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

Authors: Gaoke Wu; Bo Feng; Chengfei Wang
Journal: Mathematical Problems in Engineering
 (2022)

---

## Abstract

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.


---

## Body

## 1. 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 benefit value, SC members should coordinate with each other to resist risks and complexity together. Thus, the issue of supply chain coordination (SCC) receives much attention [1–3]. It is well known that the buyback contract is a significant 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 benefits than other strategies in the increasingly competitive environment, Lin et al. [5] constructed the operation decision and coordination strategy model in the influenza vaccine SC and obtained the optimal coordination method, and so on [6–9].The enterprises need many funds to coordinate the total SC. Therefore, the retailer will encounter the situation of insufficient funds. Trade credit is the form of deferred payment in intercompany transactions, which can relieve financial 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 profits 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 effort and used the compound contract to coordinate channel revenue. For more literatures on this, see [15–20].Due to the long-term depressed financial crisis, the risk aversion awareness of SC members has been significantly 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. The 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. Therefore, 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. Therefore, 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. The 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. The above content is the innovation and contribution of this paper.The rest of this paper is arranged as follows. Section2 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.


## 2. The Model Framework of Supply Chain

### 2.1. Problem Description, Notations, and Assumptions

Consider the SC with a well-funded supplier and a capital-constrained 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. This is regarded as a supplier-led Stackelberg game. Under the centralized decision, the buyback contract is used for SCC. To solve the above research problems, the following assumptions are required. The notations are shown in Table1.Table 1
Notations.
SymbolDescriptionSymbolDescriptionQOrder quantityrsInterest rate of trade creditwWholesale pricerfRisk-free interest ratey0Retailer’s initial fundsεResidual value of remaining goodspSelling priceαRisk aversion levelcUnit production costβBackorder ratefUnit backorder costkUnit penalty cost of shortagebBuyback priceThe required assumptions are as follows:(1)
Supplier is risk-neutral and well funded, while the retailer is risk-averse and capital constraints(2)
The market demand ofD is random, its probability density function is fx, and its distribution function is Fx(3)
When retailer is shortage, the retailer will replenish the goods to nearby enterprisesThe variables have the following relationships:rs>rf and p>f>w>b>c>ε>0. When βp−f<1−βk, 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. Therefore, suppose βp−f>1−βk.


### 2.2. The Retailer’s General Financing Model

In the process of operation, the retailer can obtain sales revenue ofpminQ,D and recovery revenue of εQ−D+. Backorder can make the retailer gain revenue, i.e., βp−fD−Q+. The penalty cost of shortage is 1−βkD−Q+. Since the retailer is capital constrained, he needs to pay interest payment wQ−y0+rs at the end. Then, the retailer’s benefit is(1)πQ=pminQ,D−wQ−wQ−y0+rs+εQ−D++βp−fD−Q+−1−βkD−Q+.


### 2.3. 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]. The CVaR criterion focuses on profits that do not meet a given target level. This criterion is commonly used in the risk management [30, 31]. Thus, the risk-averse retailer’s VaR benefit πQ is(2)VaRαπQ=supv∈R|PrπQ⩾v⩾α,where PrπQ⩾v represents the probability of the πQ more than v and VaRαπQ represents the maximum profit of the retailer with risk aversion α. At this time, we regard VaRαπQ as the targeted benefit. Therefore, the CVaR criterion for benefit πQ is(3)CVaRαπQ=11−αEπQ|πQ⩽VaRαπQ.Because this formula is hard to calculate, the researcher usually converts it into the following formula [30, 31]:(4)maxQ,vHαQ,v=v−11−αEv−πQ+.


## 2.1. Problem Description, Notations, and Assumptions

Consider the SC with a well-funded supplier and a capital-constrained 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. This is regarded as a supplier-led Stackelberg game. Under the centralized decision, the buyback contract is used for SCC. To solve the above research problems, the following assumptions are required. The notations are shown in Table1.Table 1
Notations.
SymbolDescriptionSymbolDescriptionQOrder quantityrsInterest rate of trade creditwWholesale pricerfRisk-free interest ratey0Retailer’s initial fundsεResidual value of remaining goodspSelling priceαRisk aversion levelcUnit production costβBackorder ratefUnit backorder costkUnit penalty cost of shortagebBuyback priceThe required assumptions are as follows:(1)
Supplier is risk-neutral and well funded, while the retailer is risk-averse and capital constraints(2)
The market demand ofD is random, its probability density function is fx, and its distribution function is Fx(3)
When retailer is shortage, the retailer will replenish the goods to nearby enterprisesThe variables have the following relationships:rs>rf and p>f>w>b>c>ε>0. When βp−f<1−βk, 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. Therefore, suppose βp−f>1−βk.


## 2.2. The Retailer’s General Financing Model

In the process of operation, the retailer can obtain sales revenue ofpminQ,D and recovery revenue of εQ−D+. Backorder can make the retailer gain revenue, i.e., βp−fD−Q+. The penalty cost of shortage is 1−βkD−Q+. Since the retailer is capital constrained, he needs to pay interest payment wQ−y0+rs at the end. Then, the retailer’s benefit is(1)πQ=pminQ,D−wQ−wQ−y0+rs+εQ−D++βp−fD−Q+−1−βkD−Q+.


## 2.3. 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]. The CVaR criterion focuses on profits that do not meet a given target level. This criterion is commonly used in the risk management [30, 31]. Thus, the risk-averse retailer’s VaR benefit πQ is(2)VaRαπQ=supv∈R|PrπQ⩾v⩾α,where PrπQ⩾v represents the probability of the πQ more than v and VaRαπQ represents the maximum profit of the retailer with risk aversion α. At this time, we regard VaRαπQ as the targeted benefit. Therefore, the CVaR criterion for benefit πQ is(3)CVaRαπQ=11−αEπQ|πQ⩽VaRαπQ.Because this formula is hard to calculate, the researcher usually converts it into the following formula [30, 31]:(4)maxQ,vHαQ,v=v−11−αEv−πQ+.


## 3. 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.


### 3.1. The 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. The 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.(5)πQ=pminQ,D−wQ−wQ−y0rs+εQ−D++βp−fD−Q+−1−βkD−Q+.The supplier can invest the received payment and obtain financing income. Thus, the supplier’s benefit is(6)Πw=w−cQ+wQ−y0rs+y0−cQrf.Theorem 1.
Under the decentralized decision, the optimum order quantity when the retailer adopts trade credit isQ∗=F−11−αp−w1+rs−βp−f+1−βk/p−ε−βp−f+1−βk.Proof.
Based on this expression thatminQ,D=Q−Q−D+ and D−Q+=D−Q+Q−D+, one has(7)πQ=pminQ,D−wQ−wQ−y0rs+εQ−D++βp−fD−Q+−1−βkD−Q+=p−w−βp−f+1−βkQ+βp−f−1−βkD−wQ−y0rs−p−ε−βp−f+1−βkQ−D+.
Then, the benefit function (4) can be rewritten as(8)HαQ,v=v−11−αEv−πQ+=v−11−α∫0+∞v−p−w−βp−f+1−βkQ−βp−f−1−βkx+wQ−y0rs+p−ε−βp−f+1−βkQ−x++dFx=v−11−α∫0Qv−εQ+wQ−p−εx+wQ−y0rs+dFx−11−α∫Q+∞v−p−w−βp−f+1−βkQ−βp−f−1−βkx+wQ−y0rs+dFx.
To facilitate derivation, the inner layer optimization problem is considered first, i.e.,maxvHαQ,v for fixed Q in two cases.Case 1.
v<pQ−wQ−wQ−y0rs. In this case, it is obtained by equation (8):(9)HαQ,v=v−11−α∫0v−εQ+wQ+wQ−y0rsp−εv−εQ+wQ−p−εx+wQ−y0rsdFx,and(10)∂HQ,v∂v=1−11−αFv−εQ+wQ+wQ−y0rsp−ε.
Certainly,∂HQ,v/∂v|v=εQ−wQ−wQ−y0rs=1>0. Thus, if ∂HQ,v/∂v|v=pQ−wQ−wQ−y0rs=1−FQ/1−α⩽0, i.e., Q⩾F−11−α, then the optimum solution to maxv≥0HQ,v is its stationary point, which means that(11)v∗=p−εF−11−α−wQ+εQ−wQ−y0rs.
On the other hand, if∂HQ,v/∂v|v=pQ−wQ−wQ−y0rs=1FQ/1−α>0, then HQ,v is incremented with respect to v, and it is classified into the next case discussion.Case 2.
v⩾pQ−wQ−wQ−y0rs. Under this condition, it is concluded from equation (8) that(12)HαQ,v=v−11−α∫0Qv−εQ+wQ−p−εx+wQ−y0rsdFx−11−α∫Qv−p−w−βp−f+1−βkQ+wQ−y0rsβp−f−1−βkv−p−w−βp−f+1−βkQ−βp−f−1−βkx+wQ−y0rs+dFx,and(13)∂HQ,v∂v=1−11−αFv−p−w−βp−f+1−βkQ+wQ−y0rsβp−f−1−βk.
Then, there exists sufficiently largev0 such that ∂HQ,v/∂v|v=v0=1−Fv0−p−w−βp−f+1−βkQ+wQ−y0rs/βp−f−1−βk/1−α≤0. If ∂HQ,v/∂v|v=pQ−wQ−wQ−y0rs=1−FQ/1−α>0, i.e., Q<F−11−α, then the optimum solution v∗ to maxv⩾0HQ,v satisfies that(14)1−11−αFv−p−w−βp−f+1−βkQ+wQ−y0rsβp−f−1−βk=0,which means that v∗=βp−f−1−βkF−11−α+p−w−βp−f+1−βkQ−wQ−y0rs. Otherwise, ∂HQ,v/∂v|v=pQ−wQ−wQ−y0rs=1−FQ/1−α>0, and HQ,v decreases with the increase in v, which will be classified as Case 1.
From the above discussion, for any fixedQ in maxv⩾0HQ,v, we can obtain:(15)v∗=p−εF−11−α+εQ−wQ−wQ−y0rs,ifQ⩾F−11−α,βp−f−1−βkF−11−α−wQ−y0rs+p−w−βp−f+1−βkQ,ifQ<F−11−α.
Next, we solve the outer layer optimization problem of themaxQ⩾0maxv⩾0HQ,v, i.e., maxQ⩾0HQ,v∗. In this regard, we also discuss it in two cases.Case 3.
Q⩾F−11−α. Under this condition, from the result of v∗=p−εF−11−α+εQ−wQ−wQ−y0rs, HQ,v∗ can be rewritten into(16)HQ,v∗=p−εF−11−α+εQ−wQ−wQ−y0rs−11−α∫0F−11−αp−εF−11−α−xdFx.
Since∂HQ,v∗/∂Q=ε−w1+rs<0, the optimal value point is not in this region. Thus, it is classified as the next case for discussion.Case 4.
Q<F−11−α. In this case, v∗=βp−f−1−βkF−11−α+p−w−βp−f+1−βkQ−wQ−y0rs. Hence,(17)HQ,v∗=βp−f−1−βkF−11−α+p−w−βp−f+1−βkQ−wQ−y0rs−11−α∫0Qβp−f−1−βkF−11−α+p−ε−βp−f+1−βkQ−p−εxdFx+11−α∫QF−11−αβp−f−1−βkF−11−α−xdFx,and ∂HQ,v∗/∂Q=p−w1+rs−βp−f+1−βk−p−ε−βp−f+1−βkFQ/1−α.
Then, we can obtain the optimum solutionQ to maxQ⩾0HQ,v∗, i.e.,(18)Q∗=F−11−αp−w1+rs−βp−f+1−βkp−ε−βp−f+1−βk.
For the above optimum order quantity, the following corollary can be obtained:Corollary 1.
For the parametersβ,f,k, andαinvolved in the model, it holds that(1)
The optimum order quantityQ∗ decreases with β and α(2)
The optimum order quantityQ∗ increases with f and kProof.
According to Theorem1, computing the derivative Q∗ w.r.t β,f,k, and α, respectively, yields(19)∂Q∗∂β=−1−αp−f+kw1+rs−εfQ∗p−ε−βp−f+1−βk2<0,∂Q∗∂α=−p−w1+rs−βp−f+1−βkfQ∗p−ε−βp−f+1−βk2<0,∂Q∗∂f=β1−αw1+rs−εfQ∗p−ε−βp−f+1−βk2>0,∂Q∗∂k=1−β1−αw1+rs−εfQ∗p−ε−βp−f+1−βk2>0.
This proves the assertion.
From Corollary1 (1), when the backorder rate β increases, it means that some excess demand can be satisfied, and the retailer will make fewer orders. The optimal ordering quantity Q decreases with respect to the risk aversion level α. Thus, 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.Theorem 2.
Under the decentralized decision, the optimum wholesale price is(20)p−ε−βp−f+1−βkQ∗fQ∗+c1+rf1−α1+rs1−α.
ByQ∗=F−11−αp−w1+rs−βp−f+1−βk/p−ε−βp−f+1−βk from Theorem 1, we have(21)∂Q∗∂w=−1+rs1−αp−ε−βp−f+1−βkfQ∗<0.
Using the fact from (6) that(22)∂Πw∂w=Q∗+w−c1+rf∂Q∗∂w+Q∗+w∂Q∗∂wrs,the optimum wholesale price is(23)w=p−ε−βp−f+1−βkQ∗fQ∗+c1+rf1−α1+rs1−α.
For the above optimal wholesale price, there are the following properties.Corollary 2.
For the parametersα, rs, β, and f involved in the model, one has(1)
Increase in the optimum wholesale pricew∗ with α,f(2)
Decrease in the optimum wholesale pricew∗ with rs,βProof.
According to Theorem2, computing the derivative w∗ with respect to α, rs, β, and f, respectively, we can get(24)∂w∗∂α=p−ε−βp−f+1−βkQ∗fQ∗1+rs1−α>0,∂w∗∂rs=−1−αp−ε−βp−f+1−βkQ∗fQ∗+c1+rf1−α1+rs21−α2<0.∂w∗∂β=−p−f+kQ∗fQ∗1+rs1−α<0,∂w∗∂f=βQ∗fQ∗1+rs1−α>0.
The desired results are as follows.
From Corollary2 (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 rs 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.


### 3.2. The Supply Chain Coordination under Trade Credit

This 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(25)πcQ¯=pminQ¯,D−wQ¯−wQ¯−y0rs+bQ¯−D++βp−fD−Q¯+−1−βkD−Q¯+.As the supplier can obtain risk-free investment income from the payment received, thus the supplier’s benefit is(26)Πcw=w−cQ¯+wQ¯−y0rs+y0−cQ¯rf−b−εQ¯−D+.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. Then, the total supply chain’s benefit is(27)WcQc=pminQc,D−cQc+y0−cQcrf+εQc−D++βp−fD−Qc+−1−βkD−Qc+.Theorem 3.
For the concerned model, the supplier’s optimum buyback price isb∗=p−βp−f+1−βk−1−αp−w1+rs−βp−f+1−βkp−ε−βp−f+1−βk/p−c1+rf−βp−f+1−βk, and the optimum order quantity of SC isQc∗=F−1p−c1+rf−βp−f+1−βk/p−ε−βp−f+1−βk.Proof.
According to equation (27), by minQ,D=Q−Q−D+,D−Q+=D−Q+Q−D+, for the benefit of SC, one has(28)WcQc=pminQc,D−cQc+y0−cQcrf+εQc−D++βp−fD−Qc+−1−βkD−Qc+=p−c1+rf−βp−f+1−βkQc+βp−f−1−βkD+rfy0−p−ε−βp−f+1−βkQc−D+,EWcQc=p−c1+rf−βp−f+1−βkQc+βp−f−1−βk∫0∞xdFx+rfy0−p−ε−βp−f+1−βk∫0QcQc−xdFx.
By computing the stationary point of the function, we can obtain the total supply chain’s optimum order quantity(29)Qc∗=F−1p−c1+rf−βp−f+1−βkp−ε−βp−f+1−βk.
By similar argument to Theorem1, the retailer’s optimum order quantity is(30)Q¯∗=F−11−αp−w1+rs−βp−f+1−βkp−b−βp−f+1−βk.
To implement SCC, by settingQc∗=Q¯∗, the optimum buyback price is(31)b∗=p−βp−f+1−βk−Hp−c1+rf−βp−f+1−βk.
Among them,H=1−αp−w1+rs−βp−f+1−βkp−ε−βp−f+1−βk.
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 that0<α<1 and w>b>ε, one can obtain the range of α.Corollary 3.
The buyback contract can realize the SCC if the supplier’s wholesale price satisfies(32)cQc∗+y0rs−y0−cQc∗rf+b∗−ε∫0Qc∗Qc∗−xdFx+EΠQ∗,w∗Qc∗1+rs<w<p−βp−f+1−βkQc∗+βp−f−1−βk∫0∞xdFx+y0rs−p−b∗−βp−f+1−βk∫0Qc∗Qc∗−xdFx−EπQ∗,w∗Qc∗1+rs.Proof.
According to Theorem1, under decentralized decision, the benefit of both sides is a certain value. Therefore, we make a comparison to the relative benefit.(33)Δπ=Eπc−πQ∗,w∗=p−w−βp−f+1−βkQc∗+βp−f−1−βk∫0∞xdFx−wQc∗−y0rs−p−b∗−βp−f+1−βk∫0Qc∗Qc∗−xdFx−EπQ∗,w∗ΔΠ=EΠc−ΠQ∗,w∗=w−cQc∗+wQc∗−y0rs+y0−cQc∗rf−b∗−ε∫0Qc∗Qc∗−xdFx−EΠQ∗,w∗.
The range of wholesale prices is obtained fromΔπ>0:(34)w<p−βp−f+1−βkQc∗+βp−f−1−βk∫0∞xdFx+y0rs−p−b∗−βp−f+1−βk∫0Qc∗Qc∗−xdFx−EπQ∗,w∗Qc∗1+rs.
Then, the range of wholesale prices is obtained fromΔΠ>0:(35)w>cQc∗+y0rs−y0−cQc∗rf+b∗−ε∫0Qc∗Qc∗−xdFx+EΠQ∗,w∗Qc∗1+rs.
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.
The buyback contract can realize the SCC if the retailer risk aversion level satisfies(36)αcϵ0,1−p−w−βp−f+1−βkp−c1+rf−βp−f+1−βkp−w1+rs−βp−f+1−βkp−ε−βp−f+1−βk.
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. Through the method similar to Corollary3, we get the range of risk aversion level that can coordinate the supply chain.


## 3.1. The 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. The 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.(5)πQ=pminQ,D−wQ−wQ−y0rs+εQ−D++βp−fD−Q+−1−βkD−Q+.The supplier can invest the received payment and obtain financing income. Thus, the supplier’s benefit is(6)Πw=w−cQ+wQ−y0rs+y0−cQrf.Theorem 1.
Under the decentralized decision, the optimum order quantity when the retailer adopts trade credit isQ∗=F−11−αp−w1+rs−βp−f+1−βk/p−ε−βp−f+1−βk.Proof.
Based on this expression thatminQ,D=Q−Q−D+ and D−Q+=D−Q+Q−D+, one has(7)πQ=pminQ,D−wQ−wQ−y0rs+εQ−D++βp−fD−Q+−1−βkD−Q+=p−w−βp−f+1−βkQ+βp−f−1−βkD−wQ−y0rs−p−ε−βp−f+1−βkQ−D+.
Then, the benefit function (4) can be rewritten as(8)HαQ,v=v−11−αEv−πQ+=v−11−α∫0+∞v−p−w−βp−f+1−βkQ−βp−f−1−βkx+wQ−y0rs+p−ε−βp−f+1−βkQ−x++dFx=v−11−α∫0Qv−εQ+wQ−p−εx+wQ−y0rs+dFx−11−α∫Q+∞v−p−w−βp−f+1−βkQ−βp−f−1−βkx+wQ−y0rs+dFx.
To facilitate derivation, the inner layer optimization problem is considered first, i.e.,maxvHαQ,v for fixed Q in two cases.Case 1.
v<pQ−wQ−wQ−y0rs. In this case, it is obtained by equation (8):(9)HαQ,v=v−11−α∫0v−εQ+wQ+wQ−y0rsp−εv−εQ+wQ−p−εx+wQ−y0rsdFx,and(10)∂HQ,v∂v=1−11−αFv−εQ+wQ+wQ−y0rsp−ε.
Certainly,∂HQ,v/∂v|v=εQ−wQ−wQ−y0rs=1>0. Thus, if ∂HQ,v/∂v|v=pQ−wQ−wQ−y0rs=1−FQ/1−α⩽0, i.e., Q⩾F−11−α, then the optimum solution to maxv≥0HQ,v is its stationary point, which means that(11)v∗=p−εF−11−α−wQ+εQ−wQ−y0rs.
On the other hand, if∂HQ,v/∂v|v=pQ−wQ−wQ−y0rs=1FQ/1−α>0, then HQ,v is incremented with respect to v, and it is classified into the next case discussion.Case 2.
v⩾pQ−wQ−wQ−y0rs. Under this condition, it is concluded from equation (8) that(12)HαQ,v=v−11−α∫0Qv−εQ+wQ−p−εx+wQ−y0rsdFx−11−α∫Qv−p−w−βp−f+1−βkQ+wQ−y0rsβp−f−1−βkv−p−w−βp−f+1−βkQ−βp−f−1−βkx+wQ−y0rs+dFx,and(13)∂HQ,v∂v=1−11−αFv−p−w−βp−f+1−βkQ+wQ−y0rsβp−f−1−βk.
Then, there exists sufficiently largev0 such that ∂HQ,v/∂v|v=v0=1−Fv0−p−w−βp−f+1−βkQ+wQ−y0rs/βp−f−1−βk/1−α≤0. If ∂HQ,v/∂v|v=pQ−wQ−wQ−y0rs=1−FQ/1−α>0, i.e., Q<F−11−α, then the optimum solution v∗ to maxv⩾0HQ,v satisfies that(14)1−11−αFv−p−w−βp−f+1−βkQ+wQ−y0rsβp−f−1−βk=0,which means that v∗=βp−f−1−βkF−11−α+p−w−βp−f+1−βkQ−wQ−y0rs. Otherwise, ∂HQ,v/∂v|v=pQ−wQ−wQ−y0rs=1−FQ/1−α>0, and HQ,v decreases with the increase in v, which will be classified as Case 1.
From the above discussion, for any fixedQ in maxv⩾0HQ,v, we can obtain:(15)v∗=p−εF−11−α+εQ−wQ−wQ−y0rs,ifQ⩾F−11−α,βp−f−1−βkF−11−α−wQ−y0rs+p−w−βp−f+1−βkQ,ifQ<F−11−α.
Next, we solve the outer layer optimization problem of themaxQ⩾0maxv⩾0HQ,v, i.e., maxQ⩾0HQ,v∗. In this regard, we also discuss it in two cases.Case 3.
Q⩾F−11−α. Under this condition, from the result of v∗=p−εF−11−α+εQ−wQ−wQ−y0rs, HQ,v∗ can be rewritten into(16)HQ,v∗=p−εF−11−α+εQ−wQ−wQ−y0rs−11−α∫0F−11−αp−εF−11−α−xdFx.
Since∂HQ,v∗/∂Q=ε−w1+rs<0, the optimal value point is not in this region. Thus, it is classified as the next case for discussion.Case 4.
Q<F−11−α. In this case, v∗=βp−f−1−βkF−11−α+p−w−βp−f+1−βkQ−wQ−y0rs. Hence,(17)HQ,v∗=βp−f−1−βkF−11−α+p−w−βp−f+1−βkQ−wQ−y0rs−11−α∫0Qβp−f−1−βkF−11−α+p−ε−βp−f+1−βkQ−p−εxdFx+11−α∫QF−11−αβp−f−1−βkF−11−α−xdFx,and ∂HQ,v∗/∂Q=p−w1+rs−βp−f+1−βk−p−ε−βp−f+1−βkFQ/1−α.
Then, we can obtain the optimum solutionQ to maxQ⩾0HQ,v∗, i.e.,(18)Q∗=F−11−αp−w1+rs−βp−f+1−βkp−ε−βp−f+1−βk.
For the above optimum order quantity, the following corollary can be obtained:Corollary 1.
For the parametersβ,f,k, andαinvolved in the model, it holds that(1)
The optimum order quantityQ∗ decreases with β and α(2)
The optimum order quantityQ∗ increases with f and kProof.
According to Theorem1, computing the derivative Q∗ w.r.t β,f,k, and α, respectively, yields(19)∂Q∗∂β=−1−αp−f+kw1+rs−εfQ∗p−ε−βp−f+1−βk2<0,∂Q∗∂α=−p−w1+rs−βp−f+1−βkfQ∗p−ε−βp−f+1−βk2<0,∂Q∗∂f=β1−αw1+rs−εfQ∗p−ε−βp−f+1−βk2>0,∂Q∗∂k=1−β1−αw1+rs−εfQ∗p−ε−βp−f+1−βk2>0.
This proves the assertion.
From Corollary1 (1), when the backorder rate β increases, it means that some excess demand can be satisfied, and the retailer will make fewer orders. The optimal ordering quantity Q decreases with respect to the risk aversion level α. Thus, 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.Theorem 2.
Under the decentralized decision, the optimum wholesale price is(20)p−ε−βp−f+1−βkQ∗fQ∗+c1+rf1−α1+rs1−α.
ByQ∗=F−11−αp−w1+rs−βp−f+1−βk/p−ε−βp−f+1−βk from Theorem 1, we have(21)∂Q∗∂w=−1+rs1−αp−ε−βp−f+1−βkfQ∗<0.
Using the fact from (6) that(22)∂Πw∂w=Q∗+w−c1+rf∂Q∗∂w+Q∗+w∂Q∗∂wrs,the optimum wholesale price is(23)w=p−ε−βp−f+1−βkQ∗fQ∗+c1+rf1−α1+rs1−α.
For the above optimal wholesale price, there are the following properties.Corollary 2.
For the parametersα, rs, β, and f involved in the model, one has(1)
Increase in the optimum wholesale pricew∗ with α,f(2)
Decrease in the optimum wholesale pricew∗ with rs,βProof.
According to Theorem2, computing the derivative w∗ with respect to α, rs, β, and f, respectively, we can get(24)∂w∗∂α=p−ε−βp−f+1−βkQ∗fQ∗1+rs1−α>0,∂w∗∂rs=−1−αp−ε−βp−f+1−βkQ∗fQ∗+c1+rf1−α1+rs21−α2<0.∂w∗∂β=−p−f+kQ∗fQ∗1+rs1−α<0,∂w∗∂f=βQ∗fQ∗1+rs1−α>0.
The desired results are as follows.
From Corollary2 (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 rs 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.


## 3.2. The Supply Chain Coordination under Trade Credit

This 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(25)πcQ¯=pminQ¯,D−wQ¯−wQ¯−y0rs+bQ¯−D++βp−fD−Q¯+−1−βkD−Q¯+.As the supplier can obtain risk-free investment income from the payment received, thus the supplier’s benefit is(26)Πcw=w−cQ¯+wQ¯−y0rs+y0−cQ¯rf−b−εQ¯−D+.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. Then, the total supply chain’s benefit is(27)WcQc=pminQc,D−cQc+y0−cQcrf+εQc−D++βp−fD−Qc+−1−βkD−Qc+.Theorem 3.
For the concerned model, the supplier’s optimum buyback price isb∗=p−βp−f+1−βk−1−αp−w1+rs−βp−f+1−βkp−ε−βp−f+1−βk/p−c1+rf−βp−f+1−βk, and the optimum order quantity of SC isQc∗=F−1p−c1+rf−βp−f+1−βk/p−ε−βp−f+1−βk.Proof.
According to equation (27), by minQ,D=Q−Q−D+,D−Q+=D−Q+Q−D+, for the benefit of SC, one has(28)WcQc=pminQc,D−cQc+y0−cQcrf+εQc−D++βp−fD−Qc+−1−βkD−Qc+=p−c1+rf−βp−f+1−βkQc+βp−f−1−βkD+rfy0−p−ε−βp−f+1−βkQc−D+,EWcQc=p−c1+rf−βp−f+1−βkQc+βp−f−1−βk∫0∞xdFx+rfy0−p−ε−βp−f+1−βk∫0QcQc−xdFx.
By computing the stationary point of the function, we can obtain the total supply chain’s optimum order quantity(29)Qc∗=F−1p−c1+rf−βp−f+1−βkp−ε−βp−f+1−βk.
By similar argument to Theorem1, the retailer’s optimum order quantity is(30)Q¯∗=F−11−αp−w1+rs−βp−f+1−βkp−b−βp−f+1−βk.
To implement SCC, by settingQc∗=Q¯∗, the optimum buyback price is(31)b∗=p−βp−f+1−βk−Hp−c1+rf−βp−f+1−βk.
Among them,H=1−αp−w1+rs−βp−f+1−βkp−ε−βp−f+1−βk.
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 that0<α<1 and w>b>ε, one can obtain the range of α.Corollary 3.
The buyback contract can realize the SCC if the supplier’s wholesale price satisfies(32)cQc∗+y0rs−y0−cQc∗rf+b∗−ε∫0Qc∗Qc∗−xdFx+EΠQ∗,w∗Qc∗1+rs<w<p−βp−f+1−βkQc∗+βp−f−1−βk∫0∞xdFx+y0rs−p−b∗−βp−f+1−βk∫0Qc∗Qc∗−xdFx−EπQ∗,w∗Qc∗1+rs.Proof.
According to Theorem1, under decentralized decision, the benefit of both sides is a certain value. Therefore, we make a comparison to the relative benefit.(33)Δπ=Eπc−πQ∗,w∗=p−w−βp−f+1−βkQc∗+βp−f−1−βk∫0∞xdFx−wQc∗−y0rs−p−b∗−βp−f+1−βk∫0Qc∗Qc∗−xdFx−EπQ∗,w∗ΔΠ=EΠc−ΠQ∗,w∗=w−cQc∗+wQc∗−y0rs+y0−cQc∗rf−b∗−ε∫0Qc∗Qc∗−xdFx−EΠQ∗,w∗.
The range of wholesale prices is obtained fromΔπ>0:(34)w<p−βp−f+1−βkQc∗+βp−f−1−βk∫0∞xdFx+y0rs−p−b∗−βp−f+1−βk∫0Qc∗Qc∗−xdFx−EπQ∗,w∗Qc∗1+rs.
Then, the range of wholesale prices is obtained fromΔΠ>0:(35)w>cQc∗+y0rs−y0−cQc∗rf+b∗−ε∫0Qc∗Qc∗−xdFx+EΠQ∗,w∗Qc∗1+rs.
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.
The buyback contract can realize the SCC if the retailer risk aversion level satisfies(36)αcϵ0,1−p−w−βp−f+1−βkp−c1+rf−βp−f+1−βkp−w1+rs−βp−f+1−βkp−ε−βp−f+1−βk.
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. Through the method similar to Corollary3, we get the range of risk aversion level that can coordinate the supply chain.


## 4. Numerical Experiments

This section makes numerical experiments to test the model’s effectiveness. First, the market demandD 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,rf=0.02,rs=0.04,y0=300,α=0.15,andβ=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 Figure1(a), when the SCC under the buyback contract is adopted, the increased profits of both sides are linear functions of the wholesale price w. The 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 wε6.6,11.9, 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.Figure 1
The impact of some parameters on the relative benefit.To increase comparability, we takew=8,w=9.3,andw=12, respectively. Figure 1(b) shows that with the increase in interest rate rs, the relative benefit of the supplier is greater than zero and increasing, while the relative benefit greater than zero of the retailer is rsw=8ε0,0.55,rsw=9.3ε0,0.33,rsw=12ε0,0.03. This 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 rs that can achieve a win-win situation is gradually narrowing. Further, if we take w=9.3, then the easiest point to implement is rs=0.04.From Figure2(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. The supplier’s benefit decreases with the increase in backorder rate β which is the same as above. The 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
The impact of some parameters on the benefit.Figure2(b) shows that the retailer’s benefit decreases with unit backorder cost f under decentralized decision. 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. The reason why the supplier’s benefit increases with the increase in unit backorder cost f is the same as above. The 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 Figure2(c), under decentralized decision-making, since the retailer is risk-averse and the supplier is risk-neutral, the influence of risk aversion level α on the order quantity Q is greater than that on the wholesale price w. This 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 decision-making when each element changes.


## 5. Conclusions

This 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. The impact of unit backorder cost on the SC is just contrary to the above conclusion. The 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. The 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.



---

*Source: 6055105-2022-05-17.xml*