## 3 Combination Weighting Based on Improved AHP-Entropy Method  
### Exponential Extension AHP  
The analytic hierarchy process (AHP) is a systematic, hierarchical method that combines qualitative and quantitative aspects to determine subjective weights. However, considering the subjectivity, complexity in the calculation, and difficulty of reflecting the ambiguity of experts' thinking in the traditional AHP, scholars have proposed some methods to improve this process; these include a five-point scale, a nine-point scale, exponential scale, and interval scale [11]. The exponential scale method is developed based on the nine-point scale method, and it has good preservation properties. It can accurately and objectively reflect the original ordinal relationship of each comparison object and applies to ranking problems under a single criterion and multiple criteria, as specifically described in Table 1.  
Figure 1: A multi-level evaluation system for EPS performance. Based on the existing research, it is known that the index scale method in AHP can achieve a higher accuracy during the determination of weights, while the interval scale method can expand the scale value from point to space, thus quantifying the fuzziness of expert evaluation thinking. Therefore, this article comprehensively applies these two improvement methods to ensure the scientific and rational allocation of indicators. The specific steps are as follows [12,13]:  
1. Construct the judgment matrix of the exponential interval type The EPS evaluation system is divided by hierarchy, and many relevant experts are invited to score and judge the importance of indicators at the same level according to Table 1. Then, the relative scale between two factors in the \(n\)th layer is \(a_{ij}^{\prime}\), and \(a_{ij}^{\prime}\times a_{ji}^{\prime}=1\). Finally, with reference to the expert opinions, the extension interval \((a_{ij}^{\prime}-\lambda,a_{ij}^{\prime}+\lambda)\) formed by a certain coefficient of variation \(\lambda\) is added or subtracted, that is, the judgment matrix is \(A=(a_{ij})_{m\times n}\). The specific expression is as follows: \[A= \begin{pmatrix}a_{11}&\dots&a_{1n}\\ \vdots&\ddots&\vdots\\ a_{m1}&\dots&a_{mn}\end{pmatrix} _{m\times n}=\begin{pmatrix} \begin{pmatrix}a_{ 11}^{-},a_{11}^{+}&\dots&\begin{pmatrix}a_{1n^{\prime}}^{-},a_{1n}^{+}\\ \vdots&\ddots&\vdots\\ a_{m1}^{+},a_{m1}^{+}\end{pmatrix}&\dots&\begin{pmatrix}a_{mn}^{-},a_{mn}^{+} \end{pmatrix}_{m\times n}\end{pmatrix} _{m\times n}\] (1) \[a_{ij}=(a_{ij}^{-},\,a_{ij}^{+})\] is the number of extensible intervals, which are expressed as the upper and lower ends of the column \(j\) element in the first row \(i\) of the judgment matrix. Therefore, judgment matrix \(A=\) <\(A^{-}\), \(A^{+}\)> of the \(n\)th layer, upper and lower endpoint matrix is expressed as \(A^{-}=[a_{ij}^{-}]_{m\times n}\) and \(A^{+}=[a_{ij}^{+}]_{m\times n}\). 2. Consistency check According to \(A^{-}=[a_{ij}^{-}]_{m\times n}\) and \(A^{+}=[a_{ij}^{+}]_{m\times n}\), the values of \(k\) and \(m\) are solved by Equation (2). \[\left\{ \begin{array}{l}k=\sqrt{\sum\limits_{j=1}^{n}\left(1/\sum\limits_{i =1}^{n}a_{ij}^{+}\right)}\\ m=\sqrt{\sum\limits_{j=1}^{n}\left(1/\sum\limits_{i=1}^{n}a_{ij}^{-}\right)} \end{array} \right.\] (2) If \(0\leq k\leq 1\leq m\), the consistency of the extension judgment matrix meets the requirements. Otherwise, the judgment matrix needs to be corrected or experts need to be invited again until the requirements are met. 3. Calculate the weight vector After determining the final judgment matrix \(A=\) <\(A^{-}\), \(A^{+}\)>, the normalized eigenvector \(z^{-}\) and \(z^{+}\) corresponding to the maximum eigenvector \(A^{-}\) and \(A^{+}\) with positive components can be calculated. The eigenvector of the extension judgment matrix can be solved using the root method. Thus, the weight vector is as follows: