# The Optimal Pricing of Dual-Channel Supply Chain with the Third Party Product Recovery and Sales Effort

Authors: Limin Wang; Qiankun Song; Zhenjiang Zhao
Journal: Complexity
 (2020)

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## Abstract

The optimal pricing of dual-channel supply chain with the third party product recovery and sales effort is considered in this paper. The optimal selling pricing of direct channel and retail channel in the forward supply chain and the optimal collection pricing of retail channel and the third party in the backward supply chain are given for the general case under the centralized and decentralized model. Then, the effect of sales effort of the retailer and the optimal pricing strategy with sales effort under the centralized and decentralized model are provided and analyzed. Finally, the comparative analysis of four situations is carried out by numerical results.


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## Body

## 1.Introduction

In order to improve the utilization rate of resources and reduce environmental pollution, many enterprises are engaged in remanufacturing [1–4]. With the development of science and technology, products could be used for recycling and restoring their original function. Remanufacturing products could meet expectations of consumers on new products, at the same time, the process of remanufacturing could reduce resource consumption and emissions. Research has shown that, by implementing product remanufacturing strategy, enterprises can save the cost of 40%–65%. This not only saves the raw materials of the new parts but also avoids the waste of the resource [5]. The China Daily reported on January 4, 2018, since China adopted the global ban on the import of waste, the European and American countries faced a garbage crisis. Therefore, recycling and remanufacturing becomes even more important. In addition, due to changes in consumer shopping habits and the environmental protection consciousness enhancement, more and more enterprises would see the remanufacturing as a strategy to expand market share, and many companies have established a remanufacturing system [6–12]. Companies such as Caterpillar, GE Aviation, and Mercedes-Benz have built remanufacturing plants in Shanghai.Because the remanufacturing has very potent prospect, many independent manufacturers have focused on this area [13–20]. Many of construction equipment retailers are gradually implementing the remanufacturing strategy, as a result the remanufacturing has lower production cost. Therefore, it is one available strategy through discounts to attract consumers when facing competitive threat on the market [21]. It is widely believed that the remanufacturing market activities are more complex than the traditional production and the sales process. For many different characteristics of remanufacturing goods, such as the uncertain sources of the product of supply chain, the process involves the forward and reverse logistics [22]. Therefore, for those who are engaged in remanufacturing enterprise, how to build a profitable closed-loop supply chain to effectively recycle used products should be considered. On the contrary, the distribution of new products and remanufactured products is one of the pressing problems.Results on remanufacturing recycling dual-channel are less researched in the existing literature. In [23], the presence of competing retailers is considered in the reverse channel. Recovery system on the management of manufacturers and retailers was modeled and studied. Huang et al. studied the optimal decision of dual recycling channels of the closed-loop supply chain and put forward the macroeconomic regulation and control policy based on the numerical analysis. The authors discussed that the dual recycling channel is more effective than the single recycling channel for more manufacturers [24]. In [25], the authors studied the manufacturer-oriented three reverse-mixed recycling channels of the closed-loop supply chain structure, namely, the manufacturer and retailer mix recycling channels, manufacturer and third-party recycling channels, and retailer and third-party mix recycling channels, coming to a conclusion that the mixture of manufacturers and retailers is the most effective recycling channels.In real life, the sales effort on the market has become a key factor in determining the selling price [26]. There are many approaches to improve the market demand for products through sales effort, such as providing the bigger shelf space, promotion, and advertising. Many research results about the effect of sales effort on achieving supply chain coordination. Ma et al. discussed the profits of retailer and manufacturer characteristics under different channel strategies and sales effort levels [27]. In [28], the authors predicted a supplier-retailer of agricultural supply chain, including market demand which depends on the sales effort, and provided service.Based on the above analysis, this paper aims to study the optimal pricing of the dual-channel supply chain with the third-party product recovery and sales effort. First of all, for the general case under the centralized and decentralized model, discuss the optimal selling pricing of the direct channel and retail channel in the forward supply chain and the optimal collection pricing of the retail channel and the third party in the backward supply chain. Then, the optimal pricing strategy with sales effort under the centralized and decentralized model is provided and analyzed. Finally, the comparative analysis of four situations is carried out by numerical results.The organizational structure of this paper is given as follows. Section1 is the introduction of this paper; Section 2 is the model description of this paper; Section 3 studies the optimal selling pricing of the direct channel and retail channel in the forward supply chain and the optimal collection pricing of the retail channel and the third party in the backward supply chain; Section 4 discusses the centralized model and decentralized model with sales effort; Section 5 carries out the comparative analysis of four situations by numerical results; Section 6 summarizes the research of this paper.


## 2. Model Description

In this paper, we consider a kind of closed-loop supply chain with a dual channel which consists of three members: one manufacturer, one retailer, and one third party. As shown in Figure1, in the forward supply chain, the manufacturer produces products at cost cr and sells products to consumer by direct channel at price pd. The manufacturer sells products to the retailer at wholesale price w and the retailer sells products to the consumer by the retail channel at price pr. In the reverse supply chain, the retailer and the third party collect the used products from the consumer at recycling prices er and et, respectively. The manufacturer recycles the used products for remanufacturing from the retailer and the third party at recycling price of e.Figure 1
The structure of the whole supply chain system.The manufacturer should make a decision about the price of the direct channel and the wholesale price to the retailer, the retailer makes a decision about the price of the retail channel and the recycling priceer, and the third party needs to consider the recycling price et to maximize their own profits, respectively. Throughout this paper, the notations are given as Table 1.Table 1
Table of notations.
NotationMeaningii=r,d,t stands for the retail channel, direct channel, and third party, respectivelypiUnit selling price of channeli, i=r,deThe collection price from the retailer and third partyeiThe collection price from the consumer through channeli, i=r,twThe wholesale of nongreen product to the retailersSales effortb1Self-price demand elasticity coefficientb2Cross-price demand elasticity coefficientτCollection rate of productsτiCollection demand of channeli, i=r,tβiCollection demand elasticity coefficient,i=1,2cnUnit produce cost of producing new productscrUnit produce cost of remanufacturing productsTo obtain the main results, we give the following assumptions.Assumption 1.
As mentioned in [29], the demand function is considered as one linear function on the direct selling price and retail selling price. Since the influence of the self-price on demand should be larger than the influence of cross price, we assume that b1>b2.Assumption 2.
For the significance of remanufacturing, the unit produce cost of producing new productscn and unit produce cost of remanufacturing products cr should satisfy that cr+e≤cn.Assumption 3.
Since that the profit of collecting products must be nonnegative for the retailer and third party to guarantee the effectiveness of recycle, there existser≤e and et≤e.Assumption 4.
Assume that the collection rateτ is a constant number, which means that, for the whole demand function D, τD would be remanufactured products and 1−τD would be new manufactured products.Assumption 5.
In this paper, similar with the market demand functions, the collection functions for the retailer and the third party are considered as linear functions on their collection price. We assume thatτr=1−ατD+β1er−β1et and τt=ατD+β1et−β2er, where α represents the basic market share to the third party, ατD stands for the proportion of consumers’ preference for the third party, and 1−ατD stands for the proportion of consumers’ preference for the retail channel so that τr+τt=τD+β1−β2er+et.Assumption 6.
According to [30], we assume that the collection cost is considered as the quadratic function of the collection rate, which is considered as 1/2kτi2 for i=r,t.
Based on Assumption1, the demand functions of the retail channel and direct channel are expressed as linear functions of self-price, cross price, and market scale, which are, respectively, shown as follows:(1)Dr=D1−b1pr+b2pd,Dd=D2−b1pd+b2pr,where D1 stands for the proportion of consumers’ preference for the retail channel and D2 stands for the proportion of consumers’ preference for the direct channel.
According to Assumption5, the collection demand functions of the retail channel and third party channel are expressed as linear functions of self-collecting price, cross-collecting price, and collecting rate, which are, respectively, shown as follows:(2)τr=1−ατD+β1er−β2et,τt=ατD+β1et−β2er,where D=D1+D2 stands for the potential whole market demand.
Accordingly, based on the above assumptions and demand functions, the profit function of the retailer is considered as follows:(3)πrpr,er=Drpr−w+τre−er−12kτr2,where Drpr−w stands for the profit of selling products to the consumer by the retail channel and τre−er represents the profit of collecting products from the manufacturer. The collection cost is considered as 1/2kτr2.
Denote that the unit produce cost of remanufacturing products iscr; then, the profit function of the manufacturer is given by(4)πdw,pd=Drw+Ddpd−1−τD−β1−β2er+etcn−τD+β1−β2er+etcr+e,where the Drw and Ddpd represents the profit of selling products to the consumer by the retail channel and direct channel, respectively. 1−τD−β1−β2er+etcn denotes the producing cost for new products and τD+β1−β2er+etcr+e stands for the recycling and remanufacturing cost for the collected products.
The collection cost of the third party is considered as1/2kτr2 so that the profit function of the third party is presented as(5)πtet=τte−et−12kτt2.


## 3. Equilibrium Analysis

### 3.1. Centralized Model

In the centralized model, all members of the supply chain system cooperatively make a decision on the unit selling price of the retail channel and direct channel and the collection price from the consumer. The profit of the whole supply chain is expressed as(6)πscpr,pd,er,et=Drpr+Ddpd+β1−β2er+etcn−cr−1−τDcn−τDcr−τrer−τtet−12kτr2−12kτt2=D1−b1pr+b2pdpr+D2−b1pd+b2prpd+β1−β2er+etcn−cr−1−τDcn−τDcr−1−ατD+β1er−β2eter−ατD+β1et−β2eret−12k1−ατD+β1er−β2et2−12kατD+β1et−β2er2.Proposition 1.
When the market is under centralized situation, if the parameters satisfy that2β1+kβ12+kβ22−2β21+kβ1>0, then there exists the unique optimal solution:(7)prc1∗=b1D1+b2D22b12−b22,pdc1∗=b2D1+b1D22b12−b22,erc1∗=2β1+kβ12+kβ22β1−β2cn−cr+kβ2ατD−1+kβ11−ατD+2β21+kβ1β1−β2cn−cr+kβ21−ατD−1+kβ1ατD2β1+kβ12+kβ222−4β221+kβ12,etc1∗=2β1+kβ12+kβ22β1−β2cn−cr+kβ21−ατD−1+kβ1ατD+2β21+kβ1β1−β2cn−cr+kβ2ατD−1+kβ11−ατD2β1+kβ12+kβ222−4β221+kβ12.Proof.
According to the profit function of whole supply chain (6), taking the first-order derivatives of πscpr,pd,er,et along the retail price pr, the direct price pd, the collecting price of retailer er, and the collecting price of the third party et, we could obtain(8)∂πscpr,pd,er,et∂pr=−2b1pr+2b2pd+D1,∂πscpr,pd,er,et∂pd=2b2pr−2b1pd+D2,∂πscpr,pd,er,et∂er=−2β1+kβ12+kβ22+2β21+kβ1et+β1−β2cn−cr+kβ2ατD−1+kβ11−ατD,∂πscpr,pd,er,et∂et=2β21+kβ1er−2β1+kβ12+kβ22et+β1−β2cn−cr+kβ21−ατD−1+kβ1ατD.
Hence, the Hessian matrix is given as(9)Hc1=−2b12b2002b2−2b10000−2β1−kβ12−kβ222β21+kβ1002β21+kβ1−2β1−kβ12−kβ22.Obviously, the second derivative satisfies that ∂2πscpr,pd,er,et/∂pr2=∂2πscpr,pd,er,et/∂pd2=−2b1<0. Based on the condition of proposition, 2β1+kβ12+kβ22−2β21+kβ1>0, it could be obtained that 2β1+kβ12+kβ222−4β221+kβ12>0. Therefore, the profit function πscpr,pd,er,et is joint concave on pr, pd, er, and et. The optimal solution could be obtained by solving the first-order condition. The proof is completed.Remark 1.
Proposition1 shows the optimal solution under centralized situation. According to the result of Proposition 1, it could be concluded that the retail selling price and direct selling price are related with the values of D1 and D2. Under the assumption that D1=1−ρa and D2=ρa, we could find prc∗>pdc∗ if 0<ρ<0.5 and prc∗<pdc∗ if 0.5<ρ<1. It demonstrates that the channel has higher selling price when the channel occupies higher market share. Furthermore, if α=0.5, then erc1∗=etc1∗=2β1+kβ12+kβ22+2β21+kβ1β1−β2cn−cr+0.5kβ2τD−0.51+kβ1τD/2β1+kβ12+kβ222−4β221+kβ12. It shows that, under the centralized decision model, if the potential demand of the retailer and the third party for the recycled products was equal, there is no competitive relationship on the behavior of the recycled products, and the price of the recycled products is consistent.


### 3.2. Decentralized Model

This section considers that manufacturers, retailers, and third parties, respectively, make sales pricing decisions and collection pricing decisions to maximize their respective profits. Consider the supply chain under the manufacturer-led Stackelberg model. According to the profit function of retailer (3), profit function of manufacturer (4), and profit function of third party (5), it has(10)πrpr,er=D1−b1pr+b2pdpr−w+1−ατD+β1er−β2ete−er−12k1−ατD+β1er−β2et2,(11)πdpd,w=D1−b1pr+b2pdw+D2−b1pr+b2pdpd+β1−β2cn−cr−eer+et−1−τDcn−τDcr+e,(12)πtet=ατD+β1et−β2ere−et−12kατD+β1et−β2er2.Proposition 2.
When the market is under decentralized situation, if the parameters of supply chain (10)–(12) satisfy the condition of β122+kβ12−β221+kβ12>0, then there exists the unique optimal solution:(13)prd1∗=3b12D1−b22D1+2b1b2D24b1D1b12−b22,pdd1∗=b2D1+b1D22b12−b22,erd1∗=β12+kβ1β1e−1+kβ11−ατD+β21+kβ1β1e−1+kβ1ατDβ122+kβ12−β221+kβ12,etd1∗=β12+kβ1β1e−1+kβ1ατD+β21+kβ1β1e−1+kβ11−ατDβ122+kβ12−β221+kβ12.Proof.
For the retailer and the third party, according to the profit functions of the retailer and the third party (10) and (12), taking the first-order derivatives of πrpr,er and πtet along the retail price pr, the collecting price of retailer er, and the collecting price of the third party et, respectively, we could obtain(14)∂πrpr,er∂pr=−2b1pr+b2pd+b1w+D1,∂πrpr,er∂er=−β12+kβ1er+β21+kβ1et+β1e−1+kβ11−ατD,∂πtet∂et=β21+kβ1er−β12+kβ1et+β1e−1+kβ1ατD.
Hence, the Hessian matrix is given as(15)Hd1=−2b1000−β12+kβ1β21+kβ10β21+kβ1−β12+kβ1.
Obviously, the second derivative satisfies that∂2πrpr,er/∂pr2=−2b1<0 and ∂2πtet/∂et2=∂2πrpr,er/∂er2=−β12+kβ1<0. Based on the condition of proposition, β122+kβ12−β221+kβ12>0. Therefore, the profit functions πrpr,er and πtet are joint concave on pr, er, and et. The optimal solution could be obtained by solving the first-order condition as follows:(16)prd1∗=12b1b2pd+b1w+D1,erd1∗=β12+kβ1β1e−1+kβ11−ατD+β21+kβ1β1e−1+kβ1ατDβ122+kβ12−β221+kβ12,etd1∗=β12+kβ1β1e−1+kβ1ατD+β21+kβ1β1e−1+kβ11−ατDβ122+kβ12−β221+kβ12.
Substitute theprd1∗, erd1∗, and etd1∗ into profit function (10), and it obtains(17)πdpd,w=D2−b1pd+b22b1b2pd+b1w+D1pd+12b2pd+b1w+D1w+β1−β2cn−cr−eer+et−1−τDcn−τDcr+e.
For the manufacturer, taking the first-order derivatives ofπdpd,w along the direct price pd and wholesale price w, we could obtain(18)∂πdpd,w∂pd=−2b1+b22b1pd+b2w+D2+b2D12b1,∂πdpd,w∂w=b2pd−b1w+D12.
The Hessian matrix is given as(19)H1d1=−2b1+b22b1b2b2−b1.
Obviously, the second derivative satisfies that∂2πdpd,w/∂pd2=−2b1+b22/b1<0, ∂2πdpd,w/∂w2=−b1<0, and H1d1=2b12−b22>0. Therefore, the profit function πdpd,w is joint concave on pd and w. The optimal solution could be obtained by solving the first-order condition as follows:(20)pdd1∗=b2D1+b1D22b12−b22,w1∗=b22D1+b1b2D22b1b12−b22+D12b1.
The proof is completed.
According to the conclusion of Proposition2, in the decentralized model, the wholesale price of the manufacturer to retailers is the same as the retail selling price of traditional channels in the centralized model, that is, w2∗=prc1∗, which leads to higher selling prices for retailers under the decentralized model. As for the collection price, the result shows the collection price under the centralized model is higher than the collection price under decentralized model for erc1∗>erd1∗ and etc1∗>etd1∗.


## 3.1. Centralized Model

In the centralized model, all members of the supply chain system cooperatively make a decision on the unit selling price of the retail channel and direct channel and the collection price from the consumer. The profit of the whole supply chain is expressed as(6)πscpr,pd,er,et=Drpr+Ddpd+β1−β2er+etcn−cr−1−τDcn−τDcr−τrer−τtet−12kτr2−12kτt2=D1−b1pr+b2pdpr+D2−b1pd+b2prpd+β1−β2er+etcn−cr−1−τDcn−τDcr−1−ατD+β1er−β2eter−ατD+β1et−β2eret−12k1−ατD+β1er−β2et2−12kατD+β1et−β2er2.Proposition 1.
When the market is under centralized situation, if the parameters satisfy that2β1+kβ12+kβ22−2β21+kβ1>0, then there exists the unique optimal solution:(7)prc1∗=b1D1+b2D22b12−b22,pdc1∗=b2D1+b1D22b12−b22,erc1∗=2β1+kβ12+kβ22β1−β2cn−cr+kβ2ατD−1+kβ11−ατD+2β21+kβ1β1−β2cn−cr+kβ21−ατD−1+kβ1ατD2β1+kβ12+kβ222−4β221+kβ12,etc1∗=2β1+kβ12+kβ22β1−β2cn−cr+kβ21−ατD−1+kβ1ατD+2β21+kβ1β1−β2cn−cr+kβ2ατD−1+kβ11−ατD2β1+kβ12+kβ222−4β221+kβ12.Proof.
According to the profit function of whole supply chain (6), taking the first-order derivatives of πscpr,pd,er,et along the retail price pr, the direct price pd, the collecting price of retailer er, and the collecting price of the third party et, we could obtain(8)∂πscpr,pd,er,et∂pr=−2b1pr+2b2pd+D1,∂πscpr,pd,er,et∂pd=2b2pr−2b1pd+D2,∂πscpr,pd,er,et∂er=−2β1+kβ12+kβ22+2β21+kβ1et+β1−β2cn−cr+kβ2ατD−1+kβ11−ατD,∂πscpr,pd,er,et∂et=2β21+kβ1er−2β1+kβ12+kβ22et+β1−β2cn−cr+kβ21−ατD−1+kβ1ατD.
Hence, the Hessian matrix is given as(9)Hc1=−2b12b2002b2−2b10000−2β1−kβ12−kβ222β21+kβ1002β21+kβ1−2β1−kβ12−kβ22.Obviously, the second derivative satisfies that ∂2πscpr,pd,er,et/∂pr2=∂2πscpr,pd,er,et/∂pd2=−2b1<0. Based on the condition of proposition, 2β1+kβ12+kβ22−2β21+kβ1>0, it could be obtained that 2β1+kβ12+kβ222−4β221+kβ12>0. Therefore, the profit function πscpr,pd,er,et is joint concave on pr, pd, er, and et. The optimal solution could be obtained by solving the first-order condition. The proof is completed.Remark 1.
Proposition1 shows the optimal solution under centralized situation. According to the result of Proposition 1, it could be concluded that the retail selling price and direct selling price are related with the values of D1 and D2. Under the assumption that D1=1−ρa and D2=ρa, we could find prc∗>pdc∗ if 0<ρ<0.5 and prc∗<pdc∗ if 0.5<ρ<1. It demonstrates that the channel has higher selling price when the channel occupies higher market share. Furthermore, if α=0.5, then erc1∗=etc1∗=2β1+kβ12+kβ22+2β21+kβ1β1−β2cn−cr+0.5kβ2τD−0.51+kβ1τD/2β1+kβ12+kβ222−4β221+kβ12. It shows that, under the centralized decision model, if the potential demand of the retailer and the third party for the recycled products was equal, there is no competitive relationship on the behavior of the recycled products, and the price of the recycled products is consistent.


## 3.2. Decentralized Model

This section considers that manufacturers, retailers, and third parties, respectively, make sales pricing decisions and collection pricing decisions to maximize their respective profits. Consider the supply chain under the manufacturer-led Stackelberg model. According to the profit function of retailer (3), profit function of manufacturer (4), and profit function of third party (5), it has(10)πrpr,er=D1−b1pr+b2pdpr−w+1−ατD+β1er−β2ete−er−12k1−ατD+β1er−β2et2,(11)πdpd,w=D1−b1pr+b2pdw+D2−b1pr+b2pdpd+β1−β2cn−cr−eer+et−1−τDcn−τDcr+e,(12)πtet=ατD+β1et−β2ere−et−12kατD+β1et−β2er2.Proposition 2.
When the market is under decentralized situation, if the parameters of supply chain (10)–(12) satisfy the condition of β122+kβ12−β221+kβ12>0, then there exists the unique optimal solution:(13)prd1∗=3b12D1−b22D1+2b1b2D24b1D1b12−b22,pdd1∗=b2D1+b1D22b12−b22,erd1∗=β12+kβ1β1e−1+kβ11−ατD+β21+kβ1β1e−1+kβ1ατDβ122+kβ12−β221+kβ12,etd1∗=β12+kβ1β1e−1+kβ1ατD+β21+kβ1β1e−1+kβ11−ατDβ122+kβ12−β221+kβ12.Proof.
For the retailer and the third party, according to the profit functions of the retailer and the third party (10) and (12), taking the first-order derivatives of πrpr,er and πtet along the retail price pr, the collecting price of retailer er, and the collecting price of the third party et, respectively, we could obtain(14)∂πrpr,er∂pr=−2b1pr+b2pd+b1w+D1,∂πrpr,er∂er=−β12+kβ1er+β21+kβ1et+β1e−1+kβ11−ατD,∂πtet∂et=β21+kβ1er−β12+kβ1et+β1e−1+kβ1ατD.
Hence, the Hessian matrix is given as(15)Hd1=−2b1000−β12+kβ1β21+kβ10β21+kβ1−β12+kβ1.
Obviously, the second derivative satisfies that∂2πrpr,er/∂pr2=−2b1<0 and ∂2πtet/∂et2=∂2πrpr,er/∂er2=−β12+kβ1<0. Based on the condition of proposition, β122+kβ12−β221+kβ12>0. Therefore, the profit functions πrpr,er and πtet are joint concave on pr, er, and et. The optimal solution could be obtained by solving the first-order condition as follows:(16)prd1∗=12b1b2pd+b1w+D1,erd1∗=β12+kβ1β1e−1+kβ11−ατD+β21+kβ1β1e−1+kβ1ατDβ122+kβ12−β221+kβ12,etd1∗=β12+kβ1β1e−1+kβ1ατD+β21+kβ1β1e−1+kβ11−ατDβ122+kβ12−β221+kβ12.
Substitute theprd1∗, erd1∗, and etd1∗ into profit function (10), and it obtains(17)πdpd,w=D2−b1pd+b22b1b2pd+b1w+D1pd+12b2pd+b1w+D1w+β1−β2cn−cr−eer+et−1−τDcn−τDcr+e.
For the manufacturer, taking the first-order derivatives ofπdpd,w along the direct price pd and wholesale price w, we could obtain(18)∂πdpd,w∂pd=−2b1+b22b1pd+b2w+D2+b2D12b1,∂πdpd,w∂w=b2pd−b1w+D12.
The Hessian matrix is given as(19)H1d1=−2b1+b22b1b2b2−b1.
Obviously, the second derivative satisfies that∂2πdpd,w/∂pd2=−2b1+b22/b1<0, ∂2πdpd,w/∂w2=−b1<0, and H1d1=2b12−b22>0. Therefore, the profit function πdpd,w is joint concave on pd and w. The optimal solution could be obtained by solving the first-order condition as follows:(20)pdd1∗=b2D1+b1D22b12−b22,w1∗=b22D1+b1b2D22b1b12−b22+D12b1.
The proof is completed.
According to the conclusion of Proposition2, in the decentralized model, the wholesale price of the manufacturer to retailers is the same as the retail selling price of traditional channels in the centralized model, that is, w2∗=prc1∗, which leads to higher selling prices for retailers under the decentralized model. As for the collection price, the result shows the collection price under the centralized model is higher than the collection price under decentralized model for erc1∗>erd1∗ and etc1∗>etd1∗.


## 4. The Optimization on Sales Effort

The decision on sales effort of the retailer is considered in this section. In practical application, the sales effort on the market has become a key factor of the retail price. More examples show that the demand function depends on sales effort. Taylor points out that the retailer’s sales effort service (mainly sales work, includes providing space or shelf space, promotion, and advertising) increases the demand for products, and the results demonstrated how to realize supply chain coordination through the sales and service effect [26]. Therefore, this section discusses and compares the optimal pricing and optimal profit with sales effort of the retailer.Denote the sales effort of the retailer ass; then, the demand functions of dual channels are expressed as follows:(21)D^r=D1−b1pr+b2pd+g1s,D^d=D2−b1pd+b2pr+g2s.Based on the demand functions, the profit function of the retailer is(22)π^rpr,er=D^rpr−w+τ^re−er−12kτ^2−12ηs2,where D^rpr−w is the profit of selling products by the retail channel and τ^re−er is the profit of collection products from consumers, 1/2kτ^r2 is the cost of collection products, and 1/2ηs2 is the cost of sales effort.The profit function of the manufacturer is(23)π^dw,pd=D^rw+D^dpd+β1−β2cn−cr−eer+et−1−τDcn−τDcr+e,where D^rw is the profit of the retail channel, D^dpd is the profit of the direct channel, 1−τDcn is the production cost of the new product, and τDcr+e is the production cost of the remanufactured product. The profit function of the third party is(24)πtet=τte−et−12kτt2.


### 4.1. Centralized Model with Sales Effort

According to the profit functions of the retailer, manufacturer, and third party (22)–(24), the profit of the whole supply chain is expressed as(25)π^scpr,pd,er,et=Drpr+Ddpd+β1−β2cn−crer+et−1−τDcn−τDcr−τrer−τtet−12kτr2−12kτt2=D1−b1pr+b2pd+g1spr+D2−b1pd+b2pr+g2spd−1−τDcn−τDcr−1−ατD+β1er−β2eter−ατD+β1et−β2eret−12k1−ατD+β1er−β2et2−12kατD+β1et−β2er2−12ηs2.Proposition 3.
When the market is under centralized situation with sales effort, if the parameters of (25) satisfy that 2b12η−2b2g1g2−b1g12+g22−2b22η>0, then there exists the unique optimal solution under the centralized model with sales effort:(26)prc2∗=b1D1+b2D22b12−b22+b1g1+b2g2b1D1+b2D2g1+b1D2+b2D1g22b12−b222ηb12−b22−b1g1+b2g2g1−b1g2+b2g1g2,pdc2∗=b1D2+b2D12b12−b22+b1g2+b2g1b1D1+b2D2g1+b1D2+b2D1g22b12−b222ηb12−b22−b1g1+b2g2g1−b1g2+b2g1g2,erc2∗=2β1+kβ12+kβ22β1−β2cn−cr+kβ2ατD−1+kβ11−ατD+2β21+kβ1β1−β2cn−cr+kβ21−ατD−1+kβ1ατD2β1+kβ12+kβ222−4β221+kβ12,etc2∗=2β1+kβ12+kβ22β1−β2cn−cr+kβ21−ατD−1+kβ1ατD+2β21+kβ1β1−β2cn−cr+kβ2ατD−1+kβ11−ατD2β1+kβ12+kβ222−4β221+kβ12,sc2∗=b1D1+b2D2g1+b1D2+b2D1g22ηb12−b22−b1g1+b2g2g1−b1g2+b2g1g2.Proof.
According to the profit function of the whole supply chain with sales effort (25), taking the first-order derivatives of π^scpr,pd,er,et,s along the retail price pr, the direct price pd, the collecting price of retailer er, the collecting price of the third party et, and the sales effort s, we could obtain(27)∂π^scpr,pd,er,et,s∂pr=−2b1pr+2b2pd+d1+g1s,∂π^scpr,pd,er,et,s∂pd=2b2pr−2b1pd+d2+g2s,∂π^scpr,pd,er,et,s∂s=g1pr+g2pd−ηs,∂π^scpr,pd,er,et,s∂er=−2β1+kβ12+kβ22er+2β21+kβ1et+β1−β2cn−cr+kβ2ατD−1+kβ11−ατD,∂π^scpr,pd,er,et,s∂et=2β21+kβ1er−2β1+kβ12+kβ22et+β1−β2cn−cr+kβ21−ατD−1+kβ1ατD.
Hence, the Hessian matrix is given as(28)Hc2=−2b12b2g12b2−2b1g2g1g2−η.
Obviously, the second derivative satisfies that∂2π^scpr,pd,er,et,s/∂pr2=∂2π^scpr,pd,er,et,s/∂pd2=−2b1<0 and ∂2π^scpr,pd,er,et/∂s2=−η<0. Based on the condition of proposition, 4b22β1+kβ12+kβ22−2β21+kβ1>0, it could be obtained that Hc2=−4b12η+4b2g1g2+2b1g12+g22+4b22η<0. Therefore, the profit function π^scpr,pd,er,et,s is joint concave on pr, pd, er, et, and s. The optimal solution could be obtained by solving the first-order condition:(29)prc2∗=b1D1+b2D22b12−b22+b1g1+b2g22b12−b22sc2∗,pdc2∗=b1D2+b2D12b12−b22+b1g2+b2g12b12−b22sc2∗,sc2∗=b1D1+b2D2g1+b1D2+b2D1g22ηb12−b22−b1g1+b2g2g1−b1g2+b2g1g2.
The proof is completed.
According to Propositions1 and 3, the optimal solution of the centralized model with and without sales effort could be obtained:(30)prc2∗−prc1∗=b1g1+b2g2b1D1+b2D2g1+b1D2+b2D1g22b12−b222ηb12−b22−b1g1+b2g2g1−b1g2+b2g1g2>0,pdc2∗−pdc1∗=b1g2+b2g1b1D1+b2D2g1+b1D2+b2D1g22b12−b222ηb12−b22−b1g1+b2g2g1−b1g2+b2g1g20.
It indicates that the cost of sales effort brings higher selling price, and the increased part is proportional to the sales effort:prc2∗−prc1∗=b1g1+b2g2/2b12−b22sc2∗ and pdc2∗−pdc1∗=b1g2+b2g1/2b12−b22sc2∗.


### 4.2. Decentralized Model with Sales Effort

This section considers that manufacturers, retailers, and third parties, respectively, make sales pricing decisions and collection pricing decisions to maximize their respective profits under the decentralized model with sales effort. Consider the supply chain under the manufacturer-led Stackelberg model. According to the profit function of retailer (22), profit function of manufacturer (23), and profit function of the third party (24), it has(31)π^rpr,er,s=D1−b1pr+b2pd+g1spr−w+1−ατD+β1er−β2ete−er−12k1−ατD+β1er−β2et2−12ηs2,(32)π^dw,pd=D1−b1pr+b2pd+g1sw+D2−b1pd+b2pr+g2spd+β1−β2cn−cr−eer+et−1−τDcn−τDcr+e,(33)π^tet=ατD+β1et−β2ere−et−12kατD+β1et−β2er2.Proposition 4.
When the market is under the decentralized model with sales effort, if the parameters of supply chain (31)–(33) satisfy the condition of 2b1η−g12>0 and 4b12η2b12−b22η−b1g12−b2g1g2−2b1b2η−b2g12−b1g1g22>0, then there exists the unique optimal solution:(34)prd2∗=b2η2b12ηφ+D1b1ηϕ+b1ηw−g122D1b1ηκ+ϕφ2b1η−g124b12ηκ−ϕ2+D1η2b1η−g12,sd2∗=b2g12b12ηφ+D1b1ηϕ−b1g12D1b1ηκ+ϕφ2b1η−g124b12ηκ−ϕ2+D1g12b1η−g12,pdd2∗=2b12ηφ+D1b1ηϕ4b12ηκ−ϕ2,wd2∗=2D1b1ηκ+ϕφ4b12ηκ−ϕ2,wd2∗=2D1b1ηκ+ϕφ4b12ηκ−ϕ2,erd2∗=β12+kβ1β1e−1+kβ11−ατD+β21+kβ1β1e−1+kβ1ατDβ122+kβ12−β221+kβ12,etd2∗=β12+kβ1β1e−1+kβ1ατD+β21+kβ1β1e−1+kβ11−ατDβ122+kβ12−β221+kβ12,where φ=D1b2η+g1g2+D22b1η−g12, κ=2b12−b22η−b1g12−b2g1g2, and ϕ=2b1b2η−b2g12−b1g1g2.Proof.
For the retailer and the third party, according to the profit functions of the retailer and the third party (31) and (33), taking the first-order derivatives of π^rpr,er,s and π^tet along the retail price pr, the collecting price of retailer er, sales effort s, and the collecting price of the third party et, respectively, we could obtain(35)∂π^rpr,er,s∂pr=−2b1pr+b2pd+b1w+g1s+D1,∂π^rpr,er,s∂s=g1pr−g1w−ηs,∂π^rpr,er,s∂er=−β12+kβ1er+β21+kβ1et+β1e−1+kβ11−ατD,∂π^tet∂et=β21+kβ1er−β12+kβ1et+β1e−1+kβ1ατD.
Based on the condition of proposition2b1η−g12>0, it obtains that(36)prd2∗=b2ηpd+b1ηw−g12w+D1η2b1η−g12,sd2∗=b2g1pd−b1g1w+D1g12b1η−g12,erd2∗=β12+kβ1β1e−1+kβ11−ατD+β21+kβ1β1e−1+kβ1ατDβ122+kβ12−β221+kβ12,etd2∗=β12+kβ1β1e−1+kβ1ατD+β21+kβ1β1e−1+kβ11−ατDβ122+kβ12−β221+kβ12.
Substitute theprd2∗, sd2∗, erd2∗, and etd2∗ into profit function (32), for the manufacturer, taking the first-order derivatives of π^dpd,w along the direct price pd and wholesale price w, and we could obtain(37)∂πdpd,w∂pd=−22b12−b22η−b1g12−b2g1g22b1η−g12pd+2b1b2η−b2g12−b1g1g22b1η−g12w+D1b2η+g1g2+D22b1η−g122b1η−g12,∂πdpd,w∂w=2b1b2η−b2g12−b1g1g22b1η−g12pd−2b12η2b1η−g12w+D1b1η2b1η−g12.
The Hessian matrix is given as(38)Hd2=−22b12−b22η−b1g12−b2g1g22b1b2η−b2g12−b1g1g22b1b2η−b2g12−b1g1g2−2b12η.
According to the condition of Proposition4, 4b12η2b12−b22η−b1g12−b2g1g2−2b1b2η−b2g12−b1g1g22>0, we know Hd2=4b12η2b12−b22η−b1g12−b2g1g2−2b1b2η−b2g12−b1g1g22>0. Therefore, the profit function π^Dpd,w is joint concave on pd and w. The optimal solution could be obtained by solving the first-order condition as follows:(39)pdd2∗=2b12ηD1b2η+g1g2+D22b1η−g12+D1b1η2b1b2η−b2g12−b1g1g24b12η2b12−b22η−b1g12−b2g1g2−2b1b2η−b2g12−b1g1g22,wd2∗=2D1b1η2b12−b22η−b1g12−b2g1g24b12η2b12−b22η−b1g12−b2g1g2−2b1b2η−b2g12−b1g1g22+2b1b2η−b2g12−b1g1g2D1b2η+g1g2+D22b1η−g124b12η2b12−b22η−b1g12−b2g1g2−2b1b2η−b2g12−b1g1g22.
The proof is completed.Remark 2.
In the study of [19], analysis for strategy of the closed-loop supply chain with dual recycling channel was carried out. Based on game theory, the authors characterized the supply chain performance in terms of the pricing decisions and the recycling strategies for both the decentralized and the centralized channel scenarios. On the contrary, the supply chain channel strategy with quality and marketing effort-dependent demand was considered in [22]. Different from the existing results, this paper provides four scenarios for different decision models and with or without sales effort. The optimal solution is derived and comparison for four scenarios is given in this paper.


## 4.1. Centralized Model with Sales Effort

According to the profit functions of the retailer, manufacturer, and third party (22)–(24), the profit of the whole supply chain is expressed as(25)π^scpr,pd,er,et=Drpr+Ddpd+β1−β2cn−crer+et−1−τDcn−τDcr−τrer−τtet−12kτr2−12kτt2=D1−b1pr+b2pd+g1spr+D2−b1pd+b2pr+g2spd−1−τDcn−τDcr−1−ατD+β1er−β2eter−ατD+β1et−β2eret−12k1−ατD+β1er−β2et2−12kατD+β1et−β2er2−12ηs2.Proposition 3.
When the market is under centralized situation with sales effort, if the parameters of (25) satisfy that 2b12η−2b2g1g2−b1g12+g22−2b22η>0, then there exists the unique optimal solution under the centralized model with sales effort:(26)prc2∗=b1D1+b2D22b12−b22+b1g1+b2g2b1D1+b2D2g1+b1D2+b2D1g22b12−b222ηb12−b22−b1g1+b2g2g1−b1g2+b2g1g2,pdc2∗=b1D2+b2D12b12−b22+b1g2+b2g1b1D1+b2D2g1+b1D2+b2D1g22b12−b222ηb12−b22−b1g1+b2g2g1−b1g2+b2g1g2,erc2∗=2β1+kβ12+kβ22β1−β2cn−cr+kβ2ατD−1+kβ11−ατD+2β21+kβ1β1−β2cn−cr+kβ21−ατD−1+kβ1ατD2β1+kβ12+kβ222−4β221+kβ12,etc2∗=2β1+kβ12+kβ22β1−β2cn−cr+kβ21−ατD−1+kβ1ατD+2β21+kβ1β1−β2cn−cr+kβ2ατD−1+kβ11−ατD2β1+kβ12+kβ222−4β221+kβ12,sc2∗=b1D1+b2D2g1+b1D2+b2D1g22ηb12−b22−b1g1+b2g2g1−b1g2+b2g1g2.Proof.
According to the profit function of the whole supply chain with sales effort (25), taking the first-order derivatives of π^scpr,pd,er,et,s along the retail price pr, the direct price pd, the collecting price of retailer er, the collecting price of the third party et, and the sales effort s, we could obtain(27)∂π^scpr,pd,er,et,s∂pr=−2b1pr+2b2pd+d1+g1s,∂π^scpr,pd,er,et,s∂pd=2b2pr−2b1pd+d2+g2s,∂π^scpr,pd,er,et,s∂s=g1pr+g2pd−ηs,∂π^scpr,pd,er,et,s∂er=−2β1+kβ12+kβ22er+2β21+kβ1et+β1−β2cn−cr+kβ2ατD−1+kβ11−ατD,∂π^scpr,pd,er,et,s∂et=2β21+kβ1er−2β1+kβ12+kβ22et+β1−β2cn−cr+kβ21−ατD−1+kβ1ατD.
Hence, the Hessian matrix is given as(28)Hc2=−2b12b2g12b2−2b1g2g1g2−η.
Obviously, the second derivative satisfies that∂2π^scpr,pd,er,et,s/∂pr2=∂2π^scpr,pd,er,et,s/∂pd2=−2b1<0 and ∂2π^scpr,pd,er,et/∂s2=−η<0. Based on the condition of proposition, 4b22β1+kβ12+kβ22−2β21+kβ1>0, it could be obtained that Hc2=−4b12η+4b2g1g2+2b1g12+g22+4b22η<0. Therefore, the profit function π^scpr,pd,er,et,s is joint concave on pr, pd, er, et, and s. The optimal solution could be obtained by solving the first-order condition:(29)prc2∗=b1D1+b2D22b12−b22+b1g1+b2g22b12−b22sc2∗,pdc2∗=b1D2+b2D12b12−b22+b1g2+b2g12b12−b22sc2∗,sc2∗=b1D1+b2D2g1+b1D2+b2D1g22ηb12−b22−b1g1+b2g2g1−b1g2+b2g1g2.
The proof is completed.
According to Propositions1 and 3, the optimal solution of the centralized model with and without sales effort could be obtained:(30)prc2∗−prc1∗=b1g1+b2g2b1D1+b2D2g1+b1D2+b2D1g22b12−b222ηb12−b22−b1g1+b2g2g1−b1g2+b2g1g2>0,pdc2∗−pdc1∗=b1g2+b2g1b1D1+b2D2g1+b1D2+b2D1g22b12−b222ηb12−b22−b1g1+b2g2g1−b1g2+b2g1g20.
It indicates that the cost of sales effort brings higher selling price, and the increased part is proportional to the sales effort:prc2∗−prc1∗=b1g1+b2g2/2b12−b22sc2∗ and pdc2∗−pdc1∗=b1g2+b2g1/2b12−b22sc2∗.


## 4.2. Decentralized Model with Sales Effort

This section considers that manufacturers, retailers, and third parties, respectively, make sales pricing decisions and collection pricing decisions to maximize their respective profits under the decentralized model with sales effort. Consider the supply chain under the manufacturer-led Stackelberg model. According to the profit function of retailer (22), profit function of manufacturer (23), and profit function of the third party (24), it has(31)π^rpr,er,s=D1−b1pr+b2pd+g1spr−w+1−ατD+β1er−β2ete−er−12k1−ατD+β1er−β2et2−12ηs2,(32)π^dw,pd=D1−b1pr+b2pd+g1sw+D2−b1pd+b2pr+g2spd+β1−β2cn−cr−eer+et−1−τDcn−τDcr+e,(33)π^tet=ατD+β1et−β2ere−et−12kατD+β1et−β2er2.Proposition 4.
When the market is under the decentralized model with sales effort, if the parameters of supply chain (31)–(33) satisfy the condition of 2b1η−g12>0 and 4b12η2b12−b22η−b1g12−b2g1g2−2b1b2η−b2g12−b1g1g22>0, then there exists the unique optimal solution:(34)prd2∗=b2η2b12ηφ+D1b1ηϕ+b1ηw−g122D1b1ηκ+ϕφ2b1η−g124b12ηκ−ϕ2+D1η2b1η−g12,sd2∗=b2g12b12ηφ+D1b1ηϕ−b1g12D1b1ηκ+ϕφ2b1η−g124b12ηκ−ϕ2+D1g12b1η−g12,pdd2∗=2b12ηφ+D1b1ηϕ4b12ηκ−ϕ2,wd2∗=2D1b1ηκ+ϕφ4b12ηκ−ϕ2,wd2∗=2D1b1ηκ+ϕφ4b12ηκ−ϕ2,erd2∗=β12+kβ1β1e−1+kβ11−ατD+β21+kβ1β1e−1+kβ1ατDβ122+kβ12−β221+kβ12,etd2∗=β12+kβ1β1e−1+kβ1ατD+β21+kβ1β1e−1+kβ11−ατDβ122+kβ12−β221+kβ12,where φ=D1b2η+g1g2+D22b1η−g12, κ=2b12−b22η−b1g12−b2g1g2, and ϕ=2b1b2η−b2g12−b1g1g2.Proof.
For the retailer and the third party, according to the profit functions of the retailer and the third party (31) and (33), taking the first-order derivatives of π^rpr,er,s and π^tet along the retail price pr, the collecting price of retailer er, sales effort s, and the collecting price of the third party et, respectively, we could obtain(35)∂π^rpr,er,s∂pr=−2b1pr+b2pd+b1w+g1s+D1,∂π^rpr,er,s∂s=g1pr−g1w−ηs,∂π^rpr,er,s∂er=−β12+kβ1er+β21+kβ1et+β1e−1+kβ11−ατD,∂π^tet∂et=β21+kβ1er−β12+kβ1et+β1e−1+kβ1ατD.
Based on the condition of proposition2b1η−g12>0, it obtains that(36)prd2∗=b2ηpd+b1ηw−g12w+D1η2b1η−g12,sd2∗=b2g1pd−b1g1w+D1g12b1η−g12,erd2∗=β12+kβ1β1e−1+kβ11−ατD+β21+kβ1β1e−1+kβ1ατDβ122+kβ12−β221+kβ12,etd2∗=β12+kβ1β1e−1+kβ1ατD+β21+kβ1β1e−1+kβ11−ατDβ122+kβ12−β221+kβ12.
Substitute theprd2∗, sd2∗, erd2∗, and etd2∗ into profit function (32), for the manufacturer, taking the first-order derivatives of π^dpd,w along the direct price pd and wholesale price w, and we could obtain(37)∂πdpd,w∂pd=−22b12−b22η−b1g12−b2g1g22b1η−g12pd+2b1b2η−b2g12−b1g1g22b1η−g12w+D1b2η+g1g2+D22b1η−g122b1η−g12,∂πdpd,w∂w=2b1b2η−b2g12−b1g1g22b1η−g12pd−2b12η2b1η−g12w+D1b1η2b1η−g12.
The Hessian matrix is given as(38)Hd2=−22b12−b22η−b1g12−b2g1g22b1b2η−b2g12−b1g1g22b1b2η−b2g12−b1g1g2−2b12η.
According to the condition of Proposition4, 4b12η2b12−b22η−b1g12−b2g1g2−2b1b2η−b2g12−b1g1g22>0, we know Hd2=4b12η2b12−b22η−b1g12−b2g1g2−2b1b2η−b2g12−b1g1g22>0. Therefore, the profit function π^Dpd,w is joint concave on pd and w. The optimal solution could be obtained by solving the first-order condition as follows:(39)pdd2∗=2b12ηD1b2η+g1g2+D22b1η−g12+D1b1η2b1b2η−b2g12−b1g1g24b12η2b12−b22η−b1g12−b2g1g2−2b1b2η−b2g12−b1g1g22,wd2∗=2D1b1η2b12−b22η−b1g12−b2g1g24b12η2b12−b22η−b1g12−b2g1g2−2b1b2η−b2g12−b1g1g22+2b1b2η−b2g12−b1g1g2D1b2η+g1g2+D22b1η−g124b12η2b12−b22η−b1g12−b2g1g2−2b1b2η−b2g12−b1g1g22.
The proof is completed.Remark 2.
In the study of [19], analysis for strategy of the closed-loop supply chain with dual recycling channel was carried out. Based on game theory, the authors characterized the supply chain performance in terms of the pricing decisions and the recycling strategies for both the decentralized and the centralized channel scenarios. On the contrary, the supply chain channel strategy with quality and marketing effort-dependent demand was considered in [22]. Different from the existing results, this paper provides four scenarios for different decision models and with or without sales effort. The optimal solution is derived and comparison for four scenarios is given in this paper.


## 5. Numerical Illustrations

This section carries on numerical example analysis of the proposed propositions by comparing the optimal decision of four propositions. Sensitivity analysis are provided to illustrate the impact of supply chain parameters on system performance. Table2 shows the basic data.Table 2
Table of basic data.
Parameterb1b2τcncrηαβ1β2keg1g2Value0.70.20.950110.510.90.20.150.80.5


### 5.1. Centralized Model and Decentralized Model

By comparing the optimal solution under the centralized model and decentralized model with and without sales effort, it could be concluded from Figure2 that the profit under the centralized model is always higher than the profit under the decentralized model. Therefore, Pareto improvement could be achieved by the revenue sharing contract.Figure 2
The optimal profit of whole supply chain under the centralized and decentralized model.
(a)(b)Figures3 and 4 show that, with the increase of the market scale, retail selling price and direct selling price will increase. It is easy to understand that the increase of the market scale leads to higher demand. Under the decentralized model, the retail selling price is higher than the retail selling price under the centralized model. The existence of wholesale price brings this result under the decentralized model. Figure 4 shows that the direct selling price of the manufacturer is independent of the decentralized model or centralized model. Figures 5 and 6 are the collection price of the retailer and the third party, respectively. With the increase of the market scale, the collection price would decrease.Figure 3
The optimal retail selling price under the centralized and decentralized model.
(a)(b)Figure 4
The optimal direct selling price under the centralized and decentralized model.
(a)(b)Figure 5
The optimal collection price of the retailer under the centralized and decentralized model.
(a)(b)Figure 6
The optimal collection price of the third party under the centralized and decentralized model.
(a)(b)


### 5.2. The Effect of Sales Effort on Optimal Pricing and Profit

The demand for different channels depends on the choice of consumers, and the sales effort of the retailer has an important influence on the market. In the process of operation and management, managers should pay attention to the channel preference of customers. As shown in Figure7, sales effort of the retailer under the centralized model could increase the whole profit of supply chain.Figure 7
The optimal profit of whole supply chain with sales effort under the centralized model.It can be seen from Figure8 that the sales effort of the retailer under the decentralized model could increase the whole profit of the supply chain and the profit of the manufacturer and retailer, respectively.Figure 8
The optimal profit with and without sales effort under the decentralized model.
(a)(b)(c)


### 5.3. Sensitivity Analysis

This section discusses the effect of the marginal cost coefficient of sales effort on profit. It can be found from Figure9 that when η increases, each member’s profit decreases and the rate of change decreases.Figure 9
The effect of the value ofη on profit under the centralized model and decentralized model.
(a)(b)(c)It can be found from Figure10 that when g1 increases, the profits of each supply chain members all increase and the rate of change increases.Figure 10
The effect of the value ofg1 on profit under the centralized model and decentralized model.
(a)(b)(c)


## 5.1. Centralized Model and Decentralized Model

By comparing the optimal solution under the centralized model and decentralized model with and without sales effort, it could be concluded from Figure2 that the profit under the centralized model is always higher than the profit under the decentralized model. Therefore, Pareto improvement could be achieved by the revenue sharing contract.Figure 2
The optimal profit of whole supply chain under the centralized and decentralized model.
(a)(b)Figures3 and 4 show that, with the increase of the market scale, retail selling price and direct selling price will increase. It is easy to understand that the increase of the market scale leads to higher demand. Under the decentralized model, the retail selling price is higher than the retail selling price under the centralized model. The existence of wholesale price brings this result under the decentralized model. Figure 4 shows that the direct selling price of the manufacturer is independent of the decentralized model or centralized model. Figures 5 and 6 are the collection price of the retailer and the third party, respectively. With the increase of the market scale, the collection price would decrease.Figure 3
The optimal retail selling price under the centralized and decentralized model.
(a)(b)Figure 4
The optimal direct selling price under the centralized and decentralized model.
(a)(b)Figure 5
The optimal collection price of the retailer under the centralized and decentralized model.
(a)(b)Figure 6
The optimal collection price of the third party under the centralized and decentralized model.
(a)(b)


## 5.2. The Effect of Sales Effort on Optimal Pricing and Profit

The demand for different channels depends on the choice of consumers, and the sales effort of the retailer has an important influence on the market. In the process of operation and management, managers should pay attention to the channel preference of customers. As shown in Figure7, sales effort of the retailer under the centralized model could increase the whole profit of supply chain.Figure 7
The optimal profit of whole supply chain with sales effort under the centralized model.It can be seen from Figure8 that the sales effort of the retailer under the decentralized model could increase the whole profit of the supply chain and the profit of the manufacturer and retailer, respectively.Figure 8
The optimal profit with and without sales effort under the decentralized model.
(a)(b)(c)


## 5.3. Sensitivity Analysis

This section discusses the effect of the marginal cost coefficient of sales effort on profit. It can be found from Figure9 that when η increases, each member’s profit decreases and the rate of change decreases.Figure 9
The effect of the value ofη on profit under the centralized model and decentralized model.
(a)(b)(c)It can be found from Figure10 that when g1 increases, the profits of each supply chain members all increase and the rate of change increases.Figure 10
The effect of the value ofg1 on profit under the centralized model and decentralized model.
(a)(b)(c)


## 6. Conclusions

This paper studies the optimal pricing of the dual-channel supply chain with the third-party product recovery and sales effort. First of all, for the general case under the centralized and decentralized model, the optimal selling pricing of the direct channel and retail channel in the forward supply chain and the optimal collection pricing of the retail channel and the third party in the backward supply chain are discussed. Then, the optimal pricing strategy with sales effort under the centralized and decentralized model is provided and analyzed. Finally, the comparative analysis of four situations is carried out by numerical results. The effect of sales effort and decision-making model on the optimal pricing, demand, and profit is demonstrated.



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*Source: 4951341-2020-06-25.xml*