# Risk Analysis of a Two-Level Supply Chain Subject to Misplaced Inventory
## 3 Decision Policies of the Supply Chain with Information Asymmetry  
### Problem Description  
We consider a two-level supply chain which consists of a supplier and a retailer subject to misplaced inventory. The supplier produces seasonal products and sells the products to the retailer. The retailer then sells the products to the end customers. The supplier decides a wholesale price for the product and the retailer decides the order quantity from the supplier. Therefore, a Stackelberg game, which is a method employed to model the hierarchical decision process, is proposed to model the interaction between the supplier and the retailer. As the Stackelberg leader, the supplier determines the wholesale price based on its impact on the retailer's order quantity, while the retailer decides his order quantity based on the demand and the wholesale price.  
The following notations and assumptions are used in this paper:  
\begin{tabular}{l l} \hline \hline \(c\) & Unit production cost \\ \(p\) & Unit selling price \\ \(s\) & Unit salvage price \\ \(D\) & Random demand \\ \(f(D)\) & Probability density function (PDF) of \(D\) \\ \(F(D)\) & Cumulative distribution function (CDF) of \(D\) \\ \(\beta\) & Ratio of items that are misplaced among the total physical inventory \\ \(\eta\) & Risk factor that reflects the degree of risk aversion \\ \(t\) & Unit variable RFID tag cost \\ \(L\) & Fixed RFID investment cost \\ \(E\) & Information sharing cost \\ \(\pi\) & Expected profit \\ \(k\) & The retailer's share of revenue \\ \(\theta\) & The retailer's share of RFID cost \\ Decision Variables & \\ \(Q\) & Order quantity \\ \(w\) & Unit wholesale price \\ \(\varepsilon\) & Scale of RFID investment \\ \hline \hline \end{tabular}  
**Assumption 1**.: _Assume that \(\beta\) portion of items are misplaced instantaneously at the beginning of the season and cannot be found until the end of the season. Normally, misplacement occurs gradually over time in real-life situations. However, this assumption is widely used in the literature of misplaced inventory for analytical tractability [25; 32]. Therefore, only (\(1-\beta\))Q amount of items are available for customers. At the end of the period, the whole unsold quantity (misplaced items and items on shelves) is salvaged._  
**Assumption 2**.: _To obtain closed-form analytical expressions, we assume that customer demand is uniformly distributed on [0, \(D_{\max}\)], which is commonly adopted in inventory management literature [25; 45]._  
**Assumption 3**.: _The effectiveness of RFID in eliminating misplaced inventory is not 100%. Most previous studies consider RFID as a perfect technology which has a 100% ability to eliminate inventory errors. However, this assumption does not hold in reality and the effectiveness of RFID for the reduction of inventory errors can be quite different [47]. The effectiveness of RFID in eliminating misplaced inventory is considered as a function of the scale of RFID investment in this study._  
**Assumption 4**.: _Assume that the retailer is risk-averse and the supplier is risk-neutral. The risk neutrality assumption of the supplier is based on the fact that, in practice, the supplier is able to diversify her risk by serving a number of independent retailers [48]. The risk-neutral supplier makes decisions to maximize her profit, whereas the CVaR measure is considered as the risk-averse retailer's objective._  
Throughout the remainder of this study, superscripts are used to indicate the concerned supply chain level (\(S\), supplier; \(R\), retailer; \(SC\), supply chain), and subscripts (\(1\)-\(3\)) are used to distinguish between three cases (Case 1, information asymmetry about inventory errors exists; Case 2, the retailer shares information about inventory errors with the supplier; Case 3, RFID is implemented in the supply chain).