# An Improved Genetic Algorithm for Solving the Semi-Soft Clustered Vehicle Routing Problem
## 6 Conclusions  
There have been many studies on the CluVRP and the SoftCluVRP in the literature. In this paper, we considered a problem that falls between these two problems regarding the assumptions of customers, namely the SemiSoftCluVRP. In the SemiSoftCluVRP, all customers are divided into clusters, and all clusters are further partitioned into hard clusters and soft clusters. Customers within the hard cluster require uninterrupted service from the same vehicle, while customers within the soft cluster only need to be served by the same vehicle. For the SemiSoftCluVRP, a two-level genetic algorithm that integrates a VND method was proposed.  
To verify the effectiveness of the algorithm proposed in this paper, we applied it to solve the CluVRP and SoftCluVRP instances from the literature, which are special cases of the SemiSoftCluVRP instances. From the results obtained, it can be seen that in 77 test instances, the proposed algorithm can achieve more than 94% of the literature results, while the four instances that did not achieve the literature results had differences of no more than 6.84% and 9.79%, respectively, which to some extent demonstrates the effectiveness of the algorithm proposed in this paper. To analyze the impact of the hard-soft attributes of clusters on the experimental results, we compared the results of SemiSoftCluVRP with those of CluVRP and SoftCluVRP. From the results obtained, it can be observed that compared to the results of CluVRP, the results of SemiSoftCluVRP can improve logistics costs by up to 6.50%; compared to the results of SoftCluVRP, the results of SemiSoftCluVRP can be increased by up to 7.52%. Therefore, in practical applications, decision-makers can provide more flexible operational options and richer decision-making references for enterprises by setting the hard-soft attributes of clusters, which can ultimately meet the actual needs of customers while reducing the logistics costs of enterprises.  
However, from the results of CluVRP and SoftCluVRP, it can be seen that there is still room for further improvement in the performance of the proposed algorithm when solving large-scale instances. In future research, firstly, we will further design high-performance solution methods tailored for the SemiSoftCluVRP, such as exploiting novel algorithmic operators and developing new hybrid algorithms. Secondly, we will explore its related variants, such as multi-depot variants, multi-objective variants that incorporate additional objectives, etc., in order to better simulate a wider range of practical problems. Thirdly, we will conduct surveys of real enterprises and solve their real-world cases to address their genuine practical challenges.  
Conceptualization, X.B.; methodology, [PERSON] and X.B.; software, Y.M.; validation, [PERSON] and X.B.; formal analysis, [PERSON] and X.B.; investigation, Y.M.; resources, X.B.; data curation, Y.M.; writing--original draft, [PERSON]; writing--review and editing, [PERSON] and X.B.; visualization, Y.M.; supervision, X.B.; project administration, X.B.; funding acquisition, X.B. All authors have read and agreed to the published version of the manuscript.  
This work was supported by the National Natural Science Foundation of China under grant number 11701363.  
**Institutional Review Board Statement:** Not applicable.  
Figure 4: Representation of the solutions of instance A-n45-C15-V3-Q100.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Index** & **Instances** & **Hard Clusters** & **Soft Clusters** \\ \hline  
12] & [2, 20, 17, 23, 25, 13, 12, 25, 13, 12] \\
74 & P-n101-C34-V2-Q400 & [27, 22, 8, 23, 14, 20, 18, 15, 29] & [12, 3, 6, 14, 5, 10, 9, 17, 7, 18, 14, 10, 17, 5] \\
75 & M-n