## 5 Conclusions  
This study investigated the relationship between the tuned values of meta-heuristic control parameters and the performance objective that is used for the tuning process. Using iterated F-race, it was shown that different values for control parameters are optimal for different performance objectives.  
Multi-objective tuning allows for multiple performance objectives to be taken into account when tuning a meta-heuristic. However, with multiple objectives being considered, a single parameter configuration can no longer be considered optimal. Some configurations result in better performance for one objective than another. Multi-objective tuning thus requires the discovery of several Pareto-optimal parameter configurations, each representing different levels of trade-off between competing performance objectives.  
This paper presented M-race, a multi-objective tuning algorithm that discovers Pareto-optimal control parameters using a racing strategy. It uses non-dominated sorting to rank candidate parameter configurations based on their performance on each objective. Results showed that M-race has the ability to discover Pareto-optimal configurations for both a particle swarm optimiser and a differential evolution algorithm on different benchmark optimisation problems. However, the quality of the generated Pareto-optimal fronts is sensitive to the initial sample of candidate configurations. Good coverage of the search space with many candidate configurations is required for decent results.  
Future work should address this sensitivity through improved sampling strategies. A promising direction is the development of an iterative version similar to iterated F-race, where new samples are drawn from an already discovered Pareto-optimal set to further explore the search space. However, the feasibility of such an approach requires careful consideration given the sensitivity of meta-heuristics to parameter variations. Additionally, investigating adaptive sampling methods that adjust coverage based on the emerging Pareto front could enhance the robustness of the tuning process. Future research could also evaluate M-race on other meta-heuristic algorithms beyond PSO and DE to further validate its generalisability across different algorithmic paradigms.  
Figure 9: Hypervolume distribution comparisons across test functions. (**a**) Distributions of hypervolume measures for the Ackley function; (**b**) distributions of hypervolume measures for the Rastrigin function; (**c**) distributions of hypervolume measures for the Rana function; (**d**) distributions of hypervolume measures for the Schwefel function.  
**Author Contributions:** Conceptualisation, C.J. and A.E.; methodology, C.J. and A.E.; software, C.J.; validation, C.J.; formal analysis, C.J.; investigation, C.J. and A.E.; resources, C.J.; data curation, C.J.; writing--original draft preparation, C.J.; writing--review and editing, A.E. and K.A.; visualisation, C.J.; supervision, A.E.; project administration, C.J., A.E. and K.A. All authors have read and agreed to the published version of the manuscript.  
**Funding:** This research received no external funding.  
**Institutional Review Board Statement:** Not applicable.  
**Informed Consent Statement:** Not applicable.  
**Data Availability Statement:** The datasets generated during and/or analysed during the current study are available from the corresponding author upon reasonable request.  
**Conflicts of Interest:** The authors declare no conflicts of interest.