# An Improved Genetic Algorithm for Solving the Semi-Soft Clustered Vehicle Routing Problem
## 5 Results and Discussion
### Computational Experiments
#### 5.3.5 Sensitivity Analysis and Management Insights  
To analyze the impact of the hard-soft attributes of clusters on the experimental results, we conducted a sensitivity analysis on the proportion of hard-soft clusters among all clusters. Let \(\rho\) denote the percentage of soft clusters among all clusters. The values of \(\rho\) are set to 0%, 25%, 50%, 75%, and 100%, respectively. It is noted that when \(\rho\) takes the values of 0% and 100%, the corresponding instances degenerate into CluVRP and SoftCluVRP instances, respectively. We selected one test instance of each type A and B for experimentation. The experimental results are shown in Figure 3.  
From the results of the above two instances, it can be seen that for the Type A instance, when \(\rho\) = 0%, the objective function value was 712. At this time, since the number of soft clusters was 0, the constraint conditions were relatively strict, resulting in a higher objective value. When \(\rho\) increased to 25%, the objective value dropped to 701. This indicates that moderately increasing the proportion of soft clusters can reduce the objective value. When \(\rho\) was further increased to 50%, the objective value remained at 701. This shows that at this stage, the further increase in the proportion of soft clusters had little effect on objective value reduction. When \(\rho\) reached 75%, the objective value was further reduced to 669, achieving a significant objective value improvement. When \(\rho\) = 100%, the objective value was 652 and continued to decrease. This indicates that when all clusters were soft, the route delivery scheme obtained was the most efficient. For the Type B instance, when \(\rho\) ranged from 0% to 50%, the objective value remained at 476 all the time. This may be due to the combined influence of factors such as customer distribution in this instance. When \(\rho\) increased to 75%, the objective value dropped to 471. This shows that at this point, the increase in the proportion of soft clusters began to have an impact on the cost. When \(\rho\) = 100%, the objective value was 469, and the objective value continued to decrease as well.  
By synthesizing the experimental results of the above two instances, it can be seen that as \(\rho\) gradually increases, the objective value generally shows a downward trend, and the extent of the decrease is related to the type of the instance. In Figure 4, we present the representation of the solutions of the instance A-n45-C15-V3-Q100 when \(\rho\) = 0%, 75%, and 100%. Here, the red solid circle represents the depot, while other solid circles of the same color represent customers from the same cluster, and the solid lines of the same color constitute the route of the same vehicle. The figure more clearly illustrates the changes in vehicle routes in the corresponding solutions after the clusters are relaxed from hard to soft.  
From the above experimental results, we can offer some managerial suggestions to provide decision-making references for managers of relevant enterprises. Since in most cases the logistics costs of enterprises can be significantly reduced when the proportion of soft clusters among all clusters is relatively high, relaxing some hard clusters into soft clusters can bring considerable economic benefits to enterprises. To relax hard clusters into soft clusters, we believe that two aspects should be considered. On the one hand, a thorough investigation of all clusters should be conducted. Based on the actual conditions of these clusters, they should initially be classified into hard clusters and soft clusters. On the other hand, through computational experiments, several key hard clusters that can significantly reduce the enterprise's logistics costs should be identified. Subsequently, measures such as cooperation or compensation should be taken to relax these key hard clusters into soft clusters as much as possible, thereby minimizing the enterprise's logistics costs.  
Figure 3: The objective function values of the test instances under different values of \(\rho\).