## 3 Research Methodology
### Application of MCDM Methods
#### 3.2.4 Technique for Order of Preference by Similarity to Ideal Solution (TOPSIS)  
Technique for Order Preference by Similarity to Ideal Solution (TOPSIS) was developed as an alternative to ELECTRE method [7]. According the concept of selecting the shortest Euclidean distance from a positive ideal solution and the farthest from a negative ideal solution [7; 48; 49]. Three TOPSIS methods have been implemented with the same decision matrix and weight calculation approach, which were used in WSM and WPM methods as follows:  
1. TOPSIS1: TOPSIS with the weight parameter, calculated based on the direct ranking are discussed in the following parts. 2. TOPSIS2: TOPSIS with the weight parameter, calculated using the pairwise comparison and the Eigenvalue approach. 3. TOPSIS3: TOPSIS with the weight parameter, calculated using the pairwise comparison and the Normalized matrix. The following steps are related to the TOPSIS methodology. Step 1: Decision Matrix:  
After defining the necessary criteria and alternatives, a decision matrix (\(m\times n\)) must be created, where \(m\) is the number of alternatives, and \(n\) is the number of criteria. Each value of the decision matrix is the performance value of an alternative w.r.t corresponding criteria. Performance value is essentially the number given for the alternative, depending on the actual value of the criterion. In this project, the performance values used to rank alternatives w.r.t to criteria are 1-9 with 1 as a best performance and 9 as the worst performance. \begin{table}
\begin{tabular}{c c} \hline
**Difference in Ranking of Parameter** & **Importance Ranking** \\ \hline
0 & 1 \\ \hline
1 & 2 \\ \hline
2 & 3 \\ \hline
3 & 5 \\ \hline
4 & 7 \\ \hline
5 & 9 \\ \hline \end{tabular}
\end{table} 
Table 13: Saaty scale based on the difference between number of plan irregularities. \begin{table}
\begin{tabular}{c c} \hline
**Difference in Ranking of Parameter** & **Importance Ranking** \\ \hline
0 & 1 \\ \hline
1 & 2 \\ \hline
2 & 3 \\ \hline
3 & 5 \\ \hline
4 & 7 \\ \hline
5 & 9 \\ \hline \end{tabular}
\end{table} 
Table 12: Saaty scale based on the difference between number of vertical irregularities. Step 2: Normalized Decision Matrix:  
The normalized decision matrix will be created by using Equation (9). \[NDM=R_{ij}=\frac{X_{ij}}{\sqrt{\sum_{i+1}^{m}X_{ij}^{2}}}, \tag{9}\]  
where \(R_{ij}\) is an entity of the normalized decision matrix, and \(X_{ij}\) indicates an entity of the decision matrix. Instead of summation of all values from a column, the maximum value in a column can also be used to normalize the values in the decision matrix. Step 3: Weighted Normalized Decision Matrix:  
This matrix will be calculated by multiplying each entity of NDM with the corresponding weight value of parameters or criteria (Equation (10)). \[V=V_{ij}=W_{j}\times R_{ij}, \tag{10}\]  
where, \(V_{ij}\) is the entity of weighted normalized decision matrix of \(i\)th alternative for \(j\)th criterion and \(W_{j}\) is the weight of \(j\)th criterion. The weights of criteria can be calculated in any of the previously explained ways. Step 4: Positive and Negative Ideal Solution (PIS and NIS):  
A positive ideal solution is the best combination of each criterion's values from the weighted normalized decision matrix and vice versa. These combinations can be found by selecting maximum or minimum values in each column of the weighted normalized decision matrix based on the corresponding criteria (Equations (11) and (12)). If the higher numbers of particular criteria negatively influence alternatives, then the minimum value will be selected as a positive ideal solution instead. The maximum amount will be selected as a perfect positive solution, and vice versa.