## 3 Determine the Weights of PQ Indices
### Determination of Subjective Weight Based on Improved AHP  
AHP is a multi-objective decision-making method founded by American operations researcher [PERSON]. Its core idea is to rank the evaluation indices according to the experience of experts [27; 28]. However, the traditional AHP usually constructs a judgment matrix and needs to pass the consistency test. In the actual calculation process, the test is often difficult to pass. As shown in Figure 4a, the portion of the red dashed box is uncontrollable. Therefore, the amount of calculation of the conventional AHP is large. Moreover, when the order of the judgment matrix is high, the amount of calculation will be larger. Therefore, experts [29] proposed using the scale method to construct the judgment matrix, as shown in Figure 4b. The improved AHP avoids the consistent test by constructing a judgment matrix, making the calculation process more straightforward and reducing the workload greatly.  
Figure 3: Decision diagram of the TOPSIS method.  
Therefore, this paper uses the comparison scale to construct the judgment matrix. It is assumed that an evaluation sample has \(n\) indices, according to the expert opinion, the \(n\) indices are sorted in descending order of importance, such as: index \(1>\) index \(2>...>\) index \(n\). By comparing the index \(j\) and the index \(j+1\) in pairs, the corresponding scale is recorded as \(b_{j}\). The specific definition of the scale is shown in Table 1.  
According to the scale assignment method described above, the judgment matrix \(R=[r_{jk}]\) is constructed, and the matrix is defined as follows:  
\begin{table}
\begin{tabular}{c c} \hline
**Scale Values \(b_{j}\)** & **Scale Definition** \\ \hline \(1\) & A and B are equally important \\ \(1.2\) & A is slightly more important than B \\ \(1.4\) & A is obviously more important than B \\ \(1.6\) & A is strongly more important than B \\ \(1.8\) & A is extremely more important than B \\ \(1.1\), \(1.3\), \(1.5\), \(1.7\) & Intermediate values of the above-mentioned judgment. \\ \hline \end{tabular}
\end{table}
Table 1: The meanings of scale values.  
Figure 4: Comparison between traditional AHP and improved AHP: (**a**) flow chart of traditional AHP; (**b**) flow chart of improved AHP.  
\(r_{jk}\) is the scale value after comparing the \(j\)th index with the \(k\)th index, where \(j\), \(k=1\), \(2\),..., \(n\); \(r_{kj}=1\)/\(r_{jk}\), where \(j\), \(k=1\), \(2\),..., \(n\); \(r_{jj}=1\), where \(j=1\), \(2\),..., \(n\).  
Based on the above definition, the judgment matrix \(R=[r_{jk}]\) can be obtained as follows:  
\[R=\left[\begin{array}{cccccccc}1&h_{1}&h_{1}h_{2}&h_{1}h_{2}h_{3}&h_{1}h_{2 }h_{3}h_{4}&h_{1}h_{2}h_{3}h_{4}h_{5}&...&h_{1}h_{2}\ldots h_{n-1}\\ 1/(b_{1})&1&b_{2}&b_{2}h_{3}&b_{2}h_{3}h_{4}&h_{2}h_{3}h_{4}h_{5}&...&h_{2}h_{3 }\ldots h_{n-1}\\ 1/(b_{1}b_{2})&1/(b_{2})&1&b_{3}&b_{3}b_{4}&b_{3}b_{4}h_{5}&...&b_{3}b_{4} \ldots h_{n-1}\\ 1/(b_{1}b_{2}b_{3})&1/(b_{2}b_{3})&1/(b_{3})&1&h_{4}&b_{4}h_{5}&...&b_{4}b_{5} \ldots h_{n-1}\\ 1/(b_{1}b_{2}b_{3}b_{4})&1/(b_{2}b_{3}b_{4})&1/(b_{3}b_{4})&1/(b_{4})&1&b_{5} &...&b_{3}b_{5}\ldots b_{n-1}\\...&...&...&...&...&...&...&...\\ 1/(b_{1}b_{2}\ldots b_{n-2})&1/(b_{2}b_{3}\ldots b_{n-2})&1/(b_{3}b_{4}\ldots b _{n-2})&1/(b_{4}b_{3}\ldots b_{n-2})&1/(b_{5}b_{6}\ldots b_{n-2})&1&...&b_{n -1}\\ 1/(b_{1}b_{2}\ldots b_{n-1})&1/(b_{2}b_{3}\ldots b_{n-1})&1/(b_{3}b_{4}\ldots b _{n-1})&1/(b_{5}b_{6}\ldots b_{n-1})&1/(b_{6}\ldots b_{n-1})&...&1\end{array}\right] \tag{1}\]  
According to the matrix R, the subjective weights of each index can be calculated, as shown in Equation (2):  
\[\alpha_{j}=\sqrt[n]{\prod_{k=1}^{n}r_{jk}}/\sum_{j=1}^{n}\sqrt[n]{\prod_{k=1 }^{n}r_{jk}}, \tag{2}\]  
where \(\alpha_{j}\) is the subjective weight of each index, \(j\), \(k=1\), \(2\),..., \(n\).