# Supply Chain Coordination with a Risk-Averse Retailer and the Call Option Contract in the Presence of a Service Requirement

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## Abstract

**:**

## 1. Introduction

- (1)
- How does the risk-averse retailer determine the order quantity to maximize CVaR about profit in a supply chain with the call option contract in the presence of a service requirement?
- (2)
- What is the supplier’s optimal production policy with the call option contract and a service requirement?
- (3)
- How do risk aversion, service requirement and contract parameters affect the retailer’s optimal order policy, the supplier’s optimal production policy and supply chain performance?
- (4)
- What is the condition for supply chain coordination with the call option contract and a service requirement?

## 2. Literature Review

## 3. Model Description

## 4. Risk-Averse Retailer’s Optimal Ordering Policy

**Theorem**

**1.**

**Proof.**

**Corollary**

**1.**

**Proof.**

**Corollary**

**2.**

**Proof.**

**Corollary**

**3.**

- (i)
- ${\mathit{CVaR}}_{\eta}\left({\pi}_{r}(D;{q}^{*})\right)$ is decreasing in o,
- (ii)
- When $\eta \beta \u2a7e\alpha $, ${\mathit{CVaR}}_{\eta}\left({\pi}_{r}(D;{q}^{*})\right)$ is decreasing in e. When $\eta \beta <\alpha $, if $\alpha <\eta $, then ${\mathit{CVaR}}_{\eta}\left({\pi}_{r}(D;{q}^{*})\right)$ is decreasing in e; if $\alpha =\eta $, then ${\mathit{CVaR}}_{\eta}\left({\pi}_{r}(D;{q}^{*})\right)$ is constant in e; otherwise, ${\mathit{CVaR}}_{\eta}\left({\pi}_{r}(D;{q}^{*})\right)$ is increasing in e.

**Proof.**

## 5. Risk-Neutral Supplier’s Optimal Production Policy

**Theorem**

**2.**

**Proof.**

**Corollary**

**4.**

**Proof.**

**Corollary**

**5.**

**Proof.**

**Corollary**

**6.**

**Proof.**

## 6. Supply Chain Coordination

**Theorem**

**3.**

**Proof.**

**Theorem**

**4.**

**Proof.**

## 7. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A

**Proof of**

**Theorem 1.**

**Proof of**

**Corollary 1.**

**Proof of**

**Corollary 2.**

**Proof of**

**Corollary 3.**

**Proof of**

**Theorem 2.**

**Proof of**

**Corollary 4.**

**Proof of**

**Corollary 5.**

**Proof of**

**Corollary 6.**

**Proof of**

**Theorem 3.**

**Proof of**

**Theorem 4.**

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Author(s) | Call Option Contract | Service Requirement | Risk Aversion | CVaR |
---|---|---|---|---|

Barnes-Schuster et al. [21] | √ | |||

Fu et al. [22] | √ | |||

Zhao et al. [7] | √ | √ | ||

Hu et al. [1] | √ | |||

Chen et al. [8] | √ | √ | ||

Luo et al. [23] | √ | |||

Wang et al. [24] | √ | |||

Zhuo et al. [25] | √ | |||

Wan and Chen [26] | √ | |||

Huang et al. [27] | √ | |||

Fan et al. [28] | √ | √ | √ | |

Liu et al. [29] | √ | √ | √ | |

Yang et al. [30] | √ | √ | ||

Wu et al. [31] | √ | √ | ||

Chen et al. [32] | √ | √ | ||

Li et al. [33] | √ | √ | ||

Wang et al. [34] | √ | √ | √ | |

Xie et al. [35] | √ | √ | ||

Zhao et al. [36] | √ | √ | ||

Chen et al. [37] | √ | √ | ||

Zhao et al. [38] | √ | √ | √ | |

Liu et al. [39] | √ | √ | ||

Ernst and Powell [40] | √ | |||

Sawik [44] | √ | √ | ||

Sethi et al. [41] | √ | |||

Chen and Shen [42] | √ | √ | ||

Jha and Shanker [43] | √ | |||

Sethi et al. [45] | √ | √ | ||

Hu and Feng [12] | √ | |||

Chen et al. [46] | √ | √ | ||

He et al. [47] | √ | |||

Chen et al. [48] | √ | |||

This paper | √ | √ | √ | √ |

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

p | Unit retail price, |

o | Unit option price, |

e | Unit exercise price, |

c | Unit production cost, |

s | Unit salvage value, $p>o+e>c>s$, |

h | Supplier’s unit shortage cost for each exercised call option |

which cannot be immediately filled, $c<h$, | |

q | Option order quantity, |

Q | Production quantity, |

D | Random demand, $E\left(D\right)=\mu $ |

$f\left(x\right)$ | Probability density function of D, |

$F\left(x\right)$ | Distribution function of D, $F\left(0\right)=0$ and ${F}^{\prime}\left(x\right)=f\left(x\right)$, |

$\overline{F}\left(x\right)$ | Tail distribution of $F\left(x\right)$, i.e., $\overline{F}\left(x\right)=1-F\left(x\right)$, |

$\alpha $ | Service requirement, $0\le \alpha \le 1$, |

$\eta $ | Risk aversion coefficient, $0<\eta \le 1$, |

${\pi}_{r}(D;q)$ | Retailer’s profit without a service requirement, |

${\mathrm{CVaR}}_{\eta}\left({\pi}_{r}(D;q)\right)$ | Retailer’s CVaR without a service requirement, |

${\pi}_{s}(D;Q)$ | Supplier’s profit without a service requirement, |

${\mathrm{E}}_{D}\left[{\pi}_{s}(D;Q)\right]$ | Supplier’s expected profit without a service requirement, |

${q}^{\beta}$ | Optimal option order quantity maximizing CVaR about profit |

without a service requirement, | |

${q}^{*}$ | Optimal option order quantity maximizing CVaR about profit, |

${Q}^{*}$ | Optimal production quantity. |

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## Share and Cite

**MDPI and ACS Style**

Zhao, H.; Wang, H.; Liu, W.; Song, S.; Liao, Y.
Supply Chain Coordination with a Risk-Averse Retailer and the Call Option Contract in the Presence of a Service Requirement. *Mathematics* **2021**, *9*, 787.
https://doi.org/10.3390/math9070787

**AMA Style**

Zhao H, Wang H, Liu W, Song S, Liao Y.
Supply Chain Coordination with a Risk-Averse Retailer and the Call Option Contract in the Presence of a Service Requirement. *Mathematics*. 2021; 9(7):787.
https://doi.org/10.3390/math9070787

**Chicago/Turabian Style**

Zhao, Han, Hui Wang, Wei Liu, Shiji Song, and Yu Liao.
2021. "Supply Chain Coordination with a Risk-Averse Retailer and the Call Option Contract in the Presence of a Service Requirement" *Mathematics* 9, no. 7: 787.
https://doi.org/10.3390/math9070787