# Integrated Decision Support Framework of Optimal Scaffolding System for Construction Projects
## 3 Implementation of the Proposed Framework
### Step 3: Implementation of Phase 2  
Phase 2 (Alternative ranking module) enables the users to determine the weight for each selected criterion. Ten respondents participated in the FAHP survey for the research. In the survey, they were asked to make pairwise comparisons between the criteria. The scaling rules for pairwise comparisons are presented in Table 1. The fuzzy evaluation matrix of the main criteria is illustrated in Table 4.  
After obtaining the pairwise comparison matrix, local weights for the main criteria were calculated. As the procedure for obtaining local weights is the same for both the main criteria and sub-criteria, a sample calculation is given below for the main criteria C1-C4. (1) Generate the geometric mean. Firstly, convert the \(n\times n\) matrix to \(n\times 1\) matrix through the multiplication of \(A_{1}\otimes A_{2}\) as shown in Equation (13). \[A_{1}\otimes A_{2}=(l_{1},m_{1},u_{1})\otimes(l_{2},m_{2},u_{2})=(l_{1}\times l _{2},\ m_{1}\times m_{2},\ u_{1}\times u_{2}) \tag{14}\]  
where \(A_{1}\) and \(A_{2}\) are two fuzzy matrixes, \(l_{1}\)_and_\(l_{2}\) are, respectively, the lower value in the first column of the corresponding matrix, \(m_{1}\) and \(m_{2}\) are, respectively, the middle value in the second column, and \(u_{1}\) and \(u_{2}\) are, respectively, the upper value in the third column. (2) Then the \(1/n\)th root of each value in the resultant matrix is calculated. For the matrix \(W\) obtained from Table 5, the root value of the first line is determined through Equation (15). \[\widetilde{r}_{1}=\left((1\times 1\times\frac{1}{3}\times 2)^{\frac{1}{4}},(1 \times 2\times\frac{1}{2}\times 3)^{\frac{1}{4}},(1\times 3\times 1\times 4)^{ \frac{1}{4}}\right)=(0.904,1.316,1.861) \tag{15}\]  
Similarly, the root value of the 2 nd, 3 rd and 4 th line can be calculated and the results are shown. \[\widetilde{r}_{2}=(0.577,0.841,1.316) \tag{16}\] \begin{table}
\begin{tabular}{c c c c c} \hline
**Criteria** & **C1** & **C2** & **C3** & **C4** \\ \hline C1 & (1, 1, 1) & (1, 2, 3) & (1/3, 1/2, 1) & (2, 3, 4) \\ C2 & (1/3, 1/2, 1) & (1, 1, 1) & (1/3, 1/2, 1) & (1, 2, 3) \\ C3 & (1, 2, 3) & (1, 2, 3) & (1, 1, 1) & (3, 4, 5) \\ C4 & (1/4, 1/3, 1/2) & (1/3, 1/2, 1) & (1/5, 1/4, 1/3) & (1, 1, 1) \\ \hline \end{tabular}
\end{table} 
Table 4: Fuzzy evaluation results of main criteria. \begin{table}
\begin{tabular}{c c c c c c} \hline
**Main Criteria** & **Local Weight** & **Sub-Criteria** & **Local Weight** & **Global Weight** & **Ranking** \\ \hline C1 & 0.2885 & C11 & 0.2415 & 0.070 & 5 \\ C1 & 0.2885 & C12 & 0.6131 & 0.177 & 1 \\  & & C13 & 0.1454 & 0.022 & 9 \\ \hline C2 & 0.1958 & C21 & 0.2140 & 0.042 & 10 \\  & & C22 & 0.1870 & 0.037 & 11 \\ C2 & 0.1958 & C23 & 0.1143 & 0.022 & 12 \\  & & C24 & 0.2367 & 0.046 & 8 \\  & & C25 & 0.2480 & 0.049 & 6 \\ \hline C3 & 0.4145 & C31 & 0.4211 & 0.175 & 2 \\ C3 & 0.4145 & C32 & 0.3707 & 0.154 & 3 \\  & & C33 & 0.2082 & 0.086 & 4 \\ \hline C4 & 0.1012 & C41 & 0.4618 & 0.047 & 7 \\  & & C42 & 0.2121 & 0.021 & 13 \\  & & C43 & 0.1282 & 0.013 & 15 \\  & & C44 & 0.1979 & 0.020 & 14 \\ \hline \end{tabular}
\end{table} 
Table 5: Global weights and ranking of criteria. \[\widetilde{\tau}_{3}=(1.316,2.000,2.590) \tag{17}\]  
\[\widetilde{\tau}_{4}=(0.359,0.452,0.639) \tag{18}\]  
The geometric mean for each line of matrix W is illustrated as Equations (15)-(18). (3) Calculate the sum of each column and reciprocate the sum value. \[\text{Column\_Sum(S}_{1})=(3.156,4.609,6.406) \tag{19}\]  
Reciprocal of S\({}_{1}\) = (1/6.406, 1/4.609, 1/3.156) = (0.156, 0.217, 0.317) (20)  
(4) Determine the center of area (COA) for the fuzzy matrix W, and then calculate the defuzzified value for matrix W.  
\[\text{Weighted Matrix}= \begin{bmatrix}\text{C}_{1}&(0.141,0.286,0.590)\\ \text{C}_{2}&(0.090,0.182,0.417)\\ \text{C}_{3}&(0.205,0.434,0.821)\\ \text{C}_{4}&(0.056,0.098,0.202)\end{bmatrix}  \tag{21}\]  
\[\text{Center of Area (COA)}=\frac{(l+m+u)}{3} \tag{22}\]  
\[\text{W1 }=\text{Defuzzified Value for W}= \begin{bmatrix}0.339\\ 0.230\\ 0.487\\ 0.119\end{bmatrix}