10976


FACTA UNIVERSITATIS  

Series: Mechanical Engineering  

https://doi.org/10.22190/FUME220829004W 

© 2020 by University of Niš, Serbia | Creative Commons License: CC BY-NC-ND 

Original scientific paper 

AN IMPROVED BARE-BONES PARTICLE SWARM 

ALGORITHM FOR MULTI-OBJECTIVE OPTIMIZATION WITH 

APPLICATION TO THE ENGINEERING STRUCTURES 

Zhaohua Wang1, Guobiao Yang1, Yinxu Sun2, Yongxin Li2,  

Fenghe Wu2 

1School of Mechanical Engineering,  

Taiyuan University of Science and Technology, Taiyuan, China 
2School of Mechanical Engineering, Yanshan University, Qinhuangdao, China 

Abstract. In this paper, an improved bare-bones multi-objective particle swarm 

algorithm is proposed to solve the multi-objective size optimization problems with 

non-linearity and constraints in structural design and optimization. Firstly, the 

development of particle individual guide and the randomness of gravity factor are 

increased by modifying the updated form of particle position. Then, the combination of 

spatial grid density and congestion distance ranking is used to maintain the external 

archive, which is divided into two parts: feasible solution set and infeasible solution set. 

Next, the global best positions are determined by increasing the probability allocation 

strategy which varies with time. The algorithmic complexity is given and the 

performance of solution ability, convergence and constraint processing are analyzed 

through standard test functions and compared with other algorithms. Next, as a case 

study, a support frame of triangle track wheel is optimized by the BB-MOPSO and 

improved BB-MOPSO. The results show that the improved algorithm improves the 

cross-region exploration, optimal solution distribution and convergence of the 

bare-bones particle swarm optimization algorithm, which can effectively solve the 

multi-objective size optimization problem with non-linearity and constraints. 

Key words: Optimization, Multi-objective, Particle swarm, BB-MOPSO 

1. INTRODUCTION 

At present, engineering structures are developing towards high reliability and 

lightweight design. In the design process, the stiffness, strength and vibration characteristics 

                                                           
Received: August 29, 2022 / Accepted January 16, 2023  

Corresponding author: Fenghe Wu 

School of Mechanical Engineering, Yanshan University, No 438, West of Hebei Avenue, Qinhuangdao, China, 

066004. 

E-mail: risingwu@ysu.edu.cn 



2 Z. Wang, G. Yang, Y. Sun, Y. Li, F. Wu 

of the structure must be taken into account at the same time [1, 2]. Therefore, the 

multi-objective optimization problem (MOP) in the process of structural design is 

becoming more and more important [3]. In the conceptual design stage, the topology 

configuration of structure is determined by the topology optimization methods [4]. Then, 

the local size parameters are further optimized to enhance the mechanical performance in 

the detailed design stage. Size optimization needs to consider many mechanical 

performances of the structure at the same time, including stiffness, strength, fatigue life, etc. 

Local extremum and non-convergence often occur in the calculation process, especially the 

stress-based optimization, which belongs to a typical nonlinear MOP with constraints. 

With the increasing complexity of multi-objective optimization problems, more and 

more intelligent algorithms are applied to size optimization. Bekdas et al. [5] and Assimi et 

al. [6] improved different intelligent algorithms to optimize the truss size with good results 

obtained. A new collision box is designed and optimized by using archive-based 

micro-genetic algorithm and improved, non-dominated sorting genetic algorithm, which 

improve the energy absorption characteristics and comprehensive crashworthiness [7]. 

Although many achievements have been made in the application of intelligent algorithms to 

size optimization, there are still problems of local extremum and non-convergence for the 

solution with multi-constraints. Evolutionary algorithm or genetic algorithm [8-9] has a 

special evaluation mechanism, and has gradually matured after several generations of 

development, but its local search ability is poor, and the selection of more parameters 

affects the quality of the solution. Simulated annealing method [10-11] draws lessons from 

the phenomenon of sudden jump of solid properties during annealing heat treatment and 

becomes a good global optimization method, but it has low efficiency and slow 

convergence speed. Ant colony algorithm [12-13] is a parallel algorithm with fast 

convergence speed and obvious advantages in dealing with complex combinatorial 

optimization problems, but it is prone to premature termination. Inspired by the predation 

behavior of birds, Particle Swarm Optimization (PSO) [14, 15] is a random iterative parallel 

algorithm, which is simple and easy to implement, has strong global search ability for 

nonlinear multi-peak problems, and has obvious advantages in solving multi-objective size 

optimization problems. However, the algorithm is limited to unconstrained optimization 

problems, and it is ignored to solve general multi-objective optimization problems with 

constraints. 

Since the PSO does not have the ability to deal with constraint problems, it is often 

necessary to transform the constraint problems with unconstrained problems. An improved 

multi-objective particle swarm optimization algorithm (MOPSO) based on the concept of 

constraint domination is proposed in [16], which perturbs particles with small probability to 

improve the diversity of the algorithm. Mohamad et al. [17] proposed a MOPSO with good 

convergence and constraint processing ability for high-dimensional problems to solve 

complex engineering problems with many optimization variables. Li et al. [18] proposed an 

adaptive particle swarm optimization algorithm with different learning strategies to solve 

the path planning problem of mobile robot under different types of constraints in complex 

environment. Xu et al. [19] proposed an improved adaptive weighted PSO and applied it to 

multi-objective optimization design of planetary gears, which solved the problem that PSO 

is not easy to converge or fall into local optimum under complex constraints. It can be seen 

that many scholars have done extensive research on PSO in solving MOP and constrained 



 An Improved Bare-Bones Particle Swarm Algorithm for Multi-Objective Optimization... 3 

problems [20], but the above algorithms need to set parameters such as learning factor and 

inertia weight, which affects the computational efficiency of PSO. 

A bare-bones particle swarm optimization algorithm (BB-PSO), using a Gaussian 

distribution of personal best positions and global best positions to update particle positions, 

was first proposed by Kennedy [21], which has the advantage of not setting inertia weights, 

learning factors and other control parameter. Zhang et al. [22] first extended the BB-PSO to 

multi-objective optimization problem, and the multi-objective economic/environmental 

scheduling problems with constraints are solved. However, the bare-bones multi-objective 

particle swarm optimization algorithm (BB-MOPSO) focuses more on the globality in 

particle update and global best positions selection, which makes the algorithm's 

optimization ability stronger, but the local development ability, boundary search ability and 

particle diversity are correspondingly reduced. Therefore, it needs to be further improved 

for the problems that there are many local extremum and the optimal solution may be 

located at the boundary when solving nonlinear multi-objective optimization with 

constraints.  

Based on the above analysis, this paper proposes an improved BB-MOPSO to solve the 

multi-objective size optimization problems with non-linearity and constraints. The MOP 

and BB-MOPSO are introduced in Section 2 and Section 3. An improved BB-MOPSO is 

proposed in Section 4. The algorithmic complexity is given and the performance of solution 

ability, convergence and constraint processing are analyzed in Section 5. As a case study, a 

support frame of triangle track wheel is optimized in Section 6.  

2. MULTI-OBJECTIVE OPTIMIZATION PROBLEM 

A classical multi-objective optimization problem can be defined as finding a decision 

variable 
* * * *

1 2
[ , , , ]

n
x x x x  that satisfies the following conditions: 

Decision space： 

 
min max

,  1, 2, ,   ( )
i i i i

x x x i n x x    . (1) 

Equality constraint： 

 ( ) 0,    1, 2, ,
j

h x j J  . (2) 

Inequality constraint： 

 ( ) 0,   1, 2, ,
k

g k K x . (3) 

Minimized objective function： 

 
1 2

( ) [ ( ), ( ), , ( )]
m

f x f x f x f x . (4) 

Function f(x) is called the objective function. Each component value of decision 

variable x is constrained by two boundary values 
min

i
x  and

max

i
x . The boundaries of all 

components of the decision variable constitute the decision space of multi-objective 

optimization. The output values of m objective functions constitute the objective space of 

multi-objective optimization. The decision variables that conform to all constraints 



4 Z. Wang, G. Yang, Y. Sun, Y. Li, F. Wu 

constitute the feasible region of multi-objective optimization, and the solution that 

conforms to Eq. (4) in the feasible region is called the optimal solution. 

Different from the single objective optimization problem, the MOP contains many 

conflicting objective functions, and it is difficult for each objective to achieve the optimal at 

the same time. Therefore, its solution becomes a solution set containing infinite elements. 

Usually, we find the effective solution of this solution set, also known as Pareto solution, 

which is defined as: if there is not x X , make ( ) ( )*f x f x , and then 
*

x is defined as an 

effective solution to the MOP. 

3. BB-MOPSO 

In the PSO, each particle represents a solution of the optimized problem. The position of 

the i-th particle in the n-dimensional space is represented as xi=[xi,1, xi,2,…, xi,n], and the 

velocity is represented as vi=[vi,1, vi,2,…, vi,n]. Each particle has an adaptation value 
determined by the optimization function, and it is known that the optimal value of the 

current particle individual is pi=[pi,1, pi,2,…, pi,n] and the global optimal value of the particle 

is gi=[g1, g2,…, gn] at the i-th iteration. At the next iteration, the particle determines the next 
motion state according to its own experience and the experience of particles in the same 

neighborhood until the end of the iteration. The velocity update equation is, 

 
, , 1 1 , , 2 2 ,

( 1) ( ) ( ( ) ( )) ( ( ) ( ))
i j i j i j i j j i j

v t wv t c r p t x t c r g t x t      . (5) 

where w is the inertia weight, c1 and c2 are learning factors. r1 and r2 are random obeying 

uniformly distributed U(0,1); j=1, 2,... , n, and i=1, 2,..., N, N is the number of particles. 
The particle position is, 

 
, , ,

( 1) ( ) ( 1)
i j i j i j

x t x t v t    . (6) 

BB-MOPSO has the same optimization idea as PSO, but it is different from traditional 

PSO. On the one hand, the control parameters such as inertia coefficient and learning factor 

are deleted to overcome the disadvantage that traditional PSO relies too much on 

parameters. The mathematical model to determine the updating position of particles is 

realized by a Gaussian sampling (normal distribution) of personal best positions and global 

best positions. The mathematical model of BB-MOPSO [18] is as follows: 

 
 1 , 1 , , ,

,

,

[ ( ) (1 ) ( )] / 2, ( ) ( )        (0,1) 0.5
( 1)

( )                                                                             others

i j i j i j i j

i j

i j

N r p t r g t p t g t U
x t

p t

      
  



. (7) 

On the other hand, PSO belongs to the single-objective optimization method, which has a 

definite optimal solution. BB-MOPSO belongs to the multi-objective optimization method. 

Due to the addition of a non-dominated relationship between each particle, the algorithm 

may get more than one set of optimal solutions (non-inferior solutions) after iteration. 

Therefore, for each particle, there will be more than one candidate point when updating the 

global best positions, and these candidate points do not dominate each other. All candidate 

points are stored in a set different from particle swarm, which is the external archive 



 An Improved Bare-Bones Particle Swarm Algorithm for Multi-Objective Optimization... 5 

mentioned above. When the particle position needs to be updated, the global best position 

of the particle is selected from the external archive, and the external archive element is also 

submitted to the decision maker as the final result of the algorithm. The flow of the 

algorithm is shown in Fig. 1, and the Tmax is the maximum number of iterations. 

 

Fig. 1 Flow chart of BB-MOPSO 

4. IMPROVED BB-MOPSO 

Multi-objective size optimization is a nonlinear multi-objective optimization problem 

with constraints. There are many local extremum, and the Pareto solution may be at the 

boundary, and the constraint relationship is complex. Although BB-MOPSO has some 

advantages for solving such problems, but the algorithm focuses more on globality in 

particle update and global best positions selection, which reduces the algorithm’s boundary 

and cross-region search ability and particle diversity. Therefore, the algorithm is difficult to 

find the global optimal solution when dealing with multi-objective size optimization 

problems.  

In this paper, the BB-MOPSO is improved from three aspects, including particle update 

mode, maintenance strategy of external archive and global best positions selection, to 

increase the algorithm's boundary and cross-region search ability and particle diversity. The 

calculation process of new algorithm is shown in Fig. 2, and the improvement of each part 

of the algorithm will be introduced in detail in sections 4.1~4.3. 

4.1 Particle Position Updates Approach 

The particle position update of BB-MOPSO (Eq. 7) is more inclined to the global best 

positions. But for the size optimization problems, the optimal solution is likely to appear 

near the constrained boundary, especially the stress-based optimization. Therefore, in the 

process of particle updating, it is necessary to increase the local development of particles. In 

this paper, the development of individual particle guides and the randomness of 

gravitational factors are increased in this paper. The new update mode of particle position is 

shown in Eq. (8). 



6 Z. Wang, G. Yang, Y. Sun, Y. Li, F. Wu 

 
 1 , 2 , , ,

,

,

[ ( ) ( )] / 2, ( ) ( )       (0,1) 0.5
( 1)

( )                                                                     others

i j i j i j i j

i j

i j

N r p t r g t p t g t U
x t

p t

     
  



, (8) 

where r1 and r2 are random numbers in the range of 0~1. 

 

Fig. 2 Flow chart of improved BB-MOPSO 

The new update mode can make the particle have a 50% probability to select the 

relevant component of the current local optimal particle in each iteration calculation. At the 

same time, the random numbers r1 and r2 can improve the possibility of particles entering 

another feasible solution region, which expands the particle search range and theoretically 

increases the boundary and cross region search ability of the algorithm. 

The updated particle xi (t+1) needs to be compared with the historical optimal particle 

pi(t) to select the personal best positions. The principle of selection is as follows: when xi 
(t+1) dominates pi(t), then pi(t+1)=xi(t+1); when xi(t+1) and pi(t) do not dominate each 

other, anyone can be selected as pi(t+1), which can increase the probability of "excellent 

particles" entering the next generation; Otherwise, pi(t+1)=pi(t). 

4.2 Maintenance Strategy of External Archive 

The constraint dominance relation is constructed by direct comparison method in 

BB-MOPSO, which is easy to fall into local optimum in multi-island problem. In order to 

reduce the possibility of local optimum, the external archive is divided into two parts: 

feasible solution set and infeasible solution set. In this way, the development of isolated 

regions and the global capability of the algorithm can be enhanced [23].  

After each update, the feasible solution set is updated first, that is, the particle set with 

constraint violation degree of 0. A new feasible solution set is composed of the newly 

obtained particles and the particles in the original feasible solution set. Pareto domination 



 An Improved Bare-Bones Particle Swarm Algorithm for Multi-Objective Optimization... 7 

relationship is analysed, and the particles that do not dominate each other are left. In order 

to ensure the diversity of learning particles, the original BB-MOPSO method uses crowded 

distances sorting [24] as the external archive maintenance strategy. It needs to calculate the 

Euclidean distance of each particle relative to other particles, which has a large 

computational complexity.  

In this paper, the combination of spatial grid and crowded distance sorting are used. The 

objective space is first divided into M1×M2×…Mk grid, k is the objective dimension, Mk is 

the grid number of each dimension. The grid position of each particle is calculated to 

determine the grid density, which can reduce the computational complexity and maintain 

the optimal solution distribution in the global range. 

The grid density can be calculated by the following method. The upper bounds F
max 

i  and 

lower bounds F
min 

i  of the objective space are given by analysing the values range of each 

dimension. The objective width of i-th dimension of each grid is 

 
max min

( ) /
i i i i

d F F M  . (9) 

Suppose the coordinate of a particle corresponding to the objective space is s=[s1, 

s2,…,sm], then the particle position in each dimension of the objective space is 

 
min

( , ) 1
i i i i

L fix s F d   , (10) 

where fix (a, b) is a rounding function.  

Each grid is numbered and placed into different grids according to the position values of 

each dimension. The number of particles put into the grids is recorded as N (j, t), which 

indicates the number of particles in the region where the j-th particle is located at time t. 

Suppose the maximum theoretical capacity of the grid is Npi, the density is 

 ( , )  /
t

i pi
N j t N  . (11) 

When the total number of feasible solutions in a certain grid is larger than the grid 

capacity, the crowding distance of each particle is calculated and Npi with larger distance as 

the new external archive particles are selected, to ensure the distribution of the solution. The 

crowding distance can be calculated by the following method. The objective values of each 

dimension are sorted from small to large, and composition of objective value sequence [fi,1, 

fi,2, …, fi,p, …, fi,sn] , then the crowding distance of a objective value is 

 
max min1

2, , 1 , -1

1 1

( ) / ( )
k k

p i p i p i p i i

i i

dy dy f f f f


 

     , (12) 

where f 
max 

i  and f 
min 

i  are represent the maximum and minimum values of the i-th objective in 

the current archive. dyi,p is the crowding distance of the p-th particle in the objective value 

of i-dimension. sn is the size of particle swarm. 

For the update of the infeasible solution set (the particle set with constraint violation 

degree greater than 0), the Pareto dominance relationship analysis is first performed to 

obtain the mutually un-dominated particles. And then, the adaptive grid technique is used to 

divide the particles. For the grid exceeding the capacity, instead of congestion distance 

sorting, Npi particles are randomly selected and re-placed into the grid. Although this 



8 Z. Wang, G. Yang, Y. Sun, Y. Li, F. Wu 

approach may reduce the diversity of infeasible solution set, it can still ensure the ability of 

the algorithm to explore the unknown feasible region and reduce the computational 

complexity. 

4.3 Global Best Positions Selection 

Whether the selection of global best positions is reasonable or not is related to the ability 

of the algorithm to accurately converge to the Pareto front. In the BB-MOPSO, the global 

best positions for each particle are selected by the tournament selection method with scale 

of 2 based on the crowding distance value. The larger the congestion distance value, the 

more likely it is selected as the global best positions. In this paper, the dynamic probability 

selection method is used to give the global best positions, that is, the global best positions 

are selected from the infeasible solution set and the feasible solution set with the probability 

of psl and 1-psl. 

 
1 2

/
sl sl sl

p p p t T   , (13) 

where psl1 and psl2 are dynamic probability constants, psl1=0.7 and psl2=0.6. t is the current 

number of iterations. T is the total number of iterations. 

In this way, the algorithm has sufficient global search ability. The diversity of learning 

particles in the early stage is improved, and the convergence in the later stage is accelerated. 

After determining the global best positions, a probability selection equation (Eq. 14) is 

constructed based on the density values of each grid, so that the probability of selecting a 

value with small density is high and that of selecting a value with large density is low. The 

global best positions selected in this way will have good distribution and convergence. 

 
1

1 / ( )
ks

t t t

i i i

i

P  


   , (14) 

where 
t

i
p  is the final choice probability. ks is total number of grids. 

4.4 Analysis of Algorithmic Complexity 

The computational cost of the improved BB-MOPSO mainly concentrates on the update 

of the reserve set. Suppose an optimization problem with M objectives and K constraints, 

the number of feasible and infeasible solutions in the new particle swarm optimization is Np 

and Nq, the feasible reserve capacity is Na, and the infeasible reserve capacity is Nb. 

Firstly, the renewal process of feasible reserve set is analysed. The size of the new 

population is Np+Na, and the number of comparisons is M×(Np+Na) to judge whether a 

particle is advantages and disadvantages. Considering the worst case (the particles in the 

population are not dominated by each other), it takes M×(Np+Na)
2 comparisons to select all 

non-inferior solutions from Np+Na particles. On the other hand, the computational 

complexity of using crowded distance to maintain the reserve set is O(M×(Np+Na)× 

log(Np+Na)) and log(Np+Na)<Np+Na [25]. As a result, the computational complexity of 

updating the feasible reserve set is O(M×(Np+Na)
2). 

Next, the update process of infeasible reserve set is analyzed. To judge whether a 

particle in the new population (new infeasible solution and infeasible reserve set) is 

dominated by the feasible solution, the number of comparisons is M×(Nq+Nb)
2. Because 



 An Improved Bare-Bones Particle Swarm Algorithm for Multi-Objective Optimization... 9 

there is no need for crowded distance, the computational complexity of updating infeasible 

reserve set is O(M×(Nq+Nb)
2). 

In summary, the computational complexity of the improved BB-MOPSO is 

O(M×(Np+Na)
2)+O(M×(Nq+Nb)

2). 

5. PERFORMANCE ANALYSIS OF IMPROVED BB-MOPSO 

The performance of BB-MOPSO have been tested by a large number of functions and 

compared with other intelligent algorithms [22]. Therefore, the performance of improved 

BB-MOPSO was compared with original BB-MOPSO, NSGA2 and SPEA2 in this paper. 

Although the improved BB-MOPSO is proposed for the problem with constraints, it does 

not change the constraint processing method. So, the classical functions of ZDT1, ZDT3 

and ZDT4 [26] are used to test the ability of the algorithm to solve convex functions, the 

ability to solve discontinuous and multi-connected domain functions, and the ability to 

solve global optimization. In addition, the function of DTLZ3 [27] is used to test the ability 

to converge to global frontier.  

In the test, the algorithm runs 30 times independently for each function. ZDT1 and 

ZDT3 are taken as 100 dimensions, ZDT4 is 30 dimensions, particle number is 100, and 

iteration number is 300. DTLZ3 is taken as 10 dimensions particle number is 500, iteration 

number is 1000, external archive capacity is 50, and grid capacity is 10. The Pareto fronts 

by each algorithm are shown in Figs. 3~6. 

   
(a)     (b) 

   
(c)     (d) 

Fig. 3 Pareto front of test function ZDT1 (a) NSGA2 (b) SPEA2 (c) BB-MOPSO (d) 

Improved BB-MOPSO 



10 Z. Wang, G. Yang, Y. Sun, Y. Li, F. Wu 

   
(a)     (b) 

   
(c)     (d) 

Fig. 4 Pareto front of test function ZDT3 (a) NSGA2 (b) SPEA2 (c) BB-MOPSO (d) 

Improved BB-MOPSO 

   
(a)      (b) 

   
(c)      (d) 

Fig. 5 Pareto front of test function ZDT4 (a) NSGA2 (b) SPEA2 (c) BB-MOPSO (d) 

Improved BB-MOPSO 



 An Improved Bare-Bones Particle Swarm Algorithm for Multi-Objective Optimization... 11 

(a)      (b)  

 (c)    (d)  

      

Fig. 6 Pareto front of test function DTLZ3 (a) NSGA2 (b) SPEA2 (c) BB-MOPSO (d) 

Improved BB-MOPSO 

As can be seen from Figs. 3-6, neither NSGA2 nor SPEA2 can find the real frontier in 

the above test functions. In addition, the solution falls into the local optimum. The improved 

algorithm is significantly different from the original algorithm in the distribution of the 

solution. The solution of the improved algorithm converges more uniformly to the frontier. 

The performance of the four algorithms is quantified by calculating the values of Spacing 

Metric (SP) and Generational Distance Metric (GD) [26, 28]. The SP reflects the 

distribution and diversity of feasible solution set in the objective space, while the GD 

reflects the distance between feasible solution and Pareto frontier, as shown in Tables 1-4, 

where the best results are given in bold.  

Table 1 Performance statistics of ZDT1 

 IM-BB-MOPSO BB-MOPSO NSGA2 SPEA2 

GDAV 1.53E-4 1.49E-4 4.99E-2 5.09E-2 

GDVAR 5.35E-5 5.31E-5 5.21E-2 5.10E-3 

GDBEST 1.53E-4 1.09E-4 1.55E-2 3.97E-2 

GDWORST 4.38E-4 4.05E-4 2.64E-1 6.04E-2 

SPAV 1.67E-2 2.42E-2 1.64E-2 2.25E-2 

SPVAR 1.70E-3 6.60E-3 3.00E-3 1.08E-2 

SPBEST 1.35E-2 1.30E-2 1.15E-2 1.22E-2 

SPWORST 1.97E-2 4.19E-2 2.46E-2 5.91E-2 



12 Z. Wang, G. Yang, Y. Sun, Y. Li, F. Wu 

Table 2 Performance statistics of ZDT3 

 IM-BB-MOPSO BB-MOPSO NSGA2 SPEA2 

GDAV 6.53E-4 7.74E-4 7.87E-2 5.31E-2 

GDVAR 4.57E-5 5.60E-5 4.64E-2 6.50E-3 

GDBEST 5.87E-4 6.78E-4 2.75E-2 4.38E-2 

GDWORST 7.10E-4 8.60E-4 2.15E-1 7.50E-2 

SPAV 2.05E-2 2.91E-2 2.07E-2 3.21E-2 

SPVAR 2.50E-3 4.70E-3 4.60E-3 1.78E-2 

SPBEST 1.54E-2 2.13E-2 1.02E-2 1.59E-2 

SPWORST 2.78E-2 4.33E-2 2.97E-2 7.62E-2 

Table 3 Performance statistics of ZDT4 

 IM-BB-MOPSO BB-MOPSO NSGA2 SPEA2 

GDAV 2.91E-4 3.38E-4 2.73E+0 6.81E+0 

GDVAR 4.46E-4 5.70E+4 1.51E+0 3.96E+0 

GDBEST 1.80E-4 2.80E-4 6.55E-1 2.50E+0 

GDWORST 4.34E-4 5.80E-4 5.58E+0 1.66E+1 

SPAV 1.80E-2 2.58E-2 1.50E-2 1.41E+0 

SPVAR 1.30E-3 7.80E-3 1.61E-2 5.37E+0 

SPBEST 1.39E-2 1.42E-2 0.00E+0 0.00E+0 

SPWORST 2.04E-2 4.36E-2 4.98E-2 2.29E+1 

Table 4 Performance statistics of DTLZ3 

 IM-BB-MOPSO BB-MOPSO NSGA2 SPEA2 

GDAV 3.70E-3 4.50E-3 7.45E+0 1.92E+1 

GDVAR 5.10E-4 7.95E-4 2.42E+0 6.12E+0 

GDBEST 2.70E-3 3.50E-3 4.01E+0 7.34E+0 

GDWORST 3.16E-3 5.50E-3 1.42E+1 2.91E+1 

SPAV 7.73E-2 8.61E-2 4.78E+0 8.50E+1 

SPVAR 4.90E-3 6.40E-3 5.01E+0 3.59E+1 

SPBEST 6.53E-2 7.82E-2 6.17E-1 2.00E+1 

SPWORST 8.94E-2 9.20E-2 1.87E+1 1.73E+2 

 

Tables show that the distribution and diversity of feasible solutions of the improved 

algorithm are much stronger than that of the original algorithm when dealing with ZDT1, 

ZTD3, ZTD4 and DTLZ3. Then, the functions of BNH, TNK and DTLZ8 [27, 29] are used 

to test the ability of constraints processing for the improved BB-MOPSO. BNH and TNK 

are set as follows: particle number is 100, iteration number is 500, maximum capacity of 

reserve set is 100, and grid capacity is 10. DTLZ8 is set as follows: particle number is 500, 



 An Improved Bare-Bones Particle Swarm Algorithm for Multi-Objective Optimization... 13 

iteration number is 1000. Similarly, the algorithm runs 30 times independently for each 

function. The Pareto front by each algorithm is shown in Figs. 7~9. 

(a)  (b)  

 (c)  (d)  

Fig. 7 Pareto front of test function BNH (a) NSGA2 (b) SPEA2 (c) BB-MOPSO (d) 

Improved BB-MOPSO 

(a)  (b)  

(c)  (d)  

Fig. 8 Pareto front of test function TNK (a) NSGA2 (b) SPEA2 (c) BB-MOPSO (d) 

Improved BB-MOPSO 



14 Z. Wang, G. Yang, Y. Sun, Y. Li, F. Wu 

(a)     (b)  

 (c)      (d)  

Fig. 9 Pareto front of test function DTLZ8 (a) NSGA2 (b) SPEA2 (c) BB-MOPSO (d) 

Improved BB-MOPSO 

In order to express the proportion of the elements that are dominant to each other in the 

solution set of the improved algorithm and other algorithms, the C measure [30] is used for 

the test functions of BNH, TNK and DTLZ8. The results are shown in Tables 5-7 and the 

best results are marked in bold. 

Table 5 Performance statistics of TNK 

 IM-BB-MOPSO BB-MOPSO NSGA2 SPEA2 

C(IM-BB) - 3.50E-1 2.32E-1 1.63E-1 

C(BB-D) 3.72E-1 - - - 

C(NSGA2) 3.25E-1 - - - 

C(SPEA2) 2.36E-1 - - - 

SPAV 6.70E-3 7.00E-3 5.60E-3 7.20E-3 

SPVAR 8.56E-4 1.00E-3 9.12E-4 1.90E-3 

SPBEST 4.50E-3 5.10E-3 2.40E-3 4.60E-3 

SPWORST 8.40E-3 9.00E-3 7.10E-3 1.47E-2 

As can be seen from Tables 5-7, the distribution of solutions and the dominant number 

of solutions of the improved algorithm are better than those of other algorithms when 

dealing with BNH, TNK and DTLZ8. Although the distribution of NAGA2 solutions is 

better when dealing with TNK, the distribution of solutions of the improved algorithm is 



 An Improved Bare-Bones Particle Swarm Algorithm for Multi-Objective Optimization... 15 

more stable. This benefits from external archive maintenance based on grid density and 

congestion distance sorting, which maintains a certain number of solutions in each region. 

Table 6 Performance statistics of BNH 

 IM-BB-MOPSO BB-MOPSO NSGA2 SPEA2 

C(IM-BB) - 7.40E-2 9.50E-2 2.47E-1 

C(BB-D) 9.97E-2 - - - 

C(NSGA2) 2.03E-1 - - - 

C(SPEA2) 3.70E-1 - - - 

SPAV 8.42E-1 8.77E-1 8.65E-1 9.40E-1 

SPVAR 4.34E-2 1.66E-1 6.75E-2 2.51E-1 

SPBEST 5.07E-1 6.20E-1 5.15E-1 6.13E-1 

SPWORST 6.97E-1 1.39E+0 7.74E-1 1.57E+0 

Table 7 Performance statistics of DTLZ8 

 IM-BB-MOPSO BB-MOPSO NSGA2 SPEA2 

C(IM-BB) - 1.50E-1 3.55E-2 3.35E-2 

C(BB-D) 1.70E-1 - - - 

C(NSGA2) 1.30E-1 - - - 

C(SPEA2) 9.40E-2 - - - 

SPAV 2.99E-2 3.11E-2 3.60E-2 3.79E-2 

SPVAR 4.44E-3 5.00E-3 9.40E-3 6.90E-3 

SPBEST 1.91E-2 2.16E-2 2.34E-2 2.37E-2 

SPWORST 4.13E-2 4.17E-2 5.80E-2 5.24E-2 

 

In order to further verify the optimization ability of the algorithm, the hybrid 

composition functions of CF4, CF5 and CF6 provided by Liang et al. [31] are used to 

calculate. The shape of three functions is shown in Fig. 10. 

 
(a)   (b)    (c) 

Fig. 10 The shape of three functions (a) CF4 (b) CF5 (c) CF6 



16 Z. Wang, G. Yang, Y. Sun, Y. Li, F. Wu 

For each test function, each algorithm is run 20 times and the maximum fitness 

evaluations are set at 50,000 for all algorithms. The results are compared as shown in Table 

8. The first six groups of data about functions of PSO, CPSO, CLPSO, CMA-ES, G3-PCX, 

DE are from reference [31]. The results show that the improved algorithm has good 

performance in dealing with complex functions. However, the stability of the solution is low 

when dealing with CF6 function. 

Table 8 Performance statistics of CF4, CF5 and CF6 

 PSO CPSO CLPSO CMA-ES G3-PCX DE IM-BBMOPSO 

CF4(AV) 3.14E+2 5.22E+2 3.22E+2 6.16E+2 4.93E+2 3.25E+2 3.10E+2 

CF4(STD) 2.01E+1 1.22E+2 2.75E+1 6.72E+2 1.42E+2 1.48E+1 7.55E+0 

CF5(AV) 8.35E+1 2.56E+2 5.37 E+0 3.59E+2 2.60E+1 1.08E+1 3.22E+0 

CF5(STD) 1.01E+0 1.76E+2 2.61E+0 1.68E+2 4.16E+1 2.60E+1 7.42E-1 

CF6(AV) 8.61E+2 8.53E+2 5.01E+2 9.00E+2 7.72E+2 4.91E+2 4.26E+2 

CF6(STD) 1.26E+2 1.28E+2 7.78E-1 8.32E-2 1.89E+2 3.95E+1 1.07E+1 

 

As can be seen from the above data, the convergence of the improved algorithm is 

slightly weaker when dealing with ZDT1. But the convergence of the improved algorithm is 

slightly stronger for dealing with ZTD3, ZTD4 and DTLZ3. When dealing with BNH, TNK 

and DTLZ8, the improved algorithm has a higher proportion of dominance. When dealing 

with CF4, CF5 and CF6 functions, the search results of the improved algorithm are best, this 

shows that the improved algorithm has a higher accuracy. This is due to the updating of 

particle location for boundary search and the strategy of selecting local particles, which 

enhances the ability of developing local particles. 

Based on the above analysis, the new algorithm proposed in this paper has better 

applicability and accuracy for multi-connected and multi-extreme complex problems, and 

can be used as a good tool for solving multi-objective size optimization problems with 

constraints. 

6. CASE ANALYSIS 

As a case study, a support frame of triangle track wheel is optimized by the new 

algorithm proposed in this paper to verify the applicability and accuracy. The structure is 

shown in Fig. 11, which can be found in reference [32].  In the process of travelling, the 

triangle track wheel has various working conditions such as climbing, crossing obstacle, 

starting, braking, turning, crossing the soft road, snow, muddy land, marsh, sand and so on. 

The support frame has different load-bearing mode under different working-conditions. 

The three wheel supported ground is the worst working condition, which is considered in 

this paper. The loads borne by the support frame include: self-gravity 4.5 kN, hydraulic 

cylinder preload 36 kN, and chassis pressure 78.4 kN. The weight is 0.366 tons. The 

stiffness and strength of the support frame are more sensitive to the change of mass, and the 

restriction relationship is more significant. Local extremum and non-convergence is easy to 

occur in the calculation process when the stress is taken as performance index, which is a 



 An Improved Bare-Bones Particle Swarm Algorithm for Multi-Objective Optimization... 17 

nonlinear MOP with constraints. In this section, multi-objective size optimization of 

support frame is carried out.  

guide 
wheels

track drive wheel

tensioning
device

tension 
wheels

support
frame

load-bearing 
wheels

(a) (b)  

Fig. 11 The structure of support frame (a) Triangle track wheel (b) Support frame 

6.1 Mathematical Model of Multi-objective Size Optimization  

The parametric model of the support frame is established as shown in Fig. 12. Seven 

important dimensions (plate thickness of local area) are marked including x1~x7, which are 

design variables.  

x5

x4
x3 x2 x1

x7

x6

 

Fig. 12 Parametric model of the support frame 

According to the design and manufacturing process [33], each size parameter of the 

support frame is constrained, and the first-order natural frequency should exceed the 

excitation frequency transmitted by the engine. The constraints can be expressed as 

  
min max

0

, 1, 2, 7
. .

( )

i i i

i

x x x i
s t

f x f

   



, (15) 

where ximin and ximax are the minimum and maximum value of xi. f(xi) is the first-order 

natural frequency of the support frame. f0 is the excitation frequency of engine, the value is 

50 Hz. 

For the support frame, the optimization objective is how to reduce the weight, 

displacement and stress. So, the objective function can be defined as shown in Eq. (16): 



18 Z. Wang, G. Yang, Y. Sun, Y. Li, F. Wu 

  
max max

min  [ ( ), ( ), ( )]
i i i

x d x V x , (16) 

where σmax(xi) and dmax(xi) are maximum stress and displacement of support frame. V(xi) is 

the weight. xi is the design parameter. 

So, the mathematical model of size optimization is established, as shown in Eq. (17), 

which belongs to a typical nonlinear MOP with constraints. 

  
max max

min max

0

Find  ( 1, 2, 3, 4, 5, 6, 7)

min   ( ), ( ), ( )

    

         ( )  

i

i i i

i i i

i

x i

x d x V x

s.t. x x x

f x f








 
 

. (17) 

In the optimization model, the stress, displacement, volume and natural frequency of the 

support frame lack a functional relationship with the design variables. Therefore, the 

approximate model of structural performance and parameters is established in the next 

section. 

6.2 Setting an Approximate Model 

The response surface methodology is a method to establish an approximate model. It 

synthesizes experimental design and mathematical model, and obtains the relationship 

between design objectives and design variables through limited experimental design of the 

set of sample points in the designated design space. It is also possible to smooth the 

response function and reduce the "numerical noise", which is conducive to faster 

convergence to the global optimum in optimization process. 

In this paper, a second-order polynomial is used to construct the response surface 

methodology. The basis function is as follows: 

  
0

11 1

 

Y
k k k

i i ii i ij i ij
ii i

i j

x x x x   
 



      , (18) 

where β is an unknown coefficient; k is the number of design variables, k=7; Y is the 

predicted response value; β0, βi and βii are deviation term coefficient, linear deviation term 

coefficient and second-order deviation term coefficient, respectively;  βij is the interaction 

coefficient. 

Four approximate models of response are obtained through experimental design as 

follows: 

  

T

1 1

T

2 2

T

3 3

T

4 4

f = X C

f = X C

f = X C

f = X C









, (19) 

where f1, f2, f3, f4 represent the approximate model of stress, natural frequency, displacement 

and volume, respectively. The parameters of approximate model are given in Table 9. 



 An Improved Bare-Bones Particle Swarm Algorithm for Multi-Objective Optimization... 19 

Table 9 Parameters of approximate model 

X C1  C2 C3 C4 

Constant 2.60E+2 3.36E+1 3.30E-1 2.80E+7 

x1 9.84E-1 3.73E-1 -1.10E-2 3.81E+5 

x2 -5.42E+0 2.44E-1 -5.00E-3 3.82E+5 

x3 2.57E+0 1.19E+0 -4.00E-3 7.96E+4 

x4 -1.04E+1 1.79E-1 3.00E-3 4.59E+4 

x5 -7.52E+0 3.81E-1 1.00E-2 5.39E+4 

x6 -2.11E+0 7.36E-1 -5.00E-3 6.87E+4 

x7 -1.82E+0 2.19E-1 2.00E-3 5.16E+4 

x12 -1.03E-1 -3.00E-3 0 4.29E+2 

x22 1.10E-1 -2.00E-3 0 4.53E+2 

x32 -7.80E-2 -2.10E-2 0 6.26E+2 

x42 2.34E-1 2.00E-3 0 4.89E+2 

x52 1.42E-1 -5.00E-3 0 5.82E+2 

x62 4.70E-2 -1.30E-2 0 9.01E+2 

x72 1.00E-2 1.00E-3 0 5.20E+2 

x1·x2 8.40E-2 -6.00E-3 0 -8.96E+1 

x1·x3 -3.90E-2 -1.00E-3 0 1.69E+1 

x1·x4 2.80E-2 1.00E-3 0 -1.70E+2 

x1·x5 2.10E-2 -1.00E-3 0 -1.41E+2 

x1·x6 1.00E-3 -1.00E-3 0 -3.64E+2 

x1·x7 -2.20E-2 0 0 -1.45E+2 

x2·x3 1.40E-2 3.00E-3 0 9.87E+0 

x2·x4 2.10E-2 1.00E-3 0 -1.74E+2 

x2·x5 4.60E-2 0 0 -1.63E+2 

x2·x6 7.00E-3 2.00E-3 0 -3.91E+2 

x2·x7 -5.00E-3 -1.00E-3 0 -1.38E+2 

x3·x4 -2.70E-2 -2.00E-3 0 -1.19E+1 

x3·x5 4.50E-2 -3.00E-3 0 1.44E+0 

x3·x6 -3.70E-2 0 0 1.01E+1 

x3·x7 -1.70E-2 -1.00E-3 0 3.03E-1 

x4·x5 4.40E-2 -1.00E-3 0 -2.86E+1 

x4·x6 6.00E-3 -1.00E-3 0 -3.97E+1 

x4·x7 5.00E-2 -2.00E-3 0 -8.88E+0 

x5·x6 3.00E-3 0 0 -6.02E+0 

x5·x7 2.00E-3 0 0 6.68E+0 

x6·x7 4.80E-2 3.00E-3 0 1.47E+1 



20 Z. Wang, G. Yang, Y. Sun, Y. Li, F. Wu 

After establishing the approximate model, it is necessary to evaluate its predictive 

power. Commonly used evaluation index are mean error (AE), maximum error (ME), root 

mean square error (RMSE), and correlation coefficient (R2). AE is the average of all errors 

and can be expressed as 

  

'

1

1

( )

AE

N

i

i

f f

N








, (20) 

where i is the i-th sample point; fi is the finite element analysis result of the i-th sample point;  

fi’ is the calculated value of the approximate model of the i-th sample point; N is the number 

of sample points. The parameters of the approximate model are given in Table 4.  

ME is the maximum value of all errors, that is max (fi -fi’). RMSE also known as 

standard error and can be expressed as: 

  

' 2

1

( )

RMSE
1

N

i i

i

f f

N p






 


, (21) 

where p is the number of terms of the polynomial. 

R2 is an index for evaluating the fitting accuracy of the approximate model to the 

experimental data. The closer R2 is to 1, the smaller the error. R2 can be described as: 

  

2 ' 2

2 1 1

2

1

( ) ( )

R

( )

N N

i i i

i i

N

i

i

f f f f

f f

 



  





 


, (22) 

where f  is the average of the finite element analysis of all sample points. 

Table 10 shows the error analysis results of approximate model for different 

performance functions. It shows that the AE of the four responses is less than 0.1, ME is less 

than 0.3, RSME is less than 0.2, R2 is greater than 0.9, which shows that the approximate 

model has high prediction accuracy. Therefore, the approximate model can be used for 

subsequent optimization design. It is not difficult to see that the accuracy of stress 

prediction is little poor, because stress is easy to produce numerical errors and has high 

non-linearity. While the volume prediction accuracy is very high, because volume function 

of structure is an explicit function of design variables in size optimization. 

Table 10 Error analysis 

Variables Mass Stress Displacement Mode 

AE 1.76E-3 5.04E-2 2.04E-2 2.85E-2 

ME 5.12E-3 2.05E-1 6.58E-2 6.52E-2 

RMSE 2.17E-3 6.42E-2 2.70E-2 3.39E-2 

R2 9.99E-1 9.23E-1 9.86E-1 9.83E-1 



 An Improved Bare-Bones Particle Swarm Algorithm for Multi-Objective Optimization... 21 

6.3 Multi-Objective Size Optimization 

In this section, the mathematical model in Eq. (17) are optimized by using the 

BB-MOPSO and the improved BB-MOPSO. The optimization results of design variables 

are shown in Table 11.  

Table 11 The results of size optimization 

Method x1 x2 x3 x4 x5 x6 x7 

BB-MOPSO 20 13.96 10 10 14.68 10 19.69 

Improved 

BB-MOPSO 
20 10 11.15 16.49 10 10 20 

 

The model is modified according to the optimization results in Table 11. A new model is 

established and the finite element analysis of the worst working conditions is carried out. 

The results using the BB-MOPSO and the improved BB-MOPSO are listed in Table 12 to 

compare the applicability and accuracy for multi-connected and multi-extreme complex 

problem.  

Table 12 The comparison of mechanical performance 

Method Mass (kg) Displacement (mm) Stress (MPa) 

Not optimized 376.65 0.217 132.90 

BB-MOPSO 337.28 0.196 73.111 

Improved BB-MOPSO 335.70 0.217 68.167 

 

From Table 12, the mechanical performance of the support frame is improved by 

optimization using BB-MOPSO and improved BB-MOPSO. However, the optimization 

model using BB-MOPSO obtains better results in displacement, but it is worse in mass and 

stress than the improved BB-MOPSO. Generally, displacement reflects the overall stiffness 

of the structure, while stress reflects the local strength. The results of Table 12 show that the 

improved BB-MOPSO has better boundary and cross-region search ability. So, the 

improved BB-MOPSO is more suitable for structures with requirements of light weight and 

high strength. 

7. CONCLUSION 

In this paper, an improved BB-MOPSO is proposed to solve the multi-objective size 

optimization problems with non-linearity and constraints in structural design and 

optimization. The ability of particle searching for boundary and cross-region is enhanced by 

modifying the updating form of particle location. The optimal solution distribution in the 

global scope is maintained by combining the spatial grid density with the ranking of 

crowding distance. The global best positions are determined by increasing the probability 

allocation strategy changing with time, and the exploration of unknown region and the 



22 Z. Wang, G. Yang, Y. Sun, Y. Li, F. Wu 

convergence of the algorithm are increased. The performance of the improved algorithm is 

verified by the test functions. As a case study, a support frame of triangle track wheel is 

optimized by the BB-MOPSO and improved BB-MOPSO. The results show that the 

improved algorithm can enhance the search ability of boundary and cross-region and 

diversity, which can effectively solve the multi-objective size optimization problem with 

non-linearity and constraints. 

Acknowledgement: This research is supported by Scientific and Technological Innovation 

Programs of Higher Education Institutions in Shanxi (2021L291), Fundamental Research Program 

of Shanxi Province (202103021223290), Taiyuan University of Science and Technology Scientific 

Research Initial Funding (20212040) and National Natural Science Foundation of China 

(52205278). 

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