# Risk Analysis of a Two-Level Supply Chain Subject to Misplaced Inventory
## Appendix A Proof of Theorem 1  
Proof. : For a general demand distribution, the retailer's optimal order quantity under the CVaR criterion was derived as follows (see [PERSON] et al. [45]):  
\[Q^{*}=\left\{ \begin{array}{ll}\frac{1}{(1-\beta)}F^{-1}\left[\frac{\eta(p(1- \beta)+s\beta-w)}{(1-\beta)(p-s)}\right],&\text{if }(p-s)\beta\leq p-w\\ 0,&\text{otherwise}\end{array} \right. \tag{10}\]  
and his performance is measured by:  
\[\pi^{\eta}(Q^{*})=\left\{ \begin{array}{ll}\frac{p-s}{\eta}\int_{0}^{(1-\beta )Q^{*}}DdF(D),&\text{if }(p-s)\beta\leq p-w\\ 0,&\text{otherwise}\end{array} \right. \tag{11}\]  
We get the retailer's optimal order quantity when the demand is uniformly distributed on \([0,D_{\text{max}}]\), as follows:  
\[Q_{1}^{*}=\left\{ \begin{array}{ll}\frac{\eta D_{\text{max}}((p-s)(1-\beta)+s -w_{1})}{(1-\beta)^{2}(p-s)},&\text{if }\beta\leq\frac{p-w_{1}}{p-s}\\ 0,&\text{otherwise}\end{array} \right.. \tag{12}\]  
When there are no inventory errors (i.e., \(\beta\) = 0), the retailer's optimal order quantity can be expressed as:  
\[Q_{0}^{*}=\frac{\eta D_{\text{max}}(p-w_{0})}{(p-s)}. \tag{13}\]  
The supplier has no information about the misplacement errors, and thereby makes her decision based on \(Q_{0}^{*}\). We substitute Equation (13) into Equation (4), and derive the second-order derivative of \(\pi_{1}^{5}\) with respect to \(w_{0}\), which is:  
\[\frac{\partial^{2}\pi_{1}^{5}}{\partial{w_{0}}^{2}}=\frac{-2\eta D_{\text{ max}}}{(p-s)}\leq 0. \tag{14}\]  
Equation (14) shows that \(\pi_{1}^{5}\) is concave in \(w_{0}\). From the first-order condition, we obtain \(w_{0}^{*}\). \(\square\)  
Proof. : With RFID implementation, the supply chain participants' expected profit functions are expressed as:  
\[\pi_{3}^{S}(w,\epsilon)=(w_{3}-c-\phi\epsilon^{2})Q_{3}-\psi_{5}\epsilon^{2}, \tag{15}\]  
\[\pi_{3}^{R}(Q_{3},D)=-\psi_{R}\epsilon^{2}-w_{3}Q_{3}+\int_{0}^{(1-\beta+ \epsilon\beta)Q_{3}}(pD+s(Q_{3}-D))dF(D)\]  
\[+\int_{(1-\beta+\epsilon\beta)Q_{3}}^{+\infty}(p(1-\beta+\epsilon\beta)Q_{3}+s \beta(1-\epsilon)Q_{3})dF(D). \tag{16}\]Similar to Theorem 1, we derive the retailer's optimal order quantity for given values of \(w_{3}\) and \(\varepsilon\), as follows:  
\[Q_{3}^{*}=\left\{ \begin{array}{ll}\frac{\eta D_{\max}(-(p-s)\beta(1-\varepsilon )+p-w_{3}^{*})}{(1-\beta+\varepsilon\beta)^{2}(p-s)},&\text{if }\beta\leq\frac{p-w_{3}^{*}}{(p-s)(1- \varepsilon)}\\ 0,&\text{otherwise}\end{array} \right.. \tag{10}\]  
We then substitute \(Q_{3}^{*}\) into Equation (11), and derive the second-order derivative of \(\pi_{3}^{S}(w,\varepsilon)\) with respect to \(w_{3}\):  
\[\partial\pi_{3}^{S}(w,\varepsilon)/\partial w^{2}=-2\eta D_{\max}/(1-\beta+ \varepsilon\beta)(p-s)\leq 0. \tag{11}\]  
which indicates that \(\pi_{3}^{S}(w,\varepsilon)\) is a concave function in \(w_{3}\) for fixed \(\varepsilon\). From the first-order condition, we get \(w_{3}^{*}(\varepsilon^{*})\). Substituting \(w_{3}^{*}(\varepsilon^{*})\) into Equation (11), we obtain \(\pi_{3}^{S}(\varepsilon)\). We then take the first- and second-order derivatives of \(\pi_{3}^{S}(\varepsilon)\) with respect to \(\varepsilon\), as follows:  
\[ \begin{array}{l}\frac{\partial\pi_{3}^{S}(\varepsilon)}{\partial\varepsilon }=\frac{\eta D_{\max}((c-s)\beta-2\phi\varepsilon(1-\beta)-\phi\varepsilon^ {2})}{2(p-s)(1-\beta+\varepsilon\beta)^{3}}\\ \times(p-c-\phi\varepsilon^{2}-(p-s)\beta(1-\varepsilon))-2\psi_{R} \varepsilon,\end{array}  \tag{12}\]  
\[ \begin{array}{l}\frac{\partial^{2}\pi_{3}^{S}(\varepsilon)}{\partial \varepsilon^{2}}=-2\psi_{R}+\frac{\eta D_{\max}}{2(p-s)(1-\beta+\varepsilon )^{3}}\times\left[\begin{array}{l}A_{2}(1-\beta)-3\beta^{2}(c-s)(p-c-(p-s) \beta)\\ -2\beta\varepsilon(A_{2}+\phi(1-\beta)((p-s)\beta-p-2c+3s)\\ +\phi\varepsilon^{2}(3(-(p-s)\beta^{2}-2\phi\beta+2\phi)(1-\beta)+\\ ((p-s)\beta-p-2c+3s)\beta^{2})\\ +4\phi^{2}\beta(1-\beta)\varepsilon^{3}+\phi^{2}\beta^{2}\varepsilon^{4}) \end{array}\right].\end{array}  \tag{13}\]  
where \(A_{2}=(p-s)\beta^{2}(c-s-2\phi)+2\phi(2p-c-s)\beta-2\phi(p-c)\).