120597119876

120597119876

120597119908+120597119864Π119872

120597119908

= (119908 minus 119888) 119865 (119908119876 minus 119861)120597119876

120597119908

+ [119876119865 (119908119876 minus 119861) +119861119888

1199082119865 (119908119876 minus 119861)

+119888

1199082int119908119876minus119861

0

119909 119889119865 (119909)]

(10)

1205972119876

1205971199082= minus(

119865 (119908119876 minus 119861) [1 minus (119908 minus 119904)119876ℎ (119908119876 minus 119861)]

(1 minus 119904) 119891 (119876) minus (119908 minus 119904)119908119891 (119908119876 minus 119861))

1015840

119908

(11)

then

(119865 (119908119876 minus 119861) [1 minus (119908 minus 119904)119876ℎ (119908119876 minus 119861)]

(1 minus 119904) 119891 (119876) minus (119908 minus 119904)119908119891 (119908119876 minus 119861))

1015840

119908

=119865 (119908119876 minus 119861) [1 minus (119908 minus 119904)119876ℎ (119908119876 minus 119861)]

1015840

119908

[(1 minus 119904) 119891 (119876) minus (119908 minus 119904)119908119891 (119908119876 minus 119861)]2

times [(1 minus 119904) 119891 (119876) minus (119908 minus 119904)119908119891 (119908119876 minus 119861)]

minus[(1 minus 119904) 119891 (119876) minus (119908 minus 119904)119908119891 (119908119876 minus 119861)]

1015840

119908

[(1 minus 119904) 119891 (119876) minus (119908 minus 119904)119908119891 (119908119876 minus 119861)]2

times 119865 (119908119876 minus 119861) [1 minus (119908 minus 119904)119876ℎ (119908119876 minus 119861)]

=minus2119876119891 (119908119876 minus 119861) minus (119908 minus 119904)11987621198911015840 (119908119876 minus 119861)

[(1 minus 119904) 119891 (119876) minus (119908 minus 119904)119908119891 (119908119876 minus 119861)]2

times [(1 minus 119904) 119891 (119876) minus (119908 minus 119904)119908119891 (119908119876 minus 119861)]

Journal of Applied Mathematics 5

minus[minus (2119908 minus 119904) 119891 (119908119876 minus 119861) minus (119908 minus 119904)1199081198761198911015840 (119908119876 minus 119861)]

[(1 minus 119904) 119891 (119876) minus (119908 minus 119904)119908119891 (119908119876 minus 119861)]2

times 119865 (119908119876 minus 119861) [1 minus (119908 minus 119904)119876ℎ (119908119876 minus 119861)]

=1

119908119876

minus2119908119876119891 (119908119876 minus 119861) minus (119908 minus 119904)11990811987621198911015840 (119908119876 minus 119861)

[(1 minus 119904) 119891 (119876) minus (119908 minus 119904)119908119891 (119908119876 minus 119861)]2

times [(1 minus 119904)119876119891 (119876) minus (119908 minus 119904)119908119876119891 (119908119876 minus 119861)]

+1

119908119876[ (2119908 minus 119904)119876119891 (119908119876 minus 119861)

+ (119908 minus 119904)1199081198762

1198911015840

(119908119876 minus 119861)]

times ([(1 minus 119904) 119891 (119876) minus (119908 minus 119904)119908119891 (119908119876 minus 119861)]2

)minus1

times [119908119865 (119908119876 minus 119861) minus (119908 minus 119904)119908119876119891 (119908119876 minus 119861)]

=1

119908119876

minus 2119908119876119891 (119908119876 minus 119861) + (119908 minus 119904)11990811987621198911015840 (119908119876 minus 119861)

[(1 minus 119904) 119891 (119876) minus (119908 minus 119904)119908119891 (119908119876 minus 119861)]2

times [(1 minus 119904)119876119891 (119876) minus (119908 minus 119904)119908119876119891 (119908119876 minus 119861)]

+1

119908119876[2119908119876119891 (119908119876 minus 119861) + (119908 minus 119904)119908119876

2

1198911015840

(119908119876 minus 119861)

minus 119904119876119891 (119908119876 minus 119861)]

times ([(1 minus 119904) 119891 (119876) minus (119908 minus 119904)119908119891 (119908119876 minus 119861)]2

)minus1

times [(119908 minus 119904) 119865 (119908119876 minus 119861) minus (119908 minus 119904)119908119876119891 (119908119876 minus 119861)

+119904119865 (119908119876 minus 119861)]

=1

119908119876[(1 minus 119904) 119891 (119876) minus (119908 minus 119904)119908119891 (119908119876 minus 119861)]2times Δ1

+1

119908119876[(1 minus 119904) 119891 (119876) minus (119908 minus 119904)119908119891 (119908119876 minus 119861)]2Δ2

(12)

where

Δ1= minus [2119908119876119891 (119908119876 minus 119861) + (119908 minus 119904)119908119876

2

1198911015840

(119908119876 minus 119861)]

times [(1 minus 119904)119876119891 (119876) minus (119908 minus 119904)119908119876119891 (119908119876 minus 119861)]

+ [2119908119876119891 (119908119876 minus 119861) + (119908 minus 119904)1199081198762

1198911015840

(119908119876 minus 119861)]

times [(119908 minus 119904) 119865 (119908119876 minus 119861) minus (119908 minus 119904)119908119876119891 (119908119876 minus 119861)]

= [2119908119876119891 (119908119876 minus 119861) + (119908 minus 119904)1199081198762

1198911015840

(119908119876 minus 119861)]

times (1 minus 119904) 119865 (119908119876 minus 119861) 1 minus 119876ℎ (119876)

(13)

Since the demand distribution has the property of IFR Δ1gt

0Δ2= minus119904119876119891 (119908119876 minus 119861)

times [119908119865 (119908119876 minus 119861) minus (119908 minus 119904)119908119876119891 (119908119876 minus 119861)]

+ 119904119865 (119908119876 minus 119861)

times [(2119908 minus 119904)119876119891 (119908119876 minus 119861) + (119908 minus 119904)1199081198762

1198911015840

(119908119876 minus 119861)]

= (119908 minus 119904)1199081199041198912

(119908119876 minus 119861) + (119908 minus 119904) 119904119876119891 (119908119876 minus 119861)

times 119865 (119908119876 minus 119861) [1 minus 119908119876ℎ (119908119876 minus 119861)] gt 0

(14)

So 12059721198761205971199082 lt 0

Theorem 4 Under buy back guarantee contract the optimalwholesale price of the core enterprise 119908lowast can be expressed asfollows

(119908 minus 119888) 119865 (119908119876 minus 119861)120597119876

120597119908

+ [119876119865 (119908119876 minus 119861) +119861119888

1199082119865 (119908119876 minus 119861) +

119888

1199082int119908119876minus119861

0

119909 119889119865 (119909)]

= 0

(15)

Proof Consider1198892119864Π

119872

1198891199082= 2119865 (119908119876 minus 119861)

120597119876

120597119908[1 minus (119908 minus 119888)119876ℎ (119908119876 minus 119861)]

minus (119908 minus 119888)119908119891 (119908119876 minus 119861) (120597119876

120597119908)2

+ (119908 minus 119888) 119865 (119908119876 minus 119861)1205972119876

1205971199082

minus (1 minus119888

119908)1198762

119891 (119908119876 minus 119861) minus 2119861119888

1199083119865 (119908119876 minus 119861)

minus 2119888

1199083int119908119876minus119861

0

119909119891 (119909) 119889119909 lt 0

(16)

From Lemma 3 the optimal wholesale price of the coreenterprise can be expressed as follows

(119908 minus 119888) 119865 (119908119876 minus 119861)120597119876

120597119908+ 119876119865 (119908119876 minus 119861) +

119861119888

1199082119865 (119908119876 minus 119861)

+119888

1199082int119908119876minus119861

0

119909 119889119865 (119909) = 0

(17)

5 Numerical Example

Figure 2 gives the relation between the distributorrsquos initialcapital and the optimal ordering quantity We assume that

6 Journal of Applied Mathematics

0 50 100 150 200 250 300 350 400 450 5000

500

1000

1500

2000

B

Q

Retailerrsquos optimal ordering level under buy back

Retailerrsquos optimal ordering level under trade credit contractThe optimal ordering level in traditional newsvendor modelRetailerrsquos optimal ordering level without financing

guarantee contract

Figure 2 Relation between the optimal ordering quantities withdifferent financial modes

Table 1 Optimal ordering quantity of the distributor and the totalprofit of supply chain system

119861

119876

119876NF 1198760

119876119879 119876lowast 119876119862

Π

ΠNF Π0

Π119879 Πlowast Π119862

119861 = 5063 405 935 1406 176630 159 243 404 321

119861 = 100125 405 709 1237 176658 159 236 374 321

119861 = 200250 405 405 866 176689 159 192 294 321

119861 = 400405 405 405 405 1766158 160 192 159 321

119901 = 1 119908 = 08 and 119904 = 04 and 120585simexponential (11000)From Figure 2 we know that the optimal ordering quantity119876lowast is bigger than 119876119879 the optimal ordering quantity of thetrade credit contractThe optimal ordering quantities119876lowast and119876119879 are both bigger than 119876NF the optimal ordering quantityof the retailer without financing service It also indicates thatthe 119876lowast decreases in the initial budget 119861 of the retailer Theseconclusions are all consistent withTheorem 1

Let ΠNF Π0 Π119879 Πlowast and Π119862 be the total expected

profits of supply chain without system finance the traditionalnewsvendormodel traded credit financing buy back guaran-tee financing and the centralized decision without financeWith the change of the initial capital let 119861 = 50 100 200 and400 Table 1 gives the optimal ordering quantity of the retailerand the total profit of supply chain system

6 Conclusions

Working under the assumption of stochastic market demandand a distributor that is capital constraints this paper hasconsidered the optimal ordering quantity of the distributor

and the wholesale price of the core enterprise with buy backguarantee contract in a single stage supply chain systemWe compare the optimal ordering quantities of the differ-ent financing modes trade credit contract the traditionalnewsvendor model and buy back guarantee contract Thestudy has highlighted how the capital-constrained distributorcould be financed well facing different financing modes andthe research results show that the different financing modesbring different expected profits for supply chain systemwith the capital-constrained distributor The results couldbe applied for the distributor and the core enterprise whenthey make decisions in reality The paper has also raised thepotential for further research into finance contact of supplychain where more than one distributor and core enterprisecan also be considered

Conflict of Interests

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

Acknowledgments

This work was financially supported by the Key Project oftheNationalNatural Science Foundation of China (71131003)Project of the National Natural Science Foundation of China(71371075) China Postdoctoral Science Foundation fundedProject (2013M542183)GuangdongProvinceNatural ScienceFund (S2013040014920) of China and Research Fund for theDoctoral ProgramofGuangdongUniversity of Technology ofChina (13ZK0155)

References

[1] J A Buzacott and R Q Zhang ldquoInventory management withasset-based financingrdquo Management Science vol 50 no 9 pp1274ndash1292 2004

[2] X D Xu and J R Birge ldquoJoint production and financing deci-sions modeling and analysisrdquoWorking PaperTheUniversity ofChicago Graduate School of Business 2004

[3] X D Xu and J R Birge ldquoOperational decisions capitalstructure and managerial compensation a news vendor per-spectiverdquo Working Paper The University of Chicago GraduateSchool of Business 2005

[4] R Swinney G Cachon and S Netessine ldquoCapacity investmentby competitive startupsrdquo Working Paper The Wharton SchoolUniversity of Pennsylvania 2005

[5] L Li M Shubik and M Sobel ldquoControl of dividends capitalsubscriptions and physical inventoriesrdquo Working Paper YaleSchool of Management 2005

[6] V Babich G Aydin P Brunet J Keppo and R Saigal ldquoRiskfinancing and optimal number of suppliersrdquo Working PaperIOE University of Michigan Ann Arbor Mich USA 2005

[7] Y Xu and J Zhang ldquoOn the selection of supply chain coordi-nating contracts the role of capital constraintrdquo Working PaperUniversity of Miami 2007

[8] M Dada and Q Hu ldquoFinancing newsvendor inventoryrdquo Oper-ations Research Letters vol 36 no 5 pp 569ndash573 2008

Journal of Applied Mathematics 7

[9] D Gupta ldquoTechnical note financing the newsvendorrdquoWorkingPaper Industrial and Systems Engineering University of Min-nesota 2008

[10] G Lai L G Debo and K Sycara ldquoSharing inventory risk insupply chain the implication of financial constraintrdquo Omegavol 37 no 4 pp 811ndash825 2009

[11] P Kouvelis and H W Zhao ldquoFinancing the NewsvendorSupplier vs Bank and the Structure of Optimal Trade CreditContractsrdquo Working paper Olin Business School WashingtonUniversity September 2009

[12] P Kouvelis and W H Zhao ldquoThe newsvendor problem andprice-only contract when bankruptcy costs existrdquo WorkingPaper Olin Business School Washington University 2010

[13] X F Chen and G Wan ldquoThe effect of financing on a budget-constrained supply chain under wholesale price contractrdquoAsia-Pacific Journal of Operational Research vol 28 no 4 pp 457ndash485 2011

[14] X F Chen and A Wang ldquoTrade credit contract with limitedliability in the supply chain with budget constraintsrdquo Annals ofOperations Research vol 196 pp 153ndash165 2012

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

4 Journal of Applied Mathematics

According to the assumption we have119908 isin [119904 1] Suppose1198722(119908) = (119908 minus 119904)119865(119908119876 minus 119861) then1198721015840

2(119908) = 119865(119908119876 minus 119861)[1 minus

(119908 minus 119904)119876ℎ(119908119876 minus 119861)]119865(119908119876 minus 119861) and [1 minus (119908 minus 119904)119876ℎ(119908119876 minus 119861)] both decrease

in 119908 as the IFR property of random demand 120585 Then1198722(119908)

is unimodal in 119908 and achieves the maximum at 1 = (119908 minus119904)119876ℎ(119908119876minus119861) We have119872

2(119904) = 0 and119872

2(1) = (1 minus 119904)119865(119876minus

119861) gt (1 minus 119904)119865(119876)We conclude that for 119908 isin [119904 1] (1 minus 119904)119865(119876) = (119908 minus

119904)119865(119908119876 minus 119861) can only be satisfied on the positive slope of1198722(119908) so 1198721015840

2(119908) gt 0 [1 minus (119908 minus 119904)119876ℎ(119908119876 minus 119861)] gt 0 and

120597119876120597119908 lt 0 It is easy to get 120597119876120597119861 = minusℎ(119908119876 minus 119861)(ℎ(119876) minus119908ℎ(119908119876 minus 119861)) lt 0 This then indicates that Theorem 1 iscorrect

Recall that in the traditional newsvendor model (Hadleyand Whitin 1963) with no capital constraints the optimalordering quantity of the retailer is (1) That is a first-orderoptimality condition that requires the marginal revenue of anextra unit (1 minus 119904)119865(119876

0) to be equal to the cost of this extra

unit (119908 minus 119904)Under buy back guarantee contract the first-order opti-

mality condition satisfies (6) Since (119908minus 119904) ge (119908minus 119904)119865(119908119876lowast minus119861) it follows that a distributor (retailer) has a lower marginalcost in this case and 119876lowast ge 119876

0 It is because that the low

demand reduces the distributorrsquos downside risk Under thetrade credit contract the distributorrsquos (retailerrsquos) optimalordering quantity satisfies (4) since (119908 minus 119904)119865[((119908 minus 119904)(1 minus119904))119876119879 minus 119861(1 minus 119904)] ge (119908 minus 119904)119865(119908119876lowast minus 119861) it follows that 119876lowast ge119876119879 ge 119876

0ge 119876NF In particular 119904 = 0 the optimal ordering

quantity of distributor (retailer) satisfies119865(119876)minus119908119865(119908119876minus119861) =0 It is equal to the optimal ordering quantity under tradecredit contract [14] At this time the incentive effect of buyback guarantee contract and the trade credit contract is thesame Furthermore Theorem 1 also shows that the optimalordering quantity decreases in 119861 the reason why 120597119876120597119861 lt 0is that the distributor with a low budget has little stake atrisk when choosing its ordering quantity in the sense thatits potential losses are bounded by 119861 on the other handthe reason is the incentive effect of the buy back guaranteecontract

Theorem 2 Under buy back guarantee contract the optimalorder quantity of distributor 119876lowast increases in 119904

Proof Let 119866(119876119908) = (1 minus 119904)119865(119876) minus (119908 minus 119904)119865(119908119876 minus 119861)Then

120597119876

120597119904= minus

[119865 (119876) minus 119865 (119908119876 minus 119861)]

(119908 minus 119904) 119865 (119908119876 minus 119861) [ℎ (119876) minus 119908ℎ (119908119876 minus 119861)]gt 0

(7)

The increase of 119904 reduces the random market risk of dis-tributor faces and increases the expected profit of distributorso the distributor will increase the optimal ordering quantityto raise profit

42 The Core Enterprise The expected profit of the coreenterprise can be written as follows

119864Π119872= int119908119876minus119861

0

119876 minus[(119908119876 minus 119861) minus 119909]

119908 (119908 minus 119888) 119891 (119909) 119889119909

+ int+infin

119908119876minus119861

(119908 minus 119888)119876119891 (119909) 119889119909

(8)

Lemma 3 The optimal ordering quantity of distributor 119876satisfies 12059721198761205971199082 lt 0

Proof Let

120597119876

120597119908= minus

1 minus (119908 minus 119904)119876ℎ (119908119876 minus 119861)

(119908 minus 119904) ℎ (119876) minus 119908ℎ (119908119876 minus 119861) (9)

Then

119889119864Π119872

119889119908=120597119864Π119872

120597119876

120597119876

120597119908+120597119864Π119872

120597119908

= (119908 minus 119888) 119865 (119908119876 minus 119861)120597119876

120597119908

+ [119876119865 (119908119876 minus 119861) +119861119888

1199082119865 (119908119876 minus 119861)

+119888

1199082int119908119876minus119861

0

119909 119889119865 (119909)]

(10)

1205972119876

1205971199082= minus(

119865 (119908119876 minus 119861) [1 minus (119908 minus 119904)119876ℎ (119908119876 minus 119861)]

(1 minus 119904) 119891 (119876) minus (119908 minus 119904)119908119891 (119908119876 minus 119861))

1015840

119908

(11)

then

(119865 (119908119876 minus 119861) [1 minus (119908 minus 119904)119876ℎ (119908119876 minus 119861)]

(1 minus 119904) 119891 (119876) minus (119908 minus 119904)119908119891 (119908119876 minus 119861))

1015840

119908

=119865 (119908119876 minus 119861) [1 minus (119908 minus 119904)119876ℎ (119908119876 minus 119861)]

1015840

119908

[(1 minus 119904) 119891 (119876) minus (119908 minus 119904)119908119891 (119908119876 minus 119861)]2

times [(1 minus 119904) 119891 (119876) minus (119908 minus 119904)119908119891 (119908119876 minus 119861)]

minus[(1 minus 119904) 119891 (119876) minus (119908 minus 119904)119908119891 (119908119876 minus 119861)]

1015840

119908

[(1 minus 119904) 119891 (119876) minus (119908 minus 119904)119908119891 (119908119876 minus 119861)]2

times 119865 (119908119876 minus 119861) [1 minus (119908 minus 119904)119876ℎ (119908119876 minus 119861)]

=minus2119876119891 (119908119876 minus 119861) minus (119908 minus 119904)11987621198911015840 (119908119876 minus 119861)

[(1 minus 119904) 119891 (119876) minus (119908 minus 119904)119908119891 (119908119876 minus 119861)]2

times [(1 minus 119904) 119891 (119876) minus (119908 minus 119904)119908119891 (119908119876 minus 119861)]

Journal of Applied Mathematics 5

minus[minus (2119908 minus 119904) 119891 (119908119876 minus 119861) minus (119908 minus 119904)1199081198761198911015840 (119908119876 minus 119861)]

[(1 minus 119904) 119891 (119876) minus (119908 minus 119904)119908119891 (119908119876 minus 119861)]2

times 119865 (119908119876 minus 119861) [1 minus (119908 minus 119904)119876ℎ (119908119876 minus 119861)]

=1

119908119876

minus2119908119876119891 (119908119876 minus 119861) minus (119908 minus 119904)11990811987621198911015840 (119908119876 minus 119861)

[(1 minus 119904) 119891 (119876) minus (119908 minus 119904)119908119891 (119908119876 minus 119861)]2

times [(1 minus 119904)119876119891 (119876) minus (119908 minus 119904)119908119876119891 (119908119876 minus 119861)]

+1

119908119876[ (2119908 minus 119904)119876119891 (119908119876 minus 119861)

+ (119908 minus 119904)1199081198762

1198911015840

(119908119876 minus 119861)]

times ([(1 minus 119904) 119891 (119876) minus (119908 minus 119904)119908119891 (119908119876 minus 119861)]2

)minus1

times [119908119865 (119908119876 minus 119861) minus (119908 minus 119904)119908119876119891 (119908119876 minus 119861)]

=1

119908119876

minus 2119908119876119891 (119908119876 minus 119861) + (119908 minus 119904)11990811987621198911015840 (119908119876 minus 119861)

[(1 minus 119904) 119891 (119876) minus (119908 minus 119904)119908119891 (119908119876 minus 119861)]2

times [(1 minus 119904)119876119891 (119876) minus (119908 minus 119904)119908119876119891 (119908119876 minus 119861)]

+1

119908119876[2119908119876119891 (119908119876 minus 119861) + (119908 minus 119904)119908119876

2

1198911015840

(119908119876 minus 119861)

minus 119904119876119891 (119908119876 minus 119861)]

times ([(1 minus 119904) 119891 (119876) minus (119908 minus 119904)119908119891 (119908119876 minus 119861)]2

)minus1

times [(119908 minus 119904) 119865 (119908119876 minus 119861) minus (119908 minus 119904)119908119876119891 (119908119876 minus 119861)

+119904119865 (119908119876 minus 119861)]

=1

119908119876[(1 minus 119904) 119891 (119876) minus (119908 minus 119904)119908119891 (119908119876 minus 119861)]2times Δ1

+1

119908119876[(1 minus 119904) 119891 (119876) minus (119908 minus 119904)119908119891 (119908119876 minus 119861)]2Δ2

(12)

where

Δ1= minus [2119908119876119891 (119908119876 minus 119861) + (119908 minus 119904)119908119876

2

1198911015840

(119908119876 minus 119861)]

times [(1 minus 119904)119876119891 (119876) minus (119908 minus 119904)119908119876119891 (119908119876 minus 119861)]

+ [2119908119876119891 (119908119876 minus 119861) + (119908 minus 119904)1199081198762

1198911015840

(119908119876 minus 119861)]

times [(119908 minus 119904) 119865 (119908119876 minus 119861) minus (119908 minus 119904)119908119876119891 (119908119876 minus 119861)]

= [2119908119876119891 (119908119876 minus 119861) + (119908 minus 119904)1199081198762

1198911015840

(119908119876 minus 119861)]

times (1 minus 119904) 119865 (119908119876 minus 119861) 1 minus 119876ℎ (119876)

(13)

Since the demand distribution has the property of IFR Δ1gt

0Δ2= minus119904119876119891 (119908119876 minus 119861)

times [119908119865 (119908119876 minus 119861) minus (119908 minus 119904)119908119876119891 (119908119876 minus 119861)]

+ 119904119865 (119908119876 minus 119861)

times [(2119908 minus 119904)119876119891 (119908119876 minus 119861) + (119908 minus 119904)1199081198762

1198911015840

(119908119876 minus 119861)]

= (119908 minus 119904)1199081199041198912

(119908119876 minus 119861) + (119908 minus 119904) 119904119876119891 (119908119876 minus 119861)

times 119865 (119908119876 minus 119861) [1 minus 119908119876ℎ (119908119876 minus 119861)] gt 0

(14)

So 12059721198761205971199082 lt 0

Theorem 4 Under buy back guarantee contract the optimalwholesale price of the core enterprise 119908lowast can be expressed asfollows

(119908 minus 119888) 119865 (119908119876 minus 119861)120597119876

120597119908

+ [119876119865 (119908119876 minus 119861) +119861119888

1199082119865 (119908119876 minus 119861) +

119888

1199082int119908119876minus119861

0

119909 119889119865 (119909)]

= 0

(15)

Proof Consider1198892119864Π

119872

1198891199082= 2119865 (119908119876 minus 119861)

120597119876

120597119908[1 minus (119908 minus 119888)119876ℎ (119908119876 minus 119861)]

minus (119908 minus 119888)119908119891 (119908119876 minus 119861) (120597119876

120597119908)2

+ (119908 minus 119888) 119865 (119908119876 minus 119861)1205972119876

1205971199082

minus (1 minus119888

119908)1198762

119891 (119908119876 minus 119861) minus 2119861119888

1199083119865 (119908119876 minus 119861)

minus 2119888

1199083int119908119876minus119861

0

119909119891 (119909) 119889119909 lt 0

(16)

From Lemma 3 the optimal wholesale price of the coreenterprise can be expressed as follows

(119908 minus 119888) 119865 (119908119876 minus 119861)120597119876

120597119908+ 119876119865 (119908119876 minus 119861) +

119861119888

1199082119865 (119908119876 minus 119861)

+119888

1199082int119908119876minus119861

0

119909 119889119865 (119909) = 0

(17)

5 Numerical Example

Figure 2 gives the relation between the distributorrsquos initialcapital and the optimal ordering quantity We assume that

6 Journal of Applied Mathematics

0 50 100 150 200 250 300 350 400 450 5000

500

1000

1500

2000

B

Q

Retailerrsquos optimal ordering level under buy back

Retailerrsquos optimal ordering level under trade credit contractThe optimal ordering level in traditional newsvendor modelRetailerrsquos optimal ordering level without financing

guarantee contract

Figure 2 Relation between the optimal ordering quantities withdifferent financial modes

Table 1 Optimal ordering quantity of the distributor and the totalprofit of supply chain system

119861

119876

119876NF 1198760

119876119879 119876lowast 119876119862

Π

ΠNF Π0

Π119879 Πlowast Π119862

119861 = 5063 405 935 1406 176630 159 243 404 321

119861 = 100125 405 709 1237 176658 159 236 374 321

119861 = 200250 405 405 866 176689 159 192 294 321

119861 = 400405 405 405 405 1766158 160 192 159 321

119901 = 1 119908 = 08 and 119904 = 04 and 120585simexponential (11000)From Figure 2 we know that the optimal ordering quantity119876lowast is bigger than 119876119879 the optimal ordering quantity of thetrade credit contractThe optimal ordering quantities119876lowast and119876119879 are both bigger than 119876NF the optimal ordering quantityof the retailer without financing service It also indicates thatthe 119876lowast decreases in the initial budget 119861 of the retailer Theseconclusions are all consistent withTheorem 1

Let ΠNF Π0 Π119879 Πlowast and Π119862 be the total expected

profits of supply chain without system finance the traditionalnewsvendormodel traded credit financing buy back guaran-tee financing and the centralized decision without financeWith the change of the initial capital let 119861 = 50 100 200 and400 Table 1 gives the optimal ordering quantity of the retailerand the total profit of supply chain system

6 Conclusions

Working under the assumption of stochastic market demandand a distributor that is capital constraints this paper hasconsidered the optimal ordering quantity of the distributor

and the wholesale price of the core enterprise with buy backguarantee contract in a single stage supply chain systemWe compare the optimal ordering quantities of the differ-ent financing modes trade credit contract the traditionalnewsvendor model and buy back guarantee contract Thestudy has highlighted how the capital-constrained distributorcould be financed well facing different financing modes andthe research results show that the different financing modesbring different expected profits for supply chain systemwith the capital-constrained distributor The results couldbe applied for the distributor and the core enterprise whenthey make decisions in reality The paper has also raised thepotential for further research into finance contact of supplychain where more than one distributor and core enterprisecan also be considered

Conflict of Interests

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

Acknowledgments

This work was financially supported by the Key Project oftheNationalNatural Science Foundation of China (71131003)Project of the National Natural Science Foundation of China(71371075) China Postdoctoral Science Foundation fundedProject (2013M542183)GuangdongProvinceNatural ScienceFund (S2013040014920) of China and Research Fund for theDoctoral ProgramofGuangdongUniversity of Technology ofChina (13ZK0155)

References

[1] J A Buzacott and R Q Zhang ldquoInventory management withasset-based financingrdquo Management Science vol 50 no 9 pp1274ndash1292 2004

[2] X D Xu and J R Birge ldquoJoint production and financing deci-sions modeling and analysisrdquoWorking PaperTheUniversity ofChicago Graduate School of Business 2004

[3] X D Xu and J R Birge ldquoOperational decisions capitalstructure and managerial compensation a news vendor per-spectiverdquo Working Paper The University of Chicago GraduateSchool of Business 2005

[4] R Swinney G Cachon and S Netessine ldquoCapacity investmentby competitive startupsrdquo Working Paper The Wharton SchoolUniversity of Pennsylvania 2005

[5] L Li M Shubik and M Sobel ldquoControl of dividends capitalsubscriptions and physical inventoriesrdquo Working Paper YaleSchool of Management 2005

[6] V Babich G Aydin P Brunet J Keppo and R Saigal ldquoRiskfinancing and optimal number of suppliersrdquo Working PaperIOE University of Michigan Ann Arbor Mich USA 2005

[7] Y Xu and J Zhang ldquoOn the selection of supply chain coordi-nating contracts the role of capital constraintrdquo Working PaperUniversity of Miami 2007

[8] M Dada and Q Hu ldquoFinancing newsvendor inventoryrdquo Oper-ations Research Letters vol 36 no 5 pp 569ndash573 2008

Journal of Applied Mathematics 7

[9] D Gupta ldquoTechnical note financing the newsvendorrdquoWorkingPaper Industrial and Systems Engineering University of Min-nesota 2008

[10] G Lai L G Debo and K Sycara ldquoSharing inventory risk insupply chain the implication of financial constraintrdquo Omegavol 37 no 4 pp 811ndash825 2009

[11] P Kouvelis and H W Zhao ldquoFinancing the NewsvendorSupplier vs Bank and the Structure of Optimal Trade CreditContractsrdquo Working paper Olin Business School WashingtonUniversity September 2009

[12] P Kouvelis and W H Zhao ldquoThe newsvendor problem andprice-only contract when bankruptcy costs existrdquo WorkingPaper Olin Business School Washington University 2010

[13] X F Chen and G Wan ldquoThe effect of financing on a budget-constrained supply chain under wholesale price contractrdquoAsia-Pacific Journal of Operational Research vol 28 no 4 pp 457ndash485 2011

[14] X F Chen and A Wang ldquoTrade credit contract with limitedliability in the supply chain with budget constraintsrdquo Annals ofOperations Research vol 196 pp 153ndash165 2012

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 5

minus[minus (2119908 minus 119904) 119891 (119908119876 minus 119861) minus (119908 minus 119904)1199081198761198911015840 (119908119876 minus 119861)]

[(1 minus 119904) 119891 (119876) minus (119908 minus 119904)119908119891 (119908119876 minus 119861)]2

times 119865 (119908119876 minus 119861) [1 minus (119908 minus 119904)119876ℎ (119908119876 minus 119861)]

=1

119908119876

minus2119908119876119891 (119908119876 minus 119861) minus (119908 minus 119904)11990811987621198911015840 (119908119876 minus 119861)

[(1 minus 119904) 119891 (119876) minus (119908 minus 119904)119908119891 (119908119876 minus 119861)]2

times [(1 minus 119904)119876119891 (119876) minus (119908 minus 119904)119908119876119891 (119908119876 minus 119861)]

+1

119908119876[ (2119908 minus 119904)119876119891 (119908119876 minus 119861)

+ (119908 minus 119904)1199081198762

1198911015840

(119908119876 minus 119861)]

times ([(1 minus 119904) 119891 (119876) minus (119908 minus 119904)119908119891 (119908119876 minus 119861)]2

)minus1

times [119908119865 (119908119876 minus 119861) minus (119908 minus 119904)119908119876119891 (119908119876 minus 119861)]

=1

119908119876

minus 2119908119876119891 (119908119876 minus 119861) + (119908 minus 119904)11990811987621198911015840 (119908119876 minus 119861)

[(1 minus 119904) 119891 (119876) minus (119908 minus 119904)119908119891 (119908119876 minus 119861)]2

times [(1 minus 119904)119876119891 (119876) minus (119908 minus 119904)119908119876119891 (119908119876 minus 119861)]

+1

119908119876[2119908119876119891 (119908119876 minus 119861) + (119908 minus 119904)119908119876

2

1198911015840

(119908119876 minus 119861)

minus 119904119876119891 (119908119876 minus 119861)]

times ([(1 minus 119904) 119891 (119876) minus (119908 minus 119904)119908119891 (119908119876 minus 119861)]2

)minus1

times [(119908 minus 119904) 119865 (119908119876 minus 119861) minus (119908 minus 119904)119908119876119891 (119908119876 minus 119861)

+119904119865 (119908119876 minus 119861)]

=1

119908119876[(1 minus 119904) 119891 (119876) minus (119908 minus 119904)119908119891 (119908119876 minus 119861)]2times Δ1

+1

119908119876[(1 minus 119904) 119891 (119876) minus (119908 minus 119904)119908119891 (119908119876 minus 119861)]2Δ2

(12)

where

Δ1= minus [2119908119876119891 (119908119876 minus 119861) + (119908 minus 119904)119908119876

2

1198911015840

(119908119876 minus 119861)]

times [(1 minus 119904)119876119891 (119876) minus (119908 minus 119904)119908119876119891 (119908119876 minus 119861)]

+ [2119908119876119891 (119908119876 minus 119861) + (119908 minus 119904)1199081198762

1198911015840

(119908119876 minus 119861)]

times [(119908 minus 119904) 119865 (119908119876 minus 119861) minus (119908 minus 119904)119908119876119891 (119908119876 minus 119861)]

= [2119908119876119891 (119908119876 minus 119861) + (119908 minus 119904)1199081198762

1198911015840

(119908119876 minus 119861)]

times (1 minus 119904) 119865 (119908119876 minus 119861) 1 minus 119876ℎ (119876)

(13)

Since the demand distribution has the property of IFR Δ1gt

0Δ2= minus119904119876119891 (119908119876 minus 119861)

times [119908119865 (119908119876 minus 119861) minus (119908 minus 119904)119908119876119891 (119908119876 minus 119861)]

+ 119904119865 (119908119876 minus 119861)

times [(2119908 minus 119904)119876119891 (119908119876 minus 119861) + (119908 minus 119904)1199081198762

1198911015840

(119908119876 minus 119861)]

= (119908 minus 119904)1199081199041198912

(119908119876 minus 119861) + (119908 minus 119904) 119904119876119891 (119908119876 minus 119861)

times 119865 (119908119876 minus 119861) [1 minus 119908119876ℎ (119908119876 minus 119861)] gt 0

(14)

So 12059721198761205971199082 lt 0

Theorem 4 Under buy back guarantee contract the optimalwholesale price of the core enterprise 119908lowast can be expressed asfollows

(119908 minus 119888) 119865 (119908119876 minus 119861)120597119876

120597119908

+ [119876119865 (119908119876 minus 119861) +119861119888

1199082119865 (119908119876 minus 119861) +

119888

1199082int119908119876minus119861

0

119909 119889119865 (119909)]

= 0

(15)

Proof Consider1198892119864Π

119872

1198891199082= 2119865 (119908119876 minus 119861)

120597119876

120597119908[1 minus (119908 minus 119888)119876ℎ (119908119876 minus 119861)]

minus (119908 minus 119888)119908119891 (119908119876 minus 119861) (120597119876

120597119908)2

+ (119908 minus 119888) 119865 (119908119876 minus 119861)1205972119876

1205971199082

minus (1 minus119888

119908)1198762

119891 (119908119876 minus 119861) minus 2119861119888

1199083119865 (119908119876 minus 119861)

minus 2119888

1199083int119908119876minus119861

0

119909119891 (119909) 119889119909 lt 0

(16)

From Lemma 3 the optimal wholesale price of the coreenterprise can be expressed as follows

(119908 minus 119888) 119865 (119908119876 minus 119861)120597119876

120597119908+ 119876119865 (119908119876 minus 119861) +

119861119888

1199082119865 (119908119876 minus 119861)

+119888

1199082int119908119876minus119861

0

119909 119889119865 (119909) = 0

(17)

5 Numerical Example

Figure 2 gives the relation between the distributorrsquos initialcapital and the optimal ordering quantity We assume that

6 Journal of Applied Mathematics

0 50 100 150 200 250 300 350 400 450 5000

500

1000

1500

2000

B

Q

Retailerrsquos optimal ordering level under buy back

Retailerrsquos optimal ordering level under trade credit contractThe optimal ordering level in traditional newsvendor modelRetailerrsquos optimal ordering level without financing

guarantee contract

Figure 2 Relation between the optimal ordering quantities withdifferent financial modes

Table 1 Optimal ordering quantity of the distributor and the totalprofit of supply chain system

119861

119876

119876NF 1198760

119876119879 119876lowast 119876119862

Π

ΠNF Π0

Π119879 Πlowast Π119862

119861 = 5063 405 935 1406 176630 159 243 404 321

119861 = 100125 405 709 1237 176658 159 236 374 321

119861 = 200250 405 405 866 176689 159 192 294 321

119861 = 400405 405 405 405 1766158 160 192 159 321

119901 = 1 119908 = 08 and 119904 = 04 and 120585simexponential (11000)From Figure 2 we know that the optimal ordering quantity119876lowast is bigger than 119876119879 the optimal ordering quantity of thetrade credit contractThe optimal ordering quantities119876lowast and119876119879 are both bigger than 119876NF the optimal ordering quantityof the retailer without financing service It also indicates thatthe 119876lowast decreases in the initial budget 119861 of the retailer Theseconclusions are all consistent withTheorem 1

Let ΠNF Π0 Π119879 Πlowast and Π119862 be the total expected

profits of supply chain without system finance the traditionalnewsvendormodel traded credit financing buy back guaran-tee financing and the centralized decision without financeWith the change of the initial capital let 119861 = 50 100 200 and400 Table 1 gives the optimal ordering quantity of the retailerand the total profit of supply chain system

6 Conclusions

Working under the assumption of stochastic market demandand a distributor that is capital constraints this paper hasconsidered the optimal ordering quantity of the distributor

and the wholesale price of the core enterprise with buy backguarantee contract in a single stage supply chain systemWe compare the optimal ordering quantities of the differ-ent financing modes trade credit contract the traditionalnewsvendor model and buy back guarantee contract Thestudy has highlighted how the capital-constrained distributorcould be financed well facing different financing modes andthe research results show that the different financing modesbring different expected profits for supply chain systemwith the capital-constrained distributor The results couldbe applied for the distributor and the core enterprise whenthey make decisions in reality The paper has also raised thepotential for further research into finance contact of supplychain where more than one distributor and core enterprisecan also be considered

Conflict of Interests

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

Acknowledgments

This work was financially supported by the Key Project oftheNationalNatural Science Foundation of China (71131003)Project of the National Natural Science Foundation of China(71371075) China Postdoctoral Science Foundation fundedProject (2013M542183)GuangdongProvinceNatural ScienceFund (S2013040014920) of China and Research Fund for theDoctoral ProgramofGuangdongUniversity of Technology ofChina (13ZK0155)

References

[1] J A Buzacott and R Q Zhang ldquoInventory management withasset-based financingrdquo Management Science vol 50 no 9 pp1274ndash1292 2004

[2] X D Xu and J R Birge ldquoJoint production and financing deci-sions modeling and analysisrdquoWorking PaperTheUniversity ofChicago Graduate School of Business 2004

[3] X D Xu and J R Birge ldquoOperational decisions capitalstructure and managerial compensation a news vendor per-spectiverdquo Working Paper The University of Chicago GraduateSchool of Business 2005

[4] R Swinney G Cachon and S Netessine ldquoCapacity investmentby competitive startupsrdquo Working Paper The Wharton SchoolUniversity of Pennsylvania 2005

[5] L Li M Shubik and M Sobel ldquoControl of dividends capitalsubscriptions and physical inventoriesrdquo Working Paper YaleSchool of Management 2005

[6] V Babich G Aydin P Brunet J Keppo and R Saigal ldquoRiskfinancing and optimal number of suppliersrdquo Working PaperIOE University of Michigan Ann Arbor Mich USA 2005

[7] Y Xu and J Zhang ldquoOn the selection of supply chain coordi-nating contracts the role of capital constraintrdquo Working PaperUniversity of Miami 2007

[8] M Dada and Q Hu ldquoFinancing newsvendor inventoryrdquo Oper-ations Research Letters vol 36 no 5 pp 569ndash573 2008

Journal of Applied Mathematics 7

[9] D Gupta ldquoTechnical note financing the newsvendorrdquoWorkingPaper Industrial and Systems Engineering University of Min-nesota 2008

[10] G Lai L G Debo and K Sycara ldquoSharing inventory risk insupply chain the implication of financial constraintrdquo Omegavol 37 no 4 pp 811ndash825 2009

[11] P Kouvelis and H W Zhao ldquoFinancing the NewsvendorSupplier vs Bank and the Structure of Optimal Trade CreditContractsrdquo Working paper Olin Business School WashingtonUniversity September 2009

[12] P Kouvelis and W H Zhao ldquoThe newsvendor problem andprice-only contract when bankruptcy costs existrdquo WorkingPaper Olin Business School Washington University 2010

[13] X F Chen and G Wan ldquoThe effect of financing on a budget-constrained supply chain under wholesale price contractrdquoAsia-Pacific Journal of Operational Research vol 28 no 4 pp 457ndash485 2011

[14] X F Chen and A Wang ldquoTrade credit contract with limitedliability in the supply chain with budget constraintsrdquo Annals ofOperations Research vol 196 pp 153ndash165 2012

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

0 50 100 150 200 250 300 350 400 450 5000

500

1000

1500

2000

B

Q

Retailerrsquos optimal ordering level under buy back

Retailerrsquos optimal ordering level under trade credit contractThe optimal ordering level in traditional newsvendor modelRetailerrsquos optimal ordering level without financing

guarantee contract

Figure 2 Relation between the optimal ordering quantities withdifferent financial modes

Table 1 Optimal ordering quantity of the distributor and the totalprofit of supply chain system

119861

119876

119876NF 1198760

119876119879 119876lowast 119876119862

Π

ΠNF Π0

Π119879 Πlowast Π119862

119861 = 5063 405 935 1406 176630 159 243 404 321

119861 = 100125 405 709 1237 176658 159 236 374 321

119861 = 200250 405 405 866 176689 159 192 294 321

119861 = 400405 405 405 405 1766158 160 192 159 321

119901 = 1 119908 = 08 and 119904 = 04 and 120585simexponential (11000)From Figure 2 we know that the optimal ordering quantity119876lowast is bigger than 119876119879 the optimal ordering quantity of thetrade credit contractThe optimal ordering quantities119876lowast and119876119879 are both bigger than 119876NF the optimal ordering quantityof the retailer without financing service It also indicates thatthe 119876lowast decreases in the initial budget 119861 of the retailer Theseconclusions are all consistent withTheorem 1

Let ΠNF Π0 Π119879 Πlowast and Π119862 be the total expected

profits of supply chain without system finance the traditionalnewsvendormodel traded credit financing buy back guaran-tee financing and the centralized decision without financeWith the change of the initial capital let 119861 = 50 100 200 and400 Table 1 gives the optimal ordering quantity of the retailerand the total profit of supply chain system

6 Conclusions

Working under the assumption of stochastic market demandand a distributor that is capital constraints this paper hasconsidered the optimal ordering quantity of the distributor

and the wholesale price of the core enterprise with buy backguarantee contract in a single stage supply chain systemWe compare the optimal ordering quantities of the differ-ent financing modes trade credit contract the traditionalnewsvendor model and buy back guarantee contract Thestudy has highlighted how the capital-constrained distributorcould be financed well facing different financing modes andthe research results show that the different financing modesbring different expected profits for supply chain systemwith the capital-constrained distributor The results couldbe applied for the distributor and the core enterprise whenthey make decisions in reality The paper has also raised thepotential for further research into finance contact of supplychain where more than one distributor and core enterprisecan also be considered

Conflict of Interests

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

Acknowledgments

This work was financially supported by the Key Project oftheNationalNatural Science Foundation of China (71131003)Project of the National Natural Science Foundation of China(71371075) China Postdoctoral Science Foundation fundedProject (2013M542183)GuangdongProvinceNatural ScienceFund (S2013040014920) of China and Research Fund for theDoctoral ProgramofGuangdongUniversity of Technology ofChina (13ZK0155)

References

[1] J A Buzacott and R Q Zhang ldquoInventory management withasset-based financingrdquo Management Science vol 50 no 9 pp1274ndash1292 2004

[2] X D Xu and J R Birge ldquoJoint production and financing deci-sions modeling and analysisrdquoWorking PaperTheUniversity ofChicago Graduate School of Business 2004

[3] X D Xu and J R Birge ldquoOperational decisions capitalstructure and managerial compensation a news vendor per-spectiverdquo Working Paper The University of Chicago GraduateSchool of Business 2005

[4] R Swinney G Cachon and S Netessine ldquoCapacity investmentby competitive startupsrdquo Working Paper The Wharton SchoolUniversity of Pennsylvania 2005

[5] L Li M Shubik and M Sobel ldquoControl of dividends capitalsubscriptions and physical inventoriesrdquo Working Paper YaleSchool of Management 2005

[6] V Babich G Aydin P Brunet J Keppo and R Saigal ldquoRiskfinancing and optimal number of suppliersrdquo Working PaperIOE University of Michigan Ann Arbor Mich USA 2005

[7] Y Xu and J Zhang ldquoOn the selection of supply chain coordi-nating contracts the role of capital constraintrdquo Working PaperUniversity of Miami 2007

[8] M Dada and Q Hu ldquoFinancing newsvendor inventoryrdquo Oper-ations Research Letters vol 36 no 5 pp 569ndash573 2008

Journal of Applied Mathematics 7

[9] D Gupta ldquoTechnical note financing the newsvendorrdquoWorkingPaper Industrial and Systems Engineering University of Min-nesota 2008

[10] G Lai L G Debo and K Sycara ldquoSharing inventory risk insupply chain the implication of financial constraintrdquo Omegavol 37 no 4 pp 811ndash825 2009

[11] P Kouvelis and H W Zhao ldquoFinancing the NewsvendorSupplier vs Bank and the Structure of Optimal Trade CreditContractsrdquo Working paper Olin Business School WashingtonUniversity September 2009

[12] P Kouvelis and W H Zhao ldquoThe newsvendor problem andprice-only contract when bankruptcy costs existrdquo WorkingPaper Olin Business School Washington University 2010

[13] X F Chen and G Wan ldquoThe effect of financing on a budget-constrained supply chain under wholesale price contractrdquoAsia-Pacific Journal of Operational Research vol 28 no 4 pp 457ndash485 2011

[14] X F Chen and A Wang ldquoTrade credit contract with limitedliability in the supply chain with budget constraintsrdquo Annals ofOperations Research vol 196 pp 153ndash165 2012

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

[9] D Gupta ldquoTechnical note financing the newsvendorrdquoWorkingPaper Industrial and Systems Engineering University of Min-nesota 2008

[10] G Lai L G Debo and K Sycara ldquoSharing inventory risk insupply chain the implication of financial constraintrdquo Omegavol 37 no 4 pp 811ndash825 2009

[11] P Kouvelis and H W Zhao ldquoFinancing the NewsvendorSupplier vs Bank and the Structure of Optimal Trade CreditContractsrdquo Working paper Olin Business School WashingtonUniversity September 2009

[12] P Kouvelis and W H Zhao ldquoThe newsvendor problem andprice-only contract when bankruptcy costs existrdquo WorkingPaper Olin Business School Washington University 2010

[13] X F Chen and G Wan ldquoThe effect of financing on a budget-constrained supply chain under wholesale price contractrdquoAsia-Pacific Journal of Operational Research vol 28 no 4 pp 457ndash485 2011

[14] X F Chen and A Wang ldquoTrade credit contract with limitedliability in the supply chain with budget constraintsrdquo Annals ofOperations Research vol 196 pp 153ndash165 2012

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