# Deep-Mining Backtracking Search Optimization Algorithm Guided by Collective Wisdom

Authors: Zheng Li; Zhongbo Hu; Yongfei Miao; Zenggang Xiong; Xinlin Xu; Canyun Dai
Journal: Mathematical Problems in Engineering
 (2019)

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## Abstract

The backtracking search optimization algorithm (BSA) is a recently proposed evolutionary algorithm with simple structure and well global exploration capability, which has been widely used to solve optimization problems. However, the exploitation capability of the BSA is poor. This paper proposes a deep-mining backtracking search optimization algorithm guided by collective wisdom (MBSAgC) to improve its performance. The proposed algorithm develops two learning mechanisms, i.e., a novel topological opposition-based learning operator and a linear combination strategy, by deeply mining the winner-tendency of collective wisdom. The topological opposition-based learning operator guides MBSAgC to search the vertices in a hypercube about the best individual. The linear combination strategy contains a difference vector guiding individuals learning from the best individual. In addition, in order to balance the overall performance, MBSAgC simulates the clusterity-tendency strategy of collective wisdom to develop another difference vector in the above linear combination strategy. The vector guides individuals to learn from the mean value of the current generation. The performance of MBSAgC is tested on CEC2005 benchmark functions (including 10-dimension and 30-dimension), CEC2014 benchmark functions, and a test suite composed of five engineering design problems. The experimental results of MBSAgC are very competitive compared with those of the original BSA and state-of-the-art algorithms.


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## Body

## 1. Introduction

In the real world, there are many complex problems that cannot be solved by traditional optimization algorithms. The emergence of global optimization algorithms overcomes the shortcomings of traditional optimization techniques to some extent. Global optimization algorithms mainly include two categories: one is swarm intelligence algorithms (SIs), and the other is evolutionary algorithms (EAs). SIs generally simulate the intelligent behavior of groups of creatures, such as particle swarm optimization (PSO) [1], the artificial bee colony (ABC) algorithm [2], brain storm optimization (BSO) algorithm [3–7], and ant colony optimization (ACO) [8]. EAs are inspired by the concepts of evolution in nature, like genetic algorithm (GA) [9], differential evolution (DE) [10–12], covariance matrix adaptation evolution strategy (CMAES) [13], and backtracking search optimization algorithm (BSA) [14].The BSA was first introduced by Civicioglu in 2013, which is a population-based evolutionary algorithm for solving real-valued optimization problems. Unlike other evolutionary algorithms, the BSA has a memory for storing a population from a randomly chosen previous generation. The memory allows the BSA to take an advantage of experiences gained from the previous evolution process. Meanwhile, the BSA possesses two more advantages of a simple structure and only one control parameter (mixrate). These advantages make it become a promising newcomer among evolutionary algorithms. And so far, it has been successfully applied to digital image processing [15, 16], power systems [17–19], energy and environmental management [20–22], artificial neural networks [23–26], induction motor [27, 28], antenna arrays [29, 30], and so on.However, the BSA has a big shortcoming that its local exploitation capability is very poor. As a result, many improvement research studies have been developed to overcome the drawback. The major modified research studies include the following five categories: (1) modified BSA algorithms by developing hybridization mechanisms [31–39], (2) modified BSA algorithms by developing novel initialization mechanisms [40–46], (3) modified BSA algorithms by developing novel reproduction (mutation or crossover) mechanisms [47–55], (4) modified BSA algorithms by developing novel selection mechanisms [56–60], and (5) modified BSA algorithms by developing novel control mechanisms for parameters (including the control factor F which is randomly generated in the original BSA) [61–69].The reproduction mechanism is the core content of the BSA. The mutation operator of the original BSA is determined by Gauss coefficient of variation and historical population, so it has a good global exploration capability in the early stage. However, due to the lack of guidance of favorable factors, the convergence speed of the algorithm is relatively slow. Many researchers have modified the reproduction mechanisms of the BSA to improve the performance of the BSA. Vitayasak et al. [48] modified the BSA in three mechanisms: (i) applied a control mechanism after the crossover process, (ii) changed the process of generating mapped value, and (iii) combined (i) and (ii). Li et al. [4] modified the mutation process of the BSA by introducing the optimal learning evolutionary process. Zhao et al. [50] proposed an improved BSA method which combined with the differential evolution algorithm and the breeder genetic algorithm mutation operator for constrained optimization problems. Das et al. [51] combined BSA mutation operator with DE current−to−best1 mutation strategy for interference suppression of linear antenna arrays. Chen et al. [52] developed a learning guidance strategy to modify the BSA mutation strategy. And two years later, Chen et al. [53] proposed a new mutation operator with knowledge learning and adaptive control parameter. Zhao et al. [54] introduced population control factor and optimal learning strategy into the mutation process of the BSA to improve the performance of algorithm. Yu et al. [55] proposed a multiple learning strategy to replace reproduction mechanism of the original BSA. These modified reproduction mechanisms all had satisfactory results in the BSA.Inspired from the winner-tendency behavior and clusterity-tendency behavior of collective wisdom, this paper deeply mines the guided information of the best individual (as a winner) to improve the exploitation capability of the BSA and uses the mean value of individuals (as a clustering center) to balance the algorithmic overall performance. Along the motivation, this paper proposes a modified BSA, deep-mining backtracking search optimization algorithm guided by collective wisdom (MBSAgC). The main contributions of the proposed algorithm are as follows.(i)
Deeply Mine the Winner-Tendency of Collective Wisdom. MBSAgC applies the winner-tendency behavior to design two strategies, i.e., a novel topological opposition-based learning and a linear combination, which generate offspring around the best individual of a current generation to enhance the exploitation capability. Especially, the topological opposition-based learning operator, which is the core innovation of the proposed algorithm, guides the algorithm to search the vertices in a hypercube about the best individual. Compared with the original opposition-based learning strategy, the new operator broadens its search region around the best individual.(ii)
Appropriately Apply the Clusterity-Tendency of Collective Wisdom. For balancing the overall capability, MBSAgC also adds a learning strategy in the above linear combination. The strategy can guide individuals to learn from the mean value of the current generation.(iii)
Keep the Simple Structure of the Original BSA. MBSAgC calculates only one function evaluation (FE) for an individual per generation and also do not add any parameter needed to set by users. It still keeps the simple structure of the original BSA.The remainder of this paper is organized as follows. The basic BSA is introduced in Section2. Section 3 describes in detail the proposed MBSAgC. Section 4 analyzes and discusses the novel topological opposition-based learning operator and the searching mechanism of MBSAgC. Simulation experiments are then performed in Section 5. Finally, conclusions are given in Section 6.


## 2. Basic BSA

BSA is one of population-based evolutionary algorithms. It consists of the following five basic operators: initialization, selection-I, mutation, crossover, and selection-II.


### 2.1. Initialization

BSA generates an initial populationP and an old population OldP by the following formula:(1)Pi,j∼ULowj,Upj,OldPi,j∼ULowj,Upj,where i=1,2,…,N, N is the population size, and j=1,2…,D, D is the problem dimension. U represents the uniform distribution, Lowj and Upj are the lower boundary and upper boundary of the jth dimension of the ith individual, respectively.


### 2.2. Selection-I

The BSA always implements selection-I at the beginning of each iteration. This process aims to select a new OldP which is used for calculating the search direction. And the OldP is reselected through “if-then” rule as follows:(2)ifa<bthenOldP=P,wherea,b∼U0,1.Botha and b are random numbers between 0 and 1. To ensure that the BSA has a memory, this operation reselects a population to be a new OldP. After this, the OldP is randomly permuted by using the following formula:(3)OldP=permutingOldP.


### 2.3. Mutation

The operation of mutation is used to generate the initial form of the trial populationT, and the operator of this process is described as follows:(4)Mut=P+F⋅OldP−P,where F is the control factor which controls the amplitude of the search direction; the value F=3⋅randn, where randn is a random number from 0 to 1 that submits to standardized normal distribution.


### 2.4. Crossover

The final form of the trial populationT is constructed by the crossover process of the BSA. There are two strategies in this operation, and the BSA uses two strategies to achieve the selection of individual elements in each iteration. Both strategies generate a different N⋅D map (binary integer-value matrix) to select the elements of individuals that have to be manipulated.One strategy uses mix-rate (mix-rate parameter) to control the number of elements of individuals that will be handled by usingmix−rate⋅randr⋅D (max-rate = 1 and randr is a random number on 0 to 1). The other strategy uses randi (D) to operate the only one element which is randomly selected from individuals, where randi is a random integration on 0 to D.Both strategies are used equiprobably to handle the elements of individuals through the “if-then” rule: ifmapi,j=1, then, T is updated with Ti,j=Pi,j. Otherwise, T is updated with Ti,j=Muti,j, where i∈1,2,3,…,N and j∈1,2,3,…,D.At the end of crossover process, if some individuals inT have overflowed the lower boundary or upper boundary, they will need to be regenerated by using Equation (1).


### 2.5. Selection-II

In this state, we select individuals with higher fitness values from populationT and P according to a greedy selection mechanism, which is used to update the population P. If Pi fitness value is better than Ti, then Pi is retained, whereas Pi is replaced by Ti. This process is described as follows:(5)Pi=Ti,if fitnessTi<fitnessPi,Pi,otherwise.


## 2.1. Initialization

BSA generates an initial populationP and an old population OldP by the following formula:(1)Pi,j∼ULowj,Upj,OldPi,j∼ULowj,Upj,where i=1,2,…,N, N is the population size, and j=1,2…,D, D is the problem dimension. U represents the uniform distribution, Lowj and Upj are the lower boundary and upper boundary of the jth dimension of the ith individual, respectively.


## 2.2. Selection-I

The BSA always implements selection-I at the beginning of each iteration. This process aims to select a new OldP which is used for calculating the search direction. And the OldP is reselected through “if-then” rule as follows:(2)ifa<bthenOldP=P,wherea,b∼U0,1.Botha and b are random numbers between 0 and 1. To ensure that the BSA has a memory, this operation reselects a population to be a new OldP. After this, the OldP is randomly permuted by using the following formula:(3)OldP=permutingOldP.


## 2.3. Mutation

The operation of mutation is used to generate the initial form of the trial populationT, and the operator of this process is described as follows:(4)Mut=P+F⋅OldP−P,where F is the control factor which controls the amplitude of the search direction; the value F=3⋅randn, where randn is a random number from 0 to 1 that submits to standardized normal distribution.


## 2.4. Crossover

The final form of the trial populationT is constructed by the crossover process of the BSA. There are two strategies in this operation, and the BSA uses two strategies to achieve the selection of individual elements in each iteration. Both strategies generate a different N⋅D map (binary integer-value matrix) to select the elements of individuals that have to be manipulated.One strategy uses mix-rate (mix-rate parameter) to control the number of elements of individuals that will be handled by usingmix−rate⋅randr⋅D (max-rate = 1 and randr is a random number on 0 to 1). The other strategy uses randi (D) to operate the only one element which is randomly selected from individuals, where randi is a random integration on 0 to D.Both strategies are used equiprobably to handle the elements of individuals through the “if-then” rule: ifmapi,j=1, then, T is updated with Ti,j=Pi,j. Otherwise, T is updated with Ti,j=Muti,j, where i∈1,2,3,…,N and j∈1,2,3,…,D.At the end of crossover process, if some individuals inT have overflowed the lower boundary or upper boundary, they will need to be regenerated by using Equation (1).


## 2.5. Selection-II

In this state, we select individuals with higher fitness values from populationT and P according to a greedy selection mechanism, which is used to update the population P. If Pi fitness value is better than Ti, then Pi is retained, whereas Pi is replaced by Ti. This process is described as follows:(5)Pi=Ti,if fitnessTi<fitnessPi,Pi,otherwise.


## 3. Deep-Mining BSA Guided by Collective Wisdom (MBSAgC)

The collective wisdom here refers to the behaviors of a group of people learning from each other. Generally speaking, people tend to learn from the best individual. This behavior could be called awinner-tendency in this paper. Meanwhile, there are lots of compromises when people make serious decisions. These compromises could be called a clusterity-tendency behavior.Inspired by the above two behaviors of collective wisdom, we designed three strategies to improve the overall performance of the BSA. The winner-tendency behavior is simulated to design two strategies, including a novel topological opposition-based learning operator and a linear combination strategy, which guide population to learn from the best individual of the current generation. The two strategies tend to improve the exploitation capability of the BSA. The rest of the strategy, which simulates the clusterity-tendency behavior to balance the algorithmic search, is designed to learn from the mean value of the current generation. And then, the three strategies are used to design an improved BSA, deep-mining backtracking search optimization algorithm guided by collective wisdom, abbreviated as MBSAgC.Unlike the basic BSA, the proposed MBSAgC executes a function-optimized task by looping thetopological opposition-based learning operator and an improved mutation operator for updating the population. The improved mutation operator is designed by integrating the last two strategies into the original mutation of the BSA.


### 3.1. Topological Opposition-Based Learning Operator (TOBL)

The topological opposition-based learning operator (TOBL), generated by deeply mining the winner-tendency behavior, is identified by the process of learning from the location of the best individual of a current population. It not only does not increase the computational overhead for function evaluations but also expands the searching space of opposite points. The proposed TOBL is defined as follows (the detailed analysis of TOBL will be given in the next section).Definition 1.
(topological opposition-based learning operator). LetPi=Pi1,Pi2,…,Pij,…,PiD be a point (vector) in a search space of D-dimensions, where Pij∈Lowj,Upj, its opposite point OPi=OPi1,OPi2,…,OPij,…,OPiD is(6)OPij=Lowj+Upj−Pij,j=1,2,…,D.
Then the topological opposite pointTPi=TPi1,TPi2,…,TPij,…,TPiD of Pi can be defined as follows:(7)TPij=OPij,if absPbest,j−Pij>absPbest,j−OPij,Pij,otherwise,where Pbest,j is the jth dimension of the best individual Pbest of a population P.


### 3.2. Improved Mutation Operator

In the mutation of the basic BSA, the position of an individual is updated by the historical information and current information. The offspring do not get any knowledge from the best individual or the mean position of the population. MBSAgC proposes an improved mutation operator which guides individuals to learn from the best and the mean individuals of a population. The individualsIMuti using the improved mutation operator is described as follows:(8)IMuti=Pi+F⋅OldPi−Pi+randr1⋅Pbest−Pi+randr2⋅Pmean−Pi,where Pmean is the mean individual of the population; randr1andrandr2 denote two differential random numbers on 0 to 1.The pseudocode for MBSAgC is shown in Algorithm1.Algorithm 1:The pseudocode of MBSAgC.
Input: N,D,Gmax,Low,Up
Output: Globalminimum
(1)
%Gmax is the maximum number of iterations. Globalminimum records the current minimum.
(2)
%Initialization process ofPand OldP
(3)
for i=1:N do
(4)
for j=1:D do
(5)
Pi,j=rand⋅Upj−Lowj+lowj OldPi,j=rand⋅Upj−Lowj+Lowj
(6)
end
(7)
end
(8)
for g=1:Gmax do
(9)
%Topological opposition-based learning (TOBL)
(10)
for i=1:N do
(11)
for j=1:D do
(12)
OPij=Lowj+Upj−Pij
(13)
if absPbest,j−Pij>absOPbest,j−OPij then
(14)
Pij=OPij
(15)
end
(16)
end
(17)
end
(18)
%Selection-I
(19)
for a=randr;b=randrdo
(20)
if a<b then
(21)
OldP=P
(22)
end
(23)
end
(24)
OldP=permutingOldP
(25)
%Improved mutation
(26)
for i=1:Ndo
(27)
IMuti=Pi+F⋅OldPi−Pi+randr⋅Pbest−Pi+randr⋅Pmean−Pi
(28)
end
(29)
%Crossover
(30)
if c<dc,d∼U0,1 then
(31)
for i=1:N do
(32)
mapi,u1:mix⋅rate⋅D=0u=permuting1,2,3,…,D
(33)
end
(34)
end
(35)
fori=1:N do
(36)
mapi,randiD=0
(37)
end
(38)
T=IMut
(39)
for i=1:N do
(40)
forj=1:D do
(41)
if mapi,j=1 then
(42)
Ti,j=Pi,j
(43)
end
(44)
end
(45)
end
(46)
%Boundary control mechanism
(47)
for i=1:N do
(48)
forj=1:D do
(49)
if Ti,j<Lowi,j or Ti,j>Upi,j then
(50)
Ti,j=Lowj+randr⋅Upj−Lowj
(51)
end
(52)
end
(53)
end
(54)
%Selection-II
(55)
for i=1:N do
(56)
if fitnessTi<fitnessPi then
(57)
fitnessPi=fitnessTi
(58)
Pi=Ti
(59)
end
(60)
end
(61)
if fitnessPbest<globalminmum then
(62)
globalminmum=fitnessPbest
(63)
end
(64)
end


## 3.1. Topological Opposition-Based Learning Operator (TOBL)

The topological opposition-based learning operator (TOBL), generated by deeply mining the winner-tendency behavior, is identified by the process of learning from the location of the best individual of a current population. It not only does not increase the computational overhead for function evaluations but also expands the searching space of opposite points. The proposed TOBL is defined as follows (the detailed analysis of TOBL will be given in the next section).Definition 1.
(topological opposition-based learning operator). LetPi=Pi1,Pi2,…,Pij,…,PiD be a point (vector) in a search space of D-dimensions, where Pij∈Lowj,Upj, its opposite point OPi=OPi1,OPi2,…,OPij,…,OPiD is(6)OPij=Lowj+Upj−Pij,j=1,2,…,D.
Then the topological opposite pointTPi=TPi1,TPi2,…,TPij,…,TPiD of Pi can be defined as follows:(7)TPij=OPij,if absPbest,j−Pij>absPbest,j−OPij,Pij,otherwise,where Pbest,j is the jth dimension of the best individual Pbest of a population P.


## 3.2. Improved Mutation Operator

In the mutation of the basic BSA, the position of an individual is updated by the historical information and current information. The offspring do not get any knowledge from the best individual or the mean position of the population. MBSAgC proposes an improved mutation operator which guides individuals to learn from the best and the mean individuals of a population. The individualsIMuti using the improved mutation operator is described as follows:(8)IMuti=Pi+F⋅OldPi−Pi+randr1⋅Pbest−Pi+randr2⋅Pmean−Pi,where Pmean is the mean individual of the population; randr1andrandr2 denote two differential random numbers on 0 to 1.The pseudocode for MBSAgC is shown in Algorithm1.Algorithm 1:The pseudocode of MBSAgC.
Input: N,D,Gmax,Low,Up
Output: Globalminimum
(1)
%Gmax is the maximum number of iterations. Globalminimum records the current minimum.
(2)
%Initialization process ofPand OldP
(3)
for i=1:N do
(4)
for j=1:D do
(5)
Pi,j=rand⋅Upj−Lowj+lowj OldPi,j=rand⋅Upj−Lowj+Lowj
(6)
end
(7)
end
(8)
for g=1:Gmax do
(9)
%Topological opposition-based learning (TOBL)
(10)
for i=1:N do
(11)
for j=1:D do
(12)
OPij=Lowj+Upj−Pij
(13)
if absPbest,j−Pij>absOPbest,j−OPij then
(14)
Pij=OPij
(15)
end
(16)
end
(17)
end
(18)
%Selection-I
(19)
for a=randr;b=randrdo
(20)
if a<b then
(21)
OldP=P
(22)
end
(23)
end
(24)
OldP=permutingOldP
(25)
%Improved mutation
(26)
for i=1:Ndo
(27)
IMuti=Pi+F⋅OldPi−Pi+randr⋅Pbest−Pi+randr⋅Pmean−Pi
(28)
end
(29)
%Crossover
(30)
if c<dc,d∼U0,1 then
(31)
for i=1:N do
(32)
mapi,u1:mix⋅rate⋅D=0u=permuting1,2,3,…,D
(33)
end
(34)
end
(35)
fori=1:N do
(36)
mapi,randiD=0
(37)
end
(38)
T=IMut
(39)
for i=1:N do
(40)
forj=1:D do
(41)
if mapi,j=1 then
(42)
Ti,j=Pi,j
(43)
end
(44)
end
(45)
end
(46)
%Boundary control mechanism
(47)
for i=1:N do
(48)
forj=1:D do
(49)
if Ti,j<Lowi,j or Ti,j>Upi,j then
(50)
Ti,j=Lowj+randr⋅Upj−Lowj
(51)
end
(52)
end
(53)
end
(54)
%Selection-II
(55)
for i=1:N do
(56)
if fitnessTi<fitnessPi then
(57)
fitnessPi=fitnessTi
(58)
Pi=Ti
(59)
end
(60)
end
(61)
if fitnessPbest<globalminmum then
(62)
globalminmum=fitnessPbest
(63)
end
(64)
end


## 4. Analysis and Discussion of MBSAgC

In fact, there are many heuristic algorithms simulating the collective wisdom in different forms. The representative algorithms include collective decision optimization algorithm (CDOA) [70, 71], teaching-learning-based optimization (TLBO) [72], learning backtracking search optimization algorithm (LBSA) [52], and so on. Unlike these algorithms, MBSAgC proposes the TOBL operator to deeply mine the winner-tendency of collective wisdom and designs an improved mutation operator to merge multiple learning strategies. Especially, the TOBL operator broadens its search region around the best individual compared with the original opposition-based learning. Next, we analyze the search mechanism of TOBL and discuss the numerical performance of MBSAgC and BSA on several simple functions.


### 4.1. Search Mechanism of TOBL

The opposition-based learning (OBL) was originally proposed by Tizhoosh [73] in 2005. It has already attracted a recognizable interest. So far, about 500 papers have been published on the OBL concept. There are three articles summarizing the developmental stages of OBL. Al-Qunaieer et al. [74] firstly summarized the applications of OBL up to 2011. In 2012, Xu et al. [75] provided summative research on the basic concepts, theories, and applications of OBL in intelligent algorithms, typical fields, and a number of challenges since 2005. Mahdavi et al. [76] offered a global overview of research works on theories, developments, and applications of OBL from 2012 to 2018.Inspired by the above research works on OBL, this paper proposes a new improved strategy, topological OBL (TOBL). Compared with OBL, TOBL holds two advantages on computational overhead and searching space.(i)
Not Increase the Computational Overhead. As shown in Equation (7), TOBL evaluates the two candidates by comparing the distances between the two candidates and the current best individual on each corresponding dimension. Unlike OBL and the other improved OBL operators, TOBL thereby avoids the computational overhead for function evaluations.(ii)
Expand the Searching Space. Figure 1 shows the sketch of TOBL on 1-dimension, 2-dimension, and 3-dimension, respectively. Take the 3-dimension sketch as example; P11,P12,P13 is an original point. Its opposite point P11∗,P12∗,P13∗ can be obtained according to Equation (6). However, according to Equation (7), the eight vertices of the cube are all the potential topological opposite points of TOBL. At this case with the position of Pbest as shown in Figure 1(c), the topological opposite points of P11,P12,P13 is P11,P12∗,P13 (marked in blue). In fact, TOBL has 2D potential points, while OBL has only one for each original point (where D denotes the dimension of a search space).Figure 1
Individuals and their potential topological opposite points of TOBL. (a) 1-dimension. (b) 2-dimension (c) 3-dimension.
(a)
(b)
(c)Next, we simulate the original OBL and TOBL operators on the Sphere (2-dimension) and Rosenbrock (2-dimension) functions. As shown in Figure2, the simulative results demonstrate that (1) TOBL operator cannot optimize symmetric functions (Sphere) as with the original OBL and (2) for asymmetric functions (Rosenbrock), TOBL can obtain better optimization effectiveness than the original OBL.Figure 2
Simulation experiments of OBL and TOBL on two functions. (a) Sphere. (b) Rosenbrock.
(a)
(b)


### 4.2. Contrastive Analysis between MBSAgC and BSA

In order to investigate the effectiveness of the modified strategies, we test the performance of MBSAgC and BSA on five unconstrained benchmarks with different characteristics. The five benchmark functions are Sphere, Schwefel 1.2, Rosenbrock, Rastrigin, and Griewank. The details of five functions are given in Table1, where C, N, D, and Max denote characteristics, population sizes, dimensions, and maximum fitness evaluation times (FEs), respectively. “U,” “M,” “S,” and “N” are unimodal, multimodal, separable, and nonseparable, respectively. And in this section, as the iteration increases, several tests are implemented through comparing MBSAgC and BSA on the change of population P and the best function value, respectively. In the experiment, we use the same size of population (NP), dimension (D), and maximum (Max). The details of five functions and the parameter settings are given in Table 1. Figure 3 shows that the best function values for MBSAgC and BSA vary with FEs of the five benchmark functions, and Figure 4 represents that the means of population variances in MBSAgC and BSA vary with FEs.Table 1
Five unconstrained benchmarks and their corresponding parameter settings.
Function
Range
C
N
D
Max
Formulation
Griewank
(−600, 600)
M N
100
30
10,000 × D
f
x
=
1
/
4000
∑
i
=
1
D
x
i
2
−
∏
i
=
1
D
cos
x
i
/
i
+
1
Rastrigin
(−5.12, 5.12)
M S
100
30
10,000 × D
f
x
=
∑
i
=
1
D
x
i
2
−
100
cos
2
π
x
i
+
10
Rosenbrock
(−30, 30)
U N
100
30
10,000 × D
f
x
=
∑
i
=
1
D
−
1
100
x
i
+
1
−
x
i
2
2
+
x
i
−
1
2
Schwefel 1.2
(−100, 100)
U N
100
30
10,000 × D
f
x
=
∑
i
=
1
D
∑
j
=
1
i
x
j
2
Sphere
(−100, 100)
U S
100
30
10,000 × D
f
x
=
∑
i
=
1
D
x
i
2Figure 3
Best-value curves of MBSAgC and BSA. (a) Griewank. (b) Rastrigin. (c) Rosenbrock. (d) Schwefell 1.2. (e) Sphere.
(a)
(b)
(c)
(d)
(e)Figure 4
Mean-value curves of MBSAgC and BSA. (a) Griewank. (b) Rastrigin. (c) Rosenbrock. (d) Schwefell 1.2. (e) Sphere.
(a)
(b)
(c)
(d)
(e)According to the best function value in Figure3, it can be seen that with the increase of the number of evaluations, the best function value of MBSAgC decreases faster than that of the BSA. It illustrates that the convergence rate of MBSAgC is faster than that of BSA. And Figure 3 shows that MBSAgC finds better function value than the BSA in all five benchmark functions. Figure 4 shows the change in population variances of MBSAgC and BSA. We can see from Figure 4 that in the early iteration process, with the increase of evaluation times, the population variance of MBSAgC and BSA began to decrease, and compared with BSA, MBSAgC decreases faster and tended to zero earlier. The results show that MBSAgC has a faster rate of convergence than BSA.


## 4.1. Search Mechanism of TOBL

The opposition-based learning (OBL) was originally proposed by Tizhoosh [73] in 2005. It has already attracted a recognizable interest. So far, about 500 papers have been published on the OBL concept. There are three articles summarizing the developmental stages of OBL. Al-Qunaieer et al. [74] firstly summarized the applications of OBL up to 2011. In 2012, Xu et al. [75] provided summative research on the basic concepts, theories, and applications of OBL in intelligent algorithms, typical fields, and a number of challenges since 2005. Mahdavi et al. [76] offered a global overview of research works on theories, developments, and applications of OBL from 2012 to 2018.Inspired by the above research works on OBL, this paper proposes a new improved strategy, topological OBL (TOBL). Compared with OBL, TOBL holds two advantages on computational overhead and searching space.(i)
Not Increase the Computational Overhead. As shown in Equation (7), TOBL evaluates the two candidates by comparing the distances between the two candidates and the current best individual on each corresponding dimension. Unlike OBL and the other improved OBL operators, TOBL thereby avoids the computational overhead for function evaluations.(ii)
Expand the Searching Space. Figure 1 shows the sketch of TOBL on 1-dimension, 2-dimension, and 3-dimension, respectively. Take the 3-dimension sketch as example; P11,P12,P13 is an original point. Its opposite point P11∗,P12∗,P13∗ can be obtained according to Equation (6). However, according to Equation (7), the eight vertices of the cube are all the potential topological opposite points of TOBL. At this case with the position of Pbest as shown in Figure 1(c), the topological opposite points of P11,P12,P13 is P11,P12∗,P13 (marked in blue). In fact, TOBL has 2D potential points, while OBL has only one for each original point (where D denotes the dimension of a search space).Figure 1
Individuals and their potential topological opposite points of TOBL. (a) 1-dimension. (b) 2-dimension (c) 3-dimension.
(a)
(b)
(c)Next, we simulate the original OBL and TOBL operators on the Sphere (2-dimension) and Rosenbrock (2-dimension) functions. As shown in Figure2, the simulative results demonstrate that (1) TOBL operator cannot optimize symmetric functions (Sphere) as with the original OBL and (2) for asymmetric functions (Rosenbrock), TOBL can obtain better optimization effectiveness than the original OBL.Figure 2
Simulation experiments of OBL and TOBL on two functions. (a) Sphere. (b) Rosenbrock.
(a)
(b)


## 4.2. Contrastive Analysis between MBSAgC and BSA

In order to investigate the effectiveness of the modified strategies, we test the performance of MBSAgC and BSA on five unconstrained benchmarks with different characteristics. The five benchmark functions are Sphere, Schwefel 1.2, Rosenbrock, Rastrigin, and Griewank. The details of five functions are given in Table1, where C, N, D, and Max denote characteristics, population sizes, dimensions, and maximum fitness evaluation times (FEs), respectively. “U,” “M,” “S,” and “N” are unimodal, multimodal, separable, and nonseparable, respectively. And in this section, as the iteration increases, several tests are implemented through comparing MBSAgC and BSA on the change of population P and the best function value, respectively. In the experiment, we use the same size of population (NP), dimension (D), and maximum (Max). The details of five functions and the parameter settings are given in Table 1. Figure 3 shows that the best function values for MBSAgC and BSA vary with FEs of the five benchmark functions, and Figure 4 represents that the means of population variances in MBSAgC and BSA vary with FEs.Table 1
Five unconstrained benchmarks and their corresponding parameter settings.
Function
Range
C
N
D
Max
Formulation
Griewank
(−600, 600)
M N
100
30
10,000 × D
f
x
=
1
/
4000
∑
i
=
1
D
x
i
2
−
∏
i
=
1
D
cos
x
i
/
i
+
1
Rastrigin
(−5.12, 5.12)
M S
100
30
10,000 × D
f
x
=
∑
i
=
1
D
x
i
2
−
100
cos
2
π
x
i
+
10
Rosenbrock
(−30, 30)
U N
100
30
10,000 × D
f
x
=
∑
i
=
1
D
−
1
100
x
i
+
1
−
x
i
2
2
+
x
i
−
1
2
Schwefel 1.2
(−100, 100)
U N
100
30
10,000 × D
f
x
=
∑
i
=
1
D
∑
j
=
1
i
x
j
2
Sphere
(−100, 100)
U S
100
30
10,000 × D
f
x
=
∑
i
=
1
D
x
i
2Figure 3
Best-value curves of MBSAgC and BSA. (a) Griewank. (b) Rastrigin. (c) Rosenbrock. (d) Schwefell 1.2. (e) Sphere.
(a)
(b)
(c)
(d)
(e)Figure 4
Mean-value curves of MBSAgC and BSA. (a) Griewank. (b) Rastrigin. (c) Rosenbrock. (d) Schwefell 1.2. (e) Sphere.
(a)
(b)
(c)
(d)
(e)According to the best function value in Figure3, it can be seen that with the increase of the number of evaluations, the best function value of MBSAgC decreases faster than that of the BSA. It illustrates that the convergence rate of MBSAgC is faster than that of BSA. And Figure 3 shows that MBSAgC finds better function value than the BSA in all five benchmark functions. Figure 4 shows the change in population variances of MBSAgC and BSA. We can see from Figure 4 that in the early iteration process, with the increase of evaluation times, the population variance of MBSAgC and BSA began to decrease, and compared with BSA, MBSAgC decreases faster and tended to zero earlier. The results show that MBSAgC has a faster rate of convergence than BSA.


## 5. Numerical Experiments

In order to verify the performance of the proposed MBSAgC, two numerical experiments are executed in this section. The first experiment is tested on CEC2005 benchmark functions [77] (10-dimension and 30-dimension) and CEC2014 benchmark functions [78] (30-dimension). To prove the effectiveness of MBSAgC, in this experiment, LBSA, BSA, SaDE [79], CLPSO [80], PSOFIPS [81], TLBO, ETLBO [82], PSOFDR [83], BSAISA [68], OBSA, and some state-of-the-art algorithms are used as comparison algorithms.The second experiment is tested on five engineering constrained optimization problems. They are the three-bar truss design problem (TTP), pressure vessel design problem (PVP), tension/compression spring design problem (TCSP), welded beam design problem (SRP), and speed reducer design problem (SRP), respectively. The results in the tables show the best function value (Best), the worst function value (Worst), the mean function value(Mean), the standard deviation (Std), and the fitness evaluation times (FEs). The FEs is used to evaluate the performance of MBSAgC. And we compare the results of MBSAgC with other well-known algorthms.(1)
Parameter Setting. All the experiments are implemented on the MATLAB 2012a. For the first experiment, each comparison algorithm runs 30 times independently. The size of population (N) is set to 50, and the dimension variable (D) is 10 and 30, respectively. In addition, the stopping criterion for all algorithms depends on the maximum FEs, the maximum FEs is set to 5000∗D. For five engineering design problems, each actual problem needs to be run 50 times separately because those problems have different constraint conditions. Their parameters are almost different. For the TTP, we set N=20, maximum iteration Tmax is 1000; for the PVP, we set N=20, Tmax=3000; for the TCSP, we set N=20, Tmax=3000; for the WBP, we set N=20, Tmax=3000; and for the SRP, we set N=20, Tmax=2000.


### 5.1. Experiments on Benchmark Functions

#### 5.1.1. Experimental Results

(1) On 10-Dimension Functions in CEC2005. Table 2 shows the simulation results of 10-dimension functions in CEC2005. Table 2 compares the mean, standard deviations, and the best solution of the ten algorithms. With respect to the best solution, the results of Table 2 show that the proposed algorithm MBSAgC ranks first in functions F1, F2, F4, F6, F9, F12, F15, F18–F20, F22, F24, and F25. LBSA is superior to other nine algorithms on function F3. For function F3, all compared algorithms could not find the theoretical optimal solution. The rank of MBSAgC in terms of best solution is eighth, it is worse than most of others, and the best solution of LBSA is closest to the theoretical optimal value, and the standard deviation of LBSA is also the smallest among ten algorithms. The best solution of PSOFDR is superior to other algorithms in functions F10, F11, and F14. The best solution of BSAISA algorithm is better than other nine algorithms in function F7. For function F1, all of the compared algorithms get the global optimal solution. For function F2, the best solution of CLPSO and OBSA is worse than the other eight algorithms. For function F4, BSAISA, OBSA, BSA, and CLPSO could not converge to the global optimum. MBSAgC can find the theoretical optimal solution on functions F1, F2, F4, F6, F9, F12, and F15. In Table 2, in terms of the optimal solutions of twenty-five functions, thirteen functions for the proposed algorithm MBSAgC rank first. The average ranking of the proposed algorithm is 2.76; moreover, MBSAgC ranks first in the best solution. To sum up, the performance of MBSAgC is competitive with other algorithms.Table 2
Comparative results of ten algorithms for 10D problems from CEC2005.
F
AL
LBSA
BSA
CLPSO
PSOFIPS
TLBO
ETLBO
PSOFDR
BSAISA
OBSA
MBSAgC
F1
Mean
−450.000
−450.000
−450.000
−450.000
−450.000
−450.000
−439.674
−450.000
−450.000
−450.000
Std.
0.000
0.000
0.000
0.000
0.000
0.000
32.653
0.000
1.847E − 11
0.000
Best
−450.000
−450.000
−450.000
−450.000
−450.000
−450.000
−450.000
−450.000
−450.000
−450.000
Rank
1
1
1
1
1
1
1
1
1
1
F2
Mean
−450.000
−449.997
−449.676
−450.000
−450.000
−450.000
−450.000
−449.968
−446.527
−450.000
Std.
0.000
0.005
0.177
0.000
0.000
0.000
0.000
0.044
3.336
6.91E − 06
Best
−450.000
−450.000
−449.906
−450.000
−450.000
−450.000
−450.000
−450.000
−449.791
−450.000
Rank
1
1
2
1
1
1
1
1
3
1
F3
Mean
5652.945
21518.124
695430.984
114431.841
70188.530
132295.917
143065.100
31699.067
427940.36
176254.16
Std.
7686.985
18993.641
263113.140
54898.334
47419.217
98983.396
156627.508
17717.113
471417.61
126171.69
Best
251.786
3316.003
333184.289
48691.653
22503.152
23119.940
6576.186
9211.109
35974.764
39533.675
Rank
1
2
10
9
5
6
3
4
7
8
F4
Mean
−450.000
−449.800
−445.693
−450.000
−450.000
−450.000
−438.829
−444.970
−430.129
−449.999
Std.
0.000
0.306
2.683
0.000
0.000
0.000
23.550
9.926
19.385
0.001
Best
−450.000
−449.994
−449.361
−450.000
−450.000
−450.000
−450.000
−449.972
−448.989
−450.000
Rank
1
2
5
1
1
1
1
4
3
1
F5
Mean
−310.000
−310.000
−310.000
−233.009
−310.000
−310.000
−310.000
−301.311
−228.735
−309.328
Std.
0.000
0.000
0.000
33.727
0.000
0.000
0.000
7.416
64.078
1.034
Best
−310.000
−310.000
−310.000
−289.340
−310.000
−310.000
−310.000
−309.052
−304.344
−309.932
Rank
1
1
1
5
1
1
1
3
4
2
F6
Mean
390.548
392.105
391.599
394.658
392.592
392.167
399.415
391.651
404.196
397.153
Std.
1.112
2.741
1.610
0.405
3.041
2.313
18.471
2.043
11.599
7.459
Best
390.001
390.091
390.348
393.841
390.000
390.000
390.002
390.005
390.000
390.000
Rank
2
5
6
8
1
1
3
4
1
1
F7
Mean
1087.046
1087.082
1087.208
−179.810
1087.046
1087.046
1087.288
−179.935
−179.816
−179.816
Std.
0.000
0.076
0.045
0.098
0.000
0.000
0.197
0.043
0.078
0.086
Best
1087.046
1087.046
1087.046
−179.949
1087.046
1087.046
1087.266
−179.986
−179.933
−179.958
Rank
5
5
5
3
5
5
6
1
4
2
F8
Mean
−119.643
−119.681
−119.624
−119.640
−119.628
−119.648
−119.714
−119.616
−119.607
−119.796
Std.
0.053
0.085
0.081
0.090
0.087
0.066
0.075
0.070
0.072
0.070
Best
−119.728
−119.812
−119.792
−119.851
−119.831
−119.628
−119.800
−119.747
−119.722
−119.741
Rank
8
3
6
1
2
10
4
7
9
5
F9
Mean
−330.000
−330.000
−330.000
−329.333
−322.342
−322.737
−327.314
−330.000
−329.343
−329.877
Std.
0.000
0.000
0.000
0.723
2.928
2.696
1.487
0.000
0.844
0.312
Best
−330.000
−330.000
−330.000
−330.000
−326.020
−327.015
−329.005
−330.000
−330.000
−330.000
Rank
1
1
1
1
4
3
2
1
1
1
F10
Mean
−323.493
−320.479
−316.698
−317.409
−319.885
−321.513
−319.259
−319.005
−310.290
−317.531
Std.
1.987
3.542
3.125
5.428
4.132
3.593
6.612
3.914
9.254
6.979
Best
−325.724
−324.562
−322.465
−324.188
−325.025
−327.015
−328.010
−325.505
−323.468
−324.812
Rank
3
7
10
8
5
2
1
4
9
6
F11
Mean
95.188
94.889
95.319
93.633
94.002
94.632
91.962
94.957
95.001
94.381
Std.
0.746
1.196
0.572
0.536
1.652
0.997
1.023
0.848
1.271
1.627
Best
93.737
92.408
94.425
92.922
91.059
92.619
90.702
92.744
92.209
91.286
Rank
9
5
10
8
2
6
1
7
4
3
F12
Mean
−444.398
−438.290
−363.666
−422.531
1445.385
794.693
−203.253
−448.681
−197.091
−294.114
Std.
16.545
16.021
65.389
47.468
5027.194
2860.501
507.838
13.907
645.215
421.269
Best
−459.985
−455.477
−431.128
−455.719
−460.000
−460.000
−459.496
−460.000
−458.383
−460.000
Rank
2
6
7
5
1
1
3
1
4
1
F13
Mean
−129.666
−129.697
−129.564
−128.810
−129.196
−129.044
−129.352
−129.577
−129.353
−129.342
Std.
0.073
0.118
0.118
0.146
0.165
0.256
0.271
0.067
0.151
0.175
Best
−129.830
−129.932
−129.769
−129.109
−129.447
−129.384
−129.745
−129.728
−129.663
−129.625
Rank
2
1
3
10
8
9
4
5
6
7
F14
Mean
−296.902
−296.694
−296.591
−297.170
−297.215
−297.081
−297.241
−296.757
−296.517
−296.627
Std.
0.207
0.205
0.151
0.306
0.249
0.459
0.685
0.215
0.253
0.276
Best
−297.288
−297.108
−296.948
−297.642
−297.566
−297.701
−298.570
−297.395
−297.095
−297.545
Rank
7
8
10
3
4
2
1
6
9
5
F15
Mean
120.030
124.232
140.582
209.829
450.644
401.382
387.275
123.356
170.949
3.654
Std.
0.089
12.918
28.900
99.415
141.887
156.564
203.967
11.180
99.357
200.365
Best
120.000
120.000
120.002
120.460
195.411
197.129
176.891
120.000
120.003
120.000
Rank
1
1
2
4
6
7
5
1
3
1
F16
Mean
223.754
231.537
257.700
232.105
228.617
230.096
254.799
228.345
245.791
235.776
Std.
3.450
10.701
14.508
10.777
9.831
9.972
29.896
5.584
10.049
9.911
Best
218.507
209.614
223.542
221.355
219.015
217.668
222.865
215.613
221.217
218.609
Rank
4
1
10
8
6
3
9
2
7
5
F17
Mean
241.073
249.588
269.953
251.164
246.701
228.562
247.905
242.396
272.287
261.650
Std.
11.036
10.448
16.875
19.976
19.773
5.540
21.479
10.483
29.610
18.040
Best
227.764
237.089
252.680
216.798
218.722
222.809
219.166
227.236
230.774
230.880
Rank
6
9
10
1
2
4
3
5
7
8
F18
Mean
603.702
393.335
702.404
580.649
709.009
715.153
864.345
735.541
831.072
861.593
Std.
253.504
155.192
86.522
277.920
237.366
240.232
96.241
202.699
157.059
121.229
Best
310.000
310.000
559.128
310.000
310.000
393.632
700.558
310.075
310.000
310.000
Rank
1
1
4
1
1
3
5
2
1
1
F19
Mean
572.111
404.878
752.056
442.020
756.348
716.366
864.003
688.951
711.459
842.334
Std.
278.437
162.270
46.800
214.463
201.804
227.717
131.102
190.598
241.647
160.075
Best
310.000
310.000
656.476
310.000
417.269
366.021
698.725
312.483
310.009
310.000
Rank
1
1
6
1
5
4
7
3
2
1
F20
Mean
442.654
410.210
666.374
748.635
791.997
896.631
681.122
807.355
595.677
863.810
Std.
211.159
177.868
75.428
251.178
186.662
164.780
117.945
208.837
102.402
162.465
Best
310.000
310.000
566.636
310.000
310.000
486.451
693.240
310.000
314.640
310.000
Rank
1
1
4
1
1
3
5
1
2
1
F21
Mean
910.002
832.592
846.708
830.619
1110.454
931.239
1212.545
944.930
1002.404
1028.386
Std.
171.591
86.672
28.555
222.507
367.780
360.290
400.425
196.868
269.495
302.654
Best
660.019
585.919
790.540
660.000
660.000
660.000
660.000
613.723
660.000
660.000
Rank
4
1
5
3
3
3
3
2
3
3
F22
Mean
1074.270
1079.861
1141.724
1131.082
1150.209
1135.155
1177.217
924.494
961.467
1075.162
Std.
145.718
148.046
7.787
3.327
29.625
24.736
79.315
232.730
231.505
170.075
Best
660.000
660.000
1131.022
1125.586
1116.253
1110.458
1109.829
660.000
660.000
660.000
Rank
1
1
6
5
4
3
2
1
1
1
F23
Mean
829.609
918.677
906.039
1069.522
1178.115
1237.380
1431.577
998.441
1275.131
1176.097
Std.
56.624
1.797
42.468
179.455
232.518
257.118
315.479
149.953
146.070
188.499
Best
785.173
914.095
785.173
919.468
919.468
919.468
919.468
785.173
919.468
919.468
Rank
1
2
1
3
3
3
3
1
3
3
F24
Mean
460.000
460.000
696.247
588.505
460.000
460.000
1126.804
460.000
500.000
460.000
Std.
0.000
0.000
252.248
86.471
0.000
0.000
239.476
0.000
103.723
000.000
Best
460.000
460.000
460.211
460.001
460.000
460.000
760.000
460.000
460.000
460.000
Rank
1
1
3
2
1
1
4
1
1
1
F25
Mean
1079.394
1080.375
1079.655
612.785
1078.221
1076.067
1339.389
460.000
660.000
460.000
Std.
2.258
2.634
2.151
80.691
2.460
20650
217.394
0.000
164.002
000.000
Best
1075.267
1074.953
1077.376
460.000
1074.200
1071.563
1071.868
460.000
460.000
460.000
Rank
6
5
7
1
4
2
3
1
1
1
Average rank
2.84
2.88
5.4
3.76
3.08
3.4
3.24
2.76
3.84
2.76(2) On 30-Dimension Functions in CEC2005. In this section, the experiment of 30-dimension functions is carried out in CEC2005. The mean, standard deviations, and the best solutions of the error values are compared. In Table 3, all the algorithms are obtained after 30 independent runs; in addition, the optimal solution of the algorithm error value is sorted. From Table 3, it can be seen that MBSAgC is superior to other ten algorithms on F6, F7, F11, and F12 at D = 30. MBSAgC also performs best on functions F20, F21, and F23–F25. LBSA is the first ranking for the functions F1, F9, F20, F21, F23, and F24. SaDE is better than all other algorithms on functions F3, F4, F5, F10, and F16. OBSA ranks first on functions F18, F19, F21, and F23–F25. CLPSO and PSOFDR perform worse than the other nine algorithms on functions F21 and F24. The average ranking of MBSAgC is 2.56; it is also ranked first in all compared algorithms. The statistical results in Table 3 indicate that MBSAgC has a better global performance than the other algorithms, and MBSAgC has a better capability to get the best solution than other compared algorithms in 30 runs. According to the results in Table 3, it is obvious to find that MBSAgC performs better than all other comparison algorithms on 30-dimension functions in CEC2005.Table 3
Comparative results of ten algorithms for 30D problems from CEC2005.
F
AL
LBSA
BSA
SaDE
CLPSO
PSOFIPS
TLBO
ETLBO
PSOFDR
BSAISA
OBSA
MBSAgC
F1
Mean
0.000E + 00
3.995E − 16
0.000E + 00
4.839E − 10
1.057E − 06
5.836E − 26
3.138E − 27
9.33E + 02
1.270E − 13
6.253E − 14
6.442E − 14
Std.
0.000E + 00
2.267E − 16
0.000E + 00
1.409E − 10
3.274E − 07
2.769E − 26
1.661E − 27
1.632E + 03
3.559E − 14
1.734E − 14
1.965E − 14
Best
0.000E + 00
1.173E − 16
0.000E + 00
3.244E − 10
7.480E − 07
2.640E − 26
1.379E − 27
0.000E + 00
5.684E − 14
5.684E − 14
5.684E − 14
Rank
1
4
1
6
7
3
2
1
5
5
5
F2
Mean
4.995E − 02
2.644E + 03
8.734E − 03
6.355E + 03
4.560E + 02
7.480E − 05
6.632E − 05
9.374E + 01
2.545E + 02
2.005E + 03
2.010E + 00
Std.
4.852E − 02
2.038E + 02
4.026E − 03
7.050E + 02
6.497E + 01
3.424E − 05
7.342E − 05
8.638E + 01
9.556E + 01
1.270E + 03
1.614E + 00
Best
1.714E − 02
2.450E + 03
4.213E − 03
5.676E + 03
3.818E + 02
3.527E − 05
2.493E − 06
8.972E − 03
8.585E + 01
4.243E + 02
3.203E − 01
Rank
5
10
3
11
8
2
1
4
7
9
6
F3
Mean
9.240E + 05
3.069E + 06
4.385E + 05
4.814E + 07
1.265E + 07
1.344E + 06
1.331E + 06
1.757E + 06
2.331E + 06
7.732E + 06
1.974E + 06
Std.
3.486E + 05
2.791E + 05
1.529E + 05
5.129E + 06
1.170E + 06
6.881E + 05
6.881E + 05
6.131E + 05
7.157E + 05
3.513E + 06
7.342E + 05
Best
7.127E + 05
2.784E + 06
3.380E + 05
4.296E + 07
1.138E + 07
6.838E + 05
6.838E + 05
1.084E + 06
9.462E + 05
1.313E + 06
7.049E + 05
Rank
4
8
1
10
9
2
2
6
5
7
3
F4
Mean
7.097E + 02
9.404E + 03
4.769E + 02
1.613E + 04
3.019E + 03
2.128E + 03
8.611E + 02
7.308E + 02
9.381E + 03
1.759E + 04
3.427E + 03
Std.
4.298E + 02
7.848E + 02
7.790E + 02
1.800E + 03
8.707E + 02
2.915E + 03
7.453E + 02
2.258E + 02
2.302E + 03
5.206E + 03
2.863E + 03
Best
2.659E + 02
8.510E + 03
2.337E + 01
1.505E + 04
2.016E + 03
1.736E + 02
4.067E + 02
5.912E + 02
4.585E + 03
8.304E + 03
1.902E + 02
Rank
4
10
1
11
7
2
5
6
8
9
3
F5
Mean
3.029E + 03
3.334E + 03
2.427E + 03
4.315E + 03
3.104E + 03
4.546E + 03
4.161E + 03
6.156E + 03
3.906E + 03
5.079E + 03
3.551E + 03
Std.
7.939E + 02
1.767E + 02
2.731E + 02
9.405E + 02
1.055E + 02
3.140E + 02
4.601E + 02
1.750E + 03
4.184E + 02
9.557E + 02
6.652E + 02
Best
2.501E + 03
3.130E + 03
2.141E + 03
3.523E + 03
3.009E + 03
4.197E + 03
3.678E + 03
5.077E + 03
2.692E + 03
3.310E + 03
2.457E + 03
Rank
3
6
1
8
5
10
9
11
4
7
2
F6
Mean
2.958E + 01
7.803E + 01
7.807E + 01
3.974E + 01
2.891E + 01
2.297E + 01
3.895E + 01
1.638E + 07
1.299E + 02
9.279E + 01
5.594E + 01
Std.
4.229E + 01
3.027E + 01
2.212E + 00
2.123E + 01
2.546E + 00
2.780E + 00
3.169E + 01
2.646E + 07
9.030E + 01
2.766E + 01
4.434E + 01
Best
3.069E + 00
4.494E + 01
7.557E + 01
1.529E + 01
2.714E + 01
2.109E + 01
2.207E + 01
2.290E + 02
2.895E + 01
2.495E + 01
4.900E − 01
Rank
2
9
10
3
7
4
5
11
8
6
1
F7
Mean
4.696E + 03
4.696E + 03
4.696E + 03
4.696E + 03
3.846E − 01
4.696E + 03
4.696E + 03
4.918E + 03
5.994E − 01
5.078E − 01
2.142E − 02
Std.
1.575E − 12
1.720E − 08
3.750E − 12
0.000E + 00
9.646E − 02
1.575E − 12
9.905E − 13
3.844E + 02
1.723E − 01
1.931E − 01
2.577E − 02
Best
4.696E + 03
4.696E + 03
4.696E + 03
4.696E + 03
2.859E − 01
4.696E + 03
4.696E + 03
4.696E + 03
2.429E − 01
1.760E − 01
3.856E − 07
Rank
5
5
5
5
4
5
5
5
3
2
1
F8
Mean
2.096E + 01
2.298E + 01
2.097E + 01
2.093E + 01
2.093E + 01
2.098E + 01
2.089E + 01
2.092E + 01
2.089E + 01
2.095E + 01
2.096E + 01
Std.
5.689E − 02
4.595E − 02
2.665E − 02
8.812E − 02
1.807E − 02
1.689E − 02
6.181E − 02
2.802E − 02
6.723E − 02
5.909E − 02
7.067E − 02
Best
2.091E + 01
2.093E + 01
2.095E + 01
2.085E + 01
2.092E + 01
2.097E + 01
2.084E + 01
2.089E + 01
2.072E + 01
2.082E + 02
2.076E + 01
Rank
6
8
9
4
7
10
3
5
1
11
2
F9
Mean
1.184E − 15
2.227E + 00
3.317E − 01
5.082E − 04
5.904E + 01
9.618E + 01
6.567E + 01
5.164E + 01
1.327E − 01
4.961E + 00
2.820E + 00
Std.
2.051E − 15
1.295E + 00
5.744E − 01
2.078E − 04
1.441E + 01
2.185E + 01
3.011E + 01
5.605E + 00
4.320E − 01
4.199E + 00
2.996E + 00
Best
0.000E + 00
1.134E + 00
0.000E + 00
2.719E − 04
4.420E + 01
8.258E + 01
4.179E + 01
4.754E + 01
1.432E − 11
1.294E − 01
1.526E − 02
Rank
1
6
1
3
8
10
7
9
2
5
4
F10
Mean
7.969E + 01
8.281E + 01
7.975E + 01
1.647E + 02
1.822E + 02
9.228E + 01
9.913E + 01
1.141E + 02
1.101E + 02
1.575E + 02
1.358E + 02
Std.
7.575E + 00
1.541E + 01
4.349E + 01
5.872E + 00
5.261E + 00
4.150E + 01
1.259E + 01
1.892E + 01
1.696E + 01
5.376E + 01
3.937E + 01
Best
7.131E + 01
6.963E + 01
3.022E + 01
1.608E + 02
1.788E + 02
5.771E + 01
8.743E + 01
9.451E + 01
6.792E + 01
8.846E + 01
6.676E + 01
Rank
6
5
1
10
11
2
7
9
4
8
3
F11
Mean
2.833E + 01
3.091E + 01
2.997E + 01
2.798E + 01
2.879E + 01
3.467E + 01
3.714E + 01
1.803E + 01
2.666E + 01
2.586E + 01
2.198E + 01
Std.
1.680E + 00
2.377E + 00
1.669E + 00
1.205E + 00
2.252E + 00
4.680E + 00
1.629E + 00
2.803E + 00
2.365E + 00
2.798E + 00
4.908E + 00
Best
2.661E + 01
2.843E + 01
2.840E + 01
2.685E + 01
2.625E + 01
2.930E + 01
3.526E + 01
1.551E + 01
2.058E + 01
1.913E + 01
1.280E + 01
Rank
6
9
8
7
5
10
11
2
4
3
1
F12
Mean
2.976E + 03
2.801E + 04
6.056E + 03
4.411E + 04
2.555E + 04
9.523E + 03
1.550E + 04
2.882E + 04
7.015E + 03
1.536E + 04
4.162E + 03
Std.
1.281E + 03
1.173E + 04
7.626E + 03
3.443E + 03
9.809E + 03
3.881E + 03
1.420E + 04
4.351E + 04
2.924E + 03
9.177E + 03
3.474E + 03
Best
1.698E + 03
1.447E + 04
1.008E + 03
4.022E + 04
1.962E + 04
6.257E + 03
3.727E + 03
3.259E + 03
1.585E + 03
3.634E + 03
1.998E + 02
Rank
4
9
2
11
10
8
7
5
3
6
1
F13
Mean
1.883E + 00
2.524E + 00
4.792E + 00
3.409E + 00
1.358E + 01
4.248E + 00
4.063E + 00
2.870E + 00
1.882E + 00
3.382E + 00
2.930E + 00
Std.
1.203E − 01
1.192E − 01
6.707E − 01
2.156E − 01
1.025E − 01
5.438E − 01
1.202E + 00
7.919E − 01
1.674E − 01
4.748E − 01
4.052E − 01
Best
1.771E + 00
2.408E + 00
4.113E + 00
3.162E + 00
1.347E + 01
3.912E + 00
2.682E + 00
2.293E + 00
1.403E + 00
2.314E + 00
2.149E + 00
Rank
2
6
10
8
11
9
7
4
1
5
3
F14
Mean
1.286E + 01
1.309E + 01
1.309E + 01
1.312E + 01
1.280E + 01
1.289E + 01
1.309E + 01
1.185E + 01
1.281E + 01
1.291E + 01
1.277E + 01
Std.
1.284E − 01
3.739E − 01
1.846E − 01
1.258E − 01
1.223E − 01
1.513E − 01
7.225E − 02
7.922E − 01
2.286E − 01
2.685E − 01
3.944E − 01
Best
1.276E + 01
1.266E + 01
1.288E + 01
1.298E + 01
1.266E + 01
1.278E + 01
1.303E + 01
1.112E + 01
1.234E + 01
1.229E + 01
1.126E + 01
Rank
6
5
8
9
5
7
10
1
4
3
2
F15
Mean
4.015E + 02
7.111E + 01
2.354E + 02
1.214E + 02
3.544E + 02
4.713E + 02
4.746E + 02
5.258E + 02
9.242E + 01
2.367E + 02
3.054E + 02
Std.
1.013E + 02
1.159E + 01
2.047E + 02
2.545E + 01
6.430E + 01
9.623E + 01
4.142E + 01
5.548E + 01
1.230E + 02
4.901E + 01
7.386E + 01
Best
3.000E + 02
6.421E + 01
6.213E + 00
9.610E + 01
3.007E + 02
3.977E + 02
4.268E + 02
4.831E + 02
2.946E − 06
2.000E + 02
8.141E + 00
Rank
7
4
2
5
8
9
10
11
1
6
3
F16
Mean
8.450E + 01
1.484E + 02
1.051E + 02
2.018E + 02
2.507E + 02
2.400E + 02
2.652E + 02
2.413E + 02
1.122E + 02
1.892E + 02
3.217E + 02
Std.
5.658E + 00
3.611E + 01
3.010E + 01
2.174E + 01
5.874E + 01
2.269E + 02
1.175E + 02
1.107E + 02
1.453E + 01
9.878E + 01
1.745E + 02
Best
7.797E + 01
1.240E + 02
7.058E + 01
1.775E + 02
2.087E + 02
1.078E + 02
1.912E + 02
1.245E + 02
7.278E + 01
1.075E + 02
1.033E + 02
Rank
3
7
1
9
11
6
10
8
2
5
4
F17
Mean
1.511E + 02
2.117E + 02
1.663E + 02
3.001E + 02
2.812E + 02
2.639E + 02
2.404E + 02
1.932E + 02
1.706E + 02
3.262E + 02
2.957E + 02
Std.
4.484E + 01
1.541E + 01
1.345E + 01
4.652E + 01
6.842E + 01
1.708E + 02
1.690E + 02
1.022E + 02
3.320E + 01
1.391E + 02
1.052E + 02
Best
1.198E + 02
1.964E + 02
1.508 + 02
2.574E + 02
2.388E + 02
1.399E + 02
1.323E + 02
1.267E + 02
1.183E + 02
1.840E + 02
1.258E + 02
Rank
2
9
7
11
10
6
5
4
1
8
3
F18
Mean
9.186E + 02
9.153E + 02
9.116E + 02
9.098E + 02
8.323E + 02
9.148E + 02
8.771E + 02
9.388E + 02
8.444E + 02
9.148E + 02
9.312E + 02
Std.
6.806E + 00
3.761E + 00
4.342E − 01
7.681E − 01
1.686E + 00
4.459E + 00
6.679E + 01
2.396E + 01
6.401E + 01
2.253E + 01
3.644E + 01
Best
9.108E + 02
9.119E + 02
9.111E + 02
9.090E + 02
8.310E + 02
9.105E + 02
8.000E + 02
9.187E + 02
8.000E + 02
2.000E + 02
8.000E + 02
Rank
6
8
7
4
3
5
2
9
2
1
2
F19
Mean
9.137E + 02
9.161E + 02
8.741E + 02
9.100E + 02
8.323E + 02
9.594E + 02
9.289E + 02
9.449E + 02
8.467E + 02
8.632E + 02
9.210E + 02
Std.
1.897E − 01
2.038E + 00
6.415E + 01
4.197E − 01
4.396E − 01
5.067E + 01
2.414E + 01
2.678E + 01
6.268E + 01
6.018E + 01
2.448E + 01
Best
9.136E + 02
9.147E + 02
8.000E + 02
9.096E + 02
8.318E + 02
9.201E + 02
9.903E + 02
9.172E + 02
8.000E + 02
2.000E + 02
8.000E + 02
Rank
5
6
2
4
3
8
9
7
2
1
2
F20
Mean
8.779E + 02
9.149E + 02
8.737E + 02
9.088E + 02
8.321E + 02
9.475E + 02
9.216E + 02
9.501E + 02
8.822E + 02
90197E + 02
9.220E + 02
Std.
6.752E + 01
2.841E + 00
6.383E + 01
9.501E − 01
9.107E − 01
3.759E + 01
8.330E + 00
2.351E + 01
6.391E + 01
9.503E + 00
3.591E + 01
Best
8.000E + 02
9.128E + 02
8.000E + 02
9.078E + 02
8.310E + 02
9.141E + 02
9.136E + 02
9.327E + 02
8.000E + 02
9.098E + 02
8.000E + 02
Rank
1
5
1
3
2
7
6
8
1
4
1
F21
Mean
5.000E + 02
5.000E + 02
5.000E + 02
6.508E + 02
5.000E + 02
9.551E + 02
5.000E + 02
9.728E + 02
6.097E + 02
5.973E + 02
7.584E + 02
Std.
1.969E − 13
1.842E − 13
0.000E + 00
1.376E + 02
3.922E − 06
3.941E + 02
8.247E − 13
1.750E + 02
2.499E + 02
2.291E + 02
2.791E + 02
Best
5.000E + 02
5.000E + 02
5.000E + 02
5.427E + 02
5.000E + 02
5.000E + 02
5.000E + 02
8.549E + 02
5.000E + 02
5.000E + 02
5.000E + 02
Rank
1
1
1
2
1
1
1
3
1
1
1
F22
Mean
9.372E + 02
9.709E + 02
9.275E + 02
9.189E + 02
5.290E + 02
9.220E + 02
9.904E + 02
8.860E + 02
1.002E + 03
1.039E + 03
1.014E + 03
Std.
1.765E + 01
8.997E + 00
4.853E + 00
2.153E + 01
2.608E − 01
3.630E + 01
3.054E + 01
2.439E + 01
2.722E + 01
4.147E + 01
0.340E + 01
Best
9.266E + 02
9.617E + 02
9.230E + 02
8.984E + 02
5.287E + 02
8.831E + 02
8.747E + 02
8.648E + 02
9.615E + 02
9.733E + 02
9.437E + 02
Rank
7
10
6
5
1
4
3
2
9
11
8
F23
Mean
5.342E + 02
5.342E + 02
5.342E + 02
6.655E + 02
5.342E + 02
1.052E + 03
1.226E + 02
1.030E + 03
6.437E + 02
7.727E + 02
6.864E + 02
Std.
4.661E − 04
1.987E − 05
1.186E − 04
5.669E + 01
2.583E − 04
2.102E + 02
3.965E + 01
1.329E + 02
2.395E + 02
2.544E + 02
2.445E + 02
Best
5.342E + 02
5.342E + 02
5.342E + 02
6.255E + 02
5.342E + 02
8.094E + 02
1.184E + 03
8.767E + 02
5.342E + 02
5.342E + 02
5.342E + 02
Rank
1
1
1
2
1
3
5
4
1
1
1
F24
Mean
2.000E + 02
2.000E + 02
2.000E + 02
9.489E + 02
2.183E + 02
2.000E + 02
2.000E + 02
9.579E + 02
2.000E + 02
2.676E + 02
2.000E + 02
Std.
0.000E + 00
1.897E − 12
3.481E − 14
1.994E + 00
9.720E − 01
4.702E − 11
3.212E − 12
2.510E + 01
3.585E − 06
2.527E + 02
8.407E − 13
Best
2.000E + 02
2.000E + 02
2.000E + 02
9.471E + 02
2.176E + 02
2.000E + 02
2.000E + 02
9.399E + 02
2.000E + 02
2.000E + 02
2.000E + 02
Rank
1
1
1
4
2
1
1
3
1
1
1
F25
Mean
9.938E + 02
1.016E + 03
9.951E + 02
1.008E + 03
2.070E + 02
1.006E + 03
1.052E + 03
1.064E + 03
2.000E + 02
3.418E + 02
2.000E + 02
Std.
1.593E + 00
8.040E + 00
4.148E + 00
1.268E + 01
1.213E + 01
2.774E + 01
4.316E + 01
2.582E + 01
2.545E − 06
3.618E + 02
1.445E − 13
Best
9.922E + 02
1.007E + 03
9.924E + 02
1.000E + 03
2.000E + 02
9.817E + 02
1.009E + 03
1.036E + 03
2.000E + 02
2.000E + 02
2.000E + 02
Rank
3
6
4
5
1
2
7
8
1
1
1
Average rank
3.68
6.32
3.76
6.4
5.88
5.44
5.6
5.84
3.24
5.04
2.56(3) On 30-Dimension Functions in CEC2014. There are 30 benchmark functions in CEC2014. It is difficult to get the global optimal solution because their global optimum is shifted. In this part, we compare the mean, standard deviations, and the best solution of the error values between the actual solution value and the theoretical optimal value and rank the best error value of each function, and the statistical results of each comparison algorithm are given in Table 4. For all 30 functions, MBSAgC outperforms the other nine algorithms for functions F4, F5, F14, F19, and F21. PSOFIPS performs best on functions F6, F29, and F30. PSOFDR is superior to other algorithms on functions F11, F15, and F16. CLPSO performs a little unsatisfactory on all functions except function F26. LBSA ranks first in terms of the best for functions F2, F3, F8, F9, F10, F13, F17, F20, and F22. CBSA performs better than other algorithms for functions F12, F23, F27, and F28. For function F7, in terms of the best, MBSAgC and LBSA obtain the same function value; they are better than other algorithms. All algorithms get the same value in function F26 except CLBSA. For function F24 and F25, CBSA, TLBO, ETLBO, and DGSTLBO obtain the same value, and they are better than other six algorithms. The average ranking of three comparative criterions is also listed in the table. As we can see, LBSA is the best in all three criterions; MBSAgC performs slightly worse in best and Std than LBSA, but it is better than other algorithms. Overall, MBSAgC’s rank is still very competitive in CEC2014.Table 4
Comparative results of ten algorithms for 30D problems from CEC2014.
F
AL
PSOFIPS
PSOFDR
CLPSO
CBSA
CLBSA
TLBO
ETLBO
DGSTLBO
LBSA
MBSAgC
F1
Mean
1.02E + 07
2.78E + 06
2.92E + 07
4.12E + 07
1.61E + 08
8.28E + 05
3.33E + 05
1.04E + 07
2.79E + 05
1.26E + 06
Std
2.88E + 06
1.98E + 06
4.33E + 06
1.19E + 07
6.60E + 07
7.89E + 05
2.02E + 05
8.61E + 06
1.65E + 05
1.12E + 06
Best
5.60E + 06
1.32E + 05
2.14E + 07
2.17E + 07
5.69E + 07
5.77E + 04
1.37E + 05
3.15E + 06
7.85E + 04
1.27E + 05
Rank
7
4
8
9
10
1
5
6
2
3
F2
Mean
1.13E + 04
1.53E + 08
1.26E + 03
3.20E + 08
2.43E + 10
1.44E + 02
1.19E + 02
4.59E + 06
4.83E − 14
3.51E + 01
Std
5.27E + 03
3.61E + 08
7.40E + 02
6.61E + 07
7.40E + 09
1.72E + 02
1.56E + 02
1.11E + 07
1.92E − 14
5.42E + 01
Best
3.63E + 03
4.60E + 01
2.76E + 02
1.75E + 08
1.21E + 10
2.82E + 01
3.29E − 01
1.48E + 03
2.84E − 14
1.14E − 13
Rank
8
5
6
9
10
4
3
7
1
2
F3
Mean
6.94E + 03
5.40E + 02
7.67E + 02
2.37E + 03
3.09E + 04
2.39E + 03
2.14E + 03
1.44E + 01
2.27E − 13
5.67E + 02
Std
5.03E + 03
6.05E + 02
1.20E + 03
1.05E + 03
9.52E + 03
1.77E + 03
1.02E + 03
1.68E + 01
1.00E − 13
1.46E + 03
Best
1.32E + 03
1.36E + 01
5.76E + 01
9.14E + 02
1.36E + 04
2.90E + 02
5.59E + 02
6.71E − 01
1.14E − 13
1.56E + 00
Rank
9
4
5
8
10
6
7
2
1
3
F4
Mean
2.67E + 01
9.50E + 01
1.16E + 02
1.56E + 02
2.46E + 03
9.70E + 01
7.53E + 01
1.46E + 02
4.23E + 01
5.20E + 01
Std
6.37E − 01
3.24E + 01
1.33E + 01
2.13E + 01
1.56E + 03
3.36E + 01
5.88E + 00
3.78E + 01
3.57E + 01
3.63E + 01
Best
2.53E + 01
6.66E + 01
9.56E + 01
1.13E + 02
6.39E + 02
6.74E + 01
6.79E + 01
8.79E + 01
1.73E − 03
2.59E − 04
Rank
3
4
8
9
10
5
6
7
2
1
F5
Mean
2.10E + 01
2.09E + 01
2.05E + 01
4.14E + 01
7.61E + 02
2.10E + 01
2.09E + 01
2.10E + 01
2.03E + 01
2.63E − 13
Std
5.71E − 02
1.22E − 01
4.58E − 02
9.01E + 00
3.65E + 02
5.23E − 02
7.24E − 02
4.34E − 02
3.42E − 02
1.00E − 13
Best
2.09E + 01
2.06E + 01
2.04E + 01
2.51E + 01
1.58E + 02
2.09E + 01
2.08E + 01
2.09E + 01
2.03E + 01
1.14E − 13
Rank
6
4
3
7
8
6
5
6
2
1
F6
Mean
6.19E + 00
7.67E + 00
1.70E + 01
9.93E + 01
7.24E + 02
1.54E + 01
1.59E + 01
1.67E + 01
8.37E + 00
3.83E + 01
Std
2.22E + 00
1.98E + 00
6.10E − 01
1.09E + 01
2.63E + 02
2.14E + 00
2.75E + 00
3.45E + 00
3.22E + 00
2.81E + 01
Best
2.75E + 00
4.63E + 00
1.60E + 01
7.26E + 01
3.05E + 02
1.17E + 01
1.01E + 01
1.23E + 01
5.25E + 00
5.80E + 00
Rank
1
2
8
9
10
6
5
7
3
4
F7
Mean
2.56E − 03
1.33E + 01
3.92E − 03
4.00E + 00
2.41E + 02
6.74E − 02
2.75E − 02
1.01E + 00
5.41E − 03
9.84E − 03
Std
6.75E − 03
1.31E + 01
1.61E − 03
6.84E − 01
7.28E + 01
8.69E − 02
2.23E − 02
1.50E + 00
8.26E − 03
1.50E − 02
Best
7.08E − 05
1.48E − 02
1.37E − 03
1.91E + 00
9.95E + 01
7.84E − 12
1.02E − 12
1.20E − 01
1.14E − 13
1.14E − 13
Rank
4
6
5
8
9
3
2
7
1
1
F8
Mean
6.60E + 01
3.48E + 01
5.86E − 01
1.86E + 01
1.90E + 02
7.37E + 01
8.00E + 01
7.67E + 01
1.14E − 13
6.70E + 00
Std
1.22E + 01
5.70E + 00
6.20E − 01
1.86E + 00
1.89E + 01
2.11E + 01
1.88E + 01
2.45E + 01
0.00E + 00
5.26E + 00
Best
4.32E + 01
2.19E + 01
2.65E − 04
1.36E + 01
1.49E + 02
5.07E + 01
4.48E + 01
3.48E + 01
1.14E − 13
1.37E + 00
Rank
7
5
2
4
10
9
8
6
1
3
F9
Mean
1.53E + 02
5.53E + 01
7.79E + 01
9.56E + 01
1.98E + 02
7.68E + 01
8.00E + 01
9.84E + 01
4.32E + 01
8.48E + 01
Std
1.61E + 01
1.54E + 01
1.49E + 01
9.56E + 00
2.47E + 01
1.08E + 01
1.42E + 01
3.08E + 01
7.61E + 00
2.00E + 01
Best
1.21E + 02
3.51E + 01
5.52E + 01
7.30E + 01
1.49E + 02
5.87E + 01
5.87E + 01
6.41E + 01
2.70E + 01
4.24E + 01
Rank
8
2
4
7
9
5
5
6
1
3
F10
Mean
1.87E + 03
9.70E + 02
2.50E + 01
1.35E + 02
4.78E + 03
1.72E + 03
1.57E + 03
2.39E + 03
1.08E + 01
1.67E + 02
Std
4.44E + 02
3.47E + 02
5.21E + 00
2.35E + 01
4.01E + 02
7.41E + 02
4.26E + 02
4.71E + 02
5.07E + 00
8.55E + 01
Best
1.35E + 03
3.70E + 02
1.83E + 01
7.75E + 01
3.97E + 03
4.64E + 02
9.52E + 02
1.59E + 03
5.12E + 00
4.43E + 01
Rank
8
5
2
4
10
6
7
9
1
3
F11
Mean
5.76E + 03
3.08E + 03
3.24E + 03
2.80E + 03
5.08E + 03
6.71E + 03
6.46E + 03
3.39E + 03
2.31E + 03
2.67E + 03
Std
3.43E + 02
6.19E + 02
2.18E + 02
2.58E + 02
5.45E + 02
3.92E + 02
4.46E + 02
5.45E + 02
3.17E + 02
3.86E + 02
Best
5.23E + 03
1.77E + 03
2.98E + 03
2.22E + 03
4.08E + 03
6.01E + 03
5.52E + 03
2.42E + 03
1.86E + 03
1.87E + 03
Rank
8
1
6
4
7
10
9
5
2
3
F12
Mean
2.62E + 00
8.12E − 01
5.42E − 01
4.72E − 01
2.19E + 00
2.64E + 00
2.60E + 00
2.75E + 00
4.18E − 01
5.76E − 01
Std
2.82E − 01
6.51E − 01
8.53E − 02
6.66E − 02
6.22E − 01
2.47E − 01
3.32E − 01
2.62E − 01
5.15E − 02
8.85E − 02
Best
1.97E + 00
3.12E − 01
4.12E − 01
2.44E − 01
6.09E − 01
2.13E + 00
1.93E + 00
2.48E + 00
3.62E − 01
3.87E − 01
Rank
8
2
5
1
6
9
7
10
3
4
F13
Mean
3.46E − 01
4.35E − 01
4.18E − 01
3.89E − 01
3.88E + 00
4.88E − 01
4.11E − 01
4.71E − 01
2.90E − 01
3.70E − 01
Std
3.30E − 02
1.07E − 01
7.14E − 02
5.27E − 02
5.51E − 01
1.14E − 01
6.86E − 02
1.13E − 01
4.34E − 02
7.93E − 02
Best
2.89E − 01
2.55E − 01
2.77E − 01
2.63E − 01
2.74E + 00
3.50E − 01
2.97E − 01
3.09E − 01
2.10E − 01
2.22E − 01
Rank
6
3
5
4
10
9
7
8
1
2
F14
Mean
3.07E − 01
8.91E − 01
3.44E − 01
2.83E − 01
9.26E + 01
2.88E − 01
2.78E − 01
2.88E − 01
2.30E − 01
2.51E − 01
Std
3.63E − 02
3.59E − 01
3.38E − 02
3.87E − 02
2.41E + 01
4.70E − 02
4.42E − 02
4.92E − 02
2.69E − 02
4.40E − 02
Best
2.54E − 01
1.88E − 01
2.97E − 01
2.20E − 01
3.51E + 01
2.11E − 01
1.84E − 01
2.26E − 01
1.87E − 01
1.74E − 01
Rank
8
4
9
6
10
5
2
7
3
1
F15
Mean
1.58E + 01
4.49E + 00
1.09E + 01
1.99E + 01
8.76E + 03
1.80E + 01
1.89E + 01
3.75E + 01
5.93E + 00
8.76E + 00
Std
9.26E − 01
6.87E − 01
9.63E − 01
2.08E + 00
1.12E + 04
5.93E + 00
8.66E + 00
2.19E + 01
1.22E + 00
1.71E + 00
Best
1.41E + 01
3.18E + 00
9.09E + 00
1.55E + 01
5.91E + 02
1.03E + 01
9.75E + 00
1.54E + 01
3.87E + 00
5.15E + 00
Rank
7
1
4
9
10
6
5
8
2
3
F16
Mean
1.18E + 01
1.05E + 01
1.11E + 01
1.05E + 01
1.17E + 01
1.20E + 01
1.19E + 01
1.11E + 01
1.02E + 01
1.06E + 01
Std
2.50E − 01
8.64E − 01
2.31E − 01
3.26E − 01
5.87E − 01
4.35E − 01
3.94E − 01
6.62E − 01
4.55E − 01
4.12E − 01
Best
1.13E + 01
8.55E + 00
1.07E + 01
9.78E + 00
1.04E + 01
1.14E + 01
1.14E + 01
1.01E + 01
9.37E + 00
9.89E + 00
Rank
8
1
7
3
6
9
9
5
2
4
F17
Mean
3.76E + 05
1.36E + 05
2.07E + 06
3.46E + 06
4.28E + 05
1.91E + 05
2.11E + 05
1.67E + 05
4.42E + 04
1.92E + 05
Std
1.12E + 05
1.04E + 05
6.62E + 05
1.54E + 06
4.64E + 05
1.73E + 05
9.21E + 04
2.13E + 05
3.70E + 04
1.60E + 05
Best
2.44E + 05
2.69E + 04
1.07E + 06
9.41E + 05
3.61E + 04
3.29E + 04
8.02E + 04
3.54E + 04
8.62E + 03
3.99E + 04
Rank
8
2
10
9
5
3
7
4
1
6
F18
Mean
1.59E + 03
3.38E + 03
4.92E + 02
4.54E + 06
2.46E + 04
2.93E + 03
3.31E + 03
8.71E + 02
1.14E + 03
2.19E + 03
Std
8.50E + 02
4.21E + 03
9.30E + 01
1.89E + 06
1.06E + 05
2.35E + 03
4.11E + 03
1.02E + 03
1.29E + 03
2.81E + 03
Best
4.78E + 02
3.13E + 02
3.44E + 02
1.54E + 06
2.80E + 02
1.35E + 02
1.64E + 02
6.17E + 01
6.23E + 01
6.28E + 01
Rank
9
7
8
10
6
4
5
1
2
3
F19
Mean
1.19E + 01
1.57E + 01
1.01E + 01
2.04E + 01
8.25E + 01
2.12E + 01
2.12E + 01
2.71E + 01
6.31E + 00
6.90E + 00
Std
8.29E − 01
2.33E + 01
1.02E + 00
5.81E + 00
3.96E + 01
2.55E + 01
2.54E + 01
2.86E + 01
1.01E + 00
2.05E + 00
Best
1.01E + 01
4.44E + 00
8.20E + 00
1.33E + 01
2.24E + 01
5.77E + 00
6.72E + 00
9.65E + 00
4.60E + 00
3.34E + 00
Rank
8
2
6
9
10
4
5
7
3
1
F20
Mean
5.81E + 03
7.05E + 03
5.95E + 03
6.95E + 03
7.38E + 03
1.54E + 03
1.44E + 03
4.28E + 02
9.02E + 01
2.02E + 03
Std
2.71E + 03
1.27E + 04
3.35E + 03
3.35E + 03
7.07E + 03
9.69E + 02
6.69E + 02
1.77E + 02
8.20E + 01
2.99E + 03
Best
2.36E + 03
1.37E + 03
2.48E + 03
1.05E + 03
3.45E + 02
6.68E + 02
3.41E + 02
2.03E + 02
3.06E + 01
5.99E + 01
Rank
9
8
10
7
5
6
4
3
1
2
F21
Mean
1.49E + 05
6.77E + 04
3.39E + 05
6.64E + 05
4.18E + 04
9.72E + 04
1.19E + 05
2.20E + 04
9.86E + 03
6.55E + 04
Std
6.40E + 04
4.21E + 04
1.22E + 05
2.63E + 05
5.34E + 04
9.01E + 04
1.00E + 05
2.22E + 04
1.32E + 04
6.81E + 04
Best
8.47E + 04
8.85E + 03
1.25E + 05
1.26E + 05
2.10E + 03
1.89E + 04
3.16E + 04
4.09E + 03
1.22E + 03
1.19E + 03
Rank
8
5
9
10
3
6
7
4
2
1
F22
Mean
2.25E + 02
2.27E + 02
2.70E + 02
4.19E + 02
3.98E + 02
2.81E + 02
3.27E + 02
3.14E + 02
1.14E + 02
2.27E + 02
Std
7.27E + 01
9.97E + 01
5.98E + 01
1.04E + 02
1.63E + 02
1.08E + 02
1.33E + 02
1.41E + 02
7.56E + 01
1.20E + 02
Best
1.67E + 02
1.44E + 02
1.59E + 02
1.84E + 02
1.61E + 02
1.64E + 02
5.82E + 01
1.49E + 02
2.41E + 01
2.68E + 01
Rank
9
4
6
10
7
8
3
5
1
2
F23
Mean
3.14E + 02
3.18E + 02
3.15E + 02
2.26E + 02
3.87E + 02
3.15E + 02
3.15E + 02
3.15E + 02
3.15E + 02
3.15E + 02
Std
1.57E − 04
3.50E + 00
2.29E − 01
4.77E + 01
2.07E + 01
1.23E − 11
1.78E − 10
4.43E − 01
1.44E − 13
3.92E − 13
Best
3.14E + 02
3.15E + 02
3.15E + 02
2.00E + 02
3.34E + 02
3.15E + 02
3.15E + 02
3.15E + 02
3.15E + 02
3.15E + 02
Rank
2
3
3
1
4
3
3
3
3
3
F24
Mean
2.24E + 02
2.27E + 02
2.27E + 02
2.00E + 02
2.79E + 02
2.00E + 02
2.00E + 02
2.00E + 02
2.26E + 02
2.38E + 02
Std
5.46E − 01
2.58E + 00
1.01E + 00
5.57E − 04
9.88E + 00
2.20E − 03
2.19E − 03
9.68E − 04
1.28E + 00
5.98E + 00
Best
2.23E + 02
2.23E + 02
2.25E + 02
2.00E + 02
2.60E + 02
2.00E + 02
2.00E + 02
2.00E + 02
2.24E + 02
2.25E + 02
Rank
2
2
4
1
5
1
1
1
3
4
F25
Mean
2.07E + 02
2.06E + 02
2.10E + 02
2.00E + 02
2.24E + 02
2.01E + 02
2.00E + 02
2.02E + 02
2.09E + 02
2.06E + 02
Std
2.45E + 00
2.09E + 00
1.62E + 00
8.04E − 09
4.39E + 00
2.39E + 00
5.33E − 06
3.62E + 00
3.10E + 00
2.68E + 00
Best
2.04E + 02
2.04E + 02
2.08E + 02
2.00E + 02
2.16E + 02
2.00E + 02
2.00E + 02
2.00E + 02
2.04E + 02
2.04E + 02
Rank
2
2
3
1
4
1
1
1
2
2
F26
Mean
1.70E + 02
1.51E + 02
1.00E + 02
1.00E + 02
1.43E + 02
1.10E + 02
1.00E + 02
1.10E + 02
1.00E + 02
1.34E + 02
Std
4.83E + 01
5.18E + 01
3.00E − 02
4.88E − 02
4.93E + 01
3.15E + 01
1.57E − 01
3.15E + 01
7.52E − 02
4.78E + 01
Best
1.00E + 02
1.00E + 02
1.00E + 02
1.00E + 02
1.01E + 02
1.00E + 02
1.00E + 02
1.00E + 02
1.00E + 02
1.00E + 02
Rank
1
1
1
1
2
1
1
1
1
1
F27
Mean
4.34E + 02
7.12E + 02
4.38E + 02
3.62E + 02
8.61E + 02
5.47E + 02
6.56E + 02
7.94E + 02
4.34E + 02
5.37E + 02
Std
3.50E + 01
1.84E + 02
1.33E + 01
9.59E + 01
2.83E + 02
1.66E + 02
2.13E + 02
2.15E + 02
6.71E + 01
1.46E + 02
Best
3.76E + 02
4.02E + 02
4.17E + 02
2.00E + 02
4.44E + 02
4.02E + 02
4.02E + 02
4.06E + 02
3.71E + 02
4.01E + 02
Rank
3
5
7
1
8
5
5
6
2
4
F28
Mean
3.98E + 02
1.48E + 03
9.21E + 02
5.74E + 02
2.66E + 03
1.24E + 03
1.18E + 03
1.43E + 03
8.34E + 02
1.20E + 03
Std
1.16E + 01
3.70E + 02
3.47E + 01
4.65E + 02
4.48E + 02
3.51E + 02
2.30E + 02
4.37E + 02
3.87E + 01
2.91E + 02
Best
3.89E + 02
1.02E + 03
8.64E + 02
2.00E + 02
1.99E + 03
9.20E + 02
8.94E + 02
9.96E + 02
7.87E + 02
7.48E + 02
Rank
2
9
5
1
10
7
6
8
4
3
F29
Mean
2.14E + 02
6.65E + 06
3.11E + 04
3.37E + 04
7.77E + 06
3.31E + 06
3.93E + 06
3.08E + 06
1.29E + 03
1.19E + 03
Std
1.02E + 00
1.10E + 07
1.52E + 04
1.09E + 04
1.58E + 07
5.39E + 06
5.14E + 06
4.99E + 06
5.91E + 02
2.70E + 02
Best
2.12E + 02
2.17E + 03
7.11E + 03
1.12E + 04
1.45E + 03
1.25E + 03
1.44E + 03
9.91E + 02
7.75E + 02
7.84E + 02
Rank
1
8
9
10
7
5
6
4
2
3
F30
Mean
7.00E + 02
1.82E + 04
7.94E + 03
1.38E + 04
7.38E + 04
3.35E + 03
3.37E + 03
6.47E + 03
1.93E + 03
2.42E + 03
Std
8.54E + 01
3.03E + 04
2.72E + 03
3.52E + 03
4.24E + 04
1.35E + 03
1.75E + 03
3.43E + 03
5.26E + 02
4.86E + 02
Best
5.98E + 02
1.78E + 03
5.62E + 03
7.35E + 03
3.87E + 03
1.62E + 03
1.58E + 03
3.37E + 03
1.15E + 03
1.56E + 03
Rank
1
6
9
10
8
5
4
7
2
3
Average rank
Mean
4.6
5.27
4.67
5.57
8.43
5.43
5.13
5.633
1.8
3.667
Std
4.033
6.57
3.7
5.5
9.1
5.8
5.17
6.87
2.83
5.1
Best
5.7
3.9
5.9
6.07
7.63
5.27
5
5.37
1.9
2.63


#### 5.1.2. Statistical Analyses for Benchmark Results

Sign tests [84] is a common method to determine whether there is significant difference between two algorithms. In this paper, we use the two-tailed sign tests which has a significance level of 0.05 to test the differences between the results obtained by MBSAgC and other compared algorithms. The statistical results are listed in Table 5. The best solution is used as the criterion of the sign test. The signs “+,” “≈,” and “−” represent that MBSAgC performs better, almost the same, and worse than other compared algorithms, respectively. And the null hypothesis in this paper is that the performances of MBSAgC and one of the other compared algorithms are not significantly differential.Table 5
Comparisons between MBSAgC and other algorithms in sign tests.
AL
+
≈
−
Total
P
LBSA
28
18
34
80
0.619
CLPSO
68
5
7
80
0.000
PSOFIPS
54
15
11
80
0.000
TLBO
49
14
17
80
0.000
ETLBO
54
14
12
80
0.000
PSOFDR
52
8
20
80
0.000
BSAISA
19
17
14
50
0.474
OBSA
32
13
5
50
0.004
BSA
30
12
8
50
0.001
CBSA
22
1
7
30
0.003
CLBSA
29
0
1
30
0.000
DGSTLBO
23
2
5
30
0.002
SADE
10
5
10
25
0.717Thirteen algorithms are compared in the benchmark experiment. Table5 lists the analytical test results of all the algorithms. The P values of supporting the null hypothesis of sign test for ten compared algorithms (MBSAgC-CLPSO, MBSAgC-PSOFIPS, MBSAgC-TLBO, MBSAgC-ETLBO, MBSAgC-PSOFDR, MBSAgC-OBSA, MBSAgC-BSA, MBSAgC-CBSA, MBSAgC-CLBSA, and MBSAgC-DGSTLBA) are lower than significance level 0.05, so we can reject the null hypothesis. And it is illustrated that the optimization performance of MBSAgC is significantly better than those of the other ten algorithms. The P values of MBSAgC-LBSA, MBSAgC-BSAISA, and MBSAgC-SaDE are higher than 0.05; they are 0.619, 0.474, and 0.717, respectively, so we cannot reject the null hypothesis. However, it does not mean that MBSAgC is much worse than them. In the light of sign value from Table 5, for MBSAgC-LBSA, MBSAgC is worse than LBSA on thirty-four functions, and it performs better on twenty-eight functions; moreover, MBSAgC performs better on both 10D and 30D in CEC2005. MBSAgC is slightly worse than BSAISA on fourteen functions, but better than BSAISA on nineteen functions. For MBSAgC-SaDE, the number of “+” is the same as the number of “−,” but MBSAgC is better than SaDE in comprehensive ranking.


### 5.2. Experiments on Engineering Design Problems

In this part, we test MBSAgC in five well-known engineering design problems: three-bar truss design problem(TTP), pressure vessel design problem(PVP), tension/compression spring design problem(TCSP), welded beam design problem(WBP), and speed reducer design problem(SRP), respectively. The details of five design problems can be found in AppendixA. And the results in Tables 6–15 show the performance of MBSAgC and its competitive with other compared algorithms.Table 6
Comparison of best solutions for the three-bar truss design problem.
Method
DEDS
HEAA
PSO-DE
DELC
MBA
BSA
MBSAgC
X
1
0.788675
0.788680
0.788675
0.788675
0.788675
0.788675
0.788675
X
2
0.408248
0.408234
0.408248
0.408248
0.408560
0.408248
0 0.408248
g
1
X
1.77E − 08
NA
−5.29E − 11
NA
−5.29E − 11
−3.23E − 12
−1.8208E − 13
g
2
X
−1.464102
NA
−1.463748
NA
−1.463748
−1.464102
−1.464102
g
3
X
−0.535898
NA
−0.536252
NA
−0.536252
−0.535898
−0.535898
f
X
263.895843
263.895843
263.895843
263.895843
263.895852
263.895843
263.895843Table 7
Comparison of statistical results for the three-bar truss design problem.
Method
Worst
Mean
Best
Std
FEs
DEDS
263.895849
263.895843
263.895843
9.7E − 07
15000
HEAA
263.896099
263.895865
263.895843
4.9E − 05
15000
PSO-DE
263.895843
263.895843
263.895843
4.5E − 10
17600
DELC
263.895843
263.895843
263.895843
4.3E − 14
10000
MBA
263.915983
263.897996
263.895852
3.93E − 03
13280
BSA
263.895845
263.895843
263.895843
2.64E − 07
13720
MBSAgC
263.895843
263.895843
263.895843
1.83E − 10
9540Table 8
Comparison of best solutions for the pressure vessel design problem.
Method
DA-DT
MDE
CPSO
HPSO
DELC
ABC
BSA-SAε
BSA
MBSAgC
X
1
0.8125
0.8125
0.8125
0.8125
0.8125
0.8125
0.8125
0.8125
0.8125
X
2
0.4375
0.4375
0.4375
0.4375
0.4375
0.4375
0.4375
0.4375
0.4375
X
3
42.0974
42.0984
42.0913
42.0984
42.0984
42.0984
42.0984
42.0984
42.0984
X
4
176.6540
176.6360
176.7465
176.6366
176.6366
176.6366
176.6366
176.6366
176.6366
g
1
X
−3.01E − 03
0
−1.37E − 06
NA
NA
0
−9.5E − 10
−3.69E − 08
0
g
2
X
−3.58E − 02
−0.035881
−3.59E − 04
NA
NA
−0.35881
−3.59E − 2
−0.35881
−3.59E − 2
g
3
X
−24.7593
−0.000
−118.7687
NA
NA
−0.000226
−1.2E − 4
−0.095446
0
g
4
X
−63.3460
−63.3639
−63.2535
NA
NA
−63.363
−63.363
−63.2842
−63.3634
f
X
6059.9463
6059.7017
6061.0777
6059.7143
6059.7143
6059.7143
6059.7143
6059.7150
6059.7143Table 9
Comparison of best solutions for the pressure vessel design problem.
Method
Worst
Mean
Best
Std
FEs
GA-DT
6469.3220
6177.2533
6059.9643
130.9297
80000
MDE
6059.7017
6059.7017
6059.7017
1.0E − 12
24000
CPSO
6363.8041
6147.1332
6061.0777
86.45
30000
HPSO
6288.6770
6099.9323
6059.7143
86.20
81000
DELC
6059.7143
6059.7143
6059.7143
2.1E − 11
30000
PSO-DE
6059.7143
6059.7143
6059.7143
1.0E − 10
42100
ABC
NA
6245.3081
6059.7147
2.05E + 02
30000
BSA-SAε
6116.7804
6074.3682
6059.7143
1.71E + 01
80000
BSA
6771.5969
6221.2861
6059.7150
2.03E + 02
60000
MBSAgC
6374.8627
6073.6436
6059.7143
2.03E + 04
16180Table 10
Comparison of best solutions for the tension/compression spring design problem.
Method
X
1
X
2
X
3
g
1
X
g
2
X
g
3
X
g
4
X
f
X
GA-DT
0.051989
0.363965
10.890522
−1.3E − 05
−2.1E − 05
−4.061338
−0.722698
0.012681
MDE
0.051688
0.356692
11.290483
−0.000000
−0.000000
−4.053734
−0.727090
0.012665
CPSO
0.051728
0.357644
11.244543
−8.45E − 04
−1.26E − 05
−4.051300
−0.727090
0.012675
HPSO
0.051706
0.357126
11.265083
NA
NA
NA
NA
0.012665
DEDS
0.051689
0.356718
11.288965
NA
NA
NA
NA
0.012665
HEAA
0.051690
0.356729
11.288294
NA
NA
NA
NA
0.012665
DELC
0.051689
0.356718
11.288966
NA
NA
NA
NA
0.012665
ABC
0.051749
0.358179
11.203763
−0.000000
−0.000000
−4.0566663
−0.726713
0.012665
MBA
0.051656
0.35594
11.344665
0
0
−4.052248
−0.728268
0.012665
SSOC
0.051689
0.356718
11.288965
NA
NA
NA
NA
0.012665
BSA-SAε
0.051989
0.356727
11.288425
−7.70E − 09
−3.30E − 09
−4.054
−0.728
0.012665
BSA
0.51694
0.356845
11.281488
−1.05E − 07
−1.77E − 08
−4.054037
−0.727640
0.012665
MBSAgC
0.051689
0.356718
11.288954
−4.15E − 14
−1.13E − 13
−4.0538
−0.7277
0.012665Table 11
Comparison of statistical results for the tension/compression spring design problem.
Method
Worst
Mean
Best
Std
FEs
GA-DT
0.012973
0.12742
0.12681
5.90E − 05
80000
MDE
0.012674
0.012666
0.012665
2.0E − 06
24000
CPSO
0.012924
0.012730
0.012675
5.20E − 05
23000
HPSO
0.012719
0.012707
0.012665
1.58E − 05
81000
DEDS
0.012738
0.012669
0.012665
1.25E − 05
24000
HEAA
0.012665
0.012665
0.012665
1.4E − 09
24000
DELC
0.012666
0.012665
0.012665
1.3E − 07
20000
PSO-DE
0.012665
0.012665
0.012665
1.2E − 08
24950
ABC
NA
0.012709
0.012665
0.012813
30000
MBA
0.012900
0.012713
0.012665
6.30E − 05
7650
SSOC
0.012868
0.012765
0.012665
9.29E − 05
25000
BSA-SAε
0.012666
0.012665
0.012665
1.62E − 07
80000
BSA
0.012669
0.012666
0.012665
7.24E − 07
43220
MBSAgC
0.012665
0.012665
0.012665
1.25E − 07
22680Table 12
Comparison of best solutions for the welded beam design problem.
Method
GA-DT
MDE
CPSO
HPSO
ABC
MBA
BSA-SAε
BSA
BSAISA
MBSAgC
X
1
h
0.205986
0.205730
0.202369
0.205730
0.205730
0.205729
0.205730
0.205730
0.205730
0.205730
X
2
l
3.471328
3.470489
3.544214
3.470489
3.470489
3.470493
3.470489
3.470489
3.470489
3.47049
X
3
t
9.020224
9.036624
9.04821
9.036624
9.036624
9.036626
9.036624
9.036624
9.036624
9.036624
X
4
b
0.206480
0.205730
0.205723
0.205730
0.205730
0.205729
0.205730
0.205730
0.205730
0.205730
g
1
X
−0.074092
−0.000335
−12.8399796
NA
0.000000
−0.001614
−1.55E − 10
−5.32E − 07
0
−2.2628E − 09
g
2
X
−0.266227
−0.000753
−1.247467
NA
−0.000002
−0.016911
−4.30E − 09
−9.02E − 06
0
−1.8335E − 09
g
3
X
−4.95E − 04
−0.0000000
−1.49E − 03
NA
0.0000000
−2.40E − 07
−1.55E − 15
−7.86E − 12
−5.55E − 17
−2629E − 13
g
4
X
−3.430044
−3.432984
−3.429347
NA
−3.432984
−3.432982
−3.4330
−3.432984
−3.43294
−3.4330
g
5
X
−0.080986
−0.080730
−0.079381
NA
−0.080730
−0.080729
−8.07E − 02
−0.080730
−0.080730
−0.080730
g
6
X
−0.235514
−0.235540
−0.235536
NA
−0.235540
−0.235540
−0.2355
−0.235540
−0.235540
−0.2355
g
7
X
−58.666440
−0.000882
−11.681355
NA
0.000000
−0.001464
−1.85E − 10
−1.13E − 07
−5.46E − 12
−4.0927E − 10
f
X
1.728226
1.724852
1.728024
1.724852
1.724852
1.724853
1.724852
1.724852
1.724852
1.724852Table 13
Comparison of statistical results for the welded beam design problem.
Method
Worst
Mean
Best
Std
FEs
GA-DT
1.993408
1.792654
1.728226
7.47E − 02
80000
MDE
1.724854
1.724853
1.724852
NA
24000
CPSO
1.782143
1.748831
1.728024
1.29E − 02
24000
HPSO
1.814295
1749040
1.724852
4.00E − 02
81000
DELC
1.724852
1.724852
1.724852
4.1E − 13
20000
PSO-DE
1.724852
1.724852
1.724852
6.7E − 16
66600
ABC
NA
1.741913
1.724852
3.1E − 02
30000
MBA
1.724853
1.724853
1.724853
6.94E − 19
47340
SSOC
1.799332
1.746462
1.724852
2.57E − 02
25000
BSA-SAε
1.724852
1.724852
1.724852
8.11E − 10
80000
BSA
1.724854
1.724852
1.724852
2.35E − 07
45480
BSAISA
1.724854
1.724852
1.724852
2.96E − 07
29000
MBSAgC
3.121674
1.780725
1.724852
2.765E − 01
21640Table 14
Comparison of best solutions for the speed reducer design problem.
Method
MDE
DEDS
DELC
HEAA
PSO-DE
MBA
BSA
BSAISA
MBSAgC
X
1
3.500010
3.500000
3.500000
3.500000
3.500000
3.500000
3.500000
3.500000
3.500000
X
2
0.700000
0.700000
0.700000
0.700000
0.700000
0.700000
0.700000
0.700000
0.700000
X
3
17
17
17
17.000013
17.000000
17.000000
17
17
17
X
4
7.300156
7.300000
7.300000
7.300428
7.300000
7.300033
7.300000
7.300000
7.300000
X
5
7.800027
7.715320
7.715320
7.715377
7.800000
7.715772
7.715320
7.715320
7.715320
X
6
3.350221
3.350215
3.350215
3.350231
3.350215
3.350218
3.350215
3.350215
3.305215
X
7
5.286685
5.286654
5.286654
5.286664
5.286683
5.286654
5.286654
5.286654
5.286654
f
X
2996.356689
2994.471066
2994.471066
2994.488107
2996.348167
2994.482453
2994.471066
2994.471066
2994.471066Table 15
Comparison of statistical results for the speed reducer design problem.
Method
Worst
Mean
Best
Std
FEs
MDE
2996.390137
2996.367220
2996.356689
8.2E − 03
24000
DEDS
2994.471066
2994.471066
2994.471066
3.58E − 12
30000
HEAA
2994.752311
2994.613368
2994.499107
7.0E − 02
40000
DELC
2994.471066
2994.471066
2994.471066
1.9E − 12
30000
PSO-DE
2996.348204
2996.348174
2996.348167
6.4E − 06
54350
ABC
NA
2997.058412
2997.058412
0
30000
MBA
2999.652444
2996.769019
2994.482453
1.56
6300
SSOC
2996.113298
2996.113298
2996.113298
1.34E − 12
25000
BSA
2994.471066
2994.471066
2994.471066
9.87E − 11
25640
BSAISA
2994.471095
2994.471067
2994.471066
5.40E − 06
15860
MBSAgC
2998.327568
2996.438784
2994.471066
3.24E − 05
12580


#### 5.2.1. Engineering Problem and Experimental Results

(1) Three-Bar Truss Design Problem(TTP). We compare MBSAgC with the BSA, differential evolution with dynamic stochastic selection(DEDS) [85], hybrid evolutionary algorithm(HEAA) [86], hybrid particle swarm optimization with differential evolution(POS-DE) [87], differential evolution with level comparison (DELC) [88], and mine blast algorithm(MBA) [89]. The results of all eight algorithms are showed in Tables 6 and 7. And the best solution obtained by MBSAgC is x=0.788675,0.408248 with fx=263.895843 using 9540 FEs. In Table 6, we can see that only MBA does not get the best solution with the corresponding function value; the other five algorithms all reach the best solution with the corresponding function value equal to 263.895483. MBSAgC is not only lower than other six algorithms in term of FEs but also Std value is the lowest in all algorithms. Table 6 shows the comparison of the best solutions obtained by MBSAgC and other algorithms. The statistical results of all compared algorithms are listed in Table 7.From Tables6 and 7, all of the compared algorithms reach the best solution equal to 263.89583 except MBA with 263.89582. However, MBSAgC requires the lowest FEs (9540) among all algorithms. Its Std value is also better than all compared algorithms. These comparative results illustrate that MBSAgC outperforms other algorithms in terms of computational overhead and robustness for this problem.Figure5(a) shows the convergence curves of MBSAgC and BSA for three-bar truss design problem. As shown in Figure 5(a), the BSA reaches the global optimum at about 13,000 FEs, while MBSAgC only reaches the global optimum at about 9000 FEs. It is illustrated that convergence rate of MBSAgC is faster than the BSA for this problem.Figure 5
Convergence curves of MBSAgC and BSA. (a) Three-bar truss design problem (TTP). (b) Pressure vessel design problem. (c) Tension/compression spring design. (d) Welded beam design problem (WBP). (e) Speed reducer design problem (SRP).
(a)
(b)
(c)
(d)
(e)(2) Pressure Vessel Design Problem (PVP). There is a nonlinear objective function in the pressure vessel design problem, and the function has three linear and one nonlinear inequality constrains and two discrete and two continuous design variables. For this problem, BSA, BSA-SAε, DELC, PSO-DE, genetic algorithms based on dominance tournament selection (GA-DT) [90], modified differential evolution (MDE) [91], coevolutionary particle swarm optimization (CPSO) [92], hybrid particle swarm optimization (HPSO) [93], and artificial bee colony algorithm (ABC) are used as comparison algorithms. And the best solution obtained by MBSAgC is x=0.8125,0.4375.42.0984,176.6366 with fx=6059.7143 using 16180 FEs. It is the lowest FEs in all compared algorithms. Table 8 shows the comparison of the best solutions obtained by MBSAgC and other algorithms. The statistical results of all compared algorithms are listed in Table 9.As shown in Table8, MBSAgC, BSA-SAε, ABC, DELC, and HPSO reach the same function value equal to 6059.7143, which is worse than the function value 6059.7017 of MDE, but all seven algorithms get the feasible solution of this design problem. From Table 9, the solution of MDE ranks first in terms of the best solution; the best solution of MBSAgC is better than those of GA-DT, CPSO, ABC, and BSA. Though MDE is better than MBSAgC in terms of the best solution, the FEs of MBSAgC is much lower than that of MDE. It is illustrated that MBSAgC is competitive with other algorithms on this engineering problem.Figure5(b) describes the convergence curves of MBSAgC and BSA for the pressure vessel design problem. As shown in Figure 5(b), MBSAgC finds the global optimum at about 16,000 FEs and obtains a far more accurate function value than BSA. Moreover, the convergence speed of MBSAgC is much faster than that of BSA.(3) Tension/Compression Spring Design Problem (TCSP). There are three continuous variables and four nonlinear inequality constrains in the tension/compression spring design problem. MBSAgC obtains the best feasible solution at x=0.56187,0.356669,11.2889 with fx=0.012665 using 22680 FEs. And we compare the results obtained by MBSAgC with GA-DT, MDE, CPSO, HPSO, DEDS, HEAA, DELC, PSO-DE, ABC, MBA, BSA-SAε, BSAISA, and social spider optimization (SSOC) [94]. The comparison of the best solutions obtained by all comparison algorithms is presented in Table 10, and the statistical results of various algorithms are listed in Table 11.In Table10, only GA-DT and CPSO fail to get the best solution for this problem; other compared algorithms all reach the best solution value 0.012665. MBSAgC requires 22680 FEs to get the global best solution. It is worse than MBA (FEs : 7650) and DELC (FEs : 20000), but its Std is better than both of them. It can be summarized that MBSAgC outperforms other compared algorithms except MBA and DELC; in addition, the robustness of MBSAgC is much better than others, except HEAA.Figure5(c) depicts the convergence curves of MBSAgC and BSA for the tension/compression spring design problem. From Figure 5(c), it can be observed that both MBSAgC and BSA are prone to fall into a local optimum in the early iterations, but they are able to jump out of local optimum successfully. However, the convergence speed of MBSAgC is obviously faster than that of BSA.(4) Welded Beam Design Problem (WBP). The welded beam design problem is a minimum-cost problem, which has four continuous design variables and subjects to two linear and five nonlinear inequality constraints. For this problem, MBSAgC is compared with GA-DT, MDE, CPSO, HPSO, DELC, PSO-DE, ABC, MBA, BSA, BSA-SAε, SSOC, and BSAISA. The comparison of the best solutions obtained by all comparison algorithms is presented in Table 12, and the statistical results of various algorithms are listed in Table 13.As can be seen from Table13, MBSAgC has obtained the best feasible solution at x=0.205730,3.4706,9.036624,0.205730 with fx=1.724852 using 21640 FEs. Only GA-DE, CPSO, and MBA fail to find the best function value 1.724852; the other compared algorithms all find the best function values. DELC outperforms other algorithms in both Std and FEs. MBSAgC performs better than other algorithms except DELC in terms of FEs. It is illustrated that MBSAgC is competitive on computational cost, while the robustness of MBSAgC is really bad in terms of its Std vlaue (2.765E − 01), which is worse than those of most existing comparison algorithms.Figure5(d) depicts the convergence curves of MBSAgC and BSA for the welded beam design problem. Figure 5(d) shows that the convergence speed of MBSAgC is faster than that of BSA remarkably.(5) Speed Reducer Design Problem (SRP). The speed reducer design problem consists of eleven constraints and seven continuous design variables, where x3 is an integer variable. The best solution obtained by MBSAgC is x=3.500000,0.700000,17,7.300000,7.7153,3.3052,5.2867 with fx=2994.471066 using 12580 FEs. We compared the results of MBSAgC with 10 well-known algorithms: BSAISA, BSA, MDE, DEDS, HEAA, DELC, PSO-DE, ABC, MBA, and SSOC. The compared best solutions and the statistical results of all comparison algorithms are represented in Tables 14 and 15, respectively.As seen in Tables14 and 15, MBSAgC, BSAISA, BSA, DEDS, and DELC get the same optimal values equaling to 2294.471066. Other algorithms have different optimal values, but all the function values they get are feasible. MBA requires the lowest FEs (6300) to get the global optimal solution. MBSAgC has much lower FEs than other compared algorithms except MBA. Though MBA has the lowest FEs, it fails to find the best solution for this problem. This experiment is to test whether the algorithm can find the optimal solution with the minimum computational cost. Obviously, MBSAgC conforms to the experimental requirements because it requires so low FEs to find the best solution for this problem.Figure5(e) depicts the convergence curves of MBSAgC and BSA for the speed reducer design problem and shows that the convergence speed of MBSAgC is faster than that of BSA.


#### 5.2.2. Statistical Analyses for Engineering Results

In order to further analyze the comprehensive performance of the proposed MBSAgC, we list the optimal feasible solutions and the minimum FEs for all the abovementioned engineering design problems, and compare them with the proposed MBSAgC. The statistical results are shown in Table16. As can be seen from Table 16, MBSAgC can find the optimal solution of four engineering design problems. The feasible solution obtained by MBSAgC is the optimal for three-bar truss problem, and the FEs of MBSAgC is also the lowest; it is illustrated that the results of MBSAgC are competitive. For pressure vessel design problem, the feasible solution found by MBSAgC is the same as most of compared algorithms, but the FEs of MBSAgC is the minimum among all the compared algorithms. For the rest three problems, MBSAgC can find the optimal feasible solution; although the FEs of MBSAgC is not the smallest, its ranking among the all compared algorithms is competitive. Furthermore, the MBA is with a smaller FEs, but it cannot find out the current best value on speed reducer problem. So overall, about the ability to find the optimal solution and the algorithmic computational overhead, MBSAgC holds a very competitive performance on these optimization problems compared with the state-of-the-art algorithms.Table 16
Comparison for the best solution and FEs.
Engineering design
Smallest “Best”
“Best” by MBSAgC
Smallest FEs (algorithm)
FEs for MBSAgC (rank)
Three-bar truss
263.895843
263.895843
10000 (DELC)
9560 (1)
Pressure vessel
6059.7017
6059.7143
24000 (MDE)
16180 (1)
Tension/compression spring
0.012665
0.012665
7650 (MBA)
22680 (3)
Welded beam
1.724852
1.724852
20000 (DELC)
21640 (2)
Speed reducer
2994.471066
2994.471066
6300 (MBA)
12580 (2)


### 5.3. Synthetical Discussions Based on Experimental Results

Based on the above experimental results on CEC2005, CEC2014, and several engineering optimization problems, this subsection discusses the search behaviors of the proposed algorithm from three perspectives including its solution accuracy, convergence speed, and future development.


#### 5.3.1. Solution Accuracy

The “Best” values of all the previous tables are the reflection of the solution accuracy for each algorithm in solving all the test functions (or engineering optimization problems). The smaller this “Best” value, the higher the solution accuracy is. For 10-dimension functions of CEC2005, the average ranking of MBSAgC’s “Best” value equals to 2.76 (shown in Table2) which ranks first in all the compared algorithms. For 30-dimension functions of CEC2005, the average ranking of MBSAgC also is first (2.56, shown in Table 3). For 30-dimension functions of CEC2014, its average ranking is second (2.63, shown in Table 4). And MBSAgC can find the current optimum of four of the five engineering optimization problems. Overall, the solution accuracy of MBSAgC is obviously superior to the other compared algorithms.


#### 5.3.2. Convergence Speed

The “FEs” values of all the previous tables are the reflection of the convergence speed for each algorithm in solving all the test functions (or engineering optimization problems). The smaller this “FEs” value, the faster the convergence speed is. In addition, each curve of the previous convergence figures represents the variation of the current optima (or error values) over the fitness evaluation times. The variation trend of the curve also reflects the convergence speed of the compared algorithms. For the five engineering optimization problems (shown in Table16), two of “FEs” values of MBSAgC rank first, two of them rank second, and the rest one ranks third. Similar to all the compared algorithms, MBSAgC solves all the benchmark functions within the given “FEs”. And Figures 3–5 show that MBSAgC holds faster convergence speed than the original BSA. Overall, the proposed MBSAgC keeps as a strong competitiveness on the convergence speed as the compared state-of-the-arts algorithms.


#### 5.3.3. Future Development

According to the above discussions, we can draw a conclusion that the proposed MBSAgC improves the solution accuracy on a condition of keeping competitive convergence speed. This indicates that the modified strategies based on the collective wisdom, especially the winner-tendency, do enhance the exploitation capability of MBSAgC. However, we also notice that MBSAgC performs worse than LBSA on the CEC2014 benchmark functions. This also leaves a space for further development of the topological opposition-based learning operator (TOBL) which is the core of all the modified strategies of MBSAgC.


## 5.1. Experiments on Benchmark Functions

### 5.1.1. Experimental Results

(1) On 10-Dimension Functions in CEC2005. Table 2 shows the simulation results of 10-dimension functions in CEC2005. Table 2 compares the mean, standard deviations, and the best solution of the ten algorithms. With respect to the best solution, the results of Table 2 show that the proposed algorithm MBSAgC ranks first in functions F1, F2, F4, F6, F9, F12, F15, F18–F20, F22, F24, and F25. LBSA is superior to other nine algorithms on function F3. For function F3, all compared algorithms could not find the theoretical optimal solution. The rank of MBSAgC in terms of best solution is eighth, it is worse than most of others, and the best solution of LBSA is closest to the theoretical optimal value, and the standard deviation of LBSA is also the smallest among ten algorithms. The best solution of PSOFDR is superior to other algorithms in functions F10, F11, and F14. The best solution of BSAISA algorithm is better than other nine algorithms in function F7. For function F1, all of the compared algorithms get the global optimal solution. For function F2, the best solution of CLPSO and OBSA is worse than the other eight algorithms. For function F4, BSAISA, OBSA, BSA, and CLPSO could not converge to the global optimum. MBSAgC can find the theoretical optimal solution on functions F1, F2, F4, F6, F9, F12, and F15. In Table 2, in terms of the optimal solutions of twenty-five functions, thirteen functions for the proposed algorithm MBSAgC rank first. The average ranking of the proposed algorithm is 2.76; moreover, MBSAgC ranks first in the best solution. To sum up, the performance of MBSAgC is competitive with other algorithms.Table 2
Comparative results of ten algorithms for 10D problems from CEC2005.
F
AL
LBSA
BSA
CLPSO
PSOFIPS
TLBO
ETLBO
PSOFDR
BSAISA
OBSA
MBSAgC
F1
Mean
−450.000
−450.000
−450.000
−450.000
−450.000
−450.000
−439.674
−450.000
−450.000
−450.000
Std.
0.000
0.000
0.000
0.000
0.000
0.000
32.653
0.000
1.847E − 11
0.000
Best
−450.000
−450.000
−450.000
−450.000
−450.000
−450.000
−450.000
−450.000
−450.000
−450.000
Rank
1
1
1
1
1
1
1
1
1
1
F2
Mean
−450.000
−449.997
−449.676
−450.000
−450.000
−450.000
−450.000
−449.968
−446.527
−450.000
Std.
0.000
0.005
0.177
0.000
0.000
0.000
0.000
0.044
3.336
6.91E − 06
Best
−450.000
−450.000
−449.906
−450.000
−450.000
−450.000
−450.000
−450.000
−449.791
−450.000
Rank
1
1
2
1
1
1
1
1
3
1
F3
Mean
5652.945
21518.124
695430.984
114431.841
70188.530
132295.917
143065.100
31699.067
427940.36
176254.16
Std.
7686.985
18993.641
263113.140
54898.334
47419.217
98983.396
156627.508
17717.113
471417.61
126171.69
Best
251.786
3316.003
333184.289
48691.653
22503.152
23119.940
6576.186
9211.109
35974.764
39533.675
Rank
1
2
10
9
5
6
3
4
7
8
F4
Mean
−450.000
−449.800
−445.693
−450.000
−450.000
−450.000
−438.829
−444.970
−430.129
−449.999
Std.
0.000
0.306
2.683
0.000
0.000
0.000
23.550
9.926
19.385
0.001
Best
−450.000
−449.994
−449.361
−450.000
−450.000
−450.000
−450.000
−449.972
−448.989
−450.000
Rank
1
2
5
1
1
1
1
4
3
1
F5
Mean
−310.000
−310.000
−310.000
−233.009
−310.000
−310.000
−310.000
−301.311
−228.735
−309.328
Std.
0.000
0.000
0.000
33.727
0.000
0.000
0.000
7.416
64.078
1.034
Best
−310.000
−310.000
−310.000
−289.340
−310.000
−310.000
−310.000
−309.052
−304.344
−309.932
Rank
1
1
1
5
1
1
1
3
4
2
F6
Mean
390.548
392.105
391.599
394.658
392.592
392.167
399.415
391.651
404.196
397.153
Std.
1.112
2.741
1.610
0.405
3.041
2.313
18.471
2.043
11.599
7.459
Best
390.001
390.091
390.348
393.841
390.000
390.000
390.002
390.005
390.000
390.000
Rank
2
5
6
8
1
1
3
4
1
1
F7
Mean
1087.046
1087.082
1087.208
−179.810
1087.046
1087.046
1087.288
−179.935
−179.816
−179.816
Std.
0.000
0.076
0.045
0.098
0.000
0.000
0.197
0.043
0.078
0.086
Best
1087.046
1087.046
1087.046
−179.949
1087.046
1087.046
1087.266
−179.986
−179.933
−179.958
Rank
5
5
5
3
5
5
6
1
4
2
F8
Mean
−119.643
−119.681
−119.624
−119.640
−119.628
−119.648
−119.714
−119.616
−119.607
−119.796
Std.
0.053
0.085
0.081
0.090
0.087
0.066
0.075
0.070
0.072
0.070
Best
−119.728
−119.812
−119.792
−119.851
−119.831
−119.628
−119.800
−119.747
−119.722
−119.741
Rank
8
3
6
1
2
10
4
7
9
5
F9
Mean
−330.000
−330.000
−330.000
−329.333
−322.342
−322.737
−327.314
−330.000
−329.343
−329.877
Std.
0.000
0.000
0.000
0.723
2.928
2.696
1.487
0.000
0.844
0.312
Best
−330.000
−330.000
−330.000
−330.000
−326.020
−327.015
−329.005
−330.000
−330.000
−330.000
Rank
1
1
1
1
4
3
2
1
1
1
F10
Mean
−323.493
−320.479
−316.698
−317.409
−319.885
−321.513
−319.259
−319.005
−310.290
−317.531
Std.
1.987
3.542
3.125
5.428
4.132
3.593
6.612
3.914
9.254
6.979
Best
−325.724
−324.562
−322.465
−324.188
−325.025
−327.015
−328.010
−325.505
−323.468
−324.812
Rank
3
7
10
8
5
2
1
4
9
6
F11
Mean
95.188
94.889
95.319
93.633
94.002
94.632
91.962
94.957
95.001
94.381
Std.
0.746
1.196
0.572
0.536
1.652
0.997
1.023
0.848
1.271
1.627
Best
93.737
92.408
94.425
92.922
91.059
92.619
90.702
92.744
92.209
91.286
Rank
9
5
10
8
2
6
1
7
4
3
F12
Mean
−444.398
−438.290
−363.666
−422.531
1445.385
794.693
−203.253
−448.681
−197.091
−294.114
Std.
16.545
16.021
65.389
47.468
5027.194
2860.501
507.838
13.907
645.215
421.269
Best
−459.985
−455.477
−431.128
−455.719
−460.000
−460.000
−459.496
−460.000
−458.383
−460.000
Rank
2
6
7
5
1
1
3
1
4
1
F13
Mean
−129.666
−129.697
−129.564
−128.810
−129.196
−129.044
−129.352
−129.577
−129.353
−129.342
Std.
0.073
0.118
0.118
0.146
0.165
0.256
0.271
0.067
0.151
0.175
Best
−129.830
−129.932
−129.769
−129.109
−129.447
−129.384
−129.745
−129.728
−129.663
−129.625
Rank
2
1
3
10
8
9
4
5
6
7
F14
Mean
−296.902
−296.694
−296.591
−297.170
−297.215
−297.081
−297.241
−296.757
−296.517
−296.627
Std.
0.207
0.205
0.151
0.306
0.249
0.459
0.685
0.215
0.253
0.276
Best
−297.288
−297.108
−296.948
−297.642
−297.566
−297.701
−298.570
−297.395
−297.095
−297.545
Rank
7
8
10
3
4
2
1
6
9
5
F15
Mean
120.030
124.232
140.582
209.829
450.644
401.382
387.275
123.356
170.949
3.654
Std.
0.089
12.918
28.900
99.415
141.887
156.564
203.967
11.180
99.357
200.365
Best
120.000
120.000
120.002
120.460
195.411
197.129
176.891
120.000
120.003
120.000
Rank
1
1
2
4
6
7
5
1
3
1
F16
Mean
223.754
231.537
257.700
232.105
228.617
230.096
254.799
228.345
245.791
235.776
Std.
3.450
10.701
14.508
10.777
9.831
9.972
29.896
5.584
10.049
9.911
Best
218.507
209.614
223.542
221.355
219.015
217.668
222.865
215.613
221.217
218.609
Rank
4
1
10
8
6
3
9
2
7
5
F17
Mean
241.073
249.588
269.953
251.164
246.701
228.562
247.905
242.396
272.287
261.650
Std.
11.036
10.448
16.875
19.976
19.773
5.540
21.479
10.483
29.610
18.040
Best
227.764
237.089
252.680
216.798
218.722
222.809
219.166
227.236
230.774
230.880
Rank
6
9
10
1
2
4
3
5
7
8
F18
Mean
603.702
393.335
702.404
580.649
709.009
715.153
864.345
735.541
831.072
861.593
Std.
253.504
155.192
86.522
277.920
237.366
240.232
96.241
202.699
157.059
121.229
Best
310.000
310.000
559.128
310.000
310.000
393.632
700.558
310.075
310.000
310.000
Rank
1
1
4
1
1
3
5
2
1
1
F19
Mean
572.111
404.878
752.056
442.020
756.348
716.366
864.003
688.951
711.459
842.334
Std.
278.437
162.270
46.800
214.463
201.804
227.717
131.102
190.598
241.647
160.075
Best
310.000
310.000
656.476
310.000
417.269
366.021
698.725
312.483
310.009
310.000
Rank
1
1
6
1
5
4
7
3
2
1
F20
Mean
442.654
410.210
666.374
748.635
791.997
896.631
681.122
807.355
595.677
863.810
Std.
211.159
177.868
75.428
251.178
186.662
164.780
117.945
208.837
102.402
162.465
Best
310.000
310.000
566.636
310.000
310.000
486.451
693.240
310.000
314.640
310.000
Rank
1
1
4
1
1
3
5
1
2
1
F21
Mean
910.002
832.592
846.708
830.619
1110.454
931.239
1212.545
944.930
1002.404
1028.386
Std.
171.591
86.672
28.555
222.507
367.780
360.290
400.425
196.868
269.495
302.654
Best
660.019
585.919
790.540
660.000
660.000
660.000
660.000
613.723
660.000
660.000
Rank
4
1
5
3
3
3
3
2
3
3
F22
Mean
1074.270
1079.861
1141.724
1131.082
1150.209
1135.155
1177.217
924.494
961.467
1075.162
Std.
145.718
148.046
7.787
3.327
29.625
24.736
79.315
232.730
231.505
170.075
Best
660.000
660.000
1131.022
1125.586
1116.253
1110.458
1109.829
660.000
660.000
660.000
Rank
1
1
6
5
4
3
2
1
1
1
F23
Mean
829.609
918.677
906.039
1069.522
1178.115
1237.380
1431.577
998.441
1275.131
1176.097
Std.
56.624
1.797
42.468
179.455
232.518
257.118
315.479
149.953
146.070
188.499
Best
785.173
914.095
785.173
919.468
919.468
919.468
919.468
785.173
919.468
919.468
Rank
1
2
1
3
3
3
3
1
3
3
F24
Mean
460.000
460.000
696.247
588.505
460.000
460.000
1126.804
460.000
500.000
460.000
Std.
0.000
0.000
252.248
86.471
0.000
0.000
239.476
0.000
103.723
000.000
Best
460.000
460.000
460.211
460.001
460.000
460.000
760.000
460.000
460.000
460.000
Rank
1
1
3
2
1
1
4
1
1
1
F25
Mean
1079.394
1080.375
1079.655
612.785
1078.221
1076.067
1339.389
460.000
660.000
460.000
Std.
2.258
2.634
2.151
80.691
2.460
20650
217.394
0.000
164.002
000.000
Best
1075.267
1074.953
1077.376
460.000
1074.200
1071.563
1071.868
460.000
460.000
460.000
Rank
6
5
7
1
4
2
3
1
1
1
Average rank
2.84
2.88
5.4
3.76
3.08
3.4
3.24
2.76
3.84
2.76(2) On 30-Dimension Functions in CEC2005. In this section, the experiment of 30-dimension functions is carried out in CEC2005. The mean, standard deviations, and the best solutions of the error values are compared. In Table 3, all the algorithms are obtained after 30 independent runs; in addition, the optimal solution of the algorithm error value is sorted. From Table 3, it can be seen that MBSAgC is superior to other ten algorithms on F6, F7, F11, and F12 at D = 30. MBSAgC also performs best on functions F20, F21, and F23–F25. LBSA is the first ranking for the functions F1, F9, F20, F21, F23, and F24. SaDE is better than all other algorithms on functions F3, F4, F5, F10, and F16. OBSA ranks first on functions F18, F19, F21, and F23–F25. CLPSO and PSOFDR perform worse than the other nine algorithms on functions F21 and F24. The average ranking of MBSAgC is 2.56; it is also ranked first in all compared algorithms. The statistical results in Table 3 indicate that MBSAgC has a better global performance than the other algorithms, and MBSAgC has a better capability to get the best solution than other compared algorithms in 30 runs. According to the results in Table 3, it is obvious to find that MBSAgC performs better than all other comparison algorithms on 30-dimension functions in CEC2005.Table 3
Comparative results of ten algorithms for 30D problems from CEC2005.
F
AL
LBSA
BSA
SaDE
CLPSO
PSOFIPS
TLBO
ETLBO
PSOFDR
BSAISA
OBSA
MBSAgC
F1
Mean
0.000E + 00
3.995E − 16
0.000E + 00
4.839E − 10
1.057E − 06
5.836E − 26
3.138E − 27
9.33E + 02
1.270E − 13
6.253E − 14
6.442E − 14
Std.
0.000E + 00
2.267E − 16
0.000E + 00
1.409E − 10
3.274E − 07
2.769E − 26
1.661E − 27
1.632E + 03
3.559E − 14
1.734E − 14
1.965E − 14
Best
0.000E + 00
1.173E − 16
0.000E + 00
3.244E − 10
7.480E − 07
2.640E − 26
1.379E − 27
0.000E + 00
5.684E − 14
5.684E − 14
5.684E − 14
Rank
1
4
1
6
7
3
2
1
5
5
5
F2
Mean
4.995E − 02
2.644E + 03
8.734E − 03
6.355E + 03
4.560E + 02
7.480E − 05
6.632E − 05
9.374E + 01
2.545E + 02
2.005E + 03
2.010E + 00
Std.
4.852E − 02
2.038E + 02
4.026E − 03
7.050E + 02
6.497E + 01
3.424E − 05
7.342E − 05
8.638E + 01
9.556E + 01
1.270E + 03
1.614E + 00
Best
1.714E − 02
2.450E + 03
4.213E − 03
5.676E + 03
3.818E + 02
3.527E − 05
2.493E − 06
8.972E − 03
8.585E + 01
4.243E + 02
3.203E − 01
Rank
5
10
3
11
8
2
1
4
7
9
6
F3
Mean
9.240E + 05
3.069E + 06
4.385E + 05
4.814E + 07
1.265E + 07
1.344E + 06
1.331E + 06
1.757E + 06
2.331E + 06
7.732E + 06
1.974E + 06
Std.
3.486E + 05
2.791E + 05
1.529E + 05
5.129E + 06
1.170E + 06
6.881E + 05
6.881E + 05
6.131E + 05
7.157E + 05
3.513E + 06
7.342E + 05
Best
7.127E + 05
2.784E + 06
3.380E + 05
4.296E + 07
1.138E + 07
6.838E + 05
6.838E + 05
1.084E + 06
9.462E + 05
1.313E + 06
7.049E + 05
Rank
4
8
1
10
9
2
2
6
5
7
3
F4
Mean
7.097E + 02
9.404E + 03
4.769E + 02
1.613E + 04
3.019E + 03
2.128E + 03
8.611E + 02
7.308E + 02
9.381E + 03
1.759E + 04
3.427E + 03
Std.
4.298E + 02
7.848E + 02
7.790E + 02
1.800E + 03
8.707E + 02
2.915E + 03
7.453E + 02
2.258E + 02
2.302E + 03
5.206E + 03
2.863E + 03
Best
2.659E + 02
8.510E + 03
2.337E + 01
1.505E + 04
2.016E + 03
1.736E + 02
4.067E + 02
5.912E + 02
4.585E + 03
8.304E + 03
1.902E + 02
Rank
4
10
1
11
7
2
5
6
8
9
3
F5
Mean
3.029E + 03
3.334E + 03
2.427E + 03
4.315E + 03
3.104E + 03
4.546E + 03
4.161E + 03
6.156E + 03
3.906E + 03
5.079E + 03
3.551E + 03
Std.
7.939E + 02
1.767E + 02
2.731E + 02
9.405E + 02
1.055E + 02
3.140E + 02
4.601E + 02
1.750E + 03
4.184E + 02
9.557E + 02
6.652E + 02
Best
2.501E + 03
3.130E + 03
2.141E + 03
3.523E + 03
3.009E + 03
4.197E + 03
3.678E + 03
5.077E + 03
2.692E + 03
3.310E + 03
2.457E + 03
Rank
3
6
1
8
5
10
9
11
4
7
2
F6
Mean
2.958E + 01
7.803E + 01
7.807E + 01
3.974E + 01
2.891E + 01
2.297E + 01
3.895E + 01
1.638E + 07
1.299E + 02
9.279E + 01
5.594E + 01
Std.
4.229E + 01
3.027E + 01
2.212E + 00
2.123E + 01
2.546E + 00
2.780E + 00
3.169E + 01
2.646E + 07
9.030E + 01
2.766E + 01
4.434E + 01
Best
3.069E + 00
4.494E + 01
7.557E + 01
1.529E + 01
2.714E + 01
2.109E + 01
2.207E + 01
2.290E + 02
2.895E + 01
2.495E + 01
4.900E − 01
Rank
2
9
10
3
7
4
5
11
8
6
1
F7
Mean
4.696E + 03
4.696E + 03
4.696E + 03
4.696E + 03
3.846E − 01
4.696E + 03
4.696E + 03
4.918E + 03
5.994E − 01
5.078E − 01
2.142E − 02
Std.
1.575E − 12
1.720E − 08
3.750E − 12
0.000E + 00
9.646E − 02
1.575E − 12
9.905E − 13
3.844E + 02
1.723E − 01
1.931E − 01
2.577E − 02
Best
4.696E + 03
4.696E + 03
4.696E + 03
4.696E + 03
2.859E − 01
4.696E + 03
4.696E + 03
4.696E + 03
2.429E − 01
1.760E − 01
3.856E − 07
Rank
5
5
5
5
4
5
5
5
3
2
1
F8
Mean
2.096E + 01
2.298E + 01
2.097E + 01
2.093E + 01
2.093E + 01
2.098E + 01
2.089E + 01
2.092E + 01
2.089E + 01
2.095E + 01
2.096E + 01
Std.
5.689E − 02
4.595E − 02
2.665E − 02
8.812E − 02
1.807E − 02
1.689E − 02
6.181E − 02
2.802E − 02
6.723E − 02
5.909E − 02
7.067E − 02
Best
2.091E + 01
2.093E + 01
2.095E + 01
2.085E + 01
2.092E + 01
2.097E + 01
2.084E + 01
2.089E + 01
2.072E + 01
2.082E + 02
2.076E + 01
Rank
6
8
9
4
7
10
3
5
1
11
2
F9
Mean
1.184E − 15
2.227E + 00
3.317E − 01
5.082E − 04
5.904E + 01
9.618E + 01
6.567E + 01
5.164E + 01
1.327E − 01
4.961E + 00
2.820E + 00
Std.
2.051E − 15
1.295E + 00
5.744E − 01
2.078E − 04
1.441E + 01
2.185E + 01
3.011E + 01
5.605E + 00
4.320E − 01
4.199E + 00
2.996E + 00
Best
0.000E + 00
1.134E + 00
0.000E + 00
2.719E − 04
4.420E + 01
8.258E + 01
4.179E + 01
4.754E + 01
1.432E − 11
1.294E − 01
1.526E − 02
Rank
1
6
1
3
8
10
7
9
2
5
4
F10
Mean
7.969E + 01
8.281E + 01
7.975E + 01
1.647E + 02
1.822E + 02
9.228E + 01
9.913E + 01
1.141E + 02
1.101E + 02
1.575E + 02
1.358E + 02
Std.
7.575E + 00
1.541E + 01
4.349E + 01
5.872E + 00
5.261E + 00
4.150E + 01
1.259E + 01
1.892E + 01
1.696E + 01
5.376E + 01
3.937E + 01
Best
7.131E + 01
6.963E + 01
3.022E + 01
1.608E + 02
1.788E + 02
5.771E + 01
8.743E + 01
9.451E + 01
6.792E + 01
8.846E + 01
6.676E + 01
Rank
6
5
1
10
11
2
7
9
4
8
3
F11
Mean
2.833E + 01
3.091E + 01
2.997E + 01
2.798E + 01
2.879E + 01
3.467E + 01
3.714E + 01
1.803E + 01
2.666E + 01
2.586E + 01
2.198E + 01
Std.
1.680E + 00
2.377E + 00
1.669E + 00
1.205E + 00
2.252E + 00
4.680E + 00
1.629E + 00
2.803E + 00
2.365E + 00
2.798E + 00
4.908E + 00
Best
2.661E + 01
2.843E + 01
2.840E + 01
2.685E + 01
2.625E + 01
2.930E + 01
3.526E + 01
1.551E + 01
2.058E + 01
1.913E + 01
1.280E + 01
Rank
6
9
8
7
5
10
11
2
4
3
1
F12
Mean
2.976E + 03
2.801E + 04
6.056E + 03
4.411E + 04
2.555E + 04
9.523E + 03
1.550E + 04
2.882E + 04
7.015E + 03
1.536E + 04
4.162E + 03
Std.
1.281E + 03
1.173E + 04
7.626E + 03
3.443E + 03
9.809E + 03
3.881E + 03
1.420E + 04
4.351E + 04
2.924E + 03
9.177E + 03
3.474E + 03
Best
1.698E + 03
1.447E + 04
1.008E + 03
4.022E + 04
1.962E + 04
6.257E + 03
3.727E + 03
3.259E + 03
1.585E + 03
3.634E + 03
1.998E + 02
Rank
4
9
2
11
10
8
7
5
3
6
1
F13
Mean
1.883E + 00
2.524E + 00
4.792E + 00
3.409E + 00
1.358E + 01
4.248E + 00
4.063E + 00
2.870E + 00
1.882E + 00
3.382E + 00
2.930E + 00
Std.
1.203E − 01
1.192E − 01
6.707E − 01
2.156E − 01
1.025E − 01
5.438E − 01
1.202E + 00
7.919E − 01
1.674E − 01
4.748E − 01
4.052E − 01
Best
1.771E + 00
2.408E + 00
4.113E + 00
3.162E + 00
1.347E + 01
3.912E + 00
2.682E + 00
2.293E + 00
1.403E + 00
2.314E + 00
2.149E + 00
Rank
2
6
10
8
11
9
7
4
1
5
3
F14
Mean
1.286E + 01
1.309E + 01
1.309E + 01
1.312E + 01
1.280E + 01
1.289E + 01
1.309E + 01
1.185E + 01
1.281E + 01
1.291E + 01
1.277E + 01
Std.
1.284E − 01
3.739E − 01
1.846E − 01
1.258E − 01
1.223E − 01
1.513E − 01
7.225E − 02
7.922E − 01
2.286E − 01
2.685E − 01
3.944E − 01
Best
1.276E + 01
1.266E + 01
1.288E + 01
1.298E + 01
1.266E + 01
1.278E + 01
1.303E + 01
1.112E + 01
1.234E + 01
1.229E + 01
1.126E + 01
Rank
6
5
8
9
5
7
10
1
4
3
2
F15
Mean
4.015E + 02
7.111E + 01
2.354E + 02
1.214E + 02
3.544E + 02
4.713E + 02
4.746E + 02
5.258E + 02
9.242E + 01
2.367E + 02
3.054E + 02
Std.
1.013E + 02
1.159E + 01
2.047E + 02
2.545E + 01
6.430E + 01
9.623E + 01
4.142E + 01
5.548E + 01
1.230E + 02
4.901E + 01
7.386E + 01
Best
3.000E + 02
6.421E + 01
6.213E + 00
9.610E + 01
3.007E + 02
3.977E + 02
4.268E + 02
4.831E + 02
2.946E − 06
2.000E + 02
8.141E + 00
Rank
7
4
2
5
8
9
10
11
1
6
3
F16
Mean
8.450E + 01
1.484E + 02
1.051E + 02
2.018E + 02
2.507E + 02
2.400E + 02
2.652E + 02
2.413E + 02
1.122E + 02
1.892E + 02
3.217E + 02
Std.
5.658E + 00
3.611E + 01
3.010E + 01
2.174E + 01
5.874E + 01
2.269E + 02
1.175E + 02
1.107E + 02
1.453E + 01
9.878E + 01
1.745E + 02
Best
7.797E + 01
1.240E + 02
7.058E + 01
1.775E + 02
2.087E + 02
1.078E + 02
1.912E + 02
1.245E + 02
7.278E + 01
1.075E + 02
1.033E + 02
Rank
3
7
1
9
11
6
10
8
2
5
4
F17
Mean
1.511E + 02
2.117E + 02
1.663E + 02
3.001E + 02
2.812E + 02
2.639E + 02
2.404E + 02
1.932E + 02
1.706E + 02
3.262E + 02
2.957E + 02
Std.
4.484E + 01
1.541E + 01
1.345E + 01
4.652E + 01
6.842E + 01
1.708E + 02
1.690E + 02
1.022E + 02
3.320E + 01
1.391E + 02
1.052E + 02
Best
1.198E + 02
1.964E + 02
1.508 + 02
2.574E + 02
2.388E + 02
1.399E + 02
1.323E + 02
1.267E + 02
1.183E + 02
1.840E + 02
1.258E + 02
Rank
2
9
7
11
10
6
5
4
1
8
3
F18
Mean
9.186E + 02
9.153E + 02
9.116E + 02
9.098E + 02
8.323E + 02
9.148E + 02
8.771E + 02
9.388E + 02
8.444E + 02
9.148E + 02
9.312E + 02
Std.
6.806E + 00
3.761E + 00
4.342E − 01
7.681E − 01
1.686E + 00
4.459E + 00
6.679E + 01
2.396E + 01
6.401E + 01
2.253E + 01
3.644E + 01
Best
9.108E + 02
9.119E + 02
9.111E + 02
9.090E + 02
8.310E + 02
9.105E + 02
8.000E + 02
9.187E + 02
8.000E + 02
2.000E + 02
8.000E + 02
Rank
6
8
7
4
3
5
2
9
2
1
2
F19
Mean
9.137E + 02
9.161E + 02
8.741E + 02
9.100E + 02
8.323E + 02
9.594E + 02
9.289E + 02
9.449E + 02
8.467E + 02
8.632E + 02
9.210E + 02
Std.
1.897E − 01
2.038E + 00
6.415E + 01
4.197E − 01
4.396E − 01
5.067E + 01
2.414E + 01
2.678E + 01
6.268E + 01
6.018E + 01
2.448E + 01
Best
9.136E + 02
9.147E + 02
8.000E + 02
9.096E + 02
8.318E + 02
9.201E + 02
9.903E + 02
9.172E + 02
8.000E + 02
2.000E + 02
8.000E + 02
Rank
5
6
2
4
3
8
9
7
2
1
2
F20
Mean
8.779E + 02
9.149E + 02
8.737E + 02
9.088E + 02
8.321E + 02
9.475E + 02
9.216E + 02
9.501E + 02
8.822E + 02
90197E + 02
9.220E + 02
Std.
6.752E + 01
2.841E + 00
6.383E + 01
9.501E − 01
9.107E − 01
3.759E + 01
8.330E + 00
2.351E + 01
6.391E + 01
9.503E + 00
3.591E + 01
Best
8.000E + 02
9.128E + 02
8.000E + 02
9.078E + 02
8.310E + 02
9.141E + 02
9.136E + 02
9.327E + 02
8.000E + 02
9.098E + 02
8.000E + 02
Rank
1
5
1
3
2
7
6
8
1
4
1
F21
Mean
5.000E + 02
5.000E + 02
5.000E + 02
6.508E + 02
5.000E + 02
9.551E + 02
5.000E + 02
9.728E + 02
6.097E + 02
5.973E + 02
7.584E + 02
Std.
1.969E − 13
1.842E − 13
0.000E + 00
1.376E + 02
3.922E − 06
3.941E + 02
8.247E − 13
1.750E + 02
2.499E + 02
2.291E + 02
2.791E + 02
Best
5.000E + 02
5.000E + 02
5.000E + 02
5.427E + 02
5.000E + 02
5.000E + 02
5.000E + 02
8.549E + 02
5.000E + 02
5.000E + 02
5.000E + 02
Rank
1
1
1
2
1
1
1
3
1
1
1
F22
Mean
9.372E + 02
9.709E + 02
9.275E + 02
9.189E + 02
5.290E + 02
9.220E + 02
9.904E + 02
8.860E + 02
1.002E + 03
1.039E + 03
1.014E + 03
Std.
1.765E + 01
8.997E + 00
4.853E + 00
2.153E + 01
2.608E − 01
3.630E + 01
3.054E + 01
2.439E + 01
2.722E + 01
4.147E + 01
0.340E + 01
Best
9.266E + 02
9.617E + 02
9.230E + 02
8.984E + 02
5.287E + 02
8.831E + 02
8.747E + 02
8.648E + 02
9.615E + 02
9.733E + 02
9.437E + 02
Rank
7
10
6
5
1
4
3
2
9
11
8
F23
Mean
5.342E + 02
5.342E + 02
5.342E + 02
6.655E + 02
5.342E + 02
1.052E + 03
1.226E + 02
1.030E + 03
6.437E + 02
7.727E + 02
6.864E + 02
Std.
4.661E − 04
1.987E − 05
1.186E − 04
5.669E + 01
2.583E − 04
2.102E + 02
3.965E + 01
1.329E + 02
2.395E + 02
2.544E + 02
2.445E + 02
Best
5.342E + 02
5.342E + 02
5.342E + 02
6.255E + 02
5.342E + 02
8.094E + 02
1.184E + 03
8.767E + 02
5.342E + 02
5.342E + 02
5.342E + 02
Rank
1
1
1
2
1
3
5
4
1
1
1
F24
Mean
2.000E + 02
2.000E + 02
2.000E + 02
9.489E + 02
2.183E + 02
2.000E + 02
2.000E + 02
9.579E + 02
2.000E + 02
2.676E + 02
2.000E + 02
Std.
0.000E + 00
1.897E − 12
3.481E − 14
1.994E + 00
9.720E − 01
4.702E − 11
3.212E − 12
2.510E + 01
3.585E − 06
2.527E + 02
8.407E − 13
Best
2.000E + 02
2.000E + 02
2.000E + 02
9.471E + 02
2.176E + 02
2.000E + 02
2.000E + 02
9.399E + 02
2.000E + 02
2.000E + 02
2.000E + 02
Rank
1
1
1
4
2
1
1
3
1
1
1
F25
Mean
9.938E + 02
1.016E + 03
9.951E + 02
1.008E + 03
2.070E + 02
1.006E + 03
1.052E + 03
1.064E + 03
2.000E + 02
3.418E + 02
2.000E + 02
Std.
1.593E + 00
8.040E + 00
4.148E + 00
1.268E + 01
1.213E + 01
2.774E + 01
4.316E + 01
2.582E + 01
2.545E − 06
3.618E + 02
1.445E − 13
Best
9.922E + 02
1.007E + 03
9.924E + 02
1.000E + 03
2.000E + 02
9.817E + 02
1.009E + 03
1.036E + 03
2.000E + 02
2.000E + 02
2.000E + 02
Rank
3
6
4
5
1
2
7
8
1
1
1
Average rank
3.68
6.32
3.76
6.4
5.88
5.44
5.6
5.84
3.24
5.04
2.56(3) On 30-Dimension Functions in CEC2014. There are 30 benchmark functions in CEC2014. It is difficult to get the global optimal solution because their global optimum is shifted. In this part, we compare the mean, standard deviations, and the best solution of the error values between the actual solution value and the theoretical optimal value and rank the best error value of each function, and the statistical results of each comparison algorithm are given in Table 4. For all 30 functions, MBSAgC outperforms the other nine algorithms for functions F4, F5, F14, F19, and F21. PSOFIPS performs best on functions F6, F29, and F30. PSOFDR is superior to other algorithms on functions F11, F15, and F16. CLPSO performs a little unsatisfactory on all functions except function F26. LBSA ranks first in terms of the best for functions F2, F3, F8, F9, F10, F13, F17, F20, and F22. CBSA performs better than other algorithms for functions F12, F23, F27, and F28. For function F7, in terms of the best, MBSAgC and LBSA obtain the same function value; they are better than other algorithms. All algorithms get the same value in function F26 except CLBSA. For function F24 and F25, CBSA, TLBO, ETLBO, and DGSTLBO obtain the same value, and they are better than other six algorithms. The average ranking of three comparative criterions is also listed in the table. As we can see, LBSA is the best in all three criterions; MBSAgC performs slightly worse in best and Std than LBSA, but it is better than other algorithms. Overall, MBSAgC’s rank is still very competitive in CEC2014.Table 4
Comparative results of ten algorithms for 30D problems from CEC2014.
F
AL
PSOFIPS
PSOFDR
CLPSO
CBSA
CLBSA
TLBO
ETLBO
DGSTLBO
LBSA
MBSAgC
F1
Mean
1.02E + 07
2.78E + 06
2.92E + 07
4.12E + 07
1.61E + 08
8.28E + 05
3.33E + 05
1.04E + 07
2.79E + 05
1.26E + 06
Std
2.88E + 06
1.98E + 06
4.33E + 06
1.19E + 07
6.60E + 07
7.89E + 05
2.02E + 05
8.61E + 06
1.65E + 05
1.12E + 06
Best
5.60E + 06
1.32E + 05
2.14E + 07
2.17E + 07
5.69E + 07
5.77E + 04
1.37E + 05
3.15E + 06
7.85E + 04
1.27E + 05
Rank
7
4
8
9
10
1
5
6
2
3
F2
Mean
1.13E + 04
1.53E + 08
1.26E + 03
3.20E + 08
2.43E + 10
1.44E + 02
1.19E + 02
4.59E + 06
4.83E − 14
3.51E + 01
Std
5.27E + 03
3.61E + 08
7.40E + 02
6.61E + 07
7.40E + 09
1.72E + 02
1.56E + 02
1.11E + 07
1.92E − 14
5.42E + 01
Best
3.63E + 03
4.60E + 01
2.76E + 02
1.75E + 08
1.21E + 10
2.82E + 01
3.29E − 01
1.48E + 03
2.84E − 14
1.14E − 13
Rank
8
5
6
9
10
4
3
7
1
2
F3
Mean
6.94E + 03
5.40E + 02
7.67E + 02
2.37E + 03
3.09E + 04
2.39E + 03
2.14E + 03
1.44E + 01
2.27E − 13
5.67E + 02
Std
5.03E + 03
6.05E + 02
1.20E + 03
1.05E + 03
9.52E + 03
1.77E + 03
1.02E + 03
1.68E + 01
1.00E − 13
1.46E + 03
Best
1.32E + 03
1.36E + 01
5.76E + 01
9.14E + 02
1.36E + 04
2.90E + 02
5.59E + 02
6.71E − 01
1.14E − 13
1.56E + 00
Rank
9
4
5
8
10
6
7
2
1
3
F4
Mean
2.67E + 01
9.50E + 01
1.16E + 02
1.56E + 02
2.46E + 03
9.70E + 01
7.53E + 01
1.46E + 02
4.23E + 01
5.20E + 01
Std
6.37E − 01
3.24E + 01
1.33E + 01
2.13E + 01
1.56E + 03
3.36E + 01
5.88E + 00
3.78E + 01
3.57E + 01
3.63E + 01
Best
2.53E + 01
6.66E + 01
9.56E + 01
1.13E + 02
6.39E + 02
6.74E + 01
6.79E + 01
8.79E + 01
1.73E − 03
2.59E − 04
Rank
3
4
8
9
10
5
6
7
2
1
F5
Mean
2.10E + 01
2.09E + 01
2.05E + 01
4.14E + 01
7.61E + 02
2.10E + 01
2.09E + 01
2.10E + 01
2.03E + 01
2.63E − 13
Std
5.71E − 02
1.22E − 01
4.58E − 02
9.01E + 00
3.65E + 02
5.23E − 02
7.24E − 02
4.34E − 02
3.42E − 02
1.00E − 13
Best
2.09E + 01
2.06E + 01
2.04E + 01
2.51E + 01
1.58E + 02
2.09E + 01
2.08E + 01
2.09E + 01
2.03E + 01
1.14E − 13
Rank
6
4
3
7
8
6
5
6
2
1
F6
Mean
6.19E + 00
7.67E + 00
1.70E + 01
9.93E + 01
7.24E + 02
1.54E + 01
1.59E + 01
1.67E + 01
8.37E + 00
3.83E + 01
Std
2.22E + 00
1.98E + 00
6.10E − 01
1.09E + 01
2.63E + 02
2.14E + 00
2.75E + 00
3.45E + 00
3.22E + 00
2.81E + 01
Best
2.75E + 00
4.63E + 00
1.60E + 01
7.26E + 01
3.05E + 02
1.17E + 01
1.01E + 01
1.23E + 01
5.25E + 00
5.80E + 00
Rank
1
2
8
9
10
6
5
7
3
4
F7
Mean
2.56E − 03
1.33E + 01
3.92E − 03
4.00E + 00
2.41E + 02
6.74E − 02
2.75E − 02
1.01E + 00
5.41E − 03
9.84E − 03
Std
6.75E − 03
1.31E + 01
1.61E − 03
6.84E − 01
7.28E + 01
8.69E − 02
2.23E − 02
1.50E + 00
8.26E − 03
1.50E − 02
Best
7.08E − 05
1.48E − 02
1.37E − 03
1.91E + 00
9.95E + 01
7.84E − 12
1.02E − 12
1.20E − 01
1.14E − 13
1.14E − 13
Rank
4
6
5
8
9
3
2
7
1
1
F8
Mean
6.60E + 01
3.48E + 01
5.86E − 01
1.86E + 01
1.90E + 02
7.37E + 01
8.00E + 01
7.67E + 01
1.14E − 13
6.70E + 00
Std
1.22E + 01
5.70E + 00
6.20E − 01
1.86E + 00
1.89E + 01
2.11E + 01
1.88E + 01
2.45E + 01
0.00E + 00
5.26E + 00
Best
4.32E + 01
2.19E + 01
2.65E − 04
1.36E + 01
1.49E + 02
5.07E + 01
4.48E + 01
3.48E + 01
1.14E − 13
1.37E + 00
Rank
7
5
2
4
10
9
8
6
1
3
F9
Mean
1.53E + 02
5.53E + 01
7.79E + 01
9.56E + 01
1.98E + 02
7.68E + 01
8.00E + 01
9.84E + 01
4.32E + 01
8.48E + 01
Std
1.61E + 01
1.54E + 01
1.49E + 01
9.56E + 00
2.47E + 01
1.08E + 01
1.42E + 01
3.08E + 01
7.61E + 00
2.00E + 01
Best
1.21E + 02
3.51E + 01
5.52E + 01
7.30E + 01
1.49E + 02
5.87E + 01
5.87E + 01
6.41E + 01
2.70E + 01
4.24E + 01
Rank
8
2
4
7
9
5
5
6
1
3
F10
Mean
1.87E + 03
9.70E + 02
2.50E + 01
1.35E + 02
4.78E + 03
1.72E + 03
1.57E + 03
2.39E + 03
1.08E + 01
1.67E + 02
Std
4.44E + 02
3.47E + 02
5.21E + 00
2.35E + 01
4.01E + 02
7.41E + 02
4.26E + 02
4.71E + 02
5.07E + 00
8.55E + 01
Best
1.35E + 03
3.70E + 02
1.83E + 01
7.75E + 01
3.97E + 03
4.64E + 02
9.52E + 02
1.59E + 03
5.12E + 00
4.43E + 01
Rank
8
5
2
4
10
6
7
9
1
3
F11
Mean
5.76E + 03
3.08E + 03
3.24E + 03
2.80E + 03
5.08E + 03
6.71E + 03
6.46E + 03
3.39E + 03
2.31E + 03
2.67E + 03
Std
3.43E + 02
6.19E + 02
2.18E + 02
2.58E + 02
5.45E + 02
3.92E + 02
4.46E + 02
5.45E + 02
3.17E + 02
3.86E + 02
Best
5.23E + 03
1.77E + 03
2.98E + 03
2.22E + 03
4.08E + 03
6.01E + 03
5.52E + 03
2.42E + 03
1.86E + 03
1.87E + 03
Rank
8
1
6
4
7
10
9
5
2
3
F12
Mean
2.62E + 00
8.12E − 01
5.42E − 01
4.72E − 01
2.19E + 00
2.64E + 00
2.60E + 00
2.75E + 00
4.18E − 01
5.76E − 01
Std
2.82E − 01
6.51E − 01
8.53E − 02
6.66E − 02
6.22E − 01
2.47E − 01
3.32E − 01
2.62E − 01
5.15E − 02
8.85E − 02
Best
1.97E + 00
3.12E − 01
4.12E − 01
2.44E − 01
6.09E − 01
2.13E + 00
1.93E + 00
2.48E + 00
3.62E − 01
3.87E − 01
Rank
8
2
5
1
6
9
7
10
3
4
F13
Mean
3.46E − 01
4.35E − 01
4.18E − 01
3.89E − 01
3.88E + 00
4.88E − 01
4.11E − 01
4.71E − 01
2.90E − 01
3.70E − 01
Std
3.30E − 02
1.07E − 01
7.14E − 02
5.27E − 02
5.51E − 01
1.14E − 01
6.86E − 02
1.13E − 01
4.34E − 02
7.93E − 02
Best
2.89E − 01
2.55E − 01
2.77E − 01
2.63E − 01
2.74E + 00
3.50E − 01
2.97E − 01
3.09E − 01
2.10E − 01
2.22E − 01
Rank
6
3
5
4
10
9
7
8
1
2
F14
Mean
3.07E − 01
8.91E − 01
3.44E − 01
2.83E − 01
9.26E + 01
2.88E − 01
2.78E − 01
2.88E − 01
2.30E − 01
2.51E − 01
Std
3.63E − 02
3.59E − 01
3.38E − 02
3.87E − 02
2.41E + 01
4.70E − 02
4.42E − 02
4.92E − 02
2.69E − 02
4.40E − 02
Best
2.54E − 01
1.88E − 01
2.97E − 01
2.20E − 01
3.51E + 01
2.11E − 01
1.84E − 01
2.26E − 01
1.87E − 01
1.74E − 01
Rank
8
4
9
6
10
5
2
7
3
1
F15
Mean
1.58E + 01
4.49E + 00
1.09E + 01
1.99E + 01
8.76E + 03
1.80E + 01
1.89E + 01
3.75E + 01
5.93E + 00
8.76E + 00
Std
9.26E − 01
6.87E − 01
9.63E − 01
2.08E + 00
1.12E + 04
5.93E + 00
8.66E + 00
2.19E + 01
1.22E + 00
1.71E + 00
Best
1.41E + 01
3.18E + 00
9.09E + 00
1.55E + 01
5.91E + 02
1.03E + 01
9.75E + 00
1.54E + 01
3.87E + 00
5.15E + 00
Rank
7
1
4
9
10
6
5
8
2
3
F16
Mean
1.18E + 01
1.05E + 01
1.11E + 01
1.05E + 01
1.17E + 01
1.20E + 01
1.19E + 01
1.11E + 01
1.02E + 01
1.06E + 01
Std
2.50E − 01
8.64E − 01
2.31E − 01
3.26E − 01
5.87E − 01
4.35E − 01
3.94E − 01
6.62E − 01
4.55E − 01
4.12E − 01
Best
1.13E + 01
8.55E + 00
1.07E + 01
9.78E + 00
1.04E + 01
1.14E + 01
1.14E + 01
1.01E + 01
9.37E + 00
9.89E + 00
Rank
8
1
7
3
6
9
9
5
2
4
F17
Mean
3.76E + 05
1.36E + 05
2.07E + 06
3.46E + 06
4.28E + 05
1.91E + 05
2.11E + 05
1.67E + 05
4.42E + 04
1.92E + 05
Std
1.12E + 05
1.04E + 05
6.62E + 05
1.54E + 06
4.64E + 05
1.73E + 05
9.21E + 04
2.13E + 05
3.70E + 04
1.60E + 05
Best
2.44E + 05
2.69E + 04
1.07E + 06
9.41E + 05
3.61E + 04
3.29E + 04
8.02E + 04
3.54E + 04
8.62E + 03
3.99E + 04
Rank
8
2
10
9
5
3
7
4
1
6
F18
Mean
1.59E + 03
3.38E + 03
4.92E + 02
4.54E + 06
2.46E + 04
2.93E + 03
3.31E + 03
8.71E + 02
1.14E + 03
2.19E + 03
Std
8.50E + 02
4.21E + 03
9.30E + 01
1.89E + 06
1.06E + 05
2.35E + 03
4.11E + 03
1.02E + 03
1.29E + 03
2.81E + 03
Best
4.78E + 02
3.13E + 02
3.44E + 02
1.54E + 06
2.80E + 02
1.35E + 02
1.64E + 02
6.17E + 01
6.23E + 01
6.28E + 01
Rank
9
7
8
10
6
4
5
1
2
3
F19
Mean
1.19E + 01
1.57E + 01
1.01E + 01
2.04E + 01
8.25E + 01
2.12E + 01
2.12E + 01
2.71E + 01
6.31E + 00
6.90E + 00
Std
8.29E − 01
2.33E + 01
1.02E + 00
5.81E + 00
3.96E + 01
2.55E + 01
2.54E + 01
2.86E + 01
1.01E + 00
2.05E + 00
Best
1.01E + 01
4.44E + 00
8.20E + 00
1.33E + 01
2.24E + 01
5.77E + 00
6.72E + 00
9.65E + 00
4.60E + 00
3.34E + 00
Rank
8
2
6
9
10
4
5
7
3
1
F20
Mean
5.81E + 03
7.05E + 03
5.95E + 03
6.95E + 03
7.38E + 03
1.54E + 03
1.44E + 03
4.28E + 02
9.02E + 01
2.02E + 03
Std
2.71E + 03
1.27E + 04
3.35E + 03
3.35E + 03
7.07E + 03
9.69E + 02
6.69E + 02
1.77E + 02
8.20E + 01
2.99E + 03
Best
2.36E + 03
1.37E + 03
2.48E + 03
1.05E + 03
3.45E + 02
6.68E + 02
3.41E + 02
2.03E + 02
3.06E + 01
5.99E + 01
Rank
9
8
10
7
5
6
4
3
1
2
F21
Mean
1.49E + 05
6.77E + 04
3.39E + 05
6.64E + 05
4.18E + 04
9.72E + 04
1.19E + 05
2.20E + 04
9.86E + 03
6.55E + 04
Std
6.40E + 04
4.21E + 04
1.22E + 05
2.63E + 05
5.34E + 04
9.01E + 04
1.00E + 05
2.22E + 04
1.32E + 04
6.81E + 04
Best
8.47E + 04
8.85E + 03
1.25E + 05
1.26E + 05
2.10E + 03
1.89E + 04
3.16E + 04
4.09E + 03
1.22E + 03
1.19E + 03
Rank
8
5
9
10
3
6
7
4
2
1
F22
Mean
2.25E + 02
2.27E + 02
2.70E + 02
4.19E + 02
3.98E + 02
2.81E + 02
3.27E + 02
3.14E + 02
1.14E + 02
2.27E + 02
Std
7.27E + 01
9.97E + 01
5.98E + 01
1.04E + 02
1.63E + 02
1.08E + 02
1.33E + 02
1.41E + 02
7.56E + 01
1.20E + 02
Best
1.67E + 02
1.44E + 02
1.59E + 02
1.84E + 02
1.61E + 02
1.64E + 02
5.82E + 01
1.49E + 02
2.41E + 01
2.68E + 01
Rank
9
4
6
10
7
8
3
5
1
2
F23
Mean
3.14E + 02
3.18E + 02
3.15E + 02
2.26E + 02
3.87E + 02
3.15E + 02
3.15E + 02
3.15E + 02
3.15E + 02
3.15E + 02
Std
1.57E − 04
3.50E + 00
2.29E − 01
4.77E + 01
2.07E + 01
1.23E − 11
1.78E − 10
4.43E − 01
1.44E − 13
3.92E − 13
Best
3.14E + 02
3.15E + 02
3.15E + 02
2.00E + 02
3.34E + 02
3.15E + 02
3.15E + 02
3.15E + 02
3.15E + 02
3.15E + 02
Rank
2
3
3
1
4
3
3
3
3
3
F24
Mean
2.24E + 02
2.27E + 02
2.27E + 02
2.00E + 02
2.79E + 02
2.00E + 02
2.00E + 02
2.00E + 02
2.26E + 02
2.38E + 02
Std
5.46E − 01
2.58E + 00
1.01E + 00
5.57E − 04
9.88E + 00
2.20E − 03
2.19E − 03
9.68E − 04
1.28E + 00
5.98E + 00
Best
2.23E + 02
2.23E + 02
2.25E + 02
2.00E + 02
2.60E + 02
2.00E + 02
2.00E + 02
2.00E + 02
2.24E + 02
2.25E + 02
Rank
2
2
4
1
5
1
1
1
3
4
F25
Mean
2.07E + 02
2.06E + 02
2.10E + 02
2.00E + 02
2.24E + 02
2.01E + 02
2.00E + 02
2.02E + 02
2.09E + 02
2.06E + 02
Std
2.45E + 00
2.09E + 00
1.62E + 00
8.04E − 09
4.39E + 00
2.39E + 00
5.33E − 06
3.62E + 00
3.10E + 00
2.68E + 00
Best
2.04E + 02
2.04E + 02
2.08E + 02
2.00E + 02
2.16E + 02
2.00E + 02
2.00E + 02
2.00E + 02
2.04E + 02
2.04E + 02
Rank
2
2
3
1
4
1
1
1
2
2
F26
Mean
1.70E + 02
1.51E + 02
1.00E + 02
1.00E + 02
1.43E + 02
1.10E + 02
1.00E + 02
1.10E + 02
1.00E + 02
1.34E + 02
Std
4.83E + 01
5.18E + 01
3.00E − 02
4.88E − 02
4.93E + 01
3.15E + 01
1.57E − 01
3.15E + 01
7.52E − 02
4.78E + 01
Best
1.00E + 02
1.00E + 02
1.00E + 02
1.00E + 02
1.01E + 02
1.00E + 02
1.00E + 02
1.00E + 02
1.00E + 02
1.00E + 02
Rank
1
1
1
1
2
1
1
1
1
1
F27
Mean
4.34E + 02
7.12E + 02
4.38E + 02
3.62E + 02
8.61E + 02
5.47E + 02
6.56E + 02
7.94E + 02
4.34E + 02
5.37E + 02
Std
3.50E + 01
1.84E + 02
1.33E + 01
9.59E + 01
2.83E + 02
1.66E + 02
2.13E + 02
2.15E + 02
6.71E + 01
1.46E + 02
Best
3.76E + 02
4.02E + 02
4.17E + 02
2.00E + 02
4.44E + 02
4.02E + 02
4.02E + 02
4.06E + 02
3.71E + 02
4.01E + 02
Rank
3
5
7
1
8
5
5
6
2
4
F28
Mean
3.98E + 02
1.48E + 03
9.21E + 02
5.74E + 02
2.66E + 03
1.24E + 03
1.18E + 03
1.43E + 03
8.34E + 02
1.20E + 03
Std
1.16E + 01
3.70E + 02
3.47E + 01
4.65E + 02
4.48E + 02
3.51E + 02
2.30E + 02
4.37E + 02
3.87E + 01
2.91E + 02
Best
3.89E + 02
1.02E + 03
8.64E + 02
2.00E + 02
1.99E + 03
9.20E + 02
8.94E + 02
9.96E + 02
7.87E + 02
7.48E + 02
Rank
2
9
5
1
10
7
6
8
4
3
F29
Mean
2.14E + 02
6.65E + 06
3.11E + 04
3.37E + 04
7.77E + 06
3.31E + 06
3.93E + 06
3.08E + 06
1.29E + 03
1.19E + 03
Std
1.02E + 00
1.10E + 07
1.52E + 04
1.09E + 04
1.58E + 07
5.39E + 06
5.14E + 06
4.99E + 06
5.91E + 02
2.70E + 02
Best
2.12E + 02
2.17E + 03
7.11E + 03
1.12E + 04
1.45E + 03
1.25E + 03
1.44E + 03
9.91E + 02
7.75E + 02
7.84E + 02
Rank
1
8
9
10
7
5
6
4
2
3
F30
Mean
7.00E + 02
1.82E + 04
7.94E + 03
1.38E + 04
7.38E + 04
3.35E + 03
3.37E + 03
6.47E + 03
1.93E + 03
2.42E + 03
Std
8.54E + 01
3.03E + 04
2.72E + 03
3.52E + 03
4.24E + 04
1.35E + 03
1.75E + 03
3.43E + 03
5.26E + 02
4.86E + 02
Best
5.98E + 02
1.78E + 03
5.62E + 03
7.35E + 03
3.87E + 03
1.62E + 03
1.58E + 03
3.37E + 03
1.15E + 03
1.56E + 03
Rank
1
6
9
10
8
5
4
7
2
3
Average rank
Mean
4.6
5.27
4.67
5.57
8.43
5.43
5.13
5.633
1.8
3.667
Std
4.033
6.57
3.7
5.5
9.1
5.8
5.17
6.87
2.83
5.1
Best
5.7
3.9
5.9
6.07
7.63
5.27
5
5.37
1.9
2.63


### 5.1.2. Statistical Analyses for Benchmark Results

Sign tests [84] is a common method to determine whether there is significant difference between two algorithms. In this paper, we use the two-tailed sign tests which has a significance level of 0.05 to test the differences between the results obtained by MBSAgC and other compared algorithms. The statistical results are listed in Table 5. The best solution is used as the criterion of the sign test. The signs “+,” “≈,” and “−” represent that MBSAgC performs better, almost the same, and worse than other compared algorithms, respectively. And the null hypothesis in this paper is that the performances of MBSAgC and one of the other compared algorithms are not significantly differential.Table 5
Comparisons between MBSAgC and other algorithms in sign tests.
AL
+
≈
−
Total
P
LBSA
28
18
34
80
0.619
CLPSO
68
5
7
80
0.000
PSOFIPS
54
15
11
80
0.000
TLBO
49
14
17
80
0.000
ETLBO
54
14
12
80
0.000
PSOFDR
52
8
20
80
0.000
BSAISA
19
17
14
50
0.474
OBSA
32
13
5
50
0.004
BSA
30
12
8
50
0.001
CBSA
22
1
7
30
0.003
CLBSA
29
0
1
30
0.000
DGSTLBO
23
2
5
30
0.002
SADE
10
5
10
25
0.717Thirteen algorithms are compared in the benchmark experiment. Table5 lists the analytical test results of all the algorithms. The P values of supporting the null hypothesis of sign test for ten compared algorithms (MBSAgC-CLPSO, MBSAgC-PSOFIPS, MBSAgC-TLBO, MBSAgC-ETLBO, MBSAgC-PSOFDR, MBSAgC-OBSA, MBSAgC-BSA, MBSAgC-CBSA, MBSAgC-CLBSA, and MBSAgC-DGSTLBA) are lower than significance level 0.05, so we can reject the null hypothesis. And it is illustrated that the optimization performance of MBSAgC is significantly better than those of the other ten algorithms. The P values of MBSAgC-LBSA, MBSAgC-BSAISA, and MBSAgC-SaDE are higher than 0.05; they are 0.619, 0.474, and 0.717, respectively, so we cannot reject the null hypothesis. However, it does not mean that MBSAgC is much worse than them. In the light of sign value from Table 5, for MBSAgC-LBSA, MBSAgC is worse than LBSA on thirty-four functions, and it performs better on twenty-eight functions; moreover, MBSAgC performs better on both 10D and 30D in CEC2005. MBSAgC is slightly worse than BSAISA on fourteen functions, but better than BSAISA on nineteen functions. For MBSAgC-SaDE, the number of “+” is the same as the number of “−,” but MBSAgC is better than SaDE in comprehensive ranking.


## 5.1.1. Experimental Results

(1) On 10-Dimension Functions in CEC2005. Table 2 shows the simulation results of 10-dimension functions in CEC2005. Table 2 compares the mean, standard deviations, and the best solution of the ten algorithms. With respect to the best solution, the results of Table 2 show that the proposed algorithm MBSAgC ranks first in functions F1, F2, F4, F6, F9, F12, F15, F18–F20, F22, F24, and F25. LBSA is superior to other nine algorithms on function F3. For function F3, all compared algorithms could not find the theoretical optimal solution. The rank of MBSAgC in terms of best solution is eighth, it is worse than most of others, and the best solution of LBSA is closest to the theoretical optimal value, and the standard deviation of LBSA is also the smallest among ten algorithms. The best solution of PSOFDR is superior to other algorithms in functions F10, F11, and F14. The best solution of BSAISA algorithm is better than other nine algorithms in function F7. For function F1, all of the compared algorithms get the global optimal solution. For function F2, the best solution of CLPSO and OBSA is worse than the other eight algorithms. For function F4, BSAISA, OBSA, BSA, and CLPSO could not converge to the global optimum. MBSAgC can find the theoretical optimal solution on functions F1, F2, F4, F6, F9, F12, and F15. In Table 2, in terms of the optimal solutions of twenty-five functions, thirteen functions for the proposed algorithm MBSAgC rank first. The average ranking of the proposed algorithm is 2.76; moreover, MBSAgC ranks first in the best solution. To sum up, the performance of MBSAgC is competitive with other algorithms.Table 2
Comparative results of ten algorithms for 10D problems from CEC2005.
F
AL
LBSA
BSA
CLPSO
PSOFIPS
TLBO
ETLBO
PSOFDR
BSAISA
OBSA
MBSAgC
F1
Mean
−450.000
−450.000
−450.000
−450.000
−450.000
−450.000
−439.674
−450.000
−450.000
−450.000
Std.
0.000
0.000
0.000
0.000
0.000
0.000
32.653
0.000
1.847E − 11
0.000
Best
−450.000
−450.000
−450.000
−450.000
−450.000
−450.000
−450.000
−450.000
−450.000
−450.000
Rank
1
1
1
1
1
1
1
1
1
1
F2
Mean
−450.000
−449.997
−449.676
−450.000
−450.000
−450.000
−450.000
−449.968
−446.527
−450.000
Std.
0.000
0.005
0.177
0.000
0.000
0.000
0.000
0.044
3.336
6.91E − 06
Best
−450.000
−450.000
−449.906
−450.000
−450.000
−450.000
−450.000
−450.000
−449.791
−450.000
Rank
1
1
2
1
1
1
1
1
3
1
F3
Mean
5652.945
21518.124
695430.984
114431.841
70188.530
132295.917
143065.100
31699.067
427940.36
176254.16
Std.
7686.985
18993.641
263113.140
54898.334
47419.217
98983.396
156627.508
17717.113
471417.61
126171.69
Best
251.786
3316.003
333184.289
48691.653
22503.152
23119.940
6576.186
9211.109
35974.764
39533.675
Rank
1
2
10
9
5
6
3
4
7
8
F4
Mean
−450.000
−449.800
−445.693
−450.000
−450.000
−450.000
−438.829
−444.970
−430.129
−449.999
Std.
0.000
0.306
2.683
0.000
0.000
0.000
23.550
9.926
19.385
0.001
Best
−450.000
−449.994
−449.361
−450.000
−450.000
−450.000
−450.000
−449.972
−448.989
−450.000
Rank
1
2
5
1
1
1
1
4
3
1
F5
Mean
−310.000
−310.000
−310.000
−233.009
−310.000
−310.000
−310.000
−301.311
−228.735
−309.328
Std.
0.000
0.000
0.000
33.727
0.000
0.000
0.000
7.416
64.078
1.034
Best
−310.000
−310.000
−310.000
−289.340
−310.000
−310.000
−310.000
−309.052
−304.344
−309.932
Rank
1
1
1
5
1
1
1
3
4
2
F6
Mean
390.548
392.105
391.599
394.658
392.592
392.167
399.415
391.651
404.196
397.153
Std.
1.112
2.741
1.610
0.405
3.041
2.313
18.471
2.043
11.599
7.459
Best
390.001
390.091
390.348
393.841
390.000
390.000
390.002
390.005
390.000
390.000
Rank
2
5
6
8
1
1
3
4
1
1
F7
Mean
1087.046
1087.082
1087.208
−179.810
1087.046
1087.046
1087.288
−179.935
−179.816
−179.816
Std.
0.000
0.076
0.045
0.098
0.000
0.000
0.197
0.043
0.078
0.086
Best
1087.046
1087.046
1087.046
−179.949
1087.046
1087.046
1087.266
−179.986
−179.933
−179.958
Rank
5
5
5
3
5
5
6
1
4
2
F8
Mean
−119.643
−119.681
−119.624
−119.640
−119.628
−119.648
−119.714
−119.616
−119.607
−119.796
Std.
0.053
0.085
0.081
0.090
0.087
0.066
0.075
0.070
0.072
0.070
Best
−119.728
−119.812
−119.792
−119.851
−119.831
−119.628
−119.800
−119.747
−119.722
−119.741
Rank
8
3
6
1
2
10
4
7
9
5
F9
Mean
−330.000
−330.000
−330.000
−329.333
−322.342
−322.737
−327.314
−330.000
−329.343
−329.877
Std.
0.000
0.000
0.000
0.723
2.928
2.696
1.487
0.000
0.844
0.312
Best
−330.000
−330.000
−330.000
−330.000
−326.020
−327.015
−329.005
−330.000
−330.000
−330.000
Rank
1
1
1
1
4
3
2
1
1
1
F10
Mean
−323.493
−320.479
−316.698
−317.409
−319.885
−321.513
−319.259
−319.005
−310.290
−317.531
Std.
1.987
3.542
3.125
5.428
4.132
3.593
6.612
3.914
9.254
6.979
Best
−325.724
−324.562
−322.465
−324.188
−325.025
−327.015
−328.010
−325.505
−323.468
−324.812
Rank
3
7
10
8
5
2
1
4
9
6
F11
Mean
95.188
94.889
95.319
93.633
94.002
94.632
91.962
94.957
95.001
94.381
Std.
0.746
1.196
0.572
0.536
1.652
0.997
1.023
0.848
1.271
1.627
Best
93.737
92.408
94.425
92.922
91.059
92.619
90.702
92.744
92.209
91.286
Rank
9
5
10
8
2
6
1
7
4
3
F12
Mean
−444.398
−438.290
−363.666
−422.531
1445.385
794.693
−203.253
−448.681
−197.091
−294.114
Std.
16.545
16.021
65.389
47.468
5027.194
2860.501
507.838
13.907
645.215
421.269
Best
−459.985
−455.477
−431.128
−455.719
−460.000
−460.000
−459.496
−460.000
−458.383
−460.000
Rank
2
6
7
5
1
1
3
1
4
1
F13
Mean
−129.666
−129.697
−129.564
−128.810
−129.196
−129.044
−129.352
−129.577
−129.353
−129.342
Std.
0.073
0.118
0.118
0.146
0.165
0.256
0.271
0.067
0.151
0.175
Best
−129.830
−129.932
−129.769
−129.109
−129.447
−129.384
−129.745
−129.728
−129.663
−129.625
Rank
2
1
3
10
8
9
4
5
6
7
F14
Mean
−296.902
−296.694
−296.591
−297.170
−297.215
−297.081
−297.241
−296.757
−296.517
−296.627
Std.
0.207
0.205
0.151
0.306
0.249
0.459
0.685
0.215
0.253
0.276
Best
−297.288
−297.108
−296.948
−297.642
−297.566
−297.701
−298.570
−297.395
−297.095
−297.545
Rank
7
8
10
3
4
2
1
6
9
5
F15
Mean
120.030
124.232
140.582
209.829
450.644
401.382
387.275
123.356
170.949
3.654
Std.
0.089
12.918
28.900
99.415
141.887
156.564
203.967
11.180
99.357
200.365
Best
120.000
120.000
120.002
120.460
195.411
197.129
176.891
120.000
120.003
120.000
Rank
1
1
2
4
6
7
5
1
3
1
F16
Mean
223.754
231.537
257.700
232.105
228.617
230.096
254.799
228.345
245.791
235.776
Std.
3.450
10.701
14.508
10.777
9.831
9.972
29.896
5.584
10.049
9.911
Best
218.507
209.614
223.542
221.355
219.015
217.668
222.865
215.613
221.217
218.609
Rank
4
1
10
8
6
3
9
2
7
5
F17
Mean
241.073
249.588
269.953
251.164
246.701
228.562
247.905
242.396
272.287
261.650
Std.
11.036
10.448
16.875
19.976
19.773
5.540
21.479
10.483
29.610
18.040
Best
227.764
237.089
252.680
216.798
218.722
222.809
219.166
227.236
230.774
230.880
Rank
6
9
10
1
2
4
3
5
7
8
F18
Mean
603.702
393.335
702.404
580.649
709.009
715.153
864.345
735.541
831.072
861.593
Std.
253.504
155.192
86.522
277.920
237.366
240.232
96.241
202.699
157.059
121.229
Best
310.000
310.000
559.128
310.000
310.000
393.632
700.558
310.075
310.000
310.000
Rank
1
1
4
1
1
3
5
2
1
1
F19
Mean
572.111
404.878
752.056
442.020
756.348
716.366
864.003
688.951
711.459
842.334
Std.
278.437
162.270
46.800
214.463
201.804
227.717
131.102
190.598
241.647
160.075
Best
310.000
310.000
656.476
310.000
417.269
366.021
698.725
312.483
310.009
310.000
Rank
1
1
6
1
5
4
7
3
2
1
F20
Mean
442.654
410.210
666.374
748.635
791.997
896.631
681.122
807.355
595.677
863.810
Std.
211.159
177.868
75.428
251.178
186.662
164.780
117.945
208.837
102.402
162.465
Best
310.000
310.000
566.636
310.000
310.000
486.451
693.240
310.000
314.640
310.000
Rank
1
1
4
1
1
3
5
1
2
1
F21
Mean
910.002
832.592
846.708
830.619
1110.454
931.239
1212.545
944.930
1002.404
1028.386
Std.
171.591
86.672
28.555
222.507
367.780
360.290
400.425
196.868
269.495
302.654
Best
660.019
585.919
790.540
660.000
660.000
660.000
660.000
613.723
660.000
660.000
Rank
4
1
5
3
3
3
3
2
3
3
F22
Mean
1074.270
1079.861
1141.724
1131.082
1150.209
1135.155
1177.217
924.494
961.467
1075.162
Std.
145.718
148.046
7.787
3.327
29.625
24.736
79.315
232.730
231.505
170.075
Best
660.000
660.000
1131.022
1125.586
1116.253
1110.458
1109.829
660.000
660.000
660.000
Rank
1
1
6
5
4
3
2
1
1
1
F23
Mean
829.609
918.677
906.039
1069.522
1178.115
1237.380
1431.577
998.441
1275.131
1176.097
Std.
56.624
1.797
42.468
179.455
232.518
257.118
315.479
149.953
146.070
188.499
Best
785.173
914.095
785.173
919.468
919.468
919.468
919.468
785.173
919.468
919.468
Rank
1
2
1
3
3
3
3
1
3
3
F24
Mean
460.000
460.000
696.247
588.505
460.000
460.000
1126.804
460.000
500.000
460.000
Std.
0.000
0.000
252.248
86.471
0.000
0.000
239.476
0.000
103.723
000.000
Best
460.000
460.000
460.211
460.001
460.000
460.000
760.000
460.000
460.000
460.000
Rank
1
1
3
2
1
1
4
1
1
1
F25
Mean
1079.394
1080.375
1079.655
612.785
1078.221
1076.067
1339.389
460.000
660.000
460.000
Std.
2.258
2.634
2.151
80.691
2.460
20650
217.394
0.000
164.002
000.000
Best
1075.267
1074.953
1077.376
460.000
1074.200
1071.563
1071.868
460.000
460.000
460.000
Rank
6
5
7
1
4
2
3
1
1
1
Average rank
2.84
2.88
5.4
3.76
3.08
3.4
3.24
2.76
3.84
2.76(2) On 30-Dimension Functions in CEC2005. In this section, the experiment of 30-dimension functions is carried out in CEC2005. The mean, standard deviations, and the best solutions of the error values are compared. In Table 3, all the algorithms are obtained after 30 independent runs; in addition, the optimal solution of the algorithm error value is sorted. From Table 3, it can be seen that MBSAgC is superior to other ten algorithms on F6, F7, F11, and F12 at D = 30. MBSAgC also performs best on functions F20, F21, and F23–F25. LBSA is the first ranking for the functions F1, F9, F20, F21, F23, and F24. SaDE is better than all other algorithms on functions F3, F4, F5, F10, and F16. OBSA ranks first on functions F18, F19, F21, and F23–F25. CLPSO and PSOFDR perform worse than the other nine algorithms on functions F21 and F24. The average ranking of MBSAgC is 2.56; it is also ranked first in all compared algorithms. The statistical results in Table 3 indicate that MBSAgC has a better global performance than the other algorithms, and MBSAgC has a better capability to get the best solution than other compared algorithms in 30 runs. According to the results in Table 3, it is obvious to find that MBSAgC performs better than all other comparison algorithms on 30-dimension functions in CEC2005.Table 3
Comparative results of ten algorithms for 30D problems from CEC2005.
F
AL
LBSA
BSA
SaDE
CLPSO
PSOFIPS
TLBO
ETLBO
PSOFDR
BSAISA
OBSA
MBSAgC
F1
Mean
0.000E + 00
3.995E − 16
0.000E + 00
4.839E − 10
1.057E − 06
5.836E − 26
3.138E − 27
9.33E + 02
1.270E − 13
6.253E − 14
6.442E − 14
Std.
0.000E + 00
2.267E − 16
0.000E + 00
1.409E − 10
3.274E − 07
2.769E − 26
1.661E − 27
1.632E + 03
3.559E − 14
1.734E − 14
1.965E − 14
Best
0.000E + 00
1.173E − 16
0.000E + 00
3.244E − 10
7.480E − 07
2.640E − 26
1.379E − 27
0.000E + 00
5.684E − 14
5.684E − 14
5.684E − 14
Rank
1
4
1
6
7
3
2
1
5
5
5
F2
Mean
4.995E − 02
2.644E + 03
8.734E − 03
6.355E + 03
4.560E + 02
7.480E − 05
6.632E − 05
9.374E + 01
2.545E + 02
2.005E + 03
2.010E + 00
Std.
4.852E − 02
2.038E + 02
4.026E − 03
7.050E + 02
6.497E + 01
3.424E − 05
7.342E − 05
8.638E + 01
9.556E + 01
1.270E + 03
1.614E + 00
Best
1.714E − 02
2.450E + 03
4.213E − 03
5.676E + 03
3.818E + 02
3.527E − 05
2.493E − 06
8.972E − 03
8.585E + 01
4.243E + 02
3.203E − 01
Rank
5
10
3
11
8
2
1
4
7
9
6
F3
Mean
9.240E + 05
3.069E + 06
4.385E + 05
4.814E + 07
1.265E + 07
1.344E + 06
1.331E + 06
1.757E + 06
2.331E + 06
7.732E + 06
1.974E + 06
Std.
3.486E + 05
2.791E + 05
1.529E + 05
5.129E + 06
1.170E + 06
6.881E + 05
6.881E + 05
6.131E + 05
7.157E + 05
3.513E + 06
7.342E + 05
Best
7.127E + 05
2.784E + 06
3.380E + 05
4.296E + 07
1.138E + 07
6.838E + 05
6.838E + 05
1.084E + 06
9.462E + 05
1.313E + 06
7.049E + 05
Rank
4
8
1
10
9
2
2
6
5
7
3
F4
Mean
7.097E + 02
9.404E + 03
4.769E + 02
1.613E + 04
3.019E + 03
2.128E + 03
8.611E + 02
7.308E + 02
9.381E + 03
1.759E + 04
3.427E + 03
Std.
4.298E + 02
7.848E + 02
7.790E + 02
1.800E + 03
8.707E + 02
2.915E + 03
7.453E + 02
2.258E + 02
2.302E + 03
5.206E + 03
2.863E + 03
Best
2.659E + 02
8.510E + 03
2.337E + 01
1.505E + 04
2.016E + 03
1.736E + 02
4.067E + 02
5.912E + 02
4.585E + 03
8.304E + 03
1.902E + 02
Rank
4
10
1
11
7
2
5
6
8
9
3
F5
Mean
3.029E + 03
3.334E + 03
2.427E + 03
4.315E + 03
3.104E + 03
4.546E + 03
4.161E + 03
6.156E + 03
3.906E + 03
5.079E + 03
3.551E + 03
Std.
7.939E + 02
1.767E + 02
2.731E + 02
9.405E + 02
1.055E + 02
3.140E + 02
4.601E + 02
1.750E + 03
4.184E + 02
9.557E + 02
6.652E + 02
Best
2.501E + 03
3.130E + 03
2.141E + 03
3.523E + 03
3.009E + 03
4.197E + 03
3.678E + 03
5.077E + 03
2.692E + 03
3.310E + 03
2.457E + 03
Rank
3
6
1
8
5
10
9
11
4
7
2
F6
Mean
2.958E + 01
7.803E + 01
7.807E + 01
3.974E + 01
2.891E + 01
2.297E + 01
3.895E + 01
1.638E + 07
1.299E + 02
9.279E + 01
5.594E + 01
Std.
4.229E + 01
3.027E + 01
2.212E + 00
2.123E + 01
2.546E + 00
2.780E + 00
3.169E + 01
2.646E + 07
9.030E + 01
2.766E + 01
4.434E + 01
Best
3.069E + 00
4.494E + 01
7.557E + 01
1.529E + 01
2.714E + 01
2.109E + 01
2.207E + 01
2.290E + 02
2.895E + 01
2.495E + 01
4.900E − 01
Rank
2
9
10
3
7
4
5
11
8
6
1
F7
Mean
4.696E + 03
4.696E + 03
4.696E + 03
4.696E + 03
3.846E − 01
4.696E + 03
4.696E + 03
4.918E + 03
5.994E − 01
5.078E − 01
2.142E − 02
Std.
1.575E − 12
1.720E − 08
3.750E − 12
0.000E + 00
9.646E − 02
1.575E − 12
9.905E − 13
3.844E + 02
1.723E − 01
1.931E − 01
2.577E − 02
Best
4.696E + 03
4.696E + 03
4.696E + 03
4.696E + 03
2.859E − 01
4.696E + 03
4.696E + 03
4.696E + 03
2.429E − 01
1.760E − 01
3.856E − 07
Rank
5
5
5
5
4
5
5
5
3
2
1
F8
Mean
2.096E + 01
2.298E + 01
2.097E + 01
2.093E + 01
2.093E + 01
2.098E + 01
2.089E + 01
2.092E + 01
2.089E + 01
2.095E + 01
2.096E + 01
Std.
5.689E − 02
4.595E − 02
2.665E − 02
8.812E − 02
1.807E − 02
1.689E − 02
6.181E − 02
2.802E − 02
6.723E − 02
5.909E − 02
7.067E − 02
Best
2.091E + 01
2.093E + 01
2.095E + 01
2.085E + 01
2.092E + 01
2.097E + 01
2.084E + 01
2.089E + 01
2.072E + 01
2.082E + 02
2.076E + 01
Rank
6
8
9
4
7
10
3
5
1
11
2
F9
Mean
1.184E − 15
2.227E + 00
3.317E − 01
5.082E − 04
5.904E + 01
9.618E + 01
6.567E + 01
5.164E + 01
1.327E − 01
4.961E + 00
2.820E + 00
Std.
2.051E − 15
1.295E + 00
5.744E − 01
2.078E − 04
1.441E + 01
2.185E + 01
3.011E + 01
5.605E + 00
4.320E − 01
4.199E + 00
2.996E + 00
Best
0.000E + 00
1.134E + 00
0.000E + 00
2.719E − 04
4.420E + 01
8.258E + 01
4.179E + 01
4.754E + 01
1.432E − 11
1.294E − 01
1.526E − 02
Rank
1
6
1
3
8
10
7
9
2
5
4
F10
Mean
7.969E + 01
8.281E + 01
7.975E + 01
1.647E + 02
1.822E + 02
9.228E + 01
9.913E + 01
1.141E + 02
1.101E + 02
1.575E + 02
1.358E + 02
Std.
7.575E + 00
1.541E + 01
4.349E + 01
5.872E + 00
5.261E + 00
4.150E + 01
1.259E + 01
1.892E + 01
1.696E + 01
5.376E + 01
3.937E + 01
Best
7.131E + 01
6.963E + 01
3.022E + 01
1.608E + 02
1.788E + 02
5.771E + 01
8.743E + 01
9.451E + 01
6.792E + 01
8.846E + 01
6.676E + 01
Rank
6
5
1
10
11
2
7
9
4
8
3
F11
Mean
2.833E + 01
3.091E + 01
2.997E + 01
2.798E + 01
2.879E + 01
3.467E + 01
3.714E + 01
1.803E + 01
2.666E + 01
2.586E + 01
2.198E + 01
Std.
1.680E + 00
2.377E + 00
1.669E + 00
1.205E + 00
2.252E + 00
4.680E + 00
1.629E + 00
2.803E + 00
2.365E + 00
2.798E + 00
4.908E + 00
Best
2.661E + 01
2.843E + 01
2.840E + 01
2.685E + 01
2.625E + 01
2.930E + 01
3.526E + 01
1.551E + 01
2.058E + 01
1.913E + 01
1.280E + 01
Rank
6
9
8
7
5
10
11
2
4
3
1
F12
Mean
2.976E + 03
2.801E + 04
6.056E + 03
4.411E + 04
2.555E + 04
9.523E + 03
1.550E + 04
2.882E + 04
7.015E + 03
1.536E + 04
4.162E + 03
Std.
1.281E + 03
1.173E + 04
7.626E + 03
3.443E + 03
9.809E + 03
3.881E + 03
1.420E + 04
4.351E + 04
2.924E + 03
9.177E + 03
3.474E + 03
Best
1.698E + 03
1.447E + 04
1.008E + 03
4.022E + 04
1.962E + 04
6.257E + 03
3.727E + 03
3.259E + 03
1.585E + 03
3.634E + 03
1.998E + 02
Rank
4
9
2
11
10
8
7
5
3
6
1
F13
Mean
1.883E + 00
2.524E + 00
4.792E + 00
3.409E + 00
1.358E + 01
4.248E + 00
4.063E + 00
2.870E + 00
1.882E + 00
3.382E + 00
2.930E + 00
Std.
1.203E − 01
1.192E − 01
6.707E − 01
2.156E − 01
1.025E − 01
5.438E − 01
1.202E + 00
7.919E − 01
1.674E − 01
4.748E − 01
4.052E − 01
Best
1.771E + 00
2.408E + 00
4.113E + 00
3.162E + 00
1.347E + 01
3.912E + 00
2.682E + 00
2.293E + 00
1.403E + 00
2.314E + 00
2.149E + 00
Rank
2
6
10
8
11
9
7
4
1
5
3
F14
Mean
1.286E + 01
1.309E + 01
1.309E + 01
1.312E + 01
1.280E + 01
1.289E + 01
1.309E + 01
1.185E + 01
1.281E + 01
1.291E + 01
1.277E + 01
Std.
1.284E − 01
3.739E − 01
1.846E − 01
1.258E − 01
1.223E − 01
1.513E − 01
7.225E − 02
7.922E − 01
2.286E − 01
2.685E − 01
3.944E − 01
Best
1.276E + 01
1.266E + 01
1.288E + 01
1.298E + 01
1.266E + 01
1.278E + 01
1.303E + 01
1.112E + 01
1.234E + 01
1.229E + 01
1.126E + 01
Rank
6
5
8
9
5
7
10
1
4
3
2
F15
Mean
4.015E + 02
7.111E + 01
2.354E + 02
1.214E + 02
3.544E + 02
4.713E + 02
4.746E + 02
5.258E + 02
9.242E + 01
2.367E + 02
3.054E + 02
Std.
1.013E + 02
1.159E + 01
2.047E + 02
2.545E + 01
6.430E + 01
9.623E + 01
4.142E + 01
5.548E + 01
1.230E + 02
4.901E + 01
7.386E + 01
Best
3.000E + 02
6.421E + 01
6.213E + 00
9.610E + 01
3.007E + 02
3.977E + 02
4.268E + 02
4.831E + 02
2.946E − 06
2.000E + 02
8.141E + 00
Rank
7
4
2
5
8
9
10
11
1
6
3
F16
Mean
8.450E + 01
1.484E + 02
1.051E + 02
2.018E + 02
2.507E + 02
2.400E + 02
2.652E + 02
2.413E + 02
1.122E + 02
1.892E + 02
3.217E + 02
Std.
5.658E + 00
3.611E + 01
3.010E + 01
2.174E + 01
5.874E + 01
2.269E + 02
1.175E + 02
1.107E + 02
1.453E + 01
9.878E + 01
1.745E + 02
Best
7.797E + 01
1.240E + 02
7.058E + 01
1.775E + 02
2.087E + 02
1.078E + 02
1.912E + 02
1.245E + 02
7.278E + 01
1.075E + 02
1.033E + 02
Rank
3
7
1
9
11
6
10
8
2
5
4
F17
Mean
1.511E + 02
2.117E + 02
1.663E + 02
3.001E + 02
2.812E + 02
2.639E + 02
2.404E + 02
1.932E + 02
1.706E + 02
3.262E + 02
2.957E + 02
Std.
4.484E + 01
1.541E + 01
1.345E + 01
4.652E + 01
6.842E + 01
1.708E + 02
1.690E + 02
1.022E + 02
3.320E + 01
1.391E + 02
1.052E + 02
Best
1.198E + 02
1.964E + 02
1.508 + 02
2.574E + 02
2.388E + 02
1.399E + 02
1.323E + 02
1.267E + 02
1.183E + 02
1.840E + 02
1.258E + 02
Rank
2
9
7
11
10
6
5
4
1
8
3
F18
Mean
9.186E + 02
9.153E + 02
9.116E + 02
9.098E + 02
8.323E + 02
9.148E + 02
8.771E + 02
9.388E + 02
8.444E + 02
9.148E + 02
9.312E + 02
Std.
6.806E + 00
3.761E + 00
4.342E − 01
7.681E − 01
1.686E + 00
4.459E + 00
6.679E + 01
2.396E + 01
6.401E + 01
2.253E + 01
3.644E + 01
Best
9.108E + 02
9.119E + 02
9.111E + 02
9.090E + 02
8.310E + 02
9.105E + 02
8.000E + 02
9.187E + 02
8.000E + 02
2.000E + 02
8.000E + 02
Rank
6
8
7
4
3
5
2
9
2
1
2
F19
Mean
9.137E + 02
9.161E + 02
8.741E + 02
9.100E + 02
8.323E + 02
9.594E + 02
9.289E + 02
9.449E + 02
8.467E + 02
8.632E + 02
9.210E + 02
Std.
1.897E − 01
2.038E + 00
6.415E + 01
4.197E − 01
4.396E − 01
5.067E + 01
2.414E + 01
2.678E + 01
6.268E + 01
6.018E + 01
2.448E + 01
Best
9.136E + 02
9.147E + 02
8.000E + 02
9.096E + 02
8.318E + 02
9.201E + 02
9.903E + 02
9.172E + 02
8.000E + 02
2.000E + 02
8.000E + 02
Rank
5
6
2
4
3
8
9
7
2
1
2
F20
Mean
8.779E + 02
9.149E + 02
8.737E + 02
9.088E + 02
8.321E + 02
9.475E + 02
9.216E + 02
9.501E + 02
8.822E + 02
90197E + 02
9.220E + 02
Std.
6.752E + 01
2.841E + 00
6.383E + 01
9.501E − 01
9.107E − 01
3.759E + 01
8.330E + 00
2.351E + 01
6.391E + 01
9.503E + 00
3.591E + 01
Best
8.000E + 02
9.128E + 02
8.000E + 02
9.078E + 02
8.310E + 02
9.141E + 02
9.136E + 02
9.327E + 02
8.000E + 02
9.098E + 02
8.000E + 02
Rank
1
5
1
3
2
7
6
8
1
4
1
F21
Mean
5.000E + 02
5.000E + 02
5.000E + 02
6.508E + 02
5.000E + 02
9.551E + 02
5.000E + 02
9.728E + 02
6.097E + 02
5.973E + 02
7.584E + 02
Std.
1.969E − 13
1.842E − 13
0.000E + 00
1.376E + 02
3.922E − 06
3.941E + 02
8.247E − 13
1.750E + 02
2.499E + 02
2.291E + 02
2.791E + 02
Best
5.000E + 02
5.000E + 02
5.000E + 02
5.427E + 02
5.000E + 02
5.000E + 02
5.000E + 02
8.549E + 02
5.000E + 02
5.000E + 02
5.000E + 02
Rank
1
1
1
2
1
1
1
3
1
1
1
F22
Mean
9.372E + 02
9.709E + 02
9.275E + 02
9.189E + 02
5.290E + 02
9.220E + 02
9.904E + 02
8.860E + 02
1.002E + 03
1.039E + 03
1.014E + 03
Std.
1.765E + 01
8.997E + 00
4.853E + 00
2.153E + 01
2.608E − 01
3.630E + 01
3.054E + 01
2.439E + 01
2.722E + 01
4.147E + 01
0.340E + 01
Best
9.266E + 02
9.617E + 02
9.230E + 02
8.984E + 02
5.287E + 02
8.831E + 02
8.747E + 02
8.648E + 02
9.615E + 02
9.733E + 02
9.437E + 02
Rank
7
10
6
5
1
4
3
2
9
11
8
F23
Mean
5.342E + 02
5.342E + 02
5.342E + 02
6.655E + 02
5.342E + 02
1.052E + 03
1.226E + 02
1.030E + 03
6.437E + 02
7.727E + 02
6.864E + 02
Std.
4.661E − 04
1.987E − 05
1.186E − 04
5.669E + 01
2.583E − 04
2.102E + 02
3.965E + 01
1.329E + 02
2.395E + 02
2.544E + 02
2.445E + 02
Best
5.342E + 02
5.342E + 02
5.342E + 02
6.255E + 02
5.342E + 02
8.094E + 02
1.184E + 03
8.767E + 02
5.342E + 02
5.342E + 02
5.342E + 02
Rank
1
1
1
2
1
3
5
4
1
1
1
F24
Mean
2.000E + 02
2.000E + 02
2.000E + 02
9.489E + 02
2.183E + 02
2.000E + 02
2.000E + 02
9.579E + 02
2.000E + 02
2.676E + 02
2.000E + 02
Std.
0.000E + 00
1.897E − 12
3.481E − 14
1.994E + 00
9.720E − 01
4.702E − 11
3.212E − 12
2.510E + 01
3.585E − 06
2.527E + 02
8.407E − 13
Best
2.000E + 02
2.000E + 02
2.000E + 02
9.471E + 02
2.176E + 02
2.000E + 02
2.000E + 02
9.399E + 02
2.000E + 02
2.000E + 02
2.000E + 02
Rank
1
1
1
4
2
1
1
3
1
1
1
F25
Mean
9.938E + 02
1.016E + 03
9.951E + 02
1.008E + 03
2.070E + 02
1.006E + 03
1.052E + 03
1.064E + 03
2.000E + 02
3.418E + 02
2.000E + 02
Std.
1.593E + 00
8.040E + 00
4.148E + 00
1.268E + 01
1.213E + 01
2.774E + 01
4.316E + 01
2.582E + 01
2.545E − 06
3.618E + 02
1.445E − 13
Best
9.922E + 02
1.007E + 03
9.924E + 02
1.000E + 03
2.000E + 02
9.817E + 02
1.009E + 03
1.036E + 03
2.000E + 02
2.000E + 02
2.000E + 02
Rank
3
6
4
5
1
2
7
8
1
1
1
Average rank
3.68
6.32
3.76
6.4
5.88
5.44
5.6
5.84
3.24
5.04
2.56(3) On 30-Dimension Functions in CEC2014. There are 30 benchmark functions in CEC2014. It is difficult to get the global optimal solution because their global optimum is shifted. In this part, we compare the mean, standard deviations, and the best solution of the error values between the actual solution value and the theoretical optimal value and rank the best error value of each function, and the statistical results of each comparison algorithm are given in Table 4. For all 30 functions, MBSAgC outperforms the other nine algorithms for functions F4, F5, F14, F19, and F21. PSOFIPS performs best on functions F6, F29, and F30. PSOFDR is superior to other algorithms on functions F11, F15, and F16. CLPSO performs a little unsatisfactory on all functions except function F26. LBSA ranks first in terms of the best for functions F2, F3, F8, F9, F10, F13, F17, F20, and F22. CBSA performs better than other algorithms for functions F12, F23, F27, and F28. For function F7, in terms of the best, MBSAgC and LBSA obtain the same function value; they are better than other algorithms. All algorithms get the same value in function F26 except CLBSA. For function F24 and F25, CBSA, TLBO, ETLBO, and DGSTLBO obtain the same value, and they are better than other six algorithms. The average ranking of three comparative criterions is also listed in the table. As we can see, LBSA is the best in all three criterions; MBSAgC performs slightly worse in best and Std than LBSA, but it is better than other algorithms. Overall, MBSAgC’s rank is still very competitive in CEC2014.Table 4
Comparative results of ten algorithms for 30D problems from CEC2014.
F
AL
PSOFIPS
PSOFDR
CLPSO
CBSA
CLBSA
TLBO
ETLBO
DGSTLBO
LBSA
MBSAgC
F1
Mean
1.02E + 07
2.78E + 06
2.92E + 07
4.12E + 07
1.61E + 08
8.28E + 05
3.33E + 05
1.04E + 07
2.79E + 05
1.26E + 06
Std
2.88E + 06
1.98E + 06
4.33E + 06
1.19E + 07
6.60E + 07
7.89E + 05
2.02E + 05
8.61E + 06
1.65E + 05
1.12E + 06
Best
5.60E + 06
1.32E + 05
2.14E + 07
2.17E + 07
5.69E + 07
5.77E + 04
1.37E + 05
3.15E + 06
7.85E + 04
1.27E + 05
Rank
7
4
8
9
10
1
5
6
2
3
F2
Mean
1.13E + 04
1.53E + 08
1.26E + 03
3.20E + 08
2.43E + 10
1.44E + 02
1.19E + 02
4.59E + 06
4.83E − 14
3.51E + 01
Std
5.27E + 03
3.61E + 08
7.40E + 02
6.61E + 07
7.40E + 09
1.72E + 02
1.56E + 02
1.11E + 07
1.92E − 14
5.42E + 01
Best
3.63E + 03
4.60E + 01
2.76E + 02
1.75E + 08
1.21E + 10
2.82E + 01
3.29E − 01
1.48E + 03
2.84E − 14
1.14E − 13
Rank
8
5
6
9
10
4
3
7
1
2
F3
Mean
6.94E + 03
5.40E + 02
7.67E + 02
2.37E + 03
3.09E + 04
2.39E + 03
2.14E + 03
1.44E + 01
2.27E − 13
5.67E + 02
Std
5.03E + 03
6.05E + 02
1.20E + 03
1.05E + 03
9.52E + 03
1.77E + 03
1.02E + 03
1.68E + 01
1.00E − 13
1.46E + 03
Best
1.32E + 03
1.36E + 01
5.76E + 01
9.14E + 02
1.36E + 04
2.90E + 02
5.59E + 02
6.71E − 01
1.14E − 13
1.56E + 00
Rank
9
4
5
8
10
6
7
2
1
3
F4
Mean
2.67E + 01
9.50E + 01
1.16E + 02
1.56E + 02
2.46E + 03
9.70E + 01
7.53E + 01
1.46E + 02
4.23E + 01
5.20E + 01
Std
6.37E − 01
3.24E + 01
1.33E + 01
2.13E + 01
1.56E + 03
3.36E + 01
5.88E + 00
3.78E + 01
3.57E + 01
3.63E + 01
Best
2.53E + 01
6.66E + 01
9.56E + 01
1.13E + 02
6.39E + 02
6.74E + 01
6.79E + 01
8.79E + 01
1.73E − 03
2.59E − 04
Rank
3
4
8
9
10
5
6
7
2
1
F5
Mean
2.10E + 01
2.09E + 01
2.05E + 01
4.14E + 01
7.61E + 02
2.10E + 01
2.09E + 01
2.10E + 01
2.03E + 01
2.63E − 13
Std
5.71E − 02
1.22E − 01
4.58E − 02
9.01E + 00
3.65E + 02
5.23E − 02
7.24E − 02
4.34E − 02
3.42E − 02
1.00E − 13
Best
2.09E + 01
2.06E + 01
2.04E + 01
2.51E + 01
1.58E + 02
2.09E + 01
2.08E + 01
2.09E + 01
2.03E + 01
1.14E − 13
Rank
6
4
3
7
8
6
5
6
2
1
F6
Mean
6.19E + 00
7.67E + 00
1.70E + 01
9.93E + 01
7.24E + 02
1.54E + 01
1.59E + 01
1.67E + 01
8.37E + 00
3.83E + 01
Std
2.22E + 00
1.98E + 00
6.10E − 01
1.09E + 01
2.63E + 02
2.14E + 00
2.75E + 00
3.45E + 00
3.22E + 00
2.81E + 01
Best
2.75E + 00
4.63E + 00
1.60E + 01
7.26E + 01
3.05E + 02
1.17E + 01
1.01E + 01
1.23E + 01
5.25E + 00
5.80E + 00
Rank
1
2
8
9
10
6
5
7
3
4
F7
Mean
2.56E − 03
1.33E + 01
3.92E − 03
4.00E + 00
2.41E + 02
6.74E − 02
2.75E − 02
1.01E + 00
5.41E − 03
9.84E − 03
Std
6.75E − 03
1.31E + 01
1.61E − 03
6.84E − 01
7.28E + 01
8.69E − 02
2.23E − 02
1.50E + 00
8.26E − 03
1.50E − 02
Best
7.08E − 05
1.48E − 02
1.37E − 03
1.91E + 00
9.95E + 01
7.84E − 12
1.02E − 12
1.20E − 01
1.14E − 13
1.14E − 13
Rank
4
6
5
8
9
3
2
7
1
1
F8
Mean
6.60E + 01
3.48E + 01
5.86E − 01
1.86E + 01
1.90E + 02
7.37E + 01
8.00E + 01
7.67E + 01
1.14E − 13
6.70E + 00
Std
1.22E + 01
5.70E + 00
6.20E − 01
1.86E + 00
1.89E + 01
2.11E + 01
1.88E + 01
2.45E + 01
0.00E + 00
5.26E + 00
Best
4.32E + 01
2.19E + 01
2.65E − 04
1.36E + 01
1.49E + 02
5.07E + 01
4.48E + 01
3.48E + 01
1.14E − 13
1.37E + 00
Rank
7
5
2
4
10
9
8
6
1
3
F9
Mean
1.53E + 02
5.53E + 01
7.79E + 01
9.56E + 01
1.98E + 02
7.68E + 01
8.00E + 01
9.84E + 01
4.32E + 01
8.48E + 01
Std
1.61E + 01
1.54E + 01
1.49E + 01
9.56E + 00
2.47E + 01
1.08E + 01
1.42E + 01
3.08E + 01
7.61E + 00
2.00E + 01
Best
1.21E + 02
3.51E + 01
5.52E + 01
7.30E + 01
1.49E + 02
5.87E + 01
5.87E + 01
6.41E + 01
2.70E + 01
4.24E + 01
Rank
8
2
4
7
9
5
5
6
1
3
F10
Mean
1.87E + 03
9.70E + 02
2.50E + 01
1.35E + 02
4.78E + 03
1.72E + 03
1.57E + 03
2.39E + 03
1.08E + 01
1.67E + 02
Std
4.44E + 02
3.47E + 02
5.21E + 00
2.35E + 01
4.01E + 02
7.41E + 02
4.26E + 02
4.71E + 02
5.07E + 00
8.55E + 01
Best
1.35E + 03
3.70E + 02
1.83E + 01
7.75E + 01
3.97E + 03
4.64E + 02
9.52E + 02
1.59E + 03
5.12E + 00
4.43E + 01
Rank
8
5
2
4
10
6
7
9
1
3
F11
Mean
5.76E + 03
3.08E + 03
3.24E + 03
2.80E + 03
5.08E + 03
6.71E + 03
6.46E + 03
3.39E + 03
2.31E + 03
2.67E + 03
Std
3.43E + 02
6.19E + 02
2.18E + 02
2.58E + 02
5.45E + 02
3.92E + 02
4.46E + 02
5.45E + 02
3.17E + 02
3.86E + 02
Best
5.23E + 03
1.77E + 03
2.98E + 03
2.22E + 03
4.08E + 03
6.01E + 03
5.52E + 03
2.42E + 03
1.86E + 03
1.87E + 03
Rank
8
1
6
4
7
10
9
5
2
3
F12
Mean
2.62E + 00
8.12E − 01
5.42E − 01
4.72E − 01
2.19E + 00
2.64E + 00
2.60E + 00
2.75E + 00
4.18E − 01
5.76E − 01
Std
2.82E − 01
6.51E − 01
8.53E − 02
6.66E − 02
6.22E − 01
2.47E − 01
3.32E − 01
2.62E − 01
5.15E − 02
8.85E − 02
Best
1.97E + 00
3.12E − 01
4.12E − 01
2.44E − 01
6.09E − 01
2.13E + 00
1.93E + 00
2.48E + 00
3.62E − 01
3.87E − 01
Rank
8
2
5
1
6
9
7
10
3
4
F13
Mean
3.46E − 01
4.35E − 01
4.18E − 01
3.89E − 01
3.88E + 00
4.88E − 01
4.11E − 01
4.71E − 01
2.90E − 01
3.70E − 01
Std
3.30E − 02
1.07E − 01
7.14E − 02
5.27E − 02
5.51E − 01
1.14E − 01
6.86E − 02
1.13E − 01
4.34E − 02
7.93E − 02
Best
2.89E − 01
2.55E − 01
2.77E − 01
2.63E − 01
2.74E + 00
3.50E − 01
2.97E − 01
3.09E − 01
2.10E − 01
2.22E − 01
Rank
6
3
5
4
10
9
7
8
1
2
F14
Mean
3.07E − 01
8.91E − 01
3.44E − 01
2.83E − 01
9.26E + 01
2.88E − 01
2.78E − 01
2.88E − 01
2.30E − 01
2.51E − 01
Std
3.63E − 02
3.59E − 01
3.38E − 02
3.87E − 02
2.41E + 01
4.70E − 02
4.42E − 02
4.92E − 02
2.69E − 02
4.40E − 02
Best
2.54E − 01
1.88E − 01
2.97E − 01
2.20E − 01
3.51E + 01
2.11E − 01
1.84E − 01
2.26E − 01
1.87E − 01
1.74E − 01
Rank
8
4
9
6
10
5
2
7
3
1
F15
Mean
1.58E + 01
4.49E + 00
1.09E + 01
1.99E + 01
8.76E + 03
1.80E + 01
1.89E + 01
3.75E + 01
5.93E + 00
8.76E + 00
Std
9.26E − 01
6.87E − 01
9.63E − 01
2.08E + 00
1.12E + 04
5.93E + 00
8.66E + 00
2.19E + 01
1.22E + 00
1.71E + 00
Best
1.41E + 01
3.18E + 00
9.09E + 00
1.55E + 01
5.91E + 02
1.03E + 01
9.75E + 00
1.54E + 01
3.87E + 00
5.15E + 00
Rank
7
1
4
9
10
6
5
8
2
3
F16
Mean
1.18E + 01
1.05E + 01
1.11E + 01
1.05E + 01
1.17E + 01
1.20E + 01
1.19E + 01
1.11E + 01
1.02E + 01
1.06E + 01
Std
2.50E − 01
8.64E − 01
2.31E − 01
3.26E − 01
5.87E − 01
4.35E − 01
3.94E − 01
6.62E − 01
4.55E − 01
4.12E − 01
Best
1.13E + 01
8.55E + 00
1.07E + 01
9.78E + 00
1.04E + 01
1.14E + 01
1.14E + 01
1.01E + 01
9.37E + 00
9.89E + 00
Rank
8
1
7
3
6
9
9
5
2
4
F17
Mean
3.76E + 05
1.36E + 05
2.07E + 06
3.46E + 06
4.28E + 05
1.91E + 05
2.11E + 05
1.67E + 05
4.42E + 04
1.92E + 05
Std
1.12E + 05
1.04E + 05
6.62E + 05
1.54E + 06
4.64E + 05
1.73E + 05
9.21E + 04
2.13E + 05
3.70E + 04
1.60E + 05
Best
2.44E + 05
2.69E + 04
1.07E + 06
9.41E + 05
3.61E + 04
3.29E + 04
8.02E + 04
3.54E + 04
8.62E + 03
3.99E + 04
Rank
8
2
10
9
5
3
7
4
1
6
F18
Mean
1.59E + 03
3.38E + 03
4.92E + 02
4.54E + 06
2.46E + 04
2.93E + 03
3.31E + 03
8.71E + 02
1.14E + 03
2.19E + 03
Std
8.50E + 02
4.21E + 03
9.30E + 01
1.89E + 06
1.06E + 05
2.35E + 03
4.11E + 03
1.02E + 03
1.29E + 03
2.81E + 03
Best
4.78E + 02
3.13E + 02
3.44E + 02
1.54E + 06
2.80E + 02
1.35E + 02
1.64E + 02
6.17E + 01
6.23E + 01
6.28E + 01
Rank
9
7
8
10
6
4
5
1
2
3
F19
Mean
1.19E + 01
1.57E + 01
1.01E + 01
2.04E + 01
8.25E + 01
2.12E + 01
2.12E + 01
2.71E + 01
6.31E + 00
6.90E + 00
Std
8.29E − 01
2.33E + 01
1.02E + 00
5.81E + 00
3.96E + 01
2.55E + 01
2.54E + 01
2.86E + 01
1.01E + 00
2.05E + 00
Best
1.01E + 01
4.44E + 00
8.20E + 00
1.33E + 01
2.24E + 01
5.77E + 00
6.72E + 00
9.65E + 00
4.60E + 00
3.34E + 00
Rank
8
2
6
9
10
4
5
7
3
1
F20
Mean
5.81E + 03
7.05E + 03
5.95E + 03
6.95E + 03
7.38E + 03
1.54E + 03
1.44E + 03
4.28E + 02
9.02E + 01
2.02E + 03
Std
2.71E + 03
1.27E + 04
3.35E + 03
3.35E + 03
7.07E + 03
9.69E + 02
6.69E + 02
1.77E + 02
8.20E + 01
2.99E + 03
Best
2.36E + 03
1.37E + 03
2.48E + 03
1.05E + 03
3.45E + 02
6.68E + 02
3.41E + 02
2.03E + 02
3.06E + 01
5.99E + 01
Rank
9
8
10
7
5
6
4
3
1
2
F21
Mean
1.49E + 05
6.77E + 04
3.39E + 05
6.64E + 05
4.18E + 04
9.72E + 04
1.19E + 05
2.20E + 04
9.86E + 03
6.55E + 04
Std
6.40E + 04
4.21E + 04
1.22E + 05
2.63E + 05
5.34E + 04
9.01E + 04
1.00E + 05
2.22E + 04
1.32E + 04
6.81E + 04
Best
8.47E + 04
8.85E + 03
1.25E + 05
1.26E + 05
2.10E + 03
1.89E + 04
3.16E + 04
4.09E + 03
1.22E + 03
1.19E + 03
Rank
8
5
9
10
3
6
7
4
2
1
F22
Mean
2.25E + 02
2.27E + 02
2.70E + 02
4.19E + 02
3.98E + 02
2.81E + 02
3.27E + 02
3.14E + 02
1.14E + 02
2.27E + 02
Std
7.27E + 01
9.97E + 01
5.98E + 01
1.04E + 02
1.63E + 02
1.08E + 02
1.33E + 02
1.41E + 02
7.56E + 01
1.20E + 02
Best
1.67E + 02
1.44E + 02
1.59E + 02
1.84E + 02
1.61E + 02
1.64E + 02
5.82E + 01
1.49E + 02
2.41E + 01
2.68E + 01
Rank
9
4
6
10
7
8
3
5
1
2
F23
Mean
3.14E + 02
3.18E + 02
3.15E + 02
2.26E + 02
3.87E + 02
3.15E + 02
3.15E + 02
3.15E + 02
3.15E + 02
3.15E + 02
Std
1.57E − 04
3.50E + 00
2.29E − 01
4.77E + 01
2.07E + 01
1.23E − 11
1.78E − 10
4.43E − 01
1.44E − 13
3.92E − 13
Best
3.14E + 02
3.15E + 02
3.15E + 02
2.00E + 02
3.34E + 02
3.15E + 02
3.15E + 02
3.15E + 02
3.15E + 02
3.15E + 02
Rank
2
3
3
1
4
3
3
3
3
3
F24
Mean
2.24E + 02
2.27E + 02
2.27E + 02
2.00E + 02
2.79E + 02
2.00E + 02
2.00E + 02
2.00E + 02
2.26E + 02
2.38E + 02
Std
5.46E − 01
2.58E + 00
1.01E + 00
5.57E − 04
9.88E + 00
2.20E − 03
2.19E − 03
9.68E − 04
1.28E + 00
5.98E + 00
Best
2.23E + 02
2.23E + 02
2.25E + 02
2.00E + 02
2.60E + 02
2.00E + 02
2.00E + 02
2.00E + 02
2.24E + 02
2.25E + 02
Rank
2
2
4
1
5
1
1
1
3
4
F25
Mean
2.07E + 02
2.06E + 02
2.10E + 02
2.00E + 02
2.24E + 02
2.01E + 02
2.00E + 02
2.02E + 02
2.09E + 02
2.06E + 02
Std
2.45E + 00
2.09E + 00
1.62E + 00
8.04E − 09
4.39E + 00
2.39E + 00
5.33E − 06
3.62E + 00
3.10E + 00
2.68E + 00
Best
2.04E + 02
2.04E + 02
2.08E + 02
2.00E + 02
2.16E + 02
2.00E + 02
2.00E + 02
2.00E + 02
2.04E + 02
2.04E + 02
Rank
2
2
3
1
4
1
1
1
2
2
F26
Mean
1.70E + 02
1.51E + 02
1.00E + 02
1.00E + 02
1.43E + 02
1.10E + 02
1.00E + 02
1.10E + 02
1.00E + 02
1.34E + 02
Std
4.83E + 01
5.18E + 01
3.00E − 02
4.88E − 02
4.93E + 01
3.15E + 01
1.57E − 01
3.15E + 01
7.52E − 02
4.78E + 01
Best
1.00E + 02
1.00E + 02
1.00E + 02
1.00E + 02
1.01E + 02
1.00E + 02
1.00E + 02
1.00E + 02
1.00E + 02
1.00E + 02
Rank
1
1
1
1
2
1
1
1
1
1
F27
Mean
4.34E + 02
7.12E + 02
4.38E + 02
3.62E + 02
8.61E + 02
5.47E + 02
6.56E + 02
7.94E + 02
4.34E + 02
5.37E + 02
Std
3.50E + 01
1.84E + 02
1.33E + 01
9.59E + 01
2.83E + 02
1.66E + 02
2.13E + 02
2.15E + 02
6.71E + 01
1.46E + 02
Best
3.76E + 02
4.02E + 02
4.17E + 02
2.00E + 02
4.44E + 02
4.02E + 02
4.02E + 02
4.06E + 02
3.71E + 02
4.01E + 02
Rank
3
5
7
1
8
5
5
6
2
4
F28
Mean
3.98E + 02
1.48E + 03
9.21E + 02
5.74E + 02
2.66E + 03
1.24E + 03
1.18E + 03
1.43E + 03
8.34E + 02
1.20E + 03
Std
1.16E + 01
3.70E + 02
3.47E + 01
4.65E + 02
4.48E + 02
3.51E + 02
2.30E + 02
4.37E + 02
3.87E + 01
2.91E + 02
Best
3.89E + 02
1.02E + 03
8.64E + 02
2.00E + 02
1.99E + 03
9.20E + 02
8.94E + 02
9.96E + 02
7.87E + 02
7.48E + 02
Rank
2
9
5
1
10
7
6
8
4
3
F29
Mean
2.14E + 02
6.65E + 06
3.11E + 04
3.37E + 04
7.77E + 06
3.31E + 06
3.93E + 06
3.08E + 06
1.29E + 03
1.19E + 03
Std
1.02E + 00
1.10E + 07
1.52E + 04
1.09E + 04
1.58E + 07
5.39E + 06
5.14E + 06
4.99E + 06
5.91E + 02
2.70E + 02
Best
2.12E + 02
2.17E + 03
7.11E + 03
1.12E + 04
1.45E + 03
1.25E + 03
1.44E + 03
9.91E + 02
7.75E + 02
7.84E + 02
Rank
1
8
9
10
7
5
6
4
2
3
F30
Mean
7.00E + 02
1.82E + 04
7.94E + 03
1.38E + 04
7.38E + 04
3.35E + 03
3.37E + 03
6.47E + 03
1.93E + 03
2.42E + 03
Std
8.54E + 01
3.03E + 04
2.72E + 03
3.52E + 03
4.24E + 04
1.35E + 03
1.75E + 03
3.43E + 03
5.26E + 02
4.86E + 02
Best
5.98E + 02
1.78E + 03
5.62E + 03
7.35E + 03
3.87E + 03
1.62E + 03
1.58E + 03
3.37E + 03
1.15E + 03
1.56E + 03
Rank
1
6
9
10
8
5
4
7
2
3
Average rank
Mean
4.6
5.27
4.67
5.57
8.43
5.43
5.13
5.633
1.8
3.667
Std
4.033
6.57
3.7
5.5
9.1
5.8
5.17
6.87
2.83
5.1
Best
5.7
3.9
5.9
6.07
7.63
5.27
5
5.37
1.9
2.63


## 5.1.2. Statistical Analyses for Benchmark Results

Sign tests [84] is a common method to determine whether there is significant difference between two algorithms. In this paper, we use the two-tailed sign tests which has a significance level of 0.05 to test the differences between the results obtained by MBSAgC and other compared algorithms. The statistical results are listed in Table 5. The best solution is used as the criterion of the sign test. The signs “+,” “≈,” and “−” represent that MBSAgC performs better, almost the same, and worse than other compared algorithms, respectively. And the null hypothesis in this paper is that the performances of MBSAgC and one of the other compared algorithms are not significantly differential.Table 5
Comparisons between MBSAgC and other algorithms in sign tests.
AL
+
≈
−
Total
P
LBSA
28
18
34
80
0.619
CLPSO
68
5
7
80
0.000
PSOFIPS
54
15
11
80
0.000
TLBO
49
14
17
80
0.000
ETLBO
54
14
12
80
0.000
PSOFDR
52
8
20
80
0.000
BSAISA
19
17
14
50
0.474
OBSA
32
13
5
50
0.004
BSA
30
12
8
50
0.001
CBSA
22
1
7
30
0.003
CLBSA
29
0
1
30
0.000
DGSTLBO
23
2
5
30
0.002
SADE
10
5
10
25
0.717Thirteen algorithms are compared in the benchmark experiment. Table5 lists the analytical test results of all the algorithms. The P values of supporting the null hypothesis of sign test for ten compared algorithms (MBSAgC-CLPSO, MBSAgC-PSOFIPS, MBSAgC-TLBO, MBSAgC-ETLBO, MBSAgC-PSOFDR, MBSAgC-OBSA, MBSAgC-BSA, MBSAgC-CBSA, MBSAgC-CLBSA, and MBSAgC-DGSTLBA) are lower than significance level 0.05, so we can reject the null hypothesis. And it is illustrated that the optimization performance of MBSAgC is significantly better than those of the other ten algorithms. The P values of MBSAgC-LBSA, MBSAgC-BSAISA, and MBSAgC-SaDE are higher than 0.05; they are 0.619, 0.474, and 0.717, respectively, so we cannot reject the null hypothesis. However, it does not mean that MBSAgC is much worse than them. In the light of sign value from Table 5, for MBSAgC-LBSA, MBSAgC is worse than LBSA on thirty-four functions, and it performs better on twenty-eight functions; moreover, MBSAgC performs better on both 10D and 30D in CEC2005. MBSAgC is slightly worse than BSAISA on fourteen functions, but better than BSAISA on nineteen functions. For MBSAgC-SaDE, the number of “+” is the same as the number of “−,” but MBSAgC is better than SaDE in comprehensive ranking.


## 5.2. Experiments on Engineering Design Problems

In this part, we test MBSAgC in five well-known engineering design problems: three-bar truss design problem(TTP), pressure vessel design problem(PVP), tension/compression spring design problem(TCSP), welded beam design problem(WBP), and speed reducer design problem(SRP), respectively. The details of five design problems can be found in AppendixA. And the results in Tables 6–15 show the performance of MBSAgC and its competitive with other compared algorithms.Table 6
Comparison of best solutions for the three-bar truss design problem.
Method
DEDS
HEAA
PSO-DE
DELC
MBA
BSA
MBSAgC
X
1
0.788675
0.788680
0.788675
0.788675
0.788675
0.788675
0.788675
X
2
0.408248
0.408234
0.408248
0.408248
0.408560
0.408248
0 0.408248
g
1
X
1.77E − 08
NA
−5.29E − 11
NA
−5.29E − 11
−3.23E − 12
−1.8208E − 13
g
2
X
−1.464102
NA
−1.463748
NA
−1.463748
−1.464102
−1.464102
g
3
X
−0.535898
NA
−0.536252
NA
−0.536252
−0.535898
−0.535898
f
X
263.895843
263.895843
263.895843
263.895843
263.895852
263.895843
263.895843Table 7
Comparison of statistical results for the three-bar truss design problem.
Method
Worst
Mean
Best
Std
FEs
DEDS
263.895849
263.895843
263.895843
9.7E − 07
15000
HEAA
263.896099
263.895865
263.895843
4.9E − 05
15000
PSO-DE
263.895843
263.895843
263.895843
4.5E − 10
17600
DELC
263.895843
263.895843
263.895843
4.3E − 14
10000
MBA
263.915983
263.897996
263.895852
3.93E − 03
13280
BSA
263.895845
263.895843
263.895843
2.64E − 07
13720
MBSAgC
263.895843
263.895843
263.895843
1.83E − 10
9540Table 8
Comparison of best solutions for the pressure vessel design problem.
Method
DA-DT
MDE
CPSO
HPSO
DELC
ABC
BSA-SAε
BSA
MBSAgC
X
1
0.8125
0.8125
0.8125
0.8125
0.8125
0.8125
0.8125
0.8125
0.8125
X
2
0.4375
0.4375
0.4375
0.4375
0.4375
0.4375
0.4375
0.4375
0.4375
X
3
42.0974
42.0984
42.0913
42.0984
42.0984
42.0984
42.0984
42.0984
42.0984
X
4
176.6540
176.6360
176.7465
176.6366
176.6366
176.6366
176.6366
176.6366
176.6366
g
1
X
−3.01E − 03
0
−1.37E − 06
NA
NA
0
−9.5E − 10
−3.69E − 08
0
g
2
X
−3.58E − 02
−0.035881
−3.59E − 04
NA
NA
−0.35881
−3.59E − 2
−0.35881
−3.59E − 2
g
3
X
−24.7593
−0.000
−118.7687
NA
NA
−0.000226
−1.2E − 4
−0.095446
0
g
4
X
−63.3460
−63.3639
−63.2535
NA
NA
−63.363
−63.363
−63.2842
−63.3634
f
X
6059.9463
6059.7017
6061.0777
6059.7143
6059.7143
6059.7143
6059.7143
6059.7150
6059.7143Table 9
Comparison of best solutions for the pressure vessel design problem.
Method
Worst
Mean
Best
Std
FEs
GA-DT
6469.3220
6177.2533
6059.9643
130.9297
80000
MDE
6059.7017
6059.7017
6059.7017
1.0E − 12
24000
CPSO
6363.8041
6147.1332
6061.0777
86.45
30000
HPSO
6288.6770
6099.9323
6059.7143
86.20
81000
DELC
6059.7143
6059.7143
6059.7143
2.1E − 11
30000
PSO-DE
6059.7143
6059.7143
6059.7143
1.0E − 10
42100
ABC
NA
6245.3081
6059.7147
2.05E + 02
30000
BSA-SAε
6116.7804
6074.3682
6059.7143
1.71E + 01
80000
BSA
6771.5969
6221.2861
6059.7150
2.03E + 02
60000
MBSAgC
6374.8627
6073.6436
6059.7143
2.03E + 04
16180Table 10
Comparison of best solutions for the tension/compression spring design problem.
Method
X
1
X
2
X
3
g
1
X
g
2
X
g
3
X
g
4
X
f
X
GA-DT
0.051989
0.363965
10.890522
−1.3E − 05
−2.1E − 05
−4.061338
−0.722698
0.012681
MDE
0.051688
0.356692
11.290483
−0.000000
−0.000000
−4.053734
−0.727090
0.012665
CPSO
0.051728
0.357644
11.244543
−8.45E − 04
−1.26E − 05
−4.051300
−0.727090
0.012675
HPSO
0.051706
0.357126
11.265083
NA
NA
NA
NA
0.012665
DEDS
0.051689
0.356718
11.288965
NA
NA
NA
NA
0.012665
HEAA
0.051690
0.356729
11.288294
NA
NA
NA
NA
0.012665
DELC
0.051689
0.356718
11.288966
NA
NA
NA
NA
0.012665
ABC
0.051749
0.358179
11.203763
−0.000000
−0.000000
−4.0566663
−0.726713
0.012665
MBA
0.051656
0.35594
11.344665
0
0
−4.052248
−0.728268
0.012665
SSOC
0.051689
0.356718
11.288965
NA
NA
NA
NA
0.012665
BSA-SAε
0.051989
0.356727
11.288425
−7.70E − 09
−3.30E − 09
−4.054
−0.728
0.012665
BSA
0.51694
0.356845
11.281488
−1.05E − 07
−1.77E − 08
−4.054037
−0.727640
0.012665
MBSAgC
0.051689
0.356718
11.288954
−4.15E − 14
−1.13E − 13
−4.0538
−0.7277
0.012665Table 11
Comparison of statistical results for the tension/compression spring design problem.
Method
Worst
Mean
Best
Std
FEs
GA-DT
0.012973
0.12742
0.12681
5.90E − 05
80000
MDE
0.012674
0.012666
0.012665
2.0E − 06
24000
CPSO
0.012924
0.012730
0.012675
5.20E − 05
23000
HPSO
0.012719
0.012707
0.012665
1.58E − 05
81000
DEDS
0.012738
0.012669
0.012665
1.25E − 05
24000
HEAA
0.012665
0.012665
0.012665
1.4E − 09
24000
DELC
0.012666
0.012665
0.012665
1.3E − 07
20000
PSO-DE
0.012665
0.012665
0.012665
1.2E − 08
24950
ABC
NA
0.012709
0.012665
0.012813
30000
MBA
0.012900
0.012713
0.012665
6.30E − 05
7650
SSOC
0.012868
0.012765
0.012665
9.29E − 05
25000
BSA-SAε
0.012666
0.012665
0.012665
1.62E − 07
80000
BSA
0.012669
0.012666
0.012665
7.24E − 07
43220
MBSAgC
0.012665
0.012665
0.012665
1.25E − 07
22680Table 12
Comparison of best solutions for the welded beam design problem.
Method
GA-DT
MDE
CPSO
HPSO
ABC
MBA
BSA-SAε
BSA
BSAISA
MBSAgC
X
1
h
0.205986
0.205730
0.202369
0.205730
0.205730
0.205729
0.205730
0.205730
0.205730
0.205730
X
2
l
3.471328
3.470489
3.544214
3.470489
3.470489
3.470493
3.470489
3.470489
3.470489
3.47049
X
3
t
9.020224
9.036624
9.04821
9.036624
9.036624
9.036626
9.036624
9.036624
9.036624
9.036624
X
4
b
0.206480
0.205730
0.205723
0.205730
0.205730
0.205729
0.205730
0.205730
0.205730
0.205730
g
1
X
−0.074092
−0.000335
−12.8399796
NA
0.000000
−0.001614
−1.55E − 10
−5.32E − 07
0
−2.2628E − 09
g
2
X
−0.266227
−0.000753
−1.247467
NA
−0.000002
−0.016911
−4.30E − 09
−9.02E − 06
0
−1.8335E − 09
g
3
X
−4.95E − 04
−0.0000000
−1.49E − 03
NA
0.0000000
−2.40E − 07
−1.55E − 15
−7.86E − 12
−5.55E − 17
−2629E − 13
g
4
X
−3.430044
−3.432984
−3.429347
NA
−3.432984
−3.432982
−3.4330
−3.432984
−3.43294
−3.4330
g
5
X
−0.080986
−0.080730
−0.079381
NA
−0.080730
−0.080729
−8.07E − 02
−0.080730
−0.080730
−0.080730
g
6
X
−0.235514
−0.235540
−0.235536
NA
−0.235540
−0.235540
−0.2355
−0.235540
−0.235540
−0.2355
g
7
X
−58.666440
−0.000882
−11.681355
NA
0.000000
−0.001464
−1.85E − 10
−1.13E − 07
−5.46E − 12
−4.0927E − 10
f
X
1.728226
1.724852
1.728024
1.724852
1.724852
1.724853
1.724852
1.724852
1.724852
1.724852Table 13
Comparison of statistical results for the welded beam design problem.
Method
Worst
Mean
Best
Std
FEs
GA-DT
1.993408
1.792654
1.728226
7.47E − 02
80000
MDE
1.724854
1.724853
1.724852
NA
24000
CPSO
1.782143
1.748831
1.728024
1.29E − 02
24000
HPSO
1.814295
1749040
1.724852
4.00E − 02
81000
DELC
1.724852
1.724852
1.724852
4.1E − 13
20000
PSO-DE
1.724852
1.724852
1.724852
6.7E − 16
66600
ABC
NA
1.741913
1.724852
3.1E − 02
30000
MBA
1.724853
1.724853
1.724853
6.94E − 19
47340
SSOC
1.799332
1.746462
1.724852
2.57E − 02
25000
BSA-SAε
1.724852
1.724852
1.724852
8.11E − 10
80000
BSA
1.724854
1.724852
1.724852
2.35E − 07
45480
BSAISA
1.724854
1.724852
1.724852
2.96E − 07
29000
MBSAgC
3.121674
1.780725
1.724852
2.765E − 01
21640Table 14
Comparison of best solutions for the speed reducer design problem.
Method
MDE
DEDS
DELC
HEAA
PSO-DE
MBA
BSA
BSAISA
MBSAgC
X
1
3.500010
3.500000
3.500000
3.500000
3.500000
3.500000
3.500000
3.500000
3.500000
X
2
0.700000
0.700000
0.700000
0.700000
0.700000
0.700000
0.700000
0.700000
0.700000
X
3
17
17
17
17.000013
17.000000
17.000000
17
17
17
X
4
7.300156
7.300000
7.300000
7.300428
7.300000
7.300033
7.300000
7.300000
7.300000
X
5
7.800027
7.715320
7.715320
7.715377
7.800000
7.715772
7.715320
7.715320
7.715320
X
6
3.350221
3.350215
3.350215
3.350231
3.350215
3.350218
3.350215
3.350215
3.305215
X
7
5.286685
5.286654
5.286654
5.286664
5.286683
5.286654
5.286654
5.286654
5.286654
f
X
2996.356689
2994.471066
2994.471066
2994.488107
2996.348167
2994.482453
2994.471066
2994.471066
2994.471066Table 15
Comparison of statistical results for the speed reducer design problem.
Method
Worst
Mean
Best
Std
FEs
MDE
2996.390137
2996.367220
2996.356689
8.2E − 03
24000
DEDS
2994.471066
2994.471066
2994.471066
3.58E − 12
30000
HEAA
2994.752311
2994.613368
2994.499107
7.0E − 02
40000
DELC
2994.471066
2994.471066
2994.471066
1.9E − 12
30000
PSO-DE
2996.348204
2996.348174
2996.348167
6.4E − 06
54350
ABC
NA
2997.058412
2997.058412
0
30000
MBA
2999.652444
2996.769019
2994.482453
1.56
6300
SSOC
2996.113298
2996.113298
2996.113298
1.34E − 12
25000
BSA
2994.471066
2994.471066
2994.471066
9.87E − 11
25640
BSAISA
2994.471095
2994.471067
2994.471066
5.40E − 06
15860
MBSAgC
2998.327568
2996.438784
2994.471066
3.24E − 05
12580


### 5.2.1. Engineering Problem and Experimental Results

(1) Three-Bar Truss Design Problem(TTP). We compare MBSAgC with the BSA, differential evolution with dynamic stochastic selection(DEDS) [85], hybrid evolutionary algorithm(HEAA) [86], hybrid particle swarm optimization with differential evolution(POS-DE) [87], differential evolution with level comparison (DELC) [88], and mine blast algorithm(MBA) [89]. The results of all eight algorithms are showed in Tables 6 and 7. And the best solution obtained by MBSAgC is x=0.788675,0.408248 with fx=263.895843 using 9540 FEs. In Table 6, we can see that only MBA does not get the best solution with the corresponding function value; the other five algorithms all reach the best solution with the corresponding function value equal to 263.895483. MBSAgC is not only lower than other six algorithms in term of FEs but also Std value is the lowest in all algorithms. Table 6 shows the comparison of the best solutions obtained by MBSAgC and other algorithms. The statistical results of all compared algorithms are listed in Table 7.From Tables6 and 7, all of the compared algorithms reach the best solution equal to 263.89583 except MBA with 263.89582. However, MBSAgC requires the lowest FEs (9540) among all algorithms. Its Std value is also better than all compared algorithms. These comparative results illustrate that MBSAgC outperforms other algorithms in terms of computational overhead and robustness for this problem.Figure5(a) shows the convergence curves of MBSAgC and BSA for three-bar truss design problem. As shown in Figure 5(a), the BSA reaches the global optimum at about 13,000 FEs, while MBSAgC only reaches the global optimum at about 9000 FEs. It is illustrated that convergence rate of MBSAgC is faster than the BSA for this problem.Figure 5
Convergence curves of MBSAgC and BSA. (a) Three-bar truss design problem (TTP). (b) Pressure vessel design problem. (c) Tension/compression spring design. (d) Welded beam design problem (WBP). (e) Speed reducer design problem (SRP).
(a)
(b)
(c)
(d)
(e)(2) Pressure Vessel Design Problem (PVP). There is a nonlinear objective function in the pressure vessel design problem, and the function has three linear and one nonlinear inequality constrains and two discrete and two continuous design variables. For this problem, BSA, BSA-SAε, DELC, PSO-DE, genetic algorithms based on dominance tournament selection (GA-DT) [90], modified differential evolution (MDE) [91], coevolutionary particle swarm optimization (CPSO) [92], hybrid particle swarm optimization (HPSO) [93], and artificial bee colony algorithm (ABC) are used as comparison algorithms. And the best solution obtained by MBSAgC is x=0.8125,0.4375.42.0984,176.6366 with fx=6059.7143 using 16180 FEs. It is the lowest FEs in all compared algorithms. Table 8 shows the comparison of the best solutions obtained by MBSAgC and other algorithms. The statistical results of all compared algorithms are listed in Table 9.As shown in Table8, MBSAgC, BSA-SAε, ABC, DELC, and HPSO reach the same function value equal to 6059.7143, which is worse than the function value 6059.7017 of MDE, but all seven algorithms get the feasible solution of this design problem. From Table 9, the solution of MDE ranks first in terms of the best solution; the best solution of MBSAgC is better than those of GA-DT, CPSO, ABC, and BSA. Though MDE is better than MBSAgC in terms of the best solution, the FEs of MBSAgC is much lower than that of MDE. It is illustrated that MBSAgC is competitive with other algorithms on this engineering problem.Figure5(b) describes the convergence curves of MBSAgC and BSA for the pressure vessel design problem. As shown in Figure 5(b), MBSAgC finds the global optimum at about 16,000 FEs and obtains a far more accurate function value than BSA. Moreover, the convergence speed of MBSAgC is much faster than that of BSA.(3) Tension/Compression Spring Design Problem (TCSP). There are three continuous variables and four nonlinear inequality constrains in the tension/compression spring design problem. MBSAgC obtains the best feasible solution at x=0.56187,0.356669,11.2889 with fx=0.012665 using 22680 FEs. And we compare the results obtained by MBSAgC with GA-DT, MDE, CPSO, HPSO, DEDS, HEAA, DELC, PSO-DE, ABC, MBA, BSA-SAε, BSAISA, and social spider optimization (SSOC) [94]. The comparison of the best solutions obtained by all comparison algorithms is presented in Table 10, and the statistical results of various algorithms are listed in Table 11.In Table10, only GA-DT and CPSO fail to get the best solution for this problem; other compared algorithms all reach the best solution value 0.012665. MBSAgC requires 22680 FEs to get the global best solution. It is worse than MBA (FEs : 7650) and DELC (FEs : 20000), but its Std is better than both of them. It can be summarized that MBSAgC outperforms other compared algorithms except MBA and DELC; in addition, the robustness of MBSAgC is much better than others, except HEAA.Figure5(c) depicts the convergence curves of MBSAgC and BSA for the tension/compression spring design problem. From Figure 5(c), it can be observed that both MBSAgC and BSA are prone to fall into a local optimum in the early iterations, but they are able to jump out of local optimum successfully. However, the convergence speed of MBSAgC is obviously faster than that of BSA.(4) Welded Beam Design Problem (WBP). The welded beam design problem is a minimum-cost problem, which has four continuous design variables and subjects to two linear and five nonlinear inequality constraints. For this problem, MBSAgC is compared with GA-DT, MDE, CPSO, HPSO, DELC, PSO-DE, ABC, MBA, BSA, BSA-SAε, SSOC, and BSAISA. The comparison of the best solutions obtained by all comparison algorithms is presented in Table 12, and the statistical results of various algorithms are listed in Table 13.As can be seen from Table13, MBSAgC has obtained the best feasible solution at x=0.205730,3.4706,9.036624,0.205730 with fx=1.724852 using 21640 FEs. Only GA-DE, CPSO, and MBA fail to find the best function value 1.724852; the other compared algorithms all find the best function values. DELC outperforms other algorithms in both Std and FEs. MBSAgC performs better than other algorithms except DELC in terms of FEs. It is illustrated that MBSAgC is competitive on computational cost, while the robustness of MBSAgC is really bad in terms of its Std vlaue (2.765E − 01), which is worse than those of most existing comparison algorithms.Figure5(d) depicts the convergence curves of MBSAgC and BSA for the welded beam design problem. Figure 5(d) shows that the convergence speed of MBSAgC is faster than that of BSA remarkably.(5) Speed Reducer Design Problem (SRP). The speed reducer design problem consists of eleven constraints and seven continuous design variables, where x3 is an integer variable. The best solution obtained by MBSAgC is x=3.500000,0.700000,17,7.300000,7.7153,3.3052,5.2867 with fx=2994.471066 using 12580 FEs. We compared the results of MBSAgC with 10 well-known algorithms: BSAISA, BSA, MDE, DEDS, HEAA, DELC, PSO-DE, ABC, MBA, and SSOC. The compared best solutions and the statistical results of all comparison algorithms are represented in Tables 14 and 15, respectively.As seen in Tables14 and 15, MBSAgC, BSAISA, BSA, DEDS, and DELC get the same optimal values equaling to 2294.471066. Other algorithms have different optimal values, but all the function values they get are feasible. MBA requires the lowest FEs (6300) to get the global optimal solution. MBSAgC has much lower FEs than other compared algorithms except MBA. Though MBA has the lowest FEs, it fails to find the best solution for this problem. This experiment is to test whether the algorithm can find the optimal solution with the minimum computational cost. Obviously, MBSAgC conforms to the experimental requirements because it requires so low FEs to find the best solution for this problem.Figure5(e) depicts the convergence curves of MBSAgC and BSA for the speed reducer design problem and shows that the convergence speed of MBSAgC is faster than that of BSA.


### 5.2.2. Statistical Analyses for Engineering Results

In order to further analyze the comprehensive performance of the proposed MBSAgC, we list the optimal feasible solutions and the minimum FEs for all the abovementioned engineering design problems, and compare them with the proposed MBSAgC. The statistical results are shown in Table16. As can be seen from Table 16, MBSAgC can find the optimal solution of four engineering design problems. The feasible solution obtained by MBSAgC is the optimal for three-bar truss problem, and the FEs of MBSAgC is also the lowest; it is illustrated that the results of MBSAgC are competitive. For pressure vessel design problem, the feasible solution found by MBSAgC is the same as most of compared algorithms, but the FEs of MBSAgC is the minimum among all the compared algorithms. For the rest three problems, MBSAgC can find the optimal feasible solution; although the FEs of MBSAgC is not the smallest, its ranking among the all compared algorithms is competitive. Furthermore, the MBA is with a smaller FEs, but it cannot find out the current best value on speed reducer problem. So overall, about the ability to find the optimal solution and the algorithmic computational overhead, MBSAgC holds a very competitive performance on these optimization problems compared with the state-of-the-art algorithms.Table 16
Comparison for the best solution and FEs.
Engineering design
Smallest “Best”
“Best” by MBSAgC
Smallest FEs (algorithm)
FEs for MBSAgC (rank)
Three-bar truss
263.895843
263.895843
10000 (DELC)
9560 (1)
Pressure vessel
6059.7017
6059.7143
24000 (MDE)
16180 (1)
Tension/compression spring
0.012665
0.012665
7650 (MBA)
22680 (3)
Welded beam
1.724852
1.724852
20000 (DELC)
21640 (2)
Speed reducer
2994.471066
2994.471066
6300 (MBA)
12580 (2)


## 5.2.1. Engineering Problem and Experimental Results

(1) Three-Bar Truss Design Problem(TTP). We compare MBSAgC with the BSA, differential evolution with dynamic stochastic selection(DEDS) [85], hybrid evolutionary algorithm(HEAA) [86], hybrid particle swarm optimization with differential evolution(POS-DE) [87], differential evolution with level comparison (DELC) [88], and mine blast algorithm(MBA) [89]. The results of all eight algorithms are showed in Tables 6 and 7. And the best solution obtained by MBSAgC is x=0.788675,0.408248 with fx=263.895843 using 9540 FEs. In Table 6, we can see that only MBA does not get the best solution with the corresponding function value; the other five algorithms all reach the best solution with the corresponding function value equal to 263.895483. MBSAgC is not only lower than other six algorithms in term of FEs but also Std value is the lowest in all algorithms. Table 6 shows the comparison of the best solutions obtained by MBSAgC and other algorithms. The statistical results of all compared algorithms are listed in Table 7.From Tables6 and 7, all of the compared algorithms reach the best solution equal to 263.89583 except MBA with 263.89582. However, MBSAgC requires the lowest FEs (9540) among all algorithms. Its Std value is also better than all compared algorithms. These comparative results illustrate that MBSAgC outperforms other algorithms in terms of computational overhead and robustness for this problem.Figure5(a) shows the convergence curves of MBSAgC and BSA for three-bar truss design problem. As shown in Figure 5(a), the BSA reaches the global optimum at about 13,000 FEs, while MBSAgC only reaches the global optimum at about 9000 FEs. It is illustrated that convergence rate of MBSAgC is faster than the BSA for this problem.Figure 5
Convergence curves of MBSAgC and BSA. (a) Three-bar truss design problem (TTP). (b) Pressure vessel design problem. (c) Tension/compression spring design. (d) Welded beam design problem (WBP). (e) Speed reducer design problem (SRP).
(a)
(b)
(c)
(d)
(e)(2) Pressure Vessel Design Problem (PVP). There is a nonlinear objective function in the pressure vessel design problem, and the function has three linear and one nonlinear inequality constrains and two discrete and two continuous design variables. For this problem, BSA, BSA-SAε, DELC, PSO-DE, genetic algorithms based on dominance tournament selection (GA-DT) [90], modified differential evolution (MDE) [91], coevolutionary particle swarm optimization (CPSO) [92], hybrid particle swarm optimization (HPSO) [93], and artificial bee colony algorithm (ABC) are used as comparison algorithms. And the best solution obtained by MBSAgC is x=0.8125,0.4375.42.0984,176.6366 with fx=6059.7143 using 16180 FEs. It is the lowest FEs in all compared algorithms. Table 8 shows the comparison of the best solutions obtained by MBSAgC and other algorithms. The statistical results of all compared algorithms are listed in Table 9.As shown in Table8, MBSAgC, BSA-SAε, ABC, DELC, and HPSO reach the same function value equal to 6059.7143, which is worse than the function value 6059.7017 of MDE, but all seven algorithms get the feasible solution of this design problem. From Table 9, the solution of MDE ranks first in terms of the best solution; the best solution of MBSAgC is better than those of GA-DT, CPSO, ABC, and BSA. Though MDE is better than MBSAgC in terms of the best solution, the FEs of MBSAgC is much lower than that of MDE. It is illustrated that MBSAgC is competitive with other algorithms on this engineering problem.Figure5(b) describes the convergence curves of MBSAgC and BSA for the pressure vessel design problem. As shown in Figure 5(b), MBSAgC finds the global optimum at about 16,000 FEs and obtains a far more accurate function value than BSA. Moreover, the convergence speed of MBSAgC is much faster than that of BSA.(3) Tension/Compression Spring Design Problem (TCSP). There are three continuous variables and four nonlinear inequality constrains in the tension/compression spring design problem. MBSAgC obtains the best feasible solution at x=0.56187,0.356669,11.2889 with fx=0.012665 using 22680 FEs. And we compare the results obtained by MBSAgC with GA-DT, MDE, CPSO, HPSO, DEDS, HEAA, DELC, PSO-DE, ABC, MBA, BSA-SAε, BSAISA, and social spider optimization (SSOC) [94]. The comparison of the best solutions obtained by all comparison algorithms is presented in Table 10, and the statistical results of various algorithms are listed in Table 11.In Table10, only GA-DT and CPSO fail to get the best solution for this problem; other compared algorithms all reach the best solution value 0.012665. MBSAgC requires 22680 FEs to get the global best solution. It is worse than MBA (FEs : 7650) and DELC (FEs : 20000), but its Std is better than both of them. It can be summarized that MBSAgC outperforms other compared algorithms except MBA and DELC; in addition, the robustness of MBSAgC is much better than others, except HEAA.Figure5(c) depicts the convergence curves of MBSAgC and BSA for the tension/compression spring design problem. From Figure 5(c), it can be observed that both MBSAgC and BSA are prone to fall into a local optimum in the early iterations, but they are able to jump out of local optimum successfully. However, the convergence speed of MBSAgC is obviously faster than that of BSA.(4) Welded Beam Design Problem (WBP). The welded beam design problem is a minimum-cost problem, which has four continuous design variables and subjects to two linear and five nonlinear inequality constraints. For this problem, MBSAgC is compared with GA-DT, MDE, CPSO, HPSO, DELC, PSO-DE, ABC, MBA, BSA, BSA-SAε, SSOC, and BSAISA. The comparison of the best solutions obtained by all comparison algorithms is presented in Table 12, and the statistical results of various algorithms are listed in Table 13.As can be seen from Table13, MBSAgC has obtained the best feasible solution at x=0.205730,3.4706,9.036624,0.205730 with fx=1.724852 using 21640 FEs. Only GA-DE, CPSO, and MBA fail to find the best function value 1.724852; the other compared algorithms all find the best function values. DELC outperforms other algorithms in both Std and FEs. MBSAgC performs better than other algorithms except DELC in terms of FEs. It is illustrated that MBSAgC is competitive on computational cost, while the robustness of MBSAgC is really bad in terms of its Std vlaue (2.765E − 01), which is worse than those of most existing comparison algorithms.Figure5(d) depicts the convergence curves of MBSAgC and BSA for the welded beam design problem. Figure 5(d) shows that the convergence speed of MBSAgC is faster than that of BSA remarkably.(5) Speed Reducer Design Problem (SRP). The speed reducer design problem consists of eleven constraints and seven continuous design variables, where x3 is an integer variable. The best solution obtained by MBSAgC is x=3.500000,0.700000,17,7.300000,7.7153,3.3052,5.2867 with fx=2994.471066 using 12580 FEs. We compared the results of MBSAgC with 10 well-known algorithms: BSAISA, BSA, MDE, DEDS, HEAA, DELC, PSO-DE, ABC, MBA, and SSOC. The compared best solutions and the statistical results of all comparison algorithms are represented in Tables 14 and 15, respectively.As seen in Tables14 and 15, MBSAgC, BSAISA, BSA, DEDS, and DELC get the same optimal values equaling to 2294.471066. Other algorithms have different optimal values, but all the function values they get are feasible. MBA requires the lowest FEs (6300) to get the global optimal solution. MBSAgC has much lower FEs than other compared algorithms except MBA. Though MBA has the lowest FEs, it fails to find the best solution for this problem. This experiment is to test whether the algorithm can find the optimal solution with the minimum computational cost. Obviously, MBSAgC conforms to the experimental requirements because it requires so low FEs to find the best solution for this problem.Figure5(e) depicts the convergence curves of MBSAgC and BSA for the speed reducer design problem and shows that the convergence speed of MBSAgC is faster than that of BSA.


## 5.2.2. Statistical Analyses for Engineering Results

In order to further analyze the comprehensive performance of the proposed MBSAgC, we list the optimal feasible solutions and the minimum FEs for all the abovementioned engineering design problems, and compare them with the proposed MBSAgC. The statistical results are shown in Table16. As can be seen from Table 16, MBSAgC can find the optimal solution of four engineering design problems. The feasible solution obtained by MBSAgC is the optimal for three-bar truss problem, and the FEs of MBSAgC is also the lowest; it is illustrated that the results of MBSAgC are competitive. For pressure vessel design problem, the feasible solution found by MBSAgC is the same as most of compared algorithms, but the FEs of MBSAgC is the minimum among all the compared algorithms. For the rest three problems, MBSAgC can find the optimal feasible solution; although the FEs of MBSAgC is not the smallest, its ranking among the all compared algorithms is competitive. Furthermore, the MBA is with a smaller FEs, but it cannot find out the current best value on speed reducer problem. So overall, about the ability to find the optimal solution and the algorithmic computational overhead, MBSAgC holds a very competitive performance on these optimization problems compared with the state-of-the-art algorithms.Table 16
Comparison for the best solution and FEs.
Engineering design
Smallest “Best”
“Best” by MBSAgC
Smallest FEs (algorithm)
FEs for MBSAgC (rank)
Three-bar truss
263.895843
263.895843
10000 (DELC)
9560 (1)
Pressure vessel
6059.7017
6059.7143
24000 (MDE)
16180 (1)
Tension/compression spring
0.012665
0.012665
7650 (MBA)
22680 (3)
Welded beam
1.724852
1.724852
20000 (DELC)
21640 (2)
Speed reducer
2994.471066
2994.471066
6300 (MBA)
12580 (2)


## 5.3. Synthetical Discussions Based on Experimental Results

Based on the above experimental results on CEC2005, CEC2014, and several engineering optimization problems, this subsection discusses the search behaviors of the proposed algorithm from three perspectives including its solution accuracy, convergence speed, and future development.


### 5.3.1. Solution Accuracy

The “Best” values of all the previous tables are the reflection of the solution accuracy for each algorithm in solving all the test functions (or engineering optimization problems). The smaller this “Best” value, the higher the solution accuracy is. For 10-dimension functions of CEC2005, the average ranking of MBSAgC’s “Best” value equals to 2.76 (shown in Table2) which ranks first in all the compared algorithms. For 30-dimension functions of CEC2005, the average ranking of MBSAgC also is first (2.56, shown in Table 3). For 30-dimension functions of CEC2014, its average ranking is second (2.63, shown in Table 4). And MBSAgC can find the current optimum of four of the five engineering optimization problems. Overall, the solution accuracy of MBSAgC is obviously superior to the other compared algorithms.


### 5.3.2. Convergence Speed

The “FEs” values of all the previous tables are the reflection of the convergence speed for each algorithm in solving all the test functions (or engineering optimization problems). The smaller this “FEs” value, the faster the convergence speed is. In addition, each curve of the previous convergence figures represents the variation of the current optima (or error values) over the fitness evaluation times. The variation trend of the curve also reflects the convergence speed of the compared algorithms. For the five engineering optimization problems (shown in Table16), two of “FEs” values of MBSAgC rank first, two of them rank second, and the rest one ranks third. Similar to all the compared algorithms, MBSAgC solves all the benchmark functions within the given “FEs”. And Figures 3–5 show that MBSAgC holds faster convergence speed than the original BSA. Overall, the proposed MBSAgC keeps as a strong competitiveness on the convergence speed as the compared state-of-the-arts algorithms.


### 5.3.3. Future Development

According to the above discussions, we can draw a conclusion that the proposed MBSAgC improves the solution accuracy on a condition of keeping competitive convergence speed. This indicates that the modified strategies based on the collective wisdom, especially the winner-tendency, do enhance the exploitation capability of MBSAgC. However, we also notice that MBSAgC performs worse than LBSA on the CEC2014 benchmark functions. This also leaves a space for further development of the topological opposition-based learning operator (TOBL) which is the core of all the modified strategies of MBSAgC.


## 5.3.1. Solution Accuracy

The “Best” values of all the previous tables are the reflection of the solution accuracy for each algorithm in solving all the test functions (or engineering optimization problems). The smaller this “Best” value, the higher the solution accuracy is. For 10-dimension functions of CEC2005, the average ranking of MBSAgC’s “Best” value equals to 2.76 (shown in Table2) which ranks first in all the compared algorithms. For 30-dimension functions of CEC2005, the average ranking of MBSAgC also is first (2.56, shown in Table 3). For 30-dimension functions of CEC2014, its average ranking is second (2.63, shown in Table 4). And MBSAgC can find the current optimum of four of the five engineering optimization problems. Overall, the solution accuracy of MBSAgC is obviously superior to the other compared algorithms.


## 5.3.2. Convergence Speed

The “FEs” values of all the previous tables are the reflection of the convergence speed for each algorithm in solving all the test functions (or engineering optimization problems). The smaller this “FEs” value, the faster the convergence speed is. In addition, each curve of the previous convergence figures represents the variation of the current optima (or error values) over the fitness evaluation times. The variation trend of the curve also reflects the convergence speed of the compared algorithms. For the five engineering optimization problems (shown in Table16), two of “FEs” values of MBSAgC rank first, two of them rank second, and the rest one ranks third. Similar to all the compared algorithms, MBSAgC solves all the benchmark functions within the given “FEs”. And Figures 3–5 show that MBSAgC holds faster convergence speed than the original BSA. Overall, the proposed MBSAgC keeps as a strong competitiveness on the convergence speed as the compared state-of-the-arts algorithms.


## 5.3.3. Future Development

According to the above discussions, we can draw a conclusion that the proposed MBSAgC improves the solution accuracy on a condition of keeping competitive convergence speed. This indicates that the modified strategies based on the collective wisdom, especially the winner-tendency, do enhance the exploitation capability of MBSAgC. However, we also notice that MBSAgC performs worse than LBSA on the CEC2014 benchmark functions. This also leaves a space for further development of the topological opposition-based learning operator (TOBL) which is the core of all the modified strategies of MBSAgC.


## 6. Conclusions

In order to enhance the exploitation capability of BSA, this paper proposes an improved BSA algorithm, deep-mining backtracking search optimization algorithm guided by collective wisdom (MBSAgC). MBSAgC designs three learning mechanisms by deeply mining collective wisdom. The three mechanisms are assembled into two novel reproduction operators, i.e., a topological opposite-based learning operator (TOBL) and an improved mutation operator. The searching mechanisms of the novel TOBL operator and MBSAgC are analyzed and discussed in detail. The theoretical analysis demonstrates that the TOBL operator searches the vertices in a hypercube about the best individual. This broadens its search region of the original opposition-based learning strategy. In addition, the proposed algorithm does not increase any parameter to be artificial set. The numerical results on CEC2005, CEC2014 benchmark functions, and five engineering constrained design problems demonstrate the superiority of the proposed algorithm. The advantages of the searching mechanisms and experimental results also illustrate that TOBL is promising as an improved operator to enhance the exploitation capability of EAs.



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*Source: 2540102-2019-12-26.xml*