# American Institute of Mathematical Sciences

doi: 10.3934/jimo.2021024

## Effects of disruption risk on a supply chain with a risk-averse retailer

 1 Department of Mathematics, East China University of Science and Technology, Shanghai 200237, China 2 Department of Electronic Business, South China University of Technology, Guangzhou, Guangdong 510006, China 3 School of Management, Fudan University, Shanghai 200433, China

* Corresponding author: Wei Wang

Received  February 2020 Revised  December 2020 Published  February 2021

Fund Project: The authors contributed equally. This work was partially supported by National Nature Science Foundation of China (No. 71531005, No. 72072036), the Fundamental Research Funds for the Central Universities, SCUT (No. D2193040), Guangdong University Characteristic Innovation Project (No. 2017WTSCX002) and Guangdong Natural Science Foundation Doctoral Research Project (No. B6180990)

This paper studies a supply chain consisting of two unreliable suppliers and a retailer, where the two suppliers' default risks are correlated. We use a mean-variance function to characterize the retailer's risk aversion. In the case of exogenous wholesale prices, we find that the retailer's risk aversion has a non-monotonic effect on its total ordering quantity. We also show that when the suppliers' default correlation increases, the retailer's total ordering quantity is non-increasing. In the case of endogenous wholesale prices, we find that the profits of the suppliers and the retailer are non-monotonic in retailer's risk aversion level or suppliers' default correlation. As risk aversion level increases, the retailer becomes less sensitive to wholesale prices. Finally, the numerical results indicate that when the suppliers' delivery rates are different, the supplier with a low delivery rate can benefit from the retailer's risk aversion under certain conditions.

Citation: Min Li, Jiahua Zhang, Yifan Xu, Wei Wang. Effects of disruption risk on a supply chain with a risk-averse retailer. Journal of Industrial & Management Optimization, doi: 10.3934/jimo.2021024
##### References:

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##### References:
Retailer's optimal order quantities given $(w_1, w_2)$
Retailer's optimal order quantities given $(w_1, w_2)$ with different risk aversion levels
Retailer's optimal order quantities given $(w_1, w_2)$ with different default correlations
Retailer's optimal order quantities with the increase of the default correlation or the risk aversion level
Suppliers' profits with the increase of the default correlation for different risk aversion levels
Suppliers' profits with the increase of the risk aversion level for different default correlations
Examples for the optimal wholesale prices in $\Omega_2$ and $\Omega_3$
Retailer's optimal order quantities given $(w_1, w_2)$ when $\alpha>\beta$
Suppliers' profits with the increase of the risk aversion level
Suppliers' profits with the increase of the default correlation
Equilibrium results when $\alpha = 0.5$ and $p_{00} = \beta$
 $k$ $w_1^*$ $w_2*$ $Q_1^*$ $Q_2^*$ $\pi_{S_1}^*$ $\pi_{S_2}^*$ 0.00031 1.38 0 100 100 138 0 0.00041 1.268 0.044 96.6463 3.3537 122.5476 0.1476 0.00061 1.5613 0.3907 79.9863 20.0137 124.8853 7.8187 0.00071 1.708 0.564 75.1761 24.8239 128.4007 14.0007 0.00081 1.6 0.48 69.1358 24.6914 110.6173 11.8519
 $k$ $w_1^*$ $w_2*$ $Q_1^*$ $Q_2^*$ $\pi_{S_1}^*$ $\pi_{S_2}^*$ 0.00031 1.38 0 100 100 138 0 0.00041 1.268 0.044 96.6463 3.3537 122.5476 0.1476 0.00061 1.5613 0.3907 79.9863 20.0137 124.8853 7.8187 0.00071 1.708 0.564 75.1761 24.8239 128.4007 14.0007 0.00081 1.6 0.48 69.1358 24.6914 110.6173 11.8519
Equilibrium results when $\alpha = 0.9$ and $k = 0.0034$
 $p_{00}$ $w_1^*$ $w_2*$ $Q_1^*$ $Q_2^*$ $\pi_{S_1}^*$ $\pi_{S_2}^*$ 0.271 4.4964 1.4750 73.4740 10.3295 330.3661 15.2359 0.272 4.4926 1.4499 73.4243 10.1556 329.8676 14.7248 0.273 8.9749 0.7090 59.1954 41.6294 531.2713 29.5148 0.274 8.9815 0.683 59.0538 41.5197 530.3940 28.3595 0.275 8.9881 0.6571 58.9137 41.4115 529.5238 27.2100
 $p_{00}$ $w_1^*$ $w_2*$ $Q_1^*$ $Q_2^*$ $\pi_{S_1}^*$ $\pi_{S_2}^*$ 0.271 4.4964 1.4750 73.4740 10.3295 330.3661 15.2359 0.272 4.4926 1.4499 73.4243 10.1556 329.8676 14.7248 0.273 8.9749 0.7090 59.1954 41.6294 531.2713 29.5148 0.274 8.9815 0.683 59.0538 41.5197 530.3940 28.3595 0.275 8.9881 0.6571 58.9137 41.4115 529.5238 27.2100
Equilibrium results when $\alpha = 0.7$ and $p_{00} = 0.27$
 $k$ $w_1^*$ $w_2*$ $Q_1^*$ $Q_2^*$ $\pi_{S_1}^*$ $\pi_{S_2}^*$ 0.0009 3.2376 0.3073 74.3118 59.4039 240.5899 18.2533 0.0010 3.3548 0.3184 69.3784 55.4602 232.7506 17.6585 0.0011 3.2083 0.9583 74.9360 22.3835 240.4198 21.4508 0.0012 3.2083 0.9583 68.7430 20.5336 220.5504 19.6781 0.0013 3.2083 0.9583 63.4954 18.9662 203.7145 18.1759
 $k$ $w_1^*$ $w_2*$ $Q_1^*$ $Q_2^*$ $\pi_{S_1}^*$ $\pi_{S_2}^*$ 0.0009 3.2376 0.3073 74.3118 59.4039 240.5899 18.2533 0.0010 3.3548 0.3184 69.3784 55.4602 232.7506 17.6585 0.0011 3.2083 0.9583 74.9360 22.3835 240.4198 21.4508 0.0012 3.2083 0.9583 68.7430 20.5336 220.5504 19.6781 0.0013 3.2083 0.9583 63.4954 18.9662 203.7145 18.1759
Equilibrium results when $\alpha = 0.5$ and $p_{00} = \beta$
 $k$ $g_1$ $g_2$ $c$ $\pi_{S_1}^*$ $\pi_{S_2}^*$ 0.0004 0.5 0.5 0.15 130 0 0.0005 0.5 0.5 0.15 128.4028 1.7361 0.0006 0.5 0.5 0.15 130.0208 6.0208 0.0007 0.5 0.5 0.15 133.0972 11.7639 0.0008 0.5 0.5 0.15 118.2447 11.1747
 $k$ $g_1$ $g_2$ $c$ $\pi_{S_1}^*$ $\pi_{S_2}^*$ 0.0004 0.5 0.5 0.15 130 0 0.0005 0.5 0.5 0.15 128.4028 1.7361 0.0006 0.5 0.5 0.15 130.0208 6.0208 0.0007 0.5 0.5 0.15 133.0972 11.7639 0.0008 0.5 0.5 0.15 118.2447 11.1747
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