# Pricing Decisions and Game Analysis on Advanced Delivery and Cross-Channel Return in a Dual-Channel Supply Chain System

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Literature Review

## 3. The Model

- Which scenario is preferred for the online retailer whether choose the advanced delivery strategy?
- Which scenario is preferred for the online retailer and physical store to adopt a cross-channel return strategy together?
- How does the cost change in two retailers influence the pricing and profit of the supply chain in four cases?
- How does the perceived value added by the online retailer to consumers influence the profits of supply chain members?

#### 3.1. Assumptions and Denotations

- Usually, the customer should provide a deposit for their desired goods until they receive them. If consumers are dissatisfied with the goods or return them due to quality problems, these products are no longer sold during this sales season. Their values are recorded as 0;
- When returning products across channels, the online retailer shall bear the return expenses of the physical store and the freight back to the online retailer’s warehouse because they are different retailers. Furthermore, it shall also give additional subsidies to the physical store, which is recorded as e;
- The online retailer does not consider operating costs, generally;
- We assume that the total return rate of customers who buy products from the online retailer remains unchanged. Only the proportion of returns from the original channel will be apportioned after cross-channel returns are adopted;
- The physical store does not provide a return service for the goods purchased by customers from the physical store;
- The manufacturer’s manufacturing cost is 0.

#### 3.2. The Model

#### 3.2.1. The Scenario of (N, N)

#### 3.2.2. The Scenario of (Y, N)

#### 3.2.3. The Scenario (N, Y)

#### 3.2.4. The Scenario (Y, Y)

## 4. Equilibrium Analysis and Discussions

#### 4.1. Equilibrium Outcomes

**Proposition**

**1.**

**Proposition**

**2.**

**Proposition**

**3.**

**Proposition**

**4.**

#### 4.2. Impact of Physical Store Operation Costs on Supply Chain

**Proposition**

**5.**

#### 4.3. Impact of Operating Costs on Total Demand

**Proposition**

**6.**

#### 4.4. The Impact of Transportation Costs of Online Retailer on Supply Chain

**Proposition**

**7.**

#### 4.5. Impact of Transportation Cost on Total Demand

**Proposition**

**8.**

**Proposition**

**9.**

## 5. Numerical Examples

#### 5.1. The Impact of Transportation Costs on the Profits of the Online Retailer

_{1}= 0.1, u = 0.2, v

_{4}= 0.05, v

_{3}= 0.05, e = 0.02, θ = 0.1, and ϕ = 0.3, because it should be satisfied with u more than 2c

_{t}, c

_{t}taking [0.015, 0.09]. We calculate the profits of the online retailer in four cases when the transportation costs take different values in Table 2. When the transportation costs increase, the profits of the online retailer will decrease. In the four cases studied, the profits of the online retailer decrease significantly with adopting two strategies. The reason for this is that the transportation costs increase under the adoption of the advanced delivery strategy. The profits of the unpaid balances are further reduced, and this affects the overall profits of the online retailer. When the online retailer adopts the cross-channel return service, the transportation costs increase and the costs of the online retailer recovering returns from the physical store further increases. This affects overall profits. Therefore, the adoption of the two strategies has the greatest impact on the online retailer’s profits.

#### 5.2. The Impact of Unpaid Balance Ratio on the Profits of the Online Retailer

_{1}= 0.1, β = 0.7, ε = 0.25, c

_{t}= 0.08, u = 0.2, v

_{4}= 0.05, v

_{3}= 0.05, e = 0.02, and ϕ = 0.3, and θ belongs to [0, 0.5]. wherein it adopts only the advanced delivery and only the cross-channel return strategy, receptively, which are shown in Figure 5 and Figure 6.

#### 5.3. The Impact of Cross-Channel Return Rate on the Profits of the Online Retailer

_{1}= 0.1, β = 0.7, ε = 0.25, c

_{t}= 0.08, u = 0.2, v

_{4}= 0.05, v

_{3}= 0.05, e = 0.02, and θ = 0.1, and ϕ belongs [0.1, 0.5]. We can obtain the profits of the physical store and the online retailer in the scenarios wherein the online retailer adopts a cross-channel return strategy without and with an advanced delivery strategy, which is shown in Figure 7 and Figure 8.

#### 5.4. Impact of Perceived Value of Advanced Delivery on the Profits of Supply Chain Members

#### 5.4.1. The Online Retailer Does Not Adopt Cross-Channel Return Strategy

_{t}= 0.08, u = 0.2, e = 0.02, θ = 0.1, c

_{1}= 0.08, and ϕ = 0.3, and we obtain the profits of each member of the supply chain and the whole supply chain without cross-channel return, as shown in Figure 9, Figure 10 and Figure 11.

_{3}< 0.02. Currently, Proposition 1 shows that the price increase is small, and the change in total demand is also very small. The change in defaulting customers is small. Proposition 3 shows that when wholesale prices decrease, the online retailer’s profits are more significant, and the overall profit in the supply chain is greater under the adoption of the advanced delivery strategy. If the online retailer adopts the advanced delivery strategy, the loss of demand for the physical store is small. Due to the reduction of wholesale prices, the physical store can increase retail prices to increase its profits. On the contrary, the manufacturers’ demand will decrease, and changing wholesale prices cannot recover profits by increasing demand. If $0.02\le {v}_{3}<0.07$, the total demand will rise, and the manufacturer will increase wholesale prices for its own interests. Due to the rise in the manufacturer’s wholesale prices and the online retailers’ adoption of the advanced delivery strategy to attract more customers, the physical store can only choose to increase retail prices in order to make up for the loss of profits. The online retailer will thus lose customers due to these price increases. At this time, the increased profits cannot compensate for the loss of defaulting customers. The profits are not as good as when the strategy is not adopted. The overall profit of the supply chain is not as good as the overall profit when the advanced delivery is not adopted. If ${v}_{3}\ge 0.07$, the online retailer will increase its demand through high perceived value and maintain a steady increase in demand by raising retail prices again. The increased profits resulting from this exceed the losses caused by defaulting customers, and the overall supply chain profits are also better than those without adopting the strategy.

#### 5.4.2. Online Retailer Adopts Cross-Channel Return Strategy

_{t}= 0.08, u = 0.2, e = 0.02, θ = 0.1, c

_{1}= 0.08, v

_{4}= 0.2, and ϕ = 0.3, and we obtain the profits of each member of the supply chain and the whole supply chain without the adoption of the cross-channel return strategy, as shown in Figure 12, Figure 13 and Figure 14.

## 6. Conclusions and Future Research

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## Appendix A

_{1}(p

_{1}) concave in p

_{1}; $\frac{{\partial}^{2}{{\pi}_{2}}^{NN}}{\partial {{p}_{2}}^{2}}=-\frac{2+2\left(-1+\theta \right)\epsilon}{\left(1-\beta \right)\beta}<0$, which shows π

_{2}(p

_{2}) concave in p

_{2}, and $\frac{{\partial}^{2}{{\pi}_{m}}^{NN}}{\partial {w}^{2}}=-\frac{2\left[2+\beta +\beta \left(-1+\theta \right)\epsilon \right]}{\left(4-\beta \right)\beta \left[1+\left(-1+\theta \right)\epsilon \right]}<0$, which shows π

_{m}(w) concave in w. Therefore, let $\frac{\partial {{\pi}_{m}}^{NN}}{\partial w}=0$, and we find the value of w and substitute it into the solution to get the other values in Table A1. The optimal decisions in scenario (Y, N), scenario (N, Y), and scenario (Y, Y) are listed in Table A2, Table A3 and Table A4, respectively.

(N, N) | |
---|---|

${w}^{N{N}^{*}}$ | $\frac{\beta \left(3-{c}_{1}\right)\left(1+\left(-1+\theta \right)\epsilon \right)+2\left(-{c}_{t}+\theta u\right)}{2\left(2+\beta +\beta \left(-1+\theta \right)\epsilon \right)}$. |

${{p}_{1}}^{N{N}^{*}}$ | $\frac{\begin{array}{l}8+8{c}_{1}-2{c}_{t}+2\theta u+2(-1+\theta )(4+4{c}_{1}-2{c}_{t}+2\theta u)\epsilon -4{\left(\beta +\beta (-1+\theta )\epsilon \right)}^{2}\\ +\left(1+(-1+\theta )\epsilon \right)\left(5+{c}_{1}+2{c}_{t}-2\theta u+2(5+{c}_{1})(-1+\theta )\epsilon \right)\beta \end{array}}{2\left(4-\beta \right)\left(1+\left(-1+\theta \right)\epsilon \right)\left(2+\beta +\beta \left(-1+\theta \right)\epsilon \right)}$. |

${{p}_{2}}^{N{N}^{*}}$ | $\frac{\left(-2{\beta}^{2}+\beta \left(5+{c}_{1}\right)\right)\left(1+\left(-1+\theta \right)\epsilon \right)+2\left({c}_{t}-\theta u\right)}{2\left(4-\beta \right)\left(1+\left(-1+\theta \right)\epsilon \right)}$. |

(Y, N) | |
---|---|

${w}^{Y{N}^{*}}$ | $\frac{\beta \left(3-{c}_{1}\right)\left(1+(-1+\theta )\epsilon \right)+2\left(-{c}_{t}-2{c}_{t}\theta +\theta u+{v}_{3}-{v}_{3}\epsilon +{v}_{3}\theta \epsilon \right)}{2\left(2+\beta +\beta \left(-1+\theta \right)\epsilon \right)}$. |

${{p}_{1}}^{Y{N}^{*}}$ | $\frac{\begin{array}{l}4{\left(\beta +\beta \left(-1+\theta \right)\epsilon \right)}^{2}-\left(1+(-1+\theta )\epsilon \right)\left(5+{c}_{1}+2{c}_{t}-2\theta u+2(5+{c}_{1})(-1+\theta )\epsilon \right)\beta \\ -\left(4{c}_{t}-2{v}_{3}\right)\beta \left(1+\left(-1+\theta \right)\epsilon \right)+2\left(\begin{array}{l}-4-4c1+ct+2ct\theta -\theta u-2v3+\left(-1+\theta \right)\\ \left(-4-4c1+2ct+4ct\theta -2\theta u-3v3\right)\epsilon -2v3{\left(-1+\theta \right)}^{2}{\epsilon}^{2}\end{array}\right)\end{array}}{2\left(4-\beta \right)\left(1+\left(-1+\theta \right)\epsilon \right)\left(2+\beta +\beta \left(-1+\theta \right)\epsilon \right)}.$ |

${{p}_{2}}^{Y{N}^{*}}$ | $\frac{\left(-2{\beta}^{2}+\beta \left(5+{c}_{1}-2{v}_{3}\right)\right)\left(1+\left(-1+\theta \right)\epsilon \right)+2\left({c}_{t}+2{c}_{t}\theta -\theta u+3{v}_{3}-3{v}_{3}\epsilon +3{v}_{3}\theta \epsilon \right)}{2\left(4-\beta \right)\left(1+\left(-1+\theta \right)\epsilon \right)}$. |

(N, Y) | |
---|---|

${w}^{N{Y}^{*}}$ | $\frac{2(-{c}_{t}+\theta u+{v}_{4})+2(-1+\theta )\epsilon \left({v}_{4}+\left({c}_{t}+e\right)\varphi \right)+\beta \left((-1+\theta )\epsilon \right)\left(3-{c}_{1}+e(-1+\theta )\epsilon \varphi \right)}{2\left(2+\beta +\beta \left(-1+\theta \right)\epsilon \right)}$. |

${{p}_{1}}^{N{Y}^{*}}$ | $\frac{\begin{array}{l}2(4+4{c}_{1}-{c}_{t}+\theta u+{v}_{4})+6(-1+\theta ){v}_{4}\epsilon +2(-1+\theta )(4+4{c}_{1}-2{c}_{t}+2\theta u)\epsilon +4{v}_{4}(-1+\theta {)}^{2}{\epsilon}^{2}\\ -4{\left(\beta +\beta (-1+\theta )\epsilon \right)}^{2}+2(-1+\theta )\epsilon \varphi \left({c}_{t}-3e+2({c}_{t}-e)(-1+\theta )\epsilon \right)+\beta \left(1+(-1+\theta )\epsilon \right)\\ \left(5+{c}_{1}+2{c}_{t}-2\theta u-2{v}_{4}+2{c}_{1}\left(-1+\theta \right)\epsilon +2\left(-1+\theta \right)\left(5-{v}_{4}\right)\epsilon -\left(-1+\theta \right)\epsilon \left(2{c}_{t}+3e+2e\left(-1+\theta \right)\epsilon \right)\varphi \right).\end{array}}{2\left(4-\beta \right)\left(1+\left(-1+\theta \right)\epsilon \right)\left(2+\beta +\beta \left(-1+\theta \right)\epsilon \right)}$ |

${{p}_{2}}^{N{Y}^{*}}$ | $\frac{\begin{array}{l}-2\left({c}_{t}-\theta u+3{v}_{4}\right)+2{\beta}^{2}\left(1+\left(-1+\theta \right)\epsilon \right)+2\left(-1+\theta \right)\epsilon \left(-3v4+\left(ct+e\right)\varphi \right)\\ -\beta \left(1+\left(-1+\theta \right)\epsilon \right)\left(5+{c}_{1}-2{v}_{4}-e\left(-1+\theta \right)\epsilon \varphi \right)\end{array}}{2\left(-4+\beta \right)\left(1+(-1+\theta )\epsilon \right)}$. |

(Y, Y) | |
---|---|

${w}^{Y{Y}^{*}}$ | $\frac{\begin{array}{l}2\left(-{c}_{t}\left(1+2\theta \right)+\theta u+{v}_{3}+{v}_{4}\right)+2\left(-1+\theta \right)\epsilon \left({v}_{3}+{v}_{4}+\left({c}_{t}+e\right)\varphi \right)+\beta \left(1+\left(-1+\theta \right)\epsilon \right)\\ \left(3-{c}_{1}+e\left(-1+\theta \right)\epsilon \varphi \right)\end{array}}{2\left(2+\beta +\beta \left(-1+\theta \right)\epsilon \right)}$ |

${{p}_{1}}^{Y{Y}^{*}}$ | $\frac{\begin{array}{l}-2\theta u-2\left(4+4{c}_{1}-{c}_{t}+{v}_{3}+{v}_{4}-4\epsilon \right)+8{c}_{1}\epsilon +6({v}_{3}+{v}_{4})\epsilon -2\theta \left(4+4{c}_{1}+2(-1+\theta )u+3{v}_{3}+3{v}_{4}\right)\epsilon \\ -4{\left(-1+\theta \right)}^{2}\left(v3+v4\right){\epsilon}^{2}+4{\left(\beta +\beta (-1+\theta )\epsilon \right)}^{2}+4ct\left(\theta +(-1+\theta )(1+2\theta )\epsilon \right)-\\ 2ct\left(-1+\theta \right)\epsilon \left(1+2(-1+\theta )\epsilon \right)\varphi +2e(-1+\theta )\epsilon \left(3+2(-1+\theta )\epsilon \right)\varphi -\beta \left(1+\left(-1+\theta \right)\epsilon \right)\\ \left(\begin{array}{l}5+{c}_{1}+2{c}_{t}+4{c}_{t}\theta -2\theta u-2{v}_{3}-2{v}_{4}-10\epsilon +2\epsilon \left({c}_{1}\left(-1+\theta \right)+{v}_{3}+{v}_{4}-\theta \left(-5+{v}_{3}+{v}_{4}\right)\right)\\ -\left(-1+\theta \right)\epsilon \left(2{c}_{t}+e\left(3+2\left(-1+\theta \right)\epsilon \right)\right)\varphi \end{array}\right)\end{array}}{2\left(-4+\beta \right)\left(1+\left(-1+\theta \right)\epsilon \right)\left(2+\beta +\beta \left(-1+\theta \right)\epsilon \right)}$ |

${{p}_{2}}^{Y{Y}^{*}}$ | $\frac{\begin{array}{l}-2\left({c}_{t}+2{c}_{t}\theta -\theta u+3\left({v}_{3}+{v}_{4}\right)\right)+2{\beta}^{2}\left(1+\left(-1+\theta \right)\epsilon \right)+2\left(-1+\theta \right)\epsilon \\ \left(-3\left({v}_{3}+{v}_{4}\right)+\left({c}_{t}+e\right)\varphi \right)-\beta \left(1+\left(-1+\theta \right)\epsilon \right)\left(5+{c}_{1}-2{v}_{3}-2{v}_{4}-e\left(-1+\theta \right)\epsilon \varphi \right)\end{array}}{2\left(4-\beta \right)\left(1+\left(-1+\theta \right)\epsilon \right)}$ |

## Appendix B. Proof of Proposition 1

## Appendix C. Proof of Proposition 2

## Appendix D. Proof of Proposition 3

## Appendix E. Proof of Proposition 4

## Appendix F. Proof of Proposition 5

## Appendix G. Proof of Proposition 6

## Appendix H. Proof of Proposition 7

^{*}and 1. We obtain this when $\theta \le 1/\left(2-\beta \right)$, then $0<{\epsilon}^{*}<1$. If $\theta >1/\left(2-\beta \right)$, then ${\epsilon}^{*}>1$. On this basis, we obtain the following conclusions:

## Appendix I. Proof of Proposition 9

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**Figure 5.**The impact of unpaid balance on the profits of online retailer without cross-channel return.

**Figure 9.**The influence of customer perceived value on the physical store and manufacturer when online retailer does not adopt cross-channel return.

**Figure 10.**The influence of customer perceived value on the online retailer without cross-channel return.

**Figure 11.**The influence of customer perceived value on the supply chain when online retailer does not adopt cross-channel return strategy.

**Figure 12.**The influence of customer perceived value on the physical store and manufacturer if online retailer adopts cross-channel return strategy.

**Figure 13.**The influence of customer perceived value on the online retailer adopting cross-channel return strategy.

**Figure 14.**The influence of customer perceived value on the supply chain when online retailer adopts cross-channel return strategy.

Notation | Notation Description |
---|---|

Decision variables | |

w | Wholesale price for the product |

p_{1} | Selling price for a product of the physical store |

p_{2} | Selling price for a product of the online retailer |

Parameters | |

ε | The proportion of the returned product |

ϕ | The proportion of the cross-channel return product |

β | Matching probability of online retailer’ products to consumers |

V | The perceived value of consumers (evenly distributed within (0, 1)) |

v_{3} | The increase in the perceived value of consumers when online retailer adopts an advanced delivery strategy |

v_{4} | The increase in the perceived value of consumers when online retailer adopts a cross-channel return strategy |

U_{1} | The surplus value of products purchased by consumers from the physical store |

U_{2} | The surplus value of products purchased by consumers from the online retailer |

θ | The proportion of consumers who do not pay the balance after paying the deposit |

c_{1} | Operating costs of the physical store |

c_{t} | The transportation costs paid by the online retailer’ |

u | The deposits paid by consumers |

e | The subsidies for the physical store for cross-channel returns from the online retailer |

π_{1} | The profit of the physical store |

π_{2} | The profit of the online retailer |

π_{m} | The profit of the manufacturer |

D_{1} | The demand for the physical store |

D_{2} | The demand for the online retailer |

${c}_{t}$ | ${{\pi}_{2}}^{N{N}^{*}}$ | ${{\pi}_{2}}^{Y{N}^{*}}$ | ${{\pi}_{2}}^{N{Y}^{*}}$ | ${{\pi}_{2}}^{Y{Y}^{*}}$ |
---|---|---|---|---|

0.015 | 0.00431 | 0.00758 | 0.00621 | 0.01005 |

0.03 | 0.00288 | 0.00528 | 0.00427 | 0.00712 |

0.045 | 0.00173 | 0.00339 | 0.00269 | 0.00469 |

0.06 | 0.00088 | 0.00192 | 0.00147 | 0.00277 |

0.075 | 0.00031 | 0.00086 | 0.00062 | 0.00135 |

0.09 | 0.00003 | 0.00022 | 0.00013 | 0.00044 |

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## Share and Cite

**MDPI and ACS Style**

Zhang, R.; Lu, Y.; Liu, B.
Pricing Decisions and Game Analysis on Advanced Delivery and Cross-Channel Return in a Dual-Channel Supply Chain System. *Systems* **2023**, *11*, 155.
https://doi.org/10.3390/systems11030155

**AMA Style**

Zhang R, Lu Y, Liu B.
Pricing Decisions and Game Analysis on Advanced Delivery and Cross-Channel Return in a Dual-Channel Supply Chain System. *Systems*. 2023; 11(3):155.
https://doi.org/10.3390/systems11030155

**Chicago/Turabian Style**

Zhang, Rong, Yuhao Lu, and Bin Liu.
2023. "Pricing Decisions and Game Analysis on Advanced Delivery and Cross-Channel Return in a Dual-Channel Supply Chain System" *Systems* 11, no. 3: 155.
https://doi.org/10.3390/systems11030155