# 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 10: Associate the solutions in the solution set \(S_{t}\) with the reference points. Step 11: Calculate the niche count of each reference point \(j\) in the reference point set \(Z^{r}\), select elements according to the niche count to construct the population \(P_{t+1}\), and select \(K\) solutions from \(F_{l}\) to join the population \(P_{t+1}\). Step 12: Reserve the existing population, randomly take out half of the solutions and store them in the archive set \(D_{t+1}\). Let \(gen=gen+1\); if \(gen<Gen_{max}\), go to step 4. if \(gen=Gen_{max}\), output the final solution set \(P_{find}\), and go to step 13. Step 13: Save the current environment information and the final optimal solution \(P_{final}\) to the optimal solution set \(bestpop\), and add it to \(bestpop_{final}(t)\). Let \(t=t+1\); if \(t<t_{max}\), go to step 2. If \(t=t_{max}\), output the final optimal solution set. The Robust Dynamic Nondominated Sorting Multi-Objective Genetic Algorithm Based on Density Fuzzy C-Means Method DFCM-RDNSGA-III  
In the paper [24], it is pointed out that the main challenge of solving the Pareto front distribution problem is the unknown distribution of Pareto optimal fronts in multi-objective optimization algorithms. The paper [25] pointed out that clustering algorithms can help multi-objective optimization algorithms detect the distribution of solutions in the objective space and guide the search to the Pareto fronts. However, the randomness of initial cluster generation and the difficulty of determining the number of initial cluster centers occur when using fuzzy c-means clustering algorithm. Therefore, a density-based clustering method (DBSCAN) [26] is adopted in this paper to conduct density clustering of the initial cluster centers when initializing the cluster centers. The reasons to adopt this method can be explained from three aspects. Firstly, the spatial characteristics of data points can be reflected, and the rationality of cluster center distribution can be ensured when using density clustering, therefore improving the quality of the initial cluster centers of the fuzzy c-means clustering algorithm. Secondly, the number of cluster centers does not need to be set in advance because cluster centers can be generated automatically according to the specific density distribution of the population, which can solve the problem of setting the number of cluster centers by trial and error in fuzzy c-means clustering algorithm. In addition, through the fuzzy c-means clustering algorithm, the given initial cluster center set is continuously optimized, and the position of the cluster center is adjusted; therefore the shape information of more real Pareto fronts can be obtained. Based on the RDNSGA-III algorithm in Section 4.1 and the DFCM algorithm proposed in paper [26], a robust dynamic nondominated sorting multi-objective genetic algorithm based on density fuzzy c-means method DFCM-RDNSGA-III is proposed in this section. The major improvement of this algorithm is that, in the following description of the DFCM-RDNSGA-III algorithm, specifically in Steps 9 and 10, the difficulty of determining Pareto fronts during the algorithm optimization process is solved based on density fuzzy c-means method DFCM. Specific distribution of the Pareto fronts can be acquired through adopting this clustering method. By describing the Pareto fronts through cluster centers, the distribution of the solutions over the objective space is detected, and the Pareto fronts can be tracked continuously [27]. Therefore, the search direction can be guided towards the Pareto fronts, leading to an increase in population diversity and convergence speed. Additionally, the details of the improvement are presented in sub-algorithm 1 and sub-algorithm 2 in this section. The flowchart of the proposed algorithm DFCM-NSGA-III is given in Figure 1. The framework of the proposed algorithm DFCM-NSGA-III will be described first, then sub-algorithm 1 and sub-algorithm 2 used in the proposed algorithm will be discussed later.