## 3 Research Methods
### Evaluation of the Application Maturity of Enterprise AI Technology
#### 3.2.1 ANP Method  
At present, many methods are used to determine the weights of evaluation indicators; the commonly used comprehensive evaluation methods include the expert scoring method, the analytic hierarchy process, data envelopment analysis, the artificial neural network evaluation method, and the grey comprehensive evaluation method. The expert scoring method and analytic hierarchy process are simple to perform; however, they are highly subjective and do not consider correlations between elements. The data envelopment analysis method is applied to evaluate the relative effectiveness of multi-input and multi-output methods. The artificial neural network evaluation calculation process is more complicated, and the grey comprehensive evaluation is mostly used in element correlation analysis. Based on the above analysis, the ANP, as a multicriteria decision-making method, can integrate the knowledge and experience of experts through the comprehensive use of qualitative and quantitative indicators, and is especially suitable for dealing with complex engineering problems with multiple objectives and multiple levels as well as complex correlations among indicators, such as evaluation indicators of the artificial intelligence technology application maturity of enterprises. The ANP is a multi-attribute decision model developed by the famous American operations research scientist [PERSON]. Its unique feature is that the ANP allows for the consideration of the interdependence and feedback between system elements [46]. With the development of the analytic hierarchy process (AHP), the ANP contains interdependencies within and between hierarchies, which objectively reflect the relative weights of various influencing factors. In complex environments, the ANP is more practical than the AHP is. Under the ANP method, the system elements are composed of two types of elements: control layer elements, including the overall objective and decision criteria, and network layer elements, which may be interrelated and influence each other, forming a network structure composed of individual element groups and influencing relationships. Thus, in the ANP, not only does the importance of the decision criteria determine the importance of the elements, but the importance of the elements also determines the importance of the individual criteria. Considering that the evaluation of the application maturity of enterprise AI technology is a complex system problem, it is more scientific and reasonable to evaluate the evaluation index model of the application maturity of enterprise AI technology using the ANP method in this study, which can consider the complex relationships among various indexes of the application maturity of enterprise AI technology. The basic steps of the ANP method are as follows:  
(1) Establish the ANP structure model. The decision problem of the system is analyzed, and an ANP structural model of the network structure is established. (2) Construct the hypermatrix of the ANP structure. Suppose that the control layer has \(n\) criteria, denoted as \(B_{s}\) (\(s\) = 1, 2, \(\ldots\), \(n\)), and the network layer has \(m\) element groups, denoted as \(C_{i}\) (\(i\) = 1, 2, \(\ldots\), \(m\)), where each element group contains \(n_{i}\) elements, denoted as \(C_{i1}\), \(C_{i2}\), \(\ldots\), \(C_{mi}\). Then, by a pairwise comparison of elements and between elements, the judgment matrix is constructed, the normalized eigenvector is calculated, and the supermatrix W is calculated. (3) A weighted supermatrix of the ANP structure is constructed. Each column of the supermatrix is normalized, the normalized eigenvector \((a_{1k},a_{2k},\ldots,a_{mk})^{\mathrm{T}}\) is calculated, the weighted matrix A is summarized, and then the weighted supermatrix is obtained. The calculation formula is as follows:  
\[\text{A}= \begin{bmatrix}a_{11}&a_{12}&\cdots&a_{1m}\\ a_{21}&a_{22}&\cdots&a_{2m}\\ \vdots&\vdots&\vdots&\vdots\\ a_{m1}&a_{m2}&\cdots&a_{mm}\\ \end{bmatrix}