Advances in Engineering Software 199 (2025) 103797

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Advances in Engineering Software

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Research paper

Multi-objective optimization of automotive seat frames using

machine learning
Haifeng Chen , Ping Yu , Jiangqi Long *
College of Mechanical and Electrical Engineering, Wenzhou University, Chashan Street, Ouhai District, Wenzhou 325035, China

A R T I C L E I N F O A B S T R A C T

Keywords: The optimal design of automobile seats plays an important role in passenger safety in high-speed accidents. In
Seat frame order to enhance the accuracy of the prediction of the input variables and output response of the seat, a hybrid
Machine learning machine learning prediction model that combines the improved gray wolf optimizer (IGWO) and back propa-
Improved grey wolf optimizer
gation neural network (BPNN) has been proposed, and the prediction effect of the model was validated using the
Multi-objective optimization
seat simulation data. Initially, based on the experimental data, finite element models were developed for eight
Multi-criteria decision-making
typical working conditions of automobile seats and their accuracy was validated. Subsequently, the energy ab-
sorption to mass ratio method was employed to screen the design variables, resulting in the selection of 17
thickness variables and 15 material variables. Thereafter, the gray wolf optimizer (GWO) algorithm underwent
enhancement through the incorporation of the dynamic leadership hierarchy (DLH) mechanism and the revision
of the positional formula, yielding the IGWO algorithm. Following this, the IGWO algorithm was applied to
optimize the hyperparameters of BPNN, culminating in the establishment of the IGWO-BPNN model. Ultimately,
the seat multi-objective optimization design process was addressed using multi-objective gray wolf optimizer
(MOGWO) to achieve the Pareto frontier, while the decision-making was conducted using the combined
compromise solution (CoCoSo) method to determine the best trade-off solution. Furthermore, the effectiveness of
the proposed optimal design method is evidenced by comparing the baseline design, simulation analysis, and
optimal design methods. The results indicate that the optimized automotive seat frame achieves a reduction in
cost by 20.7 % and mass by 22.9 %, simultaneously maintaining safety performance. Consequently, the proposed
optimization design methodology is demonstrated to be highly effective for the multi-objective optimization
design of automotive seat frames.

1. Introduction worldwide. Zhang et al. [1] employed a topology optimization method

and optimized the material distribution and force transfer paths of the
As an essential component of automobiles, seats not only perform a seat backrest frame by considering multiple seat conditions. Shan et al.
vital protective function but also provide a comfortable driving envi- [2] employed grey relational analysis (GRA) in combination with the
ronment for their occupants. Nonetheless, with the continuous upgrades optimization coefficient of variation (OCV) method to optimize the
in safety and comfort standards for automotive seats, the supplemental materials and thicknesses of seat design components, and conducted a
structural elements of seat enhancements have eventually led to a sig- detailed comparative analysis using multi-criteria decision making
nificant surge in their mass. Therefore, optimizing the design of the seat (MCDM) methods. Ju and Jeon [3] applied discrete material and
frame becomes imperative. thickness optimization methods for high strength and advanced high
Optimization of the automotive seat frame is a critically important strength steels to design the frame strength of automotive seats more
research field in the contemporary automotive industry. Optimization effectively. Wang et al. [4] introduced a hierarchical optimization
not only reduces the overall weight of the seat but also enhances the approach for seats, segmenting the process into three stages and ulti-
material utilization efficiency, lowers the manufacturing cost, and en- mately employing a GRA to address the seat frame optimization prob-
sures safety performance. Consequently, the design optimization of lem. Liao et al. [5] introduced an enhanced version of the gray Euclidean
automotive seats has attracted extensive attention from scholars relationship analysis (GERA) method, based on Euclidean distance,

* Corresponding author.
E-mail address: longjiangqi@163.com (J. Long).

https://doi.org/10.1016/j.advengsoft.2024.103797
Received 1 June 2024; Received in revised form 22 September 2024; Accepted 15 October 2024
Available online 24 October 2024
0965-9978/© 2024 Elsevier Ltd. All rights are reserved, including those for text and data mining, AI training, and similar technologies.
H. Chen et al. Advances in Engineering Software 199 (2025) 103797

which confirmed the optimal layering scheme for the carbon fiber seats. Secondly, the optimization of automotive seat frames is consid-
reinforced polymers (CFRP) backrest skeleton of automotive seats. Dai ered a multi-input multi-output system. As input parameters vary, the
et al. [6] conducted a contribution analysis to screen the design vari- output response tends to be nonlinear, discrete, and increasingly com-
ables and employed a combination of optimal Latin hypercube sampling plex. In studying such multi-input multi-output systems, traditional
(OLHS), response surface method (RSM) surrogate model, methods like linear regression, response surface methodology, and
non-dominated sorting genetic algorithm-II (NSGA- II), and MCDM surrogate modeling techniques are becoming increasingly inadequate.
method to optimize the car seat frame. Li et al. [7] introduced a novel Simultaneously, machine learning, a crucial component of artificial in-
structural damage identification method based on an agent-assisted telligence, has seen rapid development across diverse research fields in
evolutionary optimization algorithm, and constructed an integrated recent years. Notably, neural networks (NN) and support vector ma-
prediction model with stronger generalization ability and higher accu- chines have demonstrated outstanding capabilities in handling
racy. Kalita et al. [8] introduced a novel hybrid multi-objective opti- nonlinear regression and classification challenges [14]. These tech-
mization method called TOPSIS-PR-GWO for parameter optimization of niques are increasingly being applied to structural optimization and
abrasive jetting process. Zhang et al. [9] designed the CFRP seat skeleton performance prediction by scholars globally. For instance, Huang et al.
with the aim of enhancing structural safety and reducing weight, which [15] proposed a comprehensive prognostic method based on deep
resulted in significant improvements in seat frame performance. convolutional neural networks (CNN) for predicting the remaining ser-
Generally, the aforementioned scholars have focused on the lightweight vice life of rolling bearings, which was experimentally validated through
design of automotive seats through both structural optimization and the a case study. Alcantara et al. [16] employed deep neural networks
application of novel materials. Several scholars have utilized a local (DNN) to directly estimate prediction intervals for solar and wind dis-
optimization method [10], based on MCDM [11], to optimize the trict energy forecasts, and the results indicate that DNN predictions are
automotive seat frame. Other scholars have focused on optimizing al- highly effective. Yang et al. [17] employed the back propagation neural
gorithms for global optimization by constructing surrogate models [12, network (BPNN) training method to model the relationship between the
13]. These two approaches represent the primary methodologies in the lateral displacement response of the seat belt and various random pa-
current field of multi-objective optimization problem (MOOP). Fig. 1 rameters, offering a theoretical foundation for the parametric reliability
presents a flow chart illustrating the two principal optimization methods design of the seat belt guide ring. Yang et al. [18] employed an artificial
in the field of MOOP. neural network (ANN) to map the conformational parameters of
However, owing to the continuous improvement in safety and superstorms to their reflection coefficients, and achieved the
comfort performance standards for automotive seats, there has been an quasi-periodic distribution through training a dynamic graph convolu-
increase in auxiliary structural components, leading to a rise in overall tional neural network (DGCNN). Afzal et al. [19] conducted predictions
weight. Consequently, the results obtained solely by local optimization of cooling and heating loads by comparing three ANN frameworks with
methods are insufficient to meet the current lightweight requirements of a regression model. Following the selection of the best network, four

Fig. 1. Two principal optimization methods in MOOP.

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optimization techniques were employed to tune the hyperparameters of et al. [37,38] proposed an efficient finite element model based on the
the selected network, aiming to develop the best hybrid model. Schrader first-order shear deformation theory and combined with a genetic al-
and Schauer [20] applied a feed-forward NN for multi-directional design gorithm to analyze the effect of mass ratio variations on the optimal
control of automotive plastic crash components, demonstrating the NN’s design, and proposed an ideal stacking sequence for the maximum fre-
exceptional predictive capabilities. Huang et al. [21] utilized a BPNN to quency spacing value. Various metamodeling methods and main results
identify the location of signal sources in a metal plate. The findings are also discussed. Compared to other MOOA, the multi-objective gray
reveal that the trained BPNN can accurately classify the sound source wolf optimizer (MOGWO) stands out due to its faster convergence,
areas in two different environments. While the use of BPNN models for effective distribution of Pareto frontiers, and its robustness and reli-
predicting the structural performance characteristics of seats can indeed ability. As a result, the MOGWO algorithm is particularly promising and
reduce experimental costs, it also presents some challenges. For has seen widespread application across various research fields in finding
instance, these models are highly sensitive to their initial parameters Pareto frontier solutions. For instance, Makhadmeh et al. [39] con-
[22] during the learning process and exhibit poor generalization capa- ducted a review of MOGWO, demonstrating that this algorithm excels in
bilities, which can lead to inaccurate predictions when applied to new finding Pareto-optimal solutions. Salari et al. [40] achieved optimal
data. To alleviate these limitations, researchers have been exploring the system performance by implementing MOGWO. Additionally, simula-
combination of BPNN optimization algorithms to improve prediction tion results indicate that MOGWO surpasses other optimization tech-
accuracy and model robustness. Nguyen-Ngoc et al. [23] presented a niques in solving optimization problems. In summary, compared with
new method for SHM damage detection in truss bridges by combining a classical algorithms, such as multi-objective particle swarm optimiza-
DNN model with an evolved artificial rabbit optimization (EVARO) al- tion (MOPSO) [41–43], MOGWO exhibits enhanced global search
gorithm. Tran et al. [24] developed a combination of ANN and Balanced capability, superior distribution of frontier solutions, increased robust-
Composite Motion Optimization (BCMO) to solve optimization problems ness and reliability, thereby yielding better solutions and improved ac-
and predict stochastic vibration and buckling behaviors of functionally curacy. Additionally, the algorithm is advantageous due to its
gradient porous microplates with material property uncertainties. Bai straightforward programming, comprehensibility, and rapid conver-
et al. [25] proposed a BSLO-based ANN prediction model for predicting gence [44]. Consequently, this study focuses on the global optimization
the diameter of fibers written by melt electrostatic spinning, and further of automotive seat frames through the application of machine learning
verified the applicability of BSLO to practical problems. Pal and Subham prediction models and MOOA [45–47]. Specifically, the IGWO-BPNN
et al. [26,27] proposed a GWO to find the optimal stacking sequence and machine learning prediction model, coupled with the MOGWO algo-
the optimal layup angle to maximize the fundamental frequency of the rithm as a multi-objective optimization tool for automotive seat design,
shell. The GWO-based finite element model is compared to six different is employed to address the Pareto frontier problem.
metaheuristics and the results show that GWO consistently outperforms After obtaining a set of Pareto frontier solutions without prior pref-
these competing algorithms. Wang et al. [28] utilized the GWO algo- erence, determining how to select the optimal solution from the Pareto
rithm to tune the parameters of the kernel extreme learning machine frontier solutions represents both a central focus and a significant
(KELM). An effective GWO-KELM model was developed for bankruptcy challenge of our research, with MCDM playing a pivotal role. Opricovic
prediction. Ultimately, the predictive ability of the model was validated [48], who developed the Visekriterijumsko KOmpromisno Rangiranje
through comparative analysis. Mosavi et al. [29] utilized the GWO al- (VIKOR) method for addressing discrete MCDM problems, Yoon and
gorithm for NN training, and the findings indicated that GWO generally Hwang [49,50] adopted the Technique for Order Preference by Simi-
outperforms or is comparable to other algorithms in most scenarios. Xu larity to an Ideal Solution (TOPSIS) method, which identifies the solu-
et al. [30] introduced an arithmetic performance prediction algorithm tion closest to the ideal and farthest from the negative ideal. Another
based on the GWO-BPNN for generating training data. Compared with method, multi-objective optimization ratio analysis (MOORA), boasts a
extreme learning machine (ELM), locally weighted linear regression broad range of applications across various disciplines and industries.
(LWLR), and support vector machines (SVMs), the findings revealed that Occasionally, the logical integration of these two methods can be syn-
the GWO-BPNN method yielded the most accurate performance pre- ergized into a unified structure. For instance, the weighted product
dictions. However, GWO also exhibits the typical challenges associated method (WPM) and the weighted sum method (WSM) can be combined
with meta-heuristic algorithms, such as premature convergence and to form a tool known as weighted aggregated sum product assessment
slow computational progress. To enhance the GWO algorithm, Dong (WASPAS) [51,52]. Numerous studies have elucidated the advantages of
et al. [31] refined it by optimizing the initial population position, this method. This method validates the final performance scores of al-
convergence factor, and iterative weights. Yu et al. [32] chose a selec- ternatives through linear relationships, and power and multiplicative
tion of grey wolf individuals closer to the optimal solution as the initial aggregation [53]. This study introduces a combined compromise solu-
population using a lens imaging reverse learning strategy. Additionally, tion (CoCoSo) method that incorporates an aggregation strategy. To
an adaptive tuning strategy and a re-selection strategy for optimal in- enhance the flexibility of the results, a distance metric derived from the
dividuals were incorporated during position updating to enhance search gray relational coefficient (GRC) and the target has been considered.
efficacy and avoid local optima. Bi et al. [33] enhanced the GWO al- Consequently, the weights of the alternatives are incorporated into three
gorithm by incorporating nonlinear convergence factors, leading to the equations during the decision-making process. In the final stage, an
development of the improved grey wolf optimization (IGWO) algorithm. aggregation multiplication rule is applied to determine the ranking of
Wei et al. [34] employed particle swarm optimization (PSO) to generate the alternatives and conclude the decision-making process.
a range of initial positions, followed by the use of GWO to update the The remainder of the paper is structured as follows: Section 2 dis-
populations’ current positions in the discrete search space, thereby cusses the finite element (FE) modeling and experimental validation,
enhancing the GWO algorithm’s performance. Section 3 presents the proposed optimization strategy and method for
Furthermore, with the increasing complexity of optimization prob- the seat frame, Section 4 delves into the intricate optimization process of
lems and more stringent optimization requirements, single-objective the seat frame, Section 5 compares and analyzes the optimal trade-off
optimization algorithms prove inadequate for solving multi-objective solution under various optimization methods, and finally, Section 6
optimization problems (MOOP). Moreover, employing multi-objective concludes the paper.
optimization algorithms (MOOA) is crucial in the process of identi-
fying solutions on the Pareto front [35]. For example, Qing et al. [36]
used in-situ vibration measurements, finite element model updating
(FEMU), and the Improved Artificial Fish Swarm Algorithm (IAFSA) to
assess the condition of steel-tube concrete arch bridges. Pal and Kalita

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2. Modeling and verification scenarios, while the safety performance indices primarily relate to the
strain measures of structural components and the torque of bolts at
2.1. Seat working condition model critical locations. The lightweight evaluation indices primarily indicate
the total mass of the automobile seat, and the economic evaluation
An accurate and efficient FE model is essential for numerical analysis indices refer specifically to the total material cost of the selected design
and optimization. Consequently, this study selects eight typical working components of the automobile seat. The maximum torque exerted on the
conditions to optimize the automotive seat backrest, in accordance with seat’s backrest adjuster in crash situations serves as an indicator of seat
international automotive seat safety standards and industry regulations. safety performance, and the maximum displacement of the dummy at
These conditions include the 50 dummy rear crash (50RC), 50 dummy point H is utilized as a measure of comfort performance.
frontal crash (50FC), headrest static strength test (HSST), seat backrest
strength test (SBST), seatbelt anchorage test (SAT), antisubmarine pan
test (APT), front ultimate load (FUL), and rear ultimate load (RUL).
Among these conditions, 50FC and 50RC working conditions represent
dynamic test scenarios to assess the overall performance of the seat,
albeit with differing emphases. The 50FC working conditions examines
whether the seat basin and other components provide effective protec-
tion to the dummy, whereas the 50RC working condition focuses on the
support provided by the backrest and headrest. The HSST, SBST, SAT,
APT, FUL, and RUL constitute static test conditions designed to evaluate
the performance of specific parts of the seat. The FE model’s validity is
confirmed by comparing the seat damage and changes at key measure-
ment points between numerical analysis and experimental tests. Fig. 2
(a) presents a detailed diagram of the FE model under these working
conditions. The detailed construction process for each seat finite
element model case is outlined in Ref. [6]. Furthermore, LS-DYNA is
selected as the primary computational software for this study.

2.2. Lightweight design responses and measurement point layout

Considerations for the lightweight design of automobile seats

encompass comfort performance indices, safety performance indices,
lightweight evaluation indices, and economic evaluation indices. Ac-
cording to the international safety standard for automotive seats, key
Fig. 3. Distribution of measurement points for each property of the automo-
vehicle comfort performance indices include the maximum displace-
tive seat.
ment and acceleration of the dummy’s internal H point in collision

Fig. 2. Simulation modeling and experimental performance testing of automotive seats.

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This study specifies seven measurement points, as illustrated in

Fig. 3, with points A, B, C, H, and E designated to measure displacement
in APT, FUL and RUL, HSST, 50FC and 50RC, and SAT, respectively.
Point K is used to estimate the torque at the angle adjusters on both sides
of the seat in 50RC, while Ab assesses the angle variation of the seat
backrest during SBST. With point A representing the center of the lower
cross-tube at the seat’s front, point B marking the center of the upper
side of the backrest’s round tube, point C denoting the midpoint on
either side of the headrest tube, and point H correlating with the rotation
point where the dummy’s thigh connects to the torso, point E identifies
the center of the seat belt latch, and point K represents the fixed point of
the angle adjuster.
During the optimization of car seats, the failure of components often
increases seat safety risks. Therefore, it is imperative that no seat frame
components, encompassing structural elements and angle adjuster
connection bolts, should fail. To more effectively monitor component
failures, this study introduces the concept of a strain index, which is
defined as follows:
εp
Fp = (1) Fig. 4. Comparison of simulation and experimental measurement point
Ep
result errors.
where Fp represents the strain index of p component, εp is the plastic
strain of p component, and Ep is the post-fracture elongation of material condition.
of p component. By synthesizing the data from Table 1 and Fig. 4, the following
In this paper, if Fp is greater than 1, the seat component is at risk of conclusions can be drawn: In this study, the accuracy of the finite
failure. To ensure a relatively safe design margin for automotive seat element model for each working condition of the automobile seat is
structures, the upper limit of Fp is set to 0.95. Therefore, if Fp exceeds maintained within an error margin of approximately 8 %. Specifically,
0.95, the seat structure is deemed to have failed. the error accuracy is notably higher in the three indices of A_H1, A_H2,
and T_KL. Given that A_H1 and A_H2 represent the acceleration of the
dummy at point H under the car seat’s front and rear crash conditions,
2.3. Seat FE model test validation they are strong nonlinear indices, making precise error control chal-
lenging. Additionally, T_KL, which measures the torque on the left bolt
Verifying the accuracy of the finite element model is a crucial aspect of the car seat backrest adjuster under rear-bump conditions, is another
of the seat optimization design process. In this study, the FE model’s nonlinear index, with an error of 6.648 %. Overall, the fitting error for
accuracy is maintained within an 8 % error margin by benchmarking the finite element model of the automotive seat frame in this study is
key points’ displacement, acceleration, and torque values against those maintained at approximately 8 % across all working conditions,
obtained from actual seat tests. Fig. 2 illustrates both the simulation and rendering it suitable for subsequent multi-objective optimization of the
physical performance test models under the selected working condi- seat frame.
tions. Table 1 presents a comparative analysis between simulation and
experimental results, including maximum displacement, maximum 3. Optimization strategy and methodology
torque at critical areas, and maximum acceleration and displacement at
the H-point of the dummy. These results are visually depicted in Fig. 4. 3.1. Optimization strategy
D_H1 and D_H2 represent the total displacement at point H in 50FC and
50RC conditions, respectively; D_B1 and D_B2 denote the total Initially, the FE model of the seat frame is established. This FE
displacement at point B in FUL and RUL conditions, respectively; D_A analysis model is constructed to reflect eight typical working conditions,
represents the total displacement at point A in APT condition; D_E and in compliance with international seat safety standards and enterprise
D_C represent the total displacement at point E and C in SAT and HSST standards. Subsequently, the performance indexes of key measurement
conditions, respectively; T_KL and T_KR represent the torque at point K points under various working conditions are obtained through static and
under 50RC conditions; A_H1 and A_H2 denote the maximum acceler- dynamic crash experiments of the seat, and the accuracy of the FE model
ation at point H under 50FC and 50RC conditions, respectively; Ab is verified against the experimental data.
represents the angle change value of the seat backrest under SBST Following this, methods such as the ratio of energy absorption to
mass, the integrated structure-material multi-objective lightweight
Table 1 optimization model, the IGWO-BPNN hybrid prediction modeling
Comparison of simulation and experimental measurement point results. method, the MOGWO algorithm, and CoCoSo are integrated into a
No Simulation Experiment Error ( %) multi-objective lightweight optimization procedure. Initially, the light-
weight design responses, including the constraints and objectives of the
D_H1 (mm) 208.364 196.617 5.638
D_H2 (mm) 133.846 126.060 5.817 optimization problem, are derived from both dynamic and static seat
D_B1 (mm) 64.816 62.348 3.807 safety performance tests. Next, the final design variables are identified
D_B2 (mm) 69.532 68.788 1.07 and selected using the ratio of energy absorption to mass method.
D_A (mm) 51.496 49.340 4.187 Subsequently, a design of experiment (DOE) is conducted between the
D_E (mm) 35.416 33.686 4.885
D_C (mm) 62.151 60.136 3.242
final design variables and the responses using the OLHS method. This
T_KL (Nm) 493.119 460.335 6.648 process derives a subset of inputs from the design variables and outputs
T_KR (Nm) 650.900 611.993 5.977 from the lightweight design responses, which are utilized to construct
Ab (◦ ) 17.63 16.875 4.282 the surrogate models via the IGWO-BPNN hybrid prediction modeling
A_H1 (g) 55.436 51.164 7.706
method. The MOGWO algorithm is subsequently employed to search for
A_H2 (g) 28.461 26.188 7.986

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Pareto-optimal solutions to the multi-objective lightweight design GWO searches the solution space by simulating the hunting process
problem. Subsequently, the CoCoSo method is applied to determine the of the gray wolf, which consists of three main phases [47]:
best compromise solution from the obtained Pareto solutions, followed Surrounding the prey: Gray wolf packs approach prey by constantly
by the verification of the lightweight design. Fig. 5 presents a flowchart adjusting their position. Each wolf adjusts its position according to the
summarizing the proposed optimization design method. distance to the prey and the behavior of the leader wolves (α, β, δ).
Finally, the superiority of the method presented in this paper is Tracking prey: The grey wolf gradually approaches its prey through
verified through a comparative analysis with different methods. The the leadership of the alpha wolf, beta wolf, and delta wolf, and its po-
analysis is conducted from three perspectives: first, in optimizing BPNN, sition is updated based on the average weight of these three wolves.
various heuristic algorithms are utilized to optimize the BPNN param- Attack prey:
eters, accompanied by a comparative analysis. Subsequently, in the In the later stage of optimization, when the prey is surrounded, the
realm of MOOA, commonly used methods like the NSGA-II, MOPSO, and grey wolf approaches the prey more accurately by reducing the value of
others, are compared with MOGWO. Finally, regarding MCDM methods, parameter A until the optimal solution is found.
VIKOR and GRA methods are employed to compare and analyze → ⃒→ →
⃒
→ ⃒
⃒
decision-making results with CoCoSo, thereby exploring the superiority D = ⃒ C ⋅ X p(t) − X (t)⃒ (2)
of various approaches.
→ → →→
X (t + 1) = X p(t) − A ⋅ D (3)
3.2. Methodology
The aforementioned mathematical equations elegantly delineate the
3.2.1. Grey wolf optimizer encircling behavior of a pack of grey wolves. In these equations, t rep-
→ → →
The GWO algorithm is inspired by the hunting behavior of gray wolf resents the current iteration, while A and C are coefficient vectors; X p
packs and was proposed by Mirjalili et al. [54] in 2014. Gray wolf packs →
denotes the position vector of the prey, and X signifies the position
have a well-defined social hierarchy and demonstrate effective hunting vector of a grey wolf.
strategies through cooperative hunting. GWO simulates this natural → →
The vectors A and C are calculated as follows:
behavior and applies it to an optimization problem. The key mechanism
of GWO is to simulate a gray wolf pack by tracking, encircling, and →
A = 2→
a ⋅→
r 1− →
a (4)
attacking its prey. In GWO, gray wolves are divided into four levels, as
illustrated in Fig. 6: α (head wolf): responsible for decision making, →
C = 2⋅→
r2 (5)
usually the position of the optimal solution in the group. β (deputy alpha
wolf): assists the alpha wolf in decision making, usually the position of
where elements of → a linearly decrease from 2 to 0 over the course of
the suboptimal solution. δ (sub-sub-lead wolf): obeys the instructions of
iterations and r1, r2 are random vectors in [0,1]. Furthermore, Search
α and β, usually the position of the third best solution. ω (ordinary wolf): →
obeys the commands of the above three wolves and explores the solution agents tend to diverge from the prey when | A | > 1 and converge to-
→
space. wards the prey when | A | < 1.

Fig. 5. Optimization strategy.

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Fig. 6. Leadership hierarchy.

To mathematically simulate the hunting behavior of grey wolves, it is problems, the main advantages of using the IGWO in conjunction with
assumed that α, β, δ possesses superior knowledge about the potential the multi-objective optimization problem for automotive seats are as
location of the prey. In this context, α is identified as the best candidate follows:
solution, followed by β, δ. Consequently, the first three optimal solutions (1) Enhancing global search capabilities
obtained are retained, compelling other search agents ω to update their The IGWO improves global search capability by enhancing popula-
positions in accordance with the position of α, β, δ, as detailed subse- tion diversity and improving the leader selection mechanism. This is
quently. The positional updates of the wolves are illustrated in Fig. 7. especially critical to address the complexity in multi-objective optimi-
⎧ ⃒→ → zation of car seats.
→ →⃒⃒
⎪
⎪
⎪ D α =
⃒
⃒ C 1⋅ X α − X⃒ (2) Accelerated convergence
⎪
⎨→ ⃒→ →
⃒ →⃒⃒ The IGWO optimizes the search strategy of the algorithm to achieve a
D β = ⃒ C 2⋅ X β − X ⃒ (6)
⎪
⎪ ⃒→ ⃒ more balanced exploration and exploitation by dynamically adjusting
⎪ → → →
⎩ D δ = ⃒⃒ C 3⋅ X δ − X ⃒⃒
⎪ the weighting parameters. In this way, IGWO is able to perform fuller
global exploration in the early stages and accelerate the convergence to
⎧
→ → → (→ ) the optimal solution in the later stages. This mechanism is crucial for
⎪ X1 = Xα − A 1⋅ D α
⎪
⎪
⎪
⎨→ quickly finding efficient solutions in multi-objective optimization of car
→ → (→ ) seats, thus reducing computational time and resources.
X2 = Xβ − A 2⋅ D β (7)
⎪ (3) Improved multi-objective processing
⎪
⎪ → → → (→ )
⎩ X3 = Xδ −
⎪ A 3⋅ D δ
In the multi-objective optimization design of automotive seats,
weight and cost are two objectives that often constrain each other.
→
→ → →
X1 + X2 + X3 IGWO is able to better handle conflicts between multiple objectives and
X (t + 1) = (8) find better Pareto frontier solutions through an improved search
3
mechanism.
3.2.2. Improved grey wolf optimizer (4) Improved robustness and stability
The BPNN network has some defects in the data fitting process, such In the multi-objective optimization of automotive seats, the design
as over-reliance on the gradient descent method, which leads the results parameters have greater complexity and constraints, while IGWO is able
to fall into local optimal solutions. Sensitivity to initial weights leads to to maintain stable optimization results under complex constraints
poor model training. Slow convergence during complex nonlinear model through its flexibility and adaptability.
fitting, and BPNN has multiple hyperparameters (e.g., learning rate, (5) Adaptation to complex multi-objective environment
number of hidden layers, number of nodes, etc.), which require complex Multi-objective optimization of automotive seats needs to deal with
adjustments to find the optimal configuration. To address the above multiple design objectives and constraints. The IGWO can better cope

Fig. 7. Update of position in GWO.

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with these complex constraints through its improved adaptability, the process is applied to all individuals, the number of iterations is
which is suitable for multi-objective optimization problems with many incremented by 1, continuing until the iterative search reaches the pre-
design variables and obvious objective conflicts. This adaptability gives defined number of iterations. The pseudo-code for the proposed
IGWO a significant advantage in the multi-objective optimization of seat improved gray wolf optimizer (I-GWO) algorithm is as follows:
design, especially when weight, cost and safety are taken into account.
To summarize, IGWO has stronger global search capability, faster 3.2.3. IGWO-BPNN prediction model
convergence speed, ability to handle multi-objective optimization, and The process of building the IGWO-BPNN prediction model is a sys-
higher robustness and stability than traditional GWO. In the multi- tematic step combining the IGWO and BPNN. IGWO optimizes the
objective optimization problem of automotive seats, IGWO can better weights and biases of the neural network to make the BPNN more pre-
achieve the balance of complex design objectives and enhance the dictive in specific applications. that make the BPNN more predictive in
optimization efficiency through these improvements. Specific specific applications. The process includes model design, algorithm
improvement steps are as follows. design and optimization solution, and the steps of implementation are as
Initially, the parameters are set with n wolves being randomly follows:
distributed in the search region [lj, uj]. The entire wolf population is Step 1: Initialize all relevant algorithm parameters. This includes
represented in a matrix ’Pop’, which consists of N rows and D columns, parameters such as the BPNN parameters, the number of gray wolves
with D denoting the problem’s dimension. The fitness function is utilized (Pop), and the maximum number of iterations.
to compute Xi(t). Step 2: Initialize the positions of the α, β and δ wolves, and subse-
quently compute the individual best fitness and position obtained by
Xij = lj + randj[0, 1] × (uj − lj), i ∈ [1, N], j ∈ [1, D] (9)
each wolf. In this study, root mean square error (RMSE) has been chosen
In the GWO algorithm, α, β, and δ represent the three best wolves. As as the fitness function. Its expression can be found in Eq. (15).
the coefficients and the positions of Xα, Xβ and Xδ linearly decrease, the Step 3: The three wolves execute the position update in accordance
position of the surrounding prey is determined. The first candidate to with Eq. (7) and furthermore, compute the position Xi − GWO utilizing
move to the new position, aligned with the grey wolf Xi(t) ’, is desig- Eq. (8).
nated as Xi − GWO(t + 1). The DLH strategy introduces an additional Step 4: Determine the radius of neighboring wolves in order to
candidate position, Xi − DLH, d(t + 1), for the wolves, with the calcu- enhance their local search capability as delineated in Eq. (12), and
lation formula as follows: execute multi-neighbor learning as described in Eq. (10).
Step 5: Selection and update phase: During this phase, the fitness
Xi − DLH, d(t + 1) = Xi, d(t) + rand × (Xn, d(t) − Xr, d(t)) (10)
values of the two candidates, Xi − GWO(t + 1) and Xi − DLH(t + 1), are
compared utilizing Eq. (13), and a subsequent Pop update is performed.
whereXn, d(t) is a wolf nearby, Xr, d(t)is a random wolf in the
Step 6: If the number of iterations surpasses the maximum number of
population.
evolutions, or if the predetermined accuracy requirement set by the user
The distance calculation formula is as follows:
is met, the operation terminates; otherwise, execution reverts to step 3.
Mi(t) = {Xj(t)|Di(Xi(t), Xj(t)) ≤ Ri(t), Xj(t) ∈ I} (11) Step 7: The individuals, resulting from the optimization of the IGWO
algorithm, are subsequently decomposed into the connection weights
Ri(t) =‖ Xi(t) − Xi − GWO(t + 1) ‖ (12) and thresholds of the BPNN. These components serve as the initial
weights and thresholds of the prediction model. Through the training of
where Ri(t) is the Euclidean distance between Xi(t) and Xi − GWO(t + 1), the BPNN prediction network, one can obtain the optimal solution for
Mi(t) is the distance of Ri(t) of wolves adjacent to Xi(t), Di is the network prediction.
Euclidean distance between Xi(t) and Xj(t). Fig. 8 depicts the flowchart of the IGWO-BPNN, detailing its specific
{ } optimization process
Xi − GWO(t + 1), f(Xi − GWO) < f(Xi − DLH)
Xi(t + 1) = (13) The test dataset serves to validate the predictive ability of the opti-
Xi − DLH(t + 1), otherwise
mized BPNN model; insufficient data can result in inaccurate fitting
To update the new positionXi(t + 1) ’, if the selected candidate’s accuracy, while excessive data might cause overfitting [55] problems.
fitness is lower than Xi(t), ’, the position Xi(t + 1) is updated to the To enable a more comprehensive and rigorous comparison, various
selected candidate’s position. If not, it remains unchanged. Finally, once statistical error analysis parameters are employed as criteria for evalu-
ating the model [56]. Important statistical parameters employed in this
study include the Pearson coefficient of determination (R2) and RMSE
Algorithm 1 [57]. These parameters are calculated according to Eqs. (14) and (15):
The improved gray wolf optimizer (I-GWO). (∑ )
n 2
Input: n, D, Maxiter i=1 (xi − yi)
R =1−
2
∑n 2
(14)
Output: The global optimum i=1 (xi − x)
1: Begin
2: Initializing (the number of gray wolves and calculating their fitness) √̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅
3: For iter = 2 to Maxiter 1∑1 2
RMSE = |xi − yi| (15)
4: Find Xα, Xβ and Xδ n n

5: For i= 1 to n
where n is the total number of datasets, xiand yi represents the predicted
→ → →
6: Computing X 1, X 2, X 3 by using Eq. (7)
7: Computing Xi − GWO(t + 1) by using Eq. (8) value and the target value, respectively, and x is the average of the target
8: Computing Ri(t) by Eq. (12)
value.
9: Constructing neighborhood Xi(t) with radius Ri by Eq. (11)
10: For d= 1to D
11: Xi − DLH, d(t + 1) = Xi, d(t) + rand × (Xn, d(t) − Xr, d(t)) 3.2.4. Multi-objective gray wolf optimization algorithm
12: End for After building the IGWO-BPNN prediction model, the process of
13: Selecting best (Xi − GWO(t + 1),Xi − DLH(t + 1))
solving its Pareto frontier solution by MOGWO helps to achieve the
14: Updating Pop
15: End for
optimization of multiple conflicting objectives by ensuring that one
16: End for objective will not be significantly degraded while improving the others
17: Return the global optimum as much as possible. MOGWO is an extension of the single-objective
18: End GWO to multi-objective optimization problems by means of the

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Fig. 8. IGWO-BPNN machine learning model flowchart.

dynamic tuning and the nondominated ordering mechanism. By opti- computational time of the neural network, including the training time
mizing multiple objective functions at the same time, the best compro- and prediction time
mise between different objectives (Pareto frontier) is found and the (3) Non-dominated ordering
optimal compromise between different objectives is formed. The first Define non-dominated solution: in multi-objective optimization, a
step in performing multi-objective optimization of automotive seats is to solution A is said to be non-dominated if no other solution can simul-
define the multi-objective optimization problem. These objectives are taneously outperform A on all objectives. specifically, a solution A may
often in conflict with each other, e.g., reducing cost may require outperform other solutions on some objectives, but is inferior to other
increasing the safety risk of the seat, while improving the quality of the solutions on some objectives, and no solution can simultaneously
seat to provide seat safety performance may increase the cost. Therefore, outperform A on all objectives.
multiple objectives need to be optimized to find the best trade-off while Non-dominated ordering: the wolves are sorted according to non-
ensuring safety. The specific steps for solving the Pareto frontier in dominated relationships to generate different levels of non-dominated
MOGWO are as follows, and Fig. 9 illustrates the flowchart of the IGWO- solution sets. The sorting process is as follows: For each wolf, compare
BPNN-MOGWO algorithm. its performance on all objective functions. If the current wolf is not
(1) Wolf initialization significantly outperformed by other wolves on all objectives, it is labeled
The position of each wolf represents a candidate solution, and the as a non-dominated solution. The set of non-dominated solutions at the
dimension of the solution corresponds to the weight and bias combi- first level is called the Pareto frontier solution set.
nation of the BPNN. The number of initial wolf packs is N, and the po- (4) Calculation of the degree of crowding
sition of each wolf is initialized randomly. Each individual solution of In order to ensure diversity in the solution set of the Pareto front, the
the wolf pack corresponds to a weight and bias set of the BPNN model. degree of crowding is calculated for each non-dominated solution. A
(2) Calculate the degree of adaptation for each wolf larger distance indicates that the solution is more sparsely distributed in
Calculate the prediction error of the trained neural network based on the solution set, implying a greater diversity of solutions in the solution
the current wolf position (i.e., BPNN weights and bias) on the test set, set.
calculate the complexity of the neural network, and measure the (5) Updating wolf positions

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Fig. 9. Flowchart of the IGWO-BPNN-MOGWO algorithm.

Based on the results of non-dominated ordering, the optimal three rule, while the other calculates the distance to the weighted power of the
non-dominated solutions are selected as α-wolf, β-wolf and δ-wolf. The comparable sequence. To validate the ranking index, three distinct
other ω wolves update their positions according to the positions of these measures (aggregation strategies) are defined for each alternative. Ul-
three leader wolves. timately, an equation yields a ranking score for each strategy, and the
(6) Iteration and Convergence Judgment strategy with the highest score is designated as the combined compro-
The wolf pack iterates continuously, updating the position in each mise solution.
iteration and continuously filtering the optimal solution set based on (1) The initial decision-making matrix is established as follows:
non-dominated ordering and congestion calculation. When the ⎡ ⎤
x11 x12 ⋯ x1n
maximum number of iterations is reached or the Pareto frontier solution ⎢ x21 x22 ⋯ x2n ⎥
set is stable and no longer changes, iteration is stopped. xij = ⎢
⎣ ⋯
⎥; i = 1, 2, …, m; j = 1, 2, …, n. (16)
⋯ ⋯ ⋯ ⎦
(7) Generation of Pareto front solution set xm1 xm2 ⋯ xmn
Through multiple rounds of iterations, MOGWO gradually converges
to the Pareto-optimal solution set for multiple objectives. Each solution (2) The normalization of criteria values is accomplished based on
in the Pareto frontier solution set is an optimal compromise solution that compromise normalization equation:
optimizes one objective without significantly deteriorating the other xij − ⏟⏞⏞⏟
min xij
objectives. rij = i
; forbenefitcriterion (17)
max xij min xij
⏟̅⏞⏞̅⏟i − ⏟⏞⏞⏟ i
3.2.5. Combined compromise solution method
The combined compromise solution method’s key contribution is the max xij − xij
⏟̅⏞⏞̅⏟
introduction of a weight aggregation process within the gray relation- rij = i
; forcostcriterion (18)
max xij − ⏟⏞⏞⏟
⏟̅⏞⏞̅⏟ min xij
ship generation method. Initially, this method measures the distance of i i

each performance rating from the ideal performance rating. This (3) For each alternative, Si is defined as the sum of the weighted
approach resembles the VIKOR method but employs a slightly altered comparability sequence, and Pi as the total of the power weight of
formula. The divergence from VIKOR in this method originates from the comparability sequences. This formulation accounts for both the
unique application of the weights’ power in the aggregation process. aggregate of the weighted comparability sequence and the overall
This aspect results in a more robust distance measurement approach, amount of the power weight for each alternative:
which appears to be beneficial for modeling. The proposed method n
∑
utilizes comparable sequences and aggregates the weights in two Si = (ωjrij) (19)
distinct manners. One method follows the conventional multiplication j=1

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The Si value is calculated using the grey relational grade (GRG) the optimal combination of design variables. The specific approach is as
approach: follows:
n
Step 1: Conduct a simulation analysis for all safety tests.
∑
Pi = (rij)ωj (20) Step 2: Calculate the energy absorption and the mass of components
j=1 for each safety test.
The Pi value is also determined in accordance with the WASPAS Eξi − Bi
Rξi − Bi = (25)
multiplicative. GBi
(4) Relative weights of the alternatives are computed using specified
aggregation strategies. This step involves employing three appraisal where ξi denotes the codes of safety tests, Rξi − Bi represents the ratio of
score strategies to generate relative weights for the options, as delin- energy absorption to mass for the Bi component in ξi safety test, Eξi − Bi
eated in Eqs. (21)–(23): indicates the energy absorption value of the Bi component in the ξi safety
test, and GBi is indicative of the mass of the Bi component.
Pi + Si Step 3: Compare the values of the ratio of energy absorption to mass
kia = ∑m (21)
i=1 (Pi + Si) for different components, selecting those with a higher ratio for opti-
mization in the collision safety test.
Si Pi
kib = + (22) For simplicity, an initial selection of 23 parts is made, labeled from
min Si ⏟⏞⏞⏟
min Pi
⏟⏞⏞⏟i i P1 to P23 (Fig. 10(a)), including components such as the headrest tube,
backrest upper square tube, backrest side member, backrest recliner
kic =
λ(Si) + (1 − λ)(Pi)
(23) lower bracket, among others. Fig. 11 displays the values of energy ab-
max Si + (1 − λ)⏟̅⏞⏞̅⏟
λ⏟̅⏞⏞̅⏟ max Pi sorption and the mass ratio for the 23 parts under various working
i i
conditions. The analysis yields the following conclusions: (1) Parts with
Eq. (21) is interpreted to represent the arithmetic mean of the sums left-right symmetry demonstrate varying energy absorption under
of the WSM and WPM scores. In contrast, Eq. (22) delineates the sum of identical seat test conditions, rendering mirror design variables unsuit-
relative scores of WSM and WPM compared to the best. Eq. (23) signifies able for optimizing automobile seat frames. (2) The same part absorbs
the balanced compromise between WSM and WPM model scores. Within varying amounts of energy under distinct working conditions, high-
Eq. (23), the parameter λ (typically set to 0.5) is selected by the decision- lighting varying focuses for each seat test condition. (3) The effective-
makers. However, the adaptability and robustness of the proposed ness of a single optimization scheme in one seat test condition does not
CoCoSo method can accommodate other values for λ. ensure its universal applicability to other conditions. Even an optimal
(5) The final ranking of the alternatives is determined based on Ki, compromise solution for one condition might not meet the performance
values (as more significant as better): criteria of other conditions. In summary, this study has identified the
1 1 primary energy-absorbing components for each seat safety test condi-
Ki = (kiakibkic)3 + (kia + kib + kic) (24) tion. Specifically, parts P10, P14, and P21 have been selected in the
3
50FC condition, parts P11 and P19 have been chosen in the 50RC con-
4. Multi-objective optimization process dition, parts P7, P8, P9, P15, and P16 have been picked in the SBST
condition, part P1 has been singled out in the HSST condition, and parts
This section chiefly details the selection process of design variables P2, P3, and P4 have been opted for in the RUL condition. Parts P12 have
for automobile seat frames, the development of the IGWO-BPNN pre- been selected in the SAT condition, and parts P13 and P22 have been
diction model, and the utilization of the MOGWO algorithm to ascertain designated in the APT condition.
the Pareto frontier solution. In the final selection, 17 components have been chosen from the seat
frame (Fig. 10(b)), with their thicknesses denoted as design variables
from T1 to T17 and their corresponding material types defined as design
4.1. Ratio of energy absorption to mass method variables from M1 to M17.

The traditional sensitivity analysis method is limited to linear anal-

4.2. Design of experiment
ysis, and the collision-based contribution analysis method is overly
complex [58,59]. Moreover, owing to the traction effect of the seat belt,
Based on the ratio of energy absorption to mass method results, a
the loading conditions of symmetric parts are not identical. These
total of 17 thickness variables and 15 material variables were selected.
symmetric parts do not equally contribute to the strength of the overall
seat frame under varying safety conditions. Therefore, it is imperative The specific material types and some mechanical property parameters of
the candidate materials, such as yield strength, tensile strength, and
that the mirrored design components are distinguished and optimized
separately. In conclusion, to address this challenge, we propose the elongation at break, are detailed in Table 2. Furthermore, the elongation
at break contributes to determining the material strain index, and the
energy absorption to mass ratio method as a directional approach to
identify the final lightweight design components of the seat frame. Its price is employed to calculate the total cost of the seat material. Table 3
enumerates the experimental ranges for the thickness and material
main advantages are the following. (1) Optimization for improving en-
ergy absorption efficiency: This method helps to improve crash safety variables. The thickness variable is defined within a range of values
corresponding to 60 %–120 % above and below the initial thickness and
and optimize energy absorption while reducing seat weight. (2)
Balancing weight and safety:It helps us balance safety and weight by is characterized as a discrete variable with a step size of 0.1 mm. Ulti-
mately, the OLHS method was employed to design 300 sets of sample
finding solutions that maximize energy absorption and minimize weight
when selecting design variables. (3) Optimizing material and structural points based on the optimization ranges of material-thickness outlined
in Table 3. The output response is determined using LS-DYNA simulation
design: The Energy Absorption Mass Ratio (EAMR) methodology guides
material selection and structural optimization to ensure efficient energy software. The detailed data pertaining to the specific seat input-output
results are presented in Table 4.
absorption and lightweight design. (4) Improve feasibility and economy:
The method optimizes design variables to reduce weight and cost,
achieving economic efficiency and feasibility of the design. (5) Critical 4.3. IGWO-BPNN machine learning predictive models
multi-objective optimization: The energy absorption mass ratio provides
a key metric for multi-objective optimization, helping to quickly identify In this section, we employ the IGWO-BPNN machine learning model

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H. Chen et al. Advances in Engineering Software 199 (2025) 103797

Fig. 10. Comparison of design variables before and after ratio of energy absorption to mass method.

Fig. 11. Ratio of energy absorption to mass of design variables.

to train the seat frame experimental samples. The inputs of the models learning process, the R2 values of the four objective responses: D_H1,
are categorized into two types: one being the material variable and the D_H2, quality, and cost stand at 0.94101, 0.94037, 0.99920, 0.99808,
other, the thickness variable; the outputs consist of the evaluation respectively; and the RMSE values are 1.60006, 0.88528, 0.00262, and
indices of the seat frames. Among them, there are two types of evalua- 0.04958, respectively. This indicates that the fitting accuracy of the
tion indices: one is linear indices (cost, quality) and the other is prediction model meets the necessary requirements, enabling the sub-
nonlinear indices (displacement, acceleration, torque). We restructured sequent step of solving the Pareto solution set using a MOOA.
the dataset in Table 4, allocating 80 % for training the IGWO-BPNN and
20 % for testing and evaluating the predictive models’ accuracy using
4.4. Multi-objective lightweight design process
RMSE, and R2. Figs. 12 and 13 depicts the scatterplot of the fitting ac-
curacy of the IGWO-BPNN model for the four targets under the test set.
Subsequently, the thickness and material type of the selected parts
As can be seen from Figs. 12 and 13, After the IGWO-BPNN machine
are utilized as design variables; the maximum displacement, maximum

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Table 2 ⎧
Material mechanical parameter. ⎪
⎪
⎪ find(x, y) = (x1, x2, ⋯, x17, y1, y2, ⋯y15)
⎪
⎪
⎪
⎪
Material Material Yield Tensile post-fracture Price ⎪
⎪ ming(x, y) = {gcost(x, y), gmass(x, y), gD H1(x, y), gD H2(x, y)}
no name strength strength elongation (¥/kg)
⎨
(MPa) (MPa) (%) s.t.Dv(x, y) ≤ DUv , v = 1, 2, ⋯, 7 (26)
⎪
⎪
⎪
⎪ Mc(x, y) ≤ MUc , c = 1, 2, 3
101 DC01 195 353 0.27 4.28 ⎪
⎪
⎪
102 Q195 195 465 0.278 4.5
⎪
⎪
⎩ Fk(x, y) ≤ FkU , k = 1, 2, ⋯, 8
103 Q235 235 521 0.223 4.3
104 SAPH440 305 572 0.278 5.02
105 QSTE340TM 340 580 0.191 4.88 where x1, x2, ⋅⋅⋅, x17 and y1, y2, ⋅⋅⋅y15 signify the thickness and material
106 Q345 345 710 0.199 4.9 of components, gcost(x, y) and gmass(x, y) represent the total material
107 QSTE380TM 380 520 0.191 4.92
cost and total mass of selected seat design variables, gD H1(x, y) and
108 QSTE420TM 420 665 0.191 5.08
109 B410LA 485 690 0.157 5.1 gD H2(x, y) denote the maximum displacement at point H under the
110 S500MC 500 616 0.113 5.17 50FC and 50RC working conditions, Dv(x, y) and Mc(x, y) represent the
111 S550MC 550 672 0.113 5.3 maximum displacement and maximum torque at the key points in crit-
112 QSTE600TM 600 782 0.14 5.41
ical locations under each working condition, DUv and MUc signify the
113 SAPH780 677 974 0.131 5.48
114 HC700/ 702 1180 0.077 5.56 upper performance limits of Dv(x, y) and Mc(x, y) under working con-
980DP ditions v and c. Fk(x, y) denote the strain index of the structural com-
115 42CrMo4 930 1210 0.113 9.5 ponents under each working condition, FkU denote the upper strain
indexes limits of Fk(x, y) under working conditions k.
After configuring various parameters for MOGWO (Table 5), the
Table 3 machine learning model developed earlier is resolved, and the Pareto
Range of values for part material-thickness. frontier solution is subsequently obtained, as illustrated in Fig. 14.
Thickness Original Optimization Material Origin Optimization
variable value/ range variables level range
mm 4.5. Multi-criteria decision-making process
T1 2.0 60–120 % M1 103 106–110
T2 1.5 60–120 % M2 106 103–107 In the study of Pareto frontier solutions, selecting optimal trade-off
T3 1.2 60–120 % M3 109 107–110 solutions represents a long-standing challenge. In Section 3.2.5 of this
T4 1.2 60–120 % M4 109 107–110 paper, the elaboration focuses on the proposed MCDM strategy and its
T5 1.9 60–120 % M5 109 106–110
T6 2 60–120 % M6 109 107–110
computational procedure. Now, we implement the method outlined in
T7 1.5 60–120 % M7 110 106–111 Section 2.2.5 to identify the optimal solutions. First, as illustrated in
T8 1.7 60–120 % M8 109 106–110 Table 6, the design responses are weighted using the best-worst method
T9 2.6 60–120 % M9 108 105–109 (BWM) technique, and the comprehensive computational procedure of
T10 3 60–120 % M10 109 105–110
the BWM method is described in Ref. [60]. It should be emphasized that
T11 2.8 60–120 % M11 108 104–108
T12 2.9 60–120 % M12 110 106–111 in the BWM approach, the importance of the design responses is prior-
T13 2 60–120 % M13 115 - itized in the order of total material cost, total mass, and maximum
T14 2 60–120 % M14 108 104–109 displacement. Subsequently, the CoCoSo method is employed to select
T15 1.3 60–120 % M15 107 104–108 the 16th group as the optimal trade-off solution from 200 groups of
T16 0.8 60–120 % M16 101 -
T17 1.2 60–120 % M17 110 105–111
Pareto frontier solutions, as illustrated in Fig. 15.
To ensure the accuracy of the optimal trade-off solution obtained by
IGWO-BPNN machine learning, combined with the MOGWO algorithm
torque, and strain index of the key parts at each key point of the seat and the CoCoSo decision-making method, the design variables of the
working condition are considered as constraints. Furthermore, the dis- optimal solution are reintroduced into the FE model of each seat for
placements of the dummy’s H-points under the seat front and rear crash validation. Table 7 demonstrates the performance of comparing the
states, the total mass of the seat, and the total material cost of the optimized automotive seat frame with the baseline design in terms of
selected design parts are established as the four conflicting optimization cost and mass reduction. Table 8 provides a detailed comparison of the
objectives. Therefore, the finalized multi-objective lightweight design CoCoSo decision-making results with the simulation validation results,
model for the automotive seat frame is presented as follows: including the optimization objectives and constraints.
From Table 8, it is observed that the results of the IGWO-BPNN
machine learning prediction model proposed in this study maintain an
error margin within 5 % compared to the results obtained from the

Table 4
Results of the experimental design.
No Design of experiment Experiment results

T1/mm … T17/mm M1 … M17 Cost/¥ Mass/kg … D_H1/mm D_H2/mm

1 1.8 … 1.2 109 … 110 ​ 18.565 3.730 … 199.190 132.390

2 1.6 … 0.8 109 … 105 ​ 16.657 3.347 … 215.942 137.307
3 1.9 … 0.7 106 … 108 ​ 18.956 3.852 … 195.710 131.203
4 2.1 … 0.9 110 … 107 ​ 19.878 4.028 … 196.233 128.936
5 2 … 0.8 110 … 111 ​ 20.005 4.026 … 191.701 130.085
6 1.7 … 1.1 109 … 106 ​ 16.646 3.379 … 199.811 146.060
7 1.4 … 0.7 108 … 105 ​ 18.661 3.851 … 206.288 136.724
8 1.7 … 1 109 … 110 ​ 19.758 4.040 … 193.271 131.320
… … … … … … … ​ … … … … …
300 1.6 … 1.3 108 … 109 ​ 19.511 3.935 … 190.881 133.715

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H. Chen et al. Advances in Engineering Software 199 (2025) 103797

Fig. 12. Results of the R2 accuracy fitting for seat optimization objectives.

Fig. 13. Results of the RMSE accuracy fitting for seat optimization objectives.

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H. Chen et al. Advances in Engineering Software 199 (2025) 103797

Table 5
Parameters of MOGWO.
Parameter value Parameter value

Grey wolves 50 α 0.1

Archive size 200 β 4 Table 6
Maximum generation 200 δ 2 Seat output response weighting values.
Weighting method Quotas

simulation analysis, and the errors of the four objective functions are Mass Cost D_H1 D_H2

consistently maintained at approximately 1 %. The constraints are BWM 0.3255 0.4186 0.1395 0.1164
categorized into two segments: one involves the displacement of key
points and the torque in each working condition of the seat, while the
other encompasses the strain index of some major stressed parts. This
indicates that the accuracy of the constraint indices is higher than that of
the objective functions, yet the discrepancy between the simulation re-
sults and the prediction results remains below 5 %.

4.6. Simulation analysis and experimental verification

Multi-objective optimization of automotive seats is a multi-objective

optimization with passenger safety in mind, and therefore, validating
the safety of the final optimized design is an essential and non-negligible
part of the process. In this study, the safety verification of the optimized
design scheme is carried out through two aspects, on the one hand, the
seat optimization design scheme is simulated and analyzed for eight seat
working conditions, by determining whether the strain index of its main
stress components fails and whether the indexes such as the displace-
ment of the critical points satisfy the regulatory requirements. Secondly,
considering the actual cost and irresistible factors, one physical exper-
iment is carried out to verify the seat, namely dynamic seat conditions.
Combining the above two points, the safety performance of the opti-
Fig. 15. Design information for the final model of the seat.
mized design is fully verified.
Among the 8 types of automotive seat experimental conditions
selected in this study, each seat experimental condition examines a Table 7
different seat withstanding capacity. For example, the front and rear Performance of optimized automotive seat frame compared to baseline design.
collision of the seat are global test conditions, but their focuses are Type Quotas Baseline design COCOSO Change ( %)
different. The front collision test focuses on whether the sitting basin and
- Mass/kg 4.429 3.411 22.98
other parts provide effective protection for the dummy, while the rear - Cost/¥ 21.280 16.874 20.7
collision emphasizes the supporting effect of the backrest and headrest. 50FC D_H1/mm 208.364 202.789 2.67
Therefore, in order to fully validate the safety of the optimized seat, we 50RC D_H2/mm 133.846 132.591 0.93
verified the strain indices of different seat components under different

Fig. 14. Pareto frontier solutions for optimization.

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H. Chen et al. Advances in Engineering Software 199 (2025) 103797

Table 8
Comparison of predicted results with FE simulation results.
Type Conditions Quotas FE CoCoSo Change Error ( %)

Optimization objectives 50FC D_H1/mm 204.911 202.789 -2.122 1.036

50RC D_H2/mm 134.095 132.591 -1.504 1.122
- Mass/kg 3.385 3.411 0.026 0.768
- Cost/¥ 16.831 16.874 0.043 0.255
Optimization constraints APT D_A/mm 55.381 57.102 1.712 3.107
FUL D_B1/mm 70.781 68.503 -2.278 3.218
RUL D_B2/mm 80.386 78.658 -1.728 2.15
HSST D_C/mm 60.165 62.119 1.954 3.247
SAT D_E/mm 36.686 34.957 -1.729 4.713
50RC T_KL/Nm 485.672 505.478 19.806 4.078
T_KR/Nm 648.663 623.569 -25.094 3.869

seat operating conditions. And whether the seat performance under shows the specific physics experiment.
different seat conditions meets the regulatory and industry Fig. 16 shows the specific physical experiment procedure. The results
requirements. of the specific physical experiments are presented in Table 11, from
Fig. 15 shows the final seat part thickness and material (top thick- which the following conclusions can be drawn: the seat belt was well
ness, bottom material) obtained by solving the COCOSO multi-criteria restrained and the floor effectively limited the movement of the lower
decision method. It is imported into the seat finite element model for limbs of the dummy under the FC50 crash conditions, with no seat belt
multi-case validation to extract the component strain indices and critical tearing, unbuckling, excessive displacement of the dummy’s body, or
point displacements. Table 9 lists the strain indices of the components of excessive rotation of the neck. Some structural deformation occurred in
the final optimized design of the seat for each operating condition. the seat skeleton, but the seat belt and backrest locking device could
Table 10 shows the regulatory requirements for the final optimized work normally. This shows that when the car encounters a high-speed
design of the seat components for the eight seating conditions. collision, although the skeleton has undergone some degree of defor-
Simulation analysis shows that the final optimized design derived mation, the backrest locking device can still work normally and ensure
from the multi-objective optimization strategy proposed in this study the safety of passengers.
has a strain index of less than 1 for the seat components under eight In summary, although the skeleton was deformed to some extent in
typical automotive seat working condition tests, and also passes the the physical experiment of the automobile seat, no fracture traces were
corresponding regulatory requirements, and the results show good found, and the seatbelt and backrest locking device were able to work
performance and optimization effect. normally. The experimental results show that the optimization design
However, in order to further verify the practical application value of scheme shows good performance in practical application, which verifies
the optimization scheme and enhance the persuasive power of this its effectiveness and feasibility in multi-objective optimization. This
study, we conducted actual crash physics experiments on the seat ac- provides strong experimental evidence for further promoting the
cording to the industry guideline SMTC-3-421-162 experimental speci- application of the multi-objective optimization strategy proposed in this
fication, which ensures the reliability and validity of the optimized study.
design in actual use by detecting the test data of the seatbelt protector,
dummy situation and seat skeleton in the real environment. As the 5. Comparison and analysis
function of the car seat is mainly reflected in the car seat can effectively
protect the safety of passengers when the car is involved in a high-speed 5.1. Comparison of machine learning predictive models
collision. Therefore, this study chooses the 50FC condition as the test
condition for verifying the final optimized design in the dynamic crash To more effectively validate the superiority of the IGWO-BPNN
physics experiment of automotive seats. Before the beginning of the prediction model for seat frames, which integrates material-thickness-
experiment, we marked the key components and determined the safety performance, additional intelligent optimization algorithms were
performance of the seat by observing the specific conditions of the employed to optimize the BP neural network. Following extensive
dummy, seat belt, and seat parts before and after the collision. Fig. 16 analysis and research, the BPNN, GWO-BPNN, and northern goshawk

Table 9
Strain indices of the final optimized solution for all working conditions.
part number 50FC 50RC SAT APT SBST HSST FUL RUL

P1 0.000 0.000 0.000 0.000 0.411 0.866 0.000 0.000

P2 0.126 0.513 0.002 0.348 0.369 0.833 0.903 0.836
P3 0.000 0.000 0.064 0.004 0.023 0.000 0.722 0.837
P4 0.004 0.000 0.000 0.126 0.539 0.000 0.792 0.945
P5 0.031 0.123 0.000 0.323 0.924 0.007 0.198 0.133
P6 0.046 0.165 0.000 0.063 0.324 0.000 0.307 0.323
P7 0.321 0.000 0.000 0.655 0.456 0.000 0.056 0.211
P8 0.520 0.022 0.106 0.701 0.701 0.006 0.005 0.531
P9 0.793 0.428 0.801 0.622 0.124 0.585 0.095 0.478
P10 0.916 0.019 0.717 0.286 0.398 0.000 0.000 0.392
P11 0.912 0.050 0.050 0.110 0.023 0.000 0.005 0.214
P12 0.261 0.011 0.000 0.072 0.023 0.000 0.000 0.412
P13 0.026 0.356 0.650 0.348 0.754 0.000 0.021 0.327
P14 0.015 0.635 0.028 0.222 0.103 0.000 0.006 0.188
P15 0.239 0.824 0.768 0.225 0.626 0.024 0.162 0.266
P16 0.567 0.861 0.029 0.923 0.306 0.000 0.224 0.144
P17 0.921 0.017 0.001 0.715 0.000 0.000 0.000 0.249

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H. Chen et al. Advances in Engineering Software 199 (2025) 103797

Table 10
Regulatory requirements for the final optimized solution under all operating conditions.
Regulation Seating conditions Require Simulation Pass or not

SMTC-3–421–162 50F/RC No cracks or fastener damage to the frame No cracks, damage P

SMTC-3–421–163
GB14167–2013 SAT No cracks or fastener damage to the frame No cracks, damage P
GB11550–2009 HSST D_C (mm)<102mm 84.632 P
SMTC-3–421–163 SBST Ab (◦ )<50◦ 30◦ P
SMTC-3–421–164 APT No cracks or fastener damage to the frame No cracks, damage P
SMTC-3–421–104–2011 FUL D_B1 (mm)<80mm 68.612 P
RUL D_B2 (mm)<80 mm 72.611 P

Fig. 16. FC50 physics experiment results for car seats.

Table 11
Seat physics experiment verification results.
Regulation Seating conditions Require Experiment Pass or not

SMTC-3–421–162 50FC Seatbelt restraint No seat belt tearing, unbuckling P

Dummy body position Dummy body displacement normal P
Excessive neck rotation Dummy body displacement normal P
Seat frame Seat frame deformed, but not broken P

Optimization (NGO)-BPNN machine learning prediction models were models, the single BPNN model exhibits the lowest accuracy. While the
chosen. To ensure a valid comparison, the intelligent optimization al- GWO-BPNN and NGO-BPNN models demonstrate improved fitting ac-
gorithms were configured with identical parameters in terms of popu- curacies, the IGWO-BPNN model remains the most effective for pre-
lation size and number of iterations; the fitting objective was the same as dicting automotive seat frames with integrated material-thickness-
that of the IGWO-BPNN algorithm, and the fitting accuracy was assessed property in this study.
using R2 and RMSE between the actual and predicted values of the test
set. The fitting accuracies of the various prediction models are displayed
5.2. Comparison of different multi-objective optimization algorithms
in Fig. 18.
The following conclusions can be drawn from Fig. 17: (1) Regarding
To verify the optimization potential and accuracy of the multi-
the four evaluation indices, regardless of the prediction model
objective Gray Wolf algorithm in optimizing automotive seat frames,
employed, the fitting accuracy for quality and cost indices is consistently
several multi-objective optimization algorithms have been employed in
higher than that of D_H1 and D_H2. This is attributed to the fact that the
this study to address the problem. It should be noted that to maintain the
latter are nonlinear indices, which are inherently more challenging to fit
accuracy and comparability of the results, the IGWO-BPNN machine
than linear indices like quality and cost. (2) Concerning the prediction
learning prediction model with the highest fitting accuracy, as proposed

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H. Chen et al. Advances in Engineering Software 199 (2025) 103797

Fig. 17. Comparison of different machine learning fitting accuracies.

in this study, is utilized across all prediction models. Furthermore, the solution in the context of the MOOP associated with the seat frame,
maximum number of iterations and the population size for these opti- utilizing the CoCoSo decision-making method. Fig. 19 illustrates a
mization algorithms are aligned with the MOGWO algorithm of this detailed comparison of the final results derived from the different multi-
study, and all employ the CoCoSo decision-making method to address objective optimization algorithms presented in Table 12.
the Pareto solution sets obtained by the various multi-objective opti- When considering Fig. 18, Table 12, and Fig. 19 collectively, it be-
mization algorithms. The introduced multi-objective optimization al- comes evident that various MOOA exhibit distinct qualities, diversity,
gorithms primarily comprise three types: the MOPSO algorithm, NSGA- robustness, and stability in their solutions to the Pareto frontier.
II, and the multi-objective whale optimization (MOSWO) algorithm. Furthermore, in the final results derived using the CoCoSo method to
Fig. 18 illustrates the Pareto solution sets obtained by the various multi- address the Pareto frontier solutions of different algorithms, it is
objective optimization algorithms, and Table 12 presents the results apparent that the values of the four objective functions each possess
obtained from different MOOA aimed at identifying the Pareto front specific strengths and weaknesses. Specifically, for the D_H1 index, the

Fig. 18. Comparison of Pareto frontier solutions for different optimization algorithms.

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H. Chen et al. Advances in Engineering Software 199 (2025) 103797

MOSWO algorithm achieves the optimal result of 20.1081cm. Regarding experimental validation. The results show that the optimized automo-
the D_H2 index, the NSGA-II algorithm secures the lowest value of tive seat skeleton reduces the cost by 20.7 % and the mass by 22.9 %
13.1257cm. However, for the cost and mass indices, the MOGWO al- while maintaining the safety performance. Here, we will discuss in
gorithm utilized in this study offers the most advantageous lightweight further detail the practical application implications of the findings of
and economic solution. Among the other three optimization algorithms, this research, especially regarding the following points:
MOSWO appears to have the lowest cost and mass. In conclusion, the
MOGWO algorithm employed in this study emerges as the superior 6. Discussion and conclusion
choice for providing the most efficient lightweight and economical so-
lution in addressing the MOOP challenge for automotive seat frames. 6.1. Discussion

In this research, in order to improve the accuracy of the seat input

5.3. Comparison of different decision-making methods
variables and output response prediction, a hybrid machine learning
prediction model with an IGWO and BPNN is proposed, which solves the
In the field of multi-criteria decision making, a large number of
multi-objective optimal design process of the seat in conjunction with
decision-making methods have been applied in engineering research.
the MOGWO in order to achieve a Pareto frontier while using a CoCoSo
Different decision-making methods give different results. In the field of
methodology for decision making to determine the best compromise
automotive seat frame optimization, the smallest possible cost and
solution. In addition, the effectiveness of the proposed optimal design
quality are the priority factors to be considered under the premise of
methodology is demonstrated by comparing the baseline design, simu-
guaranteeing the satisfaction of safety performance. In order to verify
lation analysis, optimal design methodology and physical experimental
the effectiveness and accuracy of CoCoSo decision making method in
validation. The results show that the optimized automotive seat skeleton
automotive seat frame optimization, this study introduces different
reduces the cost by 20.7 % and the mass by 22.9 % while maintaining
decision-making methods for Pareto front solution planning, the main
the safety performance. Here, we will discuss in further detail the
methods introduced are gray relational analysis (GRA) and VIKOR
practical application implications of the findings of this research, espe-
methods. It is to be noted that in order to maintain the accuracy and
cially regarding the following points: Cost and weight reductions
comparability of the results, the values of the objective weights in
directly improve the productivity and economics of seating. By opti-
Table 6 and the Pareto frontier solutions obtained by MOGWO solving
mizing the design and use of materials, fewer raw materials are required
are used. Table 13 shows the results obtained from different decision-
for the production process, with a corresponding reduction in
making methods. Fig. 20 compares the results in Table 13 in detail.
manufacturing energy consumption, resulting in improved resource
From Table 13 and Fig. 20, it is evident that the final results derived
utilization. For mass-production automakers, this means lower produc-
from various decision-making methods vary significantly. Considering
tion costs and higher profit margins. At the same time, reduced seat
only the seat comfort performance indices D_H1 and D_H2, the solution
weight simplifies the manufacturing process, reducing the complexity of
yielded by GRA is superior, as it significantly enhances the displacement
material handling during transportation and installation, which further
of the H point of the dummy under crash conditions of the seat. Sec-
enhances the efficiency of the manufacturing process.
ondly, regarding the lightweight aspect, the VIKOR decision-making
approach stands out with a 23.685 % weight reduction compared to
(1) The impact of the manufacturing process:
the initial design, although its comfort performance index shows mini-
mal optimization. Economically, the CoCoSo method results in a saving
Cost and weight reductions directly improve the productivity and
of 20.705 % in material costs compared to the original design. Its eco-
economics of seating. By optimizing the design and use of materials,
nomic efficiency is unparalleled.
fewer raw materials are required for the production process, with a
In summary, the results across various decision-making methods are
corresponding reduction in manufacturing energy consumption, result-
distinct. For optimal comfort performance, this study advocates for the
ing in improved resource utilization. For mass-production automakers,
GRA method; for significant lightweighting, the VIKOR method; and for
this means lower production costs and higher profit margins. At the
cost minimization, the CoCoSo method is recommended. From the
same time, reduced seat weight simplifies the manufacturing process,
perspective of a seating company, the primary goal is minimizing costs,
reducing the complexity of material handling during transportation and
followed by ensuring mass and comfort performance. Consequently, the
installation, which further enhances the efficiency of the manufacturing
CoCoSo decision scheme outlined in this study demonstrates substantial
process.
economic and lightweighting benefits in the optimization of automotive
seat frames. In this research, in order to improve the accuracy of the seat
(2) Performance in high-speed crashes:
input variables and output response prediction, a hybrid machine
learning prediction model with an improved Gray Wolf Optimizer
The safety of automotive seats is critical in high-speed crashes. While
(IGWO) and Back Propagation Neural Network (BPNN) is proposed,
weight reduction may be a concern for crash safety, through multi-
which solves the multi-objective optimal design process of the seat in
objective design optimization, seats can be reduced in weight while
conjunction with the Multi-Objective Gray Wolf Optimizer (MOGWO) in
maintaining or improving their crash-absorbing properties. Proper se-
order to achieve a Pareto frontier while using a Combined Compromise
lection of lightweight, high-strength materials and structural design can
Solution (CoCoSo) methodology for decision making to determine the
ensure passenger safety in a crash while maintaining seat stiffness and
best compromise solution. In addition, the effectiveness of the proposed
energy absorption.
optimal design methodology is demonstrated by comparing the baseline
design, simulation analysis, optimal design methodology and physical
(3) Long-term safety and durability:

Table 12 In addition to the impact on crash safety, weight and cost reductions
Comparison of results of different optimization algorithms.
can have a profound effect on the long-term safety and durability of the
MOOA D_H1/cm D_H2/cm Mass/kg Cost/¥ seat. Lightweight designs, often with greater material efficiency and
MOPSO 20.1081 13.2255 3.623 17.652 better structural design, can increase the service life of a seat. Problems
NAGA-II 20.3593 13.1257 3.508 17.056 such as material fatigue and wear and tear that seats may face during
MOSWO 19.9795 13.3510 3.412 16.904 long-term use can be overcome by optimizing material selection and
MOGWO 20.2789 13.2591 3.411 16.874
manufacturing processes. This not only reduces costs, but also improves

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H. Chen et al. Advances in Engineering Software 199 (2025) 103797

Fig. 19. Comparison of results of different optimization algorithms.

Table 13
Comparison of results of different MCDM methods.
Type Quotas Original CoCoSo Change ( %) GRA Change ( %) VIKOR Change ( %)

50FC D_H1/cm 20.8364 20.2789 2.677 20.0605 3.724 20.6795 0.753

50RC D_H2/cm 13.3846 13.2591 0.938 13.0313 2.640 13.4010 0.123
- Cost/¥ 21.280 16.874 20.705 17.718 16.739 16.996 20.132
- Mass/kg 4.429 3.411 22.988 3.573 19.327 3.380 23.685

Fig. 20. Comparison of results of different MCDM methods.

the overall quality and long-term safety performance of the product. proposed. Based on the findings of this paper, the following conclusions
In summary, the reduction of cost and weight is not just an can be drawn.
improvement in economic efficiency, it can also significantly affect the (1) The established FE model demonstrates considerable accuracy, as
manufacturing process and the safety and durability of the seat in long- evidenced by rigorous comparisons with physical tests under various car
term use. By optimizing the design and material selection, it is ensured seat safety test conditions. Consequently, the optimized design derived
that lightweighting is achieved without compromising or even from this model is deemed reliable.
enhancing its overall performance. (2) The proposed method of screening design variables through the
energy absorption to mass ratio is characterized by its simplicity and
efficiency. Moreover, the selected design variables exert a significant
6.2. Conclusion impact on seat performance, making the optimization of such parts
particularly meaningful.
In this study, a strategy for the optimal design of automotive seat (3) The DLH mechanism has been incorporated into the GWO
frames that integrates machine learning, MOOP, and MCDM is

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H. Chen et al. Advances in Engineering Software 199 (2025) 103797

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