## 3 Determination of Commonality Indexes Weights and Calculation of Comprehensive Score
### The Calculation of Weights for Commonality Indexes  
Since the commonality indexes such as benefits, resource development, and development potential are indexes shared by the five DSER, and the preference of the decision makers for these indexes is more obvious, the Analytic Hierarchy Process (AHP) is used to evaluate the commonality indexes.  
On one hand, the AHP takes the subjective experience judgement of expert scoring into account. On the other hand, the expert judgement is transformed into a mathematical model for quantitative calculation, so that the proportion of each index in the company evaluation index can be calculated. The combination of analysis and calculation is extremely useful for highlighting corporate evaluation in different periods.  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{**User Type**} & \multirow{2}{*}{**User Representative**} & \multicolumn{5}{c}{**Quantized Value**} \\ \cline{3-6}  & & **Response** & **Response** & **Response** & **Response** \\  & & **Willingness** & **Capacity** & **Success Rate** & **Cost** \\ \hline \multirow{2}{*}{Public user} & office, school, & \multirow{2}{*}{1–2} & \multirow{2}{*}{1} & \multirow{2}{*}{1–2} & \multirow{2}{*}{1} \\  & hospital & & & & \\ \hline \multirow{2}{*}{Commercial user} & supermarket, & \multirow{2}{*}{1–2} & \multirow{2}{*}{3} & \multirow{2}{*}{1–2} & \multirow{2}{*}{3–4} \\  & shopping mall, hotel & & & & \\ \hline \multirow{4}{*}{Industrial user} & spinning & 4 & 4 & 3–4 & 2–3 \\  & cement & 3–4 & 4 & 3–4 & 2–3 \\ \cline{1-1}  & steel & 1–2 & 2–3 & 1–2 & 5 \\ \cline{1-1}  & mechanical & 1–2 & 3–4 & 1–2 & 4–5 \\ \hline \multirow{2}{*}{Resident user} & Residential area & 4–5 & 1 & 4–5 & 1 \\ \hline \hline \end{tabular}
\end{table}
Table 7: The rule of quantifying users’ response capacity.  
The basic idea of the AHP is to build the problem hierarchically based on the decision goal (as is shown in Figure 4). The highest level is the target level, several intermediate levels are the criterion level, and the bottom level is the various options selected for solving the problem, which is called the plan level [24; 25].  
The weights of indexes are determined by AHP as follows:  
1. Compare pairs of indexes in the same level, and refer to the number 1-9 and its reciprocal as a scale to define the judgment matrix \(A\), as shown in Table 8.
2. Calculate consistency ratios and test consistency \[CR=\frac{CI}{RI}\] (4) \[CI=\frac{\lambda_{\max}-n}{n-1}\] (5)  
where \(CI\)--Consistency indicator; \(RI\)--Random consistency indicator; \(CR\)--Test coefficient; \(\lambda_{\max}\)--The maximum eigenvalue of the judgment matrix \(A\); \(n\)--The order of the judgment matrix \(A\).  
The value of \(RI\) is shown in Table 9  
Figure 4: Hierarchical structure.  
\begin{table}
\begin{tabular}{c c} \hline \hline
**Scale** & **Meaning** \\ \hline
1 & two factors of equal importance \\
3 & two factors, the former is slightly more important than the latter \\
5 & two factors, the former is obviously more important than the latter \\
7 & two factors, the former is mightily more important than the latter \\
9 & two factors, the former is extremely more important than the latter \\
2, 4, 6, 8 & the middle value of the above adjacent judgment \\ \hline reciprocal & If the ratio of the importance of factor \(i\) to factor \(j\) is \(a_{ij}\), the ratio of the \\  & importance of factor \(j\) to the factor \(i\) is \(a_{ji}=1/a_{ij}\) \\ \hline \hline \end{tabular}
\end{table}
Table 8: The definition of the scale in the matrix.  
When CR < 0.10, the consistency of the judgment matrix is considered acceptable, otherwise the judgment matrix should be properly modified.  
(3) The calculation of weight vector \(W\)  
The weight vector \(W\) in AHP is calculated by the eigenvector method, and the weight vector \(W\) is multiplied by the judgement matrix \(A\)  
\[AW=\lambda_{\text{max}}W \tag{6}\]