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Received August 3, 2019, accepted August 11, 2019, date of publication August 16, 2019, date of current version September 6, 2019.

Digital Object Identifier 10.1109/ACCESS.2019.2935741

Optimal Financing and Production Decisions for a Supply Chain With Buyer-Backed Purchase Order Financing Contract YUAN CAO 1, JI-HONG ZHANG2, AND XIAO-YU MA2 1Antai College of Economics and Management, Shanghai Jiao Tong University, Shanghai 200030, China 2International Business School, Beijing Foreign Studies University, Beijing 100089, China

Corresponding author: Xiao-Yu Ma ([email protected])

This work was supported by the National Natural Science Foundation of China under Grant 71602011 and Grant 71371032.

ABSTRACT Motivated by the practice fromWalmart in China, we investigate Buyer-backed Purchase Order Financing (BPOF) contract by considering a supply chain consisting of one capital-constrained supplier and one well-capitalized retailer. The supplier is a small enterprise and cannot get financing from banks independently. In order to get funds for the supplier’s production, three parties (the retailer, supplier and bank) reach a BPOF contract. In this contract, the retailer provides credit guarantee for the supplier’s loan. As the loan guarantor, the retailer determines the amount of credit guarantee. The bank provides the supplier with the loan as the retailer requests. Then, the supplier gets the loan and determines production quantity. Finally, the market demand is realized and the retailer places order to the supplier. After the supplier delivers products and gets order payment, he can repay the loan. According to BPOF contract, if the order payment is insufficient to repay the loan, the retailer has to compensate the bank for the loss. To identify the optimal decisions for the retailer and supplier, we develop a two-stage Stackelberg game, in which the retailer is leader and the supplier is follower. We find that BPOF contract can help mitigate the supplier’s financial distress and improve the retailer’s revenue. In addition, we show that how the optimal decisions are affected by the supplier’s initial capital and supply chainmembers’ inventory risk. Our study further gives suggestions to supply chain management practices: small suppliers should take advantage of the retailer’s high credit for financing, and the retailers who cooperate with small suppliers should maintain a good credit rating.

INDEX TERMS Budget constrained, buyer-backed purchase order financing contract, newsvendor model, supply chain finance.

I. INTRODUCTION This study is motivated by the practice that we observe from Walmart. In China, Walmart has tens of thousands of upstream suppliers, many of which are small enterprises. Due to asymmetry of information and lack of collateral, small enterprises are often confronted with difficulties in financing from financial institutions. As a result, these small suppliers are budget-constrained in production. The shortage of funds incurs great risks to the breakdown of production process and insufficient supply for Walmart. In other words, small sup- pliers’ difficulty in financing not only generates huge losses for suppliers themselves, but creates pressure for downstream

The associate editor coordinating the review of this article and approving it for publication was Omar Khadeer Hussain.

retailers such as Walmart. Although Walmart could replace all small suppliers with large and well-capitalized suppliers to ensure the sufficient supply, it is not a good choice for three reasons. Firstly, the price advantage of small suppliers is critical for Walmart to achieve its low price competitive advantage. Secondly, switching to new suppliers may bring new costs for Walmart, which offset the benefits from new suppliers to some degree. Thirdly, some small suppliers have near-monopoly in specific varieties of products. As a con- sequence, Walmart does not search for new large suppliers. Nevertheless, Walmart has the motivation to help its existing small suppliers on financing and funding before production.

In real practice, Walmart assists its small suppliers through Buyer-backed Purchase Order Financing (BPOF) contract. To be specific, Walmart provides credit guarantee for its

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Y. Cao et al.: Optimal Financing and Production Decisions for a Supply Chain With BPOF Contract

small suppliers to Industrial and Commercial Bank of China (ICBC), one of the biggest commercial banks in China.

It is difficult for these small suppliers to get financing from financial institutions independently. However, they can get loans from ICBC with Walmart’s credit guarantee, because Walmart has a high credit rating, and these small suppli- ers have a long-term business relationship with Walmart. Through BPOF contract, Walmart’s credit guarantee serves as a form of collateral for small suppliers’ bank loan [1]. When the loan is due, the small suppliers repay the bank loan by the purchase order payment with Walmart. Since the market demand is random, Walmart cannot inform its suppliers the exact order amount until the actual demand arrives. Hence, the order payment that suppliers can get is uncertain and may not sufficient to cover the loan’s principal and interest. However, ICBC has no need to worry about this uncertainty, as Walmart has guaranteed to compensate ICBC’s loss in BPOF contract. To conclude, BPOF contract enables small suppliers to get financing for production from banks by utilizing the advantage of retailer’s high credit rating (see Fig. 1 and Fig. 2).

FIGURE 1. Supply chain finance framework without buyer-backed purchase order financing (BPOF) contract.

FIGURE 2. Supply chain finance framework with buyer-backed purchase order financing (BPOF) contract.

In China, small andmedium-sized enterprises (SMEs) play a significant role in economic development. According to Financing SMEs and Entrepreneurs 2019: An OECD Score- board [2], SMEs account for more than 98% of all enterprises

in China. Moreover, they contributed 58% of GDP and 50% of tax revenue in 2018. However, due to asymmetric infor- mation, little collateral, and high default risk, financing has often been identified as an obstacle for SMEs [3], [4]. The rejection rate for SMEs’ bank loan application is nearly three times that for large enterprises. Even if a SME’s bank loan application is accepted, on average only 53.1% of funding amounts requested are finally granted. To promote SMEs’ financing and funding, various innovations have emerged [5], [6]. In this paper, we try to contribute to this landscape by investigating Buyer-backed Purchase Order Financing contract.

In BPOF contract, a series of questions need to be answered. For Walmart, the first and the most natural ques- tion is: Should Walmart offer a credit guarantee? Once the contingent compensation is realized, this would mean unnec- essary cost for Walmart. Therefore, how should Walmart balance the benefit from adequate product supply and cost from offering credit guarantee? If Walmart initiates a BPOF contract, other questions arise: how much credit guarantee should Walmart offer? What factors affect Walmart’s optimal decision? For the supplier, as Walmart cannot inform the exact order amount until the actual demand arrives, how many products should the supplier produce in advance? If the supplier produces few but the realized demand is high, it loses sales. If the supplier produces many but the realized demand is low, it incurs unnecessary production costs. In this paper, we try to answer these questions and identify the optimal decisions for both Walmart and suppliers.

Although BPOF contract is derived fromWalmart practice in China, it is applicable to a wide variety of industries that have financing needs for small enterprises’ production. In fact, BPOF contract is gradually expanding as a conven- tional supply chain financing contract in China’s practices. However, little research has been done on this novel financing scheme. This paper tries to fill this gap.

The remainder of this paper is organized as follows. After a relevant literature review in the next section, we explain settings and establish a two-periodmodel with BPOF contract in Section III. In Section IV and V, based on the magni- tude of supplier’s initial capital, we analyze the model and derive the optimal decisions under two cases, respectively. Finally, Section VI concludes this paper and offers some future research directions.

For convenience, we will use ‘‘he’’ and ‘‘she’’ to represent the supplier and retailer involved in the supply chain, respec- tively, in the remainder sections.

II. LITERATURE REVIEW Our research relates to two research streams – financing and funding for SMEs, and supply chain finance. We review the two areas, respectively, in this section.

In the past two decades, numerous attempts have been made to find a way to assess SMEs’ credit risk and to reduce information asymmetry between SMEs and financial institutions [7]. Yoshino and Taghizadeh-Hesary [8] collect

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11 financial ratios of 1363 SMEs in Asia, and employ prin- cipal component analysis and cluster analysis with the data. By doing this, they provide a scheme for assigning credit ratings to SMEs. In addition, nationwide SME databases have been established in several countries, such as Credit Risk Database (CRD) in Japan and Public Credit Registries in Malaysia. Kuwahara et al. [9] analyze the characteristics of Japan’s CRD and conclude that CRD can create reliable scoring models for assessment of SMEs’ risk. Furthermore, the problem that how to provide financing for SMEs in a reasonable way has also been studied [5]. Evidence suggests that credit guarantee scheme can reduce the risk of green finance and induce private participation in green finance and investment, because part of the risk is alleviated by the gov- ernment [10]. Moreover, a theoretical model has been devel- oped to determine the government’s optimal credit guarantee ratio [1]. The research concludes that the optimal credit guar- antee ratio is affected by three factors: government policy, macroeconomic conditions, and banks’ behavior.

Most of the above researches consider SMEs’ financing from the perspectives of banks or governments. Motivated by Walmart’s practice in China, we suppose that the SMEs’ financing problem can also be considered in the viewpoint of the supply chain. In a supply chain, a member enterprise has incentives to help other member enterprises to get financing. Our paper strives to contribute to this landscape by investigat- ing a new financing contract, Buyer-backed Purchase Order Financing (BPOF) contract.

Another relevant stream of literature is on supply chain finance. The history of supply chain finance can be tracked back to the 1970s [11]. For example, [12] study the effect of trade credit and inventories on the net cash flow gener- ated in business operations. Haley and Higgins [13] consider the basic lot-size model and study the relationship between inventory policy and trade credit policy. Later, as the issue of exploring corporates’ financial flows associated with supply chains attracted more and more attention, scholars started to investigate ‘‘supply chain finance’’ problem in a system- atic way. Buzacott and Zhang [14] were the first to demon- strate the importance of joint consideration of production and financing decisions. To do so, they model the available cash in each period as a function of assets and liabilities that may be updated periodically according to the dynamics of the pro- duction activities. In order to characterize the optimization of the cost of capital with supply chains, Pfohl and Gomm [15] propose a conceptual framework and a mathematical model of supply chain finance. Their model shows that supply chain finance is profitable for both supply chain members and capital markets under certain general conditions. In addition, based on the framework they proposed, all supply chain financing instruments can be classified into two categories, ‘‘internal financing’’ and ‘‘external financing’’.

‘‘Internal financing’’ refers to financing among supply chain members without participation of financial institutions. The most widely used internal financing instrument is trade credit. Trade credit is the credit extended by a seller to a buyer

when the goods or services are bought on credit. It facilitates the purchase without immediate payment [16]. Gupta and Wang [17] investigate the retailer’s operations with random demand, and prove that the structure of the optimal policy is not affected by credit terms. Chang [18] view trade-credit from a supplier’s perspective, and present it as a tool for supply chain coordination.

In contrast, ‘‘external financing’’ means that suppliers or retailers in a supply chain complete financing in presence of financial institutions. Kouvelis and Xu [19] develop a supply chain theory of factoring and reverse factoring, and show that in what condition these post-shipment financing schemes should be adopted. Lekkakos and Serrano [20] study the impact of reverse factoring schemes on suppliers’ operational decisions and performance. The authors model a supplier’s inventory replenishment problem as a multi-stage dynamic program and derive the supplier’s optimal inventory policy for two cases: no access to external financing, and access to external financing through reverse factoring or traditional factoring. Bi et al. [21] consider a supply chain consisting of a well-capitalized manufacturer and a capital-constrained retailer. To help the retailer get financing for a purchase order, the manufacturer promises to pay the lender a proportion of the retailer’s loan if the retailer goes broke.

Some scholars compare internal financing instruments with external financing instruments, in order to find the retailer’s and supplier’s preferences. For example, Cai et al. [22] study the roles of bank and trade credits with a capital constrained retailer. Either in a single-credit scenario or in a dual-credit scenario, they evaluate the retailer’s opti- mal order quantity and the creditors’ optimal credit limits and interest rates. In addition, they use empirical panel data to verify the substitutability and complementarity between the two credits. Tang and Cai [23] compare the reverse factoring with direct prepayment. They examine a supply chain with one capital-constrained supplier and one well-capitalized retailer, and find that the retailer prefers reverse factoring than direct prepayment if the retailer can reach a long credit term. Kouvelis and Zhao [24] consider a supply chain where both the retailer and supplier are capital constrained. The retailer can use both trade credits and short-term bank loans, and the supplier can use the retailer’s early payment and/or short-term bank loans. They show the retailer’s and supplier’s preferences under different conditions.

Different with existed literature, our research has two main contributions. First, most literature on supply chain finance, especially on external financing, view the retailer as capital-constrained and the supplier as a well-capitalized big enterprise. In contrast, we consider another scenario where the supplier is a small enterprise with budget constraint and the retailer is well-capitalized. We try to investigate events and decision process in this scenario and identify the optimal decisions for the retailer and supplier.

Second, we study a novel financing scheme, Buyer-backed Purchase Order Financing (BPOF) contract. BPOF contract is based on credit guarantee scheme, and the guarantee itself is a

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form of collateral. In specific, BPOF contract is different from factoring or reverse factoring, which are two popular financ- ing contracts in industry. Compared with factoring, BPOF contract aims to promote suppliers financing rather than retailers. Compared with reverse factoring, BPOF contract requires the supplier to repay bank loans by purchase order rather than accounts receivable. These differences distinguish BPOF contract and may bring new findings.

III. SETTINGS AND MODEL We consider a stylized supply chain consisting of one capital-constrained supplier and one well-capitalized retailer. The supplier produces a type of product at unit cost, c, and sells to end customers via the retailer. Suppose all the infor- mation of the supplier and retailer is common knowledge to the supply chain members. The supplier and retailer are both risk-neutral; they make the optimal decisions to maximize their expected profits, respectively. The transaction between the supplier and retailer is based on a wholesale price contract and a pull system. That is, the retailer purchases the products from the supplier at a wholesale price,w, and then sells to cus- tomers at a retail price, p. In the pull system, the retailer places order to the supplier immediately when the random market demandD is realized. Since the supplier is budget constrained with a limited initial capital, B, his production capacity is only B/c units of products unless he gets financing before production. However, the supplier cannot get financing from banks independently, because he is a small enterprise with low credit rating.

Since the inadequate supply may decrease the retailer’s profits, the retailer has incentive to help the supplier get financing before production. To this end, the retailer, sup- plier and bank reach a Buyer-backed Purchase Order Financ- ing (BPOF) contract. In this contract, the retailer provides credit guarantee for the supplier’s loan. Since the retailer is an established firm with high credit rating, the bank is willing to provide loan to the supplier with the retailer’s guarantee. As the loan guarantor, the retailer determines the amount of credit guarantee, M , and the bank provides the supplier a loan as the retailer requests. For the convenience of description, we will refer to M as ‘‘the retailer’s credit guarantee’’ or ‘‘credit guarantee’’ below. Then, the supplier gets the loan and determines production quantity, Q. Finally, the market demand, D, is realized and the retailer places order to the supplier. As the supplier has inventory Q, he can deliver the retailer min(Q,D) products and get order payment wmin(Q,D) from the retailer. Fig. 3 shows the sequence of events in our model.

Suppose that the loan interest rate is r . All c, w, p, B and r are assumed to be exogenous, and D is assumed as a continu- ous random variable, whose PDF and CDF are f (·) and F(·), respectively. As a convention, we let F̄(·) = 1 − F(·) and F−1(·) [or F̄−1(·)] be the inverse function of F(·) [or F̄(·)]. We denote x+ := max{x, 0}. See Table 1 for the notations of our analysis. Moreover, we make the following assumption.

FIGURE 3. Sequence of events in the supply chain with buyer-backed purchase order financing (BPOF) contract.

TABLE 1. Notation.

Assumption 1: r ≤ min ( w− c c

, p− w w

) .

Assumption 1 implies that the interest rate is no greater than either the supplier’s profit margin or the retailer’s profit margin. It is reasonable because otherwise the supplier and/or retailer would not engage in business operations, which leads to a meaningless situation in our model.

When the loan is due, the supplier needs to repayM (1+ r) to the bank. Ifwmin(Q,D) ≥ M (1+r), the supplier is able to pay off the loan. If wmin(Q,D) < M (1+ r), the retailer has to compensate the bank’s loss. For the retailer, the potential risk to compensate is a cost of providing credit guarantee for the supplier.

Note that if the supplier’s production quantity is so small that wmin(Q,D) < M (1+ r) always holds, then the retailer has to compensate no matter how large the realized market demand is. Hence, it is reasonable that the retailer requires a lower bound to be imposed on the supplier’s production

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quantity, to ensure that the supplier is able to repay the loan when the market demand is large. DenoteQL(M ) as the lower bound, then

QL(M ) = M (1+ r)

w . (1)

We establish a model based on a two-stage Stackelberg game, where the retailer is leader and the supplier is follower. The retailer determines credit guarantee,M , and the supplier determines production quantity, Q. In the following, we will investigate each player’s optimal decision in reverse order. That is, we first study the supplier’s production quantity decision, and then the retailer’s credit guarantee decision. For convenience, we herein follow this sequence to set up our model.

A. SUPPLIER’S PRODUCTION QUANTITY Given the retailer’s credit guarantee, M , the supplier deter- mines his optimal production quantity, Q∗, to maximize his expected profit. The supplier’s expected profit is

5S (Q|M ) = E { M + [wmin(Q,D)−M (1+ r) ]+ − cQ

} = (w− c)Q− w

∫ Q QL (M )

F(D) dD−Mr, (2)

which is concave, and themaximizer isQP = F̄−1(c/w). This is consistent with the supplier’s optimal production quantity in the newsvendor model with no budget constraint.

Even with BPOF contract, the supplier still has no power to produce infinitely. Denote by

QU (M ) = B+M c

(3)

his production capacity. Considering the bounds imposed on the supplier’s production quantity, Q∗ ∈ [QL(M ),QU (M )] should always be ensured.

To characterize the optimal production quantity decision, we define two critical values, M1 and M2 , as

M1 := cQP − B and M2 := wQP

1+ r .

It is easily to show M1 ≤ cQP ≤ M2 ≤ wQP and the following equivalent conditions,

QU (M ) ≤ QP ⇐⇒ M ≤ M1, QL(M ) ≤ QP < QU (M ) ⇐⇒ M1 < M ≤ M2, QP < QL(M ) ⇐⇒ M > M2.

(4)

B. RETAILER’S CREDIT GUARANTEE Anticipating the impact of credit guarantee on the supplier’s optimal production quantity, the retailer needs to choose her optimal credit guarantee, M∗, to maximize her expected profit,

5R(M )

= E { (p−w) min(Q∗,D)− [M (1+ r)−wmin(Q∗,D) ]+

}

= (p−w)

[ Q∗−

∫ Q∗ 0

F(D) dD

] − w

∫ QL (M ) 0

F(D) dD.

(5)

In the following, we analyze the supply chain members’ optimal decisions. When B ≥ cQP, the supplier has the capability to produce QP units of products without BPOF contract. Hence, we regard the supplier’s initial capital as high if B ≥ cQP, and low otherwise. Our analysis will be done in the two cases, respectively.

For convenience, we make the following definition. Definition 1: For a gap function 1(x1, x2) = 5R(x1) −

5R(x2) whose value can be both positive and negative, define a function δ(x1, x2) as

δ(x1, x2) =

{ x1, if1(x1, x2) ≥ 0, x2, if1(x1, x2) < 0.

(6)

IV. SUPPLIER’S INITIAL CAPITAL IS HIGH A. SUPPLIER’S OPTIMAL PRODUCTION QUANTITY Theorem 1: Given the retailer’s credit guarantee, M,

the supplier’s optimal production quantity is

Q∗(M ) =

{ QP, if M ≤ M2, QL(M ), if M > M2.

(7)

Proof: Since B ≥ cQP, by (4), QP ≤ QU (M ) always holds. Then, if M ≤ M2, we have QL(M ) ≤ QP ≤ QU (M ), and thus Q∗ = QP. If M > M2, we have QP ≤ QL(M ) ≤ QU (M ), so5S (Q|M ) is decreasing on [QL(M ),QU (M )], and thus Q∗ = QL(M ). The proof is complete. �

B. RETAILER’S OPTIMAL CREDIT GUARANTEE By substituting Q∗ implied by Theorem 1 into (5), the retailer’s profit function becomes:

5R(M )

=

 (p−w)

[ QP−

∫ QP 0 F(D) dD

] −w

∫ QL (M ) 0 F(D) dD,

if M ≤ M2,

(p−w)QL(M )− p ∫ QL (M ) 0 F(D) dD,

if M > M2.

(8)

It can be verified that 5R(M ) is continuous in M . Its first order derivative is

5′R(M ) =

{ −(1+ r)F(QL(M )), if M ≤ M2, (1+ r)[(p/w)F̄(QL(M ))− 1], if M > M2.

(9)

Clearly,5′R(M ) is always non-positive in the interval [0,M2] and decreasing in the interval [M2,+∞). Therefore, 5R(M ) is non-increasing in [0,M2] and concave in [M2,+∞). For convenience, we denote two potential maximizer of the retailer’s expected profit as

M3 = 0 and M4 = w

1+ r F̄−1

( w p

) (10)

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In order to identify the unique optimal credit guarantee, we need to compare the retailer’s profits5R(M3) and5R(M4) within their respective sub-intervals. By substituting (10) into (5), we have5R(M3) = (p− w)

[ QP −

∫ QP 0 F(D) dD

] ,

5R(M4) = (p− w)QH − p ∫ QH 0 F(D) dD,

(11)

where QH = QL(M4) = F̄−1(w/p). For compar- ison, we define 11(M3,M4) = 5R(M3) − 5R(M4). If 11(M3,M4) > 0, then M3 is the unique optimal credit guarantee. If11(M3,M4) < 0, thenM4 is the unique optimal credit guarantee. If 11(M3,M4) = 0, then M3 and M4 are both the optimal credit guarantee. Lemma 1: By definition 1, 11(M3,M4) is well defined. Proof: To observe the value of 11(M3,M4), we regard

11(M3,M4) as a function of c and p, namely,

11(c, p) = (p− w)

[ QP − QH −

∫ QP 0

F(D) dD

]

+ p ∫ QH 0

F(D) dD, (12)

where QP = F̄−1(c/w), and QH = QL(M4) = F̄−1(w/p). In the following, we firstly show that 11(M3,M4) > 0 can hold.

Obviously, 11(c, p) is continuous with c and p. Suppose c = w/λ, p→ λw+, and λ > 1. Then we have QH → QP+, which results in 11(c, p)→ w

∫ QP 0 F(D) dD > 0.

Then, we show that 11(M3,M4) < 0 can hold, too. Suppose c→ w−, it suffices to show R1(M3)→ 0+. Define an auxiliary function

h(Q) = (p− w)Q− p ∫ Q 0 F(D) dD. (13)

By the first order condition, we have

h′(Q) = pF̄−1(Q)− w⇒ Q∗ = F̄−1(w/p). (14)

Moreover, since h′′(Q) = −pf (Q) < 0, h(Q) is concave and thus h(Q∗) > h(0) = 0, which indicates R2(M4) > 0. Therefore, we have 11(c, p) < 0. The proof is complete. � Theorem 2: The retailer’s optimal credit guarantee is

M∗ =

{ 0, if pc ≤ w2, δ1(M3,M4), if pc > w2.

(15)

Proof: Given pc ≤ w2, we have M4 ≤ M2. Therefore, 5R(M ) is non-increasing in the whole interval [0,+∞). As a consequence, the optimal credit guarantee, M∗, can always be found at M = 0. Given pc > w2, we have M4 > M2. Therefore, 5R(M )

is non-increasing in interval [0,M2) and [M4,+∞), whereas increasing in interval [M2,M4]. By Lemma 1, the retailer’s optimal decision is M∗ = δ1(M3,M4). The proof is complete. �

Recall that QP is the optimal solution for the newsvendor model without capital constraints. When B ≥ cQP, the sup- plier has sufficient initial capital to produce QP. By Theo- rem 2, the retailer’s optimal credit guarantee M∗ is either 0 or M4. When M∗ = 0, our model reduces to the traditional newsvendor model and the transaction between the supplier and retailer retains a pull system in which the selling of goods is triggered by the supplier’s production. When M∗ = M4, the retailer plays an active role in sharing inventory risk with the supplier, and the transaction in the supply chain turns to a push system. To illustrate this change with BPOF contract, suppose

p/w � 1 and c/w close to 1, then QP is close to 0. In tra- ditional newsvendor model, the supplier faces the extremely high inventory risk and both the supplier and retailer suffer great losses. By Theorem 2, the retailer chooses to provide the credit guarantee M4, and thus the production quantity turns to be F̄−1(w/p), which implies that the retailer becomes the newsvendor and the inventory risk has been transferred to the retailer.

As was shown, although BPOF contract is meant to miti- gate the supplier’s financial distress, it helps the supply chain to transfer the risk and get coordinated. With BPOF contract, the retailer bears a part of loss when the market demand is realized to be low.

V. SUPPLIER’S INITIAL CAPITAL IS LOW A. SUPPLIER’S OPTIMAL PRODUCTION QUANTITY Theorem 3: Given the retailer’s credit guarantee, M,

the supplier’s optimal production quantity is

Q∗(M ) =

 QU (M ), if M ≤ M1. QP, if M1 < M ≤ M2, QL(M ), if M > M2.

(16)

Proof: Recall the definitionM1 := cQP−B. Since B ≥ cQP,M1 > 0 always holds. Similar to the proof in Theorem 1, 5S (Q|M ) is concave. Its maximizer is QP if QP is within the feasible interval [QL(M ),QU (M )] and the end points if not. The proof is complete. �

B. RETAILER’S OPTIMAL CREDIT GUARANTEE By Theorem 3, (5) can be transformed to

5R(M )=



(p−w) [ QU (M )−

∫ QU (M ) 0 F(D) dD

] −w

∫ QL (M ) 0 F(D) dD, if M ≤ M1,

(p−w) [ QP−

∫ QP 0 F(D) dD

] −w

∫ QL (M ) 0 F(D) dD, if M1 < M ≤ M2,

(p−w)QL(M )−p ∫ QL (M ) 0 F(D) dD, if M>M2.

(17)

It can be readily verified that 5S (M ) is continuous at M1 and M2, respectively. Hence, 5S (M ) is continuous in

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[0,+∞). The first order derivative is

5′R(M ) =



p− w c

F̄(QU (M ))− (1+ r)F(QL(M )),

if M ≤ M1, −(1+ r)F(QL(M )),

if M1 < M ≤ M2,

(1+ r)[ p w F̄(QL(M ))− 1],

if M > M2.

(18)

To characterize the optimal credit guarantee decision, we define

M5 : = {M | p− w c

F̄(QU (M ))− (1+ r)F(QL(M )) = 0},

B1 : = cQP − w

1+ r F−1

( p− w w(1+ r)

) (19)

Lemma 2: For M5 and B1, the following equivalent condi- tion holds:

M5 < M1 ⇐⇒ B < B1 . Proof: For convenience, we define the auxiliary func-

tion T (M ) as

T (M ) = p− w c

F̄(QU (M ))− (1+ r)F(QL(M )). (20)

It suffices to show the corresponding first order derivative is negative.

T ′(M ) = − p− w c2

f (QU (M ))− (1+ r)2

w f (QL(M )) < 0

(21)

By the definitions of M5 and B1, we have

B < B1 ⇐⇒ M1 > w

1+ r F−1

( p− w w(1+ r)

) ⇐⇒ T (M1) > 0⇐⇒ M5 < M1 .

The proof is complete. � Lemma 3: If pc > w2, then B1 < 0. Proof: Given pc > w2, we have

B1 = cQP − w

1+ r F−1

( p− w w(1+ r)

) < cQP −

w 1+ r

F−1 ( p− w p

) < cQP −

w 1+ r

F−1 ( w− c w

) =

( c−

w 1+ r

) QP

< 0,

where the first and the last inequality hold due to the Assump- tion 1.

The proof is complete. �

When the retailer’s credit guarantee is M1, her expected profit is

5R(M1)

= (p−w)

[ QP−

∫ QP 0

F(D) dD

] − w

∫ QL (M1) 0

F(D) dD.

(22)

Similar to the definition of 11, we define 12(M1,M4) = 5R(M1)−5R(M4), in order to identify which one ofM1 and M4 is the optimal credit guarantee. Lemma 4: By Definition 1, 12 is well defined. Proof: Similar to the proof in Lemma 1, we regard

12(M1,M4) as a function of B, c, and p, namely,

12(B, c, p)

= (p− w)

[ QP − QH −

∫ QP 0

F(D) dD

]

+ p ∫ QH 0

F(D) dD− w ∫ QL (M1) 0

F(D) dD, (23)

where QP = F̄−1(c/w), and QH = QL(M4) = F̄−1(w/p). In the following, we firstly show that 12(M1,M4) > 0 can hold.

Obviously,12(B, c, p) is continuous with B, c and p. Given pc > w2 and B → cQP−, with an analogous proof showing 11(c, p) > 0, we have 12(B, c, p) > 0.

Then, we show that11(M3,M4) < 0 can hold, too. Recall the auxiliary function defined by (13). Given pc > w2, we have h(QP) − h(QH ) < 0 since QH is the maximizer of the function h(Q) and QP 6= QH . Suppose B = 0, we have M1 = cQP and

12(B, c, p) < h(QP)− h(QH ) < 0, (24)

where the first inequality holds due to the Assumption 1. The proof is complete. � For convenience, we define the critical point B2 = {B|12 = 0}. Then the equivalent condition

12 > 0⇐⇒ B > B2 (25)

always holds, since5R(M1) is increasing with B and5R(M4) is irrelevant with B. Theorem 4: The retailer’s optimal credit guarantee is (i). if pc ≤ w2, then

M∗ =

{ M5, if B < B1, M1, if B ≥ B1.

(26)

(ii). if pc > w2, then

M∗ =

{ M4, if B < B2, M1, if B ≥ B2.

(27)

Proof: Recall the proof in Theorem 2. Given pc ≤ w2, we have M4 ≤ M2 and that 5R(M ) is non-increasing in interval [M1,+∞). As a consequence, the optimal credit

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Y. Cao et al.: Optimal Financing and Production Decisions for a Supply Chain With BPOF Contract

guarantee, M∗, can always be found in interval [0,M1]. By Lemma 2, we have

M∗ =

{ M5, if B < B1, M1, if B ≥ B1.

(28)

Given pc > w2, we have M4 > M2. Furthermore, by Lemma 2 and Lemma 3, we have M5 < M1. There- fore,5R(M ) is increasing in interval [0,M5] and [M4,+∞), whereas non-increasing in interval [M5,M4]. Due to the con- tinuity of 5R(M ), by Lemma 4 and (25), we can conclude that

M∗ =

{ M4, if B < B2, M1, if B ≥ B2.

(29)

The proof is complete. � As Theorem 4 is shown, when the supplier’s initial capital

is low, the retailer’s optimal decision is always to provide credit guarantee, and the amount of credit guarantee is differ- ent under different conditions. Recall the definitions of three values, M1, M4, M5. M1 is the gap between the supplier’s initial capital and the funds needed to produceQP. Moreover, we have M4 > M1 when the retailer’s optimal credit guar- antee is M4, and M5 < M1 when the retailer’s optimal credit guarantee is M5. Such a result reveals some interesting findings. On the

condition that the supplier’s initial capital is not sufficient to produce QP units of products, if the supplier’s initial cap- ital is relatively high (but not as high as cQP), the retailer should offer the credit guarantee M1 to make the supplier produce exactly QP. However, if the supplier’s initial capital is relatively low, the optimal credit guarantee for the retailer is not M1 anymore. Instead, the retailer should make dif- ferent decisions according to the relationship between the retailer’s inventory risk and the supplier’s inventory risk. If the retailer’s inventory risk is lower than the supplier’s, the retailer’s optimal credit guarantee should increase from M1 to M4. In other words, the retailer offers more credit guarantee than what needed to produce QP. On contrary, if the retailer’s inventory risk is no lower than the supplier’s, then the retailer should decrease the credit guarantee from M1 toM5. This phenomenon can be explained by Theorem 2, which states that if the supplier faces high inventory risk, the retailer should replace the supplier to be the newsvendor.

VI. CONCLUSION REMARKS In this paper, we investigate the novel Buyer-backed Pur- chase Order Financing (BPOF) contract in which the large retailer helps the small budget-constrained supplier raise funds for production by providing credit guarantee to the bank. We build a two-stage Stackelberg game model, where the retailer first determines the amount of credit guaran- tee, and then the supplier determines production quantity. By assuming an extremely risk-averse single bank in the model, we characterize the optimal decisions for the retailer and supplier. Through analysis, we found that: (i) BPOF

contract can help mitigate the small supplier’s financial dis- tress and improve the retailer’s revenue. (ii) if the supplier’s initial capital is less than a certain threshold, the retailer should provide credit guarantee to help the supplier get financing. Both the supplier and retailer benefit from BPOF contract. With BPOF contract, the budget-constrained sup- plier’s production quantity can even increase to the level as the capital-unconstrained supplier has, under certain circum- stances. (iii) Even if the supplier’s initial capital is high, the retailer should still initiate BPOF contract, provided that the inventory risk of retailer is greatly lower than that of the supplier.

The policy implication of our research lies in that it enriches the present initiatives to promote SMEs’ financing. In China, banks are generally reluctant to lend to small enter- prises. Our study provides new insights to tackle this problem, if small enterprises are suppliers in a supply chain and they have large downstream retailers. To be specific, for small suppliers, seeking for the large retailer’s credit guarantee is a feasible way to get bank loan, since the retailer shares a common motivation to support the supplier’s production. Through BPOF contract, the small supplier’s risk is shared by the retailer who provides credit guarantee. Credit guarantee scheme is the core of BPOF contract, and the guarantee itself is a form of collateral. If the retailer has a poor credit rating, banks will not provide loans [25]. The supplier’s shortage of funds incurs insufficient product supply for the retailer, and in turn, brings a loss to the retailer’s profit. Therefore, maintaining a good credit rating is critical to large retailers who cooperate with small suppliers.

In the future research, vast opportunities around this novel BPOF contract exist. First, we assume that the retailer, as the loan guarantor, has to compensate all of the bank’s loss. In reality, some banks may not that risk-averse and the retailer may need to compensate only a part of the loss. With this change, the optimal decisions for supply chain members may different from our results. Therefore, it is meaningful to extend our model to the case with partial buyer-backed financing. Second, we consider the interest rate as exogenous parameter. Intuitively, if the bank actively engages in BPOF contract, then the bank has incentive to adjust the interest rate to motivate a larger amount of loan. In this regard, treating the interest rate as bank’s decision variable would be another extension. Third, we consider a single financing instrument in our model. It is worth considering the scenario with multiple choices in the financing market, such as factoring and reverse factoring. It should be interesting to compare different financ- ing instruments in the viewpoint of a well-capitalized retailer.

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YUAN CAO is currently pursuing the Ph.D. degree in management science and engineering from Shanghai Jiao Tong University, Shanghai, China. His research interests include supply chain finance, production and service operations man- agement, and optimization.

JI-HONG ZHANG received the Ph.D. degree in operations research and control theory from the Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing, China, in 2002. He is currently a Professor with the Department of Management Science and Engi- neering, International Business School, Beijing Foreign Studies University. His research interests include logistics and supply chain management, optimization and decision making, and data statis-

tics and analysis. He was awarded special allowances from the State Council of China for outstanding contributions.

XIAO-YU MA received the Ph.D. degree in man- agement science and engineering from the School of Economics and Management, Tsinghua Uni- versity, Beijing, China, in 2015. She was a Vis- iting Scholar with the University of California at Berkeley, Berkeley, USA. She is currently an Associate Professor with the Department of Man- agement Science and Engineering, International Business School, Beijing Foreign Studies Univer- sity. Her research interests include optimization,

service operations management, and revenue management. She is also a Reviewer of The International Journal of Management Science (OMEGA) and Asia-Pacific Journal of Operational Research.

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