# Application of the Improved PSO-Based Extended Domain Method in Engineering

Authors: Bin Bai; Zhi-wei Guo; Qi-liang Wu; Junyi Zhang; Yan-chao Cui
Journal: Mathematical Problems in Engineering
 (2020)

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## Abstract

The standard particle swarm optimization (PSO) algorithm is the boundary constraints of simple variables, which can hardly be directly applied in the constrained optimization. Furthermore, the standard PSO algorithm often fails to obtain the global optimal solution when the dimensionality is high for unconstrained optimization. Thus, an improved PSO-based extended domain method (IPSO-EDM) is proposed to solve engineering optimization problems. The core idea of this method is that the original feasible region is expanded in the constrained optimization which is transformed into the unconstrained optimization by combining the ergodicity of chaos optimization and the evolutionary variation to realize global search. In addition, to verify the effectiveness of the IPSO-EDM, an unconstrained optimization case study, four constrained optimization case studies, and one engineering example are investigated. The results indicate that the computational accuracy of the IPSO-EDM is comparable to that provided by the existing literature, and the computational efficiency of the IPSO-EDM is significantly improved. Meanwhile, this method has conspicuous global search ability and stability in engineering optimization.


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## Body

## 1. Introduction

Optimization began in the 17th century, which originated from differential and integral calculus invented by Newton and Leibnitz. Then, optimization algorithms [1–5] were rapidly developed, such as artificial neural network, simulated annealing, genetic algorithm, ant colony optimization, and particle swarm optimization (PSO). All these methods were widely used in different fields [6–12], such as chemical engineering, biomedicine, navigation, robot, automobile, architecture, and aerospace.Actually, the mechanical engineering optimization can be expressed as continuity interval constraint optimization. To investigate this problem, some traditional gradient methodologies [13–15] were investigated such as the penalty function and Lagrange multiplier. Although the theory of these methods is impeccable, the objective function and the constraint condition must be differentiable. However, the constraint function and objective function are nondifferentiable and discontinuous implicit functions in practical engineering. Consequently, a new optimization method was developed to study this problem, which plays an important role for the development of engineering optimization design.Initially, the PSO was presented by Kennedy and Eberhart [16] to investigate the flight behaviour of birds, which was termed as the global PSO algorithm. This method has been extended to different kinds of fields. For instance, Xue et al. [17] used analytical method with modified PSO to establish the subdomain model and optimize cogging torque. Han et al. [18] adopted an adaptive gradient multiobjective PSO to improve the computational performance for a mechanism. Yi et al. [19] presented a parallel chaotic local search algorithm to solve constrained engineering design problems. Park et al. [20] used chaotic sequences with conventional linearly decreasing inertia weights to increase the exploitation capability. However, the essence of these methods is not improved, thus some scholars renewed the equation of the global PSO. For instance, Phung et al. [21] proposed a discrete PSO algorithm to solve the extended travelling salesman problem for robotic inspection. Wang and Zhang [22] presented an optimization method to describe a planar parallel 3-DOF nano positioner in modelling. Nickabadi et al. [23] successfully regarded the adaptive inertia weight factor as the feedback parameter to ascertain the situation of the particles in the search space. Mojarrad and Nayeripour [24] used a fuzzy adaptive PSO to solve the non-convex economic dispatch problems. Khan et al. [25] proposed a modified PSO algorithm to avoid the premature convergence and strengthen its robustness. This algorithm can adaptively update parameters to keep the diversity of the swarm. However, the convergence speed of inertia weight method is not very fast, so another method is developed. Liang et al. [26] proposed a fuzzy multilevel algorithm-based PSO to optimize support vector regression machine and realize a fuzzy multilevel drilling leak risk evaluation system. Tian et al. [27] utilized sigmoid-based acceleration coefficients to avoid premature convergence and entrapment into local optima when they handled complex multimodal problems. Hsieh et al. [28] developed a discrete cooperative coevolving PSO algorithm to study the influence of detour distance constraints on the carpooling performance. Zahara et al. [29, 30] put forward the hybrid Nelder-Mead-PSO algorithm for unconstrained optimization, and then they presented embedded constraint handling methods for dealing with constraints. Liu et al. [31] adopted a bottleneck objective learning strategy for many-objective optimization to improve convergence on all objectives. Xia et al. [32] proposed a triple archives PSO to deal with the selecting proper exemplars and designing an efficient learning model. Wang et al. [33] investigated the evolutionary computation community for large-scale optimization.The above investigations on the PSO algorithm indicate that some constrained points are not located in the feasible region but outside the feasible region; however, these excluded constrained points are closer to the boundary than the points located in the feasible region. Obviously, this is unreasonable. Based on the above research studies, a new methodology named improved particle swarm optimization-based extended domain method (IPSO-EDM) is proposed to investigate engineering optimization.In the following,Section 2 describes the basic theory of the IPSO-EDM, including standard PSO, improved PSO, and the algorithm principle. Section3 gives an unconstrained optimization case study, four numerical constrained optimization case studies, and one engineering case study to verify the effectiveness of the IPSO-EDM. Section 4 gives the conclusions.


## 2. Improved Particle Swarm Optimization

### 2.1. Standard PSO

Assuming that the particle swarm is composed ofM particles in an N-dimensional space, the position of the ith particle of the kth iteration is expressed as Xik=xi1k,xi2k,…,xiNkT, the flight speed is described as Vik=vi1k,vi2k,…,viNkT, the local optimal position is Pi=pi1,pi2,…,piNT, and the global optimal location is Pg=pg1,pg2,…,pgNT. Each particle updates its speed and position according to the following equation, which are expressed as(1)vijk+1=vijk+c1r1j×pijk−xijk+c2r2j×pgjk−xijk,xijk+1=xijk+vijk+1,where j is the jth component of the ith particle, c1andc2 are positive constants and they are called learning factor, c1 adjusts the step length for particles to their optimal location, and c2 adjusts the step length for particles to the global optimal position, and r1j and r2j are, respectively, random numbers that obey a uniform distribution, and their values are within 0,1.To prevent particles from flying out of the search space in the optimization, the velocity and position are limited asvij∈vjmin,vjmax and xij∈xjmin,xjmax. Meanwhile, the inertia weight w is involved. This method is termed as the standard PSO algorithm. The update equations are expressed as(2)vijk+1=wk×vijk+c1r1j×pijk−xijk+c2r2j×pgjk−xijk,xijk+1=xijk+vijk+1.The standard PSO does not have too many requirements on the objective function and constraint function, which can conduct constraint optimization, and these constraints limit the range of each variable interval value such asxj∈xjmin,xjmax. This is different from the gradient method, but it becomes more complicated if there are equality or inequality constraints in the optimization. One obvious reason is that the feasible region has changed from a hypercube to a less regular region, and the variables are not independent on each other. For this reason, the standard PSO must be improved to deal with the constraints.


### 2.2. Improved PSO

#### 2.2.1. Constraint Methods of PSO

At present, there are four typical methods to deal with constraints:(1)
Discriminant function method: inequality and equality constraints are used as discriminant functions to determine whether the search points are feasible points in the optimization process. It will be discarded or modified to be a feasible point during search if the search point is not a feasible point. Thus, this method has strict restrictions to the search point, and it is very difficult to generate the initial feasible point when the feasible region composed by equality and inequality constraints is small.(2)
Penalty function method: the optimization and constraint functions are combined to form the penalty function. The original constrained optimization with equality and inequality constraints has become an unconstrained optimization with the penalty function. However, the disadvantage of this method is that the penalty factor must be chosen correctly; otherwise, it can hardly obtain the optimal solution.(3)
Multiobjective optimization method: optimization and constraint functions are, respectively, used as the new optimization targets. However, solving the multiobjective optimization is more difficult than solving a single-objective optimization in many cases.(4)
“Competitive selection” method: deals with the feasible particles (the design points represented by particles satisfy all constraint requirements) and infeasible particles (some or all design points represented by particles dissatisfy the optimization constraint requirements) in PSO. However, it is not appropriate to deem that feasible particles are superior to infeasible ones in competitive selection, and the infeasible region can be extended into a feasible region to find the optimal point.


#### 2.2.2. Core Idea of the “Competitive Selection” Constraint

The “competitive selection” constraint can be summarized in three aspects:(1)
All feasible particles are superior to infeasible particles(2)
The particle with a better objective function is selected for two feasible particles(3)
The advantages and disadvantages of the particle are judged according to the degree of constraint violation for two infeasible particles; the lesser the degree of the constraint, the larger the violationIn Figure1(a), unconstrained optimal point A is in the feasible region, and the constraint is useless to the whole optimization; essentially, the constrained optimal point is the unconstrained optimal point. However, the unconstrained optimal point is not located in the feasible region but at its boundary for some optimization. For example, point B is the constrained optimal point, point C is in the feasible region, and point D is outside the feasible region in Figure 1(b). According to the competitive selection, point C is superior to D. However, point D is closer to B than C; thus, the optimal information provided by D is superior to C. As a result, point D is labelled as suboptimal, which is not appropriate. So, the algorithm must be improved. Figure 1(c) shows an improved method, which expands the original feasible region to include points such as D which is unfeasible but close to the feasible region and can provide better function information and is taken as the feasible point. This methodology is termed as extended domain method (EDM).Figure 1
Relationship among the optimal point, feasible domain, and extended domain: (a) optimal point in the feasible region; (b) optimal point outside the feasible region; (c) optimal point in the extended domain.
(a)(b)(c)In addition, the new speed depends on the current speed in the updated formula for standard PSO. However, the algorithm is random, and it is impossible to predict and control the size of particle speed. To solve this problem, a control variableξ is introduced in the extended domain, and the result is shown in Figure 2,where x∗ is the optimal location, x is the current particle position, pg is the optimal historical position of the particle swarm, px is the optimal historical position of the current particle, x^ is the optimal location of the current particle swarm, and vl and vs are, respectively, the largest and smallest speeds.Figure 2
Renewal schematic diagram of the particle position.Figure2 shows that the particle with higher speed may miss a better position, while the particle with smaller speed improves the position, but it is not the optimal position. Thus, the current particle speed is expected to be controlled. One method is to use current location information of the particle swarm to determine its speed. The difference value between the particle swarm and the particle in the current optimal position is used to determine current particle velocity. The results indicate that the position xx^ determined using this method is better than the positions xl and xs. In fact, the constrained optimization is transformed into an unconstrained optimization by the EDM, so a scientific and reasonable unconstrained optimization method needs to be developed.


#### 2.2.3. Unconstrained PSO

Firstly, the initial particle swarm position and velocity are determined in a random way in standard PSO. Generally, the designers expect the particles in the particle swarm space can better reflect the information which is studied in the initial state. One of the direct ways is to generate many particles which fill in the whole search space. However, this will consume a lot of computational resources in subsequent iterations. The PSO algorithm will lose its characteristics of group cooperation if the number of particles in the particle swarm space is too small, which is meaningless. Usually, the particle swarm with dozens of particles can solve the complex optimization problem, but random arrangements of particles often result in a “cluster” in an area. To enable the particles to be relatively “evenly” distributed in the search space, the uniform design method based on statistical theory is used to initialize the particle swarm space. The random distribution and uniform distribution of the initial particle swarm space are shown in Figure3.Figure 3
Distribution diagram of the initial particle swarm space: (a) random distribution; (b) uniform distribution.
(a)(b)Secondly, the standard PSO algorithm itself cannot obtain global information of the objective function, and it is easy to fall into the local optimal. Generally, the particle can jump out of the local optimal and give new global information by dynamically adjusting the inertia factor, but simply adjusting the inertia factor is not enough for the complex multipeak function. To handle this problem, the ergodicity of chaos optimization [22] is combined with evolutionary variation to realize global search. In this method, logistic chaotic system equation is applied in the PSO algorithm and obeys Chebyshev distribution, as shown in Figure 4.Figure 4
Probability density curve of the logistic chaotic sequence.Figure4 indicates that the middle value of the logistic chaotic sequence is relatively uniform, while the probability of both sides is relatively large. It means that the chance of finding the global optimal point will be reduced using the logistic chaotic sequence if the global optimal point is not at ends of the design variable. Therefore, it is very necessary to find a chaotic sequence, which can not only maintain ergodicity but also keep uniform statistical distribution. According to this problem, evolutionary variation strategy is developed. Generally, this strategy introduces a mutation operator to change the design variable. This method can help the particle swarm escape local optimal and maintain its overall vitality and prevent the particle swarm from falling into the condition of “precocity” at the earlier iteration. Based on the above research, the idea of uniform design, chaos optimization, and evolutionary variation is introduced into unconstrained PSO, and this method is improved to realize global search and local search.Firstly, the uniform design of the Halton sequence [34] is adopted to initialize the particle swarm. Assume that the particle position scope of the jth design variable is xjmin,xjmax, whose component values of the ith particle are hij, and the position xij of the initialized particle swarm is expressed as(3)xij0=xjmin+xjmax−xjmin×hij.Similarly, the velocityvij for the jth component values of the ith particle is expressed as(4)vij0=vjmin+vjmax−vjmin×hij.In the process of particle swarm evolution, the points around the optimal position still need to be searched to get a better position when the optimal position of the particle swarm is found. This is called as the chaos optimization search method, which is expressed as(5)zl+1=1−μ×zl2.However, the chaotic sequence generated by equation (5) is not uniform in statistics; therefore, the chaotic sequence needs to be transformed as follows:(6)tl=arccoszlπ.Equation (6) not only satisfies the ergodicity of the chaotic sequence but also complies with the uniform distribution in 0,1. The uniform logistic chaotic sequence and original logistic chaotic sequence of the frequency curve and ergodic graph are shown in Figures 5 and 6. The point set of the chaotic sequence is 1e4.Figure 5
Uniform logistic chaotic sequence: (a) frequency curve; (b) ergodic graph.
(a)(b)Figure 6
Original logistic chaotic sequence: (a) frequency curve; (b) ergodic graph.
(a)(b)The optimal position of the particle swarm is denoted asPgk after the kth evolution, and the ith component is denoted as pgik; then, the initial value of the chaotic sequence is expressed as(7)zi0=pgik−ximinximax−ximin.According to equations (5) and (6), the chaotic sequence zil,l=1,2,…,m, is generated, and its coordinate value corresponding to the original space is obtained via reverse transformation, which is expressed as(8)p¯il=ximin+ximax−ximin×zil,where l is the number of iterations of the chaotic sequence and k is the number of iterations.However, the chaos optimization can hardly make particle swarm get rid of local optimal, and the evolutionary variation is used to help it jump out local optimal. The mutation operator plays a key role in evolutionary variation. The Gaussian operator and Cauchy operator are two commonly used mutation operators. The global searching ability of the Cauchy mutation is stronger than that of the Gaussian mutation, but Cauchy mutation may produce large stride length in search, which means its local search ability is not as good as that of the Gaussian mutation. Therefore, according to the characteristics of Cauchy variation and Gaussian variation, the “coarse tune” of the particle swarm is obtained through the Cauchy mutation at first, and then its “fine tune” is obtained through the Gaussian mutation. The mutation formula is described as(9)zk∗=zk+βg−1×r,where zk and zk∗ represent the value of the kth design variable before and after mutation, respectively; r is a random number; β is the contraction coefficient and β=0.1; and g is the number of mutations.The design variable value can hardly exceed its interval after mutation because of its range limitation in optimization process. The design variable value will be mutated again when its value is not in the definition domain after mutation until its solution is convergence. The maximum number of mutations is stipulated asq when the mutation value still does not satisfy the value range in the procedure. Then, the mutation can be judged according to the absolute value of the difference between the mutation value and interval endpoint value, namely, the mutation value is the left endpoint value and vice versa if the variation value is close to the left endpoint.


#### 2.2.4. Algorithm Principle

Generally, the optimization model is described as(10)minfxs.tgix≤0,i=1,2,…,Nhjx=0,j=1,2,…,Mxk∈xkmin,xkmax,k=1,2,…,n,where f⋅ is the objective function; g⋅ is the inequality constraint function; and h⋅ is the equality constraint function.To divide the particle swarm space into the feasible domain and unfeasible domain, a constraint conflict functionViox is constructed, which is defined as(11)Viox=∑i=1Nmaxgix,0+∑j=1Mhjx.According to equation (11), an arbitrary feasible point satisfies Viox=0, and an arbitrary unfeasible point satisfies Viox>0; meanwhile, Viox can also describe the degree of constraint violation of the unfeasible point.To control the size of the extended domain and particle speed, the control variableξ is defined in the following four situations for any two given search points x1 and x2:(1)
x1 is superior to x2 when x1 and x2 are both in the extended domain, i.e., Viox1<ξ,Viox2<ξ and fx1<fx2(2)
x1 is superior to x2 when Viox1=Viox2 and fx1<fx2(3)
x1 is superior to x2 when x1 and x2 are both outside the extended domain, i.e., Viox1>ξ, Viox2>ξ and Viox1<Viox2(4)
x1 is superior to x2 when x1 is in the extended domain, Viox1<ξ, and x2 is outside the extended domain, Viox2>ξThe control variableξ of the extended domain is gradually changing with particle swarm evolution, and the strategy is expressed as(12)ξk=ξ0×exp−β×k,where k is particle swarm evolution algebra and ξ0 is the initial control variable of the extended domain, which is defined as(13)ξ0=121N∑i=1NVioxi+minx∈x0Viox,where x0 is the set composed by the initial particle swarm.Whenx0 evolves to the kth generation, stipulating ξK=ξK, β can be expressed as(14)β=logξ0/ξKK.The control variableξ can be expressed as(15)ξk=ξ0×exp−β×k,1≤k≤K,0,K≤k≤kmax.To make the particle swarm fast gather to the optimal point in the direction, its update strategy is written as(16)vijk+1=wk×x^jk−xijk×sgnvij+r×pijk−xijk+1−r×pgjk−xijk,xijk+1=xijk+vijk+1,where x^ is the optimal position of particles in the current generation; x^jk−xijk controls step length; and sgnvij determines the motion direction of the particle.The following limit policy is adopted when the particle swarm crosses the border in the process of updating, which is denoted as(17)xijk=x¯jk+r×xjmin−x¯jk,xijk<xjmin,xijk=x¯jk+r×xjmax−x¯jk,xijk>xjmax,where r is the uniform-distributed random number and x¯ is the locational average of the particle swarm.To include points such asD in Figure 1(b), the EDM expands the original feasible region and provides preferable information, which is more reasonable.


## 2.1. Standard PSO

Assuming that the particle swarm is composed ofM particles in an N-dimensional space, the position of the ith particle of the kth iteration is expressed as Xik=xi1k,xi2k,…,xiNkT, the flight speed is described as Vik=vi1k,vi2k,…,viNkT, the local optimal position is Pi=pi1,pi2,…,piNT, and the global optimal location is Pg=pg1,pg2,…,pgNT. Each particle updates its speed and position according to the following equation, which are expressed as(1)vijk+1=vijk+c1r1j×pijk−xijk+c2r2j×pgjk−xijk,xijk+1=xijk+vijk+1,where j is the jth component of the ith particle, c1andc2 are positive constants and they are called learning factor, c1 adjusts the step length for particles to their optimal location, and c2 adjusts the step length for particles to the global optimal position, and r1j and r2j are, respectively, random numbers that obey a uniform distribution, and their values are within 0,1.To prevent particles from flying out of the search space in the optimization, the velocity and position are limited asvij∈vjmin,vjmax and xij∈xjmin,xjmax. Meanwhile, the inertia weight w is involved. This method is termed as the standard PSO algorithm. The update equations are expressed as(2)vijk+1=wk×vijk+c1r1j×pijk−xijk+c2r2j×pgjk−xijk,xijk+1=xijk+vijk+1.The standard PSO does not have too many requirements on the objective function and constraint function, which can conduct constraint optimization, and these constraints limit the range of each variable interval value such asxj∈xjmin,xjmax. This is different from the gradient method, but it becomes more complicated if there are equality or inequality constraints in the optimization. One obvious reason is that the feasible region has changed from a hypercube to a less regular region, and the variables are not independent on each other. For this reason, the standard PSO must be improved to deal with the constraints.


## 2.2. Improved PSO

### 2.2.1. Constraint Methods of PSO

At present, there are four typical methods to deal with constraints:(1)
Discriminant function method: inequality and equality constraints are used as discriminant functions to determine whether the search points are feasible points in the optimization process. It will be discarded or modified to be a feasible point during search if the search point is not a feasible point. Thus, this method has strict restrictions to the search point, and it is very difficult to generate the initial feasible point when the feasible region composed by equality and inequality constraints is small.(2)
Penalty function method: the optimization and constraint functions are combined to form the penalty function. The original constrained optimization with equality and inequality constraints has become an unconstrained optimization with the penalty function. However, the disadvantage of this method is that the penalty factor must be chosen correctly; otherwise, it can hardly obtain the optimal solution.(3)
Multiobjective optimization method: optimization and constraint functions are, respectively, used as the new optimization targets. However, solving the multiobjective optimization is more difficult than solving a single-objective optimization in many cases.(4)
“Competitive selection” method: deals with the feasible particles (the design points represented by particles satisfy all constraint requirements) and infeasible particles (some or all design points represented by particles dissatisfy the optimization constraint requirements) in PSO. However, it is not appropriate to deem that feasible particles are superior to infeasible ones in competitive selection, and the infeasible region can be extended into a feasible region to find the optimal point.


### 2.2.2. Core Idea of the “Competitive Selection” Constraint

The “competitive selection” constraint can be summarized in three aspects:(1)
All feasible particles are superior to infeasible particles(2)
The particle with a better objective function is selected for two feasible particles(3)
The advantages and disadvantages of the particle are judged according to the degree of constraint violation for two infeasible particles; the lesser the degree of the constraint, the larger the violationIn Figure1(a), unconstrained optimal point A is in the feasible region, and the constraint is useless to the whole optimization; essentially, the constrained optimal point is the unconstrained optimal point. However, the unconstrained optimal point is not located in the feasible region but at its boundary for some optimization. For example, point B is the constrained optimal point, point C is in the feasible region, and point D is outside the feasible region in Figure 1(b). According to the competitive selection, point C is superior to D. However, point D is closer to B than C; thus, the optimal information provided by D is superior to C. As a result, point D is labelled as suboptimal, which is not appropriate. So, the algorithm must be improved. Figure 1(c) shows an improved method, which expands the original feasible region to include points such as D which is unfeasible but close to the feasible region and can provide better function information and is taken as the feasible point. This methodology is termed as extended domain method (EDM).Figure 1
Relationship among the optimal point, feasible domain, and extended domain: (a) optimal point in the feasible region; (b) optimal point outside the feasible region; (c) optimal point in the extended domain.
(a)(b)(c)In addition, the new speed depends on the current speed in the updated formula for standard PSO. However, the algorithm is random, and it is impossible to predict and control the size of particle speed. To solve this problem, a control variableξ is introduced in the extended domain, and the result is shown in Figure 2,where x∗ is the optimal location, x is the current particle position, pg is the optimal historical position of the particle swarm, px is the optimal historical position of the current particle, x^ is the optimal location of the current particle swarm, and vl and vs are, respectively, the largest and smallest speeds.Figure 2
Renewal schematic diagram of the particle position.Figure2 shows that the particle with higher speed may miss a better position, while the particle with smaller speed improves the position, but it is not the optimal position. Thus, the current particle speed is expected to be controlled. One method is to use current location information of the particle swarm to determine its speed. The difference value between the particle swarm and the particle in the current optimal position is used to determine current particle velocity. The results indicate that the position xx^ determined using this method is better than the positions xl and xs. In fact, the constrained optimization is transformed into an unconstrained optimization by the EDM, so a scientific and reasonable unconstrained optimization method needs to be developed.


### 2.2.3. Unconstrained PSO

Firstly, the initial particle swarm position and velocity are determined in a random way in standard PSO. Generally, the designers expect the particles in the particle swarm space can better reflect the information which is studied in the initial state. One of the direct ways is to generate many particles which fill in the whole search space. However, this will consume a lot of computational resources in subsequent iterations. The PSO algorithm will lose its characteristics of group cooperation if the number of particles in the particle swarm space is too small, which is meaningless. Usually, the particle swarm with dozens of particles can solve the complex optimization problem, but random arrangements of particles often result in a “cluster” in an area. To enable the particles to be relatively “evenly” distributed in the search space, the uniform design method based on statistical theory is used to initialize the particle swarm space. The random distribution and uniform distribution of the initial particle swarm space are shown in Figure3.Figure 3
Distribution diagram of the initial particle swarm space: (a) random distribution; (b) uniform distribution.
(a)(b)Secondly, the standard PSO algorithm itself cannot obtain global information of the objective function, and it is easy to fall into the local optimal. Generally, the particle can jump out of the local optimal and give new global information by dynamically adjusting the inertia factor, but simply adjusting the inertia factor is not enough for the complex multipeak function. To handle this problem, the ergodicity of chaos optimization [22] is combined with evolutionary variation to realize global search. In this method, logistic chaotic system equation is applied in the PSO algorithm and obeys Chebyshev distribution, as shown in Figure 4.Figure 4
Probability density curve of the logistic chaotic sequence.Figure4 indicates that the middle value of the logistic chaotic sequence is relatively uniform, while the probability of both sides is relatively large. It means that the chance of finding the global optimal point will be reduced using the logistic chaotic sequence if the global optimal point is not at ends of the design variable. Therefore, it is very necessary to find a chaotic sequence, which can not only maintain ergodicity but also keep uniform statistical distribution. According to this problem, evolutionary variation strategy is developed. Generally, this strategy introduces a mutation operator to change the design variable. This method can help the particle swarm escape local optimal and maintain its overall vitality and prevent the particle swarm from falling into the condition of “precocity” at the earlier iteration. Based on the above research, the idea of uniform design, chaos optimization, and evolutionary variation is introduced into unconstrained PSO, and this method is improved to realize global search and local search.Firstly, the uniform design of the Halton sequence [34] is adopted to initialize the particle swarm. Assume that the particle position scope of the jth design variable is xjmin,xjmax, whose component values of the ith particle are hij, and the position xij of the initialized particle swarm is expressed as(3)xij0=xjmin+xjmax−xjmin×hij.Similarly, the velocityvij for the jth component values of the ith particle is expressed as(4)vij0=vjmin+vjmax−vjmin×hij.In the process of particle swarm evolution, the points around the optimal position still need to be searched to get a better position when the optimal position of the particle swarm is found. This is called as the chaos optimization search method, which is expressed as(5)zl+1=1−μ×zl2.However, the chaotic sequence generated by equation (5) is not uniform in statistics; therefore, the chaotic sequence needs to be transformed as follows:(6)tl=arccoszlπ.Equation (6) not only satisfies the ergodicity of the chaotic sequence but also complies with the uniform distribution in 0,1. The uniform logistic chaotic sequence and original logistic chaotic sequence of the frequency curve and ergodic graph are shown in Figures 5 and 6. The point set of the chaotic sequence is 1e4.Figure 5
Uniform logistic chaotic sequence: (a) frequency curve; (b) ergodic graph.
(a)(b)Figure 6
Original logistic chaotic sequence: (a) frequency curve; (b) ergodic graph.
(a)(b)The optimal position of the particle swarm is denoted asPgk after the kth evolution, and the ith component is denoted as pgik; then, the initial value of the chaotic sequence is expressed as(7)zi0=pgik−ximinximax−ximin.According to equations (5) and (6), the chaotic sequence zil,l=1,2,…,m, is generated, and its coordinate value corresponding to the original space is obtained via reverse transformation, which is expressed as(8)p¯il=ximin+ximax−ximin×zil,where l is the number of iterations of the chaotic sequence and k is the number of iterations.However, the chaos optimization can hardly make particle swarm get rid of local optimal, and the evolutionary variation is used to help it jump out local optimal. The mutation operator plays a key role in evolutionary variation. The Gaussian operator and Cauchy operator are two commonly used mutation operators. The global searching ability of the Cauchy mutation is stronger than that of the Gaussian mutation, but Cauchy mutation may produce large stride length in search, which means its local search ability is not as good as that of the Gaussian mutation. Therefore, according to the characteristics of Cauchy variation and Gaussian variation, the “coarse tune” of the particle swarm is obtained through the Cauchy mutation at first, and then its “fine tune” is obtained through the Gaussian mutation. The mutation formula is described as(9)zk∗=zk+βg−1×r,where zk and zk∗ represent the value of the kth design variable before and after mutation, respectively; r is a random number; β is the contraction coefficient and β=0.1; and g is the number of mutations.The design variable value can hardly exceed its interval after mutation because of its range limitation in optimization process. The design variable value will be mutated again when its value is not in the definition domain after mutation until its solution is convergence. The maximum number of mutations is stipulated asq when the mutation value still does not satisfy the value range in the procedure. Then, the mutation can be judged according to the absolute value of the difference between the mutation value and interval endpoint value, namely, the mutation value is the left endpoint value and vice versa if the variation value is close to the left endpoint.


### 2.2.4. Algorithm Principle

Generally, the optimization model is described as(10)minfxs.tgix≤0,i=1,2,…,Nhjx=0,j=1,2,…,Mxk∈xkmin,xkmax,k=1,2,…,n,where f⋅ is the objective function; g⋅ is the inequality constraint function; and h⋅ is the equality constraint function.To divide the particle swarm space into the feasible domain and unfeasible domain, a constraint conflict functionViox is constructed, which is defined as(11)Viox=∑i=1Nmaxgix,0+∑j=1Mhjx.According to equation (11), an arbitrary feasible point satisfies Viox=0, and an arbitrary unfeasible point satisfies Viox>0; meanwhile, Viox can also describe the degree of constraint violation of the unfeasible point.To control the size of the extended domain and particle speed, the control variableξ is defined in the following four situations for any two given search points x1 and x2:(1)
x1 is superior to x2 when x1 and x2 are both in the extended domain, i.e., Viox1<ξ,Viox2<ξ and fx1<fx2(2)
x1 is superior to x2 when Viox1=Viox2 and fx1<fx2(3)
x1 is superior to x2 when x1 and x2 are both outside the extended domain, i.e., Viox1>ξ, Viox2>ξ and Viox1<Viox2(4)
x1 is superior to x2 when x1 is in the extended domain, Viox1<ξ, and x2 is outside the extended domain, Viox2>ξThe control variableξ of the extended domain is gradually changing with particle swarm evolution, and the strategy is expressed as(12)ξk=ξ0×exp−β×k,where k is particle swarm evolution algebra and ξ0 is the initial control variable of the extended domain, which is defined as(13)ξ0=121N∑i=1NVioxi+minx∈x0Viox,where x0 is the set composed by the initial particle swarm.Whenx0 evolves to the kth generation, stipulating ξK=ξK, β can be expressed as(14)β=logξ0/ξKK.The control variableξ can be expressed as(15)ξk=ξ0×exp−β×k,1≤k≤K,0,K≤k≤kmax.To make the particle swarm fast gather to the optimal point in the direction, its update strategy is written as(16)vijk+1=wk×x^jk−xijk×sgnvij+r×pijk−xijk+1−r×pgjk−xijk,xijk+1=xijk+vijk+1,where x^ is the optimal position of particles in the current generation; x^jk−xijk controls step length; and sgnvij determines the motion direction of the particle.The following limit policy is adopted when the particle swarm crosses the border in the process of updating, which is denoted as(17)xijk=x¯jk+r×xjmin−x¯jk,xijk<xjmin,xijk=x¯jk+r×xjmax−x¯jk,xijk>xjmax,where r is the uniform-distributed random number and x¯ is the locational average of the particle swarm.To include points such asD in Figure 1(b), the EDM expands the original feasible region and provides preferable information, which is more reasonable.


## 2.2.1. Constraint Methods of PSO

At present, there are four typical methods to deal with constraints:(1)
Discriminant function method: inequality and equality constraints are used as discriminant functions to determine whether the search points are feasible points in the optimization process. It will be discarded or modified to be a feasible point during search if the search point is not a feasible point. Thus, this method has strict restrictions to the search point, and it is very difficult to generate the initial feasible point when the feasible region composed by equality and inequality constraints is small.(2)
Penalty function method: the optimization and constraint functions are combined to form the penalty function. The original constrained optimization with equality and inequality constraints has become an unconstrained optimization with the penalty function. However, the disadvantage of this method is that the penalty factor must be chosen correctly; otherwise, it can hardly obtain the optimal solution.(3)
Multiobjective optimization method: optimization and constraint functions are, respectively, used as the new optimization targets. However, solving the multiobjective optimization is more difficult than solving a single-objective optimization in many cases.(4)
“Competitive selection” method: deals with the feasible particles (the design points represented by particles satisfy all constraint requirements) and infeasible particles (some or all design points represented by particles dissatisfy the optimization constraint requirements) in PSO. However, it is not appropriate to deem that feasible particles are superior to infeasible ones in competitive selection, and the infeasible region can be extended into a feasible region to find the optimal point.


## 2.2.2. Core Idea of the “Competitive Selection” Constraint

The “competitive selection” constraint can be summarized in three aspects:(1)
All feasible particles are superior to infeasible particles(2)
The particle with a better objective function is selected for two feasible particles(3)
The advantages and disadvantages of the particle are judged according to the degree of constraint violation for two infeasible particles; the lesser the degree of the constraint, the larger the violationIn Figure1(a), unconstrained optimal point A is in the feasible region, and the constraint is useless to the whole optimization; essentially, the constrained optimal point is the unconstrained optimal point. However, the unconstrained optimal point is not located in the feasible region but at its boundary for some optimization. For example, point B is the constrained optimal point, point C is in the feasible region, and point D is outside the feasible region in Figure 1(b). According to the competitive selection, point C is superior to D. However, point D is closer to B than C; thus, the optimal information provided by D is superior to C. As a result, point D is labelled as suboptimal, which is not appropriate. So, the algorithm must be improved. Figure 1(c) shows an improved method, which expands the original feasible region to include points such as D which is unfeasible but close to the feasible region and can provide better function information and is taken as the feasible point. This methodology is termed as extended domain method (EDM).Figure 1
Relationship among the optimal point, feasible domain, and extended domain: (a) optimal point in the feasible region; (b) optimal point outside the feasible region; (c) optimal point in the extended domain.
(a)(b)(c)In addition, the new speed depends on the current speed in the updated formula for standard PSO. However, the algorithm is random, and it is impossible to predict and control the size of particle speed. To solve this problem, a control variableξ is introduced in the extended domain, and the result is shown in Figure 2,where x∗ is the optimal location, x is the current particle position, pg is the optimal historical position of the particle swarm, px is the optimal historical position of the current particle, x^ is the optimal location of the current particle swarm, and vl and vs are, respectively, the largest and smallest speeds.Figure 2
Renewal schematic diagram of the particle position.Figure2 shows that the particle with higher speed may miss a better position, while the particle with smaller speed improves the position, but it is not the optimal position. Thus, the current particle speed is expected to be controlled. One method is to use current location information of the particle swarm to determine its speed. The difference value between the particle swarm and the particle in the current optimal position is used to determine current particle velocity. The results indicate that the position xx^ determined using this method is better than the positions xl and xs. In fact, the constrained optimization is transformed into an unconstrained optimization by the EDM, so a scientific and reasonable unconstrained optimization method needs to be developed.


## 2.2.3. Unconstrained PSO

Firstly, the initial particle swarm position and velocity are determined in a random way in standard PSO. Generally, the designers expect the particles in the particle swarm space can better reflect the information which is studied in the initial state. One of the direct ways is to generate many particles which fill in the whole search space. However, this will consume a lot of computational resources in subsequent iterations. The PSO algorithm will lose its characteristics of group cooperation if the number of particles in the particle swarm space is too small, which is meaningless. Usually, the particle swarm with dozens of particles can solve the complex optimization problem, but random arrangements of particles often result in a “cluster” in an area. To enable the particles to be relatively “evenly” distributed in the search space, the uniform design method based on statistical theory is used to initialize the particle swarm space. The random distribution and uniform distribution of the initial particle swarm space are shown in Figure3.Figure 3
Distribution diagram of the initial particle swarm space: (a) random distribution; (b) uniform distribution.
(a)(b)Secondly, the standard PSO algorithm itself cannot obtain global information of the objective function, and it is easy to fall into the local optimal. Generally, the particle can jump out of the local optimal and give new global information by dynamically adjusting the inertia factor, but simply adjusting the inertia factor is not enough for the complex multipeak function. To handle this problem, the ergodicity of chaos optimization [22] is combined with evolutionary variation to realize global search. In this method, logistic chaotic system equation is applied in the PSO algorithm and obeys Chebyshev distribution, as shown in Figure 4.Figure 4
Probability density curve of the logistic chaotic sequence.Figure4 indicates that the middle value of the logistic chaotic sequence is relatively uniform, while the probability of both sides is relatively large. It means that the chance of finding the global optimal point will be reduced using the logistic chaotic sequence if the global optimal point is not at ends of the design variable. Therefore, it is very necessary to find a chaotic sequence, which can not only maintain ergodicity but also keep uniform statistical distribution. According to this problem, evolutionary variation strategy is developed. Generally, this strategy introduces a mutation operator to change the design variable. This method can help the particle swarm escape local optimal and maintain its overall vitality and prevent the particle swarm from falling into the condition of “precocity” at the earlier iteration. Based on the above research, the idea of uniform design, chaos optimization, and evolutionary variation is introduced into unconstrained PSO, and this method is improved to realize global search and local search.Firstly, the uniform design of the Halton sequence [34] is adopted to initialize the particle swarm. Assume that the particle position scope of the jth design variable is xjmin,xjmax, whose component values of the ith particle are hij, and the position xij of the initialized particle swarm is expressed as(3)xij0=xjmin+xjmax−xjmin×hij.Similarly, the velocityvij for the jth component values of the ith particle is expressed as(4)vij0=vjmin+vjmax−vjmin×hij.In the process of particle swarm evolution, the points around the optimal position still need to be searched to get a better position when the optimal position of the particle swarm is found. This is called as the chaos optimization search method, which is expressed as(5)zl+1=1−μ×zl2.However, the chaotic sequence generated by equation (5) is not uniform in statistics; therefore, the chaotic sequence needs to be transformed as follows:(6)tl=arccoszlπ.Equation (6) not only satisfies the ergodicity of the chaotic sequence but also complies with the uniform distribution in 0,1. The uniform logistic chaotic sequence and original logistic chaotic sequence of the frequency curve and ergodic graph are shown in Figures 5 and 6. The point set of the chaotic sequence is 1e4.Figure 5
Uniform logistic chaotic sequence: (a) frequency curve; (b) ergodic graph.
(a)(b)Figure 6
Original logistic chaotic sequence: (a) frequency curve; (b) ergodic graph.
(a)(b)The optimal position of the particle swarm is denoted asPgk after the kth evolution, and the ith component is denoted as pgik; then, the initial value of the chaotic sequence is expressed as(7)zi0=pgik−ximinximax−ximin.According to equations (5) and (6), the chaotic sequence zil,l=1,2,…,m, is generated, and its coordinate value corresponding to the original space is obtained via reverse transformation, which is expressed as(8)p¯il=ximin+ximax−ximin×zil,where l is the number of iterations of the chaotic sequence and k is the number of iterations.However, the chaos optimization can hardly make particle swarm get rid of local optimal, and the evolutionary variation is used to help it jump out local optimal. The mutation operator plays a key role in evolutionary variation. The Gaussian operator and Cauchy operator are two commonly used mutation operators. The global searching ability of the Cauchy mutation is stronger than that of the Gaussian mutation, but Cauchy mutation may produce large stride length in search, which means its local search ability is not as good as that of the Gaussian mutation. Therefore, according to the characteristics of Cauchy variation and Gaussian variation, the “coarse tune” of the particle swarm is obtained through the Cauchy mutation at first, and then its “fine tune” is obtained through the Gaussian mutation. The mutation formula is described as(9)zk∗=zk+βg−1×r,where zk and zk∗ represent the value of the kth design variable before and after mutation, respectively; r is a random number; β is the contraction coefficient and β=0.1; and g is the number of mutations.The design variable value can hardly exceed its interval after mutation because of its range limitation in optimization process. The design variable value will be mutated again when its value is not in the definition domain after mutation until its solution is convergence. The maximum number of mutations is stipulated asq when the mutation value still does not satisfy the value range in the procedure. Then, the mutation can be judged according to the absolute value of the difference between the mutation value and interval endpoint value, namely, the mutation value is the left endpoint value and vice versa if the variation value is close to the left endpoint.


## 2.2.4. Algorithm Principle

Generally, the optimization model is described as(10)minfxs.tgix≤0,i=1,2,…,Nhjx=0,j=1,2,…,Mxk∈xkmin,xkmax,k=1,2,…,n,where f⋅ is the objective function; g⋅ is the inequality constraint function; and h⋅ is the equality constraint function.To divide the particle swarm space into the feasible domain and unfeasible domain, a constraint conflict functionViox is constructed, which is defined as(11)Viox=∑i=1Nmaxgix,0+∑j=1Mhjx.According to equation (11), an arbitrary feasible point satisfies Viox=0, and an arbitrary unfeasible point satisfies Viox>0; meanwhile, Viox can also describe the degree of constraint violation of the unfeasible point.To control the size of the extended domain and particle speed, the control variableξ is defined in the following four situations for any two given search points x1 and x2:(1)
x1 is superior to x2 when x1 and x2 are both in the extended domain, i.e., Viox1<ξ,Viox2<ξ and fx1<fx2(2)
x1 is superior to x2 when Viox1=Viox2 and fx1<fx2(3)
x1 is superior to x2 when x1 and x2 are both outside the extended domain, i.e., Viox1>ξ, Viox2>ξ and Viox1<Viox2(4)
x1 is superior to x2 when x1 is in the extended domain, Viox1<ξ, and x2 is outside the extended domain, Viox2>ξThe control variableξ of the extended domain is gradually changing with particle swarm evolution, and the strategy is expressed as(12)ξk=ξ0×exp−β×k,where k is particle swarm evolution algebra and ξ0 is the initial control variable of the extended domain, which is defined as(13)ξ0=121N∑i=1NVioxi+minx∈x0Viox,where x0 is the set composed by the initial particle swarm.Whenx0 evolves to the kth generation, stipulating ξK=ξK, β can be expressed as(14)β=logξ0/ξKK.The control variableξ can be expressed as(15)ξk=ξ0×exp−β×k,1≤k≤K,0,K≤k≤kmax.To make the particle swarm fast gather to the optimal point in the direction, its update strategy is written as(16)vijk+1=wk×x^jk−xijk×sgnvij+r×pijk−xijk+1−r×pgjk−xijk,xijk+1=xijk+vijk+1,where x^ is the optimal position of particles in the current generation; x^jk−xijk controls step length; and sgnvij determines the motion direction of the particle.The following limit policy is adopted when the particle swarm crosses the border in the process of updating, which is denoted as(17)xijk=x¯jk+r×xjmin−x¯jk,xijk<xjmin,xijk=x¯jk+r×xjmax−x¯jk,xijk>xjmax,where r is the uniform-distributed random number and x¯ is the locational average of the particle swarm.To include points such asD in Figure 1(b), the EDM expands the original feasible region and provides preferable information, which is more reasonable.


## 3. Example

### 3.1. Unconstrained Optimization

Four test functions are investigated to verify the effectiveness of chaotic methodology. The comparison of PSO, CPSO, and IPSO-EDM is shown in Tables1–4. Firstly, the mathematic models are established as follows.(1)
Sphere function:(18)f1x=∑i=1nxi2,xi∈−100,100.(2)
Rastrigin function:(19)f2x=∑i=1nxi2−10cos2πxi+10,xi∈−5.12,5.12.(3)
Rosenbrock function:(20)f3x=∑i=1n100xi+1−xi22+xi−12,xi∈−30,30.(4)
Ackley function:(21)f4x=−20exp−0.21n∑i=1nxi2−exp1n∑i=1ncos2πxi+20+e,xi∈−32.768,32.768.Table 1
Experimental result of the Sphere function.
f1PSOCPSOIPSO-EDMDimensionalityn = 10n = 20n = 10n = 20n = 10n = 20Worst value4.4369e – 497.4023e − 203.0672e − 486.0385e − 223.3696e − 709.4993e − 39Mean value6.5256e – 506.2447e − 211.5934e − 497.9355e − 231.7190e − 717.0559e − 40Optimal value1.1728e − 531.5012e − 241.5102e − 542.1277e − 2700Standard variance1.1743e − 491.6707e − 206.8481e − 491.5866e − 227.5283e − 712.2945e − 39Table 2
Experimental result of the Rastrigin function.
f2PSOCPSOIPSO-EDMDimensionalityn = 10n = 20n = 10n = 20n = 10n = 20Worst value5.969727.858801.8847e − 1000Mean value3.034616.267609.4753e − 1200Optimal value06.96470000Standard variance1.53045.316804.2132e − 1100Table 3
Experimental result of the Rosenbrock function.
f3PSOCPSOIPSO-EDMDimensionalityn = 10n = 20n = 10n = 20n = 10n = 20Worst value22.19173.0299e312.200376.56622.99134.9399Mean value4.3672174.96612.494920.58550.25300.2822Optimal value0.09764.74830.00140.19269.9854e − 49.9068e − 4Standard variance4.6971672.35082.733026.61680.69191.0986Table 4
Experimental result of the Ackley function.
f4PSOCPSOIPSO-EDMDimensionalityn = 10n = 20n = 10n = 20n = 10n = 20Worst value2.6645e − 151.0484e − 102.6645e − 151.0991e − 116.2172e − 151.3323e − 14Mean value2.6645e − 153.9470e − 112.6645e − 151.8865e − 122.8422e − 157.1054e − 15Optimal value2.6645e − 151.1324e − 122.6645e − 151.0925e − 132.6645e − 152.6645e − 15Standard variance03.6101e − 1103.0165e − 127.9441e − 162.7938e − 15Note. The scale of the particle swarm is 50, the change interval of inertia factor ω∈0.4,0.9, earning factors c1=c2=2, the number of iterations is 2000 times, the number of chaotic iterations is 1000 times, and the maximum number of evolutionary variations allowed is 1000 times.It is seen from Tables1–4 that four test functions are used to verify the effectiveness of the IPSO-EDM. As seen in Tables 1–4, the worst value, average value, optimal value, and standard variance in the same dimension are, respectively, calculated via PSO, CPSO, and IPSO-EDM, and the results calculated by the IPSO-EDM are the minimum values. Thus, the computational accuracy of the IPSO-EDM is the highest. In short, the IPSO-EDM has optimal computational accuracy compared with PSO and CPSO.


### 3.2. Constrained Optimization

#### 3.2.1. Numerical Case Studies

To test the effectiveness of the IPSO-EDM, 3 test case studies are investigated, the scale of the particle swarm is 50, the number of iterations is 1000 times, andξK=1e−15,K=900. The statistical results of the optimal function and constraint conflict function are listed in Tables 5 and 6. The comparison of different methods is shown in Tables 7–10, where “N/A” means not available. Firstly, the mathematic models are established as follows.G1:(22)minfx=5∑i=14xi−5∑i=14xi2−∑i=513xis.t.g1x=2x1+2x2+x10+x11−10≤0g2x=2x1+2x3+x10+x12−10≤0g3x=2x2+2x3+x11+x12−10≤0g4x=−8x1+x10≤0g5x=−8x2+x11≤0g6x=−8x3+x12≤0g7x=−2x4−x5+x10≤0g8x=−2x6−x7+x11≤0g9x=−2x8−x9+x12≤0,where0≤xi≤1i=1,2,…,9,13and0≤xi≤100i=10,11,12.G2:(23)minfx=5.3578547x32+0.8356891x1x5+37.293239x1−40792.141s.t.g1x=85.334407+0.0056858x2x5+0.0006262x1x4−0.0022053x3x5−92≤0g2x=−85.334407−0.0056858x2x5−0.0006262x1x4+0.0022053x3x5≤0g3x=80.51249+0.0071317x2x5+0.0029955x1x2+0.0021813x32−110≤0g4x=−80.51249−0.0071317x2x5−0.0029955x1x2−0.0021813x32+90≤0g5x=9.300961+0.0047026x3x5+0.0012547x1x3+0.0019085x3x4−25≤0g6x=−9.300961−0.0047026x3x5−0.0012547x1x3−0.0019085x3x4+20≤0,where78≤x1≤102,33≤x2≤45,and27≤xi≤45i=3,4,5.G3:(24)minfx=3x1+0.000001x13+2x2+0.0000023x23s.t.g1x=−x4+x3−0.55≤0g2x=−x3+x4−0.55≤0h3x=1000sin−x3−0.25+1000sin−x4−0.25+894.8−x1=0h4x=1000sinx3−0.25+1000sinx3−x4−0.25+894.8−x2=0h5x=1000sinx4−0.25+1000sinx4−x3−0.25+1294.8=0,where0≤x1≤1200,0≤x2≤1200,−0.55≤x3≤0.55,and−0.55≤x4≤0.55.Table 5
Statistics of the optimal value with the IPSO-EDM.
Test examplesOptimal valueMean valueWorst valueStandard varianceG1−15−15−151.8724e − 15G2−3.066e4−3.066e4−3.066e40.0019G35.1265e35.1482e35.1871e320.3178Table 6
Statistics of the constraint conflict function with the IPSO-EDM.
Test examplesOptimal valueMean valueWorst valueStandard varianceG104.8850e − 162.6645e − 151.0352e − 15G201.7764e − 151.4211e − 144.5094e − 15G308.8510e − 40.00610.0020Table 7
Comparison of the optimal results obtained by different methods.
Test examplesHM [35]ASCHEA [36]SR [37]IPSO-EDMEDPSO [21]MPSO [25]G1−14.7886−15.0000−15.0000−15.0000−15.0000−14.9863G2−30665.5−30665.5−30665.5−30665.5−30665.5−30665.5G3N/A5126.55126.45126.55126.55126.5Table 8
Comparison of average results obtained by different methods.
Test examplesHM [35]ASCHEA [36]SR [37]IPSO-EDMEDPSO [21]MPSO [25]G1−14.7082−14.8400−15.0000−15.0000−15.0000−14.9986G2−30665.3−30665.5−30665.5−30665.5−30665.5−30665.5G3N/A5141.75128.95148.25148.15149.7Table 9
Comparison of the worst results obtained by different methods.
Test examplesHM [35]ASCHEA [36]SR [37]IPSO-EDMEDPSO [21]MPSO [25]G1−14.6154N/A−15.0000−15.0000−15.0000−14.9992G2−30645.9N/A−30665.5−30665.5−30665.5−30665.5G3N/AN/A5142.4725187.1345148.15146.5Table 10
Comparison of computational efficiency of different methods.
Test examplesHM [35]ASCHEA [36]SR [37]IPSO-EDMEDPSO [21]MPSO [25]ηIP_H (%)ηIP_A (%)ηIP_S (%)ηIP_E (%)ηIP_M (%)G11400000150000035000050000600009000096.4296.6785.7116.6744.44G21400000150000035000050000600009000096.4296.6785.7116.6744.44G31400000150000035000050000600009000096.4296.6785.7116.6744.44Note. ηIP_H is the improved computational efficiency by the IPSO-EDM compared with that by HM; ηIP_A is the improved computational efficiency of the IPSO-EDM compared with that by ASCHEA; ηIP_S is the improved computational efficiency by the IPSO-EDM compared with that by SR; ηIP_E is the improved computational efficiency by the IPSO-EDM compared with that by EDPSO; and ηIP_M is the improved computational efficiency by the IPSO-EDM compared with that by MPSO.Three numerical case studies are used to verify the effectiveness of the proposed IPSO-EDM. As seen in Tables7–10, the worst value, average value, and the best value are, respectively, calculated via the IPSO-EDM and five famous methods, i.e., HM, ASCHEA, SR, EDPSO, and MPSO. The investigation indicates that the computational accuracy of the IPSO-EDM is comparable to HM, ASCHEA, SR, EDPSO, and MPSO. Furthermore, this research manifests that the computational efficiency of the IPSO-EDM is significantly improved compared with the other five methods (HM, ASCHEA, SR, EDPSO, and MPSO) by measuring the product of the group size and algorithm cycle times. It can be seen from Table 10 that the computational efficiency of the IPSO-EDM is the largest compared with the other five methods, i.e., the product of population size and algorithm cycle times of the IPSO-EDM is the least among all methods.To verify the effectiveness of the presented algorithm, the new parameters such asβ and Viox are investigated. The performance function is written as(25)gX1,X2=X13+X23−18,where X1∼N10,52andX2∼N9.9,52.The iteration process ofβ and Viox is calculated by HL, W-G, and IPSO-EDM, which are shown in Figure 7.Figure 7
Iteration process ofβ and Viox.Figure7 indicates that Viox of the standard HL method is very large, and its convergence value is 674.2829, but convergence values of W-G and IPSO-EDM are, respectively, 2.42e − 2 and 5.14e − 4, which manifests feasible points of HL are very few, i.e., the accuracy is very low. Meanwhile, it can be seen that the curve of Viox obtained is fairly close each other by W-G and IPSO-EDM, but the convergence value obtained via the IPSO-EDM is 5.14e − 4 which is smaller than 2.42e − 2 obtained by W-G, and this means the accuracy of IPSO-EDM is higher than that of W-G. Meanwhile, the convergence value of β via HL is 1.1657, and they are 2.22572 and 2.22599 by W-G and IPSO-EDM. Thus, the computational accuracy and efficiency of the IPSO-EDM are optimal and of HL are the worst.


#### 3.2.2. Engineering Case Study

Many researchers use the finite element method to study engineering [38], but they do not optimize it. This method can also be used in practical engineering, for instance, this is a welded beam structure, which is shown in Figure 8.Figure 8
Welded beam schematic diagram.The optimization goal is to seek four design variables, i.e.,x1h, x2l, x3t, and x4b, which satisfy the constraints of shear stress τ, bending stress σ, bending load Pc of the welding rod, deviation δ, and the boundary condition, and the total manufacturing cost of the welding rod is the minimum. The mathematical model is described as(26)minfx=1.104712x12x2+0.04811x3x414+x2s.t.g1x=τx−13600≤0g2x=σx−30000≤0g3x=x1−x4≤0g4x=0.10471x12+0.04811x3x414+x2−5≤0g5x=0.125−x1≤0g6x=δx−0.25≤0g7x=P−Pcx≤0,where(27)τx=τ′2+2τ′τ″x22R+τ″2,τ′x=P2x1x2,τ″=QRJ,Qx=PL+x22,Rx=x224+x1+x222,Jx=22x1x2x2212+x1+x322,σx=6PLx4x32,δx=4PL3Ex33x4,Pcx=4.013Ex32x46/36L21−x32LE4G.Note thatP=6000,L=14,E=30×106,G=12×106,0.1≤x1≤2,0.1≤x2≤10,0.1≤x3≤10,and0.1≤x4≤2.The comparison results obtained by the IPSO-EDM and the other methods are shown in Table11.Table 11
Comparison of the calculation results by different methods for the welding beam.
Methodsx1hx2lx3tx4bfxCoello [39]0.2088003.4205008.9975000.2100001.748309Coello and Montes [40]0.2059863.4713289.0202240.2064801.728226Coello and Becerra [41]0.2057003.4705009.0366000.2057001.724852IPSO-EDM0.2057303.4704899.0366240.2057301.724852CPSO [27]0.2057313.4705829.0368390.2056801.724237MethodsOptimal valueMean valueWorst valueStandard varianceCoello [39]1.7483091.7719731.7858350.001122Coello and Montes [40]1.7282261.7926541.9934080.074713Coello and Becerra [41]1.7248521.9718093.1797090.443131IPSO-EDM1.7248521.7386201.9596060.048555CPSO [27]1.7243811.7388101.9587210.047891The optimal results obtained by the IPSO-EDM are equivalent to those provided by the existing literature, which verifies the accuracy of this method. However, the computational efficiency of the proposed IPSO-EDM is higher than the methods in the literature in Table10.


## 3.1. Unconstrained Optimization

Four test functions are investigated to verify the effectiveness of chaotic methodology. The comparison of PSO, CPSO, and IPSO-EDM is shown in Tables1–4. Firstly, the mathematic models are established as follows.(1)
Sphere function:(18)f1x=∑i=1nxi2,xi∈−100,100.(2)
Rastrigin function:(19)f2x=∑i=1nxi2−10cos2πxi+10,xi∈−5.12,5.12.(3)
Rosenbrock function:(20)f3x=∑i=1n100xi+1−xi22+xi−12,xi∈−30,30.(4)
Ackley function:(21)f4x=−20exp−0.21n∑i=1nxi2−exp1n∑i=1ncos2πxi+20+e,xi∈−32.768,32.768.Table 1
Experimental result of the Sphere function.
f1PSOCPSOIPSO-EDMDimensionalityn = 10n = 20n = 10n = 20n = 10n = 20Worst value4.4369e – 497.4023e − 203.0672e − 486.0385e − 223.3696e − 709.4993e − 39Mean value6.5256e – 506.2447e − 211.5934e − 497.9355e − 231.7190e − 717.0559e − 40Optimal value1.1728e − 531.5012e − 241.5102e − 542.1277e − 2700Standard variance1.1743e − 491.6707e − 206.8481e − 491.5866e − 227.5283e − 712.2945e − 39Table 2
Experimental result of the Rastrigin function.
f2PSOCPSOIPSO-EDMDimensionalityn = 10n = 20n = 10n = 20n = 10n = 20Worst value5.969727.858801.8847e − 1000Mean value3.034616.267609.4753e − 1200Optimal value06.96470000Standard variance1.53045.316804.2132e − 1100Table 3
Experimental result of the Rosenbrock function.
f3PSOCPSOIPSO-EDMDimensionalityn = 10n = 20n = 10n = 20n = 10n = 20Worst value22.19173.0299e312.200376.56622.99134.9399Mean value4.3672174.96612.494920.58550.25300.2822Optimal value0.09764.74830.00140.19269.9854e − 49.9068e − 4Standard variance4.6971672.35082.733026.61680.69191.0986Table 4
Experimental result of the Ackley function.
f4PSOCPSOIPSO-EDMDimensionalityn = 10n = 20n = 10n = 20n = 10n = 20Worst value2.6645e − 151.0484e − 102.6645e − 151.0991e − 116.2172e − 151.3323e − 14Mean value2.6645e − 153.9470e − 112.6645e − 151.8865e − 122.8422e − 157.1054e − 15Optimal value2.6645e − 151.1324e − 122.6645e − 151.0925e − 132.6645e − 152.6645e − 15Standard variance03.6101e − 1103.0165e − 127.9441e − 162.7938e − 15Note. The scale of the particle swarm is 50, the change interval of inertia factor ω∈0.4,0.9, earning factors c1=c2=2, the number of iterations is 2000 times, the number of chaotic iterations is 1000 times, and the maximum number of evolutionary variations allowed is 1000 times.It is seen from Tables1–4 that four test functions are used to verify the effectiveness of the IPSO-EDM. As seen in Tables 1–4, the worst value, average value, optimal value, and standard variance in the same dimension are, respectively, calculated via PSO, CPSO, and IPSO-EDM, and the results calculated by the IPSO-EDM are the minimum values. Thus, the computational accuracy of the IPSO-EDM is the highest. In short, the IPSO-EDM has optimal computational accuracy compared with PSO and CPSO.


## 3.2. Constrained Optimization

### 3.2.1. Numerical Case Studies

To test the effectiveness of the IPSO-EDM, 3 test case studies are investigated, the scale of the particle swarm is 50, the number of iterations is 1000 times, andξK=1e−15,K=900. The statistical results of the optimal function and constraint conflict function are listed in Tables 5 and 6. The comparison of different methods is shown in Tables 7–10, where “N/A” means not available. Firstly, the mathematic models are established as follows.G1:(22)minfx=5∑i=14xi−5∑i=14xi2−∑i=513xis.t.g1x=2x1+2x2+x10+x11−10≤0g2x=2x1+2x3+x10+x12−10≤0g3x=2x2+2x3+x11+x12−10≤0g4x=−8x1+x10≤0g5x=−8x2+x11≤0g6x=−8x3+x12≤0g7x=−2x4−x5+x10≤0g8x=−2x6−x7+x11≤0g9x=−2x8−x9+x12≤0,where0≤xi≤1i=1,2,…,9,13and0≤xi≤100i=10,11,12.G2:(23)minfx=5.3578547x32+0.8356891x1x5+37.293239x1−40792.141s.t.g1x=85.334407+0.0056858x2x5+0.0006262x1x4−0.0022053x3x5−92≤0g2x=−85.334407−0.0056858x2x5−0.0006262x1x4+0.0022053x3x5≤0g3x=80.51249+0.0071317x2x5+0.0029955x1x2+0.0021813x32−110≤0g4x=−80.51249−0.0071317x2x5−0.0029955x1x2−0.0021813x32+90≤0g5x=9.300961+0.0047026x3x5+0.0012547x1x3+0.0019085x3x4−25≤0g6x=−9.300961−0.0047026x3x5−0.0012547x1x3−0.0019085x3x4+20≤0,where78≤x1≤102,33≤x2≤45,and27≤xi≤45i=3,4,5.G3:(24)minfx=3x1+0.000001x13+2x2+0.0000023x23s.t.g1x=−x4+x3−0.55≤0g2x=−x3+x4−0.55≤0h3x=1000sin−x3−0.25+1000sin−x4−0.25+894.8−x1=0h4x=1000sinx3−0.25+1000sinx3−x4−0.25+894.8−x2=0h5x=1000sinx4−0.25+1000sinx4−x3−0.25+1294.8=0,where0≤x1≤1200,0≤x2≤1200,−0.55≤x3≤0.55,and−0.55≤x4≤0.55.Table 5
Statistics of the optimal value with the IPSO-EDM.
Test examplesOptimal valueMean valueWorst valueStandard varianceG1−15−15−151.8724e − 15G2−3.066e4−3.066e4−3.066e40.0019G35.1265e35.1482e35.1871e320.3178Table 6
Statistics of the constraint conflict function with the IPSO-EDM.
Test examplesOptimal valueMean valueWorst valueStandard varianceG104.8850e − 162.6645e − 151.0352e − 15G201.7764e − 151.4211e − 144.5094e − 15G308.8510e − 40.00610.0020Table 7
Comparison of the optimal results obtained by different methods.
Test examplesHM [35]ASCHEA [36]SR [37]IPSO-EDMEDPSO [21]MPSO [25]G1−14.7886−15.0000−15.0000−15.0000−15.0000−14.9863G2−30665.5−30665.5−30665.5−30665.5−30665.5−30665.5G3N/A5126.55126.45126.55126.55126.5Table 8
Comparison of average results obtained by different methods.
Test examplesHM [35]ASCHEA [36]SR [37]IPSO-EDMEDPSO [21]MPSO [25]G1−14.7082−14.8400−15.0000−15.0000−15.0000−14.9986G2−30665.3−30665.5−30665.5−30665.5−30665.5−30665.5G3N/A5141.75128.95148.25148.15149.7Table 9
Comparison of the worst results obtained by different methods.
Test examplesHM [35]ASCHEA [36]SR [37]IPSO-EDMEDPSO [21]MPSO [25]G1−14.6154N/A−15.0000−15.0000−15.0000−14.9992G2−30645.9N/A−30665.5−30665.5−30665.5−30665.5G3N/AN/A5142.4725187.1345148.15146.5Table 10
Comparison of computational efficiency of different methods.
Test examplesHM [35]ASCHEA [36]SR [37]IPSO-EDMEDPSO [21]MPSO [25]ηIP_H (%)ηIP_A (%)ηIP_S (%)ηIP_E (%)ηIP_M (%)G11400000150000035000050000600009000096.4296.6785.7116.6744.44G21400000150000035000050000600009000096.4296.6785.7116.6744.44G31400000150000035000050000600009000096.4296.6785.7116.6744.44Note. ηIP_H is the improved computational efficiency by the IPSO-EDM compared with that by HM; ηIP_A is the improved computational efficiency of the IPSO-EDM compared with that by ASCHEA; ηIP_S is the improved computational efficiency by the IPSO-EDM compared with that by SR; ηIP_E is the improved computational efficiency by the IPSO-EDM compared with that by EDPSO; and ηIP_M is the improved computational efficiency by the IPSO-EDM compared with that by MPSO.Three numerical case studies are used to verify the effectiveness of the proposed IPSO-EDM. As seen in Tables7–10, the worst value, average value, and the best value are, respectively, calculated via the IPSO-EDM and five famous methods, i.e., HM, ASCHEA, SR, EDPSO, and MPSO. The investigation indicates that the computational accuracy of the IPSO-EDM is comparable to HM, ASCHEA, SR, EDPSO, and MPSO. Furthermore, this research manifests that the computational efficiency of the IPSO-EDM is significantly improved compared with the other five methods (HM, ASCHEA, SR, EDPSO, and MPSO) by measuring the product of the group size and algorithm cycle times. It can be seen from Table 10 that the computational efficiency of the IPSO-EDM is the largest compared with the other five methods, i.e., the product of population size and algorithm cycle times of the IPSO-EDM is the least among all methods.To verify the effectiveness of the presented algorithm, the new parameters such asβ and Viox are investigated. The performance function is written as(25)gX1,X2=X13+X23−18,where X1∼N10,52andX2∼N9.9,52.The iteration process ofβ and Viox is calculated by HL, W-G, and IPSO-EDM, which are shown in Figure 7.Figure 7
Iteration process ofβ and Viox.Figure7 indicates that Viox of the standard HL method is very large, and its convergence value is 674.2829, but convergence values of W-G and IPSO-EDM are, respectively, 2.42e − 2 and 5.14e − 4, which manifests feasible points of HL are very few, i.e., the accuracy is very low. Meanwhile, it can be seen that the curve of Viox obtained is fairly close each other by W-G and IPSO-EDM, but the convergence value obtained via the IPSO-EDM is 5.14e − 4 which is smaller than 2.42e − 2 obtained by W-G, and this means the accuracy of IPSO-EDM is higher than that of W-G. Meanwhile, the convergence value of β via HL is 1.1657, and they are 2.22572 and 2.22599 by W-G and IPSO-EDM. Thus, the computational accuracy and efficiency of the IPSO-EDM are optimal and of HL are the worst.


### 3.2.2. Engineering Case Study

Many researchers use the finite element method to study engineering [38], but they do not optimize it. This method can also be used in practical engineering, for instance, this is a welded beam structure, which is shown in Figure 8.Figure 8
Welded beam schematic diagram.The optimization goal is to seek four design variables, i.e.,x1h, x2l, x3t, and x4b, which satisfy the constraints of shear stress τ, bending stress σ, bending load Pc of the welding rod, deviation δ, and the boundary condition, and the total manufacturing cost of the welding rod is the minimum. The mathematical model is described as(26)minfx=1.104712x12x2+0.04811x3x414+x2s.t.g1x=τx−13600≤0g2x=σx−30000≤0g3x=x1−x4≤0g4x=0.10471x12+0.04811x3x414+x2−5≤0g5x=0.125−x1≤0g6x=δx−0.25≤0g7x=P−Pcx≤0,where(27)τx=τ′2+2τ′τ″x22R+τ″2,τ′x=P2x1x2,τ″=QRJ,Qx=PL+x22,Rx=x224+x1+x222,Jx=22x1x2x2212+x1+x322,σx=6PLx4x32,δx=4PL3Ex33x4,Pcx=4.013Ex32x46/36L21−x32LE4G.Note thatP=6000,L=14,E=30×106,G=12×106,0.1≤x1≤2,0.1≤x2≤10,0.1≤x3≤10,and0.1≤x4≤2.The comparison results obtained by the IPSO-EDM and the other methods are shown in Table11.Table 11
Comparison of the calculation results by different methods for the welding beam.
Methodsx1hx2lx3tx4bfxCoello [39]0.2088003.4205008.9975000.2100001.748309Coello and Montes [40]0.2059863.4713289.0202240.2064801.728226Coello and Becerra [41]0.2057003.4705009.0366000.2057001.724852IPSO-EDM0.2057303.4704899.0366240.2057301.724852CPSO [27]0.2057313.4705829.0368390.2056801.724237MethodsOptimal valueMean valueWorst valueStandard varianceCoello [39]1.7483091.7719731.7858350.001122Coello and Montes [40]1.7282261.7926541.9934080.074713Coello and Becerra [41]1.7248521.9718093.1797090.443131IPSO-EDM1.7248521.7386201.9596060.048555CPSO [27]1.7243811.7388101.9587210.047891The optimal results obtained by the IPSO-EDM are equivalent to those provided by the existing literature, which verifies the accuracy of this method. However, the computational efficiency of the proposed IPSO-EDM is higher than the methods in the literature in Table10.


## 3.2.1. Numerical Case Studies

To test the effectiveness of the IPSO-EDM, 3 test case studies are investigated, the scale of the particle swarm is 50, the number of iterations is 1000 times, andξK=1e−15,K=900. The statistical results of the optimal function and constraint conflict function are listed in Tables 5 and 6. The comparison of different methods is shown in Tables 7–10, where “N/A” means not available. Firstly, the mathematic models are established as follows.G1:(22)minfx=5∑i=14xi−5∑i=14xi2−∑i=513xis.t.g1x=2x1+2x2+x10+x11−10≤0g2x=2x1+2x3+x10+x12−10≤0g3x=2x2+2x3+x11+x12−10≤0g4x=−8x1+x10≤0g5x=−8x2+x11≤0g6x=−8x3+x12≤0g7x=−2x4−x5+x10≤0g8x=−2x6−x7+x11≤0g9x=−2x8−x9+x12≤0,where0≤xi≤1i=1,2,…,9,13and0≤xi≤100i=10,11,12.G2:(23)minfx=5.3578547x32+0.8356891x1x5+37.293239x1−40792.141s.t.g1x=85.334407+0.0056858x2x5+0.0006262x1x4−0.0022053x3x5−92≤0g2x=−85.334407−0.0056858x2x5−0.0006262x1x4+0.0022053x3x5≤0g3x=80.51249+0.0071317x2x5+0.0029955x1x2+0.0021813x32−110≤0g4x=−80.51249−0.0071317x2x5−0.0029955x1x2−0.0021813x32+90≤0g5x=9.300961+0.0047026x3x5+0.0012547x1x3+0.0019085x3x4−25≤0g6x=−9.300961−0.0047026x3x5−0.0012547x1x3−0.0019085x3x4+20≤0,where78≤x1≤102,33≤x2≤45,and27≤xi≤45i=3,4,5.G3:(24)minfx=3x1+0.000001x13+2x2+0.0000023x23s.t.g1x=−x4+x3−0.55≤0g2x=−x3+x4−0.55≤0h3x=1000sin−x3−0.25+1000sin−x4−0.25+894.8−x1=0h4x=1000sinx3−0.25+1000sinx3−x4−0.25+894.8−x2=0h5x=1000sinx4−0.25+1000sinx4−x3−0.25+1294.8=0,where0≤x1≤1200,0≤x2≤1200,−0.55≤x3≤0.55,and−0.55≤x4≤0.55.Table 5
Statistics of the optimal value with the IPSO-EDM.
Test examplesOptimal valueMean valueWorst valueStandard varianceG1−15−15−151.8724e − 15G2−3.066e4−3.066e4−3.066e40.0019G35.1265e35.1482e35.1871e320.3178Table 6
Statistics of the constraint conflict function with the IPSO-EDM.
Test examplesOptimal valueMean valueWorst valueStandard varianceG104.8850e − 162.6645e − 151.0352e − 15G201.7764e − 151.4211e − 144.5094e − 15G308.8510e − 40.00610.0020Table 7
Comparison of the optimal results obtained by different methods.
Test examplesHM [35]ASCHEA [36]SR [37]IPSO-EDMEDPSO [21]MPSO [25]G1−14.7886−15.0000−15.0000−15.0000−15.0000−14.9863G2−30665.5−30665.5−30665.5−30665.5−30665.5−30665.5G3N/A5126.55126.45126.55126.55126.5Table 8
Comparison of average results obtained by different methods.
Test examplesHM [35]ASCHEA [36]SR [37]IPSO-EDMEDPSO [21]MPSO [25]G1−14.7082−14.8400−15.0000−15.0000−15.0000−14.9986G2−30665.3−30665.5−30665.5−30665.5−30665.5−30665.5G3N/A5141.75128.95148.25148.15149.7Table 9
Comparison of the worst results obtained by different methods.
Test examplesHM [35]ASCHEA [36]SR [37]IPSO-EDMEDPSO [21]MPSO [25]G1−14.6154N/A−15.0000−15.0000−15.0000−14.9992G2−30645.9N/A−30665.5−30665.5−30665.5−30665.5G3N/AN/A5142.4725187.1345148.15146.5Table 10
Comparison of computational efficiency of different methods.
Test examplesHM [35]ASCHEA [36]SR [37]IPSO-EDMEDPSO [21]MPSO [25]ηIP_H (%)ηIP_A (%)ηIP_S (%)ηIP_E (%)ηIP_M (%)G11400000150000035000050000600009000096.4296.6785.7116.6744.44G21400000150000035000050000600009000096.4296.6785.7116.6744.44G31400000150000035000050000600009000096.4296.6785.7116.6744.44Note. ηIP_H is the improved computational efficiency by the IPSO-EDM compared with that by HM; ηIP_A is the improved computational efficiency of the IPSO-EDM compared with that by ASCHEA; ηIP_S is the improved computational efficiency by the IPSO-EDM compared with that by SR; ηIP_E is the improved computational efficiency by the IPSO-EDM compared with that by EDPSO; and ηIP_M is the improved computational efficiency by the IPSO-EDM compared with that by MPSO.Three numerical case studies are used to verify the effectiveness of the proposed IPSO-EDM. As seen in Tables7–10, the worst value, average value, and the best value are, respectively, calculated via the IPSO-EDM and five famous methods, i.e., HM, ASCHEA, SR, EDPSO, and MPSO. The investigation indicates that the computational accuracy of the IPSO-EDM is comparable to HM, ASCHEA, SR, EDPSO, and MPSO. Furthermore, this research manifests that the computational efficiency of the IPSO-EDM is significantly improved compared with the other five methods (HM, ASCHEA, SR, EDPSO, and MPSO) by measuring the product of the group size and algorithm cycle times. It can be seen from Table 10 that the computational efficiency of the IPSO-EDM is the largest compared with the other five methods, i.e., the product of population size and algorithm cycle times of the IPSO-EDM is the least among all methods.To verify the effectiveness of the presented algorithm, the new parameters such asβ and Viox are investigated. The performance function is written as(25)gX1,X2=X13+X23−18,where X1∼N10,52andX2∼N9.9,52.The iteration process ofβ and Viox is calculated by HL, W-G, and IPSO-EDM, which are shown in Figure 7.Figure 7
Iteration process ofβ and Viox.Figure7 indicates that Viox of the standard HL method is very large, and its convergence value is 674.2829, but convergence values of W-G and IPSO-EDM are, respectively, 2.42e − 2 and 5.14e − 4, which manifests feasible points of HL are very few, i.e., the accuracy is very low. Meanwhile, it can be seen that the curve of Viox obtained is fairly close each other by W-G and IPSO-EDM, but the convergence value obtained via the IPSO-EDM is 5.14e − 4 which is smaller than 2.42e − 2 obtained by W-G, and this means the accuracy of IPSO-EDM is higher than that of W-G. Meanwhile, the convergence value of β via HL is 1.1657, and they are 2.22572 and 2.22599 by W-G and IPSO-EDM. Thus, the computational accuracy and efficiency of the IPSO-EDM are optimal and of HL are the worst.


## 3.2.2. Engineering Case Study

Many researchers use the finite element method to study engineering [38], but they do not optimize it. This method can also be used in practical engineering, for instance, this is a welded beam structure, which is shown in Figure 8.Figure 8
Welded beam schematic diagram.The optimization goal is to seek four design variables, i.e.,x1h, x2l, x3t, and x4b, which satisfy the constraints of shear stress τ, bending stress σ, bending load Pc of the welding rod, deviation δ, and the boundary condition, and the total manufacturing cost of the welding rod is the minimum. The mathematical model is described as(26)minfx=1.104712x12x2+0.04811x3x414+x2s.t.g1x=τx−13600≤0g2x=σx−30000≤0g3x=x1−x4≤0g4x=0.10471x12+0.04811x3x414+x2−5≤0g5x=0.125−x1≤0g6x=δx−0.25≤0g7x=P−Pcx≤0,where(27)τx=τ′2+2τ′τ″x22R+τ″2,τ′x=P2x1x2,τ″=QRJ,Qx=PL+x22,Rx=x224+x1+x222,Jx=22x1x2x2212+x1+x322,σx=6PLx4x32,δx=4PL3Ex33x4,Pcx=4.013Ex32x46/36L21−x32LE4G.Note thatP=6000,L=14,E=30×106,G=12×106,0.1≤x1≤2,0.1≤x2≤10,0.1≤x3≤10,and0.1≤x4≤2.The comparison results obtained by the IPSO-EDM and the other methods are shown in Table11.Table 11
Comparison of the calculation results by different methods for the welding beam.
Methodsx1hx2lx3tx4bfxCoello [39]0.2088003.4205008.9975000.2100001.748309Coello and Montes [40]0.2059863.4713289.0202240.2064801.728226Coello and Becerra [41]0.2057003.4705009.0366000.2057001.724852IPSO-EDM0.2057303.4704899.0366240.2057301.724852CPSO [27]0.2057313.4705829.0368390.2056801.724237MethodsOptimal valueMean valueWorst valueStandard varianceCoello [39]1.7483091.7719731.7858350.001122Coello and Montes [40]1.7282261.7926541.9934080.074713Coello and Becerra [41]1.7248521.9718093.1797090.443131IPSO-EDM1.7248521.7386201.9596060.048555CPSO [27]1.7243811.7388101.9587210.047891The optimal results obtained by the IPSO-EDM are equivalent to those provided by the existing literature, which verifies the accuracy of this method. However, the computational efficiency of the proposed IPSO-EDM is higher than the methods in the literature in Table10.


## 4. Conclusions

The standard PSO algorithm is improved from the perspective of engineering application to solve engineering optimization problems.(1)
The original feasible region is expanded, and some points that are closer to the constrained optimal point in the feasible region are contained as feasible points, which provide preferable function information compared with the points in the original feasible region. This approach uses the current location information of the particles and particle swarm to determine the speed of the particles. In short, the difference value in the current optimal positions of the particle swarm and the particle is used to determine the current particle velocity. The optimal location of obtained points using the proposed method is better than that obtained using the standard PSO.(2)
The constrained optimization is transformed into the unconstrained optimization by combining the ergodicity of chaos optimization and the evolutionary variation to realize global search. The logistic chaotic system equation is applied in the PSO algorithm, and the mutation operator is introduced in the evolutionary variation strategy to escape local optimal and maintain its vitality of the particle swarm, which prevents the particle swarm from falling into the condition of “precocity” at the earlier iteration.(3)
An unconstrained optimization case study, four numerical case studies, and one engineering case study are used to verify the effectiveness of the IPSO-EDM. The worst value, average value, and optimal value are, respectively, calculated via the IPSO-EDM and compared with other methods. The investigation indicates that the computational accuracy of the IPSO-EDM is comparable to that provided by the existing literature; however, the computational efficiency of the IPSO-EDM is significantly improved.(4)
Though the PSO method is improved and six case studies are investigated to prove the effectiveness of this method, yet the numerical case studies are relatively simple, and the design variables of the engineering case study are small. Moreover, only the deterministic optimization is studied. Thus, the nondeterministic optimization will be researched, and the random variable will be subject to normal distribution, exponential distribution, or Weibull distribution. Accordingly, the proposed method can be expanded to wide-spread engineering application fields. Further studies will focus on the IPSO-EDM considering random variables of different distributions including normal distribution, exponential distribution, and Weibull distribution to deal with actual operation of the machines.



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