# Dual-Channel Supply Chain Pricing Decisions under Discounted Advertising Value

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## Abstract

**:**

## 1. Introduction

#### 1.1. Backgrounds and Motivations

#### 1.2. Research Questions and Main Findings

- (i)
- What is the impact of advertising strategies in different sales channels on the profitability of a dual-channel retailer?
- (ii)
- What should advertise levels and pricing decisions be for a dual-channel retailer across different sales channels?
- (iii)
- Given optimal advertising levels and pricing decisions, which is more profitable for retailers, online or offline sales channels?

#### 1.3. Contributions and Paper Arrangements

## 2. Literature Review

#### 2.1. Dual-Channel Operation

#### 2.2. Advertising Strategies

## 3. The Problems and Assumptions

_{i}($i=t,l$, where t denotes the online sales channel and l denotes the offline sales channel). The market demand for the product is D

_{i}. The consumers’ valuation for the product is x. This paper assumes that x is uniformly distributed between the signs 0 and 1, i.e., $x~U\left(0,1\right)$. The probability density function is f(x), and the cumulative distribution function is F(x).

_{i}. The cost to the retailer to achieve this level of advertising is ${C}_{i}=k{e}_{i}{}^{2}/2$. This quadratic function is widely used in operations studies [37,38,39]. Advertising increases the consumer’s utility. However, because consumers cannot touch and feel the product when they buy it online, the effect of advertising is discounted by a factor of β. Then we can derive consumers’ utility when purchasing online and offline. Given the above reasons, this paper explores the retailer’s pricing and advertisement strategies when selling the product online and offline. The notation used in this paper is shown in Table 1.

## 4. Models and Analysis

#### 4.1. Online Sales Model

#### 4.2. Offline Sales Model

#### 4.3. Sensitivity Analysis

**Proposition**

**1.**

- (i)
- ${p}_{t}^{*}$decreases with$k$, increases with${c}_{t}$,$n$, and$\beta $.
- (ii)
- ${e}_{t}^{*}$decreases with$k$and${c}_{t}$, increases with$n$and$\beta $.
- (iii)
- ${\pi}_{t}^{*}$decreases with$k$and${c}_{t}$, increases with$n$and$\beta $.

**Proposition**

**2.**

- (i)
- ${p}_{l}^{*}$decreases with$k$and$n$; increases with${c}_{l}$.
- (ii)
- ${e}_{l}^{*}$decreases with$k$,$n$, and${c}_{l}$.
- (iii)
- ${\pi}_{l}^{*}$decreases with$k$,$n$, and${c}_{l}$.

## 5. The Retailer’s Channel Decision

**Proposition**

**3.**

## 6. Extended Models: Advertising Effect

**Proposition**

**4.**

## 7. Numerical Analysis

## 8. Conclusions

#### 8.1. Conclusion Remarks

#### 8.2. Managerial Insights

#### 8.3. Future Research

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A

**Proof**

**of Proposition 1.**

**Proof**

**of Proposition 2.**

**Proof**

**of Proposition 3.**

**Proof**

**of Proposition 4.**

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**Figure 2.**Variation of optimal profit with advertising level cost factor for the online sales channel.

**Figure 3.**Variation in optimal profit with the level of advertising discounts for the online sales channel.

**Figure 4.**Variation of optimal profit with advertising level cost factor for the offline sales channels.

**Figure 5.**Optimal profit for an offline sales channel as a function of the cost of maintaining that channel.

Notations | Explanation |
---|---|

${U}_{i}$ | The consumers’ utility when purchasing the product, where $i=t,l$. |

${D}_{i}$ | The demand for the product, where $i=t,l$. |

$n$ | The number of consumers. |

${e}_{i}$ | The advertisement levels, where $i=t,l$. |

${c}_{i}$ | The cost for sale channels, where $i=t,l$. |

${C}_{i}$ | The cost for the advertisement, where $i=t,l$. |

$k$ | The cost coefficient for the advertisements, where $0<k<1$. |

$\beta $ | The discount factor for the online advertisements, where $0<\beta <1$. |

${p}_{i}$ | The retail prices, where $i=t,l$. |

$x$ | Consumers’ valuation for the product, where $x~U\left(0,1\right)$. |

${\pi}_{i}$ | The retailer’s profits, where $i=t,l$. |

$\Delta \pi $ | The difference in profits between online and offline channels. |

$\mathit{k}$ | $\mathit{\beta}$ | $\mathit{n}$ | ${\mathit{c}}_{\mathit{t}}$ | ${\mathit{c}}_{\mathit{l}}$ | |
---|---|---|---|---|---|

${p}_{t}^{*}\uparrow $ | $\downarrow $ | $\uparrow $ | $\uparrow $ | $\uparrow $ | $-$ |

${e}_{t}^{*}\uparrow $ | $\downarrow $ | $\uparrow $ | $\uparrow $ | $\downarrow $ | $-$ |

${\pi}_{t}{}^{*}\uparrow $ | $\downarrow $ | $\uparrow $ | $\uparrow $ | $\downarrow $ | $-$ |

${p}_{l}^{*}\uparrow $ | $\downarrow $ | $-$ | $\downarrow $ | $\uparrow $ | |

${e}_{l}^{*}\uparrow $ | $\downarrow $ | $-$ | $\downarrow $ | $\downarrow $ | |

${\pi}_{l}{}^{*}\uparrow $ | $\downarrow $ | $-$ | $\downarrow $ | $\downarrow $ |

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**MDPI and ACS Style**

Yan, K.; Liu, S.; Zuo, M.; Zheng, J.; Xu, Y.
Dual-Channel Supply Chain Pricing Decisions under Discounted Advertising Value. *Systems* **2022**, *10*, 76.
https://doi.org/10.3390/systems10030076

**AMA Style**

Yan K, Liu S, Zuo M, Zheng J, Xu Y.
Dual-Channel Supply Chain Pricing Decisions under Discounted Advertising Value. *Systems*. 2022; 10(3):76.
https://doi.org/10.3390/systems10030076

**Chicago/Turabian Style**

Yan, Ke, Shuai Liu, Min Zuo, Jiamin Zheng, and Yadong Xu.
2022. "Dual-Channel Supply Chain Pricing Decisions under Discounted Advertising Value" *Systems* 10, no. 3: 76.
https://doi.org/10.3390/systems10030076