*2.2. Suspension Structure Optimization Based on Non-Dominated Sorting Genetic Algorithm III (NSGA-III) and Technique for Order Preference by Similarity to Ideal Solution (TOPSIS)*

In single-objective optimization problems, since only one objective function needs to be considered optimal, the optimal solution can be found by some common mathematical methods. In multi-objective optimization problems, however, there may be certain

constraints between the various objective functions. When one of the objectives reaches optimality, the performance of the other objectives may be unacceptable. Therefore, solving the multi-objective problem for the optimal solution is very difficult. Further, in the solution of multi-objective optimization problems, the non-dominated solutions set, the Pareto set, is usually used to represent the acceptable better solution. The solutions of the Pareto set are non-dominated by any other solution, and are considered as equally optimal solutions. With the TOPSIS method, it is possible to select the optimal solution needed from the non-dominated solution set.

#### 2.2.1. Non-Dominated Sorting Genetic Algorithm III (NSGA-III)

The non-dominated genetic sorting algorithm is a multi-objective evolutionary algorithm based on a genetic algorithm that simulates the evolutionary behaviour of natural populations in terms of "survival of the fittest" and the reproduction of offspring.

In natural selection, individuals in a population that are well adapted to nature are more likely to survive and thus reproduce their offspring, passing on the genes for good traits from generation to generation. The process of reproduction produces offspring with different traits from their parents due to crossover and variation, ensuring a diversity of traits in the population. Through multiple generations of evolution, the individuals that eventually survive in the population are those that are well adapted to nature.

Compared to NSGA, NSGA-II reduces the computational complexity of nondominated sorting, introduces and elitism strategy which could help prevent the loss of good solutions, and doesnot require specifying the sharing parameter. With the above improvements, the iterative convergence speed of NSGA-II is improved, and the computational complexity is reduced from *O*(*MN*3) to *O*(*MN*2). NSGA-II, when faced with high-dimensional multi-objective optimization problems with more than three objectives, suffers from the shortcomings of convergence. Compared to NSGA-II, NSGA-III, based on the reference point selection mechanism, is effective in reducing computational complexity and improving convergence for high-dimensional multi-objective optimization problems with a large number of objectives. NSGA-III improves on the population update selection mechanism of NSGA-II by providing and adaptively updating a number of well-distributed reference points to help maintain diversity among population members. With these improvements, the computational complexity of NSGA-III is the greater of *O*(*MN*2) or *O*(*N*<sup>2</sup> log*M*−<sup>2</sup> *N*).

#### 2.2.2. Technique for Order Preference by Similarity to Ideal Solution (TOPSIS)

The TOPSIS method is a ranking method that approximates an ideal solution and makes full use of the information from raw data to accurately reflect the gaps between individual evaluation objects. It is a method of ranking a limited number of evaluation objects according to their proximity to an idealised target, and is an evaluation of the relative merits of the available objects. Therefore, TOPSIS is a common and effective method for multi-objective decision analysis.

In this study, since there are a finite number of evaluation objects in the non-dominated solution set, and each object has four known evaluation indicators, the TOPSIS method of scoring is as follows.

Assume that there are n solutions in the non-dominated solution set, i.e., there are n evaluation objects and four evaluation indicators, consisting of a standardised matrix.

$$Z = \begin{bmatrix} z\_{11} & z\_{12} & z\_{13} & z\_{14} \\ z\_{21} & z\_{22} & z\_{23} & z\_{24} \\ \vdots & \vdots & \vdots & \vdots \\ z\_{n1} & z\_{n2} & z\_{n3} & z\_{n4} \end{bmatrix} \tag{17}$$

Define the minimum value

$$\mathbf{Z}^- = (\mathbf{Z}\_1^-, \mathbf{Z}\_2^-, \mathbf{Z}\_3^-, \mathbf{Z}\_4^-) = (\min\{z\_{11}, z\_{21}, \dots, z\_{n1}\}, \min\{z\_{12}, z\_{22}, \dots, z\_{n2}\}, \min\{z\_{13}, z\_{23}, \dots, z\_{n3}\}, \min\{z\_{14}, z\_{24}, \dots, z\_{n4}\}) = (\min\{z\_{11}, z\_{21}, \dots, z\_{n1}\}, \min\{z\_{12}, z\_{22}, \dots, z\_{n2}\}, \min\{z\_{13}, z\_{23}, \dots, z\_{n3}\}, \min\{z\_{14}, z\_{24}, \dots, z\_{n4}\}) = (\min\{z\_{11}, z\_{22}, \dots, z\_{n3}\}, \min\{z\_{13}, z\_{24}, \dots, z\_{n4}\}, \min\{z\_{14}, z\_{23}, \dots, z\_{n4}\}) = (\min\{z\_{11}, z\_{22}, \dots, z\_{n3}\}, \min\{z\_{12}, z\_{23}, \dots, z\_{n3}\}, \min\{z\_{13}, z\_{24}, \dots, z\_{n4}\}) = (\min\{z\_{11}, z\_{22}, \dots, z\_{n3}\}, \min\{z\_{14}, z\_{23}, \dots, z\_{n4}\}, \min\{z\_{15}, z\_{26}, \dots, z\_{n4}\}) = (\min\{z\_{11}, z\_{22}, \dots, z\_{n4}\}, \min\{z\_{15}, z\_{25}, \dots, z\_{n5}\})$$

*Di*

Define the distance of the *i*(*i* = 1, 2, ··· , *n*) evaluation object from the minimum value − = 4 ∑ (*Zj* <sup>−</sup> − *zij*) 2

*j*=1 The unnormalised score of the *i*(*i* = 1, 2, ··· , *n*) evaluation object can be calculated *Si* = <sup>1</sup> *Di* −

Normalised scores for the *<sup>i</sup>*(*<sup>i</sup>* <sup>=</sup> 1, 2, ··· , *<sup>n</sup>*) evaluation objects *Si* <sup>=</sup> *Si <sup>n</sup>* ∑ *Si* .

The individual with the highest score based on the normalised score is selected as the optimal solution.

*i*=1

#### 2.2.3. Parameters Optimization of the Suspension System Based on NSGA-III and TOPSIS

This section describes the use of NSGA-III to optimize the structural parameters of a robotic suspension system and to select the optimal solution in a non-dominated solutions set using the TOPSIS method. The objective of the optimized robotic suspension system problem is to find optimal values for the system parameters (damper length, cross arm length, stiffness and damping coefficient), aiming to minimise the four evaluation metrics developed (total suspension mass, suspension centre of mass height, suspension fluctuation coefficient and suspension compression coefficient) to obtain the highest score. Algorithm 1 describes the process of optimizing the optimum parameter values for the quarter passive robot suspension system.

#### **Algorithm 1: Parameters Optimization of the Suspension System Based on NSGA-III and TOPSIS**


#### **3. Results and Discussion**

To investigate the advantages of multi-objective evolutionary algorithms in multiobjective optimization problems, we compared the performance of NSGA-III with some multi-objective evolutionary algorithms. At the same time, we built a quarter suspension model in Adams software to perform a dynamics analysis of the parameter optimization results and compare the time response of the parameter combinations in the optimal solution set for the suspension dynamics analysis.

### *3.1. Comparison of Multi-Objective Evolutionary Algorithms' (MOEA) Results*

The non-dominated ranking genetic algorithm II (NSGA-II), the covariance matrix adaptive evolution strategy (CMA-ES), the third generation generalised difference algorithm (GDE3), the decomposition-based multi-objective evolution algorithm (MOEAD), the multi-objective particle swarm optimization algorithm (OMOPSO), the speed-constrained multi-objective particle swarm optimization algorithm (SMOPSO) and the strength Pareto Evolutionary Algorithm (SPEA2) as representatives of multi-objective evolutionary algorithms were compared with NSGA-III. The parameter settings for the experimental tests are shown in Table 2.


**Table 2.** Parameter setting of the optimization test.

The differential equation model based on Equations (8)–(12) was optimally solved within the parameter intervals shown in Table 2 using the Wolfram Language NDSolve function and the multi-objective evolutionary algorithm described above. NDSolve function is a general-purpose numerical solver for differential equations. It can solve many ordinary differential equations (ODEs) and some partial differential equations (PDEs). The set of Pareto solutions obtained from each evolutionary algorithm solution was selected using the TOPSIS method to find the optimal solution. The optimal solutions obtained by the eight evolutionary algorithms were integrated and scored again using the TOPSIS method.

The experimental results of the optimization of the multi-objective dynamics model using NSGA-III on a quarter suspension model compared to other multi-objective evolutionary algorithms are shown in Tables 3–5. Each algorithm was run five times and standard deviations analysed to determine the stability of the algorithm. The experimental results show that the NSGA-III achieved optimal evaluation metrics for best, worst and average results with the lowest standard deviation. Meanwhile, among the 40 optimal solution sets obtained by the eight multi-objective evolutionary algorithms, the optimal solution obtained by the NSGA-III achieved the highest score in the TOPSIS composite score. This indicates that the NSGA-III outperforms other comparative algorithms in terms of solution quality and robustness in solving multi-objective suspension optimization design problems. A comparison of the computational time of NSGA-III with other multi-objective evolutionary algorithms is presented in Table 6. All optimizations were conducted on a laptop with Intel i7-6700HQ (2.60 GHz) and 16 GB RAM. Both NSGA-III and MOEAD took longer than the others in terms of computation time. However, there was no real-time requirement for the task of suspension design. Therefore, the average computation time of 183.31 s for NSGA-III is acceptable, which does not affect the excellent performance of NSGA-III in accomplishing the design problem targeted in our study.


**Table 3.** Optimum score of the NSGA-III in comparison with other evolutionary algorithms.

**Table 4.** Based on TOPSIS method, the optimal solution sets integrated by eight evolutionary algorithms were comprehensively scored, and the top five solutions are listed.



**Table 5.** The top five scoring solutions correspond to the four design objective values based on Equations (13)–(16), respectively.

**Table 6.** Computational time of the NSGA-III in comparison with other evolutionary algorithms.

