I Introduction
Many optimisation problems can be formulated as searching for the best element or solution from all feasible solutions with respect to one or several criteria. While conventional optimisation algorithms like gradientbased methods are rooted in a solid mathematical formulation, they typically fail to provide satisfactory results and often stagnate in local optima [1],[2],[3].
Populationbased metaheuristic algorithms are able to address this issue [4]
and can be divided into three general categories, namely evolutionary, swarmbased, and physicsbased approaches. Evolutionary algorithms, such as the genetic algorithm (GA)
[5], are inspired by biological evolutionary processes. Swarmbased algorithms, e.g. particle swarm optimization (PSO)
[6] and artificial bee colony (ABC) [7], imitate the social behaviour of animals, while physicsbased algorithms, including the gravitational search algorithm (GSA) [8] and the mine blast algorithm (MBA) [9], are based on the laws of physics.Human Mental Search (HMS) [10] is a relatively recent populationbased metaheuristic algorithm that is based on the concept of exploring the search space of online auctions. Here, mental search, a main part of the algorithm, employs Levy flight to explore the vicinity of candidate solutions. HMS has been shown to solve effectively a wide range of optimisation problems, including unimodal, multimodal, highdimensional, rotated, shifted, and complex functions [10], as well as various machine vision applications including multilevel thresholding [11, 12], colour quantisation [13, 14], image segmentation [15], and image clustering [16].
HMS algorithm has three main operators, mental search to explore the vicinity of candidate solutions based on a Levy flight distribution, grouping to cluster the current population to find a promising region, and movement to move candidate solutions towards the promising region. Several improvements to HMS have been recently introduced, including leveraging a random clustering strategy [17], and grouping in both search and objective space [18].
In the grouping phase of HMS, a clustering algorithm is employed and the cluster with the best mean objective function value is selected as the winner cluster. However, this approach has a tendency of getting stuck in a local optimum. To address this issue, in this paper, we propose a novel HMS algorithm, MCSHMS, that is based on multicluster selection. Here, the best candidate solutions in each cluster have a chance of being selected. We further employ a more efficient onestep means algorithm for clustering. Extensive experiments on various benchmark functions with different characteristics show that MCSHMS outperforms HMS as well as other populationbased metaheuristic algorithms.
Ii Human Mental Search
HMS is a metaheuristic algorithm inspired by the way the human mind searches. Each candidate solution, called a bid in HMS, is initially randomly generated. HMS then proceeds, iteratively, based on three main operators, mental search, grouping, and movement.
During mental search, a number of new bids are obtained as
(1) 
where is a current bid, and
(2) 
where represents the number of objective function evaluations thus far, is the maximum number of function evaluations, is a random integer number, and is the best bid found so far. and
are two random numbers with normal distributions as
(3) 
and
(4) 
where is a standard gamma function.
The grouping operator uses the means algorithm to cluster similar bids in the population. Then, for each cluster the mean objective function value is calculated and the cluster with the best value is selected as the winner cluster to represent a promising area in search space.
Bids in the other clusters then towards the identified promising area by
(5) 
where is the best solution in the promising area, is a constant, is a random number in , indicates the current iteration, and subscript indicates the th element of a bid.
Iii Proposed MCSHMS Algorithm
One of the drawbacks of the standard HMS algorithm is that selecting the cluster with the best mean objective functions does not necessarily give the best choice for the promising area because the cluster might be placed far away from the global optimum. Therefore, choosing the best cluster and its best bid may limit diversification and may cause the next generations to be misled to a less successful area. Therefore, in this paper, we introduce a novel modification of HMS to tackle this issue and find a better promising area.
Our idea is to select a few promising bids in a memory and use these. Specifically, instead of selecting the best bid in the cluster with the best mean objective function, we consider the best bid of each cluster as illustrated in Fig 1. The memory keeps only the bids with the best objective function value of each cluster, and thus the length of equals the number of clusters.
In the next step, we randomly select a bid from as the target for the movement operator and thus the corresponding cluster as the promising area in search space. In contrast to standard HMS, this mechanism leads to improved exploration since the target is selected from the whole search space, and can also prevent early convergence.
In addition, we address the relative inefficiency of HMS due to its application of means in the grouping process. Inspired by [19], In MCSHMS we replace means with a onestep means algorithm. That is only a single iteration of means is conducted, leading to quicker execution.
Iv Experimental Results
For evaluation, we use the 30 CEC 2017 benchmark functions [20] which include unimodal (F1 to F3), multimodal (F4 to F10), hybrid multimodal (F11 to F20) and composite (F21 to F30) test functions. Besides comparing MCSHMS with standard HMS, we also benchmark it against a number of other populationbased metaheuristic algorithms, namely particle swarm optimisation (PSO) [6], the covariance matrix adaptionevolution strategy (CMAES) [21], artificial bee colony (ABC) [7], the grey wolf optimiser (GWO) [22], moth flame optimisation (MFO) [23], and the whale optimisation algorithm (WOA) [24].
As dimensionality of the search space we use 30, 50 and 100. The employed parameters of the various algorithms are given in Table I.
algorithm  parameter  values 
PSO  inertia constant  1 to 0 
cognitive constant  2  
social constant  2  
CMAES  50  
ABC  limit  × 
GWO  no parameters  
MFO  no parameters  
WOA  1  
HMS  number of clusters  5 
minimum mental processes  2  
maximum mental processes  5  
1  
MCSHMS  same settings as HMS 
PSO  CMAES  ABC  GWO  MFO  WOA  HMS  MCSHMS  

F1  7.16E+07  2.11E+10  2.91E+07  1.75E+09  8.80E+09  8.43E+07  4.55E+06  1.37E+04 
F2  8.13E+11  3.33E+42  3.93E+41  4.09E+28  1.16E+39  1.70E+32  4.19E+24  6.08E+20 
F3  3.24E+02  2.11E+05  3.68E+05  3.94E+04  1.08E+05  1.99E+05  1.09E+04  1.33E+04 
F4  7.54E+01  3.62E+03  1.19E+02  1.86E+02  6.64E+02  2.02E+02  1.21E+02  9.96E+01 
F5  2.12E+02  3.31E+02  2.39E+02  9.59E+01  1.98E+02  2.78E+02  1.19E+02  9.64E+01 
F6  5.21E+01  6.30E+01  5.20E+00  7.58E+00  3.04E+01  6.82E+01  1.01E+01  3.52E+00 
F7  2.47E+02  1.81E+02  2.78E+02  1.53E+02  3.83E+02  5.06E+02  1.46E+02  1.30E+02 
F8  1.57E+02  2.69E+02  2.47E+02  8.14E+01  1.90E+02  2.31E+02  1.07E+02  9.82E+01 
F9  4.64E+03  1.66E+03  3.08E+03  9.48E+02  5.76E+03  7.99E+03  1.39E+03  7.98E+02 
F10  4.83E+03  7.03E+03  8.17E+03  3.22E+03  4.63E+03  5.46E+03  3.71E+03  3.50E+03 
F11  1.53E+02  1.67E+04  9.40E+03  8.31E+02  3.64E+03  1.98E+03  2.28E+02  1.42E+02 
F12  1.45E+07  4.32E+09  4.97E+08  6.48E+07  2.23E+08  8.52E+07  7.50E+06  2.86E+06 
F13  1.87E+06  3.89E+09  2.57E+06  1.23E+07  4.86E+07  2.07E+05  4.54E+04  3.23E+04 
F14  1.41E+04  5.45E+06  2.98E+05  2.57E+05  1.30E+05  2.39E+06  4.27E+04  3.91E+04 
F15  1.56E+05  4.90E+08  1.40E+06  1.58E+06  4.53E+04  1.26E+05  7.90E+03  5.82E+03 
F16  1.26E+03  3.25E+03  2.34E+03  9.59E+02  1.57E+03  2.06E+03  9.13E+02  1.08E+03 
F17  5.43E+02  1.91E+03  1.14E+03  2.66E+02  7.10E+02  9.24E+02  4.28E+02  3.69E+02 
F18  2.17E+05  3.35E+07  1.46E+07  1.13E+06  3.06E+06  4.17E+06  5.33E+05  4.73E+05 
F19  5.96E+05  4.42E+08  8.37E+04  8.96E+05  9.48E+06  4.38E+06  1.35E+04  1.44E+04 
F20  5.96E+02  8.25E+02  9.94E+02  3.85E+02  6.57E+02  8.42E+02  3.28E+02  3.38E+02 
F21  4.05E+02  5.51E+02  4.46E+02  2.84E+02  3.97E+02  4.79E+02  3.20E+02  3.04E+02 
F22  3.13E+03  7.49E+03  8.25E+03  2.07E+03  4.10E+03  5.14E+03  3.73E+03  2.41E+03 
F23  8.44E+02  7.46E+02  6.07E+02  4.54E+02  5.18E+02  7.37E+02  5.04E+02  4.76E+02 
F24  8.17E+02  7.71E+02  6.83E+02  5.12E+02  5.78E+02  8.13E+02  5.85E+02  5.74E+02 
F25  3.97E+02  1.40E+03  4.20E+02  4.74E+02  8.20E+02  4.96E+02  3.98E+02  3.89E+02 
F26  2.16E+03  5.62E+03  3.15E+03  2.07E+03  3.06E+03  5.21E+03  2.49E+03  2.23E+03 
F27  5.19E+02  6.99E+02  5.00E+02  5.46E+02  5.46E+02  7.24E+02  5.19E+02  5.13E+02 
F28  4.44E+02  3.80E+03  5.00E+02  5.82E+02  1.36E+03  5.73E+02  5.15E+02  4.84E+02 
F29  1.40E+03  2.83E+03  2.02E+03  9.50E+02  1.19E+03  2.17E+03  9.06E+02  8.39E+02 
F30  1.88E+06  4.88E+08  4.99E+05  6.71E+06  1.08E+06  2.04E+07  3.38E+04  1.41E+04 
PSO  CMAES  ABC  GWO  MFO  WOA  HMS  MCSHMS  

F1  2.57E+08  4.64E+10  2.81E+09  5.59E+09  3.75E+10  4.96E+08  2.99E+08  5.03E+05 
F2  2.66E+25  5.07E+79  5.48E+80  7.96E+54  6.94E+76  1.25E+70  1.42E+53  1.22E+45 
F3  9.48E+03  3.75E+05  7.71E+05  9.42E+04  1.67E+05  1.85E+05  3.48E+04  3.71E+04 
F4  1.62E+02  7.61E+03  3.24E+03  7.14E+02  4.06E+03  5.30E+02  3.04E+02  2.20E+02 
F5  3.94E+02  3.08E+02  5.49E+02  2.07E+02  4.62E+02  4.70E+02  2.38E+02  2.27E+02 
F6  6.95E+01  6.80E+01  4.93E+01  1.76E+01  4.78E+01  8.30E+01  1.87E+01  1.16E+01 
F7  5.01E+02  2.20E+02  6.50E+02  3.59E+02  1.13E+03  1.06E+03  3.00E+02  3.31E+02 
F8  4.15E+02  5.17E+02  5.51E+02  2.19E+02  4.79E+02  4.49E+02  2.54E+02  2.38E+02 
F9  2.47E+04  1.35E+04  4.27E+04  8.16E+03  1.71E+04  2.76E+04  6.46E+03  6.91E+03 
F10  8.82E+03  1.32E+04  1.50E+04  6.50E+03  7.47E+03  1.01E+04  7.53E+03  6.73E+03 
F11  3.59E+02  6.45E+04  5.59E+04  2.97E+03  1.28E+04  1.17E+03  8.25E+02  5.58E+02 
F12  1.07E+08  2.24E+10  1.05E+10  5.44E+08  4.63E+09  5.79E+08  1.07E+08  1.94E+07 
F13  1.47E+07  1.29E+10  6.17E+07  1.05E+08  8.80E+08  5.55E+06  1.17E+05  2.51E+04 
F14  1.17E+05  2.17E+07  4.29E+06  9.45E+05  1.08E+06  1.97E+06  2.85E+05  1.68E+05 
F15  3.05E+06  2.51E+09  1.28E+07  1.56E+07  8.65E+06  1.08E+06  3.24E+04  1.43E+04 
F16  2.15E+03  5.37E+03  5.11E+03  1.46E+03  2.73E+03  3.81E+03  2.10E+03  2.05E+03 
F17  1.68E+03  1.05E+03  3.12E+03  1.06E+03  2.30E+03  2.55E+03  1.51E+03  1.39E+03 
F18  1.50E+06  1.27E+08  6.83E+07  5.10E+06  2.96E+06  1.41E+07  1.78E+06  1.29E+06 
F19  2.78E+06  1.10E+09  2.55E+04  3.16E+06  5.93E+07  5.13E+06  3.12E+04  1.44E+04 
F20  1.30E+03  1.62E+03  2.50E+03  8.97E+02  1.40E+03  1.70E+03  1.10E+03  1.08E+03 
F21  6.61E+02  7.99E+02  7.52E+02  3.94E+02  6.41E+02  8.43E+02  4.91E+02  4.71E+02 
F22  9.09E+03  1.43E+04  1.51E+04  6.78E+03  8.28E+03  1.01E+04  7.85E+03  6.99E+03 
F23  1.55E+03  1.14E+03  9.78E+02  6.64E+02  8.41E+02  1.35E+03  8.14E+02  7.58E+02 
F24  1.21E+03  1.14E+03  1.09E+03  7.24E+02  8.16E+02  1.30E+03  8.78E+02  8.30E+02 
F25  5.12E+02  2.72E+03  2.22E+03  1.06E+03  3.08E+03  9.11E+02  6.34E+02  5.65E+02 
F26  6.09E+03  8.76E+03  6.33E+03  3.95E+03  5.82E+03  1.05E+04  5.47E+03  4.60E+03 
F27  1.07E+03  1.12E+03  5.00E+02  8.99E+02  8.75E+02  1.69E+03  7.22E+02  6.95E+02 
F28  4.97E+02  6.77E+03  5.00E+02  1.48E+03  5.11E+03  1.20E+03  7.09E+02  5.87E+02 
F29  2.55E+03  1.04E+04  6.25E+03  1.62E+03  2.31E+03  4.85E+03  1.67E+03  1.31E+03 
F30  5.58E+07  2.43E+09  3.38E+08  1.09E+08  1.29E+08  1.35E+08  2.31E+06  1.51E+06 
PSO  CMAES  ABC  GWO  MFO  WOA  HMS  MCSHMS  

F1  1.07E+09  4.20E+10  3.61E+11  4.25E+10  1.24E+11  1.98E+07  1.51E+10  2.89E+07 
F2  8.45E+71  1.04E+166  2.96E+182  4.30E+133  7.99E+159  1.15E+148  6.53E+134  2.10E+109 
F3  2.36E+05  8.50E+05  2.22E+06  2.45E+05  6.55E+05  7.55E+05  1.14E+05  1.51E+05 
F4  3.87E+02  1.71E+04  1.53E+05  3.36E+03  2.38E+04  6.69E+02  2.01E+03  4.34E+02 
F5  1.09E+03  1.07E+03  2.03E+03  6.41E+02  1.20E+03  9.25E+02  6.78E+02  6.48E+02 
F6  8.60E+01  1.44E+01  1.39E+02  3.65E+01  7.32E+01  7.93E+01  3.32E+01  2.24E+01 
F7  1.25E+03  8.65E+02  9.19E+03  1.25E+03  4.04E+03  2.57E+03  9.67E+02  1.13E+03 
F8  1.20E+03  1.27E+03  2.08E+03  6.33E+02  1.27E+03  1.12E+03  7.14E+02  7.20E+02 
F9  6.32E+04  3.25E+04  1.93E+05  3.27E+04  4.21E+04  3.54E+04  2.22E+04  2.05E+04 
F10  2.20E+04  3.04E+04  3.26E+04  1.57E+04  1.68E+04  1.95E+04  1.79E+04  1.60E+04 
F11  2.44E+03  4.51E+05  8.88E+05  5.33E+04  1.32E+05  1.17E+04  1.35E+04  1.08E+04 
F12  8.27E+08  5.27E+10  1.19E+11  7.40E+09  3.48E+10  7.23E+08  1.52E+09  1.71E+08 
F13  5.08E+07  1.16E+10  3.13E+09  4.15E+08  4.43E+09  6.20E+04  3.78E+07  3.73E+04 
F14  1.57E+06  1.15E+08  8.61E+07  4.95E+06  8.14E+06  1.45E+06  2.07E+06  1.58E+06 
F15  1.52E+07  5.73E+09  5.70E+08  1.02E+08  1.81E+09  6.21E+04  5.00E+05  1.24E+04 
F16  5.85E+03  1.20E+04  1.52E+04  4.42E+03  6.36E+03  8.43E+03  6.06E+03  5.17E+03 
F17  4.32E+03  4.22E+04  2.97E+04  3.54E+03  6.44E+03  5.47E+03  5.25E+03  3.98E+03 
F18  3.18E+06  1.39E+08  2.86E+08  4.83E+06  1.63E+07  1.88E+06  5.67E+06  3.21E+06 
F19  2.67E+07  4.44E+09  7.29E+06  1.15E+08  7.69E+08  1.33E+07  1.54E+06  1.44E+04 
F20  3.77E+03  5.14E+03  6.57E+03  3.28E+03  3.78E+03  4.23E+03  3.65E+03  3.42E+03 
F21  1.69E+03  1.44E+03  2.56E+03  8.69E+02  1.55E+03  1.79E+03  1.10E+03  1.05E+03 
F22  2.44E+04  3.11E+04  3.34E+04  1.74E+04  1.84E+04  2.17E+04  1.99E+04  1.73E+04 
F23  2.91E+03  1.84E+03  3.16E+03  1.23E+03  1.49E+03  2.44E+03  1.41E+03  1.12E+03 
F24  3.04E+03  2.44E+03  5.16E+03  1.75E+03  1.94E+03  3.55E+03  1.95E+03  1.67E+03 
F25  9.45E+02  8.27E+03  8.55E+04  3.42E+03  1.12E+04  1.13E+03  1.59E+03  1.01E+03 
F26  1.71E+04  1.93E+04  4.96E+04  1.22E+04  1.53E+04  2.82E+04  1.58E+04  1.28E+04 
F27  6.43E+02  2.00E+03  5.00E+02  1.33E+03  1.28E+03  2.27E+03  8.75E+02  7.95E+02 
F28  6.38E+02  1.91E+04  5.00E+02  4.73E+03  1.62E+04  9.13E+02  2.37E+03  3.18E+03 
F29  6.94E+03  1.34E+04  6.10E+04  5.28E+03  8.22E+03  1.11E+04  5.04E+03  4.30E+03 
F30  1.07E+08  1.02E+10  3.27E+09  1.03E+09  2.33E+09  1.83E+08  1.02E+07  8.14E+04 
Since the employed methods are stochastic, we run all algorithms 25 times and report the mean over these runs in terms of the difference between the optimal value and that returned by an algorithm. The obtained results are given in Tables II, III, and IV for , and , respectively, and show impressively that MCSHMS not only provides superior performance compared to standard HMS but that it also outperforms all the other algorithms.
Its superiority compared to standard HMS is further illustrated in Figs. 2, 3, and 4 which compared the rankings of HMS and MCSHMS. From there, we can see that for the vast majority of cases, MCSHMS is ranked top or second and that in particular, MCSHMS gives the best result for 24, 25 and 27 functions for , and , respectively.
avg. rank  best rank  worst rank  std.dev.  

PSO  3.73  1  8  1.95 
CMAES  7.40  4  8  1.08 
ABC  5.50  1  8  1.91 
GWO  3.37  1  7  1.92 
MFO  5.30  3  7  1.27 
WOA  6.20  3  8  1.17 
HMS  2.77  1  4  0.92 
MCSHMS  1.73  1  3  0.63 
show, for each algorithm, the average rank, best rank, worst rank and standard deviation over the 30 benchmark functions for
, and , respectively.avg. rank  best rank  worst rank  std.dev.  

PSO  3.83  1  8  2.02 
CMAES  6.70  1  8  1.86 
ABC  6.23  1  8  1.86 
GWO  3.27  1  7  1.95 
MFO  5.30  2  8  1.44 
WOA  5.93  3  8  1.46 
HMS  2.87  1  4  0.67 
MCSHMS  1.87  1  3  0.67 
avg. rank  best rank  worst rank  std.dev.  

PSO  3.90  1  7  1.90 
CMAES  6.23  1  8  1.87 
ABC  7.13  1  8  1.93 
GWO  3.47  1  6  1.84 
MFO  5.60  3  7  1.31 
WOA  4.43  1  8  1.94 
HMS  3.30  1  5  0.94 
MCSHMS  1.93  1  5  0.89 
As can be seen from there, MCSHMS clearly yields the best performance for all dimensionalities, and by a significant margin. Table VIII further illustrates this by a pairwise comparison of MCSHMS with all other methods. It is evident from there that in the large majority of cases, MCSHMS outperforms the other approaches.
MCSHMS vs.  better  worse  better  worse  better  worse 

PSO  23  7  23  7  22  8 
CMAES  30  0  28  2  28  2 
ABC  29  1  28  2  28  2 
GWO  20  10  19  11  22  8 
MFO  30  0  29  1  30  0 
WOA  30  0  30  0  26  4 
HMS  26  4  27  3  26  4 
Last not least, we conduct a twosided Wilcoxon signed rank test to see if the differences between MCSHMS and its opponents are statistically significant and give the results in Table IX. With all value below 0.05, it is apparent that MCSHMS statistically outperforms all other algorithms.
MCSHMS vs. PSO  0.0148  0.0034  0.0132 

MCSHMS vs. CMAES  1.73E06  4.73E06  2.35E06 
MCSHMS vs. ABC  2.13E06  2.88E06  4.29E06 
MCSHMS vs. GWO  0.0082  0.0057  8.31E04 
MCSHMS vs. MFO  1.73E06  1.92E06  1.73E06 
MCSHMS vs. WOA  1.73E06  1.73E06  0.003 
MCSHMS vs. HMS  5.29E04  1.15E04  7.51E05 
V Conclusions
In this paper, we have introduced MCSHMS, an enhanced version of the human mental search (HMS) optimisation algorithm. MCSHMS employs a multicluster strategy in HMS’s grouping phase which allows for improved exploration ability, while also using a onestep means algorithm for clustering to speed up the algorithm’s execution. Based on a set of experiments carried out on the CEC 2017 test functions, we show that MCSHMS not only significantly improves the standard HMS algorithm but that it also outperforms a number of other populationbased metaheuristics.
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