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Mohammed, H.M., Abdul, Z.K., Rashid, T.A., Alsadoon, A. and Bacanin, N. (2021), "A new K-means gray wolf algorithm for engineering problems", World Journal of Engineering. https://doi.org/10.1108/WJE-10-2020-0527

Figure 1 Mohammed, H.M., Abdul, Z.K., Rashid, T.A., Alsadoon, A. and Bacanin, N. (2021), "A new K-means gray wolf algorithm for engineering problems", World Journal of Engineering. https://doi.org/10.1108/WJE-10-2020-0527

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Variable a is the controlling parameter that changes the coefficient A’. This technique helps the omega wolf to decide to chase the prey or run away from it. Therefore, if the value of |A | is greater than 1, it means that the omega wolf tries to discover new search space. However, if the value of |A| is less than 1, the omega wolf would follow and approach the prey. Thus, this mechanism is called local search. When the hunting mechanism is achieved by doing the above producer, the grey wolf begins to avoid any movement of the prey to attack it properly. This mechanism is done by decreasing the value of a which is between 2 and 0. The value of A is also reduced by the value a and it is in the range of [-1,1] [2]. Besides the importance of optimization algorithms, data analysis is considered a great subject that can be worked on by researchers. Therefore, one of the well-known exploratory data analysis techniques is clustering that has been used to get an intuition about the structure of the data. K-means is one of the most popular unsupervised algorithms and the standard K-means algorithm was firstly presented by Stuart Lloyd in 1957 [68]. The algorithm can cluster the data into subgroups. Somehow, data points in each cluster are very similar compared to the data points in different clusters
Variable a is the controlling parameter that changes the coefficient A’. This technique helps the omega wolf to decide to chase the prey or run away from it. Therefore, if the value of |A | is greater than 1, it means that the omega wolf tries to discover new search space. However, if the value of |A| is less than 1, the omega wolf would follow and approach the prey. Thus, this mechanism is called local search. When the hunting mechanism is achieved by doing the above producer, the grey wolf begins to avoid any movement of the prey to attack it properly. This mechanism is done by decreasing the value of a which is between 2 and 0. The value of A is also reduced by the value a and it is in the range of [-1,1] [2]. Besides the importance of optimization algorithms, data analysis is considered a great subject that can be worked on by researchers. Therefore, one of the well-known exploratory data analysis techniques is clustering that has been used to get an intuition about the structure of the data. K-means is one of the most popular unsupervised algorithms and the standard K-means algorithm was firstly presented by Stuart Lloyd in 1957 [68]. The algorithm can cluster the data into subgroups. Somehow, data points in each cluster are very similar compared to the data points in different clusters
Table 1 Comparison of GWO and KMGWO using CEC2019 Benchmark Test Functions. KMGWO also is compared to other common metaheuristic algorithms, such as Cat Swarm Optimizer (CSO), WOA-BAT [73], GWO, and WOA. As it is clear from Table 1, KMGWO is better than GWO. Therefore, it would be great if KMGWO is compared to other common algorithms. Consequently, Table 2 shows that KMGWO and WOA-BAT are competitive because KMGWO performs well in 4 functions out of 10. however, WOA-BAT has the second-best rank due to having the best results in 3 functions out of 10. The third best is CSO [74], which is superior in 2 functions while WOA is efficient in only 1 function. The worst algorithm is GWO which does not have effective resu ts compared to the other four algorithms.
Table 1 Comparison of GWO and KMGWO using CEC2019 Benchmark Test Functions. KMGWO also is compared to other common metaheuristic algorithms, such as Cat Swarm Optimizer (CSO), WOA-BAT [73], GWO, and WOA. As it is clear from Table 1, KMGWO is better than GWO. Therefore, it would be great if KMGWO is compared to other common algorithms. Consequently, Table 2 shows that KMGWO and WOA-BAT are competitive because KMGWO performs well in 4 functions out of 10. however, WOA-BAT has the second-best rank due to having the best results in 3 functions out of 10. The third best is CSO [74], which is superior in 2 functions while WOA is efficient in only 1 function. The worst algorithm is GWO which does not have effective resu ts compared to the other four algorithms.
Table 2 Comparison of CSO, WOA-BAT, GWO, WOA, and KMGWO using CEC2019 Benchmark Test Functions. To evaluate the average results that were achieved in Table 2, differences between each algorithm with KMGWO are calculated to find statistical results. Statistical results indicate that whether the results are significant or not. Table 3 shows that KMGWO has significant function results compared to CSO, WOA- BAT, GWO, and WOA. Overall, it can be said that KMGWO could achieve significant value against the above algorithms because of using clustering methods.
Table 2 Comparison of CSO, WOA-BAT, GWO, WOA, and KMGWO using CEC2019 Benchmark Test Functions. To evaluate the average results that were achieved in Table 2, differences between each algorithm with KMGWO are calculated to find statistical results. Statistical results indicate that whether the results are significant or not. Table 3 shows that KMGWO has significant function results compared to CSO, WOA- BAT, GWO, and WOA. Overall, it can be said that KMGWO could achieve significant value against the above algorithms because of using clustering methods.
Table 3 Statistical Results of KMGWO (P-value ) against CSO, WOA-BAT, GWO, and WOA. Mohammed, H.M., Abdul, Z.K., Rashid, T.A., Alsadoon, A. and Bacanin, N. (2021), "A new K-means gray wolf algorithm for engineering problems", World Journal of Engineering. https://doi.org/10.1108/WJE-10-2020-0527
Table 3 Statistical Results of KMGWO (P-value ) against CSO, WOA-BAT, GWO, and WOA. Mohammed, H.M., Abdul, Z.K., Rashid, T.A., Alsadoon, A. and Bacanin, N. (2021), "A new K-means gray wolf algorithm for engineering problems", World Journal of Engineering. https://doi.org/10.1108/WJE-10-2020-0527
Figure 2 Pressure Vessel Design Problem [78]. Mohammed, H.M., Abdul, Z.K., Rashid, T.A., Alsadoon, A. and Bacanin, N. (2021), "A new K-means gray wolf algorithm for engineering problems", World Journal of Engineering. https://doi.org/10.1108/WJE-10-2020-0527
Figure 2 Pressure Vessel Design Problem [78]. Mohammed, H.M., Abdul, Z.K., Rashid, T.A., Alsadoon, A. and Bacanin, N. (2021), "A new K-means gray wolf algorithm for engineering problems", World Journal of Engineering. https://doi.org/10.1108/WJE-10-2020-0527
Table 4 Comparison WOA, WOAGWO, FDO, and KMGWO for Pressure Vessel Design
Table 4 Comparison WOA, WOAGWO, FDO, and KMGWO for Pressure Vessel Design
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