# A Hybrid Global Optimization Algorithm Based on Wind Driven Optimization and Differential Evolution

Authors: Zongfan Bao; Yongquan Zhou; Liangliang Li; Mingzhi Ma
Journal: Mathematical Problems in Engineering
 (2015)

---

## Abstract

This paper presents a new hybrid global optimization algorithm, which is based on the wind driven optimization (WDO) and differential evolution (DE), named WDO-DE algorithm. The WDO-DE algorithm is based on a double population evolution strategy, the individuals in a population evolved by wind driven optimization algorithm, and a population of individuals evolved from difference operation. The populations of individuals both in WDO and DE employ an information sharing mechanism to implement coevolution. This paper chose fifteen benchmark functions to have a test. The experimental results show that the proposed algorithm can be feasible in both low-dimensional and high-dimensional cases. Compared to GA-PSO, WDO, DE, PSO, and BA algorithm, the convergence speed and precision of WDO-DE are higher. This hybridization showed a better optimization performance and robustness and significantly improves the original WDO algorithm.


---

## Body

## 1. Introduction

Nature always makes people produce a lot of inspiration. In the past, many natural heuristic algorithms have been proposed for solving real-world and large scale problems which are very difficult to solve by traditional methods, for example, genetic algorithm (GA) [1], particle swarm optimization (PSO) [2, 3], ant colony optimization (ACO) [4], differential evolution (DE) [5], firefly algorithm (FA) [6], bat algorithm (BA) [7], cuckoo search (CS) [8], flower pollination algorithm (FPA) [9], wind driven optimization (WDO) [10].Because some heuristic algorithms are not very satisfactory in many respects, in the recent years, according to the characteristics of some algorithms, the heuristic algorithm combined with each other is an increasingly popular strategy to improve algorithms. These hybridizations have been shown to be effective global optimization algorithms and have been applied to solve application problems. For example, a hybrid of genetic algorithm (GA) and particle swarm optimization (PSO) is applied to recurrent neural/fuzzy network design [11]. A hybrid metaheuristic differential evolution (DE) and cuckoo search (CS) algorithm is implemented to solve the UCAV path planning problem [12]. A hybrid particle swarm optimization (PSO) and ant colony optimization (ACO) algorithm is implemented to solve hierarchical classification [13].The wind driven optimization (WDO) is a novel nature-inspired technique. It is a population based iterative heuristic process. The WDO was proposed by Bayraktar et al. in 2010 [10]. The inspiration of the WDO derives from the atmosphere, where wind blows in an attempt to balance the atmospheric pressure. Due to the fact that the WDO has few parameters in need of control and it is very easy to be carried out, it has been paid more attention by the academic community. But, in the early stage of solving optimization problems, the convergence speed of the WDO is quicker than others; when all the individuals are close to the optimal one in the late stage of solving optimization problems, it will lead to the loss of population diversity, and it is easy to fall into local optimum in finding better solutions.In 1997, Storn and Price have introduced the DE algorithm [5]. In the next time, the DE algorithm has been applied to solve optimization problems in diverse fields. After the population initialization, the population through mutation, crossover, and selection operators generates new population, it is able to maintain diversity of population, which can achieve the search of global optimal solution after several iterations. But it also has flaws. Its disadvantages contain the slow pace of convergence, and it is easy to fall into local optimum.WDO algorithm can easily suffer from the premature convergence when solving global optimization problems. It is an important method for relieving the premature convergence to control the population diversity. In order to overcome the deficiencies of a single algorithm in solving a global optimization problem, in this paper, we propose a new hybrid global optimization algorithm based on the wind driven optimization and differential evolution. This evolution strategy allows WDO and DE algorithms to give full play to their respective advantages. DE algorithm enables keeping the diversity of the population. This can be a good remedy defect of WDO algorithm, so it can avoid falling into a local optimum due to the loss of population diversity. And WDO algorithm converges faster; it can be good to make up for the shortcomings of DE algorithm in convergence speed, utilizing the individual of DE algorithm to guide the evolution of the individual of WDO algorithm, which reduces the risk of falling into a local optimal solution. It is not only to ensure the accuracy of the algorithm, but also to guarantee the speed of solving problems. Finally, the fifteen benchmark functions are tested; the experimental results show that the proposed algorithm can be feasible in both low-dimensional and high-dimensional cases. Compared to GA-PSO, WDO, DE, PSO, and BA algorithm, the convergence speed and precision of WDO-DE are higher. This hybridization showed a better optimization performance and robustness and significantly improves the original WDO algorithm.


## 2. A Brief Introduction on WDO and DE Algorithm

### 2.1. Wind Driven Optimization

The inspiration of the proposed WDO derives from the atmosphere. In the atmosphere, wind blows in an attempt to balance the imbalance of pressure. It flows from high pressure areas to low pressure areas at a velocity. Depending on the above analysis, some theoretical assumptions are formulated in derivation of the WDO algorithm. The starting point of WDO algorithm is Newton’s second law of motion, which is used to provide accurate results for the analysis of atmospheric motion in the Lagrangian description [14, 15] (1)
ρ
α
→
=
∑
F
→
i
,where α
→ is the acceleration, ρ is the air density for an infinitesimal air parcel, and F
→
i are all the forces acting on the air parcel. In order to let air pressure establish the equation relationship with the air parcel’s density and temperature, the ideal gas law is given by(2)
P
=
ρ
R
T
,where P is the pressure, R is the universal gas constant, and T is the temperature.The cause of the air movement is due to the combination of many forces, mainly including gravitational force (F
→
G), pressure gradient force (F
→
P
G), Coriolis force (F
→
C), and friction force (F
→
F). The physical equations of the above mentioned forces are as follows:(3)
F
→
G
=
ρ
δ
V
g
→
,
F
→
P
G
=
-
∇
P
δ
V
,
F
→
C
=
-
2
Ω
×
u
→
,
F
→
F
=
-
ρ
α
u
→
,where δ
V is finite volume of the air, g
→ represents the gravitational acceleration, ∇
P represents the pressure gradient, Ω is rotation of the Earth, u
→ represents the velocity vector of the wind, and α is the friction coefficient.The forces mentioned above can be added to (1). The equation can be described as in(4)
ρ
Δ
u
→
Δ
t
=
ρ
δ
V
g
→
+
-
∇
P
δ
V
+
-
2
Ω
×
u
→
+
-
ρ
α
u
→
,where the acceleration α
→ in (1) is rewritten as α
→
=
Δ
u
→
/
Δ
t; for simplicity, set Δ
t
=
1; for an infinitesimal air parcel, set δ
V
=
1, which simplifies (4) to(5)
ρ
Δ
u
→
=
ρ
g
→
+
-
∇
P
+
-
2
Ω
×
u
→
+
-
ρ
α
u
→
.On the basis of (2), the density ρ can be written in terms of the pressure; thus (5) can be rewritten as(6)
Δ
u
→
=
g
→
+
-
∇
P
R
T
P
cur
+
-
2
Ω
×
u
→
R
T
P
cur
+
-
α
u
→
,where P
cur is the pressure of current location. It is assumed in the WDO algorithm that velocity and position of the air parcel are changing at each iteration. Thus, Δ
u
→ can be written as Δ
u
→
=
u
→
new
-
u
→
cur, where u
→
new represents the velocity in next iteration and u
→
cur is the velocity at the current iteration. g
→ and ∇
P are vectors, they can be broken down in direction and magnitude as g
→
=
g
0
-
x
cur, -
∇
P
=
P
opt
-
P
cur
x
opt
-
x
cur, P
opt is the optimum pressure point that has been found so far, x
opt is the optimum location that has been found so far, and x
cur is the current location; updating (6) with the new equations, (6) can be rewritten as(7)
u
→
new
=
1
-
α
u
→
cur
-
g
x
cur
+
R
T
P
cur
P
opt
-
P
cur
x
opt
-
x
cur
+
-
2
Ω
×
u
→
R
T
P
cur
.Finally, there are three additional substitutions needed. Firstly, the influence of the Coriolis forceΩ
×
u
→ is replaced by the velocity influence from another dimension u
→
cur
other
d
i
m. Secondly, all the coefficients are combined together; that is, c
=
-
2
R
T. Thirdly, in some cases where the pressure is extremely large, the updated velocities are too large to become meaningless and the efficiency of the WDO algorithm will be reduced. So the actual pressure value is replaced by rank among all air parcels based on their pressure values, the resulting equation of updating the velocity can be described as in (8), and the equation of updating the location can be described as in (9):(8)
u
→
new
=
1
-
α
u
→
cur
-
g
x
cur
+
R
T
1
-
1
i
x
opt
-
x
cur
+
c
u
→
cur
other
d
i
m
i
,
(9)
x
→
new
=
x
→
cur
+
u
→
new
×
Δ
t
,where i is the ranking among all air parcels and x
→
new represents the new location for the next iteration.WDO is similar to other nature-inspired optimization algorithms, but compared to other optimization algorithms, the code of WDO is more simple and easy to implement; it has less few control variables that need adjustment.


### 2.2. Differential Evolution

Differential evolution is introduced by Storn and Price in 1997 [5]. DE is an effective and simple global optimization algorithm.First of all, a population is generated randomly, it may be represented asX
i
,
G
i
=
1,2
,
…
,
N
P
=
x
i
,
G
1
,
x
i
,
G
2
,
…
,
x
i
,
G
D, NP is the number of population, D is the number of dimensions, and G denotes the generation of the population. There are three operators—mutation, crossover, and selection. Then the original population will through three operators generate a new population. The main progress of DE in detail is as follows.(1) Mutation. After initialization, mutation operation is used to generate the mutant vectors V
i
,
G
=
v
i
,
G
1
,
v
i
,
G
2
,
…
,
v
i
,
G
D. Mutation operation usually has the following five most frequently implemented strategies.DE/rand/1:(10)
V
i
,
G
=
X
r
1
i
,
G
+
F
·
X
r
2
i
,
G
-
X
r
3
i
,
G
.DE/best/1:(11)
V
i
,
G
=
X
best
,
G
+
F
·
X
r
1
i
,
G
-
X
r
2
i
,
G
.DE/current-to-best/1:(12)
V
i
,
G
=
X
i
,
G
+
F
·
X
best
,
G
-
X
i
,
G
+
F
·
X
r
1
i
,
G
-
X
r
2
i
,
G
.DE/best/2:(13)
V
i
,
G
=
X
best
,
G
+
F
·
X
r
1
i
,
G
-
X
r
2
i
,
G
+
F
·
X
r
3
i
,
G
-
X
r
4
i
,
G
.DE/rand/2:(14)
V
i
,
G
=
X
r
1
i
,
G
+
F
·
X
r
2
i
,
G
-
X
r
3
i
,
G
+
F
·
X
r
4
i
,
G
-
X
r
5
i
,
G
.The subscripts r
1
i
,
r
2
i
,
r
3
i
,
r
4
i
,
r
5
i are mutually exclusive integers within the range 1
,
N
P and F is the scale factor of difference vector.(2) Crossover. After the mutation operation is completed, DE will utilize crossover operation to generate a trial vector U
i
,
G
=
u
i
,
G
1
,
u
i
,
G
2
,
…
,
u
i
,
G
D:(15)
u
i
,
G
j
=
v
i
,
G
j
if
rand
j
0,1
≤
C
R
or
j
=
j
rand
,
x
i
,
G
j
others
,where C
R
∈
0,1; it is a crossover constant. j
rand is a random integer within the range 1
,
D.(3) Selection. After completing the first two operators, according the fitness value of trial vectors, DE utilizes selection operation to select the best one for the next generation:(16)
X
i
,
G
+
1
=
U
i
,
G
if
f
U
i
,
G
≤
f
X
i
,
G
,
X
i
,
G
otherwise
.After gradual iteration, DE can achieve the search of global optimal solution.


## 2.1. Wind Driven Optimization

The inspiration of the proposed WDO derives from the atmosphere. In the atmosphere, wind blows in an attempt to balance the imbalance of pressure. It flows from high pressure areas to low pressure areas at a velocity. Depending on the above analysis, some theoretical assumptions are formulated in derivation of the WDO algorithm. The starting point of WDO algorithm is Newton’s second law of motion, which is used to provide accurate results for the analysis of atmospheric motion in the Lagrangian description [14, 15] (1)
ρ
α
→
=
∑
F
→
i
,where α
→ is the acceleration, ρ is the air density for an infinitesimal air parcel, and F
→
i are all the forces acting on the air parcel. In order to let air pressure establish the equation relationship with the air parcel’s density and temperature, the ideal gas law is given by(2)
P
=
ρ
R
T
,where P is the pressure, R is the universal gas constant, and T is the temperature.The cause of the air movement is due to the combination of many forces, mainly including gravitational force (F
→
G), pressure gradient force (F
→
P
G), Coriolis force (F
→
C), and friction force (F
→
F). The physical equations of the above mentioned forces are as follows:(3)
F
→
G
=
ρ
δ
V
g
→
,
F
→
P
G
=
-
∇
P
δ
V
,
F
→
C
=
-
2
Ω
×
u
→
,
F
→
F
=
-
ρ
α
u
→
,where δ
V is finite volume of the air, g
→ represents the gravitational acceleration, ∇
P represents the pressure gradient, Ω is rotation of the Earth, u
→ represents the velocity vector of the wind, and α is the friction coefficient.The forces mentioned above can be added to (1). The equation can be described as in(4)
ρ
Δ
u
→
Δ
t
=
ρ
δ
V
g
→
+
-
∇
P
δ
V
+
-
2
Ω
×
u
→
+
-
ρ
α
u
→
,where the acceleration α
→ in (1) is rewritten as α
→
=
Δ
u
→
/
Δ
t; for simplicity, set Δ
t
=
1; for an infinitesimal air parcel, set δ
V
=
1, which simplifies (4) to(5)
ρ
Δ
u
→
=
ρ
g
→
+
-
∇
P
+
-
2
Ω
×
u
→
+
-
ρ
α
u
→
.On the basis of (2), the density ρ can be written in terms of the pressure; thus (5) can be rewritten as(6)
Δ
u
→
=
g
→
+
-
∇
P
R
T
P
cur
+
-
2
Ω
×
u
→
R
T
P
cur
+
-
α
u
→
,where P
cur is the pressure of current location. It is assumed in the WDO algorithm that velocity and position of the air parcel are changing at each iteration. Thus, Δ
u
→ can be written as Δ
u
→
=
u
→
new
-
u
→
cur, where u
→
new represents the velocity in next iteration and u
→
cur is the velocity at the current iteration. g
→ and ∇
P are vectors, they can be broken down in direction and magnitude as g
→
=
g
0
-
x
cur, -
∇
P
=
P
opt
-
P
cur
x
opt
-
x
cur, P
opt is the optimum pressure point that has been found so far, x
opt is the optimum location that has been found so far, and x
cur is the current location; updating (6) with the new equations, (6) can be rewritten as(7)
u
→
new
=
1
-
α
u
→
cur
-
g
x
cur
+
R
T
P
cur
P
opt
-
P
cur
x
opt
-
x
cur
+
-
2
Ω
×
u
→
R
T
P
cur
.Finally, there are three additional substitutions needed. Firstly, the influence of the Coriolis forceΩ
×
u
→ is replaced by the velocity influence from another dimension u
→
cur
other
d
i
m. Secondly, all the coefficients are combined together; that is, c
=
-
2
R
T. Thirdly, in some cases where the pressure is extremely large, the updated velocities are too large to become meaningless and the efficiency of the WDO algorithm will be reduced. So the actual pressure value is replaced by rank among all air parcels based on their pressure values, the resulting equation of updating the velocity can be described as in (8), and the equation of updating the location can be described as in (9):(8)
u
→
new
=
1
-
α
u
→
cur
-
g
x
cur
+
R
T
1
-
1
i
x
opt
-
x
cur
+
c
u
→
cur
other
d
i
m
i
,
(9)
x
→
new
=
x
→
cur
+
u
→
new
×
Δ
t
,where i is the ranking among all air parcels and x
→
new represents the new location for the next iteration.WDO is similar to other nature-inspired optimization algorithms, but compared to other optimization algorithms, the code of WDO is more simple and easy to implement; it has less few control variables that need adjustment.


## 2.2. Differential Evolution

Differential evolution is introduced by Storn and Price in 1997 [5]. DE is an effective and simple global optimization algorithm.First of all, a population is generated randomly, it may be represented asX
i
,
G
i
=
1,2
,
…
,
N
P
=
x
i
,
G
1
,
x
i
,
G
2
,
…
,
x
i
,
G
D, NP is the number of population, D is the number of dimensions, and G denotes the generation of the population. There are three operators—mutation, crossover, and selection. Then the original population will through three operators generate a new population. The main progress of DE in detail is as follows.(1) Mutation. After initialization, mutation operation is used to generate the mutant vectors V
i
,
G
=
v
i
,
G
1
,
v
i
,
G
2
,
…
,
v
i
,
G
D. Mutation operation usually has the following five most frequently implemented strategies.DE/rand/1:(10)
V
i
,
G
=
X
r
1
i
,
G
+
F
·
X
r
2
i
,
G
-
X
r
3
i
,
G
.DE/best/1:(11)
V
i
,
G
=
X
best
,
G
+
F
·
X
r
1
i
,
G
-
X
r
2
i
,
G
.DE/current-to-best/1:(12)
V
i
,
G
=
X
i
,
G
+
F
·
X
best
,
G
-
X
i
,
G
+
F
·
X
r
1
i
,
G
-
X
r
2
i
,
G
.DE/best/2:(13)
V
i
,
G
=
X
best
,
G
+
F
·
X
r
1
i
,
G
-
X
r
2
i
,
G
+
F
·
X
r
3
i
,
G
-
X
r
4
i
,
G
.DE/rand/2:(14)
V
i
,
G
=
X
r
1
i
,
G
+
F
·
X
r
2
i
,
G
-
X
r
3
i
,
G
+
F
·
X
r
4
i
,
G
-
X
r
5
i
,
G
.The subscripts r
1
i
,
r
2
i
,
r
3
i
,
r
4
i
,
r
5
i are mutually exclusive integers within the range 1
,
N
P and F is the scale factor of difference vector.(2) Crossover. After the mutation operation is completed, DE will utilize crossover operation to generate a trial vector U
i
,
G
=
u
i
,
G
1
,
u
i
,
G
2
,
…
,
u
i
,
G
D:(15)
u
i
,
G
j
=
v
i
,
G
j
if
rand
j
0,1
≤
C
R
or
j
=
j
rand
,
x
i
,
G
j
others
,where C
R
∈
0,1; it is a crossover constant. j
rand is a random integer within the range 1
,
D.(3) Selection. After completing the first two operators, according the fitness value of trial vectors, DE utilizes selection operation to select the best one for the next generation:(16)
X
i
,
G
+
1
=
U
i
,
G
if
f
U
i
,
G
≤
f
X
i
,
G
,
X
i
,
G
otherwise
.After gradual iteration, DE can achieve the search of global optimal solution.


## 3. WDO-DE Algorithm

This section focuses on the rationale of the WDO-DE algorithm. WDO and DE are based on the population of the global search techniques. And the WDO-DE is based on a double population evolution strategy [16]. The individuals both in WDO and DE employ an information sharing mechanism to implement coevolution. The strategy makes WDO-DE enjoy the advantages of two algorithms. It can maintain diversity of the populations, and the WDO-DE algorithm has the capability to jump out of the local optimal solution. Based on the above description, the main procedure of WDO-DE is as shown in Algorithm 1.Algorithm 1:WDO-DE.
Step 1. Initialize parameters.
N (Population size); G (Max number of generations);
Parameters of WDO: RT (RT coefficient);α (The friction coefficient); max
⁡
V (Maximum allowed speed); g (Gravitational
constant);c (Constant in the update equation).
Parameters of DE:F (Mutation scale factor); P
c (Crossover probability).
Step 2. Initialization populations.
Step 2.1. Generate one initial population with N
/
2 air particles, each air particle assign random location P
i
,
G
w
d
o and velocity
V
i
,
G
w
d
o, evaluation the population and identify the best solution of WDO algorithm P
b
e
s
t
,
G
w
d
o;
Step 2.2. Generate one initial population P
i
,
G
d
e with N
/
2 individuals, evaluation the population and identify the best
solution P
b
e
s
t
,
G
d
e.
Step 3. Identify the best solution P
b
e
s
t among all particles in WDO and DE.
Step 4. While stopping criterion is not satisfied
Step 4.1. Running process of the WDO algorithm
for i
=
1 to the N
/
2 do
P
b
e
s
t
,
G
w
d
o
=
P
b
e
s
t
Generate the trial velocity according to  (8)
Generate the trial location U
i
,
G by  (9)
Evaluate the trial location U
i
,
G
If f
U
i
,
G
≤
f
P
i
,
G
w
d
o
P
i
,
G
+
1
w
d
o
=
U
i
,
G,  f
P
i
,
G
+
1
w
d
o
=
f
U
i
,
G
Iff
U
i
,
G
≤
f
P
b
e
s
t
,
G
w
d
o
P
b
e
s
t
,
G
w
d
o
=
U
i
,
G,  f
P
b
e
s
t
,
G
w
d
o
=
f
U
i
,
G
End if
End if
End for
Step 4.2. Running process of the DE algorithm
for i
=
1 to the N
/
2 do
P
b
e
s
t
,
G
d
e
=
P
b
e
s
t
Generate V
i
,
G using (11)
Generate the trial vector U
i
,
G by (15)
Evaluate the trial vector U
i
,
G
If f
U
i
,
G
≤
f
P
i
,
G
d
e
P
i
,
G
+
1
d
e
=
U
i
,
G,  f
P
i
,
G
+
1
d
e
=
f
U
i
,
G
Iff
U
i
,
G
≤
f
P
b
e
s
t
,
G
d
e
P
b
e
s
t
,
G
d
e
=
U
i
,
G,  f
P
b
e
s
t
,
G
d
e
=
f
U
i
,
G
End if
End if
End for
Step 4.3. Identify the best solution P
b
e
s
t
If f
P
b
e
s
t
,
G
w
d
o
≤
f
P
b
e
s
t
,
G
d
e
P
b
e
s
t
=
P
b
e
s
t
,
G
w
d
o
Else
P
b
e
s
t
=
P
b
e
s
t
,
G
d
e
End if
Step 4.4. Increment the generation count G
=
G
+
1
Step 5. End while


## 4. Experimental Results

### 4.1. Experimental Setup

All algorithms are implemented in MATLAB R2012a, and experiments are performed on a Pentium 3.00 GHz Processor with 4.0 GB of memory, Windows 7 operating system.


### 4.2. Benchmark Test Functions

To test the performance of WDO-DE algorithm, we use 15 benchmark functions [17, 18] which have been widely used in the test. Among these benchmarks, part I contains the nine high-dimensional functions and part II contains six low-dimensional functions. Table 1 has shown the benchmark functions.Table 1
Description of the benchmark function used in our experiment.
Category
Test functions
Name
Range of search
Minimum value
Ι
f
1
x
=
∑
i
=
1
n
x
i
2
Sphere
[−100, 100]
0
f
2
x
=
-
20
exp
⁡
-
1
5
1
n
∑
i
=
1
n
x
i
2
-
exp
⁡
1
n
∑
i
=
1
n
c
o
s
2
π
x
i
+
20
+
e
Ackley
[−32, 32]
0
f
3
x
=
1
4000
∑
i
=
1
n
x
i
2
-
∏
i
=
1
n
cos
⁡
x
i
i
+
1
Griewank
[−600, 600]
0
f
4
x
=
-
cos
⁡
2
π
∑
i
=
1
n
x
i
2
+
0.1
∑
i
=
1
n
x
i
2
+
1
Salomon
[−100, 100]
0
f
5
x
=
-
1
n
∑
i
=
1
n
x
i
sin
⁡
x
i
Normalized Schwefel
[−512, 512]
−418.9829
f
6
x
=
∑
i
=
1
n
i
x
i
4
+
r
a
n
d
0,1
Quartic function, that is, noise
[−1.28, 1.28]
0
f
7
x
=
∑
i
=
1
n
∑
j
=
1
i
x
j
2
Rotated hyperellipsoid
[−100, 100]
0
f
8
x
=
10
n
+
∑
i
=
1
n
x
i
2
-
10
cos
⁡
2
π
x
i
Rastrigin
[−5.12, 5.12]
0
f
9
x
=
∑
i
=
1
n
x
i
sin
⁡
x
i
+
0.1
x
i
Alpine
[−10, 10]
0
II
f
10
x
=
x
2
-
5.1
4
π
2
x
1
2
+
5
π
x
1
-
6
2
+
10
1
-
1
8
π
cos
⁡
x
1
+
10
Branin
[−5, 15]
0.397887
f
11
x
=
-
cos
⁡
x
1
cos
⁡
x
2
exp
⁡
-
x
1
-
π
2
+
x
2
-
π
2
Easom
[−100, 100]
−1
f
12
x
=
1
+
x
1
+
x
2
+
1
2
19
-
14
x
1
+
3
x
1
2
-
14
x
2
+
6
x
1
x
2
+
3
x
2
2        ×
30
+
2
x
1
-
3
x
2
2
18
-
32
x
1
+
12
x
1
2
+
48
x
2
-
36
x
1
x
2
+
27
x
2
2
Goldstein and Price
[−2, 2]
3
f
13
x
=
4
-
2.1
x
1
2
+
x
1
4
3
x
1
2
+
x
1
x
2
+
-
4
+
4
x
2
2
x
2
2
Six Hump Camel back
[−5, 5]
−1.0316
f
14
x
=
-
∑
i
=
1
4
α
i
exp
⁡
-
∑
j
=
1
3
A
i
j
x
i
-
P
i
j
2, α
=
1.0,1.2,3.0,3.2
T, A
=
3.0
10
30
;
0.1
10
35
;
3.0
10
30
;
0.1
10
35,
P
=
0.3689
0.117
0.2673
;
0.4699
0.4387
0.747
;
0.1091
0.8732
0.5547
;
0.0381
0.5743
0.8828
Hartmann (3,4)
[
0,1
]
−3.86278
f
15
x
=
-
∑
i
=
1
n
sin
⁡
x
i
sin
2
m
⁡
i
x
i
2
π
Michalewicz
[
0
,
p
i
]
−9.66015


### 4.3. Parameters Setting

In this section, the parameters setting are presented. Bayraktar et al. did a lot of research for the parameters set of WDO [19]. The parameters set of WDO and DE is based on the practical experience to take the appropriate value. Tables 2 and 3 represent the necessary parameters in our experiment [20].Table 2
The parameters set of WDO.
Parameters
Value
Population size
50
Max number of iterations
1000
RT coefficient
1
Constants in the update equation
0.8
Maximum allowed speed
0.3
Gravitational constant
0.6
Coriolis effect
0.7Table 3
The parameters set of DE.
Parameters
Value
Population size
50
Max number of iterations
1000
Mutation scale factor
0.2
Crossover probability
0.03


### 4.4. Algorithm Performance Comparison

In this section, in order to test the performance of WDO-DE algorithm, WDO-DE algorithm has been compared with the algorithms GA-PSO, WDO, DE, BA, and PSO in low dimension and high dimension. The mean results, standard deviation (Std.) results, the optimal fitness value, the worst fitness value, and rank results between the algorithms of 50 independent runs forf
1
~
f
15 are shown in Table 4. Bold and italicized results mean that WDO-DE is better. Population size of other algorithms is 100. Max number of iterations of all tests is 1000.Table 4
Comparison of performance of algorithms in low dimension.
Test functions
Result
Algorithm
PSO
BA
DE
WDO
GAPSO
WDO-DE
f
1 (D
=
30)
Mean
1.198565069
0.001259031
1.15546
E
-
12
0
8.938
E
-
112
0
Std.
0.986982066
0.000164979
3.96904
E
-
13
0
6.2926
E
-
111
0
Worst
4.623331223
0.001579358
2.31909
E
-
12
0
4.4499
E
-
110
0
Best
0.077133013
0.000941037
6.51508
E
-
13
0
1.7446
E
-
133
0
Rank
6
5
4
1
3
1
f
2 (D
=
30)
Mean
3.23151972
18.95562594
3.29757
E
-
07
8.88178
E
-
16
8.88178
E
-
16
8.88178
E
-
16
Std.
0.928231321
0.240934834
4.98931
E
-
08
0
0
0
Worst
4.958505686
19.31856935
4.15132
E
-
07
8.88178
E
-
16
8.88178
E
-
16
8.88178
E
-
16
Best
1.51521671
18.06057448
2.41907
E
-
07
8.88178
E
-
16
8.88178
E
-
16
8.88178
E
-
16
Rank
5
6
5
1
1
1
f
3 (D
=
30)
Mean
0.617836326
293.8529401
9.92471
E
-
11
0
0
0
Std.
0.243522453
31.91402142
2.04264
E
-
10
0
0
0
Worst
1.060834843
364.9175036
1.24088
E
-
09
0
0
0
Best
0.118835584
231.9851842
6.60083
E
-
12
0
0
0
Rank
5
6
4
1
1
1
f
4 (D
=
30)
Mean
1.399873346
25.41987335
0.703241831
0.011986233
2.22495
E
-
05
0
Std.
0.257935287
1.292363757
0.060062328
0.032788391
2.93452
E
-
05
0
Worst
2.099873346
27.79987335
0.800431238
0.099920546
0.000122472
0
Best
0.699873346
22.59987335
0.599891695
0
0
0
Rank
5
6
4
3
2
1
f
5 (D
=
30)
Mean
-
265.951733
-
272.200713
-
418.982887
-
89.1395986
-
4.234983
-
366.236378
Std.
30.56577124
27.19605457
8.05405
E
-
12
14.48624741
3.304955999
12.75651852
Worst
-
202.664961
-
208.394008
-
418.982887
-
60.8635682
-
1.84114041
-
332.945962
Best
-
346.349165
-
328.828387
-
418.982887
-
149.39382
-
14.3267589
-
397.157289
Rank
4
3
1
5
6
2
f
6 (D
=
30)
Mean
1.39
E
-
01
0.005067
0.035133193
1.02406
E
-
05
0.001098329
4.81416
E
-
06
Std.
3.80
E
-
01
0.001999654
0.006615916
1.03384
E
-
05
0.000919425
5.23045
E
-
06
Worst
2.75
E
+
00
0.013185707
0.053066164
4.44353
E
-
05
0.004461194
2.57286
E
-
05
Best
2.30
E
-
02
0.002422956
0.019201609
7.08496
E
-
08
2.49604
E
-
05
6.73402
E
-
08
Rank
6
4
5
2
3
1
f
7 (D
=
30)
Mean
120.2398214
20.32158962
11826.12725
0
1.5811
E
-
114
0
Std.
71.10080897
67.06165947
1857.70219
0
1.0078
E
-
113
0
Worst
273.1473491
408.7930042
15721.31651
0
7.097
E
-
113
0
Best
20.21367214
0.004650193
8201.656294
0
6.0955
E
-
138
0
Rank
5
4
6
1
3
1
f
8 (D
=
30)
Mean
68.83078526
188.2789504
9.98004
E
-
08
0
0
0
Std.
18.36191672
28.50077605
1.1162
E
-
07
0
0
0
Worst
139.4625725
249.9566
6.78839
E
-
07
0
0
0
Best
38.07588886
103.6807268
1.22403
E
-
08
0
0
0
Rank
5
6
4
1
1
1
f
9 (D
=
30)
Mean
3.257708808
5.701336524
0.000124894
0
3.10916
E
-
59
0
Worst
1.752511346
2.663533038
4.35934
E
-
05
0
1.26282
E
-
58
0
Best
7.423856504
13.2677015
0.000220385
0
6.7331
E
-
58
0
Worst
0.503649602
1.083438608
5.01768
E
-
05
0
4.23177
E
-
67
0
Rank
5
6
4
1
3
1
f
10 (D
=
2)
Mean
0.397887358
0.397887361
0.397887358
0.507404422
0.411708093
0.397887358
Std.
3.36448
E
-
16
4.03531
E
-
09
3.36448
E
-
16
0.134086235
0.008420321
3.36448
E
-
16
Worst
0.397887358
0.397887373
0.397887358
1.01191027
0.427364689
0.397887358
Best
0.397887358
0.397887358
0.397887358
0.407529279
0.398014086
0.397887358
Rank
1
4
1
6
5
1
f
11 (D
=
2)
Mean
-
1
-
0.04000324
-
1
-
0.96070704
-
0.99999566
-
1
Std.
0
0.197947995
0
0.039519354
7.81918
E
-
06
0
Worst
-
1
0
-
1
-
0.82029884
-
0.99995866
-
1
Best
-
1
-
1
-
1
-
0.99846688
-
1
-
1
Rank
1
6
1
5
4
1
f
12 (D
=
2)
Mean
3
7.320000769
3
7.838633652
3
3
Std.
4.29203
E
-
15
9.998857372
3.99781
E
-
15
4.897011481
4.12124
E
-
15
3.48119
E
-
15
Worst
3
30.00000257
3
19.47699347
3
3
Best
3
3.000000005
3
3.024309606
3
3
Rank
4
5
2
6
3
1
f
13 (D
=
2)
Mean
-
1.03162845
-
0.95001198
-
1.03162845
-
1.02310481
-
1.03162845
-
1.03162845
Std.
2.24299
E
-
16
0.247335237
2.24299
E
-
16
0.010122307
2.24299
E
-
16
3.58878
E
-
16
Worst
-
1.03162845
-
0.21546376
-
1.03162845
-
0.97714288
-
1.03162845
-
1.03162845
Best
-
1.03162845
-
1.03162845
-
1.03162845
-
1.03147363
-
1.03162845
-
1.03162845
Rank
1
6
1
5
1
4
f
14 (D
=
3)
Mean
-
3.86199199
-
3.86277843
-
3.86277979
-
3.73158405
-
3.86277979
-
3.86277979
Std.
0.002387392
8.83669
E
-
07
4.93458
E
-
15
0.076497205
4.93458
E
-
15
4.18232
E
-
15
Worst
-
3.8549018
-
3.86277528
-
3.86277979
-
3.52239202
-
3.86277979
-
3.86277979
Best
-
3.86277979
-
3.86277961
-
3.86277979
-
3.85565215
-
3.86277979
-
3.86277979
Rank
5
4
2
6
2
1
f
15 (D
=
10)
Mean
-
7.67025031
-
5.6907234
-
9.66015172
-
5.16498045
-
6.53572784
-
9.56777535
Std.
0.781635349
1.010252541
4.39534
E
-
16
0.511107324
0.820311453
0.060299252
Worst
-
5.60672009
-
3.1283389
-
9.66015172
-
3.93161423
-
4.57905773
-
9.32718849
Best
-
9.0444038
-
8.11751567
-
9.66015172
-
6.35730534
-
8.47351627
-
9.64648266
Rank
3
5
1
6
4
2
Average rank
4.06
5.06
3
3.33
2.8
1.33
Overall rank
5
6
3
4
2
1For the low-dimensional case, according to Table4, test results of WDO-DE are better than the other algorithms except f
5, f
13, and f
15. For f
5 and f
15, the DE algorithm gives the better results. Although the result of f
13 function is worse than GA-PSO, DE, and PSO algorithm, it has already reached the theoretical optimal value. What is more, WDO-DE can find the theoretical optimum values for twelve benchmark functions (f
1
~
f
4, f
7
~
f
14) and has a very strong robustness. The novel hybrid global optimization algorithm is better than the original algorithm.In the last, we calculated the average rank based on these fifteen functions’ ranking [18]. Then, we rank the average rank and obtain the overall rank. From the average rank of each algorithm, we can learn that WDO-DE is very robust and efficient.For the benchmark functionf
11, the solution of WDO-DE algorithm is the closest to the theoretical optimal solution; for the benchmark function f
12
~
f
14, the original WDO algorithm cannot be close to the theoretical optimal solution; however the WDO-DE algorithm is close to the theoretical optimal solution (Table 5). And we can find that the convergence speed of the WDO-DE algorithm is quicker than other algorithms. In benchmark functions f
11 and f
13, the WDO-DE algorithm has converged within 100 generations. And in benchmark functions f
12 and f
14, the WDO-DE algorithm also has converged within 200 generations.Table 5
Comparison of optimal solution of algorithms in low dimension.
Test functions
Numbers of
Algorithm
Theoretical optimal
iterations
PSO
BA
DE
WDO
GAPSO
WDO-DE
solution
50
(3.129031847760726, 3.134768665032161)
(3.141553103558010,  −14.166037126201127)
(3.171893641024008, 3.131824693854047)
(0.0312746919874603, 0.0313915996157868)
(3.10191140474814, 3.10191140474814)
(3.14140267832352, 3.14201661395468)
100
(3.140173624854166, 3.142584336878753)
(3.141553103558010,  −14.166037126201127)
(3.141594887583755, 3.141597340960960)
(0.031336085130091, 0.0314633376198137)
(3.1630840677713, 3.1630840677713)
(3.14159264694396, 3.14159265459054)
f
11 (D
=
2)
200
(3.141597860814148, 3.141592422127699)
(3.141553103558010,  −14.166037126201127)
(3.141592746546202, 3.141592336946040)
(0.031336085130091, 0.0314633376198137)
(3.14537991099944, 3.14537991099944)
(3.14159264694396, 3.14159265459054)
(π
,
π)
500
(3.141592649491149, 3.141592658550973)
(3.141553103558010,  −14.166037126201127)
(3.141592746546202, 3.141592336946040)
(0.031336085130091, 0.0314633376198137)
(3.14380966948824, 3.14380966948824)
(3.14159264694396, 3.14159265459054)
1000
(3.141592649491149, 3.141592658550973)
(3.141643909560600,  −14.166039618443593)
(3.141592746546202, 3.141592336946040)
(0.031336085130091, 0.0314633376198137)
(3.14241959184766, 3.14241959184766)
(3.14159264694396, 3.14159265459054)
50
(−0.000274778503056,  −1.000060523567713)
(−0.000239117085666,  −1.000035528802250)
(−0.000019254003131,  −1.000054781208720)
(0.0282917779333518,  −0.448857319887312)
(-
1.48794259472975
e
-
05,  −0.999978563799561)
(3.40024382036861
e
-
07,  −1.00000055778692)
100
(−0.000274778503056,  −1.000060523567713)
(−0.000170307371325,  −1.000123380488044)
(−0.000000003959147,  −0.999999996915653)
(0.0282917779333518,  −0.448857319887312)
(-
1.93260130983304
e
-
09,  −0.999999998297379)
(-
1.81504593815985
e  −  09,  −1.00000000299309)
f
12 (D
=
2)
200
(0.000000218751157,  −0.999999566676768)
(−0.000109683118045,  −1.000005656513546)
(0.000000000326953,  −1.000000001047802)
(0.0282917779333518,  −0.448857319887312)
(1.72540643351734
e
-
09,  −1.00000000037874)
(4.59636719429074
e
-
10,  −1.00000000221316)
(0, −1)
500
(−0.000000003246821,  −1.000000000641675)
(0.000010008823714,  −1.000028736158479)
(−0.000000000328438,  −1.000000001077549)
(0.0282917779333518,  −0.448857319887312)
(1.72540643351734
e
-
09,  −1.00000000037874)
(4.59636719429074
e
-
10,  −1.00000000221316)
1000
(−0.000000001007721,  −0.999999999978878)
(0.000010008823714,  −1.000028736158479)
(−0.000000005887967,  −1.000000003780139)
(0.0282917779333518,  −0.448857319887312)
(5.07174141681265
e
-
09,  −0.999999995696904)
(4.59636719429074
e
-
10,  −1.00000000221316)
50
(−0.090537017481104, 0.713861505393922)
(0.089934924086771,  −0.712819423269000)
(−0.089842014825500, 0.712656400621485)
(0.0150932106229768,  −0.139673825192312)
(−0.0898421518832255, 0.712656330384267)
(−0.0898489505500046, 0.712661110362659)
(0.0898, −0.7126)and (−0.0898, 0.7126)
100
(−0.089579213574223, 0.712890938477391)
(0.089669828873280,  −0.712754046171649)
(−0.089842014825500, 0.712656400621485)
(0.0150932106229768,  −0.139673825192312)
(−0.0898420125705887, 0.712656401610307)
(−0.0898420134082198, 0.71265640418782)
f
13 (D
=
2)
200
(−0.089839286217455, 0.712655891026959)
(0.089744843168425,  −0.712675910202981)
(−0.089842014825500, 0.712656400621485)
(0.0150932106229768,  −0.139673825192312)
(−0.0898420139216395, 0.712656402819083)
(−0.0898420134082198, 0.71265640418782)
500
(−0.089842013891696, 0.712656404507059)
(0.089744843168425,  −0.712675910202981)
(−0.089842014825500, 0.712656400621485)
(0.0150932106229768,  −0.139673825192312)
(−0.0898420139216395, 0.712656402819083)
(−0.0898420134082198, 0.71265640418782)
1000
(−0.089842013891696, 0.712656404507059)
(0.089806395772405,  −0.712611171310040)
(−0.089842014825500, 0.712656400621485)
(0.0150932106229768,  −0.139673825192312)
(−0.0898420139216395, 0.712656402819083)
(−0.0898420134082198, 0.71265640418782)
50
(0, 0.555683378893354, 0.853290531318392)
(0.113849817052731, 0.555603849176208, 0.852558071570959)
(0.111559931280739, 0.555576225589055, 0.852504656091373)
(0.344200172275741, 0.191217218742259, 0.719829205828109)
(0.114453784913746, 0.555632415452041, 0.852564399900014)
(0.114490931509466, 0.555654876198981, 0.852559993186743)
100
(0, 0.555636156053527, 0.853073594865928)
(0.113849817052731, 0.555603849176208, 0.852558071570959)
(0.114589394834657, 0.555648863896550, 0.852546999524024)
(0.344200172275741, 0.191217218742259, 0.719829205828109)
(0.114588883123683, 0.555648895909646, 0.852546984663178)
(0.114588896201262, 0.555648895689609, 0.852546983169442)
f
14 (D
=
3)
200
(0, 0.555678232858826, 0.853059878827053)
(0.115127379365939, 0.555614295721604, 0.852539734887561)
(0.114588865815882, 0.555648895017175, 0.852546985276585)
(0.344200172275741, 0.191217218742259, 0.719829205828109)
(0.114588890069778, 0.555648895154575, 0.852546983865954)
(0.114588880154869, 0.555648895361092, 0.852546984294326)
(0.114614, 0.555649, 0.852547)
500
(0, 0.555678096423990, 0.853059974727304)
(0.115127379365939, 0.555614295721604, 0.852539734887561)
(0.114588865815882, 0.555648895017175, 0.852546985276585)
(0.344200172275741, 0.191217218742259, 0.719829205828109)
(0.114588890069778, 0.555648895154575, 0.852546983865954)
(0.114588880154869, 0.555648895361092, 0.852546984294326)
1000
(0, 0.555678096423990, 0.853059974727304)
(0.115127379365939, 0.555614295721604, 0.852539734887561)
(0.114588865815882, 0.555648895017175, 0.852546985276585)
(0.344200172275741, 0.191217218742259, 0.719829205828109)
(0.114588890069778, 0.555648895154575, 0.852546983865954)
(0.114588880154869, 0.555648895361092, 0.852546984294326)Meanwhile, Figures1–15 have shown the evolutionary process of fitness value (the vertical axis is logarithmic fitness value). And Figures 16–30 are the ANOVA tests of the global minimum. As can be seen from Figures 1–15, WDO-DE algorithm can converge within the maximum number of iterations except f
5 and f
15, and it has a faster global convergence speed in many functions and higher convergence precision. From the evolutionary process of fitness value it can be seen that the WDO-DE algorithm has a strong ability to find the optimal solutions. Moreover, as seen from Figures 16–30, we can learn that WDO-DE is the most robust in these algorithms. Therefore, WDO-DE is an effective and feasible solution for optimization problems in low-dimensional case.Figure 1
Comparison of performance of algorithms for minimization off
1 (D
=
30).Figure 2
Comparison of performance of algorithms for minimization off
2 (D
=
30).Figure 3
Comparison of performance of algorithms for minimization off
3 (D
=
30).Figure 4
Comparison of performance of algorithms for minimization off
4 (D
=
30).Figure 5
Comparison of performance of algorithms for minimization off
5 (D
=
30).Figure 6
Comparison of performance of algorithms for minimization off
6 (D
=
30).Figure 7
Comparison of performance of algorithms for minimization off
7 (D
=
30).Figure 8
Comparison of performance of algorithms for minimization off
8 (D
=
30).Figure 9
Comparison of performance of algorithms for minimization off
9 (D
=
30).Figure 10
Comparison of performance of algorithms for minimization off
10 (D
=
2).Figure 11
Comparison of performance of algorithms for minimization off
11 (D
=
2).Figure 12
Comparison of performance of algorithms for minimization off
12 (D
=
2).Figure 13
Comparison of performance of algorithms for minimization off
13 (D
=
2).Figure 14
Comparison of performance of algorithms for minimization off
14 (D
=
3).Figure 15
Comparison of performance of algorithms for minimization off
15 (D
=
10).Figure 16
ANOVA tests of the global minimum values forf
1 (D
=
30).Figure 17
ANOVA tests of the global minimum values forf
2 (D
=
30).Figure 18
ANOVA tests of the global minimum values forf
3 (D
=
30).Figure 19
ANOVA tests of the global minimum values forf
4 (D
=
30).Figure 20
ANOVA tests of the global minimum values forf
5 (D
=
30).Figure 21
ANOVA tests of the global minimum values forf
6 (D
=
30).Figure 22
ANOVA tests of the global minimum values forf
7 (D
=
30).Figure 23
ANOVA tests of the global minimum values forf
8 (D
=
30).Figure 24
ANOVA tests of the global minimum values forf
9 (D
=
30).Figure 25
ANOVA tests of the global minimum values forf
10 (D
=
30).Figure 26
ANOVA tests of the global minimum values forf
11 (D
=
2).Figure 27
ANOVA tests of the global minimum values forf
12 (D
=
2).Figure 28
ANOVA tests of the globalminimum values forf
13 (D
=
2).Figure 29
ANOVA tests of the global minimum values forf
14 (D
=
2).Figure 30
ANOVA tests of the global minimum values forf
15 (D
=
3).In order to test the optimization ability of the algorithms in high-dimensional space, this paper selects several different dimensions for tests [15]. Among them, f
3 and f
5 were set to 100 dimensions, f
4 and f
7 set to 300 dimensions, f
1 and f
8 set to 500 dimensions, and f
2, f
6, and f
9 set to 1000 dimensions. In all the tests, the max number of iterations is 1000, and the set of other parameters is the same. The mean results, standard deviation (Std.) results, the optimal fitness value, the worst fitness value, and rank results between the algorithms of 50 independent runs for f
1
~
f
9 are shown in Table 6.Table 6
Comparison of performance of algorithms in high dimension.
Test functions
Result
Algorithm
PSO
BA
DE
WDO
GAPSO
WDO-DE
f
1 (D
=
500)
Mean
5942.812635
690529.75
7833.289879
0
3.8474
E
-
114
0
Std.
2010.942412
16105.47575
307.7206962
0
2.6148
E
-
113
0
Worst
12148.04486
721605.363
8350.202931
0
1.8493
E
-
112
0
Best
1477.889332
637825.2912
7208.53767
0
5.6298
E
-
132
0
Rank
4
6
5
1
3
1
f
2 (D
=
1000)
Mean
6.021999854
19.58454162
6.556491165
8.88178
E
-
16
8.88178
E
-
16
8.88178
E
-
16
Std.
0.963645094
0.072035802
0.086504851
0
0
0
Worst
7.69115243
20.05614049
6.718429507
8.88178
E
-
16
8.88178
E
-
16
8.88178
E
-
16
Best
3.224297469
19.5300771
6.318110244
8.88178
E
-
16
8.88178
E
-
16
8.88178
E
-
16
Rank
4
6
5
1
1
1
f
3 (D
=
100)
Mean
5.151497492
1774.455594
0.038171893
0
0
0
Std.
2.161430833
103.0300286
0.006252419
0
0
0
Worst
13.35711745
1927.77745
0.051473105
0
0
0
Best
1.782030264
1418.804786
0.025514473
0
0
0
Rank
5
6
4
1
1
1
f
4 (D
=
300)
Mean
7.811873346
93.93587335
23.2983014
0.01799082
0.000134214
0
Std.
1.358065852
1.176428355
0.51684138
0.038789028
0.00022591
0
Worst
12.59987335
96.09987335
24.38078634
0.100231302
0.001189194
0
Best
5.099873346
90.29987335
22.11997471
0
3.67969
E
-
07
0
Rank
4
6
5
3
2
1
f
5 (D
=
100)
Mean
−188.353807
−225.492975
−415.031229
−49.9853541
−2.07455021
−266.617912
Std.
24.35490997
13.25383666
1.616785061
9.085364316
1.865239717
16.69768835
Worst
−110.804763
−196.772214
−409.751888
−29.977684
−0.9468723
−229.62838
Best
−231.936138
−260.621897
−417.256124
−71.336055
−8.7435685
−294.826296
Rank
4
3
1
5
6
2
f
6 (D
=
1000)
Mean
6096.609068
23.0239015
1384.433152
9.50848
E
-
06
0.001216642
6.70813
E
-
06
Std.
3404.371174
1.515569145
74.20351119
1.03931
E
-
05
0.000874723
7.15954
E
-
06
Worst
14502.3679
27.87497256
1594.24402
4.67328
E
-
05
0.003668518
3.54361
E
-
05
Best
457.2497659
19.88875782
1219.60675
1.46012
E
-
07
1.20405
E
-
05
3.50446
E
-
07
Rank
6
4
5
2
3
1
f
7 (D
=
300)
Mean
97529.8314
570894.7164
1506962.5
1.94072
E
-
26
1.1197
E
-
114
0
Std.
49025.64187
99081.42372
90799.2443
1.3723
E
-
25
6.0113
E
-
114
0
Worst
254179.9951
781652.5658
1740971.299
9.70362
E
-
25
3.9713
E
-
113
0
Best
39232.88764
383943.7522
1315295.793
0
2.0359
E
-
133
0
Rank
4
5
6
3
2
1
f
8 (D
=
500)
Mean
3250.396382
4161.811265
3678.744509
0
0
0
Std.
478.6226817
131.4154437
45.71686166
0
0
0
Worst
4235.766671
4449.362401
3755.802488
0
0
0
Best
2253.591512
3822.595502
3563.581287
0
0
0
Rank
4
6
5
1
1
1
f
9 (D
=
1000)
Mean
201.0787949
703.6914404
751.9162928
0
2.40675
E
-
58
0
Worst
51.1605331
35.62374915
15.21931169
0
8.18033
E
-
58
0
Best
385.1517001
765.0787636
779.6415024
0
4.24827
E
-
57
0
Worst
87.92253474
626.8641799
720.1039204
0
1.30498
E
-
65
0
Rank
4
5
6
1
3
1
Average rank
4.33
5.22
4.67
2
2.44
1.11
Overall rank
4
6
5
2
3
1For the high-dimensional case, as seen from Table6, test results of WDO-DE are better than the other algorithms except f
5. For f
5, the results of f
5 function are secondary to DE algorithm; although the DE algorithm gives better results, it has not reached the theoretical optimal value. WDO-DE can find the theoretical optimum values for seven benchmark functions (f
1
~
f
4, f
7
~
f
9) and has a very strong robustness. This indicates that WDO-DE is very robust and efficient.The same as before, we calculated the average rank based on these nine functions’ ranking. Then, we rank the average rank and obtain the overall rank. From the average rank of each algorithm, we can learn that WDO-DE is very robust and efficient.The same as before, Figures31–39 have shown the evolutionary process of fitness value (the vertical axis is logarithmic fitness value). And Figures 40–48 are the ANOVA tests of the global minimum. As can be seen from Figures 31–39, WDO-DE algorithm has the fastest global convergence speed and the highest convergence precision in all of these functions only except f
5. From the evolutionary process of fitness value it can be seen that the WDO-DE algorithm has a strong ability to find the optimal solutions. Moreover, as seen from Figures 40–48, we can learn that WDO-DE is the most robust in these algorithms. Therefore, WDO-DE is also an effective and feasible solution for optimization problems in high-dimensional case.Figure 31
Comparison of performance of algorithms for minimization off
1 (D
=
500).Figure 32
Comparison of performance of algorithms for minimization off
2 (D
=
1000).Figure 33
Comparison of performance of algorithms for minimization off
3 (D
=
100).Figure 34
Comparison of performance of algorithms for minimization off
4 (D
=
300).Figure 35
Comparison of performance of algorithms for minimization off
5 (D
=
100).Figure 36
Comparison of performance of algorithms for minimization off
6 (D
=
1000).Figure 37
Comparison of performance of algorithms for minimization off
7 (D
=
300).Figure 38
Comparison of performance of algorithms for minimization off
8 (D
=
500).Figure 39
Comparison of performance of algorithms for minimization off
9 (D
=
1000).Figure 40
ANOVA tests of the global minimum values forf
1 (D
=
500).Figure 41
ANOVA tests of the global minimum values forf
2 (D
=
1000).Figure 42
ANOVA tests of the global minimum values forf
3 (D
=
100).Figure 43
ANOVA tests of the global minimum values forf
4 (D
=
300).Figure 44
ANOVA tests of the global minimum values forf
5 (D
=
100).Figure 45
ANOVA tests of the global minimum values forf
6 (D
=
1000).Figure 46
ANOVA tests of the global minimum values forf
7 (D
=
300).Figure 47
ANOVA tests of the global minimum values forf
8 (D
=
500).Figure 48
ANOVA tests of the global minimum values forf
9 (D
=
1000).


## 4.1. Experimental Setup

All algorithms are implemented in MATLAB R2012a, and experiments are performed on a Pentium 3.00 GHz Processor with 4.0 GB of memory, Windows 7 operating system.


## 4.2. Benchmark Test Functions

To test the performance of WDO-DE algorithm, we use 15 benchmark functions [17, 18] which have been widely used in the test. Among these benchmarks, part I contains the nine high-dimensional functions and part II contains six low-dimensional functions. Table 1 has shown the benchmark functions.Table 1
Description of the benchmark function used in our experiment.
Category
Test functions
Name
Range of search
Minimum value
Ι
f
1
x
=
∑
i
=
1
n
x
i
2
Sphere
[−100, 100]
0
f
2
x
=
-
20
exp
⁡
-
1
5
1
n
∑
i
=
1
n
x
i
2
-
exp
⁡
1
n
∑
i
=
1
n
c
o
s
2
π
x
i
+
20
+
e
Ackley
[−32, 32]
0
f
3
x
=
1
4000
∑
i
=
1
n
x
i
2
-
∏
i
=
1
n
cos
⁡
x
i
i
+
1
Griewank
[−600, 600]
0
f
4
x
=
-
cos
⁡
2
π
∑
i
=
1
n
x
i
2
+
0.1
∑
i
=
1
n
x
i
2
+
1
Salomon
[−100, 100]
0
f
5
x
=
-
1
n
∑
i
=
1
n
x
i
sin
⁡
x
i
Normalized Schwefel
[−512, 512]
−418.9829
f
6
x
=
∑
i
=
1
n
i
x
i
4
+
r
a
n
d
0,1
Quartic function, that is, noise
[−1.28, 1.28]
0
f
7
x
=
∑
i
=
1
n
∑
j
=
1
i
x
j
2
Rotated hyperellipsoid
[−100, 100]
0
f
8
x
=
10
n
+
∑
i
=
1
n
x
i
2
-
10
cos
⁡
2
π
x
i
Rastrigin
[−5.12, 5.12]
0
f
9
x
=
∑
i
=
1
n
x
i
sin
⁡
x
i
+
0.1
x
i
Alpine
[−10, 10]
0
II
f
10
x
=
x
2
-
5.1
4
π
2
x
1
2
+
5
π
x
1
-
6
2
+
10
1
-
1
8
π
cos
⁡
x
1
+
10
Branin
[−5, 15]
0.397887
f
11
x
=
-
cos
⁡
x
1
cos
⁡
x
2
exp
⁡
-
x
1
-
π
2
+
x
2
-
π
2
Easom
[−100, 100]
−1
f
12
x
=
1
+
x
1
+
x
2
+
1
2
19
-
14
x
1
+
3
x
1
2
-
14
x
2
+
6
x
1
x
2
+
3
x
2
2        ×
30
+
2
x
1
-
3
x
2
2
18
-
32
x
1
+
12
x
1
2
+
48
x
2
-
36
x
1
x
2
+
27
x
2
2
Goldstein and Price
[−2, 2]
3
f
13
x
=
4
-
2.1
x
1
2
+
x
1
4
3
x
1
2
+
x
1
x
2
+
-
4
+
4
x
2
2
x
2
2
Six Hump Camel back
[−5, 5]
−1.0316
f
14
x
=
-
∑
i
=
1
4
α
i
exp
⁡
-
∑
j
=
1
3
A
i
j
x
i
-
P
i
j
2, α
=
1.0,1.2,3.0,3.2
T, A
=
3.0
10
30
;
0.1
10
35
;
3.0
10
30
;
0.1
10
35,
P
=
0.3689
0.117
0.2673
;
0.4699
0.4387
0.747
;
0.1091
0.8732
0.5547
;
0.0381
0.5743
0.8828
Hartmann (3,4)
[
0,1
]
−3.86278
f
15
x
=
-
∑
i
=
1
n
sin
⁡
x
i
sin
2
m
⁡
i
x
i
2
π
Michalewicz
[
0
,
p
i
]
−9.66015


## 4.3. Parameters Setting

In this section, the parameters setting are presented. Bayraktar et al. did a lot of research for the parameters set of WDO [19]. The parameters set of WDO and DE is based on the practical experience to take the appropriate value. Tables 2 and 3 represent the necessary parameters in our experiment [20].Table 2
The parameters set of WDO.
Parameters
Value
Population size
50
Max number of iterations
1000
RT coefficient
1
Constants in the update equation
0.8
Maximum allowed speed
0.3
Gravitational constant
0.6
Coriolis effect
0.7Table 3
The parameters set of DE.
Parameters
Value
Population size
50
Max number of iterations
1000
Mutation scale factor
0.2
Crossover probability
0.03


## 4.4. Algorithm Performance Comparison

In this section, in order to test the performance of WDO-DE algorithm, WDO-DE algorithm has been compared with the algorithms GA-PSO, WDO, DE, BA, and PSO in low dimension and high dimension. The mean results, standard deviation (Std.) results, the optimal fitness value, the worst fitness value, and rank results between the algorithms of 50 independent runs forf
1
~
f
15 are shown in Table 4. Bold and italicized results mean that WDO-DE is better. Population size of other algorithms is 100. Max number of iterations of all tests is 1000.Table 4
Comparison of performance of algorithms in low dimension.
Test functions
Result
Algorithm
PSO
BA
DE
WDO
GAPSO
WDO-DE
f
1 (D
=
30)
Mean
1.198565069
0.001259031
1.15546
E
-
12
0
8.938
E
-
112
0
Std.
0.986982066
0.000164979
3.96904
E
-
13
0
6.2926
E
-
111
0
Worst
4.623331223
0.001579358
2.31909
E
-
12
0
4.4499
E
-
110
0
Best
0.077133013
0.000941037
6.51508
E
-
13
0
1.7446
E
-
133
0
Rank
6
5
4
1
3
1
f
2 (D
=
30)
Mean
3.23151972
18.95562594
3.29757
E
-
07
8.88178
E
-
16
8.88178
E
-
16
8.88178
E
-
16
Std.
0.928231321
0.240934834
4.98931
E
-
08
0
0
0
Worst
4.958505686
19.31856935
4.15132
E
-
07
8.88178
E
-
16
8.88178
E
-
16
8.88178
E
-
16
Best
1.51521671
18.06057448
2.41907
E
-
07
8.88178
E
-
16
8.88178
E
-
16
8.88178
E
-
16
Rank
5
6
5
1
1
1
f
3 (D
=
30)
Mean
0.617836326
293.8529401
9.92471
E
-
11
0
0
0
Std.
0.243522453
31.91402142
2.04264
E
-
10
0
0
0
Worst
1.060834843
364.9175036
1.24088
E
-
09
0
0
0
Best
0.118835584
231.9851842
6.60083
E
-
12
0
0
0
Rank
5
6
4
1
1
1
f
4 (D
=
30)
Mean
1.399873346
25.41987335
0.703241831
0.011986233
2.22495
E
-
05
0
Std.
0.257935287
1.292363757
0.060062328
0.032788391
2.93452
E
-
05
0
Worst
2.099873346
27.79987335
0.800431238
0.099920546
0.000122472
0
Best
0.699873346
22.59987335
0.599891695
0
0
0
Rank
5
6
4
3
2
1
f
5 (D
=
30)
Mean
-
265.951733
-
272.200713
-
418.982887
-
89.1395986
-
4.234983
-
366.236378
Std.
30.56577124
27.19605457
8.05405
E
-
12
14.48624741
3.304955999
12.75651852
Worst
-
202.664961
-
208.394008
-
418.982887
-
60.8635682
-
1.84114041
-
332.945962
Best
-
346.349165
-
328.828387
-
418.982887
-
149.39382
-
14.3267589
-
397.157289
Rank
4
3
1
5
6
2
f
6 (D
=
30)
Mean
1.39
E
-
01
0.005067
0.035133193
1.02406
E
-
05
0.001098329
4.81416
E
-
06
Std.
3.80
E
-
01
0.001999654
0.006615916
1.03384
E
-
05
0.000919425
5.23045
E
-
06
Worst
2.75
E
+
00
0.013185707
0.053066164
4.44353
E
-
05
0.004461194
2.57286
E
-
05
Best
2.30
E
-
02
0.002422956
0.019201609
7.08496
E
-
08
2.49604
E
-
05
6.73402
E
-
08
Rank
6
4
5
2
3
1
f
7 (D
=
30)
Mean
120.2398214
20.32158962
11826.12725
0
1.5811
E
-
114
0
Std.
71.10080897
67.06165947
1857.70219
0
1.0078
E
-
113
0
Worst
273.1473491
408.7930042
15721.31651
0
7.097
E
-
113
0
Best
20.21367214
0.004650193
8201.656294
0
6.0955
E
-
138
0
Rank
5
4
6
1
3
1
f
8 (D
=
30)
Mean
68.83078526
188.2789504
9.98004
E
-
08
0
0
0
Std.
18.36191672
28.50077605
1.1162
E
-
07
0
0
0
Worst
139.4625725
249.9566
6.78839
E
-
07
0
0
0
Best
38.07588886
103.6807268
1.22403
E
-
08
0
0
0
Rank
5
6
4
1
1
1
f
9 (D
=
30)
Mean
3.257708808
5.701336524
0.000124894
0
3.10916
E
-
59
0
Worst
1.752511346
2.663533038
4.35934
E
-
05
0
1.26282
E
-
58
0
Best
7.423856504
13.2677015
0.000220385
0
6.7331
E
-
58
0
Worst
0.503649602
1.083438608
5.01768
E
-
05
0
4.23177
E
-
67
0
Rank
5
6
4
1
3
1
f
10 (D
=
2)
Mean
0.397887358
0.397887361
0.397887358
0.507404422
0.411708093
0.397887358
Std.
3.36448
E
-
16
4.03531
E
-
09
3.36448
E
-
16
0.134086235
0.008420321
3.36448
E
-
16
Worst
0.397887358
0.397887373
0.397887358
1.01191027
0.427364689
0.397887358
Best
0.397887358
0.397887358
0.397887358
0.407529279
0.398014086
0.397887358
Rank
1
4
1
6
5
1
f
11 (D
=
2)
Mean
-
1
-
0.04000324
-
1
-
0.96070704
-
0.99999566
-
1
Std.
0
0.197947995
0
0.039519354
7.81918
E
-
06
0
Worst
-
1
0
-
1
-
0.82029884
-
0.99995866
-
1
Best
-
1
-
1
-
1
-
0.99846688
-
1
-
1
Rank
1
6
1
5
4
1
f
12 (D
=
2)
Mean
3
7.320000769
3
7.838633652
3
3
Std.
4.29203
E
-
15
9.998857372
3.99781
E
-
15
4.897011481
4.12124
E
-
15
3.48119
E
-
15
Worst
3
30.00000257
3
19.47699347
3
3
Best
3
3.000000005
3
3.024309606
3
3
Rank
4
5
2
6
3
1
f
13 (D
=
2)
Mean
-
1.03162845
-
0.95001198
-
1.03162845
-
1.02310481
-
1.03162845
-
1.03162845
Std.
2.24299
E
-
16
0.247335237
2.24299
E
-
16
0.010122307
2.24299
E
-
16
3.58878
E
-
16
Worst
-
1.03162845
-
0.21546376
-
1.03162845
-
0.97714288
-
1.03162845
-
1.03162845
Best
-
1.03162845
-
1.03162845
-
1.03162845
-
1.03147363
-
1.03162845
-
1.03162845
Rank
1
6
1
5
1
4
f
14 (D
=
3)
Mean
-
3.86199199
-
3.86277843
-
3.86277979
-
3.73158405
-
3.86277979
-
3.86277979
Std.
0.002387392
8.83669
E
-
07
4.93458
E
-
15
0.076497205
4.93458
E
-
15
4.18232
E
-
15
Worst
-
3.8549018
-
3.86277528
-
3.86277979
-
3.52239202
-
3.86277979
-
3.86277979
Best
-
3.86277979
-
3.86277961
-
3.86277979
-
3.85565215
-
3.86277979
-
3.86277979
Rank
5
4
2
6
2
1
f
15 (D
=
10)
Mean
-
7.67025031
-
5.6907234
-
9.66015172
-
5.16498045
-
6.53572784
-
9.56777535
Std.
0.781635349
1.010252541
4.39534
E
-
16
0.511107324
0.820311453
0.060299252
Worst
-
5.60672009
-
3.1283389
-
9.66015172
-
3.93161423
-
4.57905773
-
9.32718849
Best
-
9.0444038
-
8.11751567
-
9.66015172
-
6.35730534
-
8.47351627
-
9.64648266
Rank
3
5
1
6
4
2
Average rank
4.06
5.06
3
3.33
2.8
1.33
Overall rank
5
6
3
4
2
1For the low-dimensional case, according to Table4, test results of WDO-DE are better than the other algorithms except f
5, f
13, and f
15. For f
5 and f
15, the DE algorithm gives the better results. Although the result of f
13 function is worse than GA-PSO, DE, and PSO algorithm, it has already reached the theoretical optimal value. What is more, WDO-DE can find the theoretical optimum values for twelve benchmark functions (f
1
~
f
4, f
7
~
f
14) and has a very strong robustness. The novel hybrid global optimization algorithm is better than the original algorithm.In the last, we calculated the average rank based on these fifteen functions’ ranking [18]. Then, we rank the average rank and obtain the overall rank. From the average rank of each algorithm, we can learn that WDO-DE is very robust and efficient.For the benchmark functionf
11, the solution of WDO-DE algorithm is the closest to the theoretical optimal solution; for the benchmark function f
12
~
f
14, the original WDO algorithm cannot be close to the theoretical optimal solution; however the WDO-DE algorithm is close to the theoretical optimal solution (Table 5). And we can find that the convergence speed of the WDO-DE algorithm is quicker than other algorithms. In benchmark functions f
11 and f
13, the WDO-DE algorithm has converged within 100 generations. And in benchmark functions f
12 and f
14, the WDO-DE algorithm also has converged within 200 generations.Table 5
Comparison of optimal solution of algorithms in low dimension.
Test functions
Numbers of
Algorithm
Theoretical optimal
iterations
PSO
BA
DE
WDO
GAPSO
WDO-DE
solution
50
(3.129031847760726, 3.134768665032161)
(3.141553103558010,  −14.166037126201127)
(3.171893641024008, 3.131824693854047)
(0.0312746919874603, 0.0313915996157868)
(3.10191140474814, 3.10191140474814)
(3.14140267832352, 3.14201661395468)
100
(3.140173624854166, 3.142584336878753)
(3.141553103558010,  −14.166037126201127)
(3.141594887583755, 3.141597340960960)
(0.031336085130091, 0.0314633376198137)
(3.1630840677713, 3.1630840677713)
(3.14159264694396, 3.14159265459054)
f
11 (D
=
2)
200
(3.141597860814148, 3.141592422127699)
(3.141553103558010,  −14.166037126201127)
(3.141592746546202, 3.141592336946040)
(0.031336085130091, 0.0314633376198137)
(3.14537991099944, 3.14537991099944)
(3.14159264694396, 3.14159265459054)
(π
,
π)
500
(3.141592649491149, 3.141592658550973)
(3.141553103558010,  −14.166037126201127)
(3.141592746546202, 3.141592336946040)
(0.031336085130091, 0.0314633376198137)
(3.14380966948824, 3.14380966948824)
(3.14159264694396, 3.14159265459054)
1000
(3.141592649491149, 3.141592658550973)
(3.141643909560600,  −14.166039618443593)
(3.141592746546202, 3.141592336946040)
(0.031336085130091, 0.0314633376198137)
(3.14241959184766, 3.14241959184766)
(3.14159264694396, 3.14159265459054)
50
(−0.000274778503056,  −1.000060523567713)
(−0.000239117085666,  −1.000035528802250)
(−0.000019254003131,  −1.000054781208720)
(0.0282917779333518,  −0.448857319887312)
(-
1.48794259472975
e
-
05,  −0.999978563799561)
(3.40024382036861
e
-
07,  −1.00000055778692)
100
(−0.000274778503056,  −1.000060523567713)
(−0.000170307371325,  −1.000123380488044)
(−0.000000003959147,  −0.999999996915653)
(0.0282917779333518,  −0.448857319887312)
(-
1.93260130983304
e
-
09,  −0.999999998297379)
(-
1.81504593815985
e  −  09,  −1.00000000299309)
f
12 (D
=
2)
200
(0.000000218751157,  −0.999999566676768)
(−0.000109683118045,  −1.000005656513546)
(0.000000000326953,  −1.000000001047802)
(0.0282917779333518,  −0.448857319887312)
(1.72540643351734
e
-
09,  −1.00000000037874)
(4.59636719429074
e
-
10,  −1.00000000221316)
(0, −1)
500
(−0.000000003246821,  −1.000000000641675)
(0.000010008823714,  −1.000028736158479)
(−0.000000000328438,  −1.000000001077549)
(0.0282917779333518,  −0.448857319887312)
(1.72540643351734
e
-
09,  −1.00000000037874)
(4.59636719429074
e
-
10,  −1.00000000221316)
1000
(−0.000000001007721,  −0.999999999978878)
(0.000010008823714,  −1.000028736158479)
(−0.000000005887967,  −1.000000003780139)
(0.0282917779333518,  −0.448857319887312)
(5.07174141681265
e
-
09,  −0.999999995696904)
(4.59636719429074
e
-
10,  −1.00000000221316)
50
(−0.090537017481104, 0.713861505393922)
(0.089934924086771,  −0.712819423269000)
(−0.089842014825500, 0.712656400621485)
(0.0150932106229768,  −0.139673825192312)
(−0.0898421518832255, 0.712656330384267)
(−0.0898489505500046, 0.712661110362659)
(0.0898, −0.7126)and (−0.0898, 0.7126)
100
(−0.089579213574223, 0.712890938477391)
(0.089669828873280,  −0.712754046171649)
(−0.089842014825500, 0.712656400621485)
(0.0150932106229768,  −0.139673825192312)
(−0.0898420125705887, 0.712656401610307)
(−0.0898420134082198, 0.71265640418782)
f
13 (D
=
2)
200
(−0.089839286217455, 0.712655891026959)
(0.089744843168425,  −0.712675910202981)
(−0.089842014825500, 0.712656400621485)
(0.0150932106229768,  −0.139673825192312)
(−0.0898420139216395, 0.712656402819083)
(−0.0898420134082198, 0.71265640418782)
500
(−0.089842013891696, 0.712656404507059)
(0.089744843168425,  −0.712675910202981)
(−0.089842014825500, 0.712656400621485)
(0.0150932106229768,  −0.139673825192312)
(−0.0898420139216395, 0.712656402819083)
(−0.0898420134082198, 0.71265640418782)
1000
(−0.089842013891696, 0.712656404507059)
(0.089806395772405,  −0.712611171310040)
(−0.089842014825500, 0.712656400621485)
(0.0150932106229768,  −0.139673825192312)
(−0.0898420139216395, 0.712656402819083)
(−0.0898420134082198, 0.71265640418782)
50
(0, 0.555683378893354, 0.853290531318392)
(0.113849817052731, 0.555603849176208, 0.852558071570959)
(0.111559931280739, 0.555576225589055, 0.852504656091373)
(0.344200172275741, 0.191217218742259, 0.719829205828109)
(0.114453784913746, 0.555632415452041, 0.852564399900014)
(0.114490931509466, 0.555654876198981, 0.852559993186743)
100
(0, 0.555636156053527, 0.853073594865928)
(0.113849817052731, 0.555603849176208, 0.852558071570959)
(0.114589394834657, 0.555648863896550, 0.852546999524024)
(0.344200172275741, 0.191217218742259, 0.719829205828109)
(0.114588883123683, 0.555648895909646, 0.852546984663178)
(0.114588896201262, 0.555648895689609, 0.852546983169442)
f
14 (D
=
3)
200
(0, 0.555678232858826, 0.853059878827053)
(0.115127379365939, 0.555614295721604, 0.852539734887561)
(0.114588865815882, 0.555648895017175, 0.852546985276585)
(0.344200172275741, 0.191217218742259, 0.719829205828109)
(0.114588890069778, 0.555648895154575, 0.852546983865954)
(0.114588880154869, 0.555648895361092, 0.852546984294326)
(0.114614, 0.555649, 0.852547)
500
(0, 0.555678096423990, 0.853059974727304)
(0.115127379365939, 0.555614295721604, 0.852539734887561)
(0.114588865815882, 0.555648895017175, 0.852546985276585)
(0.344200172275741, 0.191217218742259, 0.719829205828109)
(0.114588890069778, 0.555648895154575, 0.852546983865954)
(0.114588880154869, 0.555648895361092, 0.852546984294326)
1000
(0, 0.555678096423990, 0.853059974727304)
(0.115127379365939, 0.555614295721604, 0.852539734887561)
(0.114588865815882, 0.555648895017175, 0.852546985276585)
(0.344200172275741, 0.191217218742259, 0.719829205828109)
(0.114588890069778, 0.555648895154575, 0.852546983865954)
(0.114588880154869, 0.555648895361092, 0.852546984294326)Meanwhile, Figures1–15 have shown the evolutionary process of fitness value (the vertical axis is logarithmic fitness value). And Figures 16–30 are the ANOVA tests of the global minimum. As can be seen from Figures 1–15, WDO-DE algorithm can converge within the maximum number of iterations except f
5 and f
15, and it has a faster global convergence speed in many functions and higher convergence precision. From the evolutionary process of fitness value it can be seen that the WDO-DE algorithm has a strong ability to find the optimal solutions. Moreover, as seen from Figures 16–30, we can learn that WDO-DE is the most robust in these algorithms. Therefore, WDO-DE is an effective and feasible solution for optimization problems in low-dimensional case.Figure 1
Comparison of performance of algorithms for minimization off
1 (D
=
30).Figure 2
Comparison of performance of algorithms for minimization off
2 (D
=
30).Figure 3
Comparison of performance of algorithms for minimization off
3 (D
=
30).Figure 4
Comparison of performance of algorithms for minimization off
4 (D
=
30).Figure 5
Comparison of performance of algorithms for minimization off
5 (D
=
30).Figure 6
Comparison of performance of algorithms for minimization off
6 (D
=
30).Figure 7
Comparison of performance of algorithms for minimization off
7 (D
=
30).Figure 8
Comparison of performance of algorithms for minimization off
8 (D
=
30).Figure 9
Comparison of performance of algorithms for minimization off
9 (D
=
30).Figure 10
Comparison of performance of algorithms for minimization off
10 (D
=
2).Figure 11
Comparison of performance of algorithms for minimization off
11 (D
=
2).Figure 12
Comparison of performance of algorithms for minimization off
12 (D
=
2).Figure 13
Comparison of performance of algorithms for minimization off
13 (D
=
2).Figure 14
Comparison of performance of algorithms for minimization off
14 (D
=
3).Figure 15
Comparison of performance of algorithms for minimization off
15 (D
=
10).Figure 16
ANOVA tests of the global minimum values forf
1 (D
=
30).Figure 17
ANOVA tests of the global minimum values forf
2 (D
=
30).Figure 18
ANOVA tests of the global minimum values forf
3 (D
=
30).Figure 19
ANOVA tests of the global minimum values forf
4 (D
=
30).Figure 20
ANOVA tests of the global minimum values forf
5 (D
=
30).Figure 21
ANOVA tests of the global minimum values forf
6 (D
=
30).Figure 22
ANOVA tests of the global minimum values forf
7 (D
=
30).Figure 23
ANOVA tests of the global minimum values forf
8 (D
=
30).Figure 24
ANOVA tests of the global minimum values forf
9 (D
=
30).Figure 25
ANOVA tests of the global minimum values forf
10 (D
=
30).Figure 26
ANOVA tests of the global minimum values forf
11 (D
=
2).Figure 27
ANOVA tests of the global minimum values forf
12 (D
=
2).Figure 28
ANOVA tests of the globalminimum values forf
13 (D
=
2).Figure 29
ANOVA tests of the global minimum values forf
14 (D
=
2).Figure 30
ANOVA tests of the global minimum values forf
15 (D
=
3).In order to test the optimization ability of the algorithms in high-dimensional space, this paper selects several different dimensions for tests [15]. Among them, f
3 and f
5 were set to 100 dimensions, f
4 and f
7 set to 300 dimensions, f
1 and f
8 set to 500 dimensions, and f
2, f
6, and f
9 set to 1000 dimensions. In all the tests, the max number of iterations is 1000, and the set of other parameters is the same. The mean results, standard deviation (Std.) results, the optimal fitness value, the worst fitness value, and rank results between the algorithms of 50 independent runs for f
1
~
f
9 are shown in Table 6.Table 6
Comparison of performance of algorithms in high dimension.
Test functions
Result
Algorithm
PSO
BA
DE
WDO
GAPSO
WDO-DE
f
1 (D
=
500)
Mean
5942.812635
690529.75
7833.289879
0
3.8474
E
-
114
0
Std.
2010.942412
16105.47575
307.7206962
0
2.6148
E
-
113
0
Worst
12148.04486
721605.363
8350.202931
0
1.8493
E
-
112
0
Best
1477.889332
637825.2912
7208.53767
0
5.6298
E
-
132
0
Rank
4
6
5
1
3
1
f
2 (D
=
1000)
Mean
6.021999854
19.58454162
6.556491165
8.88178
E
-
16
8.88178
E
-
16
8.88178
E
-
16
Std.
0.963645094
0.072035802
0.086504851
0
0
0
Worst
7.69115243
20.05614049
6.718429507
8.88178
E
-
16
8.88178
E
-
16
8.88178
E
-
16
Best
3.224297469
19.5300771
6.318110244
8.88178
E
-
16
8.88178
E
-
16
8.88178
E
-
16
Rank
4
6
5
1
1
1
f
3 (D
=
100)
Mean
5.151497492
1774.455594
0.038171893
0
0
0
Std.
2.161430833
103.0300286
0.006252419
0
0
0
Worst
13.35711745
1927.77745
0.051473105
0
0
0
Best
1.782030264
1418.804786
0.025514473
0
0
0
Rank
5
6
4
1
1
1
f
4 (D
=
300)
Mean
7.811873346
93.93587335
23.2983014
0.01799082
0.000134214
0
Std.
1.358065852
1.176428355
0.51684138
0.038789028
0.00022591
0
Worst
12.59987335
96.09987335
24.38078634
0.100231302
0.001189194
0
Best
5.099873346
90.29987335
22.11997471
0
3.67969
E
-
07
0
Rank
4
6
5
3
2
1
f
5 (D
=
100)
Mean
−188.353807
−225.492975
−415.031229
−49.9853541
−2.07455021
−266.617912
Std.
24.35490997
13.25383666
1.616785061
9.085364316
1.865239717
16.69768835
Worst
−110.804763
−196.772214
−409.751888
−29.977684
−0.9468723
−229.62838
Best
−231.936138
−260.621897
−417.256124
−71.336055
−8.7435685
−294.826296
Rank
4
3
1
5
6
2
f
6 (D
=
1000)
Mean
6096.609068
23.0239015
1384.433152
9.50848
E
-
06
0.001216642
6.70813
E
-
06
Std.
3404.371174
1.515569145
74.20351119
1.03931
E
-
05
0.000874723
7.15954
E
-
06
Worst
14502.3679
27.87497256
1594.24402
4.67328
E
-
05
0.003668518
3.54361
E
-
05
Best
457.2497659
19.88875782
1219.60675
1.46012
E
-
07
1.20405
E
-
05
3.50446
E
-
07
Rank
6
4
5
2
3
1
f
7 (D
=
300)
Mean
97529.8314
570894.7164
1506962.5
1.94072
E
-
26
1.1197
E
-
114
0
Std.
49025.64187
99081.42372
90799.2443
1.3723
E
-
25
6.0113
E
-
114
0
Worst
254179.9951
781652.5658
1740971.299
9.70362
E
-
25
3.9713
E
-
113
0
Best
39232.88764
383943.7522
1315295.793
0
2.0359
E
-
133
0
Rank
4
5
6
3
2
1
f
8 (D
=
500)
Mean
3250.396382
4161.811265
3678.744509
0
0
0
Std.
478.6226817
131.4154437
45.71686166
0
0
0
Worst
4235.766671
4449.362401
3755.802488
0
0
0
Best
2253.591512
3822.595502
3563.581287
0
0
0
Rank
4
6
5
1
1
1
f
9 (D
=
1000)
Mean
201.0787949
703.6914404
751.9162928
0
2.40675
E
-
58
0
Worst
51.1605331
35.62374915
15.21931169
0
8.18033
E
-
58
0
Best
385.1517001
765.0787636
779.6415024
0
4.24827
E
-
57
0
Worst
87.92253474
626.8641799
720.1039204
0
1.30498
E
-
65
0
Rank
4
5
6
1
3
1
Average rank
4.33
5.22
4.67
2
2.44
1.11
Overall rank
4
6
5
2
3
1For the high-dimensional case, as seen from Table6, test results of WDO-DE are better than the other algorithms except f
5. For f
5, the results of f
5 function are secondary to DE algorithm; although the DE algorithm gives better results, it has not reached the theoretical optimal value. WDO-DE can find the theoretical optimum values for seven benchmark functions (f
1
~
f
4, f
7
~
f
9) and has a very strong robustness. This indicates that WDO-DE is very robust and efficient.The same as before, we calculated the average rank based on these nine functions’ ranking. Then, we rank the average rank and obtain the overall rank. From the average rank of each algorithm, we can learn that WDO-DE is very robust and efficient.The same as before, Figures31–39 have shown the evolutionary process of fitness value (the vertical axis is logarithmic fitness value). And Figures 40–48 are the ANOVA tests of the global minimum. As can be seen from Figures 31–39, WDO-DE algorithm has the fastest global convergence speed and the highest convergence precision in all of these functions only except f
5. From the evolutionary process of fitness value it can be seen that the WDO-DE algorithm has a strong ability to find the optimal solutions. Moreover, as seen from Figures 40–48, we can learn that WDO-DE is the most robust in these algorithms. Therefore, WDO-DE is also an effective and feasible solution for optimization problems in high-dimensional case.Figure 31
Comparison of performance of algorithms for minimization off
1 (D
=
500).Figure 32
Comparison of performance of algorithms for minimization off
2 (D
=
1000).Figure 33
Comparison of performance of algorithms for minimization off
3 (D
=
100).Figure 34
Comparison of performance of algorithms for minimization off
4 (D
=
300).Figure 35
Comparison of performance of algorithms for minimization off
5 (D
=
100).Figure 36
Comparison of performance of algorithms for minimization off
6 (D
=
1000).Figure 37
Comparison of performance of algorithms for minimization off
7 (D
=
300).Figure 38
Comparison of performance of algorithms for minimization off
8 (D
=
500).Figure 39
Comparison of performance of algorithms for minimization off
9 (D
=
1000).Figure 40
ANOVA tests of the global minimum values forf
1 (D
=
500).Figure 41
ANOVA tests of the global minimum values forf
2 (D
=
1000).Figure 42
ANOVA tests of the global minimum values forf
3 (D
=
100).Figure 43
ANOVA tests of the global minimum values forf
4 (D
=
300).Figure 44
ANOVA tests of the global minimum values forf
5 (D
=
100).Figure 45
ANOVA tests of the global minimum values forf
6 (D
=
1000).Figure 46
ANOVA tests of the global minimum values forf
7 (D
=
300).Figure 47
ANOVA tests of the global minimum values forf
8 (D
=
500).Figure 48
ANOVA tests of the global minimum values forf
9 (D
=
1000).


## 5. Conclusion and Future Research

In this paper, we present a new hybrid global optimization algorithm called WDO-DE, which is based on the wind driven optimization (WDO) and differential evolution (DE) for solving optimization problems. We use 15 benchmark functions which contain unimodal, multimodal, low-dimensional, and high-dimensional unconstrained test functions to test the performance of WDO-DE algorithm. The WDO-DE algorithm can converge within the maximum number of iterations in most functions. In comparison with the GA-PSO, WDO, DE, BA, and PSO, the WDO-DE algorithm is more effective in finding better solutions and the convergence speed and precision of WDO-DE are higher. It is an effective and reliable global optimization algorithm.Although in this paper the hybrid WDO-DE algorithm was implemented only for function optimization, in the field of optimization, there are still many aspects worthy of our study. Firstly, the hybrid algorithm proposed in this paper is based on the continuous space optimization. The future research may concentrate on discrete WDO algorithm. We can utilize many discretized strategies to discretize WDO algorithm to solve a problem characterized by discrete-valued design variables. Secondly, in engineering application, production management, and national defense construction, many optimization problems are multiobjective optimization problems, which are widely used in practical engineering. We would apply our proposed hybrid approach to solve multiobjective optimization problem. Lastly, we will learn more algorithms which have better optimization performance and analyze their characteristics. We would develop new hybrid approaches to solve the optimization problems raised above.



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*Source: 389630-2015-10-25.xml*