# Fusion Global-Local-Topology Particle Swarm Optimization for Global Optimization Problems

Authors: Zahra Beheshti; Siti Mariyam Shamsuddin; Sarina Sulaiman
Journal: Mathematical Problems in Engineering
 (2014)

---

## Abstract

In recent years, particle swarm optimization (PSO) has been extensively applied in various optimization problems because of its structural and implementation simplicity. However, the PSO can sometimes find local optima or exhibit slow convergence speed when solving complex multimodal problems. To address these issues, an improved PSO scheme called fusion global-local-topology particle swarm optimization (FGLT-PSO) is proposed in this study. The algorithm employs both global and local topologies in PSO to jump out of the local optima. FGLT-PSO is evaluated using twenty (20) unimodal and multimodal nonlinear benchmark functions and its performance is compared with several well-known PSO algorithms. The experimental results showed that the proposed method improves the performance of PSO algorithm in terms of solution accuracy and convergence speed.


---

## Body

## 1. Introduction

PSO is a population-based metaheuristic algorithm introduced by Kennedy and Eberhart [1] in 1995. The algorithm imitates the social behavior of bird flocking or fish schooling to find the global best solution. Due to the simple concept, having a few parameters and being easy to implement, PSO has received much more attention to solve real-world optimization problems [2–6] in recent years. Nevertheless, PSO may easily get trapped in local optima when solving complex multimodal problems [7]. Hence, a number of variant PSO algorithms have been proposed in the literature to avoid the local optima and to find the best solution promptly.The algorithm applies two different topologies to find a good solution: global and local topologies. In global topology, the position of each particle is affected by the best-fitness particles of the entire population in the search space while each particle is influenced by the best-fitness particles of its neighborhood in the local topology. Kennedy and Mendes proposed local (ring) topological structure PSO (LPSO) [8] and the Von Neumann topological structure PSO (VPSO) [9]. Mendes et al. [10] introduced the fully informed particle swarm (FIPS) algorithm and Ratnaweera et al. [11] suggested self-organizing hierarchical particle swarm optimizer with time-varying acceleration coefficients (HPSO-TVAC). Other researchers presented the several variants of PSO algorithms such as dynamic multiswarm PSO (DMS-PSO) [12], comprehensive learning PSO (CLPSO) [13], median-oriented particle swarm optimization (MPSO) [14], centripetal accelerated particle swarm optimization (CAPSO) [15], quadratic interpolation PSO (QIPSO) [16], quantum-behaved particle swarm optimization (QPSO) [17], and adaptive particle swarm optimization (APSO) [18].Although the aforementioned algorithms have obtained satisfactory results, there are still some disadvantages in their utilization. For example, LPSO presents a slow convergence rate in unimodal functions [14, 15] or CLPSO is not good for solving unimodal problems [13]. Moreover, some of the algorithms have a better performance than PSO but their structures are not as simple as PSO.To overcome the disadvantages, this study introduces fusion global-local-topology particle swarm optimization (FGLT-PSO). The proposed algorithm performs a global search over the entire search space with a fast convergence speed using hybridizing two local and global topologies in PSO to jump out from local optima.The remainder of this paper is organized as follows. In Section2, a brief review of PSO is provided followed by some well-known PSO algorithms. The proposed algorithm is described in Section 3 in detail. In Section 4, FGLT-PSO is used to solve several benchmark functions and its performance is compared with the other PSO algorithms in the literature. Finally, conclusions and the future research directions are presented in Section 5.


## 2. Particle Swarm Optimization (PSO)

### 2.1.  PSO Framework

The PSO algorithm is a population-based metaheuristic algorithm that applies two approaches of global exploration and local exploitation to find the optimum solution. The exploration is the ability of expanding search space, where the exploitation is the ability of finding the optima around a good solution. The algorithm is initialized by creating a swarm, that is, population of particles, with random positions. Every particle is shown as a vector(
X
→
i
,
V
→
i
,
P
→
i
) in a d
-dimensional search space where X
→
i and V
→
i are the position and velocity, respectively, and P
→
i is the personal best position (
P
best
) found by the ith particle:
(1)
X
→
i
=
(
x
i
1
,
x
i
2
,
…
,
x
i
d
)
for
i
=
1,2
,
…
,
N
,
V
→
i
=
(
v
i
1
,
v
i
2
,
…
,
v
i
d
)
for
i
=
1,2
,
…
,
N
,
P
→
i
=
(
p
i
1
,
p
i
2
,
…
,
p
i
d
)
for
i
=
1,2
,
…
,
N
.
In addition, the best position obtained by the entire population (P
→
g) is computed to update the particle velocity:
(2)
P
→
g
=
(
p
g
1
,
p
g
2
,
…
,
p
g
d
)
.
Based on P
→
i and P
→
g, the next velocity and position of the ith particle are computed using (3) and (4) as follows:
(3)
v
i
d
(
t
+
1
)
=
w
×
v
i
d
(
t
)
+
C
1
×
rand
1
×
(
p
i
d
(
t
)
-
x
i
d
(
t
)
)
+
C
2
×
rand
2
×
(
p
g
d
(
t
)
-
x
i
d
(
t
)
)
,
(4)
x
i
d
(
t
+
1
)
=
x
i
d
(
t
)
+
v
i
d
(
t
+
1
)
,
where v
i
d
(
t
+
1
) and v
i
d
(
t
) are the next and current velocity of the ith particle, respectively. w is inertia weight, C
1 and C
2 are acceleration coefficients, and rand1 and rand2 are random numbers in the interval [
0
,
1
]. N is the number of particles; x
i
d
(
t
+
1
) and x
i
d
(
t
) are the next and current position of the ith particle.Also,|
v
i
d
(
t
+
1
)
|
<
v
max
⁡ and v
max
⁡ is set to a constant bounded based on the search space bound. A larger value of w encourages global exploration (searching new areas), while a smaller value provides a local exploitation.In (3), the second and the third terms are called cognition and social term, respectively. The two models applied to choose P
→
g are known as g
best (for global topology) and l
best (for local topology) models. In this paper, the g
best model and l
best model are called PSO and LPSO, respectively.


### 2.2. Improved PSO Algorithms

Since Kennedy and Eberhart introduced PSO algorithm, the algorithm and its improved schemes have been extensively applied in many problems [20–25]. Many researchers have proposed the variants of modified PSO through swarm topology [8, 9], parameter selection [19, 26], combining PSO with other evolutionary computation (EC) techniques [27, 28], integration of its self-adaptation [29], and so on.LPSO [8] and VPSO [9] were proposed based on a local topology to avoid premature convergence rate in solving multimodal problems. FIPS algorithm [10] is another PSO algorithm which uses the information of the entire neighborhood to guide the particles for finding the best solution. Dynamic multiswarm PSO (DMS-PSO) [12] was suggested by Liang and Suganthan to dynamically enhance the topological structure. Ratnaweera et al. [11] proposed HPSO-TVAC algorithm based on linearly time-varying acceleration coefficients where a larger C
1 and a smaller C
2 are set at the beginning and gradually reversed throughout the search. Liang et al. [13] presented comprehensive learning particle swarm optimization (CLPSO) which focused on avoiding the local optima by encouraging each particle to learn its behavior from other particles on different dimensions.In another research, a selection operator for PSO was first introduced by Angeline [30]. It is similar to what was used in a genetic algorithm (GA). Other researchers used a part of crossover [31] and mutation [29] operations from GA into PSO. Pant et al. proposed a quadratic crossover operator to PSO algorithm called quadratic interpolation PSO (QIPSO) [16]. An adaptive fuzzy particle swarm optimization (AFPSO) [19] was proposed to utilize fuzzy inferences for adjusting acceleration coefficients. Meanwhile, the quadratic crossover operator [16] was used in the proposed AFPSO algorithm (AFPSO-QI) [19] to have better performance in solving multimodal problems. Zhan et al. presented an adaptive particle swarm optimization (APSO) [18] using a real-time evolutionary state estimation procedure and an elitist learning strategy. A variant of PSO algorithm based on orthogonal learning strategy (OLPSO) [32] was introduced to guide particles for discovering useful information from their personal best positions and from their neighborhood’s best position in order to fly in better directions. Gao et al. [33] used PSO with chaotic opposition-based population initialization and stochastic search technique to solve complex multimodal problems. The algorithm called CSPSO finds new solutions in the neighborhoods of the previous best positions in order to escape from local optima in multimodal functions. Beheshti et al. proposed median-oriented particle swarm optimization (MPSO) [14] and centripetal accelerated particle swarm optimization (CAPSO) [15] based on Newton’s laws of motion to accelerate the learning and convergence of optimization problems.


## 2.1.  PSO Framework

The PSO algorithm is a population-based metaheuristic algorithm that applies two approaches of global exploration and local exploitation to find the optimum solution. The exploration is the ability of expanding search space, where the exploitation is the ability of finding the optima around a good solution. The algorithm is initialized by creating a swarm, that is, population of particles, with random positions. Every particle is shown as a vector(
X
→
i
,
V
→
i
,
P
→
i
) in a d
-dimensional search space where X
→
i and V
→
i are the position and velocity, respectively, and P
→
i is the personal best position (
P
best
) found by the ith particle:
(1)
X
→
i
=
(
x
i
1
,
x
i
2
,
…
,
x
i
d
)
for
i
=
1,2
,
…
,
N
,
V
→
i
=
(
v
i
1
,
v
i
2
,
…
,
v
i
d
)
for
i
=
1,2
,
…
,
N
,
P
→
i
=
(
p
i
1
,
p
i
2
,
…
,
p
i
d
)
for
i
=
1,2
,
…
,
N
.
In addition, the best position obtained by the entire population (P
→
g) is computed to update the particle velocity:
(2)
P
→
g
=
(
p
g
1
,
p
g
2
,
…
,
p
g
d
)
.
Based on P
→
i and P
→
g, the next velocity and position of the ith particle are computed using (3) and (4) as follows:
(3)
v
i
d
(
t
+
1
)
=
w
×
v
i
d
(
t
)
+
C
1
×
rand
1
×
(
p
i
d
(
t
)
-
x
i
d
(
t
)
)
+
C
2
×
rand
2
×
(
p
g
d
(
t
)
-
x
i
d
(
t
)
)
,
(4)
x
i
d
(
t
+
1
)
=
x
i
d
(
t
)
+
v
i
d
(
t
+
1
)
,
where v
i
d
(
t
+
1
) and v
i
d
(
t
) are the next and current velocity of the ith particle, respectively. w is inertia weight, C
1 and C
2 are acceleration coefficients, and rand1 and rand2 are random numbers in the interval [
0
,
1
]. N is the number of particles; x
i
d
(
t
+
1
) and x
i
d
(
t
) are the next and current position of the ith particle.Also,|
v
i
d
(
t
+
1
)
|
<
v
max
⁡ and v
max
⁡ is set to a constant bounded based on the search space bound. A larger value of w encourages global exploration (searching new areas), while a smaller value provides a local exploitation.In (3), the second and the third terms are called cognition and social term, respectively. The two models applied to choose P
→
g are known as g
best (for global topology) and l
best (for local topology) models. In this paper, the g
best model and l
best model are called PSO and LPSO, respectively.


## 2.2. Improved PSO Algorithms

Since Kennedy and Eberhart introduced PSO algorithm, the algorithm and its improved schemes have been extensively applied in many problems [20–25]. Many researchers have proposed the variants of modified PSO through swarm topology [8, 9], parameter selection [19, 26], combining PSO with other evolutionary computation (EC) techniques [27, 28], integration of its self-adaptation [29], and so on.LPSO [8] and VPSO [9] were proposed based on a local topology to avoid premature convergence rate in solving multimodal problems. FIPS algorithm [10] is another PSO algorithm which uses the information of the entire neighborhood to guide the particles for finding the best solution. Dynamic multiswarm PSO (DMS-PSO) [12] was suggested by Liang and Suganthan to dynamically enhance the topological structure. Ratnaweera et al. [11] proposed HPSO-TVAC algorithm based on linearly time-varying acceleration coefficients where a larger C
1 and a smaller C
2 are set at the beginning and gradually reversed throughout the search. Liang et al. [13] presented comprehensive learning particle swarm optimization (CLPSO) which focused on avoiding the local optima by encouraging each particle to learn its behavior from other particles on different dimensions.In another research, a selection operator for PSO was first introduced by Angeline [30]. It is similar to what was used in a genetic algorithm (GA). Other researchers used a part of crossover [31] and mutation [29] operations from GA into PSO. Pant et al. proposed a quadratic crossover operator to PSO algorithm called quadratic interpolation PSO (QIPSO) [16]. An adaptive fuzzy particle swarm optimization (AFPSO) [19] was proposed to utilize fuzzy inferences for adjusting acceleration coefficients. Meanwhile, the quadratic crossover operator [16] was used in the proposed AFPSO algorithm (AFPSO-QI) [19] to have better performance in solving multimodal problems. Zhan et al. presented an adaptive particle swarm optimization (APSO) [18] using a real-time evolutionary state estimation procedure and an elitist learning strategy. A variant of PSO algorithm based on orthogonal learning strategy (OLPSO) [32] was introduced to guide particles for discovering useful information from their personal best positions and from their neighborhood’s best position in order to fly in better directions. Gao et al. [33] used PSO with chaotic opposition-based population initialization and stochastic search technique to solve complex multimodal problems. The algorithm called CSPSO finds new solutions in the neighborhoods of the previous best positions in order to escape from local optima in multimodal functions. Beheshti et al. proposed median-oriented particle swarm optimization (MPSO) [14] and centripetal accelerated particle swarm optimization (CAPSO) [15] based on Newton’s laws of motion to accelerate the learning and convergence of optimization problems.


## 3. FGLT-PSO: The Proposed Method

### 3.1. FGLT-PSO Algorithm

FGLT-PSO tends to overcome the disadvantages of PSO by avoiding local optima and accelerating convergence speed. According to [14, 15], PSO has shown a better performance than LPSO in unimodal problems and LPSO illustrates good results in multimodal problems. Hence, both local and global topologies are hybridized in FGLT-PSO to increase the convergence rate and to avoid trapping into local optima.In FGLT-PSO algorithm, each particle uses the best position found by its neighbors (P
→
l
best) to update the particles’ velocities:
(5)
P
→
l
best
=
(
p
l
best
1
,
p
l
best
2
,
…
,
p
l
best
d
)
,
(6)
v
i
d
(
t
+
1
)
=
w
(
t
)
×
v
i
d
(
t
)
+
C
1
(
t
)
×
rand
1
×
(
p
i
d
(
t
)
-
x
i
d
(
t
)
)
+
C
2
(
t
)
×
rand
2
×
(
p
l
best
d
(
t
)
-
x
i
d
(
t
)
)
.
The next position of each particle is computed based on the current position, x
i
d
(
t
), the next velocity, v
i
d
(
t
+
1
), and the best position found by the swarm, P
→
g
best, as follows:
(7)
x
i
d
(
t
+
1
)
=
x
i
d
(
t
)
+
v
i
d
(
t
+
1
)
+
C
3
(
t
)
×
rand
3
×
(
p
g
best
d
(
t
)
-
x
i
d
(
t
)
)
,
(8)
P
→
g
best
=
(
p
g
best
1
,
p
g
best
2
,
…
,
p
g
best
d
)
.
In (6), w is computed as
(9)
w
(
t
)
=
w
max
⁡
-
t
×
(
w
max
⁡
-
w
min
⁡
)
T
.
Also, C
1
(
t
)
,
C
2
(
t
), and C
3
(
t
) are acceleration coefficients and modified according to (10):
(10)
C
j
(
t
)
=
C
j
min
⁡
+
t
×
(
C
j
max
⁡
-
C
j
min
⁡
)
T
,
j
=
1,2
,
3
,
where t and T are the current iteration and the number of maximum iterations, respectively.The second term in (6) is called the cognition term, and the third terms in (6) and (7) are named the social terms. In (7), |
x
i
d
(
t
+
1
)
|
<
x
max
⁡ and x
max
⁡ is set to a constant based on the search space bound.


### 3.2. Analysis of FGLT-PSO

A metaheuristic algorithm explores new spaces to avoid trapping in a local optimum in the initial steps. Due to the poor exploration in the standard PSO (PSO), it can sometimes find local optima in multimodal problems. Sometimes, if a particle falls into a local optimum, it will not be able to get out of it. That is, ifP
→
g
best obtained through the population lies in a local optimum while the current position and the personal best position of particle i are in the same local optimum, the second and third terms of (3) tend to zero and w decreases linearly to near zero. Consequently, the next velocity of particle i tends to zero, and its next position in (4) does not change; thus, the particle remains in the local optimum. Hence, the main aim in FGLT-PSO is to overcome the poor exploration and to increase the convergence rate by combining the local and global searches as shown in Figure 1. The particles move in the search space based on the best solutions found by their neighbors (P
→
l
best) and the swarm (P
→
g
best). At the beginning, the particles search new spaces. By lapse of iterations, the exploration should fade out and the exploitation should fade in. It means the particles accelerate to the good solution and make search around it to find the best solution.Figure 1
Graphical representation of the particle movement using FGLT-PSO algorithm.


## 3.1. FGLT-PSO Algorithm

FGLT-PSO tends to overcome the disadvantages of PSO by avoiding local optima and accelerating convergence speed. According to [14, 15], PSO has shown a better performance than LPSO in unimodal problems and LPSO illustrates good results in multimodal problems. Hence, both local and global topologies are hybridized in FGLT-PSO to increase the convergence rate and to avoid trapping into local optima.In FGLT-PSO algorithm, each particle uses the best position found by its neighbors (P
→
l
best) to update the particles’ velocities:
(5)
P
→
l
best
=
(
p
l
best
1
,
p
l
best
2
,
…
,
p
l
best
d
)
,
(6)
v
i
d
(
t
+
1
)
=
w
(
t
)
×
v
i
d
(
t
)
+
C
1
(
t
)
×
rand
1
×
(
p
i
d
(
t
)
-
x
i
d
(
t
)
)
+
C
2
(
t
)
×
rand
2
×
(
p
l
best
d
(
t
)
-
x
i
d
(
t
)
)
.
The next position of each particle is computed based on the current position, x
i
d
(
t
), the next velocity, v
i
d
(
t
+
1
), and the best position found by the swarm, P
→
g
best, as follows:
(7)
x
i
d
(
t
+
1
)
=
x
i
d
(
t
)
+
v
i
d
(
t
+
1
)
+
C
3
(
t
)
×
rand
3
×
(
p
g
best
d
(
t
)
-
x
i
d
(
t
)
)
,
(8)
P
→
g
best
=
(
p
g
best
1
,
p
g
best
2
,
…
,
p
g
best
d
)
.
In (6), w is computed as
(9)
w
(
t
)
=
w
max
⁡
-
t
×
(
w
max
⁡
-
w
min
⁡
)
T
.
Also, C
1
(
t
)
,
C
2
(
t
), and C
3
(
t
) are acceleration coefficients and modified according to (10):
(10)
C
j
(
t
)
=
C
j
min
⁡
+
t
×
(
C
j
max
⁡
-
C
j
min
⁡
)
T
,
j
=
1,2
,
3
,
where t and T are the current iteration and the number of maximum iterations, respectively.The second term in (6) is called the cognition term, and the third terms in (6) and (7) are named the social terms. In (7), |
x
i
d
(
t
+
1
)
|
<
x
max
⁡ and x
max
⁡ is set to a constant based on the search space bound.


## 3.2. Analysis of FGLT-PSO

A metaheuristic algorithm explores new spaces to avoid trapping in a local optimum in the initial steps. Due to the poor exploration in the standard PSO (PSO), it can sometimes find local optima in multimodal problems. Sometimes, if a particle falls into a local optimum, it will not be able to get out of it. That is, ifP
→
g
best obtained through the population lies in a local optimum while the current position and the personal best position of particle i are in the same local optimum, the second and third terms of (3) tend to zero and w decreases linearly to near zero. Consequently, the next velocity of particle i tends to zero, and its next position in (4) does not change; thus, the particle remains in the local optimum. Hence, the main aim in FGLT-PSO is to overcome the poor exploration and to increase the convergence rate by combining the local and global searches as shown in Figure 1. The particles move in the search space based on the best solutions found by their neighbors (P
→
l
best) and the swarm (P
→
g
best). At the beginning, the particles search new spaces. By lapse of iterations, the exploration should fade out and the exploitation should fade in. It means the particles accelerate to the good solution and make search around it to find the best solution.Figure 1
Graphical representation of the particle movement using FGLT-PSO algorithm.


## 4. Experimental Results

In this section, the FGLT-PSO algorithm is compared with some well-known PSO algorithms. The algorithms are tested using various unimodal and multimodal functions in different dimensions. Several benchmark functions [34, 35] are selected to evaluate the performance of proposed method.


### 4.1. Benchmark Functions

Twenty (20) minimization functions are applied in the experimental study including unimodal, multimodal, rotated, shifted, and shifted-rotated functions as detailed in Table1. In the table,Range and n are the feasible bound and the dimension of each function, respectively. F
opt is the optimum value of function. Among the benchmarks, functions (
1
)
–
(
5
) are unimodal functions and functions (
6
)
–
(
9
) are in the class of multimodal functions. Functions (
10
)
–
(
14
) are rotated and functions (
15
)
–
(
18
) are shifted unimodal and multimodal functions. Two functions (19) and (20) are shifted-rotated multimodal functions.Table 1
Dimensions, ranges, and global optimum values of test functions used in the experiments.
Test function
Dimension (n)
[Range]
n
X
opt
F
opt
F
1
(
x
)
10/30/50
[
-
100,100
]
n
0
0
F
2
(
x
)
10/30/50
[
-
10,10
]
n
0
0
F
3
(
x
)
10/30/50
[
-
100,100
]
n
0
0
F
4
(
x
)
10/30/50
[
-
1.28,1.28
]
n
0
0
F
5
(
x
)
10/30/50
[
-
10,10
]
n
1
0
F
6
(
x
)
10/30/50
[
-
500,500
]
n
[
420.96
]
n
-
418.9829
×
n
F
7
(
x
)
10/30/50
[
-
32,32
]
n
0
0
F
8
(
x
)
10/30/50
[
-
50,50
]
n
0
0
F
9
(
x
)
10/30/50
[
-
5.12,5.12
]
n
0
0
F
10
(
x
)
10/30/50
[
-
10,10
]
n
0
0
F
11
(
x
)
10/30/50
[
-
5,5
]
n
0
0
F
12
(
x
)
10/30/50
[
-
5.12,5.12
]
n
0
0
F
13
(
x
)
10/30/50
[
-
0.5,0.5
]
n
0
0
F
14
(
x
)
10/30/50
[
-
100,100
]
n
0
0
F
15
(
x
)
10/30/50
[
-
100,100
]
n
0
−450
F
16
(
x
)
10/30/50
[
-
600,600
]
n
0
−180
F
17
(
x
)
10/30/50
[
-
100,100
]
n
1
390
F
18
(
x
)
10/30/50
[
-
5.2,5.2
]
n
0
−330
F
19
(
x
)
10/30/50
[
-
32,32
]
n
0
−140
F
20
(
x
)
10/30/50
[
-
5.2,5.2
]
n
0
−330In unimodal functions, the convergence rate of search algorithm is more interesting than the final results because other methods have been designed to optimize these kinds of functions. In multimodal functions, finding an optimal (or a good near-global optimal) solution is important. These functions are more difficult to optimize because the number of local optima exponentially increases as the dimension increases. Therefore, the search algorithms should not become trapped in a local optimum and should be able to obtain good solutions.The rotation of function increases the function complexity. It does not affect the shape of function. The variableY
→ is computed using an orthogonal matrix M [36] and applied to obtain the fitness value of rotated function as follows:
(11)
Y
→
=
M
×
X
→
.In shifted functions, the global optimumX
*
→
=
(
x
1
*
,
x
2
*
,
…
,
x
n
*
) is shifted to the new position O
→
=
(
o
1
,
o
2
,
…
,
o
n
). All the test functions are shown as follows.(1) Sphere Model (unimodal function). Consider
(12)
F
1
(
x
)
=
∑
i
=
1
n
x
i
2
.(2) Shifted’s Schwefel’s Problem (unimodal function). Consider
(13)
F
2
(
x
)
=
∑
i
=
1
n
|
x
i
|
+
∏
i
=
1
n
|
x
i
|
.(3) Schwefel’s Problem 1.2 (unimodal function). Consider
(14)
F
3
(
x
)
=
∑
i
=
1
n
(
∑
j
=
1
i
x
j
)
2
.(4) Quartic Function, That Is, Noise (unimodal function). Consider
(15)
F
4
(
x
)
=
∑
i
=
1
n
i
x
i
4
+
random
[
0,1
)
.(5) Rosenbrock’s Function (multimodal function). Consider
(16)
F
5
(
x
)
=
∑
i
=
1
n
-
1
[
100
(
x
i
2
-
x
i
+
1
)
2
+
(
x
i
-
1
)
2
]
.
F
5 is unimodal in a 2-dimension or 3-dimension search space but can be treated as a multimodal function in high-dimensional cases.(6) Generalized Schwefel’s Problem 2.26 (multimodal function). Consider
(17)
F
6
(
x
)
=
∑
i
=
1
n
-
x
i
sin
(
|
x
i
|
)
.(7) Ackley’s Function (multimodal function). Consider
(18)
F
7
(
x
)
=
-
20
exp
⁡
(
-
0.2
1
n
∑
i
=
1
n
x
i
2
)
F
7
(
x
)
=
-
exp
⁡
(
1
n
∑
i
=
1
n
cos
⁡
2
π
x
i
)
+
20
+
e
.(8) Generalized Penalized Function (multimodal function). Consider
(19)
F
8
(
x
)
=
π
n
{
10
sin
⁡
(
π
y
1
)
+
∑
i
=
1
n
-
1
(
y
i
-
1
)
2
F
8
(
x
)
=
π
n
×
[
1
+
10
sin
2
(
π
y
i
+
1
)
]
+
(
y
n
-
1
)
2
∑
i
=
1
n
-
1
(
y
i
-
1
)
2
[
1
+
10
sin
2
(
π
y
i
+
1
)
]
}
F
8
(
x
)
=
+
∑
i
=
1
n
u
(
x
i
,
10,100,4
)
y
i
=
1
+
x
i
+
1
4
,
u
(
x
i
,
a
,
k
,
m
)
=
{
k
(
x
i
-
a
)
m
x
i
>
a
0
-
a
≤
x
i
≤
a
k
(
-
x
i
-
a
)
m
x
i
<
-
a
.(9) Noncontinuous Rastrigin’s Function (multimodal function). Consider
(20)
F
9
(
x
)
=
∑
i
=
1
n
[
y
i
2
-
10
cos
⁡
⁡
(
2
π
y
i
)
+
10
]
,
where
y
i
=
{
x
i
|
x
i
|
≤
0.5
round
(
2
x
i
)
2
|
x
i
|
≥
0.5
.(10) Rotated Schwefel’s Problem 2.22 (unimodal function). Consider
(21)
F
10
(
x
)
=
∑
i
=
1
n
|
y
i
|
+
∏
i
=
1
n
|
y
i
|
,
y
=
M
×
x
.(11) Rotated
2
n
Minima Function (multimodal function). Consider
(22)
F
11
(
x
)
=
1
n
∑
i
=
1
n
(
y
i
4
-
16
y
i
2
+
5
y
i
)
,
y
=
M
×
x
.(12) Rotated Rastrigin’s Function (multimodal function). Consider
(23)
F
12
(
x
)
=
∑
i
=
1
n
[
y
i
2
-
10
cos
⁡
⁡
(
2
π
y
i
)
+
10
]
,
y
=
M
×
x
.(13) Rotated Weierstrass’ Function (multimodal function). Consider
(24)
F
13
(
x
)
=
∑
i
=
1
n
(
∑
k
=
0
k
max
⁡
[
a
k
cos
⁡
(
2
π
b
k
(
y
i
+
0.5
)
)
]
)
F
13
(
x
)
=
-
n
∑
k
=
0
k
max
⁡
[
a
k
cos
⁡
(
2
π
b
k
×
0.5
)
]
,
y
=
M
×
x
,
a
=
0.5
,
b
=
3
,
k
max
⁡
=
20
.(14) Rotated Salomon’s Function (multimodal function). Consider
(25)
F
14
(
x
)
=
1
-
cos
⁡
(
2
π
∑
i
=
1
n
y
i
2
)
+
0.1
∑
i
=
1
n
y
i
2
,
y
=
M
×
x
.(15) Shifted Schwefel’s Problem 2.21 (unimodal function). Consider
(26)
F
15
(
x
)
=
max
⁡
i
{
|
z
i
|
,
1
≤
i
≤
n
}
+
f
bias
15
,
max
⁡
i
f
bias
15
=
-
450
,
z
=
x
-
o
.(16) Shifted Generalized Griewank’s Function (multimodal function). Consider
(27)
F
16
(
x
)
=
1
4000
∑
i
=
1
n
z
i
2
-
∏
i
=
1
n
cos
⁡
(
z
i
i
)
+
1
+
f
bias
16
,
p
k
p
k
p
k
p
k
p
k
p
k
f
bias
16
=
-
180
,
z
=
x
-
o
.(17) Shifted Rosenbrock’s Function (multimodal function). Consider
(28)
F
17
(
x
)
=
∑
i
=
1
n
-
1
[
100
(
z
i
2
-
z
i
+
1
)
2
+
(
z
i
-
1
)
2
]
+
f
bias
17
,
[
10
(
9
-
z
i
+
1
)
2
]
f
bias
17
=
390
,
z
=
x
-
o
+
1
.(18) Shifted Rastrigin’s Function (multimodal function). Consider
(29)
F
18
(
x
)
=
∑
i
=
1
n
[
z
i
2
-
10
cos
⁡
⁡
(
2
π
z
i
)
+
10
]
+
f
bias
18
,
[
z
i
2
-
i
+
10
]
f
bias
18
=
-
330
,
z
=
x
-
o
.(19) Shifted Rotated Ackley’s Function (multimodal function). Consider
(30)
F
19
(
x
)
=
-
20
exp
⁡
(
-
0.2
1
n
∑
i
=
1
n
z
i
2
)
F
19
(
x
)
=
-
exp
⁡
(
1
n
∑
i
=
1
n
cos
⁡
2
π
z
i
)
+
20
+
e
+
f
bias
19
,
where f
bias
19
=
-
140
;
z
=
(
x
-
o
)
×
M
′ is a linear transformation matrix with condition number = 100.(20) Shifted Rotated Rastrigin’s Function (multimodal function). Consider
(31)
F
20
(
x
)
=
∑
i
=
1
n
[
z
i
2
-
10
cos
⁡
⁡
(
2
π
z
i
)
+
10
]
+
f
bias
20
,
where f
bias
20
=
-
330
;
z
=
(
x
-
o
)
×
M
′ is a linear transformation matrix with condition number = 2.


### 4.2. Results of FGLT-PSO

The results of FGLT-PSO are provided in three sections. In Section4.2.1, the acceleration coefficients C
1
,
C
2, and C
3 in the proposed method are changed according to (10) and in Section 4.2.2, these factors are constant. In these sections, FGLT-PSO is evaluated using the benchmark functions with dimensions 10, 30, and 50. The number of maximum iterations is set at 5000 for n
=
10, at 10000 for n
=
30, and at 15000 for n
=
50. The population size is set to 50 (N
=
50). Also, w decreases linearly from 0.9 to 0.4.In Section4.2.3, the results of FGLT-PSO are compared with those of several well-known PSO algorithms from [19] on the common functions. In this section, the population size is set to 30 (N
=
30), n is 30, and the number of maximum iterations is set at 10000.The ring topology is used as the neighborhood structure in thel
best model for the FGLT-PSO and LPSO algorithms and the number of neighbours for each particle is three. The algorithms are run independently 30 times for the benchmark functions and the results are averaged.


#### 4.2.1. The Results of Proposed Method with Variable Acceleration Coefficients

Four algorithms of FGLT-PSO, PSO, LPSO, and QIPSO are randomly initialized and run on benchmark functions. The average best solution, the standard deviation (SD), and the median of the best solution in the last iteration are reported in Tables2, 3, 6, 7, 10, and 11. The best results from among the algorithms are shown in bold numbers. In the tables, the algorithms are ranked based on the average best results.Table 2
Minimization results for the unimodal and multimodal functions (maximum iteration = 5000 andn
=
10).
Function
FGLT-PSO
PSO
LPSO
QIPSO
F
1
Avg. best solution
0.000e + 000
1.405e − 135
1.452e − 063
1.526e − 138
SD
0.000e + 000
6.789e − 135
5.681e − 063
5.723e − 138
Median best solution
0.000e + 000
1.243e − 139
7.381e − 065
3.961e − 142
Avg. iteration for finding the best solution
2646
5000
5000
5000
F
2
Avg. best solution
2.741e − 273
2.992e − 077
4.130e − 038
6.663e − 079
SD
0.000e + 000
1.490e − 076
7.564e − 038
1.234e − 078
Median best solution
6.578e − 277
3.849e − 079
9.945e − 039
1.019e − 079
Avg. iteration for finding the best solution
4952
5000
4999
5000
F
3
Avg. best solution
1.637e − 116
3.215e − 044
5.110e − 014
3.919e − 044
SD
8.969e − 116
1.565e − 043
1.126e − 013
2.117e − 043
Median best solution
2.369e − 127
1.857e − 048
5.320e − 015
3.570e − 048
Avg. iteration for finding the best solution
4630
4999
4999
4999
F
4
Avg. best solution
3.209e − 004
4.738e − 004
1.269e − 003
6.514e − 004
SD
3.006e − 004
2.057e − 004
4.861e − 004
3.241e − 004
Median best solution
2.114e − 004
4.595e − 004
1.197e − 003
5.797e − 004
Avg. iteration for finding the best solution
3715
4404
4414
4459
F
5
Avg. best solution
2.973e − 001
1.820e + 000
1.661e + 000
2.040e + 000
SD
1.015e + 000
1.336e + 000
1.281e + 000
1.622e + 000
Median best solution
4.614e − 004
2.014e + 000
1.727e + 000
1.862e + 000
Avg. iteration for finding the best solution
4537
4811
4988
4837
F
6
Avg. best solution
−3.910e + 003
−3.452e + 003
−3.892e + 003
−3.424e + 003
SD
1.184e + 002
2.307e + 002
1.843e + 002
2.775e + 002
Median best solution
−3.953e + 003
−3.475e + 003
−3.834e + 003
−3.475e + 003
Avg. iteration for finding the best solution
2974
2726
3918
2635
F
7
Avg. best solution
4.441e − 015
4.322e − 015
4.441e − 015
4.441e − 015
SD
0.000e + 000
6.486e − 016
0.000e + 000
0.000e + 000
Median best solution
4.441e − 015
4.441e − 015
4.441e − 015
4.441e − 015
Avg. iteration for finding the best solution
376
3147
3772
3109
F
8
Avg. best solution
4.712e − 032
4.712e − 032
4.712e − 032
4.712e − 032
SD
1.670e − 047
1.670e − 047
1.670e − 047
1.670e − 047
Median best solution
4.712e − 032
4.712e − 032
4.712e − 032
4.712e − 032
Avg. iteration for finding the best solution
420
3166
3895
2687
F
9
Avg. best solution
9.495e − 002
9.667e − 001
1.574e + 000
0.000e + 000
SD
2.720e − 001
4.560e + 000
1.065e + 000
0.000e + 000
Median best solution
9.236e − 008
0.000e + 000
2.000e + 000
0.000e + 000
Avg. iteration for finding the best solution
4856
3357
4366
3401
Avg. rank
1.1
2.9
3.4
2.6
Final rank
1
3
4
2
Algorithms
FGLT-PSO
PSO
LPSO
QIPSOTable 3
Minimization results for the rotated and shifted unimodal and multimodal functions (maximum iteration = 5000 andn
=
10).
Function
FGLT-PSO
PSO
LPSO
QIPSO
F
10
Avg. best solution
7.056e −208
4.585e −059
2.358e −021
1.127e −058
SD
0.000e +000
8.736e −059
7.425e −021
5.346e −058
Median best solution
1.930e −209
7.785e −060
3.417e −022
1.023e −060
Avg. iteration for finding the best solution
4741
4999
4997
4999
F
11
Avg. best solution
−6.602e +001
−6.645e +001
−6.616e +001
−6.613e +001
SD
1.736e +000
1.175e +000
9.989e −001
1.720e +000
Median best solution
−6.606e +001
−6.614e +001
−6.607e +001
−6.588e +001
Avg. iteration for finding the best solution
3408
2930
3099
3185
F
12
Avg. best solution
1.604e +001
2.352e +001
2.319e +001
2.278e +001
SD
2.974e +000
3.762e +000
4.012e +000
4.387e +000
Median best solution
1.589e +001
2.358e +001
2.351e +001
2.240e +001
Avg. iteration for finding the best solution
2394
3656
3786
3660
F
13
Avg. best solution
0.000e +000
1.351e +000
7.438e −008
7.957e −001
SD
0.000e +000
3.079e +000
4.072e −007
2.428e +000
Median best solution
0.000e +000
0.000e +000
0.000e +000
0.000e +000
Avg. iteration for finding the best solution
448
3192
4877
3446
F
14
Avg. best solution
1.165e −001
1.032e −001
9.987e −002
1.132e −001
SD
3.790e −002
1.826e −002
0.000e +000
3.457e −002
Median best solution
9.987e −002
9.987e −002
9.987e −002
9.987e −002
Avg. iteration for finding the best solution
1998
2916
3268
2795
F
15
Avg. best solution
−4.500e +002
−4.396e +002
−4.480e +002
−4.406e +002
SD
8.039e −014
7.334e +000
2.470e +000
7.991e +000
Median best solution
−4.500e +002
−4.450e +002
−4.500e +002
−4.450e +002
Avg. iteration for finding the best solution
1324
757
3561
1213
F
16
Avg. best solution
−1.798e +002
−1.745e +002
−1.793e +002
−1.766e +002
SD
3.789e −001
8.186e +000
5.222e −001
3.309e +000
Median best solution
−1.800e +002
−1.780e +002
−1.789e +002
−1.780e +002
Avg. iteration for finding the best solution
2922
2592
3398
2711
F
17
Avg. best solution
3.915e +002
9.048e +006
6.752e +004
6.596e +006
SD
6.315e +000
1.678e +007
3.676e +005
1.638e +007
Median best solution
3.900e +002
2.014e +006
3.933e +002
4.867e +002
Avg. iteration for finding the best solution
4831
3983
4884
4090
F
18
Avg. best solution
−3.287e +002
−3.215e +002
−3.247e +002
−3.233e +002
SD
1.113e +000
5.979e +000
3.144e +000
4.822e +000
Median best solution
−3.290e +002
−3.225e +002
−3.250e +002
−3.225e +002
Avg. iteration for finding the best solution
4809
3193
4509
3331
F
19
Avg. best solution
−119.744
−119.745
−119.804
−119.746
SD
5.750e −002
7.142e −002
5.558e −002
6.911e −002
Median best solution
−119.731
−119.748
−119.804
−119.73
Avg. iteration for finding the best solution
1849
2240
2072
2414
F
20
Avg. best solution
−3.196e +002
−3.091e +002
−3.137e +002
−3.078e +002
SD
4.236e +000
7.174e +000
5.338e +000
7.995e +000
Median best solution
−3.196e +002
−3.102e +002
−3.130e +002
−3.079e +002
Avg. iteration for finding the best solution
3252
3315
4721
3479
Avg. rank
1.8
3.2
2.1
2.9
Final rank
1
4
2
3
Algorithms
FGLT-PSO
PSO
LPSO
QIPSOMoreover, Wilcoxon’s rank sum test [37] is conducted in order to determine whether the results obtained by the FGLT-PSO are different from those generated by other algorithms with a statistical significance. The tests are shown in Tables 4, 5, 8, 9, 12, and 13, where h
-value = 1 indicates the case in which proposed algorithm significantly outperformed the compared algorithm with 95% certainty, h
-value = −1 represents that the compared algorithm is significantly better than the proposed algorithm, and h
-value = 0 denotes that the results of the two considered algorithms are not significantly different. In these tables, rows 1 (better), 0 (same), and −1 (worse) give the number of functions that the FGLT-PSO performs significantly better than, almost the same as, and significantly worse than the compared algorithm, respectively.Table 4
Comparison of FGLT-PSO with PSO, LPSO, and QIPSO for the unimodal and multimodal functions using Wilcoxon’s rank sum test (n
=
10).
Function
Wilcoxon’s rank sum test
PSO
LPSO
QIPSO
F
1
P value
1.2118e −012
1.2118e −012
1.2118e −012
h-value
1
1
1
z-value
−7.10402
−7.10402
−7.10402
F
2
P value
3.0199e −011
3.0199e −011
3.0199e −011
h-value
1
1
1
z-value
−6.6456
−6.6456
−6.6456
F
3
P value
3.0199e −011
3.0199e −011
3.0199e −011
h-value
1
1
1
z-value
−6.6456
−6.6456
−6.6456
F
4
P value
1.0576e −003
6.1210e −010
3.5923e −005
h-value
1
1
1
z-value
−3.27475
−6.18728
−4.13225
F
5
P value
1.5581e −008
1.8731e −007
1.0095e −008
h-value
1
1
1
z-value
−5.65504
−5.21151
−5.72912
F
6
P value
6.3474e −011
1.2724e −001
9.3829e −010
h-value
1
0
1
z-value
−6.53532
−1.52509
−6.11957
F
7
P value
3.3371e −001
—
—
h-value
0
0
0
z-value
0.966667
—
—
F
8
P value
—
—
—
h-value
0
0
0
z-value
—
—
—
F
9
P value
1.9097e −005
1.6703e −006
6.2470e −010
h-value
−1
1
−1
z-value
4.27519
−4.7897
6.18407
1 (better)
6
6
6
0 (same)
2
3
2
−1 (worse)
1
0
1Table 5
Comparison of FGLT-PSO with PSO, LPSO, and QIPSO for the rotated and shifted unimodal and multimodal functions using Wilcoxon’s rank sum test (n
=
10).
Function
Wilcoxon’s rank sum test
PSO
LPSO
QIPSO
F
10
P value
3.0199e −011
3.0199e −011
3.0199e −011
h-value
1
1
1
z-value
−6.6456
−6.6456
−6.6456
F
11
P value
4.1191e −001
7.3940e −001
8.4180e −001
h-value
0
0
0
z-value
0.820536
0.33265
0.19959
F
12
P value
1.8567e −009
1.8500e −008
5.5329e −008
h-value
1
1
1
z-value
−6.00987
−5.62547
−5.43328
F
13
P value
2.1577e −002
2.1577e −002
8.1523e −002
h-value
1
1
0
z-value
−2.29773
−2.29773
−1.74192
F
14
P value
1.1439e −001
1.4425e −004
7.7181e −001
h-value
0
−1
0
z-value
1.57878
3.80076
0.290007
F
15
P value
1.5553e −011
1.2883e −011
1.7966e −008
h-value
1
1
1
z-value
−6.74264
−6.76995
−5.63053
F
16
P value
1.2755e −010
1.9667e −003
5.5338e −010
h-value
1
1
1
z-value
−6.43006
−3.09521
−6.20317
F
17
P value
7.3270e −011
3.6443e −008
5.0030e −010
h-value
1
1
1
z-value
−6.51381
−5.50727
−6.21901
F
18
P value
5.7512e −006
3.5043e −007
4.0269e −005
h-value
1
1
1
z-value
−4.53533
−5.09408
−4.10593
F
19
P value
9.2344e −001
2.3885e −004
8.6499e −001
h-value
0
−1
0
z-value
0.0960988
3.67393
0.170021
F
20
P value
5.2991e −008
9.2051e −005
4.6756e −008
h-value
1
1
1
z-value
−5.44097
−3.91064
−5.46322
1 (better)
8
8
7
0 (same)
3
1
4
−1 (worse)
0
2
0Table 6
Minimization results for the unimodal and multimodal functions (maximum iteration = 10000 andn
=
30).
Function
FGLT-PSO
PSO
LPSO
QIPSO
F
1
Avg. best solution
0.000e +000
5.397e −069
2.747e −028
3.502e −069
SD
0.000e +000
1.161e −068
4.078e −028
1.664e −068
Median best solution
0.000e +000
2.321e −070
1.047e −028
2.727e −072
F
2
Avg. best solution
7.598e −110
7.000e +000
3.849e −028
2.333e +000
SD
4.085e −109
7.944e +000
9.015e −028
4.302e +000
Median best solution
1.684e −136
1.000e +001
1.331e −031
1.188e −071
F
3
Avg. best solution
5.959e −020
1.189e +004
1.399e +002
5.733e +003
SD
1.762e −019
6.605e +003
9.380e +001
5.484e +003
Median best solution
5.483e −021
1.083e +004
1.254e +002
5.000e +003
F
4
Avg. best solution
6.415e −003
1.320e −003
9.664e −003
1.635e −003
SD
2.858e −003
8.008e −004
3.621e −003
9.357e −004
Median best solution
5.941e −003
1.088e −003
9.106e −003
1.333e −003
F
5
Avg. best solution
5.718e −001
1.129e +003
1.751e +001
3.858e +002
SD
1.225e +000
3.023e +003
2.041e +001
1.832e +003
Median best solution
1.707e −001
7.428e +001
9.494e +000
2.139e +001
F
6
Avg. best solution
−1.048e +004
−9.095e +003
−9.858e +003
−9.227e +003
SD
5.996e +002
7.444e +002
4.872e +002
7.524e +002
Median best solution
−1.058e +004
−9.051e +003
−9.828e +003
−9.232e +003
F
7
Avg. best solution
7.771e −002
1.013e −014
2.931e −014
8.941e −015
SD
2.015e −001
3.312e −015
1.433e −014
2.457e −015
Median best solution
7.994e −015
7.994e −015
2.576e −014
7.994e −015
F
8
Avg. best solution
5.164e −003
1.037e −002
1.160e −025
1.571e −032
SD
2.274e −002
3.163e −002
3.552e −025
5.567e −048
Median best solution
1.571e −032
1.591e −032
3.737e −027
1.571e −032
F
9
Avg. best solution
2.136e +001
5.827e +001
4.281e +001
4.310e +001
SD
8.612e +000
3.080e +001
2.064e +001
2.901e +001
Median best solution
2.212e +001
5.450e +001
3.753e +001
3.500e +001
Avg. rank
1.8
3.3
2.6
2.3
Final rank
1
4
3
2
Algorithms
FGLT-PSO
PSO
LPSO
QIPSOTable 7
Minimization results for the rotated and shifted unimodal and multimodal functions (maximum iteration = 10000 andn
=
30).
Function
FGLT-PSO
PSO
LPSO
QIPSO
F
10
Avg. best solution
8.707e − 020
3.422e + 002
7.818e + 000
5.318e + 001
SD
4.547e − 019
1.556e + 003
2.173e + 001
1.986e + 002
Median best solution
4.028e − 043
1.332e − 006
2.267e − 003
8.579e − 020
F
11
Avg. best solution
−4.963e + 001
−4.981e + 001
−4.933e + 001
−4.993e + 001
SD
1.098e + 000
1.324e + 000
1.417e + 000
1.489e + 000
Median best solution
−4.930e + 001
−4.985e + 001
−4.927e + 001
−4.968e + 001
F
12
Avg. best solution
1.514e + 002
1.935e + 002
1.902e + 002
1.886e + 002
SD
8.330e + 000
1.509e + 001
1.246e + 001
1.233e + 001
Median best solution
1.511e + 002
1.978e + 002
1.899e + 002
1.850e + 002
F
13
Avg. best solution
7.160e − 001
3.101e + 001
1.185e + 001
2.046e + 001
SD
9.412e − 001
1.248e + 001
1.136e + 001
1.804e + 001
Median best solution
2.067e − 001
3.613e + 001
5.997e + 000
3.319e + 001
F
14
Avg. best solution
4.065e − 001
3.465e − 001
3.632e − 001
3.599e − 001
SD
1.230e − 001
5.074e − 002
5.561e − 002
7.240e − 002
Median best solution
3.999e − 001
2.999e − 001
3.999e − 001
3.499e − 001
F
15
Avg. best solution
−4.280e + 002
−4.044e + 002
−4.175e + 002
−4.500e + 002
SD
7.441e + 000
7.592e + 000
8.860e + 000
2.151e − 006
Median best solution
−4.266e + 002
−4.056e + 002
−4.155e + 002
−4.500e + 002
F
16
Avg. best solution
−1.794e + 002
−8.340e + 001
−1.705e + 002
−9.806e + 001
SD
1.112e + 000
4.560e + 001
4.344e + 000
3.839e + 001
Median best solution
−1.798e + 002
−9.269e + 001
−1.704e + 002
−1.053e + 002
F
17
Avg. best solution
4.225e + 002
1.658e + 009
8.168e + 007
1.549e + 009
SD
5.410e + 001
1.409e + 009
5.644e + 007
1.599e + 009
Median best solution
3.940e + 002
1.136e + 009
6.932e + 007
1.205e + 009
F
18
Avg. best solution
−2.874e + 002
−2.174e + 002
−2.455e + 002
−2.219e + 002
SD
1.680e + 001
2.897e + 001
1.196e + 001
2.614e + 001
Median best solution
−2.902e + 002
−2.199e + 002
−2.458e + 002
−2.227e + 002
F
19
Avg. best solution
−1.1912e + 002
−1.1912e + 002
−1.1915e + 002
−1.1913e + 002
SD
5.597e − 002
5.015e − 002
5.377e − 002
6.957e − 002
Median best solution
−1.1911e + 002
−1.1911e + 002
−1.1915e + 002
−1.1913e + 002
F
20
Avg. best solution
−2.367e + 002
−1.237e + 002
−1.499e + 002
−1.375e + 002
SD
2.561e + 001
3.612e + 001
2.719e + 001
4.160e + 001
Median best solution
−2.427e + 002
−1.256e + 002
−1.481e + 002
−1.374e + 002
Avg. rank
1.8
3.5
2.4
2.4
Final rank
1
3
2
2
Algorithms
FGLT-PSO
PSO
LPSO
QIPSOTable 8
Comparison of FGLT-PSO with PSO, LPSO, and QIPSO for the unimodal and multimodal functions using Wilcoxon’s rank sum test (n
=
30).
Function
Wilcoxon’s rank sum test
PSO
LPSO
QIPSO
F
1
P value
1.2118e −012
1.2118e −012
1.2118e −012
h-value
1
1
1
z-value
−7.10402
−7.10402
−7.10402
F
2
P value
2.8991e −011
3.0199e −011
2.9822e −011
h-value
1
1
1
z-value
−6.65161
−6.6456
−6.64745
F
3
P value
3.0199e −011
3.0199e −011
3.0199e −011
h-value
1
1
1
z-value
−6.6456
−6.6456
−6.6456
F
4
P value
1.2057e −010
2.2539e −004
1.9568e −010
h-value
−1
1
−1
z-value
6.43862
−3.68871
6.3647
F
5
P value
1.2018e −008
1.2472e −004
5.0922e −008
h-value
1
1
1
z-value
−5.69948
−3.83667
−5.44806
F
6
P value
1.2018e −008
1.2472e −004
5.0922e −008
h-value
1
1
1
z-value
−5.69948
−3.83667
−5.44806
F
7
P value
4.2942e −002
3.7277e −001
2.8314e −003
h-value
−1
0
−1
z-value
2.02428
−0.891289
2.98547
F
8
P value
1.7374e −001
2.4305e −002
6.6167e −004
h-value
0
1
−1
z-value
−1.36028
−2.25228
3.40499
F
9
P value
1.9990e −006
2.7726e −005
7.2926e −004
h-value
1
1
1
z-value
−4.75352
−4.19138
−3.37834
1 (better)
6
8
6
0 (same)
1
1
0
−1 (worse)
2
0
3Table 9
Comparison of FGLT-PSO with PSO, LPSO, and QIPSO for the rotated and shifted unimodal and multimodal functions using Wilcoxon’s rank sum test (n
=
30).
Function
Wilcoxon’s rank sum test
PSO
LPSO
QIPSO
F
10
P value
3.2922e −010
1.4041e −010
3.1451e −009
h-value
1
1
1
z-value
−6.28435
−6.41545
−5.92384
F
11
P value
1.0233e −001
3.0418e −001
4.4642e −001
h-value
0
0
0
z-value
1.63368
−1.02752
0.761398
F
12
P value
1.0937e −010
1.4643e −010
3.0199e −011
h-value
1
1
1
z-value
−6.4534
−6.40905
−6.6456
F
13
P value
1.4288e −008
1.3281e −010
1.1736e −003
h-value
1
1
1
z-value
−5.66991
−6.42392
−3.24523
F
14
P value
1.3470e −003
6.7707e −003
3.5527e −003
h-value
−1
−1
−1
z-value
3.20578
2.70792
2.91537
F
15
P value
2.8840e −010
4.7657e −005
2.8252e −011
h-value
1
1
−1
z-value
−6.30489
−4.06683
6.65541
F
16
P value
2.9543e −011
5.3712e −011
2.9543e −011
h-value
1
1
1
z-value
−6.64883
−6.56027
−6.64883
F
17
P value
3.0199e −11
3.0199e −11
3.0199e −11
h-value
1
1
1
z-value
−6.6456
−6.6456
−6.6456
F
18
P value
1.0937e −010
1.2870e −009
1.4643e −010
h-value
1
1
1
z-value
−6.4534
−6.06901
−6.40905
F
19
P value
8.7663e −001
4.0595e −002
7.7312e −001
h-value
0
1
0
z-value
0.155236
2.04764
0.288296
F
20
P value
3.0199e −011
7.3891e −011
4.9752e −011
h-value
1
1
1
z-value
−6.6456
−6.51254
−6.57168
1 (better)
8
9
7
0 (same)
2
1
2
−1 (worse)
1
1
2Table 10
Minimization results for the unimodal and multimodal functions (maximum iteration = 15000 andn
=
50).
Function
FGLT-PSO
PSO
LPSO
QIPSO
F
1
Avg. best solution
5.239e −232
7.5857e −049
7.297e −020
2.446e −049
SD
0.000e +000
2.1243e −048
7.695e −020
8.478e −049
Median best solution
2.541e −251
4.5415e −050
4.843e −020
1.395e −050
F
2
Avg. best solution
9.246e −075
3.400e +001
2.194e −015
1.367e +001
SD
3.646e −074
1.714e +001
1.439e −015
1.159e +001
Median best solution
2.684e −080
3.000e +001
1.881e −015
1.000e +001
F
3
Avg. best solution
1.098e −008
4.167e +004
2.775e +004
3.585e +004
SD
2.405e −008
1.404e +004
8.532e +003
1.967e +004
Median best solution
5.928e −009
4.002e +004
2.859e +004
3.434e +004
F
4
Avg. best solution
5.070e −002
6.015e +000
6.452e −002
2.055e −002
SD
2.802e −002
5.352e +000
1.488e −002
4.363e −003
Median best solution
4.205e −002
5.385e +000
6.409e −002
1.971e −002
F
5
Avg. best solution
6.128e +000
1.568e +003
6.024e +001
4.700e +002
SD
5.716e +000
3.409e +003
3.032e +001
1.817e +003
Median best solution
5.080e +000
8.179e +001
7.397e +001
7.942e +001
F
6
Avg. best solution
−1.505e +004
−1.364e +004
−1.470e +004
−1.339e +004
SD
9.481e +002
9.581e +002
8.491e +002
1.025e +003
Median best solution
−1.515e +004
−1.349e +004
−1.445e +004
−1.343e +004
F
7
Avg. best solution
1.299e +000
2.049e +000
1.145e −005
1.782e −014
SD
5.560e −001
4.661e +000
6.223e −005
3.695e −015
Median best solution
1.282e +000
2.220e −014
1.782e −009
1.510e −014
F
8
Avg. best solution
1.061e −001
1.451e −002
3.831e −012
2.074e −003
SD
1.935e −001
3.535e −002
1.984e −011
1.136e −002
Median best solution
2.081e −005
1.308e −032
2.881e −015
9.423e −033
F
9
Avg. best solution
4.759e +001
1.980e +002
1.872e +002
1.546e +002
SD
2.294e +001
4.888e +001
3.958e +001
5.192e +001
Median best solution
4.636e +001
1.890e +002
1.850e +002
1.475e +002
Avg. rank
1.7
3.7
2.3
2.3
Final rank
1
3
2
2
Algorithms
FGLT-PSO
PSO
LPSO
QIPSOTable 11
Minimization results for the rotated and shifted unimodal and multimodal functions (maximum iteration = 15000 andn
=
50).
Function
FGLT-PSO
PSO
LPSO
QIPSO
F
10
Avg. best solution
9.936e −031
6.202e +009
5.150e +003
3.989e +006
SD
5.442e −030
2.576e +010
1.153e +004
1.825e +007
Median best solution
1.121e −048
1.763e +005
1.184e +002
5.637e −007
F
11
Avg. best solution
−4.401e +001
−4.328e +001
−4.357e +001
−4.385e +001
SD
1.043e +000
1.331e +000
9.431e −001
1.329e +000
Median best solution
−4.391e +001
−4.344e +001
−4.339e +001
−4.367e +001
F
12
Avg. best solution
3.155e +002
4.234e +002
3.995e +002
4.045e +002
SD
1.166e +001
3.689e +001
1.813e +001
2.468e +001
Median best solution
3.171e +002
4.236e +002
3.995e +002
4.062e +002
F
13
Avg. best solution
1.660e +001
6.640e +001
5.210e +001
4.946e +001
SD
6.875e +000
2.371e +000
1.517e +001
2.599e +001
Median best solution
1.759e +001
6.697e +001
6.163e +001
6.361e +001
F
14
Avg. best solution
7.599e −001
5.899e −001
8.067e −001
6.099e −001
SD
3.103e −001
8.030e −002
9.046e −002
8.847e −002
Median best solution
6.999e −001
5.999e −001
7.999e −001
5.999e −001
F
15
Avg. best solution
−3.976e +002
−3.795e +002
−4.019e +002
−4.499e +002
SD
5.964e +000
1.974e +001
6.096e +000
1.849e −003
Median best solution
−3.974e +002
−3.815e +002
−4.021e +002
−4.50e +002
F
16
Avg. best solution
−1.791e +002
8.313e +001
−1.551e +002
6.894e +001
SD
1.212e +000
9.121e +001
1.216e +001
7.896e +001
Median best solution
−1.797e +002
7.911e +001
−1.565e +002
7.481e +001
F
17
Avg. best solution
4.260e +002
8.648e +009
3.341e +008
8.196e +009
SD
7.188e +001
4.442e +009
2.597e +008
5.588e +009
Median best solution
3.977e +002
8.332e +009
2.599e +008
6.614e +009
F
18
Avg. best solution
−1.920e +002
−4.497e +001
−9.832e +001
−3.612e +001
SD
2.634e +001
4.206e +001
1.963e +001
4.934e +001
Median best solution
−1.944e +002
−4.804e +001
−9.679e +001
−2.665e +001
F
19
Avg. best solution
−1.189e +002
−1.189e +002
−1.190e +002
−1.189e +002
SD
4.303e −002
4.364e −002
5.121e −002
5.576e −002
Median best solution
−1.189e +002
−1.189e +002
−1.190e +002
−1.189e +002
F
20
Avg. best solution
−1.016e +002
1.537e +002
5.974e +001
1.082e +002
SD
4.898e +001
8.906e +001
5.117e +001
9.020e +001
Median best solution
−1.004e +002
1.539e +002
6.594e +001
1.035e +002
Avg. rank
1.5
3.6
2.3
2.7
Final rank
1
4
2
3
Algorithms
FGLT-PSO
PSO
LPSO
QIPSOTable 12
Comparison of FGLT-PSO with PSO, LPSO, and QIPSO for the unimodal and multimodal functions using Wilcoxon’s rank sum test (n
=
50).
Function
Wilcoxon’s rank sum test
PSO
LPSO
QIPSO
F
1
P value
3.0199e −011
3.0199e −011
3.0199e −011
h-value
1
1
1
z-value
−6.6456
−6.6456
−6.6456
F
2
P value
2.9673e −011
3.0199e −011
2.9229e −011
h-value
1
1
1
z-value
−6.64819
−6.6456
−6.65041
F
3
P value
3.0199e −011
3.0199e −011
3.0199e −011
h-value
1
1
1
z-value
−6.6456
−6.6456
−6.6456
F
4
P value
2.5306e −004
3.9881e −004
3.1589e −010
h-value
1
1
−1
z-value
−3.65915
−3.54087
6.29077
F
5
P value
6.1210e −010
9.7555e −010
1.2870e −009
h-value
1
1
1
z-value
−6.18728
−6.11336
−6.06901
F
6
P value
2.8790e −006
9.3341e −002
2.5711e −007
h-value
1
0
1
z-value
−4.67927
−1.67803
−5.15244
F
7
P value
7.7050e −006
3.0199e −011
1.4811e −011
h-value
−1
−1
−1
z-value
4.47322
6.6456
6.74974
F
8
P value
7.7087e −002
8.2796e −003
1.4193e −007
h-value
0
−1
−1
z-value
1.76784
2.64045
5.26273
F
9
P value
3.3384e −011
3.3384e −011
4.6159e −010
h-value
1
1
1
z-value
−6.63081
−6.63081
−6.23164
1 (better)
7
6
6
0 (same)
1
1
0
−1 (worse)
1
2
3Table 13
Comparison of FGLT-PSO with PSO, LPSO, and QIPSO for the rotated and shifted unimodal and multimodal functions using Wilcoxon’s rank sum test (n
=
50).
Function
Wilcoxon’s rank sum test
PSO
LPSO
QIPSO
F
10
P value
3.0199e −011
3.0199e −011
3.0199e −011
h-value
1
1
1
z-value
−6.6456
−6.6456
−6.6456
F
11
P value
3.0317e −002
6.5671e −002
6.7350e −001
h-value
1
0
0
z-value
−2.16592
−1.84066
−0.421356
F
12
P value
3.0199e −011
3.0199e −011
3.0199e −011
h-value
1
1
1
z-value
−6.6456
−6.6456
−6.6456
F
13
P value
3.0199e −011
1.2057e −010
1.8916e −004
h-value
1
1
1
z-value
−6.6456
−6.43862
−3.73307
F
14
P value
7.3131e −003
3.0749e −002
3.0494e −002
h-value
−1
1
−1
z-value
2.68224
−2.16031
2.16361
F
15
P value
2.2256e −004
5.4038e −004
2.7547e −011
h-value
1
−1
−1
z-value
−3.69193
3.4599
6.65912
F
16
P value
3.0142e −011
3.0123e −011
3.0142e −011
h-value
1
1
1
z-value
−6.64588
−6.64597
−6.64588
F
17
P value
3.0199e −011
3.0199e −011
3.0199e −011
h-value
1
1
1
z-value
−6.6456
−6.6456
−6.6456
F
18
P value
3.0199e −011
4.0772e −011
3.0199e −011
h-value
1
1
1
z-value
−6.6456
−6.60125
−6.6456
F
19
P value
2.3399e −001
5.8737e −004
9.2344e −001
h-value
0
−1
0
z-value
1.19015
3.43738
0.0960988
F
20
P value
3.3384e −011
1.3289e −010
8.9934e −011
h-value
1
1
1
z-value
−6.63081
−6.42383
−6.48297
1 (better)
9
8
7
0 (same)
1
1
2
−1 (worse)
1
2
2The acceleration coefficientsC
1, C
2, and C
3 are updated based on (10). Their minimum and maximum values are as follows: C
1
min
⁡
=
0.5
,
C
1
max
⁡
=
2
,
C
2
min
⁡
=
1
,
C
2
max
⁡
=
2
,
C
3
min
⁡
=
0.5, and C
3
max
⁡ = 1.5.In these tables, the benchmark functions are divided to two categories:(
1
) unimodal and multimodal functions and (
2
) rotated, shifted, and shifted-rotated unimodal and multimodal functions. The experimental results demonstrate that FGLT-PSO performs superior results for most of the functions in all tested dimensions.Tables2 and 3 show the experimental results for all benchmark functions with dimension n
=
10. As illustrated, the FGLT-PSO algorithm surpasses the PSO, LPSO, and QIPSO algorithms in minimizing functions (
1
)
–
(
6
), (
10
), (
12
), (
13
), (
15
)
–
(
18
), and (20). Moreover, the proposed method provides significant improvements in functions (
1
), (
2
), (
3
), (
10
), (
13
), (
17
), and (
18
). In these functions, the convergent results attain the optimal (or good near optimal) solutions. In Tables 2 and 3, the average iteration for finding the best solution is computed. The average iteration is the required iterations to find the best solution by each algorithm. As shown, FGLT-PSO finds the best solutions faster than the other algorithms in the majority of functions. Also, it is noticeable that the FGLT-PSO algorithm achieves the best solution in a considerably lower iteration in functions (
7
) and (
8
). In these functions, the algorithms show identical results.According to Wilcoxon’s rank sum test in Tables4 and 5 for n
=
10, the results of the FGLT-PSO are statistically significantly different from the three compared algorithms. The superior convergence rate of FGLT-PSO is shown in Figure 2. The results in this figure illustrate that FGLT-PSO tends to find the global optimum in F
1 and F
17 faster than PSO, LPSO, and QIPSO and obtains the highest accuracy for these functions from among all the algorithms.Figure 2
Convergence performance of FGLT-PSO, PSO, LPSO, and QIPSO for the test functionsF
1 and F
17 (n
=
10).The minimization results of the benchmark functions with dimensionn
=
30 are presented in Tables 6 and 7. As seen in these tables and Tables 8 and 9, the FGLT-PSO outperforms the PSO, LPSO, and QIPSO algorithms in functions (
1
), (
2
), (
3
), (
5
), (
6
), (
9
), (
10
), (
12
), (
13
), (
16
), (
17
), (
18
), and (
20
). The largest difference in performance between the proposed algorithm with PSO, LPSO, and QIPSO occurs for the functions (
1
), (
2
), (
3
), (
5
), (
16
), (
17
), (
18
), and (
20
). Figure 3 illustrates the progress of the average best solution over 30 runs for F
3 and F
18. As demonstrated, the FGLT-PSO shows a higher convergence rate than the other algorithms.Figure 3
Convergence performance of FGLT-PSO, PSO, LPSO, and QIPSO for the test functionsF
3 and F
18 (n
=
30).Tables10 and 11 present the results of algorithms for the test functions with dimension n
=
50. As illustrated in these tables and regarding the results of Wilcoxon’s rank sum in Tables 12 and 13, the performance of proposed method is the best in most of the functions especially for functions (
1
), (
2
), (
3
), (
5
), (
6
), (
9
), (
10
), (
11
), (
12
), (
13
), (
16
), (
17
), (
18
), and (
20
). The superior convergence rate of FGLT-PSO is shown in Figure 4. The results in this figure show that the FGLT-PSO performs the best for the test functions F
16 and F
20 with n
=
50.Figure 4
Convergence performance of FGLT-PSO, PSO, LPSO, and QIPSO for the test functionsF
16 and F
20 (n
=
50).In addition, it is considerable that the PSO, LPSO, and QIPSO algorithms return the results far from the global optima as the dimension increases. This problem is clear in the functions(
3
), (
5
), (
9
), (
10
), (
16
), (
17
), (
18
), and (
20
) in Tables 10 and 11 with n
=
50. These all indicate that the proposed algorithm, FGLT-PSO, is more powerful and robust than the others for solving unimodal and multimodal functions.


#### 4.2.2. The Results of Proposed Method with Constant Acceleration Coefficients

In this section, theC
1, C
2, and C
3 as acceleration coefficients are set at constant values to compare with the presented results in the Section 4.2.1. The coefficient of cognition term (C
1) and social terms (C
2 and C
3) are considered as C
1
=
2 and C
2
=
C
3
=
1. Table 14 shows the results of proposed method for the benchmark functions with dimensions 10, 30, and 50. As seen, the FGLT-PSO with the constant acceleration coefficients performs well in most of the functions. As the dimension increases, the FGLT-PSO with the variable acceleration coefficients (Section 4.2.1) shows the better performance than the constant one for functions (
1
), (
2
), (
3
), (
5
), (
17
), and (
20
). Also, the FGLT-PSO with the constant acceleration coefficients presents a better performance for functions (
7
), (
13
), and (
18
).Table 14
The results of FGLT-PSO with constant acceleration coefficients and different dimensions (N
=
50).
Functions
Iteration = 5000,n
=
10
Iteration = 10000,n
=
30
Iteration = 15000,n
=
50
Avg. best solution ± SD
Avg. best solution ± SD
Avg. best solution ± SD
F
1
0.000e +000  ±  0.000e +000
1.936e −067 ± 1.060e −066
1.687e −038 ± 5.602e −038
F
2
6.678e −160 ± 3.658e −159
4.922e −049 ± 2.305e −048
7.064e −032 ± 3.167e −031
F
3
7.520e −035 ± 3.579e −034
2.026e −003 ± 9.321e −003
2.062e +000 ± 2.616e +000
F
4
1.621e −004 ± 8.583e −005
9.078e −004 ± 3.306e −004
1.930e −003 ± 6.461e −004
F
5
3.086e +000 ± 1.772e +000
3.542e +001 ± 2.864e +001
7.854e +001 ± 4.435e +001
F
6
−3.957e +003 ± 1.375e +002
−1.052e +004 ± 4.069e +002
−1.643e +004 ± 7.233e +002
F
7
4.086e −015 ± 1.084e −015
5.507e −015 ± 1.656e −015
9.484e −002 ± 3.612e −001
F
8
4.712e −032 ± 1.670e −047
4.492e −002 ± 1.379e −001
7.063e −002 ± 1.508e −001
F
9
2.567e +000 ± 9.714e −001
2.040e +001 ± 6.100e +000
5.183e +001 ± 1.117e +001
F
10
6.599e −249 ± 0.000e +000
3.539e −066 ± 1.939e −065
4.923e −019 ± 2.696e −018
F
11
−6.620e +001 ± 1.111e +000
−5.036e +001 ± 9.041e −001
−4.482e +001 ± 9.061e −001
F
12
1.659e +001 ± 4.479e +000
1.602e +002 ± 1.096e +001
3.369e +002 ± 1.585e +001
F
13
0.000e +000 ± 0.000e +000
0.000e +000 ± 0.000e +000
1.5294e −007 ± 1.1896e −007
F
14
9.987e −002 ± 2.247e −017
3.932e −001 ± 1.081e −001
3.065e −001 ± 2.537e −002
F
15
−4.500e +002 ± 1.493e −014
−4.268e +002 ± 1.016e +001
−4.420e +002 ± 3.137e +000
F
16
−1.799e +002 ± 2.005e −001
−1.787e +002 ± 3.851e +000
−1.775e +002 ± 3.055e +000
F
17
4.001e +002 ± 2.060e +001
4.661e +002 ± 1.083e +002
5.748e +005 ± 1.022e +006
F
18
−3.283e +002 ± 8.796e −001
−2.956e +002 ± 8.540e +000
−2.396e +002 ± 1.763e +001
F
19
−1.1974e +002 ± 5.990e −002
−1.191e +002 ± 6.798e −002
−1.189e +002 ± 3.710e −002
F
20
−3.196e +002 ± 4.136e +000
−1.994e +002 ± 2.829e +001
−2.684e +001 ± 4.506e +001


#### 4.2.3. Comparison with the Other PSO Algorithms

In this section, several well-known PSO algorithms are selected to assess the performance of proposed algorithm for the benchmarks. The PSO, QIPSO, FIPS, DMS-PSO, CLPSO, AFPSO, and AFPSO-QI algorithms are considered for the comparison. The details of these algorithms are listed in Table15. The FGLT-PSO is run 30 times and the average best solutions and the SD of results for eight common multimodal benchmark functions are compared with the reported results by [19] as illustrated in Table 16. The maximum iteration is 10000, n
=
30, and N
=
30. As seen, the FGLT-PSO provides better results than the other algorithms for the majority of functions (functions (
11
), (
12
), (
13
), (
14
), (
17
), and (
20
)) and has the first rank.Table 15
Some well-known PSO algorithms in the literature.
Algorithm
Topology
Parameter settings
PSO
Global star
ω: 0.9–0.4, C
1
=
C
2
=
2.0
QIPSO
Global star
ω: 0.9–0.4, C
1
=
C
2
=
2.0
FIPS
Local U-ring
χ = 0.729,
∑
c
i = 4.1
DMS-PSO
Dynamic multiswarm
ω: 0.9–0.2, C
1
=
C
2
=
2.0
,
m
=
3
,
R
=
5
CLPSO
Comprehensive learning
ω: 0.9–0.4, C
=
1.49445
,
m
=
7
AFPSO
Global star
ω: 0.9–0.4, C
1
,
C
2 are based on fuzzy rule [19]
AFPSO-QI
Global star
ω: 0.9–0.4, C
1
,
C
2 are based on fuzzy rule [19]
FGLT-PSO
Global star and local ring
ω: 0.9–0.4, C
1: 0.5–2, C
2: 1-2, C
3: 0.5–1.5Table 16
Comparison results of eight PSO algorithms [19] with FGLT-PSO for eight benchmark functions (maximum iteration = 10000, n
=
30, and N
=
30).
PSOs
Functions
Avg. best solution ± SD
Avg. best solution ± SD
Avg. best solution ± SD
F
11
F
12
F
13
PSO
−4.529e +001 ± 1.911e +000
3.202e +002 ± 1.470e +001
3.835e +001 ± 1.482e +000
QIPSO
−3.332e +001 ± 1.781e +000
3.175e +002 ± 2.324e +001
4.077e +001 ± 2.015e +000
FIPS
−1.955e +001 ± 8.477e +000
4.341e +002 ± 3.499e +001
4.155e +001 ± 1.363e +000
DMS-PSO
−4.572e +001 ± 1.703e +000
2.837e +002 ± 1.606e +001
3.632e +001 ± 1.225e +000
CLPSO
−4.529e +001 ± 1.269e +000
2.633e +002 ± 1.196e +001
3.496e +001 ± 1.768e +000
AFPSO
−4.547e +001 ± 1.608e +000
2.663e +002 ± 1.200e +001
3.609e +001 ± 2.540e +000
AFPSO-QI
−4.678e +001 ± 1.212e +000
2.533e +002 ± 1.263e +001
3.135e +001 ± 3.301e +000
FGLT-PSO
−4.920e +001 ± 1.091e +000
1.580e +002 ± 1.065e +001
9.933e +000 ± 3.737e +000
F
14
F
17
F
18
PSO
1.703e +001 ± 2.554e +000
3.217e +009 ± 3.880e +009
−1.953e +002 ± 3.282e +001
QIPSO
1.520e +001 ± 1.319e +000
2.347e +009 ± 1.872e +009
−1.963e +002 ± 2.964e +001
FIPS
2.660e +001 ± 1.417e +000
1.340e +003 ± 2.044e +003
−2.173e +002 ± 3.076e +001
DMS-PSO
1.292e +001 ± 1.328e +000
3.362e +008 ± 3.089e +008
−2.456e +002 ± 1.293e +001
CLPSO
1.194e +001 ± 1.365e +000
5.943e +002 ± 5.069e +001
−2.605e +002 ± 7.359e +000
AFPSO
1.038e +001 ± 1.379e +000
9.700e +007 ± 1.197e +008
−2.718e +002 ± 1.072e +001
AFPSO-QI
8.462e +000 ± 9.477e −001
8.832e +007 ± 9.793e +007
−2.736e +002 ± 9.667e +000
FGLT-PSO
5.899e −001 ± 2.820e −001
5.204e +002 ± 1.328e +002
−2.658e +002 ± 2.078e +001
F
19
F
20
Avg. rank
Final rank
PSO
−1.191e +002 ± 7.093e −002
−1.118e +002 ± 4.289e +001
8
8
QIPSO
−1.191e +002 ± 5.677e −001
−1.152e +002 ± 3.390e +001
6.6
7
FIPS
−1.199e +002 ± 3.239e −002
−1.437e +002 ± 5.164e +001
5.6
6
DMS-PSO
−1.192e +002 ± 6.117e −002
−1.910e +002 ± 1.994e +001
4.4
5
CLPSO
−1.190e +002 ± 3.781e −002
−1.312e +002 ± 2.437e +001
4.3
4
AFPSO
−1.197e +002 ± 4.280e −002
−1.267e +002 ± 2.727e +001
3.9
3
AFPSO-QI
−1.198e +002 ± 3.854e −001
−1.339e +002 ± 2.208e +001
2.4
2
FGLT-PSO
−1.191e +002 ± 4.805e −002
−2.166e +002 ± 3.117e +001
1.8
1


## 4.1. Benchmark Functions

Twenty (20) minimization functions are applied in the experimental study including unimodal, multimodal, rotated, shifted, and shifted-rotated functions as detailed in Table1. In the table,Range and n are the feasible bound and the dimension of each function, respectively. F
opt is the optimum value of function. Among the benchmarks, functions (
1
)
–
(
5
) are unimodal functions and functions (
6
)
–
(
9
) are in the class of multimodal functions. Functions (
10
)
–
(
14
) are rotated and functions (
15
)
–
(
18
) are shifted unimodal and multimodal functions. Two functions (19) and (20) are shifted-rotated multimodal functions.Table 1
Dimensions, ranges, and global optimum values of test functions used in the experiments.
Test function
Dimension (n)
[Range]
n
X
opt
F
opt
F
1
(
x
)
10/30/50
[
-
100,100
]
n
0
0
F
2
(
x
)
10/30/50
[
-
10,10
]
n
0
0
F
3
(
x
)
10/30/50
[
-
100,100
]
n
0
0
F
4
(
x
)
10/30/50
[
-
1.28,1.28
]
n
0
0
F
5
(
x
)
10/30/50
[
-
10,10
]
n
1
0
F
6
(
x
)
10/30/50
[
-
500,500
]
n
[
420.96
]
n
-
418.9829
×
n
F
7
(
x
)
10/30/50
[
-
32,32
]
n
0
0
F
8
(
x
)
10/30/50
[
-
50,50
]
n
0
0
F
9
(
x
)
10/30/50
[
-
5.12,5.12
]
n
0
0
F
10
(
x
)
10/30/50
[
-
10,10
]
n
0
0
F
11
(
x
)
10/30/50
[
-
5,5
]
n
0
0
F
12
(
x
)
10/30/50
[
-
5.12,5.12
]
n
0
0
F
13
(
x
)
10/30/50
[
-
0.5,0.5
]
n
0
0
F
14
(
x
)
10/30/50
[
-
100,100
]
n
0
0
F
15
(
x
)
10/30/50
[
-
100,100
]
n
0
−450
F
16
(
x
)
10/30/50
[
-
600,600
]
n
0
−180
F
17
(
x
)
10/30/50
[
-
100,100
]
n
1
390
F
18
(
x
)
10/30/50
[
-
5.2,5.2
]
n
0
−330
F
19
(
x
)
10/30/50
[
-
32,32
]
n
0
−140
F
20
(
x
)
10/30/50
[
-
5.2,5.2
]
n
0
−330In unimodal functions, the convergence rate of search algorithm is more interesting than the final results because other methods have been designed to optimize these kinds of functions. In multimodal functions, finding an optimal (or a good near-global optimal) solution is important. These functions are more difficult to optimize because the number of local optima exponentially increases as the dimension increases. Therefore, the search algorithms should not become trapped in a local optimum and should be able to obtain good solutions.The rotation of function increases the function complexity. It does not affect the shape of function. The variableY
→ is computed using an orthogonal matrix M [36] and applied to obtain the fitness value of rotated function as follows:
(11)
Y
→
=
M
×
X
→
.In shifted functions, the global optimumX
*
→
=
(
x
1
*
,
x
2
*
,
…
,
x
n
*
) is shifted to the new position O
→
=
(
o
1
,
o
2
,
…
,
o
n
). All the test functions are shown as follows.(1) Sphere Model (unimodal function). Consider
(12)
F
1
(
x
)
=
∑
i
=
1
n
x
i
2
.(2) Shifted’s Schwefel’s Problem (unimodal function). Consider
(13)
F
2
(
x
)
=
∑
i
=
1
n
|
x
i
|
+
∏
i
=
1
n
|
x
i
|
.(3) Schwefel’s Problem 1.2 (unimodal function). Consider
(14)
F
3
(
x
)
=
∑
i
=
1
n
(
∑
j
=
1
i
x
j
)
2
.(4) Quartic Function, That Is, Noise (unimodal function). Consider
(15)
F
4
(
x
)
=
∑
i
=
1
n
i
x
i
4
+
random
[
0,1
)
.(5) Rosenbrock’s Function (multimodal function). Consider
(16)
F
5
(
x
)
=
∑
i
=
1
n
-
1
[
100
(
x
i
2
-
x
i
+
1
)
2
+
(
x
i
-
1
)
2
]
.
F
5 is unimodal in a 2-dimension or 3-dimension search space but can be treated as a multimodal function in high-dimensional cases.(6) Generalized Schwefel’s Problem 2.26 (multimodal function). Consider
(17)
F
6
(
x
)
=
∑
i
=
1
n
-
x
i
sin
(
|
x
i
|
)
.(7) Ackley’s Function (multimodal function). Consider
(18)
F
7
(
x
)
=
-
20
exp
⁡
(
-
0.2
1
n
∑
i
=
1
n
x
i
2
)
F
7
(
x
)
=
-
exp
⁡
(
1
n
∑
i
=
1
n
cos
⁡
2
π
x
i
)
+
20
+
e
.(8) Generalized Penalized Function (multimodal function). Consider
(19)
F
8
(
x
)
=
π
n
{
10
sin
⁡
(
π
y
1
)
+
∑
i
=
1
n
-
1
(
y
i
-
1
)
2
F
8
(
x
)
=
π
n
×
[
1
+
10
sin
2
(
π
y
i
+
1
)
]
+
(
y
n
-
1
)
2
∑
i
=
1
n
-
1
(
y
i
-
1
)
2
[
1
+
10
sin
2
(
π
y
i
+
1
)
]
}
F
8
(
x
)
=
+
∑
i
=
1
n
u
(
x
i
,
10,100,4
)
y
i
=
1
+
x
i
+
1
4
,
u
(
x
i
,
a
,
k
,
m
)
=
{
k
(
x
i
-
a
)
m
x
i
>
a
0
-
a
≤
x
i
≤
a
k
(
-
x
i
-
a
)
m
x
i
<
-
a
.(9) Noncontinuous Rastrigin’s Function (multimodal function). Consider
(20)
F
9
(
x
)
=
∑
i
=
1
n
[
y
i
2
-
10
cos
⁡
⁡
(
2
π
y
i
)
+
10
]
,
where
y
i
=
{
x
i
|
x
i
|
≤
0.5
round
(
2
x
i
)
2
|
x
i
|
≥
0.5
.(10) Rotated Schwefel’s Problem 2.22 (unimodal function). Consider
(21)
F
10
(
x
)
=
∑
i
=
1
n
|
y
i
|
+
∏
i
=
1
n
|
y
i
|
,
y
=
M
×
x
.(11) Rotated
2
n
Minima Function (multimodal function). Consider
(22)
F
11
(
x
)
=
1
n
∑
i
=
1
n
(
y
i
4
-
16
y
i
2
+
5
y
i
)
,
y
=
M
×
x
.(12) Rotated Rastrigin’s Function (multimodal function). Consider
(23)
F
12
(
x
)
=
∑
i
=
1
n
[
y
i
2
-
10
cos
⁡
⁡
(
2
π
y
i
)
+
10
]
,
y
=
M
×
x
.(13) Rotated Weierstrass’ Function (multimodal function). Consider
(24)
F
13
(
x
)
=
∑
i
=
1
n
(
∑
k
=
0
k
max
⁡
[
a
k
cos
⁡
(
2
π
b
k
(
y
i
+
0.5
)
)
]
)
F
13
(
x
)
=
-
n
∑
k
=
0
k
max
⁡
[
a
k
cos
⁡
(
2
π
b
k
×
0.5
)
]
,
y
=
M
×
x
,
a
=
0.5
,
b
=
3
,
k
max
⁡
=
20
.(14) Rotated Salomon’s Function (multimodal function). Consider
(25)
F
14
(
x
)
=
1
-
cos
⁡
(
2
π
∑
i
=
1
n
y
i
2
)
+
0.1
∑
i
=
1
n
y
i
2
,
y
=
M
×
x
.(15) Shifted Schwefel’s Problem 2.21 (unimodal function). Consider
(26)
F
15
(
x
)
=
max
⁡
i
{
|
z
i
|
,
1
≤
i
≤
n
}
+
f
bias
15
,
max
⁡
i
f
bias
15
=
-
450
,
z
=
x
-
o
.(16) Shifted Generalized Griewank’s Function (multimodal function). Consider
(27)
F
16
(
x
)
=
1
4000
∑
i
=
1
n
z
i
2
-
∏
i
=
1
n
cos
⁡
(
z
i
i
)
+
1
+
f
bias
16
,
p
k
p
k
p
k
p
k
p
k
p
k
f
bias
16
=
-
180
,
z
=
x
-
o
.(17) Shifted Rosenbrock’s Function (multimodal function). Consider
(28)
F
17
(
x
)
=
∑
i
=
1
n
-
1
[
100
(
z
i
2
-
z
i
+
1
)
2
+
(
z
i
-
1
)
2
]
+
f
bias
17
,
[
10
(
9
-
z
i
+
1
)
2
]
f
bias
17
=
390
,
z
=
x
-
o
+
1
.(18) Shifted Rastrigin’s Function (multimodal function). Consider
(29)
F
18
(
x
)
=
∑
i
=
1
n
[
z
i
2
-
10
cos
⁡
⁡
(
2
π
z
i
)
+
10
]
+
f
bias
18
,
[
z
i
2
-
i
+
10
]
f
bias
18
=
-
330
,
z
=
x
-
o
.(19) Shifted Rotated Ackley’s Function (multimodal function). Consider
(30)
F
19
(
x
)
=
-
20
exp
⁡
(
-
0.2
1
n
∑
i
=
1
n
z
i
2
)
F
19
(
x
)
=
-
exp
⁡
(
1
n
∑
i
=
1
n
cos
⁡
2
π
z
i
)
+
20
+
e
+
f
bias
19
,
where f
bias
19
=
-
140
;
z
=
(
x
-
o
)
×
M
′ is a linear transformation matrix with condition number = 100.(20) Shifted Rotated Rastrigin’s Function (multimodal function). Consider
(31)
F
20
(
x
)
=
∑
i
=
1
n
[
z
i
2
-
10
cos
⁡
⁡
(
2
π
z
i
)
+
10
]
+
f
bias
20
,
where f
bias
20
=
-
330
;
z
=
(
x
-
o
)
×
M
′ is a linear transformation matrix with condition number = 2.


## 4.2. Results of FGLT-PSO

The results of FGLT-PSO are provided in three sections. In Section4.2.1, the acceleration coefficients C
1
,
C
2, and C
3 in the proposed method are changed according to (10) and in Section 4.2.2, these factors are constant. In these sections, FGLT-PSO is evaluated using the benchmark functions with dimensions 10, 30, and 50. The number of maximum iterations is set at 5000 for n
=
10, at 10000 for n
=
30, and at 15000 for n
=
50. The population size is set to 50 (N
=
50). Also, w decreases linearly from 0.9 to 0.4.In Section4.2.3, the results of FGLT-PSO are compared with those of several well-known PSO algorithms from [19] on the common functions. In this section, the population size is set to 30 (N
=
30), n is 30, and the number of maximum iterations is set at 10000.The ring topology is used as the neighborhood structure in thel
best model for the FGLT-PSO and LPSO algorithms and the number of neighbours for each particle is three. The algorithms are run independently 30 times for the benchmark functions and the results are averaged.


### 4.2.1. The Results of Proposed Method with Variable Acceleration Coefficients

Four algorithms of FGLT-PSO, PSO, LPSO, and QIPSO are randomly initialized and run on benchmark functions. The average best solution, the standard deviation (SD), and the median of the best solution in the last iteration are reported in Tables2, 3, 6, 7, 10, and 11. The best results from among the algorithms are shown in bold numbers. In the tables, the algorithms are ranked based on the average best results.Table 2
Minimization results for the unimodal and multimodal functions (maximum iteration = 5000 andn
=
10).
Function
FGLT-PSO
PSO
LPSO
QIPSO
F
1
Avg. best solution
0.000e + 000
1.405e − 135
1.452e − 063
1.526e − 138
SD
0.000e + 000
6.789e − 135
5.681e − 063
5.723e − 138
Median best solution
0.000e + 000
1.243e − 139
7.381e − 065
3.961e − 142
Avg. iteration for finding the best solution
2646
5000
5000
5000
F
2
Avg. best solution
2.741e − 273
2.992e − 077
4.130e − 038
6.663e − 079
SD
0.000e + 000
1.490e − 076
7.564e − 038
1.234e − 078
Median best solution
6.578e − 277
3.849e − 079
9.945e − 039
1.019e − 079
Avg. iteration for finding the best solution
4952
5000
4999
5000
F
3
Avg. best solution
1.637e − 116
3.215e − 044
5.110e − 014
3.919e − 044
SD
8.969e − 116
1.565e − 043
1.126e − 013
2.117e − 043
Median best solution
2.369e − 127
1.857e − 048
5.320e − 015
3.570e − 048
Avg. iteration for finding the best solution
4630
4999
4999
4999
F
4
Avg. best solution
3.209e − 004
4.738e − 004
1.269e − 003
6.514e − 004
SD
3.006e − 004
2.057e − 004
4.861e − 004
3.241e − 004
Median best solution
2.114e − 004
4.595e − 004
1.197e − 003
5.797e − 004
Avg. iteration for finding the best solution
3715
4404
4414
4459
F
5
Avg. best solution
2.973e − 001
1.820e + 000
1.661e + 000
2.040e + 000
SD
1.015e + 000
1.336e + 000
1.281e + 000
1.622e + 000
Median best solution
4.614e − 004
2.014e + 000
1.727e + 000
1.862e + 000
Avg. iteration for finding the best solution
4537
4811
4988
4837
F
6
Avg. best solution
−3.910e + 003
−3.452e + 003
−3.892e + 003
−3.424e + 003
SD
1.184e + 002
2.307e + 002
1.843e + 002
2.775e + 002
Median best solution
−3.953e + 003
−3.475e + 003
−3.834e + 003
−3.475e + 003
Avg. iteration for finding the best solution
2974
2726
3918
2635
F
7
Avg. best solution
4.441e − 015
4.322e − 015
4.441e − 015
4.441e − 015
SD
0.000e + 000
6.486e − 016
0.000e + 000
0.000e + 000
Median best solution
4.441e − 015
4.441e − 015
4.441e − 015
4.441e − 015
Avg. iteration for finding the best solution
376
3147
3772
3109
F
8
Avg. best solution
4.712e − 032
4.712e − 032
4.712e − 032
4.712e − 032
SD
1.670e − 047
1.670e − 047
1.670e − 047
1.670e − 047
Median best solution
4.712e − 032
4.712e − 032
4.712e − 032
4.712e − 032
Avg. iteration for finding the best solution
420
3166
3895
2687
F
9
Avg. best solution
9.495e − 002
9.667e − 001
1.574e + 000
0.000e + 000
SD
2.720e − 001
4.560e + 000
1.065e + 000
0.000e + 000
Median best solution
9.236e − 008
0.000e + 000
2.000e + 000
0.000e + 000
Avg. iteration for finding the best solution
4856
3357
4366
3401
Avg. rank
1.1
2.9
3.4
2.6
Final rank
1
3
4
2
Algorithms
FGLT-PSO
PSO
LPSO
QIPSOTable 3
Minimization results for the rotated and shifted unimodal and multimodal functions (maximum iteration = 5000 andn
=
10).
Function
FGLT-PSO
PSO
LPSO
QIPSO
F
10
Avg. best solution
7.056e −208
4.585e −059
2.358e −021
1.127e −058
SD
0.000e +000
8.736e −059
7.425e −021
5.346e −058
Median best solution
1.930e −209
7.785e −060
3.417e −022
1.023e −060
Avg. iteration for finding the best solution
4741
4999
4997
4999
F
11
Avg. best solution
−6.602e +001
−6.645e +001
−6.616e +001
−6.613e +001
SD
1.736e +000
1.175e +000
9.989e −001
1.720e +000
Median best solution
−6.606e +001
−6.614e +001
−6.607e +001
−6.588e +001
Avg. iteration for finding the best solution
3408
2930
3099
3185
F
12
Avg. best solution
1.604e +001
2.352e +001
2.319e +001
2.278e +001
SD
2.974e +000
3.762e +000
4.012e +000
4.387e +000
Median best solution
1.589e +001
2.358e +001
2.351e +001
2.240e +001
Avg. iteration for finding the best solution
2394
3656
3786
3660
F
13
Avg. best solution
0.000e +000
1.351e +000
7.438e −008
7.957e −001
SD
0.000e +000
3.079e +000
4.072e −007
2.428e +000
Median best solution
0.000e +000
0.000e +000
0.000e +000
0.000e +000
Avg. iteration for finding the best solution
448
3192
4877
3446
F
14
Avg. best solution
1.165e −001
1.032e −001
9.987e −002
1.132e −001
SD
3.790e −002
1.826e −002
0.000e +000
3.457e −002
Median best solution
9.987e −002
9.987e −002
9.987e −002
9.987e −002
Avg. iteration for finding the best solution
1998
2916
3268
2795
F
15
Avg. best solution
−4.500e +002
−4.396e +002
−4.480e +002
−4.406e +002
SD
8.039e −014
7.334e +000
2.470e +000
7.991e +000
Median best solution
−4.500e +002
−4.450e +002
−4.500e +002
−4.450e +002
Avg. iteration for finding the best solution
1324
757
3561
1213
F
16
Avg. best solution
−1.798e +002
−1.745e +002
−1.793e +002
−1.766e +002
SD
3.789e −001
8.186e +000
5.222e −001
3.309e +000
Median best solution
−1.800e +002
−1.780e +002
−1.789e +002
−1.780e +002
Avg. iteration for finding the best solution
2922
2592
3398
2711
F
17
Avg. best solution
3.915e +002
9.048e +006
6.752e +004
6.596e +006
SD
6.315e +000
1.678e +007
3.676e +005
1.638e +007
Median best solution
3.900e +002
2.014e +006
3.933e +002
4.867e +002
Avg. iteration for finding the best solution
4831
3983
4884
4090
F
18
Avg. best solution
−3.287e +002
−3.215e +002
−3.247e +002
−3.233e +002
SD
1.113e +000
5.979e +000
3.144e +000
4.822e +000
Median best solution
−3.290e +002
−3.225e +002
−3.250e +002
−3.225e +002
Avg. iteration for finding the best solution
4809
3193
4509
3331
F
19
Avg. best solution
−119.744
−119.745
−119.804
−119.746
SD
5.750e −002
7.142e −002
5.558e −002
6.911e −002
Median best solution
−119.731
−119.748
−119.804
−119.73
Avg. iteration for finding the best solution
1849
2240
2072
2414
F
20
Avg. best solution
−3.196e +002
−3.091e +002
−3.137e +002
−3.078e +002
SD
4.236e +000
7.174e +000
5.338e +000
7.995e +000
Median best solution
−3.196e +002
−3.102e +002
−3.130e +002
−3.079e +002
Avg. iteration for finding the best solution
3252
3315
4721
3479
Avg. rank
1.8
3.2
2.1
2.9
Final rank
1
4
2
3
Algorithms
FGLT-PSO
PSO
LPSO
QIPSOMoreover, Wilcoxon’s rank sum test [37] is conducted in order to determine whether the results obtained by the FGLT-PSO are different from those generated by other algorithms with a statistical significance. The tests are shown in Tables 4, 5, 8, 9, 12, and 13, where h
-value = 1 indicates the case in which proposed algorithm significantly outperformed the compared algorithm with 95% certainty, h
-value = −1 represents that the compared algorithm is significantly better than the proposed algorithm, and h
-value = 0 denotes that the results of the two considered algorithms are not significantly different. In these tables, rows 1 (better), 0 (same), and −1 (worse) give the number of functions that the FGLT-PSO performs significantly better than, almost the same as, and significantly worse than the compared algorithm, respectively.Table 4
Comparison of FGLT-PSO with PSO, LPSO, and QIPSO for the unimodal and multimodal functions using Wilcoxon’s rank sum test (n
=
10).
Function
Wilcoxon’s rank sum test
PSO
LPSO
QIPSO
F
1
P value
1.2118e −012
1.2118e −012
1.2118e −012
h-value
1
1
1
z-value
−7.10402
−7.10402
−7.10402
F
2
P value
3.0199e −011
3.0199e −011
3.0199e −011
h-value
1
1
1
z-value
−6.6456
−6.6456
−6.6456
F
3
P value
3.0199e −011
3.0199e −011
3.0199e −011
h-value
1
1
1
z-value
−6.6456
−6.6456
−6.6456
F
4
P value
1.0576e −003
6.1210e −010
3.5923e −005
h-value
1
1
1
z-value
−3.27475
−6.18728
−4.13225
F
5
P value
1.5581e −008
1.8731e −007
1.0095e −008
h-value
1
1
1
z-value
−5.65504
−5.21151
−5.72912
F
6
P value
6.3474e −011
1.2724e −001
9.3829e −010
h-value
1
0
1
z-value
−6.53532
−1.52509
−6.11957
F
7
P value
3.3371e −001
—
—
h-value
0
0
0
z-value
0.966667
—
—
F
8
P value
—
—
—
h-value
0
0
0
z-value
—
—
—
F
9
P value
1.9097e −005
1.6703e −006
6.2470e −010
h-value
−1
1
−1
z-value
4.27519
−4.7897
6.18407
1 (better)
6
6
6
0 (same)
2
3
2
−1 (worse)
1
0
1Table 5
Comparison of FGLT-PSO with PSO, LPSO, and QIPSO for the rotated and shifted unimodal and multimodal functions using Wilcoxon’s rank sum test (n
=
10).
Function
Wilcoxon’s rank sum test
PSO
LPSO
QIPSO
F
10
P value
3.0199e −011
3.0199e −011
3.0199e −011
h-value
1
1
1
z-value
−6.6456
−6.6456
−6.6456
F
11
P value
4.1191e −001
7.3940e −001
8.4180e −001
h-value
0
0
0
z-value
0.820536
0.33265
0.19959
F
12
P value
1.8567e −009
1.8500e −008
5.5329e −008
h-value
1
1
1
z-value
−6.00987
−5.62547
−5.43328
F
13
P value
2.1577e −002
2.1577e −002
8.1523e −002
h-value
1
1
0
z-value
−2.29773
−2.29773
−1.74192
F
14
P value
1.1439e −001
1.4425e −004
7.7181e −001
h-value
0
−1
0
z-value
1.57878
3.80076
0.290007
F
15
P value
1.5553e −011
1.2883e −011
1.7966e −008
h-value
1
1
1
z-value
−6.74264
−6.76995
−5.63053
F
16
P value
1.2755e −010
1.9667e −003
5.5338e −010
h-value
1
1
1
z-value
−6.43006
−3.09521
−6.20317
F
17
P value
7.3270e −011
3.6443e −008
5.0030e −010
h-value
1
1
1
z-value
−6.51381
−5.50727
−6.21901
F
18
P value
5.7512e −006
3.5043e −007
4.0269e −005
h-value
1
1
1
z-value
−4.53533
−5.09408
−4.10593
F
19
P value
9.2344e −001
2.3885e −004
8.6499e −001
h-value
0
−1
0
z-value
0.0960988
3.67393
0.170021
F
20
P value
5.2991e −008
9.2051e −005
4.6756e −008
h-value
1
1
1
z-value
−5.44097
−3.91064
−5.46322
1 (better)
8
8
7
0 (same)
3
1
4
−1 (worse)
0
2
0Table 6
Minimization results for the unimodal and multimodal functions (maximum iteration = 10000 andn
=
30).
Function
FGLT-PSO
PSO
LPSO
QIPSO
F
1
Avg. best solution
0.000e +000
5.397e −069
2.747e −028
3.502e −069
SD
0.000e +000
1.161e −068
4.078e −028
1.664e −068
Median best solution
0.000e +000
2.321e −070
1.047e −028
2.727e −072
F
2
Avg. best solution
7.598e −110
7.000e +000
3.849e −028
2.333e +000
SD
4.085e −109
7.944e +000
9.015e −028
4.302e +000
Median best solution
1.684e −136
1.000e +001
1.331e −031
1.188e −071
F
3
Avg. best solution
5.959e −020
1.189e +004
1.399e +002
5.733e +003
SD
1.762e −019
6.605e +003
9.380e +001
5.484e +003
Median best solution
5.483e −021
1.083e +004
1.254e +002
5.000e +003
F
4
Avg. best solution
6.415e −003
1.320e −003
9.664e −003
1.635e −003
SD
2.858e −003
8.008e −004
3.621e −003
9.357e −004
Median best solution
5.941e −003
1.088e −003
9.106e −003
1.333e −003
F
5
Avg. best solution
5.718e −001
1.129e +003
1.751e +001
3.858e +002
SD
1.225e +000
3.023e +003
2.041e +001
1.832e +003
Median best solution
1.707e −001
7.428e +001
9.494e +000
2.139e +001
F
6
Avg. best solution
−1.048e +004
−9.095e +003
−9.858e +003
−9.227e +003
SD
5.996e +002
7.444e +002
4.872e +002
7.524e +002
Median best solution
−1.058e +004
−9.051e +003
−9.828e +003
−9.232e +003
F
7
Avg. best solution
7.771e −002
1.013e −014
2.931e −014
8.941e −015
SD
2.015e −001
3.312e −015
1.433e −014
2.457e −015
Median best solution
7.994e −015
7.994e −015
2.576e −014
7.994e −015
F
8
Avg. best solution
5.164e −003
1.037e −002
1.160e −025
1.571e −032
SD
2.274e −002
3.163e −002
3.552e −025
5.567e −048
Median best solution
1.571e −032
1.591e −032
3.737e −027
1.571e −032
F
9
Avg. best solution
2.136e +001
5.827e +001
4.281e +001
4.310e +001
SD
8.612e +000
3.080e +001
2.064e +001
2.901e +001
Median best solution
2.212e +001
5.450e +001
3.753e +001
3.500e +001
Avg. rank
1.8
3.3
2.6
2.3
Final rank
1
4
3
2
Algorithms
FGLT-PSO
PSO
LPSO
QIPSOTable 7
Minimization results for the rotated and shifted unimodal and multimodal functions (maximum iteration = 10000 andn
=
30).
Function
FGLT-PSO
PSO
LPSO
QIPSO
F
10
Avg. best solution
8.707e − 020
3.422e + 002
7.818e + 000
5.318e + 001
SD
4.547e − 019
1.556e + 003
2.173e + 001
1.986e + 002
Median best solution
4.028e − 043
1.332e − 006
2.267e − 003
8.579e − 020
F
11
Avg. best solution
−4.963e + 001
−4.981e + 001
−4.933e + 001
−4.993e + 001
SD
1.098e + 000
1.324e + 000
1.417e + 000
1.489e + 000
Median best solution
−4.930e + 001
−4.985e + 001
−4.927e + 001
−4.968e + 001
F
12
Avg. best solution
1.514e + 002
1.935e + 002
1.902e + 002
1.886e + 002
SD
8.330e + 000
1.509e + 001
1.246e + 001
1.233e + 001
Median best solution
1.511e + 002
1.978e + 002
1.899e + 002
1.850e + 002
F
13
Avg. best solution
7.160e − 001
3.101e + 001
1.185e + 001
2.046e + 001
SD
9.412e − 001
1.248e + 001
1.136e + 001
1.804e + 001
Median best solution
2.067e − 001
3.613e + 001
5.997e + 000
3.319e + 001
F
14
Avg. best solution
4.065e − 001
3.465e − 001
3.632e − 001
3.599e − 001
SD
1.230e − 001
5.074e − 002
5.561e − 002
7.240e − 002
Median best solution
3.999e − 001
2.999e − 001
3.999e − 001
3.499e − 001
F
15
Avg. best solution
−4.280e + 002
−4.044e + 002
−4.175e + 002
−4.500e + 002
SD
7.441e + 000
7.592e + 000
8.860e + 000
2.151e − 006
Median best solution
−4.266e + 002
−4.056e + 002
−4.155e + 002
−4.500e + 002
F
16
Avg. best solution
−1.794e + 002
−8.340e + 001
−1.705e + 002
−9.806e + 001
SD
1.112e + 000
4.560e + 001
4.344e + 000
3.839e + 001
Median best solution
−1.798e + 002
−9.269e + 001
−1.704e + 002
−1.053e + 002
F
17
Avg. best solution
4.225e + 002
1.658e + 009
8.168e + 007
1.549e + 009
SD
5.410e + 001
1.409e + 009
5.644e + 007
1.599e + 009
Median best solution
3.940e + 002
1.136e + 009
6.932e + 007
1.205e + 009
F
18
Avg. best solution
−2.874e + 002
−2.174e + 002
−2.455e + 002
−2.219e + 002
SD
1.680e + 001
2.897e + 001
1.196e + 001
2.614e + 001
Median best solution
−2.902e + 002
−2.199e + 002
−2.458e + 002
−2.227e + 002
F
19
Avg. best solution
−1.1912e + 002
−1.1912e + 002
−1.1915e + 002
−1.1913e + 002
SD
5.597e − 002
5.015e − 002
5.377e − 002
6.957e − 002
Median best solution
−1.1911e + 002
−1.1911e + 002
−1.1915e + 002
−1.1913e + 002
F
20
Avg. best solution
−2.367e + 002
−1.237e + 002
−1.499e + 002
−1.375e + 002
SD
2.561e + 001
3.612e + 001
2.719e + 001
4.160e + 001
Median best solution
−2.427e + 002
−1.256e + 002
−1.481e + 002
−1.374e + 002
Avg. rank
1.8
3.5
2.4
2.4
Final rank
1
3
2
2
Algorithms
FGLT-PSO
PSO
LPSO
QIPSOTable 8
Comparison of FGLT-PSO with PSO, LPSO, and QIPSO for the unimodal and multimodal functions using Wilcoxon’s rank sum test (n
=
30).
Function
Wilcoxon’s rank sum test
PSO
LPSO
QIPSO
F
1
P value
1.2118e −012
1.2118e −012
1.2118e −012
h-value
1
1
1
z-value
−7.10402
−7.10402
−7.10402
F
2
P value
2.8991e −011
3.0199e −011
2.9822e −011
h-value
1
1
1
z-value
−6.65161
−6.6456
−6.64745
F
3
P value
3.0199e −011
3.0199e −011
3.0199e −011
h-value
1
1
1
z-value
−6.6456
−6.6456
−6.6456
F
4
P value
1.2057e −010
2.2539e −004
1.9568e −010
h-value
−1
1
−1
z-value
6.43862
−3.68871
6.3647
F
5
P value
1.2018e −008
1.2472e −004
5.0922e −008
h-value
1
1
1
z-value
−5.69948
−3.83667
−5.44806
F
6
P value
1.2018e −008
1.2472e −004
5.0922e −008
h-value
1
1
1
z-value
−5.69948
−3.83667
−5.44806
F
7
P value
4.2942e −002
3.7277e −001
2.8314e −003
h-value
−1
0
−1
z-value
2.02428
−0.891289
2.98547
F
8
P value
1.7374e −001
2.4305e −002
6.6167e −004
h-value
0
1
−1
z-value
−1.36028
−2.25228
3.40499
F
9
P value
1.9990e −006
2.7726e −005
7.2926e −004
h-value
1
1
1
z-value
−4.75352
−4.19138
−3.37834
1 (better)
6
8
6
0 (same)
1
1
0
−1 (worse)
2
0
3Table 9
Comparison of FGLT-PSO with PSO, LPSO, and QIPSO for the rotated and shifted unimodal and multimodal functions using Wilcoxon’s rank sum test (n
=
30).
Function
Wilcoxon’s rank sum test
PSO
LPSO
QIPSO
F
10
P value
3.2922e −010
1.4041e −010
3.1451e −009
h-value
1
1
1
z-value
−6.28435
−6.41545
−5.92384
F
11
P value
1.0233e −001
3.0418e −001
4.4642e −001
h-value
0
0
0
z-value
1.63368
−1.02752
0.761398
F
12
P value
1.0937e −010
1.4643e −010
3.0199e −011
h-value
1
1
1
z-value
−6.4534
−6.40905
−6.6456
F
13
P value
1.4288e −008
1.3281e −010
1.1736e −003
h-value
1
1
1
z-value
−5.66991
−6.42392
−3.24523
F
14
P value
1.3470e −003
6.7707e −003
3.5527e −003
h-value
−1
−1
−1
z-value
3.20578
2.70792
2.91537
F
15
P value
2.8840e −010
4.7657e −005
2.8252e −011
h-value
1
1
−1
z-value
−6.30489
−4.06683
6.65541
F
16
P value
2.9543e −011
5.3712e −011
2.9543e −011
h-value
1
1
1
z-value
−6.64883
−6.56027
−6.64883
F
17
P value
3.0199e −11
3.0199e −11
3.0199e −11
h-value
1
1
1
z-value
−6.6456
−6.6456
−6.6456
F
18
P value
1.0937e −010
1.2870e −009
1.4643e −010
h-value
1
1
1
z-value
−6.4534
−6.06901
−6.40905
F
19
P value
8.7663e −001
4.0595e −002
7.7312e −001
h-value
0
1
0
z-value
0.155236
2.04764
0.288296
F
20
P value
3.0199e −011
7.3891e −011
4.9752e −011
h-value
1
1
1
z-value
−6.6456
−6.51254
−6.57168
1 (better)
8
9
7
0 (same)
2
1
2
−1 (worse)
1
1
2Table 10
Minimization results for the unimodal and multimodal functions (maximum iteration = 15000 andn
=
50).
Function
FGLT-PSO
PSO
LPSO
QIPSO
F
1
Avg. best solution
5.239e −232
7.5857e −049
7.297e −020
2.446e −049
SD
0.000e +000
2.1243e −048
7.695e −020
8.478e −049
Median best solution
2.541e −251
4.5415e −050
4.843e −020
1.395e −050
F
2
Avg. best solution
9.246e −075
3.400e +001
2.194e −015
1.367e +001
SD
3.646e −074
1.714e +001
1.439e −015
1.159e +001
Median best solution
2.684e −080
3.000e +001
1.881e −015
1.000e +001
F
3
Avg. best solution
1.098e −008
4.167e +004
2.775e +004
3.585e +004
SD
2.405e −008
1.404e +004
8.532e +003
1.967e +004
Median best solution
5.928e −009
4.002e +004
2.859e +004
3.434e +004
F
4
Avg. best solution
5.070e −002
6.015e +000
6.452e −002
2.055e −002
SD
2.802e −002
5.352e +000
1.488e −002
4.363e −003
Median best solution
4.205e −002
5.385e +000
6.409e −002
1.971e −002
F
5
Avg. best solution
6.128e +000
1.568e +003
6.024e +001
4.700e +002
SD
5.716e +000
3.409e +003
3.032e +001
1.817e +003
Median best solution
5.080e +000
8.179e +001
7.397e +001
7.942e +001
F
6
Avg. best solution
−1.505e +004
−1.364e +004
−1.470e +004
−1.339e +004
SD
9.481e +002
9.581e +002
8.491e +002
1.025e +003
Median best solution
−1.515e +004
−1.349e +004
−1.445e +004
−1.343e +004
F
7
Avg. best solution
1.299e +000
2.049e +000
1.145e −005
1.782e −014
SD
5.560e −001
4.661e +000
6.223e −005
3.695e −015
Median best solution
1.282e +000
2.220e −014
1.782e −009
1.510e −014
F
8
Avg. best solution
1.061e −001
1.451e −002
3.831e −012
2.074e −003
SD
1.935e −001
3.535e −002
1.984e −011
1.136e −002
Median best solution
2.081e −005
1.308e −032
2.881e −015
9.423e −033
F
9
Avg. best solution
4.759e +001
1.980e +002
1.872e +002
1.546e +002
SD
2.294e +001
4.888e +001
3.958e +001
5.192e +001
Median best solution
4.636e +001
1.890e +002
1.850e +002
1.475e +002
Avg. rank
1.7
3.7
2.3
2.3
Final rank
1
3
2
2
Algorithms
FGLT-PSO
PSO
LPSO
QIPSOTable 11
Minimization results for the rotated and shifted unimodal and multimodal functions (maximum iteration = 15000 andn
=
50).
Function
FGLT-PSO
PSO
LPSO
QIPSO
F
10
Avg. best solution
9.936e −031
6.202e +009
5.150e +003
3.989e +006
SD
5.442e −030
2.576e +010
1.153e +004
1.825e +007
Median best solution
1.121e −048
1.763e +005
1.184e +002
5.637e −007
F
11
Avg. best solution
−4.401e +001
−4.328e +001
−4.357e +001
−4.385e +001
SD
1.043e +000
1.331e +000
9.431e −001
1.329e +000
Median best solution
−4.391e +001
−4.344e +001
−4.339e +001
−4.367e +001
F
12
Avg. best solution
3.155e +002
4.234e +002
3.995e +002
4.045e +002
SD
1.166e +001
3.689e +001
1.813e +001
2.468e +001
Median best solution
3.171e +002
4.236e +002
3.995e +002
4.062e +002
F
13
Avg. best solution
1.660e +001
6.640e +001
5.210e +001
4.946e +001
SD
6.875e +000
2.371e +000
1.517e +001
2.599e +001
Median best solution
1.759e +001
6.697e +001
6.163e +001
6.361e +001
F
14
Avg. best solution
7.599e −001
5.899e −001
8.067e −001
6.099e −001
SD
3.103e −001
8.030e −002
9.046e −002
8.847e −002
Median best solution
6.999e −001
5.999e −001
7.999e −001
5.999e −001
F
15
Avg. best solution
−3.976e +002
−3.795e +002
−4.019e +002
−4.499e +002
SD
5.964e +000
1.974e +001
6.096e +000
1.849e −003
Median best solution
−3.974e +002
−3.815e +002
−4.021e +002
−4.50e +002
F
16
Avg. best solution
−1.791e +002
8.313e +001
−1.551e +002
6.894e +001
SD
1.212e +000
9.121e +001
1.216e +001
7.896e +001
Median best solution
−1.797e +002
7.911e +001
−1.565e +002
7.481e +001
F
17
Avg. best solution
4.260e +002
8.648e +009
3.341e +008
8.196e +009
SD
7.188e +001
4.442e +009
2.597e +008
5.588e +009
Median best solution
3.977e +002
8.332e +009
2.599e +008
6.614e +009
F
18
Avg. best solution
−1.920e +002
−4.497e +001
−9.832e +001
−3.612e +001
SD
2.634e +001
4.206e +001
1.963e +001
4.934e +001
Median best solution
−1.944e +002
−4.804e +001
−9.679e +001
−2.665e +001
F
19
Avg. best solution
−1.189e +002
−1.189e +002
−1.190e +002
−1.189e +002
SD
4.303e −002
4.364e −002
5.121e −002
5.576e −002
Median best solution
−1.189e +002
−1.189e +002
−1.190e +002
−1.189e +002
F
20
Avg. best solution
−1.016e +002
1.537e +002
5.974e +001
1.082e +002
SD
4.898e +001
8.906e +001
5.117e +001
9.020e +001
Median best solution
−1.004e +002
1.539e +002
6.594e +001
1.035e +002
Avg. rank
1.5
3.6
2.3
2.7
Final rank
1
4
2
3
Algorithms
FGLT-PSO
PSO
LPSO
QIPSOTable 12
Comparison of FGLT-PSO with PSO, LPSO, and QIPSO for the unimodal and multimodal functions using Wilcoxon’s rank sum test (n
=
50).
Function
Wilcoxon’s rank sum test
PSO
LPSO
QIPSO
F
1
P value
3.0199e −011
3.0199e −011
3.0199e −011
h-value
1
1
1
z-value
−6.6456
−6.6456
−6.6456
F
2
P value
2.9673e −011
3.0199e −011
2.9229e −011
h-value
1
1
1
z-value
−6.64819
−6.6456
−6.65041
F
3
P value
3.0199e −011
3.0199e −011
3.0199e −011
h-value
1
1
1
z-value
−6.6456
−6.6456
−6.6456
F
4
P value
2.5306e −004
3.9881e −004
3.1589e −010
h-value
1
1
−1
z-value
−3.65915
−3.54087
6.29077
F
5
P value
6.1210e −010
9.7555e −010
1.2870e −009
h-value
1
1
1
z-value
−6.18728
−6.11336
−6.06901
F
6
P value
2.8790e −006
9.3341e −002
2.5711e −007
h-value
1
0
1
z-value
−4.67927
−1.67803
−5.15244
F
7
P value
7.7050e −006
3.0199e −011
1.4811e −011
h-value
−1
−1
−1
z-value
4.47322
6.6456
6.74974
F
8
P value
7.7087e −002
8.2796e −003
1.4193e −007
h-value
0
−1
−1
z-value
1.76784
2.64045
5.26273
F
9
P value
3.3384e −011
3.3384e −011
4.6159e −010
h-value
1
1
1
z-value
−6.63081
−6.63081
−6.23164
1 (better)
7
6
6
0 (same)
1
1
0
−1 (worse)
1
2
3Table 13
Comparison of FGLT-PSO with PSO, LPSO, and QIPSO for the rotated and shifted unimodal and multimodal functions using Wilcoxon’s rank sum test (n
=
50).
Function
Wilcoxon’s rank sum test
PSO
LPSO
QIPSO
F
10
P value
3.0199e −011
3.0199e −011
3.0199e −011
h-value
1
1
1
z-value
−6.6456
−6.6456
−6.6456
F
11
P value
3.0317e −002
6.5671e −002
6.7350e −001
h-value
1
0
0
z-value
−2.16592
−1.84066
−0.421356
F
12
P value
3.0199e −011
3.0199e −011
3.0199e −011
h-value
1
1
1
z-value
−6.6456
−6.6456
−6.6456
F
13
P value
3.0199e −011
1.2057e −010
1.8916e −004
h-value
1
1
1
z-value
−6.6456
−6.43862
−3.73307
F
14
P value
7.3131e −003
3.0749e −002
3.0494e −002
h-value
−1
1
−1
z-value
2.68224
−2.16031
2.16361
F
15
P value
2.2256e −004
5.4038e −004
2.7547e −011
h-value
1
−1
−1
z-value
−3.69193
3.4599
6.65912
F
16
P value
3.0142e −011
3.0123e −011
3.0142e −011
h-value
1
1
1
z-value
−6.64588
−6.64597
−6.64588
F
17
P value
3.0199e −011
3.0199e −011
3.0199e −011
h-value
1
1
1
z-value
−6.6456
−6.6456
−6.6456
F
18
P value
3.0199e −011
4.0772e −011
3.0199e −011
h-value
1
1
1
z-value
−6.6456
−6.60125
−6.6456
F
19
P value
2.3399e −001
5.8737e −004
9.2344e −001
h-value
0
−1
0
z-value
1.19015
3.43738
0.0960988
F
20
P value
3.3384e −011
1.3289e −010
8.9934e −011
h-value
1
1
1
z-value
−6.63081
−6.42383
−6.48297
1 (better)
9
8
7
0 (same)
1
1
2
−1 (worse)
1
2
2The acceleration coefficientsC
1, C
2, and C
3 are updated based on (10). Their minimum and maximum values are as follows: C
1
min
⁡
=
0.5
,
C
1
max
⁡
=
2
,
C
2
min
⁡
=
1
,
C
2
max
⁡
=
2
,
C
3
min
⁡
=
0.5, and C
3
max
⁡ = 1.5.In these tables, the benchmark functions are divided to two categories:(
1
) unimodal and multimodal functions and (
2
) rotated, shifted, and shifted-rotated unimodal and multimodal functions. The experimental results demonstrate that FGLT-PSO performs superior results for most of the functions in all tested dimensions.Tables2 and 3 show the experimental results for all benchmark functions with dimension n
=
10. As illustrated, the FGLT-PSO algorithm surpasses the PSO, LPSO, and QIPSO algorithms in minimizing functions (
1
)
–
(
6
), (
10
), (
12
), (
13
), (
15
)
–
(
18
), and (20). Moreover, the proposed method provides significant improvements in functions (
1
), (
2
), (
3
), (
10
), (
13
), (
17
), and (
18
). In these functions, the convergent results attain the optimal (or good near optimal) solutions. In Tables 2 and 3, the average iteration for finding the best solution is computed. The average iteration is the required iterations to find the best solution by each algorithm. As shown, FGLT-PSO finds the best solutions faster than the other algorithms in the majority of functions. Also, it is noticeable that the FGLT-PSO algorithm achieves the best solution in a considerably lower iteration in functions (
7
) and (
8
). In these functions, the algorithms show identical results.According to Wilcoxon’s rank sum test in Tables4 and 5 for n
=
10, the results of the FGLT-PSO are statistically significantly different from the three compared algorithms. The superior convergence rate of FGLT-PSO is shown in Figure 2. The results in this figure illustrate that FGLT-PSO tends to find the global optimum in F
1 and F
17 faster than PSO, LPSO, and QIPSO and obtains the highest accuracy for these functions from among all the algorithms.Figure 2
Convergence performance of FGLT-PSO, PSO, LPSO, and QIPSO for the test functionsF
1 and F
17 (n
=
10).The minimization results of the benchmark functions with dimensionn
=
30 are presented in Tables 6 and 7. As seen in these tables and Tables 8 and 9, the FGLT-PSO outperforms the PSO, LPSO, and QIPSO algorithms in functions (
1
), (
2
), (
3
), (
5
), (
6
), (
9
), (
10
), (
12
), (
13
), (
16
), (
17
), (
18
), and (
20
). The largest difference in performance between the proposed algorithm with PSO, LPSO, and QIPSO occurs for the functions (
1
), (
2
), (
3
), (
5
), (
16
), (
17
), (
18
), and (
20
). Figure 3 illustrates the progress of the average best solution over 30 runs for F
3 and F
18. As demonstrated, the FGLT-PSO shows a higher convergence rate than the other algorithms.Figure 3
Convergence performance of FGLT-PSO, PSO, LPSO, and QIPSO for the test functionsF
3 and F
18 (n
=
30).Tables10 and 11 present the results of algorithms for the test functions with dimension n
=
50. As illustrated in these tables and regarding the results of Wilcoxon’s rank sum in Tables 12 and 13, the performance of proposed method is the best in most of the functions especially for functions (
1
), (
2
), (
3
), (
5
), (
6
), (
9
), (
10
), (
11
), (
12
), (
13
), (
16
), (
17
), (
18
), and (
20
). The superior convergence rate of FGLT-PSO is shown in Figure 4. The results in this figure show that the FGLT-PSO performs the best for the test functions F
16 and F
20 with n
=
50.Figure 4
Convergence performance of FGLT-PSO, PSO, LPSO, and QIPSO for the test functionsF
16 and F
20 (n
=
50).In addition, it is considerable that the PSO, LPSO, and QIPSO algorithms return the results far from the global optima as the dimension increases. This problem is clear in the functions(
3
), (
5
), (
9
), (
10
), (
16
), (
17
), (
18
), and (
20
) in Tables 10 and 11 with n
=
50. These all indicate that the proposed algorithm, FGLT-PSO, is more powerful and robust than the others for solving unimodal and multimodal functions.


### 4.2.2. The Results of Proposed Method with Constant Acceleration Coefficients

In this section, theC
1, C
2, and C
3 as acceleration coefficients are set at constant values to compare with the presented results in the Section 4.2.1. The coefficient of cognition term (C
1) and social terms (C
2 and C
3) are considered as C
1
=
2 and C
2
=
C
3
=
1. Table 14 shows the results of proposed method for the benchmark functions with dimensions 10, 30, and 50. As seen, the FGLT-PSO with the constant acceleration coefficients performs well in most of the functions. As the dimension increases, the FGLT-PSO with the variable acceleration coefficients (Section 4.2.1) shows the better performance than the constant one for functions (
1
), (
2
), (
3
), (
5
), (
17
), and (
20
). Also, the FGLT-PSO with the constant acceleration coefficients presents a better performance for functions (
7
), (
13
), and (
18
).Table 14
The results of FGLT-PSO with constant acceleration coefficients and different dimensions (N
=
50).
Functions
Iteration = 5000,n
=
10
Iteration = 10000,n
=
30
Iteration = 15000,n
=
50
Avg. best solution ± SD
Avg. best solution ± SD
Avg. best solution ± SD
F
1
0.000e +000  ±  0.000e +000
1.936e −067 ± 1.060e −066
1.687e −038 ± 5.602e −038
F
2
6.678e −160 ± 3.658e −159
4.922e −049 ± 2.305e −048
7.064e −032 ± 3.167e −031
F
3
7.520e −035 ± 3.579e −034
2.026e −003 ± 9.321e −003
2.062e +000 ± 2.616e +000
F
4
1.621e −004 ± 8.583e −005
9.078e −004 ± 3.306e −004
1.930e −003 ± 6.461e −004
F
5
3.086e +000 ± 1.772e +000
3.542e +001 ± 2.864e +001
7.854e +001 ± 4.435e +001
F
6
−3.957e +003 ± 1.375e +002
−1.052e +004 ± 4.069e +002
−1.643e +004 ± 7.233e +002
F
7
4.086e −015 ± 1.084e −015
5.507e −015 ± 1.656e −015
9.484e −002 ± 3.612e −001
F
8
4.712e −032 ± 1.670e −047
4.492e −002 ± 1.379e −001
7.063e −002 ± 1.508e −001
F
9
2.567e +000 ± 9.714e −001
2.040e +001 ± 6.100e +000
5.183e +001 ± 1.117e +001
F
10
6.599e −249 ± 0.000e +000
3.539e −066 ± 1.939e −065
4.923e −019 ± 2.696e −018
F
11
−6.620e +001 ± 1.111e +000
−5.036e +001 ± 9.041e −001
−4.482e +001 ± 9.061e −001
F
12
1.659e +001 ± 4.479e +000
1.602e +002 ± 1.096e +001
3.369e +002 ± 1.585e +001
F
13
0.000e +000 ± 0.000e +000
0.000e +000 ± 0.000e +000
1.5294e −007 ± 1.1896e −007
F
14
9.987e −002 ± 2.247e −017
3.932e −001 ± 1.081e −001
3.065e −001 ± 2.537e −002
F
15
−4.500e +002 ± 1.493e −014
−4.268e +002 ± 1.016e +001
−4.420e +002 ± 3.137e +000
F
16
−1.799e +002 ± 2.005e −001
−1.787e +002 ± 3.851e +000
−1.775e +002 ± 3.055e +000
F
17
4.001e +002 ± 2.060e +001
4.661e +002 ± 1.083e +002
5.748e +005 ± 1.022e +006
F
18
−3.283e +002 ± 8.796e −001
−2.956e +002 ± 8.540e +000
−2.396e +002 ± 1.763e +001
F
19
−1.1974e +002 ± 5.990e −002
−1.191e +002 ± 6.798e −002
−1.189e +002 ± 3.710e −002
F
20
−3.196e +002 ± 4.136e +000
−1.994e +002 ± 2.829e +001
−2.684e +001 ± 4.506e +001


### 4.2.3. Comparison with the Other PSO Algorithms

In this section, several well-known PSO algorithms are selected to assess the performance of proposed algorithm for the benchmarks. The PSO, QIPSO, FIPS, DMS-PSO, CLPSO, AFPSO, and AFPSO-QI algorithms are considered for the comparison. The details of these algorithms are listed in Table15. The FGLT-PSO is run 30 times and the average best solutions and the SD of results for eight common multimodal benchmark functions are compared with the reported results by [19] as illustrated in Table 16. The maximum iteration is 10000, n
=
30, and N
=
30. As seen, the FGLT-PSO provides better results than the other algorithms for the majority of functions (functions (
11
), (
12
), (
13
), (
14
), (
17
), and (
20
)) and has the first rank.Table 15
Some well-known PSO algorithms in the literature.
Algorithm
Topology
Parameter settings
PSO
Global star
ω: 0.9–0.4, C
1
=
C
2
=
2.0
QIPSO
Global star
ω: 0.9–0.4, C
1
=
C
2
=
2.0
FIPS
Local U-ring
χ = 0.729,
∑
c
i = 4.1
DMS-PSO
Dynamic multiswarm
ω: 0.9–0.2, C
1
=
C
2
=
2.0
,
m
=
3
,
R
=
5
CLPSO
Comprehensive learning
ω: 0.9–0.4, C
=
1.49445
,
m
=
7
AFPSO
Global star
ω: 0.9–0.4, C
1
,
C
2 are based on fuzzy rule [19]
AFPSO-QI
Global star
ω: 0.9–0.4, C
1
,
C
2 are based on fuzzy rule [19]
FGLT-PSO
Global star and local ring
ω: 0.9–0.4, C
1: 0.5–2, C
2: 1-2, C
3: 0.5–1.5Table 16
Comparison results of eight PSO algorithms [19] with FGLT-PSO for eight benchmark functions (maximum iteration = 10000, n
=
30, and N
=
30).
PSOs
Functions
Avg. best solution ± SD
Avg. best solution ± SD
Avg. best solution ± SD
F
11
F
12
F
13
PSO
−4.529e +001 ± 1.911e +000
3.202e +002 ± 1.470e +001
3.835e +001 ± 1.482e +000
QIPSO
−3.332e +001 ± 1.781e +000
3.175e +002 ± 2.324e +001
4.077e +001 ± 2.015e +000
FIPS
−1.955e +001 ± 8.477e +000
4.341e +002 ± 3.499e +001
4.155e +001 ± 1.363e +000
DMS-PSO
−4.572e +001 ± 1.703e +000
2.837e +002 ± 1.606e +001
3.632e +001 ± 1.225e +000
CLPSO
−4.529e +001 ± 1.269e +000
2.633e +002 ± 1.196e +001
3.496e +001 ± 1.768e +000
AFPSO
−4.547e +001 ± 1.608e +000
2.663e +002 ± 1.200e +001
3.609e +001 ± 2.540e +000
AFPSO-QI
−4.678e +001 ± 1.212e +000
2.533e +002 ± 1.263e +001
3.135e +001 ± 3.301e +000
FGLT-PSO
−4.920e +001 ± 1.091e +000
1.580e +002 ± 1.065e +001
9.933e +000 ± 3.737e +000
F
14
F
17
F
18
PSO
1.703e +001 ± 2.554e +000
3.217e +009 ± 3.880e +009
−1.953e +002 ± 3.282e +001
QIPSO
1.520e +001 ± 1.319e +000
2.347e +009 ± 1.872e +009
−1.963e +002 ± 2.964e +001
FIPS
2.660e +001 ± 1.417e +000
1.340e +003 ± 2.044e +003
−2.173e +002 ± 3.076e +001
DMS-PSO
1.292e +001 ± 1.328e +000
3.362e +008 ± 3.089e +008
−2.456e +002 ± 1.293e +001
CLPSO
1.194e +001 ± 1.365e +000
5.943e +002 ± 5.069e +001
−2.605e +002 ± 7.359e +000
AFPSO
1.038e +001 ± 1.379e +000
9.700e +007 ± 1.197e +008
−2.718e +002 ± 1.072e +001
AFPSO-QI
8.462e +000 ± 9.477e −001
8.832e +007 ± 9.793e +007
−2.736e +002 ± 9.667e +000
FGLT-PSO
5.899e −001 ± 2.820e −001
5.204e +002 ± 1.328e +002
−2.658e +002 ± 2.078e +001
F
19
F
20
Avg. rank
Final rank
PSO
−1.191e +002 ± 7.093e −002
−1.118e +002 ± 4.289e +001
8
8
QIPSO
−1.191e +002 ± 5.677e −001
−1.152e +002 ± 3.390e +001
6.6
7
FIPS
−1.199e +002 ± 3.239e −002
−1.437e +002 ± 5.164e +001
5.6
6
DMS-PSO
−1.192e +002 ± 6.117e −002
−1.910e +002 ± 1.994e +001
4.4
5
CLPSO
−1.190e +002 ± 3.781e −002
−1.312e +002 ± 2.437e +001
4.3
4
AFPSO
−1.197e +002 ± 4.280e −002
−1.267e +002 ± 2.727e +001
3.9
3
AFPSO-QI
−1.198e +002 ± 3.854e −001
−1.339e +002 ± 2.208e +001
2.4
2
FGLT-PSO
−1.191e +002 ± 4.805e −002
−2.166e +002 ± 3.117e +001
1.8
1


## 4.2.1. The Results of Proposed Method with Variable Acceleration Coefficients

Four algorithms of FGLT-PSO, PSO, LPSO, and QIPSO are randomly initialized and run on benchmark functions. The average best solution, the standard deviation (SD), and the median of the best solution in the last iteration are reported in Tables2, 3, 6, 7, 10, and 11. The best results from among the algorithms are shown in bold numbers. In the tables, the algorithms are ranked based on the average best results.Table 2
Minimization results for the unimodal and multimodal functions (maximum iteration = 5000 andn
=
10).
Function
FGLT-PSO
PSO
LPSO
QIPSO
F
1
Avg. best solution
0.000e + 000
1.405e − 135
1.452e − 063
1.526e − 138
SD
0.000e + 000
6.789e − 135
5.681e − 063
5.723e − 138
Median best solution
0.000e + 000
1.243e − 139
7.381e − 065
3.961e − 142
Avg. iteration for finding the best solution
2646
5000
5000
5000
F
2
Avg. best solution
2.741e − 273
2.992e − 077
4.130e − 038
6.663e − 079
SD
0.000e + 000
1.490e − 076
7.564e − 038
1.234e − 078
Median best solution
6.578e − 277
3.849e − 079
9.945e − 039
1.019e − 079
Avg. iteration for finding the best solution
4952
5000
4999
5000
F
3
Avg. best solution
1.637e − 116
3.215e − 044
5.110e − 014
3.919e − 044
SD
8.969e − 116
1.565e − 043
1.126e − 013
2.117e − 043
Median best solution
2.369e − 127
1.857e − 048
5.320e − 015
3.570e − 048
Avg. iteration for finding the best solution
4630
4999
4999
4999
F
4
Avg. best solution
3.209e − 004
4.738e − 004
1.269e − 003
6.514e − 004
SD
3.006e − 004
2.057e − 004
4.861e − 004
3.241e − 004
Median best solution
2.114e − 004
4.595e − 004
1.197e − 003
5.797e − 004
Avg. iteration for finding the best solution
3715
4404
4414
4459
F
5
Avg. best solution
2.973e − 001
1.820e + 000
1.661e + 000
2.040e + 000
SD
1.015e + 000
1.336e + 000
1.281e + 000
1.622e + 000
Median best solution
4.614e − 004
2.014e + 000
1.727e + 000
1.862e + 000
Avg. iteration for finding the best solution
4537
4811
4988
4837
F
6
Avg. best solution
−3.910e + 003
−3.452e + 003
−3.892e + 003
−3.424e + 003
SD
1.184e + 002
2.307e + 002
1.843e + 002
2.775e + 002
Median best solution
−3.953e + 003
−3.475e + 003
−3.834e + 003
−3.475e + 003
Avg. iteration for finding the best solution
2974
2726
3918
2635
F
7
Avg. best solution
4.441e − 015
4.322e − 015
4.441e − 015
4.441e − 015
SD
0.000e + 000
6.486e − 016
0.000e + 000
0.000e + 000
Median best solution
4.441e − 015
4.441e − 015
4.441e − 015
4.441e − 015
Avg. iteration for finding the best solution
376
3147
3772
3109
F
8
Avg. best solution
4.712e − 032
4.712e − 032
4.712e − 032
4.712e − 032
SD
1.670e − 047
1.670e − 047
1.670e − 047
1.670e − 047
Median best solution
4.712e − 032
4.712e − 032
4.712e − 032
4.712e − 032
Avg. iteration for finding the best solution
420
3166
3895
2687
F
9
Avg. best solution
9.495e − 002
9.667e − 001
1.574e + 000
0.000e + 000
SD
2.720e − 001
4.560e + 000
1.065e + 000
0.000e + 000
Median best solution
9.236e − 008
0.000e + 000
2.000e + 000
0.000e + 000
Avg. iteration for finding the best solution
4856
3357
4366
3401
Avg. rank
1.1
2.9
3.4
2.6
Final rank
1
3
4
2
Algorithms
FGLT-PSO
PSO
LPSO
QIPSOTable 3
Minimization results for the rotated and shifted unimodal and multimodal functions (maximum iteration = 5000 andn
=
10).
Function
FGLT-PSO
PSO
LPSO
QIPSO
F
10
Avg. best solution
7.056e −208
4.585e −059
2.358e −021
1.127e −058
SD
0.000e +000
8.736e −059
7.425e −021
5.346e −058
Median best solution
1.930e −209
7.785e −060
3.417e −022
1.023e −060
Avg. iteration for finding the best solution
4741
4999
4997
4999
F
11
Avg. best solution
−6.602e +001
−6.645e +001
−6.616e +001
−6.613e +001
SD
1.736e +000
1.175e +000
9.989e −001
1.720e +000
Median best solution
−6.606e +001
−6.614e +001
−6.607e +001
−6.588e +001
Avg. iteration for finding the best solution
3408
2930
3099
3185
F
12
Avg. best solution
1.604e +001
2.352e +001
2.319e +001
2.278e +001
SD
2.974e +000
3.762e +000
4.012e +000
4.387e +000
Median best solution
1.589e +001
2.358e +001
2.351e +001
2.240e +001
Avg. iteration for finding the best solution
2394
3656
3786
3660
F
13
Avg. best solution
0.000e +000
1.351e +000
7.438e −008
7.957e −001
SD
0.000e +000
3.079e +000
4.072e −007
2.428e +000
Median best solution
0.000e +000
0.000e +000
0.000e +000
0.000e +000
Avg. iteration for finding the best solution
448
3192
4877
3446
F
14
Avg. best solution
1.165e −001
1.032e −001
9.987e −002
1.132e −001
SD
3.790e −002
1.826e −002
0.000e +000
3.457e −002
Median best solution
9.987e −002
9.987e −002
9.987e −002
9.987e −002
Avg. iteration for finding the best solution
1998
2916
3268
2795
F
15
Avg. best solution
−4.500e +002
−4.396e +002
−4.480e +002
−4.406e +002
SD
8.039e −014
7.334e +000
2.470e +000
7.991e +000
Median best solution
−4.500e +002
−4.450e +002
−4.500e +002
−4.450e +002
Avg. iteration for finding the best solution
1324
757
3561
1213
F
16
Avg. best solution
−1.798e +002
−1.745e +002
−1.793e +002
−1.766e +002
SD
3.789e −001
8.186e +000
5.222e −001
3.309e +000
Median best solution
−1.800e +002
−1.780e +002
−1.789e +002
−1.780e +002
Avg. iteration for finding the best solution
2922
2592
3398
2711
F
17
Avg. best solution
3.915e +002
9.048e +006
6.752e +004
6.596e +006
SD
6.315e +000
1.678e +007
3.676e +005
1.638e +007
Median best solution
3.900e +002
2.014e +006
3.933e +002
4.867e +002
Avg. iteration for finding the best solution
4831
3983
4884
4090
F
18
Avg. best solution
−3.287e +002
−3.215e +002
−3.247e +002
−3.233e +002
SD
1.113e +000
5.979e +000
3.144e +000
4.822e +000
Median best solution
−3.290e +002
−3.225e +002
−3.250e +002
−3.225e +002
Avg. iteration for finding the best solution
4809
3193
4509
3331
F
19
Avg. best solution
−119.744
−119.745
−119.804
−119.746
SD
5.750e −002
7.142e −002
5.558e −002
6.911e −002
Median best solution
−119.731
−119.748
−119.804
−119.73
Avg. iteration for finding the best solution
1849
2240
2072
2414
F
20
Avg. best solution
−3.196e +002
−3.091e +002
−3.137e +002
−3.078e +002
SD
4.236e +000
7.174e +000
5.338e +000
7.995e +000
Median best solution
−3.196e +002
−3.102e +002
−3.130e +002
−3.079e +002
Avg. iteration for finding the best solution
3252
3315
4721
3479
Avg. rank
1.8
3.2
2.1
2.9
Final rank
1
4
2
3
Algorithms
FGLT-PSO
PSO
LPSO
QIPSOMoreover, Wilcoxon’s rank sum test [37] is conducted in order to determine whether the results obtained by the FGLT-PSO are different from those generated by other algorithms with a statistical significance. The tests are shown in Tables 4, 5, 8, 9, 12, and 13, where h
-value = 1 indicates the case in which proposed algorithm significantly outperformed the compared algorithm with 95% certainty, h
-value = −1 represents that the compared algorithm is significantly better than the proposed algorithm, and h
-value = 0 denotes that the results of the two considered algorithms are not significantly different. In these tables, rows 1 (better), 0 (same), and −1 (worse) give the number of functions that the FGLT-PSO performs significantly better than, almost the same as, and significantly worse than the compared algorithm, respectively.Table 4
Comparison of FGLT-PSO with PSO, LPSO, and QIPSO for the unimodal and multimodal functions using Wilcoxon’s rank sum test (n
=
10).
Function
Wilcoxon’s rank sum test
PSO
LPSO
QIPSO
F
1
P value
1.2118e −012
1.2118e −012
1.2118e −012
h-value
1
1
1
z-value
−7.10402
−7.10402
−7.10402
F
2
P value
3.0199e −011
3.0199e −011
3.0199e −011
h-value
1
1
1
z-value
−6.6456
−6.6456
−6.6456
F
3
P value
3.0199e −011
3.0199e −011
3.0199e −011
h-value
1
1
1
z-value
−6.6456
−6.6456
−6.6456
F
4
P value
1.0576e −003
6.1210e −010
3.5923e −005
h-value
1
1
1
z-value
−3.27475
−6.18728
−4.13225
F
5
P value
1.5581e −008
1.8731e −007
1.0095e −008
h-value
1
1
1
z-value
−5.65504
−5.21151
−5.72912
F
6
P value
6.3474e −011
1.2724e −001
9.3829e −010
h-value
1
0
1
z-value
−6.53532
−1.52509
−6.11957
F
7
P value
3.3371e −001
—
—
h-value
0
0
0
z-value
0.966667
—
—
F
8
P value
—
—
—
h-value
0
0
0
z-value
—
—
—
F
9
P value
1.9097e −005
1.6703e −006
6.2470e −010
h-value
−1
1
−1
z-value
4.27519
−4.7897
6.18407
1 (better)
6
6
6
0 (same)
2
3
2
−1 (worse)
1
0
1Table 5
Comparison of FGLT-PSO with PSO, LPSO, and QIPSO for the rotated and shifted unimodal and multimodal functions using Wilcoxon’s rank sum test (n
=
10).
Function
Wilcoxon’s rank sum test
PSO
LPSO
QIPSO
F
10
P value
3.0199e −011
3.0199e −011
3.0199e −011
h-value
1
1
1
z-value
−6.6456
−6.6456
−6.6456
F
11
P value
4.1191e −001
7.3940e −001
8.4180e −001
h-value
0
0
0
z-value
0.820536
0.33265
0.19959
F
12
P value
1.8567e −009
1.8500e −008
5.5329e −008
h-value
1
1
1
z-value
−6.00987
−5.62547
−5.43328
F
13
P value
2.1577e −002
2.1577e −002
8.1523e −002
h-value
1
1
0
z-value
−2.29773
−2.29773
−1.74192
F
14
P value
1.1439e −001
1.4425e −004
7.7181e −001
h-value
0
−1
0
z-value
1.57878
3.80076
0.290007
F
15
P value
1.5553e −011
1.2883e −011
1.7966e −008
h-value
1
1
1
z-value
−6.74264
−6.76995
−5.63053
F
16
P value
1.2755e −010
1.9667e −003
5.5338e −010
h-value
1
1
1
z-value
−6.43006
−3.09521
−6.20317
F
17
P value
7.3270e −011
3.6443e −008
5.0030e −010
h-value
1
1
1
z-value
−6.51381
−5.50727
−6.21901
F
18
P value
5.7512e −006
3.5043e −007
4.0269e −005
h-value
1
1
1
z-value
−4.53533
−5.09408
−4.10593
F
19
P value
9.2344e −001
2.3885e −004
8.6499e −001
h-value
0
−1
0
z-value
0.0960988
3.67393
0.170021
F
20
P value
5.2991e −008
9.2051e −005
4.6756e −008
h-value
1
1
1
z-value
−5.44097
−3.91064
−5.46322
1 (better)
8
8
7
0 (same)
3
1
4
−1 (worse)
0
2
0Table 6
Minimization results for the unimodal and multimodal functions (maximum iteration = 10000 andn
=
30).
Function
FGLT-PSO
PSO
LPSO
QIPSO
F
1
Avg. best solution
0.000e +000
5.397e −069
2.747e −028
3.502e −069
SD
0.000e +000
1.161e −068
4.078e −028
1.664e −068
Median best solution
0.000e +000
2.321e −070
1.047e −028
2.727e −072
F
2
Avg. best solution
7.598e −110
7.000e +000
3.849e −028
2.333e +000
SD
4.085e −109
7.944e +000
9.015e −028
4.302e +000
Median best solution
1.684e −136
1.000e +001
1.331e −031
1.188e −071
F
3
Avg. best solution
5.959e −020
1.189e +004
1.399e +002
5.733e +003
SD
1.762e −019
6.605e +003
9.380e +001
5.484e +003
Median best solution
5.483e −021
1.083e +004
1.254e +002
5.000e +003
F
4
Avg. best solution
6.415e −003
1.320e −003
9.664e −003
1.635e −003
SD
2.858e −003
8.008e −004
3.621e −003
9.357e −004
Median best solution
5.941e −003
1.088e −003
9.106e −003
1.333e −003
F
5
Avg. best solution
5.718e −001
1.129e +003
1.751e +001
3.858e +002
SD
1.225e +000
3.023e +003
2.041e +001
1.832e +003
Median best solution
1.707e −001
7.428e +001
9.494e +000
2.139e +001
F
6
Avg. best solution
−1.048e +004
−9.095e +003
−9.858e +003
−9.227e +003
SD
5.996e +002
7.444e +002
4.872e +002
7.524e +002
Median best solution
−1.058e +004
−9.051e +003
−9.828e +003
−9.232e +003
F
7
Avg. best solution
7.771e −002
1.013e −014
2.931e −014
8.941e −015
SD
2.015e −001
3.312e −015
1.433e −014
2.457e −015
Median best solution
7.994e −015
7.994e −015
2.576e −014
7.994e −015
F
8
Avg. best solution
5.164e −003
1.037e −002
1.160e −025
1.571e −032
SD
2.274e −002
3.163e −002
3.552e −025
5.567e −048
Median best solution
1.571e −032
1.591e −032
3.737e −027
1.571e −032
F
9
Avg. best solution
2.136e +001
5.827e +001
4.281e +001
4.310e +001
SD
8.612e +000
3.080e +001
2.064e +001
2.901e +001
Median best solution
2.212e +001
5.450e +001
3.753e +001
3.500e +001
Avg. rank
1.8
3.3
2.6
2.3
Final rank
1
4
3
2
Algorithms
FGLT-PSO
PSO
LPSO
QIPSOTable 7
Minimization results for the rotated and shifted unimodal and multimodal functions (maximum iteration = 10000 andn
=
30).
Function
FGLT-PSO
PSO
LPSO
QIPSO
F
10
Avg. best solution
8.707e − 020
3.422e + 002
7.818e + 000
5.318e + 001
SD
4.547e − 019
1.556e + 003
2.173e + 001
1.986e + 002
Median best solution
4.028e − 043
1.332e − 006
2.267e − 003
8.579e − 020
F
11
Avg. best solution
−4.963e + 001
−4.981e + 001
−4.933e + 001
−4.993e + 001
SD
1.098e + 000
1.324e + 000
1.417e + 000
1.489e + 000
Median best solution
−4.930e + 001
−4.985e + 001
−4.927e + 001
−4.968e + 001
F
12
Avg. best solution
1.514e + 002
1.935e + 002
1.902e + 002
1.886e + 002
SD
8.330e + 000
1.509e + 001
1.246e + 001
1.233e + 001
Median best solution
1.511e + 002
1.978e + 002
1.899e + 002
1.850e + 002
F
13
Avg. best solution
7.160e − 001
3.101e + 001
1.185e + 001
2.046e + 001
SD
9.412e − 001
1.248e + 001
1.136e + 001
1.804e + 001
Median best solution
2.067e − 001
3.613e + 001
5.997e + 000
3.319e + 001
F
14
Avg. best solution
4.065e − 001
3.465e − 001
3.632e − 001
3.599e − 001
SD
1.230e − 001
5.074e − 002
5.561e − 002
7.240e − 002
Median best solution
3.999e − 001
2.999e − 001
3.999e − 001
3.499e − 001
F
15
Avg. best solution
−4.280e + 002
−4.044e + 002
−4.175e + 002
−4.500e + 002
SD
7.441e + 000
7.592e + 000
8.860e + 000
2.151e − 006
Median best solution
−4.266e + 002
−4.056e + 002
−4.155e + 002
−4.500e + 002
F
16
Avg. best solution
−1.794e + 002
−8.340e + 001
−1.705e + 002
−9.806e + 001
SD
1.112e + 000
4.560e + 001
4.344e + 000
3.839e + 001
Median best solution
−1.798e + 002
−9.269e + 001
−1.704e + 002
−1.053e + 002
F
17
Avg. best solution
4.225e + 002
1.658e + 009
8.168e + 007
1.549e + 009
SD
5.410e + 001
1.409e + 009
5.644e + 007
1.599e + 009
Median best solution
3.940e + 002
1.136e + 009
6.932e + 007
1.205e + 009
F
18
Avg. best solution
−2.874e + 002
−2.174e + 002
−2.455e + 002
−2.219e + 002
SD
1.680e + 001
2.897e + 001
1.196e + 001
2.614e + 001
Median best solution
−2.902e + 002
−2.199e + 002
−2.458e + 002
−2.227e + 002
F
19
Avg. best solution
−1.1912e + 002
−1.1912e + 002
−1.1915e + 002
−1.1913e + 002
SD
5.597e − 002
5.015e − 002
5.377e − 002
6.957e − 002
Median best solution
−1.1911e + 002
−1.1911e + 002
−1.1915e + 002
−1.1913e + 002
F
20
Avg. best solution
−2.367e + 002
−1.237e + 002
−1.499e + 002
−1.375e + 002
SD
2.561e + 001
3.612e + 001
2.719e + 001
4.160e + 001
Median best solution
−2.427e + 002
−1.256e + 002
−1.481e + 002
−1.374e + 002
Avg. rank
1.8
3.5
2.4
2.4
Final rank
1
3
2
2
Algorithms
FGLT-PSO
PSO
LPSO
QIPSOTable 8
Comparison of FGLT-PSO with PSO, LPSO, and QIPSO for the unimodal and multimodal functions using Wilcoxon’s rank sum test (n
=
30).
Function
Wilcoxon’s rank sum test
PSO
LPSO
QIPSO
F
1
P value
1.2118e −012
1.2118e −012
1.2118e −012
h-value
1
1
1
z-value
−7.10402
−7.10402
−7.10402
F
2
P value
2.8991e −011
3.0199e −011
2.9822e −011
h-value
1
1
1
z-value
−6.65161
−6.6456
−6.64745
F
3
P value
3.0199e −011
3.0199e −011
3.0199e −011
h-value
1
1
1
z-value
−6.6456
−6.6456
−6.6456
F
4
P value
1.2057e −010
2.2539e −004
1.9568e −010
h-value
−1
1
−1
z-value
6.43862
−3.68871
6.3647
F
5
P value
1.2018e −008
1.2472e −004
5.0922e −008
h-value
1
1
1
z-value
−5.69948
−3.83667
−5.44806
F
6
P value
1.2018e −008
1.2472e −004
5.0922e −008
h-value
1
1
1
z-value
−5.69948
−3.83667
−5.44806
F
7
P value
4.2942e −002
3.7277e −001
2.8314e −003
h-value
−1
0
−1
z-value
2.02428
−0.891289
2.98547
F
8
P value
1.7374e −001
2.4305e −002
6.6167e −004
h-value
0
1
−1
z-value
−1.36028
−2.25228
3.40499
F
9
P value
1.9990e −006
2.7726e −005
7.2926e −004
h-value
1
1
1
z-value
−4.75352
−4.19138
−3.37834
1 (better)
6
8
6
0 (same)
1
1
0
−1 (worse)
2
0
3Table 9
Comparison of FGLT-PSO with PSO, LPSO, and QIPSO for the rotated and shifted unimodal and multimodal functions using Wilcoxon’s rank sum test (n
=
30).
Function
Wilcoxon’s rank sum test
PSO
LPSO
QIPSO
F
10
P value
3.2922e −010
1.4041e −010
3.1451e −009
h-value
1
1
1
z-value
−6.28435
−6.41545
−5.92384
F
11
P value
1.0233e −001
3.0418e −001
4.4642e −001
h-value
0
0
0
z-value
1.63368
−1.02752
0.761398
F
12
P value
1.0937e −010
1.4643e −010
3.0199e −011
h-value
1
1
1
z-value
−6.4534
−6.40905
−6.6456
F
13
P value
1.4288e −008
1.3281e −010
1.1736e −003
h-value
1
1
1
z-value
−5.66991
−6.42392
−3.24523
F
14
P value
1.3470e −003
6.7707e −003
3.5527e −003
h-value
−1
−1
−1
z-value
3.20578
2.70792
2.91537
F
15
P value
2.8840e −010
4.7657e −005
2.8252e −011
h-value
1
1
−1
z-value
−6.30489
−4.06683
6.65541
F
16
P value
2.9543e −011
5.3712e −011
2.9543e −011
h-value
1
1
1
z-value
−6.64883
−6.56027
−6.64883
F
17
P value
3.0199e −11
3.0199e −11
3.0199e −11
h-value
1
1
1
z-value
−6.6456
−6.6456
−6.6456
F
18
P value
1.0937e −010
1.2870e −009
1.4643e −010
h-value
1
1
1
z-value
−6.4534
−6.06901
−6.40905
F
19
P value
8.7663e −001
4.0595e −002
7.7312e −001
h-value
0
1
0
z-value
0.155236
2.04764
0.288296
F
20
P value
3.0199e −011
7.3891e −011
4.9752e −011
h-value
1
1
1
z-value
−6.6456
−6.51254
−6.57168
1 (better)
8
9
7
0 (same)
2
1
2
−1 (worse)
1
1
2Table 10
Minimization results for the unimodal and multimodal functions (maximum iteration = 15000 andn
=
50).
Function
FGLT-PSO
PSO
LPSO
QIPSO
F
1
Avg. best solution
5.239e −232
7.5857e −049
7.297e −020
2.446e −049
SD
0.000e +000
2.1243e −048
7.695e −020
8.478e −049
Median best solution
2.541e −251
4.5415e −050
4.843e −020
1.395e −050
F
2
Avg. best solution
9.246e −075
3.400e +001
2.194e −015
1.367e +001
SD
3.646e −074
1.714e +001
1.439e −015
1.159e +001
Median best solution
2.684e −080
3.000e +001
1.881e −015
1.000e +001
F
3
Avg. best solution
1.098e −008
4.167e +004
2.775e +004
3.585e +004
SD
2.405e −008
1.404e +004
8.532e +003
1.967e +004
Median best solution
5.928e −009
4.002e +004
2.859e +004
3.434e +004
F
4
Avg. best solution
5.070e −002
6.015e +000
6.452e −002
2.055e −002
SD
2.802e −002
5.352e +000
1.488e −002
4.363e −003
Median best solution
4.205e −002
5.385e +000
6.409e −002
1.971e −002
F
5
Avg. best solution
6.128e +000
1.568e +003
6.024e +001
4.700e +002
SD
5.716e +000
3.409e +003
3.032e +001
1.817e +003
Median best solution
5.080e +000
8.179e +001
7.397e +001
7.942e +001
F
6
Avg. best solution
−1.505e +004
−1.364e +004
−1.470e +004
−1.339e +004
SD
9.481e +002
9.581e +002
8.491e +002
1.025e +003
Median best solution
−1.515e +004
−1.349e +004
−1.445e +004
−1.343e +004
F
7
Avg. best solution
1.299e +000
2.049e +000
1.145e −005
1.782e −014
SD
5.560e −001
4.661e +000
6.223e −005
3.695e −015
Median best solution
1.282e +000
2.220e −014
1.782e −009
1.510e −014
F
8
Avg. best solution
1.061e −001
1.451e −002
3.831e −012
2.074e −003
SD
1.935e −001
3.535e −002
1.984e −011
1.136e −002
Median best solution
2.081e −005
1.308e −032
2.881e −015
9.423e −033
F
9
Avg. best solution
4.759e +001
1.980e +002
1.872e +002
1.546e +002
SD
2.294e +001
4.888e +001
3.958e +001
5.192e +001
Median best solution
4.636e +001
1.890e +002
1.850e +002
1.475e +002
Avg. rank
1.7
3.7
2.3
2.3
Final rank
1
3
2
2
Algorithms
FGLT-PSO
PSO
LPSO
QIPSOTable 11
Minimization results for the rotated and shifted unimodal and multimodal functions (maximum iteration = 15000 andn
=
50).
Function
FGLT-PSO
PSO
LPSO
QIPSO
F
10
Avg. best solution
9.936e −031
6.202e +009
5.150e +003
3.989e +006
SD
5.442e −030
2.576e +010
1.153e +004
1.825e +007
Median best solution
1.121e −048
1.763e +005
1.184e +002
5.637e −007
F
11
Avg. best solution
−4.401e +001
−4.328e +001
−4.357e +001
−4.385e +001
SD
1.043e +000
1.331e +000
9.431e −001
1.329e +000
Median best solution
−4.391e +001
−4.344e +001
−4.339e +001
−4.367e +001
F
12
Avg. best solution
3.155e +002
4.234e +002
3.995e +002
4.045e +002
SD
1.166e +001
3.689e +001
1.813e +001
2.468e +001
Median best solution
3.171e +002
4.236e +002
3.995e +002
4.062e +002
F
13
Avg. best solution
1.660e +001
6.640e +001
5.210e +001
4.946e +001
SD
6.875e +000
2.371e +000
1.517e +001
2.599e +001
Median best solution
1.759e +001
6.697e +001
6.163e +001
6.361e +001
F
14
Avg. best solution
7.599e −001
5.899e −001
8.067e −001
6.099e −001
SD
3.103e −001
8.030e −002
9.046e −002
8.847e −002
Median best solution
6.999e −001
5.999e −001
7.999e −001
5.999e −001
F
15
Avg. best solution
−3.976e +002
−3.795e +002
−4.019e +002
−4.499e +002
SD
5.964e +000
1.974e +001
6.096e +000
1.849e −003
Median best solution
−3.974e +002
−3.815e +002
−4.021e +002
−4.50e +002
F
16
Avg. best solution
−1.791e +002
8.313e +001
−1.551e +002
6.894e +001
SD
1.212e +000
9.121e +001
1.216e +001
7.896e +001
Median best solution
−1.797e +002
7.911e +001
−1.565e +002
7.481e +001
F
17
Avg. best solution
4.260e +002
8.648e +009
3.341e +008
8.196e +009
SD
7.188e +001
4.442e +009
2.597e +008
5.588e +009
Median best solution
3.977e +002
8.332e +009
2.599e +008
6.614e +009
F
18
Avg. best solution
−1.920e +002
−4.497e +001
−9.832e +001
−3.612e +001
SD
2.634e +001
4.206e +001
1.963e +001
4.934e +001
Median best solution
−1.944e +002
−4.804e +001
−9.679e +001
−2.665e +001
F
19
Avg. best solution
−1.189e +002
−1.189e +002
−1.190e +002
−1.189e +002
SD
4.303e −002
4.364e −002
5.121e −002
5.576e −002
Median best solution
−1.189e +002
−1.189e +002
−1.190e +002
−1.189e +002
F
20
Avg. best solution
−1.016e +002
1.537e +002
5.974e +001
1.082e +002
SD
4.898e +001
8.906e +001
5.117e +001
9.020e +001
Median best solution
−1.004e +002
1.539e +002
6.594e +001
1.035e +002
Avg. rank
1.5
3.6
2.3
2.7
Final rank
1
4
2
3
Algorithms
FGLT-PSO
PSO
LPSO
QIPSOTable 12
Comparison of FGLT-PSO with PSO, LPSO, and QIPSO for the unimodal and multimodal functions using Wilcoxon’s rank sum test (n
=
50).
Function
Wilcoxon’s rank sum test
PSO
LPSO
QIPSO
F
1
P value
3.0199e −011
3.0199e −011
3.0199e −011
h-value
1
1
1
z-value
−6.6456
−6.6456
−6.6456
F
2
P value
2.9673e −011
3.0199e −011
2.9229e −011
h-value
1
1
1
z-value
−6.64819
−6.6456
−6.65041
F
3
P value
3.0199e −011
3.0199e −011
3.0199e −011
h-value
1
1
1
z-value
−6.6456
−6.6456
−6.6456
F
4
P value
2.5306e −004
3.9881e −004
3.1589e −010
h-value
1
1
−1
z-value
−3.65915
−3.54087
6.29077
F
5
P value
6.1210e −010
9.7555e −010
1.2870e −009
h-value
1
1
1
z-value
−6.18728
−6.11336
−6.06901
F
6
P value
2.8790e −006
9.3341e −002
2.5711e −007
h-value
1
0
1
z-value
−4.67927
−1.67803
−5.15244
F
7
P value
7.7050e −006
3.0199e −011
1.4811e −011
h-value
−1
−1
−1
z-value
4.47322
6.6456
6.74974
F
8
P value
7.7087e −002
8.2796e −003
1.4193e −007
h-value
0
−1
−1
z-value
1.76784
2.64045
5.26273
F
9
P value
3.3384e −011
3.3384e −011
4.6159e −010
h-value
1
1
1
z-value
−6.63081
−6.63081
−6.23164
1 (better)
7
6
6
0 (same)
1
1
0
−1 (worse)
1
2
3Table 13
Comparison of FGLT-PSO with PSO, LPSO, and QIPSO for the rotated and shifted unimodal and multimodal functions using Wilcoxon’s rank sum test (n
=
50).
Function
Wilcoxon’s rank sum test
PSO
LPSO
QIPSO
F
10
P value
3.0199e −011
3.0199e −011
3.0199e −011
h-value
1
1
1
z-value
−6.6456
−6.6456
−6.6456
F
11
P value
3.0317e −002
6.5671e −002
6.7350e −001
h-value
1
0
0
z-value
−2.16592
−1.84066
−0.421356
F
12
P value
3.0199e −011
3.0199e −011
3.0199e −011
h-value
1
1
1
z-value
−6.6456
−6.6456
−6.6456
F
13
P value
3.0199e −011
1.2057e −010
1.8916e −004
h-value
1
1
1
z-value
−6.6456
−6.43862
−3.73307
F
14
P value
7.3131e −003
3.0749e −002
3.0494e −002
h-value
−1
1
−1
z-value
2.68224
−2.16031
2.16361
F
15
P value
2.2256e −004
5.4038e −004
2.7547e −011
h-value
1
−1
−1
z-value
−3.69193
3.4599
6.65912
F
16
P value
3.0142e −011
3.0123e −011
3.0142e −011
h-value
1
1
1
z-value
−6.64588
−6.64597
−6.64588
F
17
P value
3.0199e −011
3.0199e −011
3.0199e −011
h-value
1
1
1
z-value
−6.6456
−6.6456
−6.6456
F
18
P value
3.0199e −011
4.0772e −011
3.0199e −011
h-value
1
1
1
z-value
−6.6456
−6.60125
−6.6456
F
19
P value
2.3399e −001
5.8737e −004
9.2344e −001
h-value
0
−1
0
z-value
1.19015
3.43738
0.0960988
F
20
P value
3.3384e −011
1.3289e −010
8.9934e −011
h-value
1
1
1
z-value
−6.63081
−6.42383
−6.48297
1 (better)
9
8
7
0 (same)
1
1
2
−1 (worse)
1
2
2The acceleration coefficientsC
1, C
2, and C
3 are updated based on (10). Their minimum and maximum values are as follows: C
1
min
⁡
=
0.5
,
C
1
max
⁡
=
2
,
C
2
min
⁡
=
1
,
C
2
max
⁡
=
2
,
C
3
min
⁡
=
0.5, and C
3
max
⁡ = 1.5.In these tables, the benchmark functions are divided to two categories:(
1
) unimodal and multimodal functions and (
2
) rotated, shifted, and shifted-rotated unimodal and multimodal functions. The experimental results demonstrate that FGLT-PSO performs superior results for most of the functions in all tested dimensions.Tables2 and 3 show the experimental results for all benchmark functions with dimension n
=
10. As illustrated, the FGLT-PSO algorithm surpasses the PSO, LPSO, and QIPSO algorithms in minimizing functions (
1
)
–
(
6
), (
10
), (
12
), (
13
), (
15
)
–
(
18
), and (20). Moreover, the proposed method provides significant improvements in functions (
1
), (
2
), (
3
), (
10
), (
13
), (
17
), and (
18
). In these functions, the convergent results attain the optimal (or good near optimal) solutions. In Tables 2 and 3, the average iteration for finding the best solution is computed. The average iteration is the required iterations to find the best solution by each algorithm. As shown, FGLT-PSO finds the best solutions faster than the other algorithms in the majority of functions. Also, it is noticeable that the FGLT-PSO algorithm achieves the best solution in a considerably lower iteration in functions (
7
) and (
8
). In these functions, the algorithms show identical results.According to Wilcoxon’s rank sum test in Tables4 and 5 for n
=
10, the results of the FGLT-PSO are statistically significantly different from the three compared algorithms. The superior convergence rate of FGLT-PSO is shown in Figure 2. The results in this figure illustrate that FGLT-PSO tends to find the global optimum in F
1 and F
17 faster than PSO, LPSO, and QIPSO and obtains the highest accuracy for these functions from among all the algorithms.Figure 2
Convergence performance of FGLT-PSO, PSO, LPSO, and QIPSO for the test functionsF
1 and F
17 (n
=
10).The minimization results of the benchmark functions with dimensionn
=
30 are presented in Tables 6 and 7. As seen in these tables and Tables 8 and 9, the FGLT-PSO outperforms the PSO, LPSO, and QIPSO algorithms in functions (
1
), (
2
), (
3
), (
5
), (
6
), (
9
), (
10
), (
12
), (
13
), (
16
), (
17
), (
18
), and (
20
). The largest difference in performance between the proposed algorithm with PSO, LPSO, and QIPSO occurs for the functions (
1
), (
2
), (
3
), (
5
), (
16
), (
17
), (
18
), and (
20
). Figure 3 illustrates the progress of the average best solution over 30 runs for F
3 and F
18. As demonstrated, the FGLT-PSO shows a higher convergence rate than the other algorithms.Figure 3
Convergence performance of FGLT-PSO, PSO, LPSO, and QIPSO for the test functionsF
3 and F
18 (n
=
30).Tables10 and 11 present the results of algorithms for the test functions with dimension n
=
50. As illustrated in these tables and regarding the results of Wilcoxon’s rank sum in Tables 12 and 13, the performance of proposed method is the best in most of the functions especially for functions (
1
), (
2
), (
3
), (
5
), (
6
), (
9
), (
10
), (
11
), (
12
), (
13
), (
16
), (
17
), (
18
), and (
20
). The superior convergence rate of FGLT-PSO is shown in Figure 4. The results in this figure show that the FGLT-PSO performs the best for the test functions F
16 and F
20 with n
=
50.Figure 4
Convergence performance of FGLT-PSO, PSO, LPSO, and QIPSO for the test functionsF
16 and F
20 (n
=
50).In addition, it is considerable that the PSO, LPSO, and QIPSO algorithms return the results far from the global optima as the dimension increases. This problem is clear in the functions(
3
), (
5
), (
9
), (
10
), (
16
), (
17
), (
18
), and (
20
) in Tables 10 and 11 with n
=
50. These all indicate that the proposed algorithm, FGLT-PSO, is more powerful and robust than the others for solving unimodal and multimodal functions.


## 4.2.2. The Results of Proposed Method with Constant Acceleration Coefficients

In this section, theC
1, C
2, and C
3 as acceleration coefficients are set at constant values to compare with the presented results in the Section 4.2.1. The coefficient of cognition term (C
1) and social terms (C
2 and C
3) are considered as C
1
=
2 and C
2
=
C
3
=
1. Table 14 shows the results of proposed method for the benchmark functions with dimensions 10, 30, and 50. As seen, the FGLT-PSO with the constant acceleration coefficients performs well in most of the functions. As the dimension increases, the FGLT-PSO with the variable acceleration coefficients (Section 4.2.1) shows the better performance than the constant one for functions (
1
), (
2
), (
3
), (
5
), (
17
), and (
20
). Also, the FGLT-PSO with the constant acceleration coefficients presents a better performance for functions (
7
), (
13
), and (
18
).Table 14
The results of FGLT-PSO with constant acceleration coefficients and different dimensions (N
=
50).
Functions
Iteration = 5000,n
=
10
Iteration = 10000,n
=
30
Iteration = 15000,n
=
50
Avg. best solution ± SD
Avg. best solution ± SD
Avg. best solution ± SD
F
1
0.000e +000  ±  0.000e +000
1.936e −067 ± 1.060e −066
1.687e −038 ± 5.602e −038
F
2
6.678e −160 ± 3.658e −159
4.922e −049 ± 2.305e −048
7.064e −032 ± 3.167e −031
F
3
7.520e −035 ± 3.579e −034
2.026e −003 ± 9.321e −003
2.062e +000 ± 2.616e +000
F
4
1.621e −004 ± 8.583e −005
9.078e −004 ± 3.306e −004
1.930e −003 ± 6.461e −004
F
5
3.086e +000 ± 1.772e +000
3.542e +001 ± 2.864e +001
7.854e +001 ± 4.435e +001
F
6
−3.957e +003 ± 1.375e +002
−1.052e +004 ± 4.069e +002
−1.643e +004 ± 7.233e +002
F
7
4.086e −015 ± 1.084e −015
5.507e −015 ± 1.656e −015
9.484e −002 ± 3.612e −001
F
8
4.712e −032 ± 1.670e −047
4.492e −002 ± 1.379e −001
7.063e −002 ± 1.508e −001
F
9
2.567e +000 ± 9.714e −001
2.040e +001 ± 6.100e +000
5.183e +001 ± 1.117e +001
F
10
6.599e −249 ± 0.000e +000
3.539e −066 ± 1.939e −065
4.923e −019 ± 2.696e −018
F
11
−6.620e +001 ± 1.111e +000
−5.036e +001 ± 9.041e −001
−4.482e +001 ± 9.061e −001
F
12
1.659e +001 ± 4.479e +000
1.602e +002 ± 1.096e +001
3.369e +002 ± 1.585e +001
F
13
0.000e +000 ± 0.000e +000
0.000e +000 ± 0.000e +000
1.5294e −007 ± 1.1896e −007
F
14
9.987e −002 ± 2.247e −017
3.932e −001 ± 1.081e −001
3.065e −001 ± 2.537e −002
F
15
−4.500e +002 ± 1.493e −014
−4.268e +002 ± 1.016e +001
−4.420e +002 ± 3.137e +000
F
16
−1.799e +002 ± 2.005e −001
−1.787e +002 ± 3.851e +000
−1.775e +002 ± 3.055e +000
F
17
4.001e +002 ± 2.060e +001
4.661e +002 ± 1.083e +002
5.748e +005 ± 1.022e +006
F
18
−3.283e +002 ± 8.796e −001
−2.956e +002 ± 8.540e +000
−2.396e +002 ± 1.763e +001
F
19
−1.1974e +002 ± 5.990e −002
−1.191e +002 ± 6.798e −002
−1.189e +002 ± 3.710e −002
F
20
−3.196e +002 ± 4.136e +000
−1.994e +002 ± 2.829e +001
−2.684e +001 ± 4.506e +001


## 4.2.3. Comparison with the Other PSO Algorithms

In this section, several well-known PSO algorithms are selected to assess the performance of proposed algorithm for the benchmarks. The PSO, QIPSO, FIPS, DMS-PSO, CLPSO, AFPSO, and AFPSO-QI algorithms are considered for the comparison. The details of these algorithms are listed in Table15. The FGLT-PSO is run 30 times and the average best solutions and the SD of results for eight common multimodal benchmark functions are compared with the reported results by [19] as illustrated in Table 16. The maximum iteration is 10000, n
=
30, and N
=
30. As seen, the FGLT-PSO provides better results than the other algorithms for the majority of functions (functions (
11
), (
12
), (
13
), (
14
), (
17
), and (
20
)) and has the first rank.Table 15
Some well-known PSO algorithms in the literature.
Algorithm
Topology
Parameter settings
PSO
Global star
ω: 0.9–0.4, C
1
=
C
2
=
2.0
QIPSO
Global star
ω: 0.9–0.4, C
1
=
C
2
=
2.0
FIPS
Local U-ring
χ = 0.729,
∑
c
i = 4.1
DMS-PSO
Dynamic multiswarm
ω: 0.9–0.2, C
1
=
C
2
=
2.0
,
m
=
3
,
R
=
5
CLPSO
Comprehensive learning
ω: 0.9–0.4, C
=
1.49445
,
m
=
7
AFPSO
Global star
ω: 0.9–0.4, C
1
,
C
2 are based on fuzzy rule [19]
AFPSO-QI
Global star
ω: 0.9–0.4, C
1
,
C
2 are based on fuzzy rule [19]
FGLT-PSO
Global star and local ring
ω: 0.9–0.4, C
1: 0.5–2, C
2: 1-2, C
3: 0.5–1.5Table 16
Comparison results of eight PSO algorithms [19] with FGLT-PSO for eight benchmark functions (maximum iteration = 10000, n
=
30, and N
=
30).
PSOs
Functions
Avg. best solution ± SD
Avg. best solution ± SD
Avg. best solution ± SD
F
11
F
12
F
13
PSO
−4.529e +001 ± 1.911e +000
3.202e +002 ± 1.470e +001
3.835e +001 ± 1.482e +000
QIPSO
−3.332e +001 ± 1.781e +000
3.175e +002 ± 2.324e +001
4.077e +001 ± 2.015e +000
FIPS
−1.955e +001 ± 8.477e +000
4.341e +002 ± 3.499e +001
4.155e +001 ± 1.363e +000
DMS-PSO
−4.572e +001 ± 1.703e +000
2.837e +002 ± 1.606e +001
3.632e +001 ± 1.225e +000
CLPSO
−4.529e +001 ± 1.269e +000
2.633e +002 ± 1.196e +001
3.496e +001 ± 1.768e +000
AFPSO
−4.547e +001 ± 1.608e +000
2.663e +002 ± 1.200e +001
3.609e +001 ± 2.540e +000
AFPSO-QI
−4.678e +001 ± 1.212e +000
2.533e +002 ± 1.263e +001
3.135e +001 ± 3.301e +000
FGLT-PSO
−4.920e +001 ± 1.091e +000
1.580e +002 ± 1.065e +001
9.933e +000 ± 3.737e +000
F
14
F
17
F
18
PSO
1.703e +001 ± 2.554e +000
3.217e +009 ± 3.880e +009
−1.953e +002 ± 3.282e +001
QIPSO
1.520e +001 ± 1.319e +000
2.347e +009 ± 1.872e +009
−1.963e +002 ± 2.964e +001
FIPS
2.660e +001 ± 1.417e +000
1.340e +003 ± 2.044e +003
−2.173e +002 ± 3.076e +001
DMS-PSO
1.292e +001 ± 1.328e +000
3.362e +008 ± 3.089e +008
−2.456e +002 ± 1.293e +001
CLPSO
1.194e +001 ± 1.365e +000
5.943e +002 ± 5.069e +001
−2.605e +002 ± 7.359e +000
AFPSO
1.038e +001 ± 1.379e +000
9.700e +007 ± 1.197e +008
−2.718e +002 ± 1.072e +001
AFPSO-QI
8.462e +000 ± 9.477e −001
8.832e +007 ± 9.793e +007
−2.736e +002 ± 9.667e +000
FGLT-PSO
5.899e −001 ± 2.820e −001
5.204e +002 ± 1.328e +002
−2.658e +002 ± 2.078e +001
F
19
F
20
Avg. rank
Final rank
PSO
−1.191e +002 ± 7.093e −002
−1.118e +002 ± 4.289e +001
8
8
QIPSO
−1.191e +002 ± 5.677e −001
−1.152e +002 ± 3.390e +001
6.6
7
FIPS
−1.199e +002 ± 3.239e −002
−1.437e +002 ± 5.164e +001
5.6
6
DMS-PSO
−1.192e +002 ± 6.117e −002
−1.910e +002 ± 1.994e +001
4.4
5
CLPSO
−1.190e +002 ± 3.781e −002
−1.312e +002 ± 2.437e +001
4.3
4
AFPSO
−1.197e +002 ± 4.280e −002
−1.267e +002 ± 2.727e +001
3.9
3
AFPSO-QI
−1.198e +002 ± 3.854e −001
−1.339e +002 ± 2.208e +001
2.4
2
FGLT-PSO
−1.191e +002 ± 4.805e −002
−2.166e +002 ± 3.117e +001
1.8
1


## 5. Conclusions

In this study, a fusion global-local-topology PSO algorithm (FGLT-PSO) has been presented to extend the search capability and to improve convergent efficiency by combining local and global topologies. The algorithm is a global search algorithm with several advantages. The benefits of algorithm can be summarized as the following: FGLT-PSO has a simple concept and structure; it is easy to implement and is not sensitive to increase of the dimension.A set of standard benchmarks, including unimodal, multimodal, rotated, shifted, and shifted-rotated unimodal and multimodal functions, have been used to evaluate the proposed algorithm. The average best results obtained by the FGLT-PSO have been compared with PSO, LPSO, QIPSO, FIPS, DMS-PSO, CLPSO, AFPSO, and AFPSO-QI. The experimental results show that the proposed FGLT-PSO algorithm enhances the accuracy of results compared with the other algorithms.



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*Source: 907386-2014-06-25.xml*