# Lévy-Flight Moth-Flame Algorithm for Function Optimization and Engineering Design Problems

Authors: Zhiming Li; Yongquan Zhou; Sen Zhang; Junmin Song
Journal: Mathematical Problems in Engineering
 (2016)

---

## Abstract

The moth-flame optimization (MFO) algorithm is a novel nature-inspired heuristic paradigm. The main inspiration of this algorithm is the navigation method of moths in nature called transverse orientation. Moths fly in night by maintaining a fixed angle with respect to the moon, a very effective mechanism for travelling in a straight line for long distances. However, these fancy insects are trapped in a spiral path around artificial lights. Aiming at the phenomenon that MFO algorithm has slow convergence and low precision, an improved version of MFO algorithm based on Lévy-flight strategy, which is named as LMFO, is proposed. Lévy-flight can increase the diversity of the population against premature convergence and make the algorithm jump out of local optimum more effectively. This approach is helpful to obtain a better trade-off between exploration and exploitation ability of MFO, thus, which can make LMFO faster and more robust than MFO. And a comparison with ABC, BA, GGSA, DA, PSOGSA, and MFO on 19 unconstrained benchmark functions and 2 constrained engineering design problems is tested. These results demonstrate the superior performance of LMFO.


---

## Body

## 1. Introduction

Optimization is a process of finding the best possible solution(s) for a given problem. In real world, many problems can be viewed as optimization problems. Since the complexity of problems increases, the need for new optimization techniques becomes more evident than before. Over the past several decades, some kinds of methods have been proposed to solve optimization problems and have made great progress. For example, mathematical optimization techniques used to be the only tool for optimizing problems before the proposal of heuristic optimization techniques. However, these methods need to know the property of optimization problem, such as continuity or differentiability. In recent years, metaheuristic optimization algorithms have become more and more popular in optimization techniques. Some popular algorithms in this field are Genetic Algorithms (GA) [1, 2], Particle Swarm Optimization (PSO) [3], Ant Colony Optimization (ACO) [4], Evolutionary Strategy (ES) [5], Differential Evolution (DE) [6], and Evolutionary Programming (EP) [7]. The application of these algorithms can be found in different branches of science and industry as well. Despite the merits of these optimizers, there is a fundamental question here whether there is any optimizer for solving all optimization problems. According to the No-Free-Lunch (NFL) theorem [8] for optimization, researchers are allowed to develop new algorithms solving optimization problems more effectively. Some of the latest algorithms are Artificial Bee Colony (ABC) algorithm [9], Bat Algorithm (BA) [10], Cuckoo Search (CS) algorithm [11], Cuckoo Optimization Algorithm (COA) [12], Gravitational Search Algorithm (GSA) [13], Charged System Search (CSS) [14], Firefly Algorithm (FA) [15], and Ray Optimization (RO) [16], and Dragonfly Algorithm (DA) [17].Moth-flame optimization (MFO) [18] algorithm is a new metaheuristic optimization method through imitating the navigation method of moths in nature called transverse orientation. In this algorithm, moths and flames are both solutions. The inventor of this algorithm, Seyedali Mirjalili, proved that this algorithm is able to show very competitive results compared with other state-of-the-art metaheuristic optimization algorithms. However, the MFO algorithm has been in research stage so far, and convergence speed and calculation accuracy of this algorithm can be further advanced. To improve the performance of MFO, a Lévy-flight moth-flame optimization (LMFO) algorithm is proposed.We know that Lévy-flight [11, 19] has a strong ability of strengthening global search and overcoming the problem of being trapped in local minima. In order to make use of the good performance of Lévy-flight, we propose a Lévy-flight moth-flame optimization. MFO and Lévy-flight have complementary advantages, so the proposed algorithm can lead to a faster and more robust method. The proposed algorithm is verified on nineteen benchmark functions and two engineering problems.The rest of the paper is organized as follows: Section2 presents a brief introduction to MFO and Lévy-flight. An improved version of MFO algorithm, LMFO, is proposed in Section 3. The experimental results of test functions and engineering design problem are showed in Sections 4 and 5, respectively. Results and discussion are provided in Section 6. Finally, Section 7 concludes the work.


## 2. Related Works

In this section, a background about the moth-flame optimization algorithm and Lévy-flight will be provided briefly.


### 2.1. MFO Algorithm

Moth-flame optimization [18] algorithm is a new metaheuristic optimization method, which is proposed by Seyedali Mirjalili and based on the simulation of the behavior of moths for their special navigation methods in night. They utilize a mechanism called transverse orientation for navigation. In this method, a moth flies by maintaining a fixed angle with respect to the moon, which is a very effective mechanism for travelling long distance in a straight path because the moon is far away from the moth. This mechanism guarantees that moths fly along straight line in night. However, we usually observe that moths fly spirally around the lights. In fact, moths are tricked by artificial lights and show such behaviors. Since such light is extremely close to the moon, hence, maintaining a similar angle to the light source causes a spiral fly path of moths.In the MFO algorithm, the set of moths is represented in a matrixM. For all the moths, there is an array O
M for storing the corresponding fitness values. The second key components in the algorithm are flames. A matrix F similar to the moth matrix is considered. For the flames, it is also assumed that there is an array O
F for storing the corresponding fitness values.The MFO algorithm is a three-tuple that approximates the global optimal of the optimization problems and defined as follows:(1)
MFO
=
I
,
P
,
T
.I is a function that creates a random population of moths and corresponding fitness values. The methodical model of this function is as follows:(2)
I
:
ϕ
⟶
M
,
O
M
.TheP function, which is the main function, moves the moths around the search space. This function received the matrix of M and returns its updated one eventually:(3)
P
:
M
⟶
M
.TheT function returns true if the termination criterion is satisfied and false if the termination criterion is not satisfied:(4)
T
:
M
⟶
true
,
false
.WithI, P, and T, the general framework of the MFO algorithm is defined as follows:M
=
I;
whileT
(
M
) is equal to false
M
=
P
(
M
);
endAfter the initialization, theP function is iteratively run until the T function returns true. For the sake of simulating the behavior of moths mathematically, the position of each moth is updated with respect to a flame using the following equation: (5)
M
i
=
S
M
i
,
F
j
,where M
i indicate the ith moth, F
j indicates the jth flame, and S is the spiral function.Any types of spiral can be utilized here subject to the following conditions:(1)
Spiral’s initial point should start from the moth.
(2)
Spiral’s final point should be the position of the flame.
(3)
Fluctuation of the range of spiral should not exceed the search space.Considering these points, a logarithmic spiral is defined for the MFO algorithm as follows:(6)
S
M
i
,
F
j
=
D
i
·
e
b
t
·
cos
⁡
2
π
t
+
F
j
,where D
i indicates the distance of the ith moth for the jth flame, b is a constant for defining the shape of the logarithmic spiral, and t is a random number in [
-
1,1
].D is calculated as follows:(7)
D
i
=
F
j
-
M
i
,where M
i indicate the ith moth, F
j indicates the jth flame, and D
i indicates the distance of the ith moth for the jth flame.Equation (6) describes the spiral flying path of moths. From this equation, the next position of a moth is defined with respect to a flame. The t parameter in the spiral equation defines how much the next position of the moth should be close to the flame (t
=
-
1 is the closest position to the flame, while t
=
1 shows the farthest).A question that may rise here is that the position updating in (6) only requires the moths to move towards a flame, yet it causes the MFO algorithm to be trapped in local optima quickly. In order to prevent this, each moth is obliged to update its position using only one of the flames in (6). Another concern here is that the position updating of moths with respect to n different locations in the search space may degrade the exploitation of the best promising solutions. To resolve this concern, an adaptive mechanism provided the number of flames. The following formula is utilized in this regard:(8)
flame  no
=
round
N
-
l
∗
N
-
1
T
,where l is the current number of iteration, N is the maximum number of flames, and T indicates the maximum number of iterations.The gradual decrement in number of flames balances exploration and exploitation of the search space. After all, the general steps of theP function can be described in Algorithm 1.Algorithm 1:MFO algorithm.
(
1) Initialize the position of moths
(2) While (Iteration <= Max_iteration)
(3) Update flame no using (8)
(4) O
M = FitnessFunction(M);
(5) if iteration = = 1
(6)   F = sort(M);
(7)    O
F = sort(O
M);
(8) else
(9)    F = sort(M
t
-
1, M
t);
(10)   O
F = sort(M
t
-
1, M
t);
(11) end
(12) for i
=
1
:
n
(13)    for j
=
1
:
d
(14)        Update r and t
(15)        Calculate D using (7) with respect to the corresponding moth
(16)        Update M
(
i
,
j
) using (5) and (6) with respect to the corresponding moth
(17)   end
(18) endAs described in Algorithm1, the P function is executed until the T function returns true. After termination of the P function, the best moth is returned as the best obtained approximation of the optimum.Note that the Quicksort method is utilized in MFO and the sort’s computational complexity isO
(
n
log
⁡
n
) and O
(
n
2
) in the best and worst case, respectively (where n is the number of moths).


### 2.2. Lévy-Flight

Lévy-flight was originally introduced by the French mathematician in 1937 named Paul Lévy. Lévy-flight is a statistical description of motion that extends beyond the more traditional Brownian motion discovered over one hundred years earlier. A diverse range of both natural and artificial phenomena are now being described in terms of Lévy statistics [19].Generally speaking, animals looking for food is random, from one place to another place. A large number of studies have shown that flight behavior of many animals and insects has demonstrated the typical characteristics of randomness. However, the choice of the direction relies only on a mathematical model [20], which is called Lévy-flight. For instance, many studies have shown that flight behavior of many animals and insects has revealed the typical characteristics of Lévy-flight [21–24]. According to [24], we can know that fruit flies orDrosophila melanogaster explore their landscape utilizing a series of straight flight paths punctuated by a sudden 9
0
° turn, resulting in a Lévy-flight-style fitful scale-free pattern. Studies on human behavior such as the Ju/’hoansi hunter-gatherer foraging patterns [21] also show the typical feature of Lévy-flight. Pavlyukevich has used Lévy-flight in his research to present and theoretically justify a new stochastic algorithm for global optimization. Even the light can be related to Lévy-flight [20]. Subsequently, Lévy-flight have been applied to optimization and optimal search, and preliminary results show its promising capability [22, 25].


## 2.1. MFO Algorithm

Moth-flame optimization [18] algorithm is a new metaheuristic optimization method, which is proposed by Seyedali Mirjalili and based on the simulation of the behavior of moths for their special navigation methods in night. They utilize a mechanism called transverse orientation for navigation. In this method, a moth flies by maintaining a fixed angle with respect to the moon, which is a very effective mechanism for travelling long distance in a straight path because the moon is far away from the moth. This mechanism guarantees that moths fly along straight line in night. However, we usually observe that moths fly spirally around the lights. In fact, moths are tricked by artificial lights and show such behaviors. Since such light is extremely close to the moon, hence, maintaining a similar angle to the light source causes a spiral fly path of moths.In the MFO algorithm, the set of moths is represented in a matrixM. For all the moths, there is an array O
M for storing the corresponding fitness values. The second key components in the algorithm are flames. A matrix F similar to the moth matrix is considered. For the flames, it is also assumed that there is an array O
F for storing the corresponding fitness values.The MFO algorithm is a three-tuple that approximates the global optimal of the optimization problems and defined as follows:(1)
MFO
=
I
,
P
,
T
.I is a function that creates a random population of moths and corresponding fitness values. The methodical model of this function is as follows:(2)
I
:
ϕ
⟶
M
,
O
M
.TheP function, which is the main function, moves the moths around the search space. This function received the matrix of M and returns its updated one eventually:(3)
P
:
M
⟶
M
.TheT function returns true if the termination criterion is satisfied and false if the termination criterion is not satisfied:(4)
T
:
M
⟶
true
,
false
.WithI, P, and T, the general framework of the MFO algorithm is defined as follows:M
=
I;
whileT
(
M
) is equal to false
M
=
P
(
M
);
endAfter the initialization, theP function is iteratively run until the T function returns true. For the sake of simulating the behavior of moths mathematically, the position of each moth is updated with respect to a flame using the following equation: (5)
M
i
=
S
M
i
,
F
j
,where M
i indicate the ith moth, F
j indicates the jth flame, and S is the spiral function.Any types of spiral can be utilized here subject to the following conditions:(1)
Spiral’s initial point should start from the moth.
(2)
Spiral’s final point should be the position of the flame.
(3)
Fluctuation of the range of spiral should not exceed the search space.Considering these points, a logarithmic spiral is defined for the MFO algorithm as follows:(6)
S
M
i
,
F
j
=
D
i
·
e
b
t
·
cos
⁡
2
π
t
+
F
j
,where D
i indicates the distance of the ith moth for the jth flame, b is a constant for defining the shape of the logarithmic spiral, and t is a random number in [
-
1,1
].D is calculated as follows:(7)
D
i
=
F
j
-
M
i
,where M
i indicate the ith moth, F
j indicates the jth flame, and D
i indicates the distance of the ith moth for the jth flame.Equation (6) describes the spiral flying path of moths. From this equation, the next position of a moth is defined with respect to a flame. The t parameter in the spiral equation defines how much the next position of the moth should be close to the flame (t
=
-
1 is the closest position to the flame, while t
=
1 shows the farthest).A question that may rise here is that the position updating in (6) only requires the moths to move towards a flame, yet it causes the MFO algorithm to be trapped in local optima quickly. In order to prevent this, each moth is obliged to update its position using only one of the flames in (6). Another concern here is that the position updating of moths with respect to n different locations in the search space may degrade the exploitation of the best promising solutions. To resolve this concern, an adaptive mechanism provided the number of flames. The following formula is utilized in this regard:(8)
flame  no
=
round
N
-
l
∗
N
-
1
T
,where l is the current number of iteration, N is the maximum number of flames, and T indicates the maximum number of iterations.The gradual decrement in number of flames balances exploration and exploitation of the search space. After all, the general steps of theP function can be described in Algorithm 1.Algorithm 1:MFO algorithm.
(
1) Initialize the position of moths
(2) While (Iteration <= Max_iteration)
(3) Update flame no using (8)
(4) O
M = FitnessFunction(M);
(5) if iteration = = 1
(6)   F = sort(M);
(7)    O
F = sort(O
M);
(8) else
(9)    F = sort(M
t
-
1, M
t);
(10)   O
F = sort(M
t
-
1, M
t);
(11) end
(12) for i
=
1
:
n
(13)    for j
=
1
:
d
(14)        Update r and t
(15)        Calculate D using (7) with respect to the corresponding moth
(16)        Update M
(
i
,
j
) using (5) and (6) with respect to the corresponding moth
(17)   end
(18) endAs described in Algorithm1, the P function is executed until the T function returns true. After termination of the P function, the best moth is returned as the best obtained approximation of the optimum.Note that the Quicksort method is utilized in MFO and the sort’s computational complexity isO
(
n
log
⁡
n
) and O
(
n
2
) in the best and worst case, respectively (where n is the number of moths).


## 2.2. Lévy-Flight

Lévy-flight was originally introduced by the French mathematician in 1937 named Paul Lévy. Lévy-flight is a statistical description of motion that extends beyond the more traditional Brownian motion discovered over one hundred years earlier. A diverse range of both natural and artificial phenomena are now being described in terms of Lévy statistics [19].Generally speaking, animals looking for food is random, from one place to another place. A large number of studies have shown that flight behavior of many animals and insects has demonstrated the typical characteristics of randomness. However, the choice of the direction relies only on a mathematical model [20], which is called Lévy-flight. For instance, many studies have shown that flight behavior of many animals and insects has revealed the typical characteristics of Lévy-flight [21–24]. According to [24], we can know that fruit flies orDrosophila melanogaster explore their landscape utilizing a series of straight flight paths punctuated by a sudden 9
0
° turn, resulting in a Lévy-flight-style fitful scale-free pattern. Studies on human behavior such as the Ju/’hoansi hunter-gatherer foraging patterns [21] also show the typical feature of Lévy-flight. Pavlyukevich has used Lévy-flight in his research to present and theoretically justify a new stochastic algorithm for global optimization. Even the light can be related to Lévy-flight [20]. Subsequently, Lévy-flight have been applied to optimization and optimal search, and preliminary results show its promising capability [22, 25].


## 3. The Proposed LMFO Approach

In order to increase the diversity of population against premature convergence and accelerate the convergence speed, this paper proposes an improved Lévy-flight moth-flame optimization (LMFO) algorithm. Lévy-flight has the prominent properties to increase the diversity of population, sequentially, which can make the algorithm effectively jump out of the local optimum. In other words, this approach is beneficial to obtain a better trade-off between the exploration and exploitation ability of MFO. So, we let each moth perform once Lévy-flight using (9) after the position updating, which is formulated as follows [11, 26]:(9)
X
i
t
+
1
=
X
i
t
+
u
sign
⁡
rand
-
0.5
⊕
Levy
β
,where X
i
t is the ith moth or solution vector X
i at iteration t, u is a random parameter which conforms to a uniform distribution, ⊕ is the dot product (entrywise multiplications), and rand is a random number in [
0,1
]. It should be noted here that sign
⁡
[
rand
-
0.5
] takes only three values 1, 0, and -
1. And in (9) the combination of u
sign
⁡
[
rand
-
0.5
] and Lévy-flight can make moth walk more random. That is to say, to get rid of local minima and improve global search capability are ensured via this combination in the basic MFO. Lévy-flight are a kind of random walk in which the steps are determined by the step lengths, and the jumps conform to a Lévy distribution as follows [11, 27]:(10)
Levy
β
~
μ
=
t
-
1
-
β
,
0
≤
β
≤
2
.Formula (11) is calculated as Lévy random numbers:(11)
Levy
β
~
ϕ
×
μ
ν
1
/
β
,where μ and ν are both standard normal distributions, Γ is a standard Gamma function, β
=
1.5, and ϕ is defined as follows:(12)
ϕ
=
Γ
1
+
β
×
sin
⁡
π
×
β
/
2
Γ
1
+
β
/
2
×
β
×
2
β
-
1
/
2
1
/
β
.To sum up, global search ability of the proposed algorithm is strengthened using random walk with Lévy-flight to eliminate the weakness of MFO, its being trapped in local minimum is prevented, and it is observed to give more successful results particularly for unimodal and multimodal benchmark functions. Because of these features, the proposed algorithm has potential to provide superior performance compared to MFO. In following section, all kinds of benchmark functions are hired to verify the effectiveness of the proposed algorithm. The main steps of Lévy-flight moth-flame optimization can be simply presented in Algorithm2.Algorithm 2:LMFO algorithm.
(1) Initialize the position of moths
(2) While (Iteration <= Max_iteration)
(3) Update flame no using (8)
(4) O
M = FitnessFunction(M);
(5) if iteration == 1
(6)   F = sort(M);
(7)   O
F = sort(O
M);
(8) else
(9)   F = sort(M
t
-
1, M
t);
(10)   O
F = sort(M
t
-
1, M
t);
(11) end
(12) for i
=
1
:
n
(13)    for j
=
1
:
d
(14)        Update r and t
(15)        Calculate D using (7) with respect to the corresponding moth
(16)        Update M
(
i
,
j
) using (5) and (6) with respect to the corresponding moth
(17)    end
(18)    for each search agent
(19)        Update the position of the current search agent using Lévy-flight
(20)    end
(21)    Iteration = Iteration + 1;
(22) end


## 4. Simulation Experiments

### 4.1. Simulation Platform

All the algorithms are tested in MATLAB R2012a (7.14) and numerical experiment is set up on Intel Core (TM) i5-4590 Processor, 3.30 GHz, 4 GB RAM, running on Windows 7.


### 4.2. Benchmark Functions

It is common in this field to benchmark the performance of algorithm on a set of mathematical functions with known global optimal. The same process is followed, in which nineteen standard benchmark functions are employed from the literature [27, 28] as test beds for comparison. Three groups of benchmark functions with different characteristics are selected to benchmark the performance of the LMFO algorithm from different perspectives. As shown in Tables 1–3, these benchmark functions are divided into three groups: unimodal functions, multimodal functions, and fixed-dimension multimodal functions. As their names imply, unimodal functions are suitable for benchmarking the exploitation and convergence of an algorithm since they have one global optimum and no local optima. In contrary, multimodal functions have more than one optimum, which makes them more challenging than unimodal functions. One of the optima is called global optimum, and the rest are called local optima. An algorithm should avoid all the local optima to approach and approximate the global optimum. Therefore, exploration and local optima avoidance of algorithms can be benchmarked by multimodal functions. The mathematical formulation of the employed benchmark functions is presented in Tables 1, 2, and 3, respectively. In these three tables, Range represent the boundary of the function’s search space, Dim denotes the dimension of the function, and f
m
i
n is the theoretical minimum of the function.Table 1
Unimodal benchmark functions.
Name
Function
Range
Dim
f
m
i
n
Sphere
f
1
x
=
∑
i
=
1
n
x
i
2
[−100, 100]
200
0
Schwefel’s 2.22
f
2
x
=
∑
i
=
1
n
x
i
+
∏
i
=
1
n
x
i
[−10, 10]
200
0
Schwefel’s 1.2
f
3
x
=
∑
i
=
1
n
∑
j
=
1
i
x
j
2
[−100, 100]
200
0
Schwefel’s 2.21
f
4
x
=
max
i
⁡
x
i
,
1
≤
i
≤
n
[−100, 100]
200
0
Rosenbrock
f
5
x
=
∑
i
=
1
n
-
1
100
x
i
+
1
-
x
i
2
2
+
x
i
-
1
2
[−30, 30]
200
0
Step
f
6
x
=
∑
i
=
1
n
x
i
+
0.5
2
[−100, 100]
200
0
Quartic
f
7
x
=
∑
i
=
1
n
i
x
i
4
+
r
a
n
d
o
m
(
0
,
1
)
[−1.28, 1.28]
200
0
X.S.Yang-7
f
8
x
=
∑
i
=
1
n
ε
j
x
i
-
1
i,  ε
j
∈
⋃
[
0
,
1
]
[−5, 5]
200
0Table 2
Multimodal benchmark functions.
Name
Function
Range
Dim
f
m
i
n
Rastrigin
f
9
x
=
∑
i
=
1
n
x
i
2
-
10
cos
⁡
(
2
π
x
i
)
+
10
[−5.12, 5.12]
200
0
Ackley
f
10
x
=
-
20
exp
⁡
-
0.2
1
n
∑
i
=
1
n
x
i
2
-
exp
⁡
1
n
∑
i
=
1
n
cos
⁡
(
2
π
x
i
)
+
20
+
e
[−32, 32]
200
0
Griewank
f
11
=
1
4000
∑
i
=
1
n
x
i
2
-
∏
i
=
1
n
cos
⁡
x
i
i
+
1
[−600, 600]
200
0
Penalized 1
f
12
x
=
π
n
10
sin
⁡
π
y
1
+
∑
i
=
1
n
-
1
y
1
-
1
2
[
1
+
10
sin
2
⁡
π
y
i
+
1
]
+
y
n
-
1
2
+
∑
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
[−50, 50]
200
0
Penalized 2
f
13
x
=
0.1
sin
2
⁡
3
π
x
1
+
∑
i
=
1
n
x
i
-
1
2
1
+
sin
2
⁡
3
π
x
i
+
1
+
x
n
-
1
2
1
+
sin
2
⁡
2
π
x
n
+
∑
i
=
1
n
u
x
i
,
5,100,4
[−50, 50]
200
0
Alpine
f
14
x
=
∑
i
=
1
n
x
i
sin
⁡
x
i
+
0.1
x
i
[−10, 10]
200
0
Zakharov
f
15
x
=
∑
i
=
1
n
x
i
2
+
∑
i
=
1
n
0.5
i
x
i
2
+
∑
i
=
1
n
0.5
i
x
i
4
[−5, 10]
200
0Table 3
Fixed-dimension multimodal benchmark functions.
Name
Function
Range
Dim
f
m
i
n
Goldstein-Price
f
16
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
[−2, 2]
2
3
Drop Wave
f
17
x
=
-
1
+
cos
⁡
12
x
1
2
+
x
2
2
1
/
2
x
1
2
+
x
2
2
+
2
[−5.12, 5.12]
2
−1
Schaffer’s F6
f
18
x
=
sin
2
⁡
x
1
2
+
x
2
2
-
0.5
1
+
0.001
x
1
2
+
x
2
2
2
[−100, 100]
2
−1
Easom
f
19
=
-
cos
⁡
x
1
×
cos
⁡
x
2
×
e
-
(
x
1
-
π
)
2
-
(
x
2
-
π
)
2
[−2∗pi, 2∗pi]
2
−1Heuristic algorithms are stochastic optimization techniques, and therefore they have to be run more than 10 times for generating meaningful statistical results. The best obtained solution in the last iteration is calculated as the metrics of performance. The same method is selected to generate and report the results over 30 independent runs. However, average and standard deviation only compare the overall performance of algorithms.To explore the performance of the proposed LMFO algorithm, some of the recent and well-known algorithms in the literature are chosen: ABC [9], BA [10], GGSA [29], DA [17], PSOGSA [30], and MFO [18]. Note that 30 number search agents and 1000 iterations are utilized for each of the algorithms. It should be noted that selection of the number of moths (or other candidate solutions in other algorithms) should be done experimentally.In this paper, Best, Mean, Worst, and Std represent the optimal fitness value, mean fitness value, worst fitness value, and standard deviation, respectively. Experimental results are listed in Tables4, 5, and 6. The best results are denoted inbold type.Table 4
Results of unimodal benchmark functions.
Benchmark function
Result
Algorithm
ABC
BA
GGSA
DA
PSOGSA
MFO
LMFO
f
1 (D
=
200)
Best
19896.78
365717.3
16278.78
1513.435
34835.45
140219.4
1.2921E − 234
Worst
36049.23
456284
24262.84
52511.62
142227.4
240000.7
3.3309E − 183
Mean
26677.04
414807.8
19224.04
24716.82
94735.12
185588.4
1.1103E − 184
Std
4640.383
20515.24
1852.355
11046.55
25051.92
24177.67
0
f
2 (D
=
200)
Best
108.0735
1.83E + 84
149.7263
26.27986
637.56
420.5512
8.5E − 129
Worst
186.4672
3.9E + 99
189.8577
255.79923
8.06E + 40
722.4955
7.3E − 99
Mean
140.2811
1.45E + 98
165.8829
137.88594
2.69E + 39
560.1173
2.4E − 100
Std
14.5252
7.06E + 98
9.599018
56.674292
1.47E + 40
61.66749
1.3E − 99
f
3 (D
=
200)
Best
582404
823783.5
160277.3
69110.62
344077.3
462517.7
1.6E − 215
Worst
920006.2
4617625
1020727
906356.8
891332.5
1094366
6.4E − 167
Mean
757278.3
1572330
434895.9
342508.6
541491.3
772538.3
3.3E − 168
Std
72837.78
837152.1
190865.2
203490.9
142131.3
170946.5
0
f
4 (D
=
200)
Best
94.265
87.48997
20.44273
27.0486
76.25401
95.26014
6.1E − 117
Worst
97.40102
92.12282
31.30115
59.69617
98.74303
98.15999
6.18E − 82
Mean
95.8655
90.09865
27.12328
41.97914
95.69831
97.03653
2.06E − 83
Std
0.778229
1.330154
2.551274
7.80909
5.90858
0.687621
1.13E − 82
f
5 (D
=
200)
Best
16111660
88328339
2399953
4717141
2565682
3.61E + 08
196.9121
Worst
94041761
1.59E + 08
5103632
47976603
4.82E + 08
8.92E + 08
198.6685
Mean
41344878
1.33E + 08
3740872
18257995
1.35E + 08
6.14E + 08
198.2609
Std
17567952
18028179
643192.1
10933433
1.24E + 08
1.41E + 08
0.459128
f
6 (D
=
200)
Best
17165.43
384155.3
14899
4422.437
33665.58
136678.5
39.65709
Worst
30743.79
453905.5
24873
63473.4
109123.6
228511.5
42.92694
Mean
23922.99
417903
19220.47
24167.16
76561.32
180669.8
41.49763
Std
3074.406
17001.78
2314.721
12441.1
19018.55
24671.19
0.632867
f
7 (D
=
200)
Best
51.88092
0.400252
6.492429
9.23801
11.89278
1181.991
2.3E − 06
Worst
299.2207
0.744656
14.77883
181.7351
71.66514
2686.255
0.000427
Mean
157.1676
0.551746
9.090199
52.06488
19.83666
1908.788
8.56E − 05
Std
58.73575
0.072809
1.742479
39.34587
10.47516
360.7708
0.000108
f
8 (D
=
200)
Best
181.8735
172.1645
28.45978
14.59629
130.6204
168.6843
1.62807
Worst
198.3101
200.1905
40.21054
80.16251
169.9275
202.2427
1.840634
Mean
190.2888
188.036
33.67195
43.38706
153.398
187.9161
1.752264
Std
4.168661
6.867959
2.232835
13.6498
10.91374
7.601635
0.06096Table 5
Results of multimodal benchmark functions.
Benchmark function
Result
Algorithm
ABC
BA
GGSA
DA
PSOGSA
MFO
LMFO
f
9 (D
=
200)
Best
575.1608
1363.369
1563.422
730.92416
922.1832
1769.441
0
Worst
709.019
1810.465
1867.616
1890.9545
1476.272
2125.69
0
Mean
653.9316
1630.768
1752.838
1367.3899
1231.578
1951.426
0
Std
36.46425
100.7203
81.02372
280.04565
117.1903
79.01827
0
f
10 (D
=
200)
Best
12.54258
19.20527
10.52907
8.666612
19.23309
19.92131
8.88E − 16
Worst
14.8045
19.9564
11.89042
14.75436
19.96677
20.01897
8.88E − 16
Mean
13.82113
19.73614
11.21024
11.88909
19.69719
19.95433
8.88E − 16
Std
0.554162
0.260154
0.383654
1.453704
0.31107
0.019396
0
f
11 (D
=
200)
Best
142.2548
4290.404
137.6285
61.43179
654.9388
1229.64
0
Worst
316.036
5283.435
221.9276
600.151
1631.547
2048.454
0
Mean
225.3103
4997.492
171.5404
224.0744
1251.695
1543.04
0
Std
51.75854
184.8135
16.55978
104.4042
204.6207
197.6001
0
f
12 (D
=
200)
Best
7393398
3.86E + 08
28.17774
1662.384
7043398
7.03E + 08
8.88E − 16
Worst
1.67E + 08
6.85E + 08
77685.77
23172033
2.05E + 09
1.92E + 09
8.88E − 16
Mean
58731093
5.59E + 08
11043.76
2516012
7.28E + 08
1.3E + 09
8.88E − 16
Std
37177591
81914079
20355.03
4641204
5.15E + 08
3.01E + 08
0
f
13 (D
=
200)
Best
19464968
1.25E + 09
646700.5
2620391
26666517
1.68E + 09
19.4713
Worst
3.4E + 08
2.09E + 09
3319003
1.83E + 08
2.47E + 09
4.04E + 09
19.79025
Mean
1.53E + 08
1.72E + 09
1684844
27183053
9.73E + 08
2.62E + 09
19.62771
Std
82296565
1.89E + 08
724732.9
34759974
7.07E + 08
5.46E + 08
0.072549
f
14 (D
=
200)
Best
47.28706
90.66079
103.3283
15.38694
58.10706
129.0595
1.1E − 125
Worst
62.0743
168.7346
139.0044
197.7987
107.5327
216.9578
1.1E − 103
Mean
53.55967
119.4603
118.7804
116.3619
80.96392
171.3074
3.9E − 105
Std
3.438082
19.18661
8.52589
43.5411
12.93695
23.18654
2E − 104
f
15 (D
=
200)
Best
5373.354
4315.808
461.6698
1142.924
4411.244
5741.417
5E − 168
Worst
6101.204
10167.76
3585.345
4914.844
9429.568
11241.25
2.6E − 117
Mean
5810.789
5283
1327.234
3451.42
6873.222
8566.525
8.6E − 119
Std
207.7704
1065.437
640.8766
1139.356
1451.294
1527.992
4.7E − 118Table 6
Results of fixed-dimension multimodal benchmark functions.
Benchmark function
Result
Algorithm
ABC
BA
GGSA
DA
PSOGSA
MFO
LMFO
f
16 (D
=
2)
Best
3.000547
3
3
3
3
3
3
Worst
3.052848
84.00001
30
3.179715
84
3
3.000357
Mean
3.015989
15.6
7.144215
3.018584
8.4
3
3.000061
Std
0.015909
25.30177
9.250619
0.049376
20.55036
1.83E − 15
7.19E − 05
f
17 (D
=
2)
Best
−1
−1
−1
−1
−1
−1
−1
Worst
−0.98844
−0.36913
−0.93625
−0.78575
−0.93625
−0.93625
−1
Mean
−0.99731
−0.71344
−0.949
−0.94932
−0.9915
−0.96175
−1
Std
0.003196
0.175563
0.025938
0.053641
0.022043
0.031767
0
f
18 (D
=
2)
Best
−0.99909
−0.99028
−1
−1
−1
−1
−1
Worst
−0.98981
−0.54822
−0.99028
−0.92181
−0.99028
−0.99028
−1
Mean
−0.99095
−0.71504
−0.99286
−0.98153
−0.99093
−0.99061
−1
Std
0.002023
0.126856
0.004349
0.02047
0.002465
0.001774
0
f
19 (D
=
2)
Best
−1
−1
−1
−1
−1
−1
−1
Worst
−0.99991
−8.1E − 05
−8.1E − 05
−0.95227
−1
−1
−0.99918
Mean
−0.99999
−0.80002
−0.92221
−0.99656
−1
−1
−0.99973
Std
2.03E − 05
0.406805
0.257944
0.010494
0
0
0.000229Due to the stochastic nature of the algorithms, statistical tests should be conducted to confirm the significance of the results [31]. The averages and standard deviation only compare the overall performance of the algorithms, while a statistical test considers each run’s results and proves that the results are statistically significant. In order to determine whether the results of LMFO differ from the best results of ABC, BA, GGSA, DA, PSOGSA, and MFO in a statistical method, a nonparametric test which is known as Wilcoxon’s rank-sum test [32, 33] is performed at 5% significance level. Tables 7, 8, and 9 report the p values produced by Wilcoxon’s test for the pairwise comparison of the best value of six groups. Such groups are formed by ABC versus LMFO, BA versus LMFO, GGSA versus LMFO, DA versus MFO, PSOGSA versus LMFO, and MFO versus LMFO. In general, p values < 0.05 can be considered as sufficient evidence against the null hypothesis. With the statistical test, we can make sure that the results are not generated by chance. The nonparametric Wilcoxon statistical test is conducted and the calculated p values are reported as metrics of significance as well. Experimental results of p values rank-sum test are listed in Tables 7, 8, and 9.Table 7
Results ofp-values rank-sum test on unimodal benchmark functions.
f
1
f
2
f
3
f
4
f
5
f
6
f
7
f
8
ABC versus LMFO
3.02E − 11
3.02E − 11
3.02E − 11
3.02E − 11
3.02E − 11
3.02E − 11
3.02E − 11
3.02E − 11
BA versus LMFO
3.02E − 11
3.02E − 11
3.02E − 11
3.02E − 11
3.02E − 11
3.02E − 11
3.02E − 11
3.02E − 11
GGSA versus LMFO
3.02E − 11
3.02E − 11
3.02E − 11
3.02E − 11
3.02E − 11
3.02E − 11
3.02E − 11
3.02E − 11
DA versus LMFO
3.02E − 11
3.02E − 11
3.02E − 11
3.02E − 11
3.02E − 11
3.02E − 11
3.02E − 11
3.02E − 11
PSOGSA versus LMFO
3.02E − 11
3.02E − 11
3.02E − 11
3.02E − 11
3.02E − 11
3.02E − 11
3.02E − 11
3.02E − 11
MFO versus LMFO
3.02E − 11
3.02E − 11
3.02E − 11
3.02E − 11
3.02E − 11
3.02E − 11
3.02E − 11
3.02E − 11Table 8
Results ofp-values rank-sum test on multimodal benchmark functions.
f
9
f
10
f
11
f
12
f
13
f
14
f
15
ABC versus LMFO
1.21E − 12
1.21E − 12
1.21E − 12
3.02E − 11
3.02E − 11
3.02E − 11
3.02E − 11
BA versus LMFO
1.21E − 12
1.21E − 12
1.21E − 12
3.02E − 11
3.02E − 11
3.02E − 11
3.02E − 11
GGSA versus LMFO
1.21E − 12
1.21E − 12
1.21E − 12
3.02E − 11
3.02E − 11
3.02E − 11
3.02E − 11
DA versus LMFO
1.21E − 12
1.21E − 12
1.21E − 12
3.02E − 11
3.02E − 11
3.02E − 11
3.02E − 11
PSOGSA versus LMFO
1.21E − 12
1.21E − 12
1.21E − 12
3.02E − 11
3.02E − 11
3.02E − 11
3.02E − 11
MFO versus LMFO
1.21E − 12
1.21E − 12
1.21E − 12
3.02E − 11
3.02E − 11
3.02E − 11
3.02E − 11Table 9
Results ofp-values rank-sum test on fixed-dimension multimodal benchmark functions.
f
16
f
17
f
18
f
19
ABC versus LMFO
3.02E − 11
1.21E − 12
1.21E − 12
5.07E − 10
BA versus LMFO
7.29E − 03
1.21E − 12
1.21E − 12
6.77E − 05
GGSA versus LMFO
7.70E − 02
8.81E − 10
5.13E − 11
2.48E − 08
DA versus LMFO
5.19E − 02
4.56E − 10
1.45E − 11
2.96E − 06
PSOGSA versus LMFO
7.91E − 09
4.18E − 02
7.15E − 13
1.21E − 12
MFO versus LMFO
2.56E − 11
6.89E − 07
1.17E − 13
1.21E − 12


### 4.3. Unimodal Benchmark Functions

The unimodal benchmarks functions have only one global minimum and there are no local minima for them. Therefore, these kinds of functions are very suitable for benchmarking the convergence capability of algorithms. According to the results of Table4, LMFO is able to provide very competitive results. As can be seen from this table, LMFO outperforms all other algorithms in f
1~f
8. Therefore, the proposed algorithm has high performance to find the global minimum of unimodal benchmark functions. According to the p values of f
1~f
8 in Table 7, LMFO achieves significant improvement in all the unimodal benchmark functions compared to other algorithms. Hence, this proves that LMFO has better performance than other algorithms in forging for global optimum solution of unimodal benchmark functions.Figures1–8 illustrate the averaged convergence curves of all algorithms disposing unimodal benchmark functions over 30 independent runs. It can be noted here that all the convergence curves in the following subsections are also averaged curves. As may be seen from these curves, LMFO has the fastest convergence speed in all algorithms. From Table 4 and Figures 20–27, the LMFO’s Std is much smaller than other algorithms. These show that LMFO has a strong sense of stability and robust comparing from other algorithms.Figure 1
The convergence curves forf
1.Figure 2
The convergence curves forf
2.Figure 3
The convergence curves forf
3.Figure 4
The convergence curves forf
4.Figure 5
The convergence curves forf
5.Figure 6
The convergence curves forf
6.Figure 7
The convergence curves forf
7.Figure 8
The convergence curves forf
8.


### 4.4. Multimodal Benchmark Functions

In contrast to the unimodal benchmark functions, multimodal benchmark functions have many local minima with the number increasing exponentially with dimension. This makes them suitable for benchmarking the exploration ability of an algorithm. So, the final results are more important because these benchmark functions can reflect the ability of the algorithm to escape from poor local optima and obtain the global optimum. The statistical results of the algorithms on multimodal benchmark functions are presented in Table5. As the results of Best, Worst, Mean, and Std values show, LMFO is also able to provide very competitive results on the multimodal benchmark functions. These results show that the LMFO algorithm has merit in terms of exploration. According to the p values of f
9~f
15 reported in Table 8, LMFO achieves significant improvement on 200-D compared to other algorithms. When comparing LMFO and other algorithms, we can conclude that LMFO is significantly performing better with six groups of comparison algorithms. The p values of f
9~f
15 reported in Table 8 are less than 0.05, which is strong evidence against null hypothesis. Hence, this evidence demonstrates that the results of LMFO are statistically significant not occurring by coincidence.Seen from Table5 and Figures 9–15, the convergence rate of LMFO on the multimodal benchmark functions in majority cases is better than other algorithms. On the basis of Table 5, and Figures 9–15, we can draw a conclusion that the LMFO is able to avoid local minima in multimodal benchmark functions with a good convergence speed. From Table 5 and Figures 28–37, the LMFO’s Std is much smaller than other algorithms. These show that LMFO has a strong sense of stability and robust comparing from other algorithms.Figure 9
The convergence curves forf
9.Figure 10
The convergence curves forf
10.Figure 11
The convergence curves forf
11.Figure 12
The convergence curves forf
12.Figure 13
The convergence curves forf
13.Figure 14
The convergence curves forf
14.Figure 15
The convergence curves forf
15.


### 4.5. Fixed-Dimension Multimodal Benchmark Functions

For fixed-dimension multimodal benchmark functions with only a few local minima, the dimensions of the multimodal benchmark functions are also small. Under such circumstances, it is difficult to judge the performance of individual algorithm. The major difference compared with multimodal functions is that fixed-dimension multimodal functions appear to be simpler because of their low dimensions and a smaller number of local minima. In this experiment, the results of Best, Worst, Mean, and Std values of fixed-dimension multimodal benchmark functions are summarized in Table6. For all fixed-dimension multimodal functions, LMFO can give the best solution in terms of Best. As Table 6 shows, the LMFO algorithm provides the best results on two of fixed-dimension multimodal benchmark functions. The results are followed by the MFO, PSOGSA, and ABC algorithms. In addition, the p values of Wilcoxon’s rank-sum in Table 9 show that the result of LMFO in f
16 is not significantly better than DA and GGSA algorithms (5% significance level), but it is significantly different compared with ABC, MFO, PSOGSA, and BA. In the remaining functions (f
17, f
18, and f
19), however, the results of LMFO are significantly better than other algorithms. So, it can be concluded that the results of LMFO in these benchmark functions are better than ABC, BA, GGSA, DA, PSOGSA, and MFO.In addition, the convergence rate of LMFO on the fixed-dimension benchmark functions with 2-dim can be shown in Figures16–19. As can be seen from these figures, it can be claimed that LMFO has the faster convergence rate on functions f
17 and f
18. From Figures 35–38, we can find that all of the algorithms have a strong sense of stability except BA on fixed-dimension functions.Figure 16
The convergence curves forf
16.Figure 17
The convergence curves forf
17.Figure 18
The convergence curves forf
18.Figure 19
The convergence curves forf
19.Figure 20
Standard deviation forf
1.Figure 21
Standard deviation forf
2.Figure 22
Standard deviation forf
3.Figure 23
Standard deviation forf
4.Figure 24
Standard deviation forf
5.Figure 25
Standard deviation forf
6.Figure 26
Standard deviation forf
7.Figure 27
Standard deviation forf
8.Figure 28
Standard deviation forf
9.Figure 29
Standard deviation forf
10.Figure 30
Standard deviation forf
11.Figure 31
Standard deviation forf
12.Figure 32
Standard deviation forf
13.Figure 33
Standard deviation forf
14.Figure 34
Standard deviation forf
15.Figure 35
Standard deviation forf
16.Figure 36
Standard deviation forf
17.Figure 37
Standard deviation forf
18.Figure 38
Standard deviation forf
19.Overall, the results from Tables4–6, Tables 7–9, Figures 1–19, and Figures 20–38 show that the proposed method is effective in not only optimizing unimodal and multimodal functions but also optimizing fixed-dimension multimodal functions.Since constraints are one of the major challenges in solving real problems and the main objective of designing the LMFO algorithm is to solve real problems, two constrained real engineering problems are employed in the next section to further investigate the performance of the MFO algorithm and provide a comprehensive study.


## 4.1. Simulation Platform

All the algorithms are tested in MATLAB R2012a (7.14) and numerical experiment is set up on Intel Core (TM) i5-4590 Processor, 3.30 GHz, 4 GB RAM, running on Windows 7.


## 4.2. Benchmark Functions

It is common in this field to benchmark the performance of algorithm on a set of mathematical functions with known global optimal. The same process is followed, in which nineteen standard benchmark functions are employed from the literature [27, 28] as test beds for comparison. Three groups of benchmark functions with different characteristics are selected to benchmark the performance of the LMFO algorithm from different perspectives. As shown in Tables 1–3, these benchmark functions are divided into three groups: unimodal functions, multimodal functions, and fixed-dimension multimodal functions. As their names imply, unimodal functions are suitable for benchmarking the exploitation and convergence of an algorithm since they have one global optimum and no local optima. In contrary, multimodal functions have more than one optimum, which makes them more challenging than unimodal functions. One of the optima is called global optimum, and the rest are called local optima. An algorithm should avoid all the local optima to approach and approximate the global optimum. Therefore, exploration and local optima avoidance of algorithms can be benchmarked by multimodal functions. The mathematical formulation of the employed benchmark functions is presented in Tables 1, 2, and 3, respectively. In these three tables, Range represent the boundary of the function’s search space, Dim denotes the dimension of the function, and f
m
i
n is the theoretical minimum of the function.Table 1
Unimodal benchmark functions.
Name
Function
Range
Dim
f
m
i
n
Sphere
f
1
x
=
∑
i
=
1
n
x
i
2
[−100, 100]
200
0
Schwefel’s 2.22
f
2
x
=
∑
i
=
1
n
x
i
+
∏
i
=
1
n
x
i
[−10, 10]
200
0
Schwefel’s 1.2
f
3
x
=
∑
i
=
1
n
∑
j
=
1
i
x
j
2
[−100, 100]
200
0
Schwefel’s 2.21
f
4
x
=
max
i
⁡
x
i
,
1
≤
i
≤
n
[−100, 100]
200
0
Rosenbrock
f
5
x
=
∑
i
=
1
n
-
1
100
x
i
+
1
-
x
i
2
2
+
x
i
-
1
2
[−30, 30]
200
0
Step
f
6
x
=
∑
i
=
1
n
x
i
+
0.5
2
[−100, 100]
200
0
Quartic
f
7
x
=
∑
i
=
1
n
i
x
i
4
+
r
a
n
d
o
m
(
0
,
1
)
[−1.28, 1.28]
200
0
X.S.Yang-7
f
8
x
=
∑
i
=
1
n
ε
j
x
i
-
1
i,  ε
j
∈
⋃
[
0
,
1
]
[−5, 5]
200
0Table 2
Multimodal benchmark functions.
Name
Function
Range
Dim
f
m
i
n
Rastrigin
f
9
x
=
∑
i
=
1
n
x
i
2
-
10
cos
⁡
(
2
π
x
i
)
+
10
[−5.12, 5.12]
200
0
Ackley
f
10
x
=
-
20
exp
⁡
-
0.2
1
n
∑
i
=
1
n
x
i
2
-
exp
⁡
1
n
∑
i
=
1
n
cos
⁡
(
2
π
x
i
)
+
20
+
e
[−32, 32]
200
0
Griewank
f
11
=
1
4000
∑
i
=
1
n
x
i
2
-
∏
i
=
1
n
cos
⁡
x
i
i
+
1
[−600, 600]
200
0
Penalized 1
f
12
x
=
π
n
10
sin
⁡
π
y
1
+
∑
i
=
1
n
-
1
y
1
-
1
2
[
1
+
10
sin
2
⁡
π
y
i
+
1
]
+
y
n
-
1
2
+
∑
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
[−50, 50]
200
0
Penalized 2
f
13
x
=
0.1
sin
2
⁡
3
π
x
1
+
∑
i
=
1
n
x
i
-
1
2
1
+
sin
2
⁡
3
π
x
i
+
1
+
x
n
-
1
2
1
+
sin
2
⁡
2
π
x
n
+
∑
i
=
1
n
u
x
i
,
5,100,4
[−50, 50]
200
0
Alpine
f
14
x
=
∑
i
=
1
n
x
i
sin
⁡
x
i
+
0.1
x
i
[−10, 10]
200
0
Zakharov
f
15
x
=
∑
i
=
1
n
x
i
2
+
∑
i
=
1
n
0.5
i
x
i
2
+
∑
i
=
1
n
0.5
i
x
i
4
[−5, 10]
200
0Table 3
Fixed-dimension multimodal benchmark functions.
Name
Function
Range
Dim
f
m
i
n
Goldstein-Price
f
16
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
[−2, 2]
2
3
Drop Wave
f
17
x
=
-
1
+
cos
⁡
12
x
1
2
+
x
2
2
1
/
2
x
1
2
+
x
2
2
+
2
[−5.12, 5.12]
2
−1
Schaffer’s F6
f
18
x
=
sin
2
⁡
x
1
2
+
x
2
2
-
0.5
1
+
0.001
x
1
2
+
x
2
2
2
[−100, 100]
2
−1
Easom
f
19
=
-
cos
⁡
x
1
×
cos
⁡
x
2
×
e
-
(
x
1
-
π
)
2
-
(
x
2
-
π
)
2
[−2∗pi, 2∗pi]
2
−1Heuristic algorithms are stochastic optimization techniques, and therefore they have to be run more than 10 times for generating meaningful statistical results. The best obtained solution in the last iteration is calculated as the metrics of performance. The same method is selected to generate and report the results over 30 independent runs. However, average and standard deviation only compare the overall performance of algorithms.To explore the performance of the proposed LMFO algorithm, some of the recent and well-known algorithms in the literature are chosen: ABC [9], BA [10], GGSA [29], DA [17], PSOGSA [30], and MFO [18]. Note that 30 number search agents and 1000 iterations are utilized for each of the algorithms. It should be noted that selection of the number of moths (or other candidate solutions in other algorithms) should be done experimentally.In this paper, Best, Mean, Worst, and Std represent the optimal fitness value, mean fitness value, worst fitness value, and standard deviation, respectively. Experimental results are listed in Tables4, 5, and 6. The best results are denoted inbold type.Table 4
Results of unimodal benchmark functions.
Benchmark function
Result
Algorithm
ABC
BA
GGSA
DA
PSOGSA
MFO
LMFO
f
1 (D
=
200)
Best
19896.78
365717.3
16278.78
1513.435
34835.45
140219.4
1.2921E − 234
Worst
36049.23
456284
24262.84
52511.62
142227.4
240000.7
3.3309E − 183
Mean
26677.04
414807.8
19224.04
24716.82
94735.12
185588.4
1.1103E − 184
Std
4640.383
20515.24
1852.355
11046.55
25051.92
24177.67
0
f
2 (D
=
200)
Best
108.0735
1.83E + 84
149.7263
26.27986
637.56
420.5512
8.5E − 129
Worst
186.4672
3.9E + 99
189.8577
255.79923
8.06E + 40
722.4955
7.3E − 99
Mean
140.2811
1.45E + 98
165.8829
137.88594
2.69E + 39
560.1173
2.4E − 100
Std
14.5252
7.06E + 98
9.599018
56.674292
1.47E + 40
61.66749
1.3E − 99
f
3 (D
=
200)
Best
582404
823783.5
160277.3
69110.62
344077.3
462517.7
1.6E − 215
Worst
920006.2
4617625
1020727
906356.8
891332.5
1094366
6.4E − 167
Mean
757278.3
1572330
434895.9
342508.6
541491.3
772538.3
3.3E − 168
Std
72837.78
837152.1
190865.2
203490.9
142131.3
170946.5
0
f
4 (D
=
200)
Best
94.265
87.48997
20.44273
27.0486
76.25401
95.26014
6.1E − 117
Worst
97.40102
92.12282
31.30115
59.69617
98.74303
98.15999
6.18E − 82
Mean
95.8655
90.09865
27.12328
41.97914
95.69831
97.03653
2.06E − 83
Std
0.778229
1.330154
2.551274
7.80909
5.90858
0.687621
1.13E − 82
f
5 (D
=
200)
Best
16111660
88328339
2399953
4717141
2565682
3.61E + 08
196.9121
Worst
94041761
1.59E + 08
5103632
47976603
4.82E + 08
8.92E + 08
198.6685
Mean
41344878
1.33E + 08
3740872
18257995
1.35E + 08
6.14E + 08
198.2609
Std
17567952
18028179
643192.1
10933433
1.24E + 08
1.41E + 08
0.459128
f
6 (D
=
200)
Best
17165.43
384155.3
14899
4422.437
33665.58
136678.5
39.65709
Worst
30743.79
453905.5
24873
63473.4
109123.6
228511.5
42.92694
Mean
23922.99
417903
19220.47
24167.16
76561.32
180669.8
41.49763
Std
3074.406
17001.78
2314.721
12441.1
19018.55
24671.19
0.632867
f
7 (D
=
200)
Best
51.88092
0.400252
6.492429
9.23801
11.89278
1181.991
2.3E − 06
Worst
299.2207
0.744656
14.77883
181.7351
71.66514
2686.255
0.000427
Mean
157.1676
0.551746
9.090199
52.06488
19.83666
1908.788
8.56E − 05
Std
58.73575
0.072809
1.742479
39.34587
10.47516
360.7708
0.000108
f
8 (D
=
200)
Best
181.8735
172.1645
28.45978
14.59629
130.6204
168.6843
1.62807
Worst
198.3101
200.1905
40.21054
80.16251
169.9275
202.2427
1.840634
Mean
190.2888
188.036
33.67195
43.38706
153.398
187.9161
1.752264
Std
4.168661
6.867959
2.232835
13.6498
10.91374
7.601635
0.06096Table 5
Results of multimodal benchmark functions.
Benchmark function
Result
Algorithm
ABC
BA
GGSA
DA
PSOGSA
MFO
LMFO
f
9 (D
=
200)
Best
575.1608
1363.369
1563.422
730.92416
922.1832
1769.441
0
Worst
709.019
1810.465
1867.616
1890.9545
1476.272
2125.69
0
Mean
653.9316
1630.768
1752.838
1367.3899
1231.578
1951.426
0
Std
36.46425
100.7203
81.02372
280.04565
117.1903
79.01827
0
f
10 (D
=
200)
Best
12.54258
19.20527
10.52907
8.666612
19.23309
19.92131
8.88E − 16
Worst
14.8045
19.9564
11.89042
14.75436
19.96677
20.01897
8.88E − 16
Mean
13.82113
19.73614
11.21024
11.88909
19.69719
19.95433
8.88E − 16
Std
0.554162
0.260154
0.383654
1.453704
0.31107
0.019396
0
f
11 (D
=
200)
Best
142.2548
4290.404
137.6285
61.43179
654.9388
1229.64
0
Worst
316.036
5283.435
221.9276
600.151
1631.547
2048.454
0
Mean
225.3103
4997.492
171.5404
224.0744
1251.695
1543.04
0
Std
51.75854
184.8135
16.55978
104.4042
204.6207
197.6001
0
f
12 (D
=
200)
Best
7393398
3.86E + 08
28.17774
1662.384
7043398
7.03E + 08
8.88E − 16
Worst
1.67E + 08
6.85E + 08
77685.77
23172033
2.05E + 09
1.92E + 09
8.88E − 16
Mean
58731093
5.59E + 08
11043.76
2516012
7.28E + 08
1.3E + 09
8.88E − 16
Std
37177591
81914079
20355.03
4641204
5.15E + 08
3.01E + 08
0
f
13 (D
=
200)
Best
19464968
1.25E + 09
646700.5
2620391
26666517
1.68E + 09
19.4713
Worst
3.4E + 08
2.09E + 09
3319003
1.83E + 08
2.47E + 09
4.04E + 09
19.79025
Mean
1.53E + 08
1.72E + 09
1684844
27183053
9.73E + 08
2.62E + 09
19.62771
Std
82296565
1.89E + 08
724732.9
34759974
7.07E + 08
5.46E + 08
0.072549
f
14 (D
=
200)
Best
47.28706
90.66079
103.3283
15.38694
58.10706
129.0595
1.1E − 125
Worst
62.0743
168.7346
139.0044
197.7987
107.5327
216.9578
1.1E − 103
Mean
53.55967
119.4603
118.7804
116.3619
80.96392
171.3074
3.9E − 105
Std
3.438082
19.18661
8.52589
43.5411
12.93695
23.18654
2E − 104
f
15 (D
=
200)
Best
5373.354
4315.808
461.6698
1142.924
4411.244
5741.417
5E − 168
Worst
6101.204
10167.76
3585.345
4914.844
9429.568
11241.25
2.6E − 117
Mean
5810.789
5283
1327.234
3451.42
6873.222
8566.525
8.6E − 119
Std
207.7704
1065.437
640.8766
1139.356
1451.294
1527.992
4.7E − 118Table 6
Results of fixed-dimension multimodal benchmark functions.
Benchmark function
Result
Algorithm
ABC
BA
GGSA
DA
PSOGSA
MFO
LMFO
f
16 (D
=
2)
Best
3.000547
3
3
3
3
3
3
Worst
3.052848
84.00001
30
3.179715
84
3
3.000357
Mean
3.015989
15.6
7.144215
3.018584
8.4
3
3.000061
Std
0.015909
25.30177
9.250619
0.049376
20.55036
1.83E − 15
7.19E − 05
f
17 (D
=
2)
Best
−1
−1
−1
−1
−1
−1
−1
Worst
−0.98844
−0.36913
−0.93625
−0.78575
−0.93625
−0.93625
−1
Mean
−0.99731
−0.71344
−0.949
−0.94932
−0.9915
−0.96175
−1
Std
0.003196
0.175563
0.025938
0.053641
0.022043
0.031767
0
f
18 (D
=
2)
Best
−0.99909
−0.99028
−1
−1
−1
−1
−1
Worst
−0.98981
−0.54822
−0.99028
−0.92181
−0.99028
−0.99028
−1
Mean
−0.99095
−0.71504
−0.99286
−0.98153
−0.99093
−0.99061
−1
Std
0.002023
0.126856
0.004349
0.02047
0.002465
0.001774
0
f
19 (D
=
2)
Best
−1
−1
−1
−1
−1
−1
−1
Worst
−0.99991
−8.1E − 05
−8.1E − 05
−0.95227
−1
−1
−0.99918
Mean
−0.99999
−0.80002
−0.92221
−0.99656
−1
−1
−0.99973
Std
2.03E − 05
0.406805
0.257944
0.010494
0
0
0.000229Due to the stochastic nature of the algorithms, statistical tests should be conducted to confirm the significance of the results [31]. The averages and standard deviation only compare the overall performance of the algorithms, while a statistical test considers each run’s results and proves that the results are statistically significant. In order to determine whether the results of LMFO differ from the best results of ABC, BA, GGSA, DA, PSOGSA, and MFO in a statistical method, a nonparametric test which is known as Wilcoxon’s rank-sum test [32, 33] is performed at 5% significance level. Tables 7, 8, and 9 report the p values produced by Wilcoxon’s test for the pairwise comparison of the best value of six groups. Such groups are formed by ABC versus LMFO, BA versus LMFO, GGSA versus LMFO, DA versus MFO, PSOGSA versus LMFO, and MFO versus LMFO. In general, p values < 0.05 can be considered as sufficient evidence against the null hypothesis. With the statistical test, we can make sure that the results are not generated by chance. The nonparametric Wilcoxon statistical test is conducted and the calculated p values are reported as metrics of significance as well. Experimental results of p values rank-sum test are listed in Tables 7, 8, and 9.Table 7
Results ofp-values rank-sum test on unimodal benchmark functions.
f
1
f
2
f
3
f
4
f
5
f
6
f
7
f
8
ABC versus LMFO
3.02E − 11
3.02E − 11
3.02E − 11
3.02E − 11
3.02E − 11
3.02E − 11
3.02E − 11
3.02E − 11
BA versus LMFO
3.02E − 11
3.02E − 11
3.02E − 11
3.02E − 11
3.02E − 11
3.02E − 11
3.02E − 11
3.02E − 11
GGSA versus LMFO
3.02E − 11
3.02E − 11
3.02E − 11
3.02E − 11
3.02E − 11
3.02E − 11
3.02E − 11
3.02E − 11
DA versus LMFO
3.02E − 11
3.02E − 11
3.02E − 11
3.02E − 11
3.02E − 11
3.02E − 11
3.02E − 11
3.02E − 11
PSOGSA versus LMFO
3.02E − 11
3.02E − 11
3.02E − 11
3.02E − 11
3.02E − 11
3.02E − 11
3.02E − 11
3.02E − 11
MFO versus LMFO
3.02E − 11
3.02E − 11
3.02E − 11
3.02E − 11
3.02E − 11
3.02E − 11
3.02E − 11
3.02E − 11Table 8
Results ofp-values rank-sum test on multimodal benchmark functions.
f
9
f
10
f
11
f
12
f
13
f
14
f
15
ABC versus LMFO
1.21E − 12
1.21E − 12
1.21E − 12
3.02E − 11
3.02E − 11
3.02E − 11
3.02E − 11
BA versus LMFO
1.21E − 12
1.21E − 12
1.21E − 12
3.02E − 11
3.02E − 11
3.02E − 11
3.02E − 11
GGSA versus LMFO
1.21E − 12
1.21E − 12
1.21E − 12
3.02E − 11
3.02E − 11
3.02E − 11
3.02E − 11
DA versus LMFO
1.21E − 12
1.21E − 12
1.21E − 12
3.02E − 11
3.02E − 11
3.02E − 11
3.02E − 11
PSOGSA versus LMFO
1.21E − 12
1.21E − 12
1.21E − 12
3.02E − 11
3.02E − 11
3.02E − 11
3.02E − 11
MFO versus LMFO
1.21E − 12
1.21E − 12
1.21E − 12
3.02E − 11
3.02E − 11
3.02E − 11
3.02E − 11Table 9
Results ofp-values rank-sum test on fixed-dimension multimodal benchmark functions.
f
16
f
17
f
18
f
19
ABC versus LMFO
3.02E − 11
1.21E − 12
1.21E − 12
5.07E − 10
BA versus LMFO
7.29E − 03
1.21E − 12
1.21E − 12
6.77E − 05
GGSA versus LMFO
7.70E − 02
8.81E − 10
5.13E − 11
2.48E − 08
DA versus LMFO
5.19E − 02
4.56E − 10
1.45E − 11
2.96E − 06
PSOGSA versus LMFO
7.91E − 09
4.18E − 02
7.15E − 13
1.21E − 12
MFO versus LMFO
2.56E − 11
6.89E − 07
1.17E − 13
1.21E − 12


## 4.3. Unimodal Benchmark Functions

The unimodal benchmarks functions have only one global minimum and there are no local minima for them. Therefore, these kinds of functions are very suitable for benchmarking the convergence capability of algorithms. According to the results of Table4, LMFO is able to provide very competitive results. As can be seen from this table, LMFO outperforms all other algorithms in f
1~f
8. Therefore, the proposed algorithm has high performance to find the global minimum of unimodal benchmark functions. According to the p values of f
1~f
8 in Table 7, LMFO achieves significant improvement in all the unimodal benchmark functions compared to other algorithms. Hence, this proves that LMFO has better performance than other algorithms in forging for global optimum solution of unimodal benchmark functions.Figures1–8 illustrate the averaged convergence curves of all algorithms disposing unimodal benchmark functions over 30 independent runs. It can be noted here that all the convergence curves in the following subsections are also averaged curves. As may be seen from these curves, LMFO has the fastest convergence speed in all algorithms. From Table 4 and Figures 20–27, the LMFO’s Std is much smaller than other algorithms. These show that LMFO has a strong sense of stability and robust comparing from other algorithms.Figure 1
The convergence curves forf
1.Figure 2
The convergence curves forf
2.Figure 3
The convergence curves forf
3.Figure 4
The convergence curves forf
4.Figure 5
The convergence curves forf
5.Figure 6
The convergence curves forf
6.Figure 7
The convergence curves forf
7.Figure 8
The convergence curves forf
8.


## 4.4. Multimodal Benchmark Functions

In contrast to the unimodal benchmark functions, multimodal benchmark functions have many local minima with the number increasing exponentially with dimension. This makes them suitable for benchmarking the exploration ability of an algorithm. So, the final results are more important because these benchmark functions can reflect the ability of the algorithm to escape from poor local optima and obtain the global optimum. The statistical results of the algorithms on multimodal benchmark functions are presented in Table5. As the results of Best, Worst, Mean, and Std values show, LMFO is also able to provide very competitive results on the multimodal benchmark functions. These results show that the LMFO algorithm has merit in terms of exploration. According to the p values of f
9~f
15 reported in Table 8, LMFO achieves significant improvement on 200-D compared to other algorithms. When comparing LMFO and other algorithms, we can conclude that LMFO is significantly performing better with six groups of comparison algorithms. The p values of f
9~f
15 reported in Table 8 are less than 0.05, which is strong evidence against null hypothesis. Hence, this evidence demonstrates that the results of LMFO are statistically significant not occurring by coincidence.Seen from Table5 and Figures 9–15, the convergence rate of LMFO on the multimodal benchmark functions in majority cases is better than other algorithms. On the basis of Table 5, and Figures 9–15, we can draw a conclusion that the LMFO is able to avoid local minima in multimodal benchmark functions with a good convergence speed. From Table 5 and Figures 28–37, the LMFO’s Std is much smaller than other algorithms. These show that LMFO has a strong sense of stability and robust comparing from other algorithms.Figure 9
The convergence curves forf
9.Figure 10
The convergence curves forf
10.Figure 11
The convergence curves forf
11.Figure 12
The convergence curves forf
12.Figure 13
The convergence curves forf
13.Figure 14
The convergence curves forf
14.Figure 15
The convergence curves forf
15.


## 4.5. Fixed-Dimension Multimodal Benchmark Functions

For fixed-dimension multimodal benchmark functions with only a few local minima, the dimensions of the multimodal benchmark functions are also small. Under such circumstances, it is difficult to judge the performance of individual algorithm. The major difference compared with multimodal functions is that fixed-dimension multimodal functions appear to be simpler because of their low dimensions and a smaller number of local minima. In this experiment, the results of Best, Worst, Mean, and Std values of fixed-dimension multimodal benchmark functions are summarized in Table6. For all fixed-dimension multimodal functions, LMFO can give the best solution in terms of Best. As Table 6 shows, the LMFO algorithm provides the best results on two of fixed-dimension multimodal benchmark functions. The results are followed by the MFO, PSOGSA, and ABC algorithms. In addition, the p values of Wilcoxon’s rank-sum in Table 9 show that the result of LMFO in f
16 is not significantly better than DA and GGSA algorithms (5% significance level), but it is significantly different compared with ABC, MFO, PSOGSA, and BA. In the remaining functions (f
17, f
18, and f
19), however, the results of LMFO are significantly better than other algorithms. So, it can be concluded that the results of LMFO in these benchmark functions are better than ABC, BA, GGSA, DA, PSOGSA, and MFO.In addition, the convergence rate of LMFO on the fixed-dimension benchmark functions with 2-dim can be shown in Figures16–19. As can be seen from these figures, it can be claimed that LMFO has the faster convergence rate on functions f
17 and f
18. From Figures 35–38, we can find that all of the algorithms have a strong sense of stability except BA on fixed-dimension functions.Figure 16
The convergence curves forf
16.Figure 17
The convergence curves forf
17.Figure 18
The convergence curves forf
18.Figure 19
The convergence curves forf
19.Figure 20
Standard deviation forf
1.Figure 21
Standard deviation forf
2.Figure 22
Standard deviation forf
3.Figure 23
Standard deviation forf
4.Figure 24
Standard deviation forf
5.Figure 25
Standard deviation forf
6.Figure 26
Standard deviation forf
7.Figure 27
Standard deviation forf
8.Figure 28
Standard deviation forf
9.Figure 29
Standard deviation forf
10.Figure 30
Standard deviation forf
11.Figure 31
Standard deviation forf
12.Figure 32
Standard deviation forf
13.Figure 33
Standard deviation forf
14.Figure 34
Standard deviation forf
15.Figure 35
Standard deviation forf
16.Figure 36
Standard deviation forf
17.Figure 37
Standard deviation forf
18.Figure 38
Standard deviation forf
19.Overall, the results from Tables4–6, Tables 7–9, Figures 1–19, and Figures 20–38 show that the proposed method is effective in not only optimizing unimodal and multimodal functions but also optimizing fixed-dimension multimodal functions.Since constraints are one of the major challenges in solving real problems and the main objective of designing the LMFO algorithm is to solve real problems, two constrained real engineering problems are employed in the next section to further investigate the performance of the MFO algorithm and provide a comprehensive study.


## 5. LMFO for Engineering Optimization Problems

In this section, a set of two engineering problems (welded beam design and speed reducer design) is solved so as to further testify the performance of the proposed algorithm. There are some inequality constraints in real problems, so the LMFO algorithm should be capable of dealing with them during optimization. Several methods have been applied to handle constraints in the literature: penalty function, special operators, repaired algorithms, separation of objectives and constraints, and hybrid methods [34]. In this paper, penalty method is employed to handle the constraints of welded beam and speed reducer.


### 5.1. Welded Beam Design

The objective is to evaluate the optimal fabrication cost of a welded beam as shown in Figure39 [35]. The constraints of the beam are shear stress (τ), bending stress in the beam (θ), buckling load on the bar (P
c), end deflection of the beam (δ), and side constraints.Figure 39
Structure of welded beam design.This problem has four variables that are thickness of weld (h), length of attached part of bar (l), the height of the bar (t), and thickness of the bar (b), respectively. This problem is formulated as follows:(13)
Consider
x
→
=
x
1
,
x
2
,
x
3
,
x
4
=
h
,
l
,
t
,
b
,
Minimize
f
x
→
=
1.10471
x
1
2
x
2
+
0.04811
x
3
x
4
14.0
+
x
2
,
Subject to
g
1
x
→
=
τ
x
→
-
τ
m
a
x
≤
0
,
g
2
x
→
=
σ
x
→
-
σ
m
a
x
≤
0
,
g
3
x
→
=
δ
x
→
-
δ
m
a
x
≤
0
,
g
4
x
→
=
x
1
-
x
4
≤
0
,
g
5
x
→
=
P
-
P
c
x
→
≤
0
,
g
6
x
→
=
0.125
-
x
1
≤
0
,
g
7
x
→
=
1.10471
x
1
2
+
0.04811
x
3
x
4
14.0
+
x
2
-
5.0
≤
0
,
Variable range
0.1
≤
x
1
≤
2
,
0.1
≤
x
2
≤
10
,
0.1
≤
x
3
≤
10
,
0.1
≤
x
4
≤
2
,
Where
τ
x
→
=
τ
′
2
+
2
τ
′
τ
′
′
x
2
2
R
+
τ
′
′
2
,
τ
′
=
P
2
x
1
x
2
,
τ
′
′
=
M
R
J
,
M
=
P
L
+
x
2
2
,
R
=
x
2
2
4
+
x
1
+
x
3
2
2
,
J
=
2
2
x
1
x
2
x
2
2
4
+
x
1
+
x
3
2
2
,
σ
x
→
=
6
P
L
x
4
x
3
2
,
δ
x
→
=
6
P
L
3
E
x
3
2
x
4
P
c
x
→
=
4.013
E
x
3
2
x
4
6
/
36
L
2
1
-
x
3
2
L
E
4
G
,
P
=
6000
l
b
,
L
=
14
in.
,
δ
m
a
x
=
0.25
in.
,
E
=
30
×
1
6
psi
,
G
=
12
×
1
0
6
psi
,
τ
m
a
x
=
13600
psi
,
σ
m
a
x
=
30000
psi
.Mirjalili tried to solve this problem using MFO [18] and GGSA [29, 36]. Coello Coello [37] and Deb [38, 39] employed GA, whereas Lee and Geem [40] used HS to solve this problem. Richardson’s random method, simplex method, Davidon-Fletcher-Powell, and Griffith and Stewart’s successive linear approximation are the mathematical approaches that have been adopted by Ragsdell and Philips [41] for this problem. The comparison results of the welded beam design problem are shown in Table 10.Table 10
Comparison results of the welded beam design problem.
Algorithm
Optimum variables
Optimal cost
H
l
t
B
LMFO
0.2020
3.3575
9.0938
0.2061
1.7165
MFO [18]
0.2057
3.4703
9.0364
0.2057
1.72452
GGSA [29, 36]
0.215917
3.314955
8.896195
0.215917
1.770829
GA (Coello Coello) [37]
N/A
N/A
N/A
N/A
1.8245
GA (Deb) [38]
N/A
N/A
N/A
N/A
2.3800
GA (Deb) [39]
0.2489
6.1730
8.1789
0.2533
2.4331
HS (Lee and Geem) [40]
0.2442
6.2231
8.2915
0.2443
2.3807
Random [41]
0.4575
4.7313
5.0853
0.6600
4.1185
Simplex [41]
0.2792
5.6256
7.7512
0.2796
2.5307
David [41]
0.2434
6.2552
8.2915
0.2444
2.3841
APPROX [41]
0.2444
6.2189
8.2915
0.2444
2.3815The results of Table10 show that the LMFO algorithm is able to find the best optimal design compared to other algorithms. The results of LMFO are closely followed by the MFO and GGSA algorithms.


### 5.2. Speed Reducer Design

The objective function of this problem is to minimize the total weight of the speed reducer as illustrated in Figure40 [42]. The variables x
1~x
7 denote the face width (b), module of teeth (m), number of teeth in the pinion (z), length of the first shaft between bearings (l
1), length of the second shaft between bearings (l
2), the diameter of first (d
1), and second shafts (d
2), respectively. The mathematical formulation of this problem can be summarized as follows:(14)
Minimize
f
x
→
=
0.7854
x
1
x
2
2
3.3333
x
3
2
+
14.9334
x
3
-
43.0934
-
1.508
x
1
x
6
2
+
x
7
2
+
7.4777
x
6
3
+
x
7
3
+
0.7854
x
4
x
6
2
+
x
5
x
7
2
Subject to
g
1
x
→
=
27
x
1
x
2
2
x
3
-
1
≤
0
,
g
2
x
→
=
397.5
x
1
x
2
2
x
3
2
-
1
≤
0
,
g
3
x
→
=
1.93
x
4
3
x
2
x
6
4
x
3
-
1
≤
0
,
g
4
x
→
=
1.93
x
5
3
x
2
x
7
5
x
3
-
1
≤
0
,
g
5
x
→
=
745
x
4
/
x
2
x
3
2
+
16.9
×
1
0
6
1
/
2
110
x
6
3
-
1
≤
0
,
g
6
x
→
=
745
x
5
/
x
2
x
3
2
+
157.5
×
1
0
6
1
/
2
85
x
7
3
-
1
≤
0
,
g
7
x
→
=
x
2
x
3
40
-
1
≤
0
,
g
8
x
→
=
5
x
2
x
1
-
1
≤
0
,
g
9
x
→
=
x
1
12
x
2
-
1
≤
0
,
g
10
x
→
=
1.5
x
6
+
1.9
x
4
-
1
≤
0
,
g
11
x
→
=
1.1
x
6
+
1.7
x
5
-
1
≤
0
,
where
2.6
≤
x
1
≤
3.6
,
0.7
≤
x
2
≤
0.8
,
17
≤
x
3
≤
28
,
7.3
≤
x
4
≤
8.3
,
7.3
≤
x
5
≤
8.3
,
2.9
≤
x
6
≤
3.9
,
5.0
≤
x
7
≤
5.5
.Figure 40
Structure of speed reducer design.This problem has also been popular among researchers and optimized in many studies. The heuristic algorithms that have been employed to optimize this problem are Akhtar et al. [43], Mezura-Montes et al. [44], CS [11, 45], HCPS [46], SCA [47], (μ
+
λ) ES [5, 48], and ABC [9, 49]. The results of this problem are provided in Table 11. According to this table, the LMFO and HCPS algorithms can find a design with the minimum weight for this problem.Table 11
Comparison results of the speed reducer design problem.
Algorithm
Optimal values for variables
Optimum weight
b
x
1
m
x
2
z
x
3
l
1
x
4
l
2
x
5
d
1
x
6
d
2
x
7
LMFO
3.50411
0.7
17
7.3
7.33342
3.37164
5.28916
2994.656
Akhtar et al. [43]
3.506122
0.700006
17
7.549126
7.85933
3.365576
5.289773
3008.08
Mezura-Montes et al. [44]
3.506163
0.700831
17
7.460181
7.962143
3.3629
5.3090
3025.005
CS [11, 45]
3.5015
0.7
17
7.6050
7.8181
3.3520
5.2875
3000.981
HCPS [46]
3.5
0.7
17
7.3
7.71532
3.350215
5.286654
2994.47107
SCA [47]
3.5
0.7
17
7.327602
7.715321
3.350267
5.286655
2994.744241
(μ
+
λ) ES [5, 48]
3.499999
0.699999
17
7.3
7.8
3.350215
5.286683
2996.348094
ABC [9, 49]
3.499999
0.7
17
7.3
7.8
3.350215
5.287800
2997.058412


## 5.1. Welded Beam Design

The objective is to evaluate the optimal fabrication cost of a welded beam as shown in Figure39 [35]. The constraints of the beam are shear stress (τ), bending stress in the beam (θ), buckling load on the bar (P
c), end deflection of the beam (δ), and side constraints.Figure 39
Structure of welded beam design.This problem has four variables that are thickness of weld (h), length of attached part of bar (l), the height of the bar (t), and thickness of the bar (b), respectively. This problem is formulated as follows:(13)
Consider
x
→
=
x
1
,
x
2
,
x
3
,
x
4
=
h
,
l
,
t
,
b
,
Minimize
f
x
→
=
1.10471
x
1
2
x
2
+
0.04811
x
3
x
4
14.0
+
x
2
,
Subject to
g
1
x
→
=
τ
x
→
-
τ
m
a
x
≤
0
,
g
2
x
→
=
σ
x
→
-
σ
m
a
x
≤
0
,
g
3
x
→
=
δ
x
→
-
δ
m
a
x
≤
0
,
g
4
x
→
=
x
1
-
x
4
≤
0
,
g
5
x
→
=
P
-
P
c
x
→
≤
0
,
g
6
x
→
=
0.125
-
x
1
≤
0
,
g
7
x
→
=
1.10471
x
1
2
+
0.04811
x
3
x
4
14.0
+
x
2
-
5.0
≤
0
,
Variable range
0.1
≤
x
1
≤
2
,
0.1
≤
x
2
≤
10
,
0.1
≤
x
3
≤
10
,
0.1
≤
x
4
≤
2
,
Where
τ
x
→
=
τ
′
2
+
2
τ
′
τ
′
′
x
2
2
R
+
τ
′
′
2
,
τ
′
=
P
2
x
1
x
2
,
τ
′
′
=
M
R
J
,
M
=
P
L
+
x
2
2
,
R
=
x
2
2
4
+
x
1
+
x
3
2
2
,
J
=
2
2
x
1
x
2
x
2
2
4
+
x
1
+
x
3
2
2
,
σ
x
→
=
6
P
L
x
4
x
3
2
,
δ
x
→
=
6
P
L
3
E
x
3
2
x
4
P
c
x
→
=
4.013
E
x
3
2
x
4
6
/
36
L
2
1
-
x
3
2
L
E
4
G
,
P
=
6000
l
b
,
L
=
14
in.
,
δ
m
a
x
=
0.25
in.
,
E
=
30
×
1
6
psi
,
G
=
12
×
1
0
6
psi
,
τ
m
a
x
=
13600
psi
,
σ
m
a
x
=
30000
psi
.Mirjalili tried to solve this problem using MFO [18] and GGSA [29, 36]. Coello Coello [37] and Deb [38, 39] employed GA, whereas Lee and Geem [40] used HS to solve this problem. Richardson’s random method, simplex method, Davidon-Fletcher-Powell, and Griffith and Stewart’s successive linear approximation are the mathematical approaches that have been adopted by Ragsdell and Philips [41] for this problem. The comparison results of the welded beam design problem are shown in Table 10.Table 10
Comparison results of the welded beam design problem.
Algorithm
Optimum variables
Optimal cost
H
l
t
B
LMFO
0.2020
3.3575
9.0938
0.2061
1.7165
MFO [18]
0.2057
3.4703
9.0364
0.2057
1.72452
GGSA [29, 36]
0.215917
3.314955
8.896195
0.215917
1.770829
GA (Coello Coello) [37]
N/A
N/A
N/A
N/A
1.8245
GA (Deb) [38]
N/A
N/A
N/A
N/A
2.3800
GA (Deb) [39]
0.2489
6.1730
8.1789
0.2533
2.4331
HS (Lee and Geem) [40]
0.2442
6.2231
8.2915
0.2443
2.3807
Random [41]
0.4575
4.7313
5.0853
0.6600
4.1185
Simplex [41]
0.2792
5.6256
7.7512
0.2796
2.5307
David [41]
0.2434
6.2552
8.2915
0.2444
2.3841
APPROX [41]
0.2444
6.2189
8.2915
0.2444
2.3815The results of Table10 show that the LMFO algorithm is able to find the best optimal design compared to other algorithms. The results of LMFO are closely followed by the MFO and GGSA algorithms.


## 5.2. Speed Reducer Design

The objective function of this problem is to minimize the total weight of the speed reducer as illustrated in Figure40 [42]. The variables x
1~x
7 denote the face width (b), module of teeth (m), number of teeth in the pinion (z), length of the first shaft between bearings (l
1), length of the second shaft between bearings (l
2), the diameter of first (d
1), and second shafts (d
2), respectively. The mathematical formulation of this problem can be summarized as follows:(14)
Minimize
f
x
→
=
0.7854
x
1
x
2
2
3.3333
x
3
2
+
14.9334
x
3
-
43.0934
-
1.508
x
1
x
6
2
+
x
7
2
+
7.4777
x
6
3
+
x
7
3
+
0.7854
x
4
x
6
2
+
x
5
x
7
2
Subject to
g
1
x
→
=
27
x
1
x
2
2
x
3
-
1
≤
0
,
g
2
x
→
=
397.5
x
1
x
2
2
x
3
2
-
1
≤
0
,
g
3
x
→
=
1.93
x
4
3
x
2
x
6
4
x
3
-
1
≤
0
,
g
4
x
→
=
1.93
x
5
3
x
2
x
7
5
x
3
-
1
≤
0
,
g
5
x
→
=
745
x
4
/
x
2
x
3
2
+
16.9
×
1
0
6
1
/
2
110
x
6
3
-
1
≤
0
,
g
6
x
→
=
745
x
5
/
x
2
x
3
2
+
157.5
×
1
0
6
1
/
2
85
x
7
3
-
1
≤
0
,
g
7
x
→
=
x
2
x
3
40
-
1
≤
0
,
g
8
x
→
=
5
x
2
x
1
-
1
≤
0
,
g
9
x
→
=
x
1
12
x
2
-
1
≤
0
,
g
10
x
→
=
1.5
x
6
+
1.9
x
4
-
1
≤
0
,
g
11
x
→
=
1.1
x
6
+
1.7
x
5
-
1
≤
0
,
where
2.6
≤
x
1
≤
3.6
,
0.7
≤
x
2
≤
0.8
,
17
≤
x
3
≤
28
,
7.3
≤
x
4
≤
8.3
,
7.3
≤
x
5
≤
8.3
,
2.9
≤
x
6
≤
3.9
,
5.0
≤
x
7
≤
5.5
.Figure 40
Structure of speed reducer design.This problem has also been popular among researchers and optimized in many studies. The heuristic algorithms that have been employed to optimize this problem are Akhtar et al. [43], Mezura-Montes et al. [44], CS [11, 45], HCPS [46], SCA [47], (μ
+
λ) ES [5, 48], and ABC [9, 49]. The results of this problem are provided in Table 11. According to this table, the LMFO and HCPS algorithms can find a design with the minimum weight for this problem.Table 11
Comparison results of the speed reducer design problem.
Algorithm
Optimal values for variables
Optimum weight
b
x
1
m
x
2
z
x
3
l
1
x
4
l
2
x
5
d
1
x
6
d
2
x
7
LMFO
3.50411
0.7
17
7.3
7.33342
3.37164
5.28916
2994.656
Akhtar et al. [43]
3.506122
0.700006
17
7.549126
7.85933
3.365576
5.289773
3008.08
Mezura-Montes et al. [44]
3.506163
0.700831
17
7.460181
7.962143
3.3629
5.3090
3025.005
CS [11, 45]
3.5015
0.7
17
7.6050
7.8181
3.3520
5.2875
3000.981
HCPS [46]
3.5
0.7
17
7.3
7.71532
3.350215
5.286654
2994.47107
SCA [47]
3.5
0.7
17
7.327602
7.715321
3.350267
5.286655
2994.744241
(μ
+
λ) ES [5, 48]
3.499999
0.699999
17
7.3
7.8
3.350215
5.286683
2996.348094
ABC [9, 49]
3.499999
0.7
17
7.3
7.8
3.350215
5.287800
2997.058412


## 6. Results and Discussion

In this paper, an improved version of MFO algorithm based on Lévy-flight strategy, which is named as LMFO, is proposed. In order to benchmark the performance of LMFO, nineteen unconstrained benchmark functions and two constrained engineering design problems were conducted.According to the values of Best, Worst, Mean, and Std andp values in Section 4, the LMFO algorithm significantly outperforms others in terms of numerical optimization. There are several reasons of why LMFO algorithm did perform well on most of the test cases. First, Lévy-flight strategy: Lévy-flight can increase the diversity of the population and make the algorithm jump out of local optimum more effectively. This approach is helpful to make LMFO faster and more robust than MFO. Second, update mechanism of moths: in this mechanism, moths are required to update their positions with respect to the best recent feasible flames. Therefore, this approach promotes exploration of promising feasible regions and is the main reason of the superiority of the LMFO algorithm. Third, Quicksort method is utilized in LMFO algorithm. These are the reasons why LMFO performs better than other algorithms at the end of the results section. Another finding in the results is the poor performance of ABC, BA, and DA. These three algorithms belong to the class of swarm-based algorithms. In contrary to evolutionary algorithms, there is no mechanism for significant abrupt movements in the search space and this is likely to be the reason for the poor performance of ABC, BA, and DA.As we can see in Section4, the LMFO has been demonstrated to perform better than or highly competitive with the other algorithms. The advantages of LMFO involve performing simply and have few parameters to regulate. The work here proves the LMFO to be robust, powerful, and effective over all types of benchmark functions. Benchmark evaluating is a good way for testing the performance of the metaheuristic algorithms, but it also has some limitations. For example, different tuning parameter values in the optimization methods might lead to significant differences in their performance. Also, benchmark test may arrive at fully different conclusions if the termination criterion changes. If we change the population size or the number of iterations, we might draw a different conclusion.In Section5, the results show that MFO outperforms other algorithms in the majority of real case studies. Since the search space of these problems is unknown, these results are strong evidences for the applicability of LMFO in solving real problems. Due to the constrained nature of the case studies, in addition, it can be stated that the LMFO algorithm is able to optimize search spaces with infeasible regions as well. This is due to the update mechanism of moths, in which they are required to update their positions with respect to the best recent feasible flames. Therefore, this approach is the main reason of the superiority of the LMFO algorithm.In our study, nineteen benchmark functions have been applied to evaluate the performance of LMFO. We also test our proposed method on the real-world engineering problems. Moreover, we will compare LMFO with other optimization algorithms.


## 7. Conclusion and Future Works

Due to the limited performance of MFO, Lévy-flight strategy has been introduced into the standard MFO to develop a novel Lévy-flight moth-flame optimization algorithm for optimization problems. As shown in Section4, LMFO is very efficient with an almost exponential convergence rate and the results were compared to a wide range of algorithms for verification. This observation is based on the comparison of LMFO with other algorithms to solve optimization problems. The proposed algorithm proved its superior performance on nineteen benchmark functions in terms of enhanced convergence speed and modified avoidance of local minima. This paper also identified and discussed the reasons for poor performances of other algorithms. It was observed that the swarm-based algorithms suffer from low exploration, whereas the LMFO does not.Furthermore, this paper also considers solving two classical engineering problems by using the LMFO algorithm. The high level of exploration and exploitation of this algorithm were the motivations for this study. The comparative results in Section5 show that the LMFO algorithm has high performance on challenging constrained problems with unknown spaces. In this work, the LMFO makes an attempt at taking merits of the MFO and Lévy-flight in order to avoid local optimal. With both techniques combined, LMFO can balance exploration and exploitation and effectively solve complex problems and real-world engineering problems.For future works, two research directions can be recommended. Firstly, we are going to apply the LMFO to solve more real-world engineering problems. Secondly, it is recommended to develop binary and multiobjective versions of the MFO algorithm.



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*Source: 1423930-2016-08-09.xml*