DOI: 10.5545/sv-jme.2024.1185 169
© The Authors. CC BY 4.0 Int. Licencee: SV-JME  Strojniški vestnik - Journal of Mechanical Engineering   ▪   VOL 71   ▪   NO 5-6   ▪  Y 2025
Multi-Objective Optimization Design  
of the Ejector Plate for Rear-Loader Garbage Trucks
Fu-sheng Ding
1
 – Hong-ming Lyu
1
  – Jun Chen
2
 – Hao-ran Cao
1
 – Lan-xiang Zhang
1
1 Yancheng Institute of Technology, School of Automotive Engineering, China
2 Jiangsu Yueda Special Vehicle Co., Ltd., China
 hongming_lv@ycit.edu.cn
Abstract This work presents a multi-objective optimization design approach for the ejector plate, a critical component of rear-loader garbage trucks, with the 
goal of ensuring structural integrity and optimizing lightweight performance. A parametric finite element model of the ejector plate is developed with optimization 
objectives focused on minimizing mass, controlling deformation, and reducing the maximum von Mises stress. Through sensitivity analysis, seven key variables 
are selected for optimization. A Box-Behnken design (BBD) is used to systematically explore these parameters, and a Kriging surrogate model is constructed to 
approximate the objective function, with accuracy benchmarked against response surface methodology (RSM). The Non-dominated Sorting Genetic Algorithm 
II (NSGA-II) is applied to derive the optimal solution, achieving a lightweight design meeting all structural requirements. The results show that the mass of the 
ejector plate of the rear-loading waste compactor can be reduced by 6.06 % through structural optimization, while meeting the strength and deformation criteria. 
This improvement not only enhances waste transportation efficiency, but also lowers production costs and enhances material utilization.
Keywords  garbage truck, ejector plate, multi-objective optimization, NSGA-II, Kriging 
Highlights
 ▪ Optimizing the ejector plate considering the mass, maximum displacement and maximum von Mises stress of the ejector plate.
 ▪ Using sensitivity analysis to identify key design variables of the ejector plate.
 ▪ Using Kriging method to construct a highly accurate surrogate model.
 ▪ Applying the non-dominated sorting genetic algorithm II for multi-objective optimization of the ejector plate.
1 INTRODUCTION
As the global economy grows and urbanization accelerates, the 
generation of household waste has increased significantly. This 
trend is putting immense pressure on urban environmental health 
and poses challenges for sustainable urban development [1]. Given 
the substantial volume of urban waste, efficient transportation and 
disposal methods have become critical concerns for municipalities 
and related departments [2]. Rear loader garbage trucks are specialized 
vehicles designed to operate with waste compaction transfer stations. 
Their widespread adoption by sanitation and municipal departments 
can be attributed to their large loading capacity and effective 
sealing. The ejector plate, a key component of these trucks, plays a 
vital role in the loading and unloading of waste. Different types of 
waste impose varying demands on the strength of the ejector plate. 
During the compaction and loading/unloading processes, the ejector 
plate is subjected to forces from both the hydraulic cylinder and 
the waste itself, resulting in varying degrees of deformation. Such 
deformation can reduce the compaction ratio within the truck's box 
body, ultimately diminishing loading and unloading efficiency and 
affecting the truck's overall waste capacity. Therefore, research into 
the ejector plate is crucial for the effective design of garbage trucks.
The three dimensional (3D) model of the ejector plate was 
developed in SolidWorks software, which was then subjected to 
finite element analysis (FEA) to evaluate its stress and displacement 
distributions under operational conditions [3]. The garbage truck 
loading mechanism served as the research focus, with parametric 
analysis conducted to establish the functional relationships among 
compression-filling force, pusher stroke, and installation angle 
[4]. The garbage truck manipulator was digitally prototyped in 
SolidWorks for performance enhancement, followed by a multi-
domain simulation workflow: multi-body dynamic (MBD) analysis in 
ADAMS determined operational load spectra under various working 
conditions, FEA in HyperWorks assessed structural integrity, and 
topology optimization integrated with genetic algorithms achieved 
16.73 % mass reduction while maintaining mechanical performance 
[5].
Lightweighting vehicles is an essential strategy for saving energy 
and reducing emissions [6,7]. Reducing the weight of vehicles 
can reduce energy consumption while improving dynamics and 
braking performance [8,9]. The main approaches to lightweight 
design include structural optimization, process optimization and 
material lightweighting [10-12]. Structural optimization can be 
further categorized into size optimization, shape optimization, and 
topology optimization [13]. Currently, size and shape optimization 
are widely used in engineering applications [14,15]. Size optimization 
can be divided into discrete and continuous categories based on the 
continuity of design variables. Continuous size optimization results 
typically require rounding to fit available size parameters, making 
discrete size optimization more suitable for practical project needs 
[16]. Furthermore, discrete optimization allows for the simultaneous 
optimization of multiple variables, potentially yielding superior 
results.
In the realm of optimal design algorithms, Goldberg was the first 
to apply genetic algorithms (GA) to multi-objective optimization in 
1989 [17]. Subsequent research has built upon this foundation. For 
instance, Paz et al. [18] utilized multivariate adaptive regression 
spline techniques combined with multi-objective genetic algorithms 
to enhance the energy absorption properties of automotive 
components while reducing mass. Velea et al. [19] conducted 
multi-objective optimization of a composite body for electric 
vehicles, considering weight, cost, and stiffness. Duan et al. [20] 

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applied a multi-objective particle swarm optimization algorithm for 
lightweight design of body-in-white structures to meet reliability 
requirements. Jiang et al [21] employed the Kriging surrogate model 
alongside the non-dominated sorting genetic algorithm II (NSGA-II) 
for the multi-objective optimal design of suspension arms and torsion 
beams, achieving weight reduction without compromising structural 
or vehicle performance. Xie et al. [22] implemented a multi-objective 
optimization method incorporating the TOPSIS method for the 
lightweight design of commercial vehicle cabs, successfully meeting 
design requirements while achieving weight reduction.
The lightweight design of the ejector plate is directly related to 
the load capacity and reliability of the garbage truck. Therefore, 
pursuing a lightweight design for the ejector plate while considering 
both structural performance and load capacity is essential. This paper 
focuses on the ejector plate of a specific garbage truck, utilizing 
3D design software for parametric modeling and conducting finite 
element analysis based on actual working conditions. A sensitivity 
analysis of mass, displacement, and equivalent force concerning 
fourteen key parameters is performed, identifying 7 parameters 
with greater sensitivity as design variables. These variables are then 
optimized using a combination of the Kriging surrogate model and 
the NSGA-II algorithm to achieve the lightweight design of the 
ejector plate.
2 METHODS AND MATERIALS
2.1 Working Principle of the Ejector Plate
As shown in Fig. 1, the rear loader garbage truck consists primarily 
of several components, including the vehicle chassis, box body, 
hydraulic cylinder, packing mechanism, and ejector plate, as 
illustrated in Fig. 2. The box body is mounted on and rigidly 
connected to the chassis frame. At the rear of the truck, the packing 
mechanism is responsible for loading and initially compacting the 
waste. The waste is then pushed into the box body by the action of a 
scraper on the packing mechanism. After each compaction, both the 
compacted waste and the newly loaded waste are gradually pushed 
toward the rear of the box body, with the ejector plate continuously 
moving back as new waste is added.
Fig. 1. Construction of the rear-loader garbage truck; 1 chassis, 2 cylinder  
3 ejector plate, 4 box body, and 5 precompressor
2.2 Design Requirements for the Ejector Plate
The ejector plate consists of the front plate, frame skeleton, guide 
rail skeleton, cylinder support, and other components. As a critical 
load-bearing element of rear-loader garbage trucks, the mechanical 
properties of the ejector plate significantly impact the overall 
performance of the vehicle. During operation, the ejector plate 
experiences the combined effects of hydraulic cylinder thrust, friction 
between the ejector plate and the guide rail, and the extrusion pressure 
from the refuse. The deformation of the ejector plate skeleton directly 
affects the gap between the ejector plate and the cargo area, which, in 
turn, influences the complete discharge of the waste.
Therefore, the design of the ejector plate must ensure that 
the deformation of the frame skeleton remains below 10 mm. In 
addition, under maximum external load conditions, the stress on all 
components should not exceed 355 MPa. To meet the performance 
requirements, the design of the ejector plate should also prioritize 
minimizing weight and maintaining high quality.
Fig. 2. 3D model of the ejector plate
2.3 Optimization Flow for the Ejector plate
This study employs a combination of the finite element method 
(FEM), Kriging Surrogate Model, and GA for optimization. The 
finite element method is utilized to calculate the maximum working 
load conditions of the ejector plate. Based on the results of the 
finite element analysis, a lightweight multi-objective structural 
optimization program is proposed which focuses on minimizing the 
total deformation, mass and von Mises stress. The Kriging surrogate 
model is known for its high accuracy, adaptability and scalability, 
making it widely applicable in the structural optimization design 
of various mechanical components [23,24]. Genetic algorithms 
are favored for multi-objective optimization due to their broad 
adaptability, global search capability, parallelism, and independence 
from derivative information [25]. The optimization process is 
illustrated in Fig. 3.
The specific steps are as follows: 
•	 Parametric modeling: The ejector plate is modeled parametrically 
in 3D modeling software, and its finite element model is created in 
HyperWorks software, followed by static analysis.
•	 Sensitivity analysis: 14 parameters within the ejector plate skel-
eton are analyzed for sensitivity, leading to the selection of 7 pa-
rameters that significantly influence the mass, total displacement, 
and von Mises stress of the ejector plate skeleton for further anal-
ysis.
•	 Experimental design: The 7 highly sensitive parameters are sam-
pled and tested using the Box-Behnken experimental design meth-
od.
•	 Surrogate model generation: A Kriging surrogate model is devel-
oped based on the experimental design parameters.
•	 Optimization: A multi-objective genetic algorithm (NSGA-II) is 
employed to obtain the optimal set of solutions and to validate the 
appropriateness of the selected optimization parameters.

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2.4 Finite Element Analysis of the Ejector plate 
2.4.1 Finite Element Model 
When modeling, capturing the essence of the research object and 
appropriately simplifying the original model are essential for 
improving simulation efficiency and quality. The rubber buffer 
blocks on the ejector plate, and the sealing plates on both sides serve 
auxiliary roles that have minimal impact on the forces in the model, 
allowing them to be omitted from the modeling process.
The focus is on the overall structural characteristics of the ejector 
plate rather than localized welding issues. Rigid units are employed 
to simulate the weld relationships between different components, 
facilitating the transfer of forces. This simplified approach enhances 
the fidelity of the finite element results while significantly improving 
solution speed, enabling a more efficient design process that can 
accommodate a large number of experimental simulations.
In the ejector plate assembly, the cylinder support is a casting 
and is simulated using 8 nodes hexahedron elements. The other 
components are sheet metal parts, characterized by smaller 
dimensions in the thickness direction and larger dimensions in 
length and width, and are simulated using 4 nodes shell elements. 
Considering the enterprise's specifications for mesh size, as well 
as the need to maintain computational accuracy while minimizing 
computational cost, an 8 mm mesh was adopted for the model. Upon 
completing the meshing of the ejector plate, the model consists of 
117,823 nodes and 117,117 elements, as shown in Fig. 4.
The material of the ejector plate is high-quality structural carbon 
steel (Q355B), with properties including a Young's modulus of 2.1 × 
105 MPa, a density of 7850 kg/m³, and a Poisson's ratio of 0.3, and a 
minimum yield strength of 355 MPa.
As one of the primary load-bearing components of the truck, the 
ejector plate is subjected to the thrust from the hydraulic cylinder, 
the force exerted by the waste on its front plate during loading, 
and the support force from the guide rail. Due to the considerable 
stroke length of the ejector plate, a multi-stage hydraulic cylinder 
is employed; the shorter the stroke of the hydraulic cylinder, the 
greater the thrust produced. According to the hydraulic cylinder's 
operating behavior, the ejector plate experiences the greatest load and 
deformation when the truck is nearly full of waste and the ejector 
plate is close to its innermost position within the box body.
The hydraulic system operates at a pressure of 17.6 MPa. The 
ejector plate cylinder is a 3-stages rod cylinder, with the diameters of 
the rods measuring 90 mm (1
st
 stage), 70 mm (2
nd
 stage), and 50 mm 
(3
rd
 stage). The maximum thrusts exerted by these rod sections are 
calculated to be 111.9 kN, 67.7 kN, and 34.5 kN, respectively. When 
waste is loaded into the box body, the force exerted by the waste on 
the ejector plate must overcome both the hydraulic cylinder's force 
and the friction between the ejector plate and the guide rail before 
the ejector plate can move inward. Thus, the ejector plate experiences 
maximum load when the hydraulic cylinder's thrust is 111.9 kN. The 
ejector plate moves slowly within the box body, neglecting the effects 
of dynamic loading. The friction force, which cannot be directly 
measured, is assumed to be approximately 12 kN (about 10 % of the 
maximum thrust). During slow movement, the force from the waste 
on the front plate is balanced by the hydraulic cylinder's force and 
Parametric modeling of 
ejection panel
Finite element analysis
BBD to obtain initial 
sample points 
Determine optimised 
design parameters
Build Kriging surrogate 
model
Build optimisation objectives and 
constraints
The NSGA-II algorithms 
is used to obtain the 
Pareto optimal solution set
Judging the accuracy of 
surrogate model
Revise judge 
optimisation 
results
Close
N
Y
N
Y
Start
Define design variables
Sensitivity analysis of 
design variables
Perform simulations to 
calculate data
Redefine the 
range of 
objective 
funtions
Fig. 3. Flow chart for optimization of the ejector plate structure

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the friction between the ejector plate and the guide rail, resulting in a 
pressure of 0.06 MPa acting on the front plate.
Fig. 4. Mesh model of the ejector plate 
Fig. 5. Constraints of the ejector plate 
Fig. 6. Distribution of the load on ejector plate 
In accordance with actual working conditions, fixed displacement 
constraints in the y-direction (forward and backward) are applied at 
the cylinder support of the ejector plate. Displacement constraints in 
the x-direction (left and right) are applied to the lateral friction blocks 
of the ejector plate guide, and z-direction (vertical) displacement 
constraints are applied to the upper and lower friction blocks of the 
guide, as shown in Fig. 5. A uniform compressive pressure load of 
0.06 MPa is applied to the side of the front plate in contact with the 
waste. Considering that the waste cannot be loaded to the top of the 
box body, this load is applied only up to a height of 960 mm from the 
bottom plate, as illustrated in Fig. 6.
Taking into account the working conditions of the ejector plate, 
overload protection, personnel protection, material properties, etc., a 
safety factor of 1 is selected, and the allowable stress of the material 
is 355 MPa.
2.4.2 FEA Result Analysis
The finite element analysis yields to the static deformation and 
von Mises stress contours for the ejector plate and its skeleton, as 
illustrated in Fig. 7 through Fig. 10.
Fig. 7. Static deformation of the ejector plate 
Fig. 8. Static deformation of the ejector plate skeleton
As shown in Fig. 7, the maximum deformation of the ejector plate 
is approximately 21.4 mm, which occurs in the central and upper 
areas of the front plate. This deformation corresponds to the actual 
working conditions. The ejector plate has a width of 1800 mm and 
the front plate is made of a 2.3mm thick steel plate. Although this 
design results in a relatively low stiffness, the significant deformation 
of the front plate does not affect its performance, provided that the 
deformation of the ram skeleton remains within acceptable limits.
Figure 8 indicates that the maximum deformation of the ejector 
plate skeleton is approximately 7.36 mm, concentrated in the middle 
of the upper tube, with progressively smaller deformations further 
down. This deformation is within the allowable limit of 10 mm 
specified in the design, demonstrating that the current ejector plate 
meets the stiffness requirements and possesses a degree of stiffness 
redundancy.

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Fig. 9. von Mises stress distribution in the ejector plate 
Fig. 10. von Mises stress distribution in the ejector plate skeleton
From Figure 9, it can be seen that the maximum von Mises stress 
of the ejector plate is 304.7 MPa, which is located in the middle and 
upper sections of the front plate.
In Figure 10, the von Mises stress contour for the ejector plate 
skeleton shows a maximum stress of 292 MPa, which is located in 
the lower sections of the side tube and below the design allowable 
stress limit of 355MPa. 
To address the potential influence of mesh size on the simulation 
results, a systematic mesh sensitivity study was carried out 
specifically on the side tube (identified as the critical region with the 
highest stress concentration). Three different mesh configurations 
were tested: 8 mm, 6 mm, and 4 mm average element size. The 
corresponding results are summarized in Table 1 below.
Table 1. Mesh sensitivity analysis for the side tube
Mesh size [mm] 8 6 4
Max stress [MPa] 291.7 295.5 296.9
Stress increment [MPa] - +3.8 +5.2
Maximum stress difference - +1.3 % +1.8 %
Number of side tube elements 2760 4450 9905
From Table 1, it can be seen that as the mesh is refined from 
8 mm to 4 mm, the maximum stress exhibits a monotonic but 
diminishing increase, with a total variation of 1.8 % (5.2 MPa 
absolute difference), and the number of side tube elements shows 
an increase in geometric multiples, with a total variation of 258.9 % 
(7145 elements difference). The relative stress change between 
successive refinements (6 mm vs. 8 mm: 1.3 %; 4 mm vs. 6 mm: 
0.5 %) demonstrates convergence behavior, indicating that further 
mesh refinement beyond 6mm yields marginal improvements in 
accuracy. The sub-2% discrepancy between the 8 mm and 4 mm 
meshes falls within typical engineering tolerance thresholds for such 
analyses [26], confirming that the 8 mm mesh provides sufficient 
accuracy while maintaining computational efficiency.
The results of finite element analysis indicates that the current 
ejector plate skeleton meets the strength requirements and has a 
significant material surplus, allowing for potential lightweighting 
studies of the ejector plate structure.
2.5 Sensitivity analysis 
2.5.1 Determination of Design Variables and objective functions
According to the structural characteristics and the stress distribution 
of the ejector plate’s skeleton , the upper tube thickness P
1
, reinforced 
tube	 Ⅰ	 thickness	P
2
, side tube thickness P
3
, side steel tube thickness 
P
4
, angle iron thickness P
5
, bracket thickness P
6
, bottom skeleton 
thickness P
7
,	 reinforced	 tube	 Ⅱ	 thickness	 P
8
, reinforced steel tube 
thickness P
9
, angle brace thickness P
10
, bottom plate thickness P
11
, 
rail angle brace plate thickness P
12
, rail skeleton thickness P
13
, and 
rail support plate thickness P
14
 are used as the candidate design 
variables. The maximum static deformation D, maximum von Mises 
stress S, and mass M are used as objective functions. The position of 
each design variable of the ejector plate’s skeleton is shown in Fig. 
11.
Fig. 11. Position of each design variable
2.5.2 Sensitivity analysis of design variables
Sensitivity analysis is a widely used tool in structural optimization, 
aimed at identifying design variables that significantly impact 
structural performance. The sensitivity value associated with 
changes in a design variable intuitively indicates both the magnitude 
and direction of its effect on performance, allowing for the rapid 
screening of critical design variables for optimization [27].
This analysis typically examines how the output response of a 
system is affected by its input parameters. For complex mathematical 
models, determining higher-order sensitivities can be challenging. 
Therefore, conventional sensitivity analysis focuses on the first-order 
partial derivatives of the design response with respect to the design 
variables. The mathematical expression is given by:
sen
gX
x
gX
x
ii
() ()
,









 (1)
where g(X) is the system performance index, X is the system design 
parameter vector, and x
i
 is the ith parameter in the system design 
parameter vector.

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The sensitivity of the 14 design variables in the ejector plate 
structure corresponding to the three performance indicators of mass, 
maximum von Mises stress and total deformation are calculated, as 
shown in Fig. 12, respectively.
Fig. 12. Sensitivity of the candidate design variable 
From Figure 12, we can see that: 
1. The significance of all variables for total mass is positively 
correlated, with P
7
, P
9
, and P
12
 having the greatest significance 
for total mass.
2. P
3
, P
12
, P
13
, and P
14
 exhibit the most significant negative 
correlation with maximum von Mises stress, while the other 
variables show less significant correlations.
3. All variables show a negative correlation with total displacement. 
P
1
, P
2
, P
3
, P
4
, P
7
, P
11
 and P
12
 are most significantly negatively 
correlated with maximum deformation, while the remaining 
variables have a lesser impact.
Considering the sensitivity of each design parameter to the 
response value, 7 parameters are selected as design variables for the 
final size optimization. The initial values of each parameter and their 
allowable range of variation are presented in Table 2.
Table 2. Initial values and range of the ejector plate design variables [mm]
Design variables Parameters Original Value range
Upper tube P
1
3 2~4
Reinforced tube I P
2
3.2 2~4.4
Side tube P
3
3.2 2~4.4
Side steel tube P
4
3.2 2~4.4
Bottom skeleton P
7
5 4~6
Guide rail angle brace plate P
12
3.2 2~4.4
Guide rail skeleton P
13
4 3~5
2.6 Surrogate Model
A surrogate model is a mathematical representation constructed 
from a finite set of data obtained through computational or physical 
experiments. Complex models that require significant computational 
resources can be approximated using finite element-based models or 
less computationally intensive statistical models as surrogate models 
[28]. In optimization contexts, where evaluating objective and/or 
constraint functions demands considerable computational effort, a 
surrogate model effectively replace these functions [29]. The original 
model can be accurately predicted only if a highly precise surrogate 
model is developed.
The design of experiments (DOE) is a crucial step in creating a 
surrogate model, utilizing mathematical analyses such as probability 
theory and linear generation to identify reasonable discrete sample 
points. The distribution of these sample points within the design 
space significantly influences the construction of the Kriging 
surrogate model, making the selection of an appropriate experimental 
design method critical for subsequent optimization. Commonly 
used experimental design methods include central composite design 
(CCD), Box-Behnken design (BBD), Latin hypercube design (LHS), 
and optimal space filling design (OSF). For this study, the Box-
Behnken design method was chosen due to its advantages of requiring 
fewer trials and not necessitating the measurement of vertex points 
compared to other design methods.
2.6.1 Box-Behnken Design (BBD)
Using the upper tube thickness P
1
,	 the	 reinforced	 tube	 Ⅰ	 thickness	P
2
, 
the side tube thickness P
3
, the side steel tube thickness P
4
, the bottom 
skeleton thickness P
7
, the rail angle brace thickness P
12
, and the rail 
skeleton thickness P
13
 as independent variables, and the total mass 
(M) of the ejector plate, the maximum displacement (D), and the 
maximum von Mises stress (S) as the evaluation indices, a total of 62 
experimental samples and their corresponding response values were 
extracted, as shown in Table 3.
Table 3. Test samples and response values for BBD tests
Tests
P
1
 
[mm]
P
2
 
[mm]
P
3
 
[mm]
P
4
 
[mm]
P
7
 
[mm]
P
12
 
[mm]
P
13
 
[mm]
M 
[kg]
D 
[mm]
S 
[MPa]
1 3 3.2 3.2 2 4 2 4 169.4 7.856 414.3
2 3 3.2 3.2 4.4 4 2 4 174.8 7.509 418.4
3 3 3.2 3.2 2 6 2 4 178.9 7.685 415.3
4 3 3.2 3.2 4.4 6 2 4 184.2 7.341 419.4
5 3 3.2 3.2 2 4 4.4 4 175.7 7.530 224.5
6 3 3.2 3.2 4.4 4 4.4 4 181.0 7.195 226.6
7 3 3.2 3.2 2 6 4.4 4 185.1 7.386 224.4
8 3 3.2 3.2 4.4 6 4.4 4 190.5 7.052 227.0
… … … … … … … … … … …
60 3 3.2 3.2 3.2 5 3.2 4 180.0 7.36 291.7
61 3 3.2 3.2 3.2 5 3.2 4 180.0 7.36 291.7
62 3 3.2 3.2 3.2 5 3.2 4 180.0 7.36 291.7
2.6.2 Kriging Surrogate Model
Kriging refers to a surrogate model based on Gaussian process 
modeling, which first originated in a geostatistical paper by Krige 
[30], and is now one of the most widely used surrogate modeling 
methods. The Kriging surrogate model can be expressed as follows:
yx fx Zx () () () ,  (2)
where y(x) is an unknown function of the optimization objective, f(x) 
is a polynomial function on the variable x, and Z(x) is a stochastic 
function mainly used to correct the error of the global model.
The expression for f(x) is:
fx xx xx
ii
i
k
ii i
i
k
ij ij
ji
k
i
k
() ,  
  

  
 
0
1
2
11 1
1
 (3)
where k is the number of design variables, in this paper k = 7, β is the 
coefficient to be determined for each equation.
Z(x) is a random function that follows a Gaussian normal 
distribution with zero mean.
EZx () ,   0 (4)
VZx () .  
2
 (5)
Z(x) creates a ‘local’ deviation that satisfies Eq. (4) with minimum 
variance to obtain the best valuation of the response surface of 
the kriging model interpolated to the N sample data points. The 
covariance matrix of is given by the following equation:

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Structural Design
CovZxZ xR rxx
ij ij
() ,()[ (, )] , 






2
 (6)
where R is the N × N symmetric positive definite correlation matrix; 
r(x
i
, x
j
) is a Gaussian correlation function, which can represent the 
spatial correlation between any two sample points x
i
 and x
j
. The 
expression is as follows:
rxxx x
ij k
k
M
ki kj
(, )e xp ,  









1
2
 (7)
where θ
k
 is the unknown parameter used for fitting; M is the number 
of design variables; x
ki
 and x
kj
 are the components of the k
th
 sampling 
point x
i
, x
i
.
Based on the unbiasedness of Z(x) and the minimum variance 
of the estimate, the relevant parameter θ
k
 given by the maximum 
possible estimate, i.e. when θ > 0, is derived to maximize the 
following equation:
A
nR x
l

 ln() ln ()
,


2
2
 (8)
where n
l
 is the number of response values, σ

2
 is the variance 
estimate, and |R(x)| is the correlation value between the point to be 
measured and the sampling point. In other words, turn to find the 
minimum value. 
min( )( ).
x
m
xR x


0
1
2
  (9)
2.6.3 Surrogate Model Accuracy Validation
The surrogate model may have some errors in the process of 
establishment, in order to verify the accuracy of the established 
surrogate model, the deterministic coefficient R
2
, root mean square 
error (RMSE) are introduced. The value of R
2
 ranges between [0, 1], 
when the value of R
2
 tends to be closer to 1 and the value of RMSE 
tends to be closer to 1, they indicate that the accuracy of the fitted 
surrogate model is higher, and vice versa is lower. The formulas for 
calculating are as follows: 
R
yy
yy
ii
i
n
i i
i
n
2
2
1
2
1
1 














, (10)
RMSE
N
yy
ii
i
N










1
2
1
,
 (11)
where y
i
 are true values of the test point, y
i

 predicted values of the 
surrogate model for the test point, y
i
 average of true values, and N 
number of sample points.
The fitting accuracy of the Kriging surrogate model and response 
surface methodology (RSM) surrogate model are obtained by solving 
Eq. (10) and Eq. (11), as shown in Fig. 13, and the specific values 
are shown in Table 4. It can be seen that the fitting accuracy values 
between the three objective functions of the ejector plate and the 7 
key design variables are all greater than 0.90, while the coefficient 
of determination R
2
 of the general engineering requirements for the 
response surface model should not be less than 0.9 [31]. 
Table 4 shows the specific index values of the accuracy test results 
of each surrogate model.
Table 4. Accuracy of objective function
Performance
Kriging surrogate model RSM surrogate model
R
2
RMSE R
2
RMSE
mass 0.9505 0.0538 0.9468 0.0583
displacement 0.9659 0.0411 0.9503 0.0623
stress 0.9327 0.0601 0.9031 0.0736
The results show that both Kriging model and RSM model exhibit 
high accuracy, with R² values exceeding 0.9 and RMSE values falling 
below 0.08. Overall, the Kriging model slightly exceeds the RSM 
model in terms of accuracy. Therefore, the Kriging surrogate model 
will be used for the subsequent multi-objective optimization.
2.7 Multi-Objective Optimization of the Ejector Plate
2.7.1 Mathematical Modeling
In order to achieve the purpose of lightweighting of the ejector plate, 
according to the Kriging surrogate model established by fitting, 
the mass of the ejector plate M, the maximum static deformation 
of the ejector plate skeleton D and the maximum von Mises stress 
S are taken as the optimization objective function, the maximum 
static deformation of the ejector plate skeleton D and the maximum 
von Mises stress S are taken as the constraints, and a total of 7 key 
dimensions of the ejector plate obtained from screening P
1
~P
4
, P
7
, 
P
12
 and P
13
 are taken as the independent variables of the objective 
function, so that the following multi-objective optimization 
mathematical model can be established. The multi-objective 
optimization mathematical model of ejector plate is as follows:
 Find x
i
 (i = 1~4, 7, 12, 13)
 min: {M(x)}
 max: {D(x), S(x)}  ,          (12)
 subject to    D(x )	 ≤	 10	mm,			
   S(x )	 ≤	 355	MPa,			
   x
imin
	≤ 	x
i
	≤ 	x
imax
Fig.13. Surrogate model accuracy validation; a) mass b) displacement c) Von Mises stress

Structural Design
176   ▪   SV-JME   ▪   VOL 71   ▪   NO 5-6 ▪   Y 2025
where x is the set of design variables; M(x) is the mass of the ejector 
plate [kg]; D(x) is the displacement of the ejector plate skeleton 
[mm]; S(x) is the von Mises stress of the ejector plate skeleton [MPa]; 
x
imin
 and x
imax
 are the lower and upper bounds of each design variable.
2.7.2 Multi-Objective Optimization Using NSGA-II
NSGA-II is one of the most widely used and effective multi-objective 
evolutionary algorithms, originally proposed by Deb [25]. Unlike 
traditional genetic algorithms, NSGA-II introduces a fast non-
dominated sorting method, an elite maintenance strategy, and an 
efficient congestion distance estimation process. These enhancements 
significantly accelerate iterative convergence, reduce computational 
complexity, and ensure population diversity [32].
In this paper, the NSGA-II algorithm is employed to address the 
multi-objective optimization problem. The population size is set to 
120, with a maximum of 200 iterations. The algorithm utilizes a 
crossover probability of 0.9, a crossover distribution exponent of 
10.0, and a mutation distribution exponent of 20.0.
3 RESULTS AND DISCUSSION
After 240,001 iterations, 742 non-dominated solutions were 
obtained. The Pareto front for the multi-objective optimization of 
the lightweight design of the ejector plate, produced by the NSGA-
II algorithm, is shown in Fig. 14. In multi-objective optimization 
problems, the global optimal solution is typically not unique; instead, 
it consists of multiple optimal solutions, collectively known as the 
Pareto optimal solution set. After solving the developed response 
surface model, three sets of candidate solutions were identified, as 
presented in Table 5.
Fig. 14. Pareto solution set
Table 5 displays the optimized design parameters suggested by the 
software platform after problem resolution. Considering its superior 
performance in minimizing mass, displacement, and stress, Scheme 
3 is identified as the most favorable design for the ejector plate. The 
optimal design variables obtained after optimization were rounded 
for correction, and the finite element model of the ejector plate was 
re-analyzed for static characteristics based on this set of parameters. 
The performance indices corresponding to the design sizes of the 
ejector plate before and after optimization are summarized in Table 6.
Table 6. Comparison of the design variables before and after optimization
Design variables Original Optimal Variation Error
P
1
 [mm] 3.0 2.3 –0.7
P
2
 [mm] 3.2 2.5 –0.7
P
3
 [mm] 3.2 3.0 –0.2
P
4
 [mm] 3.2 2.5 –0.7
P
7
 [mm] 5.0 4.0 –1.0
P
12
 [mm] 3.2 3.2 0
P
13
 [mm] 4 3.5 –0.5
M [kg] 180.0 169.1 –10.9 6.06 %
D [mm] 7.36 8.20 +0.84 11.4 %
S [MPa] 291.7 345.8 +54.1 18.5 %
As seen from Table 6, it demonstrates that the maximum 
displacement of the ejector plate skeleton has increased by 0.84 mm 
from 7.36 mm before optimization to 8.20 mm after optimization 
and the maximum equivalent stress of the ejector plate skeleton has 
increased by 54.1 MPa from 291.7 MPa before optimization to 345.8 
MPa after optimization. Although both indicators have increased, 
they meet the design requirements. After optimization, the ejector 
plate's total mass decreased from 180 kg to 169.1 kg, achieving a 
reduction of 10.9 kg (6.06 %) compared to the original design. The 
optimization effect is clear, achieving the intended optimization 
goals. The effectiveness of the proposed scheme has been validated 
in actual production, as illustrated in Fig. 15. The mass of the ejector 
plate has been reduced, material utilization has improved, and safety 
is maintained.
Fig. 15. Application of the ejector plate optimization
Table 5. Candidate combinations for design parameter optimization
No
Optimized design parameters Optimal results
P
1
 [mm] P
2
 [mm] P
3
 [mm] P
4
 [mm] P
7
 [mm] P
12
 [mm] P
13
 [mm] M [kg] D [mm] S [MPa]
1 2.4278 2.5407 2.6980 2.5852 4.1747 3.1341 3.5848 167.1 8.901 355.0
2 2.4408 2.5404 2.7123 2.5228 4.1790 3.1284 3.4678 167.0 8.862 355.0
3 2.4231 2.4693 2.7431 2.5534 4.1815 3.1013 3.5779 167.0 8.853 355.0

SV-JME   ▪   VOL 71   ▪   NO 5-6 ▪   Y 2025   ▪   177
Structural Design
4 CONCLUSIONS
This work presents an innovative lightweight design framework 
for rear-loader garbage truck ejector plates, which integrates multi-
objective optimization algorithms with surrogate modeling techniques 
to achieve balanced improvements in structural performance and 
economic efficiency. Through systematic optimization, the final 
design demonstrates a total mass reduction to 169.1 kg (6.06 % 
lighter than the original design) while satisfying all operational 
constraints. The main findings can be summarized as follows:
•	 Comparative analysis of surrogate modeling approaches shows 
that the Kriging model exhibits superior prediction accuracy com-
pared to the RSM model, particularly in capturing the response 
characteristics of the ejector plate system.
•	 The implementation of global sensitivity analysis allows effective 
identification of critical design parameters that have dominant in-
fluences on key performance indicators, thereby reducing compu-
tational costs by reducing experimental design parameters.
•	 The NSGA-II evolutionary algorithm is shown to be effective in 
generating Pareto optimal solutions for push plate parameters, 
achieving convergence within 200 generations while maintaining 
solution diversity through tournament selection and simulated bi-
nary crossover mechanisms.
In particular, the developed methodology demonstrates strong 
scalability and provides a generalized framework for the optimization 
of various load-bearing components in heavy-duty vehicles, including 
but not limited to chassis structures and actuator systems. Future 
work should focus on experimental validation of optimized designs 
under real-world operating conditions and extension.
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Acknowledgements This work was supported by the National Natural 
Science Foundation of China under Grant (51875494).
Received 2024-10-05, revised 2025-04-07, 2025-04-30,  
accepted 2025-05-13 as Original Scientific Paper.

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178   ▪   SV-JME   ▪   VOL 71   ▪   NO 5-6 ▪   Y 2025
Data availability The data that support the findings of this study are 
available from the corresponding author upon reasonable request.
Author contributions Fu-sheng Ding: methodology, formal analysis, 
validation, writing – original draft; Hong-ming Lyu: supervision, project 
administration, methodology, review & editing; Jun Chen: conceptualization, 
methodology, review & editing. Hao-ran Cao: methodology, formal analysis, 
data curation. Lan-xiang Zhang: formal analysis, data curation. All authors 
have contributed significantly to this work and thoroughly reviewed and 
approved the final version of the manuscript.
AI-Assisted writing AI-based tools were employed solely to enhance the 
readability and grammatical accuracy of the manuscript.
Večkriterijska optimizacija zasnove iztisne plošče  
za smetarska vozila z zadenjskim nakladanjem
Povzetek   V  delu  je  preds ta vljena  v ečkrit erijsk a  o p timizacija  zasno v e  iztisne 
plo šče,  ključne  k o m po nent e  sme tar skih  v o zil  z  zadenjskim  nakladanjem,  z 
namenom ohranjanja trdnosti ob hkratni optimizaciji mase. Zasnovan je bil 
parame trični  mo del  s  k o nčnimi  elementi  iztisne  plo šče,  pri  čemer  so  bili  cilji 
optimizacije usmerjeni v zmanjšanje mase, maksimiranje omejitev deformacij 
t er  zmanjšanje  maksimalne  v o n  Miseso v e  nape t o s ti.  Z  anal izo   o bčutljiv o s ti 
je  bilo  identif iciranih  sedem  ključnih  spremenljivk .  Sis t ematična  zasno v a 
parame tr o v  je  bila  izv ed ena  s  po mo čjo   Bo x-Behnk eno v ega  načr t o v anja 
eksperiment o v  (BBD),  ciljna  funkcija  pa  je  bila  približana  s  Krigingo vim 
nado mes tnim  mo delo m,  k at erega  učink o vit o s t  je  bila  primerjana  z  me t o do  
o dzivne  po vršine  (RSM).  Za  do lo čit e v  o p timalne  rešitv e  je  bil  upo rabljen 
v ečkrit erijski  gene tski  algorit em  NSGA -II,  ki  o mo go ča  lahk o tnejšo   zasno v o  
o b  izpo lnje v anju  vseh  s trukturnih  zaht e v .  R ezultati  k ažejo ,  da  je  mo žno   maso 
iztisne  plo šče  sme tar sk ega  v o zila  z  zadenjskim  nakladanjem  z  o p timizacijo  
zmanjšati  za  6,06  %,  pri  čemer  so  izpo lnjeni  krit eriji  tr dno s ti  in  def o rmacij. 
T a  izbo ljša v a  ne  po v eča  le  učink o vit o s ti  transpo r ta  o dpadk o v ,  am pak  tudi 
zmanjša pr o izv o dne s tr o šk e in izbo ljša izk o riščeno s t gradiv .
Ključne besede   sme tar sk o   v o zilo ,  iztisna  plo šča,  v ečkrit erijsk a  o p timizacija, 
NSGA-II, Kriging