In a multiobjective particle swarm optimization algorithm, selection of the global best particle for each particle of the population from a set of Pareto optimal solutions has a significant impact on the convergence and diversity of solutions, especially when optimizing problems with a large number of objectives. In this paper, a new method is introduced for selecting the global best particle, which is minimum distance of point to line multiobjective particle swarm optimization (MDPLMOPSO). Using the basic concept of minimum distance of point to line and objective, the global best particle among archive members can be selected. Different test functions were used to test and compare MDPLMOPSO with CDMOPSO. The result shows that the convergence and diversity of MDPLMOPSO are relatively better than CDMOPSO. Finally, the proposed multiobjective particle swarm optimization algorithm is used for the Pareto optimal design of a fivedegreeoffreedom vehicle vibration model, which resulted in numerous effective tradeoffs among conflicting objectives, including seat acceleration, front tire velocity, rear tire velocity, relative displacement between sprung mass and front tire, and relative displacement between sprung mass and rear tire. The superiority of this work is demonstrated by comparing the obtained results with the literature.
The dynamic behavior of a vehicle is critical for determining driving performance and ride comfort. Simultaneously, it also influences the dynamic loads applied to both the road and main axles, which can cause damage to road surfaces and early failure of chassis components. However, determining the system parameters usually conflicts with the main objectives of ride comfort, driving performance, and low dynamic loads. Therefore, optimization of these parameters has been actively studied. With the development of computational capacity and computing technologies, NarimanZadeh et al. [
Optimization is becoming an important research tool in scientific research and engineering practice. The multiobjective optimization problem (MOP) is normally considered the minimizing of an objective and can be in the form of the following equation:
Here,
Multiobjective evolutionary algorithms (MOEAS) have been proposed to resolve multiobjective problems, for instance, the nondominated sorting genetic algorithm II (NSGAII) [
To improve previous solutions, a new solution called particle swarm optimization (PSO) [
In recent years, PSO has been investigated to solve MOP problems.
Ray and Liew [
Parsopoulos and Vrahatis [
Hu and Eberhart [
Currently, the Pareto Dominance method has become increasingly popular, but none of the proposals to extend PSO to solve multiobjective optimization problems used a secondary population. This may limit the algorithm performance. However, in more recent papers, these ideas have already been incorporated by other authors. The most representative proposals are the following.
Coello and Lechuga [
Hu et al. [
Fieldsend and Sing [
Mostaghim and Teich [
Li [
The Raquel and Naval Jr. [
In this paper, a minimum distance of point to line multiobjective particle swamp optimization (MDPLMOPSO) is developed. The main differences between our approach and other proposals that exist in the literature are as follows:
This algorithm adopts a bounded archive mechanism to store the global Pareto optimal solutions. The global best guide of each particle in the population was selected from the archive.
A minimum distance of point to line method was utilized to find the global best guide of each particle in the population. The selection of the global best guide of the particle swarm is crucial in a multiobjectivePSO algorithm. It affects the convergence capability of the algorithm and maintains an adequate spread of nondominated solutions. No other proposal uses the mechanism in the way it is adopted in this paper.
The mutation operator of MOPSO was adapted because of the exploratory capability it could provide the algorithm by initially performing mutation on the entire population and then rapidly decreasing its coverage over time [
This algorithm is evaluated based on representative functions along with a performance comparison with the MOPSO algorithm [
Particle swarm initialization;
Set swarm set number
Set the range for variable
Particle speed control, variable
Swarm position and speed initialization, particle creation in random,
Parameter setting evolution;
Max Iterations
Maximum and Minimum for inertia weight, value as
Learning factors
Evaluate objective function value
For
For
Evaluate
Particle normalization
Particle position initialization
Archive initialization
Store the nondominated solutions found in
If iterations cannot achieve
Find
Line
Distance
While
Particle speed and position update
where
If
Perform mutation on
Evaluate
Archive update
Insert the new nondominated solution in
Update the personal best solution of each particle in
Increment iteration counter
Cycle count increase until iteration requirements are achieved.
The flowchart of the proposed algorithm for MDPLMOPSO algorithm is shown in Figure
Flowchart of the proposed algorithm for multiobjective optimization problems.
As mentioned previously, several important MOPSO methods exist [
To overcome the disadvantages of the aforementioned methods, a MDPLMOPSO is proposed to find the global best guide for each particle. First, the basic concept of the minimum distance of point to line (MDPL) is introduced. Later, this paper explains how this method determines the global best guide for each particle of the population in the objective.
In the twodimensional coordinate system, a straight line, line
Distance of point to line 2D objective space.
Point
Similarly, in the threedimensional coordinate system, a straight line can be produced through the origin point
Distance of point to line in 3D objective space.
Point
Here, we use twoobjective optimization as an example to illustrate how to find global optimal guide.
Using the basic concept of a distance of point to line and considering the objective space, finding the global best guide
First, we draw line
Second, we calculate distance
Finally, archive particle
In other words, each particle with a minimum distance to the line of the archive member must select that archive member as the global best guide. Therefore, MDPLMOPSO can determine the most appropriate guide as its global guide for each particle in the population.
As shown in Figure
Finding the global best guide in twoobjective space.
Figure
Finding the global best guide for each particle.
To evaluate the performance of the proposed MDPLMOPSO, six representative test functions denoted as DT1, ZDT2, ZDT3, ZDT4, and ZDT6 are employed, which are detailed in Table
Characteristics of test functions.
Problem 

Bounds  Objective functions  Optimal solutions  Comments 

ZDT1  30 



Convex 


ZDT2  30 



Nonconvex 


ZDT3  30 



Convex 


ZDT4  10 



Nonconvex 


ZDT6  10 



Nonconvex 
In addition, convergence and diversity are the significant criteria for algorithm performance. Generational distance (GD) is one measure to represent convergence metric, and Spacing (
(1) Generational distance (GD): convergence measure (GD) is a measure of the convergence range of the multiobjective optimization algorithm. First, we find a set of solutions from the true Pareto optimal front in the objective space. Second, for each solution obtained with an algorithm, we compute the minimum Euclidean distance from chosen solutions on the Pareto optimal front. The root mean square is expressed as the convergence measure GD. As shown in Figure
Convergence measure GD.
(2) Diversity measure
Diversity measure
To verify the performance of the proposed algorithm, a comparison analysis is provided between the results and CDMOPSO that are conducive to archive update and maintenance as well as
Raquel and Naval Jr. [
Figures
GD results of optimal solutions for every problem.
Optimization algorithm  ZDT1  ZDT2  ZDT3  ZDT4  ZDT6 

MPDLMOPSO  0.001165 
0.000183  0.017570  0.001670  0.115910 
0.000075 
0.000025  0.001605  0.000263  0.017119  
CDMOPSO  0.001565 
0.001017  0.017990  0.002667  0.096690 
0.000243 
0.000060  0.000983  0.000347  0.009128 
Optimization algorithm  ZDT1  ZDT2  ZDT3  ZDT4  ZDT6 

MPDLMOPSO  0.001094 
0.000155  0.016820  0.001634  0.093060 
0.000074 
0.000016  0.001564  0.000253  0.012801  
CDMOPSO  0.001539 
0.000994  0.017550  0.002613  0.084730 
0.000233 
0.000059  0.000959  0.000273  0.007736 
Nondominated solutions with MDPLMOPSO and CDMOPSO on ZDT1.
MDPLMOPSO
CDMOPSO
Nondominated solutions with MDPLMOPSO and CDMOPSO on ZDT2.
MDPLMOPSO
CDMOPSO
Nondominated solutions with MDPLMOPSO and CDMOPSO on ZDT3.
MDPLMOPSO
CDMOPSO
Nondominated solutions with MDPLMOPSO and CDMOPSO on ZDT4.
MDPLMOPSO
CDMOPSO
Nondominated solutions with MDPLMOPSO and CDMOPSO on ZDT6.
MDPLMOPSO
CDMOPSO
From Figures
Table
Table
Therefore, it can be concluded that MDPLMOPSO has improved convergence and diversity performance compared with that of the CDMOPSO algorithm.
NarimanZadeh et al. [
In this paper, an improved multiobjective particle swarm optimization (MDPLMOPSO) algorithm is used to optimize the fiveDOF vehicle driving dynamics in [
A fivedegreeoffreedom vehicle with passive suspension, which is adopted from [
A fivedegreeoffreedom vibration model for a vehicle with passive suspension adopted from [
The fixed parameters and design variables of the vehicle are shown in Table
Vehicle parameters.
Parameters  Symbol  Unit  Value 

Forward tire mass 

kg  40 
Rear tire mass 

kg  35.5 
Seat mass 

kg  75 
Sprung mass 

kg  730 
Momentum inertia of sprung mass 

kg·m^{2}  130 
Forward tire relation to the center of mass 

m  1.011 
Rear tire relation to the center of mass 

m  1.803 
Forward tire stiffness coefficient 

N·m^{−1}  175500 
Rear tire stiffness coefficient 

N·m^{−1}  175500 
Seat stiffness coefficient 

N·m^{−1}  50000–150000 
Forward suspension stiffness coefficients 

N·m^{−1}  10000–20000 
Rear suspension stiffness coefficients 

N·m^{−1}  10000–20000 
Seat damping coefficient 

(N·s) m^{−1}  1000–4000 
Forward suspension damping coefficient 

(N·s)·m^{−1}  500–2000 
Rear suspension damping coefficient 

(N·s)·m^{−1}  500–2000 
Seat position in relation to the center of mass 

m  0–0.5 
Kinetic energy of vehicle vibration fivedegreeoffreedom model:
Potential energy of vehicle vibration fivedegreeoffreedom model:
Dissipation energy of vehicle vibration fivedegreeoffreedom model:
Substituting (
Then, a matrix form of the vibration model of (
Road activation vector.
It is assumed that the vehicle moves at a constant velocity
To optimize the suspension system, corresponding stiffness and damping parameters are used as the design variables, which are defined as
The five conflicting objectives are seat acceleration
The Pareto front for the vibration system is obtained using MDPLMOPSO and is shown in Figures
Obtained Pareto front for seat acceleration and forward/rear tire velocity.
Forward tire
Rear tire
Pareto front for seat acceleration and relative displacement between sprung mass and forward/rear tire.
Forward tire
Rear tire
Figure
Nondominated Pareto fronts for another chosen set of objective functions are shown in Figure
The corresponding values of objective functions and design variables of these optimum design points are provided in Tables
Objective function values and tradeoff design variables of [
Point 










A1 (by this work)  50000  10000  10000  4000  1318.9748  500  0.5  1.4796  0.1398 
B1 (by this work)  150000  20000  19288.60  1235.31  2000.000  2000  0.5  2.94514  0.13248 
C1 (by this work)  150000  15881.94  10000  4000  2000  1382.29  0.500  2.22834  0.13386 
C1_1 (by [ 
111177  10000  10117.700  3858.82  1264.710  1852.940  0.49804  2.92268  0.40883 
C1_1′ (by [ 
144263.6  10003.33  10029.590  3119.21  1232.827  1989.187  0.49635  2.91323  0.40863 
Objective function values and tradeoff design variables of [
Point 










A2 (by this work)  50000  10000  10000  1000  2000  500  0  1.81877  0.13561 
B2 (by this work)  60399.789  10000  10000  1000  500  500  0  2.50843  0.12994 
C2 (by this work)  88190.529  10013.127  10013.979  1003.96  1247.35  502.018  0.0008  2.02698  0.13234 
C2_2 (by [ 
131961  10000  10000  3400  1282.35  1613.628  0.47112  2.94568  0.43178 
C2_2′ (by [ 
139269.5  10000.420  10036.32  3485  1259.88  1511.770  0.46863  2.95293  0.43162 
Objective function values and tradeoff design variables of [
Point 










A3 (by this work)  50000  10000  10000  4000  1483.59  500  0.5  1.0652  0.01566 
B3 (by this work)  50000  20000  20000  1000  2000  500  0.0813  2.0172  0.007398 
C3 (by this work)  50000  10865.21  10865.21  3740.44  1963.46  500  0.0745  1.3981  0.008484 
C3_3 (by [ 
134706  10000.000  10078.400  3768.22  1999.95  1432.873  0.49299  2.80956  0.089026 
C3_3′ (by [ 
147210.7  10001.550  10003.770  2505.88  2000  1511.770  0.50000  2.82  0.09 
Objective function values and tradeoff design variables of [
Point 










A4 (by this work)  50000  10000  10000  4000  1308.95  500  0.5  1.47959  0.02025 
B4 (by this work)  50000  10000  20000  1967.74  1550.77  2000  0.5  1.91086  0.00819 
C4 (by this work)  50000  10000  17870.18  4000  1219.67  1118.15  0.5  1.80211  0.01241 
C4_4 (by [ 
98235.30  10000.25  19999.92  3482.94  1336.78  1999.61  0.49911  2.95040  0.05156 
C4_4′ (by [ 
146265.70  10000  18313.70  2917.65  1311.77  1994.12  0.5  2.97674  0.05345 
The time behavior of the seat acceleration of the tradeoff design points of these figures and the optimum point proposed in [
Time response of seat acceleration of design points C1, C1_1, and C1_1′.
Time response of seat acceleration of design points C2, C2_2, and C_2′.
Time response of seat acceleration of design points C3, C3_3, and C3_3′.
Time response of seat acceleration of design points C4, C4_4, and C4_4′.
In this work, an improved multiobjective particle swarm optimization algorithm was introduced. A new method was proposed to determine the global best guide particle for the population particle based on the basic concept of the minimum distance of point to line (MDPL). To consider the optimization performance of the proposal algorithm, five twoobjective test functions were used. The results were compared with the results of the CDMOPSO algorithm. The results indicate that the improved multiobjective particle swarm optimization algorithm is a successful method. Moreover, MDPLMOPSO has been used to optimally design a fivedegreeoffreedom vehicle vibration model, which resulted in numerous effective tradeoffs between conflicting objectives, including seat acceleration, front tire velocity, rear tire velocity, relative displacement between sprung mass and front tire, and relative displacement between sprung mass and rear tire. A comparison of the obtained results and the literature demonstrates the superiority of this work.
Matrices
The authors declare that they have no conflicts of interest.