# Risk Analysis of a Two-Level Supply Chain Subject to Misplaced Inventory
## 5 Supply Chain Coordination
From Equation (14), we get the retailer's optimal order quantity, as follows: \[Q_{Co}=\frac{\eta D_{\max}(-(p-s)\beta(1-\varepsilon)+p-(w_{Co}+\theta t_{ \varepsilon})/k)}{(1-\beta+\varepsilon\beta)^{2}(p-s)}.\] (21) When \(k(B-(B-c-t_{\varepsilon})/\eta)-\theta t_{\varepsilon}=w_{Co}\), we have \(Q_{Co}=Q^{SC}\), which means that the optimal decision of the centralized system is adopted. Substituting Equation (19) into (15), we obtain the supplier's expected profit after coordination, as follows: \[\pi_{Co}^{S}=(1-k)\pi^{SC}-kQ^{SC}(B-c-t_{\varepsilon})(1/\eta-1).\] (22) Comparing \(\pi_{Co}^{S}\) with \(\pi_{S}^{S}\), we derive the upper bound of \(k\), \(k_{U}=\eta/2\), which ensures that the supplier benefits from coordination. Similarly, we derive the lower bound of \(k\), \(k_{L}=\eta^{2}/(8-4\eta)\), which ensures that the retailer benefits from coordination. Since the performance of the decentralized supply chain is always improved through coordination, the supplier is able to allocate the profits so that both parties are better off (i.e., \(k\in(k_{L},k_{U})\)). Thus, coordination is achieved through the contract. 2. \(\eta<(B-c-t_{\varepsilon})/(B-s)\). In this case, the optimal order quantity of the retailer is infinite. As the Stackelberg leader, the supplier can limit the retailer's order quantity to the supply chain's optimal order quantity (i.e., \(Q_{Co}=Q^{SC}\)), and negotiate with the retailer about the values of \(k\) and \(\theta\) to make sure that the retailer benefits from coordination. In this situation, coordination is also achieved. \(\square\)  
There exist multiple sets of (\(w\), \(k\), \(\theta\)) that satisfy the required condition. In reality, the specific values of \(k\) and \(\theta\) are decided depending on the bargaining power between the retailer and the supplier. It is worth noting that when \(\theta=k\) and \(\eta=1\), the proposed cost and revenue sharing contract given by Equation (22) is specialized as the standard revenue sharing contract. In this sense, our results extend the supply chain coordination problem to the context of risk aversion and RFID costs sharing. From Theorem 4, we obtain the following corollary. **Corollary 2**. : _Under the CVaR criterion, the revenue and cost sharing contract (\(w\), \(k\), \(\theta\)) is relevant to the retailer's degree of risk aversion, where \(\eta\): \(w_{Co}\) and \(k\) are increasing in \(\eta\)._  
**Proof. ** From Equation (19), it is straightforward that \(w_{Co}\) increases in \(\eta\). We can also derive that both the upper bound and lower bound of \(k\) are increasing in \(\eta\) from Equation (20). \(\square\)  
Corollary 2 shows that when the retailer is more risk-averse (smaller \(\eta\)), he desires to share a higher portion of revenue (smaller \(k\)) to the supplier, in order to obtain a lower wholesale price (smaller \(w\)) in return. The reason for this is that the retailer's risk increases as the holding cost increases, where the holding cost is denoted as \(h\) and \(h=w-s\). Thus, the more risk-averse the retailer becomes, the lower wholesale price he desires in order to avoid the risk. We also observe that for a given \(\eta\), the wholesale price increases in \(k\) and decreases in \(\theta\), which is intuitive and useful when the participants negotiate the contract. For illustration, we consider a numerical example in the next section and show the impact of risk aversion on supply chain coordination.