# Modified Backtracking Search Optimization Algorithm Inspired by Simulated Annealing for Constrained Engineering Optimization Problems

Authors: Hailong Wang; Zhongbo Hu; Yuqiu Sun; Qinghua Su; Xuewen Xia
Journal: Computational Intelligence and Neuroscience
 (2018)

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## Abstract

The backtracking search optimization algorithm (BSA) is a population-based evolutionary algorithm for numerical optimization problems. BSA has a powerful global exploration capacity while its local exploitation capability is relatively poor. This affects the convergence speed of the algorithm. In this paper, we propose a modified BSA inspired by simulated annealing (BSAISA) to overcome the deficiency of BSA. In the BSAISA, the amplitude control factor (F) is modified based on the Metropolis criterion in simulated annealing. The redesigned F could be adaptively decreased as the number of iterations increases and it does not introduce extra parameters. A self-adaptive ε-constrained method is used to handle the strict constraints. We compared the performance of the proposed BSAISA with BSA and other well-known algorithms when solving thirteen constrained benchmarks and five engineering design problems. The simulation results demonstrated that BSAISA is more effective than BSA and more competitive with other well-known algorithms in terms of convergence speed.


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## Body

## 1. Introduction

Optimization is an essential research objective in the fields of applied mathematics and computer sciences. Optimization algorithms mainly aim to obtain the global optimum for optimization problems. There are many different kinds of optimization problems in real world. When an optimization problem has a simple and explicit gradient information or requires relatively small budgets of allowed function evaluations, the implementation of classical optimization techniques such as mathematical programming often could achieve efficient results [1]. However, many real-world engineering optimization problems may have complex, nonlinear, or nondifferentiable forms, which make them difficult to be tackled by using classical optimization techniques. The emergence of metaheuristic algorithms has overcome the deficiencies of classical optimization techniques to some extent, as they do not require gradient information and have the ability to escape from local optima. Metaheuristic algorithms are mainly inspired from a variety of natural phenomena and/or biological social behavior. Among these metaheuristic algorithms, swarm intelligence algorithms and evolutionary algorithms perhaps are the most attractive [2]. Swarm intelligence algorithms [3] generally simulate the intelligence behavior of swarms of creatures, such as particle swarm optimization (PSO) [4], ant colony optimization (ACO) [5], cuckoo search (CS) [6], and the artificial bee colony (ABC) algorithm [7]. These types of algorithms generally are developed by inspirations from a series of complex behavior processes in swarms with mutual cooperation and self-organization, in which “cooperation” is their core concept. The evolutionary algorithms (EAs) [8, 9] are inspired by the mechanism of nature evolution, in which “evolution” is the key idea. Examples of EAs include genetic algorithm (GA) [10], differential evolution (DE) [11–14], covariance matrix adaptation evolution strategy (CMAES) [15], and the backtracking search optimization algorithm (BSA) [16].BSA is an iterative population-based EA, which was first proposed by Civicioglu in 2013. BSA has three basic genetic operators: selection, mutation, and crossover. The main difference between BSA and other similar algorithms is that BSA possesses a memory for storing a population from a randomly chosen previous generation, which is used to generate the search-direction matrix for the next iteration. In addition, BSA has a simple structure, which makes it efficient, fast, and capable of solving multimodal problems. BSA has only one control parameter called themix-rate, which significantly reduces the sensitivity of the initial values to the algorithm’s parameters. Due to these characteristics, in less than 4 years, BSA has been employed successfully to solve various engineering optimization problems, such as power systems [17–19], induction motor [20, 21], antenna arrays [22, 23], digital image processing [24, 25], artificial neural networks [26–29], and energy and environmental management [30–32].However, BSA has a weak local exploitation capacity and its convergence speed is relatively slow. Thus, many studies have attempted to improve the performance of BSA and some modifications of BSA have been proposed to overcome the deficiencies. From the perspective of modified object, the modifications of BSA can be divided into the following four categories. It is noted that we consider classifying the publication into the major modification category if it has more than one modification:(i)
Modifications of the initial populations [33–38](ii)
Modifications of the reproduction operators, including the mutation and crossover operators [39–47](iii)
Modifications of the selection operators, including the local exploitation strategy [48–51](iv)
Modifications of the control factor and parameter [52–57].The research on controlling parameters of EAs is one of the most promising areas of research in evolutionary computation; even a little modification of parameters in an algorithm can make a considerable difference [58]. In the basic BSA, the value of amplitude control factor (F) is the product of three and the standard normal distribution random number (i.e., F=3·randn), which is often too large or too small according to its formulation. This may give BSA a powerful global exploration capability at the early iterations; however, it also affects the later exploitation capability of BSA. Based on these considerations, we focus mainly on the influence of the amplitude control factor (F) on the BSA, that is, the fourth of the categories defined above. Duan and Luo [52] redesigned an adaptive F based on the fitness statistics of population at each iteration. Wang et al. [53] and Tian et al. [54] proposed an adaptive F based on Maxwell-Boltzmann distribution. Askarzadeh and dos Santos Coelho [55] proposed an adaptive F based on Burger’s chaotic map. Chen et al. [56] redesigned an adaptive F by introducing two extra parameters. Nama et al. [57] proposed a new F to adaptively change in the range of (0.45,1.99) and new mix-rate to randomly change in the range of (0,1). These modifications of F have achieved good effects in the BSA.Different from the modifications ofF in BSA described above, a modified version of BSA (BSAISA) inspired by simulated annealing (SA) is proposed in this paper. In the BSAISA, F based on iterations is redesigned by learning a characteristic where SA can probabilistically accept a higher energy state and the acceptance probability decreases with the decrease in temperature. The redesigned F can adaptively decrease as the number of iterations increases without introducing extra parameters. This adaptive variation tendency provides an efficient tradeoff between early exploration and later exploitation capability. We verified the effectiveness and competitiveness of BSAISA in simulation experiments using thirteen constrained benchmarks and five engineering design problems in terms of convergence speed.The remainder of this paper is organized as follows. Section2 introduces the basic BSA. As the main contribution of this paper, a detailed explanation of BSAISA is presented in Section 3. In Section 4, we present two sets of simulation experiments in which we implemented BSAISA and BSA to solve thirteen constrained optimization and five engineering design problems. The results are compared with those obtained by other well-known algorithms in terms of the solution quality and function evaluations. Finally, we give our concluding remarks in Section 5.


## 2. Backtracking Search Optimization Algorithm (BSA)

BSA is a population-based iterative EA. BSA generates trial populations to take control of the amplitude of the search-direction matrix which provides a strong global exploration capability. BSA equiprobably uses two random crossover strategies to exchange the corresponding elements of individuals in populations and trial populations during the process of crossover. Moreover, BSA has two selection processes. One is used to select population from the current and historical populations; the other is used to select the optimal population. In general, BSA can be divided into five processes: initialization, selection I, mutation, crossover, and selection II [16].


### 2.1. Initialization

BSA generates initial populationP and initial old population oldP using (1)Pi,j~Ulowj,upj,oldPi,j~Ulowj,upj,where Pi,j and oldPi,j are the jth individual elements in the problem dimension (D) that falls in ith individual position in the population size (N), respectively, lowj and upj mean the lower boundary and the upper boundary of the jth dimension, respectively, and U is a random uniform distribution.


### 2.2. Selection I

BSA’s selection I process is the beginning of each iteration. It aims to reselect a newoldP for calculating the search direction based on population P and historical population oldP. The new oldP is reselected through the “if-then” rule in (2)ifa<bthenoldP≔P∣a,b~U0,1,where ≔ is the update operation; a and b represent random numbers between 0 and 1. The update operation (see (2)) ensures BSA has a memory. After oldP is reselected, the order of the individuals in oldP is randomly permuted by (3)oldP≔permutingoldP.


### 2.3. Mutation

The mutation operator is used for generating the initial form of trial populationM with (4)M=P+F·oldP-P,whereF is the amplitude control factor of mutation operator used to control the amplitude of the search direction. The value F=3·randn, where randn~N(0,1), and N is standard normal distribution.


### 2.4. Crossover

In this process, BSA generates the final form of trial populationT. BSA equiprobably uses two crossover strategies to manipulate the selected elements of the individuals at each iteration. Both the strategies generate different binary integer-valued matrices (map) of size N·D to select the elements of individuals that have to be manipulated.Strategy I uses the mix-rate parameter (mix-rate) to control the numbers of elements of individuals that are manipulated by usingmix-rate·rand·D, where mix-rate = 1. Strategy II manipulates the only one randomly selected element of individuals by using randi(D), where randi~U(0,D).The two strategies equiprobably are employed to manipulate the elements of individuals through the “if-then” rule: ifmapn,m=1, where n∈{1,2,…,N} and m∈{1,2,…,D}, then, T is updated with Tn,m≔Pn,m.At the end of crossover process, if some individuals inT have overflowed the allowed search space limits, they will need to be regenerated by using (1).


### 2.5. Selection II

In BSA’s selection II process, the fitness values inPi and Ti are compared, used to update the population P based on a greedy selection. If Ti have a better fitness value than Pi, then, Ti is updated to be Pi. The population P is updated by using (5)Pi=Ti,if  fitnessTi<fitnessPi,Pi,else.If the best individual ofP (Pbest) has a better fitness value than the global minimum value obtained, the global minimizer will be updated to be Pbest, and the global minimum value will be updated to be the fitness value of Pbest.


## 2.1. Initialization

BSA generates initial populationP and initial old population oldP using (1)Pi,j~Ulowj,upj,oldPi,j~Ulowj,upj,where Pi,j and oldPi,j are the jth individual elements in the problem dimension (D) that falls in ith individual position in the population size (N), respectively, lowj and upj mean the lower boundary and the upper boundary of the jth dimension, respectively, and U is a random uniform distribution.


## 2.2. Selection I

BSA’s selection I process is the beginning of each iteration. It aims to reselect a newoldP for calculating the search direction based on population P and historical population oldP. The new oldP is reselected through the “if-then” rule in (2)ifa<bthenoldP≔P∣a,b~U0,1,where ≔ is the update operation; a and b represent random numbers between 0 and 1. The update operation (see (2)) ensures BSA has a memory. After oldP is reselected, the order of the individuals in oldP is randomly permuted by (3)oldP≔permutingoldP.


## 2.3. Mutation

The mutation operator is used for generating the initial form of trial populationM with (4)M=P+F·oldP-P,whereF is the amplitude control factor of mutation operator used to control the amplitude of the search direction. The value F=3·randn, where randn~N(0,1), and N is standard normal distribution.


## 2.4. Crossover

In this process, BSA generates the final form of trial populationT. BSA equiprobably uses two crossover strategies to manipulate the selected elements of the individuals at each iteration. Both the strategies generate different binary integer-valued matrices (map) of size N·D to select the elements of individuals that have to be manipulated.Strategy I uses the mix-rate parameter (mix-rate) to control the numbers of elements of individuals that are manipulated by usingmix-rate·rand·D, where mix-rate = 1. Strategy II manipulates the only one randomly selected element of individuals by using randi(D), where randi~U(0,D).The two strategies equiprobably are employed to manipulate the elements of individuals through the “if-then” rule: ifmapn,m=1, where n∈{1,2,…,N} and m∈{1,2,…,D}, then, T is updated with Tn,m≔Pn,m.At the end of crossover process, if some individuals inT have overflowed the allowed search space limits, they will need to be regenerated by using (1).


## 2.5. Selection II

In BSA’s selection II process, the fitness values inPi and Ti are compared, used to update the population P based on a greedy selection. If Ti have a better fitness value than Pi, then, Ti is updated to be Pi. The population P is updated by using (5)Pi=Ti,if  fitnessTi<fitnessPi,Pi,else.If the best individual ofP (Pbest) has a better fitness value than the global minimum value obtained, the global minimizer will be updated to be Pbest, and the global minimum value will be updated to be the fitness value of Pbest.


## 3. Modified BSA Inspired by SA (BSAISA)

As mentioned in the introduction of this paper, the research work on the control parameters of an algorithm is very meaningful and valuable. In this paper, in order to improve BSA’s exploitation capability and convergence speed, we propose a modified version of BSA (BSAISA) where the redesign ofF is inspired by SA. The modified details of BSAISA are described in this section. First, the structure of the modified F is described, before we explain the detailed design principle for the modified F inspired by SA. Subsequently, two numerical tests are used to illustrate that the redesigned F improves the convergence speed of the algorithm. We introduce a self-adaptive ε-constrained method for handling constraints at the end of this section.


### 3.1. Structure of the Adaptive Amplitude Control FactorF

The modifiedF is a normally distributed random number, where its mean value is an exponential function and its variance is equal to 1. In BSAISA, we redesign the adaptive F to replace the original version using (6)Fi~Nexp⁡-1/ΔIi1/G,1,where i is the index of individuals, Fi is the adaptive amplitude control factor that corresponds to the ith individual, |ΔIi| is the absolute value of the difference between the objective function values of Pi and Ti (individual differences), N is the normal distribution, and G is the current iteration.According to (6), the exponential function (mean value) decreases dynamically with the change in the number of iterations (G) and the individual differences (|ΔIi|). Based on the probability density function curve of the normal distribution, the modified F can be decreased adaptively as the number of iterations increases. Another characteristic of the modifiedF is that there are not any extra parameters.


### 3.2. Design Principle of the Modified Amplitude Control FactorF

The design principle of the modifiedF is inspired by the Metropolis criterion in SA. SA is a metaheuristic optimization technique based on physical behavior in nature. SA based on the Monte Carlo method was first proposed by Metropolis et al. [59] and it was successfully introduced into the field of combinatorial optimization for solving complex optimization problems by Kirkpatrick et al. [60].The basic concept of SA derives from the process of physical annealing with solids. An annealing process occurs when a metal is heated to a molten state with a high temperature; then it is cooled slowly. If the temperature is decreased quickly, the resulting crystal will have many defects and it is just metastable; even the most stable crystalline state will be achieved at all. In other words, this may form a higher energy state than the most stable crystalline state. Therefore, in order to reach the absolute minimum energy state, the temperature needs to be decreased at a slow rate. SA simulates this process of annealing to search the global optimal solution in an optimization problem. However, accepting only the moves that lower the energy of system is like extremely rapid quenching; thus SA uses a special and effective acceptance method, that is, Metropolis criterion, which can probabilistically accept the hill-climbing moves (the higher energy moves). As a result, the energy of the system evolves into a Boltzmann distribution during the process of the simulated annealing. From this angle of view, it is no exaggeration to say that the Metropolis criterion is the core of SA.The Metropolis criterion can be expressed by the physical significance of energy, where the new energy state will be accepted when the new energy state is lower than the previous energy state, and the new energy state will be probabilistically accepted when the new energy state is higher than the previous energy state. This feature of SA can escape from being trapped in local minima especially in the early stages of the search. It can also be described as follows.(i)IfΔE=Ej-Ei≤0, then the new state j is accepted and the energy with the displaced atom is used as the starting point for the next step, where E represents the energy of the atom. Both i and j are the states of atoms, and j is the next state of i.(ii)IfΔE>0, then calculate the probability of P(ΔE)=exp⁡(-ΔE/kTc), and generate a random number θ, which is a uniform distribution over (0,1), where k is Boltzmann’s constant (in general, k=1) and Tc is the current temperature. If P(ΔE)>θ, then the new energy will be accepted; otherwise, the previous energy is used to start the next step.Analysis 1.
The Metropolis criterion states that SA has two characteristics: (1) SA can probabilistically accept the higher energy and (2) the acceptance probability of SA decreases as the temperature decreases. Therefore, SA can reject and jump out of a local minimum with a dynamic and decreasing probability to continue exploiting the other solutions in the state space. This acceptance mechanism can enrich the diversity of energy states.Analysis 2.
As shown in (4), F is used to control the amplitude of population mutation in BSA, thus F is an important factor for controlling population diversity. If F is excessively large, the diversity of the population will be too high and the convergence speed of BSA will slow down. If F is excessively small, the diversity of the population will be reduced so it will be difficult for BSA to obtain the global optimum and it may be readily trapped by a local optimum. Therefore, adaptively controlling the amplitude of F is a key to accelerating the convergence speed of the algorithm and maintaining its the population diversity.Based on Analyses1 and 2, it is clear that if F can dynamically decrease, the convergence speed of BSA will be able to accelerate while maintaining the population diversity. On the other hand, SA possesses this characteristic that its acceptance probability can be dynamically reduced. Based on these two considerations, we propose BSAISA with a redesigned F, which is inspired by SA. More specifically, the new F (see (6)) is redesigned by learning the formulation (P(ΔE)) of acceptance probability, and its formulation has been shown in the previous subsection.For the two formulas of the modifiedF and P(ΔE), the individual difference (|ΔIi|) of a population or the energy difference (ΔE) of a system will decrease as the number of iterations increases in an algorithm, and the temperature of SA tends to decrease, while the iteration of BSA tends to increase. As a result, one can observe the correspondence between modified F and P(ΔE) where the reciprocal of individual difference (1/|ΔIi|) corresponds to the energy difference (ΔE) of SA, and the reciprocal of current iteration (1/G) corresponds to the current temperature (Tc) of SA. In this way, the redesigned F can be decreased adaptively as the number of iterations increases.


### 3.3. Numerical Analysis of the Modified Amplitude Control FactorF

In order to verify that the convergence speed of the basic BSA is improved with the modifiedF, two types (unimodal and multimodal) of unconstrained benchmark functions are used to test the changing trends in F and population variances and best function values as the iterations increases, respectively. The two functions are Schwefel 1.2 and Rastrigin, and their detailed information is provided in [61]. The two functions and the user parameters including the populations (N), dimensions (D), and maximum iterations (Max) are shown in Table 1. Three groups of test results are compared in the tests including (1) the comparative curves of the mean values of the modified F and original F for Schwefel 1.2 and Rastrigin, (2) the comparative curves of the mean values of BSA and BSAISA population variances for two functions, and (3) the convergence curves of BSA and BSAISA for two functions. They are depicted in Figures 1, 2, and 3, respectively. Based on Figures 1–3, two results can be observed as follows.Table 1
Two benchmark functions and the corresponding populations, dimensions, and max iterations.
Name
Objective function
Range
N
D
Max
Schwefel 1.2
f
x
=
∑
i
=
1
D
∑
j
=
1
i
x
j
2
[
-
100,100
]
100
30
500
Rastrigin
f
x
=
∑
i
=
1
D
x
i
2
-
1
0
cos
⁡
2
π
x
i
+
10
[
-
5.12,5.12
]
100
30
500
Note. “N” means populations, “D” means dimensions, and “Max” means maximum iterations.Figure 1
Comparisons of modifiedF and original F for Schwefel 1.2 and Rastrigin.
(a)
ModifiedF on Schwefel 1.2
(b)
OriginalF on Schwefel 1.2
(c)
ModifiedF on Rastrigin
(d)
OriginalF on RastriginFigure 2
The mean values of population variances versus number of iterations using BSA and BSAISA for two functions.
(a)
Schwefel 1.2
(b)
RastriginFigure 3
Convergence curves of BSA and BSAISA for two functions.
(a)
Schwefel 1.2
(b)
Rastrigin(1) According to the trends ofF in Figure 1, both the originalF and modifiedF are subject to changes from the normal distribution. The mean value of the originalF does not tend to decrease as the number of iterations increases. By contrast, the mean value of the modifiedF exhibits a clear and fluctuating downward trend as the number of iterations increases.(2) According to Figure2, the population variances of BSAISA and BSA both exhibit a clear downward trend as the number of iterations increases. The population variances of BSAISA and BSA are almost same during the early iterations. This illustrates that the modifiedF does not reduce the population diversity in the early iterations. In the middle and later iterations, the population variances of BSAISA decrease more quickly than that of BSA. This illustrates that the modifiedF improves the convergence speed. As can be seen from Figure 3, as the number of iterations increases, the best objective function value of BSAISA drops faster than that of BSA. This shows that the modifiedF improves the convergence speed of BSA. Moreover, BSAISA can find a more accurate solution at the same computational cost.Summary. Based on the design principle and numerical analysis of the modified F, the modified F exhibits an overall fluctuating and downward trend, which matches with the concept of the acceptance probability in SA. In particular, during the early iterations, the modified F is relatively large. This allows BSAISA to search in a wide region, while maintaining the population diversity of BSAISA. As the number of iterations decreases, the modified F gradually exhibits a decreasing trend. This accelerates the convergence speed of BSAISA. In the later iterations, the modified F is relatively small. This enables BSAISA to fully search in the local region. Therefore, it can be concluded that BSAISA can adaptively control the amplitude of population mutation to change its local exploitation capacity. This may improve the convergence speed of the algorithm. Moreover, the modified F does not introduce extra parameters, so it does not increase the sensitivity of BSAISA to the parameters.


### 3.4. A Self-Adaptiveε-Constrained Method for Handling Constraints

In general, a constrained optimization problem can be mathematically formulated as a minimization problem, as follows:(7)minfXsubjectto:giX≤0,i=1,2,…,m,hjX=0,j=1,2,…,n,where X=(x1,x2,…,xD)∈RD is a D-dimensional vector, m is the total number of inequality constraints, andn is the total number of equality constraints. The equality constraints are transformed into inequality constraints by using |hj(X)|-δ≤0, where δ is a very small degree of violation, and δ=1E-4 in this paper. The maximization problems are transformed into minimization problems using -f(X). The constraint violation φ(X) is given by (8)φX=∑i=1mmax⁡0,giX+∑j=1nmax⁡0,hjX-δ.Several constraint-handling methods have been proposed previously, where the five most commonly used methods comprise penalty functions, feasibility and dominance rules (FAD), stochastic ranking,ε-constrained methods, and multiobjectives concepts. Among these five methods, the ε-constrained method is relatively effective and used widely. Zhang et al. [62] proposed a self-adaptive ε-constrained method (SAε) to combine with the basic BSA for constrained problems. It has been verified that the SAε has a stronger search efficiency and convergence than the fixed ε-constrained method and FAD. In this paper, the SAε is used to combine with BSAISA for constrained optimization problems, which comprises the following two rules: (1) if the constraint violations of two solutions are smaller than a given ε value or two solutions have the same constraint violations, the solution with a better objective function value is preferred and (2) if not, the solution with a smaller constraint violation is preferred. SAε could be expressed by the following equations: (9)thebetterone=fX1fX1<fX2,ifφX1,φX2≤ε,fX1<fX2,ifφX1=φX2,φX1≤φX2,otherwise,where ε is a positive value that represents a tolerance related to constraint violation. The self-adaptive ε value is formulated as the following equation: (10)ε0=φP0θ,(11)ε10=ε0,(12)ε2t=φTtθ,ifε0>Th1,ε0,else,(13)ε1t=ε2t,ifε2t>Th2,ε2t<ε1t-1,ε1t-1,else,(14)εt=ε1t1-tTccp,ift≤Tc0,else,where t is the number of the current iterations. φ(P0θ) is the constraint violation of the top θth individual in the initial population. φ(Ttθ) is the constraint violation of the top θth individual in the trial population at the current iteration. cp and Tc are control parameters. Th1 and Th2 are threshold values. εt is related to the iteration t and functions ε1(t) and ε2(t).Firstly,ε0 is set as φ(P0θ). If the initial value ε0 is bigger than Th1, φ(Ttθ) will be assigned to ε2(t); otherwise ε0 will be assigned to ε2(t). Then, if ε2(t)<ε1(t-1) and ε2(t) is bigger than Th2, ε2(t) will be assigned to ε1(t); otherwise ε1(t-1) will be assigned to ε1(t). Finally, εt is updated as (14). The detailed information of SAε can be acquired from [62], and the related parameter settings of SAε (the same as [62]) are presented in Table 2.Table 2
The parameter setting of SAε.
Tc=0.2∗Tmax
θ=0.3∗N
cp=5
Th1=10
T
h
2
=
2
Note. “Tmax” means the maximum iterations; “N” means populations.To illustrate the changing trend of the self-adaptiveε value vividly, BSAISA with SAε is used to solve a well-known benchmark constrained function G10 in [61]. The related parameters are set as N=30, θ=0.3N, cp=5, Tc=2333, Th1=10, and Th2=2. The changing trend of ε value is shown in Figure 4. Three sampling points, that is, ε(500)=0.6033, ε(1000)=0.1227, and ε(2000)=1.194E-4, are marked in Figure 4. As shown in Figure 4, it can be observed that ε value declines very fast at first. After it is smaller than about 2, it declines as an exponential way. This changing trend of ε value could help algorithm to sufficiently search infeasible domains near feasible domains.Figure 4
Plot ofε value with iteration.The pseudocode for BSAISA is showed in Pseudocode1. In Pseudocode 1, the modified adaptive F is shown in lines (14)–(16). When BSAISA deals with constrained optimization problems, the code in line (8) and line (40) in Pseudocode 1 should consider objective function value and constraint violation simultaneously, and SAε is applied to choose a better solution or best solution in line (42) and lines (47)-(48).Pseudocode 1:Pseudocode of BSAISA.
Input:  ObjFun, N, D, max-iteration, mix-rate, low1:D, up1:D
Output:  globalminimum, globalminimizer
//rand~U(0,1), randn~N(0,1), w=randint(⋅), randint~U(⋅)w∈1,2,…,⋅
(1) function BSAISA(ObjFun, D, N, max-iteration, low, up) //  Initialization
(2) globalminimum = inf
(3) for  i  from  1  to  N  do
(4) for  j  from  1  to  D  do
(5) Pi,j=rand⋅(upj-lowj)+lowj//  Initialization of population P
(6) oldPi,j=rand⋅(upj-lowj)+lowj//  Initialization of oldP
(7) end
∗(8)fitnessPi=ObjFunPifitnessoldPi=ObjFun(oldPi)//  Initial-fitness values of P and oldP
(9) end
(10) for  iteration  from  1  to  max-iteration  do  //  Selection-I
(11) if  a<b∣a,b~U0,1  then  oldP≔P    end
(12) oldP≔permuting(oldP)
(13) Generation of Trail-Population
•(14) ΔIi=abs(ObjFun(Pi)-ObjFun(oldPi))//ModifiedFbasedonSA
•(15) G=iteration//ModifiedFbasedonSA
•(16) Fi~Nexp-1/ΔIi1/G,1//ModifiedFbasedonSA
(17) M=P+F⋅oldP-P//  Mutation
(18) map1:N,1:D=1//  Initial-map is an N-by-D matrix of ones
(19) if  c<d∣c,d~U0,1  then  //  Crossover
(20) for  i  from  1  to  N  do
(21) mapi,u1:mix-rate⋅rand⋅D=0∣u=permuting1,2,3,…,D
(22) end
(23) else
(24) for  i  from  1  to  N  do, mapi,randiD=0,  end
(25) end
(26) T≔M//Generation of Trial Population, T
(27) for  i  from  1  to  N  do
(28) for  j  from  1  to  D  do
(29) if  mapi,j=1  then  Ti,j≔Pi,j
(30) end
(31) end
(32) for  i  from  1  to  N  do
(33) for  j  from  1  to  D  do
(34) if  (Ti,j<lowi,j)  or  (Ti,j>upi,j)  then
(35) Ti,j=rand⋅(upj-lowj)+lowj
(36) end
(37) end
(38) end
(39) end
∗
(40)
f
i
t
n
e
s
s
T
i
=
O
b
j
F
u
n
(
T
j
)   //  Seletion-II
(41) for  i  from  1  to  N  do
∗
(42)
if  fitnessTi<fitnessPi  then
(43) fitnessPi=fitnessTi
(44) Pi=Ti
(45) end
(46) end
∗
(47)
f
i
t
n
e
s
s
P
b
e
s
t
=
m
i
n
(
f
i
t
n
e
s
s
P
)
∣
b
e
s
t
∈
1
,
2
,
3
,
…
,
N
∗
(48)
if  fitnessPbest<globalminimum  then  // Export globalminimum and globalminimizer
(49) globalminimum≔fitnessPbest
(50) globalminimizer≔Pbest
(51) end
(52) end


## 3.1. Structure of the Adaptive Amplitude Control FactorF

The modifiedF is a normally distributed random number, where its mean value is an exponential function and its variance is equal to 1. In BSAISA, we redesign the adaptive F to replace the original version using (6)Fi~Nexp⁡-1/ΔIi1/G,1,where i is the index of individuals, Fi is the adaptive amplitude control factor that corresponds to the ith individual, |ΔIi| is the absolute value of the difference between the objective function values of Pi and Ti (individual differences), N is the normal distribution, and G is the current iteration.According to (6), the exponential function (mean value) decreases dynamically with the change in the number of iterations (G) and the individual differences (|ΔIi|). Based on the probability density function curve of the normal distribution, the modified F can be decreased adaptively as the number of iterations increases. Another characteristic of the modifiedF is that there are not any extra parameters.


## 3.2. Design Principle of the Modified Amplitude Control FactorF

The design principle of the modifiedF is inspired by the Metropolis criterion in SA. SA is a metaheuristic optimization technique based on physical behavior in nature. SA based on the Monte Carlo method was first proposed by Metropolis et al. [59] and it was successfully introduced into the field of combinatorial optimization for solving complex optimization problems by Kirkpatrick et al. [60].The basic concept of SA derives from the process of physical annealing with solids. An annealing process occurs when a metal is heated to a molten state with a high temperature; then it is cooled slowly. If the temperature is decreased quickly, the resulting crystal will have many defects and it is just metastable; even the most stable crystalline state will be achieved at all. In other words, this may form a higher energy state than the most stable crystalline state. Therefore, in order to reach the absolute minimum energy state, the temperature needs to be decreased at a slow rate. SA simulates this process of annealing to search the global optimal solution in an optimization problem. However, accepting only the moves that lower the energy of system is like extremely rapid quenching; thus SA uses a special and effective acceptance method, that is, Metropolis criterion, which can probabilistically accept the hill-climbing moves (the higher energy moves). As a result, the energy of the system evolves into a Boltzmann distribution during the process of the simulated annealing. From this angle of view, it is no exaggeration to say that the Metropolis criterion is the core of SA.The Metropolis criterion can be expressed by the physical significance of energy, where the new energy state will be accepted when the new energy state is lower than the previous energy state, and the new energy state will be probabilistically accepted when the new energy state is higher than the previous energy state. This feature of SA can escape from being trapped in local minima especially in the early stages of the search. It can also be described as follows.(i)IfΔE=Ej-Ei≤0, then the new state j is accepted and the energy with the displaced atom is used as the starting point for the next step, where E represents the energy of the atom. Both i and j are the states of atoms, and j is the next state of i.(ii)IfΔE>0, then calculate the probability of P(ΔE)=exp⁡(-ΔE/kTc), and generate a random number θ, which is a uniform distribution over (0,1), where k is Boltzmann’s constant (in general, k=1) and Tc is the current temperature. If P(ΔE)>θ, then the new energy will be accepted; otherwise, the previous energy is used to start the next step.Analysis 1.
The Metropolis criterion states that SA has two characteristics: (1) SA can probabilistically accept the higher energy and (2) the acceptance probability of SA decreases as the temperature decreases. Therefore, SA can reject and jump out of a local minimum with a dynamic and decreasing probability to continue exploiting the other solutions in the state space. This acceptance mechanism can enrich the diversity of energy states.Analysis 2.
As shown in (4), F is used to control the amplitude of population mutation in BSA, thus F is an important factor for controlling population diversity. If F is excessively large, the diversity of the population will be too high and the convergence speed of BSA will slow down. If F is excessively small, the diversity of the population will be reduced so it will be difficult for BSA to obtain the global optimum and it may be readily trapped by a local optimum. Therefore, adaptively controlling the amplitude of F is a key to accelerating the convergence speed of the algorithm and maintaining its the population diversity.Based on Analyses1 and 2, it is clear that if F can dynamically decrease, the convergence speed of BSA will be able to accelerate while maintaining the population diversity. On the other hand, SA possesses this characteristic that its acceptance probability can be dynamically reduced. Based on these two considerations, we propose BSAISA with a redesigned F, which is inspired by SA. More specifically, the new F (see (6)) is redesigned by learning the formulation (P(ΔE)) of acceptance probability, and its formulation has been shown in the previous subsection.For the two formulas of the modifiedF and P(ΔE), the individual difference (|ΔIi|) of a population or the energy difference (ΔE) of a system will decrease as the number of iterations increases in an algorithm, and the temperature of SA tends to decrease, while the iteration of BSA tends to increase. As a result, one can observe the correspondence between modified F and P(ΔE) where the reciprocal of individual difference (1/|ΔIi|) corresponds to the energy difference (ΔE) of SA, and the reciprocal of current iteration (1/G) corresponds to the current temperature (Tc) of SA. In this way, the redesigned F can be decreased adaptively as the number of iterations increases.


## 3.3. Numerical Analysis of the Modified Amplitude Control FactorF

In order to verify that the convergence speed of the basic BSA is improved with the modifiedF, two types (unimodal and multimodal) of unconstrained benchmark functions are used to test the changing trends in F and population variances and best function values as the iterations increases, respectively. The two functions are Schwefel 1.2 and Rastrigin, and their detailed information is provided in [61]. The two functions and the user parameters including the populations (N), dimensions (D), and maximum iterations (Max) are shown in Table 1. Three groups of test results are compared in the tests including (1) the comparative curves of the mean values of the modified F and original F for Schwefel 1.2 and Rastrigin, (2) the comparative curves of the mean values of BSA and BSAISA population variances for two functions, and (3) the convergence curves of BSA and BSAISA for two functions. They are depicted in Figures 1, 2, and 3, respectively. Based on Figures 1–3, two results can be observed as follows.Table 1
Two benchmark functions and the corresponding populations, dimensions, and max iterations.
Name
Objective function
Range
N
D
Max
Schwefel 1.2
f
x
=
∑
i
=
1
D
∑
j
=
1
i
x
j
2
[
-
100,100
]
100
30
500
Rastrigin
f
x
=
∑
i
=
1
D
x
i
2
-
1
0
cos
⁡
2
π
x
i
+
10
[
-
5.12,5.12
]
100
30
500
Note. “N” means populations, “D” means dimensions, and “Max” means maximum iterations.Figure 1
Comparisons of modifiedF and original F for Schwefel 1.2 and Rastrigin.
(a)
ModifiedF on Schwefel 1.2
(b)
OriginalF on Schwefel 1.2
(c)
ModifiedF on Rastrigin
(d)
OriginalF on RastriginFigure 2
The mean values of population variances versus number of iterations using BSA and BSAISA for two functions.
(a)
Schwefel 1.2
(b)
RastriginFigure 3
Convergence curves of BSA and BSAISA for two functions.
(a)
Schwefel 1.2
(b)
Rastrigin(1) According to the trends ofF in Figure 1, both the originalF and modifiedF are subject to changes from the normal distribution. The mean value of the originalF does not tend to decrease as the number of iterations increases. By contrast, the mean value of the modifiedF exhibits a clear and fluctuating downward trend as the number of iterations increases.(2) According to Figure2, the population variances of BSAISA and BSA both exhibit a clear downward trend as the number of iterations increases. The population variances of BSAISA and BSA are almost same during the early iterations. This illustrates that the modifiedF does not reduce the population diversity in the early iterations. In the middle and later iterations, the population variances of BSAISA decrease more quickly than that of BSA. This illustrates that the modifiedF improves the convergence speed. As can be seen from Figure 3, as the number of iterations increases, the best objective function value of BSAISA drops faster than that of BSA. This shows that the modifiedF improves the convergence speed of BSA. Moreover, BSAISA can find a more accurate solution at the same computational cost.Summary. Based on the design principle and numerical analysis of the modified F, the modified F exhibits an overall fluctuating and downward trend, which matches with the concept of the acceptance probability in SA. In particular, during the early iterations, the modified F is relatively large. This allows BSAISA to search in a wide region, while maintaining the population diversity of BSAISA. As the number of iterations decreases, the modified F gradually exhibits a decreasing trend. This accelerates the convergence speed of BSAISA. In the later iterations, the modified F is relatively small. This enables BSAISA to fully search in the local region. Therefore, it can be concluded that BSAISA can adaptively control the amplitude of population mutation to change its local exploitation capacity. This may improve the convergence speed of the algorithm. Moreover, the modified F does not introduce extra parameters, so it does not increase the sensitivity of BSAISA to the parameters.


## 3.4. A Self-Adaptiveε-Constrained Method for Handling Constraints

In general, a constrained optimization problem can be mathematically formulated as a minimization problem, as follows:(7)minfXsubjectto:giX≤0,i=1,2,…,m,hjX=0,j=1,2,…,n,where X=(x1,x2,…,xD)∈RD is a D-dimensional vector, m is the total number of inequality constraints, andn is the total number of equality constraints. The equality constraints are transformed into inequality constraints by using |hj(X)|-δ≤0, where δ is a very small degree of violation, and δ=1E-4 in this paper. The maximization problems are transformed into minimization problems using -f(X). The constraint violation φ(X) is given by (8)φX=∑i=1mmax⁡0,giX+∑j=1nmax⁡0,hjX-δ.Several constraint-handling methods have been proposed previously, where the five most commonly used methods comprise penalty functions, feasibility and dominance rules (FAD), stochastic ranking,ε-constrained methods, and multiobjectives concepts. Among these five methods, the ε-constrained method is relatively effective and used widely. Zhang et al. [62] proposed a self-adaptive ε-constrained method (SAε) to combine with the basic BSA for constrained problems. It has been verified that the SAε has a stronger search efficiency and convergence than the fixed ε-constrained method and FAD. In this paper, the SAε is used to combine with BSAISA for constrained optimization problems, which comprises the following two rules: (1) if the constraint violations of two solutions are smaller than a given ε value or two solutions have the same constraint violations, the solution with a better objective function value is preferred and (2) if not, the solution with a smaller constraint violation is preferred. SAε could be expressed by the following equations: (9)thebetterone=fX1fX1<fX2,ifφX1,φX2≤ε,fX1<fX2,ifφX1=φX2,φX1≤φX2,otherwise,where ε is a positive value that represents a tolerance related to constraint violation. The self-adaptive ε value is formulated as the following equation: (10)ε0=φP0θ,(11)ε10=ε0,(12)ε2t=φTtθ,ifε0>Th1,ε0,else,(13)ε1t=ε2t,ifε2t>Th2,ε2t<ε1t-1,ε1t-1,else,(14)εt=ε1t1-tTccp,ift≤Tc0,else,where t is the number of the current iterations. φ(P0θ) is the constraint violation of the top θth individual in the initial population. φ(Ttθ) is the constraint violation of the top θth individual in the trial population at the current iteration. cp and Tc are control parameters. Th1 and Th2 are threshold values. εt is related to the iteration t and functions ε1(t) and ε2(t).Firstly,ε0 is set as φ(P0θ). If the initial value ε0 is bigger than Th1, φ(Ttθ) will be assigned to ε2(t); otherwise ε0 will be assigned to ε2(t). Then, if ε2(t)<ε1(t-1) and ε2(t) is bigger than Th2, ε2(t) will be assigned to ε1(t); otherwise ε1(t-1) will be assigned to ε1(t). Finally, εt is updated as (14). The detailed information of SAε can be acquired from [62], and the related parameter settings of SAε (the same as [62]) are presented in Table 2.Table 2
The parameter setting of SAε.
Tc=0.2∗Tmax
θ=0.3∗N
cp=5
Th1=10
T
h
2
=
2
Note. “Tmax” means the maximum iterations; “N” means populations.To illustrate the changing trend of the self-adaptiveε value vividly, BSAISA with SAε is used to solve a well-known benchmark constrained function G10 in [61]. The related parameters are set as N=30, θ=0.3N, cp=5, Tc=2333, Th1=10, and Th2=2. The changing trend of ε value is shown in Figure 4. Three sampling points, that is, ε(500)=0.6033, ε(1000)=0.1227, and ε(2000)=1.194E-4, are marked in Figure 4. As shown in Figure 4, it can be observed that ε value declines very fast at first. After it is smaller than about 2, it declines as an exponential way. This changing trend of ε value could help algorithm to sufficiently search infeasible domains near feasible domains.Figure 4
Plot ofε value with iteration.The pseudocode for BSAISA is showed in Pseudocode1. In Pseudocode 1, the modified adaptive F is shown in lines (14)–(16). When BSAISA deals with constrained optimization problems, the code in line (8) and line (40) in Pseudocode 1 should consider objective function value and constraint violation simultaneously, and SAε is applied to choose a better solution or best solution in line (42) and lines (47)-(48).Pseudocode 1:Pseudocode of BSAISA.
Input:  ObjFun, N, D, max-iteration, mix-rate, low1:D, up1:D
Output:  globalminimum, globalminimizer
//rand~U(0,1), randn~N(0,1), w=randint(⋅), randint~U(⋅)w∈1,2,…,⋅
(1) function BSAISA(ObjFun, D, N, max-iteration, low, up) //  Initialization
(2) globalminimum = inf
(3) for  i  from  1  to  N  do
(4) for  j  from  1  to  D  do
(5) Pi,j=rand⋅(upj-lowj)+lowj//  Initialization of population P
(6) oldPi,j=rand⋅(upj-lowj)+lowj//  Initialization of oldP
(7) end
∗(8)fitnessPi=ObjFunPifitnessoldPi=ObjFun(oldPi)//  Initial-fitness values of P and oldP
(9) end
(10) for  iteration  from  1  to  max-iteration  do  //  Selection-I
(11) if  a<b∣a,b~U0,1  then  oldP≔P    end
(12) oldP≔permuting(oldP)
(13) Generation of Trail-Population
•(14) ΔIi=abs(ObjFun(Pi)-ObjFun(oldPi))//ModifiedFbasedonSA
•(15) G=iteration//ModifiedFbasedonSA
•(16) Fi~Nexp-1/ΔIi1/G,1//ModifiedFbasedonSA
(17) M=P+F⋅oldP-P//  Mutation
(18) map1:N,1:D=1//  Initial-map is an N-by-D matrix of ones
(19) if  c<d∣c,d~U0,1  then  //  Crossover
(20) for  i  from  1  to  N  do
(21) mapi,u1:mix-rate⋅rand⋅D=0∣u=permuting1,2,3,…,D
(22) end
(23) else
(24) for  i  from  1  to  N  do, mapi,randiD=0,  end
(25) end
(26) T≔M//Generation of Trial Population, T
(27) for  i  from  1  to  N  do
(28) for  j  from  1  to  D  do
(29) if  mapi,j=1  then  Ti,j≔Pi,j
(30) end
(31) end
(32) for  i  from  1  to  N  do
(33) for  j  from  1  to  D  do
(34) if  (Ti,j<lowi,j)  or  (Ti,j>upi,j)  then
(35) Ti,j=rand⋅(upj-lowj)+lowj
(36) end
(37) end
(38) end
(39) end
∗
(40)
f
i
t
n
e
s
s
T
i
=
O
b
j
F
u
n
(
T
j
)   //  Seletion-II
(41) for  i  from  1  to  N  do
∗
(42)
if  fitnessTi<fitnessPi  then
(43) fitnessPi=fitnessTi
(44) Pi=Ti
(45) end
(46) end
∗
(47)
f
i
t
n
e
s
s
P
b
e
s
t
=
m
i
n
(
f
i
t
n
e
s
s
P
)
∣
b
e
s
t
∈
1
,
2
,
3
,
…
,
N
∗
(48)
if  fitnessPbest<globalminimum  then  // Export globalminimum and globalminimizer
(49) globalminimum≔fitnessPbest
(50) globalminimizer≔Pbest
(51) end
(52) end


## 4. Experimental Studies

In this section, two sets of simulation experiments were executed to evaluate the effectiveness of the proposed BSAISA. The first experiment set performed on 13 well-known benchmark constrained functions taken from [63] (see Appendix A). These thirteen benchmarks contain different properties as shown in Table 3, including the number of variables (D), objective function types, the feasibility ratio (ρ), constraint types and number, and the number of active constraints in the optimum solution. The second experiment is conducted on 5 engineering constrained optimization problems chosen from [64] (see Appendix B). These five problems are the 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. These engineering problems include objective functions and constraints of various types and natures (quadratic, cubic, polynomial, and nonlinear) with various number of design variables (continuous, integer, mixed, and discrete).Table 3
Characters of the 13 benchmark functions.
Fun.
D
Type
ρ (%)
LI
NI
LE
NE
Active
g01
13
Quadratic
0.0003
9
0
0
0
6
g02
20
Nonlinear
99.9973
1
1
0
0
1
g03
10
Nonlinear
0.0000
0
0
0
1
1
g04
5
Quadratic
27.0079
0
6
0
0
2
g05
4
Nonlinear
0.0000
2
0
0
3
3
g06
2
Nonlinear
0.0057
0
2
0
0
2
g07
10
Quadratic
0.0003
3
5
0
0
6
g08
2
Nonlinear
0.8581
0
2
0
0
0
g09
7
Nonlinear
0.5199
0
4
0
0
2
g10
8
Linear
0.0020
3
3
0
0
3
g11
2
Quadratic
0.0973
0
0
0
1
1
g12
3
Quadratic
4.7679
0
9
3
0
0
0
g13
5
Nonlinear
0.0000
0
0
1
2
3
Note. “D” is the number of variables. “ρ” represents feasibility ratio. “LI,” “NI,” “LE,” and “NE” represent linear inequality, nonlinear inequality, linear equality, and nonlinear equality, respectively. “Active” represents the number of active constraints at the global optimum.The recorded experimental results include the best function value (Best), the worst function value (Worst), the mean function value (Mean), the standard deviation (Std), the best solution (variables of best function value), the corresponding constraint value, and the number of function evaluations (FEs). The number of function evaluations can be considered as a convergence rate or a computational cost.In order to evaluate the performance of BSAISA in terms of convergence speed, the FEs are considered as the best FEs corresponding to the obtained best solution in this paper. The calculation of FEs are the product of population sizes (N) and the number of iterations (Ibest) at which the best function value is first obtained (i.e., FEs=N∗Ibest). For example, if 2500 is the maximum number of iterations for one minimization problem, f(1999)=0.0126653,f(2000)=0.0126652,f(2500)=f(2000)=0.0126652, the Ibest value should be 2000. However, BSAISA needs to evaluate the initial historical population (oldP), so its actual FEs should be plus N (i.e., FEs=N∗Ibest+N).


### 4.1. Parameter Settings

For the first experiment, the main parameters for 13 benchmark constrained functions are the same as the following: population size (N) is set as 30; the maximum number of iterations (Tmax) is set as 11665. Therefore, BSAISA’s maximum number of function evaluations (MFEs) should equal to 34,9980 (nearly 35,0000). The 13 benchmarks were executed by using 30 independent runs.For the 5 real-world engineering design problems, we use slightly different parameter settings since each problem has different natures, that is, TTP (N=20,Tmax=1000), PVP (N=20,Tmax=3000), TCSP (N=20,Tmax=3000), WBP (N=20,Tmax=3000), SRP (N=20,Tmax=2000). The 6 engineering problems were performed using 50 independent runs.The user parameters of all experiments are presented in Table4. Termination condition may be the maximum number of iterations, CPU time, or an allowable tolerance value between the last result and the best known function value for most metaheuristics. In this paper, the maximum number of iterations is considered as termination condition. All experiments were conducted on a computer with a Intel(R) Core(TM) i5-4590 CPU @ 3.30 GHz and 4 GB RAM.Table 4
User parameters used for all experiments.
Problem
G01–G13
TTP
PVP
TCSP
WBP
SRP
N
30
20
20
20
20
20
T
m
a
x
11665
1000
3000
3000
3000
2000
Runs
30
50
50
50
50
50
Note. The expression “runs” denotes the number of independent runs.


### 4.2. Simulation on Constrained Benchmark Problems

In this section, BSAISA and BSA are performed on the 13 benchmarks simultaneously. Their statistical results obtained from 30 independent runs are listed in Tables5 and 6, including the best known function value (Best Known) and the obtained Best/Mean/Worst/Std values as well as the FEs. The best known values for all the test problems are derived from [62]. The best known values found by algorithms are highlighted inbold. From Tables 5 and 6, it can be seen that BSAISA is able to find the known optimal function values for G01, G04, G05, G06, G08, G09, G011, G012, and G013; however, BSA fails to find the best known function value on G09 and G13. For the rest of the functions, BSAISA obtains the results very close to the best known function values. Moreover, BSAISA requires fewer FEs than BSA on G01, G03, G04, G05, G06, G07, G08, G09, G011, and G012. Although the FEs of BSAISA are slightly worse than that of BSA for G02, G10, and G13, BSAISA finds more accurate best function values than BSA for G02, G03, G07, G09, G10, and G13.Table 5
The statistical results of BSAISA for 13 constrained benchmarks.
Fun.
Known optimal
Best
Mean
Worst
Std
FEs
G01
-
15
−15
-
15
-
15.000000
8.0
8
E
-
16
84,630
G02
-
0.803619
-
0.803599
-
0.787688
-
0.758565
1.1
4
E
-
02
349,500
G03
-
1.000500
-
1.000498
-
1.000481
-
1.000441
1.3
5
E
-
05
58,560
G04
-
30665.538672
−30665.538672
-
30665.538672
-
30665.538672
1.0
9
E
-
11
121,650
G05
5126.496714
5126.496714
5126.496714
5126.496714
5.8
5
E
-
13
238,410
G06
-
6961.813876
−6961.813876
-
6961.813876
-
6961.813876
1.8
5
E
-
12
89,550
G07
24.306209
24.307381
24.400881
24.758205
1.0
2
E
-
01
15,060
G08
-
0.0958250
−0.0958250
-
0.086683
-
0.027263
2.3
7
E
-
02
30,930
G09
680.630057
680.630057
680.633025
680.680043
9.1
1
E
-
03
347,760
G10
7049.248021
7049.249056
7081.241789
7326.853581
6.2
4
E
+
01
346,980
G11
0.749900
0.749900
0.749900
0.749900
1.1
3
E
-
16
87,870
G12
-
1
−1
-
1
-
1
0
5430
G13
0.0539415
0.0539415
0.1030000
0.4594033
9.8
0
E
-
02
349,800
Note. “Known optimal” denotes the best known function values in the literatures. “Bold” means the algorithm has found the best known function values. The same as Table 6.Table 6
The statistical results of the basic BSA on 13 constrained benchmarks.
Fun.
Known optimal
Best
Mean
Worst
Std
FEs
G01
−15
−15
−15
−15.000000
6.6
0
E
-
16
99,300
G02
−0.803619
−0.803255
−0.792760
−0.749326
9.1
0
E
-
03
344,580
G03
−1.000500
−1.000488
−0.998905
−0.990600
2.6
6
E
-
03
348,960
G04
−30665.538672
−30665.538672
−30665.538672
−30665.538672
1.1
1
E
-
11
272,040
G05
5126.496714
5126.496714
5144.041363
5275.384724
3.9
9
E
+
01
299,220
G06
−6961.813876
−6961.813876
−6961.813876
−6961.813876
1.8
5
E
-
12
111,450
G07
24.306209
24.307607
24.344626
24.399896
1.9
3
E
-
02
347,250
G08
−0.0958250
−0.0958250
−0.0958250
−0.0958250
2.8
2
E
-
17
73,440
G09
680.630057
680.630058
680.630352
680.632400
5.6
3
E
-
04
348,900
G10
7049.248021
7049.278543
7053.573853
7080.192700
7.2
8
E
+
00
340,020
G11
0.749900
0.749900
0.749900
0.749900
1.1
3
E
-
16
113,250
G12
−1
−1
−1
−1
0
13,590
G13
0.0539415
0.0539420
0.1816986
0.5477657
1.5
0
E
-
01
347,400To further compare BSAISA and BSA, the function value convergence curves of 13 functions that have significant differences have been plotted, as shown in Figures5 and 6. The horizontal axis represents the number of iterations, while the vertical axis represents the difference of the objective function value and the best known value. For G02, G04, G06, G08, G11, and G12, it is obvious that the convergence speed of BSAISA is faster than that of BSA, and its best function value is also better than that of BSA. For G01, G03, G05, G07, G09, G10, and G13, it can be observed that the objective function values of BSAISA and BSA fluctuate in the early iterations, and they decrease as the number of iterations increases during the middle and late stages. This illustrates that both the algorithms are able to escape the local optimum under different degrees, and the convergence curve of BSAISA drops still faster than that of BSA. Therefore, the experiment results demonstrate that the convergence speed of the basic BSA is improved with our modified F.Figure 5
The convergence curves of the first 9 functions by BSAISA and BSA.
(a)
Convergence curve on G01
(b)
Convergence curve on G02
(c)
Convergence curve on G03
(d)
Convergence curve on G04
(e)
Convergence curve on G05
(f)
Convergence curve on G06
(g)
Convergence curve on G07
(h)
Convergence curve on G08
(i)
Convergence curve on G09Figure 6
The convergence curves of the latter 4 functions by BSAISA and BSA.
(a)
Convergence curve on G10
(b)
Convergence curve on G11
(c)
Convergence curve on G12
(d)
Convergence curve on G13In order to further verify the competitiveness of BSAISA in aspect of convergence speed, we compared BSAISA with some classic and state-of-the-art approaches in terms of best function value and function evaluations. The best function value and the corresponding FEs of each algorithm on 13 benchmarks are presented in Table7, where the optimal results are inbold on each function. These compared algorithms are listed below:(1)
Stochastic ranking (SR) [63](2)
Filter simulated annealing (FSA) [65](3)
Cultured differential evolution (CDE) [66](4)
Agent based memetic algorithm (AMA) [64](5)
Modified artificial bee colony (MABC) algorithm [67](6)
Rough penalty genetic algorithm (RPGA) [68](7)
BSA combined self-adaptiveε constrained method (BSA-SAε) [62].Table 7
Comparison of the best values and FEs obtained by BSAISA and other algorithms.
Alg.
BSAISA
SR
FSA
CDE
AMA
MABC
RPGA
BSA-SAε
Fun.
Best (FEs)
Best (FEs)
Best (FEs)
Best (FEs)
Best (FEs)
Best (FEs)
Best (FEs)
Best (FEs)
G01
−15
−15.000
−14.993316
−15.000000
−15.000
−15.000
−15.000
−15.000000
(84,630)
(148,200)
(205,748)
(100,100)
(350,000)
(350,000)
(350,000)
(350,000)
G02
−0.8036
−0.8035
−0.7549
−0.8036
−0.8035
−0.8036
−0.8036
−0.8036
(349,500)
(217,200)
(227,832)
(100,100)
(350,000)
(350,000)
(350,000)
(350,000)
G03
−1.000498
−1.000
−1.0000015
−0.995413
−1.000
−1.000
−1.000
−1.000498
(58,560)
(229,200)
(314,938)
(100,100)
(350,000)
(350,000)
(350,000)
(350,000)
G04
−30665.539
−30665.539
−30665.538
−30665.539
−30665.538
−30665.539
−30665.539
−30665.539
(121,650)
(88,200)
(86,154)
(100,100)
(350,000)
(350,000)
(350,000)
(350,000)
G05
5126.497
5126.497
5126.4981
5126.571
5126.512
5126.487
5126.544
5126.497
(238,410)
(51,600)
(47,661)
(100,100)
(350,000)
(350,000)
(350,000)
(350,000)
G06
−6961.814
−6961.814
−6961.814
−6961.814
−6961.807
−6961.814
−6961.814
−6961.814
(89,550)
(118,000)
(44,538)
(100,100)
(350,000)
(350,000)
(350,000)
(350,000)
G07
24.307
24.307
24.311
24.306
24.315
24.324
24.333
24.306
(15,060)
(143,000)
(404,501)
(100,100)
(350,000)
(350,000)
(350,000)
(350,000)
G08
−0.095825
−0.095825
−0.095825
−0.095825
−0.095825
−0.095825
−0.095825
−0.095825
(30,930)
(76,200)
(56,476)
(100,100)
(350,000)
(350,000)
(350,000)
(350,000)
G09
680.630057
680.630
680.63008
680.630057
680.645
680.631
680.631
680.6301
(347,760)
(111,400)
(324,569)
(100,100)
(350,000)
(350,000)
(350,000)
(350,000)
G10
7049.249
7054.316
7059.864
7049.248
7281.957
7058.823
7049.861
7049.278
(346,980)
(128,400)
(243,520)
(100,100)
(350,000)
(350,000)
(350,000)
(350,000)
G11
0.749900
0.750
0.749999
0.749900
0.750
0.750
0.749
0.749900
(87,870)
(11,400)
(23,722)
(100,100)
(350,000)
(350,000)
(350,000)
(350,000)
G12
−1
−1.000000
−1.000000
−1.000000
−1.000
−1.000
NA
−1.000000
(5430)
(16,400)
(59,355)
(100,100)
(350,000)
(350,000)
(350,000)
G13
0.0539415
0.053957
0.0539498
0.056180
0.053947
0.757
NA
0.0539415
(349,800)
(69,800)
(120,268)
(100,100)
(350,000)
(350,000)
(350,000)
Nu.
6
+
10
1
+
5
1
+
3
4
+
10
0
+
3
1
+
7
0
+
5
0
+
10
RK
1
5
7
2
8
4
6
3
Note. “NA” means not available. The same as Tables 8, 10, 11, 12, 13, 14, 15, and 17. “RK” represents the comprehensive ranking of each algorithm on the set of benchmarks. “Nu.” represents the sum of the number of winners and the number of the optimal function values for each algorithm on the set of benchmarks.Table 8
Comparison of best solutions for the three-bar truss design problem.
Method
DEDS
HEAA
PSO-DE
DELC
MBA
BSA
BSAISA
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
0.408248
g
1
(
X
)
1.7
7
E
-
08
NA
-
5.29
E
-
11
NA
-
5.2
9
E
-
11
-
3.2
3
E
-
12
0
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
263.895843To compare these algorithms synthetically, a simple evaluation mechanism is used. It can be explained as the best function value (Best) is preferred, and the function evaluations (FEs) are secondary. More specifically, (1) if one algorithm has a betterBest than those of others on a function, there is no need to considerFEs and the algorithm is superior to other algorithms on this function. (2) If two or more algorithms have found the optimalBest on a function, the algorithm with the lowestFEs is considered as the winner on the function. (3) Record the number of winners and the number of the optimal function values for each algorithm on the set of benchmarks, and then give the sort for all algorithms.From Table7, it can be observed that the rank of these 8 algorithms is as follows: BSAISA, CDE, BSA-SAε, MABC, SR, RPGA, FSA, and AMA. Among the 13 benchmarks, BSAISA wins on 6 functions and it is able to find the optimal values of 10 functions. This is better than all other algorithms, thus BSAISA ranks the first. The second algorithm CDE performs better on G02, G07, G09, and G10 than BSAISA but worse on G01, G03, G08, G11, G12, and G13. BSA-SAε obtains the optimal function values of 10 functions but requires more function evaluations than BSAISA and CDE, so it should rank the third. MABC ranks the fourth. It obtains the optimal function values of 7 functions, which are fewer in number than those of the former three algorithms. Both SR and RPGA have found the same number of the optimal function values, while the former is the winner on G04, so SR is slightly better than RPGA. As for the last two algorithms, FSA and AMA just perform well on three functions, while FSA is the winner on G06, so FSA is slightly better than AMA.Based on the above comparison, it can be concluded that BSAISA is effective and competitive in terms of convergence speed.


### 4.3. Simulation on Engineering Design Problems

In order to assess the optimization performance of BSAISA in real-world engineering constrained optimization problems, 5 well-known engineering constrained design problems including three-bar truss design, pressure vessel design, tension/compression spring design, welded beam design, and speed reducer design are considered in the second experiment.


#### 4.3.1. Three-Bar Truss Design Problem (TTP)

The three-bar truss problem is one of the engineering minimization test problems for constrained algorithms. The best feasible solution is obtained by BSAISA atx = (0.788675, 0.408248) with the objective function value f(x) = 263.895843 using 8940 FEs. The comparison of the best solutions obtained from BSAISA, BSA, differential evolution with dynamic stochastic selection (DEDS) [69], hybrid evolutionary algorithm (HEAA) [70], hybrid particle swarm optimization with differential evolution (POS-DE) [71], differential evolution with level comparison (DELC) [72], and mine blast algorithm (MBA) [73] is presented in Table 8. Their statistical results are listed in Table 9.Table 9
Comparison of statistical results for the three-bar truss design problem.
Method
Worst
Mean
Best
Std
FEs
DEDS
263.895849
263.895843
263.895843
9.7
E
-
07
15,000
HEAA
263.896099
263.895865
263.895843
4.9
E
-
05
15,000
PSO-DE
263.895843
263.895843
263.895843
4.5
E
-
10
17,600
DELC
263.895843
263.895843
263.895843
4.3
E
-
14
10,000
MBA
263.915983
263.897996
263.895852
3.9
3
E
-
03
13,280
BSA
263.895845
263.895843
263.895843
2.6
4
E
-
07
13,720
BSAISA
263.895843
263.895843
263.895843
5.7
5
E
-
13
8940Table 10
Comparison of best solutions for the pressure vessel design problem.
Method
GA-DT
MDE
CPSO
HPSO
DELC
ABC
BSA-SAε
BSA
BSAISA
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
)
-
2.0
1
E
-
03
0
-
1.3
7
E
-
06
NA
NA
0
-
9.5
E
-
10
-
3.6
9
E
-
08
0
g
2
(
X
)
-
3.5
8
E
-
02
-
0.035881
-
3.5
9
E
-
04
NA
NA
-
0.035881
-
3.5
9
E
-
2
-
0.035881
-
0.035881
g
3
(
X
)
-
24.7593
-
0.0000
-
118.7687
NA
NA
-
0.000226
-
1.2
E
-
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 11
Comparison of statistical results for the pressure vessel design problem.
Method
Worst
Mean
Best
Std
FEs
GA-DT
6469.3220
6177.2533
6059.9463
130.9297
80,000
MDE
6059.7017
6059.7017
6059.7017
1.0
E
-
12
24,000
CPSO
6363.8041
6147.1332
6061.0777
86.45
30,000
HPSO
6288.6770
6099.9323
6059.7143
86.20
81,000
DELC
6059.7143
6059.7143
6059.7143
2.1
E
-
11
30,000
PSO-DE
6059.7143
6059.7143
6059.7143
1.0
E
-
10
42,100
ABC
NA
6245.3081
6059.7147
2.0
5
E
+
02
30,000
BSA-SAε
6116.7804
6074.3682
6059.7143
1.7
1
E
+
01
80,000
BSA
6771.5969
6221.2861
6059.7150
2.0
3
E
+
02
60,000
BSAISA
7198.0054
6418.1935
6059.7143
3.0
4
E
+
02
16,320Table 12
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
-
1.3
E
-
05
-
2.1
E
-
05
-
4.061338
-
0.722698
0.012681
MDE
0.051688
0.356692
11.290483
-
0.000000
-
0.000000
-
4.053734
-
0.727747
0.012665
CPSO
0.051728
0.357644
11.244543
-
8.4
5
E
-
04
-
1.2
6
E
-
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
-
4.056663
-
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
-
7.7
0
E
-
09
-
3.3
0
E
-
09
-
4.054
-
0.728
0.012665
BSA
0.051694
0.356845
11.281488
-
1.0
5
E
-
07
-
1.7
7
-
0
8
-
4.054037
-
0.727640
0.012665
BSAISA
0.051687
0.356669
11.291824
-
6.3
8
E
-
10
-
1.5
3
E
-
09
-
4.053689
-
0.727763
0.012665Table 13
Comparison of statistical results for the tension compression spring design problem.
Method
Worst
Mean
Best
Std
FEs
GA-DT
0.012973
0.012742
0.012681
5.9
0
E
-
05
80,000
MDE
0.012674
0.012666
0.012665
2.0
E
-
6
24,000
CPSO
0.012924
0.012730
0.012675
5.2
0
E
-
05
23,000
HPSO
0.012719
0.012707
0.012665
1.5
8
E
-
05
81,000
DEDS
0.012738
0.012669
0.012665
1.2
5
E
-
05
24,000
HEAA
0.012665
0.012665
0.012665
1.4
E
-
09
24,000
DELC
0.012666
0.012665
0.012665
1.3
E
-
07
20,000
PSO-DE
0.012665
0.012665
0.012665
1.2
E
-
08
24,950
ABC
NA
0.012709
0.012665
0.012813
30,000
MBA
0.012900
0.012713
0.012665
6.3
0
E
-
05
7650
SSOC
0.012868
0.012765
0.012665
9.2
9
E
-
05
25,000
BSA-SAε
0.012666
0.012665
0.012665
1.6
2
E
-
07
80,000
BSA
0.012669
0.012666
0.012665
7.2
4
E
-
07
43,220
BSAISA
0.012668
0.012666
0.012665
4.9
0
E
-
07
9440Table 14
Comparison of best solutions for the welded beam design problem.
Method
GA-DT
MDE
CPSO
HPSO
ABC
MBA
BSA-SAε
BSA
BSAISA
X
1
(
h
)
0.205986
0.205730
0.202369
0.205730
0.205730
0.205729
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
X
3
(
t
)
9.020224
9.036624
9.04821
9.036624
9.036624
9.036626
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
g
1
(
X
)
-
0.074092
-
0.000335
-
12.839796
NA
0.000000
-
0.001614
-
1.5
5
E
-
10
-
5.3
2
E
-
07
0
g
2
(
X
)
-
0.266227
-
0.000753
-
1.247467
NA
-
0.000002
-
0.016911
-
4.3
0
E
-
09
-
9.0
2
E
-
06
0
g
3
(
X
)
-
4.9
5
E
-
04
-
0.000000
-
1.4
9
E
-
03
NA
0.000000
-
2.4
0
E
-
07
-
1.5
5
E
-
15
-
7.8
6
E
-
12
-
5.5
5
E
-
17
g
4
(
X
)
-
3.430044
-
3.432984
-
3.429347
NA
-
3.432984
-
3.432982
-
3.4330
-
3.432984
-
3.432984
g
5
(
X
)
-
0.080986
-
0.080730
-
0.079381
NA
-
0.080730
-
0.080729
-
8.0
7
E
-
02
-
0.080730
-
0.080730
g
6
(
X
)
-
0.235514
-
0.235540
-
0.235536
NA
-
0.235540
-
0.235540
-
0.2355
-
0.235540
-
0.235540
g
7
(
X
)
-
58.666440
-
0.000882
-
11.681355
NA
0.000000
-
0.001464
-
1.8
5
E
-
10
-
1.1
3
E
-
07
-
5.4
6
E
-
12
f
(
X
)
1.728226
1.724852
1.728024
1.724852
1.724852
1.724853
1.724852
1.724852
1.724852Table 15
Comparison of statistical results for the welded beam design problem.
Method
Worst
Mean
Best
Std
FEs
GA-DT
1.993408
1.792654
1.728226
7.4
7
E
-
02
80,000
MDE
1.724854
1.724853
1.724852
NA
24,000
CPSO
1.782143
1.748831
1.728024
1.2
9
E
-
02
24,000
HPSO
1.814295
1.749040
1.724852
4.0
0
E
-
02
81,000
DELC
1.724852
1.724852
1.724852
4.1
E
-
13
20,000
PSO-DE
1.724852
1.724852
1.724852
6.7
E
-
16
66,600
ABC
NA
1.741913
1.724852
3.1
E
-
02
30,000
MBA
1.724853
1.724853
1.724853
6.9
4
E
-
19
47,340
SSOC
1.799332
1.746462
1.724852
2.5
7
E
-
2
25,000
BSA-SAε
1.724852
1.724852
1.724852
8.1
1
E
-
10
80,000
BSA
1.724854
1.724852
1.724852
2.3
5
E
-
07
45,480
BSAISA
1.724854
1.724852
1.724852
2.9
6
E
-
07
29,000From Tables8 and 9, BSAISA, BSA, DEDS, HEAA, PSO-DE, and DELC all reach the best solution with the corresponding function value equal to 263.895843 except MBA with 263.895852. However, BSAISA requires the lowest FEs (only 8940) among all algorithms. Its Std value is better than BSA, DEDS, HEAA, PSO-DE, and MBA except DELC. These comparative results indicate that BSAISA outperforms other algorithms in terms of computational cost and robustness for this problem.Figure7 depicts the convergence curves of BSAISA and BSA for the three-bar truss design problem, where the value of F(x∗) on the vertical axis equals 263.895843. As shown in Figure 7, BSA achieves the global optimum at about 700 iterations, while BSAISA only reaches the global optimum at about 400 iterations. It can be concluded that the convergence speed of BSAISA is faster than that of BSA for this problem.Figure 7
Convergence curves of BSAISA and BSA for the three-bar truss design problem.


#### 4.3.2. Pressure Vessel Design Problem (PVP)

The pressure vessel design problem has a nonlinear objective function with three linear and one nonlinear inequality constraints and two discrete and two continuous design variables. The values of the two discrete variables (x1,x2) should be the integer multiples of 0.0625. The best feasible solution is obtained by BSAISA at x = (0.8750, 0.4375, 42.0984, 176.6366) with the objective function value f(x) = 6059.7143 using 31,960 FEs.For this problem, BSAISA is compared with nine algorithms: BSA, BSA-SAε [62], DELC, POS-DE, genetic algorithms based on dominance tournament selection (GA-DT) [73], modified differential evolution (MDE) [74], coevolutionary particle swarm optimization (CPSO) [75], hybrid particle swarm optimization (HPSO) [76], and artificial bee colony algorithm (ABC) [77]. The comparison of the best solutions obtained by BSAISA and other reported algorithms is presented in Table 10. The statistical results of various algorithms are listed in Table 11.As shown Table10, the obtained solution sets of all algorithms satisfy the constraints for this problem. BSAISA, BSA-SAε, ABC, DELC, and HPSO find the same considerable good objective function value 6059.7143, which is slightly worse than MDE’s function value 6059.7143. It is worth mentioning that MBA’s best solution was obtained at x = (0.7802, 0.3856, 40.4292, 198.4964) with f(x) = 5889.3216 and the corresponding constraint values equal to gi(x) = (0, 0, −86.3645, −41.5035) in [78]. Though MBA finds a far better function value than that of MDE, its obtained variables (i.e., 0.7802 and 0.3856) are not integer multiples of 0.0625. So they are not listed in Table 10 to ensure a fair comparison. From Table 11, except for MDE with the function value of 6059.7016, BSAISA offers better function value results compared to GA-DT, CPSO, ABC, and BSA. Besides that, BSAISA is far superior to other algorithms in terms of FEs. Unfortunately, the obtained Std value of BSAISA is relatively poor compared with others for this problem.Figure8 describes the convergence curves of BSAISA and BSA for the pressure vessel design problem, where the value of F(x∗) on the vertical axis equals 6059.7143. As shown in Figure 8, BSAISA is able to find the global optimum at about 800 iterations and obtains a far more accurate function value than that of BSA. Moreover, the convergence speed of BSAISA is much faster than that of BSA.Figure 8
Convergence curves of BSAISA and BSA for the pressure vessel design problem.


#### 4.3.3. Tension Compression Spring Design Problem (TCSP)

This design optimization problem has three continuous variables and four nonlinear inequality constraints. The best feasible solution is obtained by BSAISA atx = (0.051687, 0.356669, 11.291824) with f(x) = 0.012665 using 9440 FEs. This problem has been solved by other methods as follows: GA-DT, MDE, CPSO, HPSO, DEDS, HEAA, DELC, POS-DE, ABC, MBA, BSA-SAε, and Social Spider Optimization (SSOC) [79]. The comparison of the best solutions obtained from various algorithms is presented in Table 12. Their statistical results are listed in Table 13.From Tables12 and 13, the vast majority of algorithms can find the best function value 0.012665 for this problem, while GA-DT and CPSO fail to find it. With regard to the computational cost (FEs), BSAISA only requires 9440 FEs when it reaches the global optimum, which is superior to all other algorithms except MBA with 7650 FEs. However, the Worst and Mean and Std values of BSAISA are better than those of MBA. Consequently, for this problem, it can be concluded that BSA has the obvious superiority in terms of FEs over all other algorithms except MBA. Moreover, BSAISA has a stronger robustness when compared with MBA alone.Figure9 depicts the convergence curves of BSAISA and BSA for the tension compression spring design problem, where the value of F(x∗) on the vertical axis equals 0.012665. From Figure 9 it can be observed that both BSAISA and BSA fall into a local optimum in the early iterations but they are able to successfully escape from the local optimum. However, the convergence speed of BSAISA is obviously faster than that of BSA.Figure 9
Convergence curves of BSAISA and BSA for the tension compression spring design problem.


#### 4.3.4. Welded Beam Design Problem (WBP)

The welded beam problem is a minimum cost problem with four continuous design variables and subject to two linear and five nonlinear inequality constraints. The best feasible solution is obtained by BSAISA atx = (0.205730, 3.470489, 9.036624, 0.205730) with the objective function value f(x) = 1.724852 using 29,000 FEs.For this problem, BSAISA is compared with many well-known algorithms as follows: GA-DT, MDE, CPSO, HPSO, DELC, POS-DE, ABC, MBA, BSA, BSA-SAε, and SSOC. The best solutions obtained from BSAISA and other well-known algorithms are listed in Table 14. The comparison of their statistical results is presented in Table 15.From Tables14 and 15, except that the constraint value of PSO is not available, the obtained solution sets of all algorithms satisfy the constraints for the problem. Most of algorithms including BSAISA, BSA, BSA-SAε, MDE, HPSO, DELC, POS-DE, ABC, and SSOC are able to find the best function value 1.724852, while GA-DT and CPSO and MBA fail to find it. It should be admitted that DELC is superior to all other algorithms in terms of FEs and robustness for this problem. On the other hand, except for DELC, MDE, and SSOC with FEs of 2000, 2400, and 2500, respectively, BSAISA requires fewer FEs than the remaining algorithms (excluding algorithms that do not reach the best solution). When considering the comparison of the Std values for this problem, MBA exhibits its powerful robustness and BSAISA performs better than most algorithms except MBA, DELC, PSO-DE, BSA, and BSA-SAε.It is worth mentioning that from [74] the Worst, Mean, Best, and Std value of MDE are given as 1.724854, 1.724853, 1.724852, and 1.0E-15, respectively. However, the corresponding values of DELC equal 1.724852, 1.724852, 1.724852, and 4.1E-13, respectively, where its Worst and Mean values are smaller than those of MDE while its Std is bigger than those of MDE. So we consider that the Std of MDE is probably an error data for this problem, and we replace it with NA in Table 15.Figure10 depicts the convergence curves of BSAISA and BSA for the welded beam design problem, where the value of F(x∗) on the vertical axis equals 1.724852. Figure 10 shows the convergence speed of BSAISA is faster than that of BSA remarkably.Figure 10
Convergence curves of BSAISA and BSA for the welded beam design problem.


#### 4.3.5. Speed Reducer Design Problem (SRP)

This speed reducer design problem has eleven constraints and six continuous design variables (x1,x2,x4,x5,x6,x7) and one integer variable (x3). The best solution obtained from BSAISA is x = (3.500000, 0.7000000, 17, 7.300000, 7.715320, 3.350215, 5.286654) with f(x)=2994.471066 using 15,860 FEs. The comparison of the best solutions obtained by BSAISA and other well-known algorithms is given in Table 16. The statistical results of BSAISA, BSA, MDE, DEDS, HEAA, DELC, POS-DE, ABC, MBA, and SSOC are listed in Table 17.Table 16
Comparison of best solutions for the speed reducer design problem.
Method
MDE
DEDS
DELC
HEAA
POS-DE
MBA
BSA
BSAISA
X
1
3.500010
3.500000
3.500000
3.500023
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
X
3
17
17
17
17.000013
17.000000
17.000000
17
17
X
4
7.300156
7.300000
7.300000
7.300428
7.300000
7.300033
7.300000
7.300000
X
5
7.800027
7.715320
7.715320
7.715377
7.800000
7.715772
7.715320
7.715320
X
6
3.350221
3.350215
3.350215
3.350231
3.350215
3.350218
3.350215
3.350215
X
7
5.286685
5.286654
5.286654
5.286664
5.286683
5.286654
5.286654
5.286654
f
(
X
)
2996.356689
2994.471066
2994.471066
2994.499107
2996.348167
2994.482453
2994.471066
2994.471066Table 17
Comparison of statistical results for the speed reducer design problem.
Method
Worst
Mean
Best
Std
FEs
MDE
2996.390137
2996.367220
2996.356689
8.2
E
-
03
24,000
DEDS
2994.471066
2994.471066
2994.471066
3.5
8
E
-
12
30,000
HEAA
2994.752311
2994.613368
2994.499107
7.0
E
-
02
40,000
DELC
2994.471066
2994.471066
2994.471066
1.9
E
-
12
30,000
PSO-DE
2996.348204
2996.348174
2996.348167
6.4
E
-
06
54,350
ABC
NA
2997.058412
2997.058412
0
30,000
MBA
2999.652444
2996.769019
2994.482453
1.56
6300
SSOC
2996.113298
2996.113298
2996.113298
1.3
4
E
-
12
25,000
BSA
2994.471066
2994.471066
2994.471066
9.8
7
E
-
11
25,640
BSAISA
2994.471095
2994.471067
2994.471066
5.4
0
E
-
06
15,860As shown in Tables16 and 17, the obtained solution sets of all algorithms satisfy the constraints for this problem. BSAISA, BSA, DEDS, and DELC are able to find the best function value 2994.471066 while the others do not. Among the four algorithms, DEDS, DELC, and BSA require 30000, 30000, and 25640 FEs, respectively. However, BSAISA requires only 15,860 FEs when it reaches the same best function value. MBA fails to find the best known function value; thus BSAISA is better than MBA in this problem, even though MBA has lower FEs. As for the comparison of the Std, among the four algorithms that achieve the best known function value, BSAISA is worse than the others. However, one thing that should be mentioned is that the main purpose of the experiment is to compare the convergence speed between BSAISA and other algorithms. From this point of view, it can be concluded that BSAISA has a better performance than other algorithms in terms of convergence speed.Figure11 depicts the convergence curves of BSAISA and BSA for the speed reducer design problem, where the value of F(x∗) on the vertical axis equals 2994.471066. Figure 11 shows that the convergence speed of BSAISA is faster than that of BSA.Figure 11
Convergence curves of BSAISA and BSA for the speed reducer design problem.


### 4.4. Comparisons Using Sign Test

Sign Test [80] is one of the most popular statistical methods used to determine whether two algorithms are significantly different. Recently, Miao et al. [81] utilized Sign Test method to analyze the performances between their proposed modified algorithm and the original one. In this paper, the two-tailed Sign Test with a significance level 0.05 is adopted to test the significant differences between the results obtained by different algorithms, and the test results are given in Table 18. The values of Best and FEs are two most important criterions for the evaluations of algorithms in our paper; they therefore should be chosen as the objectives of the Sign Test. The signs “+,” “≈,” and “−” represent, respectively, the fact that our BSAISA performs significantly better than, almost the same as, or significantly worse than the algorithm it is compared to. The null hypothesis herein is that the performances between BSAISA and one of the others are not significantly differential.Table 18
Comparisons between BSAISA and other algorithms in Sign Tests.
BSAISA-methods
+
≈
-
Total
p value
SR
11
1
1
13
0.006
FSA
12
0
1
13
0.003
CDE
8
0
5
13
0.581
AMA
13
0
0
13
0.000
MABC
10
2
1
13
0.012
RPGA
11
0
0
11
0.001
BSA-SAε
8
4
1
13
0.039
Note. The columns of “+,” “≈,” and “−” indicate the number of functions where BSAISA performs significantly better than, almost the same as, or significantly worse than the compared algorithm, respectively. “p value” denotes the probability value supporting the null hypothesis.As shown in Table18, the p values of supporting the null hypothesis of Sign Test for six pairs of algorithms (BSAISA-SR, BSAISA-FSA, BSAISA-AMA, BSAISA-MABC, BSAISA-RPGA, and BSAISA-BSA-SAε) are 0.006, 0.003, 0.000, 0.012, 0.001, and 0.039, respectively, and thereby we can reject the null hypothesis. This illustrates that the optimization performance of the proposed BSAISA is significantly better than those of the six algorithms. The p value of BSAISA-CDE is equal to 0.581, which shows that we cannot reject the null hypothesis. However, according to the related sign values (“+,” “≈,” and “−”) from Table 18, BSAISA is slightly worse than CDE on 5 problems but wins on another 8 problems, which illustrates that the proposed BSAISA has a relatively excellent competitiveness compared with the CDE. Generally, the statistical p values and sign values validate that BSAISA has the superiority compared to the other well-known algorithms on the constrained optimization problems.On the one hand, all experimental results suggest that the proposed method improves the convergence speed of BSA. On the other hand, the overall comparative results of BSAISA and other well-known algorithms demonstrate that BSAISA is more effective and competitive for constrained and engineering optimization problems in terms of convergence speed.


## 4.1. Parameter Settings

For the first experiment, the main parameters for 13 benchmark constrained functions are the same as the following: population size (N) is set as 30; the maximum number of iterations (Tmax) is set as 11665. Therefore, BSAISA’s maximum number of function evaluations (MFEs) should equal to 34,9980 (nearly 35,0000). The 13 benchmarks were executed by using 30 independent runs.For the 5 real-world engineering design problems, we use slightly different parameter settings since each problem has different natures, that is, TTP (N=20,Tmax=1000), PVP (N=20,Tmax=3000), TCSP (N=20,Tmax=3000), WBP (N=20,Tmax=3000), SRP (N=20,Tmax=2000). The 6 engineering problems were performed using 50 independent runs.The user parameters of all experiments are presented in Table4. Termination condition may be the maximum number of iterations, CPU time, or an allowable tolerance value between the last result and the best known function value for most metaheuristics. In this paper, the maximum number of iterations is considered as termination condition. All experiments were conducted on a computer with a Intel(R) Core(TM) i5-4590 CPU @ 3.30 GHz and 4 GB RAM.Table 4
User parameters used for all experiments.
Problem
G01–G13
TTP
PVP
TCSP
WBP
SRP
N
30
20
20
20
20
20
T
m
a
x
11665
1000
3000
3000
3000
2000
Runs
30
50
50
50
50
50
Note. The expression “runs” denotes the number of independent runs.


## 4.2. Simulation on Constrained Benchmark Problems

In this section, BSAISA and BSA are performed on the 13 benchmarks simultaneously. Their statistical results obtained from 30 independent runs are listed in Tables5 and 6, including the best known function value (Best Known) and the obtained Best/Mean/Worst/Std values as well as the FEs. The best known values for all the test problems are derived from [62]. The best known values found by algorithms are highlighted inbold. From Tables 5 and 6, it can be seen that BSAISA is able to find the known optimal function values for G01, G04, G05, G06, G08, G09, G011, G012, and G013; however, BSA fails to find the best known function value on G09 and G13. For the rest of the functions, BSAISA obtains the results very close to the best known function values. Moreover, BSAISA requires fewer FEs than BSA on G01, G03, G04, G05, G06, G07, G08, G09, G011, and G012. Although the FEs of BSAISA are slightly worse than that of BSA for G02, G10, and G13, BSAISA finds more accurate best function values than BSA for G02, G03, G07, G09, G10, and G13.Table 5
The statistical results of BSAISA for 13 constrained benchmarks.
Fun.
Known optimal
Best
Mean
Worst
Std
FEs
G01
-
15
−15
-
15
-
15.000000
8.0
8
E
-
16
84,630
G02
-
0.803619
-
0.803599
-
0.787688
-
0.758565
1.1
4
E
-
02
349,500
G03
-
1.000500
-
1.000498
-
1.000481
-
1.000441
1.3
5
E
-
05
58,560
G04
-
30665.538672
−30665.538672
-
30665.538672
-
30665.538672
1.0
9
E
-
11
121,650
G05
5126.496714
5126.496714
5126.496714
5126.496714
5.8
5
E
-
13
238,410
G06
-
6961.813876
−6961.813876
-
6961.813876
-
6961.813876
1.8
5
E
-
12
89,550
G07
24.306209
24.307381
24.400881
24.758205
1.0
2
E
-
01
15,060
G08
-
0.0958250
−0.0958250
-
0.086683
-
0.027263
2.3
7
E
-
02
30,930
G09
680.630057
680.630057
680.633025
680.680043
9.1
1
E
-
03
347,760
G10
7049.248021
7049.249056
7081.241789
7326.853581
6.2
4
E
+
01
346,980
G11
0.749900
0.749900
0.749900
0.749900
1.1
3
E
-
16
87,870
G12
-
1
−1
-
1
-
1
0
5430
G13
0.0539415
0.0539415
0.1030000
0.4594033
9.8
0
E
-
02
349,800
Note. “Known optimal” denotes the best known function values in the literatures. “Bold” means the algorithm has found the best known function values. The same as Table 6.Table 6
The statistical results of the basic BSA on 13 constrained benchmarks.
Fun.
Known optimal
Best
Mean
Worst
Std
FEs
G01
−15
−15
−15
−15.000000
6.6
0
E
-
16
99,300
G02
−0.803619
−0.803255
−0.792760
−0.749326
9.1
0
E
-
03
344,580
G03
−1.000500
−1.000488
−0.998905
−0.990600
2.6
6
E
-
03
348,960
G04
−30665.538672
−30665.538672
−30665.538672
−30665.538672
1.1
1
E
-
11
272,040
G05
5126.496714
5126.496714
5144.041363
5275.384724
3.9
9
E
+
01
299,220
G06
−6961.813876
−6961.813876
−6961.813876
−6961.813876
1.8
5
E
-
12
111,450
G07
24.306209
24.307607
24.344626
24.399896
1.9
3
E
-
02
347,250
G08
−0.0958250
−0.0958250
−0.0958250
−0.0958250
2.8
2
E
-
17
73,440
G09
680.630057
680.630058
680.630352
680.632400
5.6
3
E
-
04
348,900
G10
7049.248021
7049.278543
7053.573853
7080.192700
7.2
8
E
+
00
340,020
G11
0.749900
0.749900
0.749900
0.749900
1.1
3
E
-
16
113,250
G12
−1
−1
−1
−1
0
13,590
G13
0.0539415
0.0539420
0.1816986
0.5477657
1.5
0
E
-
01
347,400To further compare BSAISA and BSA, the function value convergence curves of 13 functions that have significant differences have been plotted, as shown in Figures5 and 6. The horizontal axis represents the number of iterations, while the vertical axis represents the difference of the objective function value and the best known value. For G02, G04, G06, G08, G11, and G12, it is obvious that the convergence speed of BSAISA is faster than that of BSA, and its best function value is also better than that of BSA. For G01, G03, G05, G07, G09, G10, and G13, it can be observed that the objective function values of BSAISA and BSA fluctuate in the early iterations, and they decrease as the number of iterations increases during the middle and late stages. This illustrates that both the algorithms are able to escape the local optimum under different degrees, and the convergence curve of BSAISA drops still faster than that of BSA. Therefore, the experiment results demonstrate that the convergence speed of the basic BSA is improved with our modified F.Figure 5
The convergence curves of the first 9 functions by BSAISA and BSA.
(a)
Convergence curve on G01
(b)
Convergence curve on G02
(c)
Convergence curve on G03
(d)
Convergence curve on G04
(e)
Convergence curve on G05
(f)
Convergence curve on G06
(g)
Convergence curve on G07
(h)
Convergence curve on G08
(i)
Convergence curve on G09Figure 6
The convergence curves of the latter 4 functions by BSAISA and BSA.
(a)
Convergence curve on G10
(b)
Convergence curve on G11
(c)
Convergence curve on G12
(d)
Convergence curve on G13In order to further verify the competitiveness of BSAISA in aspect of convergence speed, we compared BSAISA with some classic and state-of-the-art approaches in terms of best function value and function evaluations. The best function value and the corresponding FEs of each algorithm on 13 benchmarks are presented in Table7, where the optimal results are inbold on each function. These compared algorithms are listed below:(1)
Stochastic ranking (SR) [63](2)
Filter simulated annealing (FSA) [65](3)
Cultured differential evolution (CDE) [66](4)
Agent based memetic algorithm (AMA) [64](5)
Modified artificial bee colony (MABC) algorithm [67](6)
Rough penalty genetic algorithm (RPGA) [68](7)
BSA combined self-adaptiveε constrained method (BSA-SAε) [62].Table 7
Comparison of the best values and FEs obtained by BSAISA and other algorithms.
Alg.
BSAISA
SR
FSA
CDE
AMA
MABC
RPGA
BSA-SAε
Fun.
Best (FEs)
Best (FEs)
Best (FEs)
Best (FEs)
Best (FEs)
Best (FEs)
Best (FEs)
Best (FEs)
G01
−15
−15.000
−14.993316
−15.000000
−15.000
−15.000
−15.000
−15.000000
(84,630)
(148,200)
(205,748)
(100,100)
(350,000)
(350,000)
(350,000)
(350,000)
G02
−0.8036
−0.8035
−0.7549
−0.8036
−0.8035
−0.8036
−0.8036
−0.8036
(349,500)
(217,200)
(227,832)
(100,100)
(350,000)
(350,000)
(350,000)
(350,000)
G03
−1.000498
−1.000
−1.0000015
−0.995413
−1.000
−1.000
−1.000
−1.000498
(58,560)
(229,200)
(314,938)
(100,100)
(350,000)
(350,000)
(350,000)
(350,000)
G04
−30665.539
−30665.539
−30665.538
−30665.539
−30665.538
−30665.539
−30665.539
−30665.539
(121,650)
(88,200)
(86,154)
(100,100)
(350,000)
(350,000)
(350,000)
(350,000)
G05
5126.497
5126.497
5126.4981
5126.571
5126.512
5126.487
5126.544
5126.497
(238,410)
(51,600)
(47,661)
(100,100)
(350,000)
(350,000)
(350,000)
(350,000)
G06
−6961.814
−6961.814
−6961.814
−6961.814
−6961.807
−6961.814
−6961.814
−6961.814
(89,550)
(118,000)
(44,538)
(100,100)
(350,000)
(350,000)
(350,000)
(350,000)
G07
24.307
24.307
24.311
24.306
24.315
24.324
24.333
24.306
(15,060)
(143,000)
(404,501)
(100,100)
(350,000)
(350,000)
(350,000)
(350,000)
G08
−0.095825
−0.095825
−0.095825
−0.095825
−0.095825
−0.095825
−0.095825
−0.095825
(30,930)
(76,200)
(56,476)
(100,100)
(350,000)
(350,000)
(350,000)
(350,000)
G09
680.630057
680.630
680.63008
680.630057
680.645
680.631
680.631
680.6301
(347,760)
(111,400)
(324,569)
(100,100)
(350,000)
(350,000)
(350,000)
(350,000)
G10
7049.249
7054.316
7059.864
7049.248
7281.957
7058.823
7049.861
7049.278
(346,980)
(128,400)
(243,520)
(100,100)
(350,000)
(350,000)
(350,000)
(350,000)
G11
0.749900
0.750
0.749999
0.749900
0.750
0.750
0.749
0.749900
(87,870)
(11,400)
(23,722)
(100,100)
(350,000)
(350,000)
(350,000)
(350,000)
G12
−1
−1.000000
−1.000000
−1.000000
−1.000
−1.000
NA
−1.000000
(5430)
(16,400)
(59,355)
(100,100)
(350,000)
(350,000)
(350,000)
G13
0.0539415
0.053957
0.0539498
0.056180
0.053947
0.757
NA
0.0539415
(349,800)
(69,800)
(120,268)
(100,100)
(350,000)
(350,000)
(350,000)
Nu.
6
+
10
1
+
5
1
+
3
4
+
10
0
+
3
1
+
7
0
+
5
0
+
10
RK
1
5
7
2
8
4
6
3
Note. “NA” means not available. The same as Tables 8, 10, 11, 12, 13, 14, 15, and 17. “RK” represents the comprehensive ranking of each algorithm on the set of benchmarks. “Nu.” represents the sum of the number of winners and the number of the optimal function values for each algorithm on the set of benchmarks.Table 8
Comparison of best solutions for the three-bar truss design problem.
Method
DEDS
HEAA
PSO-DE
DELC
MBA
BSA
BSAISA
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
0.408248
g
1
(
X
)
1.7
7
E
-
08
NA
-
5.29
E
-
11
NA
-
5.2
9
E
-
11
-
3.2
3
E
-
12
0
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
263.895843To compare these algorithms synthetically, a simple evaluation mechanism is used. It can be explained as the best function value (Best) is preferred, and the function evaluations (FEs) are secondary. More specifically, (1) if one algorithm has a betterBest than those of others on a function, there is no need to considerFEs and the algorithm is superior to other algorithms on this function. (2) If two or more algorithms have found the optimalBest on a function, the algorithm with the lowestFEs is considered as the winner on the function. (3) Record the number of winners and the number of the optimal function values for each algorithm on the set of benchmarks, and then give the sort for all algorithms.From Table7, it can be observed that the rank of these 8 algorithms is as follows: BSAISA, CDE, BSA-SAε, MABC, SR, RPGA, FSA, and AMA. Among the 13 benchmarks, BSAISA wins on 6 functions and it is able to find the optimal values of 10 functions. This is better than all other algorithms, thus BSAISA ranks the first. The second algorithm CDE performs better on G02, G07, G09, and G10 than BSAISA but worse on G01, G03, G08, G11, G12, and G13. BSA-SAε obtains the optimal function values of 10 functions but requires more function evaluations than BSAISA and CDE, so it should rank the third. MABC ranks the fourth. It obtains the optimal function values of 7 functions, which are fewer in number than those of the former three algorithms. Both SR and RPGA have found the same number of the optimal function values, while the former is the winner on G04, so SR is slightly better than RPGA. As for the last two algorithms, FSA and AMA just perform well on three functions, while FSA is the winner on G06, so FSA is slightly better than AMA.Based on the above comparison, it can be concluded that BSAISA is effective and competitive in terms of convergence speed.


## 4.3. Simulation on Engineering Design Problems

In order to assess the optimization performance of BSAISA in real-world engineering constrained optimization problems, 5 well-known engineering constrained design problems including three-bar truss design, pressure vessel design, tension/compression spring design, welded beam design, and speed reducer design are considered in the second experiment.


### 4.3.1. Three-Bar Truss Design Problem (TTP)

The three-bar truss problem is one of the engineering minimization test problems for constrained algorithms. The best feasible solution is obtained by BSAISA atx = (0.788675, 0.408248) with the objective function value f(x) = 263.895843 using 8940 FEs. The comparison of the best solutions obtained from BSAISA, BSA, differential evolution with dynamic stochastic selection (DEDS) [69], hybrid evolutionary algorithm (HEAA) [70], hybrid particle swarm optimization with differential evolution (POS-DE) [71], differential evolution with level comparison (DELC) [72], and mine blast algorithm (MBA) [73] is presented in Table 8. Their statistical results are listed in Table 9.Table 9
Comparison of statistical results for the three-bar truss design problem.
Method
Worst
Mean
Best
Std
FEs
DEDS
263.895849
263.895843
263.895843
9.7
E
-
07
15,000
HEAA
263.896099
263.895865
263.895843
4.9
E
-
05
15,000
PSO-DE
263.895843
263.895843
263.895843
4.5
E
-
10
17,600
DELC
263.895843
263.895843
263.895843
4.3
E
-
14
10,000
MBA
263.915983
263.897996
263.895852
3.9
3
E
-
03
13,280
BSA
263.895845
263.895843
263.895843
2.6
4
E
-
07
13,720
BSAISA
263.895843
263.895843
263.895843
5.7
5
E
-
13
8940Table 10
Comparison of best solutions for the pressure vessel design problem.
Method
GA-DT
MDE
CPSO
HPSO
DELC
ABC
BSA-SAε
BSA
BSAISA
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
)
-
2.0
1
E
-
03
0
-
1.3
7
E
-
06
NA
NA
0
-
9.5
E
-
10
-
3.6
9
E
-
08
0
g
2
(
X
)
-
3.5
8
E
-
02
-
0.035881
-
3.5
9
E
-
04
NA
NA
-
0.035881
-
3.5
9
E
-
2
-
0.035881
-
0.035881
g
3
(
X
)
-
24.7593
-
0.0000
-
118.7687
NA
NA
-
0.000226
-
1.2
E
-
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 11
Comparison of statistical results for the pressure vessel design problem.
Method
Worst
Mean
Best
Std
FEs
GA-DT
6469.3220
6177.2533
6059.9463
130.9297
80,000
MDE
6059.7017
6059.7017
6059.7017
1.0
E
-
12
24,000
CPSO
6363.8041
6147.1332
6061.0777
86.45
30,000
HPSO
6288.6770
6099.9323
6059.7143
86.20
81,000
DELC
6059.7143
6059.7143
6059.7143
2.1
E
-
11
30,000
PSO-DE
6059.7143
6059.7143
6059.7143
1.0
E
-
10
42,100
ABC
NA
6245.3081
6059.7147
2.0
5
E
+
02
30,000
BSA-SAε
6116.7804
6074.3682
6059.7143
1.7
1
E
+
01
80,000
BSA
6771.5969
6221.2861
6059.7150
2.0
3
E
+
02
60,000
BSAISA
7198.0054
6418.1935
6059.7143
3.0
4
E
+
02
16,320Table 12
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
-
1.3
E
-
05
-
2.1
E
-
05
-
4.061338
-
0.722698
0.012681
MDE
0.051688
0.356692
11.290483
-
0.000000
-
0.000000
-
4.053734
-
0.727747
0.012665
CPSO
0.051728
0.357644
11.244543
-
8.4
5
E
-
04
-
1.2
6
E
-
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
-
4.056663
-
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
-
7.7
0
E
-
09
-
3.3
0
E
-
09
-
4.054
-
0.728
0.012665
BSA
0.051694
0.356845
11.281488
-
1.0
5
E
-
07
-
1.7
7
-
0
8
-
4.054037
-
0.727640
0.012665
BSAISA
0.051687
0.356669
11.291824
-
6.3
8
E
-
10
-
1.5
3
E
-
09
-
4.053689
-
0.727763
0.012665Table 13
Comparison of statistical results for the tension compression spring design problem.
Method
Worst
Mean
Best
Std
FEs
GA-DT
0.012973
0.012742
0.012681
5.9
0
E
-
05
80,000
MDE
0.012674
0.012666
0.012665
2.0
E
-
6
24,000
CPSO
0.012924
0.012730
0.012675
5.2
0
E
-
05
23,000
HPSO
0.012719
0.012707
0.012665
1.5
8
E
-
05
81,000
DEDS
0.012738
0.012669
0.012665
1.2
5
E
-
05
24,000
HEAA
0.012665
0.012665
0.012665
1.4
E
-
09
24,000
DELC
0.012666
0.012665
0.012665
1.3
E
-
07
20,000
PSO-DE
0.012665
0.012665
0.012665
1.2
E
-
08
24,950
ABC
NA
0.012709
0.012665
0.012813
30,000
MBA
0.012900
0.012713
0.012665
6.3
0
E
-
05
7650
SSOC
0.012868
0.012765
0.012665
9.2
9
E
-
05
25,000
BSA-SAε
0.012666
0.012665
0.012665
1.6
2
E
-
07
80,000
BSA
0.012669
0.012666
0.012665
7.2
4
E
-
07
43,220
BSAISA
0.012668
0.012666
0.012665
4.9
0
E
-
07
9440Table 14
Comparison of best solutions for the welded beam design problem.
Method
GA-DT
MDE
CPSO
HPSO
ABC
MBA
BSA-SAε
BSA
BSAISA
X
1
(
h
)
0.205986
0.205730
0.202369
0.205730
0.205730
0.205729
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
X
3
(
t
)
9.020224
9.036624
9.04821
9.036624
9.036624
9.036626
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
g
1
(
X
)
-
0.074092
-
0.000335
-
12.839796
NA
0.000000
-
0.001614
-
1.5
5
E
-
10
-
5.3
2
E
-
07
0
g
2
(
X
)
-
0.266227
-
0.000753
-
1.247467
NA
-
0.000002
-
0.016911
-
4.3
0
E
-
09
-
9.0
2
E
-
06
0
g
3
(
X
)
-
4.9
5
E
-
04
-
0.000000
-
1.4
9
E
-
03
NA
0.000000
-
2.4
0
E
-
07
-
1.5
5
E
-
15
-
7.8
6
E
-
12
-
5.5
5
E
-
17
g
4
(
X
)
-
3.430044
-
3.432984
-
3.429347
NA
-
3.432984
-
3.432982
-
3.4330
-
3.432984
-
3.432984
g
5
(
X
)
-
0.080986
-
0.080730
-
0.079381
NA
-
0.080730
-
0.080729
-
8.0
7
E
-
02
-
0.080730
-
0.080730
g
6
(
X
)
-
0.235514
-
0.235540
-
0.235536
NA
-
0.235540
-
0.235540
-
0.2355
-
0.235540
-
0.235540
g
7
(
X
)
-
58.666440
-
0.000882
-
11.681355
NA
0.000000
-
0.001464
-
1.8
5
E
-
10
-
1.1
3
E
-
07
-
5.4
6
E
-
12
f
(
X
)
1.728226
1.724852
1.728024
1.724852
1.724852
1.724853
1.724852
1.724852
1.724852Table 15
Comparison of statistical results for the welded beam design problem.
Method
Worst
Mean
Best
Std
FEs
GA-DT
1.993408
1.792654
1.728226
7.4
7
E
-
02
80,000
MDE
1.724854
1.724853
1.724852
NA
24,000
CPSO
1.782143
1.748831
1.728024
1.2
9
E
-
02
24,000
HPSO
1.814295
1.749040
1.724852
4.0
0
E
-
02
81,000
DELC
1.724852
1.724852
1.724852
4.1
E
-
13
20,000
PSO-DE
1.724852
1.724852
1.724852
6.7
E
-
16
66,600
ABC
NA
1.741913
1.724852
3.1
E
-
02
30,000
MBA
1.724853
1.724853
1.724853
6.9
4
E
-
19
47,340
SSOC
1.799332
1.746462
1.724852
2.5
7
E
-
2
25,000
BSA-SAε
1.724852
1.724852
1.724852
8.1
1
E
-
10
80,000
BSA
1.724854
1.724852
1.724852
2.3
5
E
-
07
45,480
BSAISA
1.724854
1.724852
1.724852
2.9
6
E
-
07
29,000From Tables8 and 9, BSAISA, BSA, DEDS, HEAA, PSO-DE, and DELC all reach the best solution with the corresponding function value equal to 263.895843 except MBA with 263.895852. However, BSAISA requires the lowest FEs (only 8940) among all algorithms. Its Std value is better than BSA, DEDS, HEAA, PSO-DE, and MBA except DELC. These comparative results indicate that BSAISA outperforms other algorithms in terms of computational cost and robustness for this problem.Figure7 depicts the convergence curves of BSAISA and BSA for the three-bar truss design problem, where the value of F(x∗) on the vertical axis equals 263.895843. As shown in Figure 7, BSA achieves the global optimum at about 700 iterations, while BSAISA only reaches the global optimum at about 400 iterations. It can be concluded that the convergence speed of BSAISA is faster than that of BSA for this problem.Figure 7
Convergence curves of BSAISA and BSA for the three-bar truss design problem.


### 4.3.2. Pressure Vessel Design Problem (PVP)

The pressure vessel design problem has a nonlinear objective function with three linear and one nonlinear inequality constraints and two discrete and two continuous design variables. The values of the two discrete variables (x1,x2) should be the integer multiples of 0.0625. The best feasible solution is obtained by BSAISA at x = (0.8750, 0.4375, 42.0984, 176.6366) with the objective function value f(x) = 6059.7143 using 31,960 FEs.For this problem, BSAISA is compared with nine algorithms: BSA, BSA-SAε [62], DELC, POS-DE, genetic algorithms based on dominance tournament selection (GA-DT) [73], modified differential evolution (MDE) [74], coevolutionary particle swarm optimization (CPSO) [75], hybrid particle swarm optimization (HPSO) [76], and artificial bee colony algorithm (ABC) [77]. The comparison of the best solutions obtained by BSAISA and other reported algorithms is presented in Table 10. The statistical results of various algorithms are listed in Table 11.As shown Table10, the obtained solution sets of all algorithms satisfy the constraints for this problem. BSAISA, BSA-SAε, ABC, DELC, and HPSO find the same considerable good objective function value 6059.7143, which is slightly worse than MDE’s function value 6059.7143. It is worth mentioning that MBA’s best solution was obtained at x = (0.7802, 0.3856, 40.4292, 198.4964) with f(x) = 5889.3216 and the corresponding constraint values equal to gi(x) = (0, 0, −86.3645, −41.5035) in [78]. Though MBA finds a far better function value than that of MDE, its obtained variables (i.e., 0.7802 and 0.3856) are not integer multiples of 0.0625. So they are not listed in Table 10 to ensure a fair comparison. From Table 11, except for MDE with the function value of 6059.7016, BSAISA offers better function value results compared to GA-DT, CPSO, ABC, and BSA. Besides that, BSAISA is far superior to other algorithms in terms of FEs. Unfortunately, the obtained Std value of BSAISA is relatively poor compared with others for this problem.Figure8 describes the convergence curves of BSAISA and BSA for the pressure vessel design problem, where the value of F(x∗) on the vertical axis equals 6059.7143. As shown in Figure 8, BSAISA is able to find the global optimum at about 800 iterations and obtains a far more accurate function value than that of BSA. Moreover, the convergence speed of BSAISA is much faster than that of BSA.Figure 8
Convergence curves of BSAISA and BSA for the pressure vessel design problem.


### 4.3.3. Tension Compression Spring Design Problem (TCSP)

This design optimization problem has three continuous variables and four nonlinear inequality constraints. The best feasible solution is obtained by BSAISA atx = (0.051687, 0.356669, 11.291824) with f(x) = 0.012665 using 9440 FEs. This problem has been solved by other methods as follows: GA-DT, MDE, CPSO, HPSO, DEDS, HEAA, DELC, POS-DE, ABC, MBA, BSA-SAε, and Social Spider Optimization (SSOC) [79]. The comparison of the best solutions obtained from various algorithms is presented in Table 12. Their statistical results are listed in Table 13.From Tables12 and 13, the vast majority of algorithms can find the best function value 0.012665 for this problem, while GA-DT and CPSO fail to find it. With regard to the computational cost (FEs), BSAISA only requires 9440 FEs when it reaches the global optimum, which is superior to all other algorithms except MBA with 7650 FEs. However, the Worst and Mean and Std values of BSAISA are better than those of MBA. Consequently, for this problem, it can be concluded that BSA has the obvious superiority in terms of FEs over all other algorithms except MBA. Moreover, BSAISA has a stronger robustness when compared with MBA alone.Figure9 depicts the convergence curves of BSAISA and BSA for the tension compression spring design problem, where the value of F(x∗) on the vertical axis equals 0.012665. From Figure 9 it can be observed that both BSAISA and BSA fall into a local optimum in the early iterations but they are able to successfully escape from the local optimum. However, the convergence speed of BSAISA is obviously faster than that of BSA.Figure 9
Convergence curves of BSAISA and BSA for the tension compression spring design problem.


### 4.3.4. Welded Beam Design Problem (WBP)

The welded beam problem is a minimum cost problem with four continuous design variables and subject to two linear and five nonlinear inequality constraints. The best feasible solution is obtained by BSAISA atx = (0.205730, 3.470489, 9.036624, 0.205730) with the objective function value f(x) = 1.724852 using 29,000 FEs.For this problem, BSAISA is compared with many well-known algorithms as follows: GA-DT, MDE, CPSO, HPSO, DELC, POS-DE, ABC, MBA, BSA, BSA-SAε, and SSOC. The best solutions obtained from BSAISA and other well-known algorithms are listed in Table 14. The comparison of their statistical results is presented in Table 15.From Tables14 and 15, except that the constraint value of PSO is not available, the obtained solution sets of all algorithms satisfy the constraints for the problem. Most of algorithms including BSAISA, BSA, BSA-SAε, MDE, HPSO, DELC, POS-DE, ABC, and SSOC are able to find the best function value 1.724852, while GA-DT and CPSO and MBA fail to find it. It should be admitted that DELC is superior to all other algorithms in terms of FEs and robustness for this problem. On the other hand, except for DELC, MDE, and SSOC with FEs of 2000, 2400, and 2500, respectively, BSAISA requires fewer FEs than the remaining algorithms (excluding algorithms that do not reach the best solution). When considering the comparison of the Std values for this problem, MBA exhibits its powerful robustness and BSAISA performs better than most algorithms except MBA, DELC, PSO-DE, BSA, and BSA-SAε.It is worth mentioning that from [74] the Worst, Mean, Best, and Std value of MDE are given as 1.724854, 1.724853, 1.724852, and 1.0E-15, respectively. However, the corresponding values of DELC equal 1.724852, 1.724852, 1.724852, and 4.1E-13, respectively, where its Worst and Mean values are smaller than those of MDE while its Std is bigger than those of MDE. So we consider that the Std of MDE is probably an error data for this problem, and we replace it with NA in Table 15.Figure10 depicts the convergence curves of BSAISA and BSA for the welded beam design problem, where the value of F(x∗) on the vertical axis equals 1.724852. Figure 10 shows the convergence speed of BSAISA is faster than that of BSA remarkably.Figure 10
Convergence curves of BSAISA and BSA for the welded beam design problem.


### 4.3.5. Speed Reducer Design Problem (SRP)

This speed reducer design problem has eleven constraints and six continuous design variables (x1,x2,x4,x5,x6,x7) and one integer variable (x3). The best solution obtained from BSAISA is x = (3.500000, 0.7000000, 17, 7.300000, 7.715320, 3.350215, 5.286654) with f(x)=2994.471066 using 15,860 FEs. The comparison of the best solutions obtained by BSAISA and other well-known algorithms is given in Table 16. The statistical results of BSAISA, BSA, MDE, DEDS, HEAA, DELC, POS-DE, ABC, MBA, and SSOC are listed in Table 17.Table 16
Comparison of best solutions for the speed reducer design problem.
Method
MDE
DEDS
DELC
HEAA
POS-DE
MBA
BSA
BSAISA
X
1
3.500010
3.500000
3.500000
3.500023
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
X
3
17
17
17
17.000013
17.000000
17.000000
17
17
X
4
7.300156
7.300000
7.300000
7.300428
7.300000
7.300033
7.300000
7.300000
X
5
7.800027
7.715320
7.715320
7.715377
7.800000
7.715772
7.715320
7.715320
X
6
3.350221
3.350215
3.350215
3.350231
3.350215
3.350218
3.350215
3.350215
X
7
5.286685
5.286654
5.286654
5.286664
5.286683
5.286654
5.286654
5.286654
f
(
X
)
2996.356689
2994.471066
2994.471066
2994.499107
2996.348167
2994.482453
2994.471066
2994.471066Table 17
Comparison of statistical results for the speed reducer design problem.
Method
Worst
Mean
Best
Std
FEs
MDE
2996.390137
2996.367220
2996.356689
8.2
E
-
03
24,000
DEDS
2994.471066
2994.471066
2994.471066
3.5
8
E
-
12
30,000
HEAA
2994.752311
2994.613368
2994.499107
7.0
E
-
02
40,000
DELC
2994.471066
2994.471066
2994.471066
1.9
E
-
12
30,000
PSO-DE
2996.348204
2996.348174
2996.348167
6.4
E
-
06
54,350
ABC
NA
2997.058412
2997.058412
0
30,000
MBA
2999.652444
2996.769019
2994.482453
1.56
6300
SSOC
2996.113298
2996.113298
2996.113298
1.3
4
E
-
12
25,000
BSA
2994.471066
2994.471066
2994.471066
9.8
7
E
-
11
25,640
BSAISA
2994.471095
2994.471067
2994.471066
5.4
0
E
-
06
15,860As shown in Tables16 and 17, the obtained solution sets of all algorithms satisfy the constraints for this problem. BSAISA, BSA, DEDS, and DELC are able to find the best function value 2994.471066 while the others do not. Among the four algorithms, DEDS, DELC, and BSA require 30000, 30000, and 25640 FEs, respectively. However, BSAISA requires only 15,860 FEs when it reaches the same best function value. MBA fails to find the best known function value; thus BSAISA is better than MBA in this problem, even though MBA has lower FEs. As for the comparison of the Std, among the four algorithms that achieve the best known function value, BSAISA is worse than the others. However, one thing that should be mentioned is that the main purpose of the experiment is to compare the convergence speed between BSAISA and other algorithms. From this point of view, it can be concluded that BSAISA has a better performance than other algorithms in terms of convergence speed.Figure11 depicts the convergence curves of BSAISA and BSA for the speed reducer design problem, where the value of F(x∗) on the vertical axis equals 2994.471066. Figure 11 shows that the convergence speed of BSAISA is faster than that of BSA.Figure 11
Convergence curves of BSAISA and BSA for the speed reducer design problem.


## 4.3.1. Three-Bar Truss Design Problem (TTP)

The three-bar truss problem is one of the engineering minimization test problems for constrained algorithms. The best feasible solution is obtained by BSAISA atx = (0.788675, 0.408248) with the objective function value f(x) = 263.895843 using 8940 FEs. The comparison of the best solutions obtained from BSAISA, BSA, differential evolution with dynamic stochastic selection (DEDS) [69], hybrid evolutionary algorithm (HEAA) [70], hybrid particle swarm optimization with differential evolution (POS-DE) [71], differential evolution with level comparison (DELC) [72], and mine blast algorithm (MBA) [73] is presented in Table 8. Their statistical results are listed in Table 9.Table 9
Comparison of statistical results for the three-bar truss design problem.
Method
Worst
Mean
Best
Std
FEs
DEDS
263.895849
263.895843
263.895843
9.7
E
-
07
15,000
HEAA
263.896099
263.895865
263.895843
4.9
E
-
05
15,000
PSO-DE
263.895843
263.895843
263.895843
4.5
E
-
10
17,600
DELC
263.895843
263.895843
263.895843
4.3
E
-
14
10,000
MBA
263.915983
263.897996
263.895852
3.9
3
E
-
03
13,280
BSA
263.895845
263.895843
263.895843
2.6
4
E
-
07
13,720
BSAISA
263.895843
263.895843
263.895843
5.7
5
E
-
13
8940Table 10
Comparison of best solutions for the pressure vessel design problem.
Method
GA-DT
MDE
CPSO
HPSO
DELC
ABC
BSA-SAε
BSA
BSAISA
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
)
-
2.0
1
E
-
03
0
-
1.3
7
E
-
06
NA
NA
0
-
9.5
E
-
10
-
3.6
9
E
-
08
0
g
2
(
X
)
-
3.5
8
E
-
02
-
0.035881
-
3.5
9
E
-
04
NA
NA
-
0.035881
-
3.5
9
E
-
2
-
0.035881
-
0.035881
g
3
(
X
)
-
24.7593
-
0.0000
-
118.7687
NA
NA
-
0.000226
-
1.2
E
-
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 11
Comparison of statistical results for the pressure vessel design problem.
Method
Worst
Mean
Best
Std
FEs
GA-DT
6469.3220
6177.2533
6059.9463
130.9297
80,000
MDE
6059.7017
6059.7017
6059.7017
1.0
E
-
12
24,000
CPSO
6363.8041
6147.1332
6061.0777
86.45
30,000
HPSO
6288.6770
6099.9323
6059.7143
86.20
81,000
DELC
6059.7143
6059.7143
6059.7143
2.1
E
-
11
30,000
PSO-DE
6059.7143
6059.7143
6059.7143
1.0
E
-
10
42,100
ABC
NA
6245.3081
6059.7147
2.0
5
E
+
02
30,000
BSA-SAε
6116.7804
6074.3682
6059.7143
1.7
1
E
+
01
80,000
BSA
6771.5969
6221.2861
6059.7150
2.0
3
E
+
02
60,000
BSAISA
7198.0054
6418.1935
6059.7143
3.0
4
E
+
02
16,320Table 12
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
-
1.3
E
-
05
-
2.1
E
-
05
-
4.061338
-
0.722698
0.012681
MDE
0.051688
0.356692
11.290483
-
0.000000
-
0.000000
-
4.053734
-
0.727747
0.012665
CPSO
0.051728
0.357644
11.244543
-
8.4
5
E
-
04
-
1.2
6
E
-
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
-
4.056663
-
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
-
7.7
0
E
-
09
-
3.3
0
E
-
09
-
4.054
-
0.728
0.012665
BSA
0.051694
0.356845
11.281488
-
1.0
5
E
-
07
-
1.7
7
-
0
8
-
4.054037
-
0.727640
0.012665
BSAISA
0.051687
0.356669
11.291824
-
6.3
8
E
-
10
-
1.5
3
E
-
09
-
4.053689
-
0.727763
0.012665Table 13
Comparison of statistical results for the tension compression spring design problem.
Method
Worst
Mean
Best
Std
FEs
GA-DT
0.012973
0.012742
0.012681
5.9
0
E
-
05
80,000
MDE
0.012674
0.012666
0.012665
2.0
E
-
6
24,000
CPSO
0.012924
0.012730
0.012675
5.2
0
E
-
05
23,000
HPSO
0.012719
0.012707
0.012665
1.5
8
E
-
05
81,000
DEDS
0.012738
0.012669
0.012665
1.2
5
E
-
05
24,000
HEAA
0.012665
0.012665
0.012665
1.4
E
-
09
24,000
DELC
0.012666
0.012665
0.012665
1.3
E
-
07
20,000
PSO-DE
0.012665
0.012665
0.012665
1.2
E
-
08
24,950
ABC
NA
0.012709
0.012665
0.012813
30,000
MBA
0.012900
0.012713
0.012665
6.3
0
E
-
05
7650
SSOC
0.012868
0.012765
0.012665
9.2
9
E
-
05
25,000
BSA-SAε
0.012666
0.012665
0.012665
1.6
2
E
-
07
80,000
BSA
0.012669
0.012666
0.012665
7.2
4
E
-
07
43,220
BSAISA
0.012668
0.012666
0.012665
4.9
0
E
-
07
9440Table 14
Comparison of best solutions for the welded beam design problem.
Method
GA-DT
MDE
CPSO
HPSO
ABC
MBA
BSA-SAε
BSA
BSAISA
X
1
(
h
)
0.205986
0.205730
0.202369
0.205730
0.205730
0.205729
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
X
3
(
t
)
9.020224
9.036624
9.04821
9.036624
9.036624
9.036626
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
g
1
(
X
)
-
0.074092
-
0.000335
-
12.839796
NA
0.000000
-
0.001614
-
1.5
5
E
-
10
-
5.3
2
E
-
07
0
g
2
(
X
)
-
0.266227
-
0.000753
-
1.247467
NA
-
0.000002
-
0.016911
-
4.3
0
E
-
09
-
9.0
2
E
-
06
0
g
3
(
X
)
-
4.9
5
E
-
04
-
0.000000
-
1.4
9
E
-
03
NA
0.000000
-
2.4
0
E
-
07
-
1.5
5
E
-
15
-
7.8
6
E
-
12
-
5.5
5
E
-
17
g
4
(
X
)
-
3.430044
-
3.432984
-
3.429347
NA
-
3.432984
-
3.432982
-
3.4330
-
3.432984
-
3.432984
g
5
(
X
)
-
0.080986
-
0.080730
-
0.079381
NA
-
0.080730
-
0.080729
-
8.0
7
E
-
02
-
0.080730
-
0.080730
g
6
(
X
)
-
0.235514
-
0.235540
-
0.235536
NA
-
0.235540
-
0.235540
-
0.2355
-
0.235540
-
0.235540
g
7
(
X
)
-
58.666440
-
0.000882
-
11.681355
NA
0.000000
-
0.001464
-
1.8
5
E
-
10
-
1.1
3
E
-
07
-
5.4
6
E
-
12
f
(
X
)
1.728226
1.724852
1.728024
1.724852
1.724852
1.724853
1.724852
1.724852
1.724852Table 15
Comparison of statistical results for the welded beam design problem.
Method
Worst
Mean
Best
Std
FEs
GA-DT
1.993408
1.792654
1.728226
7.4
7
E
-
02
80,000
MDE
1.724854
1.724853
1.724852
NA
24,000
CPSO
1.782143
1.748831
1.728024
1.2
9
E
-
02
24,000
HPSO
1.814295
1.749040
1.724852
4.0
0
E
-
02
81,000
DELC
1.724852
1.724852
1.724852
4.1
E
-
13
20,000
PSO-DE
1.724852
1.724852
1.724852
6.7
E
-
16
66,600
ABC
NA
1.741913
1.724852
3.1
E
-
02
30,000
MBA
1.724853
1.724853
1.724853
6.9
4
E
-
19
47,340
SSOC
1.799332
1.746462
1.724852
2.5
7
E
-
2
25,000
BSA-SAε
1.724852
1.724852
1.724852
8.1
1
E
-
10
80,000
BSA
1.724854
1.724852
1.724852
2.3
5
E
-
07
45,480
BSAISA
1.724854
1.724852
1.724852
2.9
6
E
-
07
29,000From Tables8 and 9, BSAISA, BSA, DEDS, HEAA, PSO-DE, and DELC all reach the best solution with the corresponding function value equal to 263.895843 except MBA with 263.895852. However, BSAISA requires the lowest FEs (only 8940) among all algorithms. Its Std value is better than BSA, DEDS, HEAA, PSO-DE, and MBA except DELC. These comparative results indicate that BSAISA outperforms other algorithms in terms of computational cost and robustness for this problem.Figure7 depicts the convergence curves of BSAISA and BSA for the three-bar truss design problem, where the value of F(x∗) on the vertical axis equals 263.895843. As shown in Figure 7, BSA achieves the global optimum at about 700 iterations, while BSAISA only reaches the global optimum at about 400 iterations. It can be concluded that the convergence speed of BSAISA is faster than that of BSA for this problem.Figure 7
Convergence curves of BSAISA and BSA for the three-bar truss design problem.


## 4.3.2. Pressure Vessel Design Problem (PVP)

The pressure vessel design problem has a nonlinear objective function with three linear and one nonlinear inequality constraints and two discrete and two continuous design variables. The values of the two discrete variables (x1,x2) should be the integer multiples of 0.0625. The best feasible solution is obtained by BSAISA at x = (0.8750, 0.4375, 42.0984, 176.6366) with the objective function value f(x) = 6059.7143 using 31,960 FEs.For this problem, BSAISA is compared with nine algorithms: BSA, BSA-SAε [62], DELC, POS-DE, genetic algorithms based on dominance tournament selection (GA-DT) [73], modified differential evolution (MDE) [74], coevolutionary particle swarm optimization (CPSO) [75], hybrid particle swarm optimization (HPSO) [76], and artificial bee colony algorithm (ABC) [77]. The comparison of the best solutions obtained by BSAISA and other reported algorithms is presented in Table 10. The statistical results of various algorithms are listed in Table 11.As shown Table10, the obtained solution sets of all algorithms satisfy the constraints for this problem. BSAISA, BSA-SAε, ABC, DELC, and HPSO find the same considerable good objective function value 6059.7143, which is slightly worse than MDE’s function value 6059.7143. It is worth mentioning that MBA’s best solution was obtained at x = (0.7802, 0.3856, 40.4292, 198.4964) with f(x) = 5889.3216 and the corresponding constraint values equal to gi(x) = (0, 0, −86.3645, −41.5035) in [78]. Though MBA finds a far better function value than that of MDE, its obtained variables (i.e., 0.7802 and 0.3856) are not integer multiples of 0.0625. So they are not listed in Table 10 to ensure a fair comparison. From Table 11, except for MDE with the function value of 6059.7016, BSAISA offers better function value results compared to GA-DT, CPSO, ABC, and BSA. Besides that, BSAISA is far superior to other algorithms in terms of FEs. Unfortunately, the obtained Std value of BSAISA is relatively poor compared with others for this problem.Figure8 describes the convergence curves of BSAISA and BSA for the pressure vessel design problem, where the value of F(x∗) on the vertical axis equals 6059.7143. As shown in Figure 8, BSAISA is able to find the global optimum at about 800 iterations and obtains a far more accurate function value than that of BSA. Moreover, the convergence speed of BSAISA is much faster than that of BSA.Figure 8
Convergence curves of BSAISA and BSA for the pressure vessel design problem.


## 4.3.3. Tension Compression Spring Design Problem (TCSP)

This design optimization problem has three continuous variables and four nonlinear inequality constraints. The best feasible solution is obtained by BSAISA atx = (0.051687, 0.356669, 11.291824) with f(x) = 0.012665 using 9440 FEs. This problem has been solved by other methods as follows: GA-DT, MDE, CPSO, HPSO, DEDS, HEAA, DELC, POS-DE, ABC, MBA, BSA-SAε, and Social Spider Optimization (SSOC) [79]. The comparison of the best solutions obtained from various algorithms is presented in Table 12. Their statistical results are listed in Table 13.From Tables12 and 13, the vast majority of algorithms can find the best function value 0.012665 for this problem, while GA-DT and CPSO fail to find it. With regard to the computational cost (FEs), BSAISA only requires 9440 FEs when it reaches the global optimum, which is superior to all other algorithms except MBA with 7650 FEs. However, the Worst and Mean and Std values of BSAISA are better than those of MBA. Consequently, for this problem, it can be concluded that BSA has the obvious superiority in terms of FEs over all other algorithms except MBA. Moreover, BSAISA has a stronger robustness when compared with MBA alone.Figure9 depicts the convergence curves of BSAISA and BSA for the tension compression spring design problem, where the value of F(x∗) on the vertical axis equals 0.012665. From Figure 9 it can be observed that both BSAISA and BSA fall into a local optimum in the early iterations but they are able to successfully escape from the local optimum. However, the convergence speed of BSAISA is obviously faster than that of BSA.Figure 9
Convergence curves of BSAISA and BSA for the tension compression spring design problem.


## 4.3.4. Welded Beam Design Problem (WBP)

The welded beam problem is a minimum cost problem with four continuous design variables and subject to two linear and five nonlinear inequality constraints. The best feasible solution is obtained by BSAISA atx = (0.205730, 3.470489, 9.036624, 0.205730) with the objective function value f(x) = 1.724852 using 29,000 FEs.For this problem, BSAISA is compared with many well-known algorithms as follows: GA-DT, MDE, CPSO, HPSO, DELC, POS-DE, ABC, MBA, BSA, BSA-SAε, and SSOC. The best solutions obtained from BSAISA and other well-known algorithms are listed in Table 14. The comparison of their statistical results is presented in Table 15.From Tables14 and 15, except that the constraint value of PSO is not available, the obtained solution sets of all algorithms satisfy the constraints for the problem. Most of algorithms including BSAISA, BSA, BSA-SAε, MDE, HPSO, DELC, POS-DE, ABC, and SSOC are able to find the best function value 1.724852, while GA-DT and CPSO and MBA fail to find it. It should be admitted that DELC is superior to all other algorithms in terms of FEs and robustness for this problem. On the other hand, except for DELC, MDE, and SSOC with FEs of 2000, 2400, and 2500, respectively, BSAISA requires fewer FEs than the remaining algorithms (excluding algorithms that do not reach the best solution). When considering the comparison of the Std values for this problem, MBA exhibits its powerful robustness and BSAISA performs better than most algorithms except MBA, DELC, PSO-DE, BSA, and BSA-SAε.It is worth mentioning that from [74] the Worst, Mean, Best, and Std value of MDE are given as 1.724854, 1.724853, 1.724852, and 1.0E-15, respectively. However, the corresponding values of DELC equal 1.724852, 1.724852, 1.724852, and 4.1E-13, respectively, where its Worst and Mean values are smaller than those of MDE while its Std is bigger than those of MDE. So we consider that the Std of MDE is probably an error data for this problem, and we replace it with NA in Table 15.Figure10 depicts the convergence curves of BSAISA and BSA for the welded beam design problem, where the value of F(x∗) on the vertical axis equals 1.724852. Figure 10 shows the convergence speed of BSAISA is faster than that of BSA remarkably.Figure 10
Convergence curves of BSAISA and BSA for the welded beam design problem.


## 4.3.5. Speed Reducer Design Problem (SRP)

This speed reducer design problem has eleven constraints and six continuous design variables (x1,x2,x4,x5,x6,x7) and one integer variable (x3). The best solution obtained from BSAISA is x = (3.500000, 0.7000000, 17, 7.300000, 7.715320, 3.350215, 5.286654) with f(x)=2994.471066 using 15,860 FEs. The comparison of the best solutions obtained by BSAISA and other well-known algorithms is given in Table 16. The statistical results of BSAISA, BSA, MDE, DEDS, HEAA, DELC, POS-DE, ABC, MBA, and SSOC are listed in Table 17.Table 16
Comparison of best solutions for the speed reducer design problem.
Method
MDE
DEDS
DELC
HEAA
POS-DE
MBA
BSA
BSAISA
X
1
3.500010
3.500000
3.500000
3.500023
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
X
3
17
17
17
17.000013
17.000000
17.000000
17
17
X
4
7.300156
7.300000
7.300000
7.300428
7.300000
7.300033
7.300000
7.300000
X
5
7.800027
7.715320
7.715320
7.715377
7.800000
7.715772
7.715320
7.715320
X
6
3.350221
3.350215
3.350215
3.350231
3.350215
3.350218
3.350215
3.350215
X
7
5.286685
5.286654
5.286654
5.286664
5.286683
5.286654
5.286654
5.286654
f
(
X
)
2996.356689
2994.471066
2994.471066
2994.499107
2996.348167
2994.482453
2994.471066
2994.471066Table 17
Comparison of statistical results for the speed reducer design problem.
Method
Worst
Mean
Best
Std
FEs
MDE
2996.390137
2996.367220
2996.356689
8.2
E
-
03
24,000
DEDS
2994.471066
2994.471066
2994.471066
3.5
8
E
-
12
30,000
HEAA
2994.752311
2994.613368
2994.499107
7.0
E
-
02
40,000
DELC
2994.471066
2994.471066
2994.471066
1.9
E
-
12
30,000
PSO-DE
2996.348204
2996.348174
2996.348167
6.4
E
-
06
54,350
ABC
NA
2997.058412
2997.058412
0
30,000
MBA
2999.652444
2996.769019
2994.482453
1.56
6300
SSOC
2996.113298
2996.113298
2996.113298
1.3
4
E
-
12
25,000
BSA
2994.471066
2994.471066
2994.471066
9.8
7
E
-
11
25,640
BSAISA
2994.471095
2994.471067
2994.471066
5.4
0
E
-
06
15,860As shown in Tables16 and 17, the obtained solution sets of all algorithms satisfy the constraints for this problem. BSAISA, BSA, DEDS, and DELC are able to find the best function value 2994.471066 while the others do not. Among the four algorithms, DEDS, DELC, and BSA require 30000, 30000, and 25640 FEs, respectively. However, BSAISA requires only 15,860 FEs when it reaches the same best function value. MBA fails to find the best known function value; thus BSAISA is better than MBA in this problem, even though MBA has lower FEs. As for the comparison of the Std, among the four algorithms that achieve the best known function value, BSAISA is worse than the others. However, one thing that should be mentioned is that the main purpose of the experiment is to compare the convergence speed between BSAISA and other algorithms. From this point of view, it can be concluded that BSAISA has a better performance than other algorithms in terms of convergence speed.Figure11 depicts the convergence curves of BSAISA and BSA for the speed reducer design problem, where the value of F(x∗) on the vertical axis equals 2994.471066. Figure 11 shows that the convergence speed of BSAISA is faster than that of BSA.Figure 11
Convergence curves of BSAISA and BSA for the speed reducer design problem.


## 4.4. Comparisons Using Sign Test

Sign Test [80] is one of the most popular statistical methods used to determine whether two algorithms are significantly different. Recently, Miao et al. [81] utilized Sign Test method to analyze the performances between their proposed modified algorithm and the original one. In this paper, the two-tailed Sign Test with a significance level 0.05 is adopted to test the significant differences between the results obtained by different algorithms, and the test results are given in Table 18. The values of Best and FEs are two most important criterions for the evaluations of algorithms in our paper; they therefore should be chosen as the objectives of the Sign Test. The signs “+,” “≈,” and “−” represent, respectively, the fact that our BSAISA performs significantly better than, almost the same as, or significantly worse than the algorithm it is compared to. The null hypothesis herein is that the performances between BSAISA and one of the others are not significantly differential.Table 18
Comparisons between BSAISA and other algorithms in Sign Tests.
BSAISA-methods
+
≈
-
Total
p value
SR
11
1
1
13
0.006
FSA
12
0
1
13
0.003
CDE
8
0
5
13
0.581
AMA
13
0
0
13
0.000
MABC
10
2
1
13
0.012
RPGA
11
0
0
11
0.001
BSA-SAε
8
4
1
13
0.039
Note. The columns of “+,” “≈,” and “−” indicate the number of functions where BSAISA performs significantly better than, almost the same as, or significantly worse than the compared algorithm, respectively. “p value” denotes the probability value supporting the null hypothesis.As shown in Table18, the p values of supporting the null hypothesis of Sign Test for six pairs of algorithms (BSAISA-SR, BSAISA-FSA, BSAISA-AMA, BSAISA-MABC, BSAISA-RPGA, and BSAISA-BSA-SAε) are 0.006, 0.003, 0.000, 0.012, 0.001, and 0.039, respectively, and thereby we can reject the null hypothesis. This illustrates that the optimization performance of the proposed BSAISA is significantly better than those of the six algorithms. The p value of BSAISA-CDE is equal to 0.581, which shows that we cannot reject the null hypothesis. However, according to the related sign values (“+,” “≈,” and “−”) from Table 18, BSAISA is slightly worse than CDE on 5 problems but wins on another 8 problems, which illustrates that the proposed BSAISA has a relatively excellent competitiveness compared with the CDE. Generally, the statistical p values and sign values validate that BSAISA has the superiority compared to the other well-known algorithms on the constrained optimization problems.On the one hand, all experimental results suggest that the proposed method improves the convergence speed of BSA. On the other hand, the overall comparative results of BSAISA and other well-known algorithms demonstrate that BSAISA is more effective and competitive for constrained and engineering optimization problems in terms of convergence speed.


## 5. Conclusions and Future Work

In this paper, we proposed a modified version of BSA inspired by the Metropolis criterion in SA (BSAISA). The Metropolis criterion may probabilistically accept a higher energy state and the acceptance probability can decrease as the temperature decreases, which motivated us to redesign the amplitude control factorF so it can adaptively decrease as the number of iterations increases. The design principle and numerical analysis of the redesigned F indicate that the change in F could accelerate the convergence speed of the algorithm by improving the local exploitation capability. Furthermore, the redesigned F does not introduce extra parameters. We successfully implemented BSAISA to solve some constrained optimization and engineering design problems. The experimental results demonstrated that BSAISA has a faster convergence speed than BSA and it can efficiently balance the capacity for global exploration and local exploitation. The comparisons of the results obtained by BSAISA and other well-known algorithms demonstrated that BSAISA is more effective and competitive for constrained and engineering optimization problems in terms of convergence speed.This paper suggests that the proposed BSAISA has a superiority in terms of convergence speed or computational cost. The downside of the proposed algorithm is, of course, that its robustness does not show enough superiority. So our future work is to further research into the robustness of BSAISA on the basis of current research. Niche technique is able to effectively maintain population diversity of evolutionary algorithms [82, 83]. How to combine BSAISA with niche technology to improve robustness of the algorithm may deserve to be studied in the future.



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