Particle swarm optimization (PSO) has attracted many researchers interested in dealing with various optimization problems, owing to its easy implementation, few tuned parameters, and acceptable performance. However, the algorithm is easy to trap in the local optima because of rapid losing of the population diversity. Therefore, improving the performance of PSO and decreasing the dependence on parameters are two important research hot points. In this paper, we present a human behaviorbased PSO, which is called HPSO. There are two remarkable differences between PSO and HPSO. First, the global worst particle was introduced into the velocity equation of PSO, which is endowed with random weight which obeys the standard normal distribution; this strategy is conducive to trade off exploration and exploitation ability of PSO. Second, we eliminate the two acceleration coefficients
Particle swarm optimization (PSO) [
Researchers have proposed various modified versions of PSO to improve its performance; however, there still are premature or lower convergence rate problems. In the PSO research, how to increase population diversity to enhance the precision of solutions and how to speed up convergence rate with least computation cost are two vital issues. Generally speaking, there are four strategies to fulfill these targets as follows.
(1) Tuning control parameters. As for inertial weight, linearly decreasing inertial weight [
(2) Hybrid PSO, which hybridizes other heuristic operators to increase population diversity. The genetic operators have been hybridized with PSO, such as selection operator [
(3) Changing the topological structure. The global and local versions of PSO are the main type of swarm topologies. The global version converges fast with the disadvantage of trapping in local optima, while the local version can obtain a better solution with slower convergence [
(4) Eliminating the velocity formula. Kennedy proposed the barebones PSO (BPSO) [
In recent years, some modified PSO have extremely enhanced the performance of PSO. For example, Zhan et al. proposed adaptive PSO (APSO) [
Though all kinds of variants of PSO have enhanced performance of PSO, there are still some problems such as hardly implement, new parameters to just, or high computation cost. So it is necessary to investigate how to trade off the exploration and exploitation ability of PSO and reduce the parameters sensitivity of the solved problems and improve the convergence accuracy and speed with the least computation cost and easy implementation. In order to carry out the targets, in this paper, the global worst position (solution) was introduced into the velocity equation of the standard PSO (SPSO), which is called impelled/penalized learning according to the corresponding weight coefficient. Meanwhile, we eliminate the two acceleration coefficients
The remainder of the paper is structured as follows. In Section
The PSO is inspired by the behavior of bird flying or fish schooling; it is firstly introduced by Kennedy and Eberhart in 1995 [
In this section, a modified version of SPSO based on human behavior, which is called HPSO, is proposed to improve the performance of SPSO. In SPSO, all particles only learn from the best particles
In HPSO, we introduce the global worst particle, who is of the worst fitness in the entire population at each iteration. It is denoted as
To simulate human behavior and make full use of the
In SPSO, the cognition and social learning terms move particle
Cognition and social terms in PSO.
Impelled/penalized term in HPSO.
Impelled learning
Penalized learning
To sum up, Figure
Evaluate fitness of all particles in
Update velocity according to (
Update position according to (
HPSO flowchart.
To evaluate the performance of HPSO, 28 minimization benchmark functions are selected [
In the experimental study, we choose 28 minimization benchmark functions, which consist of unimodal, multimodal, rotated, shifted, and shifted rotated functions. Table
Functions’ names, dimensions, ranges, and global optimum values of benchmark functions used in the experiments.
Number  Function name  Dimension ( 




Sphere model 


0 

Schwefel’s problem 2.22 


0 

Schwefel’s problem 1.2 


0 

Schwefel’s problem 2.21 


0 

Step function 


0 

Quartic function, that is, noise 


0 

Rosenbrock’s function 


0 

Schwefel’s function 


0 

Generalized Rastrigin’s function 


0 

Noncontinuous Rastrigin’s function 


0 

Ackley’s function 


0 

Generalized Griewank’s function 


0 

Weierstrass’s function 


0 

Generalized penalized function 


0 

Cosine mixture problem 




Rotated elliptic function 


0 

Rotated Schwefel’s function 


0 

Rotated Ackley’s function 


0 

Rotated Griewank’s function 


0 

Rotated Weierstrass’s function 


0 

Rotated Rastrigin’s function 


0 

Rotated Salomon’s function 


0 

Rotated Rosenbrock’s function 


0 

Shifted Rosenbrock’s function 


390 

Shifted Rastrigin’s function 


−330 

Shifted Schwefel’s problem 2.21 


−450 

Shifted rotated Ackley’s function 


−140 

Shifted rotated Weierstrass’s function 


90 
The performance on the convergence accuracy of HPSO is compared with that of SPSO. The test functions listed in Table
Experimental results obtained by SPSO and HPSO on function from
Fun  Dim  Best  Worst  Meadian  Mean  SD  Significant  


30  SPSO 



666.6686 


HPSO 







50  SPSO 


0.0078 



HPSO 







100  SPSO 






HPSO 










30  SPSO 

30.0018  10.0017  11.3364  10.0777  
HPSO 







50  SPSO  0.0329  70.0010  40.0006  37.3438  15.2918  
HPSO 







100  SPSO  51.0214  181.4054  110.5934  114.3039  29.0723  
HPSO 










30  SPSO 






HPSO 







50  SPSO 






HPSO 







100  SPSO 






HPSO 










30  SPSO  8.6091  21.2711  12.9945  13.3502  3.5341  
HPSO 







50  SPSO  24.2031  39.5127  31.0562  31.1715  4.2886  
HPSO 







100  SPSO  54.1172  75.3686  64.7834  64.2358  4.2202  
HPSO 










30  SPSO 

10001 




HPSO 







50  SPSO 

20004  4.5000 



HPSO 







100  SPSO  127  90040  40068 



HPSO 










30  SPSO  0.0344  18.8556  0.0959  3.5587  5.1400  
HPSO 







50  SPSO  0.0780  72.6594  13.6489  19.6604  19.3860  
HPSO 







100  SPSO  86.7855  381.9209  200.8146  211.9720  88.3159  
HPSO 










30  SPSO 


140.5176 



HPSO  28.6353 






50  SPSO  97.0317 

376.2306 



HPSO 







100  SPSO  706.1328 





HPSO 










30  SPSO 






HPSO 





−  
50  SPSO 






HPSO 





−  
100  SPSO 






HPSO 










30  SPSO  28.7299  160.3815  87.6754  92.5142  32.6994  
HPSO 







50  SPSO  175.2643  351.6480  260.4359  258.0518  48.4078  
HPSO 







100  SPSO  555.8950  993.3887  750.1694  749.1658  749.1658  
HPSO 










30  SPSO  61.4129  221.0445  132.7694  134.5414  33.8073  
HPSO 







50  SPSO  157.1020  440.0897  324.2632  310.3595  64.3675  
HPSO 







100  SPSO  623.5658 

804.6981  813.3435  88.5932  
HPSO 






Experimental results obtained by SPSO and HPSO on functions from
Fun  Dim  Best  Worst  Median  Mean  SD  Significant  


30  SPSO  0.0043  19.9630  0.0595  2.3935  5.4041  
HPSO 







50  SPSO  0.0598  19.9646  12.6912  10.5673  6.3042  
HPSO 







100  SPSO  15.4237  20.2143  19.5200  19.4135  0.8672  
HPSO 










30  SPSO 

90.8935  0.0178  12.0794  31.2763  
HPSO 







50  SPSO  0.0014  270.8170  0.0415  45.1971  70.1274  
HPSO 







100  SPSO  1.1140  721.0594  361.0858  376.1758  158.6584  
HPSO 










30  SPSO  0.1403  4.3952  0.3210  1.0567  1.4863  
HPSO 







50  SPSO  0.8657  15.2389  7.5828  8.2388  3.6607  
HPSO 







100  SPSO  27.6235  64.4826  49.3984  47.7138  10.0126  
HPSO 










30  SPSO 

2.2031  0.4202  0.5373  0.5730  
HPSO  0.0710  0.2803  0.1301  0.1444  0.0513  +  
50  SPSO  0.1882  6.9784  2.2774  2.3889  1.5688  
HPSO  0.1016  0.3137  0.1652  0.1702  0.0438  +  
100  SPSO  32.5063 

457.9143 



HPSO  0.1866  0.5097  0.2703  0.2736  0.0653  +  



30  SPSO  −3.0000  −2.8522  −3.0000  −2.9507  0.0709  
HPSO 







50  SPSO  −5.0000  −2.3044  −4.4827  −4.2127  0.6865  
HPSO 







100  SPSO  −7.9165  4.7637  −5.2127  −4.6977  2.8465  
HPSO 










30  SPSO 






HPSO  0 

0  390.6710 

+  
50  SPSO 






HPSO  0 

0  224.6749  873.6249  +  
100  SPSO 






HPSO  0 

0 


+  



30  SPSO 




739.7223  
HPSO 




442.4330  −  
50  SPSO 






HPSO 




669.3538  −  
100  SPSO 






HPSO 





−  



30  SPSO  20.7888  21.0951  21.0053  20.9848  0.0712  
HPSO 

21.1210  20.9931  11.2354  10.6894  +  
50  SPSO  21.0515  21.2478  21.1455  21.1436  0.0536  
HPSO 

21.2404  21.1366  12.0016  10.6745  +  
100  SPSO  21.2367  21.3931  21.3368  21.3358  0.0364  
HPSO 

21.3949  21.3545  15.6658  9.6084  +  



30  SPSO  1.0517  495.3131  273.6408  243.6176  154.3551  
HPSO 







50  SPSO  265.0558 

798.8065  786.0782  289.8401  
HPSO 







100  SPSO 




543.9053  
HPSO 










30  SPSO  22.5705  34.8494  28.6842  28.8734  3.5028  
HPSO  0  39.9834  0  3.1393  9.7817  +  
50  SPSO  45.9462  70.7399  55.5532  55.6014  5.7839  
HPSO  0  66.4051  0  2.2135  12.1239  +  
100  SPSO  106.4483  139.8394  120.6118  121.4481  7.8030  
HPSO  0  129.4941  0  8.3487  31.7918  + 
Experimental results obtained by SPSO and HPSO on functions from
Fun  Dim  Best  Worst  Median  Mean  SD  Significant  


30  SPSO  67.1541  307.3070  213.8939  203.8842  61.8125  
HPSO 







50  SPSO  158.2955  715.0245  518.1705  500.5593  135.5998  
HPSO  0  269.3463  0  8.9782  49.1757  +  
100  SPSO 






HPSO  0  582.0882  0  35.5882  136.0270  +  



30  SPSO  0.7999  14.9999  1.2522  2.9025  4.3553  
HPSO 







50  SPSO  2.0999  26.0999  13.9628  12.8291  6.9033  
HPSO 







100  SPSO  16.5013  41.9999  35.4551  33.9791  6.3075  
HPSO 










30  SPSO  81.0577 





HPSO  28.8214  28.9856  28.9323  28.9252  0.0421  +  
50  SPSO 






HPSO  48.7069  48.8900  48.8205  48.8139  0.0479  +  
100  SPSO 






HPSO  98.6590  98.8846  98.8109  98.7983  0.0545  +  



30  SPSO 






HPSO 





+  
50  SPSO 






HPSO 





+  
100  SPSO 






HPSO 





+  



30  SPSO  −229.5551  −78.6646  −176.9746  −174.7148  35.8633  
HPSO  −204.3636  −100.1465  −148.1389  −149.7299  27.1636  −  
50  SPSO  −77.4305  156.8323  22.8512  24.6168  62.2086  
HPSO  −102.9219  132.8077  −16.6107  −4.1921  58.2317  +  
100  SPSO  475.3838  860.0386  612.6947  632.8693  100.6069  
HPSO  394.3532  805.2473  581.1779  590.3932  80.6175  +  



30  SPSO  −425.5452  −331.1195  −385.1191  −387.6682  22.2647  
HPSO  −439.6877  −399.0205  −423.4928  −422.5533  11.3496  +  
50  SPSO  −399.6029  −326.6739  −379.4869  −370.8387  18.7600  
HPSO  −415.6822  −391.7124  −401.4635  −400.8395  6.5162  +  
100  SPSO  −358.3688  −300.6930  −322.8060  −324.4641  15.5861  
HPSO  −380.3478  −360.8031  −369.0319  −370.4683  5.1369  +  



30  SPSO  −119.2212  −118.8710  −119.0179  −119.0258  0.0866  
HPSO  −119.1100  −118.8700  −118.9469  −118.9589  0.0545  −  
50  SPSO  −119.0222  −118.7656  −118.8316  −118.8535  0.0603  
HPSO  −118.9117  −118.7327  −118.7780  −118.7911  0.0421  −  
100  SPSO  −118.7259  −118.6013  −118.6485  −118.6537  0.0310  
HPSO  −118.6872  −118.5986  −118.6231  −118.6289  0.0204  −  



30  SPSO  113.2663  126.0977  118.5782  119.4693  3.6330  
HPSO  114.4722  132.2305  124.3094  124.5205  4.3399  −  
50  SPSO  137.8303  153.5400  145.1433  145.1503  4.2018  
HPSO  141.9493  162.4008  153.9547  153.1087  5.4273  −  
100  SPSO  194.1222  232.4306  215.9257  215.9174  8.6772  
HPSO  212.5258  245.0126  229.4886  230.4426  7.4650  − 
From Tables
In the 9th columns of Tables
Figure
Convergence comparison of HPSO and SPSO on the selected test functions with
In this section, a comparison of HPSO with some wellknown PSO algorithms which are listed in Table
Some wellknown PSOs algorithms in the literature.
Algorithm  Year  Topology  Parameter settings 

GPSO  1998  Global star 

LPSO  2002  Local ring 

FIPS  2004  Local Uring 

HPSOTVAC  2004  Global star 

UPSO  2004  Global star 

DMSPSO  2005  Dynamic multiswarm 

VPSO  2006  Local Von Neumann 

CLPSO  2006  Comprehensive learning 

QIPSO  2007  Global star 

APSO  2009  Global star 

AFPSO  2011  Global star 

AFPSOQI  2011  Global star 

At first, we choose 10 unimodal and multimodal test functions for this evaluation. According to [
Comparison results of eight PSO algorithms [
Function  GPSO  LPSO  VPSO  FIPS  HPSOTVAC  DMSPSO  CLPSO  APSO  HPSO 



Mean 









SD 









Rank  4  8  6  7  5  3  9  2 



Mean 









SD 









Rank  3  7  5  8  6  4  9  2 



Mean 

18.60  1.44  0.77 

47.5  395 

167 
SD 

30.71  1.55  0.86 

56.4  142 

913 
Rank  3  6  5  4  2  7  9 

8 


Mean 









SD 









Rank 











Mean 









SD 









Rank  5  8  6  2  9  7  3  4 



Mean  30.7  34.90  34.09  29.98  2.39  28.1 



SD  8.68  7.25  8.07  10.92  3.71  6.42 



Rank  7  9  8  6  4  5  3  2 



Mean  15.5  30.40  21.33  35.91  1.83  32.8  0.167 


SD  7.4  9.23  9.46  9.49  2.65  6.49  0.379 


Rank  5  7  6  9  4  8  3  2 



Mean 









SD 









Rank  5  7  6  2  9  3  8  4 



Mean 









SD 









Rank  9  5  6  3  4  7  2  8 



Mean 









SD 









Rank  8  4  7  2  5 

6  3  9 


Average rank  5  6.2  5.6  4.4  4.9  4.6  5.3  2.9 

Final rank  6  9  8  3  5  4  7  2 

Then, in the next step, we choose six functions from [
Comparison results of seven PSO algorithms [
Function  SPSO  QIPSO  UPSO  FIPS  CLPSO  AFSO  AFSOQ1  HPSO 



Mean  52.30  25.61  59.40  106.1  74.39  17.93  15.69 

SD  27.35  15.98  58.05  30.54  9.77  5.63  4.47 

Rank  5  4  6  8  7  3  2 



Mean  0.534  36.38  8.70  6.40 




SD  1.74  4.66  3.08  3.04 




Rank  5  8  7  6  2  4  3 



Mean  320.2  317.5  309.5  434.1  263.3  266.3  253.3 

SD  14.70  23.24  25.88  34.99  11.96  12.00  12.63 

Rank  7  6  5  8  3  4  2 



Mean  17.03  15.20  14.29  26.60  11.94  10.38  8.46 

SD  2.55  1.32  2.15  1.42  1.37  1.38  0.948 

Rank  7  6  5  8  4  3  2 



Mean  −119.10  −119.10  −119.10  −119.90  −119.00  −119.70  −119.80  −119.05 
SD 








Rank  4  4  4  1  6  3  2  5 


Mean  115.90  121.90 

113.60  118.30  123.20  123.10  117.32 
SD  2.90  4.90  6.14  3.63  2.40 

3.01  3.65 
Rank  3  6 



8  7  4 


Average rank  5.17  5.67  4.67  5.50  4.50  4.17  3.00 

Final rank  6  8  5  7  4  3  2 

Therefore, it is worth saying that the proposed algorithm has considerably better performance than the other wellknown PSO algorithms in unimodal and multimodal highdimensional functions.
In this paper, a modified version of PSO called HPSO has been introduced to enhance the performance of SPSO. To simulate the human behavior, the global worst particle was introduced into the velocity equation of SPSO, and the learning coefficient which obeys the standard normal distribution can balance the exploration and exploitation abilities by changing the flying direction of particles. When the coefficient is positive, it is called impelled leaning coefficient, which is helpful to enhance the exploration ability. When the coefficient is negative, it is called penalized learning coefficient, which is beneficial for improving the exploitation ability. At the same time, the acceleration coefficients
The authors declare that there is no conflict of interests regarding the publication of this paper.
The project is supported by the National Natural Science Foundation of China (Grant no. 61175127) and the Science and Technology Project of Department of Education of Jiangxi Province China (Grant no. GJJ12093).