## 5 Case Study: An AHP/TOPSIS-Based Optimal Gas Station Site Selection
### Criteria Weight Computation via AHP  
In computing the factors' weights, the very first step is the hierarchical model construction. The AHP hierarchical model consists of the goal at the top or at the root level. The criteria are placed at the intermediate level, whereas the candidates or alternatives are placed at the bottom. The AHP hierarchical model for our optimal site selection problem is shown in Figure 7.  
The next step in the AHP is the construction of a pairwise comparison matrix **A** to compute the criteria priorities/weights. The matrix **A** is a \(m\)\(\times\)\(m\) real matrix, where \(m\) is the number of evaluation criteria/factors considered. Each entry \(a_{jk}\) of the matrix **A** represents the importance of the \(j\)th criterion relative to the \(k\)th criterion. If \(a_{jk}>1\), then the \(j\)th criterion is more important than the \(k\)th criterion, while if \(a_{jk}<1\), then the \(j\)th criterion is less important than the \(k\)th criterion. If two criteria have the same importance, then the entry \(a_{jk}\) is 1. The entries \(a_{jk}\) and \(a_{kj}\) satisfy the following constraint [11]:  
\[a_{jk}.a_{kj}=1\]  
The following assumptions for the four factors are derived based on the pairwise comparison matrix. Table 3 shows the pairwise comparison matrix.  
* The most significant factor is the number of competitors nearby.  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{**CID**} & \multirow{2}{*}{**Alternatives**} & \multicolumn{5}{c}{**Factors**} \\ \cline{3-6}  & & **Traffic** & **Popularity** & **Vehicle Owners** & **Competitors** \\ \cline{3-6}  & & **1000 m** & **1000 m** & **5000 m** & **3000 m** \\ \hline
1 & 40.8839, \(-\)73.8561 & 2.91667 & 178 & 28,029 & 23 \\
2 & 40.8095, \(-\)73.8807 & 3.89881 & 79 & 56,643 & 7 \\
3 & 40.8347, \(-\)73.9177 & 228.345 & 631 & 42,954 & 32 \\
4 & 40.7931, \(-\)73.9675 & 16,170.9 & 1171 & 19,815 & 7 \\
5 & 40.7586, \(-\)73.9731 & 73,897.9 & 15,316 & 25,402 & 7 \\
6 & 40.7699, \(-\)73.9271 & 2624.86 & 1276 & 47,071 & 13 \\
7 & 40.7496, \(-\)73.8692 & 416.202 & 200 & 51,613 & 16 \\
8 & 40.7231, \(-\)73.914 & 507.357 & 33 & 44,028 & 10 \\
9 & 40.6686, \(-\)73.9321 & 110.815 & 340 & 30,995 & 38 \\
10 & 40.6599, \(-\)73.8701 & 2.57738 & 122 & 33,898 & 20 \\ \hline \hline \end{tabular}
\end{table}
Table 2: Candidate sites factor computation.  
Figure 7: AHP hierarchical model.  
* The second most significant factor is nearby traffic, followed by popularity of the area.
* The least important factor found in this study is vehicle ownership in the vicinity.  
The factors' weight is computed after normalization of the pairwise comparison matrix. The normalized pairwise comparison matrix \(A_{norm}\) is derived by making the sum of the entries in each column equal to 1, i.e., each entry \(\overline{a}_{jk}\) of the matrix \(A_{norm}\) is computed as  
\[\overline{a}_{jk}=\frac{a_{jk}}{\sum_{l=1}^{m}a_{lk}}\]  
Ultimately, the criteria/factors weight vector \(\mathbf{w}\) is obtained by averaging the entries on each row of the matrix \(A_{norm}\) and can be computed as follows:  
\[w_{j}=\frac{\sum_{l=1}^{m}\overline{a}_{jl}}{m}\]  
Table 4 shows the normalized pairwise comparison matrix with the weights of the criteria. It is necessary to calculate the consistency of the pairwise comparison matrix; therefore, it is performed after criteria weight computation. The acceptance of the criteria weight \(\mathbf{w}\) depends upon the consistency ratio, which must be less than 0.1; otherwise, it is assumed that the selection of comparison matrix values are not consistent. In this situtaion, the values of the pairwise comparison matrix need to be reallocated. In order for the matrix to be consistent, please refer to [9; 25].