# Human Behavior-Based Particle Swarm Optimization

Authors: Hao Liu; Gang Xu; Gui-yan Ding; Yu-bo Sun
Journal: The Scientific World Journal
 (2014)

---

## Abstract

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 behavior-based 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 coefficientsc
1 and c
2 in the standard PSO (SPSO) to reduce the parameters sensitivity of solved problems. Experimental results on 28 benchmark functions, which consist of unimodal, multimodal, rotated, and shifted high-dimensional functions, demonstrate the high performance of the proposed algorithm in terms of convergence accuracy and speed with lower computation cost.


---

## Body

## 1. Introduction

Particle swarm optimization (PSO) [1] is a population-based intelligent algorithm, and it has been widely employed to solve various kinds of numerical and combinational optimization problems because of its simplicity, fast convergence, and high performance.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], fuzzy adaptive inertial weight [3], rand inertial weight [4], and adaptive inertial weight based on velocity information [5], they can enhance the performance of PSO. Concerning acceleration coefficients, the time-varying acceleration coefficients [6] are widely used. Clerc and Kennedy analyzed the convergence behavior by introducing constriction factor [7], which is proved to be equivalent to the inertial weight [8].(2) Hybrid PSO, which hybridizes other heuristic operators to increase population diversity. The genetic operators have been hybridized with PSO, such as selection operator [9], crossover operator [10], and mutation operator [11]. Similarly, differential evolution algorithm [12], ant colony optimization [13], and local search strategy [14] have been introduced into PSO.(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 [15]. The Von Neumann topology is helpful for solving multimodal problems and may perform better than other topologies including the global version [16].(4) Eliminating the velocity formula. Kennedy proposed the bare-bones PSO (BPSO) [17] and variants of BPSO [18, 19]. Sun et al. proposed quantum-behaved PSO (QPSO) and relative convergence analysis [20, 21].In recent years, some modified PSO have extremely enhanced the performance of PSO. For example, Zhan et al. proposed adaptive PSO (APSO) [22] and Wang et al. proposed so-called diversity enhanced particle swarm optimization with neighborhood search (DNSPSO) [23]. The former introduces an evolutionary state estimation (ESE) technique to adaptively adjust the inertia weight and acceleration coefficients. The later ones, a diversity enhancing mechanism and neighborhood-based search strategies, were employed to carry out a tradeoff between exploration and exploitation.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 coefficientsc
1 and c
2 from the SPSO to reduce the parameters sensitivity of the solved problems. The so-called HPSO has been employed to some nonlinear benchmark functions, which compose unimodal, multimodal, rotated, and shifted high-dimensional functions, to confirm its high performance by comparing with other well-known modified PSO.The remainder of the paper is structured as follows. In Section2, the standard particle swarm optimization (SPSO) is introduced. The proposed HPSO is given in Section 3. Experimental studies and discussion are provided in Section 4. Some conclusions are given in Section 5.


## 2. Standard PSO (SPSO)

The PSO is inspired by the behavior of bird flying or fish schooling; it is firstly introduced by Kennedy and Eberhart in 1995 [1] as a new heuristic algorithm. In the standard PSO (SPSO) [2], a swarm consists of a set of particles, and each particle represents a potential solution of an optimization problem. Considering the ith particle of the swarm with N particles in a D-dimensional space, its position and velocity at iteration t are denoted by X
i
(
t
)
=
(
x
i
1
(
t
)
,
x
i
2
(
t
)
,
…
,
x
i
D
(
t
)
) and V
i
(
t
)
=
(
v
i
1
(
t
)
,
v
i
2
(
t
)
,
…
,
v
i
D
(
t
)
). Then, the new velocity and position on the d-dimension of this particle at iteration t
+
1 will be calculated by using the following:
(1)
v
i
d
(
t
+
1
)
=
w
·
v
i
d
(
t
)
+
c
1
·
r
1
d
(
t
)
·
(
P
bes
t
i
d
(
t
)
-
x
i
d
(
t
)
)
+
c
2
·
r
2
d
(
t
)
·
(
G
bes
t
d
(
t
)
-
x
i
d
(
t
)
)
,
where i
=
1,2
,
…
,
N, and N is the population size; d
=
1,2
,
…
,
D, and D is the dimension of search space; r
1
d and r
2
d are two uniformly distributed random numbers in the interval [
0,1
]; acceleration coefficients c
1 and c
2 are nonnegative constants which control the influence of the cognitive and social components during the search process. P
bes
t
i
(
t
)
=
(
P
bes
t
i
1
(
t
)
,
…
,
P
bes
t
i
D
(
t
)
), called the personal best solution, represents the best solution found by the ith particle itself until iteration t; G
best
(
t
)
=
(
G
bes
t
1
(
t
)
,
…
,
G
bes
t
D
(
t
)
), called the global best solution, represents the global best solution found by all particles until iteration t. w is the inertial weight to balance the global and local search abilities of particles in the search space, which is given by
(2)
w
=
w
max
⁡
-
w
max
⁡
-
w
min
⁡
T
×
t
,
where w
max
⁡ is the initial weight, w
min
⁡ is the final weight, t is the current iteration number, and T is the maximum iteration number. Then, update particle’s position using the following:
(3)
x
i
d
(
t
+
1
)
=
x
i
d
(
t
)
+
v
i
d
(
t
+
1
)
and check x
min
⁡
d
≤
x
i
d
(
t
+
1
)
≤
x
max
⁡
d, where x
min
⁡
d and x
max
⁡
d represent lower and upper bounds of the dth variable, respectively.


## 3. Human Behavior-Based PSO (HPSO)

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 particlesP
best and G
best. Obviously, it is an ideal social condition. However, considering the human behavior, there exist some people who have bad habits or behaviors around us, at the same time, as we all known that these bad habits or behaviors will bring some effects on people around them. If we take warning from these bad habits or behaviors, it is beneficial to us. Conversely, if we learn from these bad habits or behaviors, it is harmful to us. Therefore, we must give an objective and rational view on these bad habits or behavior.In HPSO, we introduce the global worst particle, who is of the worst fitness in the entire population at each iteration. It is denoted asG
worst and defined as follows:
(4)
G
worst
(
t
)
=
argmax
{
f
(
P
bes
t
1
)
,
f
(
P
bes
t
2
)
,
…
,
f
(
P
bes
t
N
)
}
,
where f
(
·
) represents the fitness value of the corresponding particle.To simulate human behavior and make full use of theG
worst, we introduce a learning coefficient r
3, which is a random number obeying the standard normal distribution; that is, r
3
∈
N
(
0,1
). If r
3
>
0, we consider it as an impelled learning coefficient, which is helpful to enhance the “flying” velocity of the particle; therefore, it can enhance the exploration ability of particle. Conversely, if r
3
<
0, we consider it as a penalized learning coefficient, which can decrease the “flying” velocity of the particle; therefore, it is beneficial to enhance the exploitation. If r
3
=
0, it represents that these bad habits or behaviors have not effect on the particle. Meanwhile, in order to reduce the parameters sensitivity of the solved problems, we take place of the two acceleration coefficients c
1 and c
2 with two random learning coefficients r
1 and r
2, respectively. Therefore, the velocity equation has been changed as follows:
(5)
v
i
d
(
t
+
1
)
=
w
·
v
i
d
(
t
)
+
r
1
(
t
)
·
(
P
bes
t
i
d
(
t
)
-
x
i
d
(
t
)
)
+
r
2
(
t
)
·
(
G
bes
t
d
(
t
)
-
x
i
d
(
t
)
)
+
r
3
(
t
)
·
(
G
wors
t
d
(
t
)
-
x
i
d
(
t
)
)
,
where r
1 and r
2 are two random numbers in range of [
0,1
] and r
1
+
r
2
=
1. The random numbers r
1, r
2, and r
3 are the same for all d
=
1,2
,
…
,
D but different for each particle, and they are generated anew in each iteration. If v
i
d
(
t
+
1
) overflows the boundary, we set boundary value to it. Consider
(6)
v
i
d
(
t
+
1
)
=
{
v
min
⁡
d
,
if
v
i
d
(
t
+
1
)
<
v
min
⁡
d
,
v
max
⁡
d
,
if
v
i
d
(
t
+
1
)
>
v
max
⁡
d
,
v
i
d
(
t
+
1
)
,
otherwise,
where v
min
⁡
d and v
max
⁡
d are the minimum and maximum velocity of the d-dimensional search space, respectively. Similarly, if x
i
d
(
t
+
1
) flies out of the search space, we limit it to the corresponding bound value.In SPSO, the cognition and social learning terms move particlei towards good solutions based on P
bes
t
i and G
best in the search space as shown in Figure 1. This strategy makes a particle fly fast to good solutions, so it is easy to trap in local optima. From Figure 2, we can clearly observe that both impelled learning term and penalized term provide a particle with the chance to change flying direction. Therefore, the impelled/penalized term plays a key role in increasing the population diversity, which is beneficial in helping particles to escape from the local optima and enhance the convergence speed. In HPSO, the impelled/penalized learning term performs a proper tradeoff between the exploration and exploitation.Figure 1
Cognition and social terms in PSO.Impelled/penalized term in HPSO.
(a)
Impelled learning
(b)
Penalized learningTo sum up, Figure3 illustrates the flowchart of HPSO. Meanwhile, the pseudocodes of implementing the HPSO are listed as shown in Algorithm 1.Algorithm 1:HPSO.
Initialize Parameters:
N
⇐ population size;
D
⇐ the dimensionality of search space;
T
⇐ the number of maximum iteration;
w
⇐ the inertial weight;
[
x
min
⁡
d
,
x
max
⁡
d
]
⇐ the allowable position boundaries, d
=
1,2
,
…
,
D;
[
v
min
⁡
d
,
v
max
⁡
d
]
⇐ the allowable velocity boundaries, d
=
1,2
,
…
,
D;
Initialize Population:
V
i
=
(
v
i
1
,
v
i
2
,
…
,
v
i
D
), X
i
=
(
x
i
1
,
x
i
2
,
…
,
x
i
D
), i
=
1,2
,
…
,
N;
v
i
d
⇐
v
min
⁡
d
+
ran
d
i
d
·
(
v
max
⁡
d
-
v
min
⁡
d
);
x
i
d
⇐
x
min
⁡
d
+
ran
d
i
d
·
(
x
max
⁡
d
-
x
min
⁡
d
);
Initialize
P
best
,
G
best
and
G
worst
:
Evaluate fitness of all particles inX
=
{
X
1
,
X
2
,
…
,
X
N
};
P
best
⇐
X;
G
best
⇐
arg
min
⁡
{
f
(
P
bes
t
1
)
,
f
(
P
bes
t
2
)
,
…
,
f
(
P
bes
t
N
)
};
G
worst
⇐
arg
max
⁡
{
f
(
P
bes
t
1
)
,
f
(
P
bes
t
2
)
,
…
,
f
(
P
bes
t
N
)
};
For
t
=
1,2
,
…
,
T
For each particle i
=
1,2
,
…
,
N
Update velocity according to (5) and check the boundaries;
Update position according to (3) and check the boundaries;
Endfor
Evaluate fitness of all particles in
{
X
};
Update
P
best, G
best
and
G
worst;
Endfor.
Return the best solution.Figure 3
HPSO flowchart.


## 4. Experimental Studies and Discussion

To evaluate the performance of HPSO, 28 minimization benchmark functions are selected [22, 24, 25] as detailed in Section 4.1. HPSO is compared with SPSO in different search spaces and the results are given in Section 4.2. In addition, HPSO is compared with some well-known variants of PSO in Section 4.3.


### 4.1. Benchmark Functions

In the experimental study, we choose 28 minimization benchmark functions, which consist of unimodal, multimodal, rotated, shifted, and shifted rotated functions. Table1 lists the main information; please refer to papers [22, 24, 25] to obtain further detailed information about these functions. Among these functions, F
1
–
F
6 are unimodal functions. F
7 is the Rosenbrock function, which is unimodal for D
=
2 and D
=
3 but may have multiple minima in high dimension cases. F
8
–
F
15 are unrotated multimodal functions and the number of their local minima increases exponentially with the problem dimension. F
16
–
F
23 are rotated functions. F
24
–
F
26 are shifted functions and F
27 and F
28 are shifted rotated multimodal functions and O
=
(
o
1
,
o
2
,
…
,
o
D
) is a randomly generated shift vector located in the search space. To obtain a rotated function, an orthogonal matrix M [26] is considered and the rotated variable y
=
M
×
x is computed. Then, the vector y is used to evaluate the objective function value.Table 1
Functions’ names, dimensions, ranges, and global optimum values of benchmark functions used in the experiments.
Number
Function name
Dimension (D)
[
Range
]
D
F
opt
F
1
Sphere model
30
/
50
/
100
[−100, 100]
D
0
F
2
Schwefel’s problem 2.22
30
/
50
/
100
[−10, 10]
D
0
F
3
Schwefel’s problem 1.2
30
/
50
/
100
[−100, 100]
D
0
F
4
Schwefel’s problem 2.21
30
/
50
/
100
[−100, 100]
D
0
F
5
Step function
30
/
50
/
100
[−100, 100]
D
0
F
6
Quartic function, that is, noise
30
/
50
/
100
[−1.28, 1.28]
D
0
F
7
Rosenbrock’s function
30
/
50
/
100
[−10, 10]
D
0
F
8
Schwefel’s function
30
/
50
/
100
[−500, 500]
D
0
F
9
Generalized Rastrigin’s function
30
/
50
/
100
[−5.12, 5.12]
D
0
F
10
Noncontinuous Rastrigin’s function
30
/
50
/
100
[−5.12, 5.12]
D
0
F
11
Ackley’s function
30
/
50
/
100
[−32, 32]
D
0
F
12
Generalized Griewank’s function
30
/
50
/
100
[−600, 600]
D
0
F
13
Weierstrass’s function
30
/
50
/
100
[−0.5, 0.5]
D
0
F
14
Generalized penalized function
30
/
50
/
100
[−50, 50]
D
0
F
15
Cosine mixture problem
30
/
50
/
100
[−1, 1]
D
-
0.1
×
D
F
16
Rotated elliptic function
30
/
50
/
100
[−1.28, 1.28]
D
0
F
17
Rotated Schwefel’s function
30
/
50
/
100
[−500, 500]
D
0
F
18
Rotated Ackley’s function
30
/
50
/
100
[−32, 32]
D
0
F
19
Rotated Griewank’s function
30
/
50
/
100
[−600, 600]
D
0
F
20
Rotated Weierstrass’s function
30
/
50
/
100
[−0.5, 0.5]
D
0
F
21
Rotated Rastrigin’s function
30
/
50
/
100
[−5.12, 5.12]
D
0
F
22
Rotated Salomon’s function
30
/
50
/
100
[−100, 100]
D
0
F
23
Rotated Rosenbrock’s function
30
/
50
/
100
[−100, 100]
D
0
F
24
Shifted Rosenbrock’s function
30
/
50
/
100
[−100, 100]
D
390
F
25
Shifted Rastrigin’s function
30
/
50
/
100
[−5, 5]
D
−330
F
26
Shifted Schwefel’s problem 2.21
30
/
50
/
100
[−100, 100]
D
−450
F
27
Shifted rotated Ackley’s function
30
/
50
/
100
[−32, 32]
D
−140
F
28
Shifted rotated Weierstrass’s function
30
/
50
/
100
[−0.5, 0.5]
D
90


### 4.2. Comparison of HPSO with SPSO

The performance on the convergence accuracy of HPSO is compared with that of SPSO. The test functions listed in Table1 are evaluated. For a fair comparison, we set the same parameters value. Population size is set to 30 (N
=
30), upper bounds of velocity V
max
⁡
=
0.5
×
(
X
max
⁡
-
X
min
⁡
), and the corresponding lower bounds V
min
⁡
=
-
V
max
⁡, where X
min
⁡ and X
max
⁡ are the lower and upper bounds of variables, respectively. Inertia weight w is linearly decreased from 0.9 to 0.4 in SPSO and HPSO. Acceleration coefficients c
1 and c
2 in SPSO are set to 2. The two algorithms are independently run 30 times on the benchmark functions. The results in terms of the best, worst, median, mean, and standard deviation (SD) of the solutions obtained in the 30 independent runs by each algorithm in different search spaces are as shown in Tables 2, 3, and 4. At the same time, the maximum iteration T is 1000 for D
=
30, 2000 for D
=
50, and 3000 for D
=
100, respectively.Table 2
Experimental results obtained by SPSO and HPSO on function fromF
1 to F
10.
Fun
Dim
Best
Worst
Meadian
Mean
SD
Significant
F
1
30
SPSO
1.1992
e
-
04
1.0000
e
+
04
9.9690
e
-
04
666.6686
2.5371
e
+
03
HPSO
0
0
0
0
0
+
50
SPSO
9.4288
e
-
04
1.0000
e
+
04
0.0078
3.6667
e
+
03
3.6667
e
+
03
HPSO
0
0
0
0
0
+
100
SPSO
1.0013
e
+
04
7.0017
e
+
04
4.0087
e
+
04
4.0698
e
+
04
2.0974
e
+
04
HPSO
0
10000
0
333.3333
1.8257
e
+
03
+
F
2
30
SPSO
6.8555
e
-
04
30.0018
10.0017
11.3364
10.0777
HPSO
0
0
0
0
0
+
50
SPSO
0.0329
70.0010
40.0006
37.3438
15.2918
HPSO
0
0
0
0
0
+
100
SPSO
51.0214
181.4054
110.5934
114.3039
29.0723
HPSO
0
0
0
0
0
+
F
3
30
SPSO
6.4613
e
+
03
3.7311
e
+
04
2.2333
e
+
04
2.1337
e
+
04
6.7035
e
+
03
HPSO
0
5
.
1779
e
+
03
0
172.5975
945.3557
+
50
SPSO
4.0023
e
+
04
1.0191
e
+
05
6.5660
e
+
04
7.0328
e
+
04
1.7603
e
+
04
HPSO
0
6
.
9787
e
+
03
0
232.6222
1
.
2741
e
+
03
+
100
SPSO
1.7694
e
+
05
3.0086
e
+
05
2.4789
e
+
05
2.4752
e
+
05
3.6623
e
+
04
HPSO
0
2
.
6987
e
+
04
0
3
.
8008
e
+
03
6
.
9150
e
+
03
+
F
4
30
SPSO
8.6091
21.2711
12.9945
13.3502
3.5341
HPSO
0
0
0
0
0
+
50
SPSO
24.2031
39.5127
31.0562
31.1715
4.2886
HPSO
0
0
0
0
0
+
100
SPSO
54.1172
75.3686
64.7834
64.2358
4.2202
HPSO
0
0
0
0
0
+
F
5
30
SPSO
0
10001
0
1.0005
e
+
03
3.0512
e
+
03
HPSO
0
0
0
0
0
+
50
SPSO
0
20004
4.5000
5.0028
e
+
03
6.8230
e
+
03
HPSO
0
0
0
0
0
+
100
SPSO
127
90040
40068
4.3086
e
+
04
2.2747
e
+
04
HPSO
0
0
0
0
0
+
F
6
30
SPSO
0.0344
18.8556
0.0959
3.5587
5.1400
HPSO
1
.
4522
e
-
04
0.0030
0.0012
0.0012
8
.
5738
e
-
04
+
50
SPSO
0.0780
72.6594
13.6489
19.6604
19.3860
HPSO
7.4623
e
-
05
0.0017
5
.
3645
e
-
04
6
.
3534
e
-
04
4
.
7283
e
-
04
+
100
SPSO
86.7855
381.9209
200.8146
211.9720
88.3159
HPSO
3
.
5210
e
-
05
0.0019
2
.
9387
e
-
04
4
.
0826
e
-
04
3
.
5395
e
-
04
+
F
7
30
SPSO
14.3237
1.0083
e
+
04
140.5176
2.4686
e
+
03
4.2581
e
+
03
HPSO
28.6353
28.9456
28.8793
28.8461
0.0932
+
50
SPSO
97.0317
9.4285
e
+
05
376.2306
3.4093
e
+
04
1.7169
e
+
05
HPSO
48.4886
48.8766
48.7600
48.7513
0.0875
+
100
SPSO
706.1328
2.8333
e
+
06
9.4375
e
+
05
8.8851
e
+
05
8.9157
e
+
05
HPSO
98.4280
98.8373
98.7133
98.7129
0.0818
+
F
8
30
SPSO
2
.
0226
e
+
03
4
.
8935
e
+
03
3
.
5787
e
+
03
3
.
6128
e
+
03
733.1063
HPSO
3.5886
e
+
03
8.0516
e
+
03
6.6047
e
+
03
6.3505
e
+
03
1.0893
e
+
03
−
50
SPSO
5
.
8499
e
+
03
9
.
7913
e
+
03
7
.
8862
e
+
03
7
.
7139
e
+
03
1
.
0101
e
+
03
HPSO
6.5496
e
+
03
1.4460
e
+
04
1.1191
e
+
04
1.0866
e
+
04
2.1757
e
+
03
−
100
SPSO
1.8110
e
+
04
2
.
4259
e
+
04
2
.
0949
e
+
04
2
.
1084
e
+
04
1
.
7384
e
+
03
HPSO
1
.
2615
e
+
04
3.1402
e
+
04
2.4302
e
+
04
2.4077
e
+
04
4.9510
e
+
03
−
F
9
30
SPSO
28.7299
160.3815
87.6754
92.5142
32.6994
HPSO
0
0
0
0
0
+
50
SPSO
175.2643
351.6480
260.4359
258.0518
48.4078
HPSO
0
0
0
0
0
+
100
SPSO
555.8950
993.3887
750.1694
749.1658
749.1658
HPSO
0
0
0
0
0
+
F
10
30
SPSO
61.4129
221.0445
132.7694
134.5414
33.8073
HPSO
0
0
0
0
0
+
50
SPSO
157.1020
440.0897
324.2632
310.3595
64.3675
HPSO
0
0
0
0
0
+
100
SPSO
623.5658
1.0433
e
+
03
804.6981
813.3435
88.5932
HPSO
0
25
0
0.8333
4.5644
+Table 3
Experimental results obtained by SPSO and HPSO on functions fromF
11 to F
20.
Fun
Dim
Best
Worst
Median
Mean
SD
Significant
F
11
30
SPSO
0.0043
19.9630
0.0595
2.3935
5.4041
HPSO
8.8818
e
-
16
8.8818
e
-
16
8.8818
e
-
16
8.8818
e
-
16
0
+
50
SPSO
0.0598
19.9646
12.6912
10.5673
6.3042
HPSO
8.8818
e
-
16
8.8818
e
-
16
8.8818
e
-
16
8.8818
e
-
16
0
+
100
SPSO
15.4237
20.2143
19.5200
19.4135
0.8672
HPSO
8.8818
e
-
16
8.8818
e
-
16
8.8818
e
-
16
8.8818
e
-
16
0
+
F
12
30
SPSO
7.0274
e
-
04
90.8935
0.0178
12.0794
31.2763
HPSO
0
0
0
0
0
+
50
SPSO
0.0014
270.8170
0.0415
45.1971
70.1274
HPSO
0
0
0
0
0
+
100
SPSO
1.1140
721.0594
361.0858
376.1758
158.6584
HPSO
0
0
0
0
0
+
F
13
30
SPSO
0.1403
4.3952
0.3210
1.0567
1.4863
HPSO
0
0
0
0
0
+
50
SPSO
0.8657
15.2389
7.5828
8.2388
3.6607
HPSO
0
0
0
0
0
+
100
SPSO
27.6235
64.4826
49.3984
47.7138
10.0126
HPSO
0
0
0
0
0
+
F
14
30
SPSO
6.4114
e
-
05
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
5.1200
e
+
08
457.9143
7.6801
e
+
07
1.5257
e
+
08
HPSO
0.1866
0.5097
0.2703
0.2736
0.0653
+
F
15
30
SPSO
−3.0000
−2.8522
−3.0000
−2.9507
0.0709
HPSO
−3
−3
−3
−3
0
+
50
SPSO
−5.0000
−2.3044
−4.4827
−4.2127
0.6865
HPSO
−5
−5
−5
−5
0
+
100
SPSO
−7.9165
4.7637
−5.2127
−4.6977
2.8465
HPSO
−10
−10
−10
−10
0
+
F
16
30
SPSO
2.3604
e
+
03
3.8233
e
+
04
3.8233
e
+
04
1.2375
e
+
04
9.2463
e
+
03
HPSO
0
5.8369
e
+
03
0
390.6710
1.2756
e
+
03
+
50
SPSO
7.1213
e
+
03
1.0427
e
+
05
3.3195
e
+
04
3.4891
e
+
04
2.2914
e
+
04
HPSO
0
4.0529
e
+
03
0
224.6749
873.6249
+
100
SPSO
6.2317
e
+
04
2.7386
e
+
05
1.4222
e
+
05
1.4697
e
+
05
5.7699
e
+
04
HPSO
0
1.9403
e
+
04
0
1.0583
e
+
03
3.8088
e
+
03
+
F
17
30
SPSO
6.7986
e
+
03
9.7587
e
+
03
8.3387
e
+
03
8.2508
e
+
03
739.7223
HPSO
8.3590
e
+
03
9.8803
e
+
03
9.0866
e
+
03
9.0790
e
+
03
442.4330
−
50
SPSO
1.3020
e
+
04
1.7080
e
+
04
1.4999
e
+
04
1.5149
e
+
04
1.0581
e
+
03
HPSO
1.5003
e
+
04
1.7349
e
+
04
1.6514
e
+
04
1.6310
e
+
04
669.3538
−
100
SPSO
2.7400
e
+
04
2.7400
e
+
04
3.1087
e
+
04
3.1149
e
+
04
2.1654
e
+
03
HPSO
3.0329
e
+
04
3.5493
e
+
04
3.4226
e
+
04
3.3586
e
+
04
1.5320
e
+
03
−
F
18
30
SPSO
20.7888
21.0951
21.0053
20.9848
0.0712
HPSO
8.8818
e
-
16
21.1210
20.9931
11.2354
10.6894
+
50
SPSO
21.0515
21.2478
21.1455
21.1436
0.0536
HPSO
8.8818
e
-
16
21.2404
21.1366
12.0016
10.6745
+
100
SPSO
21.2367
21.3931
21.3368
21.3358
0.0364
HPSO
8.8818
e
-
16
21.3949
21.3545
15.6658
9.6084
+
F
19
30
SPSO
1.0517
495.3131
273.6408
243.6176
154.3551
HPSO
0
0
0
0
0
+
50
SPSO
265.0558
1.4393
e
+
03
798.8065
786.0782
289.8401
HPSO
0
0
0
0
0
+
100
SPSO
1.9937
e
+
03
4.0158
e
+
03
2.9388
e
+
03
2.9263
e
+
03
543.9053
HPSO
0
0
0
0
0
+
F
20
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
+Table 4
Experimental results obtained by SPSO and HPSO on functions fromF
21 to F
28.
Fun
Dim
Best
Worst
Median
Mean
SD
Significant
F
21
30
SPSO
67.1541
307.3070
213.8939
203.8842
61.8125
HPSO
0
0
0
0
0
+
50
SPSO
158.2955
715.0245
518.1705
500.5593
135.5998
HPSO
0
269.3463
0
8.9782
49.1757
+
100
SPSO
1.0850
e
+
03
1.9021
e
+
03
1.5793
e
+
03
1.5669
e
+
03
190.5584
HPSO
0
582.0882
0
35.5882
136.0270
+
F
22
30
SPSO
0.7999
14.9999
1.2522
2.9025
4.3553
HPSO
0
0
0
0
0
+
50
SPSO
2.0999
26.0999
13.9628
12.8291
6.9033
HPSO
0
0
0
0
0
+
100
SPSO
16.5013
41.9999
35.4551
33.9791
6.3075
HPSO
0
0
0
0
0
+
F
23
30
SPSO
81.0577
4.0119
e
+
09
2.0685
e
+
08
6.8745
e
+
08
1.0469
e
+
09
HPSO
28.8214
28.9856
28.9323
28.9252
0.0421
+
50
SPSO
3.7253
e
+
03
2.1495
e
+
10
3.6515
e
+
09
3.6515
e
+
09
5.3957
e
+
09
HPSO
48.7069
48.8900
48.8205
48.8139
0.0479
+
100
SPSO
6.7997
e
+
09
9.2655
e
+
10
3.3160
e
+
10
3.8223
e
+
10
2.0050
e
+
10
HPSO
98.6590
98.8846
98.8109
98.7983
0.0545
+
F
24
30
SPSO
6.2312
e
+
08
2.3418
e
+
10
4.9110
e
+
09
5.8767
e
+
09
5.6099
e
+
09
HPSO
5.9432
e
+
05
6.2859
e
+
09
7.6373
e
+
06
3.7982
e
+
08
1.2316
e
+
09
+
50
SPSO
4.3540
e
+
09
3.3195
e
+
10
1.3961
e
+
10
1.6077
e
+
10
8.3270
e
+
09
HPSO
3.9454
e
+
06
8.9387
e
+
09
3.1766
e
+
07
7.0962
e
+
08
1.9565
e
+
09
+
100
SPSO
4.9031
e
+
10
1.5465
e
+
11
9.1986
e
+
10
9.7151
e
+
10
2.8460
e
+
10
HPSO
2.0551
e
+
08
5.4553
e
+
09
6.7593
e
+
08
1.1373
e
+
09
1.2367
e
+
09
+
F
25
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
+
F
26
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
+
F
27
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
−
F
28
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 Tables2–4, we can clearly observe that the convergence accuracy of HPSO is better than SPSO on the most benchmark functions. An interesting result is that HPSO can find the global optimal solutions on functions F
2, F
4, F
5, F
9, F
12, F
13, F
15, F
19, and f
22 in all search spaces; that is to say, HPSO can obtain the 100% success rate on the listed functions. Considering F
1 and F
10, though HPSO can find the global optimal solutions in all different search ranges, it only obtained the mean values 333.3333 and 0.8333, respectively, in 100-dimensional space. At the same time, HPSO offers the higher convergence accuracy on functions F
3, F
6, F
7, F
11, F
14, F
16, F
20, F
21, F
23, and F
26. However, we must observe that SPSO has higher performance on function F
8. As for F
25, SPSO has better performance in 30-dimensional search space, but HPSO has better performance in 50- and 100-dimensional search spaces. As for shifted rotated functions F
27 and F
28, both SPSO and HPSO have worst convergence accuracy. As seen, the dimension of the selected functions has great effect on SPSO. For example, considering function F
1, SPSO has mean value 666.6686, 3.6667e + 03, and 4.0698e + 04 in 30-dimensional, 50-dimensional, and 100-dimensional search spaces, respectively, while HPSO has mean values 0, 0, and 333.333 in the corresponding search space. Therefore, we also conclude that HPSO has better stability than SPSO from the data in different search spaces.In the 9th columns of Tables2–4, we report the statistical significance level of the difference of the means of the two algorithms. Note that here “+” indicates that the t value is significant at a 0.05 level of significance by two-tailed test, and “−” stands for the difference of means that is not statistically significant.Figure4 graphically presents the comparison in terms of convergence characteristics of the evolutionary processes in solving the selected benchmark functions in 30-dimensional search space with N
=
30 and T
=
1000.Convergence comparison of HPSO and SPSO on the selected test functions withD
=
30
,
N
=
30, and T
=
1000.
(a)
f
1
(b)
f
3
(c)
f
5
(d)
f
7
(e)
f
10
(f)
f
14
(g)
f
16
(h)
f
18
(i)
f
20
(j)
f
24
(k)
f
26
(l)
f
28


### 4.3. Comparison of HPSO with Other PSO Algorithms

In this section, a comparison of HPSO with some well-known PSO algorithms which are listed in Table5 is performed to evaluate the efficiency of the proposed algorithm.Table 5
Some well-known PSOs algorithms in the literature.
Algorithm
Year
Topology
Parameter settings
GPSO
1998
Global star
w: 0.9
-
0.4, c
1
=
c
2
=
2.0
LPSO
2002
Local ring
w: 0.9
-
0.4, c
1
=
c
2
=
2.0
FIPS
2004
Local Uring
χ
=
0.729, ∑
c
i
=
4.1
HPSO-TVAC
2004
Global star
w: 0.9
-
0.4, c
1
:
2.5
-
0.5, and c
2: 0.5
-
2.5
UPSO
2004
Global star
w: 0.9
-
0.4, c
1
=
c
2
=
2.0, and U
=
0.5
DMS-PSO
2005
Dynamic multiswarm
w: 0.9
-
0.2, c
1
=
c
2
=
2.0, m
=
3, and R
=
5
VPSO
2006
Local Von Neumann
w: 0.9
-
0.4, c
1
=
c
2
=
2.0
CLPSO
2006
Comprehensive learning
w: 0.9
-
0.4, c
=
1.49445, and m
=
7
QIPSO
2007
Global star
w: 0.9
-
0.4, c
1
=
c
2
=
2.0
APSO
2009
Global star
w: 0.9, c
1
=
c
2
=
2.0; δ
:
random  in
[
0.05,0.1
], σ: 1
-
0.1
AFPSO
2011
Global star
w: 0.9
-
0.4, c
1, and c
2 are based on fuzzy rule
AFPSO-QI
2011
Global star
w: 0.9
-
0.4, c
1, and c
2 are based on fuzzy ruleAt first, we choose 10 unimodal and multimodal test functions for this evaluation. According to [22], the algorithms GPSO [2], LPSO [16], VPSO [27], FIPS [28], HPSO-TVAC [6], DMS-PSO [29], CLPSO [24], and APSO [22] are considered as detailed in Table 5. The experimental results of the algorithms are directly from [22] as shown in Table 6. In this trial, the population size N
=
20, the dimension D
=
30, and the maximum fitness evaluations (FEs) were set to 2
×
1
0
5 also. The parameter configurations of the selected algorithms have been set according to their corresponding references. The inertia weight w is linearly decreased from 0.9 to 0.4 in HPSO. HPSO is independently run 30 times and the mean and SD are shown in Table 6. As seen, HPSO has the first rank among the algorithms and obtains the global minimum on functions F
1, F
2, F
5, F
9, F
10, and F
12 and gives the good near-global optima on functions F
6 and F
11. Meanwhile, HPSO has the worst performance on functions F
3 and F
14. As for F
3, APSO has the best convergence accuracy, and HPSO only wins CLPSO. Considering F
14, DMS-PSO has the best performance.Table 6
Comparison results of eight PSO algorithms [22] with HPSO on 10 functions (N
=
20, D
=
30, and FEs
=
2
×
1
0
5).
Function
GPSO
LPSO
VPSO
FIPS
HPSO-TVAC
DMS-PSO
CLPSO
APSO
HPSO
F
1
Mean
1.98
e
-
53
4.77
e
-
29
5.11
e
-
38
3.21
e
-
30
3.38
e
-
41
3.85
e
-
54
1.89
e
-
19
1.45
e
-
150
0
SD
7.08
e
-
53
1.13
e
-
28
1.91
e
-
37
3.60
e
-
30
8.50
e
-
41
1.75
e
-
53
1.49
e
-
19
5.73
e
-
150
0
Rank
4
8
6
7
5
3
9
2
1
F
2
Mean
2.51
e
-
34
2.03
e
-
20
6.29
e
-
27
1.32
e
-
17
6.9
e
-
23
2.61
e
-
29
1.01
e
-
13
5.15
e
-
84
0
SD
5.84
e
-
34
2.89
e
-
20
8.68
e
-
27
7.86
e
-
18
6.89
e
-
23
6.6
e
-
29
6.51
e
-
14
1.44
e
-
83
0
Rank
3
7
5
8
6
4
9
2
1
F
3
Mean
6.45
e
-
2
18.60
1.44
0.77
2.89
e
-
7
47.5
395
1.0
e
-
10
167
SD
9.45
e
-
2
30.71
1.55
0.86
2.97
e
-
7
56.4
142
2.13
e
-
10
913
Rank
3
6
5
4
2
7
9
1
8
F
5
Mean
0
0
0
0
0
0
0
0
0
SD
0
0
0
0
0
0
0
0
0
Rank
1
1
1
1
1
1
1
1
1
F
6
Mean
7.77
e
-
3
1.49
e
-
2
1.08
e
-
2
2.55
e
-
3
5.54
e
-
2
1.1
e
-
2
3.92
e
-
3
4.66
e
-
3
1.03
e
-
04
SD
2.42
e
-
3
5.66
e
-
3
3.24
e
-
3
6.25
e
-
4
2.08
e
-
2
3.94
e
-
3
1.14
e
-
3
1.7
e
-
3
8.99
e
-
05
Rank
5
8
6
2
9
7
3
4
1
F
9
Mean
30.7
34.90
34.09
29.98
2.39
28.1
2.57
e
-
11
5.8
e
-
15
0
SD
8.68
7.25
8.07
10.92
3.71
6.42
6.64
e
-
11
1.01
e
-
14
0
Rank
7
9
8
6
4
5
3
2
1
F
10
Mean
15.5
30.40
21.33
35.91
1.83
32.8
0.167
4.14
e
-
16
0
SD
7.4
9.23
9.46
9.49
2.65
6.49
0.379
1.45
e
-
15
0
Rank
5
7
6
9
4
8
3
2
1
F
11
Mean
1.15
e
-
14
1.85
e
-
14
1.4
e
-
14
7.69
e
-
15
2.06
e
-
10
8.52
e
-
15
2.01
e
-
12
1.11
e
-
14
8.88
e
-
16
SD
2.27
e
-
15
4.80
e
-
15
3.48
e
-
15
9.33
e
-
16
9.45
e
-
10
1.79
e
-
15
9.22
e
-
13
3.55
e
-
15
0
Rank
5
7
6
2
9
3
8
4
1
F
12
Mean
2.37
e
-
2
1.10
e
-
2
1.31
e
-
2
9.04
e
-
4
1.07
e
-
2
1.31
e
-
2
6.45
e
-
13
1.67
e
-
2
0
SD
2.57
e
-
2
1.60
e
-
2
1.35
e
-
2
2.78
e
-
3
1.14
e
-
2
1.73
e
-
2
2.07
e
-
12
2.41
e
-
2
0
Rank
9
5
6
3
4
7
2
8
1
F
14
Mean
1.04
e
-
2
2.18
e
-
30
3.46
e
-
3
1.22
e
-
31
7.07
e
-
30
2.05
e
-
32
1.59
e
-
21
3.76
e
-
31
1.70
e
-
2
SD
3.16
e
-
2
5.14
e
-
30
1.89
e
-
2
4.85
e
-
32
4.05
e
-
30
8.12
e
-
33
1.93
e
-
21
1.2
e
-
30
1.42
e
-
2
Rank
8
4
7
2
5
1
6
3
9
Average rank
5
6.2
5.6
4.4
4.9
4.6
5.3
2.9
2.5
Final rank
6
9
8
3
5
4
7
2
1Then, in the next step, we choose six functions from [25] and seven algorithms of GPSO, QIPSO [30], UPSO [31], FIPS, AFSO [25], and AFSO-Q1 [25] as detailed in Table 5. For a fair comparison, the population size N
=
30, the dimension D
=
30, and the maximum iteration T
=
1
0,000 also in HPSO, and the inertia weight w is linearly decreased from 0.9 to 0.4. HPSO is independently run 30 times and the mean and SD are shown in Table 7. As seen, HPSO shows better performance and has the first rank. HPSO finds the global optimal solution on functions F
9, F
13, F
21, and F
22. FIPS and UPSO have better convergence accuracy on functions F
27 and F
28, respectively.Table 7
Comparison results of seven PSO algorithms [25] with HPSO on six functions (N
=
30, D
=
30, and T = 10,000).
Function
SPSO
QIPSO
UPSO
FIPS
CLPSO
AFSO
AFSO-Q1
HPSO
F
9
Mean
52.30
25.61
59.40
106.1
74.39
17.93
15.69
0
SD
27.35
15.98
58.05
30.54
9.77
5.63
4.47
0
Rank
5
4
6
8
7
3
2
1
F
13
Mean
0.534
36.38
8.70
6.40
1.39
e
-
03
4.52
e
-
03
1.50
e
-
03
0
SD
1.74
4.66
3.08
3.04
3.28
e
-
04
9.20
e
-
03
3.48
e
-
03
0
Rank
5
8
7
6
2
4
3
1
F
21
Mean
320.2
317.5
309.5
434.1
263.3
266.3
253.3
0
SD
14.70
23.24
25.88
34.99
11.96
12.00
12.63
0
Rank
7
6
5
8
3
4
2
1
F
22
Mean
17.03
15.20
14.29
26.60
11.94
10.38
8.46
0
SD
2.55
1.32
2.15
1.42
1.37
1.38
0.948
0
Rank
7
6
5
8
4
3
2
1
F
27
Mean
−119.10
−119.10
−119.10
−119.90
−119.00
−119.70
−119.80
−119.05
SD
7.09
e
-
02
5.68
e
-
02
3.24
e
-
02
3.78
e
-
02
4.28
e
-
02
3.85
e
-
02
5.45
e
-
02
5.50
e
-
02
Rank
4
4
4
1
6
3
2
5
F
28
Mean
115.90
121.90
113.20
113.60
118.30
123.20
123.10
117.32
SD
2.90
4.90
6.14
3.63
2.40
2.25
3.01
3.65
Rank
3
6
1
2
5
8
7
4
Average rank
5.17
5.67
4.67
5.50
4.50
4.17
3.00
2.17
Final rank
6
8
5
7
4
3
2
1Therefore, it is worth saying that the proposed algorithm has considerably better performance than the other well-known PSO algorithms in unimodal and multimodal high-dimensional functions.


## 4.1. Benchmark Functions

In the experimental study, we choose 28 minimization benchmark functions, which consist of unimodal, multimodal, rotated, shifted, and shifted rotated functions. Table1 lists the main information; please refer to papers [22, 24, 25] to obtain further detailed information about these functions. Among these functions, F
1
–
F
6 are unimodal functions. F
7 is the Rosenbrock function, which is unimodal for D
=
2 and D
=
3 but may have multiple minima in high dimension cases. F
8
–
F
15 are unrotated multimodal functions and the number of their local minima increases exponentially with the problem dimension. F
16
–
F
23 are rotated functions. F
24
–
F
26 are shifted functions and F
27 and F
28 are shifted rotated multimodal functions and O
=
(
o
1
,
o
2
,
…
,
o
D
) is a randomly generated shift vector located in the search space. To obtain a rotated function, an orthogonal matrix M [26] is considered and the rotated variable y
=
M
×
x is computed. Then, the vector y is used to evaluate the objective function value.Table 1
Functions’ names, dimensions, ranges, and global optimum values of benchmark functions used in the experiments.
Number
Function name
Dimension (D)
[
Range
]
D
F
opt
F
1
Sphere model
30
/
50
/
100
[−100, 100]
D
0
F
2
Schwefel’s problem 2.22
30
/
50
/
100
[−10, 10]
D
0
F
3
Schwefel’s problem 1.2
30
/
50
/
100
[−100, 100]
D
0
F
4
Schwefel’s problem 2.21
30
/
50
/
100
[−100, 100]
D
0
F
5
Step function
30
/
50
/
100
[−100, 100]
D
0
F
6
Quartic function, that is, noise
30
/
50
/
100
[−1.28, 1.28]
D
0
F
7
Rosenbrock’s function
30
/
50
/
100
[−10, 10]
D
0
F
8
Schwefel’s function
30
/
50
/
100
[−500, 500]
D
0
F
9
Generalized Rastrigin’s function
30
/
50
/
100
[−5.12, 5.12]
D
0
F
10
Noncontinuous Rastrigin’s function
30
/
50
/
100
[−5.12, 5.12]
D
0
F
11
Ackley’s function
30
/
50
/
100
[−32, 32]
D
0
F
12
Generalized Griewank’s function
30
/
50
/
100
[−600, 600]
D
0
F
13
Weierstrass’s function
30
/
50
/
100
[−0.5, 0.5]
D
0
F
14
Generalized penalized function
30
/
50
/
100
[−50, 50]
D
0
F
15
Cosine mixture problem
30
/
50
/
100
[−1, 1]
D
-
0.1
×
D
F
16
Rotated elliptic function
30
/
50
/
100
[−1.28, 1.28]
D
0
F
17
Rotated Schwefel’s function
30
/
50
/
100
[−500, 500]
D
0
F
18
Rotated Ackley’s function
30
/
50
/
100
[−32, 32]
D
0
F
19
Rotated Griewank’s function
30
/
50
/
100
[−600, 600]
D
0
F
20
Rotated Weierstrass’s function
30
/
50
/
100
[−0.5, 0.5]
D
0
F
21
Rotated Rastrigin’s function
30
/
50
/
100
[−5.12, 5.12]
D
0
F
22
Rotated Salomon’s function
30
/
50
/
100
[−100, 100]
D
0
F
23
Rotated Rosenbrock’s function
30
/
50
/
100
[−100, 100]
D
0
F
24
Shifted Rosenbrock’s function
30
/
50
/
100
[−100, 100]
D
390
F
25
Shifted Rastrigin’s function
30
/
50
/
100
[−5, 5]
D
−330
F
26
Shifted Schwefel’s problem 2.21
30
/
50
/
100
[−100, 100]
D
−450
F
27
Shifted rotated Ackley’s function
30
/
50
/
100
[−32, 32]
D
−140
F
28
Shifted rotated Weierstrass’s function
30
/
50
/
100
[−0.5, 0.5]
D
90


## 4.2. Comparison of HPSO with SPSO

The performance on the convergence accuracy of HPSO is compared with that of SPSO. The test functions listed in Table1 are evaluated. For a fair comparison, we set the same parameters value. Population size is set to 30 (N
=
30), upper bounds of velocity V
max
⁡
=
0.5
×
(
X
max
⁡
-
X
min
⁡
), and the corresponding lower bounds V
min
⁡
=
-
V
max
⁡, where X
min
⁡ and X
max
⁡ are the lower and upper bounds of variables, respectively. Inertia weight w is linearly decreased from 0.9 to 0.4 in SPSO and HPSO. Acceleration coefficients c
1 and c
2 in SPSO are set to 2. The two algorithms are independently run 30 times on the benchmark functions. The results in terms of the best, worst, median, mean, and standard deviation (SD) of the solutions obtained in the 30 independent runs by each algorithm in different search spaces are as shown in Tables 2, 3, and 4. At the same time, the maximum iteration T is 1000 for D
=
30, 2000 for D
=
50, and 3000 for D
=
100, respectively.Table 2
Experimental results obtained by SPSO and HPSO on function fromF
1 to F
10.
Fun
Dim
Best
Worst
Meadian
Mean
SD
Significant
F
1
30
SPSO
1.1992
e
-
04
1.0000
e
+
04
9.9690
e
-
04
666.6686
2.5371
e
+
03
HPSO
0
0
0
0
0
+
50
SPSO
9.4288
e
-
04
1.0000
e
+
04
0.0078
3.6667
e
+
03
3.6667
e
+
03
HPSO
0
0
0
0
0
+
100
SPSO
1.0013
e
+
04
7.0017
e
+
04
4.0087
e
+
04
4.0698
e
+
04
2.0974
e
+
04
HPSO
0
10000
0
333.3333
1.8257
e
+
03
+
F
2
30
SPSO
6.8555
e
-
04
30.0018
10.0017
11.3364
10.0777
HPSO
0
0
0
0
0
+
50
SPSO
0.0329
70.0010
40.0006
37.3438
15.2918
HPSO
0
0
0
0
0
+
100
SPSO
51.0214
181.4054
110.5934
114.3039
29.0723
HPSO
0
0
0
0
0
+
F
3
30
SPSO
6.4613
e
+
03
3.7311
e
+
04
2.2333
e
+
04
2.1337
e
+
04
6.7035
e
+
03
HPSO
0
5
.
1779
e
+
03
0
172.5975
945.3557
+
50
SPSO
4.0023
e
+
04
1.0191
e
+
05
6.5660
e
+
04
7.0328
e
+
04
1.7603
e
+
04
HPSO
0
6
.
9787
e
+
03
0
232.6222
1
.
2741
e
+
03
+
100
SPSO
1.7694
e
+
05
3.0086
e
+
05
2.4789
e
+
05
2.4752
e
+
05
3.6623
e
+
04
HPSO
0
2
.
6987
e
+
04
0
3
.
8008
e
+
03
6
.
9150
e
+
03
+
F
4
30
SPSO
8.6091
21.2711
12.9945
13.3502
3.5341
HPSO
0
0
0
0
0
+
50
SPSO
24.2031
39.5127
31.0562
31.1715
4.2886
HPSO
0
0
0
0
0
+
100
SPSO
54.1172
75.3686
64.7834
64.2358
4.2202
HPSO
0
0
0
0
0
+
F
5
30
SPSO
0
10001
0
1.0005
e
+
03
3.0512
e
+
03
HPSO
0
0
0
0
0
+
50
SPSO
0
20004
4.5000
5.0028
e
+
03
6.8230
e
+
03
HPSO
0
0
0
0
0
+
100
SPSO
127
90040
40068
4.3086
e
+
04
2.2747
e
+
04
HPSO
0
0
0
0
0
+
F
6
30
SPSO
0.0344
18.8556
0.0959
3.5587
5.1400
HPSO
1
.
4522
e
-
04
0.0030
0.0012
0.0012
8
.
5738
e
-
04
+
50
SPSO
0.0780
72.6594
13.6489
19.6604
19.3860
HPSO
7.4623
e
-
05
0.0017
5
.
3645
e
-
04
6
.
3534
e
-
04
4
.
7283
e
-
04
+
100
SPSO
86.7855
381.9209
200.8146
211.9720
88.3159
HPSO
3
.
5210
e
-
05
0.0019
2
.
9387
e
-
04
4
.
0826
e
-
04
3
.
5395
e
-
04
+
F
7
30
SPSO
14.3237
1.0083
e
+
04
140.5176
2.4686
e
+
03
4.2581
e
+
03
HPSO
28.6353
28.9456
28.8793
28.8461
0.0932
+
50
SPSO
97.0317
9.4285
e
+
05
376.2306
3.4093
e
+
04
1.7169
e
+
05
HPSO
48.4886
48.8766
48.7600
48.7513
0.0875
+
100
SPSO
706.1328
2.8333
e
+
06
9.4375
e
+
05
8.8851
e
+
05
8.9157
e
+
05
HPSO
98.4280
98.8373
98.7133
98.7129
0.0818
+
F
8
30
SPSO
2
.
0226
e
+
03
4
.
8935
e
+
03
3
.
5787
e
+
03
3
.
6128
e
+
03
733.1063
HPSO
3.5886
e
+
03
8.0516
e
+
03
6.6047
e
+
03
6.3505
e
+
03
1.0893
e
+
03
−
50
SPSO
5
.
8499
e
+
03
9
.
7913
e
+
03
7
.
8862
e
+
03
7
.
7139
e
+
03
1
.
0101
e
+
03
HPSO
6.5496
e
+
03
1.4460
e
+
04
1.1191
e
+
04
1.0866
e
+
04
2.1757
e
+
03
−
100
SPSO
1.8110
e
+
04
2
.
4259
e
+
04
2
.
0949
e
+
04
2
.
1084
e
+
04
1
.
7384
e
+
03
HPSO
1
.
2615
e
+
04
3.1402
e
+
04
2.4302
e
+
04
2.4077
e
+
04
4.9510
e
+
03
−
F
9
30
SPSO
28.7299
160.3815
87.6754
92.5142
32.6994
HPSO
0
0
0
0
0
+
50
SPSO
175.2643
351.6480
260.4359
258.0518
48.4078
HPSO
0
0
0
0
0
+
100
SPSO
555.8950
993.3887
750.1694
749.1658
749.1658
HPSO
0
0
0
0
0
+
F
10
30
SPSO
61.4129
221.0445
132.7694
134.5414
33.8073
HPSO
0
0
0
0
0
+
50
SPSO
157.1020
440.0897
324.2632
310.3595
64.3675
HPSO
0
0
0
0
0
+
100
SPSO
623.5658
1.0433
e
+
03
804.6981
813.3435
88.5932
HPSO
0
25
0
0.8333
4.5644
+Table 3
Experimental results obtained by SPSO and HPSO on functions fromF
11 to F
20.
Fun
Dim
Best
Worst
Median
Mean
SD
Significant
F
11
30
SPSO
0.0043
19.9630
0.0595
2.3935
5.4041
HPSO
8.8818
e
-
16
8.8818
e
-
16
8.8818
e
-
16
8.8818
e
-
16
0
+
50
SPSO
0.0598
19.9646
12.6912
10.5673
6.3042
HPSO
8.8818
e
-
16
8.8818
e
-
16
8.8818
e
-
16
8.8818
e
-
16
0
+
100
SPSO
15.4237
20.2143
19.5200
19.4135
0.8672
HPSO
8.8818
e
-
16
8.8818
e
-
16
8.8818
e
-
16
8.8818
e
-
16
0
+
F
12
30
SPSO
7.0274
e
-
04
90.8935
0.0178
12.0794
31.2763
HPSO
0
0
0
0
0
+
50
SPSO
0.0014
270.8170
0.0415
45.1971
70.1274
HPSO
0
0
0
0
0
+
100
SPSO
1.1140
721.0594
361.0858
376.1758
158.6584
HPSO
0
0
0
0
0
+
F
13
30
SPSO
0.1403
4.3952
0.3210
1.0567
1.4863
HPSO
0
0
0
0
0
+
50
SPSO
0.8657
15.2389
7.5828
8.2388
3.6607
HPSO
0
0
0
0
0
+
100
SPSO
27.6235
64.4826
49.3984
47.7138
10.0126
HPSO
0
0
0
0
0
+
F
14
30
SPSO
6.4114
e
-
05
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
5.1200
e
+
08
457.9143
7.6801
e
+
07
1.5257
e
+
08
HPSO
0.1866
0.5097
0.2703
0.2736
0.0653
+
F
15
30
SPSO
−3.0000
−2.8522
−3.0000
−2.9507
0.0709
HPSO
−3
−3
−3
−3
0
+
50
SPSO
−5.0000
−2.3044
−4.4827
−4.2127
0.6865
HPSO
−5
−5
−5
−5
0
+
100
SPSO
−7.9165
4.7637
−5.2127
−4.6977
2.8465
HPSO
−10
−10
−10
−10
0
+
F
16
30
SPSO
2.3604
e
+
03
3.8233
e
+
04
3.8233
e
+
04
1.2375
e
+
04
9.2463
e
+
03
HPSO
0
5.8369
e
+
03
0
390.6710
1.2756
e
+
03
+
50
SPSO
7.1213
e
+
03
1.0427
e
+
05
3.3195
e
+
04
3.4891
e
+
04
2.2914
e
+
04
HPSO
0
4.0529
e
+
03
0
224.6749
873.6249
+
100
SPSO
6.2317
e
+
04
2.7386
e
+
05
1.4222
e
+
05
1.4697
e
+
05
5.7699
e
+
04
HPSO
0
1.9403
e
+
04
0
1.0583
e
+
03
3.8088
e
+
03
+
F
17
30
SPSO
6.7986
e
+
03
9.7587
e
+
03
8.3387
e
+
03
8.2508
e
+
03
739.7223
HPSO
8.3590
e
+
03
9.8803
e
+
03
9.0866
e
+
03
9.0790
e
+
03
442.4330
−
50
SPSO
1.3020
e
+
04
1.7080
e
+
04
1.4999
e
+
04
1.5149
e
+
04
1.0581
e
+
03
HPSO
1.5003
e
+
04
1.7349
e
+
04
1.6514
e
+
04
1.6310
e
+
04
669.3538
−
100
SPSO
2.7400
e
+
04
2.7400
e
+
04
3.1087
e
+
04
3.1149
e
+
04
2.1654
e
+
03
HPSO
3.0329
e
+
04
3.5493
e
+
04
3.4226
e
+
04
3.3586
e
+
04
1.5320
e
+
03
−
F
18
30
SPSO
20.7888
21.0951
21.0053
20.9848
0.0712
HPSO
8.8818
e
-
16
21.1210
20.9931
11.2354
10.6894
+
50
SPSO
21.0515
21.2478
21.1455
21.1436
0.0536
HPSO
8.8818
e
-
16
21.2404
21.1366
12.0016
10.6745
+
100
SPSO
21.2367
21.3931
21.3368
21.3358
0.0364
HPSO
8.8818
e
-
16
21.3949
21.3545
15.6658
9.6084
+
F
19
30
SPSO
1.0517
495.3131
273.6408
243.6176
154.3551
HPSO
0
0
0
0
0
+
50
SPSO
265.0558
1.4393
e
+
03
798.8065
786.0782
289.8401
HPSO
0
0
0
0
0
+
100
SPSO
1.9937
e
+
03
4.0158
e
+
03
2.9388
e
+
03
2.9263
e
+
03
543.9053
HPSO
0
0
0
0
0
+
F
20
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
+Table 4
Experimental results obtained by SPSO and HPSO on functions fromF
21 to F
28.
Fun
Dim
Best
Worst
Median
Mean
SD
Significant
F
21
30
SPSO
67.1541
307.3070
213.8939
203.8842
61.8125
HPSO
0
0
0
0
0
+
50
SPSO
158.2955
715.0245
518.1705
500.5593
135.5998
HPSO
0
269.3463
0
8.9782
49.1757
+
100
SPSO
1.0850
e
+
03
1.9021
e
+
03
1.5793
e
+
03
1.5669
e
+
03
190.5584
HPSO
0
582.0882
0
35.5882
136.0270
+
F
22
30
SPSO
0.7999
14.9999
1.2522
2.9025
4.3553
HPSO
0
0
0
0
0
+
50
SPSO
2.0999
26.0999
13.9628
12.8291
6.9033
HPSO
0
0
0
0
0
+
100
SPSO
16.5013
41.9999
35.4551
33.9791
6.3075
HPSO
0
0
0
0
0
+
F
23
30
SPSO
81.0577
4.0119
e
+
09
2.0685
e
+
08
6.8745
e
+
08
1.0469
e
+
09
HPSO
28.8214
28.9856
28.9323
28.9252
0.0421
+
50
SPSO
3.7253
e
+
03
2.1495
e
+
10
3.6515
e
+
09
3.6515
e
+
09
5.3957
e
+
09
HPSO
48.7069
48.8900
48.8205
48.8139
0.0479
+
100
SPSO
6.7997
e
+
09
9.2655
e
+
10
3.3160
e
+
10
3.8223
e
+
10
2.0050
e
+
10
HPSO
98.6590
98.8846
98.8109
98.7983
0.0545
+
F
24
30
SPSO
6.2312
e
+
08
2.3418
e
+
10
4.9110
e
+
09
5.8767
e
+
09
5.6099
e
+
09
HPSO
5.9432
e
+
05
6.2859
e
+
09
7.6373
e
+
06
3.7982
e
+
08
1.2316
e
+
09
+
50
SPSO
4.3540
e
+
09
3.3195
e
+
10
1.3961
e
+
10
1.6077
e
+
10
8.3270
e
+
09
HPSO
3.9454
e
+
06
8.9387
e
+
09
3.1766
e
+
07
7.0962
e
+
08
1.9565
e
+
09
+
100
SPSO
4.9031
e
+
10
1.5465
e
+
11
9.1986
e
+
10
9.7151
e
+
10
2.8460
e
+
10
HPSO
2.0551
e
+
08
5.4553
e
+
09
6.7593
e
+
08
1.1373
e
+
09
1.2367
e
+
09
+
F
25
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
+
F
26
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
+
F
27
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
−
F
28
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 Tables2–4, we can clearly observe that the convergence accuracy of HPSO is better than SPSO on the most benchmark functions. An interesting result is that HPSO can find the global optimal solutions on functions F
2, F
4, F
5, F
9, F
12, F
13, F
15, F
19, and f
22 in all search spaces; that is to say, HPSO can obtain the 100% success rate on the listed functions. Considering F
1 and F
10, though HPSO can find the global optimal solutions in all different search ranges, it only obtained the mean values 333.3333 and 0.8333, respectively, in 100-dimensional space. At the same time, HPSO offers the higher convergence accuracy on functions F
3, F
6, F
7, F
11, F
14, F
16, F
20, F
21, F
23, and F
26. However, we must observe that SPSO has higher performance on function F
8. As for F
25, SPSO has better performance in 30-dimensional search space, but HPSO has better performance in 50- and 100-dimensional search spaces. As for shifted rotated functions F
27 and F
28, both SPSO and HPSO have worst convergence accuracy. As seen, the dimension of the selected functions has great effect on SPSO. For example, considering function F
1, SPSO has mean value 666.6686, 3.6667e + 03, and 4.0698e + 04 in 30-dimensional, 50-dimensional, and 100-dimensional search spaces, respectively, while HPSO has mean values 0, 0, and 333.333 in the corresponding search space. Therefore, we also conclude that HPSO has better stability than SPSO from the data in different search spaces.In the 9th columns of Tables2–4, we report the statistical significance level of the difference of the means of the two algorithms. Note that here “+” indicates that the t value is significant at a 0.05 level of significance by two-tailed test, and “−” stands for the difference of means that is not statistically significant.Figure4 graphically presents the comparison in terms of convergence characteristics of the evolutionary processes in solving the selected benchmark functions in 30-dimensional search space with N
=
30 and T
=
1000.Convergence comparison of HPSO and SPSO on the selected test functions withD
=
30
,
N
=
30, and T
=
1000.
(a)
f
1
(b)
f
3
(c)
f
5
(d)
f
7
(e)
f
10
(f)
f
14
(g)
f
16
(h)
f
18
(i)
f
20
(j)
f
24
(k)
f
26
(l)
f
28


## 4.3. Comparison of HPSO with Other PSO Algorithms

In this section, a comparison of HPSO with some well-known PSO algorithms which are listed in Table5 is performed to evaluate the efficiency of the proposed algorithm.Table 5
Some well-known PSOs algorithms in the literature.
Algorithm
Year
Topology
Parameter settings
GPSO
1998
Global star
w: 0.9
-
0.4, c
1
=
c
2
=
2.0
LPSO
2002
Local ring
w: 0.9
-
0.4, c
1
=
c
2
=
2.0
FIPS
2004
Local Uring
χ
=
0.729, ∑
c
i
=
4.1
HPSO-TVAC
2004
Global star
w: 0.9
-
0.4, c
1
:
2.5
-
0.5, and c
2: 0.5
-
2.5
UPSO
2004
Global star
w: 0.9
-
0.4, c
1
=
c
2
=
2.0, and U
=
0.5
DMS-PSO
2005
Dynamic multiswarm
w: 0.9
-
0.2, c
1
=
c
2
=
2.0, m
=
3, and R
=
5
VPSO
2006
Local Von Neumann
w: 0.9
-
0.4, c
1
=
c
2
=
2.0
CLPSO
2006
Comprehensive learning
w: 0.9
-
0.4, c
=
1.49445, and m
=
7
QIPSO
2007
Global star
w: 0.9
-
0.4, c
1
=
c
2
=
2.0
APSO
2009
Global star
w: 0.9, c
1
=
c
2
=
2.0; δ
:
random  in
[
0.05,0.1
], σ: 1
-
0.1
AFPSO
2011
Global star
w: 0.9
-
0.4, c
1, and c
2 are based on fuzzy rule
AFPSO-QI
2011
Global star
w: 0.9
-
0.4, c
1, and c
2 are based on fuzzy ruleAt first, we choose 10 unimodal and multimodal test functions for this evaluation. According to [22], the algorithms GPSO [2], LPSO [16], VPSO [27], FIPS [28], HPSO-TVAC [6], DMS-PSO [29], CLPSO [24], and APSO [22] are considered as detailed in Table 5. The experimental results of the algorithms are directly from [22] as shown in Table 6. In this trial, the population size N
=
20, the dimension D
=
30, and the maximum fitness evaluations (FEs) were set to 2
×
1
0
5 also. The parameter configurations of the selected algorithms have been set according to their corresponding references. The inertia weight w is linearly decreased from 0.9 to 0.4 in HPSO. HPSO is independently run 30 times and the mean and SD are shown in Table 6. As seen, HPSO has the first rank among the algorithms and obtains the global minimum on functions F
1, F
2, F
5, F
9, F
10, and F
12 and gives the good near-global optima on functions F
6 and F
11. Meanwhile, HPSO has the worst performance on functions F
3 and F
14. As for F
3, APSO has the best convergence accuracy, and HPSO only wins CLPSO. Considering F
14, DMS-PSO has the best performance.Table 6
Comparison results of eight PSO algorithms [22] with HPSO on 10 functions (N
=
20, D
=
30, and FEs
=
2
×
1
0
5).
Function
GPSO
LPSO
VPSO
FIPS
HPSO-TVAC
DMS-PSO
CLPSO
APSO
HPSO
F
1
Mean
1.98
e
-
53
4.77
e
-
29
5.11
e
-
38
3.21
e
-
30
3.38
e
-
41
3.85
e
-
54
1.89
e
-
19
1.45
e
-
150
0
SD
7.08
e
-
53
1.13
e
-
28
1.91
e
-
37
3.60
e
-
30
8.50
e
-
41
1.75
e
-
53
1.49
e
-
19
5.73
e
-
150
0
Rank
4
8
6
7
5
3
9
2
1
F
2
Mean
2.51
e
-
34
2.03
e
-
20
6.29
e
-
27
1.32
e
-
17
6.9
e
-
23
2.61
e
-
29
1.01
e
-
13
5.15
e
-
84
0
SD
5.84
e
-
34
2.89
e
-
20
8.68
e
-
27
7.86
e
-
18
6.89
e
-
23
6.6
e
-
29
6.51
e
-
14
1.44
e
-
83
0
Rank
3
7
5
8
6
4
9
2
1
F
3
Mean
6.45
e
-
2
18.60
1.44
0.77
2.89
e
-
7
47.5
395
1.0
e
-
10
167
SD
9.45
e
-
2
30.71
1.55
0.86
2.97
e
-
7
56.4
142
2.13
e
-
10
913
Rank
3
6
5
4
2
7
9
1
8
F
5
Mean
0
0
0
0
0
0
0
0
0
SD
0
0
0
0
0
0
0
0
0
Rank
1
1
1
1
1
1
1
1
1
F
6
Mean
7.77
e
-
3
1.49
e
-
2
1.08
e
-
2
2.55
e
-
3
5.54
e
-
2
1.1
e
-
2
3.92
e
-
3
4.66
e
-
3
1.03
e
-
04
SD
2.42
e
-
3
5.66
e
-
3
3.24
e
-
3
6.25
e
-
4
2.08
e
-
2
3.94
e
-
3
1.14
e
-
3
1.7
e
-
3
8.99
e
-
05
Rank
5
8
6
2
9
7
3
4
1
F
9
Mean
30.7
34.90
34.09
29.98
2.39
28.1
2.57
e
-
11
5.8
e
-
15
0
SD
8.68
7.25
8.07
10.92
3.71
6.42
6.64
e
-
11
1.01
e
-
14
0
Rank
7
9
8
6
4
5
3
2
1
F
10
Mean
15.5
30.40
21.33
35.91
1.83
32.8
0.167
4.14
e
-
16
0
SD
7.4
9.23
9.46
9.49
2.65
6.49
0.379
1.45
e
-
15
0
Rank
5
7
6
9
4
8
3
2
1
F
11
Mean
1.15
e
-
14
1.85
e
-
14
1.4
e
-
14
7.69
e
-
15
2.06
e
-
10
8.52
e
-
15
2.01
e
-
12
1.11
e
-
14
8.88
e
-
16
SD
2.27
e
-
15
4.80
e
-
15
3.48
e
-
15
9.33
e
-
16
9.45
e
-
10
1.79
e
-
15
9.22
e
-
13
3.55
e
-
15
0
Rank
5
7
6
2
9
3
8
4
1
F
12
Mean
2.37
e
-
2
1.10
e
-
2
1.31
e
-
2
9.04
e
-
4
1.07
e
-
2
1.31
e
-
2
6.45
e
-
13
1.67
e
-
2
0
SD
2.57
e
-
2
1.60
e
-
2
1.35
e
-
2
2.78
e
-
3
1.14
e
-
2
1.73
e
-
2
2.07
e
-
12
2.41
e
-
2
0
Rank
9
5
6
3
4
7
2
8
1
F
14
Mean
1.04
e
-
2
2.18
e
-
30
3.46
e
-
3
1.22
e
-
31
7.07
e
-
30
2.05
e
-
32
1.59
e
-
21
3.76
e
-
31
1.70
e
-
2
SD
3.16
e
-
2
5.14
e
-
30
1.89
e
-
2
4.85
e
-
32
4.05
e
-
30
8.12
e
-
33
1.93
e
-
21
1.2
e
-
30
1.42
e
-
2
Rank
8
4
7
2
5
1
6
3
9
Average rank
5
6.2
5.6
4.4
4.9
4.6
5.3
2.9
2.5
Final rank
6
9
8
3
5
4
7
2
1Then, in the next step, we choose six functions from [25] and seven algorithms of GPSO, QIPSO [30], UPSO [31], FIPS, AFSO [25], and AFSO-Q1 [25] as detailed in Table 5. For a fair comparison, the population size N
=
30, the dimension D
=
30, and the maximum iteration T
=
1
0,000 also in HPSO, and the inertia weight w is linearly decreased from 0.9 to 0.4. HPSO is independently run 30 times and the mean and SD are shown in Table 7. As seen, HPSO shows better performance and has the first rank. HPSO finds the global optimal solution on functions F
9, F
13, F
21, and F
22. FIPS and UPSO have better convergence accuracy on functions F
27 and F
28, respectively.Table 7
Comparison results of seven PSO algorithms [25] with HPSO on six functions (N
=
30, D
=
30, and T = 10,000).
Function
SPSO
QIPSO
UPSO
FIPS
CLPSO
AFSO
AFSO-Q1
HPSO
F
9
Mean
52.30
25.61
59.40
106.1
74.39
17.93
15.69
0
SD
27.35
15.98
58.05
30.54
9.77
5.63
4.47
0
Rank
5
4
6
8
7
3
2
1
F
13
Mean
0.534
36.38
8.70
6.40
1.39
e
-
03
4.52
e
-
03
1.50
e
-
03
0
SD
1.74
4.66
3.08
3.04
3.28
e
-
04
9.20
e
-
03
3.48
e
-
03
0
Rank
5
8
7
6
2
4
3
1
F
21
Mean
320.2
317.5
309.5
434.1
263.3
266.3
253.3
0
SD
14.70
23.24
25.88
34.99
11.96
12.00
12.63
0
Rank
7
6
5
8
3
4
2
1
F
22
Mean
17.03
15.20
14.29
26.60
11.94
10.38
8.46
0
SD
2.55
1.32
2.15
1.42
1.37
1.38
0.948
0
Rank
7
6
5
8
4
3
2
1
F
27
Mean
−119.10
−119.10
−119.10
−119.90
−119.00
−119.70
−119.80
−119.05
SD
7.09
e
-
02
5.68
e
-
02
3.24
e
-
02
3.78
e
-
02
4.28
e
-
02
3.85
e
-
02
5.45
e
-
02
5.50
e
-
02
Rank
4
4
4
1
6
3
2
5
F
28
Mean
115.90
121.90
113.20
113.60
118.30
123.20
123.10
117.32
SD
2.90
4.90
6.14
3.63
2.40
2.25
3.01
3.65
Rank
3
6
1
2
5
8
7
4
Average rank
5.17
5.67
4.67
5.50
4.50
4.17
3.00
2.17
Final rank
6
8
5
7
4
3
2
1Therefore, it is worth saying that the proposed algorithm has considerably better performance than the other well-known PSO algorithms in unimodal and multimodal high-dimensional functions.


## 5. Conclusion

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 coefficientsc
1 and c
2 have been replaced with two random numbers, whose sum is equal to  1 in [
0,1
]; this strategy decreases the dependence on parameters of the solved problems. The proposed algorithm has been evaluated on 28 benchmark functions including unimodal, unrotated multimodal, rotated, shifted, and shifted rotated functions, and the experimental results confirm the high performance of HPSO on the main functions. However, as seen, HPSO has the worst performance on shifted rotated functions, so it is worth researching how to enhance the performance of HPSO on shifted rotated functions in the future. Meanwhile, applying HPSO to solve real-world problems is also a research field.



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*Source: 194706-2014-04-17.xml*