## 3 Research Methodology
### Application of MCDM Methods
#### 3.2.3 Analytical Hierarchy Process (AHP)  
In this method, the calculation of weights is done by using pairwise comparison in which the performance values were given in hierarchical order using saaty scale of [9, 10, 11, 12]. \[A_{AHP-score}=max\sum_{i}^{m}a_{ij}w_{j},\ \ \text{for}\ \ i=1,2,3,\ldots,n. \tag{8}\]  
Three AHP methods have been conducted using the seismic data with different weight as discussed below:  
1. AHP1: AHP with weight of parameters and Eigenvalues of buildings had been calculated using the Eigen value approach. 2. AHP2: AHP with weight of parameters and Eigenvalues of buildings had calculated using another value approach. 3. AHP3: AHP with weight of parameters and Eigenvalues of buildings had calculated using the Eigen value approach; here the values were normalized with the maximum value instead of making a sum of all values. Steps involved in AHP:Step 1: Pairwise Comparison Matrix for Parameters:  
In making of a pairwise comparison matrix, each parameter should be differentiated with another one. Table 11 shows the pairwise comparison between parameters according to the saaty scale. The saaty scale has ranged from 9 to 1/9. For example, if a soil type has the rank 5 and the seismic zone as 9 then their difference 4 is taken into consideration and based on that the saaty scale gives a final value as 5. It indicates an importance of the seismic zone when it compared to the type of existing soil. Where the soil type has compared to the seismic zone then the reciprocal of 5 occurs. To construct a pairwise structural comparison matrix, the rank-based comparison has been done against each other with different parameters. According to this, 8 different matrices have been formed of the size 28 using 8 different parameters. The saaty scale has been used to provide a ranking system to each building with respect to following parameters:  
* Seismic zone and type of soil: From the data, all structures have similar seismic zone and soil type, so both parameters have equal significance. Therefore, a value '1' has been finalised. * Building type: The data has C2 and C3 classified buildings. The C3 type of buildings have been considered as insignificant compared to the C2 type, due to masonry infills. Hence, the intermediate rank between 3 and 5, that is, 4 has been specified after both type of buildings had been compared against each other. * Vertical irregularity: The importance ranking given to such buildings which have more vertical irregularities and it has given by taking the difference between number of irregularities as shown in Table 12.
* Plan irregularity: The ranking was designated as similar to the vertical irregularities and considered as presented in Table 13.
* Code parameter: Majority of structures were constructed in the middle of the pre-code and post-benchmark year and remaining ones after the post-benchmark year. So, both categories of structures were differentiated from one another and as per the saaty scale, rank 5 has been specified to the buildings which were constructed after the post-benchmark year, and for the rest 1 was specified. * No. of stories: Buildings were divided into three categories as (a) 1 to 3 stories, (b) 4 to 7 stories, and (c) more than 7 stories. The higher importance rank was designated to the high-rise structure; here for more than 4- or 5-story structures and for short storey (1, 2, 3) the specified rank was 3 and 1, respectively. * Apparent building quality: The importance rank was determined as per the quality of the structures, as higher ranks designated to poor quality structures and vise versa. For example, importance ranks 7, 5, and 1 have been specified for poor, moderate and good quality constructions. Mostly the importance rank 1 indicates to the apparent building quality. \begin{table}
\begin{tabular}{c c} \hline
**Difference in Ranking of Parameter** & **Satty Scale Value** \\ \hline
0 & 1 \\ \hline
1 & 2 \\ \hline
2 & 3 \\ \hline
3 & 5 \\ \hline
4 & 7 \\ \hline
5 & 9 \\ \hline \end{tabular}
\end{table}