# Risk Analysis of a Two-Level Supply Chain Subject to Misplaced Inventory
## 4 Decision Policies of the Supply Chain with RFID Implementation
### The Optimal Decisions in Case 3: RFID Is Implemented in the Supply Chain  
Currently, RFID is not extensively used because of its associated high costs, which includes variable tag costs and fixed costs (such as hardware and software costs, training costs, and infrastructure maintenance and upgrade costs). In the market, various types of RFID systems are available: e.g., RFID tags may cost as much as 100 dollars or as little as 7 cents, depending on the type of tag (passive tag or active tag), its efficiency, and its applications. Most of the previous studies consider RFID as a perfect technology which can eliminate all of the 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.  
Therefore, the trade-off between the benefits and costs of RFID investment becomes a critical issue for decision makers. In this study, we explore the supply chain's optimal scale for RFID when it has misplacement errors.  
The effectiveness of RFID in eliminating misplaced inventory is considered as a function of the scale of RFID investment. For a given scale \(\varepsilon\left(\varepsilon\in[0,1]\right)\), \(\varepsilon\beta Q\) items are prevented from misplacement and become available to meet customers' demand. The variable cost of RFID investment is denoted as \(t_{\varepsilon}\), and the fixed cost for the retailer (supplier) is denoted as \(L_{R}\) (\(L_{S}\)). We set \(t_{\varepsilon}=\phi\varepsilon^{2}\) and \(L_{R}=\psi_{R}\varepsilon^{2}\) (\(L_{S}=\psi_{S}\varepsilon^{2}\)), where \(\phi\), \(\psi_{R}\) and \(\psi_{S}\) are investment parameters. This quadratic assumption is widely used in the literature to describe the decreasing returns of investment [52; 53]. We consider the situation when the variable cost of RFID investment is incurred by the supplier and the two parties bear their fixed costs respectively. This is a prevailing situation when RFID is implemented in the supply chain. For example, in 2005, Wal-Mart required its top 100 suppliers to tag all their pallets and cases and to bear the variable tag cost. RFID implementation also leads to large gains for the suppliers, boosts a long term partner relationship with the retailer, provides timely and accurate information, and helps to speed up the operational processes [46].  
The supplier decides the wholesale price and the scale of RFID investment, by taking into account their influences on the order quantity of the retailer.  
**Theorem 3**.: _When the supply chain invests in RFID for the reduction of inventory errors, there exists an optimal scale of RFID investment, \(\varepsilon^{*}\), and the corresponding wholesale price is \(w_{3}^{*}(\varepsilon^{*})=(p+c+\phi\varepsilon^{*2}-(p-s)\beta(1-\varepsilon ^{*}))/2\). Thus, the order quantity of the retailer is given by:_  
\[Q_{3}^{*}(\varepsilon^{*})=\left\{\begin{array}{ll}\frac{\eta D_{\max}(p-c- \phi\varepsilon^{*2}-(p-s)\beta(1-\varepsilon^{*}))}{2(1-\beta+\varepsilon^{ *}\beta)^{2}(p-s)},&\text{if }\beta\leq\frac{p-c-\phi\varepsilon^{*2}}{(p-s)(1- \varepsilon^{*})}\\ 0,&\text{otherwise}\end{array}\right.. \tag{13}\]  
**Proof.** Please refer to the Appendix A for the proof.  
Using the optimality properties of Theorem 3, a simple searching algorithm may be employed to obtain the optimal scale of RFID investment, \(\varepsilon^{*}\).  
With RFID implementation, the retailer's performance is \(\tau_{3}^{\eta}=(p-s)(1-\beta+\varepsilon^{*}\beta)^{2}(Q_{3}^{*})^{2}/(2\eta D _{\max})\), and the supplier's expected profit is \(\tau_{3}^{S}=(p-s)(1-\beta+\varepsilon^{*}\beta)^{2}(Q_{3}^{*})^{2}/(\eta D _{\max})-\psi_{S}\varepsilon^{2}\).