## 4 Service and Communication Technology Adaptation Method
### Determination of Index Weights Based on the FAHP-Improved CRITIC Combined Weighting Model
#### 4.2.2 Objective Weighting  
Objective weighting methods determine the objective weight of selected indexes by exploring objective factors such as volatility, amount of information, and conflict among all alternative solutions. The CRITIC method adopted in this paper is an objective weighting method that considers the variability and conflict of index data. Variability reflects the differences of an evaluation index across various alternatives. The greater the variability, the greater the differentiation capability of the index data within the adaptability evaluation system, and thus, its objective importance is higher. Conflict reflects the correlation between an evaluation index and other indexes. Lower conflict means a higher correlation with other indexes. When an index has a similar influence on the adaptability evaluation system as other indexes, its objective importance is considered lower. Additionally, variability represents the objective attributes of the index data itself, while conflict represents the objective attributes between index data. The specific calculation steps of the CRITIC method are as follows. Step 1. Construction of evaluation matrices and standardization. Construct the original evaluation matrix \(X\) from the \(m\) alternatives at the solution layer and the \(n\) indexes at the index layer of the adaptability evaluation system:  
\[X= \begin{bmatrix}x_{11}&x_{12}&\cdots&x_{1n}\\ x_{21}&x_{22}&\cdots&x_{2n}\\ \vdots&\vdots&\ddots&\vdots\\ x_{m1}&x_{m2}&\cdots&x_{mn}\end{bmatrix}  \tag{9}\]  
where \(x_{ij}\) represents the performance data of the \(j\)-th index of the \(i\)-th alternative, where \(i=1,2,\ldots,m\) and \(j=1,2,\ldots,n\). According to the influence of each index on the adaptation results, the indexes can be categorized into benefit-type and cost-type indexes. For benefit-type indexes, the values of the indexes are directly proportional to their adaptation results, while cost-type indexes are inversely proportional. The purpose of data standardization is to make different kinds of indexes have the same impact on the results. For benefit-type indexes, carry out the following steps:  
\[x^{\prime}_{ij}=\frac{x_{ij}-\min(x_{j})}{\max(x_{j})-\min(x_{j})} \tag{10}\]For cost-type indexes, carry out the following steps:  
\[x^{\prime}_{ij}=\frac{\max(x_{j})-x_{ij}}{\max(x_{j})-\min(x_{j})} \tag{11}\]  
Finally, we obtain the standard evaluation matrix \(X^{\prime}\):  
\[\mathbf{X^{\prime}}= \begin{bmatrix}x^{\prime}_{11}&x^{\prime}_{12}&\dots&x^{\prime }_{1n}\\ x^{\prime}_{21}&x^{\prime}_{22}&\dots&x^{\prime}_{2n}\\ \vdots&\vdots&\ddots&\vdots\\ x^{\prime}_{m1}&x^{\prime}_{m2}&\dots&x^{\prime}_{mm}\end{bmatrix}  \tag{12}\]  
Step 2. Calculation of objective weights. The CRITIC method considers volatility and conflict, represented by the standard deviation and correlation coefficient, respectively. The greater the standard deviation of the index data, the higher its volatility, and thus, the higher its weight. The smaller the correlation coefficient between indexes, the greater the conflict and, consequently, the greater the weight. According to the standard evaluation matrix \(X^{\prime}\), the calculation formula for the standard deviation of each index data and the correlation coefficient between indexes is as follows:  
\[\sigma_{j}=\sqrt{\frac{\sum_{i=1}^{m}(x^{\prime}_{ij}-\overline{x}_{j})^{2}}{m}} \tag{13}\]  
\[r_{ij}=\text{cov}(X^{\prime}_{i},X^{\prime}_{j})/(\sigma_{i},\sigma_{j}) \tag{14}\]  
where \(\sigma_{j}\) represents the standard deviation of the \(j\)-th index data, \(\overline{x}_{j}\) represents the mean of the \(j\)-th index data, \(r_{y}\) represents the correlation coefficient between the \(i\)-th and \(j\)-th evaluation indexes, and \(X^{\prime}_{i}\) and \(X^{\prime}_{j}\) are the \(i\)-th row and \(j\)-th row, respectively, of the standard evaluation matrix \(X^{\prime}\). The information-carrying capacity of volatility and conflict for each index is defined as \(C_{j}\), as shown in the Formula (15).