## 5 Case Study: An AHP/TOPSIS-Based Optimal Gas Station Site Selection
### Ranking the Alternatives Using TOPSIS  
Once the criteria weights are determined with the help of AHP [11], TOPSIS is utilized to assign ranks to each alternative (gas station candidate sites). According to the researchers in [10], TOPSIS is a multi-criteria decision analysis method (MCDM), which assists in selecting the best option among a finite set of decision alternatives. A step-by-step procedure is shown below for computing the ranking of the alternatives. Step 1: This step makes use of TOPSIS to make an \(n\times m\) evaluation matrix \(E\) consisting of \(n\) alternatives and \(m\) criteria. Without losing generality, 10 alternatives (gas station candidate sites as shown in Figure 6) and 4 criteria are chosen. Table 5 presents a TOPSIS evaluation matrix of the 10 candidate sites and 4 factors. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Factors** & **Competitors** & **Traffic** & **Popularity** & **V Owners** & **Factors Weight** \\ \hline Competitors & 0.676691 & 0.779221 & 0.529411 & 0.473684 & **0.614752121** \\ Traffic & 0.135338 & 0.155844 & 0.352941 & 0.315789 & **0.239978288** \\ Popularity & 0.112781 & 0.038961 & 0.088235 & 0.157895 & **0.099468256** \\ V Owners* & 0.07518 & 0.025974 & 0.029411 & 0.052631 & **0.045801335** \\ \hline \hline \end{tabular}  
* V Owners: Vehicle owners.  
\end{table} 
Table 4: Normalized pairwise comparison matrix with factors weights. Step 2: In this step, the matrix \(\mathbf{E}\) is normalized to form the matrix \(\mathbf{E_{norm}}\), where each entry \(\overline{e}_{ij}\) of \(\mathbf{E_{norm}}\) is computed as  
\[\overline{e}_{ij}=\frac{e_{ij}}{\sqrt{\sum_{k=1}^{n}e_{kj}^{2}}}\]  
where \(i=1,2,...,n\) and \(j=1,2,...,m\). Step 3: In this step, the weighted normalized decision matrix \(\mathbf{E_{weighted}}\) is obtained by multiplying the criteria weights \(w_{j}\) (computed in Section 5.3) to the corresponding criteria values. Hence, each entry \(e_{ij}^{w}\) of \(\mathbf{E_{weighted}}\) is computed as \(\overline{e}_{ij}.w_{j}\). The weighted normalized decision matrix is represented by the grey columns in Table 5.  
Step 4: Next, we determine the worst alternative (\(V_{j}^{-}\)) and the best alternative (\(V_{j}^{+}\)) for each column in \(\mathbf{E_{weighted}}\) as follows:  
\[V_{j}^{-}=\{(max_{i}\,e_{ij}^{w}|j\in J_{-}),(min_{i}\,e_{ij}^{w}|j\in J_{+})\}\]  
\[V_{j}^{+}=\{(min_{i}\,e_{ij}^{w}|j\in J_{-}),(max_{i}\,e_{ij}^{w}|j\in J_{+})\}\]  
where \(i=1,2,...,n\), \(J_{+}=\{j=1,2,...,m\}\) is associated with the criteria having a positive impact and \(J_{-}=\{j=1,2,...,m\}\) is associated with the criteria having a negative impact. Step 5: Next, we need to compute the Euclidean distance (\(L^{2}\) - distance) between the target alternative \(i\) and the best alternative \(V_{j}^{+}\) and between the target alternative \(i\) and the worst alternative \(V_{j}^{-}\), denoted by \(S_{i}^{+}\) and \(S_{i}^{-}\), respectively, and given as follows:  
\[S_{i}^{+}=\sqrt{\sum_{j=1}^{m}(e_{ij}^{w}-V_{j}^{+})^{2}},i=1,2,...,n\]  
\[S_{i}^{-}=\sqrt{\sum_{j=1}^{m}(e_{ij}^{w}-V_{j}^{-})^{2}},i=1,2,...,n\]  
Step 6: Finally, the performance score (\(P_{i}\)) of each \(i\)th alternative is computed using the following equation. \[P_{i}=\frac{S_{i}^{-}}{S_{i}^{-}+S_{i}^{+}}\]