# Risk Analysis of a Two-Level Supply Chain Subject to Misplaced Inventory
## 5 Supply Chain Coordination  
In this section, we analyze how the cost savings from RFID implementation should be allocated among supply chain participants by a revenue and cost sharing contract. We also show that, under this contract, the supply chain is coordinated and both parties are better off. In this section, the subscript \"\(Co\)\" is used to indicate the decision policies when the supply chain is coordinated. Under a standard revenue sharing contract [54], the retailer keeps \(k\) portion of all revenue (\(k\in(0,1)\)), while the supplier takes (\(1-k\)) portion of the retailer's revenue and offers a low wholesale price, which is \(w=kc\), in order to induce the retailer to place an order that maximizes the supply chain's profit (i.e., \(Q_{Co}=\ Q^{SC}\)). With RFID implementation, the additional RFID costs also need to be shared. Hence, we propose a revenue and cost sharing contract \((w,k,\theta)\) to coordinate the supply chain. The variable cost is shared between the two parties using an arbitrary proportion, while the supplier bears (\(1-k\)) portion of the retailer's fixed cost for RFID adoption. The retailer shares \(\theta\) portion of the variable RFID costs and the supplier pays (\(1-\theta\)) portion of the variable RFID costs. We investigate how the revenue sharing policy, together with the expense sharing scheme on RFID costs, can coordinate the supply chain. Under the revenue and cost sharing contract \((w,k,\theta)\), the expected profit functions of the two parties are given by:  
\[ \begin{split}\pi^{R}_{Co}(Q_{Co},D)&=-(w_{Co}+ \theta t_{\epsilon})Q_{Co}-kL_{R}+k\int_{0}^{(1-\beta+\varepsilon)Q_{Co}}(pD+ s(Q_{Co}-D))dF(D)\\ &\quad+k\int_{(1-\beta+\varepsilon\beta)Q_{Co}}^{+\infty}(p(1- \beta+\varepsilon\beta)Q_{Co}+s\beta(1-\varepsilon)Q_{Co})dF(D),\end{split}  \tag{14}\]  
\[ \begin{split}\pi^{S}_{Co}&=(w_{Co}-c-(1-\theta)t_{ \epsilon})Q_{Co}-L_{S}-(1-k)L_{R}\\ &\quad+(1-k)\int_{0}^{(1-\beta+\varepsilon\beta)Q_{Co}}(pD+s(Q_ {Co}-D))dF(D)\\ &\quad+(1-k)\int_{(1-\beta+\varepsilon\beta)Q_{Co}}^{+\infty}(p( 1-\beta+\varepsilon\beta)Q_{Co}+s\beta(1-\varepsilon)Q_{Co})dF(D).\end{split}  \tag{15}\]  
Thus, we get the supply chain's expected profit function, which is:  
\[ \begin{split}\pi^{SC}_{Co}(Q_{Co},D)&=-(c+t_{ \epsilon})Q_{Co}-L_{S}-L_{R}+\int_{0}^{(1-\beta+\varepsilon\beta)Q_{Co}}(pD+ s(Q_{Co}-D))dF(D)\\ &\quad+\int_{(1-\beta+\varepsilon\beta)Q_{Co}}^{+\infty}(p(1- \beta+\varepsilon\beta)Q_{Co}+s\beta(1-\varepsilon)Q_{Co})dF(D).\end{split}  \tag{16}\]  
When it is cost-effective for the supply chain to adopt RFID technology, we get \(p-(p-s)\beta(1-\varepsilon)-c-t_{\epsilon}>0\). The decisions of the supply chain are equivalent to that of a centralized system, where the supplier and the retailer seek to jointly maximize their total profit. We derive the optimal order quantity and the expected profit of the centralized supply chain as follows:  
\[Q^{SC}=\frac{D_{\max}(-(p-s)\beta(1-\varepsilon)+p-c-t_{\epsilon})}{(1-\beta +\varepsilon\beta)^{2}(p-s)}, \tag{17}\]  
\[\pi^{SC}=(p-s)(1-\beta+\varepsilon\beta)^{2}(Q^{SC})^{2}/(2D_{\max})-L_{S}-L_{ R}. \tag{18}\]  
**Theorem 4**. : _With RFID implementation, a revenue and cost sharing contract \((w,k,\theta)\) can coordinate the supply chain consisting of a risk-averse retailer and a risk-neutral supplier, where the parameters satisfy:_  
\[k(B-\frac{(B-c-t_{\epsilon})}{\eta})-\theta t_{\epsilon}=w_{Co}, \tag{19}\]  
\[\frac{\eta^{2}}{(8-4\eta)}\leq k\leq\frac{\eta}{2}, \tag{20}\]  
_where \(B=p-(p-s)\beta(1-\varepsilon)\)._  
**Proof. ** From Equation (14), it is worth noting that when \(w_{Co}+\theta t_{\varepsilon}<k\)s (i.e., \(\eta<(B-c-t_{\varepsilon})/(B-s)\)), the retailer will always benefit from each ordering and thereby he tends to order as much as possible, even beyond the optimal order quantity of the supply chain. Therefore, we consider two cases of \(\eta\):  
1. \(\eta\geq(B-c-t_{\varepsilon})/(B-s)\).