## 3 Inventory Location Identification Framework in MSR
### Indentification of Inventory Location
To better understand this function, we can illustrate the concrete steps using the example shown in Figure 3. Let us assume that the mapping information of the reference tags is stored in the LS database in advance. As the width of each zone is 2 m, if we attach the reference tags to the center of each zone, then the distance between two adjacent reference tags becomes 2 m (i.e., \(T\) = 2 m). According to Equation (4), \(Rh\) is calculated as 1 m, that is, the measurement distance of the RFID reader is determined to be within 1 m. Let us then set the tag recognition interval of the RFID reader, recognizing speed, and \(\theta\) to 2 times/sec (= 0.5 sec), 1 m/sec, and 90\({}^{\circ}\), respectively. As mentioned in Section 2.1, the recognition interval means the frequency at which the RFID reader generates radio-waves to scan the tag and the recognizing speed means the pace at which the RFID reader travels to scan the tag. The outcomes of recognized stock tags could stem from the combination of the number of reference tags and \(R_{ij}\), displayed in Table 1. If there are several \(R_{ij}\), then only the first recognized \(R_{ij}\) has to be considered. Distinguishing \(R_{ij}\) among \(J_{cj}\) is very important because the former is used to locate the stocks by referring to the mapping information of the reference tag in the same zone. When we distinguish \(R_{11}\), \(R_{21}\), \(R_{31}\), and \(R_{41}\), as presented in Table 1, \(J_{11}\), \(J_{41}\), \(J_{71}\), and \(J_{101}\), which have only one reference tag, can be selected. Here, while \(J_{101}\), \(J_{111}\), and \(J_{121}\) have only one reference tag, \"D\" can be considered \(R_{41}\), the first recognized, and \(J_{101}\) should be selected as \(R_{41}\). The stock items in each zone are identified based on \(R_{ij}\). Then, we can substitute \(R_{ij}\) into \(G_{ij}\), and exclude the reference tag and \(G_{i-1j}\), which is an already recognized set of stock, from \(R_{ij}\). For example, {A, A1, A2, A3, A4} of \(R_{11}\) is substituted into \(G_{11}\), and then \(R_{11}\) is replaced with {A1, A2, A3, A4} by excluding the reference tag A and \(G_{01}\), which has a null value. Then, we can set \(c\) to 1, and increase \(c\) by 1 and exclude \(J_{cj}\) from \(R_{ij}\), while \(c\) equals \(C\). If the number of stock items remaining in \(R_{ij}\) is equal to \(N\), then \(G_{ij}\) becomes \(R_{ij}\), and the process is repeated on the next column or row. If the location of the stock items in each zone is identified for all rows and columns in the rack, then the algorithm terminates. For example, {A1, A2, A3, A4, A5} of \(J_{21}\) are excluded from {A1, A2, A3, A4, A5} of \(R_{11}\). The number of stock items remaining in \(R_{11}\) is zero and is smaller than \(N\) = 3. Thus, {A, A2, A3, A4, A5, A6, B} of \(J_{31}\) are excluded from {A1, A2, A3, A4, A5} of \(R_{11}\) continuously. As the remaining stock in \(R_{11}\) is {A1} and its number is smaller than \(N\), {B, A3, A4, A5, A6, A7} of \(J_{41}\) are excluded from \(R_{11}\) continuously. As the remaining stock in \(R_{11}\) is {A1, A2} and its number is smaller than \(N\), {B, C, A4, A5, A6, A7, A8} of \(J_{51}\) are excluded from \(R_{11}\) continuously. As [A1, A2, A3] remains in \(R_{11}\) and its number is equal to \(N\), \(R_{11}\) is assigned to \(G_{11}\). The items in \(G_{11}\) are identified as {A1, A2, A3}; that is, stocks A1, A2, and A3 are located in \(RC_{11}\). Subsequently, to identify the stock in \(RC_{21}\), the reference tag is excluded and {A1, A2, A3} of \(G_{11}\) are preferentially excluded from \(R_{21}\); then \(R_{21}\) is {A4, A5, A6, A7}. As the number of stock items in \(R_{21}\) is not equal to \(N\), \(J_{51}\) is excluded from \(R_{21}\), and the remaining stock in \(R_{21}\) is null. As the number of stock items in \(R_{21}\) is smaller than \(N\), \(J_{61}\) is excluded from \(R_{21}\), and then the remaining stock in \(R_{21}\) is {A4}.