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# Optimization and coordination in a service-constrained supply chain with the bidirectional option contract under conditional value-at-risk

• *Corresponding author: Bangdong Sun

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.

Mathematics Subject Classification: Primary: 90B05; Secondary: 91B42.

 Citation:

• Figure 1.  The decision process

Figure 2.  Impact of the service requirement $\alpha$ on $q^{*}$, $q_{0}^{*}$ and $q_{1}^{\gamma}-q_{2}^{*}$.

Figure 3.  Impact of service requirement $\alpha$ on ${\rm CVaR_{\eta}}(\pi_{r}(X; q_{1}^{\gamma}, q_{2}^{*}))$ and ${\rm CVaR_{\eta}}(\pi_{r}(X; q_{0}^{*}))$.

Figure 4.  Impact of service requirement $\alpha$ on ${\rm E}[\pi_{s}(Q^{*})]$ and ${\rm E}[\pi_{s}(Q_{0}^{*})]$.

Table 1.  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 $\surd$ $\surd$ $\surd$ $\surd$

Table 2.  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
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