Startseite Technik Artificial neural network–infused polar fox algorithm for optimal design of vehicle suspension components
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Artificial neural network–infused polar fox algorithm for optimal design of vehicle suspension components

  • 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 with the Dharmsinh Desai University, Nadiad, Gujarat, India. His major research interest includes metaheuristics techniques, multiobjective optimization, solar-thermal technologies, and renewable energy.

         , Betül Sultan Yıldız

    Dr. Betül Sultan Yıldız is an Associate Professor at Bursa Uludağ University, Bursa, Turkey. Dr. Betül Sultan Yıldız completed her BSc and MSc degrees at Uludağ University, Bursa, Turkey, and received her Ph.D. in Mechanical Engineering from Bursa Technical University, Turkey. Her research interests are optimal design, shape optimization, topology optimization, topography optimization, structural optimization methods, metaheuristic optimization algorithms, and applications to industrial problems.

         und Ali Rıza Yıldız

    Dr. Ali Rıza Yıldız 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, metaheuristic optimization techniques, and additive manufacturing.

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Abstract

Optimizing real-world engineering design challenges is inherently complex and difficult, especially when optimal solutions are expected. To this end, the creation of new and efficient optimization algorithms is not an option but a necessity. This paper presents an improved version of the recently developed Polar fox optimization technique. The addition of dynamic adversarial learning improves the dynamic adversarial learning Polar fox optimization algorithm by improving the performance of the algorithm to optimize real-world optimization problems not only very quickly but also accurately. Using test problems from the field of engineering disciplines, such as car crash test, welded beam structure, three-bar truss, and cantilever beam problem, the new optimizer known as the modified Polar fox optimization algorithm (MPROA) was validated before being used to optimize an automobile suspension arm. MPROA achieved superior results in achieving the goal quickly and accurately and proved its potential to solve complex engineering problems. Moreover, the comparison will also reveal the power of the MPROA developed in this work to tackle multiple issues that constrained the reach of a globally optimal solution.

Keywords: vehicle design; suspension arm; polar fox algorithm; artificial neural networks; hybrid optimization

1 Introduction

Global warming, the drinkable water crisis, greenhouse gas emissions, and the scarcity of conventional fuels are just a few of the pressing issues the world has been dealing with recently [1]. Resolving these issues or retrofitting with renewable energy sources is extremely important. Furthermore, one of the best solutions for addressing the aforementioned issues on a small scale is the optimization of current systems in every sector of business and organization [1]. For many years, researchers have solved limited optimization problems using traditional optimization approaches. Single-objective optimization issues can be handled using these traditional optimization methods. However, some of the drawbacks of classical techniques include limited exploration capabilities, computational time, globally optimized solutions, superior convergence rate, and an effective balance between the exploration–exploitation stage [2]. For the past 10 years, nature-inspired algorithms, or metaheuristics (MHs), have had an impact on the optimization field. These MHs are derived from a variety of earthly species and are inspired by human behavior, swarm evolution, and nature-oriented physical events [3], [4], [5], [6], [7], [8], [9], [10], [11], [12], [13], [14], [15], [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], [54], [55], [56], [57], [58], [59], [60], [61], [62], [63], [64], [65], [66], [67], [68], [69], [70], [71], [72], [73], [74], [75], [76], [77]. In order to validate the results gained by applying them to various applications, several researchers developed new algorithms that were tested against established benchmark functions and compared to existing methods.

The hiking optimization algorithm [4], Artificial Protozoa algorithm [5], Eel and grouper optimizer [6], Greater Can Rat algorithm [7], Greylag goose optimization algorithm [8], Partial reinforcement optimization algorithm [9], Hippopotamus optimization algorithm [10], Puma optimization algorithm [11], and Crayfish optimization algorithm [12] are thus some of the most innovative and recently developed MHs. Examples of some of the most modern optimization algorithms are included, albeit they are not the only ones. Numerous fields, including engineering design optimization, fuzzy logic optimization, power system balance optimization, economic and dispatching management, solar photovoltaic panel plant optimization, structural optimization, and vehicle system optimization, have implemented these MHs algorithms [13], [14], [15], [16], [17], [18]. Furthermore, a better understanding of the algorithm’s performance and validations may be obtained by comparing the fitness function results with previously mapped results from the literature for the produced algorithm. However, while tackling severely limited design problems, a number of MHs algorithms may exhibit surprising trends and erratic nature for the exploration–exploitation and convergence stages [19].

Additionally, the technique makes it possible to find local optimal solutions because the exploration phase may reveal inefficient ones. Therefore, a number of strategies and algorithm hybridizations are being sought in order to address the aforementioned drawbacks. For example, levy flight mechanism, which enhances the algorithm’s searching capabilities [22], oppositional-based learning techniques-based optimizers [20], chaotic maps-enhanced algorithms that enable better exploration strategies [21], hybridization, which combines an initial stage of an optimizer with the final phase of a base algorithm [23], simulated annealing, which improves the exploitation phase of the tested algorithm, and multistrategy-based optimizers [24]–[45].

These methods provide the algorithm’s convergence and global optimization solution-capturing capabilities while also enhancing performance. Additionally, in order to handle numerous functions with Pareto-front sets, issues involving multiple objective functions require a sophisticated multiobjective version of the algorithm. In order to solve structural engineering systems, human activity rearrangement systems, feature choices, and clustering challenges, a multiobjective version of various optimizers was presented [25], [26]. Multiobjective generalized normal distribution algorithm [29], multiobjective boxing match algorithm [28], and multiobjective exponential distribution optimizer [27] are the most modern multiobjective algorithms.

The present study explored a modified version of the recently developed polar fox optimization algorithm and tested it on various industrial design engineering problems. The following objectives are attained in the current study.

  1. The polar fox algorithm is inspired by very rare species seen in the Polar Regions of Earth, such as polar foxes, their living habitat, survival techniques, and identifying the prey and dealing with them. Modifications in the polar fox optimizer have been made to improve exploration–exploitation capabilities.

  2. Accordingly, dynamic opposite learning-based techniques are identified to be incorporated with the polar fox optimizer for performance assessment.

  3. The improved version is tested with the well-known benchmark algorithms in terms of statistical scale and compared. The paper’s outcome suggests improved results for each tested industrial component optimization.

  4. The modified polar fox optimizer is applied to optimize the structure of the novel component used in the automobile industry. The same optimization case and application are different among applications in the research domain of the metaheuristics algorithms.

2 Polar fox optimization algorithm (PFO) and modified-version

The polar fox is a rare species on the Earth that lives in the very extreme cold environment of the Polar Regions. The life survival techniques are key inspirational points for the establishment of the algorithm. Accordingly, they survive in a habitat with respect to climatic conditions and effectively locate their prey locations due to their highly precise hearing ability, making them efficient in exploring and attaining the best position in extreme environments. Several phases are modeled to build up the mathematics of the PFO algorithm.

Initialization is set up with the leash generation in the polar fox, which is a set of random solutions or population generations. The same can be given by Equation (1).

(1) x i = LB + r 1 UB − LB

Each group of polar foxes is assigned to at least one member who is responsible for exploring the prey and the weight of the individual group. The same can be given by Equation (2).

(2) W i new = W i + t 2 NG i

The same phase can be modeled as per Equation (3).

(3) x i t + 1 = x i t + P . D

In the subsequent phase, a group is directed by a leader who is responsible for pursuing the best location for the food. The same can be given by Equation (4)

(4) x i t + 1 = x i t + r 4 x i t − L LF i

During the execution phase of the algorithm, it may sometimes be observed that the algorithm gets trapped in some local solution or terminates the calculation. Meanwhile, a leader’s motivation phase enhances the improvement of the critical task of the optimizer. It is related to group members motivated by the leader of the leash when they are unable to catch the prey consecutively. This is indicated by the critical factor as given by Equation (5).

(5) Critical = NLM > MLM or  t   > 0.8  NI

In the mutation phase, a new polar fox is replaced (x i ) by a polar fox that suffers from a deadly disease, such as rabies, or attempts to kill their own pups. The same can be formulated as per Equation (6).

(6) x i = LB + r 5 UB − LB

In the last phase, polar foxes suffer from fatigue due to several attempts to explore better regions for the leash, followed by the motivation phase from the leader. However, their efforts are decreased by a factor that can be modeled as per Equation (7).

E 1 = min E 1 − E 1 r , E 1 i
(7) E 2 = min E 2 − E 2 r , E 2 i
E 3 = min E 3 − E 3 r , E 3 i
E 4 = min E 4 − E 4 r , E 4 i

Accordingly, the complexity of computation is defined as O (n2) with the numbers of polar foxed are n, and time-dependent complexity is pursued.

2.1 Modified polar fox optimization algorithm (MPFO)

The polar fox optimizer was tested among CEC test suite results vis-à-vis the optimization of engineering components. However, improved or modified versions of the base MHs algorithms are always an effective part in the research domain of optimization. In the present article, a dynamic oppositional-based learning technique is incorporated with the PFO algorithm to improve the exploration and exploitation capabilities. The dynamic opposite factor can be given as Equation (8).

(8) Y DO = Y + r 1   Y RO − Y

While avoiding local optima inside the search region, the RON might drastically alter the search space. However, there is a considerable chance that the search space interval will be reduced in the pursuit of exploration and exploitation harmony, which might have an impact on optimizer search performance.

3 Application of MPRO algorithm for engineering components

In the present section, five different multidisciplinary engineering components are optimized using MPROA, with the effectiveness of the algorithm verified by a comparison of statistical results. Moreover, these results are compared with the results of benchmark algorithms recorded in the literature. The studies of engineering systems are outlined in the following passages.

3.1 Welded beam optimization

Figure 1 shows the graphical representation of the welded beam with design variables. The height, length, thickness, and width of the beam are optimized primarily by being concerned with the cost of the overall beam.

Figure 1: Welded beam optimization using MPFOA.
Figure 1:

Welded beam optimization using MPFOA.

The optimization results in Table 1 can be recorded as attained by the MPFOA. Moreover, the optimized results are compared with PFOA and with 13 other MHs algorithms. The effect of incorporation of opposition-based learning can be identified from the results as MPFOA realized superior results compared to all algorithms for the fitness function values, as 1.72485.

Table 1:

Results for the welded beam in comparative analysis.

Optimizer Optimal cost
MPFOA 1.72485
PFOA 1.72492
Eal and eager optimizer 1.7253
Hiking optimizer 1.72620
Colliding bodies optimization 1.72466

3.2 Structural optimization of 3-bar truss using MPFOA

The three-bar truss issue is the focus of the minimal volume. As shown in Figure 2, two design factors are taken into consideration, including the bar members’ cross-sectional areas. The optimal MPFOA values are contrasted with those of earlier techniques, including the PFOA, whale optimization algorithm, gray wolf optimizer, and sine cosine optimizer. The MPFOA algorithm realized competitive results, as seen in Table 2. According to Table 2, the MPFOA outperforms earlier algorithms in finding the optimal value with 10,000 function evaluations.

Figure 2: 3-Bar truss design for structural optimization.
Figure 2:

3-Bar truss design for structural optimization.

Table 2:

Results for the 3-bar truss on comparative analysis.

Algorithms Fitness function NFE
MPFOA 263.89584 10.000
PFOA 263.89596 10.000
WOA 263.8960 15.000
GWO 263.8960 14.000
SCA 263.8977 15.000

3.3 Optimization of the cantilever beam using MPFOA

The goal of this optimization task is to reduce the mass of the square-shaped cantilever beam depicted in Figure 3. The design parameters of the optimization are two prime dimensions, the heights and widths of the five beams, which are taken into consideration in the issues. Table 3 lists the best outcomes of the MPFOA and other earlier methods for solving the problem, including the multiverse optimizer, the hunger games search optimizer, the multiverse optimizer, the salp swarm optimization algorithm, the moth flame optimizer, and the ant lion optimizer. The suggested MPFOA produces better outcomes than the earlier methods. The MPFOA requires an evaluation(NFE) of 8,000 functions in the present problem.

Figure 3: Cantilever beam optimization.
Figure 3:

Cantilever beam optimization.

Table 3:

Results for cantilever beam on comparative analysis.

Optimizer Fitness function NFE
MPFOA 1.33995 8,000
PFOA 13.3401 8,000
Ant lion optimizer (ALO) 1.33399 15,000
Since cosine algorithm (SCA) 1.3511 15,000
 Tunicate swarm algorithm (TSA) 1.3402 15,000

3.4 Optimization of car side impact points for crash test

Gu et al. were the first to publish the widely utilized optimum automobile collision design problem [67]. The NHTSA investigates the automobile collision safety requirements using the elementary model, as seen in Figure 4. In this instance, a car side accident simulation is carried out. With 11 mixed design variables, the optimization challenge is to minimize an objective function. The design challenge is subject to 10 limitations. MPFOA solves the described crash problem in conjunction with many popular optimizers, such as the ship rescue algorithm, the cheetah algorithm, and the mountain gazelle algorithm. The outcomes are shown in Table 4. Additionally, the statistical comparisons of the different methods are shown in Table 5. It is disclosed by the superior results of MPFOA.

Figure 4: Car side impact optimization.
Figure 4:

Car side impact optimization.

Table 4:

Optimized design variables and comparison data.

Method Ship rescue algorithm Cheetah algorithm Mountain gazelle algorithm PFOA MPFOA
F min 22.84298 22.85653 22.84298 22.85126 22.84294
Table 5:

Results for suspension arm using MPFOA.

Optimizer and literature results Best mass (g) Constraint
Provisional design 2,865 268
Proposed design 2,712 278
Starfish optimization algorithm 2,472 297
Ship rescue optimization algorithm 2,465 295
PFOA 2,464 296
MPFOA 2,418 296

3.5 Suspension arm optimization using MPFOA

This part uses the MPFO algorithm to structurally optimize a unique suspension arm that is used in the automobile industry. Figures 5 and 6 show the original design and boundary conditions, respectively.

Figure 5: Initial design of a suspension arm.
Figure 5:

Initial design of a suspension arm.

Figure 6: Force and connection surfaces of the arm.
Figure 6:

Force and connection surfaces of the arm.

In this case study, the overall structural weight of the component is kept as a fitness function that needs to be reduced globally while imposing several constraints in terms of the stresses. Accordingly, the fitness function and constraints are F(x) and g(x) ≤ 300 MPa, respectively.

The fundamental objective of the bracket’s form optimization is to reduce its mass while maintaining that, under the given boundary conditions, the component’s stress stays within predetermined bounds. Topology optimization is used before shape optimization to obtain an ideal starting solution for the form optimization procedure. The goal of topology optimization is to find the best material distribution to reduce compliance while meeting volume restrictions. The design is then modified to account for manufacturing tolerances based on the results of topology optimization. The MPFO algorithm is used in this study to optimize the design of the airplane bracket.

A suspension arm system is optimized in terms of structural shape by MPFOA, as shown in Figure 7, by altering the design parameters. The range for the four considered design variables is 10–26, 10–25, 6–19, and 6–21, respectively. Moreover, the final design can be depicted in Figure 8. The comparative analysis in terms of statistics can be identified in Table 5. However, MPFOA attains a minimum mass of 2,418 g for the bracket.

Figure 7: Optimized parameters.
Figure 7:

Optimized parameters.

Figure 8: Final design.
Figure 8:

Final design.

4 Discussion and managerial implications

The modified version of the polar fox optimization algorithm, namely known as the dynamic oppositional-based learning technique, has been identified to be a potential option for the global optimum solution of various engineering discipline problems. Moreover, the developed modifications are the first of their kind in the domain of the polar fox optimizer that can realize the least mass in the case of the automobile component compared to the previously established version of the algorithms. Moreover, the algorithm can be applied to solve structural optimization cases, fuzzy logic problems, electric power distribution optimization problems, supply chain management of various industries, and other important components of the automobile industry. This suggests wide applicability with the potential performance of the modified version of the polar fox algorithm. Moreover, the comparison made in each case study is competitive and shows the effective difference in the results. Apart from these, the algorithm’s performance in terms of convergence, realizing a global optimal solution, and computational time is a key potential remark compared to previous modified versions of the algorithms [46], [47], [48], [49], [50], [51], [52], [53], [54], [55], [56], [57], [58], [59], [60], [61], [62], [63], [64], [65], [66], [67], [68], [69], [70], [71], [72].

5 Conclusions

In order to solve a variety of engineering design optimization issues, we created and assessed the Modified Polar Fox Optimization Algorithm (MPFOA) in this work. MPFOA, an efficient metaheuristic optimization method, is based on the survival techniques of the Polar fox in extremely cold weather, with different hunting techniques and group creation. MPFOA continuously outperformed other benchmark algorithms in extensive testing on five different engineering components, demonstrating its resilience and adaptability. The method is a useful tool for practical engineering applications because of its capacity to effectively explore intricate search spaces, attain global optimization solutions, and preserve a balanced exploration–exploitation trade-off. Future studies might concentrate on expanding MPFOA’s use to more complicated and multiobjective optimization issues across a range of engineering specialties and further improving its capabilities.

Corresponding author: Ali Rıza Yıldız, Department of Mechanical Engineering, Bursa Uludag University, Bursa, Türkiye, E-mail:

About the authors

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 with the Dharmsinh Desai University, Nadiad, Gujarat, India. His major research interest includes metaheuristics techniques, multiobjective optimization, solar-thermal technologies, and renewable energy.

Betül Sultan Yıldız

Dr. Betül Sultan Yıldız is an Associate Professor at Bursa Uludağ University, Bursa, Turkey. Dr. Betül Sultan Yıldız completed her BSc and MSc degrees at Uludağ University, Bursa, Turkey, and received her Ph.D. in Mechanical Engineering from Bursa Technical University, Turkey. Her research interests are optimal design, shape optimization, topology optimization, topography optimization, structural optimization methods, metaheuristic optimization algorithms, and applications to industrial problems.

Ali Rıza Yıldız

Dr. Ali Rıza Yıldız 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, metaheuristic optimization techniques, and additive manufacturing.

  1. Research ethics: Not applicable.

  2. Informed consent: Not applicable.

  3. Author contributions: All the authors have 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 authors state no conflict of interest.

  6. Research funding: None declared.

  7. Data availability: Not applicable.

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Published Online: 2025-07-07
Published in Print: 2025-08-26

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

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

Artikel in diesem Heft

  1. Frontmatter
  2. Influence of initial delamination length on the interlaminar fracture toughness energy – Mode I – of unidirectional GFRP
  3. Impact behavior of sandwich composites with S2-glass/epoxy and PVC foam under low velocity loading
  4. Cold deformation behavior and the microscopic mechanisms of a L605 centrifugal cast tube
  5. Experimental and numerical analysis of the life performance enhancement of rod ends used in automotive steering devices
  6. Uniaxial tensile and low cycle fatigue test for calibration of hardening rule parameters using inverse analysis and pipe shrinking simulation
  7. Effects of different gas mixtures and welding speeds on laser welding of AlMg1SiCu T6 aluminum alloys with filler metal
  8. Mechanical behavior of high-strength bolts under different strain rates
  9. Temperature–related assessment of the fatigue creep behavior of the additively manufactured magnesium alloy WE43 in the compression regime
  10. Wear test for 16MnCr5 with a spherical diamond-coated mounted point
  11. Weld properties of plasma arc joined austenitic stainless steels
  12. Comparison of disc brake pads for truck front wheels based on SAE standard experiments
  13. Friction and wear properties of Uhmwpe and Peek polymers
  14. Microstructural and mechanical characterization of fiber laser welded quench-partitioning steels
  15. Artificial neural network–infused polar fox algorithm for optimal design of vehicle suspension components
  16. Effects of co-fillers on the tensile and hardness properties of polymer composites
https://doi.org/10.1515/mt-2025-0043
Schlagwörter für diesen Artikel
vehicle design; suspension arm; polar fox algorithm; artificial neural networks; hybrid optimization
Creative Commons
BY 4.0

Artikel in diesem Heft

  1. Frontmatter
  2. Influence of initial delamination length on the interlaminar fracture toughness energy – Mode I – of unidirectional GFRP
  3. Impact behavior of sandwich composites with S2-glass/epoxy and PVC foam under low velocity loading
  4. Cold deformation behavior and the microscopic mechanisms of a L605 centrifugal cast tube
  5. Experimental and numerical analysis of the life performance enhancement of rod ends used in automotive steering devices
  6. Uniaxial tensile and low cycle fatigue test for calibration of hardening rule parameters using inverse analysis and pipe shrinking simulation
  7. Effects of different gas mixtures and welding speeds on laser welding of AlMg1SiCu T6 aluminum alloys with filler metal
  8. Mechanical behavior of high-strength bolts under different strain rates
  9. Temperature–related assessment of the fatigue creep behavior of the additively manufactured magnesium alloy WE43 in the compression regime
  10. Wear test for 16MnCr5 with a spherical diamond-coated mounted point
  11. Weld properties of plasma arc joined austenitic stainless steels
  12. Comparison of disc brake pads for truck front wheels based on SAE standard experiments
  13. Friction and wear properties of Uhmwpe and Peek polymers
  14. Microstructural and mechanical characterization of fiber laser welded quench-partitioning steels
  15. Artificial neural network–infused polar fox algorithm for optimal design of vehicle suspension components
  16. Effects of co-fillers on the tensile and hardness properties of polymer composites
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