## 5 Conclusions  
In this study, a new HFA-K-means algorithm has been proposed and successfully implemented to cluster high-dimensional datasets automatically. The idea of the CLT was incorporated into the K-means phase of the hybrid algorithm to reduce the number of distance calculations required by the K-means algorithm. A mutation probability was also introduced into the FA phase to improve the exploitation and exploration of the FA. An initial investigative experiment conducted using another seven metaheuristic-based K-means hybrid algorithms modelled using the same framework as the proposed algorithm further confirmed the superior performance of the FA for automatic clustering, thus justifying its selection among other metaheuristic algorithms. The non-parametric mean rank test conducted on the experimental results of the eight metaheuristic-based K-means hybrid algorithms provided a further justification for the selection of FA among the other metaheuristic algorithms. The results obtained from the other computation experiments over the 20 high-dimensional datasets clearly show that the HFA-K-means algorithm demonstrates outstanding performance compared with the two classical algorithms, FA and K-means. Statistical tests were conducted on the simulation results to confirm the performance of the proposed hybrid algorithm, namely, the [PERSON] mean rank test and Wilcoxon rank-sum (a post hoc) test. A comparison of the results with other metaheuristics algorithms from the literature (GA, DE, PSO, and IWO) and other similar hybrid algorithms (SOSK-means, PSODE, FADE, and IWODE) was also performed. The proposed algorithm recorded superior results in 13 (65%) out of the 20 datasets compared with the other metaheuristics and hybrid metaheuristic algorithms. The numerical results were also compared with different FA-based hybrid algorithms from the literature, namely, FAPSO, FADE, and FATLBO, on seven high-dimensional datasets. The results showed that the proposed algorithm had superior performance in five of the seven datasets in terms of the quality of clustering solutions obtained. This indicates that the HFA-K-means performed well in 71% of the total datasets used for the comparison. The computational time for the proposed algorithm was also compared with regular FAK-means, and there was a substantial reduction in the computation time of the proposed HFAK-means over all the 20 high-dimensional datasets. These confirm that the adoption of the CLT in the K-means phase of the proposed algorithm assists in the reduction in the computation time, thereby enhancing the performance of the proposed algorithm in terms of reduced computational cost. The modified CLT-based K-means can be substituted for the standard K-means algorithm in many of the existing applications where it has been used to improve the clustering performance in various domains. For future study, the modified CLT-based K-means can be combined with other metaheuristic algorithms for respective hybridization models. Additionally, a further computation time reduction in the FA phase can be explored by incorporating Levy flights into the algorithm update phase to increase the fireflies' movement, thereby reducing the foraging time of the fireflies. Moreover, further comparative analysis and performance studies involving FA-based hybrid algorithms and K-means-based metaheuristic algorithms can be carried out. Aside these, the proposed algorithm can also be applied to other real-life applications.  
All authors contributed to the conception and design of the research work. Material preparation, experiments, and analysis were performed by A.M.I. and A.E.E. All authors have read and agreed to the published version of the manuscript.  
This research received no external funding.  
Not applicable.  
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this article.