*4.3. Discussion*

As shown in Table 4, the proposed MHTSASM algorithm could reach the best results compared with all others cluster techniques for the German towns dataset. Furthermore, it achieved the best results for all numbers of clusters for the first Bavarian postal zones dataset. TS achieved the best results for all numbers of clusters except cluster five, where it gave a bad result for the second Bavarian postal zones dataset. Cluster three gave bad results for K-M [24], TS, GA, and SA [23]. Cluster four gave better results for K-M, TS, GA, and SA than cluster three. Furthermore, GA gave good results except for cluster three. The results indicated that the proposed MHTSASM algorithm could reach better or very similar solutions to those found using the global optimization methods. Therefore, the proposed MHTSASM algorithm could deeply compute the local minimum in the clustering problem for the objective function.

As shown in Table 5, the results of the Fisher´s iris dataset indicated that the proposed MHTSASM algorithm achieved better performance than the K-M [42] technique for all numbers of clusters, except for K = 2, and better results than J-M+ for K = 2, 3. Hence, the results indicate that the results deviation found using the proposed MHTSASM algorithm from the comprehensive minimum is equal to or less than zero for all K.

As shown in Table 6, the results of numerical experiments indicate that the proposed MHTSASM algorithm performed better than results for KHM [25], PSO, and PSOKHM [26,27] for all datasets.
