## 4 Implementation of the Proposed Approach
### Step 3: Analyzing the Qualified Contractor Using the ISM Approach  
The implementation steps of the ISM are as follows:  
1. Development of the SSIM ISM is an interactive learning process involving different and directly related variables. The model can explicitly characterize the structure of a complicated problem. The first stage of the ISM method builds the contextual relationship among all factors through expert opinions. Four symbols were used to indicate the relationships among the criteria: V, A, X, and O. We can produce the following assessments considering \(I\) and \(j\) as the criteria:  
A: Criterion \(j\) causes criterion \(i\);  
V: Criterion \(i\) causes criterion \(j\);  
X: Criterion \(i\) causes criterion \(j\), and criterion j causes criterion \(i\);  
O: Criteria \(i\) and \(j\) are not related. One factor \"causes\" another indicates one leads to another. An SSIM is built on expert comments on the relationship between two factors. The experts certify the inclusion of crucial agile factors pivoting on their relationship and elucidating their intent. This process ends when all the experts approve the listed relations. In this study, the experts formed pairwise comparisons. After the pairwise comparisons were formed, we constructed the SSIM, as shown in Table 3.  
* Reachability matrix The SSIM is transformed into the initial reachability matrix, a binary matrix, by substituting 1 and 0 for V, A, X, and O, respectively. The rules for substituting 1 s and 0 s are shown in Table 4. The resulting initial reachability matrix is given in Table 5. The final reachability matrix is generated by Equations (1) and (2). \[M=S+I \tag{1}\]  
\[M_{R}=M^{k}=M^{k+1},k>1 \tag{2}\]  
where \(I\) represents the identity matrix, \(k\) symbolizes the powers, \(M_{R}\) denotes the reachability matrix, and \(M^{k}\) is the stable reachability matrix. The reachability matrix is calculated based on the addition law and Boolean multiplication (i.e., \(1\times 1=1\), \(1\times 0=0\times 1=0\), \(1+1=1\), \(1+0=0+1=1\), and \(0+0=0\)). The transitivity checks from the criteria in the ISM approach yield the final reachability matrix in Table 6. \begin{table}
\begin{tabular}{c c c c c c c c c c c c c} \hline \hline  & **Criteria** & **C12** & **C11** & **C10** & **C9** & **C8** & **C7** & **C6** & **C5** & **C4** & **C3** & **C2** & **C1** \\ \hline Financial stability (C1) & C1 & V & O & O & O & O & O & O & O & O & O & V & - \\ Credit rating (C2) & C2 & V & A & O & O & O & O & O & O & O & O & - & \\ Engineering experience (C3) & C3 & V & O & O & V & O & O & O & V & O & - & & \\ Personnel qualification (C4) & C4 & O & O & A & O & A & V & O & O & - & & & \\ Engineering ability (C5) & C5 & V & O & O & V & A & O & V & - & & & & \\ Performance and quality (C6) & C6 & V & X & A & A & A & A & - & & & & & \\ Project Management organization (C7) & C7 & O & O & O & V & A & - & & & & & & \\ Management knowledge (C8) & C8 & O & O & O & V & - & & & & & & & \\ Health and safety (C9) & C9 & V & O & O & - & & & & & & & & \\ Management accountability (C10) & C10 & O & O & - & & & & & & & & & \\ Breach of contract (C11) & C11 & V & - & & & & & & & & & & \\ Client-contractor relationships (C12) & C12 & - & & & & & & & & & & & \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Structural self-interaction matrix. 2. Level partitions  
By continuously evaluating the reachability and prior sets for each criterion, a partition of the reachability matrix was created. The reachability set included the reachable criteria. The antecedent set included the criteria and the others that could contribute to their fulfillment. The intersection of these sets was created for all criteria. The criteria could be categorized in the first level of the ISM hierarchy if they gravitated toward both the reachability set and intersection set in the same fashion. The criteria cannot be used to determine the others above their level. The next level criteria were identified through the same process.