# Finding Optimal Stations Using Euclidean Distance and Adjustable Surrounding Sphere
## 4 Proposed Methods
### Euclidean Distance + Fixed Surrounding Sphere (ED + FSS)
\[\mathit{Cut\;off\;position}\;\;=\;\;\mathit{round}(\mathit{ratio\times length \;(list)}) \tag{4}\]ratio is predefined covered area proportion of the station, length (list) is equal \(N\) where \(N\) represents the total number of locations in area. In the sorted list, we access the index of the list at the cut off position and get the value of that \(ED_{i}\) value. The value of \(ED_{i}\) is multiplied by two and defined as a diameter length of the ED + ASS method. Once, we have obtained the diameter length, and then we can continue the procedure of establishing the station network and evaluating the global maximum coverage of the station network in the same way as we did with ED + FSS. ## 5 Evaluation Criteria

In ED + FSS and ED + ASS, there is a multi-list of station networks with different first locations that must be processed to evaluate the maximum coverage percentage. Consequently, in order to find the best station network, the evaluation criteria are designed and described next. ### Coverage Percentage (COV)

The covered area of stations (\(A_{S}\)) can be determined using Equation (3). Given the list of stations \(S\) located in a study area, we can assess the k-coverage when a location is covered by at least \(k\) different stations. The parameter k is called Coverage Degree. It means at least k stations cover each location in the study area. There are previous studies of wireless network coverage that have discussed the required k value of a network. Such studies explain a proper value of k that depends on the application. For example, an application requires \(k=1\) in a monitoring environment in which fault tolerance is not important. Meanwhile, \(k>1\) should be used when stronger monitoring is required, such as in an industrial or dangerous chemical region. Furthermore, in cases requiring fault tolerance, \(k\geq 3\) is required. Therefore, it is clear that the station networks with higher k-coverage are more reliable [32; 33]. Suppose that there are four stations. The circle shapes represent covered areas of a station and \(\times\) symbols represent \(L_{i}\) for coverage degree assessment. If \(\times\) symbols are within the covered areas of one station, then we define such \(L_{i}\) as a C1. If \(\times\) symbol lies within the covered areas of two, three, and four stations, then it is denoted by C2, C3, and C4 as shown in Figure 5a-d, respectively. The area outside the circle is defined as a C0, which means it is a non-covered area. We calculate the coverage percentage using the count \(L_{i}\) in each of the coverage degrees. The coverage percentage (COV) is used to evaluate results in our proposed methods and can be calculated with Equation (5). \[COV_{percentage}=100\times\frac{\sum_{i=1}^{k}\ C_{i}}{N} \tag{5}\]

\(C_{i}\) denotes the summation of \(L_{i}\) in coverage degree, \(i\) is k-coverage number, \(k\) indicates a number of stations in the network, and \(N\) is total number of locations in study area \(A\). Figure 5: The example of coverage degree, (**a–d**) indicate the covered areas by one, two, three, and four stations. ### Weighted Coverage Degree (WCD)

Weighted coverage degree (WCD) is an additional criterion. It is used whenever COV cannot give a unique answer to the best network. The WCD corresponds to number of stations and k-coverage. For example, if the study area has four stations, here the k-coverage \(k=4\) (coverage degree: C0, C1, C2, C3, C4) and the weight has five values of coverage degree. We calculate the weight value by employing the Divide and Conquer concept. The \(\sum W_{i}\) is a value equal to one. The \(W_{k}\) is calculated from \(\sum W_{i}\) divided by two. The next weight value at \(W_{k\cdot i}\) will decrease from the previous by half, which means \(W_{k\cdot i}=W_{k}\) divided by two. The weight calculation is done continuously until the last weight at \(W_{0}\) is set equal to \(W_{1}\). Table 1 is an example of weight value generation when the coverage degree is equal to 4. The WCD can be calculated by coverage degree weighting with Equation (6).