# A Multi-Objective Evaluation Method for Smart Highway Operation and Management
## 4 Methods for Assigning Weights to Evaluation Indicators
### Group Decision-Making Method (GDM)  
The judgment matrix of an individual expert reveals a significant level of subjectivity, capable of reflecting only a specific perspective on objective phenomena. Employing equal weights for aggregation might not adequately capture the expertise of the individual expert, thus diminishing the reliability of the group decision-making results [54]. If an expert's judgments closely align with the comprehensive results of the group decision, it indicates a higher level of credibility, resulting in a correspondingly higher weight assignment. The credibility of an expert can be evaluated from two perspectives: similarity to the comprehensive results and dissimilarity from other experts. The combination of similarity and dissimilarity ultimately determines the credibility of each judgment matrix. By calculating the credibility for each judgment matrix and utilizing it as the expert's weight, this approach not only reflects each expert's evaluation of various indicators but also enhances the precision and objectivity of the group decision-making process [55]. \begin{table}
\begin{tabular}{l l l} \hline \hline
**Guideline Level B** & & **Indicator Level C** \\ \hline \multirow{4}{*}{Maintenance management (B32)} & \multirow{4}{*}{Sustainability (B324)} & Environmental impact (B3241) & Exhaust emission level (C99) \\ \cline{3-3}  & & Environmental impact (B3241) & Dust control level (C100) \\ \cline{3-3}  & & Noise pollution level (C101) \\ \cline{3-3}  & & & Application of new energy (C102) \\ \cline{3-3}  & & Maintenance process (B3242) & Road surface cleaning (C103) \\ \cline{3-3}  & & The utilization rate of green materials (C104) \\ \hline \hline \end{tabular}
\end{table} 
Table 3: \(Cont.\)Step 1: Calculation of Group Decision-Making Dissimilarity. Use the standard deviation to reflect the diversity of expert judgments, assuming there are x experts, corresponding to \(x\) judgment weight column vectors \(\gamma_{1},\gamma_{2},\ldots,\gamma_{x}\), where \(\gamma_{i}=\gamma_{k1},\ \gamma_{k2},\ldots,\gamma_{kn},\ \gamma_{kn}\) represents the judgment values of the \(k\)-th expert for \(n\) evaluation indicators. Let \(\sigma_{ki}\) be the standard deviation of the judgment made by the \(k\)-th expert for the \(i\)-th evaluation indicator, where \(k\) represents the expert code, and \(i\) represents the evaluation indicator. The definition of standard deviation is given by the following formula:  
\[\sigma_{ki}=\left|\gamma_{ki}-\frac{1}{x}\sum_{k=1}^{x}\gamma_{ki}\right|,k=1,2,\ldots,x,i=1,2,\ldots,n. \tag{5}\]  
Let \(\sigma_{k}\) represent the sum of standard deviations for each evaluation indicator by the \(k\)-th expert. It is defined as follows:  
\[\sigma_{k}=\sum_{i=1}^{n}\sigma_{ki}. \tag{6}\]  
Normalize \(\sigma_{k}\) to obtain the dissimilarity coefficient \(\lambda_{k}\), defined as follows:  
\[\lambda_{k}=\frac{\sigma_{k}}{\sum_{k=1}^{x}\sigma_{k}}. \tag{7}\]  
Step 2: Calculation of Group Decision-Making Similarity. The calculation of the similarity coefficient involves utilizing the spatial relationship between vectors to reflect the similarity of expert judgments. Specifically, the cosine of the angle between two vectors is utilized to represent their similarity. Each expert's weight vector acquired from the judgment matrix can be regarded as a column vector. With multiple experts, there are multiple column vectors, and the size of the spatial angle between any two vectors reflects the similarity between their respective judgment matrices. When two vectors are more similar, the angle between them is closer to \(0^{\circ}\), and the cosine of the angle approaches 1; conversely, as the angle approaches \(90^{\circ}\), the cosine of the angle approaches 0. The judgment weights obtained through the AHP are column vectors, denoted as \(\alpha\) and \(\beta\), where \(\alpha=\left(a_{1},a_{2},\ldots,a_{n}\right)^{T}\) and \(\beta=\left(b_{1},b_{2},\ldots,b_{n}\right)^{T}\).