On the Supply Chain Coordination with Trade Credit and Partial Backorder under CVaR Criterion

For the secondary supply chain composed of a supplier with sufficient funds and a retailer with limited funds and risk aversion, to enhance the supply chain stability and comprehensively improve benefits, the supply chain members may choose to implement a supply chain’s coordination strategy. Thus, we study the following supply chain coordination problems, i.e., the supplier provides a trade credit policy and signs a buyback contract for the surplus products with the retailer. For this, based on maximizing the benefits of supply chain members, the supply chain coordination model has been established. By analyzing two participants’ benefit, the supply chain’s optimum coordination mechanism is obtained. The theoretical results are verified by numerical experiments.


Introduction
With the growth of the economy system and Internet retail, supply chain (SC) has become further lengthened and more and more complicated. Under this circumstance, to strengthen the connection between enterprises and increase the bene t value, SC members should coordinate with each other to resist risks and complexity together. us, the issue of supply chain coordination (SCC) receives much attention [1][2][3]. It is well known that the buyback contract is a signi cant strategy to implement SCC which can coordinate operations' performance of each member and spur retailers to make larger orders by the supplier's commitment to buy back the residual goods after the sale is completed. For this, Wu [4] showed that buyback contracts might bring greater bene ts than other strategies in the increasingly competitive environment, Lin et al. [5] constructed the operation decision and coordination strategy model in the in uenza vaccine SC and obtained the optimal coordination method, and so on [6][7][8][9]. e enterprises need many funds to coordinate the total SC. erefore, the retailer will encounter the situation of insu cient funds. Trade credit is the form of deferred payment in intercompany transactions, which can relieve nancial pressure and make enterprises operate smoothly [10,11]. As for the trade credit problem under SCC, Phan et al. [12] showed that when the trade credit interest rate is comparable, SC contracts can coordinate the pro ts of all parties. Silaghi et al. [13] studied trade credit within the framework of real options and proved that trade credit is the coordination means within the framework of real options. Tsao et al. [14] considered the SCC strategy under credit risk based on information e ort and used the compound contract to coordinate channel revenue. For more literatures on this, see [15][16][17][18][19][20].
Due to the long-term depressed nancial crisis, the risk aversion awareness of SC members has been signi cantly improved [21]. For instance, clothing-related enterprises become more risk-averse in the decision to deal with the instability brought by COVID-19 [22]. Hence, the retailer's risk aversion is taken into account in the SCC model. For this, Yang et al. [23] showed that buyback strategy can coordinate the risk-averse SC. Ye et al. [24] developed a new mechanism for the sake of coordinating the SC related to agriculture with risk aversion agents in complex situations.
Backorder is also an essential part of SCC. In the operation process, some customers with unmet needs will give up the purchase, while some customers with unmet needs will be willing to wait for the replenishment of the enterprise to purchase, which is the backorder. e backorder strategy in SCC management also receives much attention in recent years; see [25,26] and references therein.
Upstream enterprises of some seasonal commodities or fast-moving products in the SC, such as Nike and Mengniu Group [27], generally cooperate with downstream enterprises through buyback contracts, and enterprises will invest excess funds or provide trade credit to obtain additional income. Due to the randomness of demand, enterprises are usually accompanied by risks and shortages. erefore, we take risk aversion and partial backorder into account to meet the actual business situation. Relevant papers rarely discuss the above contents together, and some papers do not consider the interest of trade credit strategy while others do not study the significance of backorder in decision-making and profits. erefore, our research work is of specific significance. In actual business operations, trade credit, risk aversion, partial backorder, and buyback contracts tend to coexist throughout the SC, and thus it is necessary to consider them simultaneously. From this, we consider the following SCC problem: the supplier offers a trade credit policy and two participants make a buyback contract for the surplus products. For this, based on maximizing the SC members' profits, we establish a SCC model with trade credit policy, partial backorder, and buyback contract. By analyzing the SC members' benefit, we obtain the optimal SCC mechanism. e effects of the trade credit, Conditional Value at Risk (CVaR) criterion, buyback contract, and partial backorder on the decision variables and profit are analyzed. Numerical experiments verify the theoretical achievements. e above content is the innovation and contribution of this paper. e rest of this paper is arranged as follows. Section 2 describes the model framework of this paper. Section 3 discusses SC decisions and coordination under trade credit. Numerical experiments are carried out in Section 4. Section 5 gives conclusions and extensions.

Problem Description, Notations, and Assumptions.
Consider the SC with a well-funded supplier and a capitalconstrained retailer. As the starter of a replenishment cycle, the supplier sets a wholesale price and provides him with trade credit. To this, the retailer should determine the order quantity. Meanwhile, the retailer will adopt trade credit financing in case of insufficient funds, the retailer can use the initial capital to pay partial payment to the supplier and shall pay the arrears and interest to supplier at the finish of the replenishment cycle. Under the decentralized decision, the supplier needs to choose the wholesale price for retailer to encourage them to order more products. Furthermore, the retailer decides his ordering quantity based on the wholesale price and trade credit strategy. is is regarded as a supplierled Stackelberg game. Under the centralized decision, the buyback contract is used for SCC. To solve the above research problems, the following assumptions are required. e notations are shown in Table 1. e required assumptions are as follows: (1) Supplier is risk-neutral and well funded, while the retailer is risk-averse and capital constraints (2) e market demand of D is random, its probability density function is f(x), and its distribution function is F(x) (3) When retailer is shortage, the retailer will replenish the goods to nearby enterprises e variables have the following relationships: the optimal order quantity's solution will be the sum of inverse functions of the two distribution functions, so it is difficult to solve the wholesale price further and find the sensitivity of each variables on decision variables and profits. erefore,

e Retailer's General Financing Model.
In the process of operation, the retailer can obtain sales revenue of p min(Q, D) and recovery revenue of ε(Q − D) + . Backorder can make the retailer gain revenue, i.e., Since the retailer is capital constrained, he needs to pay interest payment (wQ − y 0 ) + r s at the end. en, the retailer's benefit is (1)

Some Preliminaries about the CVaR Measure.
In order to better measure retailers' risk aversion, we use the CVaR criterion proposed by Kahneman and Tversky [28,29]. e CVaR criterion focuses on profits that do not meet a given target level. is criterion is commonly used in the risk management [30,31]. us, the risk-averse retailer's VaR benefit π(Q) is where Pr[π(Q)⩾v] represents the probability of the π(Q) more than v and VaR α [π(Q)] represents the maximum profit of the retailer with risk aversion α. At this time, we regard VaR α [π(Q)] as the targeted benefit. erefore, the CVaR criterion for benefit π(Q) is Because this formula is hard to calculate, the researcher usually converts it into the following formula [30,31]:

The Supply Chain Decision and Coordination under Trade Credit
Trade credit is a financing method to enhance the retailer's purchasing power and increase his profits when the retailer is short of funds in which the supplier first provides a certain amount of goods to the retailer and then receives payment and interest after selling the goods.

e Retailer's Optimal Decision under Trade Credit.
In this subsection, we consider the circumstance such that the supplier sets the product's wholesale price, and the retailer determines its optimum order quantity. e retailer is short of funds and applies for deferment of payment to the supplier.
According to the above description and (1), we can obtain the retailer's benefit function.
e supplier can invest the received payment and obtain financing income. us, the supplier's benefit is

Theorem 1. Under the decentralized decision, the optimum order quantity when the retailer adopts trade credit is
Proof. Based on this expression that min(Q, en, the benefit function (4) can be rewritten as To facilitate derivation, the inner layer optimization problem is considered first, i.e., max v H α (Q, v) for fixed Q in two cases.
In this case, it is obtained by equation (8): and On the other hand, if zH is incremented with respect to v, and it is classified into the next case discussion. and en, there exists sufficiently large v 0 such that decreases with the increase in v, which will be classified as Case 1. From the above discussion, for any fixed Q in max Next, we solve the outer layer optimization problem of the max Q⩾0 max v⩾0 H(Q, v), i.e., max Q⩾0 H(Q, v * ). In this regard, we also discuss it in two cases.
Since zH(Q, v * )/zQ � ε − w(1 + r s ) < 0, the optimal value point is not in this region. us, it is classified as the next case for discussion. and en, we can obtain the optimum solution Q to max Q⩾0 H(Q, v * ), i.e., For the above optimum order quantity, the following corollary can be obtained:

Corollary 1. For the parameters β, f, k, and α involved in the model, it holds that
(1) e optimum order quantity Q * decreases with β and α (2) e optimum order quantity Q * increases with f and k Proof. According to eorem 1, computing the derivative Q * w.r.t β, f, k, and α, respectively, yields is proves the assertion. From Corollary 1 (1), when the backorder rate β increases, it means that some excess demand can be satisfied, and the retailer will make fewer orders. e optimal ordering quantity Q decreases with respect to the risk aversion level α. us, to reduce the risk of loss caused by over-ordering, the retailer will make a lower order decision. From Corollary 1 (2), the increase in unit backorder cost f will lead to increased cost, while when the unit penalty cost of shortage k increases, the retailer will have a higher penalty cost of shortage. From this, to reduce the backorder cost and penalty cost of shortage, the retailer will increase orders.
For the optimum wholesale price, the following conclusion can be obtained.

Mathematical Problems in Engineering
Using the fact from (6) that the optimum wholesale price is For the above optimal wholesale price, there are the following properties.

Corollary 2. For the parameters α, r s , β, and f involved in the model, one has
(1) Increase in the optimum wholesale price w * with α, f (2) Decrease in the optimum wholesale price w * with r s , β Proof. According to eorem 2, computing the derivative w * with respect to α, r s , β, and f, respectively, we can get e desired results are as follows. From Corollary 2 (1), the increase in the risk aversion level α will prevent retailer from making a fewer order. In order to maximize profits, the supplier's optimum wholesale price w will increase with respect to the risk aversion level α. Moreover, to avoid shortage, the order quantity Q increases with respect to the unit backorder cost f. According to the game equilibrium, the supplier will raise the wholesale price w to return the order quantity Q to the most favorable one. From Corollary 2 (2), when the interest rate of trade credit r s increases, the supplier will reduce the wholesale price w to impel the retailer to choose the trade credit strategy. Moreover, the order quantity Q decreases with respect to the backorder rate β. According to the game equilibrium, the supplier will reduce the wholesale price w to return the order quantity Q to the most favorable one. e Supply Chain Coordination under Trade Credit. is subsection considers the circumstance such that the supplier buys back the remaining goods after the sale is completed to encourage retailers to order more goods.
Under this circumstance, from (1), the retailer's benefit is As the supplier can obtain risk-free investment income from the payment received, thus the supplier's benefit is According to the research of Yang et al. [23] and Chen et al. [32], the total SC can be considered as the newsvendor under the centralized decision, i.e., the risk-neutral supplier sells their products directly to the market. en, the total supply chain's benefit is

and the optimum order quantity of SC is
Proof. According to equation Mathematical Problems in Engineering By computing the stationary point of the function, we can obtain the total supply chain's optimum order quantity By similar argument to eorem 1, the retailer's optimum order quantity is To implement SCC, by setting Q * c � Q * , the optimum buyback price is Among them, According to the conclusion, the wholesale price is proportional to buyback price, i.e., when the wholesale price increases, the buyback price also increases. Further, by the fact that 0 < α < 1 and w > b > ε, one can obtain the range of α. □

Corollary 3. e buyback contract can realize the SCC if the supplier's wholesale price satisfies
Proof. According to eorem 1, under decentralized decision, the benefit of both sides is a certain value. erefore, we make a comparison to the relative benefit.
Mathematical Problems in Engineering 7 e range of wholesale prices is obtained from Δπ > 0: en, the range of wholesale prices is obtained from ΔΠ > 0: By making the profits of both sides of the supply chain under the centralized decision greater than those under the decentralized decision, we get the wholesale price range that can coordinate the supply chain.

Corollary 4. e buyback contract can realize the SCC if the retailer risk aversion level satisfies
Under the restriction of individual rationality, to ensure that the SC members agree to provide and accept the buyback contract, we must first ensure that both parties' profits are not less than the profits obtained under the decentralized decision. rough the method similar to Corollary 3, we get the range of risk aversion level that can coordinate the supply chain.

Numerical Experiments
is section makes numerical experiments to test the model's effectiveness. First, the market demand D of the model obeys the uniform distribution in [10,200], and other variables are such that p � 15, w � 10, f � 11, c � 5, ε � 2, k � 3, r f � 0.02, r s � 0.04, y 0 � 300, α � 0.15, and β � 0.5. By computation, we can obtain that under the decentralized decision, the optimum order quantity Q * � 55.724 and the optimum wholesale price w * � 9.0509.
From Figure 1(a), when the SCC under the buyback contract is adopted, the increased profits of both sides are linear functions of the wholesale price w. e retailer's profit decreases with the increase in wholesale price w, while the profits of the supplier increase with respect to wholesale price w. For wε [6.6, 11.9], the buyback contract can implement SCC, so that both sides of the SC can reach mutually beneficial results. For the sake of fairness, the most accessible point to implement is w � 9.3, where both parties increase their profits equally.
To increase comparability, we take w � 8, w � 9.3, and w � 12, respectively. Figure 1(b) shows that with the increase in interest rate r s , the relative benefit of the supplier is greater than zero and increasing, while the relative benefit greater than zero of the retailer is is indicates that the repurchase contract can reach a mutually beneficial situation in SCC within the above range. With the increase in wholesale prices, the range of interest rates r s that can achieve a win-win situation is gradually narrowing. Further, if we take w � 9.3, then the easiest point to implement is r s � 0.04. From Figure 2(a), under the decentralized decision, the retailer's benefit increases with respect to backorder rate β. Backorder is the income item of retailer's benefit. Although the increase in backorder rate β in Corollary 1 will reduce the order quantity Q, the increase in backorder rate β in Corollary 2 will reduce the wholesale price w which is another reason. e supplier's benefit decreases with the increase in backorder rate β which is the same as above. e reason why the benefit of the SC increases with the backorder rate β is that backorder is the revenue item, and the retailer's benefit increases faster than the supplier's benefit decreases. Under the centralized decision, the benefit of the retailer and SC decreases with respect to backorder rate β because the rise of backorder rate β will reduce the order quantity Q. Figure 2(b) shows that the retailer's benefit decreases with unit backorder cost f under decentralized decision. 8 Mathematical Problems in Engineering One reason is that the increase in unit backorder cost f leads to decreased backorder revenue. Although the unit backorder cost f increases with respect to the order quantity Q in Corollary 1, another reason is that the increase in unit backorder cost f will increase the wholesale price w in Corollary 2. e reason why the supplier's benefit increases with the increase in unit backorder cost f is the same as above. e SC benefit decreases with the unit backorder cost f because the revenue of backorder decreases, and the retailer's benefit decreases faster than the increase in supplier's benefit. Under the centralized decision, the benefit of the retailer and SC increases with respect to unit backorder cost f because the rise of unit backorder cost f will increase the order quantity Q. From Figure 2(c), under decentralized decision-making, since the retailer is risk-averse and the supplier is riskneutral, the influence of risk aversion level α on the order quantity Q is greater than that on the wholesale price w. is is why profits on both sides decrease as risk aversion level α increases. Under the centralized decision, the risk aversion level α has little effect on both sides.
From the above numerical experiments, the SCC with buyback contract can not only achieve a win-win situation but also perform more robust than decentralized decisionmaking when each element changes.

Conclusions
is paper investigated the capital-constrained SCC with buyback contracts and partial backorder under the CVaR criterion. For these scenarios, two models were established to maximize the supplier and retailer's benefit. According to analysis of the two parties' benefit, the SC members' optimal decisions, and the impact analysis of the trade credit, CVaR criterion and backordering to the retailer's decision were obtained. From the obtained results, we can see that as long as the risk aversion level, wholesale price and trade credit interest rate fall within a "Pareto zone," the whole SC can be coordinated, and both sides can reach mutually beneficial results, which provides management insights for the SC members. Under the decentralized decision, the retailer and the whole SC tend to have a higher backorder rate, while suppliers tend to have a lower backorder rate. Under the centralized decision, the retailer and SC tend to have a low backorder rate. e impact of unit backorder cost on the SC is just contrary to the above conclusion. e specific explanation is given in the previous section. Further, the benefit of both sides of the SC will decrease with respect to risk aversion level, which means that excessive risk aversion is not desirable. To better analyze the results and make the paper go on, we assume that the return on backorder is greater than the penalty cost of shortage, which is also one of the limitations of this paper. e extension of our research is to consider that the supplier is capital constrained. Meanwhile, the supplier can also be risk-averse, and the paper can also be combined with carbon policy to meet the real business environment.

Data Availability
e data used to support the findings of this study are available from the corresponding author upon request.

Conflicts of Interest
e authors declare that they have no conflicts of interest.