## 6 Discussion  
The results presented in the previous section showed that because the minimal expected cost savings discount parameter was considered in the decision model, it could identify a set of ridesharing participants for whom the minimal cost savings discount was guaranteed to improve user satisfaction with the ridesharing service in terms of cost savings. For example, the small example of Case 1 indicated that the minimal expected cost savings discount could be satisfied for the six ridesharing participants in the solution found by our algorithms. That is, the minimal expected cost savings discount can be guaranteed as long as a solution can be found.  
The results presented in the previous section showed that the discount-guaranteed ridesharing problem could be solved by applying different approaches. In this paper, we limited our scope to two approaches: differential evolution and particle swarm optimization. Seven DE-based algorithms and three variants of the PSO algorithm were considered as the candidate solvers to find the solutions for the discount-guaranteed ridesharing problem. Depending on the approach used, the performance and efficiency varied. The seven DE-based algorithms considered in this study followed the same three-step operation process: mutation, crossover, and selection. The differences between these seven DE algorithms lay in the mutation strategies used. The method for improving the quality of a particle in PSO is different to that in DE. In PSO-based approaches, each particle attempts to improve its  
Figure 7: Average number of generations for \(r_{D}=r_{P}=0.1\) when \(NP=50\).  
solution quality by adjusting its velocity. All three variants of the PSO algorithm considered in this study followed the same operation process to determine personal and global bests and update velocity. The differences between these three variants of the PSO algorithm lay in their methods of updating the velocity. Based on the results of our experiments, we compared the average fitness function values and the efficiency of the ten algorithms.  
In terms of performance, the average fitness function values obtained by the DE-1, DE-3, and NSDE (DE-7) algorithms were either greater than or the same as those of the other seven algorithms for all the test cases, regardless of whether the population size was 30 or 50. This indicated that the DE-1, DE-3, and NSDE (DE-7) algorithms either performed as well as or better than the other four DE algorithms and the three PSO algorithms for all the test cases, regardless of whether the population size was 30 or 50. In terms of computational efficiency, the analysis of the results indicated that the DE-7 (NSDE) algorithm and all the other DE-based algorithms were more efficient than the three variants of the PSO algorithm for all the test cases when the population size was set to 30 or 50. The efficiency of the DE-7 (NSDE) algorithm was comparable to that of the other six DE algorithms. Overall, the DE-1, DE-3, and DE-7(NSDE) algorithms were the three best choices in terms of performance and computational efficiency for the test cases in this study. This indicated that the minimal cost savings discount constraints had a significant influence on the effectiveness of the evolutionary algorithms.