# Location Optimization of Emergency Bell Based on Coverage Analysis for Crime Prevention
## 3 Materials and Methods
### EBLSCP & EBMCLP  
In this subsection, we provide a description of the variables and formulas used for the EBLSCP and EBMCLP models, which were suitable for the optimal location of emergency bells. These models were derived from LSCP and MCLP. Considering the limitations of the processing memory required to operate EBMCLP and EBLSCP, the scope of the criminal data was limited to crimes committed between 2018 and 2021. The variables used for EBLSCP and EBMCLP are described in Table 3.  
Index set \(I\) was generated by crime occurrence locations that were not covered by the existing emergency bells in each district. Set \(J\) was generated by regular points with a 25 m interval, intersecting horizonal and vertical lines to cover all possible coordinates in each district's entire area. For this study, the coverage distance D was set as a 50 m buffer from new emergency bells. Figure 5: K-means cluster by number of crimes (Num.Crime) and number of emergency bells (Num.Bell). \begin{table}
\begin{tabular}{c c} \hline \hline
**Variable** & **Description** \\ \hline \(i\) & Index of unmonitored crime occurrence location. \\ \(j\) & Index of new emergency bell candidate location. \\ \(I\) & Index set of unmonitored crime occurrence locations. \\ \(J\) & Index set of new emergency bell candidate locations. \\ \(D\) & Coverage distance (50 m buffer from new emergency bell). \\ \(d_{ij}\) & Binary parameter: 1 if the distance from new emergency bell \(j\) to crime occurrence location \(i\) is less than or equal to \(D\), 0 otherwise. \\ \(w_{i}\) & Numbers of crime occurrences at \(i\). \\ \(F\) & Number of new emergency bells to be located. \\ \(x_{j}\) & A decision variable, binary parameter: 1 if there is a new emergency bell in \(j\), 0 otherwise. \\  & A decision variable, binary parameter: 1 if unmonitored crime occurrence location \(i\) is covered by one or more new emergency bells located within \(D\), 0 otherwise. \\ \hline \hline \end{tabular}
\end{table} 
Table 3: The variables for EBMCLP and EBLSCP. Specifically, EBLSCP aims to determine the minimum number of new emergency bells required to cover all unmonitored crime occurrence coordinates. \[Minimize\sum_{j\in J}x_{j} \tag{2}\]  
Subject to  
\[\sum_{j\in J}d_{ij}x_{j}\ \geq 1\ \ \forall\ i\in I \tag{3}\]  
\[x_{j}=\{0,1\}\ \ \forall\ j\ \in J \tag{4}\]  
Equation (2) represents the objective function that minimizes the total number of emergency bells that need to be installed. The expression in Equation (3) explains the case when there is an emergency bell at each candidate location and the distance from the new emergency bell to the crime occurrence location falls within the 50 m buffer. The sum of such cases should be greater than or equal to 1. This constraint ensures that at least one new emergency bell, which covers crime location \(i\), will be installed. Equation (4) defines the role of variable \(x_{j}\) as a binary parameter, where \(x_{j}\) takes the value of 1 if there is a new emergency bell installed in location \(j\) and 0 otherwise. Meanwhile, EBMCLP aims to identify location alternatives that consider budget constraints and determine the optimal solution by limiting the number of emergency bells to provide the most efficient coverage. The model for EBMCLP applied in this study is as follows. \[Maximize\sum_{i\in I}w_{i}y_{i} \tag{5}\]  
Subject to  
\[\sum_{j\in J}d_{ij}x_{j}\ \geq\ y_{i}\ \ \forall\ i\in I \tag{6}\]  
\[\sum_{j\in J}x_{j}=F \tag{7}\]  
\[x_{j}=\{0,1\}\ \ \forall\ j\ \in J \tag{8}\]  
\[y_{i}=\{0,1\}\ \ \forall\ i\ \in I \tag{9}\]  
The objective of function (5) is to maximize the number of unmonitored crime occurrences to be covered by the optimal installation of emergency bells. To find the optimal location, the cumulative number of crime occurrences for each \(I\) was multiplied as \(w_{i}\).