# Multi-Class Freeway Congestion and Emission Based on Robust Dynamic Multi-Objective Optimization
## 3 Robust Dynamic Multi-Objective Optimization Problem in Freeway Traffic Congestion and Emission
### Robust Dynamic Nondominated Sorting Multi-Objective Genetic Algorithm RDNSGA-III
Step 2: Generate the initial cluster center set \(\{Center_{1}\}\). Calculate the initial membership degree \(\mu_{ij}\) of all solutions to the initial cluster center set \(\{Center_{1}\}\) in the current situation, \(\mu_{ij}=\frac{1}{\frac{N}{k}\left(\frac{|x_{i}-c_{j}|}{|x_{i}-x_{i}|}\right)^{ \frac{1}{m-1}}}\). \(H\) denotes the number of cluster centers; \(m(m>1)\) denotes the fuzzy allocation matrix index used to control the degree of fuzzy overlap. \(x_{i}\) refers to the \(i\)th solution and \(c_{j}\) denotes the \(j\)th cluster center. Step 3: Calculate the initial adaptiveness function \(J\) of fuzzy c-means clustering algorithm, \(J=\sum_{i=1}^{G}\sum_{j=1}^{H}\mu_{ij}\|x_{i}-c_{j}\|^{2}\), where \(G\) denotes the number of all solutions in the \(F_{l}\) and \(H\) denotes the number of cluster centers. Step 4: If the adaptiveness function \(J\) does not reach the acceptable extent or if the maximum number of iterations is satisfied, go to Step 5; otherwise, end the procedure and output the clustering result. Step 5: Update the cluster centers to form a new cluster center set \(c_{j}=\frac{\sum_{j=1}^{D_{t}}\mu_{ij}x_{i}}{\sum_{j=1}^{D_{t}}\mu_{ij}}\). Step 6: Update the membership degree \(\mu_{ij}\) and adaptiveness function \(J\) according to the new cluster centers and repeat Steps 3-5 until the output conditions are satisfied. Finally, output the final clustering results \(\{Center\}=\{c_{1},c_{2},\ldots,c_{H}\}\), i.e., the reference point set \(Z^{r}\). Step 7: Generate the solution set \(\{xc_{i}\}\) of the corresponding solutions included in the \(i\)th cluster center \(c_{i}\) according to the cluster center set \(\{Center\}\). Define the set of all the cluster centers and the solutions included as \(\{xc\}\). (2) **Sub-algorithm 2**: Solution selection method based on the density fuzzy c-means clustering algorithm DFCM. Sub-algorithm 2 provides the solution selection method based on the density fuzzy c-means clustering algorithm DFCM. The purpose is to select \(K\) solutions from \(F_{l}\), \(K=N-|P_{l+1}|\), and join them to the set \(\{rest\}\) to construct the next generation population. The specific process is described as follows. Step 1: Calculate the distance of all the solutions \(\{x_{1},\ x_{2},\ \ldots,\ x_{G}\}\) in \(F_{l}\) and each reference point in reference point set \(Z^{r}=\{c_{1},\ c_{2},\ldots,\ c_{H}\}\). \(Distance=\|x_{i}-c_{j}\|_{2}\), \(x_{i}\) denotes the \(i\)th solution in \(F_{l}\), while \(c_{j}\) denotes the \(j\)th reference point in the reference point set \(Z^{r}\). Step 2: Initialize the number of all the solutions of \(F_{l}\) distributed by \(c_{j},\ nc_{j}=0\). \(x_{i}\) being dominated by \(c_{j}\) can be denoted as \(x_{i}^{j}=\operatorname{argmin}\bigl{(}Distance\bigl{(}x_{i},\ c_{j}\bigr{)} \bigr{)}\). If \(x_{i}\) is dominated by \(c_{j}\), set \(nc_{j}=nc_{j}+1\). Step 3: For \(j=1:H\) reference points, if \(nc_{j}>0\), select the solution \(x_{i_{min}}^{j}\) among all the solutions that has the closest distance to the reference point, and join it to set \(T_{1}=T_{1}\cup\left\{x_{i_{min}}^{j}\bigl{|}argmin\bigl{(}Distance\bigl{(}x _{i}^{j},\ c_{j}\bigr{)}\bigr{)}\right\}\). If \(nc_{j}\geq 3\), except the solutions in \(T_{1}\), the remained solutions are put into the set \(T_{2}=T_{2}\cup\left\{x_{i}^{j}\bigl{|}x_{i}^{j}\
eq x_{i_{min}}^{j}\right\}\). Step 4: If the number of the solutions in \(T_{1}\) is equal to \(K(K=N-|P_{l+1}|)\), i.e., \(|T_{1}|=K\), then \(\{rest\}=T_{1}\). If the number of solutions in \(T_{2}\) is greater than or equal to \(K\) subtracted by the number of solutions in \(T_{1}\), i.e., \(|T_{2}|>K-|T_{1}|\), then randomly select the remained solutions in \(T_{2}\) and construct the solution set \(\{T_{2}rest\}\), \(\{rest\}=T_{1}\cup\{T_{2}rest\}\).