Keywords: supply chain financing, default risk, preferential credit, trade

This paper examines how preferential credit based on retailers' credit line impacts on capital-constraint retailer's operational decisions. We consider a condition of loan competition when banks and manufacturers offer preferential credit to capital-constraint retailers in the newsvendor model.

Different credit lines and discounted rates of preferential credit mainly involve in retailers' exogenous collateral and risk preference of banks and manufacturers in our model. We investigate impacts of bank financing, trade credit, and portfolio credit (financing from both bank credit and trade credit with different ratios) on retailer's inventory decision with different cases that the retailer's financing amounts exceed credit line or not. We derive the equilibrium wholesale price, expected sale price, and order quantity when retailers face with different conditions of collaterals and institutes' risk preferences facing with market risk. A debt-financed retailer favours items with trade credit compared to bank financing, especially in conditions when its sourcing demand is great and when it finances from high-risk preference institutes. Retailer prefers to using the loan with high trade credit ratio when he opts portfolio credit conditions.

Introduction

Trade credit, offered by suppliers to retailers to meet their purchasing demand, is a common component of supply chain contracts. According to the Financial Times, 90% of international merchandise trade in America was financed by trade credit, amounting to $25 trillion 1 . Boeing Company allows financing to hundreds of capital-constrained enterprises through trade credit 2 . Over the period, 2003-2012 nearly 202,696 small-and medium-sized enterprises (SMEs) across 13 European countries used trade credit for sourcing [START_REF] Mcguinness | European trade credit use and SME survival[END_REF]. Trade credit also plays a vital role in alleviating pressures from financial shortage to SMEs, especially in developing countries [START_REF] Fisman | Trade Credit, Financial Intermediary Development, and Industry Growth[END_REF][START_REF] Jacobson | Trade Credit and the Propagation of Corporate Failure: An Empirical Analysis[END_REF][START_REF] Wu | A Trade Credit Model with Asymmetric Competing Retailers[END_REF]. Feess Trade credit has some advantages compared to traditional bank financing for enterprises. It could increase order quantities for capital-constrained retailers [START_REF] Wu | A Trade Credit Model with Asymmetric Competing Retailers[END_REF]; benefits firms facing with uncertainty [START_REF] Yin | Optimal Pricing of Two Successive-Generation Products with Trade-in Options under Uncertainty[END_REF][START_REF] Wu | A loss averse competitive newsvendor problem with anchoring[END_REF]; reduce financing cost [START_REF] Melnyk | Supply chain design: issues, challenges, frameworks and solutions[END_REF]; improve buyer's bargaining power [START_REF] Klapper | Trade Credit Contracts[END_REF]; define default risk quicker [START_REF] Smith | Trade Credit and Informational Asymmetry[END_REF], and lower transaction costs [START_REF] Emery | A Pure Financial Explanation for Trade Credit[END_REF][START_REF] Gao | Optimal Stackelberg strategies for financing a supply chain through online peer-to-peer lending[END_REF] identified optimal Stackelberg strategies for peer-to-peer lending in supply chains. Most works [START_REF] Kouvelis | The Newsvendor Problem and Price-Only Contract When Bankruptcy Costs Exist[END_REF][START_REF] Chod | Inventory, Risk Shifting, and Trade Credit[END_REF][START_REF] Huang | Supplier credit guarantee loan in supply chain with financial constraint and bargaining[END_REF] consider original trade credit in supply chains to finance inventory.

In 2019 DSI annual conference, supply chain financing (SCF) attracts wide attention. Shrivastava and Rogers offered the keynote named "Technology Powering the Evolution of Supply Chain Financing" to highlight the importance of SCF to firms' development. 3 The firms operate is not isolation. They are always embedded in powerful network with other firms. Also, in this annual conference, Wetzel and trade credit based on the retailer's operating conditions. Banks would like to lend to retailers that owe high-quality assets. In this way, it could bring more revenue to themselves. Thus, banks will also offer discount bank interest rates to attract retailers to source. Questions attract us, including what would happen if retailers sourced in these competing circumstances, and how does preferential credit by manufacturers and banks to retailers affect the supply chain? Motivated by these questions, this paper aims to examine the impacts of preferential credit given by banks and manufacturers. Capital-constraint retailers with high-quality assets or belong to state-supported industries could choose bank credit or manufacturer credit facing with different discount interest rates. At the end of the transaction, retailers pay back loans and interest. However, the default will happen if retailers' revenue could not amount to its payment due to market risk, which means the condition that retailers may not sell its inventories extremely. When we consider default risk, banks would evaluate retailer's quality and design contracts, including preferential credit lines for the convenience of the retailer, according to bank risk preference to retailers' default and retailers' collateral assets, which are hard to achieve liquidity in the model. Besides, the credit line evaluated by banks can be observed by the manufacturer (we consider this a non-information asymmetry market). The manufacturer would offer a discount trade credit based on its risk preference to default risk. This paper tries to answer the questions: (1) will the bank and manufacturer risk preference affect a retailer's sourcing decision based on preferential credit? (2) how does the retailer's asset quality (collateral) impact on sourcing strategy? (3) could the preferential credit benefit the retailer when the retailer finance from bank credit or trade credit?

To answer the first question, we analyze a Stackelberg game that involves three parties: a manufacturer, a capital-constrained retailer, and a bank. We consider different risk preferences for banks and manufacturers offering credit to the retailer. Due to credit line differences, we identify two cases in bank credit. Case 1 is the situation where the bank loan exceeds the credit line. Case 2 is the situation where the bank loan is lower than the credit line. We conclude that with an increase in bank risk preference, retailer order quantity increases under Case 1. However, optimal order quantity may not change with bank risk preference in Case 2. Besides, the manufacturer's risk preference may have no effect on order quantity under trade credit. The discount interest rate leads to a higher wholesale price, which transfers the retailer's profits to the manufacturer. When we focus on portfolio credit (finance both from bank and manufacture), there are two scenarios for the retailer to source. Scenario I: bank loan is higher than the credit line under portfolio credit. Scenario II: bank loan is lower than the credit line under portfolio credit. A high manufacturer's risk preference hurts the retailer in scenario I. There is no difference for the retailer's ordering decision in scenario II. The optimal quantity for the retailer to order is between bank credit and trade credit levels.

To answer the second question, we explore the influences of the retailer's collateral based on a decentralized supply chain. In a decentralized supply chain, the retailer's profit will decrease, especially on the condition of opting trade credit. The benefit of preferential credit primarily transfers to the manufacturer, leading to the improvement of the manufacturer's profit. However, with an increase in collateral, the effect on total supply chain profit is positive.

To answer the third question, we compare the impacts of preferential credit in bank credit and trade credit. We find that benefit transferring would be extended with trade credit. The benefit from discounting interest rates completely transfers to the manufacturer under trade credit, which is also similar to bank credit in some certain.

However, the retailer could be motivated to order more quantity in trade credit. Namely, preferential credit has a positive effect on retailers.

Our paper reveals two critical managerial implications. Firstly, preferential credit cannot motivate retailers to order more under trade credit because the benefits will transfer to the manufacturer through a higher wholesale price. Secondly, the retailer's collateral cannot lead to more profit to the retailer, but it can improve total supply chain efficiency. If retailers with high-quality assets finance through banks, a higher risk preference bank would be better.

The rest of this paper is structured as follows. Relevant published literature is reviewed in Section 2. Section 3 proposes the basic model and assumptions. Stackelberg equilibrium with bank credit, trade credit, and portfolio credit are derived respectively in Section 4, Section 5, and Section 6. Besides, section 7 demonstrates managerial insights through computational study. We summarize the article in Section 8. Proofs are collected in the Appendix.

Literature review

When considering the impact of different supply chain financing schemes on asymmetric markets, our paper is related to the following research streams: models with decisions under capital constraint, comparative models of bank financing under stochastic demand, and price-setting newsvendor problem.

The first stream considers supply chain decisions under capital constraint and trade credit. [START_REF] Xu | Joint production and financing decisions: Modeling and analysis[END_REF] gave a newsvendor framework related to a capitalconstrained retailer and analyzed how capital constraint and capital structure could impact management decisions. The theory of trade credit in capital constraint was considered by [START_REF] Burkart | In-Kind Finance: A Theory of Trade Credit[END_REF] [START_REF] Chod | Inventory, Risk Shifting, and Trade Credit[END_REF] proposed a model to study how debt financing distorted a retailer's inventory decision and developed a first-best inventory decision. All these papers consider capital constraints in the supply chain no matter whether the constraint is on the retailer or the manufacturer. However, multiple source finance by a retailer is neglected, and preferential credit from heterogeneous manufacturers or lenders has not been considered in various of literature.

The second stream involves models comparing bank finance alternatives under stochastic demand in supply chains. Trade credit is compared to bank credit in recent papers such as [START_REF] Gupta | Technical note: Financing the newsvendor[END_REF] and [START_REF] Zhou | Impacts of financial collaboration in a three-party supply chain[END_REF]. Kouvelis and Zhao (2012) considered a risk-neutral supplier financing a retailer within bankruptcy risks. A supplier with an early payment discount scheme was proposed as a decision framework to analyze decisions. Their work concluded that under optimal trade credit contracts, both the supplier's profit and supply chain's efficiency gained improvements. [START_REF] Deng | Financing Multiple Heterogeneous Suppliers in Assembly Systems: Buyer Finance vs[END_REF] established a model involving financing multiple heterogeneous suppliers and demonstrated that buyer finance is better even if its own unit capital opportunity cost is higher than the bank risk-free interest rate. [START_REF] Doumpos | A Multicriteria Outranking Modeling Approach for Credit Rating[END_REF] proposed an evolutionary algorithm to assess the risk of credit portfolios in firms, which examines default risk in the supply chain. Yang and Birge (2018) demonstrated a portfolio credit impact related to bankruptcy when the retailer's cash level was low. The result demonstrated that trade credit allows the supplier to take advantage of the retailer's financial weakness, yet it may also benefit both parties (retailer and supplier). These sources investigate the bank loan effect in the supply chain but do not consider the loan competition between banks and manufacturers.

The last stream of this paper is on the price-setting newsvendor problem. Related literature to consider price factor into newsvendor problems are commonly involved in two different demand models, such as additive and multiplicative forms [START_REF] Petruzzi | Pricing and the Newsvendor Problem: A Review with Extensions[END_REF][START_REF] Granot | On the effectiveness of returns policies in the pricedependent newsvendor model[END_REF]; Jammernegg and Kischka, 2013; [START_REF] Hu | The newsvendor's joint procurement and pricing problem under price-sensitive stochastic demand and purchase price uncertainty[END_REF].

The additive demand model represents that the variance of demand is independent of price (Gallego and van Ryzin, 1994; Kyparisis and Koulamas, 2018). However, demand in multiplicative forms is a descend function with price [START_REF] Lau | How a dominant retailer might design a purchase contract for a newsvendor-type product with price-sensitive demand[END_REF]. The existing papers usually consider that the price is exogenous. However, distinctive from such papers, we endogenous the price in newsvendor problem based on a linear demand model, which means that the price could be decided based on stochastic demand and retailers' inventory policy.

In our work, we also study heterogeneous bank's and manufacturer's risk preferences to the retailer's default. The bank and manufacturer offer preferential credit to attract retailer's financing. Due to the constraint of working capital, the retailer who face with stochastic market demand could not pay back the loan and interest when risk event happens. In this case, it will cause default on their payments. The mostly related article compared to our paper is [START_REF] Kouvelis | Who Should Finance the Supply Chain? Impact of Credit Ratings on Supply Chain Decisions[END_REF]. However, Kouvelis and Zhao (2017) consider the credit rating to yield different discount interest rate financing. They ignore the risk preference's effect and the different risk preferences impacts between the bank and the manufacturer, which is significantly essential to inventory decisions. The risk-averse may impact the operational condition of retailers as many works proposed [START_REF] Gan | Channel Coordination with a Risk-Neutral Supplier and a Downside-Risk-Averse Retailer[END_REF][START_REF] Choi | Channel coordination in supply chains with agents having mean-variance objectives[END_REF]Yang et al., 2018). We consider this perspective to examine the effect of the supply chain when the default event happens.

We find that preferential credit may not particularly benefit the retailer under trade credit. With portfolio credit, the retailer's situation would be worse compared to trade credit when he faces higher manufacturers' risk preference level.

Model and notation

We now consider a supply chain system consisting of three parties: a capital constraint (hereafter referred to as "he" in the following), a manufacturer (she) which owes sufficient capital, and a competitive commercial bank (it). In our model, the retailer decides the optimal order quantity to maximize his profit. The manufacturer decides the wholesale price in the supply chain. In the product market, the retailer faces stochastic demand denoted by D with a distribution function F(D), which is continuously differentiable. The general failure rate is defined as 𝐻(𝐷) = 𝐷 𝑓(𝐷) 𝐹(𝐷) ̅̅̅̅̅̅̅ , with a hazard rate ℎ(𝐷) = 𝑓(𝐷) 𝐹(𝐷) ̅̅̅̅̅̅̅ . Assume 𝐻(𝐷) and ℎ(𝐷) are increasing in D [START_REF] Kouvelis | Financing the Newsvendor: Supplier vs. Bank, and the Structure of Optimal Trade Credit Contracts[END_REF].

Capital-constraint retailers has not enough initial capital to pay order quantity immediately. Bank credit offered by bank and trade credit offered by manufacturer are useful for capital-constraint retailer to source. Under the competitive credit market, the bank would provide preferential credit to high-quality enterprises according to their collateral in order to attract prime loans and earn profit. In other words, the bank would give loans with discount interest rates and evaluate preferential credit lines based on the retailer's mortgage asset. This mortgage asset is hard to realize liquidity to pay financing costs in our assumption. The preferential credit line (𝐿 𝐶 ) is determined by the banks, depending on collateral (TC) and mortgage rate (g). Besides, collateral is exogenous due to it is hard to realize liquidity in our assumption, which means it will not be as payment to debt. Namely, default risk exists, but there is no bankruptcy to the retailer. Furthermore, this part of the loan is convenient and easy for the retailer to lend, meaning that it has high fluidity. We assume that retailers can borrow this part of preferential credit balance to pay trade credit. Thus, when the manufacturer observes the credit balance that the retailer has, she would give a discounted trade credit to the retailer. Also, the discount rate depends on the risk preference level of bank and manufacturer to retailer's default, considered as h and e. We have the bank interest rate as follows when bank loan 𝐿 𝐵 is lower than 𝐿 𝐶 .

𝑟 𝐶,𝐵 = 𝑟 𝐵 -𝑢 × ℎ × 𝐿 𝐶 .

(1)

𝐿 𝐶 = 𝑔 × 𝑇𝐶. (2)
𝑟 𝐶,𝐵 is the discount interest rate of banks in the credit line. 𝑟 𝐵 means the bank interest rate apart from the credit line. 𝑢 is a the parameter of the discount rate, which represents the discount degree after bank's consideration for risk evaluation and preferential credit line. In our model, we use multiplication rule to reflect the discount value, which is common in supply chain finance fields [START_REF] Kouvelis | Financing the Newsvendor: Supplier vs. Bank, and the Structure of Optimal Trade Credit Contracts[END_REF].

Similarly, the manufacturer decides the discount interest rate based on normal interest rate 𝑟 𝑀 after she considering the loan amount and risk preference. Thus, we obtain the manufacturer's discount interest rate as follows:

𝑟 𝐶,𝑀 = 𝑟 𝑀 -𝑒 × 𝑢(𝑔𝑇𝐶 -𝐿 𝐵 ). (3)
Focused on this equation, 𝑟 𝐶,𝑀 is the discount interest rate of trade credit. 𝐿 𝐵 is the bank loan, and 𝑟 𝑀 means the normal trade credit interest rate. We assume that 𝑟 𝑀 ≥ 𝑟 𝐵 if the retailer does not use a bank loan. The 𝐿 𝐵 = 0, we have

𝑟 𝐶,𝑀 = 𝑟 𝑀 -𝑔 × 𝑒 × 𝑢 × 𝑇𝐶. (4)
Then, we know that the discount interest rate of trade credit only depends on the the normal trade credit interest rate (𝑟 𝑀 ), mortgage rate (g), the parameter of the discount rate (u), and collateral (TC) when the retailer only finances inventory from trade credit.

In the market, companies could decide the product's price to attract consumers, which match the demand with retailers' inventory. Thus, in our model, we relax the assumption of unchanged price and assume that the retailer could decide the price according to the relationship between expected stochastic demand (D) and order quantity (q). We consider in linear demand model, which could be written as follows.

𝑝 = 𝑎 -𝑏 × min (𝐷, 𝑞). (5)
The expected sales price

E(p) is 𝑎 -𝑏 ∫ 𝐹 ̅ (𝐷)𝑑𝐷 𝑞 0
, which is easy to prove. The retailer considers the expected sales price to maximize profit in our model. The retailer orders q units of the product at wholesale price w. With stochastic demand, when q is over-ordered and exceeds the market's demand, the overage would be sold by a buyback price s. The manufacturer has a cost of k to produce products, with E(p)> w>k>s. For the capitalconstrained retailer, the working capital is B and 𝑤𝑞 ≥ B, the retailer needs to finance from the bank or the manufacturer and repays (𝑤𝑞 -𝐵)(1 + 𝑟) in the end. We assume the retailer's quality is adequate, and collateral TC is high, and there is no retailer bankruptcy. However, default will happen when retailers could not return loans and their interest.

In this study, we derive in the Stackelberg equilibrium to examine the effect of different risk preferences and collateral levels. Key notations are collected in Table 1.

[Please insert Table 1 about here]

Bank financing

In this section, we analyze the operational supply chain decision under bank loan in two different cases. In case 1, the amount of the retailer's bank loan is in a low level and does not exceed the credit line. We define it as 𝐿 𝐵 ≤ 𝑔𝑇𝐶. For case 2, the retailer orders lots of quantity. The amount of loan is more than the credit line, which is 𝐿 𝐵 ≥ 𝑔𝑇𝐶.

According to these two different cases, the retailer decides the optimal order quantity based on different profit functions.

The retailer's problem

Before the marketing period, the capital-constrained retailer needs to pay wq to the manufacturer. However, the retailer only has initial capital T available. He lends 𝐿 𝐵 from the bank. At the end of this period, he needs to pay 𝐿 𝐵 (1+r) to the bank, and r is determined given the different cases. In the stochastic demand market, the minimum product demand is D and the maximum demand is retailers' order quantity q. When retailers can not sale the product totally, the surplus product would be recycled. The buyback price is s. Namely, the retailer's revenue depends on two aspects, the sale revenue and the buyback revenue of surplus products. It can be written as follows:

𝐸(𝑝) × 𝑚𝑖𝑛[𝑞, 𝐷] + 𝑠 × max [𝑞 -𝐷, 0]. (6)
Where 𝐸(𝑝) = 𝑎 -𝑏 ∫ 𝐹 ̅ (𝐷)𝑑𝐷 𝑞 0

.

Due to the different amounts of bank credit, we would need to consider two cases to discuss the retailer's problem.

Case 1. Bank loan does not exceed the credit line, 𝐿 𝐵 ≤ 𝐿 𝐶 .

In this condition, the bank offers the retailer with a preferential credit based on the retailer's collateral and mortgage rate. Thus, the retailer's payment to the bank at the end of the period is (𝑤𝑞 -𝑇)(1 + 𝑟 𝐶,𝐵 ). Combined with Equation ( 6), the retailer's profit is as follows:

𝐸(𝑝) × 𝑚𝑖𝑛[𝑞, 𝐷] + 𝑠 × max[𝑞 -𝐷, 0] -(𝑤𝑞 -𝑇)(1 + 𝑟 𝐶,𝐵 ) -𝑇. (7)
It is easy to verify that the expected sales price

E(p) is 𝑎 -𝑏 ∫ 𝐹 ̅ (𝐷)𝑑𝐷 𝑞 0
when we consider the expected value for Equation [START_REF] Chod | Inventory, Risk Shifting, and Trade Credit[END_REF]. Then, we take the expected value for Equation [START_REF] Choi | Channel coordination in supply chains with agents having mean-variance objectives[END_REF]. The expected profit of the retailer can be simplified as follows:

𝜋 𝑟,𝐵,1 = 𝐸(𝑝)𝑞 -(𝐸(𝑝) -𝑠) ∫ 𝐹(𝑥)𝑑𝑥 𝑞 0 -(𝑤𝑞 -𝑇)(1 + 𝑟 𝐶,𝐵 ) -𝑇. ( 8
)
Where 𝐸(𝑝) = 𝑎 -𝑏 ∫ 𝐹 ̅ (𝐷)𝑑𝐷 𝑞 0

, 𝑟 𝐶,𝐵 = 𝑟 𝐵 -𝑢 × ℎ × 𝐿 𝐶 , and 𝐿 𝐶 = 𝑔 × 𝑇𝐶.

From Equation (8), we capture the expected the profit of retailers depending on retailers' order quantity, his initial capital, buyback price, the expected sale price, the wholesale price, and the discount interest rate of banks. In order to find the optimal order quantity by maximizing the retailer's profit, we obtain the Proposition 1.

Proposition 1. In a decentralized supply chain, when the retailer is given preferential credit from a bank (with a given wholesale price), the optimal order quantity 𝑞 𝐵,1 * satisfies the first-order optimality condition of its expected profit function, which is given as follows:

𝑎 -2𝑏 ∫ 𝐹 ̅ (𝐷)𝑑𝐷 𝑞 𝐵,1 * 0 -𝑠 = 𝑤(1+𝑟 𝐵 -𝑔𝑢ℎ𝑇𝐶)-s 𝐹 ̅ (𝑞 𝐵,1 * ) . ( 9
)
When the bank loan is lower than the credit line, the optimal order quantity based on a given wholesale price can be affected by wholesale price (w), bank interest rate (𝑟 𝐵 ), risk preference level of bank (ℎ), and retailer's collateral (TC).

Corollary 1. (Effect of bank decision)

In case 1: (i) 𝑞 𝐵,1 * is increasing in bank risk preference, 𝜕𝑞 𝐵,1 * 𝜕ℎ > 0;(ii) 𝑞 𝐵,1 * is decreasing in 𝑟 𝐵 , 𝜕𝑞 𝐵,1 * 𝜕𝑟 𝐵 < 0.
The intuition behind Lemma 2 is as follows: When the bank improves the interest rate, the loan cost for the retailer is raising. Retailer orders lower order quantity; However, when retailer finances from a higher risk preference bank, which would give a higher discount, due to the cost of finance is reduced, the retailer would want to order more quantity. The intuition behind Lemma 3 is as follows: When the retailer quality is adequate, he has more collateral, and the collateral rate set by the bank is high. The bank would give a more preferential credit line to the retailer, leading to a reduction in finance cost. The retailer would like to borrow more money from the bank based on the discount interest rate, and optimal order quantity increases.

Corollary 2. (Effect of collateral) When the bank loan is lower than the preferential credit line: (i) With the increasing of retailer's collateral,𝑞

Case 2. Bank loan is higher than the credit line, 𝐿 𝐵 > 𝐿 𝑐 .

In this case, the bank offers the retailer with preferential credit based on the retailer's collateral and mortgage rate when the loan is lower than the credit line. This portion of the credit's interest rate is 𝑟 𝐶,𝐵 . After the transaction process, the retailer would pay this part of loan for 𝐿 𝐶 (1 + 𝑟 𝐶,𝐵 ) , which contains the principal and the interests. The remaining loan is 𝑤𝑞 -𝑇 -𝐿 𝐶 , which would be charged according to the normal banks' interest rate 𝑟 𝐵 . The total principal and the interests are (𝑤𝑞 -𝑇 -𝐿 𝐶 )(1 + 𝑟 𝐵 ). In the end, the payment for the retailer to the bank would be defined as follows:

(𝑤𝑞 -𝑇)(1 + 𝑟) = 𝐿 𝐶 (1 + 𝑟 𝐶,𝐵 )+(𝑤𝑞 -𝑇 -𝐿 𝐶 )(1 + 𝑟 𝐵 ). (10)
We have the expected interest rate is𝑟 = 𝐿 𝐶 𝑟 𝐶,𝐵 +(𝑤𝑞-𝑇-𝐿 𝐶 )𝑟 𝐵 𝑤𝑞-𝑇 .

As a consequence, when we combine Equation [START_REF] Doumpos | A Multicriteria Outranking Modeling Approach for Credit Rating[END_REF] and Equation ( 6), we have expected profit of retailer as follows:

𝜋 𝑟,𝐵,1 = 𝐸(𝑝)𝑞 -(𝐸(𝑝) -𝑠) ∫ 𝐹(𝑥)𝑑𝑥 𝑞 0 -(𝑤𝑞 -𝑇 -𝐿 𝐶 )(1 + 𝑟 𝐵 ) -𝑇 -𝐿 𝐶 (1 + 𝑟 𝐶,𝐵 ). ( 11
)
Where 𝐸(𝑝) = 𝑎 -𝑏 ∫ 𝐹 ̅ (𝐷)𝑑𝐷 𝑞 0

, 𝑟 𝐶,𝐵 = 𝑟 𝐵 -𝑢 × ℎ × 𝐿 𝐶 , and 𝐿 𝐶 = 𝑔 × 𝑇𝐶.

When we consider about Equation ( 11), we can obtain the optimal order quantity to achieve the retailers' operational decisions. The Proposition 2 is given as follows.

Proposition 2. In a decentralized supply chain, when the retailer is given preferential credit from a bank (with a given wholesale price), with the bank credit exceeding the credit line, the optimal order quantity 𝑞 𝐵,2 * satisfies the first-order optimality condition of its expected profit function, which is given as follows:

𝑎 -2𝑏 ∫ 𝐹 ̅ (𝐷)𝑑𝐷 𝑞 𝐵,2 * 0 -𝑠 = 𝑤(1+𝑟 𝐵 )-s 𝐹 ̅ (𝑞 𝐵,2 * ) . (12)
The Equation ( 12) decides the retailers' optimal order quantity when retailers' bank loan exceeds the credit line. The optimal order quantity is influenced by wholesale price and bank interest. There is no effect on TC, h, and g on 𝑞 𝐵,2 * . Based on Proposition 2, we know the optimal order quantity can be affected by interest rate and wholesale price, which are similar to case 1. We further obtain that optimal order quantity in case 2 decreases in 𝑟 𝐵 . However, collateral and bank risk reference levels do not affect the order decision significantly.

Proposition 3. Considering the optimal order quantity based on case 1 and case 2, we have 𝑞 𝐵,2 * ≤ 𝑞 𝐵,1 * with given wholesale price.

We give a numerical experiment to confirm our results of the effect by w, TC, and h on optimal order quantity in case 1 and case 2. Assume that the demand satisfies with uniform distribution, with D~U (0, 100), and the market scale (a) is 14, price elasticity (b) is 0.09. We give the bank's primary interest rate, 𝑟 𝐵 = 0.08. Salvage price (s) is 0.4.

Then, the parameter of the discount rate (u) equals to 0.00001. The unit cost of production of manufacturer (c) is 5. We give the mortgage rate (g) as 0.4 and the retailer's initial capital T=50. Based on different TC and h, we obtain the following Figure 1.

[Please insert Figure 1 about here]

(𝑤-𝑘)(1+𝑟 𝐵 ) ℎ(𝑞 𝐵,2 * )[𝑠-𝑤(1+𝑟 𝐵 )]-2𝑏𝐹 ̅ (𝑞 𝐵,2 * ) 2 + 𝑞 𝐵,2 * = 0. in case 2 ( 14
)
Where The Equation ( 14) decides the optimal wholesale price in different cases. We consider the intuition behind Proposition 4. It is complex to investigate the relationship between wholesale and series of exogenous affected variables (bank risk preference, bank interest rate). The result illustrates that collateral, mortgage rate, bank interest rate, and bank risk preference can influence the optimal wholesale price decision of the manufacturer in case 1. Nevertheless, only the bank interest rate may affect the wholesale price in case 2.

Trade credit financing

Trade credit play a crucial role to facilitate supply chain efficiency [START_REF] Chen | A pricing/ordering model for a dyadic supply chain with buyback guarantee financing and fairness concerns[END_REF].

In order to investigate the different impacts compared to bank finance, we show how the retailer and manufacturer would make decisions to maximize their profit by trade credit financing under a preferential credit scenario in this section.

Retailer's problem

At time zero, the retailer pays initial capital (T) to the manufacturer and borrows (wq-T) to meet orders. When the bank offers a preferential credit line to the retailer, the manufacturer can regard the retailer's trade credit as quality lending and provide a discount interest rate to the retailer for loan in line with the credit line overage. Because it is very convenient for the retailer to obtain a credit line overage from a commercial bank, the retailer can pay for trade credit by manufacturer risk-free. Thus, the trade credit's interest rate 8 can be written as 𝑟 𝑀 -𝑒𝑢(𝐿 𝐶 -𝐿 𝐵 ). Specifically, the 𝐿 𝐵 is zero when retailers only use trade credit. Namely, retailers would pay (𝑤𝑞 -𝑇)[1 + 𝑟 𝑀 -𝑒𝑢(𝐿 𝐶 -𝐿 𝐵 )] as the total principal and interests to manufacturers. Under this condition, we obtain the retailer's total payment to the manufacturer by trade credit financing as follows:

𝑇 + (𝑤𝑞 -𝑇)[1 + 𝑟 𝑀 -𝑒𝑢(𝐿 𝐶 -𝐿 𝐵 )]. (15)
Where 𝐿 𝐶 is satisfied 𝐿 𝐶 = 𝑔𝑇𝐶.

Combining Equation ( 15) with Equation ( 6), we can calculate the retailer's profit as:

𝜋 𝑟,𝑇 = 𝐸(𝑝)𝑞 -(𝐸(𝑝) -𝑠) ∫ 𝐹(𝑥)𝑑𝑥 𝑞 0 -(𝑤𝑞 -𝑇)(1 + 𝑟 𝑀 -𝑒𝑢(𝑔𝑇𝐶 -𝐿 𝐵 )) -𝑇. ( 16
)
When the credit is only from trade credit, 𝐿 𝐵 =0. The expected profit function is:

𝜋 𝑟,𝑇 = 𝐸(𝑝)𝑞 -(𝐸(𝑝) -𝑠) ∫ 𝐹(𝑥)𝑑𝑥 𝑞 0 -(𝑤𝑞 -𝑇)(1 + 𝑟 𝑀 -𝑒𝑢𝑔𝑇𝐶) -𝑇. ( 17
)
Proposition 5. In a decentralized supply chain, where the capital-constraint retailer receives preferential trade credit from the manufacturer (for a given wholesale price), the retailer's optimal order quantity𝑞 𝑇 * satisfies as follows:

𝑎 -2𝑏 ∫ 𝐹 ̅ (𝐷)𝑑𝐷 𝑞 𝑇 * 0 -𝑠 = 𝑤(1+𝑟 𝑀 -𝑒𝑔𝑢𝑇𝐶)-s 𝐹 ̅ (𝑞 𝑇 * ) . ( 18
)
Equation ( 18) reflects the retailers' optimal order quantity when retailers only use trade credit as loan. Based on Proposition 5, we obtain corollaries about different influences of manufacturers' decisions, collateral on the retailer's optimal order quantity. Considering Corollary 8 and Corollary 9, we conclude that preferential trade credit has a positive effect on the retailer's ordering decision. Financing from a high-risk preference manufacturer, the retailer will expand his ordering scale. Collateral quality will be considered when the retailer takes order decision. With high collateral quality, the order quantity can be increased.

Manufacturer's problem

When the retailer uses trade credit to borrow from the manufacturer at period zero, the manufacturer receives the initial capital of the retailer and gives a discount interest rate (𝑟 𝐶,𝑀 ) for lending based on the retailer's credit line overage. In the end, the manufacturer would receive principal and interest. As a result of maximizing profit, the manufacturer sets the wholesale price. The profit function of the manufacturer is given as follows:

𝜋 𝑚,𝑇 = (𝑤𝑞 -𝑇)(1 + 𝑟 𝐶,𝑀 ) -𝑘𝑞 -𝑇. ( 19
)
Where 𝑟 𝐶,𝑀 = 𝑟 𝑀 -𝑔 × 𝑢 × 𝑒 × 𝑇𝐶, when𝐿 𝐵 𝑒𝑞𝑢𝑎𝑙𝑠𝑧𝑒𝑟𝑜.

According to Equation [START_REF] Deng | Financing Multiple Heterogeneous Suppliers in Assembly Systems: Buyer Finance vs[END_REF], it is easy to capture the manufacturers' optimal decision about wholesale price when she needs to maximum her profit. Then, we obtain the following Proposition. The optimal wholesale price is decided by Equation [START_REF] Hu | The newsvendor's joint procurement and pricing problem under price-sensitive stochastic demand and purchase price uncertainty[END_REF].

Proposition 6. Retailer receives preferential credit from bank. On this condition, when retailer finances by trade credit, the wholesale price by manufacturer 𝑤 𝑇 * is uniquely given by the following:

(𝑤 𝑇 * (1+𝑟 𝑀 -𝑔𝑒𝑢𝑇𝐶)-𝑘)(1+𝑟 𝑀 -𝑔𝑢𝑒𝑇𝐶) ℎ(𝑞 𝑇 * )[𝑠-𝑤 𝑇 * (1+𝑟 𝑀 -𝑔𝑢𝑒𝑇𝐶)]-2𝑏𝐹 ̅ (𝑞 𝑇 * ) 2 + 𝑞 𝑇 * = 0, (20)
Where 𝑞 𝑇 * satisfies 𝑎 -2𝑏 ∫ 𝐹 ̅ (𝐷)𝑑𝐷 .

The intuition of Proposition 6 reveals that the optimal wholesale price decided by the manufacturer can be affected by the trade credit interest rate and the manufacturer's risk preference level.

Bank credit and Trade credit financing portfolio

In this section, we discuss portfolio finance by retailer. Because the discount of trade credit interest rate depends on bank loan overage and risk preference level (both bank and manufacturer), the retailer could still benefit from a cheaper trade credit interest rate based on existing bank credit overage. The loan cost could be reduced compared to finance totally by trade credit. We consider two scenarios.

Scenario I: The amounts of bank credit lower than the credit line.

In this situation, the retailer can receive a discount interest rate by bank credit and trade credit. The bank discount interest rate is as follows: 𝑟 𝐶,𝐵 = 𝑟 𝐵 -𝑔𝑢ℎ𝑇𝐶. The trade-credit interest rate is as follows: 𝑟 𝐶,𝑀 = 𝑟 𝑀 -𝑒 × 𝑢(𝑔𝑇𝐶 -𝐿 𝐵 ).

Scenario II: The amount of bank credit is higher than the credit line.

We consider the fact that the retailer loses the power to borrow a conveniently guaranteed loan from a bank when he over finances. It may not be available for retailers to source from trade credit at a discount rate. Thus, for bank loans, the part that lower than credit needs to pay by 𝑟 𝐶,𝐵 , the overage needs to pay by 𝑟 𝐵 . For trade credit, the interest rate is 𝑟 𝑀 .

We set a ratio of bank financing is 𝛼. Thus, we have 𝐿 𝐵 = 𝛼(𝑤𝑞 -𝑇); 𝐿 𝑇 = (1 -𝛼)(𝑤𝑞 -𝑇). Next, we explore the retailer's problem and the manufacturer's problem in the Scenario I and Scenario II.

Retailer's problem

Retailer finances from the bank and manufacturer simultaneously. Apart from deciding the order quantity, he needs to consider the ratio of financing portfolio and credit line to maximize profit.

Scenario I

At period zero, the total loan is𝑤𝑞 -𝑇. At the end of the period, the retailer repays the bank for 𝐿 𝐵 (1 + 𝑟 𝐶,𝐵 ),and manufacturer for 𝐿 𝑇 (1 + 𝑟 𝐶,𝑀 ). Considering Equation ( 6), the profit function of the retailer is by the following: It could be written as follows:

𝛼 * = 𝑟 𝐵 -𝑟 𝑀 +𝑔𝑢𝑇𝐶(𝑒-ℎ) 2𝑒𝑢(𝑤𝑞-𝑇) + 1 2 . ( 23
)
Equation ( 23) express the optimal α to minimize the retailer's profit. Besides, The Lemma shows that there is no existing 𝛼 for the retailer to decide in order to maximize his profit. We consider the given 𝛼 to examine the retailer's operational decision.

When we consider the maximum profit of retailers, Equation ( 24) is given, which represents the retailers' optimal inventory decision.

Proposition 7.

Retailer is given preferential credit and sources from bank credit and trade credit portfolio. When the bank loan is lower than the credit line, the optimal order quantity satisfies the first-order optimality condition of its expected profit. For a given 𝛼, 𝑞 𝑃,𝐼 * can be given as follows:

𝑞 𝑃,𝐼 * = 𝑇 𝑤 + 𝐹 ̅ (𝑞 𝑃,𝐼 * )(𝑎-2𝑏 ∫ 𝐹 ̅( 𝐷)𝑑𝐷 𝑞 𝑇 * 0 -𝑠)+𝑠-𝛼𝑤(1+𝑟 𝐵 -𝑔𝑢ℎ𝑇𝐶)-𝑤(1-𝛼)(1+𝑟 𝑀 -𝑒𝑢𝑔𝑇𝐶) 2𝑤 2 𝛼(1-𝛼)𝑒𝑢 . ( 24
)
This proposition shows that apart from bank credit interest, trade credit interest, both bank and manufacturer's risk preference level and collateral, the ratio of bank credit also affects optimal order quantity in the condition of Scenario I.

Then, numerical experiments show the different effects of 𝛼 in Figure 2. We consider banks and manufacturers to have the same risk preference, which means that h=e=10 and give TC=1200. We find that order quantity increases in 𝛼, with different given wholesale prices in (a). Order quantity decreases in wholesale price the same as to the above result.

We note that if the retailer undertakes more finance from the bank, he may improve his order quantity when the wholesale price is the same. In this case, the bank discount interest rate is 0.032, and the trade credit discount interest rate is 0.072. What about the situation where trade credit has a lower cost? We consider different manufacturer risk preference levels in (b) and interest rate in (c) to explore if the result will change.

[Please insert Figure 2 about here]

We conclude that q increases with the ratio of bank credit when the manufacturer's risk preference is under a threshold. Nevertheless, when risk preference level exceeds the threshold, as a result of the interest rate decreasing, the order quantity decreases first and then increases with 𝛼. For a given 𝛼, quantity increases in e. We find that if e=18.33, the order quantity by bank credit and trade credit is the same. Moreover, banks and manufacturers offer the same discount rate for the retailer. When e exceeds this threshold, the trade credit is more efficient.

Scenario II

For this situation, the retailer's repayment to the bank consisting of two parts. Within the credit line's part, the retailer needs to pay the bank for 𝑔𝑇𝐶(1 + 𝑟 𝐶,𝐵 ). For another part of the loan, it needs to pay (𝐿 𝐵 -𝑔𝑇𝐶)(1 + 𝑟 𝐵 ). The repayment to trade credit is 𝐿 𝑇 (1 + 𝑟 𝑀 ). Similar to Equation ( 21), the expected profit function of the retailer in this scenario could be expressed as:

𝜋 𝑟,𝑃,𝐼𝐼 = 𝐸(𝑝)𝑞 -(𝐸(𝑝) -𝑠) ∫ 𝐹(𝑥)𝑑𝑥 𝑞 0 -𝑔𝑇𝐶(1 + 𝑟 𝐶,𝐵 ) -(𝐿 𝐵 -𝑔𝑇𝐶)(1 + 𝑟 𝐵 ) - 𝐿 𝑇 (1 + 𝑟 𝑀 ) -𝑇. ( 25
)
It can be expressed as follows when we substitute 𝐿 𝐵 , 𝐿 𝑆 , and 𝑟 𝐶,𝐵 .

𝜋 𝑟,𝑃,𝐼𝐼 = 𝐸(𝑝)𝑞 -(𝐸(𝑝) -𝑠) ∫ 𝐹(𝑥)𝑑𝑥 𝑞 0 -𝑔𝑇𝐶(1 + 𝑟 𝐵 -𝑔𝑢ℎ𝑇𝐶) -(𝛼(𝑤𝑞 -𝑇) - 𝑔𝑇𝐶)(1 + 𝑟 𝐵 ) -(1 -𝛼)(𝑤𝑞 -𝑇)(1 + 𝑟 𝑀 ) -𝑇. ( 26
)
Lemma 2. There is no optimal ratio for the retailer to decide. When 𝑟 𝑀 > 𝑟 𝐵 , the retailer's profit increases in 𝛼; When 𝑟 𝑀 < 𝑟 𝐵 , the retailer's profit decreases in 𝛼.

Considering a given 𝛼, we study the management decision by the retailer. Proposition 8 is given.

Proposition 8. In a decentralized supply chain, the retailer is given preferential credit and finances by bank credit and trade credit portfolio. When the bank loan is higher than the credit line, the optimal order quantity satisfies the first-order optimality condition of its expected profit, which is given as follows:

𝑎 -2𝑏 ∫ 𝐹 ̅ (𝐷)𝑑𝐷 𝑞 𝑃,𝐼𝐼 * 0 -𝑠 = 𝑤(1+𝑟 𝑀 +𝛼(𝑟 𝐵 -𝑟 𝑀 ))-s 𝐹 ̅ (𝑞 𝑃,𝐼𝐼 * ) . ( 27
)
Equation [START_REF] Kouvelis | Who Should Finance the Supply Chain? Impact of Credit Ratings on Supply Chain Decisions[END_REF] give the optimal order quantity when retailers use trade credit and bank credit simultaneously in the condition of that bank loan is higher than credit line. It shows that the optimal order quantity in scenario II is affected by 𝑟 𝑀 , 𝑟 𝐵 , and 𝛼. There is no effect of collateral and risk preference. This Corollary shows the effect of bank financing ratio, trade credit interest rate and bank interest rate on order decision. With high cost of financing, order quantity decreases.

Manufacturer's problem

At the end of the selling period, manufacturer will receive extra interest revenue with 𝐿 𝑇 𝑟 𝐶,𝑀 in scenario I and 𝐿 𝑇 𝑟 𝑀 in scenario II. Thus, based on Equation ( 13), we obtain the manufacturer's profit function as follows:

𝜋 𝑚,𝑃,𝐼 = (𝑤 -𝑘)𝑞 + 𝐿 𝑇 𝑟 𝐶,𝑀 . Scenario I 𝜋 𝑚,𝑃,𝐼𝐼 = (𝑤 -𝑘)𝑞 + 𝐿 𝑇 𝑟 𝑀 . Scenario II [START_REF] Kyparisis | The price-setting newsvendor problem with nonnegative linear additive demand[END_REF] When we substitute 𝐿 𝑇 , 𝑟 𝐶,𝑀 into Equation ( 28), it could be written as:

𝜋 𝑚,𝑃,𝐼 = (𝑤 -𝑘)𝑞 + (1 -𝛼)(𝑤𝑞 -𝑇)(𝑟 𝑀 -𝑒𝑢 × (𝑔𝑇𝐶 -𝛼(𝑤𝑞 -𝑇))). Scenario I 𝜋 𝑚,𝑃,𝐼𝐼 = (𝑤 -𝑘)𝑞 + (1 -𝛼)(𝑤𝑞 -𝑇)𝑟 𝑀 .
Scenario II [START_REF] Lau | How a dominant retailer might design a purchase contract for a newsvendor-type product with price-sensitive demand[END_REF] The Equation ( 29) represents the manufacturers' profit in Scenario I and Scenario II when retailers finance inventory from the portfolio of bank loan and trade credit. It also helps to identify the optimal manufacturers' operational decision. The manufacturer's optimal wholesale price in two scenarios can be given respectively in Equation ( 30) and Equation [START_REF] Melnyk | Supply chain design: issues, challenges, frameworks and solutions[END_REF]. Under the condition of a decentralized supply chain, the retailer and the manufacturer decide the order quantity and wholesale price based on maximizing profit, respectively.

We study the management decision and firms' profit, considering the retailer's collateral and different discount rate loan, which is given by bank (BC), manufacturer (TC), and portfolio credit (PC).

Effect of bank risk preference level

The bank risk preference level (h) would affluence the discount bank interest rate in BC and PC, and in turn, it can change the decision by retailer. We summarize the effect of bank risk preference on optimal order quantity, wholesale price, expected sale price, retailer's profit, manufacturer's profit, and the supply chain profit with bank credit and portfolio credit in Table 2.

For bank credit, under the conditions of case 1, bank risk preference has a positive effect on the order quantity and wholesale price but leads to a decrease in expected sale price. The retailer's profit, manufacturer, and supply chain profit increase with bank risk preference. Under the conditions of case 2, order quantity, wholesale price, and expected sale price are not be affected by bank risk preference. Moreover, the manufacturer's profit is not influenced by bank risk preference. Nevertheless, the rise of banks' risk preference levels will promote the profit of the retailer and the supply chain.

[Please insert Table 2 about here]

When we consider portfolio credit, for the scenario I, the bank risk preference has a similar effect on management decisions of retailer and manufacturer to bank credit of case 1. The retailer, manufacturer, and supply chain profit all increase with bank risk preference. As to scenario II, bank risk preference's effect on operational decision is similar to bank credit of case 2.

Effect of manufacturer risk preference level

We investigate the effect of manufacturer risk preference level under trade credit and portfolio credit and analyze the results in this subsection. The result is showed in Table 3.

[Please insert Table 3 about here]

As seen from Table 3, considering trade credit, we conclude that the manufacturer's risk preference has no effect on order decision and expected sale price. However, this can improve the wholesale price given by the manufacturer. A higher risk preference manufacturer offers trade credit to the retailer, leading to dwindling in the retailer's profit.

The manufacturer's profit increases with her risk preference. The supply chain profit will not change with the manufacturer's risk preference.

Considering portfolio credit, under conditions of scenario I, we learn that the order quantity decreases with the manufacturer's risk preference. This result has a positive effect on wholesale price and expected sale price. With the rise in the manufacturer's risk preferences, both retailer and supply chain profit decrease. Manufacturer's profit increases. In scenario II, because there is no preferential trade credit, the manufacturer's risk preference has no effect on operational decisions.

Effect of collateral

The effect of the retailer's collateral will be illustrated in this subsection. We set different ratios of bank credit (𝛼 = 0.5andα = 0.8) in portfolio credit conditions. The bank risk preference (h) amounts to 5, and the manufacturer risk preference (e) is 15. We compare different finance methods with collateral change.

[Please insert Figure 3 about here]

As can be seen in Figure 3(a), the optimal order quantity increases in retailer's collateral under BC and PC condition. The quantity may not change under TC. However, the order quantity under TC is the highest than the others. For PC conditions, with the increasing ratio of bank credit, the retailer will order less quantity from the manufacturer. Figure 3(b) shows the different effects of collateral on wholesale price. The wholesale price increases dramatically with the retailer's collateral under trade credit. For other financing methods, the wholesale price charged by the manufacturer increases slightly. With the increase in 𝛼, wholesale price rises. For trade credit conditions, the wholesale price is the lowest, and it is highest in bank credit conditions. The sale price is shown in Figure 3(c).

Under TC, the sale price is the lowest with no change as collateral changing. For other financing methods, sale price decreases in collateral. With the increasing of 𝛼, sale price increases. Of interest, for bank credit in case 2 and portfolio credit in scenario II, the order quantity, wholesale price, and sale price will not change with the retailer's collateral.

[Please insert Figure 4 about here]

From Figure 4, we infer that the retailer that owes more collateral may not increase his profit with preferential credit. The discount interest rate may transfer to the manufacturer by pay for higher-order product expenses. However, the total supply chain profit will increase with collateral. As we have seen in Figure 4(a), the retailer's profit in TC financing conditions is higher than the others, although it decreases sharply with collateral. Under bank credit conditions, the retailer's profit increases slightly in case 1 but increases dramatically in case 2. The profit in case 2 is higher than that in case 1.

Further, the retailer's profit under PC condition is located between those under BC and TC conditions. With the increasing of 𝛼, the profit will decrease. From Figure 4(b), the manufacturer has the highest profit under TC and gains the lowest profit under BC. The profit of the manufacturer is increasing with collateral under all financing methods. For supply chain profit, as seen in Figure 4(c), the collateral has no effect on that under trade credit. By trade credit, the supply chain has the highest profit than others. For bank credit and portfolio credit, the supply chain profit increases with the retailer's collateral.

Furthermore, under bank credit conditions, the profit of the supply chain is the lowest.

We further discuss the other effects of the retailer's collateral on financing. Figure 5 gives collateral's influences to the expected interest rate, total interest, and total loan of retailers. It is crucial for us to explore the impacts of preferential credit from banks on management decisions.

[Please insert Figure 5 about here]

From Figure 5, we find that the expected rate of different financing methods decreases in collateral. The interest rate in trade credit is initially the highest. We conclude that although the trade credit owes the highest financing unit cost and needs to be paid the highest interest, the retailer financing by trade credit would be the most profitable. In this way, trade credit is better than bank credit. We propose that the retailer's decision needs to focus on choosing financing methods rather than only considering the interest rate and discount. The benefits of a discount loan interest rate may ultimately transfer to the manufacturer.

Conclusion

In this paper, we examine the effect of preferential credit on management decisions for capital-constraint retailers. In the competitive market, the bank offers preferential credit for high-quality retailers with collateral. The manufacturer also offers trade credit of discount interest rate based on the retailer's credit line, which is evaluated by the bank. The bank risk preference and manufacturer risk preference may affect the discount interest rate of credit. The effect of preferential credit primarily depends on the retailer's collateral and financing institute's risk preference. We investigate which financing methods would be the best for the retailer to finance, Bank credit, trade credit, or financing from both banks and manufacturers. Depending upon the loan amount compared to the credit line, financing in bank credit has two cases, and portfolio credit also has two scenarios in our model based on the relationship between credit line and bank loan. We model different situations to investigate operational decisions.

Furthermore, a quantity-discount contract to coordinate the supply chain is considered in our model.

We find that with increasing bank risk preference levels, the retailer's optimal order quantity will increase from bank credit due to the decreasing of financing cost. The retailer's profit will increase in both case 1 or case 2. Both the wholesale price and manufacturer's profit increase with bank risk preference level. The retailer's collateral has a positive effect on supply chain decisions. However, under trade credit, the manufacturer's risk preference level does not affect order strategies and retailer profit greatly. It can only improve the wholesale price and make the manufacturer's profit growth. The benefit that transfers to the manufacturer leads to harm to the retailer's profit. There are no differences in the total supply chain profit. The retailer's collateral will lead to a decreasing of the retailer's profit. But the total supply chain will increase. For portfolio credit, with the increasing of manufacturer risk preference, the retailer may order less, and the profit will be down when the part of bank loan exceeds the preferential credit line. Moreover, collateral has a negative to the retailer's decision. More collateral may not provide much profit directly to the retailer. There is no such effect in scenario I of risk preference and collateral. Consider different financing methods, and the trade credit is the most amicable for the retailer even though the financing cost is the highest.

In this paper, we investigate the effects of bank credit and trade credit under preferential credit situations in supply chain management. There are some attractive further research opportunities, including (i) credit finance with multiple manufacturers;

(ii) dual-channel supply chain situations; (iii) trade credit from asymmetric manufacturers with competition. In the future, we would consider to investigate the impact of preferential credit from asymmetric manufacturers with competition on capital-constrained retailers. We hope this paper can guide the future management of capital-constrained supply chains from the perspective of real-world managers.

retailer's expected profit function can be expressed by the following:

𝑎 -2𝑏 ∫ 𝐹 ̅ (𝐷)𝑑𝐷 𝑞 𝐵,1 * 0 -𝑠 = 𝑤(1+𝑟 𝐵 -𝑔𝑢ℎ𝑇𝐶)-s 𝐹 ̅ (𝑞 𝐵,1 * )
, when

𝜕𝜋 𝑟,𝐵,1 𝜕𝑞 = 0.
The second-order optimality condition of the retailer's expected profit function can be expressed by the following:

𝜕 2 𝜋 𝑟,𝐵,1 𝜕𝑞 2 = {ℎ(𝑞)[2𝑏 ∫ 𝐹 ̅ (𝐷)𝑑𝐷 𝑞 0 -𝑎 + 𝑠] -2𝑏𝐹 ̅ (𝑞) 2 }𝐹 ̅ (𝑞).
Because of the assumption 𝑤 > 𝑠 and 𝑟 𝐶,𝐵 = 𝑟 𝐵 -𝑔𝑢ℎ𝑇𝐶 > 0 we consider 𝑤(1 + 𝑟 𝐵 -𝑔𝑢ℎ𝑇𝐶) -s > 0. Thus, 2𝑏 ∫ 𝐹 ̅ (𝐷)𝑑𝐷 𝑞 0

-𝑎 + 𝑠 < 0. Due to ℎ(𝑞) increase in q and greater than zero. According to 𝐹 ̅ (𝑞) > 0, we have -2𝑏𝐹 ̅ (𝑞) 2 < 0. We obtain 𝜕 2 𝜋 𝑟,𝐵,1 𝜕𝑞 2 < 0, so the retailer's expected profit function is a concave with q. The value of order quantity under first-order conditions can surely maximize the retailer's profit.

Proof of Corollary 1. (Effect of bank decision)

The first derivative of q with respect to h according to ( 9) is given as follows: With given 𝑤, 𝑔, 𝑟 𝐵 ,and TC, optimal order quantity increases in bank risk preference.

The first derivative of q with respect to 𝑟 𝐵 is given as follows:

𝜕𝑞 𝐵,1 * 𝜕𝑟 𝐵 = - 𝑤 2𝑏𝐹 ̅ (𝑞 𝐵,1 * ) 2 +[𝑤(1+𝑟 𝐵 -𝑔𝑢ℎ𝑇𝐶)-𝑠]ℎ(𝑞 𝐵,1 * ) < 0.
With given 𝑤, 𝑔, ℎ,and TC, optimal order quantity decreases in bank interest rate.

Proof of Corollary 2. (Effect of collateral)

Taking the first derivative of q with respect to TC according to (9) gives 2𝑏𝐹 ̅ (𝑞 𝐵,1 * ) > 0.

Similarly, we obtain the relationship between 𝑞and collateral rate (g) by taking the first-order condition of (9).

We obtain 2𝑏𝐹 ̅ (𝑞 𝐵,1 * ) ∂𝑤 2 < 0. Thus, the manufacturer has a maximized profit when deciding the optimal wholesale price.

Proof of Proposition 5. Taking the first-and second-order derivations of the retailer's profit in trade credit scenario for q, result in the following:

𝜕𝜋 𝑟,𝑇 𝜕𝑞 = -2𝑏𝐹 ̅ (𝑞) ∫ 𝐹 ̅ (𝐷)𝑑𝐷 𝑞 0 + 𝑎𝐹 ̅ (𝑞) + 𝑠𝐹(𝑞) -(1 + 𝑟 𝑀 -𝑒𝑢𝑔𝑇𝐶)𝑤, and 𝜕 2 𝜋 𝑟,𝑇 𝜕𝑞 2 = {ℎ(𝑞)[2𝑏 ∫ 𝐹 ̅ (𝐷)𝑑𝐷 𝑞 0 -𝑎 + 𝑠] -2𝑏𝐹 ̅ (𝑞) 2 }𝐹 ̅ (𝑞).
Reminding that 2𝑏 ∫ 𝐹 ̅ (𝐷)𝑑𝐷 𝑞 0 -𝑎 + 𝑠 < 0, ℎ(𝑞) > 0, and 𝐹 ̅ (𝑞) > 0, it is easy to prove 𝜕 2 𝜋 𝑟,𝑇 𝜕𝑞 2 < 0. Thus, 𝜋 𝑟,𝑇 is concave with q. The optimal order quantity can be obtained by Taking the first derivative of 𝑞 𝑇 * respect to 𝐿 𝐶 , it can be written as follows:

𝜕𝑞 𝑇 * 𝜕(𝐿 𝐶 ) = 𝜕𝑞 𝑇
Substituting the above equations into Equation ( 19), we obtain 𝜕2 𝜋 𝑚,𝑇 𝜕𝑤 2 < 0. Hence, the manufacturer's profit function is a concave respect to w. The optimal wholesale price can be given by . As we take the second-order derivative of the retailer's profit function respect to 𝛼, it can be expressed by the following:

𝜕 2 𝜋 𝑟,𝑃,𝐼 𝜕𝛼 2 = 2𝑢𝑒(𝑤𝑞 -𝑇) 2 > 0. Thus, the profit of the retailer has a minimized value with 𝛼.

Proof of Proposition 7. From Equation ( 22), we take the first-order derivation of retailer's profit with respect to the order quantity, which can be expressed as follows: Proof of Proposition 8. We take the first derivative of retailer's profit function with respect to q, and the optimal order quantity can be written by the following: .

We have

  and Thuyn (2014) used a game-theoretic model to analyze surplus division in supply chains. Van dere Vliet et al. (2015) provided a periodic review inventory model with reverse factoring. In China, trade credit might be the only method for capital constraint enterprises to satisfy their financing demands. These SMEs' trade credit comes primarily from big Internet companies such as Baidu, Alibaba, and Tencent.

Corollary 3 .

 3 (Effect of manufacturer decision) (i) For a given wholesale price, optimal order quantity decreases in w under trade credit financing, Consider credit line 𝐿 𝐶 , 𝑞 𝑇 * increases in 𝐿 𝐶 .

22 ) 1 .

 221 𝜋 𝑟,𝑃,𝐼 = 𝐸(𝑝)𝑞 -(𝐸(𝑝) -𝑠) ∫ 𝐹(𝑥)𝑑𝑥 𝑞 0 -𝐿 𝐵 (1 + 𝑟 𝐶,𝐵 ) -𝐿 𝑇 (1 + 𝑟 𝐶,𝑀 ) -𝑇. (21) Considering 𝑟 𝐶,𝐵 = 𝑟 𝐵 -𝑔𝑢ℎ𝑇𝐶 , 𝑟 𝐶,𝑀 = 𝑟 𝑀 -𝑒 × 𝑢(𝑔𝑇𝐶 -𝐿 𝐵 ), 𝐿 𝐵 = 𝛼(𝑤𝑞 -𝑇), 𝐿 𝑇 = (1 -𝛼)(𝑤𝑞 -𝑇), Equation (21) can be written as follows: 𝜋 𝑟,𝑃,𝐼 = 𝐸(𝑝)𝑞 -(𝐸(𝑝) -𝑠) ∫ 𝐹(𝑥)𝑑𝑥 𝑞 0 -𝛼(𝑤𝑞 -𝑇)(1 + 𝑟 𝐵 -𝑔𝑢ℎ𝑇𝐶) -(1 -𝛼)(𝑤𝑞 -𝑇) (1 + 𝑟 𝑀 -𝑒 × 𝑢(𝑔𝑇𝐶 -𝛼(𝑤𝑞 -𝑇))) -𝑇. (Lemma There exists an 𝛼 that could minimize the retailer's profit. Whether the retailer would use bank or trade credit depends on risk preference level and basic interest rate.

Corollary 5 .

 5 (Effect of 𝜶) In scenario II, optimal order quantity increases in 𝛼, 𝜕𝑞 𝑃,𝐼𝐼 * 𝜕𝛼 > 0; 𝑟 𝐵 and 𝑟 𝑀 both have adverse effect to 𝑞 𝑃,𝐼𝐼 * .

Proposition 9 .

 9 In a decentralized supply chain, the retailer receives preferential credit from the bank and finances by bank credit and trade credit portfolio. The wholesale price in Scenario I (𝑤 𝑃,𝐼 7. Managerial implications In this section, numerical experiments are given to explore the operational decision in the decentralized supply chain. Due to the fact that the detailed data of trade credit contract is non-publicize, especially about the SMEs. It is hard to test the impact of preferential credit on supply chain decision with empirical distributions. In fact, many researches always consider the numerical experiments with stochastic demand distribution to exam the result (Kouvelis and Zhao, 2012; Yang and Birge, 2018; Wu et al., 2019). We assume that the demands are uniform distribution, with D~U (0, 100) and give other parameters as follows: a = 14, b = 0.09, 𝑟 𝐵 = 0.08, s = 2, g = 0.4, k= 5, T = 50.

.

  𝜕𝜋 𝑟,𝑇 𝜕𝑞 = 0 as follows: 𝑎 -2𝑏 ∫ 𝐹 ̅ (𝐷)𝑑𝐷 𝑞 𝑇 * 0 -𝑠 = 𝑤(1+𝑟 𝑀 -𝑒𝑢𝑔𝑇𝐶)-s 𝐹 ̅ (𝑞 𝑇 * ) Proof of Corollary 3. (Effect of manufacturer decision) Based on implicit function derivation rule, we take the first derivative of (18) concerning w, 𝑟 𝑀 , and 𝑒. It is easy to prove of collateral) Based on implicit function derivation rule,we take the first derivative of (18) with respect to g and 𝑇𝐶. We obtain

1 .

 1 From Equation[START_REF] Jacobson | Trade Credit and the Propagation of Corporate Failure: An Empirical Analysis[END_REF], taking the first derivative of the retailer's profit function with respect to the ratio of bank credit (𝛼), we obtain: 𝜕𝜋 𝑟,𝑃,𝐼 𝜕𝛼 = (1 + 𝑟 𝐵 -𝑔𝑢ℎ𝑇𝐶) + [-1 -𝑟 𝑀 + 𝑒𝑢(𝑔𝑇𝐶 + 𝑤𝑞 -𝑇) -2𝛼𝑒𝑢(𝑤𝑞 -𝑇)]. When 𝜕𝜋 𝑟,𝑃,𝐼 𝜕𝛼 = 0, we have 𝛼 * = 𝑟 𝐵 -𝑟 𝑀 +𝑔𝑇𝐶(𝑒-ℎ)𝑢 2𝑒𝑢(𝑤𝑞-𝑇) + 1 2

Proof of Lemma 2 . 0 .

 20 ̅ (𝑞) ∫ 𝐹 ̅ (𝐷)𝑑𝐷 𝑞 0 + 𝑎𝐹 ̅ (𝑞) -𝑠𝐹 ̅ (𝑞) + 𝑠 -(1 + 𝑟 𝐵 -𝑔𝑢ℎ𝑇𝐶)𝛼𝑤 -(1 -𝛼){𝑤[1 + 𝑟 𝑀 -𝑒𝑢[𝑔𝑇𝐶 -𝛼(𝑤𝑞 -𝑇)]] + 𝑒𝑢𝛼𝑤(𝑤𝑞 -𝑇)}. It is easy to prove 𝜕 2 𝜋 𝑟,𝑃,𝐼 𝜕𝑞 2 < 0 according to Equation (22). When 𝜕𝜋 𝑟,𝑃,𝐼 𝜕𝑞 = 0, the optimal order quantity can be expressed as follows: +𝑠-𝛼𝑤(1+𝑟 𝐵 -𝑔𝑢ℎ𝑇𝐶)-𝑤(1-𝛼)(1+𝑟 𝑀 -𝑒𝑢𝑔𝑇𝐶) 2𝑤 2 𝛼(1-𝛼)𝑒𝑢 . Taking the first-order condition of the retailer's profit with respect to 𝛼, we obtain 𝜕𝜋 𝑟,𝑃,𝐼𝐼 𝜕𝛼 = (𝑤𝑞 -𝑇)(𝑟 𝑀 -𝑟 𝐵 ). When 𝑟 𝑀 > 𝑟 𝐵 , Thus, we have 𝜕𝜋 𝑟,𝑃,𝐼𝐼 𝜕𝛼 > The profit of the retailer increases in 𝛼. When 𝑟 𝑀 < 𝑟 𝐵 , we obtain 𝜕𝜋 𝑟,𝑃,𝐼𝐼 𝜕𝛼 < 0. The retailer's profit decreases in 𝛼.

  (𝑞) -𝑠𝐹 ̅ (𝑞) + 𝑠 -(1 + 𝑟 𝐵 )𝛼𝑤 -(1 -𝛼)𝑤(1 + 𝑟 𝑀 ) = 0. This can also be expressed as follows: 𝑀 +𝛼(𝑟 𝐵 -𝑟 𝑀 ))-s 𝐹 ̅ (𝑞 𝑃,𝐼𝐼 * )

𝜕 2 𝜋 2 =Proof of Proposition 9 . 2 ( 2 } 2 . 2 < 0 .

 22922220 𝑟,𝑃,𝐼𝐼 𝜕𝑞 𝐹 ̅ (𝑞) [ 𝑠-𝑤[1+𝑟 𝑀 +𝛼(𝑟 𝐵 -𝑟 𝑀 )] 𝐹 ̅( 𝑞) ℎ(𝑞) -2𝑏𝐹 ̅ (𝑞)] < 0. Proof of Corollary 5. (Effect of 𝜶) According to Equation (27), when we take the firstorder condition of q with respect to 𝛼, 𝑟 𝐵 , and 𝑟 𝑀 , we can easily obtain Scenario I. By substituting[START_REF] Klapper | Trade Credit Contracts[END_REF] into[START_REF] Lau | How a dominant retailer might design a purchase contract for a newsvendor-type product with price-sensitive demand[END_REF], we take the firstorder condition of manufacturer's profit with respect to w, when ∂𝜋 𝑚,𝑃,𝐼 ∂w = 0. we obtain𝑞 𝑃,𝐼 * (1 + 𝑟 𝐶,𝑀 (1 -𝛼) + 𝛼𝑒𝐿 𝑇 ) + 𝛼(1+𝑟 𝐶,𝐵 )+(1-𝛼)(1+𝑟 𝐶,𝑀 )+𝛼𝑢𝑒𝐿 𝑇 +2𝑒𝑢𝛼𝑤(1-𝛼)𝑞 𝑃,𝐼 * 𝑓(𝑞 𝑃,𝐼 * )(2𝑏 ∫ 𝐹 ̅( 𝑤(1 + 𝑟 𝐶,𝑀 (1 -𝛼) + 𝛼𝑒𝑢𝐿 𝑇 ) -𝑘) = 0, where 𝑞 𝑃,+𝑠-𝛼𝑤(1+𝑟 𝐵 -𝑔𝑢ℎ𝑇𝐶)-𝑤(1-𝛼)(1+𝑟 𝑀 -𝑒𝑢𝑔𝑇𝐶) 2𝑤 2 𝛼(1-𝛼)𝑒𝑢 .Substituting the above equations into Equation (29), we obtain ∂ 2 𝜋 𝑚,𝑃,𝐼 ∂w 2 < 0. Hence, the manufacturer's profit function is a concave with respect to w. The optimal wholesale price in Scenario I can be given by 𝜕𝜋 𝑚,𝑃,𝐼 𝜕𝑤 = 0. Similarly, by substituting Equation[START_REF] Kouvelis | Who Should Finance the Supply Chain? Impact of Credit Ratings on Supply Chain Decisions[END_REF] into Equation[START_REF] Lau | How a dominant retailer might design a purchase contract for a newsvendor-type product with price-sensitive demand[END_REF], we obtain the wholesale price in Scenario II can be given by 𝜕𝜋 𝑚,𝑃,𝐼𝐼 𝜕𝑤 = 0, where𝜕 2 𝜋 𝑚,𝑃,𝐼𝐼 𝜕𝑤 2 = 2(1 + (𝑟 𝑀 -𝑒𝑢𝑔𝑇𝐶)(1 -𝛼)) + 𝜕 2 𝑞 𝑃,𝐼𝐼 * 𝜕𝑤 2 (𝑤(1 + (𝑟 𝑀 -𝑒𝑢𝑔𝑇𝐶)(1 -𝛼)) -𝑘).Based on Equation (24), we obtain follows: 𝜕𝑞 𝑃,𝐼𝐼 * 𝜕𝑤 = 1+𝑟 𝑀 +𝛼(𝑟 𝐵 -𝑟 𝑀 ) ℎ(𝑞 𝑃,𝐼𝐼 * )[𝑠-𝑤(1+𝑟 𝑀 +𝛼(𝑟 𝐵 -𝑟 𝑀 ))]-2𝑏𝐹 ̅ (𝑞 𝑃,𝑟 𝐵 -𝑟 𝑀 )+𝑟 𝑀 ]{ℎ ′ (𝑞 𝑃,𝐼𝐼 * )[𝑠-𝑤(1+𝑟 𝑀 +𝛼(𝑟 𝐵 -𝑟 𝑀 ))] 𝜕𝑞 𝑃,𝐼𝐼 * 𝜕𝑤 -ℎ(𝑞 𝑃,𝐼𝐼 * )[1+𝛼(𝑟 𝐵 -𝑟 𝑀 )+𝑟 𝑀 ]+4𝑏𝐹 ̅ (𝑞 𝑃,𝐼𝐼 * )𝑓(𝑞 𝑃,𝑠-𝑤[1+𝑟 𝑀 +𝛼(𝑟 𝐵 -𝑟 𝑀 )]]-2𝑏𝐹 ̅ (𝑞 𝑃,𝐼𝐼 * ) Substitute above into 𝜕 2 𝜋 𝑚,𝑃,𝐼𝐼 𝜕𝑤 2 . We obtain 𝜕 2 𝜋 𝑚,𝑃,𝐼𝐼𝜕𝑤 Thus, we have Equation[START_REF] Melnyk | Supply chain design: issues, challenges, frameworks and solutions[END_REF].

( a )

 a Effect of TC. (b) Effect of h.

Figure 1 . Comparison of order quantity in case 1 and case 2 (

 12 Figure 1. Comparison of order quantity in case 1 and case 2

Figure 2 .

 2 Figure 2. Effect of 𝛂

Figure 3 .

 3 Figure 3. The effect of TC on order quantity, wholesale price, and expected sale price

Figure 4 .

 4 Figure 4. The effect of TC on profit.

Figure 5 .

 5 Figure 5. The effect of TC on loan.

  ; Gupta and Dutta, 2011; Breza and Liberman, 2013; Zhan et al., 2019. Dada and Hu (2008) considered a capital constraint newsvendor model where the retailer borrows funds at a low-interest rate, leading to drop in retailer equity. Kouvels and Zhao (2011) explored retailer bankruptcy risk when the retailer was financially constrained and failed to repay the loan. With bankruptcy risk, the wholesale price would be raised. When bankruptcy risk is not present, the manufacturer may keep the wholesale price stable. Trade credit in competitive markets has been found to bring extra benefits to supply chains (Rice and Strahan, 2010; Peura et al., 2017; Chod et al., 2019).

Table 1 . Key notation

 1

	Notation	Definition
	p	Sales price
	E(p)	Expected sales price
	w	Wholesale price
	k	Unit production cost
	s	Buyback price
	T	Retailer's initial capital
	𝜋 𝑟	Expected retailer's profit
	𝜋 𝑚	Expected manufacturer's profit
	𝑟 𝐵	Bank interest rate
	𝑟 𝑀	Trade credit interest rate
	𝑟 𝐶,𝐵	Bank discount interest rate
	𝑟 𝐶,𝑀	Trade credit discount interest rate
	𝑟	Expected interest rate
	q	Ordering quantity decided by the retailer
	TC	Collateral
	g	Mortgage rate
	D	Stochastic demand
	𝐿 𝐵	Bank credit
	𝐿 𝑇	Trade credit
	𝐿 𝐶	Preferential credit line
	h	Risk preference level of bank
	e	Risk preference level of manufacturer

Table 2 . Effect of bank's risk preference on management decision with different financing

 2

	Financing methods		q w	p	𝜋 𝑟	𝜋 𝑚 𝜋 𝑠
		Case 1	↑ ↑	↓	↑	↑	↑
	Bank credit	Case 2	∽ ∽	∽ ↑	∽ ↑
		Scenario I ↑ ↑	↓	↑	↑	↑
	Portfolio credit	Scenario II ∽ ∽	∽ ↑	∽ ↑

Table 3 . Effect of the manufacturer's risk preference on management decision with different financing

 3

	Financing methods		q w	p	𝜋 𝑟	𝜋 𝑚 𝜋 𝑠
	Trade credit		∽ ↑	∽ ↓	↑	∽
		Scenario I	↓ ↑	↑	↓	↑	↓
	Portfolio credit	Scenario II ∽ ∽	∽ ∽	∽ ∽

+ 𝑞 𝐵,1 * = 0. in case 1.

+ 𝑞 𝑇 * = 0.

-Boeing-Citi-Start-Export-Import-Bank-Supplier-Financing-Program https://decisionsciences.org/two-keynotes-to-highlight-2019-conference/ -𝑠)+𝑠-𝛼𝑤(1+𝑟 𝐵 -𝑔𝑢ℎ𝑇𝐶)-𝑤(1-𝛼)(1+𝑟 𝑀 -𝑒𝑢𝑔𝑇𝐶) 2𝑤 2 𝛼(1-𝛼)𝑒𝑢 . and 𝑎 -2𝑏 ∫ 𝐹 ̅ (𝐷)𝑑𝐷 𝑞 𝑃,𝐼𝐼 * 0 -𝑠 = 𝑤(1+𝑟 𝑀 +𝛼(𝑟 𝐵 -𝑟 𝑀 ))-s 𝐹 ̅ (𝑞 𝑃,𝐼𝐼 * )

.

//www.fthttp://finance.sina.com.cn/roll/2019-01-16/doc-ihqfskcn7466096.shtml See: http://www.


Saman Vafadar
email: samanvafadar@yahoo.com

Wafa Skalli
email: wafaskalli@ensam.eu

Aurore Bonnet-Lebrun
email: aurore.bonnet-lebrun@ensam.eu

Marc Khalifé
email: marckhalife@icloud.com

Mathis Renaudin
email: mathis.renaudin@ensam.eu

Amine Hamza
email: aminehamza@hotmail.com

Laurent Gajny
email: laurent.gajny@ensam.eu

A novel dataset and deep learning-based approach for marker-less motion capture during gait

Introduction

Quantitative gait analysis is an effective tool for diagnosis and prognosis in clinical issues like cerebral palsy [START_REF] Benedetti | SIAMOC position paper on gait analysis in clinical practice: General requirements, methods and appropriateness. Results of an Italian consensus conference[END_REF]- [START_REF] Wren | Efficacy of clinical gait analysis: A systematic review[END_REF]. Also, it may assist in the diagnosis and succeeding therapy of individuals at risk of falls [START_REF] Paterson | Stride dynamics, gait variability and prospective falls risk in active community dwelling older women[END_REF], [START_REF] Toebes | Local dynamic stability and variability of gait are associated with fall history in elderly subjects[END_REF]. The current established gold standard systems for gait analysis are marker-based motion capture systems. However, their recognized interest, their routine clinical use is limited to a few highly specialized centers for various reasons: labor costs for data acquisition and processing, equipment costs, the need for a controlled laboratory environment, and the time-consuming testing procedure [START_REF] Simon | Quantification of human motion: gait analysis-benefits and limitations to its application to clinical problems[END_REF].

Because of these limitations, marker-less motion capture systems are proposed as cost-effective and easy-to-use alternatives. The marker-less systems mainly utilize two types of cameras, either depth cameras or RGB cameras, generally based on three-dimensional (3D) human pose estimation methods. Such methods refer to processing two-dimensional (2D) images, captured by a single (or multiple) depth camera or RGB camera, to locate a set of anatomical landmarks (or joint centers) in the three-dimensional capture volume.

Clark et al. [START_REF] Clark | Three-dimensional cameras and skeleton pose tracking for physical function assessment: A review of uses, validity, current developments and Kinect alternatives[END_REF] authored a recent literature review on the uses, validity, and developments of depth cameras, especially the Kinect camera. For example, for the overground gait, it was reported that the Kinect's validity for most of the kinematic gait parameters was restricted (r<0.75) [START_REF] Springer | Validity of the Kinect for Gait Assessment: A Focused Review[END_REF]. Also, based on several studies [START_REF] Pfister | Comparative abilities of Microsoft Kinect and Vicon 3D motion capture for gait analysis[END_REF], [START_REF] Xu | Accuracy of the Microsoft Kinect TM for measuring gait parameters during treadmill walking[END_REF], substantial errors were found in assessing kinematic gait parameters (RMS error > 10 degrees or Bland-Altman limits of agreement > 10 degrees).

Concerning RGB cameras, in recent years, human pose estimation methods based on artificial intelligence, especially deep learning, have demonstrated significant progress [START_REF] Cao | OpenPose: Realtime Multi-Person 2D Pose Estimation using Part Affinity Fields[END_REF]- [START_REF] Qiu | Cross View Fusion for 3D Human Pose Estimation[END_REF]. These methods estimate the three-dimensional (3D) position of joint centers (e.g., ankles, knees) using a single or a multi-view (captured by several synchronized cameras) RGB image.

The deep learning-based human pose estimation methods are mostly trained using publicly available three-dimensional human pose datasets, mainly HumanEva [START_REF] Sigal | Humaneva: Synchronized video and motion capture dataset and baseline algorithm for evaluation of articulated human motion[END_REF] and Human3.6M [START_REF] Ionescu | Human3.6M: Large Scale Datasets and Predictive Methods for 3D Human Sensing in Natural Environments[END_REF] datasets. In these datasets, the reference values for joint centers were obtained by marker-based motion capture systems. However, although very rich in terms of the number of gestures and images, they may be less adapted to train the human pose estimation methods for clinical gait study for the following reasons.

First, the subjects wore their regular clothing, and the placement of markers on the regular clothing would reduce the accuracy of joint centers' reference values. Second, the number of subjects is limited. HumanEva and Human3.6M datasets contain the reference values of only four and seven subjects. Moreover, these datasets are limited to adults and do not include the motion of pathological cases. Therefore, a human pose dataset for the specific aim of gait study should be collected.

The objective of this study is to investigate the added value of collecting a dedicated human pose dataset and comparing the performance of a marker-less motion capture system based on the stateof-the-art human pose estimation method when trained on the Human3.6M dataset alone and after finetuning on the new dataset.

Materials and Methods

Setup and calibration of the marker-less system

The gait analysis was focused on the walking part of a 4 m stand up and go test that is routinely used in clinics [START_REF] Kojima | Does the timed up and go test predict future falls among British community-dwelling older people? Prospective cohort study nested within a randomised controlled trial[END_REF]. The proposed marker-less motion capture system was composed of four RGB cameras (GoPro Hero 7 Black) -resolution 1920x1080 pixels, 100 frames per second, and linear field of view.

The cameras were fixed using specific 3D printed fasteners on two aluminum bars. The distance and angle between two cameras on a bar were respectively 75cm and 15°. Each aluminum bar was mounted on a one-meter height tripod. As shown in Figure 1, the first couple of cameras was placed in front of the subject (referred to as frontal cameras) and the second on its right side (referred to as lateral cameras). A calibration checkerboard (calib.io, 9x14, 40mm) was used for intrinsic and extrinsic calibration of the marker-less system. The classical pinhole model [START_REF] Heikkila | A four-step camera calibration procedure with implicit image correction[END_REF] was adopted due to the selection of linear field of view. Around 40 images of the checkerboard were acquired at various positions in the experiment room. The intrinsic (focal lengths, principal points, skew coefficient, and distortion parameters), extrinsic (relative position and orientation of each camera with respect to a global 3D frame) were estimated using the MATLAB Stereo Camera Calibrator App.

At this stage, a 3D frame was defined for the marker-less motion capture system, ensuring its potential for use in a realistic clinical environment. However, for validation purposes, we needed to define a common 3D frame with a reference system.

Setup and calibration of the reference system

A marker-based motion capture system (VICON system, Oxford Metrics, UK) was used as a reference for both training and evaluation. It was composed of twelve cameras. The frequency of data acquisition for data collection of the ENSAM pose dataset was 100 Hz to simplify the synchronization of the marker-less and marker-based systems. The VICON system was calibrated according to the recommendations of the supplier. The VICON 3D frame was adopted as the experiment room frame for both systems (Figure 1).

For that purpose, the calibration checkerboard was equipped with six reflective markers. The relative position of markers and checkerboard corners was measured. To this end, this calibration tool was scanned using a biplane X-ray system (EOS system, EOS imaging Paris France). It enabled the 3D reconstructions of the checkerboard and markers, thanks to an in-house software (Figure 2).

Synchronization

An Arduino board connected to the VICON-based system was used to synchronize the GoPro-based and the VICON-based systems. The Arduino generated a time-dependent signal that turned ON and OFF a set of LEDs placed in the capture space (Figure 1). The LEDs' status, recorded in every frame of the videos, was determined using image processing techniques. Cross-correlation between the retrieved signals from the GoPro cameras and VICON data helped find the temporal offset between the recorded videos and data.

Constitution of the new database

Thirty-one subjects (19 males and 12 females) participated in this study after informed consent. They all had to perform Vicon based gait analysis for research previously approved by the relevant ethics committee (CCPP 06036 and CPP 06001, Paris VI). Subjects were on average 25 years old with a large variation range (range: 6-44 years), mean height was 168 cm (125-199 cm), body mass was 65 kg (range: 30-90 kg), and Body Mass Index (BMI) was 22.5 kg/m 2 (range: 16.9-29.4 kg/m 2 ). Among this population, 22 subjects were asymptomatic, one subject had moderate scoliosis, another had a spondylolisthesis. The remaining seven subjects were children or teenagers (range: 6-15 years) suffering from X-linked hypophosphatemia (XLH). This pathology can cause short stature and bone deformities.

In-vivo data acquisition

The Vicon based gait analysis protocol, as published in [START_REF] Sauret | On the use of knee functional calibration to determine the medio-lateral axis of the femur in gait analysis: Comparison with EOS biplanar radiographs as reference[END_REF], [START_REF] Sangeux | Which method of hip joint centre localisation should be used in gait analysis?[END_REF], was used. Subjects were asked to be with their underwear. Fifty-one reflective markers were placed by an orthopedic surgeon directly on the subjects' skin, at well-defined anatomical landmarks locations. The marker-set was based on [START_REF] Leardini | A new anatomically based protocol for gait analysis in children[END_REF], [START_REF] Davis | A gait analysis data collection and reduction technique[END_REF]. Subjects equipped with markers first underwent a biplanar X-ray examination using the EOS system (EOS Imaging, Paris, France) in a standard standing posture (Figure 3) to get accurate registration between anatomic and markers coordinate systems. Then, subjects were asked to walk for ten walking trials at their comfortable self-selected walking speed, recorded by both motion capture systems. Figure 3. The 3D reconstruction of pelvis, femurs, tibias, fibulas, and markers using the biplanar X-ray images.

Data labeling

The annotation data consisted of the 3D position of 15 joint or segment centers (hereafter, for simplicity called joint centers) -head, neck, shoulders, elbows, wrists, pelvis, hips, knees, and ankles -and subjects' bounding boxes in multi-view images. The annotation data were obtained from the recorded data by the marker-based motion capture system and the EOS imaging system.

The bi-planar radiographs acquired by the EOS system were then processed as described in [START_REF] Pillet | A reference method for the evaluation of femoral head joint center location technique based on external markers[END_REF]. First, 3D shape models of the lower limbs and pelvis were reconstructed [START_REF] Chaibi | Fast 3D reconstruction of the lower limb using a parametric model and statistical inferences and clinical measurements calculation from biplanar X-rays[END_REF], [START_REF] Mitton | 3D reconstruction of the pelvis from bi-planar radiography[END_REF]. Second, the reflective markers were reconstructed by fitting markers' sphere models to the markers contours visible on the bi-planar X-ray images. These two steps helped register the bony segments' anatomical frames, constructed using the biplanar X-ray images, to the marker-based motion capture system's environment (Figure 3).

The upper limbs centers, including head, neck, shoulders, elbows, and wrists, were defined based on the markers in the marker-based motion capture environment. On the other hand, the hips, knees, and ankles centers were determined using the 3D reconstructed shape models obtained using medical imaging. These key points were first defined on the 3D model then registered in the marker-based system's environment.

Once the position of the body joint centers was entirely determined with the marker-based system, the joint centers were projected into the image planes of the marker-less system's cameras. Then, for each frame of each camera, a subject' bounding box was determined using the projected joint centers.

3D joint center detection using a convolutional neural network

The implemented deep learning-based human pose estimation method for the proposed marker-less motion capture system was presented by Iskakov et al. [START_REF] Iskakov | Learnable Triangulation of Human Pose[END_REF] that achieved state-of-the-art results on the Human3.6M dataset [START_REF] Ionescu | Human3.6M: Large Scale Datasets and Predictive Methods for 3D Human Sensing in Natural Environments[END_REF]. The pose estimation method consisted of two main parts, 2D pose estimation and 3D pose reconstruction. The input to the 2D pose estimation -based on a convolutional neural network ResNet-152 [START_REF] He | Deep Residual Learning for Image Recognition[END_REF] -was a multi-view RGB image. The 3D pose reconstruction took the output of the 2D estimation, aggregated in a 3D volume, processed by a convolutional neural network called V2V-PoseNet [START_REF] Moon | V2V-PoseNet: Voxel-to-Voxel Prediction Network for Accurate 3D Hand and Human Pose Estimation From a Single Depth Map[END_REF], and estimated the 3D joint centers.

The new human pose dataset (hereafter, referred to as ENSAM dataset) was split randomly into training (19 subjects -4 XLH, 1 Scoliosis) and test sets (12 subjects -3 XLH, 1 Spondylolisthesis). The deep learning-based pose estimation method was first trained on the Human3.6M dataset, then finetuned on the ENSAM training set. Before and after finetuning, the positions of joint centers were estimated for the ENSAM test set. Agreement between the estimated joint centers and the corresponding reference values was assessed using the Bland-Altman method by determining the mean difference (bias) and 95% limits of agreement along the axes of the global coordinate system (Figure 1). The joint position error, which determines the Euclidean distance between the estimated and reference values, was also measured. In very few cases (less than 0.5% of the total number of frames), clear outliers were detected, probably due to left-right flipped information. As these events were rare and could further be automatically detected and removed, they were arbitrarily removed.

Results

First, the human pose estimation was trained on the Human3.6M dataset and evaluated on the ENSAM test set. Then, the human pose estimation method was finetuned on the ENSAM training set and evaluated on the ENSAM test set. The difference between the estimated joint centers and the corresponding reference values, along the axes of the global coordinate system, is presented in Table 1. The mean differences (biases) were relatively high (within 49.8 mm) and became close to zero (within 3.1 mm) across all joints after finetuning on the ENSAM training set. Also, the 95% confidence intervals (2 standard deviations) were significantly reduced across all joints. For instance, along the Zaxis for the hip joint, the confidence interval was reduced from 106.9 mm to 17.4 mm.

Table 1. Bland-Altman bias and 95% confidence interval (2SD: 2 standard deviation) between the estimated and corresponding reference joint centers on the ENSAM test set

There was no statistically significant difference among the joint position error of twelve subjects (p=0.62). Also, there was no statistically significant difference between the joint position error of pathological subjects and asymptomatic subjects (p=0.58). The joint position errors across all subjects are presented in Table 2.

After finetuning on the ENSAM test set, the ankles were the most accurate (mean error = 6.2 mm).

The second and third most accurate joints were the knees (mean error = 10.3 mm) and the neck (joint position error = 11.3 mm). The least accurate joints were the shoulders (mean error = 21.1 mm). Table 2. Joint position error (Euclidean distance between the estimated joint centers and corresponding reference values) on the ENSAM test set.

Discussion

While gait analysis is of high clinical relevance, there is a lack of methods based on RGB cameras that could be accurate enough to be considered in clinical routine. While deep learning-based pose estimation methods appear attractive, accuracy is still an issue that may be related to the nonspecificity of currently used training datasets. Indeed, publicly available datasets like the Human3.6M dataset have contributed significantly to advancing the human pose estimation methods. However, these datasets are addressed to solve computer vision problems in various situations and contexts, and they are not sufficient for particular applications. Thus, a new human pose dataset (ENSAM dataset), well-adapted for gait study, was collected. In the ENSAM dataset, despite the Human3.6M dataset, the markers were placed directly on subjects' anatomical landmarks by orthopedic surgeons. The bi-planar X-ray images from subjects helped reconstruct the 3D coordinates of the bony segments and the reflective markers, thus allowing accurate registration of joint centers' coordinates to marker-based coordinate systems. This technique reduced the errors of determining joint centers, which were mostly attributed to the marker misplacement on the anatomical landmarks [START_REF] Pillet | A reference method for the evaluation of femoral head joint center location technique based on external markers[END_REF]. Also, while the Human3.6M dataset contains the reference data for only seven young adults, the ENSAM dataset contains the walking trials of thirtyone symptomatic and pathological subjects of different age groups.

This study aimed to assess the role of the ENSAM pose dataset in the advancement and evaluation of marker-less motion capture systems. Comparing the performance of the human pose estimation method on the ENSAM test set, trained on the Human3.6M dataset, before and after finetuning on the ENSAM training set, showed a drastic improvement. For instance, for the ankle joint center, the 95% confidence intervals along the axes of the global coordinate system were within 68.5 mm.

Finetuning the human pose estimation method on the ENSAM training set reduced these values to be less than or equal to 13.3 mm. These improvements may highlight the need for a well-adapted human pose dataset for training the human pose estimation methods which are the basis for the marker-less motion capture systems.

The outliers, which were excluded, were found in three walking trials of one subject. In these three walking trials, the subject was walking in the backward direction (opposite direction of walking toward frontal cameras). These high errors mostly (but not limited to) occurred because the human pose estimation method estimated the left shoulder instead of the right shoulder and vice versa. We arbitrarily manually removed this data because, in future developments, these inconsistencies in the trajectory of a joint center could be easily identified by graphical visualization and filtered out. Except for this case, results appeared as robust. There was no statistically significant difference among the 95% confidence interval of the joint position error of subjects (p=0.57).

As shown in Table 2, the joint centers were not estimated with the same level of accuracy. For instance, the ankles position error was 6.2 mm, whereas the shoulder position error was 21.1 mm.

Several factors may affect the estimation accuracy of different joint centers, including occlusion and the accuracy of joint centers' reference values.

Occlusion occurs when a joint is hidden by another body segment or joint. In the ENSAM pose dataset, in five out of ten walking trials, subjects were walking toward the frontal cameras. Thus, from the lateral cameras' view, the subjects' left elbow and wrist were mostly occluded by the subject's body and could be seen only by the frontal cameras. When a joint center was occluded, the accuracy of estimation was decreased. The mean joint position error of the right elbow and wrist was 14.1 mm and 12.8 mm, whereas the left elbow and wrist's joint position error was 18.8 mm and 16.2 mm. The accuracy of joint centers' reference values can be interpreted as the consistency of joint centers' reference values across different subjects. The shoulders' reference values were retrieved from the markers placed on the acromion that were affected by the marker placement error. The human pose estimation method was trained and assessed using these reference values. Therefore, the less accurate the reference values were, the less accurate the estimations were. The shoulders' joint position errors were the highest. Also, hip joint center error (mean: 17.7 mm) was higher than pelvis center error (mean: 14.0 mm) because the hip joint centers were retrieved from the reconstructed femurs registered to the marker-based motion capture environment using the markers placed on the thighs. The soft tissue artifact associated with the thigh was greater than the pelvis [START_REF] Leardini | Human movement analysis using stereophotogrammetry: Part 3. Soft tissue artifact assessment and compensation[END_REF].

This study had some limitations. The number of subjects is still limited and not representative of the range of subjects with or without musculoskeletal troubles (age <= 44 years, particularly lack of older adults). However, the protocol of data collection is now well defined and further data collection is in progress. Also, in this study, the performance of the proposed marker-less system was only assessed in terms of joint center positions. Work is in progress to evaluate the motion capture system in terms of clinically relevant gait parameters. Another limitation of this study was that the implemented deep learning-based human pose estimation could not estimate the foot centers' position, which is relevant in gait study. The development of the pose estimation method is in progress. Also, in this study, four RGB cameras were used. A fewer number of cameras may make the system easier-to-use. In future studies, various pose estimation methods with a fewer number of cameras should be analyzed.

Conclusion

This study proposed a new dataset that was used for training and evaluation of a marker-less motion capture system based on RGB cameras and deep learning-based human pose estimation methods.

Even with a limited number of subjects in the training set (nineteen subjects), the proposed approach demonstrated the relevance of considering a dedicated thoroughly constituted dataset. The evaluation of the marker-less system on the test set obtained promising results (mean joint position error = 14.4 mm), with potential improvements with the future extension of this dataset to a broader range of populations. Work is in progress for extensive evaluation in terms of clinically relevant gait parameters, thus paving the way for using RGB cameras-based gait analysis.

Figure 1 .

 1 Figure 1. The experimental setup showing the placement of the VICON system's cameras (Vi) and the GoPro cameras (GPi)

Figure 2 .

 2 Figure 2. Left: The calibration device equipped with six markers. Middle: The biplanar X-ray images of the calibration device. Right: The 3D reconstruction for measuring the relative position of markers and checkerboard corners.

Acknowledgments

The authors thank the ParisTech BiomecAM chair program, on subject-specific musculoskeletal modelling and in particular Société Générale and COVEA. We also thank Matthieu Boëssé for his technical help.