# Risk Analysis of a Two-Level Supply Chain Subject to Misplaced Inventory
## 3 Decision Policies of the Supply Chain with Information Asymmetry
### The Optimal Decisions in Case 1: Information Asymmetry about Inventory Errors Exists  
When there is no information sharing, the retailer realizes the inventory inaccuracy problem and conducts physical inventory audits to estimate the errors, whereas the supplier has no information about the errors. In this situation, the retailer makes his ordering decision by taking the errors into account, whereas the supplier still decides the wholesale price based on the situation of accurate inventory. With misplacement errors, the retailer's profit is given by:  
\[\text{Profit}=pMin(D,(1-\beta)Q_{1})+s(Q_{1}-Min(D,(1-\beta)Q_{1}))-wQ_{1}. \tag{1}\]  
We obtain the expected profit function of the retailer as follows:  
\[\pi_{1}^{R}(Q_{1},D)=\int_{0}^{(1-\beta)Q_{1}}\left(pD+s(Q_{1}-D)\right)dF(D)+ \int_{(1-\beta)Q_{1}}^{+\infty}\left(p(1-\beta)Q_{1}+s\beta Q_{1}\right)dF(D)- w_{1}Q_{1}. \tag{2}\]  
The retailer adopts CVaR as the decision criterion, and his objective function is given by (see [PERSON] et al. [45] for a brief description of CVaR and its application in a newsvendor problem):  
\[\min_{Q}CVaR_{\lambda}(Q)=\max_{Q_{1}}\pi_{1}^{\eta}(Q_{1},D)=\max_{Q_{1},v} \biggl{\{}v+\frac{1}{\eta}\int_{0}^{+\infty}\left[\pi_{1}^{\eta}(Q_{1},D)-v \right]^{-}\!\!f(D)dD\biggr{\}}. \tag{3}\]  
Note that \(\pi^{\eta}\) denotes the retailer's performance under the CVaR criterion and \(\eta\) denotes the degree of risk aversion. Retailers become more risk-averse as \(\eta\) decreases. Meanwhile, \(\eta=1\) indicates risk neutrality, which can be considered a special case of risk aversion. The risk-neutral supplier's objective function is given by:  
\[\pi_{1}^{\text{S}}=(w_{1}-c)Q_{0}, \tag{4}\]  
where \(Q_{0}\) is the order quantity of a retailer who has accurate inventory records. **Theorem 1**. : _When the retailer has private information about inventory errors, the supplier sets her wholesale price as:_  
\[w_{1}^{*}=w_{0}^{*}=\frac{c+p}{2}. \tag{5}\]  
_Based on the wholesale price, the retailer decides the order quantity_  
\[Q_{1}^{*}=\left\{ \begin{array}{cl}\frac{\eta D_{\max}(2(p-s)(1-\beta)+2s-p- c)}{2(1-\beta)^{2}(p-s)},&\text{if }\beta\leq\frac{p-c}{2(p-s)}\\ 0,&\text{otherwise}\end{array} \right. \tag{6}\]  
_to maximize his performance under the CVaR criterion._  
**Proof. ** Please refer to the Appendix A for the proof. The retailer maximizes his performance, which is \(\pi_{1}^{\eta}=\left(p-s\right)(1-\beta)^{2}(Q_{1}^{*})^{2}/(2\eta D_{\max})\), and the supplier maximizes her expected profit, which is \(\pi_{1}^{\text{S}}=(p-c)Q_{1}^{*}/2\). The Optimal Decisions in Case 2: The Retailer Shares Information about Inventory Errors with the Supplier  
It is well known that information sharing between the partners is beneficial to a supply chain [49]. It is also generally acknowledged that providing additional information to suppliers would offer some benefits to the suppliers [50]. In order to justify the information sharing action, such a benefit must be at least greater in value than the cost [51]. In this section, we quantify the value of information with a consideration of the information sharing cost, which is born by the supplier and denoted as \(E\). With information sharing, both parties make decisions based on the information about inventory errors, thus their expected profit functions can be expressed as:  
\[\pi_{2}^{\text{S}}=\ (w_{2}-c)Q_{2}-E, \tag{7}\]  
\[\pi_{2}^{R}(Q_{2},D)=-w_{2}Q_{2}+\int_{0}^{(1-\beta)Q_{2}}\left(pD+s(Q_{2}-D) \right)dF(D)+\int_{(1-\beta)Q_{2}}^{+\infty}\left(p(1-\beta)Q_{2}+s\beta Q_{2 }\right)dF(D). \tag{8}\]  
**Theorem 2**: _When the supply chain has public information about inventory errors, the supplier sets her wholesale price as:_  
\[w_{2}^{*}=(s+c+(p-s)(1-\beta))/2, \tag{9}\]  
_while the retailer decides his optimal order quantity as:_  
\[Q_{2}^{*}=\left\{ \begin{array}{cl}\frac{\eta D_{\max}((p-s)(1-\beta)+s-c)}{2 (1-\beta)^{2}(p-s)},&\mbox{if }\beta\leq\frac{p-c}{p-s}\\ 0,&\mbox{otherwise}\end{array}