## 4 Creation of Two-Head Self-Adaptive Metaheuristic Algorithms
In this paper, we focused on validation of the effectiveness of creating new self-adaptive differential evolution algorithms based on the old saying \"Two heads are better than one\" to solve the DGRP. Each new self-adaptive differential evolution algorithm developed in this paper is a two-head self-adaptive algorithm created based on selecting two different mutation strategies from a set of four mutation strategies. A total of six two-head self-adaptive algorithms were created. To validate whether the old saying holds for the six self-adaptive algorithms created, we compared the results obtained by applying each of the six two-head self-adaptive algorithms with the corresponding two single-head algorithms and PSO algorithm to solve several test cases. The results indicate that each two-head self-adaptive algorithm outperforms the corresponding two single-head algorithms and PSO algorithm in solving most test cases. One new finding is that the two-head self-adaptive algorithms created based on the saying are more scalable and robust than the corresponding single-head DE algorithms and \(PSO\) algorithm for problem instances with a number of decision variables no greater than 100, the problem size used in the CEC 2017 Competition for Large-Scale Global Optimization (LSGO). Another finding of this study is that the old saying that works effectively for human beings to solve problems also works effectively in the creation of new self-adaptive algorithms to solve the DGRP. In one of our previous papers, we demonstrated the validity of applying the saying \"Two heads are better than one\" to develop two-head self-adaptive algorithms to solve another instance of the ridesharing problem for trust-based shared mobility systems. The results of our previous paper and current paper are encouraging as we have demonstrated that applying the above saying to develop two-head self-adaptive algorithms to solve the ridesharing problem in trust-based shared mobility systems and the DGRP is effective. The results of this study have limitations. The strategies used in this study are mutation strategies in DE. Whether the results still hold for strategies of different metaheuristic algorithms requires further study. Although the two-head self-adaptive algorithms created in this study can work effectively for instances of problems with dimensions no greater than 100, finding the largest DGRP instance that can be solved effectively by the two-head self-adaptive algorithms and developing a method to increase the scalability of the proposed algorithms to tackle larger instances of the DGRP are challenging future research directions. Another future research direction is to study whether the way to develop two-head self-adaptive algorithms in this study is effective for other problems. Although the two-headed algorithms are proposed for single-objective optimization problems, they can be applied to solve multi-objective optimization problems based on the weighted sum method for multi-objective optimization. The other interesting future research direction is to study the effectiveness of applying the two-headed algorithms to ridesharing problems with multi-objective functions. This research was supported in part by the National Science and Technology Council, Taiwan, under Grant NSTC 111-2410-H-324-003. Not applicable. The original data presented in the study are available in [[https://drive.google.com/drive/folders/1](https://drive.google.com/drive/folders/1) COGSGVg09 bjiqpUdY29 bxptyO6 LlzXKd?usp=sharing]([https://drive.google.com/drive/folders/1](https://drive.google.com/drive/folders/1) COGSGVg09 bjiqpUdY29 bxptyO6 LlzXKd?usp=sharing). The author declares no conflicts of interest. ## References

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