Applications of Optimization Methods in Automotive and Agricultural Engineering: A Review

Abstract

The automotive and agricultural industries face increasingly stringent demands with technological advancements and rising living standards, resulting in substantially heightened engineering complexity. In this background, optimization methods become indispensable tools for solving diverse engineering challenges. This narrative review paper provides a comprehensive overview of the application and challenges of five optimization algorithms, including gradient-based optimization algorithms, heuristic algorithms, surrogate model-based optimization algorithms, Bayesian optimization algorithms, and hybrid cellular automata algorithms in two fields. To accomplish this objective, the research literature published from 2000 to the present is analyzed, focusing on automotive structural optimization, material optimization, crashworthiness, and lightweight design, as well as agricultural product inspection, mechanical parameter optimization, and ecological system optimization. A classification framework for optimization methods is established based on problem characteristics, elucidating the core strengths and limitations of each method. Cross-domain comparative studies are conducted to provide reference guidance for researchers in related fields.

1. Introduction

Guided by the ‘dual-carbon’ goal, the rapid development of modern industrial technology has accelerated the transition of the automotive and agricultural fields towards intelligence and sustainability. This process has incurred many engineering challenges, including automobile lightweight design [1,2], crashworthiness optimization [3,4], agricultural efficiency, quality improvement [5,6], and quality inspection [7]. Optimization methods have become essential for addressing engineering challenges, and their growth trajectory illustrates the progression of solution demands from straightforward linear issues to complex nonlinear systems.

As a mathematical approach for identifying optimal solutions under given constraints, optimization plays a crucial role in balancing performance, cost, and efficiency in engineering design. The five mainstream optimization methods include gradient-based optimization algorithms, heuristic optimization algorithms [8], surrogate model-based optimization algorithms [9], Bayesian optimization algorithms [10], and hybrid cellular automata algorithms [11], as shown in Figure 1. As one of five mainstream algorithms, gradient-based optimization algorithms are widely applied due to their robust mathematical foundation and excellent computing efficiency [12]. Numerous versions have been developed from the GD algorithm, including batch gradient descent (BGD), stochastic gradient descent (SGD), and mini-BGD. The academic community has proposed a succession of optimization algorithms, including Adaptive Gradient (AdaGrad), Adaptive Delta (AdaDelta), Root Mean Square Propagation (RMSProp), and Adaptive Moment Estimation (Adam), to enhance algorithm performance and stability [13,14]. These methods have emerged as fundamental optimization instruments in machine learning [15,16]. Haji et al. have presented a thorough introduction and comparison of various algorithms [17]. Nonetheless, these gradient-based optimization algorithms are prone to entrapment in local optima when addressing high-dimensional problems. In contrast, heuristic optimization algorithms, which draw inspiration from physical or natural processes, are capable of bypassing local optima without relying on gradient information. As a result, they are progressively emerging as an alternative to traditional algorithms [18,19]. Heuristic optimization algorithms such as Ant Colony Optimization (ACO) [20], Artificial Bee Colony (ABC) [21], Gray Wolf Optimizer (GWA) [22], Whale Optimization Algorithm (WOA) [23], Harris Hawk Optimization (HHO) [24], and White Shark Optimization Algorithm (WSO) [25] enhance efficiency by exploring the search space to identify solutions near the optimal solution, and are widely applied in the automotive and agricultural sectors [26,27,28]. Many researchers have performed an extensive review of heuristic algorithms [8,29]. As the complexity of modern engineering optimization problems continues to increase, researchers face three typical challenges: significant noise in objective function evaluation, high computational expenses, and the absence of analytical expressions. Traditional gradient-based optimization methods are difficult to apply, while function-value-based heuristic algorithms generally necessitate several function evaluations. To address these limitations and achieve a balance between optimization efficiency and computational cost, surrogate-based optimization (SBO) methods have been developed, which markedly decrease the number of evaluations of the actual objective function by constructing surrogate models, such as Kriging models, response surface methodology (RSM), radial basis functions (RBF), and support vector regression surrogate models (SVR). In automotive and agricultural engineering practice, these methods have achieved notable results, saving a significant amount of computational resources [30,31,32,33,34]. Based on their algorithms, Bayesian optimization (BO) further improves optimization efficiency through its three key advantages: derivative-free, facilitates global optimization, and requires fewer evaluations than other derivative-free methods. It has become the preferred method for solving three typical challenges [35], achieving a good balance between exploration and exploitation during the search process [10]. The utilization of BO is expanding across multiple research domains, including automotive material optimization, lightweight design [36], and multi-objective design in automotive engineering [37], and improving seed germination rates in agriculture [38]. Addressing problems involving multiple variables and high dimensions. Gradient-based algorithms are prone to local optimal solutions, while heuristic algorithms are time-consuming. Although surrogate models offer a viable alternative, they suffer from the curse of dimensionality as the number of variables increases. The HCA algorithm combines the local update rules of cellular automata (CA) with finite element analysis (FEA), making it suitable for solving complex nonlinear optimization problems and facilitating quick searches for optimal solutions within discrete domains. Consequently, many scholars have conducted extensive research on this topic to explore the algorithm’s extended applications [39,40,41].

As discussed, these five mainstream algorithms each possess distinct advantages in addressing challenges within the agricultural and automotive engineering domains. Numerous scholars have also conducted reviews on these algorithms to further explore their potential in tackling complex engineering problems [10,42,43]. Although existing reviews have extensively covered specific optimization methods within individual disciplines, there is a notable lack of interdisciplinary, method-oriented comparative studies. Additionally, existing studies have failed to establish a clear mapping relationship between algorithm characteristics and field-specific requirements.

This review screened literature on five categories of algorithms published in the fields of agricultural and automotive engineering over the past 25 years, since 2000. It comprehensively examines the application characteristics of the five optimization algorithms in automotive and agricultural engineering, aiming to analyze the applicability of existing algorithms to different problems, compare and clarify the performance of various algorithm types across different fields, and explore future directions for optimization development. This interdisciplinary research not only addresses gaps in existing optimization methodology studies but also offers criteria for algorithm selection in complex system optimization.

2. Methodology

This article presents a narrative review of the literature on publications selected from databases like Web of Science Core Collection databases, Scopus, and Google Scholar. To ensure that the selected literature aligns with the research focus, the search used Boolean operators to combine ‘gradient-based’, OR ‘heuristic algorithms’, OR ‘surrogate model’, OR ‘Bayesian’, OR ‘hybrid cellular automata’ AND ‘agriculture’, OR ‘automotive’ AND ‘optimization’. The scope of the study is limited to publications from 2000 to 2025. Upon identifying a pertinent study, an examination of its references was conducted to identify any additional studies missed in the initial search. This iterative process continued until no more relevant studies were uncovered. All pertinent data is collected from these studies and synthesized to address our research questions.

All pertinent data is collected from these studies and synthesized to address our research questions. The following research questions are developed to guide the systematic review:

• What optimization methods are available for automotive optimization problems?
• What optimization methods are available for agricultural optimization problems?
• What obstacles exist in applying different optimization methods to the automotive and agricultural sectors?
• What are the differences in the application of various optimization methods in the automotive and agricultural sectors?

Q1 helps us assess both the benefits and drawbacks of five algorithms in automotive optimization problems. Q2 assists us in evaluating the advantages and disadvantages of five algorithms in automotive optimization problems. Q3 facilitates our comprehension of the constraints and difficulties associated with existing approaches. Q4 help us compare and analyze the similarities and differences in how the two approaches are adopted and applied across the two fields.

During the screening process, studies unrelated to optimization methods or algorithms were excluded. The established exclusion criteria were applied during the literature retrieval and analysis phase. Exclusion criteria studies underwent a comprehensive assessment and were systematically categorized based on predefined exclusion criteria. These criteria aimed to define the scope of the systematic review and eliminate studies that did not meet the relevant standards. The subsequent list outlines the exclusion criteria:

• Articles that do not explicitly apply any of the five categories of optimization algorithms to specific problems in the automotive or agricultural sectors.
• Articles that are severely disconnected from automotive structural optimization, material optimization, collision safety, lightweight design, agricultural product testing, mechanical parameter optimization, and ecosystem optimization.
• Articles authored in languages other than English, conference papers, book chapters, reviews, surveys, Master’s theses, or PhD dissertations.
• Publications published before the year 2000.

As shown in

Table 1. Distribution of articles is based on the databases.

Database	Number of Articles That Were First Retrieved	Number of Papers Remaining After Exclusion Criteria	Number of Articles After Removing the Repeated Articles
Science Direct	764	94	87
Web of Science	923	119	112
Google Scholar	237	58	43
Total	1924	271	242

, following the application of the exclusion criteria, a total of 271 articles were ultimately chosen. Subsequently, the removal of duplicate entries from the selected databases led to the evaluation of 242 unique articles. It is worth emphasizing that only journal articles met the inclusion criteria for this review. The remaining 242 documents were categorized into five major groups based on different fields, as shown in Figure 2, and a narrative review was conducted around these classifications.

3. Literature Review

3.1. Gradient-Based Optimization Algorithms

Gradient-based optimization algorithms are the primary mathematical instruments for solving engineering problems, particularly for high computational efficiency and a strong theoretical foundation. This section systematically reviews several gradient-based algorithms and typical engineering cases to clarify the application areas of different algorithms and provide selection guidelines for readers.

3.1.1. Application of Gradient-Based Optimization in Automotive Engineering

Based on the mathematical principle that the gradient indicates the direction of steepest ascent, Cauchy [44] proposed the steepest descent method. This method laid the theoretical foundation for all subsequent gradient-based optimization algorithms. The iterative optimization algorithm based on a multipoint approximation scheme and on the steepest descent method has been used to find the optimal geometrical configuration of several energy-absorbing devices in a vehicle [45]. An improved variable-step quasi-Newton algorithm was effectively applied to optimize the drawbead restraining force in automotive panel forming, significantly enhancing design efficiency and precision [46]. However, traditional quasi-Newton methods suffer from excessive computational inefficiency in large-scale variable optimization. To address these issues, Kasac et al. proposed a BPTT-like gradient-based optimal control algorithm that significantly enhances optimization accuracy and convergence speed [47]. Furthermore, in order to overcome the drawback of genetic algorithms wasting computational resources when evaluating costly objective functions, gradient algorithms were adopted and combined with genetic algorithms to optimize the automotive electric power steering system. [48]. In 1970, the Broyden-Fletcher-Goldfarb-Shanno (BFGS) algorithm was proposed, which improves the recursive update strategy for the inverse of the Hessian matrix, achieving faster convergence than the steepest descent method. Combined with a genetic algorithm, it is applied to the drag minimization of a simplified car shape [49]. To address the memory limitations of the BFGS algorithm, the limited memory Broyden Fletcher Goldfarb Shanno (L-BFGS) algorithm was further developed and has been utilized for local optimization in prediction models for resistance spot welding of automotive body-in-white structures [50]. Constrained optimization poses significant challenges in automotive engineering applications, such as topology optimization, lightweight design, and multi-objective optimization. Among gradient-based optimization algorithms, the sequential quadratic programming (SQP) method was first proposed in 1963 and has continuously been refined [51,52]. The SQP method transforms the original nonlinear constrained optimization problem into a series of solvable quadratic programming (QP) subproblems by employing a quadratic approximation of the objective function and a linear approximation of the constraint function. This approach proves highly efficient for complex optimization problems, especially those involving highly nonlinear objective functions and substantial computational expenses. For instance, Zhu et al. [53] integrated the SQP algorithm with a multidisciplinary collaborative optimization algorithm to perform multi-objective optimization design of the body-in-white (BIW) structure, achieving significant weight reduction while ensuring performance. In another study, based on a constructed Kriging model, the SQP method was employed to obtain a quasi-global robust optimum solution of foam-filled thin-walled structures [54]. In 1984, Karmarkar [55] proposed the interior point method (IPM), which transforms constrained problems into a series of unconstrained subproblems by introducing obstacle functions or logarithmic obstacle terms, establishing a novel paradigm for solving linear and nonlinear constrained optimization. Liu et al. employed multiple methods, including IPM, NSGA-II, and multi-objective particle swarm optimization (MOPSO), to optimize the NH₃ emissions of diesel engines. The results indicated that the IPM has superior efficiency compared to other optimization algorithms [56]. In 1985, a Fortran subroutine named NLPQL was proposed to solve smooth nonlinear programming problems using sequential quadratic programming. This algorithm constructs quadratic approximations of the Lagrangian and linear approximations of the constraints to iteratively solve a series of SQP subproblems. Due to its efficiency and robustness, NLPQL has been widely adopted in automotive engineering optimization. For instance, it was employed to determine rational ply thicknesses of composite drive shafts based on response surface approximations [57]. It is also applied for topology optimization of an automotive component called the dash of BIW with constraints, achieving a 53% weight reduction [58]. More recently, NLPQL has been integrated with genetic algorithms to form a hybrid approach for constrained optimization tasks, such as in the shape optimization of automotive spring seats [59]. Furthermore, Zhou et al. [60] developed a hybrid algorithm that leverages the strengths of global and gradient-based optimization methods by combining Adaptive Simulated Annealing and Nonlinear Programming by Quadratic Lagrangian (ASA-NLPQL). Based on a surrogate model, the optimization strategy addresses the challenges of lightweight design for automotive seat skeletons, demonstrating a 22.3% reduction in seat weight. A comparison of the advantages and disadvantages of the four typical gradient-based algorithms mentioned above is shown in Figure 3.

3.1.2. Application of Gradient-Based Optimization in Agricultural Engineering

Gradient-based optimization methods have been extensively explored not only in automotive engineering but also in modern agriculture. For instance, based on a Convolutional Neural Network (CNN) model, one of the gradient-based optimization methods, named the gradient ascent method, was employed to optimize risk mitigation for crop inputs [61]. Similarly, another gradient-based optimization model has been developed to maximize agricultural profitability through the dynamic allocation of surface water and groundwater resources. This approach supports sustainable irrigation management that mitigates drought impacts while maintaining productivity [62]. In the context of agricultural IoT systems, gradient-based optimization algorithms help enhance communication efficiency by improving throughput and fairness in device-to-device networks. This enables more reliable and efficient data transmission, which is crucial for real-time monitoring and control in precision agriculture [63]. Furthermore, gradient-based optimizers such as Adam play a key role in training deep learning models for early pest detection with high accuracy. Their application allows timely and precise interventions, substantially boosting agricultural productivity and reducing crop losses [64]. Beyond pest management, the gradient-based optimization method enables precise identification of crop-specific optimal climatic conditions within deep learning models, significantly enhancing global yield prediction accuracy and supporting data-driven agricultural decisions [65].

3.1.3. Future Development of Gradient-Based Optimization Algorithms

In summary, gradient-based optimization algorithms are characterized by their rapid convergence speed and good numerical stability. Furthermore, they possess strong scalability and flexibility, enabling their integration with advanced and efficient technical methods to enhance convergence rates. Nonetheless, these algorithms are constrained by limitations, including the tendency to become trapped in local optima and the considerable complexity and computational expense involved in calculating gradient information.

With the increasing complexity of engineering systems, modern engineering challenges frequently present as high-dimensional, multi-objective, highly nonlinear, strongly constrained, and multi-physics coupled systems. The inherent limitations of gradient-based optimization algorithms prevent them from addressing all such challenges. Consequently, the future development of these algorithms will probably involve a greater focus on specific domains or particular classes of practical optimization problems within engineering, thereby leveraging their advantages to the fullest extent. In the automotive and agricultural engineering sectors, future research in gradient-based optimization will concentrate on handling high-dimensional practical engineering problems within complex constrained environments, particularly for optimization problems under linear operating conditions. Future research may explore additional promising directions. Firstly, the development of auxiliary methods, including the integration of gradient algorithms with penalty function methods for the iterative points. Secondly, maintain emphasis on employing surrogate models to approximate function values or gradient information, together with methodologies such as active learning to diminish computing expenses and enhance optimization efficiency. Furthermore, considering that particular engineering challenges are abundant in physical information, this domain knowledge can inform the trajectory of machine learning models and the update direction of optimization iterations, thereby improving the interpretability and reliability of algorithms in real-world engineering contexts.

Future research on gradient-based optimization algorithms in the engineering domain should be closely aligned with the practical demands of engineering problems, continuously advancing efficient optimization in high-dimensional and complex constrained settings, while promoting the profound integration of engineering expertise with algorithmic innovation.

3.2. Heuristic Algorithms Optimization

Gradient-based optimization algorithms are highly efficient for local searches in smooth, continuous design spaces. However, they often struggle to achieve global optimal solutions in automotive structural optimization problems characterized by multiple objectives, numerous design variables, and significant nonlinearity. The presence of bifurcation points and nonlinearities further complicates the application of gradient information. In contrast, heuristic algorithms offer a robust alternative by effectively exploring discontinuous and high-dimensional optimization spaces without relying on gradient information. These characteristics make heuristic algorithms particularly suitable for addressing complex challenges frequently encountered in automotive engineering.

3.2.1. Applications of Heuristic Algorithms in Automotive Engineering

Heuristic algorithms are mainly used to solve complex optimization problems such as structural design, crashworthiness optimization, and multi-objective optimization, as seen in Figure 4. As a classic meta-heuristic algorithm, the genetic algorithm (GA) is capable of solving linear and nonlinear optimization problems without gradient information. Due to this property, GA was widely employed in automotive engineering for solving complex nonlinear optimization challenges. For instance, it was applied to cross-sectional shape optimization of the automotive body frame [66,67]. In addition, multiple population GA has been employed to optimize the structure of electric vehicle electrical equipment [68]. Based on conventional GA, Qin et al. developed an improved and penalty-parameterless genetic algorithm (IGA) optimizer and applied it for shape optimization of 10 TWBs in the BIW side frame [69]. Results indicated that although IGA exhibits a slower convergence speed, it achieves superior solutions compared to other heuristic algorithms, such as the gray wolf optimizer (GWO) and water cycle algorithm (WCA). Further expanding the application of heuristic methods, some scholars have proposed a method combining topology, shape, and dimensional optimization to enhance structural performance under collision loading conditions. This approach consists of an outer loop, which performs the topology optimization using heuristic rules, and an inner loop, which handles the shape and sizing optimization [70].

To address optimization problems with multiple objectives and constraints, the NSGA-II algorithm has been widely adopted as an effective solution. It obtains the Pareto optimal solution set by calculating the congestion distance of parameters, successfully overcoming the local convergence limitations of earlier genetic algorithms. This method has been applied to problems with coupled multiple variables, such as material selection [75], thickness distribution [76], and cross-sectional shape [77]. Cui et al. [78] used NSGA-II to optimize the lightweight design of a car door assembly with discrete and continuous variables. They further utilized artificial neural networks (ANNs) to approximate the constraint functions, thereby reducing the run time. Similarly, NSGA-II combined with ANN was employed for the multi-objective optimization of the A-pillar cross-sectional shapes for the automobile frame [79]. Furthermore, the NSGA-II algorithm has been integrated with surrogate modeling techniques to improve efficiency in computationally expensive problems. Based on RBF models, it has been applied to optimize hot-stamped tailor rolled blank (TRB) components [80] and foam-filled crash-box [71] under axial loading. These studies demonstrated that the crashworthiness of optimized structures was improved, while the reliability of the Pareto front set was also strengthened. Additionally, multi-objective reliability-based design optimization using NSGA-II has been conducted for variable-rolled-blank and variable-cross-sectional-shape front longitudinal beam under front-impact collision [81,82]. Zhang et al. employed the NSGA-II algorithm with response surface methodology (RSM) to optimize special-shaped tubes, achieving significant improvements in crashworthiness [72]. In the domain of new structural design, Liang et al. [83] combined machine learning and the NSGA-II method to optimize bionic tubes that show potential for automotive crashworthiness. Xiong et al. [73] adopted a hybrid surrogate model that combines radial basis function neural network (RBFNN), RSM, and multi-objective particle swarm optimization (MOPSO) for the structure-material integrated multi-objective lightweight design of the front end structure of an automobile body. Similarly, Yu et al. [84] adopted the MOPSO algorithm to determine an optimized geometric distribution of TRB structures, while Gao et al. [85] employed a surrogate model-based MOPSO for optimizing the front body of an electric vehicle. MOPSO has also been applied successfully to optimize the functionally graded material-filled crash box [86] and the double-hat thin-walled structure (DHTS) [87], with results indicating significant enhancements in both crashworthiness and lightweight. Additionally, MOPSO was implemented to improve the noise, vibration, and harshness (NVH) performance of vehicle doors [74]. A multi-objective gray wolf optimizer (MOGWO) was used to achieve the seat multi-objective optimization design process, which achieves a reduction in cost by 20.7% and mass by 22.9% [88]. Beyond these methods, other meta-heuristic algorithms are also gaining traction. Chen et al. [88] proposed an improved grey wolf optimizer (IGWO) to optimize the back propagation neural network (BPNN) model for multi-objective optimization of automotive seat frames, achieving a 20.7% cost reduction and a 22.9% mass reduction while ensuring safety performance. In transmission system design, nine different meta-heuristics were employed to search for optimal design of an automatic planetary gear train [89], highlighting the expanding role of meta-heuristic methods in complex automotive engineering problems.

3.2.2. Applications of Heuristic Algorithms in Agricultural Engineering

As can be seen from Figure 4, heuristic algorithms are increasingly applied to tackle optimization challenges in complex agricultural engineering [90]. Many problems in this field exhibit nonlinear and high-dimensional characteristics, such as spectral analysis, quality detection, and resource allocation [91]. Heuristic algorithms, inspired by natural laws or collective intelligence, provide efficient mechanisms for rapidly identifying practical robust solutions in such problems [92].

In the field of agricultural product quality detection, especially with spectroscopic techniques, heuristic algorithms play a crucial role in feature selection and model optimization [93]. For instance, several studies have confirmed that models such as the least squares support vector machine (LSSVM), back propagation neural network (BPNN), and wavelet neural network (WNN) optimized by particle swarm optimization (PSO) achieve excellent prediction performance [94,95,96]. The MOPSO has been employed to enhance the reconstruction of shock vibration signals, effectively reduce test cycle time, and the cost associated with hybrid tractor bumping tests [97]. Further hybrid approaches, such as hybrid particle swarm optimization (HPSO), have been applied to optimize support vector machine (SVM) models [98], while a Two-Strategy Particle Swarm Optimization (TSPSO) was proposed to address the issues of premature convergence and easy entrapment into local optima during PSO hyperparameter optimization. It has been used to optimize the derivative model of CNN named ResNet, which is used for detecting tea quality grades [99]. Similarly, the PSO and grasshopper optimization algorithm (GOA) were applied to optimize deep belief networks (DBN) and support vector machine (SVM) for cadmium content detection, respectively [100,101]. The GWO and artificial fish swarm algorithm (AFSA) were utilized to optimize SVR models for detecting moisture content in green tea and rice seed varieties [102,103,104]. He et al. [105] utilized ACO to improve the accuracy of random forest models in detecting mites in flour, while Guo et al. used ACO to reduce the spectral complexity and improve the apple soluble solids content detection accuracy [106]. GA also plays a significant role in agricultural model optimization. Zhu et al. combined GA and partial least squares regression (PLS) to estimate the uniformity of fermentation products [107]. The GA has also been applied for the predictive optimization of the synergy interval partial least squares [108]. Bonah et al. [109] compared GA and PSO for optimizing the Support Vector Machine Regression (SVMR) in detecting foodborne bacterial pathogens, concluding that GA-SVMR outperformed PSO-SVMR in predictive accuracy. Kutsanedzie et al. [110] also confirmed superior stability in a GA-optimized model for predicting total fungi count in cocoa beans compared to an ACO-based model. Other advanced heuristics include the artificial bee colony (ABC) algorithm, which achieved 100% accuracy in a non-destructive detection model for watermelon seed viability using SVM [111], and the Harris hawks optimization (HHO) algorithm, which significantly improved the prediction accuracy and robustness of SVR models for egg quality assessment compared to GA-SVR, ABC-SVR, and GWO-SVR [112].

Heuristic algorithms play an important role not only in optimizing predictive models for agricultural detection but also in significantly enhancing the performance and efficiency of agricultural machinery. Agricultural machinery optimization aims to improve operational efficiency, adaptability, energy efficiency, and safety through technological improvements, parameter adjustments, and structural design upgrades [113,114]. The agricultural machinery design often involves multi-disciplinary coupling optimization [115]; heuristic algorithms offer flexible tools to address these complex challenges. For instance, Yin et al. [116] developed a three-arm transplanting mechanism for rice potted seedlings based on the “parameter guide” heuristic algorithm, which reduced the seedling injury rate to 0.04% and the missing planting rate to 1.4%. Cui et al. [117] optimized the pendulum suspension parameters of a large spray boom based on NSGA-II, reducing the RMS of the boom roll angle by 14.76%. Additionally, Pang et al. [118] proposed a genetic algorithm for the synchronous optimization of pipe diameter and operation frequency, reducing the irrigation cost by 19.3%.

3.2.3. Research Progress and Future Development

Traditional algorithms such as GA and PSO have performed excellently in many optimization problems. Scholars continue to draw inspiration from biological behaviors in nature, proposing a series of new heuristic algorithms, as shown in Figure 5.

The White Shark Optimizer (WSO) is a swarm intelligence algorithm proposed in 2022 based on the deep-sea foraging behavior of white sharks [119]. It simulates the process of white sharks searching, tracking, locating, and hunting for food through their sensitive sense of smell and hearing. WSO has been applied to optimize parameters in automotive component design, such as the stiffness matching of suspension systems [120], as well as to path planning of agricultural robots and optimization of irrigation systems [121]. The GWO is another nature-inspired algorithm, modeled after the social hierarchy and cooperative hunting behavior of grey wolves [122]. By imitating the way wolves track, surround, and attack prey, it gradually approaches the optimal solution. GWO was applied to battery management of new energy vehicles [123] and parameter optimization in automotive control systems [124]. The Whale Optimization Algorithm (WOA) mimics the bubble-net foraging tactic of the humpback whale [125]. This algorithm is widely employed in autonomous vehicle path planning [126] and automotive lightweight design [127], and has also been applied to maximize resource utilization efficiency in agricultural resource allocation [128]. The HHO is an optimization algorithm that simulates the cooperative hunting behavior of Harris hawks [129]. It is often used in vehicle fault diagnosis to improve maintenance efficiency [130] and in agriculture for pest and disease identification and control [131]. By optimizing the parameters of image recognition models [132], it can improve the accuracy of pest and disease detection. Furthermore, animals exhibiting unique swarming behaviors or highly efficient optimization mechanisms continue to inspire heuristic algorithms. For instance, the periodical cicada, inspired by its prime-numbered life cycle, has been applied to solve problems such as interference from periodic loads on automotive components [133] and the overlap between pest control cycles and pest breeding cycles in agriculture [134].

In the future, to address the challenge of collaborative optimization efficiency in multi-parameter automotive lightweight systems, distributed and parallel computing technologies can be employed to decompose complex optimization tasks into multiple sub-tasks for concurrent processing [135]. Furthermore, integrating novel algorithms with multiscale uncertainty analysis [136] and deepening the integration of surrogate models and algorithms [137] can enhance the robustness of lightweight design for complex structures. In the agricultural field, future agricultural machinery will develop towards intelligence and automation. Heuristic algorithms can optimize the operation parameters of agricultural machinery to achieve autonomous and collaborative operations [138]. Combined with robot technology and automatic control, heuristic algorithms can be used to achieve intelligent path planning [139] and task scheduling of agricultural machinery [140,141], reducing costs and enhancing the intelligence level of agricultural production.

3.3. Surrogate Model-Based Optimization Algorithms

Heuristic optimization algorithms often struggle with expensive cost problems due to their reliance on a large number of evaluations. In contrast, surrogate-based optimization algorithms address this issue by constructing computationally efficient surrogate models to approximate the objective function. These models effectively guide the search process. As a result, such methods require very little expensive evaluation to converge to a near-optimal solution, which significantly reduces the overall computational and experimental cost. Surrogate model-based optimization methods are widely used in various engineering fields.

3.3.1. Surrogate Model Applications in Automotive and Agricultural Engineering

Surrogate models are widely employed to alleviate the substantial computational expenses associated with computationally intensive simulations in engineering design. Common surrogate modeling techniques encompass Radial Basis Functions (RBFs) [142], (RSM) [143], Kriging [144], and Support Vector Regression (SVR) [3]. As shown in Figure 6, surrogate models have been extensively applied in automotive engineering to problems such as lightweight design [145,146,147], crashworthiness optimization [148,149,150,151], and structural design optimization [152,153,154]. In agricultural engineering, researchers have also achieved breakthrough progress using surrogate models in areas including extraction process optimization [155,156,157,158,159,160,161,162], crop processing [163,164,165,166,167], microbial colony control [168,169,170,171,172], agricultural equipment parameter optimization [173,174,175,176,177,178,179], and crop detection [180,181]. In summary, surrogate models play a crucial role in engineering applications by substantially reducing computational costs and effectively approximating complex black-box functions.

3.3.2. Advances in Surrogate Model-Based Optimization Algorithms

Surrogate models have been extensively applied in recent years to address time-consuming black-box engineering optimization problems. To reduce the computation demand, surrogate models are often used in place of the actual simulation models in design optimization. In the automotive industry, structural optimization for collision safety standards presents significant challenges in obtaining precise continuous or discrete sensitivities due to highly nonlinear characteristics. Consequently, surrogate model—based methods have gained widespread adoption in automotive design and attracted considerable industry attention. Figure 7 illustrates selected research outcomes in surrogate model-based optimization methods from recent years. Subsequent sections will introduce several commonly employed surrogate model optimization approaches.

• Kriging surrogate model-based optimization algorithms;

The Kriging model, also known as the Gaussian process model, is a predictive method based on stochastic process theory. It possesses strong local interpolation capabilities and provides variance estimates for predictions. This model excels in handling complex optimization problems involving nonlinearity, high dimensions, and small sample sizes, making it particularly suitable for applications demanding high prediction accuracy. The kriging model has been effectively applied to optimize the crashworthiness of vehicles and foam-filled thin-walled structures [54,182]. It has also been employed to enhance the intrusion performance and energy absorption capacity of bumper systems, thereby improving vehicle safety and impact management [144,183]. These applications demonstrate the value of Kriging as a robust surrogate modeling technique in simulation-driven design, where it helps reduce computational cost while maintaining high predictive fidelity in critical performance evaluations. Beyond automotive engineering, the Kriging model has also shown significant utility in agricultural applications. For instance, it has been used to significantly enhance the average deposition rate and uniformity of plant protection UAV spraying by accurately characterizing the operational parameter space and assisting multi-objective optimization. This contributes effectively to promoting pesticide reduction and sustainable agriculture [184].

2.

Response surface surrogate model-based optimization algorithms;

The RSM, which typically uses quadratic polynomials, is widely used as a metamodeling in crashworthiness optimization due to the high computational cost of finite element analysis. In 2001, Gu et al. employed a sequential quadratic programming method with mixed types of variables for the vehicle crash safety design optimisation of side impact based on RSM [185]. In 2005, RSM was integrated with spatial mapping techniques to optimize the stretching process of automotive sheet metal parts [186]. Liao et al. further applied RSM with both linear and quadratic basis functions to formulate objective functions in the crashworthiness design of vehicles [187]. To enhance the efficiency of large-scale complex problems, Shi et al. introduced a Bayesian metric approach to systematically screen the optimal response surface from multiple RSM, significantly reducing the number of computationally intensive simulations required for design optimization [188]. Addressing the crashworthiness optimization of thin-walled structures, Sun et al. adopted RSM to formulate the specific energy absorption (SEA) and peak crashing force functions [189]. More recently, the third-order RSM was identified as the optimal surrogate model for automotive seat optimization [190], and in 2023, Dai et al. employed the RSM to establish an alternative optimization design model for automotive seat frames [153]. RSM has been effectively applied across agricultural engineering filed, such as optimizing the fermentation process of selenium-enriched mulberry wine to enhance nutrient delivery and process efficiency [191], as well as efficiently resolving surface water and groundwater allocation conflicts between agricultural and ecological uses in the Heihe River Basin by replacing computationally intensive hydrological models, thereby significantly improving the sustainability and efficiency of resource management decisions [192].

3.

Radial basis function (RBF) surrogate model-based optimization algorithms;

RBF uses a series of basic functions that are symmetric and centered at each sampling point, and it was originally developed for scattered multivariate data interpolation. RBF surrogate modeling has been widely adopted in automotive engineering for its flexibility and accuracy in handling highly nonlinear responses. To address high-dimensional challenges, Li et al. [193] proposed the RBF-HDMR method, applying it to the global sensitivity analysis of key suspension parameters and lateral dynamic performance in locomotives. Similarly, Ji et al. [194] applied the L-SHADE algorithm integrated with an RBF surrogate model to achieve optimal design of a turbine blade. In a comparative study from 2005, both RSM and RBF were applied to the multi-objective optimization of a vehicle body in frontal collision. The results showed that although RSM provided an effective approximation for energy absorption, RBF generated superior predictive models using the same number of samples [195] Furthermore, an RBF-based approach was used to develop surrogate models for optimizing vehicle crashworthiness with vibration characteristics [196], while another study combined RBF with NSGA-II for multi-objective lightweight optimization of a body side structure [197]. In addition, three surrogate models, including RBF, RSM, and kriging, were compared in the accuracy of predicting SEA and PCF of functionally graded thickness honeycomb structure; the results show that RBF and RSM were more suitable in crashworthiness optimization design of the FGT honeycomb structure [198]. The RBF was combined with multi-objective evolutionary algorithms, efficiently optimized irrigation and fertilization strategies, achieving a 44% reduction in water consumption, a 37% decrease in nitrogen application, and a 7–8% increase in economic benefits in sustainable agricultural management [199].

4.

Support vector regression (SVR) surrogate model-based optimization algorithms;

Optimization methods based on Support Vector Regression (SVR) surrogate models utilize the SVR algorithm to identify the optimal hyperplane for regression fitting. Due to their outstanding generalization capability, ability to handle nonlinear problems, and favorable performance under small-sample conditions, SVR-based optimization methods have demonstrated significant advantages. It has been employed to test the performance of an optimization algorithm for the built-in rubber buffer within the structural optimization of truck air spring assemblies [200].

5.

Hybrid surrogate model-based optimization algorithms;

Single surrogate models are often constrained by their inherent construction techniques, exhibiting limitations in applicability scope and accuracy under small-sample conditions, which restricts their application in complex engineering problems. Consequently, the development of hybrid surrogate model optimization methods, which integrate multiple modeling approaches to achieve broader applicability and higher efficiency, is underway. As early as 2011, Pan et al. proposed an optimisation methodology based on multiple surrogates such as Kriging, RBF, and SVR, which dynamically selected the most accurate surrogate during each iteration. This approach was successfully applied to the lightweight design of a vehicle front-end structure under crashworthiness and vibration constraints [201]. Similarly, the ensemble of surrogates (EOS), which combines multiple approximation models for the same process, was employed to enhance vehicle structural crashworthiness predictions [202]. Further improving adaptability, an automatic surrogate selection strategy from a predefined library was proposed for vehicle design under full-frontal and offset-frontal impact conditions [203]. Many researchers have also focused on integrating multiple meta-models to improve overall predictive performance. Many researchers have also focused on integrating multiple meta-models to improve overall predictive performance [204]. To address challenges such as complex weight assignment and the limited accuracy of single surrogates in highly nonlinear multi-objective optimization, Liao et al. proposed a Modified Preference Selection Index (MPSI) and a hybrid modeling framework, demonstrated effectively in the optimization of automotive seat skeletons [205]. Another study introduced a hybrid model (SR) combining SVR and GPR, which significantly outperformed individual SVR, GPR, RSM, and Kriging models in nonlinear response prediction under small-sample conditions [3]. Peng et al. [206] introduced a pointwise weighted hybrid surrogate model based on hybrid measures (PWHSMHM) by combining RSM, Kriging, and SVR. Compared to individual models, PWHSMHM significantly enhances fitting accuracy and has been successfully applied to the optimization design of automobile front hoods. In a further application, the multi-objective automatic iterative optimisation of the body skeleton was achieved by using the RBFNN-Kriging hybrid surrogate model and NSGA-II [146]. The application of hybrid surrogate models has expanded to various automotive engineering problems, including material optimization [207], structural design [208,209]. These developments demonstrate that hybrid surrogates offer significant advantages in enhancing the robustness, accuracy, and applicability of models within complex multidisciplinary engineering systems.

3.3.3. Future Development of Surrogate Model-Based Optimization Algorithms

In the fields of automotive engineering and agricultural engineering, surrogate model-based optimization methods have been extensively employed to address expensive black-box problems. However, confronting increasingly complex engineering responses characterized by high dimensionality, strong nonlinearity, non-stationarity, and intricate structures (e.g., images, time-series data), these methods face significant challenges in terms of modeling accuracy and efficiency. Future surrogate model-based optimization approaches should focus on the following directions: Integrating advanced deep learning architectures, actively exploring the deep integration of models such as Deep Neural Networks (DNNs) and Convolutional Neural Networks (CNNs) within the surrogate modeling framework, to significantly enhance the representation capability and prediction accuracy for complex, high-dimensional, and non-stationary system responses; Improving design space management through design space partitioning, employing methods of different fidelity across the entire space and within critical regions to enhance optimization efficiency for expensive problems; and Developing more versatile and robust hybrid surrogate model optimization methods by strategically combining individual surrogate models with varying strengths and limitations, thereby expanding their applicability scope within engineering domains.

3.4. Bayesian Optimization

Bayesian optimization (BO) is a global optimization algorithm driven by a probabilistic surrogate model such as a Gaussian process and a Bayesian neural network [210,211,212]. It is particularly suitable for problems where the objective function is non-differentiable, noisy, expensive to evaluate, or lacks an analytical form. The core idea involves approximating the objective function using a probabilistic surrogate model and employing an acquisition function to balance exploration of uncertain regions with exploitation of promising regions. This approach enables efficient global optimization with a reduced number of function evaluations. Compared with traditional optimization methods such as grid search and gradient-based optimization, BO demonstrates significant advantages in machine learning [213,214], engineering design, scientific experiment optimization, robot control, and other fields.

3.4.1. Application of Bayesian Optimization in Agricultural Engineering

BO has proven to be an efficient global optimization method with significant applications across automotive and agricultural engineering. In intelligent agriculture, BO has been applied to optimize parameters for robots and mobile manipulators, enhancing operational autonomy and task efficiency [215,216]. As shown in Figure 8, BO also plays a vital role in agricultural hydrology by improving water resource management, mitigating flood risks, and enhancing drought forecasting [217].

It has been used to tune hyperparameters of Gaussian process regression models, substantially increasing the accuracy of soil moisture predictions for smart irrigation [218]. Additionally, BO effectively identifies optimal environmental parameters in controlled agricultural systems, improving seed germination rates [28]. The algorithm has been widely adopted in crop and disease management: it enables high-accuracy diagnosis of rice diseases through attention-based deep learning [219], improves wheat yield prediction models [220,221], and reduces grain loss in combine harvesters by optimizing operational parameters [222]. BO also enhances deep learning models for orange disease and ripeness detection [223], supports trait selection in plant breeding via morphological ideotype identification [224], and improves pest-infestation detection in citrus using SVM-based olfactory systems [225]. BO has been widely utilized to enhance automated decision systems across various agricultural applications through efficient hyperparameter tuning and model optimization. For instance, BO was employed to automatically select optimal hyperparameters in a deep learning pipeline, significantly improving the accuracy of pepper leaf disease detection and supporting data-driven agricultural [226]. It has also been applied to optimize the evaluation and allocation of regional crop water productivity, enabling data-driven planting structure optimization under water constraints and enhancing both agricultural efficiency and ecological sustainability in arid regions [227]. In crop recommendation systems, BO was used to fine-tune a random forest model, achieving 96% accuracy in matching suitable crops to soil conditions, thereby strengthening decision support for farmers and reducing planting risks [228]. The method also proved effective in solving multi-objective animal diet formulation problems, improving digestible energy, lysine content, and cost efficiency by 3–14% despite the high variability and evaluation costs of agricultural raw materials [229]. Further applications include tuning random forest regression models to enhance the accuracy and stability of Leaf Area Index (LAI) estimation across multiple crops and sensors for precision agriculture [230], optimizing convolutional neural networks to achieve over 92% accuracy in agricultural pest image recognition [231], and fine-tuning artificial neural networks for predicting key variables in green urea production [232]. Additionally, BO has been applied to improve the prediction of hourly photosynthetically active radiation (PARhour) in solar greenhouses using common weather data, facilitating cost-effective environmental control for small-scale farmers [233]. Bayesian optimization significantly enhances the accuracy and reliability of a GBRT-based crop selection surrogate model by automatically determining the optimal hyperparameters for the deep neural network, achieving precise classification of crop suitability based on soil characteristics [234]. In summary, BO significantly enhances the accuracy, efficiency, and resource utilization of smart agricultural practices by improving the performance of various prediction and decision-making models.

3.4.2. Application of Bayesian Optimization in Automotive Engineering

BO has become a pivotal tool in automotive engineering, driving improvements in design efficiency, performance, and cost-effectiveness. For instance, BO has been applied to optimize hybrid aerodynamic frameworks, thereby supporting vehicle lightweight initiatives [235]. It has also been applied to improve intelligent diagnostic systems through classifier fusion and parameter tuning, leading to substantial enhancements in fault detection accuracy for automotive engines [236]. Under conditions of data uncertainty, BO facilitates the automated selection of optimal surrogate models and sample sizes, markedly improving the accuracy and efficiency of large-scale design optimization for automotive safety systems, including full-frontal and offset-frontal impact simulations [203]. Furthermore, BO enables systematic selection of RSM to enhance crashworthiness design optimization while minimizing computational costs [188]. In the domain of engine fault diagnosis, BO automates the selection of classification models and hyperparameters using acoustic signals, increasing accuracy and reducing evaluation time by up to 20%, which supports real-time maintenance solutions [237]. The algorithm has also been benchmarked and refined with crash constraint handling techniques to automate controller parameter tuning, resulting in up to 16.1% improvement in closed-loop control performance compared to conventional methods [238]. Additionally, BO has been utilized to automate the shape and parameter tuning of GFRP leaf spring axle systems, effectively balancing lightweight objectives with performance constraints while reducing the number of costly finite element simulations [239]. In electric motor design, BO has been directly integrated into high-fidelity vehicle simulation frameworks, achieving a 0.13% reduction in energy consumption and identifying more efficient magnet and slot configurations [240]. BO has demonstrated considerable efficacy in addressing multi-objective and high-dimensional design challenges in automotive engineering. For instance, BO was integrated with NSGA-II to automate the multi-objective lightweight design of an automotive bumper, achieving a 53.96% weight reduction while simultaneously improving crashworthiness and energy absorption performance [241]. Similarly, BO was employed to optimize the placement of rib structures in automotive tailgate components, maximizing stiffness under strict mass constraints while minimizing the reliance on costly finite element simulations. This approach also successfully bridged design and manufacturing requirements [242]. BO was fine-tuned through an automated hyperparameter optimization approach to efficiently solve high-dimensional automotive crashworthiness problems, outperforming default configurations and conventional methods in both solution quality and convergence speed under limited simulation budgets [243]. Furthermore, it was applied to identify the optimal composition and configuration of graphene-enhanced fiber metal laminates, significantly improving tensile strength and energy absorption for lightweight, high-performance automotive structural components [244].

3.4.3. Future Development of Bayesian Optimization Algorithms

Future Bayesian optimization research will focus on the following directions: improving high-dimensional optimization capabilities, breaking through dimensional limitations through sparse models, subspace partitioning, etc.; developing efficient acquisition functions to support parallel evaluation and decoupling problems; expanding multi-fidelity strategies, using low-fidelity models and transfer learning to reduce evaluation costs; developing adaptive acquisition function selection mechanisms, improving algorithm generalization capabilities, optimizing complex real-world problems, and improving sample efficiency and the diversity of Pareto solutions. In terms of multi-objective optimization, explore diverse Pareto frontier generation methods, such as combining generative adversarial networks to predict Pareto solutions, or dynamically adjusting acquisition function weights through reinforcement learning. Cross-domain integration, such as the combination with deep learning and evolutionary algorithms, will promote the widespread application of Bayesian optimization in high-dimensional, multi-objective, and complex constraint scenarios.

3.5. Hybrid Cellular Automata Optimization

Hybrid Cellular Automata (HCA) is a computational optimization method that extends classical cellular automata (CA) by integrating global objectives into local update rules. HCA introduces a feedback mechanism to guide the system towards an optimal configuration. This enables efficient, scalable, and physically-based optimization of complex engineering problems while maintaining computational simplicity and inherent parallelism.

3.5.1. Application of Cellular Automata in Engineering Optimization

CA are fully discrete dynamical systems, first defined by John von Neumann in an attempt to simulate and explain biological self-replication abstractly. It has attracted considerable attention due to its remarkable ability to exhibit highly complex phenomena in a relatively simple setup. The main feature of CA is that information interaction is limited to the neighborhood of adjacent cells, and a unified local update rule is adopted. The framework exhibits characteristics of parallel computation and achieves an efficient combination of analysis and design by synchronous updates of field variables and design variables, which significantly reduces the computational cost of optimization design.

In the 20th century, CA was successfully applied to structural design; scholars continue to expand the boundaries of CA application, so that it shows more and more important value in engineering optimization. Kita et al. proposed a CA-based shape and topology co-optimization method that introduces CA constraints and transforms the global optimization problem into a local optimization problem. This broke through the bottleneck of local rules derived from numerical experiments [245]. Tatting et al. proposed a design rule that satisfies the static equilibrium and stress conditions for complete material failure, and applied it in combination with CA for the optimization design of two-dimensional continuous structures subjected to in–plane loads, thereby achieving topology optimization based on thickness design [246]. For Euler-Bernoulli columns with given buckling loads or natural frequencies, Abdalla et al. performed a structural optimization design based on CA under local constraints, which avoids complex global eigenvalue calculations and significantly improves computational efficiency [247]. To improve the computational efficiency of the CA algorithms, researchers upgraded the CA local update rule, which not only significantly improves computational efficiency but also eliminates the common checkerboard effect in traditional topological optimization [248], and achieved the topological optimization of truss structures [249] and planar elastic structures [250]. Subsequently, Faramarzi et al. [251] proposed a two-stage hybrid cellular automaton-linear programming (CA-LP) method, and Gholizadeh et al. combined the advantages of CA and particle swarm optimization (PSO) to propose the SCPSO algorithm, which significantly improved the speed of convergence and solution quality of solutions [252]. Aiming at the limitation that the CA method is only applicable to the same metastructure with a uniform mesh, Bouzouiki et al. [253] proposed an improved non-uniform CA algorithm that effectively solves the problem of size and topology optimization of truss structures under stress and displacement constraints. On this basis, Gan et al. proposed a novel topological optimization algorithm under dynamic loads, which integrates CA and bi-directional evolutionary structural optimization (BESO) to address common issues in nonlinear topological optimization, such as numerical oscillations. This method significantly improves computational efficiency and stability [254]. Although classical CA has demonstrated its advantages in structural topology optimization, its purely rule-driven nature suffers from insufficient global convergence. Hybrid cellular automata (HCA) significantly improve the engineering feasibility by combining CA with finite element analysis (FEA).

3.5.2. Research Progress of Hybrid Cellular Automata Methods

As a computational framework that combines discrete dynamical systems and continuous optimization methods, HCA has shown significant potential in the field of complex system modelling and multidisciplinary optimization design. It effectively integrates the parallelism advantage of CA with the global search capability of continuous optimization algorithms, and has been widely used in the design of automotive structures, lightweighting, crashworthiness optimization, and so on, as shown in Figure 9.

Tovar et al. proposed an HCA method by combining CA with FEA, which achieved simultaneous topology and shape optimization of continuum structures while balancing mass and stiffness requirements [11,255]. Mozumder et al. combined CA with nonlinear transient analysis and proposed an HCA algorithm based on uniform distribution of internal energy density (IED) to optimize thickness and cost under dynamic loading for structures manufactured by welded blank (TWB) technology, which significantly reduces the cost of the structure while improving its crashworthiness [256]. To improve the ability to withstand large deformations during structural collisions, Guo et al. presented a strain-based topology optimization method based on HCA, dividing the design domain into high-strain subdomains (HSSD) and low-strain subdomains (LSSD). The LSSD region absorbs energy through uniformly distributed internal energy, while the HSSD region controls material failure through the maximum strain criterion, effectively solving the problem of large plastic deformation [257]. Bandi et al. proposed a compliant mechanism design method based on HCA [258], which was subsequently improved to enable crashworthiness design of compliant mechanism structures under multiple working conditions. This approach provides a universal tool for automotive safety components requiring both stiffness and flexibility [259,260]. Guo et al. introduced orthogonal experimental design to analyze the influence of algorithm parameters on convergence systematically. Taking a cantilever beam as the research object, they validated the convergence of HCA in dynamic collision problems, which provided parameter selection criteria and algorithm improvement directions for crashworthiness optimization [261]. Wehrle et al. systematically evaluated topological optimization methods for transient nonlinear behavior of mechanical structures and investigated the effectiveness and limitations of methods based on HCA, establishing a systematic evaluation framework for algorithm selection in the field of transient nonlinear topological optimization [262]. Hesse et al. addressed the challenges of parameter complexity and nonlinearity in the crashworthiness design of composite structures in automotive, proposing an innovative strategy that combines surrogate models and Sobol decomposition. They used the method to determine parameter boundaries, thereby enhancing robustness [263]. Da et al. applied HCA to the design of material microstructures, calculating the equivalent properties of materials based on an energy homogenization strategy to achieve topological optimization of extreme performance materials [264]. Jia et al. proposed a multi-scale non-uniform structural topology optimization method based on HCA and K-means clustering algorithms, which significantly improves structural stiffness by implementing micro-scale non-periodic designs [265]. Raeisi et al. integrated ordered SIMP interpolation and an artificial material library to optimize multi-material structures with different steel distributions using a heuristic topology synthesis method based on HCA, significantly improving the crashworthiness of the structures [266]. Afrousheh et al. proposed a modified HCA (MHCA) method based on a variable neighborhood radius strategy. This method achieves a more uniform plastic strain distribution by intelligently adjusting the search range, significantly improving the energy absorption efficiency per unit mass of the structure [267]. Furthermore, HCA effectively enhances the load-bearing capacity, vibration resistance, and fatigue life of structures under various load conditions by integrating with the analytic hierarchy process, random vibration topology optimization method, and the dynamic-static coupled topology optimization method. This significantly expands its applicability and engineering value in structural topology optimization [268,269,270].

In the field of automotive engineering, thin-walled structures have become essential components for vehicle safety design due to their excellent lightweight potential and energy absorption capacity. In recent years, the research on the application of HCA to thin-walled structures has made remarkable progress. In 2012, Mozumder proposed an HCA method to achieve topology optimization of synthetic shells based on the uniform distribution of IED, solving problems such as design for crashworthiness, large deformations, and material plastic hardening [271]. They further co-optimized the shell-type structural material and thickness by using the HCA with the force-displacement (FD) response as the optimization objective [272]. Bandi et al. followed the topological optimization principles of compliant mechanism design and carried out progressive crush design for thin-walled tubular structures based on HCA, significantly improving the crashworthiness of the thin-walled structure [273]. Duddeck et al. defined non-uniform energy density as the optimization objective and performed topology optimization for thin-walled beams and plates based on HCA [274]. In addition, Zeng et al. [275] proposed an improved HCA method and applied it to the topology optimization of thin-walled structures. They further developed an improved version of the thin-walled structure HCA (HCATWS) method, which achieved more manufacturable optimization results compared to traditional methods [276]. Wang et al. proposed a combined variable density method with HCA. This approach was applied to optimize a multi-cell thin-walled front longitudinal beam structure, ensuring crashworthiness while reducing mass [277]. In order to solve the structural optimization problem with a large number of discrete variables, Duan et al. proposed an improved sub-region hybrid cellular automata algorithm (eHCA-VRB) to optimize the crashworthiness and thickness distribution of variable-thickness rolled blanks (VRB). The method combines outer-loop FEA for target mass determination with inner-loop thickness adjustment based on uniform IED, achieving effective optimization of VRB hat-shaped beams [278]. They further developed a sub-region hybrid cellular automata method (SHCA-T) for body skeleton thickness distribution. The outer-loop updated the field variables and mass, while the inner-loop constructed the step-type field variable updating rule, employing a PID control strategy for body thickness distribution design. This approach demonstrates excellent convergence and computational efficiency within multi-thickness variable design spaces [279]. Additionally, Duan et al. introduced nominal flow stress, continuous and discrete material variables, and proposed an optimization algorithm, which the outer loop updates the field variables and target cost through the penalty function, while the inner loop optimizes the material distribution of the thin-walled frame based on the step IED target function, significantly improving material optimization efficiency and reducing costs [280].

In the field of agricultural engineering, the modified cellular automata algorithm (Mcanda) enhances the honey-bee mating optimization (HBMO) method to efficiently optimize pipe diameter design in agricultural water distribution networks, improving hydraulic performance and resource allocation under complex operational constraints [281]. The macroscopic cellular automata (MCA) model, integrated with a deep belief network (DBN), effectively predicts spatiotemporal soil moisture distribution in agricultural fields, enabling precise irrigation scheduling and enhanced water resource management [282]. The RF-CA-Markov model enhances the prediction of agricultural water-land resource matching by simulating dynamic land use changes at adaptable grid scales, enabling precise irrigation planning and risk mitigation in water-scarce regions [283]. Cellular automata simulations demonstrate that self-organized patterns of stem inclination in sunflowers, mediated by phytochrome perception of light quality, can significantly increase crop yield under high-density conditions [284]. Cellular automata, integrated with Boosted Regression Trees, objectively simulate and predict agricultural land use changes to optimize water management and ecological planning [285]. The hybrid cellular automata model simulated and predicted future expansions of agricultural land, providing critical data for sustainable water and land resource management in the river basin [286]. The cellular automata model, integrated with surrogate-based modeling, simulates agricultural land-use decisions and conversions to assess policy impacts on farming sustainability and ecosystem services [287]. The cellular automata-Markov model effectively simulates and projects agroecology-specific cropland expansion and its environmental impacts, enabling targeted land conservation planning to sustain agricultural productivity and reduce soil erosion [288]. The cellular automata-ANN model, enhanced by hybrid predictor screening, accurately projects future cropland expansion and changes in agricultural land use, providing critical insights for sustainable resource management and environmental planning [289,290].

3.5.3. Future Development of Hybrid Cellular Automata

In the automotive field, HCA shows unique advantages in structural design, with a wide range of applications from trusses to thin-walled structures and various automotive components. By combining the adaptive local update principle of CA with FEA methods, they can flexibly address various complex operating conditions and constraints. In current research, HCA has been further extended to sub-regional cellular automata (e), which utilize “step-like” and “arbitrary-type” field variables to replace homogeneous objectives, thereby guiding structural performance toward improvement.

In the future, the potential of hybrid cellular automata can be further explored by constructing diverse cellular automata models to adapt to various irregularly shaped continuous or discontinuous bodies, combining multiple field variable construction forms to provide optimal solutions for complex engineering problems with special constrains; Expand the application scenarios of HCA from automobiles to ships, aerospace, and other fields, integrating them with existing algorithms to form a collaborative optimization framework. By utilizing algorithm nesting to compensate for deficiencies in different algorithms, the advantages of HCA can be fully leveraged, enabling parallel architecture and effectively combining surrogate model training and analysis to reduce the cost and time consumption of FEA; Further optimize cellular local update rules by incorporating more flexible control strategies to address sensitivity issues caused by multiple parameters in the optimization framework, continuously enhancing algorithm robustness; introduce multidisciplinary optimization to establish a HCA optimization platform spanning from micro to macro scales, and from materials to structures, injecting new momentum into structural optimization design across various fields.

4. Comparative Analysis of Optimization Methods

Based on the above review, this section performs a comparative analysis of five optimization methods based on 242 collected literature. Five types of algorithms are compared in terms of convergence speed, computational accuracy, and computational cost in three dimensions for three major problem classes: nonlinear problems, multi-objective problems, and high-dimensional problems. As shown in

Table 2. Comparative analysis of different optimization methods.

Method	Kinds	Convergence Speed	Accuracy	Costs
Gradient-based	Nonlinear
	Multi-objective
	High-dimensional
Heuristic	Nonlinear
	Multi-objective
	High-dimensional
Surrogate model-based	Nonlinear
	Multi-objective
	High-dimensional
Bayesian	Nonlinear
	Multi-objective
	High-dimensional
Hybrid cellular automata	Nonlinear
	Multi-objective
	High-dimensional

Note: Star ratings provide a comparative ordinal ranking: a greater number of stars denotes faster convergence, higher accuracy, and greater associated cost (e.g., is faster/higher accuracy/more costly than ). These ratings are intended for qualitative comparison only and do not represent absolute quantitative values.

.

In the field of nonlinear optimization problems, gradient-based algorithms generally exhibit rapid convergence rates and high solution accuracy, along with superior local convergence performance. However, their effectiveness is highly sensitive to the selection of initial points and step sizes. Integration with heuristic algorithms can help identify better initial solutions, while adaptive step size adjustment strategies can further enhance their performance. As a result, conjugate gradient methods and quasi-Newton methods are more widely adopted. For multi-objective optimization problems, gradient-based algorithms often need to collaborate with other methods to obtain well-distributed Pareto fronts. Heuristic algorithms, which do not rely on gradient information, are widely applicable in various engineering optimization problems. However, they typically require a large number of function evaluations to approach the global optimum. The incorporation of gradient information or other auxiliary strategies can significantly reduce the number of evaluations, thereby improving computational efficiency. In multi-objective optimization, heuristic algorithms are capable of generating well-distributed Pareto solution sets, making them one of the mainstream approaches for solving such problems. Surrogate-based optimization methods can substantially reduce computational costs, but their performance heavily depends on the accuracy of the constructed model. In research, they are often combined with heuristic algorithms to form hybrid strategies, improving both convergence efficiency and computational accuracy. However, as the dimensionality increases, these methods become susceptible to the curse of dimensionality. Hence, they are more suitable for low- to medium-dimensional nonlinear problems or serve as auxiliary strategies within other optimization frameworks. Bayesian optimization algorithms demonstrate strong global search capability and high convergence efficiency. Nevertheless, their performance is sensitive to the choice of kernel and acquisition functions. It is essential to tailor the acquisition function to the specific problem to enhance search performance. This approach is particularly suitable for multi-objective optimization, where it can achieve high-quality Pareto fronts. Hybrid cellular automata perform exceptionally well in handling nonlinear dynamic problems. They benefit from simple rule design, high parallelism, fast convergence, and high accuracy, making them widely applicable in areas such as topology optimization and vehicle crashworthiness design. In multi-objective optimization, although their current performance is slightly inferior to heuristic and Bayesian methods, they show significant potential. Under a reasonable number of constraints, effective local update rules can facilitate rapid convergence to optimal solutions while maintaining high solution quality even in high-dimensional settings. In summary, for high-dimensional optimization problems, gradient-based and surrogate-based methods are prone to the “curse of dimensionality”, and the efficiency of Bayesian optimization decreases considerably. Heuristic algorithms remain one of the primary solution approaches, while hybrid cellular automata also demonstrate considerable potential. High-dimensional optimization remains a highly challenging research direction, urgently requiring the development of more robust and efficient optimization algorithms. As evidenced by the comparative analysis above, gradient-based algorithms offer fast convergence and high solution accuracy, while heuristic algorithms demonstrate strong adaptability and are suitable for multi-objective constrained problems where gradients are unavailable, also proving effective in high-dimensional settings. To reduce expensive evaluation costs, Bayesian optimization stands as the preferred method for black-box problems, and the construction of high-fidelity surrogate models can also substantially conserve computational resources. For high-dimensional problems involving nonlinear discrete variables, hybrid cellular automata show considerable advantages. With increasing engineering complexity, problems are rarely singular in nature; thus, hybrid algorithms have emerged as a mainstream approach for problem-solving. By leveraging the complementary strengths of different algorithms, they provide robust solutions for high-dimensional complex optimization challenges.

5. Conclusions and Future Development

This review compiles applications of gradient-based optimization algorithms, heuristic algorithms, surrogate-based optimization algorithms, Bayesian optimization algorithms, and hybrid cellular automata algorithms in automotive and agricultural engineering. Given the abundance of research in this field, we could only select a limited number of articles. This review focuses on literature published from 2000 to the present, though important articles were also published prior to this period. We have reviewed these articles to identify the strengths, weaknesses, and characteristics of each algorithm for different optimization problems. As a result, this review paper can guide and assist students and researchers interested in this field.

Based on the comparative analysis of the above five types of algorithms, this section proposes the following research recommendations and future development directions to provide reference for researchers in related fields. Gradient-based optimization algorithms need to further explore the deep integration mechanism with heuristic methods and surrogate models to improve the stability and convergence efficiency of global search. In the field of heuristic algorithms, it is recommended to develop new meta-heuristic strategies by borrowing more biological behavioral mechanisms (e.g., the collaborative behavior of horned horse herds or the distributed decision-making mode of multi-armed octopus), or to integrate two or more existing meta-heuristic algorithms to build a more powerful and adaptive optimization framework suitable for high-dimensional and complex problems. Surrogate-based model optimization methods may incorporate deep learning architectures, improve design space exploration strategies, and develop hybrid surrogate-based model methods to enhance the prediction capability and optimization efficiency for high-dimensional nonlinear problems. To address the complex black-box optimization challenges, the BO method should be extended to integrate multiple acquisition function collaboration mechanisms, incorporate multi-composite intelligent modeling techniques, and achieve adaptive real-time updating of acquisition strategies in multi-objective optimization problems. These advances will significantly improve its search effectiveness and convergence stability in high-dimensional multi-objective scenarios. For complex high-dimensional multi-objective engineering problems, the advantages of HCA in structured optimization and parallel solving should be fully utilized. This includes the construction of a distributed optimization framework based on a high-fidelity surrogate model, which supports multi-algorithm collaboration and integrates heuristic or Bayesian algorithms, so as to achieve the simultaneous improvement of the computational efficiency and solution quality.