## 4 Consistency Enhancement Strategy of Ahp in Group Decision-Making
### Optimization of Aggregation Coefficient Based on Hadamard Convex Combination  
Because SIN is a complex network that spans multiple fields, some evaluation objects, such as integration test results or technical verification schemes, often require scoring by multiple experts in different professional fields to achieve a scientific and comprehensive evaluation. Moreover, this process can be attributed to the category of group decision-making. In this paper, the aggregation sequence of judgment matrix with AJJ is applied for group AHP. That is to say, the judgment matrix is aggregated first, and then the eigenvector of the comprehensive evaluation matrix is obtained.  
Common judgment matrix aggregation methods include additive aggregation based on arithmetic mean and multiplicative aggregation based on geometric mean. This paper chooses the latter as the aggregation method. Because, there is a serious flaw in the additive aggregation method: When the distance between elements of two near-coherent matrices at the same position is large, additive aggregation may increase the error of the matrix eigenvalue after aggregation, and the consistency of the aggregated matrix could not be guaranteed. Theorem 2 shows that multiplicative aggregation based on hadamard convex combination can neutralize eigenvalues better and inherit the consistency of each judgment matrix. The proving process can be found in [24].  
**Theorem 2**.: _If judgment matrices \(A_{1},A_{2},...,A_{n}\) are consistent, their hadamard convex combination are consistent._  
Hadamard convex combination defined as: \(\tilde{A}=A_{1}^{\lambda_{1}}\cdot A_{2}^{\lambda_{2}}\cdot...\cdot A_{n}^{ \lambda_{n}},\sum\limits_{i=1}^{n}\lambda_{i}=1,0<\lambda_{i}<\)1. The scoring of the experts in the group decision-making process is usually independent. On the one hand, considering the existence of authoritative experts, their scores will have a higher weight. On the other hand, it is necessary to correlate the degree of deviation of the consistency of the initial matrix with the weight of the expert, considering that there is a correlation between the cognitive degree and the consistency when the expert gives the judgment matrix. Based on the above reasons,this paper proposes a new weighting coefficient for the hadamard convex combination (also called the hadamard product), which could be described with Equation (2).  
\[\begin{split}\lambda_{\mathrm{i}}=\alpha\lambda_{i}^{1}+\beta \lambda_{i}^{2},\ \ \alpha+\beta=1,\ \ \ \alpha,\beta\in R^{+}\\ \lambda_{i}^{1}=\frac{1+i*d}{\sum\limits_{i=1}^{n}\left(1+i*d \right)},\ \lambda_{i}^{2}=\frac{\sum\limits_{i=1}^{n}CI_{\mathrm{i}}}{CI_{ \mathrm{i}}}\Bigg{/}\sum\limits_{i=1}^{n}\left(\frac{\sum\limits_{i=1}^{n}CI_{ \mathrm{i}}}{CI_{\mathrm{i}}}\right)\end{split} \tag{2}\]  
\(n\) is the number of experts, and \(\lambda_{i}^{1}\) is the authority coefficient of the i-th expert, which is usually subjectively scored by the evaluator. This paper presents a generation method based on authoritative ranking, which is equivalent to increasing the authoritative coefficient by a given step size d (we could set \(d=1/n\)). \(\lambda_{i}^{2}\) is the deviation coefficient of the i-th judgment matrix. The smaller the corresponding consistency index CI, the larger the deviation coefficient. At the same time, the distribution coefficients of \(\alpha\) and \(\beta\) also provide strong flexibility. In particular, when \(\alpha=1\) and \(\beta=0\), the hadamard coefficient is a single expert authority coefficient, while when \(\alpha=0\) and \(\beta=1\), the hadamard coefficient is determined in full compliance with the initial consistency of each expert. Such an improvement factor not only reflects the authority of experts, but also takes into account the consistency of the comprehensive matrix.