# Multi-Class Freeway Congestion and Emission Based on Robust Dynamic Multi-Objective Optimization
## 5 Simulation
### DFCM Clustering Analysis  
Fuzzy c-means clustering is an unsupervised clustering method, so the following two indicators can be considered in judging its clustering effect:  
1. The silhouette coefficient (SC) [28] is a way to evaluate the clustering effect, which can evaluate clustering results for different algorithms based on the same original data. The SC of clustering results is in the range (\(-1\), \(1\)). The greater the value, the closer the same samples are and the further the different samples are, therefore leading to a better clustering effect. 2. The Davies-Bouldin Index (DBI) [29], also known as the classification and suitability indicator, was proposed by [PERSON] and [PERSON] to evaluate the effect of the clustering algorithms. This index measures the mean value of maximum similarity of each cluster. A smaller DBI value represents that the clustering results are close within the cluster interior, and the different clusters are farther separated; that is, the smaller the distance is within the class, and the greater the distance is between the classes. The two indicators are calculated as follows:  
\[SC=\frac{b(i)-a(i)}{\max\{a(i),b(i)\}}, \tag{9}\]  
where \(a(i)\) denotes the average distance from sample \(i\) to other samples in the same cluster. The smaller it is, the more sample \(i\) should be clustered into this cluster. It can be regarded as the dissimilarity of sample \(i\) within the cluster. Similarly, \(b(i)\) denotes the average distance from sample \(i\) to all samples in other cluster centers \(\left\{c_{j}\right\}\), which can be described as the dissimilarity between sample \(i\) and other cluster centers \(\left\{c_{j}\right\}\). \[DBI=\frac{1}{N}\sum_{i=1}^{N}\underset{j\
eq 1}{\max}\Bigg{(}\frac{\overline{S_{i }}+\overline{S_{j}}}{\left\|w_{i}-w_{j}\right\|_{2}}\Bigg{)}, \tag{10}\]  
where \(N\) denotes the number of clusters. Variables \(i\) and \(j\) denote the cluster class \(i\) and cluster class \(j\). \(\overline{S_{i}}\) and \(\overline{S_{j}}\) indicate the average distance from the data of cluster class \(i\) and cluster class \(j\) to the cluster center, respectively, which represent the dispersion degree of samples in cluster class \(i\) and cluster class \(j\). \(\left\|w_{i}-w_{j}\right\|_{2}\) denotes the distance between cluster class \(i\) and cluster class \(j\). In this paper, DFCM-RDNSGA-III, a robust dynamic nondominated sorting genetic algorithm based on density fuzzy c-means clustering is proposed. The whole generated population in the iterative process of the algorithm is used as the test set. The clustering effect of the DFCM algorithm is compared with that of the basic fuzzy c-means clustering algorithm. The number of samples in the data set is \(100\), the number of iterations is \(100\), and the data dimension is \(3\). The initial number of cluster centers is set according to the  
Figure 5: Schematic diagram of traffic control strategy implementation. number of cluster centers generated by density clustering. The two algorithms are clustered 10 times, where EX is the mean of 10 times, DX is the variance of 10 times, DFCM denotes the density fuzzy c-means clustering algorithm, and FCM denotes the basic fuzzy c-means clustering algorithm. The comparison of indicators is shown in Table 4.  
It can be seen from Table 4 that the density fuzzy c-means clustering algorithm DFCM has a better clustering effect than the basic fuzzy c-means clustering algorithm FCM in terms of SC and DBI among the 10 clustering results. From the 10 running results, the average score of the DFCM algorithm in SC increased by 14%, which shows that the distance between different samples in DFCM is further and the clustering effect is better. Compared with FCM, DFCM improves 8% on DBI, which shows that DFCM has a closer distance within clusters and further distance between clusters; that is, DFCM generates a better clustering effect. Secondly, DFCM is relatively stable.