Research ArticleAnalysis of the Profits of Banks and Supply ChainEnterprises under Noncollaborative and Collaborative Financing

Xinglin Dong and Jian Pan

College of Economics and Management Shandong University of Science and Technology Qingdao 266590 China

Correspondence should be addressed to Jian Pan 13061331126163com

Received 6 July 2019 Revised 18 September 2019 Accepted 27 September 2019 Published 21 October 2019

Academic Editor Changzhi Wu

Copyright copy 2019 Xinglin Dong and Jian Panis 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 isproperly cited

In order to solve the problem of how to choose a financing strategy for supply chain enterprises with financial constraintsincluding the manufacturer and the retailer this paper puts forward two financing strategies in the bilateral supply chain withuncertain output and certain demand e two financing strategies are the noncollaborative financing (finance from a bankseparately (FBS)) and the collaborative financing (finance from a bank uniformly (FBU)) It derives the production orderformula of the supply chain enterprises under financial constraints Under the complete information according to thisformula it analyzes how the bank prices the loan interest rates and finds the optimal decisions under the two financingstrategies e following results are found (1) e manufacturerrsquos planned output is negatively correlated with the bankrsquos loaninterest ratee increased interest rates do not necessarily lead to the increased bankrsquos loan profits (2)e bankrsquos loan profit ishigher when the supply chain enterprises choose the FBS strategy (3)e FBU strategy does not necessarily make the profits ofthe manufacturer and the retailer better It is affirmative only if some parameters in the supply chain meet certain conditionse above-mentioned conclusions supply a policy guiding the supply chain enterprises with financial constraints to make achoice of the financing strategy

1 Introduction

In recent years the research on supply chain operation andcoordination in an uncertain environment has attractedmore and more attention from academic and businesscircles ere is more concern about supply chain oper-ation under the condition of uncertain demand [1ndash3] butin a customer-centric market economy there are stillmany enterprises that take order-based production [4] Inthe order-based production the enterprises arrangeproduction in accordance with the customerrsquos demandSince the production is affected by various factors such asraw materials equipment processes and employees thenumber of qualified products is uncertain but the demandis determined [5 6] For example in the contract farmingthere is a definite demand related to the order quantity inthe downstream enterprises of the supply chain But thefarmers in production have uncertain output because of

the limitation of natural conditions and other factors ofthe farmers

For another example in the semiconductor and elec-tronics industries where the production process andproduct quality requirements are more stringent the actualoutput of manufacturers is often very different from theexpected output After an order is placed by a downstreamenterprise the manufacturer cannot guarantee that theoutput will match the order volumeerefore in reality thesupply chain with output uncertainty and demand de-termination is still widespread and it often faces theproblem of capital constraints When a member of thesupply chain has funds constraints it is often unable toimplement the optimal decision-making thus ldquopulling thewhole bodyrdquo [7] Zhang [8] first proposed that the operationof the supply chain and financing issues are very importantto supply chain enterprises In view of the fact that mostscholars study the uncertainty of the output in the supply

HindawiMathematical Problems in EngineeringVolume 2019 Article ID 6263480 14 pageshttpsdoiorg10115520196263480

chain it is based on the bilateral supply chain [9] so thispaper still analyzes the supply chain financing under theuncertainty of the output and the determination of demandon this basis

Generally speaking financing channels for supply chainenterprises are divided into internal financing and externalfinancing Internal financing refers to the sources of fundswithin the supply chain mainly including the internalretained earnings the trade credits of supply chainmembersand the supply chain membersrsquo funds External financingrefers to the sources of funds outside the supply chainmainly including the stocks the bonds and the bank loansissued by the supply chain enterprises By the end of 2018540000 SMEs in China had obtained the bank loans Bankloans as the main external financing channel for the supplychain financing enterprises played an important role insolving the funding problems of the supply chain enter-prises erefore this paper mainly studies the bank loanproblem of external financing in the supply chain

Supply chain financing enterprises have different optionsfor bank loans e traditional programs include mortgagesand pledge financing With the development of supply chainfinancing the new programs rely more on the credit level ofthe core enterprises in the supply chain to provide loanservices to supply chain SMEse typical model is the creditguarantee financing for the core enterprises in the supplychain Whether it is traditional mortgage pledge financingor supply chain financing for financing enterprises in thesupply chain there are two main ways to connect with thebanks First the financing enterprise and the bank are di-rectly connected to each other For example the enterpriseobtains the bank loans through their own asset mortgagepledge and other ways Because of different credit rating andthe enterprise scale of different supply chain financing en-terprises different enterprises obtain different loan pricingfrom the banks Second the financing enterprise does notdirectly connect with the bank but obtains bank loans relyingon the supply chain core enterprise or the third-party lo-gistics enterprise to connect with the bank indirectly ebank conducts loan pricing according to the credit level ofthe core enterprise or the third-party logistics enterprise andthe financing enterprises obtain the loan interest rate con-sistent with the pricing

erefore according to different connection ways be-tween the financing supply chain enterprises and the banksthe bank loan strategies of the bilateral supply chain aredivided into noncollaborative financing (finance from abank separately (FBS)) and collaborative financing (financefrom a bank uniformly (FBU)) in this paper Under the FBSstrategy all the financing enterprises in the bilateral supplychain are directly connected to the bank e typical FBSstrategy includes the traditional mortgage the pledge fi-nancing model the warehouse receipt pledge financingmodel and the order financing model in supply chain fi-nancing Under the FBU strategy all financing enterprises inthe bilateral supply chain rely on the overall strength andcredit level of the core enterprises or third-party logisticsenterprises in the supply chain to obtain bank loans etypical FBU strategy includes the dealer the supplier

network financing model (using the credit introduction ofcore enterprises a kind of financial service provided by banksto dealers and suppliers related to core enterprises) and thebank and logistics cooperative financing model (a financialservice in which a bank cooperates with a third-party logisticscompany to provide customers with the credit through thelogistics supervision or the credit guarantee) A typical case ofthe FBU strategy is as follows

As a professional sports brand manufacturer Li NingCompany has become a core enterprise in the supply chainwith its own advantages In the supply chain a pattern hasemerged in which the upstream and downstream enterprisesjointly supply service for Li Ning Obviously the survivalstatus of upstream and downstream enterprises in the supplychain has a great impact on the development of Li Ningese upstream and downstream enterprises often find itdifficult to directly obtain bank loans because of the scale ofenterprises and credit standing level e problem of fi-nancial constraints will not only affect the technologicalinnovation and the product upgrading of the upstream anddownstream enterprises but also affect the development of LiNing indirectly In order to achieve the stable developmentof Li Ning Company Li Ning Company becomes aware ofthe importance of providing financial support for its up-stream and downstream enterprises erefore Li NingCompany and Standard Chartered Bank jointly relying onLi Ningrsquos overall strength the credit level and the businesstransactions between Li Ningrsquos business and the upstreamand downstream companies provide financing services forupstream and downstream enterprises Standard CharteredBank only cares about Li Ningrsquos credit standing strictlycontrols the coproduction process in the supply chain andprices the loan interest rates for the chain financing en-terprises according to Li Ningrsquos credit standing Obviouslyupstream and downstream financing enterprises in thesupply chain have obtained the bank loans through the FBUstrategy However in a bilateral supply chain with theuncertain output and the certain demand the financingenterprises could choose either the FBS strategy or the FBUstrategy to get the bank loans under the financial constraintsWhich financing strategy is better for supply chain enter-prises How do the banks make loan decisions with the twochoices of the financing strategies How do the two financingstrategies affect bank profit

In order to solve the above problems this paper studiesthe bilateral supply chain with the uncertain output andcertain demand is supply chain consists of a manufac-turer and a retailer In an order both of them are under thefinancing constraints and both of them can choose the FBSor the FBU strategy for the bank loans is paper analyzesthe interdependency between the bankrsquos loan interest rateand the supply chain enterprise operation decision andcompares the profit of the bank and the supply chain en-terprises in the two strategies e main results of this paperare as follows

First under the condition of both the manufacturer andthe retailer facing the financial constraints the manufac-turerrsquos optimal production formula is proved which in-dicates that the manufacturerrsquos planned output and its own

2 Mathematical Problems in Engineering

loan interest rate are inversely correlated It is proved thatthe optimal order volume of the retailer is the demand of themarket Simply put under the certain market demand al-though the manufacturerrsquos output is uncertain the orderquantity of the retailer is still the demand of the market

Second this paper puts forward the formula of the loaninterest rate under different financing strategies in the supplychain and points out that the maximization of the bankrsquos loaninterest rate does not necessarily make the maximum profitOn the contrary under certain conditions the bankrsquos loanprofit is negatively correlated with the loan interest ratewhich is contrary to our usual belief that the bankrsquos maxi-mizing loan interest rate will boost the return on loans

ird through comparing the profit of the bank underthe two financing strategies and the profit of the supply chainenterprises this study finds that the bank has higher profitunder the FBS strategy in the supply chain and the FBUstrategy is not necessarily more beneficial to supply chainenterprises In the case that the bankrsquos optimal loan interestrate to the manufacturer is at its maximum and someparameters in the supply chain meet a certain relationshipthe FBU strategy is more beneficial to the supply chain isis contrary to our usual intuition that the FBU strategy ismore beneficial to the supply chain enterprises

2 Literature Review

To solve the above-mentioned problems this study exam-ines two types of literature (1) the interface between op-erations and finance and (2) the comparison of supply chainfinancing strategies

21 Integration of Operations and Finance Some early pa-pers studied the relationship between the financing andthe operational decisions by putting the condition ofcapital constraints into traditional newsboy models Forexample Xu and Birge [10] explained the impact ofcorporate capital constraints and capital structure oninventory decisions Zhang [8] analyzed the financingdecisions of the banks and retailers at different capitallevels and found that the asset-based financing can in-crease retailersrsquo profits Kouvelis and Zhao [11] studied theimpact of bankruptcy costs on the retailersrsquo order quan-tities Later some studies further analyzed the impact ofother factors (agent costs and asymmetric information) onoperations and finances For example Boyabath andToktay [12] studied the impact of imperfect capitalmarkets on technology choices for the budget-constrainedcompanies Alan and Gaur [13] showed that the bor-rowersrsquo investment decisions may be affected by bank-ruptcy costs and asymmetric demand informationRecently some studies found that SMEsrsquo stronger partnerscould help SMEs improve their financing capabilities Forexample Sun et al [14] analyzed the interdependencebetween business decisions and financial decisions under acredit guarantee contract Bi et al [15] pointed out that thepledge of the manufacturerrsquos products would increase theorder volume of retailers and enhance the level of

cooperation in the supply chain Chen and Zhou [16]pointed out that repurchase guarantee financing can bringmore benefits to the supply chain e above study focuseson supply chain financing and operation issues for a singlecapital-constrained company In contrast our focus is onboth the manufacturer and the retailer who are financiallyconstrained We pay attention to the impact of bank fi-nancial behavior on supply chain business operation de-cisions that is the dependence of bank financial behaviorand supply chain operation decisions

22 Comparison of Different Financing Options Recentlyscholars have gradually begun to pay attention to the choiceof different financing methods in the supply chain[7 17ndash23] For example Kouvelis and Zhao [7] emphasizedthat retailers will always prefer bank financing if the beststructured solution is provided Jing et al [20] pointed outthat trade credit financing is more beneficial to enterprises atlower production costs when both banks and trade credit areviable Yang and Birge [21] pointed out that capital-con-strained retailers can choose a combination of bank fi-nancing and trade credit financing to ease the financialpressures Jin et al [24] proposed that a collaborative fi-nancing strategy is beneficial to suppliers and the entiresupply chain while for retailers noncooperative financingstrategies are preferable Our research is related to thestudies by Kouvelis et al [7] Jing et al [20] and Jin et al[24] but there are significant differences First Kouvelis andZhao [7] and Jing et al [20] considered the advantages anddisadvantages of bank financing and trade credit financingunder different conditionse two financing models belongto the internal financing and external financing of the supplychain We consider the advantages and disadvantages of theFBS strategy and FBU strategy from the perspective of theloan interest rates e FBS strategy is a bank-led financingand the FBU strategy is a combination of bank financing andtrade credit financing In fact the two financing strategies wecompare are the bank loans Second Jin et al [24] pointedout that financing strategies are beneficial to suppliers andthe entire supply chain but not to retailers We compare thedifference between the FBS strategy and the FBU strategy ofthe supply chain but the result is different from that of Jinet al [24] because we consider the differences in the profit ofthe supply chain from the perspective of the financing in-terest rate At the same time drawing on the research ofGrosfeld-Nir and Ronen [5] and Yano and Lee [6] we as-sume that the output is uncertain and the external marketdemand is determined

3 Problem Description and Assumptions

31 Operating Environment and ProductMarket In a supplychain consisting of a manufacturer and a retailer the retailerplaces orders to the manufacturer based on external marketdemand and the manufacturer arranges production plansbased on the retailerrsquos needs and their own productioncapacity e external market demand is determined emanufacturerrsquos production volume is uncertain that is

Mathematical Problems in Engineering 3

there is a certain deviation between the manufacturerrsquosplanned output and the actual production Because of theuncertainty of production there is a compensation factor inthe supply chain When the actual production of themanufacturer is higher than the order quantity of the re-tailer the retailer gives the manufacturer some compensa-tion Otherwise the manufacturer gives the retailer somecompensation e parameter settings in this study areshown in Table 1

e assumptions for the supply chain operating envi-ronment and product market are as follows (1) e actualproduction of the manufacturer is subject to a uniformdistribution with the planned output as expected (2) Supplychain enterprises are risk-neutral (3) Information is sym-metrical regardless of the cost of obtaining information frombanks and supply chain enterprises (4) e JIT method isadopted for the delivery and the manufacturerrsquos deliveryquantity is not greater than the order quantity

e manufacturerrsquos planned output q0 is deterministicbut because of the uncertainty in actual production theactual output of the manufacturer is uncertain erandom factor x is used to represent the uncertainty of themanufacturerrsquos actual output [25] where x sim U[1 minus n 1 +

n] and 0lt nlt 1 us the actual output of the manufac-turer is q0x According to assumption (1) the actualoutput of the manufacturer is subject to a uniform dis-tribution with the planned output as expected soE(q0x) q0 From assumption (2) we have wlt l1 ltp1that is the unsalable loss of the manufacturerrsquos unitproduct is between the compensation and the productprice e unsalable loss of the retailerrsquos unit product isbetween the purchase price and the selling price that isp1 lt l2 ltp2

From assumption (4) we know that the manufacturerdelivers min q0x q11113864 1113865 to the retailer and the retailerrsquos actualsales volume is min min q0x q11113864 1113865 D1113864 1113865 e supply chainoperation structure under the uncertainty of production isshown in Figure 1

32 Capital Structure and Financial Market e manufac-turer and retailer have full financial constraints in one orderand need to lend to banks e retailer is required to pay anadvance payment when placing an order to the manufac-turer e prepayment amount is proportional to the orderquantity which is kp1q1 e amount that the manufacturerneeds to make a loan to the bank is the difference between itsproduction cost and the receipt of the advance paymentq0c0 minus kp1q1

e manufacturer and retailer can choose to makeloans to the bank in two strategies One is the FBS strategyin which the manufacturer and retailer make separatecontacts with the bank and the bank gives the manu-facturer and retailer a loan interest rate that does notexceed the maximum acceptable loan rate for each en-terprise (r1 le r10 and r2 le r20) e other is the FBUstrategy where the manufacturer or retailer interfaces withthe bank as the core enterprise of the supply chain e

enterprise will give the loan to another enterprise at thesame interest rate e bankrsquos loan interest rate to themanufacturer and retailer does not exceed the minimumof the two enterprisesrsquo acceptable loan interest rate(r3 lemin r10 r201113864 1113865)

We do not consider the risk of loans because the bank-ruptcy of the supply chain enterprises is mainly due tocommercial credit loans while inmortgages and pledge loansthe collateral or pledge is often sufficient to repay their lia-bilities Sufi [26] pointed out that under the credit loan thebank will determine the credit line of the financing enterpriseerefore the probability of default of the financing enter-prisersquos loan within the credit line is acceptable by the bank Onthe contrary if the financing company needs more than thecredit line it is obvious that the actual output of the supplychain cannot meet the demand D At this time the retailerand the manufacturer will moderately reduce the orderquantity and output which can be regarded as the externalmarket demand Dprime erefore without losing the generalityit is assumed that the loan amount of the financing enterpriseis within the credit line At this time the default risk of thefinancing enterprise is within the controllable range of thebank at is under both financing strategies the funds re-quired by the manufacturer and retailer can be met by banksand the manufacturer and retailer can repay in full and ontime At the same time we assume that the bankrsquos capital costrate remains unchanged Two financing strategies for capital-constrained supply chains are shown in Figure 2

Table 1 Decision variables and model parametersIndicesi (12) Index for the manufacturer and retailer

j (SU) Index for the optimal value of FBS and FBUstrategies respectively

Decisionvariablesq0 Manufacturerrsquos planned outputq1 Retailerrsquos order quantity

riBankrsquos loan interest rate to the enterprise i

under the FBS strategy

r3Bankrsquos interest rate to the core enterprise under

the FBU strategyModelparametersc0 Unit production cost of the manufacturerli Loss of unit unsalable product of the enterprise ipi Price of unit product of the enterprise iw Coefficient of compensation priceD Market demand for productsr0 Bankrsquos capital cost rate

ri0e maximum loan interest rate acceptable by

the enterprise i

k e prepayment amount of the retailerrsquos unitproduct

Li Loan amount of the enterprise iπi Profit of the enterprise iπi+2 Bankrsquos loan profit from the enterprise ix Random factor of product productionf(x) Density function of x

4 Mathematical Problems in Engineering

4 Model Construction and Analysis

41 Operational Decisions of Supply Chain Enterprises underTwo Financing Strategies

411 Operational Decisions of Supply Chain Enterprisesunder FBS Strategy According to Section 1 we can knowthat the profit of the manufacturer is

π1 p1 min q0x q11113864 1113865 minus c0q0x + w minus l1( 1113857max q0x minus q1 01113864 1113865

minus wmax q1 minus q0x 01113864 1113865 minus q0c0 minus kq1p1( 1113857r1

(1)

where the first item is the manufacturerrsquos sales revenue thesecond item is the manufacturerrsquos production cost the thirditem is the difference between the compensation receivedand the slow-moving loss the fourth item is the compen-sation given to the retailer and the fifth item is the man-ufacturerrsquos financing cost e manufacturer expects theprofit to be

Eπ1 p1 1113946q1q0

0q0xf(x)dx + 1113946

+infin

q1q0q1f(x)dx1113890 1113891

minus c0q0 minus l1 1113946+infin

q1q0q0x minus q1( 1113857f(x)dx + w q0 minus q1( 1113857

minus q0c0 minus kq1p1( 1113857r1

(2)

Proposition 1 Under the FBS strategy the manufacturerrsquosexpected profit is a convex function and its optimal plannedoutput satisfies the following formula

1113946q1q0

0xf(x)dx 1 minus

p1 + w minus c0 1 + r1( 1113857

p1 + l1 (3)

Equation (3) shows that the manufacturerrsquos plannedoutput is negatively correlated with the bankrsquos loan in-terest rate Excessively high bank loan pricing during themanufacturer loan phase will cause manufacturers toreduce the planned output to moderately reduce loanamounts which indicates that excessive financing costswill affect the enthusiasm of financing companies Ob-viously the above conclusions are consistent with thereality Conversely if the manufacturerrsquos planned outputis positively related to the bankrsquos loan interest rate raisingthe loan interest rate will increase the manufacturerrsquosplanned output thereby indirectly increasing the loanamount and ultimately increasing the bankrsquos loan profitserefore under the conclusion that the manufacturerrsquosplanned output is positively related to the bankrsquos loaninterest rate the bank will always raise the loan interestrate which is obviously inconsistent with the actualsituation

At this time when formula (2) satisfies formula (3) themanufacturerrsquos expected profit Eπ1 is the largest

e manufacturerrsquos expected loan amount L1 is

L1 q0c0 minus kq1p1 (4)

e bankrsquos expected profit Eπ3 is

Eπ3 L1 r1 minus r0( 1113857 q0c0 minus kp1q1( 1113857 r1 minus r0( 1113857 (5)

q0 in formulas (4) and (5) satisfies formula (3)

Manufacturer Market

Uncertain production Determined demand

RetailerDeliveries min q0x q1 Deliveries min min q0x q1 D

Order quantities q1 Demand D

Figure 1 Operational structure of the supply chain under the uncertainty of production

Retailer

Market

ManufacturerBank

Retailer

Market

Manufacturer

BankRetailer

Market

Manufacturer

Bank

Or

Logistics Cash flow

Figure 2 Two financing strategies for the supply chain under capital constraints

Mathematical Problems in Engineering 5

e expected profit π2 of the retailer is

π2 p2 minus p1( 1113857min q0x q11113864 1113865 + wmax q1 minus q0x 01113864 1113865 minus wmax q0x minus q1 01113864 1113865 minus kp1q1r2 q1 ltD

p2 minus p1( 1113857min q0x q1 D1113864 1113865 + wmax q1 minus q0x 01113864 1113865 minus kp1q1r2 minus wmax q0x minus q1 01113864 1113865 minus l2 max min q0x q11113864 1113865 minus D 01113864 1113865 q1 geD

⎧⎪⎨

⎪⎩(6)

When q1 geD the first item is the difference betweenthe retailerrsquos sales revenue and the ordering cost esecond item is the compensation that the retailer receivesfrom the manufacturer because of insufficient manu-facturer output e third item is the financing cost of

the retailer e fourth item is the compensation thatthe manufacturer receives from the retailer when themanufacturerrsquos output is too large e fifth item is theloss of the retailerrsquos products e retailer expects theprofit to be

Eπ2

p2 minus p1( 1113857 q0 1113946q1q0

0xf(x)dx + q1 1113946

+infin

q1q0f(x)dx1113890 1113891 + w q1 minus q0( 1113857 minus kp1q1r2 q1 ltD

minus l2 q0 1113946q1q0

0xf(x)dx + q1 1113946

+infin

q1q0f(x)dx1113890 1113891 + w q1 minus q0( 1113857 minus kp1q1r2

+ p2 minus p1 + l2( 1113857D 1113946+infin

Dq0f(x)dx + p2 minus p1 + l2( 1113857q0 1113946

Dq0

0xf(x)dx q1 geD

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(7)

e retailer arranges the purchase according to themarket demand D

Proposition 2 Under the FBS strategy the retailer expectsthe maximum profit Eπ2 when q1 D

is proposition shows that although the manufacturerhas uncertainties in production the optimal order quantityof the retailer is still the expected market demand since theactual production of the manufacturer is subject to theuniform distribution of the planned output e reason isthat when the order quantity is greater than the expectedmarket demand the retailer has a higher risk of slow salesWhen the order quantity is less than the expected marketdemand the retailer is at risk of being out of stock

e maximum expected profit of the retailer is

Eπ2 p2 minus p1( 1113857 q0 1113946Dq0

0xf(x)dx + D 1113946

+infin

Dq0f(x)dx1113890 1113891

+ w D minus q0( 1113857 minus kp1Dr2

(8)

e expected loan amount L2 of the retailer is

L2 kp1D (9)

where k is the ratio of the prepayment of the retailerrsquos unitproduct to p0

e expected profit of the bank is

Eπ4 L2 r2 minus r0( 1113857 kp1D r2 minus r0( 1113857 (10)

412 Production and Order Decision of Supply Chain En-terprises under FBU Strategy Supply chain enterprises canchoose either the FBS strategy or the FBU strategy e FBU

strategy refers to the process in which the supply chain lendsloans to banks at all levels in the production process of theorder At this time the bank provides a unified loan interestrate to the supply chain enterprises

From Proposition 1 and Proposition 2 Proposition 3and Proposition 4 are obtained

Proposition 3 Under the FBU strategy the manufacturerrsquosexpected profit Eπ1 is a convex function of the manufac-turerrsquos planned output q0 and its optimal planned yield q0satisfies the following formula

1113946q1q0

0xf(x)dx 1 minus

p1 + w minus c0 1 + r3( 1113857

p1 + l1 (11)

Proof At this point the maximum profit of the manufac-turer is

Eπ1 p1 1113946q1q0

0q0xf(x)dx + 1113946

infin

q1q0q1f(x)dx1113890 1113891

minus c0q0 minus l1 1113946infin

q1q0q0x minus q1( 1113857f(x)dx + w q0 minus q1( 1113857

minus q0c0 minus kq1p1( 1113857r3

(12)

e manufacturerrsquos expected loan amount L1 is

L1 q0c0 minus kq1p1 (13)

e expected profit of the bank is

Eπ3 L1 r3 minus r0( 1113857 q0c0 minus kp1q1( 1113857 r3 minus r0( 1113857 (14)

q0 in equations (12)ndash(14) satisfies equation (11)

6 Mathematical Problems in Engineering

Proposition 4 Under the FBU strategy the retailer expectsthe maximum profit Eπ2 when q1 D

Proof At this point the maximum profit of the retailer is

Eπ2 p2 minus p1( 1113857 q0 1113946Dq0

0xf(x)dx + D 1113946

+infin

Dq0f(x)dx1113890 1113891

+ w D minus q0( 1113857 minus kp1Dr3

(15)

e retailer expects the loan amount L2 to be

L2 kp1D (16)

where k is the ratio of the prepayment of the retailerrsquos unitproduct to p1

e expected profit of the bank is

Eπ4 L2 r3 minus r0( 1113857 kp1D r3 minus r0( 1113857 (17)

42 Bank Loan Interest Rate Setting under Two FinancingStrategies

421 Bank Loan Interest Rate Setting under FBS StrategyAccording to equation (11) the bank expects the manu-facturerrsquos loan profit Eπ3 to be related to q0 and r1According to formula (3) the manufacturerrsquos optimalplanned output q0 is negatively correlated with the bankrsquosloan interest rate r1 erefore the bank needs to set its loaninterest rate r1 to the manufacturer so that the bankrsquos ex-pected profit for the manufacturerrsquos loan is the largest underthe premise of satisfying themanufacturerrsquos maximumprofitformula (3)

According to formula (10) the bankrsquos expected loanprofit Eπ4 to the retailer is only positively correlated with thevariable r2 so the bank needs to increase its loan interest rateto the retailer as much as possible

erefore this study proposes an interest rate settingmodel under the bankrsquos expectation of loan profitmaximization

max q0c0 minus kp1q1( 1113857 r1 minus r0( 1113857 + kp1q1 r2 minus r0( 11138571113858 1113859

st

1113946q1q0

0xf(x)dx 1 minus

p1 + w minus c0 1 + r1( 1113857

p1 + l1

q1 D r0 le r1 le r10 r0 le r2 le r20

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(18)

Proposition 5 Under the FBS strategy when the bank ex-pects the maximum loan profit the following are satisfied

(1) e bankrsquos optimal loan interest rate r2 to the retailermust be the maximum acceptable loan interest rateof the retailerat is when r2 r20 r2 is the optimalsolution of equation (18)

(2) e bankrsquos loan interest rate r1 to themanufacturer isnot necessarily the maximum value r10 that themanufacturer can accept

is proposition points out the bankrsquos loan interest ratesettingmethod under the FBS strategyemethod indicatesthat the bank should give the retailer a loan interest rate thatshould take the maximum value while giving the manu-facturer a loan interest rate that does not necessarily have totake the maximum value e reason is that the informationis symmetrical e bank knows that the bankrsquos loan interestrate to the retailer is more favorable to the bankrsquos loan profitunder the condition that the retailer determines the orderquantity In contrast the bank knows that the manufac-turerrsquos planned output is inversely proportional to the loaninterest rate and the bank needs to consider the impact ofthe loan interest rate on the manufacturerrsquos planned outputus under the FBS strategy the banks optimal loan in-terest rate to the manufacturer is not always the maximum

422 Bank Loan Interest Rate Setting under FBU StrategyUnder the FBU strategy the bank gives a uniform loaninterest rate to the supply chain e interest rate is notgreater than the maximum acceptable interest rate for themanufacturer and the retailer e bankrsquos total profit fromthe manufacturer and retailer is

Eπ3 + Eπ4 q0c0 minus kp1q1( 1113857 r3 minus r0( 1113857 + kp1q1 r3 minus r0( 1113857

q0c0 r3 minus r0( 1113857

(19)According to formula (12) the manufacturerrsquos optimal

planned output q0 is negatively correlated with the bankrsquosloan interest rate r3 erefore this study proposes that theloan interest rate setting model under the bankrsquos expectedloan profit maximization is as follows

max q0c0 r3 minus r0( 1113857

st

1113946q1q0

0xf(x)dx 1 minus

p1 + w minus c0 1 + r3( 1113857

p1 + l1

q1 D r3 lt r10 r3 lt r20

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(20)

Proposition 6 Under the FBU strategy when the bankrsquossupply chain loan interest rate is the maximum value ac-ceptable by the supply chain enterprise the bankrsquos expectedloan profit is the largest at is when rlowast3 min r10 r201113864 1113865there is

max q0c0 r3 minus r0( 11138571113858 1113859 q0c0 rlowast3 minus r0( 1113857 (21)

is proposition points out the bankrsquos loan interest ratesetting method under the FBU strategy e method statesthat the bankrsquos loan interest rate should be the maximumunder the FBU strategye reason is that the information is

Mathematical Problems in Engineering 7

symmetrical e bank knows that the loan interest rate isinversely proportional to the manufacturerrsquos planned out-put and the loan interest rate is directly proportional to thebankrsquos loan profit erefore under the FBU strategy thebank chooses to increase the loan interest rate to be morebeneficial to itself

43 Analysis of the Profit of the Bank and Supply ChainEnterprises under FBS Strategy and FBU Strategy

431 Comparison of the Bankrsquos Maximum Expected LoanProfit under the FBS Strategy and FBU Strategy Based onformula (18) under the FBS strategy and formula (20) underthe FBU strategy the following propositions are proposed

Proposition 7 e maximum loan profit of the bank underthe FBS strategy is not less than the maximum loan profit ofthe bank under the FBU strategy at is the objectivefunction value of formula (18) is not smaller than the ob-jective function value of formula (20)

is proposition indicates that bankrsquos loan profits aregreater under the FBS strategy e reason is that at thistime the bank can separately set loan interest rates to makeits own profit even greater Under the FBU strategy the bankcan only consider lending at a uniform loan interest rate Atthis time the loan profit from the manufacturer and theretailer is no higher than the loan profit under the FBSstrategy

432 Comparison of Maximum Expected Profit of SupplyChain Enterprises under the FBS Strategy and FBU StrategyBecause of complete information supply chain companiesunderstand the maximization formulas (18) and (20) of thebankrsquos loan profit and know the optimal loan interest rate ofthe bank under the two financing strategies before the loanSupply chain companies establish a better loan methodbased on their expected profit formulas to maximize theirexpected profits erefore the following proposition isproposed

Proposition 8 Under the FBS strategy when the bankrsquos loaninterest rate to the manufacturer is its acceptable maximumthat is the optimal solution of equation (18) is rS

1 r10 thefollowing are satisfied

(1) If r10 le r20 the maximum expected profit of thesupply chain under the FBU strategy is not less thanits maximum expected profit under the FBS strategy

(2) If r10 gt r20 under qS0c0 minus kp1Dgt 0 and (p2 minus p1)

1113938DqU

00 xf(x)dx minus wgt 0 the FBU strategy is more

favorable to both enterprises in the supply chain

under qS0c0 minus kp1Dgt 0 and (p2 minus p1) 1113938

DqS0

0 xf(x)

dx minus wlt 0 the FBU strategy is more favorable to themanufacturer and not good for the retailer

is proposition finds out what kind of financingstrategy the supply chain enterprise chooses to be morefavorable to itself When the bankrsquos loan interest rate to themanufacturer under the FBS strategy is the maximum valueacceptable by the manufacturer and less than the maximumvalue of the retailerrsquos acceptable loan interest rate it is moreadvantageous for supply chain enterprises to choose the FBUstrategy When the bankrsquos loan interest rate to the manu-facturer under the FBS strategy is the maximum value ac-ceptable by themanufacturer and greater than themaximumloan interest rate acceptable by the retailer we find two setsof conditions under which the manufacturerrsquos expectedprofit under the FBU strategy is higher than the expectedprofit under the FBS strategy but the retailer does not fullyearn more under the FBU strategy us Proposition 8shows that the FBU strategy is beneficial to supply chainenterprises under certain conditions but it is not alwaysbeneficial to supply chain enterprises

5 Numerical Analysis

e setting of parameters is done according to the study ofJin et al [24] We assume xsimU[1 minus n 1 + n] and give thefour types of supply chain parameters in Table 2 e bankrsquosoptimal interest rate and the expected profits of banks andsupply chain enterprises are shown in Table 3 Figures 3 and4 show the comparisons of the supply chain enterprisesrsquoprofits under two different financing strategies in groups 2and 3 in Table 3 respectively

From Table 3 and Figures 3 and 4 we get the followingresults

First Proposition 5 is confirmed As can be seen fromTable 3 under the FBS strategy the bankrsquos loan interest rateto the retailer is the maximum value of the retailerrsquos ac-ceptable loan interest rate However it can be seen from theresults of group 1 in Table 3 that under this set of supplychain parameters when the bankrsquos expected profit is thelargest the loan interest rate given to the manufacturer is0062lt 01 From the results the bankrsquos maximum loanprofit from the manufacturer is not necessarily obtained atthe maximum loan interest rate and the largest loan profitfor the retailer is obtained at the maximum loan interest rate

Second Proposition 6 is confirmed As can be seen fromthe five sets of data in Table 2 under the FBU strategy themaximum bankrsquos loan interest rate acceptable by the supplychain is min 009 01 009 Obviously in the five sets ofresults in Table 3 the bankrsquos optimal lending rate under theFBU strategy is 009 From the results when the supply chainenterprises adopt the FBU strategy the bankrsquos largest loanprofit is obtained at the largest loan interest rate

ird Proposition 7 is confirmed e results of the fivegroups in Table 3 are tested as 814gt 633 769gt 581 and807gt 581 It is clear that the bankrsquos maximum expectedprofit is higher under the FBS strategy

Fourth Proposition 8 is confirmed From the results ofthe data of groups 4 and 5 in Table 3 under rS

1 r10 andr10 le r20 both sets of results indicate that the maximumexpected profit of the manufacturer and the retailer under

8 Mathematical Problems in Engineering

the FBU strategy is not less than the maximum expectedprot under the FBS strategy is conrms the rst con-clusion of Proposition 8 From the results of the data ofgroups 2 and 3 in Table 3 under rS1 r10 and r10 gt r20 bothsets of results indicate that the maximum expected prot ofthe manufacturer under the FBU strategy is greater than that

under the FBS strategy According to Figure 3 and group 2 inTable 3 the retailerrsquos maximum expected prot is largerunder the FBU strategy than the FBS strategy In contrast inFigure 4 and group 3 in Table 3 the conclusion is reversede reason is that the second set of data satises (p2 minus p1)intDq

U0

0 xf1(x)dx minus wgt 0 and the third set of data satises

2500 2600 2700 2800 2900 3000131

1315

132

1325

133

1335

134

1345

135

Expe

cted

pro

fit o

f man

ufac

ture

r und

er F

BS (E

π 1)

FBS (2762 134233167)

FBU (2765 134462212)

131

1315

132

1325

133

1335

134

1345

135

Expe

cted

pro

fit o

f man

ufac

ture

r und

er F

BU (E

π 1)

times104 times104

q0

(a)

q0

2000 2500 3000 3500 400023

235

24

245

25

255

26

FBS

FBU

2760 2780 28002566

2567

2568

FBS (2762 256669584)

FBU (2765 256688856)

times104

times104

Eπ2

(b)

Figure 3 Prot comparison of the supply chain enterprises under the two nancing strategies in group 2 in Table 3

Table 2 Supply chain parameter settings

Types c0 p1 p2 D l1 l2 w r0 k r10 r201 3 15 20 3000 10 14 4 002 02 01 0092 3 10 20 3000 10 14 4 002 02 01 0093 3 10 15 3000 10 14 4 002 02 01 0094 3 10 20 3000 10 14 4 002 02 009 015 3 10 15 3000 10 14 4 002 02 009 01

Table 3 Prots of enterprises and banks under the FBS strategy and FBU strategy

Types Financingstrategies

Optimalloan

interest rate

Optimalproductionand orderingdecisions

Bankrsquos expectedprot

Manufacturerrsquosexpected prot

Retailerrsquos expectedprot

Total expectedprot

q0 q1

1 FBS strategy r1 0062 3023 3000 814 26627 12266 38893r2 009FBU strategy r3 009 3013 3000 633 26626 12287 38913

2 FBS strategy r1 01 2762 3000 769 13423 25667 39090r2 009FBU strategy r3 009 2765 3000 581 13446 25669 39115

3 FBS strategy r1 01 2762 3000 769 13423 13039 26462r2 009FBU strategy r3 009 2765 3000 581 13446 13034 26480

4 FBS strategy r1 009 2765 3000 807 13446 25609 39055r2 01FBU strategy r3 009 2765 3000 581 13446 25669 39115

5 FBS strategy r1 009 2765 3000 807 13446 12974 26420r2 01FBU strategy r3 009 2765 3000 581 13446 13034 26480

Mathematical Problems in Engineering 9

(p2 minus p1)intDqS00 xf1(x)dx minus wlt 0 thus conrming the

second conclusion of Proposition 8

6 Conclusion

is paper studies the external nancing problems of thebilateral supply chain with nancial constraints It startswith taking the manufacturerrsquos planned output and theretailerrsquos planned order quantity as variables builds theexpected prot model of the manufacturer and the retailerunder the FBS strategy and FBU strategy and analyzes themanufacturerrsquos optimal planned output and the retailerrsquosoptimal order quantity under the two nancing strategiesen it states on the condition of symmetry of informationhow could the bank determine the loan interest rate tomaximize the expected loan prot when the bank hasobtained the information of the planned production andplanned order quantity of the manufacturer and the retailerunder the two nancing strategies At last it proposes themaximum expected loan prot model of the bank under theFBS strategy and FBU strategy to determine the optimalloan interest rate of the bank under the two nancingstrategies And it compares the prot of the bank and theprot of the supply chain under the two nancing strategies

e research shows the following (1) Under the two -nancing strategies the manufacturerrsquos optimal plannedoutput is inversely proportional to the loan interest rate andthe retailerrsquos optimal order quantity is the market demand (2)Under the FBS strategy it is more benecial for the bank toget more expected prot when it raises the loan interest ratesfor the retailer but it does not apply to the manufacturer (3)Under the FBU strategy when the bank sets the loan interestrate at the upper limit of the acceptable line the bankrsquos ex-pected loan prot is the largest but it is less than the bankrsquosexpected loan prot under the FBS strategy (4) e FBUstrategy is more protable for both the manufacturer and the

retailer when the bankrsquos loan interest rate to the manufactureris its acceptable maximum and the loan interest rate ac-ceptable by the manufacturer is less than that by the retailer

e above-mentioned conclusions supply a policyguiding the supply chain enterprises with nancial con-straints to make a choice of the nancing strategy For thebank (1) when the bilateral supply chain enterprises withuncertain output and certain demand choose the FBSstrategy the bank should give the retailer the maximum loaninterest rate and give the manufacturer a loan interest rateaccording to formula (18) (2) when they choose the FBUstrategy the bank should give the supply chain enterprisesthe largest unied loan interest rate and (3) when supplychain enterprises choose the FBS strategy the bank makemore prots so the bank should encourage them to conductthe FBS strategy as much as possible For supply chainenterprises the optimal loan strategies of nancing enter-prises under dierent conditions are dierent so theyshould reasonably choose the optimal nancing strategybased on the bankrsquos quotation and their own acceptable loaninterest rate

For the bilateral supply chain under the FBS and FBUnancing strategies all the nancing modes are distin-guished from the perspective of obtaining the bankrsquos loaninterest rates so there is no nancing strategy superior to theFBU strategy ere are various modes under the FBUstrategy and each has its own advantages such as the dealersand supplier network nancing modes and the bank logisticscooperative nancing modes erefore various nancingmodes within the FBU strategy should be studied and an-alyzed in the future

Appendix

Proof of Proposition 1 Taking the rst-order and second-order derivatives of Eπ1 with respect to q0 it follows that

2500 2600 2700 2800 2900 3000131

1315

132

1325

133

1335

134

1345

135Ex

pect

ed p

rofit

of m

anuf

actu

rer u

nder

FBS

(Eπ 1

)

FBS (2762 134233167)

FBU (2765 134462212)

131

1315

132

1325

133

1335

134

1345

135

Expe

cted

pro

fit o

f man

ufac

ture

r und

er F

BU (E

π 1)

times104 times104

q0

(a)

q0

2000 2200 2400 2600 2800 3000124

126

128

13

132

134

136

FBS

FBU

2760 2780 28001298

13

1302

1304(2762 130394792)

(2765 130344428)

times104

times104

Eπ2

(b)

Figure 4 Prot comparison of the supply chain enterprises under the two nancing strategies in group 3 in Table 3

10 Mathematical Problems in Engineering

dEπ1

dq0 p1 + l1( 1113857 1113946

q1q0

0xf(x)dx + w minus c0 1 + r1( 1113857 minus l1

(A1)

d2Eπ1dq20

minusq21q30

p1 + l1( 1113857fq1

q01113888 1113889le 0 (A2)

When (q1q0)gt 1 + n that is q0 lt (q1(1 + n)) we havef(q1q0) 0 us (d2Eπ1dq20) 0 We obtain

dEπ1

dq0 p1 + w minus c0 1 + r1( 1113857gt 0 (A3)

So when q0 lt (q1(1 + n)) Eπ1 is a monotonically in-creasing function with a fixed slope

When (q1q0)lt 1 minus n that is q0 ge (q1(1 minus n)) we havef(q1q0) 0 us (d2Eπ1dq20) 0 We obtain

dEπ1dq0

w minus c0 1 + r1( 1113857 minus l1 lt 0 (A4)

So when (q1(1 minus n))lt q0 Eπ1 is a monotonically de-creasing function with a fixed slope

When 1 minus nle (q1q0)le 1 + n we have (d2Eπ1dq20)lt 0erefore Eπ1 is a convex function about the variable q0dEπ1dq0 is a continuous function on q0 isin [0 +infin] Whenq0 (q1(1 + n)) we obtain (dEπ1dq0)gt 0 according to(A1) When q0 (q1(1 minus n)) we obtain (dEπ1dq0)lt 0according to (A1) Hence according to the intermediatevalue theorem there must exist q0 isin [(q1(1 + n)) (q1(1 minus

n))] satisfying (dEπ1dq0) 0 At this time q0 is the onlylocal maximum point of Eπ1

In summary when q0 satisfies (dEπ1dq0) 0 that is

1113946q1q0

0xf(x)dx 1 minus

p1 + w minus c0 1 + r1( 1113857

p1 + l1 (A5)

and q0 isin [0 +infin) q0 is the only absolute maximum point ofEπ1

Proof of Proposition 2 Taking the first-order derivative ofEπ2 with respect to q1 geD it follows that

dE π2

dq1

p2 minus p1( 1113857 1113946+infin

q1q0f(x)dx + w minus kp1r2 q1 ltD

minus l2 1113946+infin

q1q0f(x)dx + w minus kp1r2 q1 geD

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(A6)

(1) q1 geD

According to Proposition 1 we have (q1q0) isin[1 minus n 1 + n] Because of xf(x)ge 0 we can obtainthat 1113938

q1q00 xf(x)dx gt 0 Let u (q1q0)gt 0 and de-

fine the function

φ1(u) 1113946U

0xf(x)dx (A7)

e first-order derivative of φ1(u) follows that

φ1prime(u) uf(u)ge 0 (A8)

Because of u isin [1 minus n 1 + n] we have φ1prime(u)gt 0us φ1(u) increases in u that is 1113938

q1q00 xf(x)dx

increases in (q1q0) Hence there exists a unique(q1q0) satisfying

1113946q1q0

0xf(x)dx 1 minus

p1 + w minus c0 1 + r1( 1113857

p1 + l1 (A9)

At this time (q1q0) is a unique constant related tothe coefficients and f(x) Hence when q1 geD(dEπ2dq1) is a unique constant If (dEπ2dq1)gt 0the retailerrsquos planned order quantity will be infinitewhich contradicts the risk-neutral assumptionHence we have (dEπ2dq1)lt 0 erefore Eπ2decreases in q1 in the region q1 isin [D +infin] and at-tains the maximum at the point q1 D

(2) q1 ltD

Because (dEπ2dq1) is a unique constant if

dEπ2dq1

p2 minus p1( 1113857 1113946+infin

q1q0f(x)dx + w minus kp1r2 lt 0

(A10)

then Eπ2 decreases in q1 in the region q1 isin [0 D) andattains the local maximum at the point q1 0 usEπ2 decreases in q1 when q1 ge 0 Eπ2 attains theabsolute maximum at the point q1 0 At this pointthe supply chain is in a state of no order We will notdiscuss this situation

In fact according to the previous literature and theactual situation of the supply chain there is often w minus kp1r2gt 0 Hence when q1 ltD we can obtain (dEπ2dq1)gt 0 andthat Eπ2 increases in q1

In summary Eπ2 attains the maximum at the pointq1 D

Proof of Proposition 3 It follows the proof of Proposition 1so we omit it

Proof of Proposition 4 It follows the proof of Proposition 2so we omit it

Proof of Proposition 5

(1) Obviously nonlinear programming (18) can bedivided into the following two planning problems

Mathematical Problems in Engineering 11

max q0c0 minus kp1q1( 1113857 r1 minus r0( 1113857

st

1113946q1q0

0xf(x)dx 1 minus

p1 + w minus c0 1 + r1( 1113857

p1 + l1

q1 D r0 le r1 le r10

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(A11)

max kp1q1 r2 minus r0( 1113857

st r0 le r2 le r20(A12)

In linear programming (A12) the objective functionincreases in r2 in the region r2 isin [r0 r20] us weattain the optimum of equation (A12) at the pointr2 r20 It is also the optimum of nonlinear pro-gramming (18)

(2) In nonlinear programming (A11) let

m1 q0( 1113857 q0c0 minus kp1D( 1113857 r1 minus r0( 1113857

st 1113946q1q0

0xf(x)dx 1 minus

p1 + w minus c0 1 + r1( 1113857

p1 + l1

m2 r1( 1113857 q0c0 minus kp1D( 1113857 r1 minus r0( 1113857

st 1113946q1q0

0xf(x)dx 1 minus

p1 + w minus c0 1 + r1( 1113857

p1 + l1

(A13)

according to

1113946q1q0

0xf(x)dx 1 minus

p1 + w minus c0 1 + r1( 1113857

p1 + l1 (A14)

We obtain that r1 decreases in q0 us when (dm1dq0)gt 0 we have (dm2dr1)lt 0 When (dm1dq0)lt 0we have (dm2dr1)gt 0

Taking the first-order derivative of m1 with respect to q0it follows thatdm1

dq0 p1 + l1( 1113857

middot 1113946q1q0

0xf(x)dx minus

p1 + l1( 1113857 q0c0 minus kp1D( 1113857q21f q1q0( 1113857

c0q30

minus r0c0 minus l1 minus c0 + w

(A15)

Under different parameters of the supply chain formula(A15) can satisfy both (dm1dq0)gt 0 and (dm1dq0)lt 0erefore there is not always (dm2dr1)gt 0 that is theoptimal of r1 is not always r10 In numerical analysis we giveexamples to support this conclusion

Proof of Proposition 6 Let

g q0 r3( 1113857 q0c0 r3 minus r0( 1113857 (A16)

Since the relationship between q0 and r3 satisfies theconstraint of (20) equation (A16) can also express equation(A17) as follows

g1 q0( 1113857 q0 p1 + l1( 1113857 1113946q1q0

0xf(x)dx + w minus l1 minus c0 1 + r0( 11138571113890 1113891

(A17)

Taking the first-order and second-order derivatives ofg1(q0) with respect to q0 it follows that

g1prime q0( 1113857 p1 + l1( 1113857 1113946q1q0

0xf(x)dx + w minus l1 minus c0 1 + r0( 11138571113890 1113891

minusq21q20

fq1

q01113888 1113889 p1 + l1( 1113857

gPrime1 q0( 1113857 p1 + l1( 1113857q21q30

fq1

q01113888 1113889gt 0

q1

q0isin [1 minus n 1 + n]

(A18)

Obviously when (q1q0) isin [1 minus n 1 + n] that isq0 isin [(q1(1 + n)) (q1(1 minus n))] g1prime(q0) increases in q0According to the proof of Proposition 1 the manufacturerattains the optimum in the region q0 isin [(q1(1 + n)) (q1(1 minus n))] When q0 (q1(1 minus n)) we can obtain

g1prime q0( 1113857 w minus l1 minus c0 1 + r0( 1113857 minus(1 minus n)2

2np1 + l1( 1113857lt 0

(A19)

12 Mathematical Problems in Engineering

us g1prime(q0) is negative in the region q0 isin [(q1(1 + n))

(q1(1 minus n))] g1(q0) decreases in q0 in the regionq0 isin [(q1(1 + n)) (q1(1 minus n))] Since

1113946q1q0

0xf1(x)dx 1 minus

p1 + w minus c0 1 + r3( 1113857

p1 + l1

st q0 isinq1

1 + n

q1

1 minus n1113876 1113877

(A20)

we obtain that r3 decreases in q0 in the region q0 isin [(q1(1 + n)) (q1(1 minus n))] Hence g1(q0) increases in r3 in theregion q0 isin [(q1(1 + n)) (q1(1 minus n))]

According to formulas (A3) and (A4) and the con-clusions of Proposition 1 when r1 isin [((w minus l1 minus c0)c0)((p1 + w minus c0)c0)] we have q0 isin [(q1(1 + n)) (q1(1 minus n))] Since we assume that the manufacturer makesdecisions based on the maximum expected profit there mustbe r10 isin [((w minus l1 minus c0)c0) ((p1 + w minus c0)c0)] which maxi-mizes the expected profit of the manufacturer under the FBSstrategy Similarly there must be min r10 r201113864 1113865 isin [((w minus l1 minus

c0)c0) ((p1 + w minus c0)c0)] which maximizes the expectedprofit of the manufacturer under the FBU strategy us wehave q0 isin [(q1(1 + n)) (q1(1 minus n))] when r3 isin [r0 minr10 r201113864 1113865] Since g1(q0) decreases in q0 in the region q0 isin

[(q1(1 + n)) (q1(1 minus n))] g1(q0) increases in r3 in theregion r3 isin [r0 min r10 r201113864 1113865] erefore g1(q0) attains themaximum at the point rlowast3 min r10 r201113864 1113865 We can obtainmax[q0c0(r3 minus r0)] q0c0(rlowast3 minus r0)

Proof of Proposition 7 If formula (20) attains the optimumat the point q0 qU

0 and r3 rU3 let q0 qU

0 r1 rU3 and

r2 rU3 in formula (18) Obviously they satisfy the con-

straint of formula (18) At this time the objective function offormula (18) is equal to the maximum of formula (20) whichmeans that the optimal of formula (20) must satisfy formula(18)erefore the maximum of formula (18) is not less thanthe maximum of formula (20)

Proof of Proposition 8

(1) When r10 le r20 we have rS1 r10 min r10 r201113864 1113865 rU

3according to Proposition 5 and rS

1 r10 us whenq1 D and r1 r3 it is obvious that q0 is equal in thetwo equations according to equations (3) and (11)Hence under qS

0 qU0 qS

1 qU1 and r1 r3 we can

obtain that EπS1 EπU

1 according to equations (2) and(12)When r10 le r20 we have rU

3 r10 le r20 rS2 accord-

ing to Proposition 5 and rS1 r10 Hence under

qS0 qU

0 qS1 qU

1 and r2 ge r3 we can obtain thatEπS

2 leEπU2 according to equations (8) and (15)

erefore when rS1 r10 and r10 le r20 we obtain that

EπS1 EπU

1 and EπS2 leEπU

2 which means the ex-pected profit of the manufacturer and retailer underthe FBU strategy is not less than the profit under theFBS strategy

(2) r10 gt r20

We know that r1 decreases in q0 in the regionq0 isin [(q1(1 + n)) (q1(1 minus n))] Since rS

1 r10 andr10 gt r20 we have q0 isin [qS

0 +infin) Substituting equa-tion (3) in (2) and taking the first-order derivative ofEπ1 with respect to q0 the following can be obtaineddEπ1

dq0

p1 + l1( 1113857q21c0q

30

fq1

q01113888 1113889 q0c0 minus kp1D( 1113857 (A21)

Since qS0c0 minus kp1Dgt 0 and q0 isin [qS

0 +infin) we have(dEπ1dq0)gt 0 When the supply chain is under theFBU strategy we have rU

3 min r10 r201113864 1113865 r20according to Proposition 6 and r10 gt r20us we haverU3 r20 lt r10 rS

1 and qU0 gt qS

0 erefore we canobtain that EπU

1 gtEπS1 when (dEπ1dq0)gt 0 and

qU0 gt qS

0 In other words we have EπU1 gtEπS

1 whenrS1 r10 qS

0c0 minus kp1D gt 0 and r10 gt r20

Taking the first-order and second-order derivatives ofEπ2 with respect to q0 the following can be obtained

dEπ2

dq0 p2 minus p1( 1113857 1113946

Dq0

0xf(x)dx minus w

d2Eπ2

dq20 minus p2 minus p1( 1113857

D2

q30f

D

q01113888 1113889lt 0

(A22)

When r1 r10 and r10 gt r20 we have rU3 lt rS

1 and qU0 gt qS

0Since (d2Eπ2dq20)lt 0 we obtain that (dEπ2dq0) decreasesin q0 When q0 isin [qS

0 qU0 ] we have

mindEπ2

dq0 p2 minus p1( 1113857 1113946

DqU0

0xf(x)dx minus w

maxdEπ2

dq0 p2 minus p1( 1113857 1113946

DqS0

0xf(x)dx minus w

(A23)

When (p2 minus p1) 1113938DqU

00 xf(x)dx minus wgt 0 and q0 isin [qS

0

qU0 ] we have (dEπ2dq0)gt 0 and EπU

2 gtEπS2 When (p2 minus

p1) 1113938DqS

00 xf(x)dx minus wlt 0 and q0 isin [qS

0 qU0 ] we have (dEπ2

dq0)lt 0 and EπU2 ltEπS

2In summary when (p2 minus p1) 1113938

DqU0

0 xf(x)dx minus wgt 0 andqS0c0 minus kp1Dgt 0 we have EπU

1 gtEπS1 and EπU

2 gtEπS2 When

(p2 minus p1) 1113938DqS

00 xf(x)dx minus wlt 0 and qS

0c0 minus kp1Dgt 0 wehave EπU

1 gtEπS1 and EπU

2 ltEπS2

Data Availability

e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

e authors declare that they have no conflicts of interest

Acknowledgments

is research was funded by the Ministry of Education ofHumanities and Social Science Project of China (grantnumber 17YJAZH018)

Mathematical Problems in Engineering 13

References

[1] S Asian and X Nie ldquoCoordination in supply chains withuncertain demand and disruption risks existence analysisand insightsrdquo IEEE Transactions on Systems Man and Cy-bernetics Systems vol 44 no 9 pp 1139ndash1154 2014

[2] A Roy S S Sana and K Chaudhuri ldquoOptimal Pricing ofcompeting retailers under uncertain demand-a two layersupply chain modelrdquo Annals of Operations Research vol 260no 1-2 pp 481ndash500 2015

[3] B C Giri and S Bardhan ldquoCoordinating a supply chain underuncertain demand and random yield in presence of supplydisruptionrdquo International Journal of Production Researchvol 53 no 16 pp 5070ndash5084 2015

[4] A Arreola-Risa and G A Decroix ldquoMake-to-order versusmake-to-stock in a production-inventory system with generalproduction timesrdquo IIE Transactions vol 30 no 8 pp 705ndash713 1998

[5] A Grosfeld-Nir and B Ronen ldquoA single bottleneck systemwith binomial yields and rigid demandrdquoManagement Sciencevol 39 no 5 pp 650ndash653 1993

[6] C A Yano and H L Lee ldquoLot sizing with random yields areviewrdquoOperations Research vol 43 no 2 pp 311ndash334 1995

[7] P Kouvelis and W Zhao ldquoFinancing the newsvendor sup-plier vs Bank and the structure of optimal trade creditcontractsrdquo Operations Research vol 60 no 3 pp 566ndash5802012

[8] B R Q Zhang ldquoInventory management with asset-basedfinancingrdquoManagement Science vol 50 no 9 pp 1274ndash12922004

[9] J Ma and L Xie ldquoe comparison and complex analysis ondual-channel supply chain under different channel powerstructures and uncertain demandrdquo Nonlinear Dynamicsvol 83 no 3 pp 1379ndash1393 2016

[10] X Xu and J R Birge ldquoJoint production and financing de-cisions modeling and analysisrdquo Operations Managementvol 28 no 12 2011

[11] P Kouvelis and W Zhao ldquoe newsvendor problem andprice-only contract when bankruptcy costs existrdquo Productionand Operations Management vol 20 no 6 pp 921ndash936 2010

[12] O Boyabath and L B Toktay ldquoStochastic capacity investmentand flexible vs Dedicated technology choice in imperfectcapital marketsrdquo Management Science vol 57 no 12pp 2163ndash2179 2011

[13] Y Alan and V Gaur ldquoOperational investment and capitalstructure under asset-based lendingrdquo Manufacturing amp Ser-vice Operations Management vol 20 no 4 2018

[14] B Sun H Zhang N Yan et al ldquoA partial credit guaranteecontract in a capital-constrained supply chain financingequilibrium and coordinating strategyrdquo International Journalof Production Economics vol 173 pp 122ndash133 2016

[15] G Bi Y Fei X Yuan et al ldquoJoint operational and financialcollaboration in a capital-constrained supply chain undermanufacturer collateralrdquo Asia-Pacific Journal of OperationalResearch vol 35 no 3 Article ID 1850010 2018

[16] J Chen Y W Zhou and Y Zhong ldquoA pricingorderingmodel for a dyadic supply chain with buyback guarantee fi-nancing and fairness concernsrdquo International Journal ofProduction Research vol 55 no 7 pp 1ndash18 2018

[17] G G Cai X Chen and Z Xiao ldquoe roles of bank and tradecredits theoretical analysis and empirical evidencerdquo Pro-duction and Operations Management vol 23 no 4pp 583ndash598 2014

[18] X F Chen ldquoA model of trade credit in a capital-constraineddistribution channelrdquo International Journal of ProductionEconomics vol 159 pp 347ndash357 2015

[19] J Chod ldquoInventory risk shifting and trade creditrdquo Man-agement Science vol 63 no 10 pp 3207ndash3225 2017

[20] B Jing X Chen and G Cai ldquoEquilibrium financing in adistribution channel with capital constraintrdquo Production andOperations Management vol 21 no 6 pp 1090ndash1101 2012

[21] S A Yang and J R Birge ldquoTrade credit risk sharing andinventory financing portfoliosrdquo Management Science vol 64no 8 2018

[22] J Zhou ldquoEssays in interfaces of operations and finance insupply chainsrdquo PhD thesis University of RochesterRochester NY USA 2009

[23] Y W Zhou B Cao Y Zhong et al ldquoOptimal advertisingordering policy and finance mode selection for a capital-constrained retailer with stochastic demandrdquo Journal of theOperational Research Society vol 68 no 12 pp 1620ndash16322017

[24] W Jin Q Zhang and J Luo ldquoNon-collaborative and col-laborative financing in a bilateral supply chain with capitalconstraintsrdquo Omega vol 88 pp 210ndash222 2018

[25] B Keren ldquoe single-period inventory problem extension torandom yield from the perspective of the supply chainrdquoOmega vol 37 no 4 pp 801ndash810 2009

[26] A Sufi ldquoBank lines of credit in corporate finance an empiricalanalysisrdquo Review of Financial Studies vol 22 no 3pp 1057ndash1088 2009

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loan interest rate are inversely correlated It is proved thatthe optimal order volume of the retailer is the demand of themarket Simply put under the certain market demand al-though the manufacturerrsquos output is uncertain the orderquantity of the retailer is still the demand of the market

Second this paper puts forward the formula of the loaninterest rate under different financing strategies in the supplychain and points out that the maximization of the bankrsquos loaninterest rate does not necessarily make the maximum profitOn the contrary under certain conditions the bankrsquos loanprofit is negatively correlated with the loan interest ratewhich is contrary to our usual belief that the bankrsquos maxi-mizing loan interest rate will boost the return on loans

ird through comparing the profit of the bank underthe two financing strategies and the profit of the supply chainenterprises this study finds that the bank has higher profitunder the FBS strategy in the supply chain and the FBUstrategy is not necessarily more beneficial to supply chainenterprises In the case that the bankrsquos optimal loan interestrate to the manufacturer is at its maximum and someparameters in the supply chain meet a certain relationshipthe FBU strategy is more beneficial to the supply chain isis contrary to our usual intuition that the FBU strategy ismore beneficial to the supply chain enterprises

2 Literature Review

To solve the above-mentioned problems this study exam-ines two types of literature (1) the interface between op-erations and finance and (2) the comparison of supply chainfinancing strategies

21 Integration of Operations and Finance Some early pa-pers studied the relationship between the financing andthe operational decisions by putting the condition ofcapital constraints into traditional newsboy models Forexample Xu and Birge [10] explained the impact ofcorporate capital constraints and capital structure oninventory decisions Zhang [8] analyzed the financingdecisions of the banks and retailers at different capitallevels and found that the asset-based financing can in-crease retailersrsquo profits Kouvelis and Zhao [11] studied theimpact of bankruptcy costs on the retailersrsquo order quan-tities Later some studies further analyzed the impact ofother factors (agent costs and asymmetric information) onoperations and finances For example Boyabath andToktay [12] studied the impact of imperfect capitalmarkets on technology choices for the budget-constrainedcompanies Alan and Gaur [13] showed that the bor-rowersrsquo investment decisions may be affected by bank-ruptcy costs and asymmetric demand informationRecently some studies found that SMEsrsquo stronger partnerscould help SMEs improve their financing capabilities Forexample Sun et al [14] analyzed the interdependencebetween business decisions and financial decisions under acredit guarantee contract Bi et al [15] pointed out that thepledge of the manufacturerrsquos products would increase theorder volume of retailers and enhance the level of

cooperation in the supply chain Chen and Zhou [16]pointed out that repurchase guarantee financing can bringmore benefits to the supply chain e above study focuseson supply chain financing and operation issues for a singlecapital-constrained company In contrast our focus is onboth the manufacturer and the retailer who are financiallyconstrained We pay attention to the impact of bank fi-nancial behavior on supply chain business operation de-cisions that is the dependence of bank financial behaviorand supply chain operation decisions

22 Comparison of Different Financing Options Recentlyscholars have gradually begun to pay attention to the choiceof different financing methods in the supply chain[7 17ndash23] For example Kouvelis and Zhao [7] emphasizedthat retailers will always prefer bank financing if the beststructured solution is provided Jing et al [20] pointed outthat trade credit financing is more beneficial to enterprises atlower production costs when both banks and trade credit areviable Yang and Birge [21] pointed out that capital-con-strained retailers can choose a combination of bank fi-nancing and trade credit financing to ease the financialpressures Jin et al [24] proposed that a collaborative fi-nancing strategy is beneficial to suppliers and the entiresupply chain while for retailers noncooperative financingstrategies are preferable Our research is related to thestudies by Kouvelis et al [7] Jing et al [20] and Jin et al[24] but there are significant differences First Kouvelis andZhao [7] and Jing et al [20] considered the advantages anddisadvantages of bank financing and trade credit financingunder different conditionse two financing models belongto the internal financing and external financing of the supplychain We consider the advantages and disadvantages of theFBS strategy and FBU strategy from the perspective of theloan interest rates e FBS strategy is a bank-led financingand the FBU strategy is a combination of bank financing andtrade credit financing In fact the two financing strategies wecompare are the bank loans Second Jin et al [24] pointedout that financing strategies are beneficial to suppliers andthe entire supply chain but not to retailers We compare thedifference between the FBS strategy and the FBU strategy ofthe supply chain but the result is different from that of Jinet al [24] because we consider the differences in the profit ofthe supply chain from the perspective of the financing in-terest rate At the same time drawing on the research ofGrosfeld-Nir and Ronen [5] and Yano and Lee [6] we as-sume that the output is uncertain and the external marketdemand is determined

3 Problem Description and Assumptions

31 Operating Environment and ProductMarket In a supplychain consisting of a manufacturer and a retailer the retailerplaces orders to the manufacturer based on external marketdemand and the manufacturer arranges production plansbased on the retailerrsquos needs and their own productioncapacity e external market demand is determined emanufacturerrsquos production volume is uncertain that is

Mathematical Problems in Engineering 3

there is a certain deviation between the manufacturerrsquosplanned output and the actual production Because of theuncertainty of production there is a compensation factor inthe supply chain When the actual production of themanufacturer is higher than the order quantity of the re-tailer the retailer gives the manufacturer some compensa-tion Otherwise the manufacturer gives the retailer somecompensation e parameter settings in this study areshown in Table 1

e assumptions for the supply chain operating envi-ronment and product market are as follows (1) e actualproduction of the manufacturer is subject to a uniformdistribution with the planned output as expected (2) Supplychain enterprises are risk-neutral (3) Information is sym-metrical regardless of the cost of obtaining information frombanks and supply chain enterprises (4) e JIT method isadopted for the delivery and the manufacturerrsquos deliveryquantity is not greater than the order quantity

e manufacturerrsquos planned output q0 is deterministicbut because of the uncertainty in actual production theactual output of the manufacturer is uncertain erandom factor x is used to represent the uncertainty of themanufacturerrsquos actual output [25] where x sim U[1 minus n 1 +

n] and 0lt nlt 1 us the actual output of the manufac-turer is q0x According to assumption (1) the actualoutput of the manufacturer is subject to a uniform dis-tribution with the planned output as expected soE(q0x) q0 From assumption (2) we have wlt l1 ltp1that is the unsalable loss of the manufacturerrsquos unitproduct is between the compensation and the productprice e unsalable loss of the retailerrsquos unit product isbetween the purchase price and the selling price that isp1 lt l2 ltp2

From assumption (4) we know that the manufacturerdelivers min q0x q11113864 1113865 to the retailer and the retailerrsquos actualsales volume is min min q0x q11113864 1113865 D1113864 1113865 e supply chainoperation structure under the uncertainty of production isshown in Figure 1

32 Capital Structure and Financial Market e manufac-turer and retailer have full financial constraints in one orderand need to lend to banks e retailer is required to pay anadvance payment when placing an order to the manufac-turer e prepayment amount is proportional to the orderquantity which is kp1q1 e amount that the manufacturerneeds to make a loan to the bank is the difference between itsproduction cost and the receipt of the advance paymentq0c0 minus kp1q1

e manufacturer and retailer can choose to makeloans to the bank in two strategies One is the FBS strategyin which the manufacturer and retailer make separatecontacts with the bank and the bank gives the manu-facturer and retailer a loan interest rate that does notexceed the maximum acceptable loan rate for each en-terprise (r1 le r10 and r2 le r20) e other is the FBUstrategy where the manufacturer or retailer interfaces withthe bank as the core enterprise of the supply chain e

enterprise will give the loan to another enterprise at thesame interest rate e bankrsquos loan interest rate to themanufacturer and retailer does not exceed the minimumof the two enterprisesrsquo acceptable loan interest rate(r3 lemin r10 r201113864 1113865)

We do not consider the risk of loans because the bank-ruptcy of the supply chain enterprises is mainly due tocommercial credit loans while inmortgages and pledge loansthe collateral or pledge is often sufficient to repay their lia-bilities Sufi [26] pointed out that under the credit loan thebank will determine the credit line of the financing enterpriseerefore the probability of default of the financing enter-prisersquos loan within the credit line is acceptable by the bank Onthe contrary if the financing company needs more than thecredit line it is obvious that the actual output of the supplychain cannot meet the demand D At this time the retailerand the manufacturer will moderately reduce the orderquantity and output which can be regarded as the externalmarket demand Dprime erefore without losing the generalityit is assumed that the loan amount of the financing enterpriseis within the credit line At this time the default risk of thefinancing enterprise is within the controllable range of thebank at is under both financing strategies the funds re-quired by the manufacturer and retailer can be met by banksand the manufacturer and retailer can repay in full and ontime At the same time we assume that the bankrsquos capital costrate remains unchanged Two financing strategies for capital-constrained supply chains are shown in Figure 2

Table 1 Decision variables and model parametersIndicesi (12) Index for the manufacturer and retailer

j (SU) Index for the optimal value of FBS and FBUstrategies respectively

Decisionvariablesq0 Manufacturerrsquos planned outputq1 Retailerrsquos order quantity

riBankrsquos loan interest rate to the enterprise i

under the FBS strategy

r3Bankrsquos interest rate to the core enterprise under

the FBU strategyModelparametersc0 Unit production cost of the manufacturerli Loss of unit unsalable product of the enterprise ipi Price of unit product of the enterprise iw Coefficient of compensation priceD Market demand for productsr0 Bankrsquos capital cost rate

ri0e maximum loan interest rate acceptable by

the enterprise i

k e prepayment amount of the retailerrsquos unitproduct

Li Loan amount of the enterprise iπi Profit of the enterprise iπi+2 Bankrsquos loan profit from the enterprise ix Random factor of product productionf(x) Density function of x

4 Mathematical Problems in Engineering

4 Model Construction and Analysis

41 Operational Decisions of Supply Chain Enterprises underTwo Financing Strategies

411 Operational Decisions of Supply Chain Enterprisesunder FBS Strategy According to Section 1 we can knowthat the profit of the manufacturer is

π1 p1 min q0x q11113864 1113865 minus c0q0x + w minus l1( 1113857max q0x minus q1 01113864 1113865

minus wmax q1 minus q0x 01113864 1113865 minus q0c0 minus kq1p1( 1113857r1

(1)

where the first item is the manufacturerrsquos sales revenue thesecond item is the manufacturerrsquos production cost the thirditem is the difference between the compensation receivedand the slow-moving loss the fourth item is the compen-sation given to the retailer and the fifth item is the man-ufacturerrsquos financing cost e manufacturer expects theprofit to be

Eπ1 p1 1113946q1q0

0q0xf(x)dx + 1113946

+infin

q1q0q1f(x)dx1113890 1113891

minus c0q0 minus l1 1113946+infin

q1q0q0x minus q1( 1113857f(x)dx + w q0 minus q1( 1113857

minus q0c0 minus kq1p1( 1113857r1

(2)

Proposition 1 Under the FBS strategy the manufacturerrsquosexpected profit is a convex function and its optimal plannedoutput satisfies the following formula

1113946q1q0

0xf(x)dx 1 minus

p1 + w minus c0 1 + r1( 1113857

p1 + l1 (3)

Equation (3) shows that the manufacturerrsquos plannedoutput is negatively correlated with the bankrsquos loan in-terest rate Excessively high bank loan pricing during themanufacturer loan phase will cause manufacturers toreduce the planned output to moderately reduce loanamounts which indicates that excessive financing costswill affect the enthusiasm of financing companies Ob-viously the above conclusions are consistent with thereality Conversely if the manufacturerrsquos planned outputis positively related to the bankrsquos loan interest rate raisingthe loan interest rate will increase the manufacturerrsquosplanned output thereby indirectly increasing the loanamount and ultimately increasing the bankrsquos loan profitserefore under the conclusion that the manufacturerrsquosplanned output is positively related to the bankrsquos loaninterest rate the bank will always raise the loan interestrate which is obviously inconsistent with the actualsituation

At this time when formula (2) satisfies formula (3) themanufacturerrsquos expected profit Eπ1 is the largest

e manufacturerrsquos expected loan amount L1 is

L1 q0c0 minus kq1p1 (4)

e bankrsquos expected profit Eπ3 is

Eπ3 L1 r1 minus r0( 1113857 q0c0 minus kp1q1( 1113857 r1 minus r0( 1113857 (5)

q0 in formulas (4) and (5) satisfies formula (3)

Manufacturer Market

Uncertain production Determined demand

RetailerDeliveries min q0x q1 Deliveries min min q0x q1 D

Order quantities q1 Demand D

Figure 1 Operational structure of the supply chain under the uncertainty of production

Retailer

Market

ManufacturerBank

Retailer

Market

Manufacturer

BankRetailer

Market

Manufacturer

Bank

Or

Logistics Cash flow

Figure 2 Two financing strategies for the supply chain under capital constraints

Mathematical Problems in Engineering 5

e expected profit π2 of the retailer is

π2 p2 minus p1( 1113857min q0x q11113864 1113865 + wmax q1 minus q0x 01113864 1113865 minus wmax q0x minus q1 01113864 1113865 minus kp1q1r2 q1 ltD

p2 minus p1( 1113857min q0x q1 D1113864 1113865 + wmax q1 minus q0x 01113864 1113865 minus kp1q1r2 minus wmax q0x minus q1 01113864 1113865 minus l2 max min q0x q11113864 1113865 minus D 01113864 1113865 q1 geD

⎧⎪⎨

⎪⎩(6)

When q1 geD the first item is the difference betweenthe retailerrsquos sales revenue and the ordering cost esecond item is the compensation that the retailer receivesfrom the manufacturer because of insufficient manu-facturer output e third item is the financing cost of

the retailer e fourth item is the compensation thatthe manufacturer receives from the retailer when themanufacturerrsquos output is too large e fifth item is theloss of the retailerrsquos products e retailer expects theprofit to be

Eπ2

p2 minus p1( 1113857 q0 1113946q1q0

0xf(x)dx + q1 1113946

+infin

q1q0f(x)dx1113890 1113891 + w q1 minus q0( 1113857 minus kp1q1r2 q1 ltD

minus l2 q0 1113946q1q0

0xf(x)dx + q1 1113946

+infin

q1q0f(x)dx1113890 1113891 + w q1 minus q0( 1113857 minus kp1q1r2

+ p2 minus p1 + l2( 1113857D 1113946+infin

Dq0f(x)dx + p2 minus p1 + l2( 1113857q0 1113946

Dq0

0xf(x)dx q1 geD

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(7)

e retailer arranges the purchase according to themarket demand D

Proposition 2 Under the FBS strategy the retailer expectsthe maximum profit Eπ2 when q1 D

is proposition shows that although the manufacturerhas uncertainties in production the optimal order quantityof the retailer is still the expected market demand since theactual production of the manufacturer is subject to theuniform distribution of the planned output e reason isthat when the order quantity is greater than the expectedmarket demand the retailer has a higher risk of slow salesWhen the order quantity is less than the expected marketdemand the retailer is at risk of being out of stock

e maximum expected profit of the retailer is

Eπ2 p2 minus p1( 1113857 q0 1113946Dq0

0xf(x)dx + D 1113946

+infin

Dq0f(x)dx1113890 1113891

+ w D minus q0( 1113857 minus kp1Dr2

(8)

e expected loan amount L2 of the retailer is

L2 kp1D (9)

where k is the ratio of the prepayment of the retailerrsquos unitproduct to p0

e expected profit of the bank is

Eπ4 L2 r2 minus r0( 1113857 kp1D r2 minus r0( 1113857 (10)

412 Production and Order Decision of Supply Chain En-terprises under FBU Strategy Supply chain enterprises canchoose either the FBS strategy or the FBU strategy e FBU

strategy refers to the process in which the supply chain lendsloans to banks at all levels in the production process of theorder At this time the bank provides a unified loan interestrate to the supply chain enterprises

From Proposition 1 and Proposition 2 Proposition 3and Proposition 4 are obtained

Proposition 3 Under the FBU strategy the manufacturerrsquosexpected profit Eπ1 is a convex function of the manufac-turerrsquos planned output q0 and its optimal planned yield q0satisfies the following formula

1113946q1q0

0xf(x)dx 1 minus

p1 + w minus c0 1 + r3( 1113857

p1 + l1 (11)

Proof At this point the maximum profit of the manufac-turer is

Eπ1 p1 1113946q1q0

0q0xf(x)dx + 1113946

infin

q1q0q1f(x)dx1113890 1113891

minus c0q0 minus l1 1113946infin

q1q0q0x minus q1( 1113857f(x)dx + w q0 minus q1( 1113857

minus q0c0 minus kq1p1( 1113857r3

(12)

e manufacturerrsquos expected loan amount L1 is

L1 q0c0 minus kq1p1 (13)

e expected profit of the bank is

Eπ3 L1 r3 minus r0( 1113857 q0c0 minus kp1q1( 1113857 r3 minus r0( 1113857 (14)

q0 in equations (12)ndash(14) satisfies equation (11)

6 Mathematical Problems in Engineering

Proposition 4 Under the FBU strategy the retailer expectsthe maximum profit Eπ2 when q1 D

Proof At this point the maximum profit of the retailer is

Eπ2 p2 minus p1( 1113857 q0 1113946Dq0

0xf(x)dx + D 1113946

+infin

Dq0f(x)dx1113890 1113891

+ w D minus q0( 1113857 minus kp1Dr3

(15)

e retailer expects the loan amount L2 to be

L2 kp1D (16)

where k is the ratio of the prepayment of the retailerrsquos unitproduct to p1

e expected profit of the bank is

Eπ4 L2 r3 minus r0( 1113857 kp1D r3 minus r0( 1113857 (17)

42 Bank Loan Interest Rate Setting under Two FinancingStrategies

421 Bank Loan Interest Rate Setting under FBS StrategyAccording to equation (11) the bank expects the manu-facturerrsquos loan profit Eπ3 to be related to q0 and r1According to formula (3) the manufacturerrsquos optimalplanned output q0 is negatively correlated with the bankrsquosloan interest rate r1 erefore the bank needs to set its loaninterest rate r1 to the manufacturer so that the bankrsquos ex-pected profit for the manufacturerrsquos loan is the largest underthe premise of satisfying themanufacturerrsquos maximumprofitformula (3)

According to formula (10) the bankrsquos expected loanprofit Eπ4 to the retailer is only positively correlated with thevariable r2 so the bank needs to increase its loan interest rateto the retailer as much as possible

erefore this study proposes an interest rate settingmodel under the bankrsquos expectation of loan profitmaximization

max q0c0 minus kp1q1( 1113857 r1 minus r0( 1113857 + kp1q1 r2 minus r0( 11138571113858 1113859

st

1113946q1q0

0xf(x)dx 1 minus

p1 + w minus c0 1 + r1( 1113857

p1 + l1

q1 D r0 le r1 le r10 r0 le r2 le r20

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(18)

Proposition 5 Under the FBS strategy when the bank ex-pects the maximum loan profit the following are satisfied

(1) e bankrsquos optimal loan interest rate r2 to the retailermust be the maximum acceptable loan interest rateof the retailerat is when r2 r20 r2 is the optimalsolution of equation (18)

(2) e bankrsquos loan interest rate r1 to themanufacturer isnot necessarily the maximum value r10 that themanufacturer can accept

is proposition points out the bankrsquos loan interest ratesettingmethod under the FBS strategyemethod indicatesthat the bank should give the retailer a loan interest rate thatshould take the maximum value while giving the manu-facturer a loan interest rate that does not necessarily have totake the maximum value e reason is that the informationis symmetrical e bank knows that the bankrsquos loan interestrate to the retailer is more favorable to the bankrsquos loan profitunder the condition that the retailer determines the orderquantity In contrast the bank knows that the manufac-turerrsquos planned output is inversely proportional to the loaninterest rate and the bank needs to consider the impact ofthe loan interest rate on the manufacturerrsquos planned outputus under the FBS strategy the banks optimal loan in-terest rate to the manufacturer is not always the maximum

422 Bank Loan Interest Rate Setting under FBU StrategyUnder the FBU strategy the bank gives a uniform loaninterest rate to the supply chain e interest rate is notgreater than the maximum acceptable interest rate for themanufacturer and the retailer e bankrsquos total profit fromthe manufacturer and retailer is

Eπ3 + Eπ4 q0c0 minus kp1q1( 1113857 r3 minus r0( 1113857 + kp1q1 r3 minus r0( 1113857

q0c0 r3 minus r0( 1113857

(19)According to formula (12) the manufacturerrsquos optimal

planned output q0 is negatively correlated with the bankrsquosloan interest rate r3 erefore this study proposes that theloan interest rate setting model under the bankrsquos expectedloan profit maximization is as follows

max q0c0 r3 minus r0( 1113857

st

1113946q1q0

0xf(x)dx 1 minus

p1 + w minus c0 1 + r3( 1113857

p1 + l1

q1 D r3 lt r10 r3 lt r20

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(20)

Proposition 6 Under the FBU strategy when the bankrsquossupply chain loan interest rate is the maximum value ac-ceptable by the supply chain enterprise the bankrsquos expectedloan profit is the largest at is when rlowast3 min r10 r201113864 1113865there is

max q0c0 r3 minus r0( 11138571113858 1113859 q0c0 rlowast3 minus r0( 1113857 (21)

is proposition points out the bankrsquos loan interest ratesetting method under the FBU strategy e method statesthat the bankrsquos loan interest rate should be the maximumunder the FBU strategye reason is that the information is

Mathematical Problems in Engineering 7

symmetrical e bank knows that the loan interest rate isinversely proportional to the manufacturerrsquos planned out-put and the loan interest rate is directly proportional to thebankrsquos loan profit erefore under the FBU strategy thebank chooses to increase the loan interest rate to be morebeneficial to itself

43 Analysis of the Profit of the Bank and Supply ChainEnterprises under FBS Strategy and FBU Strategy

431 Comparison of the Bankrsquos Maximum Expected LoanProfit under the FBS Strategy and FBU Strategy Based onformula (18) under the FBS strategy and formula (20) underthe FBU strategy the following propositions are proposed

Proposition 7 e maximum loan profit of the bank underthe FBS strategy is not less than the maximum loan profit ofthe bank under the FBU strategy at is the objectivefunction value of formula (18) is not smaller than the ob-jective function value of formula (20)

is proposition indicates that bankrsquos loan profits aregreater under the FBS strategy e reason is that at thistime the bank can separately set loan interest rates to makeits own profit even greater Under the FBU strategy the bankcan only consider lending at a uniform loan interest rate Atthis time the loan profit from the manufacturer and theretailer is no higher than the loan profit under the FBSstrategy

432 Comparison of Maximum Expected Profit of SupplyChain Enterprises under the FBS Strategy and FBU StrategyBecause of complete information supply chain companiesunderstand the maximization formulas (18) and (20) of thebankrsquos loan profit and know the optimal loan interest rate ofthe bank under the two financing strategies before the loanSupply chain companies establish a better loan methodbased on their expected profit formulas to maximize theirexpected profits erefore the following proposition isproposed

Proposition 8 Under the FBS strategy when the bankrsquos loaninterest rate to the manufacturer is its acceptable maximumthat is the optimal solution of equation (18) is rS

1 r10 thefollowing are satisfied

(1) If r10 le r20 the maximum expected profit of thesupply chain under the FBU strategy is not less thanits maximum expected profit under the FBS strategy

(2) If r10 gt r20 under qS0c0 minus kp1Dgt 0 and (p2 minus p1)

1113938DqU

00 xf(x)dx minus wgt 0 the FBU strategy is more

favorable to both enterprises in the supply chain

under qS0c0 minus kp1Dgt 0 and (p2 minus p1) 1113938

DqS0

0 xf(x)

dx minus wlt 0 the FBU strategy is more favorable to themanufacturer and not good for the retailer

is proposition finds out what kind of financingstrategy the supply chain enterprise chooses to be morefavorable to itself When the bankrsquos loan interest rate to themanufacturer under the FBS strategy is the maximum valueacceptable by the manufacturer and less than the maximumvalue of the retailerrsquos acceptable loan interest rate it is moreadvantageous for supply chain enterprises to choose the FBUstrategy When the bankrsquos loan interest rate to the manu-facturer under the FBS strategy is the maximum value ac-ceptable by themanufacturer and greater than themaximumloan interest rate acceptable by the retailer we find two setsof conditions under which the manufacturerrsquos expectedprofit under the FBU strategy is higher than the expectedprofit under the FBS strategy but the retailer does not fullyearn more under the FBU strategy us Proposition 8shows that the FBU strategy is beneficial to supply chainenterprises under certain conditions but it is not alwaysbeneficial to supply chain enterprises

5 Numerical Analysis

e setting of parameters is done according to the study ofJin et al [24] We assume xsimU[1 minus n 1 + n] and give thefour types of supply chain parameters in Table 2 e bankrsquosoptimal interest rate and the expected profits of banks andsupply chain enterprises are shown in Table 3 Figures 3 and4 show the comparisons of the supply chain enterprisesrsquoprofits under two different financing strategies in groups 2and 3 in Table 3 respectively

From Table 3 and Figures 3 and 4 we get the followingresults

First Proposition 5 is confirmed As can be seen fromTable 3 under the FBS strategy the bankrsquos loan interest rateto the retailer is the maximum value of the retailerrsquos ac-ceptable loan interest rate However it can be seen from theresults of group 1 in Table 3 that under this set of supplychain parameters when the bankrsquos expected profit is thelargest the loan interest rate given to the manufacturer is0062lt 01 From the results the bankrsquos maximum loanprofit from the manufacturer is not necessarily obtained atthe maximum loan interest rate and the largest loan profitfor the retailer is obtained at the maximum loan interest rate

Second Proposition 6 is confirmed As can be seen fromthe five sets of data in Table 2 under the FBU strategy themaximum bankrsquos loan interest rate acceptable by the supplychain is min 009 01 009 Obviously in the five sets ofresults in Table 3 the bankrsquos optimal lending rate under theFBU strategy is 009 From the results when the supply chainenterprises adopt the FBU strategy the bankrsquos largest loanprofit is obtained at the largest loan interest rate

ird Proposition 7 is confirmed e results of the fivegroups in Table 3 are tested as 814gt 633 769gt 581 and807gt 581 It is clear that the bankrsquos maximum expectedprofit is higher under the FBS strategy

Fourth Proposition 8 is confirmed From the results ofthe data of groups 4 and 5 in Table 3 under rS

1 r10 andr10 le r20 both sets of results indicate that the maximumexpected profit of the manufacturer and the retailer under

8 Mathematical Problems in Engineering

the FBU strategy is not less than the maximum expectedprot under the FBS strategy is conrms the rst con-clusion of Proposition 8 From the results of the data ofgroups 2 and 3 in Table 3 under rS1 r10 and r10 gt r20 bothsets of results indicate that the maximum expected prot ofthe manufacturer under the FBU strategy is greater than that

under the FBS strategy According to Figure 3 and group 2 inTable 3 the retailerrsquos maximum expected prot is largerunder the FBU strategy than the FBS strategy In contrast inFigure 4 and group 3 in Table 3 the conclusion is reversede reason is that the second set of data satises (p2 minus p1)intDq

U0

0 xf1(x)dx minus wgt 0 and the third set of data satises

2500 2600 2700 2800 2900 3000131

1315

132

1325

133

1335

134

1345

135

Expe

cted

pro

fit o

f man

ufac

ture

r und

er F

BS (E

π 1)

FBS (2762 134233167)

FBU (2765 134462212)

131

1315

132

1325

133

1335

134

1345

135

Expe

cted

pro

fit o

f man

ufac

ture

r und

er F

BU (E

π 1)

times104 times104

q0

(a)

q0

2000 2500 3000 3500 400023

235

24

245

25

255

26

FBS

FBU

2760 2780 28002566

2567

2568

FBS (2762 256669584)

FBU (2765 256688856)

times104

times104

Eπ2

(b)

Figure 3 Prot comparison of the supply chain enterprises under the two nancing strategies in group 2 in Table 3

Table 2 Supply chain parameter settings

Types c0 p1 p2 D l1 l2 w r0 k r10 r201 3 15 20 3000 10 14 4 002 02 01 0092 3 10 20 3000 10 14 4 002 02 01 0093 3 10 15 3000 10 14 4 002 02 01 0094 3 10 20 3000 10 14 4 002 02 009 015 3 10 15 3000 10 14 4 002 02 009 01

Table 3 Prots of enterprises and banks under the FBS strategy and FBU strategy

Types Financingstrategies

Optimalloan

interest rate

Optimalproductionand orderingdecisions

Bankrsquos expectedprot

Manufacturerrsquosexpected prot

Retailerrsquos expectedprot

Total expectedprot

q0 q1

1 FBS strategy r1 0062 3023 3000 814 26627 12266 38893r2 009FBU strategy r3 009 3013 3000 633 26626 12287 38913

2 FBS strategy r1 01 2762 3000 769 13423 25667 39090r2 009FBU strategy r3 009 2765 3000 581 13446 25669 39115

3 FBS strategy r1 01 2762 3000 769 13423 13039 26462r2 009FBU strategy r3 009 2765 3000 581 13446 13034 26480

4 FBS strategy r1 009 2765 3000 807 13446 25609 39055r2 01FBU strategy r3 009 2765 3000 581 13446 25669 39115

5 FBS strategy r1 009 2765 3000 807 13446 12974 26420r2 01FBU strategy r3 009 2765 3000 581 13446 13034 26480

Mathematical Problems in Engineering 9

(p2 minus p1)intDqS00 xf1(x)dx minus wlt 0 thus conrming the

second conclusion of Proposition 8

6 Conclusion

is paper studies the external nancing problems of thebilateral supply chain with nancial constraints It startswith taking the manufacturerrsquos planned output and theretailerrsquos planned order quantity as variables builds theexpected prot model of the manufacturer and the retailerunder the FBS strategy and FBU strategy and analyzes themanufacturerrsquos optimal planned output and the retailerrsquosoptimal order quantity under the two nancing strategiesen it states on the condition of symmetry of informationhow could the bank determine the loan interest rate tomaximize the expected loan prot when the bank hasobtained the information of the planned production andplanned order quantity of the manufacturer and the retailerunder the two nancing strategies At last it proposes themaximum expected loan prot model of the bank under theFBS strategy and FBU strategy to determine the optimalloan interest rate of the bank under the two nancingstrategies And it compares the prot of the bank and theprot of the supply chain under the two nancing strategies

e research shows the following (1) Under the two -nancing strategies the manufacturerrsquos optimal plannedoutput is inversely proportional to the loan interest rate andthe retailerrsquos optimal order quantity is the market demand (2)Under the FBS strategy it is more benecial for the bank toget more expected prot when it raises the loan interest ratesfor the retailer but it does not apply to the manufacturer (3)Under the FBU strategy when the bank sets the loan interestrate at the upper limit of the acceptable line the bankrsquos ex-pected loan prot is the largest but it is less than the bankrsquosexpected loan prot under the FBS strategy (4) e FBUstrategy is more protable for both the manufacturer and the

retailer when the bankrsquos loan interest rate to the manufactureris its acceptable maximum and the loan interest rate ac-ceptable by the manufacturer is less than that by the retailer

e above-mentioned conclusions supply a policyguiding the supply chain enterprises with nancial con-straints to make a choice of the nancing strategy For thebank (1) when the bilateral supply chain enterprises withuncertain output and certain demand choose the FBSstrategy the bank should give the retailer the maximum loaninterest rate and give the manufacturer a loan interest rateaccording to formula (18) (2) when they choose the FBUstrategy the bank should give the supply chain enterprisesthe largest unied loan interest rate and (3) when supplychain enterprises choose the FBS strategy the bank makemore prots so the bank should encourage them to conductthe FBS strategy as much as possible For supply chainenterprises the optimal loan strategies of nancing enter-prises under dierent conditions are dierent so theyshould reasonably choose the optimal nancing strategybased on the bankrsquos quotation and their own acceptable loaninterest rate

For the bilateral supply chain under the FBS and FBUnancing strategies all the nancing modes are distin-guished from the perspective of obtaining the bankrsquos loaninterest rates so there is no nancing strategy superior to theFBU strategy ere are various modes under the FBUstrategy and each has its own advantages such as the dealersand supplier network nancing modes and the bank logisticscooperative nancing modes erefore various nancingmodes within the FBU strategy should be studied and an-alyzed in the future

Appendix

Proof of Proposition 1 Taking the rst-order and second-order derivatives of Eπ1 with respect to q0 it follows that

2500 2600 2700 2800 2900 3000131

1315

132

1325

133

1335

134

1345

135Ex

pect

ed p

rofit

of m

anuf

actu

rer u

nder

FBS

(Eπ 1

)

FBS (2762 134233167)

FBU (2765 134462212)

131

1315

132

1325

133

1335

134

1345

135

Expe

cted

pro

fit o

f man

ufac

ture

r und

er F

BU (E

π 1)

times104 times104

q0

(a)

q0

2000 2200 2400 2600 2800 3000124

126

128

13

132

134

136

FBS

FBU

2760 2780 28001298

13

1302

1304(2762 130394792)

(2765 130344428)

times104

times104

Eπ2

(b)

Figure 4 Prot comparison of the supply chain enterprises under the two nancing strategies in group 3 in Table 3

10 Mathematical Problems in Engineering

dEπ1

dq0 p1 + l1( 1113857 1113946

q1q0

0xf(x)dx + w minus c0 1 + r1( 1113857 minus l1

(A1)

d2Eπ1dq20

minusq21q30

p1 + l1( 1113857fq1

q01113888 1113889le 0 (A2)

When (q1q0)gt 1 + n that is q0 lt (q1(1 + n)) we havef(q1q0) 0 us (d2Eπ1dq20) 0 We obtain

dEπ1

dq0 p1 + w minus c0 1 + r1( 1113857gt 0 (A3)

So when q0 lt (q1(1 + n)) Eπ1 is a monotonically in-creasing function with a fixed slope

When (q1q0)lt 1 minus n that is q0 ge (q1(1 minus n)) we havef(q1q0) 0 us (d2Eπ1dq20) 0 We obtain

dEπ1dq0

w minus c0 1 + r1( 1113857 minus l1 lt 0 (A4)

So when (q1(1 minus n))lt q0 Eπ1 is a monotonically de-creasing function with a fixed slope

When 1 minus nle (q1q0)le 1 + n we have (d2Eπ1dq20)lt 0erefore Eπ1 is a convex function about the variable q0dEπ1dq0 is a continuous function on q0 isin [0 +infin] Whenq0 (q1(1 + n)) we obtain (dEπ1dq0)gt 0 according to(A1) When q0 (q1(1 minus n)) we obtain (dEπ1dq0)lt 0according to (A1) Hence according to the intermediatevalue theorem there must exist q0 isin [(q1(1 + n)) (q1(1 minus

n))] satisfying (dEπ1dq0) 0 At this time q0 is the onlylocal maximum point of Eπ1

In summary when q0 satisfies (dEπ1dq0) 0 that is

1113946q1q0

0xf(x)dx 1 minus

p1 + w minus c0 1 + r1( 1113857

p1 + l1 (A5)

and q0 isin [0 +infin) q0 is the only absolute maximum point ofEπ1

Proof of Proposition 2 Taking the first-order derivative ofEπ2 with respect to q1 geD it follows that

dE π2

dq1

p2 minus p1( 1113857 1113946+infin

q1q0f(x)dx + w minus kp1r2 q1 ltD

minus l2 1113946+infin

q1q0f(x)dx + w minus kp1r2 q1 geD

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(A6)

(1) q1 geD

According to Proposition 1 we have (q1q0) isin[1 minus n 1 + n] Because of xf(x)ge 0 we can obtainthat 1113938

q1q00 xf(x)dx gt 0 Let u (q1q0)gt 0 and de-

fine the function

φ1(u) 1113946U

0xf(x)dx (A7)

e first-order derivative of φ1(u) follows that

φ1prime(u) uf(u)ge 0 (A8)

Because of u isin [1 minus n 1 + n] we have φ1prime(u)gt 0us φ1(u) increases in u that is 1113938

q1q00 xf(x)dx

increases in (q1q0) Hence there exists a unique(q1q0) satisfying

1113946q1q0

0xf(x)dx 1 minus

p1 + w minus c0 1 + r1( 1113857

p1 + l1 (A9)

At this time (q1q0) is a unique constant related tothe coefficients and f(x) Hence when q1 geD(dEπ2dq1) is a unique constant If (dEπ2dq1)gt 0the retailerrsquos planned order quantity will be infinitewhich contradicts the risk-neutral assumptionHence we have (dEπ2dq1)lt 0 erefore Eπ2decreases in q1 in the region q1 isin [D +infin] and at-tains the maximum at the point q1 D

(2) q1 ltD

Because (dEπ2dq1) is a unique constant if

dEπ2dq1

p2 minus p1( 1113857 1113946+infin

q1q0f(x)dx + w minus kp1r2 lt 0

(A10)

then Eπ2 decreases in q1 in the region q1 isin [0 D) andattains the local maximum at the point q1 0 usEπ2 decreases in q1 when q1 ge 0 Eπ2 attains theabsolute maximum at the point q1 0 At this pointthe supply chain is in a state of no order We will notdiscuss this situation

In fact according to the previous literature and theactual situation of the supply chain there is often w minus kp1r2gt 0 Hence when q1 ltD we can obtain (dEπ2dq1)gt 0 andthat Eπ2 increases in q1

In summary Eπ2 attains the maximum at the pointq1 D

Proof of Proposition 3 It follows the proof of Proposition 1so we omit it

Proof of Proposition 4 It follows the proof of Proposition 2so we omit it

Proof of Proposition 5

(1) Obviously nonlinear programming (18) can bedivided into the following two planning problems

Mathematical Problems in Engineering 11

max q0c0 minus kp1q1( 1113857 r1 minus r0( 1113857

st

1113946q1q0

0xf(x)dx 1 minus

p1 + w minus c0 1 + r1( 1113857

p1 + l1

q1 D r0 le r1 le r10

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(A11)

max kp1q1 r2 minus r0( 1113857

st r0 le r2 le r20(A12)

In linear programming (A12) the objective functionincreases in r2 in the region r2 isin [r0 r20] us weattain the optimum of equation (A12) at the pointr2 r20 It is also the optimum of nonlinear pro-gramming (18)

(2) In nonlinear programming (A11) let

m1 q0( 1113857 q0c0 minus kp1D( 1113857 r1 minus r0( 1113857

st 1113946q1q0

0xf(x)dx 1 minus

p1 + w minus c0 1 + r1( 1113857

p1 + l1

m2 r1( 1113857 q0c0 minus kp1D( 1113857 r1 minus r0( 1113857

st 1113946q1q0

0xf(x)dx 1 minus

p1 + w minus c0 1 + r1( 1113857

p1 + l1

(A13)

according to

1113946q1q0

0xf(x)dx 1 minus

p1 + w minus c0 1 + r1( 1113857

p1 + l1 (A14)

We obtain that r1 decreases in q0 us when (dm1dq0)gt 0 we have (dm2dr1)lt 0 When (dm1dq0)lt 0we have (dm2dr1)gt 0

Taking the first-order derivative of m1 with respect to q0it follows thatdm1

dq0 p1 + l1( 1113857

middot 1113946q1q0

0xf(x)dx minus

p1 + l1( 1113857 q0c0 minus kp1D( 1113857q21f q1q0( 1113857

c0q30

minus r0c0 minus l1 minus c0 + w

(A15)

Under different parameters of the supply chain formula(A15) can satisfy both (dm1dq0)gt 0 and (dm1dq0)lt 0erefore there is not always (dm2dr1)gt 0 that is theoptimal of r1 is not always r10 In numerical analysis we giveexamples to support this conclusion

Proof of Proposition 6 Let

g q0 r3( 1113857 q0c0 r3 minus r0( 1113857 (A16)

Since the relationship between q0 and r3 satisfies theconstraint of (20) equation (A16) can also express equation(A17) as follows

g1 q0( 1113857 q0 p1 + l1( 1113857 1113946q1q0

0xf(x)dx + w minus l1 minus c0 1 + r0( 11138571113890 1113891

(A17)

Taking the first-order and second-order derivatives ofg1(q0) with respect to q0 it follows that

g1prime q0( 1113857 p1 + l1( 1113857 1113946q1q0

0xf(x)dx + w minus l1 minus c0 1 + r0( 11138571113890 1113891

minusq21q20

fq1

q01113888 1113889 p1 + l1( 1113857

gPrime1 q0( 1113857 p1 + l1( 1113857q21q30

fq1

q01113888 1113889gt 0

q1

q0isin [1 minus n 1 + n]

(A18)

Obviously when (q1q0) isin [1 minus n 1 + n] that isq0 isin [(q1(1 + n)) (q1(1 minus n))] g1prime(q0) increases in q0According to the proof of Proposition 1 the manufacturerattains the optimum in the region q0 isin [(q1(1 + n)) (q1(1 minus n))] When q0 (q1(1 minus n)) we can obtain

g1prime q0( 1113857 w minus l1 minus c0 1 + r0( 1113857 minus(1 minus n)2

2np1 + l1( 1113857lt 0

(A19)

12 Mathematical Problems in Engineering

us g1prime(q0) is negative in the region q0 isin [(q1(1 + n))

(q1(1 minus n))] g1(q0) decreases in q0 in the regionq0 isin [(q1(1 + n)) (q1(1 minus n))] Since

1113946q1q0

0xf1(x)dx 1 minus

p1 + w minus c0 1 + r3( 1113857

p1 + l1

st q0 isinq1

1 + n

q1

1 minus n1113876 1113877

(A20)

we obtain that r3 decreases in q0 in the region q0 isin [(q1(1 + n)) (q1(1 minus n))] Hence g1(q0) increases in r3 in theregion q0 isin [(q1(1 + n)) (q1(1 minus n))]

According to formulas (A3) and (A4) and the con-clusions of Proposition 1 when r1 isin [((w minus l1 minus c0)c0)((p1 + w minus c0)c0)] we have q0 isin [(q1(1 + n)) (q1(1 minus n))] Since we assume that the manufacturer makesdecisions based on the maximum expected profit there mustbe r10 isin [((w minus l1 minus c0)c0) ((p1 + w minus c0)c0)] which maxi-mizes the expected profit of the manufacturer under the FBSstrategy Similarly there must be min r10 r201113864 1113865 isin [((w minus l1 minus

c0)c0) ((p1 + w minus c0)c0)] which maximizes the expectedprofit of the manufacturer under the FBU strategy us wehave q0 isin [(q1(1 + n)) (q1(1 minus n))] when r3 isin [r0 minr10 r201113864 1113865] Since g1(q0) decreases in q0 in the region q0 isin

[(q1(1 + n)) (q1(1 minus n))] g1(q0) increases in r3 in theregion r3 isin [r0 min r10 r201113864 1113865] erefore g1(q0) attains themaximum at the point rlowast3 min r10 r201113864 1113865 We can obtainmax[q0c0(r3 minus r0)] q0c0(rlowast3 minus r0)

Proof of Proposition 7 If formula (20) attains the optimumat the point q0 qU

0 and r3 rU3 let q0 qU

0 r1 rU3 and

r2 rU3 in formula (18) Obviously they satisfy the con-

straint of formula (18) At this time the objective function offormula (18) is equal to the maximum of formula (20) whichmeans that the optimal of formula (20) must satisfy formula(18)erefore the maximum of formula (18) is not less thanthe maximum of formula (20)

Proof of Proposition 8

(1) When r10 le r20 we have rS1 r10 min r10 r201113864 1113865 rU

3according to Proposition 5 and rS

1 r10 us whenq1 D and r1 r3 it is obvious that q0 is equal in thetwo equations according to equations (3) and (11)Hence under qS

0 qU0 qS

1 qU1 and r1 r3 we can

obtain that EπS1 EπU

1 according to equations (2) and(12)When r10 le r20 we have rU

3 r10 le r20 rS2 accord-

ing to Proposition 5 and rS1 r10 Hence under

qS0 qU

0 qS1 qU

1 and r2 ge r3 we can obtain thatEπS

2 leEπU2 according to equations (8) and (15)

erefore when rS1 r10 and r10 le r20 we obtain that

EπS1 EπU

1 and EπS2 leEπU

2 which means the ex-pected profit of the manufacturer and retailer underthe FBU strategy is not less than the profit under theFBS strategy

(2) r10 gt r20

We know that r1 decreases in q0 in the regionq0 isin [(q1(1 + n)) (q1(1 minus n))] Since rS

1 r10 andr10 gt r20 we have q0 isin [qS

0 +infin) Substituting equa-tion (3) in (2) and taking the first-order derivative ofEπ1 with respect to q0 the following can be obtaineddEπ1

dq0

p1 + l1( 1113857q21c0q

30

fq1

q01113888 1113889 q0c0 minus kp1D( 1113857 (A21)

Since qS0c0 minus kp1Dgt 0 and q0 isin [qS

0 +infin) we have(dEπ1dq0)gt 0 When the supply chain is under theFBU strategy we have rU

3 min r10 r201113864 1113865 r20according to Proposition 6 and r10 gt r20us we haverU3 r20 lt r10 rS

1 and qU0 gt qS

0 erefore we canobtain that EπU

1 gtEπS1 when (dEπ1dq0)gt 0 and

qU0 gt qS

0 In other words we have EπU1 gtEπS

1 whenrS1 r10 qS

0c0 minus kp1D gt 0 and r10 gt r20

Taking the first-order and second-order derivatives ofEπ2 with respect to q0 the following can be obtained

dEπ2

dq0 p2 minus p1( 1113857 1113946

Dq0

0xf(x)dx minus w

d2Eπ2

dq20 minus p2 minus p1( 1113857

D2

q30f

D

q01113888 1113889lt 0

(A22)

When r1 r10 and r10 gt r20 we have rU3 lt rS

1 and qU0 gt qS

0Since (d2Eπ2dq20)lt 0 we obtain that (dEπ2dq0) decreasesin q0 When q0 isin [qS

0 qU0 ] we have

mindEπ2

dq0 p2 minus p1( 1113857 1113946

DqU0

0xf(x)dx minus w

maxdEπ2

dq0 p2 minus p1( 1113857 1113946

DqS0

0xf(x)dx minus w

(A23)

When (p2 minus p1) 1113938DqU

00 xf(x)dx minus wgt 0 and q0 isin [qS

0

qU0 ] we have (dEπ2dq0)gt 0 and EπU

2 gtEπS2 When (p2 minus

p1) 1113938DqS

00 xf(x)dx minus wlt 0 and q0 isin [qS

0 qU0 ] we have (dEπ2

dq0)lt 0 and EπU2 ltEπS

2In summary when (p2 minus p1) 1113938

DqU0

0 xf(x)dx minus wgt 0 andqS0c0 minus kp1Dgt 0 we have EπU

1 gtEπS1 and EπU

2 gtEπS2 When

(p2 minus p1) 1113938DqS

00 xf(x)dx minus wlt 0 and qS

0c0 minus kp1Dgt 0 wehave EπU

1 gtEπS1 and EπU

2 ltEπS2

Data Availability

e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

e authors declare that they have no conflicts of interest

Acknowledgments

is research was funded by the Ministry of Education ofHumanities and Social Science Project of China (grantnumber 17YJAZH018)

Mathematical Problems in Engineering 13

References

[1] S Asian and X Nie ldquoCoordination in supply chains withuncertain demand and disruption risks existence analysisand insightsrdquo IEEE Transactions on Systems Man and Cy-bernetics Systems vol 44 no 9 pp 1139ndash1154 2014

[2] A Roy S S Sana and K Chaudhuri ldquoOptimal Pricing ofcompeting retailers under uncertain demand-a two layersupply chain modelrdquo Annals of Operations Research vol 260no 1-2 pp 481ndash500 2015

[3] B C Giri and S Bardhan ldquoCoordinating a supply chain underuncertain demand and random yield in presence of supplydisruptionrdquo International Journal of Production Researchvol 53 no 16 pp 5070ndash5084 2015

[4] A Arreola-Risa and G A Decroix ldquoMake-to-order versusmake-to-stock in a production-inventory system with generalproduction timesrdquo IIE Transactions vol 30 no 8 pp 705ndash713 1998

[5] A Grosfeld-Nir and B Ronen ldquoA single bottleneck systemwith binomial yields and rigid demandrdquoManagement Sciencevol 39 no 5 pp 650ndash653 1993

[6] C A Yano and H L Lee ldquoLot sizing with random yields areviewrdquoOperations Research vol 43 no 2 pp 311ndash334 1995

[7] P Kouvelis and W Zhao ldquoFinancing the newsvendor sup-plier vs Bank and the structure of optimal trade creditcontractsrdquo Operations Research vol 60 no 3 pp 566ndash5802012

[8] B R Q Zhang ldquoInventory management with asset-basedfinancingrdquoManagement Science vol 50 no 9 pp 1274ndash12922004

[9] J Ma and L Xie ldquoe comparison and complex analysis ondual-channel supply chain under different channel powerstructures and uncertain demandrdquo Nonlinear Dynamicsvol 83 no 3 pp 1379ndash1393 2016

[10] X Xu and J R Birge ldquoJoint production and financing de-cisions modeling and analysisrdquo Operations Managementvol 28 no 12 2011

[11] P Kouvelis and W Zhao ldquoe newsvendor problem andprice-only contract when bankruptcy costs existrdquo Productionand Operations Management vol 20 no 6 pp 921ndash936 2010

[12] O Boyabath and L B Toktay ldquoStochastic capacity investmentand flexible vs Dedicated technology choice in imperfectcapital marketsrdquo Management Science vol 57 no 12pp 2163ndash2179 2011

[13] Y Alan and V Gaur ldquoOperational investment and capitalstructure under asset-based lendingrdquo Manufacturing amp Ser-vice Operations Management vol 20 no 4 2018

[14] B Sun H Zhang N Yan et al ldquoA partial credit guaranteecontract in a capital-constrained supply chain financingequilibrium and coordinating strategyrdquo International Journalof Production Economics vol 173 pp 122ndash133 2016

[15] G Bi Y Fei X Yuan et al ldquoJoint operational and financialcollaboration in a capital-constrained supply chain undermanufacturer collateralrdquo Asia-Pacific Journal of OperationalResearch vol 35 no 3 Article ID 1850010 2018

[16] J Chen Y W Zhou and Y Zhong ldquoA pricingorderingmodel for a dyadic supply chain with buyback guarantee fi-nancing and fairness concernsrdquo International Journal ofProduction Research vol 55 no 7 pp 1ndash18 2018

[17] G G Cai X Chen and Z Xiao ldquoe roles of bank and tradecredits theoretical analysis and empirical evidencerdquo Pro-duction and Operations Management vol 23 no 4pp 583ndash598 2014

[18] X F Chen ldquoA model of trade credit in a capital-constraineddistribution channelrdquo International Journal of ProductionEconomics vol 159 pp 347ndash357 2015

[19] J Chod ldquoInventory risk shifting and trade creditrdquo Man-agement Science vol 63 no 10 pp 3207ndash3225 2017

[20] B Jing X Chen and G Cai ldquoEquilibrium financing in adistribution channel with capital constraintrdquo Production andOperations Management vol 21 no 6 pp 1090ndash1101 2012

[21] S A Yang and J R Birge ldquoTrade credit risk sharing andinventory financing portfoliosrdquo Management Science vol 64no 8 2018

[22] J Zhou ldquoEssays in interfaces of operations and finance insupply chainsrdquo PhD thesis University of RochesterRochester NY USA 2009

[23] Y W Zhou B Cao Y Zhong et al ldquoOptimal advertisingordering policy and finance mode selection for a capital-constrained retailer with stochastic demandrdquo Journal of theOperational Research Society vol 68 no 12 pp 1620ndash16322017

[24] W Jin Q Zhang and J Luo ldquoNon-collaborative and col-laborative financing in a bilateral supply chain with capitalconstraintsrdquo Omega vol 88 pp 210ndash222 2018

[25] B Keren ldquoe single-period inventory problem extension torandom yield from the perspective of the supply chainrdquoOmega vol 37 no 4 pp 801ndash810 2009

[26] A Sufi ldquoBank lines of credit in corporate finance an empiricalanalysisrdquo Review of Financial Studies vol 22 no 3pp 1057ndash1088 2009

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there is a certain deviation between the manufacturerrsquosplanned output and the actual production Because of theuncertainty of production there is a compensation factor inthe supply chain When the actual production of themanufacturer is higher than the order quantity of the re-tailer the retailer gives the manufacturer some compensa-tion Otherwise the manufacturer gives the retailer somecompensation e parameter settings in this study areshown in Table 1

e assumptions for the supply chain operating envi-ronment and product market are as follows (1) e actualproduction of the manufacturer is subject to a uniformdistribution with the planned output as expected (2) Supplychain enterprises are risk-neutral (3) Information is sym-metrical regardless of the cost of obtaining information frombanks and supply chain enterprises (4) e JIT method isadopted for the delivery and the manufacturerrsquos deliveryquantity is not greater than the order quantity

e manufacturerrsquos planned output q0 is deterministicbut because of the uncertainty in actual production theactual output of the manufacturer is uncertain erandom factor x is used to represent the uncertainty of themanufacturerrsquos actual output [25] where x sim U[1 minus n 1 +

n] and 0lt nlt 1 us the actual output of the manufac-turer is q0x According to assumption (1) the actualoutput of the manufacturer is subject to a uniform dis-tribution with the planned output as expected soE(q0x) q0 From assumption (2) we have wlt l1 ltp1that is the unsalable loss of the manufacturerrsquos unitproduct is between the compensation and the productprice e unsalable loss of the retailerrsquos unit product isbetween the purchase price and the selling price that isp1 lt l2 ltp2

From assumption (4) we know that the manufacturerdelivers min q0x q11113864 1113865 to the retailer and the retailerrsquos actualsales volume is min min q0x q11113864 1113865 D1113864 1113865 e supply chainoperation structure under the uncertainty of production isshown in Figure 1

32 Capital Structure and Financial Market e manufac-turer and retailer have full financial constraints in one orderand need to lend to banks e retailer is required to pay anadvance payment when placing an order to the manufac-turer e prepayment amount is proportional to the orderquantity which is kp1q1 e amount that the manufacturerneeds to make a loan to the bank is the difference between itsproduction cost and the receipt of the advance paymentq0c0 minus kp1q1

e manufacturer and retailer can choose to makeloans to the bank in two strategies One is the FBS strategyin which the manufacturer and retailer make separatecontacts with the bank and the bank gives the manu-facturer and retailer a loan interest rate that does notexceed the maximum acceptable loan rate for each en-terprise (r1 le r10 and r2 le r20) e other is the FBUstrategy where the manufacturer or retailer interfaces withthe bank as the core enterprise of the supply chain e

enterprise will give the loan to another enterprise at thesame interest rate e bankrsquos loan interest rate to themanufacturer and retailer does not exceed the minimumof the two enterprisesrsquo acceptable loan interest rate(r3 lemin r10 r201113864 1113865)

We do not consider the risk of loans because the bank-ruptcy of the supply chain enterprises is mainly due tocommercial credit loans while inmortgages and pledge loansthe collateral or pledge is often sufficient to repay their lia-bilities Sufi [26] pointed out that under the credit loan thebank will determine the credit line of the financing enterpriseerefore the probability of default of the financing enter-prisersquos loan within the credit line is acceptable by the bank Onthe contrary if the financing company needs more than thecredit line it is obvious that the actual output of the supplychain cannot meet the demand D At this time the retailerand the manufacturer will moderately reduce the orderquantity and output which can be regarded as the externalmarket demand Dprime erefore without losing the generalityit is assumed that the loan amount of the financing enterpriseis within the credit line At this time the default risk of thefinancing enterprise is within the controllable range of thebank at is under both financing strategies the funds re-quired by the manufacturer and retailer can be met by banksand the manufacturer and retailer can repay in full and ontime At the same time we assume that the bankrsquos capital costrate remains unchanged Two financing strategies for capital-constrained supply chains are shown in Figure 2

Table 1 Decision variables and model parametersIndicesi (12) Index for the manufacturer and retailer

j (SU) Index for the optimal value of FBS and FBUstrategies respectively

Decisionvariablesq0 Manufacturerrsquos planned outputq1 Retailerrsquos order quantity

riBankrsquos loan interest rate to the enterprise i

under the FBS strategy

r3Bankrsquos interest rate to the core enterprise under

the FBU strategyModelparametersc0 Unit production cost of the manufacturerli Loss of unit unsalable product of the enterprise ipi Price of unit product of the enterprise iw Coefficient of compensation priceD Market demand for productsr0 Bankrsquos capital cost rate

ri0e maximum loan interest rate acceptable by

the enterprise i

k e prepayment amount of the retailerrsquos unitproduct

Li Loan amount of the enterprise iπi Profit of the enterprise iπi+2 Bankrsquos loan profit from the enterprise ix Random factor of product productionf(x) Density function of x

4 Mathematical Problems in Engineering

4 Model Construction and Analysis

41 Operational Decisions of Supply Chain Enterprises underTwo Financing Strategies

411 Operational Decisions of Supply Chain Enterprisesunder FBS Strategy According to Section 1 we can knowthat the profit of the manufacturer is

π1 p1 min q0x q11113864 1113865 minus c0q0x + w minus l1( 1113857max q0x minus q1 01113864 1113865

minus wmax q1 minus q0x 01113864 1113865 minus q0c0 minus kq1p1( 1113857r1

(1)

where the first item is the manufacturerrsquos sales revenue thesecond item is the manufacturerrsquos production cost the thirditem is the difference between the compensation receivedand the slow-moving loss the fourth item is the compen-sation given to the retailer and the fifth item is the man-ufacturerrsquos financing cost e manufacturer expects theprofit to be

Eπ1 p1 1113946q1q0

0q0xf(x)dx + 1113946

+infin

q1q0q1f(x)dx1113890 1113891

minus c0q0 minus l1 1113946+infin

q1q0q0x minus q1( 1113857f(x)dx + w q0 minus q1( 1113857

minus q0c0 minus kq1p1( 1113857r1

(2)

Proposition 1 Under the FBS strategy the manufacturerrsquosexpected profit is a convex function and its optimal plannedoutput satisfies the following formula

1113946q1q0

0xf(x)dx 1 minus

p1 + w minus c0 1 + r1( 1113857

p1 + l1 (3)

Equation (3) shows that the manufacturerrsquos plannedoutput is negatively correlated with the bankrsquos loan in-terest rate Excessively high bank loan pricing during themanufacturer loan phase will cause manufacturers toreduce the planned output to moderately reduce loanamounts which indicates that excessive financing costswill affect the enthusiasm of financing companies Ob-viously the above conclusions are consistent with thereality Conversely if the manufacturerrsquos planned outputis positively related to the bankrsquos loan interest rate raisingthe loan interest rate will increase the manufacturerrsquosplanned output thereby indirectly increasing the loanamount and ultimately increasing the bankrsquos loan profitserefore under the conclusion that the manufacturerrsquosplanned output is positively related to the bankrsquos loaninterest rate the bank will always raise the loan interestrate which is obviously inconsistent with the actualsituation

At this time when formula (2) satisfies formula (3) themanufacturerrsquos expected profit Eπ1 is the largest

e manufacturerrsquos expected loan amount L1 is

L1 q0c0 minus kq1p1 (4)

e bankrsquos expected profit Eπ3 is

Eπ3 L1 r1 minus r0( 1113857 q0c0 minus kp1q1( 1113857 r1 minus r0( 1113857 (5)

q0 in formulas (4) and (5) satisfies formula (3)

Manufacturer Market

Uncertain production Determined demand

RetailerDeliveries min q0x q1 Deliveries min min q0x q1 D

Order quantities q1 Demand D

Figure 1 Operational structure of the supply chain under the uncertainty of production

Retailer

Market

ManufacturerBank

Retailer

Market

Manufacturer

BankRetailer

Market

Manufacturer

Bank

Or

Logistics Cash flow

Figure 2 Two financing strategies for the supply chain under capital constraints

Mathematical Problems in Engineering 5

e expected profit π2 of the retailer is

π2 p2 minus p1( 1113857min q0x q11113864 1113865 + wmax q1 minus q0x 01113864 1113865 minus wmax q0x minus q1 01113864 1113865 minus kp1q1r2 q1 ltD

p2 minus p1( 1113857min q0x q1 D1113864 1113865 + wmax q1 minus q0x 01113864 1113865 minus kp1q1r2 minus wmax q0x minus q1 01113864 1113865 minus l2 max min q0x q11113864 1113865 minus D 01113864 1113865 q1 geD

⎧⎪⎨

⎪⎩(6)

When q1 geD the first item is the difference betweenthe retailerrsquos sales revenue and the ordering cost esecond item is the compensation that the retailer receivesfrom the manufacturer because of insufficient manu-facturer output e third item is the financing cost of

the retailer e fourth item is the compensation thatthe manufacturer receives from the retailer when themanufacturerrsquos output is too large e fifth item is theloss of the retailerrsquos products e retailer expects theprofit to be

Eπ2

p2 minus p1( 1113857 q0 1113946q1q0

0xf(x)dx + q1 1113946

+infin

q1q0f(x)dx1113890 1113891 + w q1 minus q0( 1113857 minus kp1q1r2 q1 ltD

minus l2 q0 1113946q1q0

0xf(x)dx + q1 1113946

+infin

q1q0f(x)dx1113890 1113891 + w q1 minus q0( 1113857 minus kp1q1r2

+ p2 minus p1 + l2( 1113857D 1113946+infin

Dq0f(x)dx + p2 minus p1 + l2( 1113857q0 1113946

Dq0

0xf(x)dx q1 geD

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(7)

e retailer arranges the purchase according to themarket demand D

Proposition 2 Under the FBS strategy the retailer expectsthe maximum profit Eπ2 when q1 D

is proposition shows that although the manufacturerhas uncertainties in production the optimal order quantityof the retailer is still the expected market demand since theactual production of the manufacturer is subject to theuniform distribution of the planned output e reason isthat when the order quantity is greater than the expectedmarket demand the retailer has a higher risk of slow salesWhen the order quantity is less than the expected marketdemand the retailer is at risk of being out of stock

e maximum expected profit of the retailer is

Eπ2 p2 minus p1( 1113857 q0 1113946Dq0

0xf(x)dx + D 1113946

+infin

Dq0f(x)dx1113890 1113891

+ w D minus q0( 1113857 minus kp1Dr2

(8)

e expected loan amount L2 of the retailer is

L2 kp1D (9)

where k is the ratio of the prepayment of the retailerrsquos unitproduct to p0

e expected profit of the bank is

Eπ4 L2 r2 minus r0( 1113857 kp1D r2 minus r0( 1113857 (10)

412 Production and Order Decision of Supply Chain En-terprises under FBU Strategy Supply chain enterprises canchoose either the FBS strategy or the FBU strategy e FBU

strategy refers to the process in which the supply chain lendsloans to banks at all levels in the production process of theorder At this time the bank provides a unified loan interestrate to the supply chain enterprises

From Proposition 1 and Proposition 2 Proposition 3and Proposition 4 are obtained

Proposition 3 Under the FBU strategy the manufacturerrsquosexpected profit Eπ1 is a convex function of the manufac-turerrsquos planned output q0 and its optimal planned yield q0satisfies the following formula

1113946q1q0

0xf(x)dx 1 minus

p1 + w minus c0 1 + r3( 1113857

p1 + l1 (11)

Proof At this point the maximum profit of the manufac-turer is

Eπ1 p1 1113946q1q0

0q0xf(x)dx + 1113946

infin

q1q0q1f(x)dx1113890 1113891

minus c0q0 minus l1 1113946infin

q1q0q0x minus q1( 1113857f(x)dx + w q0 minus q1( 1113857

minus q0c0 minus kq1p1( 1113857r3

(12)

e manufacturerrsquos expected loan amount L1 is

L1 q0c0 minus kq1p1 (13)

e expected profit of the bank is

Eπ3 L1 r3 minus r0( 1113857 q0c0 minus kp1q1( 1113857 r3 minus r0( 1113857 (14)

q0 in equations (12)ndash(14) satisfies equation (11)

6 Mathematical Problems in Engineering

Proposition 4 Under the FBU strategy the retailer expectsthe maximum profit Eπ2 when q1 D

Proof At this point the maximum profit of the retailer is

Eπ2 p2 minus p1( 1113857 q0 1113946Dq0

0xf(x)dx + D 1113946

+infin

Dq0f(x)dx1113890 1113891

+ w D minus q0( 1113857 minus kp1Dr3

(15)

e retailer expects the loan amount L2 to be

L2 kp1D (16)

where k is the ratio of the prepayment of the retailerrsquos unitproduct to p1

e expected profit of the bank is

Eπ4 L2 r3 minus r0( 1113857 kp1D r3 minus r0( 1113857 (17)

42 Bank Loan Interest Rate Setting under Two FinancingStrategies

421 Bank Loan Interest Rate Setting under FBS StrategyAccording to equation (11) the bank expects the manu-facturerrsquos loan profit Eπ3 to be related to q0 and r1According to formula (3) the manufacturerrsquos optimalplanned output q0 is negatively correlated with the bankrsquosloan interest rate r1 erefore the bank needs to set its loaninterest rate r1 to the manufacturer so that the bankrsquos ex-pected profit for the manufacturerrsquos loan is the largest underthe premise of satisfying themanufacturerrsquos maximumprofitformula (3)

According to formula (10) the bankrsquos expected loanprofit Eπ4 to the retailer is only positively correlated with thevariable r2 so the bank needs to increase its loan interest rateto the retailer as much as possible

erefore this study proposes an interest rate settingmodel under the bankrsquos expectation of loan profitmaximization

max q0c0 minus kp1q1( 1113857 r1 minus r0( 1113857 + kp1q1 r2 minus r0( 11138571113858 1113859

st

1113946q1q0

0xf(x)dx 1 minus

p1 + w minus c0 1 + r1( 1113857

p1 + l1

q1 D r0 le r1 le r10 r0 le r2 le r20

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(18)

Proposition 5 Under the FBS strategy when the bank ex-pects the maximum loan profit the following are satisfied

(1) e bankrsquos optimal loan interest rate r2 to the retailermust be the maximum acceptable loan interest rateof the retailerat is when r2 r20 r2 is the optimalsolution of equation (18)

(2) e bankrsquos loan interest rate r1 to themanufacturer isnot necessarily the maximum value r10 that themanufacturer can accept

is proposition points out the bankrsquos loan interest ratesettingmethod under the FBS strategyemethod indicatesthat the bank should give the retailer a loan interest rate thatshould take the maximum value while giving the manu-facturer a loan interest rate that does not necessarily have totake the maximum value e reason is that the informationis symmetrical e bank knows that the bankrsquos loan interestrate to the retailer is more favorable to the bankrsquos loan profitunder the condition that the retailer determines the orderquantity In contrast the bank knows that the manufac-turerrsquos planned output is inversely proportional to the loaninterest rate and the bank needs to consider the impact ofthe loan interest rate on the manufacturerrsquos planned outputus under the FBS strategy the banks optimal loan in-terest rate to the manufacturer is not always the maximum

422 Bank Loan Interest Rate Setting under FBU StrategyUnder the FBU strategy the bank gives a uniform loaninterest rate to the supply chain e interest rate is notgreater than the maximum acceptable interest rate for themanufacturer and the retailer e bankrsquos total profit fromthe manufacturer and retailer is

Eπ3 + Eπ4 q0c0 minus kp1q1( 1113857 r3 minus r0( 1113857 + kp1q1 r3 minus r0( 1113857

q0c0 r3 minus r0( 1113857

(19)According to formula (12) the manufacturerrsquos optimal

planned output q0 is negatively correlated with the bankrsquosloan interest rate r3 erefore this study proposes that theloan interest rate setting model under the bankrsquos expectedloan profit maximization is as follows

max q0c0 r3 minus r0( 1113857

st

1113946q1q0

0xf(x)dx 1 minus

p1 + w minus c0 1 + r3( 1113857

p1 + l1

q1 D r3 lt r10 r3 lt r20

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(20)

Proposition 6 Under the FBU strategy when the bankrsquossupply chain loan interest rate is the maximum value ac-ceptable by the supply chain enterprise the bankrsquos expectedloan profit is the largest at is when rlowast3 min r10 r201113864 1113865there is

max q0c0 r3 minus r0( 11138571113858 1113859 q0c0 rlowast3 minus r0( 1113857 (21)

is proposition points out the bankrsquos loan interest ratesetting method under the FBU strategy e method statesthat the bankrsquos loan interest rate should be the maximumunder the FBU strategye reason is that the information is

Mathematical Problems in Engineering 7

symmetrical e bank knows that the loan interest rate isinversely proportional to the manufacturerrsquos planned out-put and the loan interest rate is directly proportional to thebankrsquos loan profit erefore under the FBU strategy thebank chooses to increase the loan interest rate to be morebeneficial to itself

43 Analysis of the Profit of the Bank and Supply ChainEnterprises under FBS Strategy and FBU Strategy

431 Comparison of the Bankrsquos Maximum Expected LoanProfit under the FBS Strategy and FBU Strategy Based onformula (18) under the FBS strategy and formula (20) underthe FBU strategy the following propositions are proposed

Proposition 7 e maximum loan profit of the bank underthe FBS strategy is not less than the maximum loan profit ofthe bank under the FBU strategy at is the objectivefunction value of formula (18) is not smaller than the ob-jective function value of formula (20)

is proposition indicates that bankrsquos loan profits aregreater under the FBS strategy e reason is that at thistime the bank can separately set loan interest rates to makeits own profit even greater Under the FBU strategy the bankcan only consider lending at a uniform loan interest rate Atthis time the loan profit from the manufacturer and theretailer is no higher than the loan profit under the FBSstrategy

432 Comparison of Maximum Expected Profit of SupplyChain Enterprises under the FBS Strategy and FBU StrategyBecause of complete information supply chain companiesunderstand the maximization formulas (18) and (20) of thebankrsquos loan profit and know the optimal loan interest rate ofthe bank under the two financing strategies before the loanSupply chain companies establish a better loan methodbased on their expected profit formulas to maximize theirexpected profits erefore the following proposition isproposed

Proposition 8 Under the FBS strategy when the bankrsquos loaninterest rate to the manufacturer is its acceptable maximumthat is the optimal solution of equation (18) is rS

1 r10 thefollowing are satisfied

(1) If r10 le r20 the maximum expected profit of thesupply chain under the FBU strategy is not less thanits maximum expected profit under the FBS strategy

(2) If r10 gt r20 under qS0c0 minus kp1Dgt 0 and (p2 minus p1)

1113938DqU

00 xf(x)dx minus wgt 0 the FBU strategy is more

favorable to both enterprises in the supply chain

under qS0c0 minus kp1Dgt 0 and (p2 minus p1) 1113938

DqS0

0 xf(x)

dx minus wlt 0 the FBU strategy is more favorable to themanufacturer and not good for the retailer

is proposition finds out what kind of financingstrategy the supply chain enterprise chooses to be morefavorable to itself When the bankrsquos loan interest rate to themanufacturer under the FBS strategy is the maximum valueacceptable by the manufacturer and less than the maximumvalue of the retailerrsquos acceptable loan interest rate it is moreadvantageous for supply chain enterprises to choose the FBUstrategy When the bankrsquos loan interest rate to the manu-facturer under the FBS strategy is the maximum value ac-ceptable by themanufacturer and greater than themaximumloan interest rate acceptable by the retailer we find two setsof conditions under which the manufacturerrsquos expectedprofit under the FBU strategy is higher than the expectedprofit under the FBS strategy but the retailer does not fullyearn more under the FBU strategy us Proposition 8shows that the FBU strategy is beneficial to supply chainenterprises under certain conditions but it is not alwaysbeneficial to supply chain enterprises

5 Numerical Analysis

e setting of parameters is done according to the study ofJin et al [24] We assume xsimU[1 minus n 1 + n] and give thefour types of supply chain parameters in Table 2 e bankrsquosoptimal interest rate and the expected profits of banks andsupply chain enterprises are shown in Table 3 Figures 3 and4 show the comparisons of the supply chain enterprisesrsquoprofits under two different financing strategies in groups 2and 3 in Table 3 respectively

From Table 3 and Figures 3 and 4 we get the followingresults

First Proposition 5 is confirmed As can be seen fromTable 3 under the FBS strategy the bankrsquos loan interest rateto the retailer is the maximum value of the retailerrsquos ac-ceptable loan interest rate However it can be seen from theresults of group 1 in Table 3 that under this set of supplychain parameters when the bankrsquos expected profit is thelargest the loan interest rate given to the manufacturer is0062lt 01 From the results the bankrsquos maximum loanprofit from the manufacturer is not necessarily obtained atthe maximum loan interest rate and the largest loan profitfor the retailer is obtained at the maximum loan interest rate

Second Proposition 6 is confirmed As can be seen fromthe five sets of data in Table 2 under the FBU strategy themaximum bankrsquos loan interest rate acceptable by the supplychain is min 009 01 009 Obviously in the five sets ofresults in Table 3 the bankrsquos optimal lending rate under theFBU strategy is 009 From the results when the supply chainenterprises adopt the FBU strategy the bankrsquos largest loanprofit is obtained at the largest loan interest rate

ird Proposition 7 is confirmed e results of the fivegroups in Table 3 are tested as 814gt 633 769gt 581 and807gt 581 It is clear that the bankrsquos maximum expectedprofit is higher under the FBS strategy

Fourth Proposition 8 is confirmed From the results ofthe data of groups 4 and 5 in Table 3 under rS

1 r10 andr10 le r20 both sets of results indicate that the maximumexpected profit of the manufacturer and the retailer under

8 Mathematical Problems in Engineering

the FBU strategy is not less than the maximum expectedprot under the FBS strategy is conrms the rst con-clusion of Proposition 8 From the results of the data ofgroups 2 and 3 in Table 3 under rS1 r10 and r10 gt r20 bothsets of results indicate that the maximum expected prot ofthe manufacturer under the FBU strategy is greater than that

under the FBS strategy According to Figure 3 and group 2 inTable 3 the retailerrsquos maximum expected prot is largerunder the FBU strategy than the FBS strategy In contrast inFigure 4 and group 3 in Table 3 the conclusion is reversede reason is that the second set of data satises (p2 minus p1)intDq

U0

0 xf1(x)dx minus wgt 0 and the third set of data satises

2500 2600 2700 2800 2900 3000131

1315

132

1325

133

1335

134

1345

135

Expe

cted

pro

fit o

f man

ufac

ture

r und

er F

BS (E

π 1)

FBS (2762 134233167)

FBU (2765 134462212)

131

1315

132

1325

133

1335

134

1345

135

Expe

cted

pro

fit o

f man

ufac

ture

r und

er F

BU (E

π 1)

times104 times104

q0

(a)

q0

2000 2500 3000 3500 400023

235

24

245

25

255

26

FBS

FBU

2760 2780 28002566

2567

2568

FBS (2762 256669584)

FBU (2765 256688856)

times104

times104

Eπ2

(b)

Figure 3 Prot comparison of the supply chain enterprises under the two nancing strategies in group 2 in Table 3

Table 2 Supply chain parameter settings

Types c0 p1 p2 D l1 l2 w r0 k r10 r201 3 15 20 3000 10 14 4 002 02 01 0092 3 10 20 3000 10 14 4 002 02 01 0093 3 10 15 3000 10 14 4 002 02 01 0094 3 10 20 3000 10 14 4 002 02 009 015 3 10 15 3000 10 14 4 002 02 009 01

Table 3 Prots of enterprises and banks under the FBS strategy and FBU strategy

Types Financingstrategies

Optimalloan

interest rate

Optimalproductionand orderingdecisions

Bankrsquos expectedprot

Manufacturerrsquosexpected prot

Retailerrsquos expectedprot

Total expectedprot

q0 q1

1 FBS strategy r1 0062 3023 3000 814 26627 12266 38893r2 009FBU strategy r3 009 3013 3000 633 26626 12287 38913

2 FBS strategy r1 01 2762 3000 769 13423 25667 39090r2 009FBU strategy r3 009 2765 3000 581 13446 25669 39115

3 FBS strategy r1 01 2762 3000 769 13423 13039 26462r2 009FBU strategy r3 009 2765 3000 581 13446 13034 26480

4 FBS strategy r1 009 2765 3000 807 13446 25609 39055r2 01FBU strategy r3 009 2765 3000 581 13446 25669 39115

5 FBS strategy r1 009 2765 3000 807 13446 12974 26420r2 01FBU strategy r3 009 2765 3000 581 13446 13034 26480

Mathematical Problems in Engineering 9

(p2 minus p1)intDqS00 xf1(x)dx minus wlt 0 thus conrming the

second conclusion of Proposition 8

6 Conclusion

is paper studies the external nancing problems of thebilateral supply chain with nancial constraints It startswith taking the manufacturerrsquos planned output and theretailerrsquos planned order quantity as variables builds theexpected prot model of the manufacturer and the retailerunder the FBS strategy and FBU strategy and analyzes themanufacturerrsquos optimal planned output and the retailerrsquosoptimal order quantity under the two nancing strategiesen it states on the condition of symmetry of informationhow could the bank determine the loan interest rate tomaximize the expected loan prot when the bank hasobtained the information of the planned production andplanned order quantity of the manufacturer and the retailerunder the two nancing strategies At last it proposes themaximum expected loan prot model of the bank under theFBS strategy and FBU strategy to determine the optimalloan interest rate of the bank under the two nancingstrategies And it compares the prot of the bank and theprot of the supply chain under the two nancing strategies

e research shows the following (1) Under the two -nancing strategies the manufacturerrsquos optimal plannedoutput is inversely proportional to the loan interest rate andthe retailerrsquos optimal order quantity is the market demand (2)Under the FBS strategy it is more benecial for the bank toget more expected prot when it raises the loan interest ratesfor the retailer but it does not apply to the manufacturer (3)Under the FBU strategy when the bank sets the loan interestrate at the upper limit of the acceptable line the bankrsquos ex-pected loan prot is the largest but it is less than the bankrsquosexpected loan prot under the FBS strategy (4) e FBUstrategy is more protable for both the manufacturer and the

retailer when the bankrsquos loan interest rate to the manufactureris its acceptable maximum and the loan interest rate ac-ceptable by the manufacturer is less than that by the retailer

e above-mentioned conclusions supply a policyguiding the supply chain enterprises with nancial con-straints to make a choice of the nancing strategy For thebank (1) when the bilateral supply chain enterprises withuncertain output and certain demand choose the FBSstrategy the bank should give the retailer the maximum loaninterest rate and give the manufacturer a loan interest rateaccording to formula (18) (2) when they choose the FBUstrategy the bank should give the supply chain enterprisesthe largest unied loan interest rate and (3) when supplychain enterprises choose the FBS strategy the bank makemore prots so the bank should encourage them to conductthe FBS strategy as much as possible For supply chainenterprises the optimal loan strategies of nancing enter-prises under dierent conditions are dierent so theyshould reasonably choose the optimal nancing strategybased on the bankrsquos quotation and their own acceptable loaninterest rate

For the bilateral supply chain under the FBS and FBUnancing strategies all the nancing modes are distin-guished from the perspective of obtaining the bankrsquos loaninterest rates so there is no nancing strategy superior to theFBU strategy ere are various modes under the FBUstrategy and each has its own advantages such as the dealersand supplier network nancing modes and the bank logisticscooperative nancing modes erefore various nancingmodes within the FBU strategy should be studied and an-alyzed in the future

Appendix

Proof of Proposition 1 Taking the rst-order and second-order derivatives of Eπ1 with respect to q0 it follows that

2500 2600 2700 2800 2900 3000131

1315

132

1325

133

1335

134

1345

135Ex

pect

ed p

rofit

of m

anuf

actu

rer u

nder

FBS

(Eπ 1

)

FBS (2762 134233167)

FBU (2765 134462212)

131

1315

132

1325

133

1335

134

1345

135

Expe

cted

pro

fit o

f man

ufac

ture

r und

er F

BU (E

π 1)

times104 times104

q0

(a)

q0

2000 2200 2400 2600 2800 3000124

126

128

13

132

134

136

FBS

FBU

2760 2780 28001298

13

1302

1304(2762 130394792)

(2765 130344428)

times104

times104

Eπ2

(b)

Figure 4 Prot comparison of the supply chain enterprises under the two nancing strategies in group 3 in Table 3

10 Mathematical Problems in Engineering

dEπ1

dq0 p1 + l1( 1113857 1113946

q1q0

0xf(x)dx + w minus c0 1 + r1( 1113857 minus l1

(A1)

d2Eπ1dq20

minusq21q30

p1 + l1( 1113857fq1

q01113888 1113889le 0 (A2)

When (q1q0)gt 1 + n that is q0 lt (q1(1 + n)) we havef(q1q0) 0 us (d2Eπ1dq20) 0 We obtain

dEπ1

dq0 p1 + w minus c0 1 + r1( 1113857gt 0 (A3)

So when q0 lt (q1(1 + n)) Eπ1 is a monotonically in-creasing function with a fixed slope

When (q1q0)lt 1 minus n that is q0 ge (q1(1 minus n)) we havef(q1q0) 0 us (d2Eπ1dq20) 0 We obtain

dEπ1dq0

w minus c0 1 + r1( 1113857 minus l1 lt 0 (A4)

So when (q1(1 minus n))lt q0 Eπ1 is a monotonically de-creasing function with a fixed slope

When 1 minus nle (q1q0)le 1 + n we have (d2Eπ1dq20)lt 0erefore Eπ1 is a convex function about the variable q0dEπ1dq0 is a continuous function on q0 isin [0 +infin] Whenq0 (q1(1 + n)) we obtain (dEπ1dq0)gt 0 according to(A1) When q0 (q1(1 minus n)) we obtain (dEπ1dq0)lt 0according to (A1) Hence according to the intermediatevalue theorem there must exist q0 isin [(q1(1 + n)) (q1(1 minus

n))] satisfying (dEπ1dq0) 0 At this time q0 is the onlylocal maximum point of Eπ1

In summary when q0 satisfies (dEπ1dq0) 0 that is

1113946q1q0

0xf(x)dx 1 minus

p1 + w minus c0 1 + r1( 1113857

p1 + l1 (A5)

and q0 isin [0 +infin) q0 is the only absolute maximum point ofEπ1

Proof of Proposition 2 Taking the first-order derivative ofEπ2 with respect to q1 geD it follows that

dE π2

dq1

p2 minus p1( 1113857 1113946+infin

q1q0f(x)dx + w minus kp1r2 q1 ltD

minus l2 1113946+infin

q1q0f(x)dx + w minus kp1r2 q1 geD

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(A6)

(1) q1 geD

According to Proposition 1 we have (q1q0) isin[1 minus n 1 + n] Because of xf(x)ge 0 we can obtainthat 1113938

q1q00 xf(x)dx gt 0 Let u (q1q0)gt 0 and de-

fine the function

φ1(u) 1113946U

0xf(x)dx (A7)

e first-order derivative of φ1(u) follows that

φ1prime(u) uf(u)ge 0 (A8)

Because of u isin [1 minus n 1 + n] we have φ1prime(u)gt 0us φ1(u) increases in u that is 1113938

q1q00 xf(x)dx

increases in (q1q0) Hence there exists a unique(q1q0) satisfying

1113946q1q0

0xf(x)dx 1 minus

p1 + w minus c0 1 + r1( 1113857

p1 + l1 (A9)

At this time (q1q0) is a unique constant related tothe coefficients and f(x) Hence when q1 geD(dEπ2dq1) is a unique constant If (dEπ2dq1)gt 0the retailerrsquos planned order quantity will be infinitewhich contradicts the risk-neutral assumptionHence we have (dEπ2dq1)lt 0 erefore Eπ2decreases in q1 in the region q1 isin [D +infin] and at-tains the maximum at the point q1 D

(2) q1 ltD

Because (dEπ2dq1) is a unique constant if

dEπ2dq1

p2 minus p1( 1113857 1113946+infin

q1q0f(x)dx + w minus kp1r2 lt 0

(A10)

then Eπ2 decreases in q1 in the region q1 isin [0 D) andattains the local maximum at the point q1 0 usEπ2 decreases in q1 when q1 ge 0 Eπ2 attains theabsolute maximum at the point q1 0 At this pointthe supply chain is in a state of no order We will notdiscuss this situation

In fact according to the previous literature and theactual situation of the supply chain there is often w minus kp1r2gt 0 Hence when q1 ltD we can obtain (dEπ2dq1)gt 0 andthat Eπ2 increases in q1

In summary Eπ2 attains the maximum at the pointq1 D

Proof of Proposition 3 It follows the proof of Proposition 1so we omit it

Proof of Proposition 4 It follows the proof of Proposition 2so we omit it

Proof of Proposition 5

(1) Obviously nonlinear programming (18) can bedivided into the following two planning problems

Mathematical Problems in Engineering 11

max q0c0 minus kp1q1( 1113857 r1 minus r0( 1113857

st

1113946q1q0

0xf(x)dx 1 minus

p1 + w minus c0 1 + r1( 1113857

p1 + l1

q1 D r0 le r1 le r10

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(A11)

max kp1q1 r2 minus r0( 1113857

st r0 le r2 le r20(A12)

In linear programming (A12) the objective functionincreases in r2 in the region r2 isin [r0 r20] us weattain the optimum of equation (A12) at the pointr2 r20 It is also the optimum of nonlinear pro-gramming (18)

(2) In nonlinear programming (A11) let

m1 q0( 1113857 q0c0 minus kp1D( 1113857 r1 minus r0( 1113857

st 1113946q1q0

0xf(x)dx 1 minus

p1 + w minus c0 1 + r1( 1113857

p1 + l1

m2 r1( 1113857 q0c0 minus kp1D( 1113857 r1 minus r0( 1113857

st 1113946q1q0

0xf(x)dx 1 minus

p1 + w minus c0 1 + r1( 1113857

p1 + l1

(A13)

according to

1113946q1q0

0xf(x)dx 1 minus

p1 + w minus c0 1 + r1( 1113857

p1 + l1 (A14)

We obtain that r1 decreases in q0 us when (dm1dq0)gt 0 we have (dm2dr1)lt 0 When (dm1dq0)lt 0we have (dm2dr1)gt 0

Taking the first-order derivative of m1 with respect to q0it follows thatdm1

dq0 p1 + l1( 1113857

middot 1113946q1q0

0xf(x)dx minus

p1 + l1( 1113857 q0c0 minus kp1D( 1113857q21f q1q0( 1113857

c0q30

minus r0c0 minus l1 minus c0 + w

(A15)

Under different parameters of the supply chain formula(A15) can satisfy both (dm1dq0)gt 0 and (dm1dq0)lt 0erefore there is not always (dm2dr1)gt 0 that is theoptimal of r1 is not always r10 In numerical analysis we giveexamples to support this conclusion

Proof of Proposition 6 Let

g q0 r3( 1113857 q0c0 r3 minus r0( 1113857 (A16)

Since the relationship between q0 and r3 satisfies theconstraint of (20) equation (A16) can also express equation(A17) as follows

g1 q0( 1113857 q0 p1 + l1( 1113857 1113946q1q0

0xf(x)dx + w minus l1 minus c0 1 + r0( 11138571113890 1113891

(A17)

Taking the first-order and second-order derivatives ofg1(q0) with respect to q0 it follows that

g1prime q0( 1113857 p1 + l1( 1113857 1113946q1q0

0xf(x)dx + w minus l1 minus c0 1 + r0( 11138571113890 1113891

minusq21q20

fq1

q01113888 1113889 p1 + l1( 1113857

gPrime1 q0( 1113857 p1 + l1( 1113857q21q30

fq1

q01113888 1113889gt 0

q1

q0isin [1 minus n 1 + n]

(A18)

Obviously when (q1q0) isin [1 minus n 1 + n] that isq0 isin [(q1(1 + n)) (q1(1 minus n))] g1prime(q0) increases in q0According to the proof of Proposition 1 the manufacturerattains the optimum in the region q0 isin [(q1(1 + n)) (q1(1 minus n))] When q0 (q1(1 minus n)) we can obtain

g1prime q0( 1113857 w minus l1 minus c0 1 + r0( 1113857 minus(1 minus n)2

2np1 + l1( 1113857lt 0

(A19)

12 Mathematical Problems in Engineering

us g1prime(q0) is negative in the region q0 isin [(q1(1 + n))

(q1(1 minus n))] g1(q0) decreases in q0 in the regionq0 isin [(q1(1 + n)) (q1(1 minus n))] Since

1113946q1q0

0xf1(x)dx 1 minus

p1 + w minus c0 1 + r3( 1113857

p1 + l1

st q0 isinq1

1 + n

q1

1 minus n1113876 1113877

(A20)

we obtain that r3 decreases in q0 in the region q0 isin [(q1(1 + n)) (q1(1 minus n))] Hence g1(q0) increases in r3 in theregion q0 isin [(q1(1 + n)) (q1(1 minus n))]

According to formulas (A3) and (A4) and the con-clusions of Proposition 1 when r1 isin [((w minus l1 minus c0)c0)((p1 + w minus c0)c0)] we have q0 isin [(q1(1 + n)) (q1(1 minus n))] Since we assume that the manufacturer makesdecisions based on the maximum expected profit there mustbe r10 isin [((w minus l1 minus c0)c0) ((p1 + w minus c0)c0)] which maxi-mizes the expected profit of the manufacturer under the FBSstrategy Similarly there must be min r10 r201113864 1113865 isin [((w minus l1 minus

c0)c0) ((p1 + w minus c0)c0)] which maximizes the expectedprofit of the manufacturer under the FBU strategy us wehave q0 isin [(q1(1 + n)) (q1(1 minus n))] when r3 isin [r0 minr10 r201113864 1113865] Since g1(q0) decreases in q0 in the region q0 isin

[(q1(1 + n)) (q1(1 minus n))] g1(q0) increases in r3 in theregion r3 isin [r0 min r10 r201113864 1113865] erefore g1(q0) attains themaximum at the point rlowast3 min r10 r201113864 1113865 We can obtainmax[q0c0(r3 minus r0)] q0c0(rlowast3 minus r0)

Proof of Proposition 7 If formula (20) attains the optimumat the point q0 qU

0 and r3 rU3 let q0 qU

0 r1 rU3 and

r2 rU3 in formula (18) Obviously they satisfy the con-

straint of formula (18) At this time the objective function offormula (18) is equal to the maximum of formula (20) whichmeans that the optimal of formula (20) must satisfy formula(18)erefore the maximum of formula (18) is not less thanthe maximum of formula (20)

Proof of Proposition 8

(1) When r10 le r20 we have rS1 r10 min r10 r201113864 1113865 rU

3according to Proposition 5 and rS

1 r10 us whenq1 D and r1 r3 it is obvious that q0 is equal in thetwo equations according to equations (3) and (11)Hence under qS

0 qU0 qS

1 qU1 and r1 r3 we can

obtain that EπS1 EπU

1 according to equations (2) and(12)When r10 le r20 we have rU

3 r10 le r20 rS2 accord-

ing to Proposition 5 and rS1 r10 Hence under

qS0 qU

0 qS1 qU

1 and r2 ge r3 we can obtain thatEπS

2 leEπU2 according to equations (8) and (15)

erefore when rS1 r10 and r10 le r20 we obtain that

EπS1 EπU

1 and EπS2 leEπU

2 which means the ex-pected profit of the manufacturer and retailer underthe FBU strategy is not less than the profit under theFBS strategy

(2) r10 gt r20

We know that r1 decreases in q0 in the regionq0 isin [(q1(1 + n)) (q1(1 minus n))] Since rS

1 r10 andr10 gt r20 we have q0 isin [qS

0 +infin) Substituting equa-tion (3) in (2) and taking the first-order derivative ofEπ1 with respect to q0 the following can be obtaineddEπ1

dq0

p1 + l1( 1113857q21c0q

30

fq1

q01113888 1113889 q0c0 minus kp1D( 1113857 (A21)

Since qS0c0 minus kp1Dgt 0 and q0 isin [qS

0 +infin) we have(dEπ1dq0)gt 0 When the supply chain is under theFBU strategy we have rU

3 min r10 r201113864 1113865 r20according to Proposition 6 and r10 gt r20us we haverU3 r20 lt r10 rS

1 and qU0 gt qS

0 erefore we canobtain that EπU

1 gtEπS1 when (dEπ1dq0)gt 0 and

qU0 gt qS

0 In other words we have EπU1 gtEπS

1 whenrS1 r10 qS

0c0 minus kp1D gt 0 and r10 gt r20

Taking the first-order and second-order derivatives ofEπ2 with respect to q0 the following can be obtained

dEπ2

dq0 p2 minus p1( 1113857 1113946

Dq0

0xf(x)dx minus w

d2Eπ2

dq20 minus p2 minus p1( 1113857

D2

q30f

D

q01113888 1113889lt 0

(A22)

When r1 r10 and r10 gt r20 we have rU3 lt rS

1 and qU0 gt qS

0Since (d2Eπ2dq20)lt 0 we obtain that (dEπ2dq0) decreasesin q0 When q0 isin [qS

0 qU0 ] we have

mindEπ2

dq0 p2 minus p1( 1113857 1113946

DqU0

0xf(x)dx minus w

maxdEπ2

dq0 p2 minus p1( 1113857 1113946

DqS0

0xf(x)dx minus w

(A23)

When (p2 minus p1) 1113938DqU

00 xf(x)dx minus wgt 0 and q0 isin [qS

0

qU0 ] we have (dEπ2dq0)gt 0 and EπU

2 gtEπS2 When (p2 minus

p1) 1113938DqS

00 xf(x)dx minus wlt 0 and q0 isin [qS

0 qU0 ] we have (dEπ2

dq0)lt 0 and EπU2 ltEπS

2In summary when (p2 minus p1) 1113938

DqU0

0 xf(x)dx minus wgt 0 andqS0c0 minus kp1Dgt 0 we have EπU

1 gtEπS1 and EπU

2 gtEπS2 When

(p2 minus p1) 1113938DqS

00 xf(x)dx minus wlt 0 and qS

0c0 minus kp1Dgt 0 wehave EπU

1 gtEπS1 and EπU

2 ltEπS2

Data Availability

e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

e authors declare that they have no conflicts of interest

Acknowledgments

is research was funded by the Ministry of Education ofHumanities and Social Science Project of China (grantnumber 17YJAZH018)

Mathematical Problems in Engineering 13

References

[1] S Asian and X Nie ldquoCoordination in supply chains withuncertain demand and disruption risks existence analysisand insightsrdquo IEEE Transactions on Systems Man and Cy-bernetics Systems vol 44 no 9 pp 1139ndash1154 2014

[2] A Roy S S Sana and K Chaudhuri ldquoOptimal Pricing ofcompeting retailers under uncertain demand-a two layersupply chain modelrdquo Annals of Operations Research vol 260no 1-2 pp 481ndash500 2015

[3] B C Giri and S Bardhan ldquoCoordinating a supply chain underuncertain demand and random yield in presence of supplydisruptionrdquo International Journal of Production Researchvol 53 no 16 pp 5070ndash5084 2015

[4] A Arreola-Risa and G A Decroix ldquoMake-to-order versusmake-to-stock in a production-inventory system with generalproduction timesrdquo IIE Transactions vol 30 no 8 pp 705ndash713 1998

[5] A Grosfeld-Nir and B Ronen ldquoA single bottleneck systemwith binomial yields and rigid demandrdquoManagement Sciencevol 39 no 5 pp 650ndash653 1993

[6] C A Yano and H L Lee ldquoLot sizing with random yields areviewrdquoOperations Research vol 43 no 2 pp 311ndash334 1995

[7] P Kouvelis and W Zhao ldquoFinancing the newsvendor sup-plier vs Bank and the structure of optimal trade creditcontractsrdquo Operations Research vol 60 no 3 pp 566ndash5802012

[8] B R Q Zhang ldquoInventory management with asset-basedfinancingrdquoManagement Science vol 50 no 9 pp 1274ndash12922004

[9] J Ma and L Xie ldquoe comparison and complex analysis ondual-channel supply chain under different channel powerstructures and uncertain demandrdquo Nonlinear Dynamicsvol 83 no 3 pp 1379ndash1393 2016

[10] X Xu and J R Birge ldquoJoint production and financing de-cisions modeling and analysisrdquo Operations Managementvol 28 no 12 2011

[11] P Kouvelis and W Zhao ldquoe newsvendor problem andprice-only contract when bankruptcy costs existrdquo Productionand Operations Management vol 20 no 6 pp 921ndash936 2010

[12] O Boyabath and L B Toktay ldquoStochastic capacity investmentand flexible vs Dedicated technology choice in imperfectcapital marketsrdquo Management Science vol 57 no 12pp 2163ndash2179 2011

[13] Y Alan and V Gaur ldquoOperational investment and capitalstructure under asset-based lendingrdquo Manufacturing amp Ser-vice Operations Management vol 20 no 4 2018

[14] B Sun H Zhang N Yan et al ldquoA partial credit guaranteecontract in a capital-constrained supply chain financingequilibrium and coordinating strategyrdquo International Journalof Production Economics vol 173 pp 122ndash133 2016

[15] G Bi Y Fei X Yuan et al ldquoJoint operational and financialcollaboration in a capital-constrained supply chain undermanufacturer collateralrdquo Asia-Pacific Journal of OperationalResearch vol 35 no 3 Article ID 1850010 2018

[16] J Chen Y W Zhou and Y Zhong ldquoA pricingorderingmodel for a dyadic supply chain with buyback guarantee fi-nancing and fairness concernsrdquo International Journal ofProduction Research vol 55 no 7 pp 1ndash18 2018

[17] G G Cai X Chen and Z Xiao ldquoe roles of bank and tradecredits theoretical analysis and empirical evidencerdquo Pro-duction and Operations Management vol 23 no 4pp 583ndash598 2014

[18] X F Chen ldquoA model of trade credit in a capital-constraineddistribution channelrdquo International Journal of ProductionEconomics vol 159 pp 347ndash357 2015

[19] J Chod ldquoInventory risk shifting and trade creditrdquo Man-agement Science vol 63 no 10 pp 3207ndash3225 2017

[20] B Jing X Chen and G Cai ldquoEquilibrium financing in adistribution channel with capital constraintrdquo Production andOperations Management vol 21 no 6 pp 1090ndash1101 2012

[21] S A Yang and J R Birge ldquoTrade credit risk sharing andinventory financing portfoliosrdquo Management Science vol 64no 8 2018

[22] J Zhou ldquoEssays in interfaces of operations and finance insupply chainsrdquo PhD thesis University of RochesterRochester NY USA 2009

[23] Y W Zhou B Cao Y Zhong et al ldquoOptimal advertisingordering policy and finance mode selection for a capital-constrained retailer with stochastic demandrdquo Journal of theOperational Research Society vol 68 no 12 pp 1620ndash16322017

[24] W Jin Q Zhang and J Luo ldquoNon-collaborative and col-laborative financing in a bilateral supply chain with capitalconstraintsrdquo Omega vol 88 pp 210ndash222 2018

[25] B Keren ldquoe single-period inventory problem extension torandom yield from the perspective of the supply chainrdquoOmega vol 37 no 4 pp 801ndash810 2009

[26] A Sufi ldquoBank lines of credit in corporate finance an empiricalanalysisrdquo Review of Financial Studies vol 22 no 3pp 1057ndash1088 2009

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OptimizationJournal of

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International Journal of

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Submit your manuscripts atwwwhindawicom

4 Model Construction and Analysis

41 Operational Decisions of Supply Chain Enterprises underTwo Financing Strategies

411 Operational Decisions of Supply Chain Enterprisesunder FBS Strategy According to Section 1 we can knowthat the profit of the manufacturer is

π1 p1 min q0x q11113864 1113865 minus c0q0x + w minus l1( 1113857max q0x minus q1 01113864 1113865

minus wmax q1 minus q0x 01113864 1113865 minus q0c0 minus kq1p1( 1113857r1

(1)

where the first item is the manufacturerrsquos sales revenue thesecond item is the manufacturerrsquos production cost the thirditem is the difference between the compensation receivedand the slow-moving loss the fourth item is the compen-sation given to the retailer and the fifth item is the man-ufacturerrsquos financing cost e manufacturer expects theprofit to be

Eπ1 p1 1113946q1q0

0q0xf(x)dx + 1113946

+infin

q1q0q1f(x)dx1113890 1113891

minus c0q0 minus l1 1113946+infin

q1q0q0x minus q1( 1113857f(x)dx + w q0 minus q1( 1113857

minus q0c0 minus kq1p1( 1113857r1

(2)

Proposition 1 Under the FBS strategy the manufacturerrsquosexpected profit is a convex function and its optimal plannedoutput satisfies the following formula

1113946q1q0

0xf(x)dx 1 minus

p1 + w minus c0 1 + r1( 1113857

p1 + l1 (3)

Equation (3) shows that the manufacturerrsquos plannedoutput is negatively correlated with the bankrsquos loan in-terest rate Excessively high bank loan pricing during themanufacturer loan phase will cause manufacturers toreduce the planned output to moderately reduce loanamounts which indicates that excessive financing costswill affect the enthusiasm of financing companies Ob-viously the above conclusions are consistent with thereality Conversely if the manufacturerrsquos planned outputis positively related to the bankrsquos loan interest rate raisingthe loan interest rate will increase the manufacturerrsquosplanned output thereby indirectly increasing the loanamount and ultimately increasing the bankrsquos loan profitserefore under the conclusion that the manufacturerrsquosplanned output is positively related to the bankrsquos loaninterest rate the bank will always raise the loan interestrate which is obviously inconsistent with the actualsituation

At this time when formula (2) satisfies formula (3) themanufacturerrsquos expected profit Eπ1 is the largest

e manufacturerrsquos expected loan amount L1 is

L1 q0c0 minus kq1p1 (4)

e bankrsquos expected profit Eπ3 is

Eπ3 L1 r1 minus r0( 1113857 q0c0 minus kp1q1( 1113857 r1 minus r0( 1113857 (5)

q0 in formulas (4) and (5) satisfies formula (3)

Manufacturer Market

Uncertain production Determined demand

RetailerDeliveries min q0x q1 Deliveries min min q0x q1 D

Order quantities q1 Demand D

Figure 1 Operational structure of the supply chain under the uncertainty of production

Retailer

Market

ManufacturerBank

Retailer

Market

Manufacturer

BankRetailer

Market

Manufacturer

Bank

Or

Logistics Cash flow

Figure 2 Two financing strategies for the supply chain under capital constraints

Mathematical Problems in Engineering 5

e expected profit π2 of the retailer is

π2 p2 minus p1( 1113857min q0x q11113864 1113865 + wmax q1 minus q0x 01113864 1113865 minus wmax q0x minus q1 01113864 1113865 minus kp1q1r2 q1 ltD

p2 minus p1( 1113857min q0x q1 D1113864 1113865 + wmax q1 minus q0x 01113864 1113865 minus kp1q1r2 minus wmax q0x minus q1 01113864 1113865 minus l2 max min q0x q11113864 1113865 minus D 01113864 1113865 q1 geD

⎧⎪⎨

⎪⎩(6)

When q1 geD the first item is the difference betweenthe retailerrsquos sales revenue and the ordering cost esecond item is the compensation that the retailer receivesfrom the manufacturer because of insufficient manu-facturer output e third item is the financing cost of

the retailer e fourth item is the compensation thatthe manufacturer receives from the retailer when themanufacturerrsquos output is too large e fifth item is theloss of the retailerrsquos products e retailer expects theprofit to be

Eπ2

p2 minus p1( 1113857 q0 1113946q1q0

0xf(x)dx + q1 1113946

+infin

q1q0f(x)dx1113890 1113891 + w q1 minus q0( 1113857 minus kp1q1r2 q1 ltD

minus l2 q0 1113946q1q0

0xf(x)dx + q1 1113946

+infin

q1q0f(x)dx1113890 1113891 + w q1 minus q0( 1113857 minus kp1q1r2

+ p2 minus p1 + l2( 1113857D 1113946+infin

Dq0f(x)dx + p2 minus p1 + l2( 1113857q0 1113946

Dq0

0xf(x)dx q1 geD

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(7)

e retailer arranges the purchase according to themarket demand D

Proposition 2 Under the FBS strategy the retailer expectsthe maximum profit Eπ2 when q1 D

is proposition shows that although the manufacturerhas uncertainties in production the optimal order quantityof the retailer is still the expected market demand since theactual production of the manufacturer is subject to theuniform distribution of the planned output e reason isthat when the order quantity is greater than the expectedmarket demand the retailer has a higher risk of slow salesWhen the order quantity is less than the expected marketdemand the retailer is at risk of being out of stock

e maximum expected profit of the retailer is

Eπ2 p2 minus p1( 1113857 q0 1113946Dq0

0xf(x)dx + D 1113946

+infin

Dq0f(x)dx1113890 1113891

+ w D minus q0( 1113857 minus kp1Dr2

(8)

e expected loan amount L2 of the retailer is

L2 kp1D (9)

where k is the ratio of the prepayment of the retailerrsquos unitproduct to p0

e expected profit of the bank is

Eπ4 L2 r2 minus r0( 1113857 kp1D r2 minus r0( 1113857 (10)

412 Production and Order Decision of Supply Chain En-terprises under FBU Strategy Supply chain enterprises canchoose either the FBS strategy or the FBU strategy e FBU

strategy refers to the process in which the supply chain lendsloans to banks at all levels in the production process of theorder At this time the bank provides a unified loan interestrate to the supply chain enterprises

From Proposition 1 and Proposition 2 Proposition 3and Proposition 4 are obtained

Proposition 3 Under the FBU strategy the manufacturerrsquosexpected profit Eπ1 is a convex function of the manufac-turerrsquos planned output q0 and its optimal planned yield q0satisfies the following formula

1113946q1q0

0xf(x)dx 1 minus

p1 + w minus c0 1 + r3( 1113857

p1 + l1 (11)

Proof At this point the maximum profit of the manufac-turer is

Eπ1 p1 1113946q1q0

0q0xf(x)dx + 1113946

infin

q1q0q1f(x)dx1113890 1113891

minus c0q0 minus l1 1113946infin

q1q0q0x minus q1( 1113857f(x)dx + w q0 minus q1( 1113857

minus q0c0 minus kq1p1( 1113857r3

(12)

e manufacturerrsquos expected loan amount L1 is

L1 q0c0 minus kq1p1 (13)

e expected profit of the bank is

Eπ3 L1 r3 minus r0( 1113857 q0c0 minus kp1q1( 1113857 r3 minus r0( 1113857 (14)

q0 in equations (12)ndash(14) satisfies equation (11)

6 Mathematical Problems in Engineering

Proposition 4 Under the FBU strategy the retailer expectsthe maximum profit Eπ2 when q1 D

Proof At this point the maximum profit of the retailer is

Eπ2 p2 minus p1( 1113857 q0 1113946Dq0

0xf(x)dx + D 1113946

+infin

Dq0f(x)dx1113890 1113891

+ w D minus q0( 1113857 minus kp1Dr3

(15)

e retailer expects the loan amount L2 to be

L2 kp1D (16)

where k is the ratio of the prepayment of the retailerrsquos unitproduct to p1

e expected profit of the bank is

Eπ4 L2 r3 minus r0( 1113857 kp1D r3 minus r0( 1113857 (17)

42 Bank Loan Interest Rate Setting under Two FinancingStrategies

421 Bank Loan Interest Rate Setting under FBS StrategyAccording to equation (11) the bank expects the manu-facturerrsquos loan profit Eπ3 to be related to q0 and r1According to formula (3) the manufacturerrsquos optimalplanned output q0 is negatively correlated with the bankrsquosloan interest rate r1 erefore the bank needs to set its loaninterest rate r1 to the manufacturer so that the bankrsquos ex-pected profit for the manufacturerrsquos loan is the largest underthe premise of satisfying themanufacturerrsquos maximumprofitformula (3)

According to formula (10) the bankrsquos expected loanprofit Eπ4 to the retailer is only positively correlated with thevariable r2 so the bank needs to increase its loan interest rateto the retailer as much as possible

erefore this study proposes an interest rate settingmodel under the bankrsquos expectation of loan profitmaximization

max q0c0 minus kp1q1( 1113857 r1 minus r0( 1113857 + kp1q1 r2 minus r0( 11138571113858 1113859

st

1113946q1q0

0xf(x)dx 1 minus

p1 + w minus c0 1 + r1( 1113857

p1 + l1

q1 D r0 le r1 le r10 r0 le r2 le r20

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(18)

Proposition 5 Under the FBS strategy when the bank ex-pects the maximum loan profit the following are satisfied

(1) e bankrsquos optimal loan interest rate r2 to the retailermust be the maximum acceptable loan interest rateof the retailerat is when r2 r20 r2 is the optimalsolution of equation (18)

(2) e bankrsquos loan interest rate r1 to themanufacturer isnot necessarily the maximum value r10 that themanufacturer can accept

is proposition points out the bankrsquos loan interest ratesettingmethod under the FBS strategyemethod indicatesthat the bank should give the retailer a loan interest rate thatshould take the maximum value while giving the manu-facturer a loan interest rate that does not necessarily have totake the maximum value e reason is that the informationis symmetrical e bank knows that the bankrsquos loan interestrate to the retailer is more favorable to the bankrsquos loan profitunder the condition that the retailer determines the orderquantity In contrast the bank knows that the manufac-turerrsquos planned output is inversely proportional to the loaninterest rate and the bank needs to consider the impact ofthe loan interest rate on the manufacturerrsquos planned outputus under the FBS strategy the banks optimal loan in-terest rate to the manufacturer is not always the maximum

422 Bank Loan Interest Rate Setting under FBU StrategyUnder the FBU strategy the bank gives a uniform loaninterest rate to the supply chain e interest rate is notgreater than the maximum acceptable interest rate for themanufacturer and the retailer e bankrsquos total profit fromthe manufacturer and retailer is

Eπ3 + Eπ4 q0c0 minus kp1q1( 1113857 r3 minus r0( 1113857 + kp1q1 r3 minus r0( 1113857

q0c0 r3 minus r0( 1113857

(19)According to formula (12) the manufacturerrsquos optimal

planned output q0 is negatively correlated with the bankrsquosloan interest rate r3 erefore this study proposes that theloan interest rate setting model under the bankrsquos expectedloan profit maximization is as follows

max q0c0 r3 minus r0( 1113857

st

1113946q1q0

0xf(x)dx 1 minus

p1 + w minus c0 1 + r3( 1113857

p1 + l1

q1 D r3 lt r10 r3 lt r20

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(20)

Proposition 6 Under the FBU strategy when the bankrsquossupply chain loan interest rate is the maximum value ac-ceptable by the supply chain enterprise the bankrsquos expectedloan profit is the largest at is when rlowast3 min r10 r201113864 1113865there is

max q0c0 r3 minus r0( 11138571113858 1113859 q0c0 rlowast3 minus r0( 1113857 (21)

is proposition points out the bankrsquos loan interest ratesetting method under the FBU strategy e method statesthat the bankrsquos loan interest rate should be the maximumunder the FBU strategye reason is that the information is

Mathematical Problems in Engineering 7

symmetrical e bank knows that the loan interest rate isinversely proportional to the manufacturerrsquos planned out-put and the loan interest rate is directly proportional to thebankrsquos loan profit erefore under the FBU strategy thebank chooses to increase the loan interest rate to be morebeneficial to itself

43 Analysis of the Profit of the Bank and Supply ChainEnterprises under FBS Strategy and FBU Strategy

431 Comparison of the Bankrsquos Maximum Expected LoanProfit under the FBS Strategy and FBU Strategy Based onformula (18) under the FBS strategy and formula (20) underthe FBU strategy the following propositions are proposed

Proposition 7 e maximum loan profit of the bank underthe FBS strategy is not less than the maximum loan profit ofthe bank under the FBU strategy at is the objectivefunction value of formula (18) is not smaller than the ob-jective function value of formula (20)

is proposition indicates that bankrsquos loan profits aregreater under the FBS strategy e reason is that at thistime the bank can separately set loan interest rates to makeits own profit even greater Under the FBU strategy the bankcan only consider lending at a uniform loan interest rate Atthis time the loan profit from the manufacturer and theretailer is no higher than the loan profit under the FBSstrategy

432 Comparison of Maximum Expected Profit of SupplyChain Enterprises under the FBS Strategy and FBU StrategyBecause of complete information supply chain companiesunderstand the maximization formulas (18) and (20) of thebankrsquos loan profit and know the optimal loan interest rate ofthe bank under the two financing strategies before the loanSupply chain companies establish a better loan methodbased on their expected profit formulas to maximize theirexpected profits erefore the following proposition isproposed

Proposition 8 Under the FBS strategy when the bankrsquos loaninterest rate to the manufacturer is its acceptable maximumthat is the optimal solution of equation (18) is rS

1 r10 thefollowing are satisfied

(1) If r10 le r20 the maximum expected profit of thesupply chain under the FBU strategy is not less thanits maximum expected profit under the FBS strategy

(2) If r10 gt r20 under qS0c0 minus kp1Dgt 0 and (p2 minus p1)

1113938DqU

00 xf(x)dx minus wgt 0 the FBU strategy is more

favorable to both enterprises in the supply chain

under qS0c0 minus kp1Dgt 0 and (p2 minus p1) 1113938

DqS0

0 xf(x)

dx minus wlt 0 the FBU strategy is more favorable to themanufacturer and not good for the retailer

is proposition finds out what kind of financingstrategy the supply chain enterprise chooses to be morefavorable to itself When the bankrsquos loan interest rate to themanufacturer under the FBS strategy is the maximum valueacceptable by the manufacturer and less than the maximumvalue of the retailerrsquos acceptable loan interest rate it is moreadvantageous for supply chain enterprises to choose the FBUstrategy When the bankrsquos loan interest rate to the manu-facturer under the FBS strategy is the maximum value ac-ceptable by themanufacturer and greater than themaximumloan interest rate acceptable by the retailer we find two setsof conditions under which the manufacturerrsquos expectedprofit under the FBU strategy is higher than the expectedprofit under the FBS strategy but the retailer does not fullyearn more under the FBU strategy us Proposition 8shows that the FBU strategy is beneficial to supply chainenterprises under certain conditions but it is not alwaysbeneficial to supply chain enterprises

5 Numerical Analysis

e setting of parameters is done according to the study ofJin et al [24] We assume xsimU[1 minus n 1 + n] and give thefour types of supply chain parameters in Table 2 e bankrsquosoptimal interest rate and the expected profits of banks andsupply chain enterprises are shown in Table 3 Figures 3 and4 show the comparisons of the supply chain enterprisesrsquoprofits under two different financing strategies in groups 2and 3 in Table 3 respectively

From Table 3 and Figures 3 and 4 we get the followingresults

First Proposition 5 is confirmed As can be seen fromTable 3 under the FBS strategy the bankrsquos loan interest rateto the retailer is the maximum value of the retailerrsquos ac-ceptable loan interest rate However it can be seen from theresults of group 1 in Table 3 that under this set of supplychain parameters when the bankrsquos expected profit is thelargest the loan interest rate given to the manufacturer is0062lt 01 From the results the bankrsquos maximum loanprofit from the manufacturer is not necessarily obtained atthe maximum loan interest rate and the largest loan profitfor the retailer is obtained at the maximum loan interest rate

Second Proposition 6 is confirmed As can be seen fromthe five sets of data in Table 2 under the FBU strategy themaximum bankrsquos loan interest rate acceptable by the supplychain is min 009 01 009 Obviously in the five sets ofresults in Table 3 the bankrsquos optimal lending rate under theFBU strategy is 009 From the results when the supply chainenterprises adopt the FBU strategy the bankrsquos largest loanprofit is obtained at the largest loan interest rate

ird Proposition 7 is confirmed e results of the fivegroups in Table 3 are tested as 814gt 633 769gt 581 and807gt 581 It is clear that the bankrsquos maximum expectedprofit is higher under the FBS strategy

Fourth Proposition 8 is confirmed From the results ofthe data of groups 4 and 5 in Table 3 under rS

1 r10 andr10 le r20 both sets of results indicate that the maximumexpected profit of the manufacturer and the retailer under

8 Mathematical Problems in Engineering

the FBU strategy is not less than the maximum expectedprot under the FBS strategy is conrms the rst con-clusion of Proposition 8 From the results of the data ofgroups 2 and 3 in Table 3 under rS1 r10 and r10 gt r20 bothsets of results indicate that the maximum expected prot ofthe manufacturer under the FBU strategy is greater than that

under the FBS strategy According to Figure 3 and group 2 inTable 3 the retailerrsquos maximum expected prot is largerunder the FBU strategy than the FBS strategy In contrast inFigure 4 and group 3 in Table 3 the conclusion is reversede reason is that the second set of data satises (p2 minus p1)intDq

U0

0 xf1(x)dx minus wgt 0 and the third set of data satises

2500 2600 2700 2800 2900 3000131

1315

132

1325

133

1335

134

1345

135

Expe

cted

pro

fit o

f man

ufac

ture

r und

er F

BS (E

π 1)

FBS (2762 134233167)

FBU (2765 134462212)

131

1315

132

1325

133

1335

134

1345

135

Expe

cted

pro

fit o

f man

ufac

ture

r und

er F

BU (E

π 1)

times104 times104

q0

(a)

q0

2000 2500 3000 3500 400023

235

24

245

25

255

26

FBS

FBU

2760 2780 28002566

2567

2568

FBS (2762 256669584)

FBU (2765 256688856)

times104

times104

Eπ2

(b)

Figure 3 Prot comparison of the supply chain enterprises under the two nancing strategies in group 2 in Table 3

Table 2 Supply chain parameter settings

Types c0 p1 p2 D l1 l2 w r0 k r10 r201 3 15 20 3000 10 14 4 002 02 01 0092 3 10 20 3000 10 14 4 002 02 01 0093 3 10 15 3000 10 14 4 002 02 01 0094 3 10 20 3000 10 14 4 002 02 009 015 3 10 15 3000 10 14 4 002 02 009 01

Table 3 Prots of enterprises and banks under the FBS strategy and FBU strategy

Types Financingstrategies

Optimalloan

interest rate

Optimalproductionand orderingdecisions

Bankrsquos expectedprot

Manufacturerrsquosexpected prot

Retailerrsquos expectedprot

Total expectedprot

q0 q1

1 FBS strategy r1 0062 3023 3000 814 26627 12266 38893r2 009FBU strategy r3 009 3013 3000 633 26626 12287 38913

2 FBS strategy r1 01 2762 3000 769 13423 25667 39090r2 009FBU strategy r3 009 2765 3000 581 13446 25669 39115

3 FBS strategy r1 01 2762 3000 769 13423 13039 26462r2 009FBU strategy r3 009 2765 3000 581 13446 13034 26480

4 FBS strategy r1 009 2765 3000 807 13446 25609 39055r2 01FBU strategy r3 009 2765 3000 581 13446 25669 39115

5 FBS strategy r1 009 2765 3000 807 13446 12974 26420r2 01FBU strategy r3 009 2765 3000 581 13446 13034 26480

Mathematical Problems in Engineering 9

(p2 minus p1)intDqS00 xf1(x)dx minus wlt 0 thus conrming the

second conclusion of Proposition 8

6 Conclusion

is paper studies the external nancing problems of thebilateral supply chain with nancial constraints It startswith taking the manufacturerrsquos planned output and theretailerrsquos planned order quantity as variables builds theexpected prot model of the manufacturer and the retailerunder the FBS strategy and FBU strategy and analyzes themanufacturerrsquos optimal planned output and the retailerrsquosoptimal order quantity under the two nancing strategiesen it states on the condition of symmetry of informationhow could the bank determine the loan interest rate tomaximize the expected loan prot when the bank hasobtained the information of the planned production andplanned order quantity of the manufacturer and the retailerunder the two nancing strategies At last it proposes themaximum expected loan prot model of the bank under theFBS strategy and FBU strategy to determine the optimalloan interest rate of the bank under the two nancingstrategies And it compares the prot of the bank and theprot of the supply chain under the two nancing strategies

e research shows the following (1) Under the two -nancing strategies the manufacturerrsquos optimal plannedoutput is inversely proportional to the loan interest rate andthe retailerrsquos optimal order quantity is the market demand (2)Under the FBS strategy it is more benecial for the bank toget more expected prot when it raises the loan interest ratesfor the retailer but it does not apply to the manufacturer (3)Under the FBU strategy when the bank sets the loan interestrate at the upper limit of the acceptable line the bankrsquos ex-pected loan prot is the largest but it is less than the bankrsquosexpected loan prot under the FBS strategy (4) e FBUstrategy is more protable for both the manufacturer and the

retailer when the bankrsquos loan interest rate to the manufactureris its acceptable maximum and the loan interest rate ac-ceptable by the manufacturer is less than that by the retailer

e above-mentioned conclusions supply a policyguiding the supply chain enterprises with nancial con-straints to make a choice of the nancing strategy For thebank (1) when the bilateral supply chain enterprises withuncertain output and certain demand choose the FBSstrategy the bank should give the retailer the maximum loaninterest rate and give the manufacturer a loan interest rateaccording to formula (18) (2) when they choose the FBUstrategy the bank should give the supply chain enterprisesthe largest unied loan interest rate and (3) when supplychain enterprises choose the FBS strategy the bank makemore prots so the bank should encourage them to conductthe FBS strategy as much as possible For supply chainenterprises the optimal loan strategies of nancing enter-prises under dierent conditions are dierent so theyshould reasonably choose the optimal nancing strategybased on the bankrsquos quotation and their own acceptable loaninterest rate

For the bilateral supply chain under the FBS and FBUnancing strategies all the nancing modes are distin-guished from the perspective of obtaining the bankrsquos loaninterest rates so there is no nancing strategy superior to theFBU strategy ere are various modes under the FBUstrategy and each has its own advantages such as the dealersand supplier network nancing modes and the bank logisticscooperative nancing modes erefore various nancingmodes within the FBU strategy should be studied and an-alyzed in the future

Appendix

Proof of Proposition 1 Taking the rst-order and second-order derivatives of Eπ1 with respect to q0 it follows that

2500 2600 2700 2800 2900 3000131

1315

132

1325

133

1335

134

1345

135Ex

pect

ed p

rofit

of m

anuf

actu

rer u

nder

FBS

(Eπ 1

)

FBS (2762 134233167)

FBU (2765 134462212)

131

1315

132

1325

133

1335

134

1345

135

Expe

cted

pro

fit o

f man

ufac

ture

r und

er F

BU (E

π 1)

times104 times104

q0

(a)

q0

2000 2200 2400 2600 2800 3000124

126

128

13

132

134

136

FBS

FBU

2760 2780 28001298

13

1302

1304(2762 130394792)

(2765 130344428)

times104

times104

Eπ2

(b)

Figure 4 Prot comparison of the supply chain enterprises under the two nancing strategies in group 3 in Table 3

10 Mathematical Problems in Engineering

dEπ1

dq0 p1 + l1( 1113857 1113946

q1q0

0xf(x)dx + w minus c0 1 + r1( 1113857 minus l1

(A1)

d2Eπ1dq20

minusq21q30

p1 + l1( 1113857fq1

q01113888 1113889le 0 (A2)

When (q1q0)gt 1 + n that is q0 lt (q1(1 + n)) we havef(q1q0) 0 us (d2Eπ1dq20) 0 We obtain

dEπ1

dq0 p1 + w minus c0 1 + r1( 1113857gt 0 (A3)

So when q0 lt (q1(1 + n)) Eπ1 is a monotonically in-creasing function with a fixed slope

When (q1q0)lt 1 minus n that is q0 ge (q1(1 minus n)) we havef(q1q0) 0 us (d2Eπ1dq20) 0 We obtain

dEπ1dq0

w minus c0 1 + r1( 1113857 minus l1 lt 0 (A4)

So when (q1(1 minus n))lt q0 Eπ1 is a monotonically de-creasing function with a fixed slope

When 1 minus nle (q1q0)le 1 + n we have (d2Eπ1dq20)lt 0erefore Eπ1 is a convex function about the variable q0dEπ1dq0 is a continuous function on q0 isin [0 +infin] Whenq0 (q1(1 + n)) we obtain (dEπ1dq0)gt 0 according to(A1) When q0 (q1(1 minus n)) we obtain (dEπ1dq0)lt 0according to (A1) Hence according to the intermediatevalue theorem there must exist q0 isin [(q1(1 + n)) (q1(1 minus

n))] satisfying (dEπ1dq0) 0 At this time q0 is the onlylocal maximum point of Eπ1

In summary when q0 satisfies (dEπ1dq0) 0 that is

1113946q1q0

0xf(x)dx 1 minus

p1 + w minus c0 1 + r1( 1113857

p1 + l1 (A5)

and q0 isin [0 +infin) q0 is the only absolute maximum point ofEπ1

Proof of Proposition 2 Taking the first-order derivative ofEπ2 with respect to q1 geD it follows that

dE π2

dq1

p2 minus p1( 1113857 1113946+infin

q1q0f(x)dx + w minus kp1r2 q1 ltD

minus l2 1113946+infin

q1q0f(x)dx + w minus kp1r2 q1 geD

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(A6)

(1) q1 geD

According to Proposition 1 we have (q1q0) isin[1 minus n 1 + n] Because of xf(x)ge 0 we can obtainthat 1113938

q1q00 xf(x)dx gt 0 Let u (q1q0)gt 0 and de-

fine the function

φ1(u) 1113946U

0xf(x)dx (A7)

e first-order derivative of φ1(u) follows that

φ1prime(u) uf(u)ge 0 (A8)

Because of u isin [1 minus n 1 + n] we have φ1prime(u)gt 0us φ1(u) increases in u that is 1113938

q1q00 xf(x)dx

increases in (q1q0) Hence there exists a unique(q1q0) satisfying

1113946q1q0

0xf(x)dx 1 minus

p1 + w minus c0 1 + r1( 1113857

p1 + l1 (A9)

At this time (q1q0) is a unique constant related tothe coefficients and f(x) Hence when q1 geD(dEπ2dq1) is a unique constant If (dEπ2dq1)gt 0the retailerrsquos planned order quantity will be infinitewhich contradicts the risk-neutral assumptionHence we have (dEπ2dq1)lt 0 erefore Eπ2decreases in q1 in the region q1 isin [D +infin] and at-tains the maximum at the point q1 D

(2) q1 ltD

Because (dEπ2dq1) is a unique constant if

dEπ2dq1

p2 minus p1( 1113857 1113946+infin

q1q0f(x)dx + w minus kp1r2 lt 0

(A10)

then Eπ2 decreases in q1 in the region q1 isin [0 D) andattains the local maximum at the point q1 0 usEπ2 decreases in q1 when q1 ge 0 Eπ2 attains theabsolute maximum at the point q1 0 At this pointthe supply chain is in a state of no order We will notdiscuss this situation

In fact according to the previous literature and theactual situation of the supply chain there is often w minus kp1r2gt 0 Hence when q1 ltD we can obtain (dEπ2dq1)gt 0 andthat Eπ2 increases in q1

In summary Eπ2 attains the maximum at the pointq1 D

Proof of Proposition 3 It follows the proof of Proposition 1so we omit it

Proof of Proposition 4 It follows the proof of Proposition 2so we omit it

Proof of Proposition 5

(1) Obviously nonlinear programming (18) can bedivided into the following two planning problems

Mathematical Problems in Engineering 11

max q0c0 minus kp1q1( 1113857 r1 minus r0( 1113857

st

1113946q1q0

0xf(x)dx 1 minus

p1 + w minus c0 1 + r1( 1113857

p1 + l1

q1 D r0 le r1 le r10

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(A11)

max kp1q1 r2 minus r0( 1113857

st r0 le r2 le r20(A12)

In linear programming (A12) the objective functionincreases in r2 in the region r2 isin [r0 r20] us weattain the optimum of equation (A12) at the pointr2 r20 It is also the optimum of nonlinear pro-gramming (18)

(2) In nonlinear programming (A11) let

m1 q0( 1113857 q0c0 minus kp1D( 1113857 r1 minus r0( 1113857

st 1113946q1q0

0xf(x)dx 1 minus

p1 + w minus c0 1 + r1( 1113857

p1 + l1

m2 r1( 1113857 q0c0 minus kp1D( 1113857 r1 minus r0( 1113857

st 1113946q1q0

0xf(x)dx 1 minus

p1 + w minus c0 1 + r1( 1113857

p1 + l1

(A13)

according to

1113946q1q0

0xf(x)dx 1 minus

p1 + w minus c0 1 + r1( 1113857

p1 + l1 (A14)

We obtain that r1 decreases in q0 us when (dm1dq0)gt 0 we have (dm2dr1)lt 0 When (dm1dq0)lt 0we have (dm2dr1)gt 0

Taking the first-order derivative of m1 with respect to q0it follows thatdm1

dq0 p1 + l1( 1113857

middot 1113946q1q0

0xf(x)dx minus

p1 + l1( 1113857 q0c0 minus kp1D( 1113857q21f q1q0( 1113857

c0q30

minus r0c0 minus l1 minus c0 + w

(A15)

Under different parameters of the supply chain formula(A15) can satisfy both (dm1dq0)gt 0 and (dm1dq0)lt 0erefore there is not always (dm2dr1)gt 0 that is theoptimal of r1 is not always r10 In numerical analysis we giveexamples to support this conclusion

Proof of Proposition 6 Let

g q0 r3( 1113857 q0c0 r3 minus r0( 1113857 (A16)

Since the relationship between q0 and r3 satisfies theconstraint of (20) equation (A16) can also express equation(A17) as follows

g1 q0( 1113857 q0 p1 + l1( 1113857 1113946q1q0

0xf(x)dx + w minus l1 minus c0 1 + r0( 11138571113890 1113891

(A17)

Taking the first-order and second-order derivatives ofg1(q0) with respect to q0 it follows that

g1prime q0( 1113857 p1 + l1( 1113857 1113946q1q0

0xf(x)dx + w minus l1 minus c0 1 + r0( 11138571113890 1113891

minusq21q20

fq1

q01113888 1113889 p1 + l1( 1113857

gPrime1 q0( 1113857 p1 + l1( 1113857q21q30

fq1

q01113888 1113889gt 0

q1

q0isin [1 minus n 1 + n]

(A18)

Obviously when (q1q0) isin [1 minus n 1 + n] that isq0 isin [(q1(1 + n)) (q1(1 minus n))] g1prime(q0) increases in q0According to the proof of Proposition 1 the manufacturerattains the optimum in the region q0 isin [(q1(1 + n)) (q1(1 minus n))] When q0 (q1(1 minus n)) we can obtain

g1prime q0( 1113857 w minus l1 minus c0 1 + r0( 1113857 minus(1 minus n)2

2np1 + l1( 1113857lt 0

(A19)

12 Mathematical Problems in Engineering

us g1prime(q0) is negative in the region q0 isin [(q1(1 + n))

(q1(1 minus n))] g1(q0) decreases in q0 in the regionq0 isin [(q1(1 + n)) (q1(1 minus n))] Since

1113946q1q0

0xf1(x)dx 1 minus

p1 + w minus c0 1 + r3( 1113857

p1 + l1

st q0 isinq1

1 + n

q1

1 minus n1113876 1113877

(A20)

we obtain that r3 decreases in q0 in the region q0 isin [(q1(1 + n)) (q1(1 minus n))] Hence g1(q0) increases in r3 in theregion q0 isin [(q1(1 + n)) (q1(1 minus n))]

According to formulas (A3) and (A4) and the con-clusions of Proposition 1 when r1 isin [((w minus l1 minus c0)c0)((p1 + w minus c0)c0)] we have q0 isin [(q1(1 + n)) (q1(1 minus n))] Since we assume that the manufacturer makesdecisions based on the maximum expected profit there mustbe r10 isin [((w minus l1 minus c0)c0) ((p1 + w minus c0)c0)] which maxi-mizes the expected profit of the manufacturer under the FBSstrategy Similarly there must be min r10 r201113864 1113865 isin [((w minus l1 minus

c0)c0) ((p1 + w minus c0)c0)] which maximizes the expectedprofit of the manufacturer under the FBU strategy us wehave q0 isin [(q1(1 + n)) (q1(1 minus n))] when r3 isin [r0 minr10 r201113864 1113865] Since g1(q0) decreases in q0 in the region q0 isin

[(q1(1 + n)) (q1(1 minus n))] g1(q0) increases in r3 in theregion r3 isin [r0 min r10 r201113864 1113865] erefore g1(q0) attains themaximum at the point rlowast3 min r10 r201113864 1113865 We can obtainmax[q0c0(r3 minus r0)] q0c0(rlowast3 minus r0)

Proof of Proposition 7 If formula (20) attains the optimumat the point q0 qU

0 and r3 rU3 let q0 qU

0 r1 rU3 and

r2 rU3 in formula (18) Obviously they satisfy the con-

straint of formula (18) At this time the objective function offormula (18) is equal to the maximum of formula (20) whichmeans that the optimal of formula (20) must satisfy formula(18)erefore the maximum of formula (18) is not less thanthe maximum of formula (20)

Proof of Proposition 8

(1) When r10 le r20 we have rS1 r10 min r10 r201113864 1113865 rU

3according to Proposition 5 and rS

1 r10 us whenq1 D and r1 r3 it is obvious that q0 is equal in thetwo equations according to equations (3) and (11)Hence under qS

0 qU0 qS

1 qU1 and r1 r3 we can

obtain that EπS1 EπU

1 according to equations (2) and(12)When r10 le r20 we have rU

3 r10 le r20 rS2 accord-

ing to Proposition 5 and rS1 r10 Hence under

qS0 qU

0 qS1 qU

1 and r2 ge r3 we can obtain thatEπS

2 leEπU2 according to equations (8) and (15)

erefore when rS1 r10 and r10 le r20 we obtain that

EπS1 EπU

1 and EπS2 leEπU

2 which means the ex-pected profit of the manufacturer and retailer underthe FBU strategy is not less than the profit under theFBS strategy

(2) r10 gt r20

We know that r1 decreases in q0 in the regionq0 isin [(q1(1 + n)) (q1(1 minus n))] Since rS

1 r10 andr10 gt r20 we have q0 isin [qS

0 +infin) Substituting equa-tion (3) in (2) and taking the first-order derivative ofEπ1 with respect to q0 the following can be obtaineddEπ1

dq0

p1 + l1( 1113857q21c0q

30

fq1

q01113888 1113889 q0c0 minus kp1D( 1113857 (A21)

Since qS0c0 minus kp1Dgt 0 and q0 isin [qS

0 +infin) we have(dEπ1dq0)gt 0 When the supply chain is under theFBU strategy we have rU

3 min r10 r201113864 1113865 r20according to Proposition 6 and r10 gt r20us we haverU3 r20 lt r10 rS

1 and qU0 gt qS

0 erefore we canobtain that EπU

1 gtEπS1 when (dEπ1dq0)gt 0 and

qU0 gt qS

0 In other words we have EπU1 gtEπS

1 whenrS1 r10 qS

0c0 minus kp1D gt 0 and r10 gt r20

Taking the first-order and second-order derivatives ofEπ2 with respect to q0 the following can be obtained

dEπ2

dq0 p2 minus p1( 1113857 1113946

Dq0

0xf(x)dx minus w

d2Eπ2

dq20 minus p2 minus p1( 1113857

D2

q30f

D

q01113888 1113889lt 0

(A22)

When r1 r10 and r10 gt r20 we have rU3 lt rS

1 and qU0 gt qS

0Since (d2Eπ2dq20)lt 0 we obtain that (dEπ2dq0) decreasesin q0 When q0 isin [qS

0 qU0 ] we have

mindEπ2

dq0 p2 minus p1( 1113857 1113946

DqU0

0xf(x)dx minus w

maxdEπ2

dq0 p2 minus p1( 1113857 1113946

DqS0

0xf(x)dx minus w

(A23)

When (p2 minus p1) 1113938DqU

00 xf(x)dx minus wgt 0 and q0 isin [qS

0

qU0 ] we have (dEπ2dq0)gt 0 and EπU

2 gtEπS2 When (p2 minus

p1) 1113938DqS

00 xf(x)dx minus wlt 0 and q0 isin [qS

0 qU0 ] we have (dEπ2

dq0)lt 0 and EπU2 ltEπS

2In summary when (p2 minus p1) 1113938

DqU0

0 xf(x)dx minus wgt 0 andqS0c0 minus kp1Dgt 0 we have EπU

1 gtEπS1 and EπU

2 gtEπS2 When

(p2 minus p1) 1113938DqS

00 xf(x)dx minus wlt 0 and qS

0c0 minus kp1Dgt 0 wehave EπU

1 gtEπS1 and EπU

2 ltEπS2

Data Availability

e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

e authors declare that they have no conflicts of interest

Acknowledgments

is research was funded by the Ministry of Education ofHumanities and Social Science Project of China (grantnumber 17YJAZH018)

Mathematical Problems in Engineering 13

References

[1] S Asian and X Nie ldquoCoordination in supply chains withuncertain demand and disruption risks existence analysisand insightsrdquo IEEE Transactions on Systems Man and Cy-bernetics Systems vol 44 no 9 pp 1139ndash1154 2014

[2] A Roy S S Sana and K Chaudhuri ldquoOptimal Pricing ofcompeting retailers under uncertain demand-a two layersupply chain modelrdquo Annals of Operations Research vol 260no 1-2 pp 481ndash500 2015

[3] B C Giri and S Bardhan ldquoCoordinating a supply chain underuncertain demand and random yield in presence of supplydisruptionrdquo International Journal of Production Researchvol 53 no 16 pp 5070ndash5084 2015

[4] A Arreola-Risa and G A Decroix ldquoMake-to-order versusmake-to-stock in a production-inventory system with generalproduction timesrdquo IIE Transactions vol 30 no 8 pp 705ndash713 1998

[5] A Grosfeld-Nir and B Ronen ldquoA single bottleneck systemwith binomial yields and rigid demandrdquoManagement Sciencevol 39 no 5 pp 650ndash653 1993

[6] C A Yano and H L Lee ldquoLot sizing with random yields areviewrdquoOperations Research vol 43 no 2 pp 311ndash334 1995

[7] P Kouvelis and W Zhao ldquoFinancing the newsvendor sup-plier vs Bank and the structure of optimal trade creditcontractsrdquo Operations Research vol 60 no 3 pp 566ndash5802012

[8] B R Q Zhang ldquoInventory management with asset-basedfinancingrdquoManagement Science vol 50 no 9 pp 1274ndash12922004

[9] J Ma and L Xie ldquoe comparison and complex analysis ondual-channel supply chain under different channel powerstructures and uncertain demandrdquo Nonlinear Dynamicsvol 83 no 3 pp 1379ndash1393 2016

[10] X Xu and J R Birge ldquoJoint production and financing de-cisions modeling and analysisrdquo Operations Managementvol 28 no 12 2011

[11] P Kouvelis and W Zhao ldquoe newsvendor problem andprice-only contract when bankruptcy costs existrdquo Productionand Operations Management vol 20 no 6 pp 921ndash936 2010

[12] O Boyabath and L B Toktay ldquoStochastic capacity investmentand flexible vs Dedicated technology choice in imperfectcapital marketsrdquo Management Science vol 57 no 12pp 2163ndash2179 2011

[13] Y Alan and V Gaur ldquoOperational investment and capitalstructure under asset-based lendingrdquo Manufacturing amp Ser-vice Operations Management vol 20 no 4 2018

[14] B Sun H Zhang N Yan et al ldquoA partial credit guaranteecontract in a capital-constrained supply chain financingequilibrium and coordinating strategyrdquo International Journalof Production Economics vol 173 pp 122ndash133 2016

[15] G Bi Y Fei X Yuan et al ldquoJoint operational and financialcollaboration in a capital-constrained supply chain undermanufacturer collateralrdquo Asia-Pacific Journal of OperationalResearch vol 35 no 3 Article ID 1850010 2018

[16] J Chen Y W Zhou and Y Zhong ldquoA pricingorderingmodel for a dyadic supply chain with buyback guarantee fi-nancing and fairness concernsrdquo International Journal ofProduction Research vol 55 no 7 pp 1ndash18 2018

[17] G G Cai X Chen and Z Xiao ldquoe roles of bank and tradecredits theoretical analysis and empirical evidencerdquo Pro-duction and Operations Management vol 23 no 4pp 583ndash598 2014

[18] X F Chen ldquoA model of trade credit in a capital-constraineddistribution channelrdquo International Journal of ProductionEconomics vol 159 pp 347ndash357 2015

[19] J Chod ldquoInventory risk shifting and trade creditrdquo Man-agement Science vol 63 no 10 pp 3207ndash3225 2017

[20] B Jing X Chen and G Cai ldquoEquilibrium financing in adistribution channel with capital constraintrdquo Production andOperations Management vol 21 no 6 pp 1090ndash1101 2012

[21] S A Yang and J R Birge ldquoTrade credit risk sharing andinventory financing portfoliosrdquo Management Science vol 64no 8 2018

[22] J Zhou ldquoEssays in interfaces of operations and finance insupply chainsrdquo PhD thesis University of RochesterRochester NY USA 2009

[23] Y W Zhou B Cao Y Zhong et al ldquoOptimal advertisingordering policy and finance mode selection for a capital-constrained retailer with stochastic demandrdquo Journal of theOperational Research Society vol 68 no 12 pp 1620ndash16322017

[24] W Jin Q Zhang and J Luo ldquoNon-collaborative and col-laborative financing in a bilateral supply chain with capitalconstraintsrdquo Omega vol 88 pp 210ndash222 2018

[25] B Keren ldquoe single-period inventory problem extension torandom yield from the perspective of the supply chainrdquoOmega vol 37 no 4 pp 801ndash810 2009

[26] A Sufi ldquoBank lines of credit in corporate finance an empiricalanalysisrdquo Review of Financial Studies vol 22 no 3pp 1057ndash1088 2009

14 Mathematical Problems in Engineering

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

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Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

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Dierential EquationsInternational Journal of

Volume 2018

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AnalysisInternational Journal of

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Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

e expected profit π2 of the retailer is

π2 p2 minus p1( 1113857min q0x q11113864 1113865 + wmax q1 minus q0x 01113864 1113865 minus wmax q0x minus q1 01113864 1113865 minus kp1q1r2 q1 ltD

p2 minus p1( 1113857min q0x q1 D1113864 1113865 + wmax q1 minus q0x 01113864 1113865 minus kp1q1r2 minus wmax q0x minus q1 01113864 1113865 minus l2 max min q0x q11113864 1113865 minus D 01113864 1113865 q1 geD

⎧⎪⎨

⎪⎩(6)

When q1 geD the first item is the difference betweenthe retailerrsquos sales revenue and the ordering cost esecond item is the compensation that the retailer receivesfrom the manufacturer because of insufficient manu-facturer output e third item is the financing cost of

the retailer e fourth item is the compensation thatthe manufacturer receives from the retailer when themanufacturerrsquos output is too large e fifth item is theloss of the retailerrsquos products e retailer expects theprofit to be

Eπ2

p2 minus p1( 1113857 q0 1113946q1q0

0xf(x)dx + q1 1113946

+infin

q1q0f(x)dx1113890 1113891 + w q1 minus q0( 1113857 minus kp1q1r2 q1 ltD

minus l2 q0 1113946q1q0

0xf(x)dx + q1 1113946

+infin

q1q0f(x)dx1113890 1113891 + w q1 minus q0( 1113857 minus kp1q1r2

+ p2 minus p1 + l2( 1113857D 1113946+infin

Dq0f(x)dx + p2 minus p1 + l2( 1113857q0 1113946

Dq0

0xf(x)dx q1 geD

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(7)

e retailer arranges the purchase according to themarket demand D

Proposition 2 Under the FBS strategy the retailer expectsthe maximum profit Eπ2 when q1 D

is proposition shows that although the manufacturerhas uncertainties in production the optimal order quantityof the retailer is still the expected market demand since theactual production of the manufacturer is subject to theuniform distribution of the planned output e reason isthat when the order quantity is greater than the expectedmarket demand the retailer has a higher risk of slow salesWhen the order quantity is less than the expected marketdemand the retailer is at risk of being out of stock

e maximum expected profit of the retailer is

Eπ2 p2 minus p1( 1113857 q0 1113946Dq0

0xf(x)dx + D 1113946

+infin

Dq0f(x)dx1113890 1113891

+ w D minus q0( 1113857 minus kp1Dr2

(8)

e expected loan amount L2 of the retailer is

L2 kp1D (9)

where k is the ratio of the prepayment of the retailerrsquos unitproduct to p0

e expected profit of the bank is

Eπ4 L2 r2 minus r0( 1113857 kp1D r2 minus r0( 1113857 (10)

412 Production and Order Decision of Supply Chain En-terprises under FBU Strategy Supply chain enterprises canchoose either the FBS strategy or the FBU strategy e FBU

strategy refers to the process in which the supply chain lendsloans to banks at all levels in the production process of theorder At this time the bank provides a unified loan interestrate to the supply chain enterprises

From Proposition 1 and Proposition 2 Proposition 3and Proposition 4 are obtained

Proposition 3 Under the FBU strategy the manufacturerrsquosexpected profit Eπ1 is a convex function of the manufac-turerrsquos planned output q0 and its optimal planned yield q0satisfies the following formula

1113946q1q0

0xf(x)dx 1 minus

p1 + w minus c0 1 + r3( 1113857

p1 + l1 (11)

Proof At this point the maximum profit of the manufac-turer is

Eπ1 p1 1113946q1q0

0q0xf(x)dx + 1113946

infin

q1q0q1f(x)dx1113890 1113891

minus c0q0 minus l1 1113946infin

q1q0q0x minus q1( 1113857f(x)dx + w q0 minus q1( 1113857

minus q0c0 minus kq1p1( 1113857r3

(12)

e manufacturerrsquos expected loan amount L1 is

L1 q0c0 minus kq1p1 (13)

e expected profit of the bank is

Eπ3 L1 r3 minus r0( 1113857 q0c0 minus kp1q1( 1113857 r3 minus r0( 1113857 (14)

q0 in equations (12)ndash(14) satisfies equation (11)

6 Mathematical Problems in Engineering

Proposition 4 Under the FBU strategy the retailer expectsthe maximum profit Eπ2 when q1 D

Proof At this point the maximum profit of the retailer is

Eπ2 p2 minus p1( 1113857 q0 1113946Dq0

0xf(x)dx + D 1113946

+infin

Dq0f(x)dx1113890 1113891

+ w D minus q0( 1113857 minus kp1Dr3

(15)

e retailer expects the loan amount L2 to be

L2 kp1D (16)

where k is the ratio of the prepayment of the retailerrsquos unitproduct to p1

e expected profit of the bank is

Eπ4 L2 r3 minus r0( 1113857 kp1D r3 minus r0( 1113857 (17)

42 Bank Loan Interest Rate Setting under Two FinancingStrategies

421 Bank Loan Interest Rate Setting under FBS StrategyAccording to equation (11) the bank expects the manu-facturerrsquos loan profit Eπ3 to be related to q0 and r1According to formula (3) the manufacturerrsquos optimalplanned output q0 is negatively correlated with the bankrsquosloan interest rate r1 erefore the bank needs to set its loaninterest rate r1 to the manufacturer so that the bankrsquos ex-pected profit for the manufacturerrsquos loan is the largest underthe premise of satisfying themanufacturerrsquos maximumprofitformula (3)

According to formula (10) the bankrsquos expected loanprofit Eπ4 to the retailer is only positively correlated with thevariable r2 so the bank needs to increase its loan interest rateto the retailer as much as possible

erefore this study proposes an interest rate settingmodel under the bankrsquos expectation of loan profitmaximization

max q0c0 minus kp1q1( 1113857 r1 minus r0( 1113857 + kp1q1 r2 minus r0( 11138571113858 1113859

st

1113946q1q0

0xf(x)dx 1 minus

p1 + w minus c0 1 + r1( 1113857

p1 + l1

q1 D r0 le r1 le r10 r0 le r2 le r20

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(18)

Proposition 5 Under the FBS strategy when the bank ex-pects the maximum loan profit the following are satisfied

(1) e bankrsquos optimal loan interest rate r2 to the retailermust be the maximum acceptable loan interest rateof the retailerat is when r2 r20 r2 is the optimalsolution of equation (18)

(2) e bankrsquos loan interest rate r1 to themanufacturer isnot necessarily the maximum value r10 that themanufacturer can accept

is proposition points out the bankrsquos loan interest ratesettingmethod under the FBS strategyemethod indicatesthat the bank should give the retailer a loan interest rate thatshould take the maximum value while giving the manu-facturer a loan interest rate that does not necessarily have totake the maximum value e reason is that the informationis symmetrical e bank knows that the bankrsquos loan interestrate to the retailer is more favorable to the bankrsquos loan profitunder the condition that the retailer determines the orderquantity In contrast the bank knows that the manufac-turerrsquos planned output is inversely proportional to the loaninterest rate and the bank needs to consider the impact ofthe loan interest rate on the manufacturerrsquos planned outputus under the FBS strategy the banks optimal loan in-terest rate to the manufacturer is not always the maximum

422 Bank Loan Interest Rate Setting under FBU StrategyUnder the FBU strategy the bank gives a uniform loaninterest rate to the supply chain e interest rate is notgreater than the maximum acceptable interest rate for themanufacturer and the retailer e bankrsquos total profit fromthe manufacturer and retailer is

Eπ3 + Eπ4 q0c0 minus kp1q1( 1113857 r3 minus r0( 1113857 + kp1q1 r3 minus r0( 1113857

q0c0 r3 minus r0( 1113857

(19)According to formula (12) the manufacturerrsquos optimal

planned output q0 is negatively correlated with the bankrsquosloan interest rate r3 erefore this study proposes that theloan interest rate setting model under the bankrsquos expectedloan profit maximization is as follows

max q0c0 r3 minus r0( 1113857

st

1113946q1q0

0xf(x)dx 1 minus

p1 + w minus c0 1 + r3( 1113857

p1 + l1

q1 D r3 lt r10 r3 lt r20

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(20)

Proposition 6 Under the FBU strategy when the bankrsquossupply chain loan interest rate is the maximum value ac-ceptable by the supply chain enterprise the bankrsquos expectedloan profit is the largest at is when rlowast3 min r10 r201113864 1113865there is

max q0c0 r3 minus r0( 11138571113858 1113859 q0c0 rlowast3 minus r0( 1113857 (21)

is proposition points out the bankrsquos loan interest ratesetting method under the FBU strategy e method statesthat the bankrsquos loan interest rate should be the maximumunder the FBU strategye reason is that the information is

Mathematical Problems in Engineering 7

symmetrical e bank knows that the loan interest rate isinversely proportional to the manufacturerrsquos planned out-put and the loan interest rate is directly proportional to thebankrsquos loan profit erefore under the FBU strategy thebank chooses to increase the loan interest rate to be morebeneficial to itself

43 Analysis of the Profit of the Bank and Supply ChainEnterprises under FBS Strategy and FBU Strategy

431 Comparison of the Bankrsquos Maximum Expected LoanProfit under the FBS Strategy and FBU Strategy Based onformula (18) under the FBS strategy and formula (20) underthe FBU strategy the following propositions are proposed

Proposition 7 e maximum loan profit of the bank underthe FBS strategy is not less than the maximum loan profit ofthe bank under the FBU strategy at is the objectivefunction value of formula (18) is not smaller than the ob-jective function value of formula (20)

is proposition indicates that bankrsquos loan profits aregreater under the FBS strategy e reason is that at thistime the bank can separately set loan interest rates to makeits own profit even greater Under the FBU strategy the bankcan only consider lending at a uniform loan interest rate Atthis time the loan profit from the manufacturer and theretailer is no higher than the loan profit under the FBSstrategy

432 Comparison of Maximum Expected Profit of SupplyChain Enterprises under the FBS Strategy and FBU StrategyBecause of complete information supply chain companiesunderstand the maximization formulas (18) and (20) of thebankrsquos loan profit and know the optimal loan interest rate ofthe bank under the two financing strategies before the loanSupply chain companies establish a better loan methodbased on their expected profit formulas to maximize theirexpected profits erefore the following proposition isproposed

Proposition 8 Under the FBS strategy when the bankrsquos loaninterest rate to the manufacturer is its acceptable maximumthat is the optimal solution of equation (18) is rS

1 r10 thefollowing are satisfied

(1) If r10 le r20 the maximum expected profit of thesupply chain under the FBU strategy is not less thanits maximum expected profit under the FBS strategy

(2) If r10 gt r20 under qS0c0 minus kp1Dgt 0 and (p2 minus p1)

1113938DqU

00 xf(x)dx minus wgt 0 the FBU strategy is more

favorable to both enterprises in the supply chain

under qS0c0 minus kp1Dgt 0 and (p2 minus p1) 1113938

DqS0

0 xf(x)

dx minus wlt 0 the FBU strategy is more favorable to themanufacturer and not good for the retailer

is proposition finds out what kind of financingstrategy the supply chain enterprise chooses to be morefavorable to itself When the bankrsquos loan interest rate to themanufacturer under the FBS strategy is the maximum valueacceptable by the manufacturer and less than the maximumvalue of the retailerrsquos acceptable loan interest rate it is moreadvantageous for supply chain enterprises to choose the FBUstrategy When the bankrsquos loan interest rate to the manu-facturer under the FBS strategy is the maximum value ac-ceptable by themanufacturer and greater than themaximumloan interest rate acceptable by the retailer we find two setsof conditions under which the manufacturerrsquos expectedprofit under the FBU strategy is higher than the expectedprofit under the FBS strategy but the retailer does not fullyearn more under the FBU strategy us Proposition 8shows that the FBU strategy is beneficial to supply chainenterprises under certain conditions but it is not alwaysbeneficial to supply chain enterprises

5 Numerical Analysis

e setting of parameters is done according to the study ofJin et al [24] We assume xsimU[1 minus n 1 + n] and give thefour types of supply chain parameters in Table 2 e bankrsquosoptimal interest rate and the expected profits of banks andsupply chain enterprises are shown in Table 3 Figures 3 and4 show the comparisons of the supply chain enterprisesrsquoprofits under two different financing strategies in groups 2and 3 in Table 3 respectively

From Table 3 and Figures 3 and 4 we get the followingresults

First Proposition 5 is confirmed As can be seen fromTable 3 under the FBS strategy the bankrsquos loan interest rateto the retailer is the maximum value of the retailerrsquos ac-ceptable loan interest rate However it can be seen from theresults of group 1 in Table 3 that under this set of supplychain parameters when the bankrsquos expected profit is thelargest the loan interest rate given to the manufacturer is0062lt 01 From the results the bankrsquos maximum loanprofit from the manufacturer is not necessarily obtained atthe maximum loan interest rate and the largest loan profitfor the retailer is obtained at the maximum loan interest rate

Second Proposition 6 is confirmed As can be seen fromthe five sets of data in Table 2 under the FBU strategy themaximum bankrsquos loan interest rate acceptable by the supplychain is min 009 01 009 Obviously in the five sets ofresults in Table 3 the bankrsquos optimal lending rate under theFBU strategy is 009 From the results when the supply chainenterprises adopt the FBU strategy the bankrsquos largest loanprofit is obtained at the largest loan interest rate

ird Proposition 7 is confirmed e results of the fivegroups in Table 3 are tested as 814gt 633 769gt 581 and807gt 581 It is clear that the bankrsquos maximum expectedprofit is higher under the FBS strategy

Fourth Proposition 8 is confirmed From the results ofthe data of groups 4 and 5 in Table 3 under rS

1 r10 andr10 le r20 both sets of results indicate that the maximumexpected profit of the manufacturer and the retailer under

8 Mathematical Problems in Engineering

the FBU strategy is not less than the maximum expectedprot under the FBS strategy is conrms the rst con-clusion of Proposition 8 From the results of the data ofgroups 2 and 3 in Table 3 under rS1 r10 and r10 gt r20 bothsets of results indicate that the maximum expected prot ofthe manufacturer under the FBU strategy is greater than that

under the FBS strategy According to Figure 3 and group 2 inTable 3 the retailerrsquos maximum expected prot is largerunder the FBU strategy than the FBS strategy In contrast inFigure 4 and group 3 in Table 3 the conclusion is reversede reason is that the second set of data satises (p2 minus p1)intDq

U0

0 xf1(x)dx minus wgt 0 and the third set of data satises

2500 2600 2700 2800 2900 3000131

1315

132

1325

133

1335

134

1345

135

Expe

cted

pro

fit o

f man

ufac

ture

r und

er F

BS (E

π 1)

FBS (2762 134233167)

FBU (2765 134462212)

131

1315

132

1325

133

1335

134

1345

135

Expe

cted

pro

fit o

f man

ufac

ture

r und

er F

BU (E

π 1)

times104 times104

q0

(a)

q0

2000 2500 3000 3500 400023

235

24

245

25

255

26

FBS

FBU

2760 2780 28002566

2567

2568

FBS (2762 256669584)

FBU (2765 256688856)

times104

times104

Eπ2

(b)

Figure 3 Prot comparison of the supply chain enterprises under the two nancing strategies in group 2 in Table 3

Table 2 Supply chain parameter settings

Types c0 p1 p2 D l1 l2 w r0 k r10 r201 3 15 20 3000 10 14 4 002 02 01 0092 3 10 20 3000 10 14 4 002 02 01 0093 3 10 15 3000 10 14 4 002 02 01 0094 3 10 20 3000 10 14 4 002 02 009 015 3 10 15 3000 10 14 4 002 02 009 01

Table 3 Prots of enterprises and banks under the FBS strategy and FBU strategy

Types Financingstrategies

Optimalloan

interest rate

Optimalproductionand orderingdecisions

Bankrsquos expectedprot

Manufacturerrsquosexpected prot

Retailerrsquos expectedprot

Total expectedprot

q0 q1

1 FBS strategy r1 0062 3023 3000 814 26627 12266 38893r2 009FBU strategy r3 009 3013 3000 633 26626 12287 38913

2 FBS strategy r1 01 2762 3000 769 13423 25667 39090r2 009FBU strategy r3 009 2765 3000 581 13446 25669 39115

3 FBS strategy r1 01 2762 3000 769 13423 13039 26462r2 009FBU strategy r3 009 2765 3000 581 13446 13034 26480

4 FBS strategy r1 009 2765 3000 807 13446 25609 39055r2 01FBU strategy r3 009 2765 3000 581 13446 25669 39115

5 FBS strategy r1 009 2765 3000 807 13446 12974 26420r2 01FBU strategy r3 009 2765 3000 581 13446 13034 26480

Mathematical Problems in Engineering 9

(p2 minus p1)intDqS00 xf1(x)dx minus wlt 0 thus conrming the

second conclusion of Proposition 8

6 Conclusion

is paper studies the external nancing problems of thebilateral supply chain with nancial constraints It startswith taking the manufacturerrsquos planned output and theretailerrsquos planned order quantity as variables builds theexpected prot model of the manufacturer and the retailerunder the FBS strategy and FBU strategy and analyzes themanufacturerrsquos optimal planned output and the retailerrsquosoptimal order quantity under the two nancing strategiesen it states on the condition of symmetry of informationhow could the bank determine the loan interest rate tomaximize the expected loan prot when the bank hasobtained the information of the planned production andplanned order quantity of the manufacturer and the retailerunder the two nancing strategies At last it proposes themaximum expected loan prot model of the bank under theFBS strategy and FBU strategy to determine the optimalloan interest rate of the bank under the two nancingstrategies And it compares the prot of the bank and theprot of the supply chain under the two nancing strategies

e research shows the following (1) Under the two -nancing strategies the manufacturerrsquos optimal plannedoutput is inversely proportional to the loan interest rate andthe retailerrsquos optimal order quantity is the market demand (2)Under the FBS strategy it is more benecial for the bank toget more expected prot when it raises the loan interest ratesfor the retailer but it does not apply to the manufacturer (3)Under the FBU strategy when the bank sets the loan interestrate at the upper limit of the acceptable line the bankrsquos ex-pected loan prot is the largest but it is less than the bankrsquosexpected loan prot under the FBS strategy (4) e FBUstrategy is more protable for both the manufacturer and the

retailer when the bankrsquos loan interest rate to the manufactureris its acceptable maximum and the loan interest rate ac-ceptable by the manufacturer is less than that by the retailer

e above-mentioned conclusions supply a policyguiding the supply chain enterprises with nancial con-straints to make a choice of the nancing strategy For thebank (1) when the bilateral supply chain enterprises withuncertain output and certain demand choose the FBSstrategy the bank should give the retailer the maximum loaninterest rate and give the manufacturer a loan interest rateaccording to formula (18) (2) when they choose the FBUstrategy the bank should give the supply chain enterprisesthe largest unied loan interest rate and (3) when supplychain enterprises choose the FBS strategy the bank makemore prots so the bank should encourage them to conductthe FBS strategy as much as possible For supply chainenterprises the optimal loan strategies of nancing enter-prises under dierent conditions are dierent so theyshould reasonably choose the optimal nancing strategybased on the bankrsquos quotation and their own acceptable loaninterest rate

For the bilateral supply chain under the FBS and FBUnancing strategies all the nancing modes are distin-guished from the perspective of obtaining the bankrsquos loaninterest rates so there is no nancing strategy superior to theFBU strategy ere are various modes under the FBUstrategy and each has its own advantages such as the dealersand supplier network nancing modes and the bank logisticscooperative nancing modes erefore various nancingmodes within the FBU strategy should be studied and an-alyzed in the future

Appendix

Proof of Proposition 1 Taking the rst-order and second-order derivatives of Eπ1 with respect to q0 it follows that

2500 2600 2700 2800 2900 3000131

1315

132

1325

133

1335

134

1345

135Ex

pect

ed p

rofit

of m

anuf

actu

rer u

nder

FBS

(Eπ 1

)

FBS (2762 134233167)

FBU (2765 134462212)

131

1315

132

1325

133

1335

134

1345

135

Expe

cted

pro

fit o

f man

ufac

ture

r und

er F

BU (E

π 1)

times104 times104

q0

(a)

q0

2000 2200 2400 2600 2800 3000124

126

128

13

132

134

136

FBS

FBU

2760 2780 28001298

13

1302

1304(2762 130394792)

(2765 130344428)

times104

times104

Eπ2

(b)

Figure 4 Prot comparison of the supply chain enterprises under the two nancing strategies in group 3 in Table 3

10 Mathematical Problems in Engineering

dEπ1

dq0 p1 + l1( 1113857 1113946

q1q0

0xf(x)dx + w minus c0 1 + r1( 1113857 minus l1

(A1)

d2Eπ1dq20

minusq21q30

p1 + l1( 1113857fq1

q01113888 1113889le 0 (A2)

When (q1q0)gt 1 + n that is q0 lt (q1(1 + n)) we havef(q1q0) 0 us (d2Eπ1dq20) 0 We obtain

dEπ1

dq0 p1 + w minus c0 1 + r1( 1113857gt 0 (A3)

So when q0 lt (q1(1 + n)) Eπ1 is a monotonically in-creasing function with a fixed slope

When (q1q0)lt 1 minus n that is q0 ge (q1(1 minus n)) we havef(q1q0) 0 us (d2Eπ1dq20) 0 We obtain

dEπ1dq0

w minus c0 1 + r1( 1113857 minus l1 lt 0 (A4)

So when (q1(1 minus n))lt q0 Eπ1 is a monotonically de-creasing function with a fixed slope

When 1 minus nle (q1q0)le 1 + n we have (d2Eπ1dq20)lt 0erefore Eπ1 is a convex function about the variable q0dEπ1dq0 is a continuous function on q0 isin [0 +infin] Whenq0 (q1(1 + n)) we obtain (dEπ1dq0)gt 0 according to(A1) When q0 (q1(1 minus n)) we obtain (dEπ1dq0)lt 0according to (A1) Hence according to the intermediatevalue theorem there must exist q0 isin [(q1(1 + n)) (q1(1 minus

n))] satisfying (dEπ1dq0) 0 At this time q0 is the onlylocal maximum point of Eπ1

In summary when q0 satisfies (dEπ1dq0) 0 that is

1113946q1q0

0xf(x)dx 1 minus

p1 + w minus c0 1 + r1( 1113857

p1 + l1 (A5)

and q0 isin [0 +infin) q0 is the only absolute maximum point ofEπ1

Proof of Proposition 2 Taking the first-order derivative ofEπ2 with respect to q1 geD it follows that

dE π2

dq1

p2 minus p1( 1113857 1113946+infin

q1q0f(x)dx + w minus kp1r2 q1 ltD

minus l2 1113946+infin

q1q0f(x)dx + w minus kp1r2 q1 geD

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(A6)

(1) q1 geD

According to Proposition 1 we have (q1q0) isin[1 minus n 1 + n] Because of xf(x)ge 0 we can obtainthat 1113938

q1q00 xf(x)dx gt 0 Let u (q1q0)gt 0 and de-

fine the function

φ1(u) 1113946U

0xf(x)dx (A7)

e first-order derivative of φ1(u) follows that

φ1prime(u) uf(u)ge 0 (A8)

Because of u isin [1 minus n 1 + n] we have φ1prime(u)gt 0us φ1(u) increases in u that is 1113938

q1q00 xf(x)dx

increases in (q1q0) Hence there exists a unique(q1q0) satisfying

1113946q1q0

0xf(x)dx 1 minus

p1 + w minus c0 1 + r1( 1113857

p1 + l1 (A9)

At this time (q1q0) is a unique constant related tothe coefficients and f(x) Hence when q1 geD(dEπ2dq1) is a unique constant If (dEπ2dq1)gt 0the retailerrsquos planned order quantity will be infinitewhich contradicts the risk-neutral assumptionHence we have (dEπ2dq1)lt 0 erefore Eπ2decreases in q1 in the region q1 isin [D +infin] and at-tains the maximum at the point q1 D

(2) q1 ltD

Because (dEπ2dq1) is a unique constant if

dEπ2dq1

p2 minus p1( 1113857 1113946+infin

q1q0f(x)dx + w minus kp1r2 lt 0

(A10)

then Eπ2 decreases in q1 in the region q1 isin [0 D) andattains the local maximum at the point q1 0 usEπ2 decreases in q1 when q1 ge 0 Eπ2 attains theabsolute maximum at the point q1 0 At this pointthe supply chain is in a state of no order We will notdiscuss this situation

In fact according to the previous literature and theactual situation of the supply chain there is often w minus kp1r2gt 0 Hence when q1 ltD we can obtain (dEπ2dq1)gt 0 andthat Eπ2 increases in q1

In summary Eπ2 attains the maximum at the pointq1 D

Proof of Proposition 3 It follows the proof of Proposition 1so we omit it

Proof of Proposition 4 It follows the proof of Proposition 2so we omit it

Proof of Proposition 5

(1) Obviously nonlinear programming (18) can bedivided into the following two planning problems

Mathematical Problems in Engineering 11

max q0c0 minus kp1q1( 1113857 r1 minus r0( 1113857

st

1113946q1q0

0xf(x)dx 1 minus

p1 + w minus c0 1 + r1( 1113857

p1 + l1

q1 D r0 le r1 le r10

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(A11)

max kp1q1 r2 minus r0( 1113857

st r0 le r2 le r20(A12)

In linear programming (A12) the objective functionincreases in r2 in the region r2 isin [r0 r20] us weattain the optimum of equation (A12) at the pointr2 r20 It is also the optimum of nonlinear pro-gramming (18)

(2) In nonlinear programming (A11) let

m1 q0( 1113857 q0c0 minus kp1D( 1113857 r1 minus r0( 1113857

st 1113946q1q0

0xf(x)dx 1 minus

p1 + w minus c0 1 + r1( 1113857

p1 + l1

m2 r1( 1113857 q0c0 minus kp1D( 1113857 r1 minus r0( 1113857

st 1113946q1q0

0xf(x)dx 1 minus

p1 + w minus c0 1 + r1( 1113857

p1 + l1

(A13)

according to

1113946q1q0

0xf(x)dx 1 minus

p1 + w minus c0 1 + r1( 1113857

p1 + l1 (A14)

We obtain that r1 decreases in q0 us when (dm1dq0)gt 0 we have (dm2dr1)lt 0 When (dm1dq0)lt 0we have (dm2dr1)gt 0

Taking the first-order derivative of m1 with respect to q0it follows thatdm1

dq0 p1 + l1( 1113857

middot 1113946q1q0

0xf(x)dx minus

p1 + l1( 1113857 q0c0 minus kp1D( 1113857q21f q1q0( 1113857

c0q30

minus r0c0 minus l1 minus c0 + w

(A15)

Under different parameters of the supply chain formula(A15) can satisfy both (dm1dq0)gt 0 and (dm1dq0)lt 0erefore there is not always (dm2dr1)gt 0 that is theoptimal of r1 is not always r10 In numerical analysis we giveexamples to support this conclusion

Proof of Proposition 6 Let

g q0 r3( 1113857 q0c0 r3 minus r0( 1113857 (A16)

Since the relationship between q0 and r3 satisfies theconstraint of (20) equation (A16) can also express equation(A17) as follows

g1 q0( 1113857 q0 p1 + l1( 1113857 1113946q1q0

0xf(x)dx + w minus l1 minus c0 1 + r0( 11138571113890 1113891

(A17)

Taking the first-order and second-order derivatives ofg1(q0) with respect to q0 it follows that

g1prime q0( 1113857 p1 + l1( 1113857 1113946q1q0

0xf(x)dx + w minus l1 minus c0 1 + r0( 11138571113890 1113891

minusq21q20

fq1

q01113888 1113889 p1 + l1( 1113857

gPrime1 q0( 1113857 p1 + l1( 1113857q21q30

fq1

q01113888 1113889gt 0

q1

q0isin [1 minus n 1 + n]

(A18)

Obviously when (q1q0) isin [1 minus n 1 + n] that isq0 isin [(q1(1 + n)) (q1(1 minus n))] g1prime(q0) increases in q0According to the proof of Proposition 1 the manufacturerattains the optimum in the region q0 isin [(q1(1 + n)) (q1(1 minus n))] When q0 (q1(1 minus n)) we can obtain

g1prime q0( 1113857 w minus l1 minus c0 1 + r0( 1113857 minus(1 minus n)2

2np1 + l1( 1113857lt 0

(A19)

12 Mathematical Problems in Engineering

us g1prime(q0) is negative in the region q0 isin [(q1(1 + n))

(q1(1 minus n))] g1(q0) decreases in q0 in the regionq0 isin [(q1(1 + n)) (q1(1 minus n))] Since

1113946q1q0

0xf1(x)dx 1 minus

p1 + w minus c0 1 + r3( 1113857

p1 + l1

st q0 isinq1

1 + n

q1

1 minus n1113876 1113877

(A20)

we obtain that r3 decreases in q0 in the region q0 isin [(q1(1 + n)) (q1(1 minus n))] Hence g1(q0) increases in r3 in theregion q0 isin [(q1(1 + n)) (q1(1 minus n))]

According to formulas (A3) and (A4) and the con-clusions of Proposition 1 when r1 isin [((w minus l1 minus c0)c0)((p1 + w minus c0)c0)] we have q0 isin [(q1(1 + n)) (q1(1 minus n))] Since we assume that the manufacturer makesdecisions based on the maximum expected profit there mustbe r10 isin [((w minus l1 minus c0)c0) ((p1 + w minus c0)c0)] which maxi-mizes the expected profit of the manufacturer under the FBSstrategy Similarly there must be min r10 r201113864 1113865 isin [((w minus l1 minus

c0)c0) ((p1 + w minus c0)c0)] which maximizes the expectedprofit of the manufacturer under the FBU strategy us wehave q0 isin [(q1(1 + n)) (q1(1 minus n))] when r3 isin [r0 minr10 r201113864 1113865] Since g1(q0) decreases in q0 in the region q0 isin

[(q1(1 + n)) (q1(1 minus n))] g1(q0) increases in r3 in theregion r3 isin [r0 min r10 r201113864 1113865] erefore g1(q0) attains themaximum at the point rlowast3 min r10 r201113864 1113865 We can obtainmax[q0c0(r3 minus r0)] q0c0(rlowast3 minus r0)

Proof of Proposition 7 If formula (20) attains the optimumat the point q0 qU

0 and r3 rU3 let q0 qU

0 r1 rU3 and

r2 rU3 in formula (18) Obviously they satisfy the con-

straint of formula (18) At this time the objective function offormula (18) is equal to the maximum of formula (20) whichmeans that the optimal of formula (20) must satisfy formula(18)erefore the maximum of formula (18) is not less thanthe maximum of formula (20)

Proof of Proposition 8

(1) When r10 le r20 we have rS1 r10 min r10 r201113864 1113865 rU

3according to Proposition 5 and rS

1 r10 us whenq1 D and r1 r3 it is obvious that q0 is equal in thetwo equations according to equations (3) and (11)Hence under qS

0 qU0 qS

1 qU1 and r1 r3 we can

obtain that EπS1 EπU

1 according to equations (2) and(12)When r10 le r20 we have rU

3 r10 le r20 rS2 accord-

ing to Proposition 5 and rS1 r10 Hence under

qS0 qU

0 qS1 qU

1 and r2 ge r3 we can obtain thatEπS

2 leEπU2 according to equations (8) and (15)

erefore when rS1 r10 and r10 le r20 we obtain that

EπS1 EπU

1 and EπS2 leEπU

2 which means the ex-pected profit of the manufacturer and retailer underthe FBU strategy is not less than the profit under theFBS strategy

(2) r10 gt r20

We know that r1 decreases in q0 in the regionq0 isin [(q1(1 + n)) (q1(1 minus n))] Since rS

1 r10 andr10 gt r20 we have q0 isin [qS

0 +infin) Substituting equa-tion (3) in (2) and taking the first-order derivative ofEπ1 with respect to q0 the following can be obtaineddEπ1

dq0

p1 + l1( 1113857q21c0q

30

fq1

q01113888 1113889 q0c0 minus kp1D( 1113857 (A21)

Since qS0c0 minus kp1Dgt 0 and q0 isin [qS

0 +infin) we have(dEπ1dq0)gt 0 When the supply chain is under theFBU strategy we have rU

3 min r10 r201113864 1113865 r20according to Proposition 6 and r10 gt r20us we haverU3 r20 lt r10 rS

1 and qU0 gt qS

0 erefore we canobtain that EπU

1 gtEπS1 when (dEπ1dq0)gt 0 and

qU0 gt qS

0 In other words we have EπU1 gtEπS

1 whenrS1 r10 qS

0c0 minus kp1D gt 0 and r10 gt r20

Taking the first-order and second-order derivatives ofEπ2 with respect to q0 the following can be obtained

dEπ2

dq0 p2 minus p1( 1113857 1113946

Dq0

0xf(x)dx minus w

d2Eπ2

dq20 minus p2 minus p1( 1113857

D2

q30f

D

q01113888 1113889lt 0

(A22)

When r1 r10 and r10 gt r20 we have rU3 lt rS

1 and qU0 gt qS

0Since (d2Eπ2dq20)lt 0 we obtain that (dEπ2dq0) decreasesin q0 When q0 isin [qS

0 qU0 ] we have

mindEπ2

dq0 p2 minus p1( 1113857 1113946

DqU0

0xf(x)dx minus w

maxdEπ2

dq0 p2 minus p1( 1113857 1113946

DqS0

0xf(x)dx minus w

(A23)

When (p2 minus p1) 1113938DqU

00 xf(x)dx minus wgt 0 and q0 isin [qS

0

qU0 ] we have (dEπ2dq0)gt 0 and EπU

2 gtEπS2 When (p2 minus

p1) 1113938DqS

00 xf(x)dx minus wlt 0 and q0 isin [qS

0 qU0 ] we have (dEπ2

dq0)lt 0 and EπU2 ltEπS

2In summary when (p2 minus p1) 1113938

DqU0

0 xf(x)dx minus wgt 0 andqS0c0 minus kp1Dgt 0 we have EπU

1 gtEπS1 and EπU

2 gtEπS2 When

(p2 minus p1) 1113938DqS

00 xf(x)dx minus wlt 0 and qS

0c0 minus kp1Dgt 0 wehave EπU

1 gtEπS1 and EπU

2 ltEπS2

Data Availability

e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

e authors declare that they have no conflicts of interest

Acknowledgments

is research was funded by the Ministry of Education ofHumanities and Social Science Project of China (grantnumber 17YJAZH018)

Mathematical Problems in Engineering 13

References

[1] S Asian and X Nie ldquoCoordination in supply chains withuncertain demand and disruption risks existence analysisand insightsrdquo IEEE Transactions on Systems Man and Cy-bernetics Systems vol 44 no 9 pp 1139ndash1154 2014

[2] A Roy S S Sana and K Chaudhuri ldquoOptimal Pricing ofcompeting retailers under uncertain demand-a two layersupply chain modelrdquo Annals of Operations Research vol 260no 1-2 pp 481ndash500 2015

[3] B C Giri and S Bardhan ldquoCoordinating a supply chain underuncertain demand and random yield in presence of supplydisruptionrdquo International Journal of Production Researchvol 53 no 16 pp 5070ndash5084 2015

[4] A Arreola-Risa and G A Decroix ldquoMake-to-order versusmake-to-stock in a production-inventory system with generalproduction timesrdquo IIE Transactions vol 30 no 8 pp 705ndash713 1998

[5] A Grosfeld-Nir and B Ronen ldquoA single bottleneck systemwith binomial yields and rigid demandrdquoManagement Sciencevol 39 no 5 pp 650ndash653 1993

[6] C A Yano and H L Lee ldquoLot sizing with random yields areviewrdquoOperations Research vol 43 no 2 pp 311ndash334 1995

[7] P Kouvelis and W Zhao ldquoFinancing the newsvendor sup-plier vs Bank and the structure of optimal trade creditcontractsrdquo Operations Research vol 60 no 3 pp 566ndash5802012

[8] B R Q Zhang ldquoInventory management with asset-basedfinancingrdquoManagement Science vol 50 no 9 pp 1274ndash12922004

[9] J Ma and L Xie ldquoe comparison and complex analysis ondual-channel supply chain under different channel powerstructures and uncertain demandrdquo Nonlinear Dynamicsvol 83 no 3 pp 1379ndash1393 2016

[10] X Xu and J R Birge ldquoJoint production and financing de-cisions modeling and analysisrdquo Operations Managementvol 28 no 12 2011

[11] P Kouvelis and W Zhao ldquoe newsvendor problem andprice-only contract when bankruptcy costs existrdquo Productionand Operations Management vol 20 no 6 pp 921ndash936 2010

[12] O Boyabath and L B Toktay ldquoStochastic capacity investmentand flexible vs Dedicated technology choice in imperfectcapital marketsrdquo Management Science vol 57 no 12pp 2163ndash2179 2011

[13] Y Alan and V Gaur ldquoOperational investment and capitalstructure under asset-based lendingrdquo Manufacturing amp Ser-vice Operations Management vol 20 no 4 2018

[14] B Sun H Zhang N Yan et al ldquoA partial credit guaranteecontract in a capital-constrained supply chain financingequilibrium and coordinating strategyrdquo International Journalof Production Economics vol 173 pp 122ndash133 2016

[15] G Bi Y Fei X Yuan et al ldquoJoint operational and financialcollaboration in a capital-constrained supply chain undermanufacturer collateralrdquo Asia-Pacific Journal of OperationalResearch vol 35 no 3 Article ID 1850010 2018

[16] J Chen Y W Zhou and Y Zhong ldquoA pricingorderingmodel for a dyadic supply chain with buyback guarantee fi-nancing and fairness concernsrdquo International Journal ofProduction Research vol 55 no 7 pp 1ndash18 2018

[17] G G Cai X Chen and Z Xiao ldquoe roles of bank and tradecredits theoretical analysis and empirical evidencerdquo Pro-duction and Operations Management vol 23 no 4pp 583ndash598 2014

[18] X F Chen ldquoA model of trade credit in a capital-constraineddistribution channelrdquo International Journal of ProductionEconomics vol 159 pp 347ndash357 2015

[19] J Chod ldquoInventory risk shifting and trade creditrdquo Man-agement Science vol 63 no 10 pp 3207ndash3225 2017

[20] B Jing X Chen and G Cai ldquoEquilibrium financing in adistribution channel with capital constraintrdquo Production andOperations Management vol 21 no 6 pp 1090ndash1101 2012

[21] S A Yang and J R Birge ldquoTrade credit risk sharing andinventory financing portfoliosrdquo Management Science vol 64no 8 2018

[22] J Zhou ldquoEssays in interfaces of operations and finance insupply chainsrdquo PhD thesis University of RochesterRochester NY USA 2009

[23] Y W Zhou B Cao Y Zhong et al ldquoOptimal advertisingordering policy and finance mode selection for a capital-constrained retailer with stochastic demandrdquo Journal of theOperational Research Society vol 68 no 12 pp 1620ndash16322017

[24] W Jin Q Zhang and J Luo ldquoNon-collaborative and col-laborative financing in a bilateral supply chain with capitalconstraintsrdquo Omega vol 88 pp 210ndash222 2018

[25] B Keren ldquoe single-period inventory problem extension torandom yield from the perspective of the supply chainrdquoOmega vol 37 no 4 pp 801ndash810 2009

[26] A Sufi ldquoBank lines of credit in corporate finance an empiricalanalysisrdquo Review of Financial Studies vol 22 no 3pp 1057ndash1088 2009

14 Mathematical Problems in Engineering

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MathematicsJournal of

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Mathematical Problems in Engineering

Applied MathematicsJournal of

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Probability and StatisticsHindawiwwwhindawicom Volume 2018

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Mathematical PhysicsAdvances in

Complex AnalysisJournal of

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OptimizationJournal of

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Engineering Mathematics

International Journal of

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Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

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Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

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Proposition 4 Under the FBU strategy the retailer expectsthe maximum profit Eπ2 when q1 D

Proof At this point the maximum profit of the retailer is

Eπ2 p2 minus p1( 1113857 q0 1113946Dq0

0xf(x)dx + D 1113946

+infin

Dq0f(x)dx1113890 1113891

+ w D minus q0( 1113857 minus kp1Dr3

(15)

e retailer expects the loan amount L2 to be

L2 kp1D (16)

where k is the ratio of the prepayment of the retailerrsquos unitproduct to p1

e expected profit of the bank is

Eπ4 L2 r3 minus r0( 1113857 kp1D r3 minus r0( 1113857 (17)

42 Bank Loan Interest Rate Setting under Two FinancingStrategies

421 Bank Loan Interest Rate Setting under FBS StrategyAccording to equation (11) the bank expects the manu-facturerrsquos loan profit Eπ3 to be related to q0 and r1According to formula (3) the manufacturerrsquos optimalplanned output q0 is negatively correlated with the bankrsquosloan interest rate r1 erefore the bank needs to set its loaninterest rate r1 to the manufacturer so that the bankrsquos ex-pected profit for the manufacturerrsquos loan is the largest underthe premise of satisfying themanufacturerrsquos maximumprofitformula (3)

According to formula (10) the bankrsquos expected loanprofit Eπ4 to the retailer is only positively correlated with thevariable r2 so the bank needs to increase its loan interest rateto the retailer as much as possible

erefore this study proposes an interest rate settingmodel under the bankrsquos expectation of loan profitmaximization

max q0c0 minus kp1q1( 1113857 r1 minus r0( 1113857 + kp1q1 r2 minus r0( 11138571113858 1113859

st

1113946q1q0

0xf(x)dx 1 minus

p1 + w minus c0 1 + r1( 1113857

p1 + l1

q1 D r0 le r1 le r10 r0 le r2 le r20

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(18)

Proposition 5 Under the FBS strategy when the bank ex-pects the maximum loan profit the following are satisfied

(1) e bankrsquos optimal loan interest rate r2 to the retailermust be the maximum acceptable loan interest rateof the retailerat is when r2 r20 r2 is the optimalsolution of equation (18)

(2) e bankrsquos loan interest rate r1 to themanufacturer isnot necessarily the maximum value r10 that themanufacturer can accept

is proposition points out the bankrsquos loan interest ratesettingmethod under the FBS strategyemethod indicatesthat the bank should give the retailer a loan interest rate thatshould take the maximum value while giving the manu-facturer a loan interest rate that does not necessarily have totake the maximum value e reason is that the informationis symmetrical e bank knows that the bankrsquos loan interestrate to the retailer is more favorable to the bankrsquos loan profitunder the condition that the retailer determines the orderquantity In contrast the bank knows that the manufac-turerrsquos planned output is inversely proportional to the loaninterest rate and the bank needs to consider the impact ofthe loan interest rate on the manufacturerrsquos planned outputus under the FBS strategy the banks optimal loan in-terest rate to the manufacturer is not always the maximum

422 Bank Loan Interest Rate Setting under FBU StrategyUnder the FBU strategy the bank gives a uniform loaninterest rate to the supply chain e interest rate is notgreater than the maximum acceptable interest rate for themanufacturer and the retailer e bankrsquos total profit fromthe manufacturer and retailer is

Eπ3 + Eπ4 q0c0 minus kp1q1( 1113857 r3 minus r0( 1113857 + kp1q1 r3 minus r0( 1113857

q0c0 r3 minus r0( 1113857

(19)According to formula (12) the manufacturerrsquos optimal

planned output q0 is negatively correlated with the bankrsquosloan interest rate r3 erefore this study proposes that theloan interest rate setting model under the bankrsquos expectedloan profit maximization is as follows

max q0c0 r3 minus r0( 1113857

st

1113946q1q0

0xf(x)dx 1 minus

p1 + w minus c0 1 + r3( 1113857

p1 + l1

q1 D r3 lt r10 r3 lt r20

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(20)

Proposition 6 Under the FBU strategy when the bankrsquossupply chain loan interest rate is the maximum value ac-ceptable by the supply chain enterprise the bankrsquos expectedloan profit is the largest at is when rlowast3 min r10 r201113864 1113865there is

max q0c0 r3 minus r0( 11138571113858 1113859 q0c0 rlowast3 minus r0( 1113857 (21)

is proposition points out the bankrsquos loan interest ratesetting method under the FBU strategy e method statesthat the bankrsquos loan interest rate should be the maximumunder the FBU strategye reason is that the information is

Mathematical Problems in Engineering 7

symmetrical e bank knows that the loan interest rate isinversely proportional to the manufacturerrsquos planned out-put and the loan interest rate is directly proportional to thebankrsquos loan profit erefore under the FBU strategy thebank chooses to increase the loan interest rate to be morebeneficial to itself

43 Analysis of the Profit of the Bank and Supply ChainEnterprises under FBS Strategy and FBU Strategy

431 Comparison of the Bankrsquos Maximum Expected LoanProfit under the FBS Strategy and FBU Strategy Based onformula (18) under the FBS strategy and formula (20) underthe FBU strategy the following propositions are proposed

Proposition 7 e maximum loan profit of the bank underthe FBS strategy is not less than the maximum loan profit ofthe bank under the FBU strategy at is the objectivefunction value of formula (18) is not smaller than the ob-jective function value of formula (20)

is proposition indicates that bankrsquos loan profits aregreater under the FBS strategy e reason is that at thistime the bank can separately set loan interest rates to makeits own profit even greater Under the FBU strategy the bankcan only consider lending at a uniform loan interest rate Atthis time the loan profit from the manufacturer and theretailer is no higher than the loan profit under the FBSstrategy

432 Comparison of Maximum Expected Profit of SupplyChain Enterprises under the FBS Strategy and FBU StrategyBecause of complete information supply chain companiesunderstand the maximization formulas (18) and (20) of thebankrsquos loan profit and know the optimal loan interest rate ofthe bank under the two financing strategies before the loanSupply chain companies establish a better loan methodbased on their expected profit formulas to maximize theirexpected profits erefore the following proposition isproposed

Proposition 8 Under the FBS strategy when the bankrsquos loaninterest rate to the manufacturer is its acceptable maximumthat is the optimal solution of equation (18) is rS

1 r10 thefollowing are satisfied

(1) If r10 le r20 the maximum expected profit of thesupply chain under the FBU strategy is not less thanits maximum expected profit under the FBS strategy

(2) If r10 gt r20 under qS0c0 minus kp1Dgt 0 and (p2 minus p1)

1113938DqU

00 xf(x)dx minus wgt 0 the FBU strategy is more

favorable to both enterprises in the supply chain

under qS0c0 minus kp1Dgt 0 and (p2 minus p1) 1113938

DqS0

0 xf(x)

dx minus wlt 0 the FBU strategy is more favorable to themanufacturer and not good for the retailer

is proposition finds out what kind of financingstrategy the supply chain enterprise chooses to be morefavorable to itself When the bankrsquos loan interest rate to themanufacturer under the FBS strategy is the maximum valueacceptable by the manufacturer and less than the maximumvalue of the retailerrsquos acceptable loan interest rate it is moreadvantageous for supply chain enterprises to choose the FBUstrategy When the bankrsquos loan interest rate to the manu-facturer under the FBS strategy is the maximum value ac-ceptable by themanufacturer and greater than themaximumloan interest rate acceptable by the retailer we find two setsof conditions under which the manufacturerrsquos expectedprofit under the FBU strategy is higher than the expectedprofit under the FBS strategy but the retailer does not fullyearn more under the FBU strategy us Proposition 8shows that the FBU strategy is beneficial to supply chainenterprises under certain conditions but it is not alwaysbeneficial to supply chain enterprises

5 Numerical Analysis

e setting of parameters is done according to the study ofJin et al [24] We assume xsimU[1 minus n 1 + n] and give thefour types of supply chain parameters in Table 2 e bankrsquosoptimal interest rate and the expected profits of banks andsupply chain enterprises are shown in Table 3 Figures 3 and4 show the comparisons of the supply chain enterprisesrsquoprofits under two different financing strategies in groups 2and 3 in Table 3 respectively

From Table 3 and Figures 3 and 4 we get the followingresults

First Proposition 5 is confirmed As can be seen fromTable 3 under the FBS strategy the bankrsquos loan interest rateto the retailer is the maximum value of the retailerrsquos ac-ceptable loan interest rate However it can be seen from theresults of group 1 in Table 3 that under this set of supplychain parameters when the bankrsquos expected profit is thelargest the loan interest rate given to the manufacturer is0062lt 01 From the results the bankrsquos maximum loanprofit from the manufacturer is not necessarily obtained atthe maximum loan interest rate and the largest loan profitfor the retailer is obtained at the maximum loan interest rate

Second Proposition 6 is confirmed As can be seen fromthe five sets of data in Table 2 under the FBU strategy themaximum bankrsquos loan interest rate acceptable by the supplychain is min 009 01 009 Obviously in the five sets ofresults in Table 3 the bankrsquos optimal lending rate under theFBU strategy is 009 From the results when the supply chainenterprises adopt the FBU strategy the bankrsquos largest loanprofit is obtained at the largest loan interest rate

ird Proposition 7 is confirmed e results of the fivegroups in Table 3 are tested as 814gt 633 769gt 581 and807gt 581 It is clear that the bankrsquos maximum expectedprofit is higher under the FBS strategy

Fourth Proposition 8 is confirmed From the results ofthe data of groups 4 and 5 in Table 3 under rS

1 r10 andr10 le r20 both sets of results indicate that the maximumexpected profit of the manufacturer and the retailer under

8 Mathematical Problems in Engineering

the FBU strategy is not less than the maximum expectedprot under the FBS strategy is conrms the rst con-clusion of Proposition 8 From the results of the data ofgroups 2 and 3 in Table 3 under rS1 r10 and r10 gt r20 bothsets of results indicate that the maximum expected prot ofthe manufacturer under the FBU strategy is greater than that

under the FBS strategy According to Figure 3 and group 2 inTable 3 the retailerrsquos maximum expected prot is largerunder the FBU strategy than the FBS strategy In contrast inFigure 4 and group 3 in Table 3 the conclusion is reversede reason is that the second set of data satises (p2 minus p1)intDq

U0

0 xf1(x)dx minus wgt 0 and the third set of data satises

2500 2600 2700 2800 2900 3000131

1315

132

1325

133

1335

134

1345

135

Expe

cted

pro

fit o

f man

ufac

ture

r und

er F

BS (E

π 1)

FBS (2762 134233167)

FBU (2765 134462212)

131

1315

132

1325

133

1335

134

1345

135

Expe

cted

pro

fit o

f man

ufac

ture

r und

er F

BU (E

π 1)

times104 times104

q0

(a)

q0

2000 2500 3000 3500 400023

235

24

245

25

255

26

FBS

FBU

2760 2780 28002566

2567

2568

FBS (2762 256669584)

FBU (2765 256688856)

times104

times104

Eπ2

(b)

Figure 3 Prot comparison of the supply chain enterprises under the two nancing strategies in group 2 in Table 3

Table 2 Supply chain parameter settings

Types c0 p1 p2 D l1 l2 w r0 k r10 r201 3 15 20 3000 10 14 4 002 02 01 0092 3 10 20 3000 10 14 4 002 02 01 0093 3 10 15 3000 10 14 4 002 02 01 0094 3 10 20 3000 10 14 4 002 02 009 015 3 10 15 3000 10 14 4 002 02 009 01

Table 3 Prots of enterprises and banks under the FBS strategy and FBU strategy

Types Financingstrategies

Optimalloan

interest rate

Optimalproductionand orderingdecisions

Bankrsquos expectedprot

Manufacturerrsquosexpected prot

Retailerrsquos expectedprot

Total expectedprot

q0 q1

1 FBS strategy r1 0062 3023 3000 814 26627 12266 38893r2 009FBU strategy r3 009 3013 3000 633 26626 12287 38913

2 FBS strategy r1 01 2762 3000 769 13423 25667 39090r2 009FBU strategy r3 009 2765 3000 581 13446 25669 39115

3 FBS strategy r1 01 2762 3000 769 13423 13039 26462r2 009FBU strategy r3 009 2765 3000 581 13446 13034 26480

4 FBS strategy r1 009 2765 3000 807 13446 25609 39055r2 01FBU strategy r3 009 2765 3000 581 13446 25669 39115

5 FBS strategy r1 009 2765 3000 807 13446 12974 26420r2 01FBU strategy r3 009 2765 3000 581 13446 13034 26480

Mathematical Problems in Engineering 9

(p2 minus p1)intDqS00 xf1(x)dx minus wlt 0 thus conrming the

second conclusion of Proposition 8

6 Conclusion

is paper studies the external nancing problems of thebilateral supply chain with nancial constraints It startswith taking the manufacturerrsquos planned output and theretailerrsquos planned order quantity as variables builds theexpected prot model of the manufacturer and the retailerunder the FBS strategy and FBU strategy and analyzes themanufacturerrsquos optimal planned output and the retailerrsquosoptimal order quantity under the two nancing strategiesen it states on the condition of symmetry of informationhow could the bank determine the loan interest rate tomaximize the expected loan prot when the bank hasobtained the information of the planned production andplanned order quantity of the manufacturer and the retailerunder the two nancing strategies At last it proposes themaximum expected loan prot model of the bank under theFBS strategy and FBU strategy to determine the optimalloan interest rate of the bank under the two nancingstrategies And it compares the prot of the bank and theprot of the supply chain under the two nancing strategies

e research shows the following (1) Under the two -nancing strategies the manufacturerrsquos optimal plannedoutput is inversely proportional to the loan interest rate andthe retailerrsquos optimal order quantity is the market demand (2)Under the FBS strategy it is more benecial for the bank toget more expected prot when it raises the loan interest ratesfor the retailer but it does not apply to the manufacturer (3)Under the FBU strategy when the bank sets the loan interestrate at the upper limit of the acceptable line the bankrsquos ex-pected loan prot is the largest but it is less than the bankrsquosexpected loan prot under the FBS strategy (4) e FBUstrategy is more protable for both the manufacturer and the

retailer when the bankrsquos loan interest rate to the manufactureris its acceptable maximum and the loan interest rate ac-ceptable by the manufacturer is less than that by the retailer

e above-mentioned conclusions supply a policyguiding the supply chain enterprises with nancial con-straints to make a choice of the nancing strategy For thebank (1) when the bilateral supply chain enterprises withuncertain output and certain demand choose the FBSstrategy the bank should give the retailer the maximum loaninterest rate and give the manufacturer a loan interest rateaccording to formula (18) (2) when they choose the FBUstrategy the bank should give the supply chain enterprisesthe largest unied loan interest rate and (3) when supplychain enterprises choose the FBS strategy the bank makemore prots so the bank should encourage them to conductthe FBS strategy as much as possible For supply chainenterprises the optimal loan strategies of nancing enter-prises under dierent conditions are dierent so theyshould reasonably choose the optimal nancing strategybased on the bankrsquos quotation and their own acceptable loaninterest rate

For the bilateral supply chain under the FBS and FBUnancing strategies all the nancing modes are distin-guished from the perspective of obtaining the bankrsquos loaninterest rates so there is no nancing strategy superior to theFBU strategy ere are various modes under the FBUstrategy and each has its own advantages such as the dealersand supplier network nancing modes and the bank logisticscooperative nancing modes erefore various nancingmodes within the FBU strategy should be studied and an-alyzed in the future

Appendix

Proof of Proposition 1 Taking the rst-order and second-order derivatives of Eπ1 with respect to q0 it follows that

2500 2600 2700 2800 2900 3000131

1315

132

1325

133

1335

134

1345

135Ex

pect

ed p

rofit

of m

anuf

actu

rer u

nder

FBS

(Eπ 1

)

FBS (2762 134233167)

FBU (2765 134462212)

131

1315

132

1325

133

1335

134

1345

135

Expe

cted

pro

fit o

f man

ufac

ture

r und

er F

BU (E

π 1)

times104 times104

q0

(a)

q0

2000 2200 2400 2600 2800 3000124

126

128

13

132

134

136

FBS

FBU

2760 2780 28001298

13

1302

1304(2762 130394792)

(2765 130344428)

times104

times104

Eπ2

(b)

Figure 4 Prot comparison of the supply chain enterprises under the two nancing strategies in group 3 in Table 3

10 Mathematical Problems in Engineering

dEπ1

dq0 p1 + l1( 1113857 1113946

q1q0

0xf(x)dx + w minus c0 1 + r1( 1113857 minus l1

(A1)

d2Eπ1dq20

minusq21q30

p1 + l1( 1113857fq1

q01113888 1113889le 0 (A2)

When (q1q0)gt 1 + n that is q0 lt (q1(1 + n)) we havef(q1q0) 0 us (d2Eπ1dq20) 0 We obtain

dEπ1

dq0 p1 + w minus c0 1 + r1( 1113857gt 0 (A3)

So when q0 lt (q1(1 + n)) Eπ1 is a monotonically in-creasing function with a fixed slope

When (q1q0)lt 1 minus n that is q0 ge (q1(1 minus n)) we havef(q1q0) 0 us (d2Eπ1dq20) 0 We obtain

dEπ1dq0

w minus c0 1 + r1( 1113857 minus l1 lt 0 (A4)

So when (q1(1 minus n))lt q0 Eπ1 is a monotonically de-creasing function with a fixed slope

When 1 minus nle (q1q0)le 1 + n we have (d2Eπ1dq20)lt 0erefore Eπ1 is a convex function about the variable q0dEπ1dq0 is a continuous function on q0 isin [0 +infin] Whenq0 (q1(1 + n)) we obtain (dEπ1dq0)gt 0 according to(A1) When q0 (q1(1 minus n)) we obtain (dEπ1dq0)lt 0according to (A1) Hence according to the intermediatevalue theorem there must exist q0 isin [(q1(1 + n)) (q1(1 minus

n))] satisfying (dEπ1dq0) 0 At this time q0 is the onlylocal maximum point of Eπ1

In summary when q0 satisfies (dEπ1dq0) 0 that is

1113946q1q0

0xf(x)dx 1 minus

p1 + w minus c0 1 + r1( 1113857

p1 + l1 (A5)

and q0 isin [0 +infin) q0 is the only absolute maximum point ofEπ1

Proof of Proposition 2 Taking the first-order derivative ofEπ2 with respect to q1 geD it follows that

dE π2

dq1

p2 minus p1( 1113857 1113946+infin

q1q0f(x)dx + w minus kp1r2 q1 ltD

minus l2 1113946+infin

q1q0f(x)dx + w minus kp1r2 q1 geD

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(A6)

(1) q1 geD

According to Proposition 1 we have (q1q0) isin[1 minus n 1 + n] Because of xf(x)ge 0 we can obtainthat 1113938

q1q00 xf(x)dx gt 0 Let u (q1q0)gt 0 and de-

fine the function

φ1(u) 1113946U

0xf(x)dx (A7)

e first-order derivative of φ1(u) follows that

φ1prime(u) uf(u)ge 0 (A8)

Because of u isin [1 minus n 1 + n] we have φ1prime(u)gt 0us φ1(u) increases in u that is 1113938

q1q00 xf(x)dx

increases in (q1q0) Hence there exists a unique(q1q0) satisfying

1113946q1q0

0xf(x)dx 1 minus

p1 + w minus c0 1 + r1( 1113857

p1 + l1 (A9)

At this time (q1q0) is a unique constant related tothe coefficients and f(x) Hence when q1 geD(dEπ2dq1) is a unique constant If (dEπ2dq1)gt 0the retailerrsquos planned order quantity will be infinitewhich contradicts the risk-neutral assumptionHence we have (dEπ2dq1)lt 0 erefore Eπ2decreases in q1 in the region q1 isin [D +infin] and at-tains the maximum at the point q1 D

(2) q1 ltD

Because (dEπ2dq1) is a unique constant if

dEπ2dq1

p2 minus p1( 1113857 1113946+infin

q1q0f(x)dx + w minus kp1r2 lt 0

(A10)

then Eπ2 decreases in q1 in the region q1 isin [0 D) andattains the local maximum at the point q1 0 usEπ2 decreases in q1 when q1 ge 0 Eπ2 attains theabsolute maximum at the point q1 0 At this pointthe supply chain is in a state of no order We will notdiscuss this situation

In fact according to the previous literature and theactual situation of the supply chain there is often w minus kp1r2gt 0 Hence when q1 ltD we can obtain (dEπ2dq1)gt 0 andthat Eπ2 increases in q1

In summary Eπ2 attains the maximum at the pointq1 D

Proof of Proposition 3 It follows the proof of Proposition 1so we omit it

Proof of Proposition 4 It follows the proof of Proposition 2so we omit it

Proof of Proposition 5

(1) Obviously nonlinear programming (18) can bedivided into the following two planning problems

Mathematical Problems in Engineering 11

max q0c0 minus kp1q1( 1113857 r1 minus r0( 1113857

st

1113946q1q0

0xf(x)dx 1 minus

p1 + w minus c0 1 + r1( 1113857

p1 + l1

q1 D r0 le r1 le r10

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(A11)

max kp1q1 r2 minus r0( 1113857

st r0 le r2 le r20(A12)

In linear programming (A12) the objective functionincreases in r2 in the region r2 isin [r0 r20] us weattain the optimum of equation (A12) at the pointr2 r20 It is also the optimum of nonlinear pro-gramming (18)

(2) In nonlinear programming (A11) let

m1 q0( 1113857 q0c0 minus kp1D( 1113857 r1 minus r0( 1113857

st 1113946q1q0

0xf(x)dx 1 minus

p1 + w minus c0 1 + r1( 1113857

p1 + l1

m2 r1( 1113857 q0c0 minus kp1D( 1113857 r1 minus r0( 1113857

st 1113946q1q0

0xf(x)dx 1 minus

p1 + w minus c0 1 + r1( 1113857

p1 + l1

(A13)

according to

1113946q1q0

0xf(x)dx 1 minus

p1 + w minus c0 1 + r1( 1113857

p1 + l1 (A14)

We obtain that r1 decreases in q0 us when (dm1dq0)gt 0 we have (dm2dr1)lt 0 When (dm1dq0)lt 0we have (dm2dr1)gt 0

Taking the first-order derivative of m1 with respect to q0it follows thatdm1

dq0 p1 + l1( 1113857

middot 1113946q1q0

0xf(x)dx minus

p1 + l1( 1113857 q0c0 minus kp1D( 1113857q21f q1q0( 1113857

c0q30

minus r0c0 minus l1 minus c0 + w

(A15)

Under different parameters of the supply chain formula(A15) can satisfy both (dm1dq0)gt 0 and (dm1dq0)lt 0erefore there is not always (dm2dr1)gt 0 that is theoptimal of r1 is not always r10 In numerical analysis we giveexamples to support this conclusion

Proof of Proposition 6 Let

g q0 r3( 1113857 q0c0 r3 minus r0( 1113857 (A16)

Since the relationship between q0 and r3 satisfies theconstraint of (20) equation (A16) can also express equation(A17) as follows

g1 q0( 1113857 q0 p1 + l1( 1113857 1113946q1q0

0xf(x)dx + w minus l1 minus c0 1 + r0( 11138571113890 1113891

(A17)

Taking the first-order and second-order derivatives ofg1(q0) with respect to q0 it follows that

g1prime q0( 1113857 p1 + l1( 1113857 1113946q1q0

0xf(x)dx + w minus l1 minus c0 1 + r0( 11138571113890 1113891

minusq21q20

fq1

q01113888 1113889 p1 + l1( 1113857

gPrime1 q0( 1113857 p1 + l1( 1113857q21q30

fq1

q01113888 1113889gt 0

q1

q0isin [1 minus n 1 + n]

(A18)

Obviously when (q1q0) isin [1 minus n 1 + n] that isq0 isin [(q1(1 + n)) (q1(1 minus n))] g1prime(q0) increases in q0According to the proof of Proposition 1 the manufacturerattains the optimum in the region q0 isin [(q1(1 + n)) (q1(1 minus n))] When q0 (q1(1 minus n)) we can obtain

g1prime q0( 1113857 w minus l1 minus c0 1 + r0( 1113857 minus(1 minus n)2

2np1 + l1( 1113857lt 0

(A19)

12 Mathematical Problems in Engineering

us g1prime(q0) is negative in the region q0 isin [(q1(1 + n))

(q1(1 minus n))] g1(q0) decreases in q0 in the regionq0 isin [(q1(1 + n)) (q1(1 minus n))] Since

1113946q1q0

0xf1(x)dx 1 minus

p1 + w minus c0 1 + r3( 1113857

p1 + l1

st q0 isinq1

1 + n

q1

1 minus n1113876 1113877

(A20)

we obtain that r3 decreases in q0 in the region q0 isin [(q1(1 + n)) (q1(1 minus n))] Hence g1(q0) increases in r3 in theregion q0 isin [(q1(1 + n)) (q1(1 minus n))]

According to formulas (A3) and (A4) and the con-clusions of Proposition 1 when r1 isin [((w minus l1 minus c0)c0)((p1 + w minus c0)c0)] we have q0 isin [(q1(1 + n)) (q1(1 minus n))] Since we assume that the manufacturer makesdecisions based on the maximum expected profit there mustbe r10 isin [((w minus l1 minus c0)c0) ((p1 + w minus c0)c0)] which maxi-mizes the expected profit of the manufacturer under the FBSstrategy Similarly there must be min r10 r201113864 1113865 isin [((w minus l1 minus

c0)c0) ((p1 + w minus c0)c0)] which maximizes the expectedprofit of the manufacturer under the FBU strategy us wehave q0 isin [(q1(1 + n)) (q1(1 minus n))] when r3 isin [r0 minr10 r201113864 1113865] Since g1(q0) decreases in q0 in the region q0 isin

[(q1(1 + n)) (q1(1 minus n))] g1(q0) increases in r3 in theregion r3 isin [r0 min r10 r201113864 1113865] erefore g1(q0) attains themaximum at the point rlowast3 min r10 r201113864 1113865 We can obtainmax[q0c0(r3 minus r0)] q0c0(rlowast3 minus r0)

Proof of Proposition 7 If formula (20) attains the optimumat the point q0 qU

0 and r3 rU3 let q0 qU

0 r1 rU3 and

r2 rU3 in formula (18) Obviously they satisfy the con-

straint of formula (18) At this time the objective function offormula (18) is equal to the maximum of formula (20) whichmeans that the optimal of formula (20) must satisfy formula(18)erefore the maximum of formula (18) is not less thanthe maximum of formula (20)

Proof of Proposition 8

(1) When r10 le r20 we have rS1 r10 min r10 r201113864 1113865 rU

3according to Proposition 5 and rS

1 r10 us whenq1 D and r1 r3 it is obvious that q0 is equal in thetwo equations according to equations (3) and (11)Hence under qS

0 qU0 qS

1 qU1 and r1 r3 we can

obtain that EπS1 EπU

1 according to equations (2) and(12)When r10 le r20 we have rU

3 r10 le r20 rS2 accord-

ing to Proposition 5 and rS1 r10 Hence under

qS0 qU

0 qS1 qU

1 and r2 ge r3 we can obtain thatEπS

2 leEπU2 according to equations (8) and (15)

erefore when rS1 r10 and r10 le r20 we obtain that

EπS1 EπU

1 and EπS2 leEπU

2 which means the ex-pected profit of the manufacturer and retailer underthe FBU strategy is not less than the profit under theFBS strategy

(2) r10 gt r20

We know that r1 decreases in q0 in the regionq0 isin [(q1(1 + n)) (q1(1 minus n))] Since rS

1 r10 andr10 gt r20 we have q0 isin [qS

0 +infin) Substituting equa-tion (3) in (2) and taking the first-order derivative ofEπ1 with respect to q0 the following can be obtaineddEπ1

dq0

p1 + l1( 1113857q21c0q

30

fq1

q01113888 1113889 q0c0 minus kp1D( 1113857 (A21)

Since qS0c0 minus kp1Dgt 0 and q0 isin [qS

0 +infin) we have(dEπ1dq0)gt 0 When the supply chain is under theFBU strategy we have rU

3 min r10 r201113864 1113865 r20according to Proposition 6 and r10 gt r20us we haverU3 r20 lt r10 rS

1 and qU0 gt qS

0 erefore we canobtain that EπU

1 gtEπS1 when (dEπ1dq0)gt 0 and

qU0 gt qS

0 In other words we have EπU1 gtEπS

1 whenrS1 r10 qS

0c0 minus kp1D gt 0 and r10 gt r20

Taking the first-order and second-order derivatives ofEπ2 with respect to q0 the following can be obtained

dEπ2

dq0 p2 minus p1( 1113857 1113946

Dq0

0xf(x)dx minus w

d2Eπ2

dq20 minus p2 minus p1( 1113857

D2

q30f

D

q01113888 1113889lt 0

(A22)

When r1 r10 and r10 gt r20 we have rU3 lt rS

1 and qU0 gt qS

0Since (d2Eπ2dq20)lt 0 we obtain that (dEπ2dq0) decreasesin q0 When q0 isin [qS

0 qU0 ] we have

mindEπ2

dq0 p2 minus p1( 1113857 1113946

DqU0

0xf(x)dx minus w

maxdEπ2

dq0 p2 minus p1( 1113857 1113946

DqS0

0xf(x)dx minus w

(A23)

When (p2 minus p1) 1113938DqU

00 xf(x)dx minus wgt 0 and q0 isin [qS

0

qU0 ] we have (dEπ2dq0)gt 0 and EπU

2 gtEπS2 When (p2 minus

p1) 1113938DqS

00 xf(x)dx minus wlt 0 and q0 isin [qS

0 qU0 ] we have (dEπ2

dq0)lt 0 and EπU2 ltEπS

2In summary when (p2 minus p1) 1113938

DqU0

0 xf(x)dx minus wgt 0 andqS0c0 minus kp1Dgt 0 we have EπU

1 gtEπS1 and EπU

2 gtEπS2 When

(p2 minus p1) 1113938DqS

00 xf(x)dx minus wlt 0 and qS

0c0 minus kp1Dgt 0 wehave EπU

1 gtEπS1 and EπU

2 ltEπS2

Data Availability

e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

e authors declare that they have no conflicts of interest

Acknowledgments

is research was funded by the Ministry of Education ofHumanities and Social Science Project of China (grantnumber 17YJAZH018)

Mathematical Problems in Engineering 13

References

[1] S Asian and X Nie ldquoCoordination in supply chains withuncertain demand and disruption risks existence analysisand insightsrdquo IEEE Transactions on Systems Man and Cy-bernetics Systems vol 44 no 9 pp 1139ndash1154 2014

[2] A Roy S S Sana and K Chaudhuri ldquoOptimal Pricing ofcompeting retailers under uncertain demand-a two layersupply chain modelrdquo Annals of Operations Research vol 260no 1-2 pp 481ndash500 2015

[3] B C Giri and S Bardhan ldquoCoordinating a supply chain underuncertain demand and random yield in presence of supplydisruptionrdquo International Journal of Production Researchvol 53 no 16 pp 5070ndash5084 2015

[4] A Arreola-Risa and G A Decroix ldquoMake-to-order versusmake-to-stock in a production-inventory system with generalproduction timesrdquo IIE Transactions vol 30 no 8 pp 705ndash713 1998

[5] A Grosfeld-Nir and B Ronen ldquoA single bottleneck systemwith binomial yields and rigid demandrdquoManagement Sciencevol 39 no 5 pp 650ndash653 1993

[6] C A Yano and H L Lee ldquoLot sizing with random yields areviewrdquoOperations Research vol 43 no 2 pp 311ndash334 1995

[7] P Kouvelis and W Zhao ldquoFinancing the newsvendor sup-plier vs Bank and the structure of optimal trade creditcontractsrdquo Operations Research vol 60 no 3 pp 566ndash5802012

[8] B R Q Zhang ldquoInventory management with asset-basedfinancingrdquoManagement Science vol 50 no 9 pp 1274ndash12922004

[9] J Ma and L Xie ldquoe comparison and complex analysis ondual-channel supply chain under different channel powerstructures and uncertain demandrdquo Nonlinear Dynamicsvol 83 no 3 pp 1379ndash1393 2016

[10] X Xu and J R Birge ldquoJoint production and financing de-cisions modeling and analysisrdquo Operations Managementvol 28 no 12 2011

[11] P Kouvelis and W Zhao ldquoe newsvendor problem andprice-only contract when bankruptcy costs existrdquo Productionand Operations Management vol 20 no 6 pp 921ndash936 2010

[12] O Boyabath and L B Toktay ldquoStochastic capacity investmentand flexible vs Dedicated technology choice in imperfectcapital marketsrdquo Management Science vol 57 no 12pp 2163ndash2179 2011

[13] Y Alan and V Gaur ldquoOperational investment and capitalstructure under asset-based lendingrdquo Manufacturing amp Ser-vice Operations Management vol 20 no 4 2018

[14] B Sun H Zhang N Yan et al ldquoA partial credit guaranteecontract in a capital-constrained supply chain financingequilibrium and coordinating strategyrdquo International Journalof Production Economics vol 173 pp 122ndash133 2016

[15] G Bi Y Fei X Yuan et al ldquoJoint operational and financialcollaboration in a capital-constrained supply chain undermanufacturer collateralrdquo Asia-Pacific Journal of OperationalResearch vol 35 no 3 Article ID 1850010 2018

[16] J Chen Y W Zhou and Y Zhong ldquoA pricingorderingmodel for a dyadic supply chain with buyback guarantee fi-nancing and fairness concernsrdquo International Journal ofProduction Research vol 55 no 7 pp 1ndash18 2018

[17] G G Cai X Chen and Z Xiao ldquoe roles of bank and tradecredits theoretical analysis and empirical evidencerdquo Pro-duction and Operations Management vol 23 no 4pp 583ndash598 2014

[18] X F Chen ldquoA model of trade credit in a capital-constraineddistribution channelrdquo International Journal of ProductionEconomics vol 159 pp 347ndash357 2015

[19] J Chod ldquoInventory risk shifting and trade creditrdquo Man-agement Science vol 63 no 10 pp 3207ndash3225 2017

[20] B Jing X Chen and G Cai ldquoEquilibrium financing in adistribution channel with capital constraintrdquo Production andOperations Management vol 21 no 6 pp 1090ndash1101 2012

[21] S A Yang and J R Birge ldquoTrade credit risk sharing andinventory financing portfoliosrdquo Management Science vol 64no 8 2018

[22] J Zhou ldquoEssays in interfaces of operations and finance insupply chainsrdquo PhD thesis University of RochesterRochester NY USA 2009

[23] Y W Zhou B Cao Y Zhong et al ldquoOptimal advertisingordering policy and finance mode selection for a capital-constrained retailer with stochastic demandrdquo Journal of theOperational Research Society vol 68 no 12 pp 1620ndash16322017

[24] W Jin Q Zhang and J Luo ldquoNon-collaborative and col-laborative financing in a bilateral supply chain with capitalconstraintsrdquo Omega vol 88 pp 210ndash222 2018

[25] B Keren ldquoe single-period inventory problem extension torandom yield from the perspective of the supply chainrdquoOmega vol 37 no 4 pp 801ndash810 2009

[26] A Sufi ldquoBank lines of credit in corporate finance an empiricalanalysisrdquo Review of Financial Studies vol 22 no 3pp 1057ndash1088 2009

14 Mathematical Problems in Engineering

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Hindawiwwwhindawicom Volume 2018

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Hindawiwwwhindawicom Volume 2018

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Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

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International Journal of

Hindawiwwwhindawicom Volume 2018

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Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

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Volume 2018

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Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

symmetrical e bank knows that the loan interest rate isinversely proportional to the manufacturerrsquos planned out-put and the loan interest rate is directly proportional to thebankrsquos loan profit erefore under the FBU strategy thebank chooses to increase the loan interest rate to be morebeneficial to itself

43 Analysis of the Profit of the Bank and Supply ChainEnterprises under FBS Strategy and FBU Strategy

431 Comparison of the Bankrsquos Maximum Expected LoanProfit under the FBS Strategy and FBU Strategy Based onformula (18) under the FBS strategy and formula (20) underthe FBU strategy the following propositions are proposed

Proposition 7 e maximum loan profit of the bank underthe FBS strategy is not less than the maximum loan profit ofthe bank under the FBU strategy at is the objectivefunction value of formula (18) is not smaller than the ob-jective function value of formula (20)

is proposition indicates that bankrsquos loan profits aregreater under the FBS strategy e reason is that at thistime the bank can separately set loan interest rates to makeits own profit even greater Under the FBU strategy the bankcan only consider lending at a uniform loan interest rate Atthis time the loan profit from the manufacturer and theretailer is no higher than the loan profit under the FBSstrategy

432 Comparison of Maximum Expected Profit of SupplyChain Enterprises under the FBS Strategy and FBU StrategyBecause of complete information supply chain companiesunderstand the maximization formulas (18) and (20) of thebankrsquos loan profit and know the optimal loan interest rate ofthe bank under the two financing strategies before the loanSupply chain companies establish a better loan methodbased on their expected profit formulas to maximize theirexpected profits erefore the following proposition isproposed

Proposition 8 Under the FBS strategy when the bankrsquos loaninterest rate to the manufacturer is its acceptable maximumthat is the optimal solution of equation (18) is rS

1 r10 thefollowing are satisfied

(1) If r10 le r20 the maximum expected profit of thesupply chain under the FBU strategy is not less thanits maximum expected profit under the FBS strategy

(2) If r10 gt r20 under qS0c0 minus kp1Dgt 0 and (p2 minus p1)

1113938DqU

00 xf(x)dx minus wgt 0 the FBU strategy is more

favorable to both enterprises in the supply chain

under qS0c0 minus kp1Dgt 0 and (p2 minus p1) 1113938

DqS0

0 xf(x)

dx minus wlt 0 the FBU strategy is more favorable to themanufacturer and not good for the retailer

is proposition finds out what kind of financingstrategy the supply chain enterprise chooses to be morefavorable to itself When the bankrsquos loan interest rate to themanufacturer under the FBS strategy is the maximum valueacceptable by the manufacturer and less than the maximumvalue of the retailerrsquos acceptable loan interest rate it is moreadvantageous for supply chain enterprises to choose the FBUstrategy When the bankrsquos loan interest rate to the manu-facturer under the FBS strategy is the maximum value ac-ceptable by themanufacturer and greater than themaximumloan interest rate acceptable by the retailer we find two setsof conditions under which the manufacturerrsquos expectedprofit under the FBU strategy is higher than the expectedprofit under the FBS strategy but the retailer does not fullyearn more under the FBU strategy us Proposition 8shows that the FBU strategy is beneficial to supply chainenterprises under certain conditions but it is not alwaysbeneficial to supply chain enterprises

5 Numerical Analysis

e setting of parameters is done according to the study ofJin et al [24] We assume xsimU[1 minus n 1 + n] and give thefour types of supply chain parameters in Table 2 e bankrsquosoptimal interest rate and the expected profits of banks andsupply chain enterprises are shown in Table 3 Figures 3 and4 show the comparisons of the supply chain enterprisesrsquoprofits under two different financing strategies in groups 2and 3 in Table 3 respectively

From Table 3 and Figures 3 and 4 we get the followingresults

First Proposition 5 is confirmed As can be seen fromTable 3 under the FBS strategy the bankrsquos loan interest rateto the retailer is the maximum value of the retailerrsquos ac-ceptable loan interest rate However it can be seen from theresults of group 1 in Table 3 that under this set of supplychain parameters when the bankrsquos expected profit is thelargest the loan interest rate given to the manufacturer is0062lt 01 From the results the bankrsquos maximum loanprofit from the manufacturer is not necessarily obtained atthe maximum loan interest rate and the largest loan profitfor the retailer is obtained at the maximum loan interest rate

Second Proposition 6 is confirmed As can be seen fromthe five sets of data in Table 2 under the FBU strategy themaximum bankrsquos loan interest rate acceptable by the supplychain is min 009 01 009 Obviously in the five sets ofresults in Table 3 the bankrsquos optimal lending rate under theFBU strategy is 009 From the results when the supply chainenterprises adopt the FBU strategy the bankrsquos largest loanprofit is obtained at the largest loan interest rate

ird Proposition 7 is confirmed e results of the fivegroups in Table 3 are tested as 814gt 633 769gt 581 and807gt 581 It is clear that the bankrsquos maximum expectedprofit is higher under the FBS strategy

Fourth Proposition 8 is confirmed From the results ofthe data of groups 4 and 5 in Table 3 under rS

1 r10 andr10 le r20 both sets of results indicate that the maximumexpected profit of the manufacturer and the retailer under

8 Mathematical Problems in Engineering

the FBU strategy is not less than the maximum expectedprot under the FBS strategy is conrms the rst con-clusion of Proposition 8 From the results of the data ofgroups 2 and 3 in Table 3 under rS1 r10 and r10 gt r20 bothsets of results indicate that the maximum expected prot ofthe manufacturer under the FBU strategy is greater than that

under the FBS strategy According to Figure 3 and group 2 inTable 3 the retailerrsquos maximum expected prot is largerunder the FBU strategy than the FBS strategy In contrast inFigure 4 and group 3 in Table 3 the conclusion is reversede reason is that the second set of data satises (p2 minus p1)intDq

U0

0 xf1(x)dx minus wgt 0 and the third set of data satises

2500 2600 2700 2800 2900 3000131

1315

132

1325

133

1335

134

1345

135

Expe

cted

pro

fit o

f man

ufac

ture

r und

er F

BS (E

π 1)

FBS (2762 134233167)

FBU (2765 134462212)

131

1315

132

1325

133

1335

134

1345

135

Expe

cted

pro

fit o

f man

ufac

ture

r und

er F

BU (E

π 1)

times104 times104

q0

(a)

q0

2000 2500 3000 3500 400023

235

24

245

25

255

26

FBS

FBU

2760 2780 28002566

2567

2568

FBS (2762 256669584)

FBU (2765 256688856)

times104

times104

Eπ2

(b)

Figure 3 Prot comparison of the supply chain enterprises under the two nancing strategies in group 2 in Table 3

Table 2 Supply chain parameter settings

Types c0 p1 p2 D l1 l2 w r0 k r10 r201 3 15 20 3000 10 14 4 002 02 01 0092 3 10 20 3000 10 14 4 002 02 01 0093 3 10 15 3000 10 14 4 002 02 01 0094 3 10 20 3000 10 14 4 002 02 009 015 3 10 15 3000 10 14 4 002 02 009 01

Table 3 Prots of enterprises and banks under the FBS strategy and FBU strategy

Types Financingstrategies

Optimalloan

interest rate

Optimalproductionand orderingdecisions

Bankrsquos expectedprot

Manufacturerrsquosexpected prot

Retailerrsquos expectedprot

Total expectedprot

q0 q1

1 FBS strategy r1 0062 3023 3000 814 26627 12266 38893r2 009FBU strategy r3 009 3013 3000 633 26626 12287 38913

2 FBS strategy r1 01 2762 3000 769 13423 25667 39090r2 009FBU strategy r3 009 2765 3000 581 13446 25669 39115

3 FBS strategy r1 01 2762 3000 769 13423 13039 26462r2 009FBU strategy r3 009 2765 3000 581 13446 13034 26480

4 FBS strategy r1 009 2765 3000 807 13446 25609 39055r2 01FBU strategy r3 009 2765 3000 581 13446 25669 39115

5 FBS strategy r1 009 2765 3000 807 13446 12974 26420r2 01FBU strategy r3 009 2765 3000 581 13446 13034 26480

Mathematical Problems in Engineering 9

(p2 minus p1)intDqS00 xf1(x)dx minus wlt 0 thus conrming the

second conclusion of Proposition 8

6 Conclusion

is paper studies the external nancing problems of thebilateral supply chain with nancial constraints It startswith taking the manufacturerrsquos planned output and theretailerrsquos planned order quantity as variables builds theexpected prot model of the manufacturer and the retailerunder the FBS strategy and FBU strategy and analyzes themanufacturerrsquos optimal planned output and the retailerrsquosoptimal order quantity under the two nancing strategiesen it states on the condition of symmetry of informationhow could the bank determine the loan interest rate tomaximize the expected loan prot when the bank hasobtained the information of the planned production andplanned order quantity of the manufacturer and the retailerunder the two nancing strategies At last it proposes themaximum expected loan prot model of the bank under theFBS strategy and FBU strategy to determine the optimalloan interest rate of the bank under the two nancingstrategies And it compares the prot of the bank and theprot of the supply chain under the two nancing strategies

e research shows the following (1) Under the two -nancing strategies the manufacturerrsquos optimal plannedoutput is inversely proportional to the loan interest rate andthe retailerrsquos optimal order quantity is the market demand (2)Under the FBS strategy it is more benecial for the bank toget more expected prot when it raises the loan interest ratesfor the retailer but it does not apply to the manufacturer (3)Under the FBU strategy when the bank sets the loan interestrate at the upper limit of the acceptable line the bankrsquos ex-pected loan prot is the largest but it is less than the bankrsquosexpected loan prot under the FBS strategy (4) e FBUstrategy is more protable for both the manufacturer and the

retailer when the bankrsquos loan interest rate to the manufactureris its acceptable maximum and the loan interest rate ac-ceptable by the manufacturer is less than that by the retailer

e above-mentioned conclusions supply a policyguiding the supply chain enterprises with nancial con-straints to make a choice of the nancing strategy For thebank (1) when the bilateral supply chain enterprises withuncertain output and certain demand choose the FBSstrategy the bank should give the retailer the maximum loaninterest rate and give the manufacturer a loan interest rateaccording to formula (18) (2) when they choose the FBUstrategy the bank should give the supply chain enterprisesthe largest unied loan interest rate and (3) when supplychain enterprises choose the FBS strategy the bank makemore prots so the bank should encourage them to conductthe FBS strategy as much as possible For supply chainenterprises the optimal loan strategies of nancing enter-prises under dierent conditions are dierent so theyshould reasonably choose the optimal nancing strategybased on the bankrsquos quotation and their own acceptable loaninterest rate

For the bilateral supply chain under the FBS and FBUnancing strategies all the nancing modes are distin-guished from the perspective of obtaining the bankrsquos loaninterest rates so there is no nancing strategy superior to theFBU strategy ere are various modes under the FBUstrategy and each has its own advantages such as the dealersand supplier network nancing modes and the bank logisticscooperative nancing modes erefore various nancingmodes within the FBU strategy should be studied and an-alyzed in the future

Appendix

Proof of Proposition 1 Taking the rst-order and second-order derivatives of Eπ1 with respect to q0 it follows that

2500 2600 2700 2800 2900 3000131

1315

132

1325

133

1335

134

1345

135Ex

pect

ed p

rofit

of m

anuf

actu

rer u

nder

FBS

(Eπ 1

)

FBS (2762 134233167)

FBU (2765 134462212)

131

1315

132

1325

133

1335

134

1345

135

Expe

cted

pro

fit o

f man

ufac

ture

r und

er F

BU (E

π 1)

times104 times104

q0

(a)

q0

2000 2200 2400 2600 2800 3000124

126

128

13

132

134

136

FBS

FBU

2760 2780 28001298

13

1302

1304(2762 130394792)

(2765 130344428)

times104

times104

Eπ2

(b)

Figure 4 Prot comparison of the supply chain enterprises under the two nancing strategies in group 3 in Table 3

10 Mathematical Problems in Engineering

dEπ1

dq0 p1 + l1( 1113857 1113946

q1q0

0xf(x)dx + w minus c0 1 + r1( 1113857 minus l1

(A1)

d2Eπ1dq20

minusq21q30

p1 + l1( 1113857fq1

q01113888 1113889le 0 (A2)

When (q1q0)gt 1 + n that is q0 lt (q1(1 + n)) we havef(q1q0) 0 us (d2Eπ1dq20) 0 We obtain

dEπ1

dq0 p1 + w minus c0 1 + r1( 1113857gt 0 (A3)

So when q0 lt (q1(1 + n)) Eπ1 is a monotonically in-creasing function with a fixed slope

When (q1q0)lt 1 minus n that is q0 ge (q1(1 minus n)) we havef(q1q0) 0 us (d2Eπ1dq20) 0 We obtain

dEπ1dq0

w minus c0 1 + r1( 1113857 minus l1 lt 0 (A4)

So when (q1(1 minus n))lt q0 Eπ1 is a monotonically de-creasing function with a fixed slope

When 1 minus nle (q1q0)le 1 + n we have (d2Eπ1dq20)lt 0erefore Eπ1 is a convex function about the variable q0dEπ1dq0 is a continuous function on q0 isin [0 +infin] Whenq0 (q1(1 + n)) we obtain (dEπ1dq0)gt 0 according to(A1) When q0 (q1(1 minus n)) we obtain (dEπ1dq0)lt 0according to (A1) Hence according to the intermediatevalue theorem there must exist q0 isin [(q1(1 + n)) (q1(1 minus

n))] satisfying (dEπ1dq0) 0 At this time q0 is the onlylocal maximum point of Eπ1

In summary when q0 satisfies (dEπ1dq0) 0 that is

1113946q1q0

0xf(x)dx 1 minus

p1 + w minus c0 1 + r1( 1113857

p1 + l1 (A5)

and q0 isin [0 +infin) q0 is the only absolute maximum point ofEπ1

Proof of Proposition 2 Taking the first-order derivative ofEπ2 with respect to q1 geD it follows that

dE π2

dq1

p2 minus p1( 1113857 1113946+infin

q1q0f(x)dx + w minus kp1r2 q1 ltD

minus l2 1113946+infin

q1q0f(x)dx + w minus kp1r2 q1 geD

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(A6)

(1) q1 geD

According to Proposition 1 we have (q1q0) isin[1 minus n 1 + n] Because of xf(x)ge 0 we can obtainthat 1113938

q1q00 xf(x)dx gt 0 Let u (q1q0)gt 0 and de-

fine the function

φ1(u) 1113946U

0xf(x)dx (A7)

e first-order derivative of φ1(u) follows that

φ1prime(u) uf(u)ge 0 (A8)

Because of u isin [1 minus n 1 + n] we have φ1prime(u)gt 0us φ1(u) increases in u that is 1113938

q1q00 xf(x)dx

increases in (q1q0) Hence there exists a unique(q1q0) satisfying

1113946q1q0

0xf(x)dx 1 minus

p1 + w minus c0 1 + r1( 1113857

p1 + l1 (A9)

At this time (q1q0) is a unique constant related tothe coefficients and f(x) Hence when q1 geD(dEπ2dq1) is a unique constant If (dEπ2dq1)gt 0the retailerrsquos planned order quantity will be infinitewhich contradicts the risk-neutral assumptionHence we have (dEπ2dq1)lt 0 erefore Eπ2decreases in q1 in the region q1 isin [D +infin] and at-tains the maximum at the point q1 D

(2) q1 ltD

Because (dEπ2dq1) is a unique constant if

dEπ2dq1

p2 minus p1( 1113857 1113946+infin

q1q0f(x)dx + w minus kp1r2 lt 0

(A10)

then Eπ2 decreases in q1 in the region q1 isin [0 D) andattains the local maximum at the point q1 0 usEπ2 decreases in q1 when q1 ge 0 Eπ2 attains theabsolute maximum at the point q1 0 At this pointthe supply chain is in a state of no order We will notdiscuss this situation

In fact according to the previous literature and theactual situation of the supply chain there is often w minus kp1r2gt 0 Hence when q1 ltD we can obtain (dEπ2dq1)gt 0 andthat Eπ2 increases in q1

In summary Eπ2 attains the maximum at the pointq1 D

Proof of Proposition 3 It follows the proof of Proposition 1so we omit it

Proof of Proposition 4 It follows the proof of Proposition 2so we omit it

Proof of Proposition 5

(1) Obviously nonlinear programming (18) can bedivided into the following two planning problems

Mathematical Problems in Engineering 11

max q0c0 minus kp1q1( 1113857 r1 minus r0( 1113857

st

1113946q1q0

0xf(x)dx 1 minus

p1 + w minus c0 1 + r1( 1113857

p1 + l1

q1 D r0 le r1 le r10

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(A11)

max kp1q1 r2 minus r0( 1113857

st r0 le r2 le r20(A12)

In linear programming (A12) the objective functionincreases in r2 in the region r2 isin [r0 r20] us weattain the optimum of equation (A12) at the pointr2 r20 It is also the optimum of nonlinear pro-gramming (18)

(2) In nonlinear programming (A11) let

m1 q0( 1113857 q0c0 minus kp1D( 1113857 r1 minus r0( 1113857

st 1113946q1q0

0xf(x)dx 1 minus

p1 + w minus c0 1 + r1( 1113857

p1 + l1

m2 r1( 1113857 q0c0 minus kp1D( 1113857 r1 minus r0( 1113857

st 1113946q1q0

0xf(x)dx 1 minus

p1 + w minus c0 1 + r1( 1113857

p1 + l1

(A13)

according to

1113946q1q0

0xf(x)dx 1 minus

p1 + w minus c0 1 + r1( 1113857

p1 + l1 (A14)

We obtain that r1 decreases in q0 us when (dm1dq0)gt 0 we have (dm2dr1)lt 0 When (dm1dq0)lt 0we have (dm2dr1)gt 0

Taking the first-order derivative of m1 with respect to q0it follows thatdm1

dq0 p1 + l1( 1113857

middot 1113946q1q0

0xf(x)dx minus

p1 + l1( 1113857 q0c0 minus kp1D( 1113857q21f q1q0( 1113857

c0q30

minus r0c0 minus l1 minus c0 + w

(A15)

Under different parameters of the supply chain formula(A15) can satisfy both (dm1dq0)gt 0 and (dm1dq0)lt 0erefore there is not always (dm2dr1)gt 0 that is theoptimal of r1 is not always r10 In numerical analysis we giveexamples to support this conclusion

Proof of Proposition 6 Let

g q0 r3( 1113857 q0c0 r3 minus r0( 1113857 (A16)

Since the relationship between q0 and r3 satisfies theconstraint of (20) equation (A16) can also express equation(A17) as follows

g1 q0( 1113857 q0 p1 + l1( 1113857 1113946q1q0

0xf(x)dx + w minus l1 minus c0 1 + r0( 11138571113890 1113891

(A17)

Taking the first-order and second-order derivatives ofg1(q0) with respect to q0 it follows that

g1prime q0( 1113857 p1 + l1( 1113857 1113946q1q0

0xf(x)dx + w minus l1 minus c0 1 + r0( 11138571113890 1113891

minusq21q20

fq1

q01113888 1113889 p1 + l1( 1113857

gPrime1 q0( 1113857 p1 + l1( 1113857q21q30

fq1

q01113888 1113889gt 0

q1

q0isin [1 minus n 1 + n]

(A18)

Obviously when (q1q0) isin [1 minus n 1 + n] that isq0 isin [(q1(1 + n)) (q1(1 minus n))] g1prime(q0) increases in q0According to the proof of Proposition 1 the manufacturerattains the optimum in the region q0 isin [(q1(1 + n)) (q1(1 minus n))] When q0 (q1(1 minus n)) we can obtain

g1prime q0( 1113857 w minus l1 minus c0 1 + r0( 1113857 minus(1 minus n)2

2np1 + l1( 1113857lt 0

(A19)

12 Mathematical Problems in Engineering

us g1prime(q0) is negative in the region q0 isin [(q1(1 + n))

(q1(1 minus n))] g1(q0) decreases in q0 in the regionq0 isin [(q1(1 + n)) (q1(1 minus n))] Since

1113946q1q0

0xf1(x)dx 1 minus

p1 + w minus c0 1 + r3( 1113857

p1 + l1

st q0 isinq1

1 + n

q1

1 minus n1113876 1113877

(A20)

we obtain that r3 decreases in q0 in the region q0 isin [(q1(1 + n)) (q1(1 minus n))] Hence g1(q0) increases in r3 in theregion q0 isin [(q1(1 + n)) (q1(1 minus n))]

According to formulas (A3) and (A4) and the con-clusions of Proposition 1 when r1 isin [((w minus l1 minus c0)c0)((p1 + w minus c0)c0)] we have q0 isin [(q1(1 + n)) (q1(1 minus n))] Since we assume that the manufacturer makesdecisions based on the maximum expected profit there mustbe r10 isin [((w minus l1 minus c0)c0) ((p1 + w minus c0)c0)] which maxi-mizes the expected profit of the manufacturer under the FBSstrategy Similarly there must be min r10 r201113864 1113865 isin [((w minus l1 minus

c0)c0) ((p1 + w minus c0)c0)] which maximizes the expectedprofit of the manufacturer under the FBU strategy us wehave q0 isin [(q1(1 + n)) (q1(1 minus n))] when r3 isin [r0 minr10 r201113864 1113865] Since g1(q0) decreases in q0 in the region q0 isin

[(q1(1 + n)) (q1(1 minus n))] g1(q0) increases in r3 in theregion r3 isin [r0 min r10 r201113864 1113865] erefore g1(q0) attains themaximum at the point rlowast3 min r10 r201113864 1113865 We can obtainmax[q0c0(r3 minus r0)] q0c0(rlowast3 minus r0)

Proof of Proposition 7 If formula (20) attains the optimumat the point q0 qU

0 and r3 rU3 let q0 qU

0 r1 rU3 and

r2 rU3 in formula (18) Obviously they satisfy the con-

straint of formula (18) At this time the objective function offormula (18) is equal to the maximum of formula (20) whichmeans that the optimal of formula (20) must satisfy formula(18)erefore the maximum of formula (18) is not less thanthe maximum of formula (20)

Proof of Proposition 8

(1) When r10 le r20 we have rS1 r10 min r10 r201113864 1113865 rU

3according to Proposition 5 and rS

1 r10 us whenq1 D and r1 r3 it is obvious that q0 is equal in thetwo equations according to equations (3) and (11)Hence under qS

0 qU0 qS

1 qU1 and r1 r3 we can

obtain that EπS1 EπU

1 according to equations (2) and(12)When r10 le r20 we have rU

3 r10 le r20 rS2 accord-

ing to Proposition 5 and rS1 r10 Hence under

qS0 qU

0 qS1 qU

1 and r2 ge r3 we can obtain thatEπS

2 leEπU2 according to equations (8) and (15)

erefore when rS1 r10 and r10 le r20 we obtain that

EπS1 EπU

1 and EπS2 leEπU

2 which means the ex-pected profit of the manufacturer and retailer underthe FBU strategy is not less than the profit under theFBS strategy

(2) r10 gt r20

We know that r1 decreases in q0 in the regionq0 isin [(q1(1 + n)) (q1(1 minus n))] Since rS

1 r10 andr10 gt r20 we have q0 isin [qS

0 +infin) Substituting equa-tion (3) in (2) and taking the first-order derivative ofEπ1 with respect to q0 the following can be obtaineddEπ1

dq0

p1 + l1( 1113857q21c0q

30

fq1

q01113888 1113889 q0c0 minus kp1D( 1113857 (A21)

Since qS0c0 minus kp1Dgt 0 and q0 isin [qS

0 +infin) we have(dEπ1dq0)gt 0 When the supply chain is under theFBU strategy we have rU

3 min r10 r201113864 1113865 r20according to Proposition 6 and r10 gt r20us we haverU3 r20 lt r10 rS

1 and qU0 gt qS

0 erefore we canobtain that EπU

1 gtEπS1 when (dEπ1dq0)gt 0 and

qU0 gt qS

0 In other words we have EπU1 gtEπS

1 whenrS1 r10 qS

0c0 minus kp1D gt 0 and r10 gt r20

Taking the first-order and second-order derivatives ofEπ2 with respect to q0 the following can be obtained

dEπ2

dq0 p2 minus p1( 1113857 1113946

Dq0

0xf(x)dx minus w

d2Eπ2

dq20 minus p2 minus p1( 1113857

D2

q30f

D

q01113888 1113889lt 0

(A22)

When r1 r10 and r10 gt r20 we have rU3 lt rS

1 and qU0 gt qS

0Since (d2Eπ2dq20)lt 0 we obtain that (dEπ2dq0) decreasesin q0 When q0 isin [qS

0 qU0 ] we have

mindEπ2

dq0 p2 minus p1( 1113857 1113946

DqU0

0xf(x)dx minus w

maxdEπ2

dq0 p2 minus p1( 1113857 1113946

DqS0

0xf(x)dx minus w

(A23)

When (p2 minus p1) 1113938DqU

00 xf(x)dx minus wgt 0 and q0 isin [qS

0

qU0 ] we have (dEπ2dq0)gt 0 and EπU

2 gtEπS2 When (p2 minus

p1) 1113938DqS

00 xf(x)dx minus wlt 0 and q0 isin [qS

0 qU0 ] we have (dEπ2

dq0)lt 0 and EπU2 ltEπS

2In summary when (p2 minus p1) 1113938

DqU0

0 xf(x)dx minus wgt 0 andqS0c0 minus kp1Dgt 0 we have EπU

1 gtEπS1 and EπU

2 gtEπS2 When

(p2 minus p1) 1113938DqS

00 xf(x)dx minus wlt 0 and qS

0c0 minus kp1Dgt 0 wehave EπU

1 gtEπS1 and EπU

2 ltEπS2

Data Availability

e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

e authors declare that they have no conflicts of interest

Acknowledgments

is research was funded by the Ministry of Education ofHumanities and Social Science Project of China (grantnumber 17YJAZH018)

Mathematical Problems in Engineering 13

References

[1] S Asian and X Nie ldquoCoordination in supply chains withuncertain demand and disruption risks existence analysisand insightsrdquo IEEE Transactions on Systems Man and Cy-bernetics Systems vol 44 no 9 pp 1139ndash1154 2014

[2] A Roy S S Sana and K Chaudhuri ldquoOptimal Pricing ofcompeting retailers under uncertain demand-a two layersupply chain modelrdquo Annals of Operations Research vol 260no 1-2 pp 481ndash500 2015

[3] B C Giri and S Bardhan ldquoCoordinating a supply chain underuncertain demand and random yield in presence of supplydisruptionrdquo International Journal of Production Researchvol 53 no 16 pp 5070ndash5084 2015

[4] A Arreola-Risa and G A Decroix ldquoMake-to-order versusmake-to-stock in a production-inventory system with generalproduction timesrdquo IIE Transactions vol 30 no 8 pp 705ndash713 1998

[5] A Grosfeld-Nir and B Ronen ldquoA single bottleneck systemwith binomial yields and rigid demandrdquoManagement Sciencevol 39 no 5 pp 650ndash653 1993

[6] C A Yano and H L Lee ldquoLot sizing with random yields areviewrdquoOperations Research vol 43 no 2 pp 311ndash334 1995

[7] P Kouvelis and W Zhao ldquoFinancing the newsvendor sup-plier vs Bank and the structure of optimal trade creditcontractsrdquo Operations Research vol 60 no 3 pp 566ndash5802012

[8] B R Q Zhang ldquoInventory management with asset-basedfinancingrdquoManagement Science vol 50 no 9 pp 1274ndash12922004

[9] J Ma and L Xie ldquoe comparison and complex analysis ondual-channel supply chain under different channel powerstructures and uncertain demandrdquo Nonlinear Dynamicsvol 83 no 3 pp 1379ndash1393 2016

[10] X Xu and J R Birge ldquoJoint production and financing de-cisions modeling and analysisrdquo Operations Managementvol 28 no 12 2011

[11] P Kouvelis and W Zhao ldquoe newsvendor problem andprice-only contract when bankruptcy costs existrdquo Productionand Operations Management vol 20 no 6 pp 921ndash936 2010

[12] O Boyabath and L B Toktay ldquoStochastic capacity investmentand flexible vs Dedicated technology choice in imperfectcapital marketsrdquo Management Science vol 57 no 12pp 2163ndash2179 2011

[13] Y Alan and V Gaur ldquoOperational investment and capitalstructure under asset-based lendingrdquo Manufacturing amp Ser-vice Operations Management vol 20 no 4 2018

[14] B Sun H Zhang N Yan et al ldquoA partial credit guaranteecontract in a capital-constrained supply chain financingequilibrium and coordinating strategyrdquo International Journalof Production Economics vol 173 pp 122ndash133 2016

[15] G Bi Y Fei X Yuan et al ldquoJoint operational and financialcollaboration in a capital-constrained supply chain undermanufacturer collateralrdquo Asia-Pacific Journal of OperationalResearch vol 35 no 3 Article ID 1850010 2018

[16] J Chen Y W Zhou and Y Zhong ldquoA pricingorderingmodel for a dyadic supply chain with buyback guarantee fi-nancing and fairness concernsrdquo International Journal ofProduction Research vol 55 no 7 pp 1ndash18 2018

[17] G G Cai X Chen and Z Xiao ldquoe roles of bank and tradecredits theoretical analysis and empirical evidencerdquo Pro-duction and Operations Management vol 23 no 4pp 583ndash598 2014

[18] X F Chen ldquoA model of trade credit in a capital-constraineddistribution channelrdquo International Journal of ProductionEconomics vol 159 pp 347ndash357 2015

[19] J Chod ldquoInventory risk shifting and trade creditrdquo Man-agement Science vol 63 no 10 pp 3207ndash3225 2017

[20] B Jing X Chen and G Cai ldquoEquilibrium financing in adistribution channel with capital constraintrdquo Production andOperations Management vol 21 no 6 pp 1090ndash1101 2012

[21] S A Yang and J R Birge ldquoTrade credit risk sharing andinventory financing portfoliosrdquo Management Science vol 64no 8 2018

[22] J Zhou ldquoEssays in interfaces of operations and finance insupply chainsrdquo PhD thesis University of RochesterRochester NY USA 2009

[23] Y W Zhou B Cao Y Zhong et al ldquoOptimal advertisingordering policy and finance mode selection for a capital-constrained retailer with stochastic demandrdquo Journal of theOperational Research Society vol 68 no 12 pp 1620ndash16322017

[24] W Jin Q Zhang and J Luo ldquoNon-collaborative and col-laborative financing in a bilateral supply chain with capitalconstraintsrdquo Omega vol 88 pp 210ndash222 2018

[25] B Keren ldquoe single-period inventory problem extension torandom yield from the perspective of the supply chainrdquoOmega vol 37 no 4 pp 801ndash810 2009

[26] A Sufi ldquoBank lines of credit in corporate finance an empiricalanalysisrdquo Review of Financial Studies vol 22 no 3pp 1057ndash1088 2009

14 Mathematical Problems in Engineering

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Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

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Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

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Volume 2018

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Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

the FBU strategy is not less than the maximum expectedprot under the FBS strategy is conrms the rst con-clusion of Proposition 8 From the results of the data ofgroups 2 and 3 in Table 3 under rS1 r10 and r10 gt r20 bothsets of results indicate that the maximum expected prot ofthe manufacturer under the FBU strategy is greater than that

under the FBS strategy According to Figure 3 and group 2 inTable 3 the retailerrsquos maximum expected prot is largerunder the FBU strategy than the FBS strategy In contrast inFigure 4 and group 3 in Table 3 the conclusion is reversede reason is that the second set of data satises (p2 minus p1)intDq

U0

0 xf1(x)dx minus wgt 0 and the third set of data satises

2500 2600 2700 2800 2900 3000131

1315

132

1325

133

1335

134

1345

135

Expe

cted

pro

fit o

f man

ufac

ture

r und

er F

BS (E

π 1)

FBS (2762 134233167)

FBU (2765 134462212)

131

1315

132

1325

133

1335

134

1345

135

Expe

cted

pro

fit o

f man

ufac

ture

r und

er F

BU (E

π 1)

times104 times104

q0

(a)

q0

2000 2500 3000 3500 400023

235

24

245

25

255

26

FBS

FBU

2760 2780 28002566

2567

2568

FBS (2762 256669584)

FBU (2765 256688856)

times104

times104

Eπ2

(b)

Figure 3 Prot comparison of the supply chain enterprises under the two nancing strategies in group 2 in Table 3

Table 2 Supply chain parameter settings

Types c0 p1 p2 D l1 l2 w r0 k r10 r201 3 15 20 3000 10 14 4 002 02 01 0092 3 10 20 3000 10 14 4 002 02 01 0093 3 10 15 3000 10 14 4 002 02 01 0094 3 10 20 3000 10 14 4 002 02 009 015 3 10 15 3000 10 14 4 002 02 009 01

Table 3 Prots of enterprises and banks under the FBS strategy and FBU strategy

Types Financingstrategies

Optimalloan

interest rate

Optimalproductionand orderingdecisions

Bankrsquos expectedprot

Manufacturerrsquosexpected prot

Retailerrsquos expectedprot

Total expectedprot

q0 q1

1 FBS strategy r1 0062 3023 3000 814 26627 12266 38893r2 009FBU strategy r3 009 3013 3000 633 26626 12287 38913

2 FBS strategy r1 01 2762 3000 769 13423 25667 39090r2 009FBU strategy r3 009 2765 3000 581 13446 25669 39115

3 FBS strategy r1 01 2762 3000 769 13423 13039 26462r2 009FBU strategy r3 009 2765 3000 581 13446 13034 26480

4 FBS strategy r1 009 2765 3000 807 13446 25609 39055r2 01FBU strategy r3 009 2765 3000 581 13446 25669 39115

5 FBS strategy r1 009 2765 3000 807 13446 12974 26420r2 01FBU strategy r3 009 2765 3000 581 13446 13034 26480

Mathematical Problems in Engineering 9

(p2 minus p1)intDqS00 xf1(x)dx minus wlt 0 thus conrming the

second conclusion of Proposition 8

6 Conclusion

is paper studies the external nancing problems of thebilateral supply chain with nancial constraints It startswith taking the manufacturerrsquos planned output and theretailerrsquos planned order quantity as variables builds theexpected prot model of the manufacturer and the retailerunder the FBS strategy and FBU strategy and analyzes themanufacturerrsquos optimal planned output and the retailerrsquosoptimal order quantity under the two nancing strategiesen it states on the condition of symmetry of informationhow could the bank determine the loan interest rate tomaximize the expected loan prot when the bank hasobtained the information of the planned production andplanned order quantity of the manufacturer and the retailerunder the two nancing strategies At last it proposes themaximum expected loan prot model of the bank under theFBS strategy and FBU strategy to determine the optimalloan interest rate of the bank under the two nancingstrategies And it compares the prot of the bank and theprot of the supply chain under the two nancing strategies

e research shows the following (1) Under the two -nancing strategies the manufacturerrsquos optimal plannedoutput is inversely proportional to the loan interest rate andthe retailerrsquos optimal order quantity is the market demand (2)Under the FBS strategy it is more benecial for the bank toget more expected prot when it raises the loan interest ratesfor the retailer but it does not apply to the manufacturer (3)Under the FBU strategy when the bank sets the loan interestrate at the upper limit of the acceptable line the bankrsquos ex-pected loan prot is the largest but it is less than the bankrsquosexpected loan prot under the FBS strategy (4) e FBUstrategy is more protable for both the manufacturer and the

retailer when the bankrsquos loan interest rate to the manufactureris its acceptable maximum and the loan interest rate ac-ceptable by the manufacturer is less than that by the retailer

e above-mentioned conclusions supply a policyguiding the supply chain enterprises with nancial con-straints to make a choice of the nancing strategy For thebank (1) when the bilateral supply chain enterprises withuncertain output and certain demand choose the FBSstrategy the bank should give the retailer the maximum loaninterest rate and give the manufacturer a loan interest rateaccording to formula (18) (2) when they choose the FBUstrategy the bank should give the supply chain enterprisesthe largest unied loan interest rate and (3) when supplychain enterprises choose the FBS strategy the bank makemore prots so the bank should encourage them to conductthe FBS strategy as much as possible For supply chainenterprises the optimal loan strategies of nancing enter-prises under dierent conditions are dierent so theyshould reasonably choose the optimal nancing strategybased on the bankrsquos quotation and their own acceptable loaninterest rate

For the bilateral supply chain under the FBS and FBUnancing strategies all the nancing modes are distin-guished from the perspective of obtaining the bankrsquos loaninterest rates so there is no nancing strategy superior to theFBU strategy ere are various modes under the FBUstrategy and each has its own advantages such as the dealersand supplier network nancing modes and the bank logisticscooperative nancing modes erefore various nancingmodes within the FBU strategy should be studied and an-alyzed in the future

Appendix

Proof of Proposition 1 Taking the rst-order and second-order derivatives of Eπ1 with respect to q0 it follows that

2500 2600 2700 2800 2900 3000131

1315

132

1325

133

1335

134

1345

135Ex

pect

ed p

rofit

of m

anuf

actu

rer u

nder

FBS

(Eπ 1

)

FBS (2762 134233167)

FBU (2765 134462212)

131

1315

132

1325

133

1335

134

1345

135

Expe

cted

pro

fit o

f man

ufac

ture

r und

er F

BU (E

π 1)

times104 times104

q0

(a)

q0

2000 2200 2400 2600 2800 3000124

126

128

13

132

134

136

FBS

FBU

2760 2780 28001298

13

1302

1304(2762 130394792)

(2765 130344428)

times104

times104

Eπ2

(b)

Figure 4 Prot comparison of the supply chain enterprises under the two nancing strategies in group 3 in Table 3

10 Mathematical Problems in Engineering

dEπ1

dq0 p1 + l1( 1113857 1113946

q1q0

0xf(x)dx + w minus c0 1 + r1( 1113857 minus l1

(A1)

d2Eπ1dq20

minusq21q30

p1 + l1( 1113857fq1

q01113888 1113889le 0 (A2)

When (q1q0)gt 1 + n that is q0 lt (q1(1 + n)) we havef(q1q0) 0 us (d2Eπ1dq20) 0 We obtain

dEπ1

dq0 p1 + w minus c0 1 + r1( 1113857gt 0 (A3)

So when q0 lt (q1(1 + n)) Eπ1 is a monotonically in-creasing function with a fixed slope

When (q1q0)lt 1 minus n that is q0 ge (q1(1 minus n)) we havef(q1q0) 0 us (d2Eπ1dq20) 0 We obtain

dEπ1dq0

w minus c0 1 + r1( 1113857 minus l1 lt 0 (A4)

So when (q1(1 minus n))lt q0 Eπ1 is a monotonically de-creasing function with a fixed slope

When 1 minus nle (q1q0)le 1 + n we have (d2Eπ1dq20)lt 0erefore Eπ1 is a convex function about the variable q0dEπ1dq0 is a continuous function on q0 isin [0 +infin] Whenq0 (q1(1 + n)) we obtain (dEπ1dq0)gt 0 according to(A1) When q0 (q1(1 minus n)) we obtain (dEπ1dq0)lt 0according to (A1) Hence according to the intermediatevalue theorem there must exist q0 isin [(q1(1 + n)) (q1(1 minus

n))] satisfying (dEπ1dq0) 0 At this time q0 is the onlylocal maximum point of Eπ1

In summary when q0 satisfies (dEπ1dq0) 0 that is

1113946q1q0

0xf(x)dx 1 minus

p1 + w minus c0 1 + r1( 1113857

p1 + l1 (A5)

and q0 isin [0 +infin) q0 is the only absolute maximum point ofEπ1

Proof of Proposition 2 Taking the first-order derivative ofEπ2 with respect to q1 geD it follows that

dE π2

dq1

p2 minus p1( 1113857 1113946+infin

q1q0f(x)dx + w minus kp1r2 q1 ltD

minus l2 1113946+infin

q1q0f(x)dx + w minus kp1r2 q1 geD

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(A6)

(1) q1 geD

According to Proposition 1 we have (q1q0) isin[1 minus n 1 + n] Because of xf(x)ge 0 we can obtainthat 1113938

q1q00 xf(x)dx gt 0 Let u (q1q0)gt 0 and de-

fine the function

φ1(u) 1113946U

0xf(x)dx (A7)

e first-order derivative of φ1(u) follows that

φ1prime(u) uf(u)ge 0 (A8)

Because of u isin [1 minus n 1 + n] we have φ1prime(u)gt 0us φ1(u) increases in u that is 1113938

q1q00 xf(x)dx

increases in (q1q0) Hence there exists a unique(q1q0) satisfying

1113946q1q0

0xf(x)dx 1 minus

p1 + w minus c0 1 + r1( 1113857

p1 + l1 (A9)

At this time (q1q0) is a unique constant related tothe coefficients and f(x) Hence when q1 geD(dEπ2dq1) is a unique constant If (dEπ2dq1)gt 0the retailerrsquos planned order quantity will be infinitewhich contradicts the risk-neutral assumptionHence we have (dEπ2dq1)lt 0 erefore Eπ2decreases in q1 in the region q1 isin [D +infin] and at-tains the maximum at the point q1 D

(2) q1 ltD

Because (dEπ2dq1) is a unique constant if

dEπ2dq1

p2 minus p1( 1113857 1113946+infin

q1q0f(x)dx + w minus kp1r2 lt 0

(A10)

then Eπ2 decreases in q1 in the region q1 isin [0 D) andattains the local maximum at the point q1 0 usEπ2 decreases in q1 when q1 ge 0 Eπ2 attains theabsolute maximum at the point q1 0 At this pointthe supply chain is in a state of no order We will notdiscuss this situation

In fact according to the previous literature and theactual situation of the supply chain there is often w minus kp1r2gt 0 Hence when q1 ltD we can obtain (dEπ2dq1)gt 0 andthat Eπ2 increases in q1

In summary Eπ2 attains the maximum at the pointq1 D

Proof of Proposition 3 It follows the proof of Proposition 1so we omit it

Proof of Proposition 4 It follows the proof of Proposition 2so we omit it

Proof of Proposition 5

(1) Obviously nonlinear programming (18) can bedivided into the following two planning problems

Mathematical Problems in Engineering 11

max q0c0 minus kp1q1( 1113857 r1 minus r0( 1113857

st

1113946q1q0

0xf(x)dx 1 minus

p1 + w minus c0 1 + r1( 1113857

p1 + l1

q1 D r0 le r1 le r10

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(A11)

max kp1q1 r2 minus r0( 1113857

st r0 le r2 le r20(A12)

In linear programming (A12) the objective functionincreases in r2 in the region r2 isin [r0 r20] us weattain the optimum of equation (A12) at the pointr2 r20 It is also the optimum of nonlinear pro-gramming (18)

(2) In nonlinear programming (A11) let

m1 q0( 1113857 q0c0 minus kp1D( 1113857 r1 minus r0( 1113857

st 1113946q1q0

0xf(x)dx 1 minus

p1 + w minus c0 1 + r1( 1113857

p1 + l1

m2 r1( 1113857 q0c0 minus kp1D( 1113857 r1 minus r0( 1113857

st 1113946q1q0

0xf(x)dx 1 minus

p1 + w minus c0 1 + r1( 1113857

p1 + l1

(A13)

according to

1113946q1q0

0xf(x)dx 1 minus

p1 + w minus c0 1 + r1( 1113857

p1 + l1 (A14)

We obtain that r1 decreases in q0 us when (dm1dq0)gt 0 we have (dm2dr1)lt 0 When (dm1dq0)lt 0we have (dm2dr1)gt 0

Taking the first-order derivative of m1 with respect to q0it follows thatdm1

dq0 p1 + l1( 1113857

middot 1113946q1q0

0xf(x)dx minus

p1 + l1( 1113857 q0c0 minus kp1D( 1113857q21f q1q0( 1113857

c0q30

minus r0c0 minus l1 minus c0 + w

(A15)

Under different parameters of the supply chain formula(A15) can satisfy both (dm1dq0)gt 0 and (dm1dq0)lt 0erefore there is not always (dm2dr1)gt 0 that is theoptimal of r1 is not always r10 In numerical analysis we giveexamples to support this conclusion

Proof of Proposition 6 Let

g q0 r3( 1113857 q0c0 r3 minus r0( 1113857 (A16)

Since the relationship between q0 and r3 satisfies theconstraint of (20) equation (A16) can also express equation(A17) as follows

g1 q0( 1113857 q0 p1 + l1( 1113857 1113946q1q0

0xf(x)dx + w minus l1 minus c0 1 + r0( 11138571113890 1113891

(A17)

Taking the first-order and second-order derivatives ofg1(q0) with respect to q0 it follows that

g1prime q0( 1113857 p1 + l1( 1113857 1113946q1q0

0xf(x)dx + w minus l1 minus c0 1 + r0( 11138571113890 1113891

minusq21q20

fq1

q01113888 1113889 p1 + l1( 1113857

gPrime1 q0( 1113857 p1 + l1( 1113857q21q30

fq1

q01113888 1113889gt 0

q1

q0isin [1 minus n 1 + n]

(A18)

Obviously when (q1q0) isin [1 minus n 1 + n] that isq0 isin [(q1(1 + n)) (q1(1 minus n))] g1prime(q0) increases in q0According to the proof of Proposition 1 the manufacturerattains the optimum in the region q0 isin [(q1(1 + n)) (q1(1 minus n))] When q0 (q1(1 minus n)) we can obtain

g1prime q0( 1113857 w minus l1 minus c0 1 + r0( 1113857 minus(1 minus n)2

2np1 + l1( 1113857lt 0

(A19)

12 Mathematical Problems in Engineering

us g1prime(q0) is negative in the region q0 isin [(q1(1 + n))

(q1(1 minus n))] g1(q0) decreases in q0 in the regionq0 isin [(q1(1 + n)) (q1(1 minus n))] Since

1113946q1q0

0xf1(x)dx 1 minus

p1 + w minus c0 1 + r3( 1113857

p1 + l1

st q0 isinq1

1 + n

q1

1 minus n1113876 1113877

(A20)

we obtain that r3 decreases in q0 in the region q0 isin [(q1(1 + n)) (q1(1 minus n))] Hence g1(q0) increases in r3 in theregion q0 isin [(q1(1 + n)) (q1(1 minus n))]

According to formulas (A3) and (A4) and the con-clusions of Proposition 1 when r1 isin [((w minus l1 minus c0)c0)((p1 + w minus c0)c0)] we have q0 isin [(q1(1 + n)) (q1(1 minus n))] Since we assume that the manufacturer makesdecisions based on the maximum expected profit there mustbe r10 isin [((w minus l1 minus c0)c0) ((p1 + w minus c0)c0)] which maxi-mizes the expected profit of the manufacturer under the FBSstrategy Similarly there must be min r10 r201113864 1113865 isin [((w minus l1 minus

c0)c0) ((p1 + w minus c0)c0)] which maximizes the expectedprofit of the manufacturer under the FBU strategy us wehave q0 isin [(q1(1 + n)) (q1(1 minus n))] when r3 isin [r0 minr10 r201113864 1113865] Since g1(q0) decreases in q0 in the region q0 isin

[(q1(1 + n)) (q1(1 minus n))] g1(q0) increases in r3 in theregion r3 isin [r0 min r10 r201113864 1113865] erefore g1(q0) attains themaximum at the point rlowast3 min r10 r201113864 1113865 We can obtainmax[q0c0(r3 minus r0)] q0c0(rlowast3 minus r0)

Proof of Proposition 7 If formula (20) attains the optimumat the point q0 qU

0 and r3 rU3 let q0 qU

0 r1 rU3 and

r2 rU3 in formula (18) Obviously they satisfy the con-

straint of formula (18) At this time the objective function offormula (18) is equal to the maximum of formula (20) whichmeans that the optimal of formula (20) must satisfy formula(18)erefore the maximum of formula (18) is not less thanthe maximum of formula (20)

Proof of Proposition 8

(1) When r10 le r20 we have rS1 r10 min r10 r201113864 1113865 rU

3according to Proposition 5 and rS

1 r10 us whenq1 D and r1 r3 it is obvious that q0 is equal in thetwo equations according to equations (3) and (11)Hence under qS

0 qU0 qS

1 qU1 and r1 r3 we can

obtain that EπS1 EπU

1 according to equations (2) and(12)When r10 le r20 we have rU

3 r10 le r20 rS2 accord-

ing to Proposition 5 and rS1 r10 Hence under

qS0 qU

0 qS1 qU

1 and r2 ge r3 we can obtain thatEπS

2 leEπU2 according to equations (8) and (15)

erefore when rS1 r10 and r10 le r20 we obtain that

EπS1 EπU

1 and EπS2 leEπU

2 which means the ex-pected profit of the manufacturer and retailer underthe FBU strategy is not less than the profit under theFBS strategy

(2) r10 gt r20

We know that r1 decreases in q0 in the regionq0 isin [(q1(1 + n)) (q1(1 minus n))] Since rS

1 r10 andr10 gt r20 we have q0 isin [qS

0 +infin) Substituting equa-tion (3) in (2) and taking the first-order derivative ofEπ1 with respect to q0 the following can be obtaineddEπ1

dq0

p1 + l1( 1113857q21c0q

30

fq1

q01113888 1113889 q0c0 minus kp1D( 1113857 (A21)

Since qS0c0 minus kp1Dgt 0 and q0 isin [qS

0 +infin) we have(dEπ1dq0)gt 0 When the supply chain is under theFBU strategy we have rU

3 min r10 r201113864 1113865 r20according to Proposition 6 and r10 gt r20us we haverU3 r20 lt r10 rS

1 and qU0 gt qS

0 erefore we canobtain that EπU

1 gtEπS1 when (dEπ1dq0)gt 0 and

qU0 gt qS

0 In other words we have EπU1 gtEπS

1 whenrS1 r10 qS

0c0 minus kp1D gt 0 and r10 gt r20

Taking the first-order and second-order derivatives ofEπ2 with respect to q0 the following can be obtained

dEπ2

dq0 p2 minus p1( 1113857 1113946

Dq0

0xf(x)dx minus w

d2Eπ2

dq20 minus p2 minus p1( 1113857

D2

q30f

D

q01113888 1113889lt 0

(A22)

When r1 r10 and r10 gt r20 we have rU3 lt rS

1 and qU0 gt qS

0Since (d2Eπ2dq20)lt 0 we obtain that (dEπ2dq0) decreasesin q0 When q0 isin [qS

0 qU0 ] we have

mindEπ2

dq0 p2 minus p1( 1113857 1113946

DqU0

0xf(x)dx minus w

maxdEπ2

dq0 p2 minus p1( 1113857 1113946

DqS0

0xf(x)dx minus w

(A23)

When (p2 minus p1) 1113938DqU

00 xf(x)dx minus wgt 0 and q0 isin [qS

0

qU0 ] we have (dEπ2dq0)gt 0 and EπU

2 gtEπS2 When (p2 minus

p1) 1113938DqS

00 xf(x)dx minus wlt 0 and q0 isin [qS

0 qU0 ] we have (dEπ2

dq0)lt 0 and EπU2 ltEπS

2In summary when (p2 minus p1) 1113938

DqU0

0 xf(x)dx minus wgt 0 andqS0c0 minus kp1Dgt 0 we have EπU

1 gtEπS1 and EπU

2 gtEπS2 When

(p2 minus p1) 1113938DqS

00 xf(x)dx minus wlt 0 and qS

0c0 minus kp1Dgt 0 wehave EπU

1 gtEπS1 and EπU

2 ltEπS2

Data Availability

e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

e authors declare that they have no conflicts of interest

Acknowledgments

is research was funded by the Ministry of Education ofHumanities and Social Science Project of China (grantnumber 17YJAZH018)

Mathematical Problems in Engineering 13

References

[1] S Asian and X Nie ldquoCoordination in supply chains withuncertain demand and disruption risks existence analysisand insightsrdquo IEEE Transactions on Systems Man and Cy-bernetics Systems vol 44 no 9 pp 1139ndash1154 2014

[2] A Roy S S Sana and K Chaudhuri ldquoOptimal Pricing ofcompeting retailers under uncertain demand-a two layersupply chain modelrdquo Annals of Operations Research vol 260no 1-2 pp 481ndash500 2015

[3] B C Giri and S Bardhan ldquoCoordinating a supply chain underuncertain demand and random yield in presence of supplydisruptionrdquo International Journal of Production Researchvol 53 no 16 pp 5070ndash5084 2015

[4] A Arreola-Risa and G A Decroix ldquoMake-to-order versusmake-to-stock in a production-inventory system with generalproduction timesrdquo IIE Transactions vol 30 no 8 pp 705ndash713 1998

[5] A Grosfeld-Nir and B Ronen ldquoA single bottleneck systemwith binomial yields and rigid demandrdquoManagement Sciencevol 39 no 5 pp 650ndash653 1993

[6] C A Yano and H L Lee ldquoLot sizing with random yields areviewrdquoOperations Research vol 43 no 2 pp 311ndash334 1995

[7] P Kouvelis and W Zhao ldquoFinancing the newsvendor sup-plier vs Bank and the structure of optimal trade creditcontractsrdquo Operations Research vol 60 no 3 pp 566ndash5802012

[8] B R Q Zhang ldquoInventory management with asset-basedfinancingrdquoManagement Science vol 50 no 9 pp 1274ndash12922004

[9] J Ma and L Xie ldquoe comparison and complex analysis ondual-channel supply chain under different channel powerstructures and uncertain demandrdquo Nonlinear Dynamicsvol 83 no 3 pp 1379ndash1393 2016

[10] X Xu and J R Birge ldquoJoint production and financing de-cisions modeling and analysisrdquo Operations Managementvol 28 no 12 2011

[11] P Kouvelis and W Zhao ldquoe newsvendor problem andprice-only contract when bankruptcy costs existrdquo Productionand Operations Management vol 20 no 6 pp 921ndash936 2010

[12] O Boyabath and L B Toktay ldquoStochastic capacity investmentand flexible vs Dedicated technology choice in imperfectcapital marketsrdquo Management Science vol 57 no 12pp 2163ndash2179 2011

[13] Y Alan and V Gaur ldquoOperational investment and capitalstructure under asset-based lendingrdquo Manufacturing amp Ser-vice Operations Management vol 20 no 4 2018

[14] B Sun H Zhang N Yan et al ldquoA partial credit guaranteecontract in a capital-constrained supply chain financingequilibrium and coordinating strategyrdquo International Journalof Production Economics vol 173 pp 122ndash133 2016

[15] G Bi Y Fei X Yuan et al ldquoJoint operational and financialcollaboration in a capital-constrained supply chain undermanufacturer collateralrdquo Asia-Pacific Journal of OperationalResearch vol 35 no 3 Article ID 1850010 2018

[16] J Chen Y W Zhou and Y Zhong ldquoA pricingorderingmodel for a dyadic supply chain with buyback guarantee fi-nancing and fairness concernsrdquo International Journal ofProduction Research vol 55 no 7 pp 1ndash18 2018

[17] G G Cai X Chen and Z Xiao ldquoe roles of bank and tradecredits theoretical analysis and empirical evidencerdquo Pro-duction and Operations Management vol 23 no 4pp 583ndash598 2014

[18] X F Chen ldquoA model of trade credit in a capital-constraineddistribution channelrdquo International Journal of ProductionEconomics vol 159 pp 347ndash357 2015

[19] J Chod ldquoInventory risk shifting and trade creditrdquo Man-agement Science vol 63 no 10 pp 3207ndash3225 2017

[20] B Jing X Chen and G Cai ldquoEquilibrium financing in adistribution channel with capital constraintrdquo Production andOperations Management vol 21 no 6 pp 1090ndash1101 2012

[21] S A Yang and J R Birge ldquoTrade credit risk sharing andinventory financing portfoliosrdquo Management Science vol 64no 8 2018

[22] J Zhou ldquoEssays in interfaces of operations and finance insupply chainsrdquo PhD thesis University of RochesterRochester NY USA 2009

[23] Y W Zhou B Cao Y Zhong et al ldquoOptimal advertisingordering policy and finance mode selection for a capital-constrained retailer with stochastic demandrdquo Journal of theOperational Research Society vol 68 no 12 pp 1620ndash16322017

[24] W Jin Q Zhang and J Luo ldquoNon-collaborative and col-laborative financing in a bilateral supply chain with capitalconstraintsrdquo Omega vol 88 pp 210ndash222 2018

[25] B Keren ldquoe single-period inventory problem extension torandom yield from the perspective of the supply chainrdquoOmega vol 37 no 4 pp 801ndash810 2009

[26] A Sufi ldquoBank lines of credit in corporate finance an empiricalanalysisrdquo Review of Financial Studies vol 22 no 3pp 1057ndash1088 2009

14 Mathematical Problems in Engineering

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

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Dierential EquationsInternational Journal of

Volume 2018

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AnalysisInternational Journal of

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Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

(p2 minus p1)intDqS00 xf1(x)dx minus wlt 0 thus conrming the

second conclusion of Proposition 8

6 Conclusion

is paper studies the external nancing problems of thebilateral supply chain with nancial constraints It startswith taking the manufacturerrsquos planned output and theretailerrsquos planned order quantity as variables builds theexpected prot model of the manufacturer and the retailerunder the FBS strategy and FBU strategy and analyzes themanufacturerrsquos optimal planned output and the retailerrsquosoptimal order quantity under the two nancing strategiesen it states on the condition of symmetry of informationhow could the bank determine the loan interest rate tomaximize the expected loan prot when the bank hasobtained the information of the planned production andplanned order quantity of the manufacturer and the retailerunder the two nancing strategies At last it proposes themaximum expected loan prot model of the bank under theFBS strategy and FBU strategy to determine the optimalloan interest rate of the bank under the two nancingstrategies And it compares the prot of the bank and theprot of the supply chain under the two nancing strategies

e research shows the following (1) Under the two -nancing strategies the manufacturerrsquos optimal plannedoutput is inversely proportional to the loan interest rate andthe retailerrsquos optimal order quantity is the market demand (2)Under the FBS strategy it is more benecial for the bank toget more expected prot when it raises the loan interest ratesfor the retailer but it does not apply to the manufacturer (3)Under the FBU strategy when the bank sets the loan interestrate at the upper limit of the acceptable line the bankrsquos ex-pected loan prot is the largest but it is less than the bankrsquosexpected loan prot under the FBS strategy (4) e FBUstrategy is more protable for both the manufacturer and the

retailer when the bankrsquos loan interest rate to the manufactureris its acceptable maximum and the loan interest rate ac-ceptable by the manufacturer is less than that by the retailer

e above-mentioned conclusions supply a policyguiding the supply chain enterprises with nancial con-straints to make a choice of the nancing strategy For thebank (1) when the bilateral supply chain enterprises withuncertain output and certain demand choose the FBSstrategy the bank should give the retailer the maximum loaninterest rate and give the manufacturer a loan interest rateaccording to formula (18) (2) when they choose the FBUstrategy the bank should give the supply chain enterprisesthe largest unied loan interest rate and (3) when supplychain enterprises choose the FBS strategy the bank makemore prots so the bank should encourage them to conductthe FBS strategy as much as possible For supply chainenterprises the optimal loan strategies of nancing enter-prises under dierent conditions are dierent so theyshould reasonably choose the optimal nancing strategybased on the bankrsquos quotation and their own acceptable loaninterest rate

For the bilateral supply chain under the FBS and FBUnancing strategies all the nancing modes are distin-guished from the perspective of obtaining the bankrsquos loaninterest rates so there is no nancing strategy superior to theFBU strategy ere are various modes under the FBUstrategy and each has its own advantages such as the dealersand supplier network nancing modes and the bank logisticscooperative nancing modes erefore various nancingmodes within the FBU strategy should be studied and an-alyzed in the future

Appendix

Proof of Proposition 1 Taking the rst-order and second-order derivatives of Eπ1 with respect to q0 it follows that

2500 2600 2700 2800 2900 3000131

1315

132

1325

133

1335

134

1345

135Ex

pect

ed p

rofit

of m

anuf

actu

rer u

nder

FBS

(Eπ 1

)

FBS (2762 134233167)

FBU (2765 134462212)

131

1315

132

1325

133

1335

134

1345

135

Expe

cted

pro

fit o

f man

ufac

ture

r und

er F

BU (E

π 1)

times104 times104

q0

(a)

q0

2000 2200 2400 2600 2800 3000124

126

128

13

132

134

136

FBS

FBU

2760 2780 28001298

13

1302

1304(2762 130394792)

(2765 130344428)

times104

times104

Eπ2

(b)

Figure 4 Prot comparison of the supply chain enterprises under the two nancing strategies in group 3 in Table 3

10 Mathematical Problems in Engineering

dEπ1

dq0 p1 + l1( 1113857 1113946

q1q0

0xf(x)dx + w minus c0 1 + r1( 1113857 minus l1

(A1)

d2Eπ1dq20

minusq21q30

p1 + l1( 1113857fq1

q01113888 1113889le 0 (A2)

When (q1q0)gt 1 + n that is q0 lt (q1(1 + n)) we havef(q1q0) 0 us (d2Eπ1dq20) 0 We obtain

dEπ1

dq0 p1 + w minus c0 1 + r1( 1113857gt 0 (A3)

So when q0 lt (q1(1 + n)) Eπ1 is a monotonically in-creasing function with a fixed slope

When (q1q0)lt 1 minus n that is q0 ge (q1(1 minus n)) we havef(q1q0) 0 us (d2Eπ1dq20) 0 We obtain

dEπ1dq0

w minus c0 1 + r1( 1113857 minus l1 lt 0 (A4)

So when (q1(1 minus n))lt q0 Eπ1 is a monotonically de-creasing function with a fixed slope

When 1 minus nle (q1q0)le 1 + n we have (d2Eπ1dq20)lt 0erefore Eπ1 is a convex function about the variable q0dEπ1dq0 is a continuous function on q0 isin [0 +infin] Whenq0 (q1(1 + n)) we obtain (dEπ1dq0)gt 0 according to(A1) When q0 (q1(1 minus n)) we obtain (dEπ1dq0)lt 0according to (A1) Hence according to the intermediatevalue theorem there must exist q0 isin [(q1(1 + n)) (q1(1 minus

n))] satisfying (dEπ1dq0) 0 At this time q0 is the onlylocal maximum point of Eπ1

In summary when q0 satisfies (dEπ1dq0) 0 that is

1113946q1q0

0xf(x)dx 1 minus

p1 + w minus c0 1 + r1( 1113857

p1 + l1 (A5)

and q0 isin [0 +infin) q0 is the only absolute maximum point ofEπ1

Proof of Proposition 2 Taking the first-order derivative ofEπ2 with respect to q1 geD it follows that

dE π2

dq1

p2 minus p1( 1113857 1113946+infin

q1q0f(x)dx + w minus kp1r2 q1 ltD

minus l2 1113946+infin

q1q0f(x)dx + w minus kp1r2 q1 geD

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(A6)

(1) q1 geD

According to Proposition 1 we have (q1q0) isin[1 minus n 1 + n] Because of xf(x)ge 0 we can obtainthat 1113938

q1q00 xf(x)dx gt 0 Let u (q1q0)gt 0 and de-

fine the function

φ1(u) 1113946U

0xf(x)dx (A7)

e first-order derivative of φ1(u) follows that

φ1prime(u) uf(u)ge 0 (A8)

Because of u isin [1 minus n 1 + n] we have φ1prime(u)gt 0us φ1(u) increases in u that is 1113938

q1q00 xf(x)dx

increases in (q1q0) Hence there exists a unique(q1q0) satisfying

1113946q1q0

0xf(x)dx 1 minus

p1 + w minus c0 1 + r1( 1113857

p1 + l1 (A9)

At this time (q1q0) is a unique constant related tothe coefficients and f(x) Hence when q1 geD(dEπ2dq1) is a unique constant If (dEπ2dq1)gt 0the retailerrsquos planned order quantity will be infinitewhich contradicts the risk-neutral assumptionHence we have (dEπ2dq1)lt 0 erefore Eπ2decreases in q1 in the region q1 isin [D +infin] and at-tains the maximum at the point q1 D

(2) q1 ltD

Because (dEπ2dq1) is a unique constant if

dEπ2dq1

p2 minus p1( 1113857 1113946+infin

q1q0f(x)dx + w minus kp1r2 lt 0

(A10)

then Eπ2 decreases in q1 in the region q1 isin [0 D) andattains the local maximum at the point q1 0 usEπ2 decreases in q1 when q1 ge 0 Eπ2 attains theabsolute maximum at the point q1 0 At this pointthe supply chain is in a state of no order We will notdiscuss this situation

In fact according to the previous literature and theactual situation of the supply chain there is often w minus kp1r2gt 0 Hence when q1 ltD we can obtain (dEπ2dq1)gt 0 andthat Eπ2 increases in q1

In summary Eπ2 attains the maximum at the pointq1 D

Proof of Proposition 3 It follows the proof of Proposition 1so we omit it

Proof of Proposition 4 It follows the proof of Proposition 2so we omit it

Proof of Proposition 5

(1) Obviously nonlinear programming (18) can bedivided into the following two planning problems

Mathematical Problems in Engineering 11

max q0c0 minus kp1q1( 1113857 r1 minus r0( 1113857

st

1113946q1q0

0xf(x)dx 1 minus

p1 + w minus c0 1 + r1( 1113857

p1 + l1

q1 D r0 le r1 le r10

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(A11)

max kp1q1 r2 minus r0( 1113857

st r0 le r2 le r20(A12)

In linear programming (A12) the objective functionincreases in r2 in the region r2 isin [r0 r20] us weattain the optimum of equation (A12) at the pointr2 r20 It is also the optimum of nonlinear pro-gramming (18)

(2) In nonlinear programming (A11) let

m1 q0( 1113857 q0c0 minus kp1D( 1113857 r1 minus r0( 1113857

st 1113946q1q0

0xf(x)dx 1 minus

p1 + w minus c0 1 + r1( 1113857

p1 + l1

m2 r1( 1113857 q0c0 minus kp1D( 1113857 r1 minus r0( 1113857

st 1113946q1q0

0xf(x)dx 1 minus

p1 + w minus c0 1 + r1( 1113857

p1 + l1

(A13)

according to

1113946q1q0

0xf(x)dx 1 minus

p1 + w minus c0 1 + r1( 1113857

p1 + l1 (A14)

We obtain that r1 decreases in q0 us when (dm1dq0)gt 0 we have (dm2dr1)lt 0 When (dm1dq0)lt 0we have (dm2dr1)gt 0

Taking the first-order derivative of m1 with respect to q0it follows thatdm1

dq0 p1 + l1( 1113857

middot 1113946q1q0

0xf(x)dx minus

p1 + l1( 1113857 q0c0 minus kp1D( 1113857q21f q1q0( 1113857

c0q30

minus r0c0 minus l1 minus c0 + w

(A15)

Under different parameters of the supply chain formula(A15) can satisfy both (dm1dq0)gt 0 and (dm1dq0)lt 0erefore there is not always (dm2dr1)gt 0 that is theoptimal of r1 is not always r10 In numerical analysis we giveexamples to support this conclusion

Proof of Proposition 6 Let

g q0 r3( 1113857 q0c0 r3 minus r0( 1113857 (A16)

Since the relationship between q0 and r3 satisfies theconstraint of (20) equation (A16) can also express equation(A17) as follows

g1 q0( 1113857 q0 p1 + l1( 1113857 1113946q1q0

0xf(x)dx + w minus l1 minus c0 1 + r0( 11138571113890 1113891

(A17)

Taking the first-order and second-order derivatives ofg1(q0) with respect to q0 it follows that

g1prime q0( 1113857 p1 + l1( 1113857 1113946q1q0

0xf(x)dx + w minus l1 minus c0 1 + r0( 11138571113890 1113891

minusq21q20

fq1

q01113888 1113889 p1 + l1( 1113857

gPrime1 q0( 1113857 p1 + l1( 1113857q21q30

fq1

q01113888 1113889gt 0

q1

q0isin [1 minus n 1 + n]

(A18)

Obviously when (q1q0) isin [1 minus n 1 + n] that isq0 isin [(q1(1 + n)) (q1(1 minus n))] g1prime(q0) increases in q0According to the proof of Proposition 1 the manufacturerattains the optimum in the region q0 isin [(q1(1 + n)) (q1(1 minus n))] When q0 (q1(1 minus n)) we can obtain

g1prime q0( 1113857 w minus l1 minus c0 1 + r0( 1113857 minus(1 minus n)2

2np1 + l1( 1113857lt 0

(A19)

12 Mathematical Problems in Engineering

us g1prime(q0) is negative in the region q0 isin [(q1(1 + n))

(q1(1 minus n))] g1(q0) decreases in q0 in the regionq0 isin [(q1(1 + n)) (q1(1 minus n))] Since

1113946q1q0

0xf1(x)dx 1 minus

p1 + w minus c0 1 + r3( 1113857

p1 + l1

st q0 isinq1

1 + n

q1

1 minus n1113876 1113877

(A20)

we obtain that r3 decreases in q0 in the region q0 isin [(q1(1 + n)) (q1(1 minus n))] Hence g1(q0) increases in r3 in theregion q0 isin [(q1(1 + n)) (q1(1 minus n))]

According to formulas (A3) and (A4) and the con-clusions of Proposition 1 when r1 isin [((w minus l1 minus c0)c0)((p1 + w minus c0)c0)] we have q0 isin [(q1(1 + n)) (q1(1 minus n))] Since we assume that the manufacturer makesdecisions based on the maximum expected profit there mustbe r10 isin [((w minus l1 minus c0)c0) ((p1 + w minus c0)c0)] which maxi-mizes the expected profit of the manufacturer under the FBSstrategy Similarly there must be min r10 r201113864 1113865 isin [((w minus l1 minus

c0)c0) ((p1 + w minus c0)c0)] which maximizes the expectedprofit of the manufacturer under the FBU strategy us wehave q0 isin [(q1(1 + n)) (q1(1 minus n))] when r3 isin [r0 minr10 r201113864 1113865] Since g1(q0) decreases in q0 in the region q0 isin

[(q1(1 + n)) (q1(1 minus n))] g1(q0) increases in r3 in theregion r3 isin [r0 min r10 r201113864 1113865] erefore g1(q0) attains themaximum at the point rlowast3 min r10 r201113864 1113865 We can obtainmax[q0c0(r3 minus r0)] q0c0(rlowast3 minus r0)

Proof of Proposition 7 If formula (20) attains the optimumat the point q0 qU

0 and r3 rU3 let q0 qU

0 r1 rU3 and

r2 rU3 in formula (18) Obviously they satisfy the con-

straint of formula (18) At this time the objective function offormula (18) is equal to the maximum of formula (20) whichmeans that the optimal of formula (20) must satisfy formula(18)erefore the maximum of formula (18) is not less thanthe maximum of formula (20)

Proof of Proposition 8

(1) When r10 le r20 we have rS1 r10 min r10 r201113864 1113865 rU

3according to Proposition 5 and rS

1 r10 us whenq1 D and r1 r3 it is obvious that q0 is equal in thetwo equations according to equations (3) and (11)Hence under qS

0 qU0 qS

1 qU1 and r1 r3 we can

obtain that EπS1 EπU

1 according to equations (2) and(12)When r10 le r20 we have rU

3 r10 le r20 rS2 accord-

ing to Proposition 5 and rS1 r10 Hence under

qS0 qU

0 qS1 qU

1 and r2 ge r3 we can obtain thatEπS

2 leEπU2 according to equations (8) and (15)

erefore when rS1 r10 and r10 le r20 we obtain that

EπS1 EπU

1 and EπS2 leEπU

2 which means the ex-pected profit of the manufacturer and retailer underthe FBU strategy is not less than the profit under theFBS strategy

(2) r10 gt r20

We know that r1 decreases in q0 in the regionq0 isin [(q1(1 + n)) (q1(1 minus n))] Since rS

1 r10 andr10 gt r20 we have q0 isin [qS

0 +infin) Substituting equa-tion (3) in (2) and taking the first-order derivative ofEπ1 with respect to q0 the following can be obtaineddEπ1

dq0

p1 + l1( 1113857q21c0q

30

fq1

q01113888 1113889 q0c0 minus kp1D( 1113857 (A21)

Since qS0c0 minus kp1Dgt 0 and q0 isin [qS

0 +infin) we have(dEπ1dq0)gt 0 When the supply chain is under theFBU strategy we have rU

3 min r10 r201113864 1113865 r20according to Proposition 6 and r10 gt r20us we haverU3 r20 lt r10 rS

1 and qU0 gt qS

0 erefore we canobtain that EπU

1 gtEπS1 when (dEπ1dq0)gt 0 and

qU0 gt qS

0 In other words we have EπU1 gtEπS

1 whenrS1 r10 qS

0c0 minus kp1D gt 0 and r10 gt r20

Taking the first-order and second-order derivatives ofEπ2 with respect to q0 the following can be obtained

dEπ2

dq0 p2 minus p1( 1113857 1113946

Dq0

0xf(x)dx minus w

d2Eπ2

dq20 minus p2 minus p1( 1113857

D2

q30f

D

q01113888 1113889lt 0

(A22)

When r1 r10 and r10 gt r20 we have rU3 lt rS

1 and qU0 gt qS

0Since (d2Eπ2dq20)lt 0 we obtain that (dEπ2dq0) decreasesin q0 When q0 isin [qS

0 qU0 ] we have

mindEπ2

dq0 p2 minus p1( 1113857 1113946

DqU0

0xf(x)dx minus w

maxdEπ2

dq0 p2 minus p1( 1113857 1113946

DqS0

0xf(x)dx minus w

(A23)

When (p2 minus p1) 1113938DqU

00 xf(x)dx minus wgt 0 and q0 isin [qS

0

qU0 ] we have (dEπ2dq0)gt 0 and EπU

2 gtEπS2 When (p2 minus

p1) 1113938DqS

00 xf(x)dx minus wlt 0 and q0 isin [qS

0 qU0 ] we have (dEπ2

dq0)lt 0 and EπU2 ltEπS

2In summary when (p2 minus p1) 1113938

DqU0

0 xf(x)dx minus wgt 0 andqS0c0 minus kp1Dgt 0 we have EπU

1 gtEπS1 and EπU

2 gtEπS2 When

(p2 minus p1) 1113938DqS

00 xf(x)dx minus wlt 0 and qS

0c0 minus kp1Dgt 0 wehave EπU

1 gtEπS1 and EπU

2 ltEπS2

Data Availability

e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

e authors declare that they have no conflicts of interest

Acknowledgments

is research was funded by the Ministry of Education ofHumanities and Social Science Project of China (grantnumber 17YJAZH018)

Mathematical Problems in Engineering 13

References

[1] S Asian and X Nie ldquoCoordination in supply chains withuncertain demand and disruption risks existence analysisand insightsrdquo IEEE Transactions on Systems Man and Cy-bernetics Systems vol 44 no 9 pp 1139ndash1154 2014

[2] A Roy S S Sana and K Chaudhuri ldquoOptimal Pricing ofcompeting retailers under uncertain demand-a two layersupply chain modelrdquo Annals of Operations Research vol 260no 1-2 pp 481ndash500 2015

[3] B C Giri and S Bardhan ldquoCoordinating a supply chain underuncertain demand and random yield in presence of supplydisruptionrdquo International Journal of Production Researchvol 53 no 16 pp 5070ndash5084 2015

[4] A Arreola-Risa and G A Decroix ldquoMake-to-order versusmake-to-stock in a production-inventory system with generalproduction timesrdquo IIE Transactions vol 30 no 8 pp 705ndash713 1998

[5] A Grosfeld-Nir and B Ronen ldquoA single bottleneck systemwith binomial yields and rigid demandrdquoManagement Sciencevol 39 no 5 pp 650ndash653 1993

[6] C A Yano and H L Lee ldquoLot sizing with random yields areviewrdquoOperations Research vol 43 no 2 pp 311ndash334 1995

[7] P Kouvelis and W Zhao ldquoFinancing the newsvendor sup-plier vs Bank and the structure of optimal trade creditcontractsrdquo Operations Research vol 60 no 3 pp 566ndash5802012

[8] B R Q Zhang ldquoInventory management with asset-basedfinancingrdquoManagement Science vol 50 no 9 pp 1274ndash12922004

[9] J Ma and L Xie ldquoe comparison and complex analysis ondual-channel supply chain under different channel powerstructures and uncertain demandrdquo Nonlinear Dynamicsvol 83 no 3 pp 1379ndash1393 2016

[10] X Xu and J R Birge ldquoJoint production and financing de-cisions modeling and analysisrdquo Operations Managementvol 28 no 12 2011

[11] P Kouvelis and W Zhao ldquoe newsvendor problem andprice-only contract when bankruptcy costs existrdquo Productionand Operations Management vol 20 no 6 pp 921ndash936 2010

[12] O Boyabath and L B Toktay ldquoStochastic capacity investmentand flexible vs Dedicated technology choice in imperfectcapital marketsrdquo Management Science vol 57 no 12pp 2163ndash2179 2011

[13] Y Alan and V Gaur ldquoOperational investment and capitalstructure under asset-based lendingrdquo Manufacturing amp Ser-vice Operations Management vol 20 no 4 2018

[14] B Sun H Zhang N Yan et al ldquoA partial credit guaranteecontract in a capital-constrained supply chain financingequilibrium and coordinating strategyrdquo International Journalof Production Economics vol 173 pp 122ndash133 2016

[15] G Bi Y Fei X Yuan et al ldquoJoint operational and financialcollaboration in a capital-constrained supply chain undermanufacturer collateralrdquo Asia-Pacific Journal of OperationalResearch vol 35 no 3 Article ID 1850010 2018

[16] J Chen Y W Zhou and Y Zhong ldquoA pricingorderingmodel for a dyadic supply chain with buyback guarantee fi-nancing and fairness concernsrdquo International Journal ofProduction Research vol 55 no 7 pp 1ndash18 2018

[17] G G Cai X Chen and Z Xiao ldquoe roles of bank and tradecredits theoretical analysis and empirical evidencerdquo Pro-duction and Operations Management vol 23 no 4pp 583ndash598 2014

[18] X F Chen ldquoA model of trade credit in a capital-constraineddistribution channelrdquo International Journal of ProductionEconomics vol 159 pp 347ndash357 2015

[19] J Chod ldquoInventory risk shifting and trade creditrdquo Man-agement Science vol 63 no 10 pp 3207ndash3225 2017

[20] B Jing X Chen and G Cai ldquoEquilibrium financing in adistribution channel with capital constraintrdquo Production andOperations Management vol 21 no 6 pp 1090ndash1101 2012

[21] S A Yang and J R Birge ldquoTrade credit risk sharing andinventory financing portfoliosrdquo Management Science vol 64no 8 2018

[22] J Zhou ldquoEssays in interfaces of operations and finance insupply chainsrdquo PhD thesis University of RochesterRochester NY USA 2009

[23] Y W Zhou B Cao Y Zhong et al ldquoOptimal advertisingordering policy and finance mode selection for a capital-constrained retailer with stochastic demandrdquo Journal of theOperational Research Society vol 68 no 12 pp 1620ndash16322017

[24] W Jin Q Zhang and J Luo ldquoNon-collaborative and col-laborative financing in a bilateral supply chain with capitalconstraintsrdquo Omega vol 88 pp 210ndash222 2018

[25] B Keren ldquoe single-period inventory problem extension torandom yield from the perspective of the supply chainrdquoOmega vol 37 no 4 pp 801ndash810 2009

[26] A Sufi ldquoBank lines of credit in corporate finance an empiricalanalysisrdquo Review of Financial Studies vol 22 no 3pp 1057ndash1088 2009

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dEπ1

dq0 p1 + l1( 1113857 1113946

q1q0

0xf(x)dx + w minus c0 1 + r1( 1113857 minus l1

(A1)

d2Eπ1dq20

minusq21q30

p1 + l1( 1113857fq1

q01113888 1113889le 0 (A2)

When (q1q0)gt 1 + n that is q0 lt (q1(1 + n)) we havef(q1q0) 0 us (d2Eπ1dq20) 0 We obtain

dEπ1

dq0 p1 + w minus c0 1 + r1( 1113857gt 0 (A3)

So when q0 lt (q1(1 + n)) Eπ1 is a monotonically in-creasing function with a fixed slope

When (q1q0)lt 1 minus n that is q0 ge (q1(1 minus n)) we havef(q1q0) 0 us (d2Eπ1dq20) 0 We obtain

dEπ1dq0

w minus c0 1 + r1( 1113857 minus l1 lt 0 (A4)

So when (q1(1 minus n))lt q0 Eπ1 is a monotonically de-creasing function with a fixed slope

When 1 minus nle (q1q0)le 1 + n we have (d2Eπ1dq20)lt 0erefore Eπ1 is a convex function about the variable q0dEπ1dq0 is a continuous function on q0 isin [0 +infin] Whenq0 (q1(1 + n)) we obtain (dEπ1dq0)gt 0 according to(A1) When q0 (q1(1 minus n)) we obtain (dEπ1dq0)lt 0according to (A1) Hence according to the intermediatevalue theorem there must exist q0 isin [(q1(1 + n)) (q1(1 minus

n))] satisfying (dEπ1dq0) 0 At this time q0 is the onlylocal maximum point of Eπ1

In summary when q0 satisfies (dEπ1dq0) 0 that is

1113946q1q0

0xf(x)dx 1 minus

p1 + w minus c0 1 + r1( 1113857

p1 + l1 (A5)

and q0 isin [0 +infin) q0 is the only absolute maximum point ofEπ1

Proof of Proposition 2 Taking the first-order derivative ofEπ2 with respect to q1 geD it follows that

dE π2

dq1

p2 minus p1( 1113857 1113946+infin

q1q0f(x)dx + w minus kp1r2 q1 ltD

minus l2 1113946+infin

q1q0f(x)dx + w minus kp1r2 q1 geD

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(A6)

(1) q1 geD

According to Proposition 1 we have (q1q0) isin[1 minus n 1 + n] Because of xf(x)ge 0 we can obtainthat 1113938

q1q00 xf(x)dx gt 0 Let u (q1q0)gt 0 and de-

fine the function

φ1(u) 1113946U

0xf(x)dx (A7)

e first-order derivative of φ1(u) follows that

φ1prime(u) uf(u)ge 0 (A8)

Because of u isin [1 minus n 1 + n] we have φ1prime(u)gt 0us φ1(u) increases in u that is 1113938

q1q00 xf(x)dx

increases in (q1q0) Hence there exists a unique(q1q0) satisfying

1113946q1q0

0xf(x)dx 1 minus

p1 + w minus c0 1 + r1( 1113857

p1 + l1 (A9)

At this time (q1q0) is a unique constant related tothe coefficients and f(x) Hence when q1 geD(dEπ2dq1) is a unique constant If (dEπ2dq1)gt 0the retailerrsquos planned order quantity will be infinitewhich contradicts the risk-neutral assumptionHence we have (dEπ2dq1)lt 0 erefore Eπ2decreases in q1 in the region q1 isin [D +infin] and at-tains the maximum at the point q1 D

(2) q1 ltD

Because (dEπ2dq1) is a unique constant if

dEπ2dq1

p2 minus p1( 1113857 1113946+infin

q1q0f(x)dx + w minus kp1r2 lt 0

(A10)

then Eπ2 decreases in q1 in the region q1 isin [0 D) andattains the local maximum at the point q1 0 usEπ2 decreases in q1 when q1 ge 0 Eπ2 attains theabsolute maximum at the point q1 0 At this pointthe supply chain is in a state of no order We will notdiscuss this situation

In fact according to the previous literature and theactual situation of the supply chain there is often w minus kp1r2gt 0 Hence when q1 ltD we can obtain (dEπ2dq1)gt 0 andthat Eπ2 increases in q1

In summary Eπ2 attains the maximum at the pointq1 D

Proof of Proposition 3 It follows the proof of Proposition 1so we omit it

Proof of Proposition 4 It follows the proof of Proposition 2so we omit it

Proof of Proposition 5

(1) Obviously nonlinear programming (18) can bedivided into the following two planning problems

Mathematical Problems in Engineering 11

max q0c0 minus kp1q1( 1113857 r1 minus r0( 1113857

st

1113946q1q0

0xf(x)dx 1 minus

p1 + w minus c0 1 + r1( 1113857

p1 + l1

q1 D r0 le r1 le r10

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(A11)

max kp1q1 r2 minus r0( 1113857

st r0 le r2 le r20(A12)

In linear programming (A12) the objective functionincreases in r2 in the region r2 isin [r0 r20] us weattain the optimum of equation (A12) at the pointr2 r20 It is also the optimum of nonlinear pro-gramming (18)

(2) In nonlinear programming (A11) let

m1 q0( 1113857 q0c0 minus kp1D( 1113857 r1 minus r0( 1113857

st 1113946q1q0

0xf(x)dx 1 minus

p1 + w minus c0 1 + r1( 1113857

p1 + l1

m2 r1( 1113857 q0c0 minus kp1D( 1113857 r1 minus r0( 1113857

st 1113946q1q0

0xf(x)dx 1 minus

p1 + w minus c0 1 + r1( 1113857

p1 + l1

(A13)

according to

1113946q1q0

0xf(x)dx 1 minus

p1 + w minus c0 1 + r1( 1113857

p1 + l1 (A14)

We obtain that r1 decreases in q0 us when (dm1dq0)gt 0 we have (dm2dr1)lt 0 When (dm1dq0)lt 0we have (dm2dr1)gt 0

Taking the first-order derivative of m1 with respect to q0it follows thatdm1

dq0 p1 + l1( 1113857

middot 1113946q1q0

0xf(x)dx minus

p1 + l1( 1113857 q0c0 minus kp1D( 1113857q21f q1q0( 1113857

c0q30

minus r0c0 minus l1 minus c0 + w

(A15)

Under different parameters of the supply chain formula(A15) can satisfy both (dm1dq0)gt 0 and (dm1dq0)lt 0erefore there is not always (dm2dr1)gt 0 that is theoptimal of r1 is not always r10 In numerical analysis we giveexamples to support this conclusion

Proof of Proposition 6 Let

g q0 r3( 1113857 q0c0 r3 minus r0( 1113857 (A16)

Since the relationship between q0 and r3 satisfies theconstraint of (20) equation (A16) can also express equation(A17) as follows

g1 q0( 1113857 q0 p1 + l1( 1113857 1113946q1q0

0xf(x)dx + w minus l1 minus c0 1 + r0( 11138571113890 1113891

(A17)

Taking the first-order and second-order derivatives ofg1(q0) with respect to q0 it follows that

g1prime q0( 1113857 p1 + l1( 1113857 1113946q1q0

0xf(x)dx + w minus l1 minus c0 1 + r0( 11138571113890 1113891

minusq21q20

fq1

q01113888 1113889 p1 + l1( 1113857

gPrime1 q0( 1113857 p1 + l1( 1113857q21q30

fq1

q01113888 1113889gt 0

q1

q0isin [1 minus n 1 + n]

(A18)

Obviously when (q1q0) isin [1 minus n 1 + n] that isq0 isin [(q1(1 + n)) (q1(1 minus n))] g1prime(q0) increases in q0According to the proof of Proposition 1 the manufacturerattains the optimum in the region q0 isin [(q1(1 + n)) (q1(1 minus n))] When q0 (q1(1 minus n)) we can obtain

g1prime q0( 1113857 w minus l1 minus c0 1 + r0( 1113857 minus(1 minus n)2

2np1 + l1( 1113857lt 0

(A19)

12 Mathematical Problems in Engineering

us g1prime(q0) is negative in the region q0 isin [(q1(1 + n))

(q1(1 minus n))] g1(q0) decreases in q0 in the regionq0 isin [(q1(1 + n)) (q1(1 minus n))] Since

1113946q1q0

0xf1(x)dx 1 minus

p1 + w minus c0 1 + r3( 1113857

p1 + l1

st q0 isinq1

1 + n

q1

1 minus n1113876 1113877

(A20)

we obtain that r3 decreases in q0 in the region q0 isin [(q1(1 + n)) (q1(1 minus n))] Hence g1(q0) increases in r3 in theregion q0 isin [(q1(1 + n)) (q1(1 minus n))]

According to formulas (A3) and (A4) and the con-clusions of Proposition 1 when r1 isin [((w minus l1 minus c0)c0)((p1 + w minus c0)c0)] we have q0 isin [(q1(1 + n)) (q1(1 minus n))] Since we assume that the manufacturer makesdecisions based on the maximum expected profit there mustbe r10 isin [((w minus l1 minus c0)c0) ((p1 + w minus c0)c0)] which maxi-mizes the expected profit of the manufacturer under the FBSstrategy Similarly there must be min r10 r201113864 1113865 isin [((w minus l1 minus

c0)c0) ((p1 + w minus c0)c0)] which maximizes the expectedprofit of the manufacturer under the FBU strategy us wehave q0 isin [(q1(1 + n)) (q1(1 minus n))] when r3 isin [r0 minr10 r201113864 1113865] Since g1(q0) decreases in q0 in the region q0 isin

[(q1(1 + n)) (q1(1 minus n))] g1(q0) increases in r3 in theregion r3 isin [r0 min r10 r201113864 1113865] erefore g1(q0) attains themaximum at the point rlowast3 min r10 r201113864 1113865 We can obtainmax[q0c0(r3 minus r0)] q0c0(rlowast3 minus r0)

Proof of Proposition 7 If formula (20) attains the optimumat the point q0 qU

0 and r3 rU3 let q0 qU

0 r1 rU3 and

r2 rU3 in formula (18) Obviously they satisfy the con-

straint of formula (18) At this time the objective function offormula (18) is equal to the maximum of formula (20) whichmeans that the optimal of formula (20) must satisfy formula(18)erefore the maximum of formula (18) is not less thanthe maximum of formula (20)

Proof of Proposition 8

(1) When r10 le r20 we have rS1 r10 min r10 r201113864 1113865 rU

3according to Proposition 5 and rS

1 r10 us whenq1 D and r1 r3 it is obvious that q0 is equal in thetwo equations according to equations (3) and (11)Hence under qS

0 qU0 qS

1 qU1 and r1 r3 we can

obtain that EπS1 EπU

1 according to equations (2) and(12)When r10 le r20 we have rU

3 r10 le r20 rS2 accord-

ing to Proposition 5 and rS1 r10 Hence under

qS0 qU

0 qS1 qU

1 and r2 ge r3 we can obtain thatEπS

2 leEπU2 according to equations (8) and (15)

erefore when rS1 r10 and r10 le r20 we obtain that

EπS1 EπU

1 and EπS2 leEπU

2 which means the ex-pected profit of the manufacturer and retailer underthe FBU strategy is not less than the profit under theFBS strategy

(2) r10 gt r20

We know that r1 decreases in q0 in the regionq0 isin [(q1(1 + n)) (q1(1 minus n))] Since rS

1 r10 andr10 gt r20 we have q0 isin [qS

0 +infin) Substituting equa-tion (3) in (2) and taking the first-order derivative ofEπ1 with respect to q0 the following can be obtaineddEπ1

dq0

p1 + l1( 1113857q21c0q

30

fq1

q01113888 1113889 q0c0 minus kp1D( 1113857 (A21)

Since qS0c0 minus kp1Dgt 0 and q0 isin [qS

0 +infin) we have(dEπ1dq0)gt 0 When the supply chain is under theFBU strategy we have rU

3 min r10 r201113864 1113865 r20according to Proposition 6 and r10 gt r20us we haverU3 r20 lt r10 rS

1 and qU0 gt qS

0 erefore we canobtain that EπU

1 gtEπS1 when (dEπ1dq0)gt 0 and

qU0 gt qS

0 In other words we have EπU1 gtEπS

1 whenrS1 r10 qS

0c0 minus kp1D gt 0 and r10 gt r20

Taking the first-order and second-order derivatives ofEπ2 with respect to q0 the following can be obtained

dEπ2

dq0 p2 minus p1( 1113857 1113946

Dq0

0xf(x)dx minus w

d2Eπ2

dq20 minus p2 minus p1( 1113857

D2

q30f

D

q01113888 1113889lt 0

(A22)

When r1 r10 and r10 gt r20 we have rU3 lt rS

1 and qU0 gt qS

0Since (d2Eπ2dq20)lt 0 we obtain that (dEπ2dq0) decreasesin q0 When q0 isin [qS

0 qU0 ] we have

mindEπ2

dq0 p2 minus p1( 1113857 1113946

DqU0

0xf(x)dx minus w

maxdEπ2

dq0 p2 minus p1( 1113857 1113946

DqS0

0xf(x)dx minus w

(A23)

When (p2 minus p1) 1113938DqU

00 xf(x)dx minus wgt 0 and q0 isin [qS

0

qU0 ] we have (dEπ2dq0)gt 0 and EπU

2 gtEπS2 When (p2 minus

p1) 1113938DqS

00 xf(x)dx minus wlt 0 and q0 isin [qS

0 qU0 ] we have (dEπ2

dq0)lt 0 and EπU2 ltEπS

2In summary when (p2 minus p1) 1113938

DqU0

0 xf(x)dx minus wgt 0 andqS0c0 minus kp1Dgt 0 we have EπU

1 gtEπS1 and EπU

2 gtEπS2 When

(p2 minus p1) 1113938DqS

00 xf(x)dx minus wlt 0 and qS

0c0 minus kp1Dgt 0 wehave EπU

1 gtEπS1 and EπU

2 ltEπS2

Data Availability

e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

e authors declare that they have no conflicts of interest

Acknowledgments

is research was funded by the Ministry of Education ofHumanities and Social Science Project of China (grantnumber 17YJAZH018)

Mathematical Problems in Engineering 13

References

[1] S Asian and X Nie ldquoCoordination in supply chains withuncertain demand and disruption risks existence analysisand insightsrdquo IEEE Transactions on Systems Man and Cy-bernetics Systems vol 44 no 9 pp 1139ndash1154 2014

[2] A Roy S S Sana and K Chaudhuri ldquoOptimal Pricing ofcompeting retailers under uncertain demand-a two layersupply chain modelrdquo Annals of Operations Research vol 260no 1-2 pp 481ndash500 2015

[3] B C Giri and S Bardhan ldquoCoordinating a supply chain underuncertain demand and random yield in presence of supplydisruptionrdquo International Journal of Production Researchvol 53 no 16 pp 5070ndash5084 2015

[4] A Arreola-Risa and G A Decroix ldquoMake-to-order versusmake-to-stock in a production-inventory system with generalproduction timesrdquo IIE Transactions vol 30 no 8 pp 705ndash713 1998

[5] A Grosfeld-Nir and B Ronen ldquoA single bottleneck systemwith binomial yields and rigid demandrdquoManagement Sciencevol 39 no 5 pp 650ndash653 1993

[6] C A Yano and H L Lee ldquoLot sizing with random yields areviewrdquoOperations Research vol 43 no 2 pp 311ndash334 1995

[7] P Kouvelis and W Zhao ldquoFinancing the newsvendor sup-plier vs Bank and the structure of optimal trade creditcontractsrdquo Operations Research vol 60 no 3 pp 566ndash5802012

[8] B R Q Zhang ldquoInventory management with asset-basedfinancingrdquoManagement Science vol 50 no 9 pp 1274ndash12922004

[9] J Ma and L Xie ldquoe comparison and complex analysis ondual-channel supply chain under different channel powerstructures and uncertain demandrdquo Nonlinear Dynamicsvol 83 no 3 pp 1379ndash1393 2016

[10] X Xu and J R Birge ldquoJoint production and financing de-cisions modeling and analysisrdquo Operations Managementvol 28 no 12 2011

[11] P Kouvelis and W Zhao ldquoe newsvendor problem andprice-only contract when bankruptcy costs existrdquo Productionand Operations Management vol 20 no 6 pp 921ndash936 2010

[12] O Boyabath and L B Toktay ldquoStochastic capacity investmentand flexible vs Dedicated technology choice in imperfectcapital marketsrdquo Management Science vol 57 no 12pp 2163ndash2179 2011

[13] Y Alan and V Gaur ldquoOperational investment and capitalstructure under asset-based lendingrdquo Manufacturing amp Ser-vice Operations Management vol 20 no 4 2018

[14] B Sun H Zhang N Yan et al ldquoA partial credit guaranteecontract in a capital-constrained supply chain financingequilibrium and coordinating strategyrdquo International Journalof Production Economics vol 173 pp 122ndash133 2016

[15] G Bi Y Fei X Yuan et al ldquoJoint operational and financialcollaboration in a capital-constrained supply chain undermanufacturer collateralrdquo Asia-Pacific Journal of OperationalResearch vol 35 no 3 Article ID 1850010 2018

[16] J Chen Y W Zhou and Y Zhong ldquoA pricingorderingmodel for a dyadic supply chain with buyback guarantee fi-nancing and fairness concernsrdquo International Journal ofProduction Research vol 55 no 7 pp 1ndash18 2018

[17] G G Cai X Chen and Z Xiao ldquoe roles of bank and tradecredits theoretical analysis and empirical evidencerdquo Pro-duction and Operations Management vol 23 no 4pp 583ndash598 2014

[18] X F Chen ldquoA model of trade credit in a capital-constraineddistribution channelrdquo International Journal of ProductionEconomics vol 159 pp 347ndash357 2015

[19] J Chod ldquoInventory risk shifting and trade creditrdquo Man-agement Science vol 63 no 10 pp 3207ndash3225 2017

[20] B Jing X Chen and G Cai ldquoEquilibrium financing in adistribution channel with capital constraintrdquo Production andOperations Management vol 21 no 6 pp 1090ndash1101 2012

[21] S A Yang and J R Birge ldquoTrade credit risk sharing andinventory financing portfoliosrdquo Management Science vol 64no 8 2018

[22] J Zhou ldquoEssays in interfaces of operations and finance insupply chainsrdquo PhD thesis University of RochesterRochester NY USA 2009

[23] Y W Zhou B Cao Y Zhong et al ldquoOptimal advertisingordering policy and finance mode selection for a capital-constrained retailer with stochastic demandrdquo Journal of theOperational Research Society vol 68 no 12 pp 1620ndash16322017

[24] W Jin Q Zhang and J Luo ldquoNon-collaborative and col-laborative financing in a bilateral supply chain with capitalconstraintsrdquo Omega vol 88 pp 210ndash222 2018

[25] B Keren ldquoe single-period inventory problem extension torandom yield from the perspective of the supply chainrdquoOmega vol 37 no 4 pp 801ndash810 2009

[26] A Sufi ldquoBank lines of credit in corporate finance an empiricalanalysisrdquo Review of Financial Studies vol 22 no 3pp 1057ndash1088 2009

14 Mathematical Problems in Engineering

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

max q0c0 minus kp1q1( 1113857 r1 minus r0( 1113857

st

1113946q1q0

0xf(x)dx 1 minus

p1 + w minus c0 1 + r1( 1113857

p1 + l1

q1 D r0 le r1 le r10

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(A11)

max kp1q1 r2 minus r0( 1113857

st r0 le r2 le r20(A12)

In linear programming (A12) the objective functionincreases in r2 in the region r2 isin [r0 r20] us weattain the optimum of equation (A12) at the pointr2 r20 It is also the optimum of nonlinear pro-gramming (18)

(2) In nonlinear programming (A11) let

m1 q0( 1113857 q0c0 minus kp1D( 1113857 r1 minus r0( 1113857

st 1113946q1q0

0xf(x)dx 1 minus

p1 + w minus c0 1 + r1( 1113857

p1 + l1

m2 r1( 1113857 q0c0 minus kp1D( 1113857 r1 minus r0( 1113857

st 1113946q1q0

0xf(x)dx 1 minus

p1 + w minus c0 1 + r1( 1113857

p1 + l1

(A13)

according to

1113946q1q0

0xf(x)dx 1 minus

p1 + w minus c0 1 + r1( 1113857

p1 + l1 (A14)

We obtain that r1 decreases in q0 us when (dm1dq0)gt 0 we have (dm2dr1)lt 0 When (dm1dq0)lt 0we have (dm2dr1)gt 0

Taking the first-order derivative of m1 with respect to q0it follows thatdm1

dq0 p1 + l1( 1113857

middot 1113946q1q0

0xf(x)dx minus

p1 + l1( 1113857 q0c0 minus kp1D( 1113857q21f q1q0( 1113857

c0q30

minus r0c0 minus l1 minus c0 + w

(A15)

Under different parameters of the supply chain formula(A15) can satisfy both (dm1dq0)gt 0 and (dm1dq0)lt 0erefore there is not always (dm2dr1)gt 0 that is theoptimal of r1 is not always r10 In numerical analysis we giveexamples to support this conclusion

Proof of Proposition 6 Let

g q0 r3( 1113857 q0c0 r3 minus r0( 1113857 (A16)

Since the relationship between q0 and r3 satisfies theconstraint of (20) equation (A16) can also express equation(A17) as follows

g1 q0( 1113857 q0 p1 + l1( 1113857 1113946q1q0

0xf(x)dx + w minus l1 minus c0 1 + r0( 11138571113890 1113891

(A17)

Taking the first-order and second-order derivatives ofg1(q0) with respect to q0 it follows that

g1prime q0( 1113857 p1 + l1( 1113857 1113946q1q0

0xf(x)dx + w minus l1 minus c0 1 + r0( 11138571113890 1113891

minusq21q20

fq1

q01113888 1113889 p1 + l1( 1113857

gPrime1 q0( 1113857 p1 + l1( 1113857q21q30

fq1

q01113888 1113889gt 0

q1

q0isin [1 minus n 1 + n]

(A18)

Obviously when (q1q0) isin [1 minus n 1 + n] that isq0 isin [(q1(1 + n)) (q1(1 minus n))] g1prime(q0) increases in q0According to the proof of Proposition 1 the manufacturerattains the optimum in the region q0 isin [(q1(1 + n)) (q1(1 minus n))] When q0 (q1(1 minus n)) we can obtain

g1prime q0( 1113857 w minus l1 minus c0 1 + r0( 1113857 minus(1 minus n)2

2np1 + l1( 1113857lt 0

(A19)

12 Mathematical Problems in Engineering

us g1prime(q0) is negative in the region q0 isin [(q1(1 + n))

(q1(1 minus n))] g1(q0) decreases in q0 in the regionq0 isin [(q1(1 + n)) (q1(1 minus n))] Since

1113946q1q0

0xf1(x)dx 1 minus

p1 + w minus c0 1 + r3( 1113857

p1 + l1

st q0 isinq1

1 + n

q1

1 minus n1113876 1113877

(A20)

we obtain that r3 decreases in q0 in the region q0 isin [(q1(1 + n)) (q1(1 minus n))] Hence g1(q0) increases in r3 in theregion q0 isin [(q1(1 + n)) (q1(1 minus n))]

According to formulas (A3) and (A4) and the con-clusions of Proposition 1 when r1 isin [((w minus l1 minus c0)c0)((p1 + w minus c0)c0)] we have q0 isin [(q1(1 + n)) (q1(1 minus n))] Since we assume that the manufacturer makesdecisions based on the maximum expected profit there mustbe r10 isin [((w minus l1 minus c0)c0) ((p1 + w minus c0)c0)] which maxi-mizes the expected profit of the manufacturer under the FBSstrategy Similarly there must be min r10 r201113864 1113865 isin [((w minus l1 minus

c0)c0) ((p1 + w minus c0)c0)] which maximizes the expectedprofit of the manufacturer under the FBU strategy us wehave q0 isin [(q1(1 + n)) (q1(1 minus n))] when r3 isin [r0 minr10 r201113864 1113865] Since g1(q0) decreases in q0 in the region q0 isin

[(q1(1 + n)) (q1(1 minus n))] g1(q0) increases in r3 in theregion r3 isin [r0 min r10 r201113864 1113865] erefore g1(q0) attains themaximum at the point rlowast3 min r10 r201113864 1113865 We can obtainmax[q0c0(r3 minus r0)] q0c0(rlowast3 minus r0)

Proof of Proposition 7 If formula (20) attains the optimumat the point q0 qU

0 and r3 rU3 let q0 qU

0 r1 rU3 and

r2 rU3 in formula (18) Obviously they satisfy the con-

straint of formula (18) At this time the objective function offormula (18) is equal to the maximum of formula (20) whichmeans that the optimal of formula (20) must satisfy formula(18)erefore the maximum of formula (18) is not less thanthe maximum of formula (20)

Proof of Proposition 8

(1) When r10 le r20 we have rS1 r10 min r10 r201113864 1113865 rU

3according to Proposition 5 and rS

1 r10 us whenq1 D and r1 r3 it is obvious that q0 is equal in thetwo equations according to equations (3) and (11)Hence under qS

0 qU0 qS

1 qU1 and r1 r3 we can

obtain that EπS1 EπU

1 according to equations (2) and(12)When r10 le r20 we have rU

3 r10 le r20 rS2 accord-

ing to Proposition 5 and rS1 r10 Hence under

qS0 qU

0 qS1 qU

1 and r2 ge r3 we can obtain thatEπS

2 leEπU2 according to equations (8) and (15)

erefore when rS1 r10 and r10 le r20 we obtain that

EπS1 EπU

1 and EπS2 leEπU

2 which means the ex-pected profit of the manufacturer and retailer underthe FBU strategy is not less than the profit under theFBS strategy

(2) r10 gt r20

We know that r1 decreases in q0 in the regionq0 isin [(q1(1 + n)) (q1(1 minus n))] Since rS

1 r10 andr10 gt r20 we have q0 isin [qS

0 +infin) Substituting equa-tion (3) in (2) and taking the first-order derivative ofEπ1 with respect to q0 the following can be obtaineddEπ1

dq0

p1 + l1( 1113857q21c0q

30

fq1

q01113888 1113889 q0c0 minus kp1D( 1113857 (A21)

Since qS0c0 minus kp1Dgt 0 and q0 isin [qS

0 +infin) we have(dEπ1dq0)gt 0 When the supply chain is under theFBU strategy we have rU

3 min r10 r201113864 1113865 r20according to Proposition 6 and r10 gt r20us we haverU3 r20 lt r10 rS

1 and qU0 gt qS

0 erefore we canobtain that EπU

1 gtEπS1 when (dEπ1dq0)gt 0 and

qU0 gt qS

0 In other words we have EπU1 gtEπS

1 whenrS1 r10 qS

0c0 minus kp1D gt 0 and r10 gt r20

Taking the first-order and second-order derivatives ofEπ2 with respect to q0 the following can be obtained

dEπ2

dq0 p2 minus p1( 1113857 1113946

Dq0

0xf(x)dx minus w

d2Eπ2

dq20 minus p2 minus p1( 1113857

D2

q30f

D

q01113888 1113889lt 0

(A22)

When r1 r10 and r10 gt r20 we have rU3 lt rS

1 and qU0 gt qS

0Since (d2Eπ2dq20)lt 0 we obtain that (dEπ2dq0) decreasesin q0 When q0 isin [qS

0 qU0 ] we have

mindEπ2

dq0 p2 minus p1( 1113857 1113946

DqU0

0xf(x)dx minus w

maxdEπ2

dq0 p2 minus p1( 1113857 1113946

DqS0

0xf(x)dx minus w

(A23)

When (p2 minus p1) 1113938DqU

00 xf(x)dx minus wgt 0 and q0 isin [qS

0

qU0 ] we have (dEπ2dq0)gt 0 and EπU

2 gtEπS2 When (p2 minus

p1) 1113938DqS

00 xf(x)dx minus wlt 0 and q0 isin [qS

0 qU0 ] we have (dEπ2

dq0)lt 0 and EπU2 ltEπS

2In summary when (p2 minus p1) 1113938

DqU0

0 xf(x)dx minus wgt 0 andqS0c0 minus kp1Dgt 0 we have EπU

1 gtEπS1 and EπU

2 gtEπS2 When

(p2 minus p1) 1113938DqS

00 xf(x)dx minus wlt 0 and qS

0c0 minus kp1Dgt 0 wehave EπU

1 gtEπS1 and EπU

2 ltEπS2

Data Availability

e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

e authors declare that they have no conflicts of interest

Acknowledgments

is research was funded by the Ministry of Education ofHumanities and Social Science Project of China (grantnumber 17YJAZH018)

Mathematical Problems in Engineering 13

References

[1] S Asian and X Nie ldquoCoordination in supply chains withuncertain demand and disruption risks existence analysisand insightsrdquo IEEE Transactions on Systems Man and Cy-bernetics Systems vol 44 no 9 pp 1139ndash1154 2014

[2] A Roy S S Sana and K Chaudhuri ldquoOptimal Pricing ofcompeting retailers under uncertain demand-a two layersupply chain modelrdquo Annals of Operations Research vol 260no 1-2 pp 481ndash500 2015

[3] B C Giri and S Bardhan ldquoCoordinating a supply chain underuncertain demand and random yield in presence of supplydisruptionrdquo International Journal of Production Researchvol 53 no 16 pp 5070ndash5084 2015

[4] A Arreola-Risa and G A Decroix ldquoMake-to-order versusmake-to-stock in a production-inventory system with generalproduction timesrdquo IIE Transactions vol 30 no 8 pp 705ndash713 1998

[5] A Grosfeld-Nir and B Ronen ldquoA single bottleneck systemwith binomial yields and rigid demandrdquoManagement Sciencevol 39 no 5 pp 650ndash653 1993

[6] C A Yano and H L Lee ldquoLot sizing with random yields areviewrdquoOperations Research vol 43 no 2 pp 311ndash334 1995

[7] P Kouvelis and W Zhao ldquoFinancing the newsvendor sup-plier vs Bank and the structure of optimal trade creditcontractsrdquo Operations Research vol 60 no 3 pp 566ndash5802012

[8] B R Q Zhang ldquoInventory management with asset-basedfinancingrdquoManagement Science vol 50 no 9 pp 1274ndash12922004

[9] J Ma and L Xie ldquoe comparison and complex analysis ondual-channel supply chain under different channel powerstructures and uncertain demandrdquo Nonlinear Dynamicsvol 83 no 3 pp 1379ndash1393 2016

[10] X Xu and J R Birge ldquoJoint production and financing de-cisions modeling and analysisrdquo Operations Managementvol 28 no 12 2011

[11] P Kouvelis and W Zhao ldquoe newsvendor problem andprice-only contract when bankruptcy costs existrdquo Productionand Operations Management vol 20 no 6 pp 921ndash936 2010

[12] O Boyabath and L B Toktay ldquoStochastic capacity investmentand flexible vs Dedicated technology choice in imperfectcapital marketsrdquo Management Science vol 57 no 12pp 2163ndash2179 2011

[13] Y Alan and V Gaur ldquoOperational investment and capitalstructure under asset-based lendingrdquo Manufacturing amp Ser-vice Operations Management vol 20 no 4 2018

[14] B Sun H Zhang N Yan et al ldquoA partial credit guaranteecontract in a capital-constrained supply chain financingequilibrium and coordinating strategyrdquo International Journalof Production Economics vol 173 pp 122ndash133 2016

[15] G Bi Y Fei X Yuan et al ldquoJoint operational and financialcollaboration in a capital-constrained supply chain undermanufacturer collateralrdquo Asia-Pacific Journal of OperationalResearch vol 35 no 3 Article ID 1850010 2018

[16] J Chen Y W Zhou and Y Zhong ldquoA pricingorderingmodel for a dyadic supply chain with buyback guarantee fi-nancing and fairness concernsrdquo International Journal ofProduction Research vol 55 no 7 pp 1ndash18 2018

[17] G G Cai X Chen and Z Xiao ldquoe roles of bank and tradecredits theoretical analysis and empirical evidencerdquo Pro-duction and Operations Management vol 23 no 4pp 583ndash598 2014

[18] X F Chen ldquoA model of trade credit in a capital-constraineddistribution channelrdquo International Journal of ProductionEconomics vol 159 pp 347ndash357 2015

[19] J Chod ldquoInventory risk shifting and trade creditrdquo Man-agement Science vol 63 no 10 pp 3207ndash3225 2017

[20] B Jing X Chen and G Cai ldquoEquilibrium financing in adistribution channel with capital constraintrdquo Production andOperations Management vol 21 no 6 pp 1090ndash1101 2012

[21] S A Yang and J R Birge ldquoTrade credit risk sharing andinventory financing portfoliosrdquo Management Science vol 64no 8 2018

[22] J Zhou ldquoEssays in interfaces of operations and finance insupply chainsrdquo PhD thesis University of RochesterRochester NY USA 2009

[23] Y W Zhou B Cao Y Zhong et al ldquoOptimal advertisingordering policy and finance mode selection for a capital-constrained retailer with stochastic demandrdquo Journal of theOperational Research Society vol 68 no 12 pp 1620ndash16322017

[24] W Jin Q Zhang and J Luo ldquoNon-collaborative and col-laborative financing in a bilateral supply chain with capitalconstraintsrdquo Omega vol 88 pp 210ndash222 2018

[25] B Keren ldquoe single-period inventory problem extension torandom yield from the perspective of the supply chainrdquoOmega vol 37 no 4 pp 801ndash810 2009

[26] A Sufi ldquoBank lines of credit in corporate finance an empiricalanalysisrdquo Review of Financial Studies vol 22 no 3pp 1057ndash1088 2009

14 Mathematical Problems in Engineering

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

us g1prime(q0) is negative in the region q0 isin [(q1(1 + n))

(q1(1 minus n))] g1(q0) decreases in q0 in the regionq0 isin [(q1(1 + n)) (q1(1 minus n))] Since

1113946q1q0

0xf1(x)dx 1 minus

p1 + w minus c0 1 + r3( 1113857

p1 + l1

st q0 isinq1

1 + n

q1

1 minus n1113876 1113877

(A20)

we obtain that r3 decreases in q0 in the region q0 isin [(q1(1 + n)) (q1(1 minus n))] Hence g1(q0) increases in r3 in theregion q0 isin [(q1(1 + n)) (q1(1 minus n))]

According to formulas (A3) and (A4) and the con-clusions of Proposition 1 when r1 isin [((w minus l1 minus c0)c0)((p1 + w minus c0)c0)] we have q0 isin [(q1(1 + n)) (q1(1 minus n))] Since we assume that the manufacturer makesdecisions based on the maximum expected profit there mustbe r10 isin [((w minus l1 minus c0)c0) ((p1 + w minus c0)c0)] which maxi-mizes the expected profit of the manufacturer under the FBSstrategy Similarly there must be min r10 r201113864 1113865 isin [((w minus l1 minus

c0)c0) ((p1 + w minus c0)c0)] which maximizes the expectedprofit of the manufacturer under the FBU strategy us wehave q0 isin [(q1(1 + n)) (q1(1 minus n))] when r3 isin [r0 minr10 r201113864 1113865] Since g1(q0) decreases in q0 in the region q0 isin

[(q1(1 + n)) (q1(1 minus n))] g1(q0) increases in r3 in theregion r3 isin [r0 min r10 r201113864 1113865] erefore g1(q0) attains themaximum at the point rlowast3 min r10 r201113864 1113865 We can obtainmax[q0c0(r3 minus r0)] q0c0(rlowast3 minus r0)

Proof of Proposition 7 If formula (20) attains the optimumat the point q0 qU

0 and r3 rU3 let q0 qU

0 r1 rU3 and

r2 rU3 in formula (18) Obviously they satisfy the con-

straint of formula (18) At this time the objective function offormula (18) is equal to the maximum of formula (20) whichmeans that the optimal of formula (20) must satisfy formula(18)erefore the maximum of formula (18) is not less thanthe maximum of formula (20)

Proof of Proposition 8

(1) When r10 le r20 we have rS1 r10 min r10 r201113864 1113865 rU

3according to Proposition 5 and rS

1 r10 us whenq1 D and r1 r3 it is obvious that q0 is equal in thetwo equations according to equations (3) and (11)Hence under qS

0 qU0 qS

1 qU1 and r1 r3 we can

obtain that EπS1 EπU

1 according to equations (2) and(12)When r10 le r20 we have rU

3 r10 le r20 rS2 accord-

ing to Proposition 5 and rS1 r10 Hence under

qS0 qU

0 qS1 qU

1 and r2 ge r3 we can obtain thatEπS

2 leEπU2 according to equations (8) and (15)

erefore when rS1 r10 and r10 le r20 we obtain that

EπS1 EπU

1 and EπS2 leEπU

2 which means the ex-pected profit of the manufacturer and retailer underthe FBU strategy is not less than the profit under theFBS strategy

(2) r10 gt r20

We know that r1 decreases in q0 in the regionq0 isin [(q1(1 + n)) (q1(1 minus n))] Since rS

1 r10 andr10 gt r20 we have q0 isin [qS

0 +infin) Substituting equa-tion (3) in (2) and taking the first-order derivative ofEπ1 with respect to q0 the following can be obtaineddEπ1

dq0

p1 + l1( 1113857q21c0q

30

fq1

q01113888 1113889 q0c0 minus kp1D( 1113857 (A21)

Since qS0c0 minus kp1Dgt 0 and q0 isin [qS

0 +infin) we have(dEπ1dq0)gt 0 When the supply chain is under theFBU strategy we have rU

3 min r10 r201113864 1113865 r20according to Proposition 6 and r10 gt r20us we haverU3 r20 lt r10 rS

1 and qU0 gt qS

0 erefore we canobtain that EπU

1 gtEπS1 when (dEπ1dq0)gt 0 and

qU0 gt qS

0 In other words we have EπU1 gtEπS

1 whenrS1 r10 qS

0c0 minus kp1D gt 0 and r10 gt r20

Taking the first-order and second-order derivatives ofEπ2 with respect to q0 the following can be obtained

dEπ2

dq0 p2 minus p1( 1113857 1113946

Dq0

0xf(x)dx minus w

d2Eπ2

dq20 minus p2 minus p1( 1113857

D2

q30f

D

q01113888 1113889lt 0

(A22)

When r1 r10 and r10 gt r20 we have rU3 lt rS

1 and qU0 gt qS

0Since (d2Eπ2dq20)lt 0 we obtain that (dEπ2dq0) decreasesin q0 When q0 isin [qS

0 qU0 ] we have

mindEπ2

dq0 p2 minus p1( 1113857 1113946

DqU0

0xf(x)dx minus w

maxdEπ2

dq0 p2 minus p1( 1113857 1113946

DqS0

0xf(x)dx minus w

(A23)

When (p2 minus p1) 1113938DqU

00 xf(x)dx minus wgt 0 and q0 isin [qS

0

qU0 ] we have (dEπ2dq0)gt 0 and EπU

2 gtEπS2 When (p2 minus

p1) 1113938DqS

00 xf(x)dx minus wlt 0 and q0 isin [qS

0 qU0 ] we have (dEπ2

dq0)lt 0 and EπU2 ltEπS

2In summary when (p2 minus p1) 1113938

DqU0

0 xf(x)dx minus wgt 0 andqS0c0 minus kp1Dgt 0 we have EπU

1 gtEπS1 and EπU

2 gtEπS2 When

(p2 minus p1) 1113938DqS

00 xf(x)dx minus wlt 0 and qS

0c0 minus kp1Dgt 0 wehave EπU

1 gtEπS1 and EπU

2 ltEπS2

Data Availability

e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

e authors declare that they have no conflicts of interest

Acknowledgments

is research was funded by the Ministry of Education ofHumanities and Social Science Project of China (grantnumber 17YJAZH018)

Mathematical Problems in Engineering 13

References

[1] S Asian and X Nie ldquoCoordination in supply chains withuncertain demand and disruption risks existence analysisand insightsrdquo IEEE Transactions on Systems Man and Cy-bernetics Systems vol 44 no 9 pp 1139ndash1154 2014

[2] A Roy S S Sana and K Chaudhuri ldquoOptimal Pricing ofcompeting retailers under uncertain demand-a two layersupply chain modelrdquo Annals of Operations Research vol 260no 1-2 pp 481ndash500 2015

[3] B C Giri and S Bardhan ldquoCoordinating a supply chain underuncertain demand and random yield in presence of supplydisruptionrdquo International Journal of Production Researchvol 53 no 16 pp 5070ndash5084 2015

[4] A Arreola-Risa and G A Decroix ldquoMake-to-order versusmake-to-stock in a production-inventory system with generalproduction timesrdquo IIE Transactions vol 30 no 8 pp 705ndash713 1998

[5] A Grosfeld-Nir and B Ronen ldquoA single bottleneck systemwith binomial yields and rigid demandrdquoManagement Sciencevol 39 no 5 pp 650ndash653 1993

[6] C A Yano and H L Lee ldquoLot sizing with random yields areviewrdquoOperations Research vol 43 no 2 pp 311ndash334 1995

[7] P Kouvelis and W Zhao ldquoFinancing the newsvendor sup-plier vs Bank and the structure of optimal trade creditcontractsrdquo Operations Research vol 60 no 3 pp 566ndash5802012

[8] B R Q Zhang ldquoInventory management with asset-basedfinancingrdquoManagement Science vol 50 no 9 pp 1274ndash12922004

[9] J Ma and L Xie ldquoe comparison and complex analysis ondual-channel supply chain under different channel powerstructures and uncertain demandrdquo Nonlinear Dynamicsvol 83 no 3 pp 1379ndash1393 2016

[10] X Xu and J R Birge ldquoJoint production and financing de-cisions modeling and analysisrdquo Operations Managementvol 28 no 12 2011

[11] P Kouvelis and W Zhao ldquoe newsvendor problem andprice-only contract when bankruptcy costs existrdquo Productionand Operations Management vol 20 no 6 pp 921ndash936 2010

[12] O Boyabath and L B Toktay ldquoStochastic capacity investmentand flexible vs Dedicated technology choice in imperfectcapital marketsrdquo Management Science vol 57 no 12pp 2163ndash2179 2011

[13] Y Alan and V Gaur ldquoOperational investment and capitalstructure under asset-based lendingrdquo Manufacturing amp Ser-vice Operations Management vol 20 no 4 2018

[14] B Sun H Zhang N Yan et al ldquoA partial credit guaranteecontract in a capital-constrained supply chain financingequilibrium and coordinating strategyrdquo International Journalof Production Economics vol 173 pp 122ndash133 2016

[15] G Bi Y Fei X Yuan et al ldquoJoint operational and financialcollaboration in a capital-constrained supply chain undermanufacturer collateralrdquo Asia-Pacific Journal of OperationalResearch vol 35 no 3 Article ID 1850010 2018

[16] J Chen Y W Zhou and Y Zhong ldquoA pricingorderingmodel for a dyadic supply chain with buyback guarantee fi-nancing and fairness concernsrdquo International Journal ofProduction Research vol 55 no 7 pp 1ndash18 2018

[17] G G Cai X Chen and Z Xiao ldquoe roles of bank and tradecredits theoretical analysis and empirical evidencerdquo Pro-duction and Operations Management vol 23 no 4pp 583ndash598 2014

[18] X F Chen ldquoA model of trade credit in a capital-constraineddistribution channelrdquo International Journal of ProductionEconomics vol 159 pp 347ndash357 2015

[19] J Chod ldquoInventory risk shifting and trade creditrdquo Man-agement Science vol 63 no 10 pp 3207ndash3225 2017

[20] B Jing X Chen and G Cai ldquoEquilibrium financing in adistribution channel with capital constraintrdquo Production andOperations Management vol 21 no 6 pp 1090ndash1101 2012

[21] S A Yang and J R Birge ldquoTrade credit risk sharing andinventory financing portfoliosrdquo Management Science vol 64no 8 2018

[22] J Zhou ldquoEssays in interfaces of operations and finance insupply chainsrdquo PhD thesis University of RochesterRochester NY USA 2009

[23] Y W Zhou B Cao Y Zhong et al ldquoOptimal advertisingordering policy and finance mode selection for a capital-constrained retailer with stochastic demandrdquo Journal of theOperational Research Society vol 68 no 12 pp 1620ndash16322017

[24] W Jin Q Zhang and J Luo ldquoNon-collaborative and col-laborative financing in a bilateral supply chain with capitalconstraintsrdquo Omega vol 88 pp 210ndash222 2018

[25] B Keren ldquoe single-period inventory problem extension torandom yield from the perspective of the supply chainrdquoOmega vol 37 no 4 pp 801ndash810 2009

[26] A Sufi ldquoBank lines of credit in corporate finance an empiricalanalysisrdquo Review of Financial Studies vol 22 no 3pp 1057ndash1088 2009

14 Mathematical Problems in Engineering

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

References

[1] S Asian and X Nie ldquoCoordination in supply chains withuncertain demand and disruption risks existence analysisand insightsrdquo IEEE Transactions on Systems Man and Cy-bernetics Systems vol 44 no 9 pp 1139ndash1154 2014

[2] A Roy S S Sana and K Chaudhuri ldquoOptimal Pricing ofcompeting retailers under uncertain demand-a two layersupply chain modelrdquo Annals of Operations Research vol 260no 1-2 pp 481ndash500 2015

[3] B C Giri and S Bardhan ldquoCoordinating a supply chain underuncertain demand and random yield in presence of supplydisruptionrdquo International Journal of Production Researchvol 53 no 16 pp 5070ndash5084 2015

[4] A Arreola-Risa and G A Decroix ldquoMake-to-order versusmake-to-stock in a production-inventory system with generalproduction timesrdquo IIE Transactions vol 30 no 8 pp 705ndash713 1998

[5] A Grosfeld-Nir and B Ronen ldquoA single bottleneck systemwith binomial yields and rigid demandrdquoManagement Sciencevol 39 no 5 pp 650ndash653 1993

[6] C A Yano and H L Lee ldquoLot sizing with random yields areviewrdquoOperations Research vol 43 no 2 pp 311ndash334 1995

[7] P Kouvelis and W Zhao ldquoFinancing the newsvendor sup-plier vs Bank and the structure of optimal trade creditcontractsrdquo Operations Research vol 60 no 3 pp 566ndash5802012

[8] B R Q Zhang ldquoInventory management with asset-basedfinancingrdquoManagement Science vol 50 no 9 pp 1274ndash12922004

[9] J Ma and L Xie ldquoe comparison and complex analysis ondual-channel supply chain under different channel powerstructures and uncertain demandrdquo Nonlinear Dynamicsvol 83 no 3 pp 1379ndash1393 2016

[10] X Xu and J R Birge ldquoJoint production and financing de-cisions modeling and analysisrdquo Operations Managementvol 28 no 12 2011

[11] P Kouvelis and W Zhao ldquoe newsvendor problem andprice-only contract when bankruptcy costs existrdquo Productionand Operations Management vol 20 no 6 pp 921ndash936 2010

[12] O Boyabath and L B Toktay ldquoStochastic capacity investmentand flexible vs Dedicated technology choice in imperfectcapital marketsrdquo Management Science vol 57 no 12pp 2163ndash2179 2011

[13] Y Alan and V Gaur ldquoOperational investment and capitalstructure under asset-based lendingrdquo Manufacturing amp Ser-vice Operations Management vol 20 no 4 2018

[14] B Sun H Zhang N Yan et al ldquoA partial credit guaranteecontract in a capital-constrained supply chain financingequilibrium and coordinating strategyrdquo International Journalof Production Economics vol 173 pp 122ndash133 2016

[15] G Bi Y Fei X Yuan et al ldquoJoint operational and financialcollaboration in a capital-constrained supply chain undermanufacturer collateralrdquo Asia-Pacific Journal of OperationalResearch vol 35 no 3 Article ID 1850010 2018

[16] J Chen Y W Zhou and Y Zhong ldquoA pricingorderingmodel for a dyadic supply chain with buyback guarantee fi-nancing and fairness concernsrdquo International Journal ofProduction Research vol 55 no 7 pp 1ndash18 2018

[17] G G Cai X Chen and Z Xiao ldquoe roles of bank and tradecredits theoretical analysis and empirical evidencerdquo Pro-duction and Operations Management vol 23 no 4pp 583ndash598 2014

[18] X F Chen ldquoA model of trade credit in a capital-constraineddistribution channelrdquo International Journal of ProductionEconomics vol 159 pp 347ndash357 2015

[19] J Chod ldquoInventory risk shifting and trade creditrdquo Man-agement Science vol 63 no 10 pp 3207ndash3225 2017

[20] B Jing X Chen and G Cai ldquoEquilibrium financing in adistribution channel with capital constraintrdquo Production andOperations Management vol 21 no 6 pp 1090ndash1101 2012

[21] S A Yang and J R Birge ldquoTrade credit risk sharing andinventory financing portfoliosrdquo Management Science vol 64no 8 2018

[22] J Zhou ldquoEssays in interfaces of operations and finance insupply chainsrdquo PhD thesis University of RochesterRochester NY USA 2009

[23] Y W Zhou B Cao Y Zhong et al ldquoOptimal advertisingordering policy and finance mode selection for a capital-constrained retailer with stochastic demandrdquo Journal of theOperational Research Society vol 68 no 12 pp 1620ndash16322017

[24] W Jin Q Zhang and J Luo ldquoNon-collaborative and col-laborative financing in a bilateral supply chain with capitalconstraintsrdquo Omega vol 88 pp 210ndash222 2018

[25] B Keren ldquoe single-period inventory problem extension torandom yield from the perspective of the supply chainrdquoOmega vol 37 no 4 pp 801ndash810 2009

[26] A Sufi ldquoBank lines of credit in corporate finance an empiricalanalysisrdquo Review of Financial Studies vol 22 no 3pp 1057ndash1088 2009

14 Mathematical Problems in Engineering

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom