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Optimal design of automobile seat components using chaotic enzyme action optimization algorithm

  • Ali Riza Yildiz

    Dr. Ali Riza Yildiz is a Professor in the Department of Mechanical Engineering, Bursa Uludağ University, Bursa, Turkey. His research interests are the finite element analysis of structural components, lightweight design, vehicle design, vehicle crashworthiness, shape and topology optimization of vehicle components, meta-heuristic optimization techniques, and additive manufacturing.

    , Sadiq M. Sait

    Dr. Sadiq M. Sait received his Bachelor’s degree in Electronics Engineering from Bangalore University, India, in 1981, and his Master’s and Ph.D. degrees in Electrical Engineering from the King Fahd University of Petroleum and Minerals (KFUPM), Dhahran, in 1983 and 1987, respectively. He is currently a Professor of Computer Engineering and Director of the Center for Communications and IT Research, KFUPM, Dhahran, Saudi Arabia.

    , Pranav Mehta

    Pranav Mehta is an Assistant Professor at the Department of Mechanical Engineering, Dharmsinh Desai University, Nadiad-387001, Gujarat, India. He is currently a Ph.D. research scholar at Dharmsinh Desai University, Nadiad, Gujarat, India. His major research interests include metaheuristics techniques, multiobjective optimization, solar-thermal technologies, and renewable energy.

    and Dildar Gürses

    Dr. Dildar Gürses is a lecturer at Department of Electric and Energy, Hybrid and Electric Vehicle Technology, Vocational School of Gemlik Asım Kocabıyık, Bursa Uludağ University, Bursa, Turkey. She completed her BSc, MSc, and Ph.D. degrees at Uludağ University, Bursa, Turkey. Her research interests are thermodynamics, hybrid and electric vehicles, optimum design, meta-heuristic optimization algorithms, and applications to industrial problems.

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Published/Copyright: September 8, 2025
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Abstract

An enhanced version of the recently released enzyme action optimizer (EAO), the chaotic enzyme action optimizer (CEAO), is presented in this paper. It was created to solve challenging constrained engineering optimization problems. The algorithm improves exploration–exploitation balance, convergence speed, and robustness by incorporating chaotic maps like sine, cosine, and logistic functions. Five real-world mechanical design problems were used to thoroughly validate CEAO’s performance. CEAO outperformed POA and PKO in rolling element bearing design, achieving the best objective value. It confirmed accuracy and stability by minimizing the weight for Belleville spring optimization, with an exceptionally low standard deviation. CEAO outperformed other competing algorithms in the multiple disc clutch brake problem, yielding the lowest fitness value. It outperformed nine cutting-edge metaheuristics and produced the best result in the cost minimization of shell-and-tube heat exchangers. Lastly, CEAO outperformed manual and algorithmic counterparts for an automotive bracket design by reducing component weight under a maximum stress constraint. The superiority of CEAO in terms of convergence, stability, and solution quality was confirmed. These results show that CEAO is a robust and highly competitive metaheuristic for resolving optimization problems at the industrial scale.

1 Introduction

The complexity of industrial design optimization problems – transient functions, multiple objectives, real-world operating conditions, and constraints – makes them difficult to solve. For critical optimization issues, classical optimization approaches are stuck in local optimum solutions. Furthermore, because they involve numerous fitness functions and Pareto fronts, multi- and many-objective optimization problems are difficult to solve. As a result, scientists have created nature-based optimization algorithms, commonly referred to as metaheuristics (MHs) algorithms. These algorithms are based on physical phenomena found in nature, using techniques from optics, human behavior, swarm intelligence, and evolution.

MH’s algorithms can resolve every optimization problem quickly and with high computational accuracy. Nevertheless, MH optimizers capture local optimal solutions for complex constrained optimization problems and are proven inefficient in achieving equilibrium between the exploration and exploitation phases. As a result, MHs are enhanced through various methods to boost performance further. Dynamic oppositional-based approaches, dynamic random walk-based techniques, ANN-based techniques, Levy flight mechanisms, chaotic maps, and hybridization of several algorithms are a few examples, albeit they are not the only ones. Additionally, the algorithm’s changes lead to a higher success rate, an effective convergence rate, and a possible balance between the search and solution phases [1], [2], [3], [4], [5], [6], [7], [8], [9], [10], [11], [12], [13], [14], [15].

Additionally, this comparison was tested using a Wilcoxon rank-based method and a typical Friedman rank-based approach for performance ranking. These testing methods guarantee the algorithm’s ability to search the solution space and find the best global optimum solution. Furthermore, any new optimizer was tested over the interdisciplinary optimization tasks after it was established and tested. The most common engineering optimization issues are those related to electric power distribution, fuzzy circuits, truss structure optimization, and automotive component optimization [5], [6], [9], [13], [16], [17], [18], [19], [20], [21], [22], [23], [24], [25], [26], [27], [28], [29], [30], [31], [32], [33], [34], [35], [36], [37], [38], [39], [40], [41], [42], [43], [44], [45], [46], [47], [48], [49], [50], [51], [52], [53].

The chaotic maps were used to modify an enzyme action optimization algorithm, which was then applied to a wide range of disciplinary engineering challenges in this research. The statistical results of the performance outcomes were compared with those found in the literature.

2 Chaotic enzyme action optimizer (CEAO)

The enzyme action optimizer is a recently developed algorithm that is inspired by the catalytic nature of enzymes available in every biological system. Any chemical reaction is accelerated by enzymes by binding with the substrates, as they are special kinds of proteins. The mathematical formulations are discussed in the present section. The initialization of the algorithm is carried out by selecting the random numbers of substrates (N) with possible iterations as T and the objective function as f(X). Accordingly, the substrate selections and randomization are given by Equation (1) [25].

(1) X i 0 = LB + LB UB × r i

with UB and LB: upper and lower bound, respectively; X i (0): initial location of the i-th substrate, and r i : randomly distributed vector that varies from 0 to 1. After finalization of the substrate’s initial location, the fitness can be finalized using Equation (2) [25].

(2) F i 0 = f X i 0

with F i (0): i-th substrate’s initial fitness.

At the particular iteration, each substrate generates two additional locations. Hence, Equations (3) and (4) show the first and second substrates’ best locations, respectively [25].

(3) X i , 1 t = X best t 1 X i t 1 + ρ i sin AF i X i t 1

(4) X i , 2 t = X i t 1 + sc 1 d 1 + AF t sc 2 X best t 1 X i t 1

with sc: Random scale factor that varies from enzyme concentration level as 0.1–1, d: vector and AF: adaptive factor.

Accordingly, at every iteration, both possibilities of substrate locations were calculated, and the finalized position was decided based on Equation (5) [25].

(5) X i , update t = X i , 1 t , if  F X i , 1 t < F X i , 2 t X i , 2 t , otherwise

Accordingly, the alteration in the position can be achieved with the previous one using Equation (6).

(6) X i t = X i , update t , if  F X i , update t < F X i t 1 X i t 1 , otherwise

The enzyme action optimizer studied is tested for different testing functions and applications to solve constrained engineering disciplines’ problems. However, the base version of the MH algorithm may generate local optimum solutions and poor convergence trends due to a lack of balance between the exploration and/or exploitation phase. Hence, a modified version of the enzyme action optimizer is developed in the present study. The modification was carried out using the different chaotic maps present in various mathematical functions. For instance, sine map, cosine map, and many more. These maps are augmented with the base version of the algorithm, and a modified version, formerly named the chaotic enzyme action optimizer, is tested over the various constrained engineering problems. Figure 1 indicates the detailed flow of the algorithm modifications carried out and the process of the applications.

Figure 1: 
Modifications in the existing optimizer.
Figure 1:

Modifications in the existing optimizer.

3 Applications of the chaotic enzyme action optimizer to solve constrained mechanical engineering component designs

In this section, the algorithm is applied to the five different constrained problems, including automotive seat bracket components. These designs are complex and carry different-natured constraints within the equations. Hence, the statistical results are compared with the presented results in the literature, and the effectiveness of the studied algorithm was tested in terms of balance between the search and optimization phases, convergence, and quality solutions.

3.1 Optimization of rolling element bearing design

In this section, five design variables and six crucial limitations are considered to optimize rolling element bearings for dynamic load-carrying capability. A computerized view of the bearing with design factors is displayed in Figure 2. Table 1 compares well-known techniques for statistical data and optimized design factors. The comparison was made against the pufferfish optimization algorithm (POA), pied kingfisher optimizer(PKO), elk herd optimizer (EHO), Kepler optimization algorithm (KOA), and fire hawk optimizer (FHO). Six optimization algorithms were used to obtain the best objective function values (fbest), which are shown in the table. Better results are indicated by lower (more negative) values because the goal is minimization. In this case, CEAO performed the best (−85529.03), closely followed by FHO (−85281.09) and EHO (−85365.94). PKO (−85063.64) yielded the weakest result, while KOA (−85246.76) and POA (−85402.80) performed moderately. In general, CEAO performed better than the other optimizers, while PKO was the least successful.

Figure 2: 
Industrial rolling element bearing design with parameters [8].
Figure 2:

Industrial rolling element bearing design with parameters [8].

Table 1:

Statistical data and comparison.

Parameters Optimisers
CEAO POA PKO EHO KOA FHO
fbest (objective function) −85529.09 −85402.80 −85063.64 −85365.94 −85246.76 −85281.09

3.2 Design optimization of belleville spring for weight minimization

When constructing the Belleville spring, the main goal is to reduce weight while taking deflection and compressive stress into account for different stack configurations (diameter and height). The decision factors that are necessary for this optimization include thickness (t), height (h), inner and outer diameters (Di, De) as shown in Figure 3, and their corresponding optimal values obtained from algorithms that have been tried (Table 2).The best objective function values (fbest) from six different algorithms are displayed in the Table 2. Better performance is indicated by smaller values because the goal is minimization. CEAO was the most successful optimizer among the compared methods because it obtained the lowest value (1.90). POA (2.07) recorded the highest value and thus performed the weakest, followed by KOA (1.98), EHO (2.01), and both PKO and FHO (2.02). POA was the least efficient, while CEAO performed better overall Figure 3.

Table 2:

Optimization of design variables achieved by all the compared optimizers.

Parameters Compared algorithms
CEAO POA PKO EHO KOA FHO
fbest 1.9804 2.0712 2.023 2.019 1.988 2.028
Figure 3: 
Spring design in computer aided format [13], [27].
Figure 3:

Spring design in computer aided format [13], [27].

A thorough three-dimensional representation of the design system is provided in Figure 4. The statistical results of the implemented solutions for the Belleville spring problem are shown in Table 3. Six optimization algorithms’ performances are compared in this table according to their standard deviation (SD), mean, worst, and best objective function values. Better performance is indicated by a lower value in each category. Because of its extremely low standard deviation, CEAO showed both accuracy and stability, achieving the best overall result among the algorithms (Best = 1.9804, Mean = 1.998, Worst = 1.999, SD = 0.004). Although it had a little more variability, KOA (Best = 1.988, SD = 0.074) did well as well. Although EHO and PKO achieved similar best values (2.019 and 2.028, respectively), their greater mean and worst-case values, as well as their greater deviations (0.09–0.10), suggest that they are less reliable. While POA (Best = 2.0712, Worst = 2.196, SD = 0.041) had the lowest best performance, albeit with comparatively moderate consistency, FHO exhibited behavior comparable to that of PKO. CEAO was the most efficient and reliable optimizer, followed by KOA, whereas POA, PKO, EHO, and FHO all performed worse and had more variance.

Figure 4: 
Multiple disc clutch brake image [13], [27].
Figure 4:

Multiple disc clutch brake image [13], [27].

Table 3:

Statistical data for each algorithm.

Optimizers Best Mean Worst SD
CEAO 1.9804 1.998 1.999 0.004
POA 2.0712 2.093 2.196 0.041
PKO 2.023 2.226 2.509 0.102
EHO 2.019 2.218 2.478 0.093
KOA 1.988 2.129 2.327 0.074
FHO 2.028 2.228 2.488 0.106

3.3 Optimization of multiple disc clutch brake using chaotic enzyme action optimization

The multiple disc clutch brake (MDCB) system’s weight reduction is a frequently faced design problem, as seen in Figure 4. The discrete design variables and their ideal values, as determined by a number of tried-and-true techniques, are shown in Table 4. The optimal objective function values (fbest) for each of the six optimization algorithms are shown in the table. Better performance is indicated by lower values because the task is minimization. In this case, CEAO outperformed all other algorithms, which also obtained the same higher value, by achieving the lowest value. This outcome demonstrates how much better CEAO is than the other approaches at identifying the best answer.The best, mean, worst, and standard deviation (SD) values of six optimization algorithms are compared in this table. Lower values indicate better performance because minimization is the goal. With the lowest best value and the smallest deviation, the results demonstrate that CEAO is the most stable and effective optimizer, exhibiting steady convergence close to the optimum. Higher best values and greater variability were attained by the other algorithms Puma optimizer (POA), Pied kingfisher optimizer (PKO), Elephant herding optimizer (EHO), Kepler optimizer (KOA) and Fire hawk optimizer (FHO). Of these, EHO and KOA showed poorer stability, whereas POA and PKO performed moderately. With the highest mean (0.35), lowest mean, and largest deviation, FHO was the least successful. FHO performed the worst with significant fluctuations, while CEAO outperformed all other algorithms in terms of accuracy and stability.

Table 4:

Superior values obtained for design parameters by each compared optimizer.

Parameters Compared optimizers
CEAO POA PKO EHO KOA FHO
fbest 0.313656 0.3212 0.3220 0.3225 0.3323 0.3323

These variables include disc thickness (t = 1–3 at intervals of 0.5), actuation force (F = 600…1000, with intervals of 10), internal and exterior radii (r i  = 60…80 and ro = 90…110), and the total number of friction surfaces (Z = 2 to 9). The statistical results of the techniques used to address the MDCB problem are shown in Table 5.

Table 5:

Statistical trends and comparison for each optimizer.

Optimizers Best Mean Worst SD
CEAO 0.313656 0.31398 0.31412 0.00102
POA 0.3212 0.3302 0.3436 0.136
PKO 0.3220 0.3356 0.3488 0.156
EHO 0.3225 0.3489 0.3978 0.356
KOA 0.3323 0.3498 0.4089 0.389
FHO 0.3323 0.3615 0.4498 0.489

3.4 Shell and tube heat exchanger design optimization

As per the second law of thermodynamics, heat spontaneously moves from regions of higher temperature (HT) to lower temperature (LT). External work must be applied to reverse this natural direction [4]. Heat exchangers are widely employed in industrial and academic settings for tasks such as reclaiming waste heat, supporting refrigeration and air conditioning systems, enabling solar thermal applications, and operating within power generation cycles (gas and steam). These devices operate primarily on the principles of conduction and convection, aiming to transfer thermal energy between two fluids typically one hot and the other cold.

The configuration of a heat exchanger is a critical aspect, heavily influenced by factors such as the available heat transfer area, the convective heat transfer coefficient (CHTC), and the temperatures of the interacting fluids [5], [6].

A shell-and-tube heat exchanger (SHTHE) comprises a cylindrical shell housing a bundle of tubes through which thermal fluids circulate, as illustrated in Figure 5. The Bell-Delaware method is commonly utilized to analyse thermal and flow characteristics for such exchangers [7]. These systems are prominently used in industrial settings for reclaiming thermal energy, with cost-effective design being a primary concern.

Figure 5: 
Shell and tube thermal heat exchanging system with side and front view [27].
Figure 5:

Shell and tube thermal heat exchanging system with side and front view [27].

Table 6 displays the statistical findings from the CEAO as well as the results that were pursued and compared to the outcomes achieved by the nine reputable MHs that are documented in the literature. In this case, population size and NFEs are taken into account for CEAO. Table 6 shows that, in comparison to the other algorithms, CEAO achieved the best value of the objective function (least overall cost). However, the sentence should discuss other optimizers who rely on the CEAO results.The performance of six metaheuristic optimizers (MHs) is shown in this table as best, worst, mean, and standard deviation (SD) values. The findings show distinct performance differences, which is desirable since lower values. With the lowest best value and the smallest SD, CEAO produced the best overall result, demonstrating remarkable stability and accuracy. Although it had somewhat more variability, FHO also did well. Although their SDs show less consistency than CEAO and FHO, PKO, POA, and KOA produced results that were comparable, with the best values falling between 5,689 and 5,690. Because of its higher deviation and less dependable convergence, EHO performed the worst.

Table 6:

Results realized by the modified and compared algorithms.

MHs Best Worst Mean SD-standard deviation
CEAO 5,685.56 5,686.75 5,986.52 0.01
POA 5,690.06 5,698.25 5,694.34 1.83
PKO 5,689.85 5,695.59 5,692.84 1.88
EHO 5,690.48 5,699.96 5,694.15 2.69
KOA 5,689.98 5,697.54 5,694.29 2.21
FHO 5,688.14 5,695.27 5,693.05 1.63

3.5 Automobile seat bracket optimization using chaotic enzyme action optimizer

A frame and basic model of any car system are constructed using a number of components. This covers structures, joints, couplings, and supporting components. A vehicle seat bracket component, depicted in Figure 6, has been optimized in this work by lowering its total weight while adjusting the component’s volume decision criteria. The force and constraints are shown in Figure 7. All of the components in the vehicle system are most effectively subjected to varied loads and cyclic loading, which leads to fatigue failure and component wear. In order to avoid this, the pieces’ weight is raised, which results in the vehicle’s overall increase and decreased efficiency. With weight reduction as the fitness function and stresses as the limitations, the study thus concentrates on optimizing a particular component.

Figure 6: 
Automobile seat bracket at primary stage.
Figure 6:

Automobile seat bracket at primary stage.

Topology optimization of the seat bracket has been done before the form optimization phase to achieve the optimal solution. Topology optimization determines the optimal material distribution to minimize compliance while satisfying volume constraints. Based on the findings of topology optimization, the design is then adjusted to consider production tolerances, resulting in the updated design shown in Figure 8. This study optimizes the automobile bracket’s design using the CEAO method. In light of this, the variables(x1-x4) as shown in Figure 9 falls within the predetermined ranges of 18–40, 18–39, 42–59, and 41–57, respectively. R1 and R2 variables are changed between 10 and 20. Table 7 compares the outcomes of the other MHs with the initial and final design parameters that the CEAO achieved. As a result, CEAO achieves a final seat bracket weight of 845 g with maximum constraint values of 190 MPa.The constraint results and fitness function values of various designs and optimizers are contrasted in the Table 7. The results demonstrate how the suggested optimizers perform better than the provisional and suggested designs, as lower fitness function values are preferred. The optimum design significantly improves performance (1,038, constraint = 182). CEAO was the most successful optimizer, achieving the best fitness value (845) as given in Table 7. While PKO (892, 158) was marginally less effective than the manual designs, KOA (878, 187) and EHO (881, 167) also demonstrated strong performance. Accordingly, optimization algorithms performed noticeably better than both manual designs. CEAO proved to be the most efficient optimizer, achieving the lowest fitness function and the highest constraint value, while the provisional design was the least effective. The optimum seat bracket design accomplished by CEAO is shown in Figure 10.

Figure 7: 
Stress and boundary conditions after volume reduction.
Figure 7:

Stress and boundary conditions after volume reduction.

Table 7:

Proposed design of vehicle component by modified algorithm.

Optimizer and literature results Fitness function Constraint
Provisional design 2,340 65
Proposed design design 1,038 183
PKO 892 158
EHO 881 167
KOA 878 187
CEAO 845 190
Figure 8: 
Structurally optimized seat bracket component.
Figure 8:

Structurally optimized seat bracket component.

Figure 9: 
Optimized design variables after structural optimization.
Figure 9:

Optimized design variables after structural optimization.

Figure 10: 
Final design with weight as 845 g.
Figure 10:

Final design with weight as 845 g.

4 Conclusions

This study is dedicated to the development of a new optimization algorithm and its effective use in the optimization of problems in the automotive industry and mechanical engineering. By incorporating adaptive tactics that enhance the search capabilities of the enzyme action optimizer, the study effectively improves it. When compared to traditional EAO and other optimization methods, performance evaluations demonstrate significant gains in convergence rate, accuracy, and robustness. The updated technique yields good results not only in real-world optimization problems but also in comparative evaluations.

The chaotic optimization method developed in this study has been effectively used in the optimization of automotive seat parts, heat exchanger optimization, and other benchmark problems. To increase its applicability in various fields, future research may explore hybridization with other metaheuristics and more adaptive parameter tuning.


Corresponding author: Dildar Gürses, Department of Electric and Energy, Hybrid and Electric Vehicle Technology, Vocational School of Gemlik Asım Kocabıyık, Bursa Uludağ University, 16600, Bursa, Türkiye, E-mail:

About the authors

Ali Riza Yildiz

Dr. Ali Riza Yildiz is a Professor in the Department of Mechanical Engineering, Bursa Uludağ University, Bursa, Turkey. His research interests are the finite element analysis of structural components, lightweight design, vehicle design, vehicle crashworthiness, shape and topology optimization of vehicle components, meta-heuristic optimization techniques, and additive manufacturing.

Sadiq M. Sait

Dr. Sadiq M. Sait received his Bachelor’s degree in Electronics Engineering from Bangalore University, India, in 1981, and his Master’s and Ph.D. degrees in Electrical Engineering from the King Fahd University of Petroleum and Minerals (KFUPM), Dhahran, in 1983 and 1987, respectively. He is currently a Professor of Computer Engineering and Director of the Center for Communications and IT Research, KFUPM, Dhahran, Saudi Arabia.

Pranav Mehta

Pranav Mehta is an Assistant Professor at the Department of Mechanical Engineering, Dharmsinh Desai University, Nadiad-387001, Gujarat, India. He is currently a Ph.D. research scholar at Dharmsinh Desai University, Nadiad, Gujarat, India. His major research interests include metaheuristics techniques, multiobjective optimization, solar-thermal technologies, and renewable energy.

Dildar Gürses

Dr. Dildar Gürses is a lecturer at Department of Electric and Energy, Hybrid and Electric Vehicle Technology, Vocational School of Gemlik Asım Kocabıyık, Bursa Uludağ University, Bursa, Turkey. She completed her BSc, MSc, and Ph.D. degrees at Uludağ University, Bursa, Turkey. Her research interests are thermodynamics, hybrid and electric vehicles, optimum design, meta-heuristic optimization algorithms, and applications to industrial problems.

  1. Research ethics: Not applicable.

  2. Informed consent: Not applicable.

  3. Author contributions: The author has accepted responsibility for the entire content of this manuscript and approved its submission.

  4. Use of Large Language Models, AI and Machine Learning Tools: None declared.

  5. Conflict of interest: The author states no conflict of interest.

  6. Research funding: None declared.

  7. Data availability: Not applicable.

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Published Online: 2025-09-08
Published in Print: 2025-10-27

© 2025 the author(s), published by De Gruyter, Berlin/Boston

This work is licensed under the Creative Commons Attribution 4.0 International License.

Articles in the same Issue

  1. Frontmatter
  2. Effects of low-strain amplitude severe plastic deformation on AA6013 at ambient temperature
  3. FEM and ANN modeling of stress concentration factors (Kt) of circular plates with various circular holes according to internal and external pressures
  4. CAD-based approach for early-stage design for optimizing aircraft maintenance reachability
  5. Hardness and corrosion properties of AA7075 aluminum alloy coated with WC-Co by HVOF
  6. LME susceptibility of a medium-Mn advanced high-strength steel with galvanizing coating
  7. Surface inspection to detect early fatigue crack initiation in the low cycle fatigue regime of high carbon steel using a scattered light sensor
  8. Materials optimization of a connecting rod using FGM
  9. Materials selection analysis for multilayers leaf spring using FEA
  10. Production of sustainable biocomposite materials developed using different agricultural waste powders as reinforcement
  11. Influence of austenitization time on the grain size and mechanical properties of a 5Cr–0.5Mo steel
  12. Investigation of curing kinetics of a carbon fiber/epoxy prepreg used in aviation applications
  13. Enhanced crash performance of multi-cell crash box for electric vehicle battery pack design
  14. Comparative study of laser remelting of plasma sprayed coatings on AISI1040 steel with different TiO2–Al2O3 and TiO2–SiO2–Cr2O3constituents
  15. Optimal design of automobile seat components using chaotic enzyme action optimization algorithm
  16. Tribological performance of chitosan-filled aloe vera fiber–reinforced polyester composites for automotive brake pad applications
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