# Optimal Decision in a Dual-Channel Supply Chain under Potential Information Leakage

^{*}

## Abstract

**:**

## 1. Introduction

- (1)
- Should the retailer invest in demand information acquisition?
- (2)
- Faced with the strategic supplier and retailer, what is the optimal strategy in dual-channel supply chain?
- (3)
- To the retailer, how to strategically share demand information with the supplier?
- (4)
- To the supplier, what is the credible information revelation mechanism?

## 2. The Model

#### 2.1. Dual-Channel Supply Chain Structure

**Lemma**

**1 (PBNE without information acquisition).**

**Proof.**

#### 2.2. Optimal Strategy under Potential Information Leakage

**Proposition**

**1 (PBNE under potential information leakage).**

**Proof.**

#### 2.3. Conditions for the Retailer to Conduct Market Research

**Proposition**

**2.**

**Proof.**

## 3. Numerical Examples

#### 3.1. Optimal Decision with Uniform Distribution of Demand Disturbance

#### 3.2. Impact of Adverse Selection Behavior on the Optimal Quantity

#### 3.3. Impact of Product Heterogeneity and Wholesale Price on the Distortion Threshold

#### 3.4. Discussion

## 4. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

- Cai, G. Channel Selection and Coordination in Dual-Channel Supply Chains. J. Retail.
**2010**, 86, 22–36. [Google Scholar] [CrossRef] - Xiao, T.; Jing, S.; Chen, G. Price and leadtime competition, and coordination for make-to-order supply chains. Comput. Ind. Eng.
**2014**, 68, 23–34. [Google Scholar] [CrossRef] - Zhi, P.; Toombs, L.; Yan, R. How does the added new online channel impact the supporting advertising expenditure? J. Retail. Consum. Serv.
**2014**, 21, 229–238. [Google Scholar] - Lei, M.; Liu, H.; Deng, H.; Huang, T.; Leong, G.K. Demand information sharing and channel choice in a dual-channel supply chain with multiple retailers. Int. J. Prod. Res.
**2014**, 52, 6792–6818. [Google Scholar] [CrossRef] - Anand, K.S.; Goyal, M. Strategic Information Management under Leakage in a Supply Chain. Manag. Sci.
**2009**, 55, 438–452. [Google Scholar] [CrossRef] - Kong, G.; Rajagopalan, S.; Zhang, H. Revenue Sharing and Information Leakage in a Supply Chain. Manag. Sci.
**2013**, 59, 556–572. [Google Scholar] [CrossRef] - Lee, H.L.; So, K.C.; Tang, C.S. The Value of Information Sharing in a Two-Level Supply Chain. Manag. Sci.
**2000**, 46, 626–643. [Google Scholar] [CrossRef] - Lin, F.R.; Huang, S.; Lin, S. Effects of information sharing on supply chain performance in electronic commerce. IEEE Trans. Eng. Manag.
**2002**, 49, 258–268. [Google Scholar] - Zhao, Y.; Simchilevi, D. The Value of Information Sharing in a Two-Stage Supply Chain with Production Capacity Constraints: The Infinite Horizon Case. Nav. Res. Logist.
**2003**, 50, 247–274. [Google Scholar] [CrossRef] - Ganesh, M.; Raghunathan, S.; Rajendran, C. The value of information sharing in a multi-product supply chain with product substitution. IIE Trans.
**2008**, 40, 1124–1140. [Google Scholar] [CrossRef] - Wakolbinger, T.; Cruz, J.M. Supply chain disruption risk management through strategic information acquisition and sharing and risk-sharing contracts. Int. J. Prod. Res.
**2011**, 49, 4063–4084. [Google Scholar] [CrossRef] - Zhu, X. Outsourcing management under various demand Information Sharing scenarios. Ann. Oper. Res.
**2017**, 257, 449–467. [Google Scholar] [CrossRef] - Rached, M.; Bahroun, Z.; Campagne, J.P. Decentralized decision-making with information sharing vs. centralized decision-making in supply chains. Int. J. Prod. Res.
**2016**, 54, 7274–7295. [Google Scholar] [CrossRef] - Cao, E.; Ma, Y.; Wan, C.; Lai, M. Contracting with asymmetric cost information in a dual-channel supply chain. Oper. Res. Lett.
**2013**, 41, 410–414. [Google Scholar] [CrossRef] - Li, Q.; Li, B.; Chen, P.; Hou, P. Dual-channel supply chain decisions under asymmetric information with a risk-averse retailer. Ann. Oper. Res.
**2017**, 257, 423–447. [Google Scholar] [CrossRef] - Li, B.; Chen, P.; Li, Q.; Wang, W. Dual-channel supply chain pricing decisions with a risk-averse retailer. Int. J. Prod. Res.
**2014**, 52, 7132–7147. [Google Scholar] [CrossRef] - Li, T.; Ma, J. Complexity analysis of the dual-channel supply chain model with delay decision. Nonlinear Dyn.
**2014**, 78, 2617–2626. [Google Scholar] [CrossRef] - Chiang, W.Y.K.; Chhajed, D.; Hess, J.D. Direct Marketing, Indirect Profits: A Strategic Analysis of Dual-Channel Supply-Chain Design. Manag. Sci.
**2003**, 49, 1–20. [Google Scholar] [CrossRef] [Green Version] - Sun, Y.; Liang, X.; Li, X.; Zhang, C. A Fuzzy Programming Method for Modeling Demand Uncertainty in the Capacitated Road–Rail Multimodal Routing Problem with Time Windows. Symmetry
**2019**, 11, 91. [Google Scholar] [CrossRef] - Cai, G.; Zhe, G.Z.; Zhang, M. Game theoretical perspectives on dual-channel supply chain competition with price discounts and pricing schemes. Int. J. Prod. Econ.
**2009**, 117, 80–96. [Google Scholar] [CrossRef] - Song, H.; Chao, Y.; Hui, L. Pricing and production decisions in a dual-channel supply chain when production costs are disrupted. Econ. Model.
**2013**, 30, 521–538. [Google Scholar] - Song, Z.; He, S.; An, B. Decision and Coordination in a Dual-Channel Three-Layered Green Supply Chain. Symmetry
**2018**, 10, 549. [Google Scholar] [CrossRef] - Yue, X.; Liu, J. Demand forecast sharing in a dual-channel supply chain. Eur. J. Oper. Res.
**2006**, 174, 646–667. [Google Scholar] [CrossRef] - Yan, R.; Pei, Z. Incentive-Compatible Information Sharing by Dual-Channel Retailers. Int. J. Electron. Commer.
**2012**, 17, 127–157. [Google Scholar] [CrossRef] - Hua, G.; Wang, S.; Chengc, T.C.E. Price and lead time decisions in dual-channel supply chains. Eur. J. Oper. Res.
**2010**, 205, 113–126. [Google Scholar] [CrossRef] - Chen, J.; Liang, L.; Yao, D.Q.; Sun, S. Price and quality decisions in dual-channel supply chains. Eur. J. Oper. Res.
**2016**, 259, 935–948. [Google Scholar] [CrossRef] - He, Y.; Song, H.; Zhang, P. Optimal Selling Strategy in Dual-Channel Supply Chains. In LISS 2012; Springer: Berlin/Heidelberg, Germany, 2013. [Google Scholar]
- Chiangab, W.Y.K. Managing inventories in a two-echelon dual-channel supply chain. Eur. J. Oper. Res.
**2005**, 162, 325–341. [Google Scholar] [CrossRef] - Cao, E. Coordination of dual-channel supply chains under demand disruptions management decisions. Int. J. Prod. Res.
**2014**, 52, 7114–7131. [Google Scholar] [CrossRef] - Amrouche, N.; Yan, R. A manufacturer distribution issue: How to manage an online and a traditional retailer. Ann. Oper. Res.
**2015**, 244, 1–38. [Google Scholar] [CrossRef] - Mahootchi, M. Investigating replenishment policies for centralized and decentralized supply chains using stochastic programming approach. Int. J. Prod. Res.
**2015**, 53, 41–69. [Google Scholar]

**Figure 2.**(

**a**) Optimal ordering quantity in the dual-channel supply chain under different demand disturbance; (

**b**) profits comparison of the retailer with and without demand information.

**Figure 3.**Impact of product heterogeneity $d$ and wholesale price $w$ on the distortion threshold $\hat{\theta}$. (

**a**) Combined effect based on $2{b}_{r}{b}_{s}>{d}^{2}$; (

**b**) Combined effect based on $2{b}_{r}{b}_{s}<{d}^{2}$; (

**c**) Product heterogeneity sensitivity based on $2{b}_{r}{b}_{s}>{d}^{2}$; (

**d**) Product heterogeneity sensitivity based on $2{b}_{r}{b}_{s}<{d}^{2}$; (

**e**) Wholesale price sensitivity based on $2{b}_{r}{b}_{s}>{d}^{2}$; (

**f**) Wholesale price sensitivity based on $2{b}_{r}{b}_{s}<{d}^{2}$.

$\mathbf{Demand}\text{}\mathbf{Disturbance}\text{}\mathit{\theta}$ | $\mathbf{Optimal}\text{}\mathbf{Quantity}\text{}{\widehat{\mathit{q}}}_{\mathit{r}}$ | $\mathbf{Profit}\text{}\mathbf{with}\text{}\mathbf{Demand}\text{}\mathbf{Information}{\widehat{\mathit{\pi}}}_{\mathit{r}}$ | $\mathbf{Profit}\text{}\mathbf{with}\text{}\mathbf{no}\text{}\mathbf{Demand}\text{}\mathbf{Information}\text{}{\mathit{\pi}}_{\mathit{r}}^{*}$ | Information Value |
---|---|---|---|---|

−17.7201 | 31.1399 | 208.9 | 91.2 | 117.7507 |

−1.1699 | 39.4151 | 753.7 | 753.2 | 0.5132 |

0 | 40.0000 | 800.0 | 800.0 | 0 |

3.7505 | 31.2141 | 878.5 | 950.0 | −71.5472 |

10.5887 | 32.5023 | 1116.0 | 1223.5 | −107.4981 |

25.1779 | 35.7372 | 1690.7 | 1807.1 | −116.4126 |

80/3 ($\approx $26.6667) | 53.3333 | 1866.7 | 1866.7 | 0 |

33.0701 | 56.5350 | 2532.9 | 2122.8 | 410.1118 |

37.1911 | 58.5956 | 2806.3 | 2287.6 | 518.6917 |

**Table 2.**Summary of partial derivatives of product heterogeneity and wholesale price to threshold $\widehat{\theta}$.

Notations | Mathematical Expression | Definition | Mathematical Expression |
---|---|---|---|

$\frac{\partial {q}_{r}^{*}}{\partial w}$ | $\frac{-2{b}_{s}}{4{b}_{r}{b}_{s}-2{d}^{2}}\{\begin{array}{c}<0,2{b}_{r}{b}_{s}{d}^{2}\\ 0,2{b}_{r}{b}_{s}{d}^{2}\end{array}$ | $\frac{{\partial}^{2}{q}_{r}^{*}}{\partial {w}^{2}}$ | 0 |

$\frac{\partial \widehat{\theta}}{\partial w}$ | $\frac{-4{b}_{s}d}{{\left(2{b}_{s}-d\right)}^{2}}<0$ | $\frac{{\partial}^{2}\widehat{\theta}}{\partial {w}^{2}}$ | 0 |

$\frac{\partial {q}_{r}^{*}}{\partial d}$ | $\frac{4\left(2{b}_{s}{a}_{r}-d{a}_{s}-2{b}_{s}w\right)d-{a}_{s}\left(4{b}_{r}{b}_{s}-2{d}^{2}\right)}{{\left(4{b}_{r}{b}_{s}-2{d}^{2}\right)}^{2}}<0$ | ||

$\frac{{\partial}^{2}{q}_{r}^{*}}{\partial {d}^{2}}$ | $\frac{4\left(2{b}_{s}{a}_{r}-d{a}_{s}-2{b}_{s}w\right)\left(4{b}_{r}{b}_{s}+6{d}^{2}\right)+2{a}_{s}d\left(4{b}_{r}{b}_{s}-2{d}^{2}\right)\left(d-4\right)}{{\left(4{b}_{r}{b}_{s}-2{d}^{2}\right)}^{3}}$ | ||

$\frac{\partial \widehat{\theta}}{\partial d}$ | $\widehat{\theta}+\frac{2\left[4\left(2{b}_{s}{b}_{r}-{d}^{2}\right)\frac{\partial {q}_{r}^{*}}{\partial d}-2w+2\left({a}_{s}+{a}_{r}\right)-8d{q}_{r}^{*}\right]{b}_{s}-4{a}_{s}d}{{\left(2{b}_{s}-d\right)}^{2}}>0$ | ||

$\frac{{\partial}^{2}\widehat{\theta}}{\partial {d}^{2}}$ | $\frac{\partial \widehat{\theta}}{\partial d}+\frac{\left\{2{b}_{s}\left[4\left(2{b}_{s}{b}_{r}-{d}^{2}\right)\frac{{\partial}^{2}{q}_{r}^{*}}{\partial {d}^{2}}-16d\frac{\partial {q}_{r}^{*}}{\partial d}-8{q}_{r}^{*}\right]-4{a}_{s}\right\}\left(2{b}_{s}-d\right)+}{{\left(2{b}_{s}-d\right)}^{3}}+\frac{\left\{2{b}_{s}\left[4\left(2{b}_{s}{b}_{r}-{d}^{2}\right)\frac{\partial {q}_{r}^{*}}{\partial d}-2w+2\left({a}_{s}+{a}_{r}\right)-8d{q}_{r}^{*}\right]-4{a}_{s}d\right\}}{{\left(2{b}_{s}-d\right)}^{3}}$ |

© 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Fang, D.; Ren, Q.
Optimal Decision in a Dual-Channel Supply Chain under Potential Information Leakage. *Symmetry* **2019**, *11*, 308.
https://doi.org/10.3390/sym11030308

**AMA Style**

Fang D, Ren Q.
Optimal Decision in a Dual-Channel Supply Chain under Potential Information Leakage. *Symmetry*. 2019; 11(3):308.
https://doi.org/10.3390/sym11030308

**Chicago/Turabian Style**

Fang, Debin, and Qiyu Ren.
2019. "Optimal Decision in a Dual-Channel Supply Chain under Potential Information Leakage" *Symmetry* 11, no. 3: 308.
https://doi.org/10.3390/sym11030308