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doi: 10.3934/dcdss.2022021
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## Optimization and coordination in a service-constrained supply chain with the bidirectional option contract under conditional value-at-risk

 1 Department of Finance and Audit, Army Logistics University, Chongqing 401311, China 2 Department of Automation, Tsinghua University, Beijing 100084, China 3 School of Management and Economics, Beijing Institute of Technology, Sustainable Development Research Institute for Economy and Society of Beijing, Beijing 100084, China 4 College of Information Science and Engineering, Xinjiang University, Xinjiang 830046, China

*Corresponding author: Bangdong Sun

Received  September 2021 Revised  November 2021 Early access February 2022

Fund Project: This paper is supported by Chongqing Social Sciences Planning Project (No. 2020BS53), National key Research and Development Project of China under Grants 2018YFB1702903 and 2017YFB0304102, National Natural Science Foundation of China under Grants U1660202, 91746210, 71871023, 61936009, 2018AAA0101604, Beijing Institute of Technology Research Fund Program for Young Scholars and Science and Technology Innovation Project

This paper investigates the optimal operational decisions for the risk-neutral supplier and the risk-averse retailer in the supply chain with a service requirement under the conditional value-at-risk. Specifically, the optimal order and production policies with and without the bidirectional option contract are derived. Further, this paper shows that the optimal conditional value-at-risk of the retailer is non-increasing in the service requirement, while the optimal expected profit of the supplier is non-decreasing in the service requirement. When the service requirement is binding, the optimal conditional value-at-risk of the retailer is increasing in the risk aversion, while the optimal expected profit of the supplier is decreasing in the risk aversion. In addition, it is shown that with the bidirectional option contract, the service level provided by the retailer is equivalent to (higher than) that without them when the service requirement is (not) binding. Finally, this paper demonstrates that the bidirectional option contract can mitigate the effect of risk aversion on the retailer's order quantity, benefit both the retailer and supplier, and improve the performance of the supply chain. Numerical experiments are conducted to further confirm our results.

Citation: Han Zhao, Bangdong Sun, Hui Wang, Shiji Song, Yuli Zhang, Liejun Wang. Optimization and coordination in a service-constrained supply chain with the bidirectional option contract under conditional value-at-risk. Discrete and Continuous Dynamical Systems - S, doi: 10.3934/dcdss.2022021
##### References:
 [1] H. U. Ben-Yong, X. Y.Wang and Q. Y. Peng, Comparison analysis on flexible supply contracts between unilateral options and bidirectional options, Chinese Journal of Management Science, 15 (2007), 92-97. [2] Z. Chang, S. Song, Y. Zhang, J. Ding, R. Zhang and R. Chiong, Distributionally robust single machine scheduling with risk aversion, European J. Oper. Res., 256 (2017), 261-274.  doi: 10.1016/j.ejor.2016.06.025. [3] X. Chen, G. Hao and L. Li, Channel coordination with a loss-averse retailer and option contracts, International Journal of Production Economics, 150 (2014), 52-57. [4] X. Chen, J. Luo, X. Wang and D. Yang, Supply chain risk management considering put options and service level constraints, Computers & Industrial Engineering, 140 (2020), 106228.  doi: 10.1016/j.cie.2019.106228. [5] X. Chen and Z.-J. (Max) Shen, An analysis of a supply chain with options contracts and service requirements, IIE Transactions, 44 (2012), 805-819.  doi: 10.1080/0740817X.2011.649383. [6] X. Chen, S. Shum and D. Simchi-Levi, Stable and coordinating contracts for a supply chain with multiple risk-averse suppliers, Production and Operations Management, 23 (2014), 379-392. [7] X. Chen, N. Wan and X. Wang, Flexibility and coordination in a supply chain with bidirectional option contracts and service requirement, International Journal of Production Economics, 193 (2017), 183-192. [8] C.-C. Hsieh and Y. T. Lu, Manufacturer's return policy in a two-stage supply chain with two risk-averse retailers and random demand, European J. Oper. Res., 207 (2010), 514-523.  doi: 10.1016/j.ejor.2010.04.026. [9] B. Hu and Y. Feng, Optimization and coordination of supply chain with revenue sharing contracts and service requirement under supply and demand uncertainty, International Journal of Production Economics, 183 (2017), 185-193. [10] B. Li, P. W. Hou, P. Chen and Q. H. Li, Pricing strategy and coordination in a dual channel supply chain with a risk-averse retailer, International Journal of Production Economics, 178 (2016), 154-168. [11] X. Li and P. Li, Stability of time-delay systems with impulsive control involving stabilizing delays, Automatica, 124 (2021), Paper No. 109336, 6 pp. doi: 10.1016/j.automatica.2020.109336. [12] X. Li and J. Wu, Sufficient stability conditions of nonlinear differential systems under impulsive control with state-dependent delay, IEEE Trans. Automat. Control, 63 (2018), 306-311.  doi: 10.1109/TAC.2016.2639819. [13] Y. Li, X. Xu and F. Ye, Supply chain coordination model with controllable lead time and service level constraint, Computers & Industrial Engineering, 61 (2011), 858-864.  doi: 10.1016/j.cie.2011.05.019. [14] W. Liu, H. Zhao, Y. Qiao and S. Song, The loss-averse newsvendor problem with random supply capacity, J. Ind. Manag. Optim., 13 (2017), 1417-1429.  doi: 10.3934/jimo.2016080. [15] F. A. Raman, Reducing the cost of demand uncertainty through accurate response to early sales, Operations Research, 44 (1996), 87-99. [16] T. Sawik, On the risk-averse optimization of service level in a supply chain under disruption risks, International Journal of Production Research, 54 (2016), 98-113.  doi: 10.1080/00207543.2015.1016192. [17] S. Sethi, H. Yan, H. Zhang and J. Zhou, A supply chain with a service requirement for each market signal, Wiley Online Library, 16 (2007), 322-342. [18] A. A. Taleizadeh, E. Sane-Zerang and T. Choi, The effect of marketing effort on dual-channel closed-loop supply chain systems, IEEE Transactions on Systems, Man, and Cybernetics: Systems, 48 (2018), 265-276. [19] C. Wang, J. Chen and X. Chen, The impact of customer returns and bidirectional option contract on refund price and order decisions, European J. Oper. Res., 274 (2019), 267-279.  doi: 10.1016/j.ejor.2018.09.023. [20] C. Wang and L. Wang, Fresh produce retailer's optimal ordering policies with bidirectional options, Advances in Information Sciences and Service Sciences, 5 (2013), 10. [21] L. Wang and R. Z. Wang, Impact of risk aversion on optimal decisions in supply contracts with bidirectional options, Applied Mechanics and Materials, 235 (2012), 261-266. [22] Q. Wang and D.-B. Tsao, Supply contract with bidirectional options: The buyer's perspective, International Journal of Production Economics, 101 (2006), 30-52. [23] R. Wang, S. Song and C. Wu, Coordination of supply chain with one supplier and two competing risk-averse retailers under an option contract, Math. Probl. Eng., 2016 (2016), Art. ID 1970615, 11 pp. doi: 10.1155/2016/1970615. [24] J. Wu, S. Wang, X. Chao, C. T. Ng and T. C. E. Cheng, Impact of risk aversion on optimal decisions in supply contracts, International Journal of Production Economics, 128 (2010), 569-576. [25] L. Yang, R. Tang and K. Chen, Call, put and bidirectional option contracts in agricultural supply chains with sales effort, Appl. Math. Model., 47 (2017), 1-16.  doi: 10.1016/j.apm.2017.03.002. [26] L. Yang, M. Xu, G. Yu and H. Zhang, Supply chain coordination with CVaR criterion, Asia-Pac. J. Oper. Res., 26 (2009), 135-160.  doi: 10.1142/S0217595909002109. [27] Y. Zhang, S. Song, Z. J. M. Shen and C. Wu, Robust shortest path problem with distributional uncertainty, IEEE Transactions on Intelligent Transportation Systems, 13 (2017), 1-11. [28] H. Zhao, S. Song, Y. Zhang, J. N. D. Gupta and A. G. Devlin, Optimal decisions of a supply chain with a risk-averse retailer and portfolio contracts, IEEE Access, 7 (2019), 123877-123892. [29] H. Zhao, S. Song, Y. Zhang, J. N. D. Gupta, A. G. Devlin and R. Chiong, Supply chain coordination with a risk-averse retailer and a combined buy-back and revenue sharing contract, Asia-Pac. J. Oper. Res., 36 (2019), 1950028, 23 pp. doi: 10.1142/S0217595919500283. [30] H. Zhao, S. Song, Y. Zhang, Y. Liao and F. Yue, Optimal decisions in supply chains with a call option contract under the carbon emissions tax regulation, Journal of Cleaner Production, 271 (2020), 122199.  doi: 10.1016/j.jclepro.2020.122199. [31] Y. Zhao, L. Ma, G. Xie and T. C. E. Cheng, Coordination of supply chains with bidirectional option contracts, European Journal of Operational Research, 229 (2013), 375-381. [32] Y. Zhao, S. Wang, T. C. E. Cheng, X. Yang and Z. Huang, Coordination of supply chains by option contracts: A cooperative game theory approach, European J. Oper. Res., 207 (2010), 668-675.  doi: 10.1016/j.ejor.2010.05.017.

show all references

##### References:
 [1] H. U. Ben-Yong, X. Y.Wang and Q. Y. Peng, Comparison analysis on flexible supply contracts between unilateral options and bidirectional options, Chinese Journal of Management Science, 15 (2007), 92-97. [2] Z. Chang, S. Song, Y. Zhang, J. Ding, R. Zhang and R. Chiong, Distributionally robust single machine scheduling with risk aversion, European J. Oper. Res., 256 (2017), 261-274.  doi: 10.1016/j.ejor.2016.06.025. [3] X. Chen, G. Hao and L. Li, Channel coordination with a loss-averse retailer and option contracts, International Journal of Production Economics, 150 (2014), 52-57. [4] X. Chen, J. Luo, X. Wang and D. Yang, Supply chain risk management considering put options and service level constraints, Computers & Industrial Engineering, 140 (2020), 106228.  doi: 10.1016/j.cie.2019.106228. [5] X. Chen and Z.-J. (Max) Shen, An analysis of a supply chain with options contracts and service requirements, IIE Transactions, 44 (2012), 805-819.  doi: 10.1080/0740817X.2011.649383. [6] X. Chen, S. Shum and D. Simchi-Levi, Stable and coordinating contracts for a supply chain with multiple risk-averse suppliers, Production and Operations Management, 23 (2014), 379-392. [7] X. Chen, N. Wan and X. Wang, Flexibility and coordination in a supply chain with bidirectional option contracts and service requirement, International Journal of Production Economics, 193 (2017), 183-192. [8] C.-C. Hsieh and Y. T. Lu, Manufacturer's return policy in a two-stage supply chain with two risk-averse retailers and random demand, European J. Oper. Res., 207 (2010), 514-523.  doi: 10.1016/j.ejor.2010.04.026. [9] B. Hu and Y. Feng, Optimization and coordination of supply chain with revenue sharing contracts and service requirement under supply and demand uncertainty, International Journal of Production Economics, 183 (2017), 185-193. [10] B. Li, P. W. Hou, P. Chen and Q. H. Li, Pricing strategy and coordination in a dual channel supply chain with a risk-averse retailer, International Journal of Production Economics, 178 (2016), 154-168. [11] X. Li and P. Li, Stability of time-delay systems with impulsive control involving stabilizing delays, Automatica, 124 (2021), Paper No. 109336, 6 pp. doi: 10.1016/j.automatica.2020.109336. [12] X. Li and J. Wu, Sufficient stability conditions of nonlinear differential systems under impulsive control with state-dependent delay, IEEE Trans. Automat. Control, 63 (2018), 306-311.  doi: 10.1109/TAC.2016.2639819. [13] Y. Li, X. Xu and F. Ye, Supply chain coordination model with controllable lead time and service level constraint, Computers & Industrial Engineering, 61 (2011), 858-864.  doi: 10.1016/j.cie.2011.05.019. [14] W. Liu, H. Zhao, Y. Qiao and S. Song, The loss-averse newsvendor problem with random supply capacity, J. Ind. Manag. Optim., 13 (2017), 1417-1429.  doi: 10.3934/jimo.2016080. [15] F. A. Raman, Reducing the cost of demand uncertainty through accurate response to early sales, Operations Research, 44 (1996), 87-99. [16] T. Sawik, On the risk-averse optimization of service level in a supply chain under disruption risks, International Journal of Production Research, 54 (2016), 98-113.  doi: 10.1080/00207543.2015.1016192. [17] S. Sethi, H. Yan, H. Zhang and J. Zhou, A supply chain with a service requirement for each market signal, Wiley Online Library, 16 (2007), 322-342. [18] A. A. Taleizadeh, E. Sane-Zerang and T. Choi, The effect of marketing effort on dual-channel closed-loop supply chain systems, IEEE Transactions on Systems, Man, and Cybernetics: Systems, 48 (2018), 265-276. [19] C. Wang, J. Chen and X. Chen, The impact of customer returns and bidirectional option contract on refund price and order decisions, European J. Oper. Res., 274 (2019), 267-279.  doi: 10.1016/j.ejor.2018.09.023. [20] C. Wang and L. Wang, Fresh produce retailer's optimal ordering policies with bidirectional options, Advances in Information Sciences and Service Sciences, 5 (2013), 10. [21] L. Wang and R. Z. Wang, Impact of risk aversion on optimal decisions in supply contracts with bidirectional options, Applied Mechanics and Materials, 235 (2012), 261-266. [22] Q. Wang and D.-B. Tsao, Supply contract with bidirectional options: The buyer's perspective, International Journal of Production Economics, 101 (2006), 30-52. [23] R. Wang, S. Song and C. Wu, Coordination of supply chain with one supplier and two competing risk-averse retailers under an option contract, Math. Probl. Eng., 2016 (2016), Art. ID 1970615, 11 pp. doi: 10.1155/2016/1970615. [24] J. Wu, S. Wang, X. Chao, C. T. Ng and T. C. E. Cheng, Impact of risk aversion on optimal decisions in supply contracts, International Journal of Production Economics, 128 (2010), 569-576. [25] L. Yang, R. Tang and K. Chen, Call, put and bidirectional option contracts in agricultural supply chains with sales effort, Appl. Math. Model., 47 (2017), 1-16.  doi: 10.1016/j.apm.2017.03.002. [26] L. Yang, M. Xu, G. Yu and H. Zhang, Supply chain coordination with CVaR criterion, Asia-Pac. J. Oper. Res., 26 (2009), 135-160.  doi: 10.1142/S0217595909002109. [27] Y. Zhang, S. Song, Z. J. M. Shen and C. Wu, Robust shortest path problem with distributional uncertainty, IEEE Transactions on Intelligent Transportation Systems, 13 (2017), 1-11. [28] H. Zhao, S. Song, Y. Zhang, J. N. D. Gupta and A. G. Devlin, Optimal decisions of a supply chain with a risk-averse retailer and portfolio contracts, IEEE Access, 7 (2019), 123877-123892. [29] H. Zhao, S. Song, Y. Zhang, J. N. D. Gupta, A. G. Devlin and R. Chiong, Supply chain coordination with a risk-averse retailer and a combined buy-back and revenue sharing contract, Asia-Pac. J. Oper. Res., 36 (2019), 1950028, 23 pp. doi: 10.1142/S0217595919500283. [30] H. Zhao, S. Song, Y. Zhang, Y. Liao and F. Yue, Optimal decisions in supply chains with a call option contract under the carbon emissions tax regulation, Journal of Cleaner Production, 271 (2020), 122199.  doi: 10.1016/j.jclepro.2020.122199. [31] Y. Zhao, L. Ma, G. Xie and T. C. E. Cheng, Coordination of supply chains with bidirectional option contracts, European Journal of Operational Research, 229 (2013), 375-381. [32] Y. Zhao, S. Wang, T. C. E. Cheng, X. Yang and Z. Huang, Coordination of supply chains by option contracts: A cooperative game theory approach, European J. Oper. Res., 207 (2010), 668-675.  doi: 10.1016/j.ejor.2010.05.017.
The decision process
Impact of the service requirement $\alpha$ on $q^{*}$, $q_{0}^{*}$ and $q_{1}^{\gamma}-q_{2}^{*}$.
Impact of service requirement $\alpha$ on ${\rm CVaR_{\eta}}(\pi_{r}(X; q_{1}^{\gamma}, q_{2}^{*}))$ and ${\rm CVaR_{\eta}}(\pi_{r}(X; q_{0}^{*}))$.
Impact of service requirement $\alpha$ on ${\rm E}[\pi_{s}(Q^{*})]$ and ${\rm E}[\pi_{s}(Q_{0}^{*})]$.
Comparison with contributions of different authors.
 Author(s) Firm orders Bidirectional option orders Service requirement CVaR Wang and Tsao [22] $\surd$ $\surd$ Ben-Yong et al. [1] $\surd$ $\surd$ Wang and Wang [20] $\surd$ $\surd$ Zhao et al. [31] $\surd$ $\surd$ Chen et al. [3] $\surd$ Yang et al. [25] $\surd$ $\surd$ Wang et al. [19] $\surd$ $\surd$ Wu et al. [24] $\surd$ $\surd$ Hsieh and Lu [8] $\surd$ $\surd$ Wang and Wang [21] $\surd$ $\surd$ $\surd$ Chen et al. [6] $\surd$ $\surd$ Li et al. [10] $\surd$ $\surd$ Wang et al. [23] $\surd$ $\surd$ Zhao et al. [28] $\surd$ $\surd$ Zhao et al. [30] $\surd$ $\surd$ Sethi et al. [17] $\surd$ $\surd$ Li et al. [13] $\surd$ $\surd$ Chen and Shen [5] $\surd$ $\surd$ Sawik [16] $\surd$ $\surd$ Chen et al. [7] $\surd$ $\surd$ $\surd$ Hu and Feng [9] $\surd$ Chen et al. [4] $\surd$ $\surd$ This paper $\surd$ $\surd$ $\surd$ $\surd$
 Author(s) Firm orders Bidirectional option orders Service requirement CVaR Wang and Tsao [22] $\surd$ $\surd$ Ben-Yong et al. [1] $\surd$ $\surd$ Wang and Wang [20] $\surd$ $\surd$ Zhao et al. [31] $\surd$ $\surd$ Chen et al. [3] $\surd$ Yang et al. [25] $\surd$ $\surd$ Wang et al. [19] $\surd$ $\surd$ Wu et al. [24] $\surd$ $\surd$ Hsieh and Lu [8] $\surd$ $\surd$ Wang and Wang [21] $\surd$ $\surd$ $\surd$ Chen et al. [6] $\surd$ $\surd$ Li et al. [10] $\surd$ $\surd$ Wang et al. [23] $\surd$ $\surd$ Zhao et al. [28] $\surd$ $\surd$ Zhao et al. [30] $\surd$ $\surd$ Sethi et al. [17] $\surd$ $\surd$ Li et al. [13] $\surd$ $\surd$ Chen and Shen [5] $\surd$ $\surd$ Sawik [16] $\surd$ $\surd$ Chen et al. [7] $\surd$ $\surd$ $\surd$ Hu and Feng [9] $\surd$ Chen et al. [4] $\surd$ $\surd$ This paper $\surd$ $\surd$ $\surd$ $\surd$
Notations
 $X$ Random variable representing stochastic market demand, which is a continuous and differentiable, $X\geq0$, $\text{E}(X)=\mu$ and $\text{Var}(X)=\sigma^{2}$ $f(x)$ Probability density function for $X$ $F(x)$ Distribution function for $X$, which is a strictly increasing, invertible and differentiable, $F(0)=0$ and $f(x)=F'(x)$ $p$ Retail price per unit $w$ Wholesale price per unit through a initial firm order $o$ Purchase price per unit of the bidirectional option, i.e., option price per unit $e_{1}$ Exercise price per unit of the bidirectional option, i.e., exercise price per unit $c$ Production cost per unit $v$ Salvage value per unit after the selling period $h$ Shortage cost per unit for each exercised option that cannot be filled $\alpha$ The service level commitment of the retailer, $0 <\alpha\leq1$ $q_{0}$ Decision variable representing firm order quantity without the bidirectional option contract $q_{1}$ Decision variable representing firm order quantity with the bidirectional option contract $q_{2}$ Decision variable representing option order quantity with the bidirectional option contract $q$ Decision variable representing total order quantity with the bidirectional option contract, $q=q_{1}+q_{2}$ $Q_{0}$ Decision variable representing production quantity without the bidirectional option contract $Q$ Decision variable representing production quantity with the bidirectional option contract
 $X$ Random variable representing stochastic market demand, which is a continuous and differentiable, $X\geq0$, $\text{E}(X)=\mu$ and $\text{Var}(X)=\sigma^{2}$ $f(x)$ Probability density function for $X$ $F(x)$ Distribution function for $X$, which is a strictly increasing, invertible and differentiable, $F(0)=0$ and $f(x)=F'(x)$ $p$ Retail price per unit $w$ Wholesale price per unit through a initial firm order $o$ Purchase price per unit of the bidirectional option, i.e., option price per unit $e_{1}$ Exercise price per unit of the bidirectional option, i.e., exercise price per unit $c$ Production cost per unit $v$ Salvage value per unit after the selling period $h$ Shortage cost per unit for each exercised option that cannot be filled $\alpha$ The service level commitment of the retailer, $0 <\alpha\leq1$ $q_{0}$ Decision variable representing firm order quantity without the bidirectional option contract $q_{1}$ Decision variable representing firm order quantity with the bidirectional option contract $q_{2}$ Decision variable representing option order quantity with the bidirectional option contract $q$ Decision variable representing total order quantity with the bidirectional option contract, $q=q_{1}+q_{2}$ $Q_{0}$ Decision variable representing production quantity without the bidirectional option contract $Q$ Decision variable representing production quantity with the bidirectional option contract
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