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• Published: 19 August 2024

Large-scale IoT attack detection scheme based on LightGBM and feature selection using an improved salp swarm algorithm

• Weizhe Chen 1 ,
• Hongyu Yang 1 ,
• Lihua Yin 1 &

Scientific Reports volume  14 , Article number:  19165 ( 2024 ) Cite this article

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Due to the swift advancement of the Internet of Things (IoT), there has been a significant surge in the quantity of interconnected IoT devices that send and exchange vital data across the network. Nevertheless, the frequency of attacks on the Internet of Things is steadily rising, posing a persistent risk to the security and privacy of IoT data. Therefore, it is crucial to develop a highly efficient method for detecting cyber threats on the Internet of Things. Nevertheless, several current network attack detection schemes encounter issues such as insufficient detection accuracy, the curse of dimensionality due to excessively high data dimensions, and the sluggish efficiency of complex models. Employing metaheuristic algorithms for feature selection in network data represents an effective strategy among the myriad of solutions. This study introduces a more comprehensive metaheuristic algorithm called GQBWSSA, which is an enhanced version of the Salp Swarm Algorithm with several strategy improvements. Utilizing this algorithm, a threshold voting-based feature selection framework is designed to obtain an optimized set of features. This procedure efficiently decreases the number of dimensions in the data, hence preventing the negative effects of having a high number of dimensions and effectively extracting the most significant and crucial information. Subsequently, the extracted feature data is combined with the LightGBM algorithm to form a lightweight and efficient ensemble learning scheme for IoT attack detection. The proposed enhanced metaheuristic algorithm has superior performance in feature selection compared to the recent metaheuristic algorithms, as evidenced by the experimental evaluation conducted using the NSLKDD and CICIoT2023 datasets. Compared to current popular ensemble learning solutions, the proposed overall solution exhibits excellent performance on multiple key indicators, including accuracy, precision, as well as training and detection time. Especially on the large-scale dataset CICIoT2023, the proposed scheme achieves an accuracy rate of 99.70% in binary classification and 99.41% in multi classification.

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Introduction.

With the ongoing advancements in sensor technology, computer technology, and Internet technology, the Internet of Things (IoT) has become increasingly prevalent. This technology enables communication between humans and objects, as well as between objects themselves. The proliferation of smart gadgets connected to the Internet of Things is growing exponentially. A large number of new IoT devices are launched every week and are increasingly deployed in various smart environments, including homes, buildings, enterprises and cities. Statista predicts that the global count of interconnected IoT devices will reach 75.44 billion by 2025. This growth will bring a huge impetus to the global economy and is expected to generate an economic value of approximately 11 trillion US dollars per year, accounting for 11% of the global economy 1 .

However, the rapid development of IoT technology has not only brought convenience and economic benefits, but also exposed some security issues. With the surge of IoT devices, network security management is facing unprecedented challenges 2 . These devices may become the entry point for attackers to launch large-scale network attacks, posing a serious threat to network security 3 . In 2008, malicious code Hydra began attacking routers and DSL modems 4 ; In 2014, Thingbots 5 launched attacks on smart home devices, sending at least 750000 emails using controlled smart TVs and refrigerators; In 2016, the botnet named Mirai 6 infected over 1.2 million various types of IoT devices and launched a DDoS attack on DynDNS, a major domain name resolution service provider in the United States, disrupting telecommunications networks, game servers, and multiple large websites, causing huge economic losses and marking the emergence of large-scale IoT attacks. In 2021, Gafgyt_Tor 7 utilized the onion network to facilitate anonymous communication. Following that, malware such as BlackMoon and Dark Frost emerged successively, employing increasingly intricate technical methods to infiltrate IoT devices, hence substantially amplifying the magnitude and effectiveness of IoT attacks. In June 2022, there was a significant increase in HTTPS distributed denial of service (DDoS) attacks, with customers of “Google Cloud Armor” seeing attacks that reached a maximum of 46 million requests per second 8 . This incident was documented as one of the most significant cyber danger in the world. The increasing scale of IoT attacks poses a serious threat to global network security.Hence, it is imperative to deploy a efficient, swift, and precise intelligent IoT attack detection framework.

In recent years, researchers have successfully utilized machine learning to accomplish intelligent detection of IoT attacks, thanks to the advancements in artificial intelligence. However, despite these achievements, there still exist problems in achieving intelligent and effective detection 9 . The current research in the field of IoT attack detection faces several issues and challenges that need to be addressed 10 . Firstly, the increase in network bandwidth and data volume has rendered timely processing of network data difficult, necessitating detection solutions with high efficiency and rapid response capabilities. Secondly, within extensive network data, not all features contribute to detecting attack behaviors; some features may be irrelevant or redundant, thereby increasing computational burden and reducing detection efficiency. Moreover, the high-dimensional nature of network traffic data often leads to the “curse of dimensionality” 11 augmenting model computation time and complexity. Lastly, effective operation of attack detection systems across diverse scenarios and environments demands detection solutions with robust generality and generalization capabilities, capable of adapting to various network traffic scenarios.

Metaheuristic algorithms, especially hybrid metaheuristic algorithms, represent an ideal solution for addressing the challenges in IoT attack detection 12 . By combining machine learning and deep learning models, they effectively handle the massive scale of IoT data, swiftly filtering critical information and delivering rapid responses, crucial for real-time monitoring and defense against IoT attacks. Moreover, metaheuristic algorithms excel in feature selection, identifying and retaining features that significantly contribute to attack detection while eliminating irrelevant or redundant information. This reduction in feature dimensions not only alleviates computational burdens but also enhances detection precision and efficiency. Among numerous metaheuristic algorithms, the Salp Swarm Algorithm 13 (SSA) has emerged as a fundamental choice for many researchers in the field of metaheuristic algorithm improvement, owing to its simplicity, efficiency, user-friendliness, and substantial potential for enhancement 14 . SSA exhibits strong adaptability, capable of addressing various types of optimization problems including continuous and discrete domains. Furthermore, SSA is easily integrated with other algorithms to form hybrid approaches, thereby enhancing problem-solving capabilities. Its successful applications across multiple domains validate its broad applicability and effectiveness.

According to the No Free Lunch Theorem 15 , it is known that there is no single optimization algorithm that can universally solve all problems. Similarly, SSA is not without its drawbacks: in complex problems, it may exhibit a slower rate of optimization in later stages and runs the risk of getting trapped in local optima. Given this reality, as well as the challenges and situations involved in IoT attack detection, a improved Salp Swarm Algorithm has been developed for IoT attack detection in this study.

This approach uses a mix of the good point set strategy and quantum encoding to enhance the diversity of the population during population initialization. Simultaneously, the algorithm improves its ability to search globally, speeds up convergence, and effectively avoids getting stuck in local optimal solutions by using the adaptive weight strategy, integrating the concept of random centroid into the diversity mutation mechanism, and implementing the step \(\beta\) threshold strategy. These comprehensive enhancements allow the algorithm to efficiently locate the global optimal solution.

Utilizing this improved metaheuristic optimization algorithm, a threshold voting-based feature selection framework for IoT traffic data is designed to reduce space and temporal complexity, computational training time, and eliminate redundant and unnecessary features. Based on the final selected feature data and LightGBM(an Ensemble Learning framework), a lightweight and efficient IoT attack detection scheme is developed to achieve rapid and accurate detection of IoT attack threats. This scheme significantly enhances the accuracy and effectiveness of IoT threat detection by reducing the complexity of the data and extracting crucial characteristics. It employs ensemble learning to overcome the limitations of a single classifier and minimizes the false alarm rate. This approach considers the differences between several weak classifiers, consequently enhancing the accuracy and generalization ability of IoT attack detection and overall detection performance.

The main contributions of this work are summarized as follows:

An improved Salp Swarm Algorithm has been proposed, which increases population diversity and improves population quality through the good point set strategy and quantum encoding for population initialization; And by using adaptive weights strategy to accelerate convergence speed, and by integrating the diversity variation of the centroid idea and the step \(\beta\) threshold strategy, it avoids getting stuck in local optimal solutions and achieves global optimization.

Based on the improved Salp Swarm Algorithm, a threshold voting-based feature selection framework for IoT traffic data has been designed to remove redundant and irrelevant features and reduce computational training time, space and time complexity. Combined with the LightGBM ensemble learning model, a lightweight and efficient IoT attack detection scheme prototype has been developed to achieve rapid and accurate detection of IoT attacks and threats.

The algorithm and scheme devised for this study have been subjected to rigorous testing and evaluation utilizing the NSLKDD and CICIoT2023 datasets.Specifically, through analysis of statistical metrics such as variance, standard deviation of fitness values, and visualizations including box plots, KDE plots, and population charts, it is visually evident that the improved metaheuristic algorithm demonstrates outstanding performance in feature selection. Furthermore, through Wilcoxon rank sum test analysis, the algorithm shows statistical significance compared to other methods. And across multiple evaluators, this algorithm outperforms recent academic metaheuristic approaches in terms of both accuracy and feature selection rates.

The proposed overall scheme demonstrates outstanding performance in classification tasks across two datasets, achieving exceptional metrics such as accuracy and F1-Score exceeding 99%, surpassing other popular ensemble learning schemes. Further SHAP analysis of selected features elucidates their impact on model outputs, confirming the effectiveness of the proposed approach and its potential for practical application.

Section " Literature review " provides an overview of the findings from past studies. Section " Background " provides pertinent background information and methodologies. Section " GQBWSSA: an improved salp swarm algorithm " provides a detailed explanation of the proposed improved algorithm, including its design, theoretical foundation, and expected innovative points. In Section " Ensemble learning scheme for large-scale IoT attack detection with GQBWSSA feature selection ", the working principle and detailed process of an IoT attack detection scheme based on the proposed algorithm described in Section " GQBWSSA: an improved salp swarm algorithm " will be introduced. Section " Experimental results " presents the assessment indicators employed in the experiment, outlines the dataset and parameters utilized, and offers an in-depth analysis of the experimental outcomes. Finally, Section " Discussion and conclusion " gives discussions and conclusions.

Literature review

The Internet of Things (IoT) is a significant component of the contemporary digital world. Due to the vast size of the Internet of Things (IoT) compared to traditional networks and the limited computing and storage capabilities of most IoT devices, the task of managing and maintaining IoT security becomes significantly more difficult. Recently, an increasing number of researchers have utilized machine learning techniques for network attack intrusion detection. Due to the complex and noisy characteristics of network traffic data, optimization algorithms have gradually become an optimization direction to improve detection efficiency and accuracy.

In recent years, a variety of metaheuristic-based intrusion detection schemes have garnered extensive research attention and application. Vanitha et al. 16 improved the Ant Colony Optimization (ACO) 17 algorithm and used it to accomplish feature selection to enhance the performance of different ML models such as Distance Decision Tree (DDT) 18 , Adaptive Neuro-Fuzzy Inference System (ANFIS) 19 and Mahalanobis Distance Support Vector Machine (MDSVM) 20 . By integrating these models with IACO feature selection method, they proposed a Machine Learning based Ensemble Intrusion Detection (MLEID) strategy that effectively identifies malicious events. Geetha et al. 21 have integrated the Chaotic Vortex Search algorithm (CVS) with the Fast Learning Network (FLN), proposing a novel solution named CLS-FLN that is based on metaheuristic feature selection and neural network classification models, which has achieved state-of-the-art (SOTA) performance across all compared models and datasets. However, it does not take into account the issues of detection efficiency and resource consumption, which may hinder its deployment on resource-constrained Internet of Things (IoT) devices. In order to reduce the amount of temporal complexity, Nazir and colleagues developed a cyber-attack detection model that they called TS-RF 22 . The model’s objective is to reduce the number of feature dimensions. On the UNSW-NB-15 dataset, the model’s accuracy was 83.12% and its false positive rate was 3.7%. Unfortunately, F1-Score, PR, and DR, three more crucial evaluation measures, were overlooked. Stankovic et al. introduced an innovative method utilizing a Hybrid Artificial Bee Colony (HABC) algorithm for feature selection, aimed at enhancing the performance of data classification in intrusion detection systems, particularly when dealing with the vast amounts of data generated in the Internet of Things (IoT) environment. This algorithm effectively improves the efficiency and accuracy of the Extreme Learning Machine (ELM) classifier by eliminating irrelevant features. Although its convergence speed is comparable to existing baseline methods, it demonstrates a clear advantage in terms of convergence quality. However, it is important to note that the authors have only validated this method on a single data model and have not yet fully demonstrated its generalizability and applicability across various scenarios. Bhattacharya et al. 23 proposed the enhancement of Principal Component Analysis (PCA) through the Firefly Algorithm (FA), improving dimensionality reduction performance on network intrusion detection data, and combined with XGBoost classifier, achieving efficient and accurate detection effect. However, only Accuracy is used as the classification metric in the experimental evaluation, and only one dataset is used for comparison, which can not sufficiently demonstrate the generalizability of the proposed method.

Khafaga et al. 24 combine`d the Whale Optimization Algorithm (WOA) 25 and DTO in order to enhance the performance of voting classifiers in NIDS. The authors constructed a new IoT dataset and demonstrated the superiority of their approach through experiments. Ethala et al. 26 proposed to use Spider Monkey Optimization (SMO) 27 and Hierarchical Particle Swarm Optimization (HPSO) 28 to address the problem of intrusion data classification. Thus improving the accuracy of network attack detection in the IoT environment. The study uses a random forest classifier to classify attacks and uses UNSW-NB15 and NSLKDD datasets to evaluate the proposed method for finding optimal solutions, showing that it outperforms other methods reviewed in their study. However, it requires longer processing time on large-scale datasets, which may limit its application in real-time network attack detection systems that require rapid response. Savanović et al. 29 had introduced a novel intrusion detection scheme for the Internet of Medical Things (IoMT) by integrating an improved Firefly Algorithm (FA) with the XGBoost classifier. However, the approach places a strong emphasis on interpretability and does not sufficiently consider the issue of detection efficiency. Aastha et al. 30 proposed a new optimized weighted voting ensemble model to detect DDoS attacks in SDN environments. The proposed method adopts a new hybrid meta heuristic priority algorithm to determine the optimal weight set, and designs a new dynamic fitness function to eliminate false negatives.However, this scheme does not take into account the multi class classification scenario. Geetha et al. 21 have integrated the Chaotic Vortex Search algorithm (CVS) with the Fast Learning Network (FLN), proposing a novel solution named CLS-FLN that is based on metaheuristic feature selection and neural network classification models, which has achieved state-of-the-art (SOTA) performance across all compared models and datasets. However, it does not take into account the issues of detection efficiency and resource consumption, which may hinder its deployment on resource-constrained Internet of Things (IoT) devices. Almasoud et al. 31 proposed a comprehensive framework for network intrusion detection, which used Enhanced Cat Swarm Optimization (ECSO) algorithm for feature selection, Twin Support Vector Machine (TSVM) for classification, and Mayfly Optimization (MFO) for parameter selection of TSVM. The optimal results are achieved on UNSW-NB15. However, it is difficult to prove whether the framework can solve the complex parameter and kernel function selection problem of TSVM with only one dataset comparison result, and the scalability and generalization problem of TSVM also need more research when facing large-scale datasets.

The investigative findings robustly substantiate the superior efficacy of metaheuristic algorithms in processing data pertaining to cyber-attacks. By combining machine learning and deep learning models, metaheuristic algorithms have not only demonstrated acumen in selecting the most salient feature combinations but also in markedly enhancing the performance of network intrusion detection models, which significantly bolster the accuracy and efficiency of IoT attack detection. Metaheuristic algorithms thus demonstrate immense potential and practical value in the field of network security.

An explanation of the rationale and context for the research is sought in this part. The article starts with a presentation of the original Salp Swarm Algorithm and continues with an in-depth analysis of the feature selection method and LightGBM algorithm used. The following sections will be easier to understand after this.

Salp swarm algorithm

Salp Swarm Algorithm(SSA) 13 , as a biological heuristic optimization algorithm, has demonstrated its strong potential and wide application value in multiple fields in recent years. According to Laith Abualigah et al.’s comprehensive research 14 , SSA has been successfully applied in multiple fields such as machine learning, engineering design, wireless networks, image processing, energy and power. It has demonstrated high efficiency in solving various optimization problems, including but not limited to feature selection, parameter tuning, and pattern recognition. For example, Jovanovic et al. 32 used an improved SSA algorithm to optimize the hyperparameters of the LSTM model for predicting multi-step changes in crude oil prices, improving the accuracy and robustness of the prediction model. In addition, Budimirovic et al. 33 proposed a hybrid model combining Enhanced Whale Optimization Algorithm (EWOA) and SSA for predicting the severity of lung infections in COVID-19 patients, demonstrating the potential and practicality of SSA in medical image analysis and disease diagnosis. The above research examples not only demonstrate the practicality and wide applicability of the SSA algorithm, but also reveal its potential for improvement in specific application scenarios.

Given the successful applications of SSA in many fields, in-depth understanding of its algorithm details is crucial to tap its potential. The following is a description of the SSA in detail.

In SSA, each salp in the swarm is confined to a specific range, and the positional data of each salp might be viewed as a potential solution to the problem. Let’s consider a salp swarm consisting of N individuals. The problem dealing with has D variables, which represents the search space dimension. In this case, the position of each salp can be stored in a matrix of \(N \times D\) dimensions, as represented by Eq. ( 1 ).

The target vector, denoted as F, is defined as the vector in the search space with the minimum fitness function value. The leader’s position is updated based on the target position and incorporates a certain level of randomness to effectively guide the entire environment search. This update is aimed at attracting the followers towards the target. The update of leader’s position is shown in Eq. ( 2 ).

Among them, \(X_{1,j}\) represents the j -dimensional position of the leader, \(F_{j}\) represents the j -dimensional position of the target position, \(ub_{j}\) and \(lb_{j}\) are the upper and lower limits of the search dimension j respectively, \(c_{1}\) , \(c_{2}\) , \(c_{3}\) are control parameters, \(c_{2}\) and \(c_{3}\) are random numbers between [0,1], determines the movement direction (forward or reverse) and movement step size of the j -dimensional update position (determines the direction and step size of the leader’s position update). \(c_{1}\) is the convergence factor, which is used to balance the local development and global exploration capabilities of salp individual during the iterative process of the method. Equation ( 2 ) shows that the leader’s position update is mainly determined by the target position. The convergence factor \(c_{1}\) is defined as:

Among these, t represents the present iteration count and \(T_{max}\) denotes the maximum iteration count.

Followers are updated based on the leader and are not moved randomly. The calculation for the follower’s position \(X_{i,j}\) is based on its speed, acceleration, and moving position, as stated in Newton’s law of motion:

During the iteration process, t represents the iteration time; a represents the acceleration, and \(a=(v_{final}-v_{0})/{t}\) , \(v_{0}\) represents the initial speed, and the followers’ speed are zero at the beginning of each iteration. Since the update of the follower’s position is only related to the position of its previous salp, its speed \(v_{final}=(X_{i-1,j}-X_{i,j})/t\) , so the position update formula can be further expressed as:

Among them, \(i\ge {2}\) , \(X_{i,j}\) reflect the position of the follower in j -dimensions. According to the population individual update method described above, the Salp Swarm Algorithm can be obtained.

• Feature selection

Due to the rapid increase in data volume, datasets that consist of a substantial number of features and samples are frequently encountered. However, it is important to acknowledge that not all of these properties have a substantial influence on the model’s prediction results, and some elements may be unneeded or redundant. The presence of these extraneous characteristics not only escalates the computational expenditure of model training, but can also diminish the predictive efficacy of the model. The emergence of feature selection technology was a response to this problem. It effectively reduces the dimensionality of the data, minimizes the utilization of computational resources, and improves the learning efficiency of the model by identifying and preserving the most valuable features for model prediction while deleting redundant information. Presently, the fields of data mining and machine learning currently make heavy use of feature selection as a preprocessing method. In addition to enhancing the model’s performance, good features shed light on the data’s properties and underlying structure, and are crucial for developing the algorithm and model further. One of the primary goals of feature selection is to decrease the dimensionality of the model by minimizing the number of features; this improves the model’s generalizability and decreases the likelihood of overfitting. The second one is to clarify eigenvalues and features more thoroughly 34 .

There are three main techniques to feature selection in the classical sense 35 : the filter, wrapper, and embedded methods. With the knowledge of the subsequent learning algorithm, the Wrapper approach 36 is often more effective in selecting features compared to other methods. Nonetheless, for high-dimensional datasets, the computational expense is considerably higher.

As illustrated in Fig.  1 , to assess the caliber of feature subsets, the wrapper feature selection approach makes use of the performance of a machine learning algorithm. Consequently, in the wrapper approach, the evaluation of a feature subset requires using an evaluative classifier to analyze the performance of this subset. The wrapper methodology employs a wide range of learning algorithms to evaluate features, including decision trees, the k-nearest neighbors (KNN) algorithm, Bayesian classifiers, neural networks, and support vector machines (SVM), among others. In contrast to the Filter approach, the Wrapper methodology often succeeds in discovering feature subsets that result in enhanced classification efficacy. Nonetheless, it is not without its shortcomings. For example, the necessity for the training and evaluation of an evaluative classifier with each assessment of a feature subset engenders a significant computational complexity within the algorithm. This is especially pronounced when managing extensive datasets, where the duration of execution can become prohibitively lengthy. Consequently, in confronting an expansive search space, the delineation of metaheuristic strategies becomes imperative 37 .

The process of the wrapper feature selection method.

The Gradient Boosting Decision Tree 38 (GBDT) is an iterative ensemble model that combines multiple weak base learners to generate the intended classification result. GBDT learns decision trees by fitting the negative gradient (also known as residuals) in each iteration. The GBDT model f ( x ) can be represented as the sum of decision trees:

where M is the number of trees in the model, \(\gamma _m\) is the learning rate, \(D(x;\theta _m)\) is the m -th decision tree, and \(\theta _m\) is the parameter of the tree. The m -th tree is trained to predict by minimizing the loss function \({\mathscr {L}}\) with respect to the tree parameters \(\theta _m\) .

Among them, \(y_i\) is the target variable, \(f_{m-1}(x_i)\) is the prediction of the previous tree, and N is the number of training samples. Optimization is typically accomplished using gradient descent, where the gradient of the loss function is calculated relative to the parameters of the tree.

Traditional Gradient Boosting Decision Trees (GBDT) are prone to overfitting issues, and their computational complexity is mostly attributed to the construction of decision trees. It involves examining all data instances in order to identify a feature that, when used as a split point, maximizes the amount of information gained. This process is highly time-consuming. XGBoost 39 addresses the issue of over-fitting in GBDT by incorporating regularization terms into the goal function as well as CatBoost 40 addresses the common issue of overfitting in Gradient Boosted Decision Trees (GBDT) through its distinctive Ordered Target Statistic encoding technique, which, when dealing with categorical features, prevents target leakage, thereby mitigating the tendency of the model to overfit. Despite their effectiveness in addressing overfitting, when confronted with massive datasets, both XGBoost and CatBoost can be computationally intensive, leading to prolonged training times 41 . Moreover, during the training phase, CatBoost might require a higher amount of memory resources compared to some other gradient boosting frameworks, particularly when dealing with large-scale data. To solve this problem, LightGBM 42 , as an efficient and scalable implementation of GBDT, significantly improves the training speed and maintains high accuracy by adopting a strategy called “leaf-by-leaf growth”. This strategy avoids the computational complexity of XGBoost in the tree building process by only splitting the nodes with the greatest gain in each layer, allowing the model to grow asymmetric and deeper decision trees.Compared to traditional GBDT, LightGBM exhibits more advantages in processing large-scale datasets. It adopts a decision tree algorithm based on histograms, which effectively reduces the memory consumption and computational complexity of the model by discretizing continuous eigenvalues. In addition, through gradient unilateral sampling technology, LightGBM focuses on samples with larger gradients, reducing unnecessary calculations. The introduction of parallel processing techniques, including feature parallelism, data parallelism, and voting parallelism, significantly improves the processing speed of large-scale datasets.Meanwhile, LightGBM’s cache friendly design improves memory access efficiency. In addition, LightGBM’s direct support for category features reduces the dimension explosion problem caused by one hot encoding, resulting in faster detection speed, less memory usage, and higher accuracy in training LightGBM on large-scale datasets.

GQBWSSA: an improved salp swarm algorithm

This section provides a detailed introduction to the Improved Salp Swarm Algorithm, which is an enhanced metaheuristic algorithm called GQBWSSA.

Quantum encoding-based population initialization using good point set strategy

In the original SSA algorithm, the random generation method is used when initializing the population, which is prone to uneven population distribution, resulting in reduced population diversity and low population quality, which in turn affects the convergence speed of the algorithm. In this phase, inspired by the paper by Chen et al. 43 , the good point set strategy is utilized for population initialization, and the diversity of the population is enhanced through quantum encoding.

The good point set 44 is an effective uniform sampling technique, first proposed by Mr. Hua Luogeng, and has been widely applied in various swarm intelligence optimization algorithms. According to the definition of the good point set, let \(G_D\) be the unit cube in D -dimensional Euclidean space. If \(r\in G_D\) , then:

And its deviation satisfies:

Then \(P_n(k)\) is called the good point set.Among them, r is the good point, and \(C(r,\varepsilon )\) is a constant that is only related to r and \(\varepsilon (\varepsilon >0)\) . Taking \(r=\{2cos(2\pi k/p)\},1\le k\le d\) , p is the minimum prime number that satisfies \((p-3)/2\geqslant D\) .The good point set strategy for the population individuals is initialized as:

The paper 45 has demonstrated that the weighted sum of n good points produces less errors compared to any other n points, which is especially well-suited for approximation calculations in high-dimensional domains. Figure  2 illustrates the comparison between the initialization of the good point set and the initialization of random in a two-dimensional unit search space, assuming a population size of 100. It is easy to observe that the random initialization method is very clustered in some areas and very sparse in others. In contrast, the best point set initialization is more evenly distributed. This allows the algorithm to have better traversal, which is conducive to the algorithm achieving the best global optimization effect.

Comparison example of the good point set initialization.

Quantum computing (QC) was first proposed by Richard Feynman 46 in 1985. The difference between classical computation and quantum computation is that quantum bits, or qubits, can be in a superposition state, which is a mixture of two polarization states, allowing them to hold more information. According to the characteristics of quantum computing, in this paper, the concept of quantum encoding is introduced to enhance the diversity of the population. During the initialization phase, each salp individual is encoded as probability amplitudes of qubits, allowing a single salp to represent a superposition of multiple states. The convergence of the method can be improved by the use of qubit coding, which not only allows for an increase in the variety of the starting population but also allows for further improvement.

A qubit, which stands for quantum bit, is theoretically described as the superposition of two states, denoted as:

Let \(\alpha\) and \(\beta\) represent complex vectors, which are referred to as probability amplitudes of quantum states. \(\varphi\) represents the superimposed quantum state, where the quantum state \(|\varphi \rangle\) collapses to \(|0\rangle\) with a probability of \(|\alpha |^2\) , or to \(|1\rangle\) with a probability of \(|\beta |^2\) , as a result of measurement. Additionally, this superimposed state satisfies the normalizing requirement:

Encoding and representing quantum bits can be accomplished by the utilization of matrices of sine and cosine functions, which correspond to the probability amplitudes, as will be detailed in the following:

Among them, \(cos\theta = \alpha\) , \(sin\theta = \beta\) Using probability amplitude to represent different positions of individuals, the individual \(q=X_{i,D}\) in the population can be expressed as:

Among them, D refers to the length of a qubit, and \(\theta _j(1\le j\le D)\) is randomly generated in the range of \([0, 2\pi ]\) . The search space is divided into two locations, and each of those positions indicates an optimal solution. Each individual occupies two positions, which are called Y solution and Z solution respectively, that is, \(q=(\begin{matrix}q_{y}\\ q_{z}\end{matrix})\) , expressed in the following way:

Linear transformation can be used to establish a mapping relationship between quantum space and solution space. If there is a quantum bit \([\textrm{cos}\theta ,\textrm{sin}\theta ]^{T}\) with a value range of \([-1,1]\) , corresponding solutions \(q_y\) and \(q_z\) with a value range of [ lb ,  ub ], then the following transformation exists:

In this algorithm, \(q_y\) and \(q_z\) represent the calculated good point set. They correspond to the collapse of the relevant quantum state into a definite state once a single observation has been made. Under the assumption that the present state is a sinusoidal position, the inverse transformation can be expressed as follows in order to restore the superposition state:

Subsequently, the location corresponding to an alternative state can be computed by utilizing the acquired rotational angle \(\theta\) . The initial qubit is composed of two state bits, which correspond to these two places.

The quantum encoding transformation is carried out on the initial population that is received via the initialization of the good point set. This particular transformation is carried out through a two-stage initialization technique. Through the optimization of the initialization phase, our goal is to improve the global search capabilities of the initial population in the search space and maximize the distribution quality of the initial population across the search space.

Adaptive weights

Adaptive weights are added to the original SSA algorithm in order to increase its performance in a variety of problems and stages. This is done in response to the sluggish convergence characteristic that the original algorithm exhibits. The concept of adaptive weights refers to assigning dynamically adjusted weights to different parameters or operations during the execution process of an algorithm, making the algorithm more flexible in adapting to the characteristics of the problem, thereby improving the optimization effect.

Adaptive weights help prevent algorithms from prematurely converging to local optima by modifying the weights. This allows the algorithm to more effectively escape from local optima and search for global optima. By using adaptive weights, the method’s convergence rate can be increased, allowing it to quickly approach solutions that are very close to the optimal solution. This, in turn, improves the overall efficiency of the process.

Among them, \(w_{max}=0.9\) , \(w_{min}=0.4\) , t is the current number of iterations, and T is the maximum number of iterations.

Diversity mutation and step \(\beta\) threshold strategy integrating the concept of random centroid

As a solution to the problem of the original algorithm being prone to becoming stuck in local optimal solutions, this paper enhances the SSA algorithm by introducing a diversity mutation strategy and step \(\beta\) threshold strategy that incorporates the concept of random centroid, which allows for the achievement of global optimization.

“Random centroid improvement” 47 usually refers to introducing a certain degree of randomness into the optimization algorithm, so that the algorithm can explore the search space more flexibly and improve the effect of global search. However, this improvement will also cause the original algorithm to obtain many meaningless solutions. Therefore, a random global optimal value is added for perturbation, allowing it to conduct an accurate search near the global optimal value to improve precession.

In addition, the original SSA may fall into over-exploration or over-exploitation, and the introduction of step size helps prevent falling into meaningless solutions and thus falling into local optima later in the iteration. The threshold strategy is mainly used to balance the balance between the original algorithm and mutation. It uses random numbers without using the threshold of the number of fusion iterations to prevent it from falling into local optimality in the later stage.

Among them, \(\beta _{max}=0.9\) , \(\beta _{min}=0.4\) , mean is the random centroid of the population, \(mean=\frac{1}{n}\sum _{j=1}^{n}X_{i,j}\) and rand is a random number.

In summary, the improved SSA ,i.e. GQBWSSA, proposed in this paper is depicted in the flowchart as shown in Algorithm 1. The algorithm achieves a more rational initial distribution of the population through the use of an good point set and quantum encoding. The utilization of adaptive weights is what allows for the acceleration of the convergence process that occurs during the optimization process. Additionally, the integration of a diversity mutation strategy based on the concept of random centroid and a step \(\beta\) threshold strategy enhances the algorithm’s global optimization capability. These enhancements not only make it possible for the GQBWSSA method to locate the optimal solution in a shorter amount of time, but they also considerably raise the chance of locating the global optimal solution, which is extremely useful for resolving complex feature selection issues.

Pseudocode of GQBWSSA

Ensemble learning scheme for large-scale IoT attack detection with GQBWSSA feature selection

As shown in Fig.  3 , in this section, the implementation of feature extraction based on GQBWSSA is described, and an efficient ensemble learning scheme for IoT attack detection is implemented based on the extracted features. It involves the following steps:

Data preprocessing: The DPKT feature extraction tool is used to extract features from the original traffic data. The data is then cleaned, missing values in the data are identified and processed, duplicate records are removed, and the data is standardized.

Feature Selection: Use KNN, SVM, and random forest(RF) as evaluation classifiers, and perform feature selection through GQBWSSA. Integrate the optimal feature subsets of the three evaluators, and obtain the final optimal feature subset through threshold voting.

Attack detection: Train the final selected feature data using the ensemble learning model LightGBM to construct a lightweight large-scale IoT attack detection model for the final attack detection.

Overview of detection scheme.

Data preprocessing

Raw traffic data are collected, and then the DPKT tool is used to extract raw features from these data. A series of pre-processing stages are undertaken on the extracted data to ensure the quality and applicability of the data. These steps include data cleaning, partitioning, standardization, feature deletion, missing value processing, deduplication, non-numeric feature encoding, and other similar procedures. Prior to carrying out operations such as data transformation, normalization, and feature dimensionality reduction, we create a training set and a test set from the data. The training set is utilized for the purpose of training the model, while the test set is utilized for the purpose of evaluating the performance of the model. The use of such segmentation helps to ensure that the category ratio of the test data set is comparable to that of the training data set. This helps to prevent data leakage and, as a result, more accurately simulates the actual situation that exists inside the Internet of Things network.

Threshold voting-based feature selection using GQBWSSA

Transfer function and individual discretization.

Feature selection problems belong to a category that is computationally very difficult, known as NP-hard problems. This problem can be thought of as an optimization work involving binary decisions, in which the number of possible solutions increases exponentially with the number of attributes that are being considered. In this issue, a value of 1 can be used to represent the selection of a feature, and a value of 0 to represent the exclusion of a feature. Since most optimization algorithms are designed to handle continuous variables, adjustments are needed to apply the GQBWSSA to the feature selection problem: On the one hand, it is necessary to make certain that the search range of the solution is in accordance with the binary nature of feature selection. On the other hand, it is necessary to convert the continuous positional information contained within the algorithm into binary form in order to match the binary decision-making process that is involved in feature selection.

In response to the above problems, first of all, assume that the population is initialized as X of N individuals through the good point set and quantum encoding strategy, where \(X_i\) is the individual in X as shown in Eq. (refeq24)

Subsequently, as shown in Fig.  4 and Eq. ( 25 ), the sigmoid function is used as the transfer function to map continuous interval vectors to the interval (0, 1) when initializing and updating individual positions, where \(X_{i, j} \in [lb,ub]\) .

The Transfer Function.

Following this, the transfer function is utilized in order to discretize the problem from a continuous domain [ lb ,  ub ] into a binary problem (either 0 or 1) in order to obtain the desired results. Specifically, when in the individual initialization phase, the individuals are initialized by the Initialized population based on the good point set and quantum encoding strategy combined with the transfer function, as shown in Eq. (refeq25) if the value of the feature position is less than 0.5, the position is set to 0, otherwise, it is set to 1.

Among them, \(X_{i,j}\) is the j -th dimension of the i -th individual, D is the number of features. As shown in Fig.  5 , the feature selection flag sequence is used as an individual in the optimization algorithm. Within them, a flag with a value of 1 indicates that the feature has been chosen, while a flag with a value of 0 indicates that it has not been chosen.

Binary selection sequence representation.

When the next iteration update is carried out, the discretization update process of the \(X_i\) solution is fully explained in Eq. (refeq27), where rand represents the random number, \(l(l \ge 1)\) is the current iteration number, and \(l+1\) is the next iteration number.

The fitness function

When designing optimization algorithms, a crucial consideration is the fitness function. Due to the fact that feature selection is essentially an optimization problem involving multiple objectives, when evaluating a solution, it is necessary to simultaneously consider two aspects: on the one hand, to minimize the number of features that are included in the features subset that has been chosen, and on the other hand, to enhance the accuracy of classification to the greatest extent possible. In the formulation of multi-objective problems, aggregation methods are a popular strategy.

In this technique, objectives are consolidated into a single objective formula, where preset weights determine the significance of each objective. In this paper, a fitness function that integrates two goals of feature selection has been employed. Specifically, the fitness function can be described as the process of locating a feature subset that minimizes the weighted sum of the classification error rate and the rate of selected features, as seen in Eq. ( 28 ).

Within the given context, \(fitness(X_i)\) denotes the fitness value of subset \(X_i\) , \(ACC(X_i)\) represents the classification accuracy of the selected features in subset \(X_i\) , and | SF | and | AF | indicate the number of selected features and the total number of original features, respectively.

\(\alpha\) and \(\beta\) are the weights of classification error rate and feature selection rate respectively, used to balance the relationship between two objectives, where \(\alpha \in (0,1)\) , and \(\beta =(1-\alpha )\) .

Threshold voting

Multiple classifiers, including KNN, SVM, and Random Forest, are utilized in this scheme to evaluate the fitness of the population. This is done in order to guarantee the robustness of the features that have been chosen, to boost the variety of feature selection, and to prevent either overfitting or underfitting. Subsequently, the selected feature markers are summed as scores and a score threshold is set. Only when the score is greater than or equal to the set threshold, this feature is the final selected feature, and its boolean expression is shown in Formula ( 29 ),where \(\tau\) represents the selection threshold.

In summary, the threshold voting-based feature selection framework using GQBWSSA is shown in Fig.  6 and Algorithm 2, with the pseudocode shown below.

The Flowchart of Threshold Voting-based Feature Selection using GQBWSSA.

Threshold-Voting Feature Selection Algorithm: GQBWSSA-FS .

Ensemble detection scheme for IoT attacks using LightGBM

Compared to traditional GBDT, LightGBM introduces gradient unilateral sampling and independent feature merging. Gradient unilateral sampling enables the LightGBM algorithm to not only maintain the evaluation accuracy of information gain, but also have more accurate sampling results and higher learning speed at the same sampling rate. LightGBM uses an optimization strategy based on histograms. During the training process, only histograms are used as features to construct decision trees, which effectively reduces the number of traversals of the data sample set. On the basis of the histogram algorithm, the LightGBM algorithm has been further optimized and improved, using a leaf-wise algorithm with depth restrictions.Every time from all current leaf nodes, find the leaf with the highest splitting gain for classification, and repeat the cycle to prevent overfitting of the model.The objective function of the LightGBM algorithm is shown in Eq. ( 30 ).

Among them, \({\hat{y}}_i\) is the predicted value, \(\theta _{t}\) is the structural parameter of the t -th tree, including the feature index and threshold of the split point; \(f_t(x_i;\theta _t)\) represents the prediction of sample \(x_i\) by the t -th tree; \(\Omega (f_k)\) is the regular term.

Use the logistic loss function to measure the difference between the predicted value \({\hat{y}}_i\) and the target value \(y_i\) , that is, \(l(y_i,{\hat{y}}_i)=y_i\textrm{ln}\Big (1+e^{-{\hat{y}}_i}\Big )+(1-y_i)\textrm{ln}\Big (1+e^{{\hat{y}}_i}\Big )\) . After Taylor expansion, the formula can be rewritten as:

In Eq. ( 31 ), \(g_i=\partial {\hat{y}}^{(t-1)}l(y_i,{\hat{y}}^{(t-1)})\) ; \(h_i=\partial _{{\hat{y}}^{(t-1)}}^2l(y_i,{\hat{y}}^{(t-1)})\) . Transform the iteration of the tree model into the iteration of the tree leaf nodes to obtain the objective function:

Among them, \(I_j\) represents the set of leaf nodes j , \(w_j\) is the score of leaf node j of each tree; T is the number of leaf nodes. \(\gamma\) means the minimum gain threshold for splitting leaf nodes to prevent overfitting, while \(\lambda\) regulates the strength of L2 regularization to mitigate model complexity and overfitting risk. After determining the structure of the tree, the objective function \({\tilde{L}}(\theta )^{(t)}\) can be obtained:

Experimental results

Dataset description.

To verify the effectiveness of the scheme and its ability in detecting IoT attacks, experiments were conducted using two datasets: NSLKDD 48 and CICIoT2023 49 .

As an upgraded version of the KDD Cup 99 dataset 50 , the NSLKDD is a widely used dataset for network intrusion detection. Normal network traffic as well as other forms of intrusion activity are both included in the NSLKDD dataset. A large number ofrecords of network connections are included in this collection, with each record being labeled as either normal or aberrant. Anomaly records are further subdivided into multiple attack types, including DoS, Probe, R2L, and U2R. The NSLKDD dataset has 41 features such as duration, protocol type, etc.

The CICIoT2023 dataset is an innovative and comprehensive collection of assaults on the Internet of Things (IoT). The dataset was constructed by executing 33 different attacks on an IoT network topology consisting of 105 devices. The attacks are divided into seven categories, including DDoS, DoS, Recon, Web, BruteForce, Spoofing and Mirai attacks. For the purpose of CICIoT2023, all of these assaults were carried out by hostile Internet of Things devices that targeted other Internet of Things devices, with a total of 46 features. The specific data distribution can be viewed in Table  1 .

Experimental setup

Experiments were conducted on a system that is equipped with an Intel Core i5-12400F processor, 32GB of RAM, and an NVIDIA GeForce RTX 2080 Ti graphics card featuring 11GB of video memory. Develop the scheme using the development languages Matlab and Python. Table  2 shows the scheme parameter settings and model parameter configurations involved in experiments.

To ensure the objectivity and effectiveness of the experiment, considering the constraints of computational resources and time costs involved in large-scale data computations, five independent repetitions of the experiment are opted for. This approach balances rigor in research with practical feasibility, yielding reliable data through a limited number of repetitions while minimizing resource consumption and time expenditure. Additionally, in machine learning models, result stability and validity are ensured through three-fold cross validation. We utilized a stratified sample strategy to randomly choose 70% of the data for training and validation purposes, and 30% for testing purposes, in the experiments.

Evaluation metrics

The tasks of this article are to detect classification tasks, including binary and multi classification detection tasks. When it comes to assessing the effectiveness of classification tasks, there are a number of different indications that may be used to evaluate the quality of the model. In binary classification detection jobs, Accuracy, precision, recall, and F1 score are some of the evaluation measures that are utilized. In multi classification detection tasks, four evaluation metrics are used: accuracy, weighted average accuracy, weighted average recall, and weighted average F1 Score. The confusion matrix 51 is utilized to derive the values for true positives (TP), true negatives (TN), false positives (FP), and false negatives (FN) in classification jobs, which are the basis for calculating the above evaluation metrics.

Accuracy: represents the proportion of correctly detected and classified samples to the total number of samples in the scheme. The calculation formula is \(\frac{TP+TN}{TP+TN+FP+FN}\) .

Precision: This metric assesses the ratio of correctly identified positive cases to the total number of instances classified and detected as positive by the scheme, thereby reflecting the model’s accuracy in predicting the positive class. The calculation of precision is represented by the formula \(\frac{TP}{TP+FP}\) .

Recall: Recall, also known as sensitivity, measures the proportion of actual positive instances that are correctly predicted as positive. The formula for calculating recall is: \(\frac{TP}{TP+FN}\) .

F1 score: F1 score is the harmonic mean of precision and recall, which strikes a balance between the two. The calculation formula is \(\frac{2Precision\times Recall}{Precision+Recall}\) .

Weighted Average Precision: \(WA\ Precision=\sum _{i=1}^n\frac{N_i}{N} \cdot Precision_i\) . Among them, \(N_i\) represents the number of samples in the i-th category, N represents the total number of samples, and \(Precision_i\) represents the precision of the i-th category.

Weighted Average Recall: \(WA\ Recall=\sum _{i=1}^n\frac{N_i}{N}\cdot Recall_i\) . Among them, \(N_i\) represents the number of samples in the i-th category, N represents the total number of samples, and \(Recall_i\) represents the recall of the i-th category.

Weighted Average F1 Score: \(WA\ F1\ Score=\sum _{i=1}^n\frac{N_i}{N}\cdot F1\ Score_i\) . Among them, \(N_i\) represents the number of samples in the i-th category, N represents the total number of samples, and \(F1\ Score_i\) represents the F1 score of the i-th category.

Furthermore, to assess the significance differences between the proposed algorithm and others, we employed Wilcoxon rank sum test 52 with a significance level set at 0.05. When the calculated p-value of Wilcoxon rank sum test is less than 0.05, it indicates that there is a statistically significant difference between the algorithms under examination; conversely, a p-value above this threshold(0.05) suggests insufficient evidence to conclude significant differences between the algorithms. Finally, to provide a more comprehensive description of the impact of selected features on scheme output, the SHAP 53 (SHapley Additive exPlanations) method is employed for model analysis.

Feature selection experiments

To evaluate the performance and effectiveness of GQBWSSA in feature selection for large-scale IoT attack data, we first compare it with four other popular algorithms using a random forest evaluator. The goal is to minimize the fitness value as a criterion for algorithm evaluation.

Table  3 presents the best, worst, average, standard deviation, and variance of fitness obtained from multiple independent runs of the proposed algorithm and the other four algorithms on various datasets. Additionally, Fig.  7 depicts box plots of the objective function values across the five algorithms on various datasets. According to the statistical analysis results in Table  3 , it is evident that GQBWSSA achieved the best results in terms of the best, worst, and average fitness values. Additionally, it is evident that GQBWSSA demonstrates the best stability by minimum standard deviation and variance metrics on four dataset tasks.

Furthermore, using box plots to visually represent the distribution of fitness, it is observed in Fig.  7 that the results produced by GQBWSSA exhibit a tighter central tendency compared to other methods, with smaller interquartile ranges. Despite the presence of outliers, it generally outperforms other algorithms. This indicates the robustness of the proposed method and its ability to effectively locate the global optimum solution.

The box plots of fitness of each metaheuristic feature selection algorithm on different datasets.

As illustrated by the KDE(kernel density estimation) diagrams in Fig.  8 , GQBWSSA outcomes predominantly cluster at lower Fitness values relative to competing methods. Moreover, the GQBWSSA algorithm exhibits higher density values compared to other algorithms, with its distribution demonstrating better concentration, which indicates that GQBWSSA shows robustness and stability in addressing feature selection issues related to large-scale network attack datasets.

The KDE diagrams of fitness of each metaheuristic feature selection algorithm on different datasets.

The swarm plots in Fig.  9 illustrate the fitness values of populations in the final round iteration. It is evident that the population of GQBWSSA shows greater aggregation than other algorithms, indicating a faster convergence speed within the limited number of iterations.

The swarm plots depict the fitness of the final round iteration of each metaheuristic feature selection algorithm across different datasets.

Moreover, within a non-parametric statistical testing, employing the Wilcoxon Rank Sum Test, the statistically significant differences between GQBWSSA and other algorithms are investigated. As delineated in Table  4 , the p-values derived from applying the Wilcoxon rank sum test to compare GQBWSSA against four competing algorithms across distinct tasks reveal that, in the majority of instances, GQBWSSA achieves p-values less than the threshold of 0.05. This indicates that, except on the CICIoT2023(Multi class) dataset where its performance relative to DBO is not as markedly different, GQBWSSA generally outperforms its counterparts at a statistically significant level. In conclusion, in most cases, GQBWSSA shows a significant level and is statistically significant compared to other algorithms.

Continuing, to test the algorithm’s generalizability, the performance of GQBWSSA on different evaluators is evaluated. Compared with other meta heuristic optimization feature selection schemes proposed in recent years, the feature selection results on multiple evaluators are shown in Tables  5 , 6 , 7 , 8 .

As shown in Table  5 , 6 , 7 , 8 , the proposed GQBWSSA performs better than other algorithms on three evaluators. In Table  5 , we conducted feature selection experiments comparing the proposed algorithm GQBWSSA with other optimization algorithms published in recent years on the binary NSLKDD dataset. After five independent experiments to calculate the average index, we found that the average accuracy of this scheme on all three evaluators is better than other algorithms. It is particularly noteworthy that in the Random Forest (RF) evaluator, the average accuracy of this scheme reaches 99.47%, which is 0.09% higher than the second ranked GWO, and the average fitness is 13.75% lower than GWO, indicating that this algorithm is more accurate in feature selection of binary classification task.

In Table  6 , compared with other optimization algorithms published in recent years, GQBWSSA conducts feature selection experiments on the five class NSLKDD dataset. And it is found that GQBWSSA ’s average accuracy on three evaluators was better than other algorithms. Similarly, in RF evaluator, though GQBWSSA’s average feature selection rate is second only to WOA’s 0.34, which is 0.05 higher than WOA, its average accuracy reaches 99.41% ,which is better than other four algorithms. And the average fitness of the proposed algorithm is 0.00621, which is about 19.4% reduction compared with the second ranked WOA algorithm. This indicates that the algorithm proposed in this paper is more stable in solving feature selection problems.

Overall, in the NSLKDD dataset, compared with the algorithms proposed in recent years, GQBWSSA exhibits better universality. Specifically, the average fitness and best fitness of binary and multi class feature selection always reach the best on three evaluators, indicating that the features selected by GQBWSSA are more representative.

As shown in Table  7 , in the CICIoT2023 dataset, the GQBWSSA algorithm performs better than other algorithms in binary feature selection tasks. Specifically, in the same evaluator, GQBWSSA achieved the best average accuracy. And its average fitness value is always the lowest among three evaluators. In the KNN evaluator, the average accuracy of GQBWSSA reached 99.17%, which is 0.0017 higher than the HHO algorithm. In Table  8 , GQBWSSA also dominates feature selection in the multi classification CICIoT2023 dataset, always achieving the best fitness value. However, it is worth noting that in the multi classification feature selection experiment on the CICIoT2023 dataset, although GQBWSSA outperformed other algorithms in SVM evaluators, its accuracy was only 77%.

In general, GQBWSA achieved the best fitness in the CICIoT2023 dataset, and the high accuracy and low feature selection rate achieved by GQBWSA indicate that the proposed method achieved the best classification results with only the minimum number of features. The filtering features obtained by GQBWSSA are meaningful and can produce the best results with the least number of feature values, which is crucial for reducing model complexity, mitigating dimension explosion, and reducing deployment costs.

Figures  10 , 11 , 12 , 13 depict the iterative processes of various algorithms on various evaluators in different datasets. Compared with multiple optimization algorithms, the proposed GQBWSSA algorithm outperforms other algorithms in terms of fitness values on three evaluators. And It is noted that algorithms such as GWO and WOA exhibit rapid convergence in the initial stages. However, as iterations progress, these algorithms often get trapped in local optima and fail to achieve the global optimum within the specified iteration limits. In contrast, the GQBWSSA algorithm shows less pronounced early-stage convergence but consistently reaches the optimal fitness value within a limited number of iterations, which reveals the outstanding global search capability of GQBWSSA, effectively avoiding the risk of premature convergence to local optima.Furthermore, by examining the swarm plots in Fig.  9 , it can be observed that in the final iteration, the fitness values of populations in GQBWSSA exhibit a significantly centralized trend compared to other algorithms, where fitness values appear more scattered. This indirectly proves that other algorithms require more iteration cycles to reach a global optimum state, thereby confirming the superior convergence speed of GQBWSSA.

In summary, the GQBWSSA algorithm not only demonstrates excellent global exploration capabilities but also excels in convergence speed. Moreover, it performs exceptionally well across three evaluators, showcasing its versatility and generalization capability.

Iterative curves of fitness for binary classification feature selection experiments on NSLKDD dataset.

Iterative curves of fitness for five classification feature selection experiments on NSLKDD dataset.

Iterative curves of fitness for binary classification feature selection experiments on CICIoT2023 dataset.

Iterative curves of fitness for eight classification feature selection experiments on CICIoT2023 dataset.

Comprehensive experiments on overall scheme

Compared with other ensemble learning methods, the experimental results are shown in Tables  9 , 10 , where the “time” indicator refers to the average training and detection time. When the feature selection threshold is set to 2, the GQBWSSA-FS-LightGBM scheme shows advantages in multiple performance metrics, including accuracy and precision. In the binary classification detection task, this scheme achieves an accuracy of 99.78% on the NSL-KDD dataset, which is 0.00118 higher than GQBWSSA-FS-CatBoost; It also performs well on the CICIoT2023 dataset, with an accuracy of 99.696%, which is 0.0012 higher than the second ranked scheme GQBWSSA-FS-XGBoost.In the detection of various types of attacks, this scheme performs excellently in multiple key indicators such as accuracy, weighted average precision, and weighted average recall, especially in terms of training and detection time, achieving optimal results. In addition, on the NSLKDD dataset, using GQBWSSA-FS-LightGBM for binary and multi-class classification shows minimal difference in processing time, with the latter even slightly advantageous over the former. This is due to LightGBM’s early stopping strategy, the small data size, and effective feature selection that distinguishes multiple categories well. Multi-class tasks achieve early stopping around the 400th iteration, whereas binary tasks require approximately 1300 iterations to stop. With the small data size facilitating faster processing, the time difference between binary and multi-class tasks is minimal. In fact, multi-class tasks demonstrate a slight advantage in average processing time, a conclusion validated through multiple independent experiments.

It is worth noting that although the GQBWSSA-FS-LightGBM scheme outperforms other schemes in multiple metrics in binary detection experiments on the NSL-KDD dataset, its training and detection time is longer.Compared with the other two schemes, when dealing with large-scale datasets CICIoT2023, The GQBWSSA-FS-LightGBM scheme not only has a short training and detection time, but also achieves optimal performance in multiple performance metrics.These results indicate that, the GQBWSSA-FS-LightGBM scheme is more suitable for large-scale datasets and is more in line with the actual needs of the current network environment. According to the analysis of LightGBM, it is evident that through innovative histogram-based construction and efficient leaf-wise splitting strategies, LightGBM significantly reduces memory usage while accelerating processing speed. As the dataset scales up, these optimizations lead to even more pronounced performance improvements, ensuring LightGBM’s outstanding performance and rapid responsiveness in high-dimensional big data environments. This makes it an ideal choice for handling massive volumes of information.

Analyzing the proposed scheme using SHAP and demonstrate the importance of the selected features as shown in Figs.  14 , 15 .

Based on the SHAP value computations, the results reveal that in the NSLKDD dataset’s binary classification task, “src_bytes” and “dst_host_srv_count” emerge as pivotal features, significantly impacting model outputs and demonstrating prominence in multi-class scenarios. Additionally, features such as “dst_bytes,” “count,” and “service” exhibit substantial effects within specific categories. In the CICIoT2023 dataset, key features for binary classification include “flow_duration,” “rst_count,” and “IAT,” which play crucial roles in model outputs. For multi-class tasks, “IAT” emerges as the most significant feature, closely followed by “Max” and “flow_duration.” Conversely, “SSH” and “Drate” show minimal influence, warranting further analysis.

The SHAP analysis underscores how the selected features critically bolster model accuracy and robustness, propelling the advancement of a lightweight, efficient, and precise IoT attack detection system.

The importance of selected features in the NSKDD dataset using SHAP analyzing.

The importance of selected features in the CICIoT2023 dataset using SHAP analyzing.

The training loss curves are shown in Figs.  16 , 17 . As the training progresses, the performance of the model on both the training and validation sets continues to improve, and the training loss and validation loss tend to be consistent. This indicates that the model does not exhibit overfitting and can be well generalized to new data. The smooth variation of the loss curve reflects the stability of the model training process, and also implies effective processing of noise features in the feature selection stage. These results indirectly demonstrate the effectiveness of GQBWSSA.

Training loss curve in binary classification experiments.

Training loss curve in multi classification experiments.

Discussion and conclusion

Time complexity analysis of gqbwssa.

Assuming the population size is \(N\) , the number of features is \(d\) , that is, the dimension of an individual is \(d\) , the maximum number of iterations is \(m\) , and the fitness function is represented by \(f(x)\) . According to the introduction of SSA 13 , the time complexity of the whole process of SSA is \(O(d+f(d))\) .

In the population initialization phase of GQBWSSA, suppose that the time to generate parameters is \(t_1\) , the time to generate the good point set of each dimension is \(t_2\) , the time to generate the quantum encoding of each dimension is \(t_3\) , the time to calculate the fitness is \(f(d)\) , and the time to sort the fitness value is \(t_4\) . Then the time complexity of GQBWSSA population initialization phase is as follows:

In the leader position update phase, the number of leaders is \(N/2\) . According to Eq. ( 20 ), the calculation time of \(c_1\) is \(t_5\) ,The value of \(c_1\) remains invariant throughout the same iteration cycle. The \(c_2\) and \(c_3\) are both random numbers with generation time set to \(t_6\) . And the time to generate the adaptive weight \(w\) is \(t_7\) , the time complexity of this phase is as follows:

In the follower position update phase, the number of followers is the remaining \(N/2\) . According to Eqs. ( 22 ) and ( 23 ), the follower position is updated by integrating the diversity variation of the centroid idea and the step \(\beta\) threshold strategy. Let the time \(t_8\) for \(rand\) generation and the number of populations with \(rand>0.1\) be R and the update time be \(t_9\) . The computation time of centroid mean is \(t_{10}\) , and the computation time of \(\beta\) step is \(t_{11}\) . Therefore, the time complexity of this stage is as follows:

In the boundary processing and food source location update phase, the time of boundary processing of each dimension of each salp individual, the time of calculating the fitness value of each individual, and the comparison and replacement time of the fitness value of the location of the food source are consistent with the basic algorithm. Hence, the time complexity of this phase is:

In summary, the overall time complexity of the GQBWSSA algorithm over \(m\) iterations is:

It follows that, compared to the original SSA algorithm, the time complexity of the proposed GQBWSSA is \(O(d+f(d))\) , which has the same time complexity as the original algorithm, indicating no reduction in execution efficiency.

Despite demonstrating good performance and effectiveness in the feature selection of IoT attack detection datasets, GQBWSSA still has certain limitations. Experimental findings from the CICIoT2023 dataset, focusing on multi-class feature selection tasks, reveal that although the GQBWSSA algorithm proposed in this study shows stability and effectiveness, it did not exhibit clear advantages compared to DBO in non-parametric statistical analyses and needs further analysis and optimization of the algorithm. In the experiment, it is evident that the GQBWSSA algorithm did not demonstrate notable benefits in multi-class feature selection on the CICIoT2023 dataset while utilizing the SVM evaluator,which requires additional investigation and optimization.

This paper proposes an improved salp swarm algorithm suitable for IoT attack feature selection. To begin, the population is initiated by utilizing the good point set method and quantum encoding to optimize the quality of the initial population, and the adaptable weight approach is utilized to quicken the convergence process. By integrating the concept of diversity mutation fused by the concept of random centroid with the step size \(\beta\) threshold strategy, it avoids falling into local optimal solution and accelerates the probability and speed of finding the global optimal solution. Through feature selection based on GQBWSSA, the feature dimensions are reduced, and the most effective features are extracted. The extracted feature data are combined with LightGBM to form a lightweight and efficient IoT attack detection ensemble learning scheme. In the feature selection experiments, GQBWSSA is analyzed using multiple statistical metrics, box plots, and non-parametric statistical analyses. The experimental results demonstrate that the proposed GQBWSSA outperforms other algorithms across several metrics, exhibiting superior optimization capability and stability. Moreover, it achieves better accuracy, feature selection rates, and fitness values across multiple evaluators compared to other recent metaheuristic optimization algorithms, indicating strong generalization ability and applicability. In terms of overall scheme, the proposed scheme outperforms current popular ensemble learning solutions in key indicators such as accuracy, precision, and training detection time on large-scale datasets, indicating that this solution can effectively detect large-scale Internet of Things attacks, improve network security, and achieve 99.78% and 99.73% accuracy on NSLKDD binary and multi classification tasks. It also achieves 99.70% binary and 99.41% multi classification accuracy on the large-scale dataset CICIoT2023., indicating that this solution can effectively detect large-scale IoT attacks and improve network security.Finally, the effectiveness of the selected features is verified by quantifying the impact of each feature on the output of the detection scheme through a SHAP analysis conducted on the scheme. Furthermore, an analysis and discussion have been conducted on the time complexity and potential limitations of the GQBWSSA algorithm. In subsequent research, a more detailed analysis and investigation of the current shortcomings will be conducted, and a more comprehensive SHAP interpretability analysis will be performed on the scheme. In general, by improving the SSA algorithm, a feature selection framework suitable for IoT attacks is implemented, and combined with LightGBM, it realizes lightweight and efficient detection of large-scale IoT attacks, which is of significant importance for reducing the resources and computational costs necessary for the deployment of the scheme.

Data availability

The datasets supporting the study’s findings are publicly accessible through the links: https://www.unb.ca/cic/datasets/nsl.html and https://www.kaggle.com/datasets/subhajournal/iotintrusion/data.

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This work is in part supported by the National Key R &D Program of China (No. 2022YFB3104100) and the National Science Foundation of China (No. 62102109).

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Metaheuristics for Solving Global and Engineering Optimization Problems: Review, Applications, Open Issues and Challenges

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• Essam H. Houssein   ORCID: orcid.org/0000-0002-8127-7233 1 ,
• Mahmoud Khalaf Saeed 1 ,
• Gang Hu 2 , 3 &
• Mustafa M. Al-Sayed 1  

The greatest and fastest advances in the computing world today require researchers to develop new problem-solving techniques capable of providing an optimal global solution considering a set of aspects and restrictions. Due to the superiority of the metaheuristic Algorithms (MAs) in solving different classes of problems and providing promising results, MAs need to be studied. Numerous studies of MAs algorithms in different fields exist, but in this study, a comprehensive review of MAs, its nature, types, applications, and open issues are introduced in detail. Specifically, we introduce the metaheuristics' advantages over other techniques. To obtain an entire view about MAs, different classifications based on different aspects (i.e., inspiration source, number of search agents, the updating mechanisms followed by search agents in updating their positions, and the number of primary parameters of the algorithms) are presented in detail, along with the optimization problems including both structure and different types. The application area occupies a lot of research, so in this study, the most widely used applications of MAs are presented. Finally, a great effort of this research is directed to discuss the different open issues and challenges of MAs, which help upcoming researchers to know the future directions of this active field. Overall, this study helps existing researchers understand the basic information of the metaheuristic field in addition to directing newcomers to the active areas and problems that need to be addressed in the future.

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1 Introduction

As the world moves towards competition in all fields, people need to best use the limited resources to score a better result and thus achieve a better place in the competition. In this context, optimization is strongly needed. Optimization is a process of picking up the optimal values of the optimization problem's parameters from a given set of values to achieve the desired output, which specifically means output minimization or maximization. In other words, we need to obtain the best optimal solution under a set of limitations and constraints by tuning the parameters of the problem to be addressed. As mentioned in [ 1 ], the optimization process includes a set of steps which starts with formulating the problem to be in the form of an optimization problem, constructing the objectives (cost or fitness) function, determining the decision variables and the restrictions on these variables, then simplifying the reality of the problem by generating the mathematical model that represents the problem. Finally, the problem solver seeks to generate the most acceptable solution by maximizing or minimizing the value of the objective function.

Stochastic optimization algorithms are the most promising type under the umbrella of optimization, which can be classified as heuristic Algorithms (HAs) and metaheuristic Algorithms (MAs). In simple words, stochastic optimization is the general class of algorithms that depends on the random nature in the process of getting the optimal or near-optimal solution. HAs are iterative algorithms, iterating many times seeking a better solution than the solution obtained previously. HAs are used to find a feasible and reasonable solution but may not be the optimal one. In addition, HAs do not provide any evidence of the optimality of the solution obtained. A set of issues can be found in HAs, such as being problem-dependent algorithms specifically designed for a particular problem [ 2 ]. Another challenge in HAs is immeasurable success as there is no information about how close the obtained solution is to the optimal. Finally, there is a dilemma in measuring the computational time. Disclosing these gaffes and achieving a trade-off between solution quality and computational time is a main purpose of the appearance of metaheuristics [ 3 ]. As it is used to solve different types of problems, metaheuristics are the most preferable type of these algorithms. Metaheuristics were introduced for the first time by Glover in [ 4 ].

1.1 Understanding Optimization: What and Why?

In this section, we will help researchers understand the fascinating world of optimization. First, we will present examples of what optimization is, then we will try to answer the question, "Why do we study optimization?" Finally, this section will clarify how to optimize anything you have.

What is Optimization? In simple words, optimization is the art of perfectionism—how to perfectly make something in the best way. Optimization answers the question: how to obtain the best solution for a problem while applying a set of limitations? Maximizing profit, minimizing mass, pollution minimization, noise reduction, and drag reduction are all practical examples that can be achieved by using optimization. In most cases, optimization helps in the design process as a replacement for the traditional approach, which depends mainly on trials or humans. To clarify the simplicity and practical power of optimization in the design process, a diagrammatic view of how optimization methods help in the design process is presented in Fig.  1 .

The optimization methods as a replacement of traditional method in the design process

Why Optimization? People may ask why we study optimization. In most cases, we do not have the opportunity to physically perform trials; instead, we use optimization to simulate a solution for a specific problem to see what the result of this trial would be. Hence, we can decide whether this trial is applicable or not. People may benefit from applying optimization in the industry to achieve a better position in competition under limited resources.

1.2 Paper Structure

The rest of this paper is structured as follows: various metaheuristic taxonomies and the development process are illustrated in Sects.  2 and 3 . Taxonomies of optimization problems based on many criteria and their performance assessment are introduced in Sects.  4 and 5 . The applications of metaheuristic algorithms (MAs) in different fields are presented in Sect.  6 . The open issues and challenges of MAs and the observations from the experiment are introduced in Sect.  7 . Finally, the research review is concluded in Sect.  8 . The outline of the article is illustrated in Fig.  2 .

The outline of the article

2 Metaheuristics Optimization Algorithms Taxonomies

Due to the rapid growth of the optimization field, many metaheuristic (MA) algorithms have been proposed recently. These algorithms need to be classified according to four main taxonomies: inspiration source, number of search agents, the mechanisms followed in the optimization process, and solution updating, in addition to the number of parameters included in the algorithm. In this section, these new algorithms will be classified.

2.1 Taxonomy According to the Inspiration Source

This is the most familiar and oldest classification of metaheuristic algorithms (MAs) and is suitable for studying the subcategory of MAs, which are nature-inspired metaheuristic algorithms. In general, by including the source of inspiration in the calculation, different studies use different classifications according to the inspiration, as illustrated in Table  1 . In this study, Fig.  3 shows a more comprehensive taxonomy for MAs.

The proposed classification of MAs based on the source of inspiration

As follows, the subcategories of the source of inspiration for MAs shown in Fig.  3 are illustrated in detail.

Swarm Intelligence (SI) is a self-organized system of collaborative behavior. SI has a set of characteristics, such as good communication skills between individuals, the ability to share information among its individuals, and the ability to learn from doing (adaptable beings). On the other hand, organisms do not have the ability to defend themselves against predators; they need to be in a swarm to perform the search or attack process for food. Mimicking the behavior of beings that live in flocks or herds seeking to hunt for prey or find food is the main inspiration for SI algorithms [ 8 ]. One of the most famous algorithms in this category is Particle Swarm Optimization (PSO) [ 9 ], which is inspired by mimicking the intelligent behavior of a flock of birds. Monkey Search Optimization (MSO) [ 10 ] is another example of SI algorithms that simulate the tree climbing process during the food discovery process. Hunting strategy and hierarchy-based leadership are the inspiration for Grey Wolf Optimizer (GWO) [ 11 ], Ant Colony Optimization (ACO) [ 12 ], Cuckoo Search (CS) [ 13 ], Ant Lion Optimizer (ALO) [ 14 ], and Honey Badger Algorithm (HBA) [ 15 ], which are well-known instances of SI algorithms.

Evolutionary Algorithms (EA) simulate the behavior of evolution, including recombination, mutation, crossover, and selection. EA begins by generating a random population; this population is then evaluated to choose the most fit individuals to contribute to the next generation. After several iterations, the population evolves to find the optimal solution. Genetic Algorithm (GA) [ 16 ] is the oldest algorithm in this class, mimicking Charles Darwin's theory of natural evolution. Other well-known EA methods include Differential Evolution (DE) [ 17 ], Genetic Programming (GP) [ 18 ], Coronavirus Disease Optimization Algorithm (COVIDOA) [ 19 ], Liver Cancer Algorithm [ 20 ], and Red Deer Algorithm (RDA) [ 21 ].

Human-based Algorithms (HA) is the main inspiration for this category. Mimicking the learning process between teachers and students led to the introduction of Teaching Learning-Based Optimization (TLBO) [ 22 ]. Tabu Search (TS) [ 23 ] enhances the search process through long and short memory. Other well-known HA algorithms include Group Leader Optimization Algorithm (GLO) [ 24 ], Stock Exchange Trading Optimization (SETO) [ 25 ], and Social Emotional Optimization Algorithm (SEOA) [ 26 ].

Physics-based Algorithms (PhA) are inspired by the physical laws or simulating a physical phenomenon such as gravitation, Big Bang, black hole, galaxy, and field. In other words, the physical rules are used in the process of generating new solutions. The most popular instances of this class are the Gravitational Search Algorithm (GSA) [ 27 ], Big Bang Big Chain (BBBC) [ 28 ], Heat Transfer Search [ 29 ], Henry Gas Solubility Optimization (HGSO) [ 30 ], Archimedes Optimization Algorithm [ 31 ], and Light Spectrum Optimizer (LSO) [ 32 ], which are some of the most common algorithms in the PhA category.

Chemistry based algorithms (ChAs) are algorithms that concentrate on the principle of chemical reactions such as molecular reaction, Brownian motion, molecular radiation. A list of algorithms that fall into this category are Chemical Reaction Optimization (CRO) [ 33 ], Gases Brownian Motion Optimization (GBMO) [ 34 ], Artificial Chemical Process (ACP) [ 35 ], Ions Motion Optimization Algorithm (IMOA) [ 36 ], and Thermal Exchange Optimization (TEO) [ 37 ], are common instances of the ChA category.

Math-based algorithms (MathA) Math-based optimization algorithms are algorithms that can be inspired from the mathematical theorems, concept and rules. Some algorithms fall into this group including; Gaussian Swarm Optimization (GSO) [ 38 ], Sine Cosine Algorithm (SCA) [ 39 ], Lévy flight distribution [ 40 ], Exponential Distribution Optimizer (EDO) [ 41 ], and Golden Sine Algorithm (GSA) [ 42 ], are common instances of the MathA category.

Plant-based Algorithms (PlA) The PLAs is relays on the simulation of the intelligent behavior of the plants. Specifically, a set of concepts in plant nature is used to inspire new metaheuristic optimization algorithms such as the flower flow pollination process, the phenomenon of colonization of invasive weeds in nature, the ecology and weed biology. Some algorithms fall into this group including; Flower Pollination Algorithm (FPA) [ 43 ], Invasive Weed Optimization (IWO) [ 44 ], Paddy Field Algorithm (PFA) [ 45 ], Artificial Plant Optimization Algorithm (APOA) [ 46 ], Plant Growth Optimization (PGO) [ 47 ], Root Growth Algorithm (RGA) [ 48 ], Rooted Tree Optimization (RTO) [ 49 ] are common instances of the PlA category.

Sports and Game based Algorithms (SpGA) Depending in the information and rules applied in the sports and gaming, a set of optimization algorithms can be inspired from team game strategies used in football, Basketball, and volleyball, Ludo Game. Ludo Game-Based Swarm Intelligence (LGSI) [ 50 ], Team Game Algorithm (TGA) [ 51 ], Football game algorithm (FGA) [ 52 ], World Cup Optimization (WCO) [ 53 ], Soccer League Competition (SLC) algorithm [ 54 ], and League championship algorithm (LCA) [ 55 ] are common instances of SpGA algorithms.

Miscellaneous The rest of metaheuristics optimization algorithms can be collected to be belongs to the miscellaneous class, the purpose of using the term miscellaneous is the miscellaneous ideas such as politics, Artificial thoughts, atmosphere, trade and other topics. Work occurring in clouds such as cloud movement, spread, and creation is the basic idea behind the inspiration of the Atmosphere Cloud Model Optimization Algorithm (ACMO) [ 56 ], the exchange of information in the stock market occurs, and is the basic motivation behind the Exchange Market Algorithm (EMA) [ 57 ]. The Grenade Explosion Method (GEM) [ 58 ], Passing Vehicle Search (PVS) [ 59 ], Small World Optimization (SWO) [ 60 ], Yin-Yang Pair Optimization (YYPO) algorithm [ 61 ], Political Optimizer (PO) [ 62 ], and the Great Deluge Algorithm (GDA) [ 63 ] are other examples of this category.

2.2 Taxonomy According to the Number of Search Agents

The classification according to the source of inspiration is the most familiar and is usually introduced in studies to summarize the concept of classification. However, this classification is not enough to tackle the classification process, as it does not provide any information about the internal mathematical structure or programming ideas of the algorithms. Hence a new angle of classification is used. Meta-heuristics can be categorized based on the number of search agents seeking to find the optimal into two groups of single-solution-based MAs (SMAs), and population-based MAs (PMAs). The following two paragraphs provide more information about each group. Figure  4 is a clarification view of this taxonomy.

The classification of MAs based on the number of search agents

Single-solution based MAs (SMAs) SMAs is also called Trajectory‑based algorithms (TAs) as the algorithms in this class depends on single trajectory nature in its work. In other words, in each iteration, the solution is directed to a single trajectory. The optimization procedure (searching about the optimal solution) of SMAs is started with single solution (from one search agent), later, and in the subsequent iterations, the solution is refined with the aim of achieving the optimal solution. We can say that the algorithm generates a single path to the optimal solution over the course of the iteration. For SMAs, the Simulated Annealing (SA) [ 64 ] is one of the familiar algorithms. where a single search agent moves through the design or search space of the problem being tackled. Over the course of iteration, a better solution or moves is accepted to participate in determining the optimal solution while the weak movements and solution are more likely to participate in the optimization process. Applying these actions guarantee generating an optimal path through the search space with a great probability of achieving a global optimal solution. Hill climbing (HC) reviewed in [ 65 ], Tabu Search (TS) [ 23 ], Great Deluge Algorithm (GDA) [ 63 ], Iterated Local Search (ILS) [ 66 ], and Greedy Randomized Adaptive Search Procedures (GRASP) [ 67 ] are some instances of this class.

Population-based MAs (PMAs) In contrast, and taking advantage of sharing information among agents, Collaborative work and data remembering, the PMAs is introduced. First, we can say that more than one agent is superior to a single agent in achieving the optimal solution. Specifically, a great number of search agents work together to extensively explore the search space, so we can call PMAs explorative-based algorithms. The optimization procedure starts with employing a population of search agents positioned at many distinct positions in the search space, and over the course of iterations, the population uses the advantage of sharing information to better achieve the best global solution. In simple words, a set of lines is drawn in the search space to extensively search the search space in order to obtain the best optimal solution achieved by all search agents. One of the oldest and widely used algorithms in PMAs is the Genetic Algorithm (GA), Chemical reaction optimization (CRO), Particle Swarm Optimization (PSO), Archimedes Optimization Algorithm (AOA), Sine Cosine Algorithm (SCA), Exponential Distribution Optimizer (EDO), Grey Wolf Optimizer (GWO), Ant Colony Optimization (ACO) and Honey Badger Algorithm (HBA) are some instances from this category.

In general, no class is totally better than the other where PMAs escape from the local optima dilemma in contrast to SMAs, also SMAs consume less computational time than PMAs, for a Itr number of iterations, the SMAs perform a lower number of objective function evaluation which equals 1  ×  Itr while the PMAs perform N  ×  Itr evaluation of the objective function. N here stands for the number of search agents employed by the algorithm to obtain the optimal solution. But overall, the scientists prefer to use the PMAs as it has a greater probability of achieving global optimal solution in a considerable amount of time.

2.3 Taxonomy According to Updating Mechanisms

However, the classification according to the number of search agents provides information about the internal structure of the algorithm, but it cannot be treated as a uniform classification due to the few algorithms belonging to one group while the remainder (majority) falls under the other group. In this context, we need to provide a different classification angle to achieve an acceptable degree of uniform classification. According to the most important step of any algorithm, which is the solution update process. From this prospective MAs can be classified as solution creation-based algorithms (SCBAs) and differential vector movement-based algorithms (DVMs) [ 68 ]. In the following paragraph, we introduce a simple classification based on the behavior of the algorithms. Figure  5 is a clarification view of this taxonomy.

The classification of MAs based on population update mechanisms

Solution Creation Based Algorithms (SCs) In SCs, A set of parent solutions are merged to generate the new solution, in other words no single solution is used to create the fresh solution. Furthermore, the SCs can be categorized into two subcategories which are combination-based algorithms and stigmergy-based algorithms. In combination-based algorithms several solutions are combined or crossover-ed. Genetic Algorithm (GA), Gene Expression (GE), Harmony Search (HS), Bee Colony Optimization (BCO), Cuckoo Search (CS), Dolphin Search (DS) are some examples of this subcategory. On the other hand, in strategy-based solutions different solutions are indirectly coordinated by intermediate structure to generate new solutions. Ant Colony Optimization (ACO), Termite Hill Algorithm (THA), River Formation Dynamics (RFD), Intelligence Water Drops Algorithm (IWDA), Water-Flow Optimization Algorithm (WFOA), and Virtual Bees Algorithm (VBA) are some examples of the second subcategory.

Differential Vector Movement Based Algorithms (DVMs) Applying the mutation or shifting operation on the algorithm in order to generate a new solution is called Differential Vector Movement method. The fresh generated solution needs to be fitted to the previous one to participate in the next iteration of the optimization procedure. In this context, DVMs is categorized into three subcategories. In the first subcategory, the whole population's solution is used to generate the new solution, such operation occurs in Firefly Algorithm (FA) Gravitational Search Algorithm (GSA), Central Force Optimization (CFO), Human Group Formation (HGF) and Charged System Search (CSS). In the second sub-category, a small number of solutions (neighbourhoods) in population is employed to generate a new solution such as Artificial chemical process (ACP), Thermal Exchange Optimization (PSO), Group Search Optimizer (ALO), and Group Search Optimizer (GWO). In the last sub-category, only the relevant (best/worst) solutions are employed to generate the new solution such as Differential Evolution (DE), Artificial Bee Colony (ABC), Particle Swarm Optimization (PSO), Ant Lion Optimizer (ALO), and Grey Wolf Optimizer (GWO).

2.4 Taxonomy According to Number of Parameters

To deeply consider the internal configuration of the algorithm for this type of classification. Tuning the parameter of the algorithm plays a vital role in the performance of the algorithm when solving a specific problem. As mentioned in [ 1 ], it is a complicated task to choose the best values of the parameter that scores a better solution. Furthermore, the parameters can enhance the robustness and flexibility of the MAs if they are adjusted correctly. The optimization problem plays a vital role in defining the values of parameters. From a complexity perspective, the complexity of an algorithm is affected by the number of parameters. In this context and taking into account the importance of the parameters, this classification is introduced. Kanchan Rajwar et al. in [ 68 ] first classify the MAs according to the number of primary parameters employed in the MAs as illustrated in Fig.  6 .

The classification of MAs according to the number of primary parameters

The number of parameters changes from one algorithm to another, which can be 0, 1, 3, 4, etc. For simplicity we will consider four main groups holding algorithm parameter numbers up to 3 and the rest fall into the miscellaneous group. The following paragraphs provide a detailed explanation of the five groups in this classification.

Zero‑parameter-based algorithms (ZPAs): The ZPAs do not have any parameter in their internal configuration so it also called Free‑parameter-based algorithms. The absence of parameters in ZPAs gives the user the opportunity to easily adapt the algorithm to be utilized in different optimization problems. Hence, the algorithms belong to this group considered as flexible, adaptive, and easy-to-use algorithms. Teaching–Learning-Based Optimization (TLBO) [ 22 ], Black Hole Algorithm (BH) [ 69 ], Multi-Particle Collision Algorithm (M-PCA) [ 70 ], Symbiosis Organisms Search (SOS) [ 71 ], Vortex Search Optimization (VS) [ 72 ], Forensic-Based Investigation (FBI) [ 73 ], and Lightning Attachment Procedure Optimization (LAPO) [ 74 ] are some examples of ZPAs.

Single‑parameter-based algorithms (SPAs): SPAs is the type of algorithms that own a single primary parameters in their internal configuration. So, it also is called monoparameter-based algorithms. Mostly, this single parameter has the ability to change the amount of exploration and exploitation that occurred in the algorithm. For example, in the Artificial Bee Colony (ABC) algorithm the single parameter Limit is used to determine the amount of food source left [ 75 ], in the Salp Swarm Algorithm (SSA) c1 is the parameter used to achieve a better balance between explorative and exploitative capabilities [ 76 ], and in Harris Hawks Optimizer (HHO) [ 77 ] the switch between soft and hard besiege is achieved by the magnitude value parameter E . Cuckoo Search (CS), Killer Whale Algorithm (KWO), and Social Group Optimization (SGO) are another example of this group.

Two‑parameter-based algorithms (TPAs): In TPAs only two primary parameters exist in the internal structure of the algorithm. For example, in the Grey Wolf Optimizer (GWO), the two primary parameters a and c must be adjusted. The a is adjusted to be equal to 2 to 0 , allowing the algorithm to perform a smooth transition from exploration and exploitation while the c parameter is used to allow the algorithm to reach distinct locations around the optimal agent relative to the current location, In the Marine Predators Algorithm (MPA), P and FADs are the two primary control parameters. To overstate the predator or prey move, P is adjusted, while FADs is used to manage exploration behavior. Finally, in the Whale Optimization Algorithm (WOA) the two primary parameters A and C need to be modified to perform the exploration-to-exploitation transition and to allow the algorithm to explore several positions around the optimal agent relative to the present location. Differential Evolution (DE), Simulated Annealing (SA), Grasshopper Optimization Algorithm (GOA), Political Optimizer (PO), and Artificial Chemical Reaction Optimization Algorithm (ACROA) are just a few instances of TPAs.

Three‑parameter-based algorithms (TrPAs): In TPAs only three primary parameters exist in the internal structure of the algorithm. For example, the mutation rate mr , the crossover rate cr , and the new population selection criterion are the three parameters used in the Genetic Algorithm (GA) to allow the algorithm to escape from the local optima, improve the accuracy of the solution, and generate a most fit new generation, respectively. The randomization, attractiveness, and absorption are the three parameters included in the Firefly Algorithm (FA) to manage the execution of the algorithm and the random walks of fireflies. Finally, the distance bandwidth (BW), the harmony memory considering rate (HMCR), and the pitch adjusting rate (PAR) are the three primary parameters used in Harmony Search (HS) to increase the opportunity of achieving a global search and improve the local search problem. Squirrel Search Algorithm (SSA), Krill Herd (KH), Spring Search Algorithm (SSA), Artificial Algae Algorithm (AAA), Gases Brownian Motion Optimization (GBMO), Hurricane-Based Optimization Algorithm (HOA), Orca Optimization Algorithm (OOA), Social Spider Algorithm (SSA), Water Cycle Algorithm (WCA), Equilibrium Optimizer (EO), Parasitism Predation Algorithm (PPA), and Heap-Based Optimizer (HBO) are few instances of this group.

Miscellaneous: The rest of algorithms that own over three parameters in their internal configuration fall under the category of the miscellaneous group. It is not easy to cover all three-parameter algorithms. so, only three subgroups are introduced. the first subgroup is the four parameter-based algorithms such as Ant Colony Optimization (ACO), Sine Cosine Algorithm (SSA), Archimedes Optimization Algorithm (AOA), and Gravitational Search Algorithm (GSA). The second subgroup holds algorithms that employed five primary parameters in their internal structure such as Particle Swarm Optimization (PSO), Cheetah Chase Algorithm (CCA) and Farmland Fertility Algorithm (FFA). The last subgroup is algorithms with more than five primary parameters in their internal configuration. Biogeography-Based Optimization (BBO) with six parameters, Henry Gas Solubility Optimization (HGSO) with twelve primary parameters and the Camel Algorithm (CA) with seven } primary parameters are the most familiar algorithms in this subgroup. Cheetah Chase Algorithm (CCA), Exchange Market Algorithm (EMA), and Forest Optimization Algorithm (FOA) are also instances of this subgroup.

In general algorithms with few parameter-based MAs are easy to be adapted and hence the applicability of these algorithms to handle any optimization problem will increase and, on the other hand, large parameter-based MAs cause a disability of these algorithms to handle the optimization problems, as we encounter a problem in adapting all of their parameters to be suited for problem being tackled. hence the applicability will be decreed.

2.5 Metaheuristic Algorithms Merits

The MAs have a priority to be studied by the researcher than HAs, as they have four characteristics [ 78 ], which can be summarized as follows.

Metaheuristics simplicity It is painless to inspire a MAs as we can use a natural concept, physical phenomena or an animal behavior in the inspiration process. Utilizing the merit of simplicity, the researchers Seize the opportunity to make an extension in the metaheuristics works as they develop a new method by mimicking a natural idea, use the ability of search enhancement techniques to boost the performance of an existing algorithm, or even take the advantages in two metaheuristics algorithms and generate a new metaheuristics algorithm by applying a hybridization process. Furthermore, simplicity encourages computer scientists and other researchers to easily study the existing MAs and then apply them to solve a wide range of problems.

Metaheuristics flexibility In the other techniques there is a need to modify the structure of the algorithm to be matched with the problem being solved, unlike these techniques metaheuristics flexibility virtue allows the researchers to easily apply the MAs on any problem as the MAs have the capability of treating the problem as black box, in other words it need the input(s), output(s) of a problem on hand. No effort is used in modifying the structure; all effort is directed towards formulating the problem being solved in the form of an optimization problem.

Metaheuristics stochastic nature Computing the derivation of the search space of the problem is a necessity for the gradient-based optimization techniques to achieve an optimal solution. Dissimilar to these techniques, the preponderance of MAs is considered as a derivative-free mechanism when applying the process of optimization, specifically the MAs follow a stochastic nature during the search process as they start the optimization process by employing a set of search agents to generate random solutions without computing the derivative of the search space. The collaborative work of these search agents allows the algorithm to get the optimal solution. This merit allows researchers to easily use the MAs algorithms to perfectly tackle compound, expensive, and difficult problems that suffer from the trouble of obtaining the derivative information.

According to the previous features, the research community has increased, and researchers from different fields and application areas have been using the metaheuristic optimization algorithm in their work. About 4,476 documents founded in the Scopus have used the word metaheuristics in the last decade. Figure  7 a is introduced to visualize the distribution of research studies according to the subject area, while Fig.  7 b is used to depict the number of studies generated in each year of the previous decade.

Scopus statistics from 2014 to 2023

3 Development Process of Metaheuristic Optimization Algorithms

The simplicity merit of MAs allows researchers to easily develop a large number of algorithms in different application areas. To develop a new metaheuristic algorithm, a researcher can follow one of the following development processes according to the type of algorithm that is being developed, and some processes can also be used together.

Develop a new optimization algorithm The most of work for developing an optimization algorithm done by inspire the main idea of the algorithm from a different metaphors or concepts. These metaphors or concepts are mainly a simulation of rules or processes in different disciplines such as Chemistry, Physics, Biology, Psychology, Computation, Maths, and Human. Figure  3 is used to visualize a different source of inspiration with examples in each category. In general, most metaheuristics have been designed to mimic the system of living and survival of beings such as animals, birds, and insects, in addition to mimicking natural evolution. Insects (specifically, bees and ants) are the most popular metaphor for the development of a new optimization method by researchers.

Develop a new optimization algorithm from existing one One of the most popular ways to develop a new optimization method is to benefit from the operators of a specific algorithm in enhancing the structure of another algorithm. In simple words, the operators of other algorithms can emerge into the basic structures of the algorithm to boost the performance of the previously developed algorithm, and hence use it in solving different types of problems and issues. There are many enhancement operators used in the field; one of the most used ones is opposition-based learning (OBL). OBL is a machine learning mechanism that is used to increase the performance of the optimization algorithm by considering the opposite position of the solution in the search space. Specifically, two values are computed, the main and opposite positions, according to the objective function value, one of the two values maintained in the optimization process, and the other discarded. Taking into account only the best values, the optimization process became more accurate and a high level of performance is achieved. The orthogonal learning (OL) strategy is another example of an operator used as an enhancement strategy for MAs. The OL strategy mainly improves the exploitation capabilities. For example, the OL strategy was used to improve the Archimedes optimization algorithm, the cuckoo search algorithm and the artificial bee colony optimization algorithm, respectively. Enhanced solution quality (ESQ) is another mechanism used in the MA enhancement process. The ESQ was used to improve the performance of the reptile search algorithm and the Harris Hawks optimization (HHO) algorithm, respectively. Finally, the Local Escaping Operator (LEO) is used to develop an optimized version of the MPA called the enhanced marine predator algorithm (EMPA).

Hybridizing two or more optimization algorithm As a trial for enhancing the performance and applicability of the optimization method, researchers can benefit from hybridizing two or more optimization algorithms together in order to take the main strengths of each algorithm. The idea behind hybridization is to choose one algorithm better in exploration capabilities and another better in exploitation capabilities. Many challenges are encountered when we develop a new algorithm using the hybridization process, such as how to select the algorithm and how to merge them together, and is the new algorithm better than each one separately?

As shown in the previous paragraphs, there is a different development process for developing a new optimization method, although there is a set of limitations that must be considered during the development process such as the difficulty of transforming all the concepts with details into a mathematical form, how the algorithm totally manages the change in information about the source of inspiration, in addition to how people with low familiarity with the inspiration sources develop new methods.

3.1 Criteria for Comparative Algorithms

To gauge the effectiveness of newly developed algorithms, it is crucial for research to present the process of comparing them with existing algorithms. This should include a discussion of the selection criteria for comparative algorithms and the methodology used for comparison. The selection criteria for comparative algorithms depend mainly on the nature of the algorithm and the development process followed in developing the algorithm. In all cases, comparative algorithms should contain common criteria, which are state-of-the-art algorithms, newly developed algorithms, CEC winner algorithms, and high-performance algorithms. Specifically in case of developing the algorithm using the inspiration of a phenomenon process, the comparative algorithms list must contain algorithms with the same inspiration source or concept if there exist in addition to the common criteria algorithms. In case of developing an algorithm using the restructure method (i.e., merging a new operator or strategy), the comparative algorithms must contain the basic algorithm, algorithms developed using the same strategy if exists, algorithms that contain the strategy itself, in addition to the common criteria algorithms. In the case of developing algorithms using the hybridization process, the comparative algorithm list must contain the two basic algorithms that participate in the hybridization process, in addition to the common criteria algorithms.

3.2 Novelty Claims of Metaphor-Based Methods

The different ways of developing an optimization algorithm and the simplicity merit of the metaheuristic allow researchers to easily develop a large number of MAs. But a question must be asked here: Does this inspiration convey a novelty? In this section, we will present a set of claims and myths in the inspiration process of the metaheuristic optimization algorithms. As introduced in [ 79 ] a six widely used algorithms have been analyzed to prove that all components of the six (grey wolf, moth-flame, whale, firefly, bat and ant lion) are equivalent to a component of well-known techniques such as evolutionary algorithms and particle swarm optimization. Hence the authors called these algorithms misleading or tricky optimization algorithms, as they were inspired by bestial or duplicated metaphors and did not bring any novelty or useful principles in the metaheuristics field. We will present what considerations must be taken when developing a new novel algorithm and how to judge about the novelty of the new proposed algorithm in the field of metaheuristics.

Recently, a large number of publications have developed self-proclaimed or novel metaphor-based methods, but it is not obvious why they used them and what the novelty ideas are behind these methods. The set of all negative points, criticizes about novelty claims of various metaphor-based methods, can be introduced in the following points:

The metaphor-based methods redefine a well-known concept in the field of optimization and deliver it as a new concept or under new terminologies.

weak translation of the metaphors into a mathematical model or equations, and the model cannot be used totally to reflect the metaphors correctly. Finally, the proposed algorithm does not translate the mathematical model obtained from the metaphor correctly.

There is a myth in introducing the motivations behind the use of metaphor where instead of delivering the motivations as a sound or scientific basis they use accurate motivations such as a new metaphor "has never been used before" or a new mathematical model "has never been appeared in the past". Additionally, there is no concentration on the optimization process itself and how this process is employed to introduce effective design choices.

Instead of applying the evaluations of the proposed algorithms mainly on the state-of-the-art problems, the authors of these methods depend on the comparison with other algorithms or experimental analysis of low complexity problems in evaluating the performance or applicability of the proposed algorithm.

To prevent these negative points, the authors must apply two metrics analyses of the proposed algorithm before naming it as a "novel", which are:

Usefulness: in which the author must clearly introduce what are the useful ideas that come from the metaphor and how this metaphor helps in solving the optimization problems.

Novelty: When proposing a new method in the field of metaheuristics, was this new metaphor novel used to convey ideas?

4 Optimization Problems Overview

Achieving an acceptable solution is the main goal of any algorithm. Due to the rapid expansion of the complexity of the problem, scientists need to develop new methods that can cover this rapid extension. In this context, scientists are working to formulate any problem as an optimization problem to be easily tackled by optimization algorithms, as they provide better solutions than other traditional methods. In different fields, a great number of problems are formulated as an optimization problem, such as genetic algorithms used to automatically find and classify solitary lung nodules \cite{de2014automatic}, perform a classification for web pages, mining the web content, and dynamic organizing of the web content by ant colony optimization [ 80 ], In [ 81 ] Hussein et al., use the HHO to discover and design the drug through chemical descriptor selection and chemical compound activities. Applying HHO in microchannel heat sinks to minimize entropy generation [ 82 ], COVID-19 prediction [ 83 ], finally applying image segmentation and thresholding in [ 84 , 85 ].

4.1 Basic Structure of Optimization Problems

In this section, we will try to support the readers who may not be familiar with optimization methods with the basic definitions and terms related to the optimization field. The process of solving an optimization problem using a metaheuristic algorithm starts with identifying the real-world problem, after that we move to the problem description stage in which we define the characteristics of the problem, determining the functional requirement in addition to analysis of nonfunctional requirements. After completing the problem description stage, we move to the research stage, in which the researcher first concentrates on how to mathematically formulate the problem in a mathematical form. To formulate the problem, we need to determine the design variables and parameters, formulating the objective function, determining the basic constraints on the variables, analyzing the complexity of the problem, and finally justifying the use of a metaheuristic algorithm. In the following paragraphs, the three main components which exist in any optimization problem are the objective function, the decision variables, and a set of constraints on these variables are discussed in detail.

Optimization model: Every system can be considered as a set of inputs producing one or more input, The system uses the set of constraints to minimize the number of inputs, in other words, we consider only the inputs that obey the constraint (i.e., valid inputs) and discard the other which does not match with the constraints (i.e., invalid inputs). The system performs processing on the valid inputs to produce the optimal solution, which can be evaluated using the objective function to obtain the minimum or maximum output value. In fact, the optimization algorithm will not find the optimal values of constraints; instead, it uses the constraints to produce the optimal solutions and construct the feasible solution area. The feasible solution area can be considered as the area which contains an infinite number of feasible solutions and one or more can be classified as the optimal one.

Formulating the optimization model as an optimization problem: When we solve the problem using the optimization algorithm, we look for all possible combinations of inputs. For example, if we have 3 inputs each with 10 discrete values, then we get 1000 combinations of inputs. The initial test to solve and evaluate the input is to use brute-force techniques. The brute force techniques will do better to obtain the optimal solution, but what about the large sized problems. Certainly, we will find a big problem in handling these problems using the brute-force techniques; hence searching all possible combinations for most real-world problems is impossible.

To avoid confusion for non-familiar people with the area of optimization, in this study, we will introduce the basic and most frequent terminologies used in the field:

The search space: it is the area in which all possible combinations of inputs are located.

The search landscape: it is the set of all possible combinations of inputs with their corresponding objective values.

Decision variables: it is the unknown quantities that need to be determined by assigning values to them. It is also known as the design variables. All possible values that can be assigned to these variables are named variable scope or domain. It can be mathematically as X i where i = 1,2,3…N.

The objective function: This is the equation of the decision variables. In which all the decision variables exist with different parameters. It is used to judge the quality of the solution obtained for the problem being handled. In other words, after calculating the values of the decision variables, we substitute them in the objective function to obtain the objective value. The minimum objective value is the optimal solution for minimization problems, and the maximum is the optimal solution for the maximization problem.

Mathematically the single objective optimization problem can be formulated as Eq. ( 1 ) while the Multi objective optimization problem can be formulated as Eq. ( 2 ).

The optimization problems can be categorized in different ways. Categorizing optimization problems is an important step in choosing the algorithm that provides the optimal solution. It is not easy to introduce a rigorous or comprehensive taxonomy for optimization problems. This is due to the multiplicity of the classification term. But due to the important role of this taxonomy, in this paper we present a simplified and summarized version of the available taxonomies, illustrated in Fig.  8 . In the following subsections, the different subcategories of the optimization problem are discussed in detail.

Optimization problem taxonomy

4.2 Taxonomy According to the Objective Function

In terms of the number of objectives, there are two types. If the number of objectives is greater than one, the problem is called a multi-objective optimization problem; otherwise, the problem is named a single-objective optimization problem. Usually, real-world optimization problems are multi-objective. For example, if we need to design a table, we will consider two objectives, for example, minimizing the weight and the price of the table.

Single-objective optimization Only one global optimal solution exists in single-objective optimization. The objective function only considers one objective; therefore, the best optimal solution can be easily determined by comparing the obtained solutions using basic comparison operators < , > , ≤ , ≥ , and = , the nature of this type allows the algorithm to easily tackle optimization problems. Without loss of generality, Eq. ( 1 ) is used to determine the mathematical structure of a single-objective optimization problem.

where the problem decision variable is symbolized by n , m and P exist to represent the number of inequality and equality constraints, respectively. For the i th variable, ub i and lb i are used to represent the upper and lower boundaries, respectively.

Multi-objective optimization In contrast to single objective optimization, A set (more than one) of objectives need to be optimized simultaneously in the multi-objective optimization problems. Usually, these objectives are a conflict with each other, so most of work in this type is paid to achieving a trade-off between these objectives. The set of solutions in this type is called a Pareto optimal solution. The Pareto optimal dominance is employed to compare the solutions obtained in order to determine the optimal solutions. Extra storage is needed to hold the Pareto optimal solutions. Without loss of generality, Eq. ( 2 ) is used to determine the mathematical structure of a single objective optimization problem.

where the problem decision variable is symbolized by n , m and P exist to represent the number of inequality and equality constraints respectively. For the variable i th , U i and L i are used to represent the upper and lower limits, respectively. The number of objectives is denoted by o , and the g i and h i are the i th inequality and equality constraints, respectively. In general, the clash among objectives enforces the problem designer to consider more than one criterion in the comparison of obtained solutions and therefore the classical comparison operator does not perform better, instead, the Pareto dominance Eq. ( 3 ) is used to define the best optimal solutions.

Here the two solutions are represented by the vectors x and y . The x is said to dominate y denoted as (x  ≤  y) if x has at least one better value in all objectives.

4.3 Taxonomy According to Function Form

From another angle, classification can be done according to function form. If we have a real-world optimization problem, the constraints are linear qualities and inequalities and the objective function formed as linear then the problem is said to be a linear optimization problem. In nonlinear optimization, one or both of the objective functions and constraints are nonlinear, and this is the realistic and complex one [ 86 ].

4.4 Taxonomy According to the Design Variable

According to the nature of the design variables, we can present three different types of optimization problems, as detailed in the following points.

Discrete optimization problems In discrete optimization problems the values of the design variables are discrete and in which there is a finite set of values. The shortest path problem and the minimum spanning tree problem are two instances of this type. For more details, we can mention that the discrete optimization consists of integer programming and combinatorial optimization. Integer programming deals with the formulation and solution of discrete integers (or binary integers) valued in the design variables. On the other hand, combinatorial optimization emphasizes the combinatorial origin, formulation, or solution of a problem. Mainly it seeks to achieve pairs (i.e., Assignments, groupings, orderings) of discrete and finite values under the influence of specific constraints. These pairs involve a component of solutions of potential combinatorial problem solutions [ 87 ]. In Bioinformatics, Artificial intelligence and other fields combinatorial optimization can be applied such as identifying propositional formula models or defining the 3D structure of protein, finding the shortest path in graphs, the travelling salesman problem, the knapsack problem in addition to the pin packing problem, the quadratic assignment problem which has been tackled in this study.

Continuous optimization problems In continuous optimization problems, A range of values is assigned to the design variables, so every design variable has an infinite set of values. These problems have two types constrained continuous optimization problems which there is a constant on the variables. For unconstrained continuous optimization problems there is an absence of these constraints maximization the general yield for differential amplifiers, optimization of the mechanical system of shock absorption are two examples of this type [ 88 ].

Mixed discrete–Continuous optimization problems In many problems a design variable has a mixture of discrete and continuous values, in this case we call the problem mixed discrete–continuous type. This type is the most widely used one, where numerous real-world problems are complex and possess a mixed quantitative and qualitative input. In [ 89 ], a set of instances is addressed using black-box optimization techniques.

4.5 Taxonomy According to Constraints

Furthermore, the classification can be according to the restrictions on the design variables.

Unconstrained Optimization Problem If there are no constraints on the design variables we call this type as unconstrained optimization problem, the unconstrained optimization can be viewed as iterative methods stating with initial estimation for the optimal solution then a set of iteration is used to reach for the optimal solution. Usually, the solutions were reduced iteratively to an optimal solution. In [ 90 ], Fletcher and Roger mentioned that the unconstrained optimization methods differ according to how much information the user provides, such as the gradient method, the second derivative method, and the non-derivative method.

Constrained optimization problem If there is one or more constraints, the optimization problem falls under the constrained optimization problem class. Furthermore, there are two subclasses of this type. The first subclass is equality constraint problem in which the values of the design variables are restricted to be equal to the specific value. The second subclass is inequality-constrained problem the design variables are restricted to greater / smaller than a specific value. From a formulation perspective, every equality constraint can be mathematically transformed to two inequality constraints. For example, ϕ(x) = 0 is equivalent to ϕ(x) < 0 and ϕ(x) > 0 [ 86 ]. Mainly, the constrained optimization covers three types of optimizations which are network optimization, bound constraints optimization, and the global optimization. Global optimization includes one of the most widely used problems, which is engineering design optimization problems.

5 Performance Assessment of Optimization Algorithms

First of all, we must refer to an important term, the efficiency of the algorithm, which means how the algorithm responds against finding the optimal solutions for the problem to be solved. Achieving an optimal solution is not the only purpose of a good optimization algorithm; instead, the algorithm must be high quality and achieve a better situation in the applicability process on different classes of problems. To judge the quality and applicability of the algorithm, the algorithm must be compared against a set of qualitative and quantitative measures. The good quality algorithm performs better and achieves better results when tested against qualitative and quantitative measures. In this section, we will present the whole assessment environment used to test the quality and applicability of the algorithm.

CEC Test suite CEC stands for Congress on evolutionary computation. Mainly the CEC holds a different class of problems, which may be uni-model, multi-modal, fixed-dimension multi-modal, and composite. CEC is usually used to test the performance of the algorithm and its ability to solve different classes of problems. In the art of optimization almost all studies perform the CEC function as a fitness function to test algorithm's performance itself and to compare the algorithm's performance against other algorithms.

Statistical Measures In this metric, the Best, Worst, Mean, and standard deviation are computed to the obtained solutions to judge about the quality of all solutions obtained together. The best solution is the one with a minimum value of fitness function in minimization optimization and the opposite is right for maximization optimization. The Worst is the solution which has a maximum value of fitness function on the minimization optimization and the opposite is right for maximization optimization. while the mean is used to compute the average value of all obtained solutions (obtained from executing the algorithm many times), and the small value of the mean means that the algorithm is doing better. Finally, the standard deviation or STD is the statistical measure that gives the reader insight into the differences among the obtained solutions, and the algorithm with small STD value is also better than the other with large value. There is also an important statistical measure, which is capable of measuring the whole performance of algorithms for any number of functions. This measure uses the mean rank sum value to rank the algorithms. Ranking these values in ascending order enables us to say that the algorithm with the lowest value is the best among all algorithms participating in comparison for all functions together.

Convergence curve Drawing a relation between the solutions scored by the algorithm and the number of iterations or number of function evaluations is the primary goal of the convergence curve. To summarize the behavior of the algorithm, the convergence curve is drawn to judge the speed of the algorithm in reaching the global optimal solution. For the minimization problem and to compare the performance of many algorithms. The lower convergence curve is better than the upper one. Also, we can compute how fast the algorithm converges towards the optimal solution through the rate of convergence measure.

Diversity Diversity measure is one of the measures related to the algorithm’s convergence behavior. In simple words, diversity means how the search agents of the algorithm are distributed in the search space. A high diversity value of the algorithm can be translated into a great exploration ability of the algorithm, and a low value can be translated into a great exploitation ability of the algorithm. Hence the diversity values of the algorithm must be smoothly transited from high value in the first iterations of the algorithm to low value in the rest of iterations of the algorithm. In this context, we can say that the good diversity of the algorithm leads to avoiding premature convergence and achieve a good speed in achieving the optimal solution hence score a high level of efficiency.

Trajectory diagram In order to test the behavior of a specific agent of the algorithm over the curse of iterations the trajectory diagram is used. The fluctuations of the curve are an indication of the better performance of that agent and its ability to explore and exploit the search space better.

Search history diagram To visualize the history of positions scored by the search agent during the process of optimization, the search history curve is drawn.

Exploration and exploitation The exploration and exploitation (EXPL-EXPT) curves are used to visualize the exploitative and explorative capabilities of the algorithm. Usually, the overlaps between the two curves exist to tell us about the shifting between exploration and exploitation, and therefore an EXPL-EXPT balance.

Real-world problems To test the ability of the algorithm in solving different classes of problem the real-world problem is tackled. Engineering design problems are the most widely used problems as many algorithms use the (pressure vessel, welded beam, 15/3/25/52-bar truss system, tension/compression spring…etc.) classical design problems to quiz the algorithm performance.

Operation platforms Alongside the previous measures, the algorithm quality can be affected by the environment setup in which the algorithm is executed. The good environment in both software and hardware capabilities leads to good behavior of the algorithm. In this context, we must mention that when we compare more than one algorithm to judge which is better, we must execute the algorithms in the same environment to achieve a fair comparison.

6 Metaheuristics Applications

As mentioned above, MAs have a great degree of applicability, as they operate better in solving different problems that involve a computation time restriction, a high-dimensional problem, and other kinds of problems. Specifically, MAs are capable of dealing with different classes of optimization problems in different fields. In the following subsections, the applicability of MAs in some of these fields are illustrated in detail.

6.1 IEEE Congress on Evolutionary Computation (IEEE CEC)

CEC stands for Congress on evolutionary computation. Mainly the CEC holds a different class of problems, which may be uni-model, multi-modal, fixed-dimension multimodal, and composite. CEC is usually used to test the performance of the algorithm and is considered as an indication of the capabilities of the algorithm to solve different classes of problems. In the art of optimization, almost all studies perform the CEC function as fitness functions to test the algorithm's performance itself and to compare the algorithm's performance against other algorithms. Almost a different version of the CEC test suite is introduced every year. Tables 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 , 11 , and 12 are presented below to provide the reader with the basic information on each version of the CEC benchmark function and how metaheuristic algorithms are applied to solve these benchmark functions.

6.2 Engineering Design Problems

It is easy to provide an optimal design for a simple problem that contains a small number of design variables with a small range of values. In contrast to complex problems with many components, algorithms consume a huge amount of time to develop an optimal design. For example, the mechanical problem with different components and multiple objectives and constraints. Another example for complex problems is the engineering design problems in which the design process starts with exploiting the experience of designers to guess an optimal design for any problem, but this is not the optimal direction. In order to treat this poor thinking, we need systematic work that guarantees achieving an optimal design that is better than any other human design. Automatic techniques or, in other words, metaheuristics algorithms (MAs) are used to effectively diversify the search space with large parameters, minimizing the cost, and improving the product life cycle. Mainly, the MAs tune the parameters of the problem to produce the best optimal values of the design variables, hence achieving the optimal design. Table 13 is used to highlight the work of single-objective metaheuristics optimization algorithms in solving engineering design problems; also, Table  14 is introduced to clarify the tries of multi-objective optimization algorithms in tackling engineering design problems.

6.3 NP‑Hard Problems

In NP‑hard problems the NP stands for nondeterministic polynomial time where the nondeterministic refers to nondeterministic Turing machines which apply the idea of brute-force search method. On the other hand, the polynomial is used to refer to the amount of time required to apply the quick search to get the single solution of the deterministic algorithm, or the time consumed by the nondeterministic Turing machines to perform extensive search. P is the set of all decision problems solvable in polynomial time. Specifically, the decision problem has two answers YES and No. Consequently, if all YES answers are checked in polynomial time, then the problem belongs to set of NP problems; on the other hand, co-NP is used for NO answer. If the polynomial-time solution obtained for a specific problem leads to a polynomial-time solution for all problems in the NP in this case the problem is said to be NP-hard. Also, a problem is NP-Complete iff it is NP-Hard, and it is in NP itself. Due to the high computational complexity, the exhaustive search methods do not have the ability of getting the best solution.

Quadratic Assignment Problem (QAP) As mentioned in [ 195 ] the QAP is NP-hard, as the polynomial time is not sufficient to obtain the approximate solution from optimal solution. QAP was first introduced by Koopmans and Beckmann in 1957 [ 196 ] as an extension of the linear assignment problem. QAP is considered a combinatorial optimization problem that has been considered and tackled by many research studies in the last three decades. However, the good results obtained in these studies but until now the QAP is not well solved as there is no exact algorithm capable of solving problems with more than 20 input sizes in a reasonable amount of computational time [ 197 ]. In QAP we seek to locate the facilities in its appropriate location under the condition that it is an exact-one-to-exact-one problem, that is, each site can only grasp only one facility and each facility must be placed in only one site where the distances between facilities and sites are determined. The main optimization goal of QAP is to minimize the distance and flow between each pair of facilities delegated to their relevant sites. Recently, QAP is addressed by many books, studies and reviews, as listed in Table  15 . From another angle, there are several problems that are considered as special types of QAP (Table  16 ).

The Bin Packing Problem (BPP) The Bin Packing Problem (BPP) is one of most familiar combinatorial problems that is considered as strongly NP-hard problem [ 218 ]. In BPP we need to pack a set of items m into n bins with the aim of minimizing the number of bins required to hold all items. The BPP can be mathematically formulated as in Eq.  4 [ 219 ].

where capacity of the bin y j is symbolized by C yj .

The BPP benchmark data sets consist of three different types that are commonly classified as Easy, Medium, and Hard class as mentioned in [ 220 ]. Also, the BPP appears in one, two, three, and multi-dimensional form (Table  17 ).

Travelling Salesman Problem (TSP) One of the most familiar combinatorial optimization NP-Hard is the TSP in which we need to minimize the route as possible consumed to visit all cites precisely once and return to the initial city given a list of cities and distances among them. For example, in the TSP of 20 city, we have a huge number of feasible solutions (approx.1.22 × 1017). Guess how much time is required to perform this task using exhaustive search? the answer is very long. Therefore, exhaustive searches have disabilities in tackling such problems. The use of MAs destroys this disability, as it was used to find near optimal solutions in a reasonable amount of time [ 233 ]. Vehicle routing problems (VRPs) is the general form of TSP and is a multi-objective real-world problem tackled by many MAs such as genetic algorithm (GA), particle swarm optimization (PSO) and colony optimization (ACO) as in [ 234 ].

Job Shop Scheduling (JSS) JSS is a NP-Hard problem in which the algorithm seeks to consume a polynomial time to solve it. In JSS we need to process a finite set of jobs using a limited set of machines. JSS is a general type of scheduling problem. JSS is addressed by many MAs in [ 235 , 236 , 237 ], and [ 238 ].

6.4 Medical Science

Most of medical activities (i.e., Diagnosing, imaging, treatment, and monitoring) depends in its work on the computer or electronic device that is operate using an algorithm-based software [ 68 ]. Several researchers have used GAs for edge detection of images acquired using different imaging modalities, including magnetic resonance imaging, CT, and ultrasound [ 239 ]. In [ 240 ], Pereira et al., applied a set of computational tools for mammogram segmentation to improve the detection of breast cancer using GA combined with wavelet analysis to allow the detection and segmentation of suspicious areas with 95% sensitivity. GA has been applied for feature selection to identify a region of interest in mammograms as normal or containing a mass [ 241 ]. Also GA is combined with a support vector machine to differentiate benign and malignant breast tumours in ultrasound images [ 242 ], GA is combined with diversity index to discover lung nodules by developing an automatic threshold clustering method [ 80 ]. In [ 243 ] electroencephalography signals were used to detect hypoglycemia in patients with type 1 diabetes. Depending on neural networks in conjunction with ant colony optimization (ACO) and particle swarm optimization (PSO) Suganthi and Madheswaran use a more advanced computer-aided decision support system and mammogram to group tumours and detect breast cancer stages as described in [ 244 ]. Based on artificial bee colony (ABC) algorithm Kockanat and et al., Develop a technique for demonising images using 2D impulse response digital filter as illustrated in [ 245 ].

6.5 Robotics

Robotics is a vital active research field that owns some challenges that needs to be optimized such as task performance, Decrease the robotics cost, achieve a better reliability, in addition to minimize the unit complexity over other traditional robot systems. In this context, metaheuristics can be used to tune machine learning methods to enhance the collaborative behavior of robotics. One of the most active problems in robotics is the redundant humanoid manipulator issue. The complexity of this problem comes from the existence of multiple number of degrees of freedom and complex joint structure. This problem causes difficulty in achieving an inverse kinematics solution. Scientists make an effort to formulate this problem as a minimization problem, hence the MAs can perform better in solving this problem. In [ 246 ], the multilayer perceptron neural network is trained by the exploitative and explorative capabilities of the bee’s algorithm to learn the inverse kinematics of a robot manipulator arm. To conquer the problem of multi-solution, the GA is used to achieve a global optimal solution for inverse kinematics of 7-DOF (seven degree of freedom) manipulator [ 247 ]. Also, the inverse kinematics of the seven-degree-of-freedom (7-DOF) manipulator is perfectly tackled by the particle swarm optimization algorithm (PSO) by exploiting the strong intelligent scene and collaborative behavior among particles [ 248 ]. Biogeography-based optimization (BBO) is hybrid with differential evolution (DE) and uses the merits of the hybrid migration operator and the adapted Gaussian mutation operator to solve the inverse kinematics problem of the 8-DOF redundant humanoid manipulator [ 249 ].

6.6 Finance

Metaheuristics algorithms can be one of the most promising techniques used to solve different types of problem that occur in the finance and banking activities. In the following points, we will introduce a list of the most familiar problems and how the metaheuristics used to solve these problems.

Portfolio optimization and selection problem (POSP) in this problem, investors seek to assign optimal weights to the assets of the portfolio to achieve a minimal risk of investment. In [ 250 ], the authors provide a survey to solve POSP using metaheuristics and examples. Furthermore, the three GA, TS, and SA metaheuristic algorithms are used to solve POSP. The authors of [ 251 ] use the PSO algorithm to solve the POSP version with a cardinality constraint.

Index tracking problem (ITP) The ITP is a trading strategy that can depend mainly on two processes (hold and buy). In ITP we want to simulate the behavior of the index of the stock market using a minimum number of stocks. In other words, the ITP is developed to passively simulate the performance of the stock market index. For the specific German index, the authors of [ 252 ] use the SA to minimize tracking errors. The combinatorial search is hybrid with the DE for solving the ITP. The authors of [ 253 ] compare the performance of GA with quadratic programming and propose a solution approach to minimize the returns on the index using data from the FTSE100 index. Finally, in [ 254 ] the authors conducted a set of experiments to solve a special type of ITP and noticed that there was an improvement in an index.

Options pricing problem (OPP) Speculative activities are one of the most familiar tasks in financial markets, and the option can be one of the tools for speculative activities. Due to the fast dynamic motion of the financial market, it is difficult to guess the price of the option using traditional methods, so metaheuristic algorithms can be a promising choice in that case. In order to find parameters that achieve consistency between the model and market prices, Gilli and Schumann [ 255 ] use the PSO and DE to study the pricing of the calibration option. Finally, the authors of [ 256 ] have shown that the pricing of option operations can be enhanced compared to the traditional binomial lattice method when we use the ACO algorithm.

6.7 Telecommunications Networks

The recently needs for developing complex and large computer systems lead to an urgent demand for designing and developing high quality and more extensive network design and routing techniques and optimally solving problems in such an area. Also, we can notice that most problems in telecommunications are complex and hard to solve using traditional techniques and approximate algorithms, so there is urgent need to employ metaheuristic algorithms to solve network design and routing problems. A set of nodes (i.e., computers, databases, equipment, or radio transmitters) can be connected together using a transmission link (i.e., optical fiber, copper cable, radio, or satellite links) to construct communication networks. Under a set of constraints such as reliability, throughput, delay and link capacity, we seek to achieve a minimum cost of configurations as an objective function for these networks, and many problems can be appeared such as number of nodes, number of routing paths, the frequency assignment, and the capacity installation. A large number of studies using metaheuristics in solving telecommunications problems such as Kim et al. [ 257 ] employ a SA algorithm in the mobile radio system to allocate the nominal cells of channels. To minimize the installation cost and maximize traffic, the authors in [ 258 ] use the tabu search algorithm with randomized greedy procedures to find the location of the base stations of the universal mobile-based communication system. Specifically, good approximate solutions for large and medium-sized instances are obtained by the randomized greedy procedures, and these solutions were improved by using the tabu search algorithm. Finally, a new metaheuristic algorithm developed based on the Genetic Algorithm and Ant System was proposed to achieve better and efficient solutions for real-life transportation network design problems in large real networks located in two different places (Canada, city of Winnipeg) [ 259 ].

6.8 Food Manufacturing Industry

Recently, the metaheuristics can be considered as one of the most widely used efficient decision-making techniques that can be used to solve problems in different disciplines. In this section, we will present brief information about using metaheuristics in one of these disciplines, which is the food manufacturing industry. Specifically, metaheuristics can be applied in many food processes such as thermal drying, fermentation, and distillation. In [ 260 ], the authors develop a new hybrid method based on artificial bee colony (ABC) and the record-to-record travel algorithm (RRT) for Optimizing the Traceability in the Food Industry. The proposed method is employed to solve and provide the optimal minimal solution for the batch dispersion manufacturing problem. The hybrid RRT-ABC is used in the French food industry to carry out real-world experiments (that is, sausage manufacturing) to obtain high-performance results compared to traditional methods. The Artificial Bee Colony Algorithm (ABCA) used in the development of a delivery route optimization method to achieve a fresh food distribution without decreasing the quality of the food [ 261 ]. Finally, in [ 262 ], the Simulated Annealing (SA) is hybrid with the Virus Colony Search Algorithm (VCS) to improve the quality of the result of a sustainable Closed-Loop Supply Chain Network (CLSCN) design in the olive industry.

7 Open Issues and Challenges

However, the good features and abilities of the MAs in solving a wide range of problems, like other techniques, suffers from a set of problems in the following points, we will refer to these problems.

The stochastic nature and near optimal solution As we know, generating an optimal solution is one of the main features of deterministic algorithms such as simplex method. On the contrary to that, the metaheuristics algorithms (as it is a stochastic algorithm in nature) does not guarantee optimality of the obtained solution, but it provides an acceptable solution. This is one of the significant disadvantages of MAs. It is worth mentioning that the deterministic methods (unlike the stochastic methods) face difficulty when dealing with high-complex problems (that is, high-dimensionality and non-differentiable problems). Practically, when we decide to use one of the previous two methods, we choose to gain something and give the other.

The scale-ability and expensive computational cost Practically, the MAs score great promising results in solving problems in different natures such as discrete, continuous and combinatorial problems that contain a large number of decision variables. However, when solving large-scale global optimization problems (LSGOP) the MAs consume an expensive amount of computational cost. This scalability, challenge is one of the most important challenges that researchers must consider in the future due to the great growth in the size of the optimization problems when dealing with high-dimensional machine learning and large-scale engineering problems. In this context, many strategies are developed by the researchers to cover this problem such as the parallelization, approximation and surrogate modelling, hybridization of local search and memetic algorithms, decomposing the big problems into sub-problems, and befit from the sampling techniques.

The weakness of theoretical and mathematical analysis In most sciences such as chemistry, Biology, physics and others, the mathematical analysis of a method can be computed accurately to specify how much the method costs in terms of computational cost. Unlike those sciences, in metaheuristics we encounter a challenge in computing the exact computational cost of the algorithm, the reason behind this difficulty is from mathematical perspective it is difficult to analyze why the metaheuristics algorithms are so successful. Also, researchers need to pay attention to solving problems in determining the convergence analysis of many metaheuristics’ optimization algorithms. Finally, researchers also need to develop innovative methods that allow researchers to easily analyze and compute the algorithm's cost in the case of modification and scaling up the algorithm.

Intensification and diversification trade-off The algorithm's degree of effectiveness is measured by the ability of the algorithm to transit smoothly between the exploration (that is, explore as much as possible the feasible area) and the exploitation (that is, achieving good steps towards the optimal solution's area) stages. Achieving a high degree of intensification and diversification balance is one of the most important challenges or issues in most MAs. However, some algorithms achieve an acceptable degree of trade-off between exploration and exploitation; the vast majority of MAs need to address this challenge by scoring a high level of global diversification and local intensification [ 263 ].

Large-scale real-world problem formulation Nowadays the vast majority of problems in recent fields such as data science and big data analysis tasks are considered as large-scale real-world problem (LSRP) that is due to the large number of problem components and problem dimensions. Formulating a large-scale real-world problem (LSRP) is one of the crucial issues in metaheuristic algorithms. The issue comes from the large number of optimization variables (decision variables) included in the problem, how these variables interact with each other, how much the variables or components are related to each other, and what is the effect of one variable on the other variables. Also, it is worth mentioning that the large number of variables is translated as the problem size, which affects the computational cost of the algorithm that deals with this problem.

The limitations of the No-Free-Lunch theorem One of the most fundamental theories in the field of optimization is the No-Free-Lunch theorem [ 264 ] which states that there is no universal optimizer for all kinds of problems that is the algorithm may do better in some kinds of problems and do no better for the other kinds. We cannot generalize this theory, as it has been proved for the type of single-objective optimization, but it does not hold yet for problems with continuous and infinite domains in addition to multi-objective optimization [ 265 , 266 ]. In this context, the researchers in the field of metaheuristics must answer how to apply the NFL in terms of several dimensions?

Comparing different algorithms Comparing similar algorithms through the absolute value of the objective function or number of function evaluations is a possible task. On the other hand, we encounter a problem in comparing different algorithms with different objectives through a formal theoretical analysis. Practically no fair/honest or rigorous comparisons exist in this field [ 267 ].

Parameter tuning and control The algorithm's parameter plays the most vital role in determining the performance of any optimization algorithms. The algorithm's designer can change the performance of the algorithms by applying the parameter tuning process of the algorithm. Specifically, we can say that poor tuning leads to poor performance, and the opposite is true. As mentioned in [ 268 ], it is practically not an easy task to tune the algorithm parameter and control it by changing its values. Another point we must refer to is that, for well-tuned parameters, there are no clear reasons for unchanging the values of these parameters during the optimization process. Until now, the process of parameter tuning has been implemented by applying parametric tests, while parameter control can be implemented stochastically in which the values of the parameters are picked randomly within a prespecified range. Therefore, there is an urgent need to develop automatic systematic methods to control and tune the parameters. The authors in [ 269 ] and [ 270 ] proposed a self-tuning method as a trial to encounter problems of parameter tuning and control, but with this trial, the computational cost is still expensive. Based on the previous notes, there is an urgent need to develop an automatic method that applies an adaptive change of the parameters in addition to less effect on the computational cost of the algorithm.

The lack of big data applicability Dealing with big data and developing a big data algorithm has turned into an urgent demand today as the data volume has increased dramatically with the help of automatic data collection methods. In this context, we noticed that there is no more concentration on the application of metaheuristics on big data in the current literature. There are no more studies on how to benefit from applying metaheuristics along with big data algorithms. Consequently, in this review, we inform the researchers to spend more effort and trials in developing new reliable methodologies and algorithms to solve big data problems with the help of metaheuristics.

The lack of machine learning and metaheuristics combination One of the most powerful and influential methods for making a decision and performing a predictions task is the machine learning (ML). Recently, very helpful results have been achieved by the ML techniques. So, researchers in the metaheuristics field must pay an attention to methods that benefit from the ML techniques in optimizing the work of current MAs algorithms or developing a new ML-based metaheuristics algorithms. The following points may be helpful and promising with regard to this point.

Using the new advances in reinforcement, ensemble, and deep learning in applying an automatic choice of specific problems to be handled by existing and new optimization algorithms [ 271 ].

Benefit from the capabilities of machine learning techniques in optimizing the work of the optimization field by generating an automatic model for representing the optimization problems, adjusting the analysis techniques for analyzing the search space, in addition to beating large and complex problems by decomposing them into smaller size problems [ 272 ]. In another prospective, we can use the ML capabilities in applying automatic configurations of the algorithms by allowing the ML algorithms to choose the appropriate values for the algorithm's operators, especially for metaheuristic algorithms due to a large number of parameters [ 273 ].

Shortened the gap between the metaheuristic’s algorithms and the problem domain knowledge Treating the problem as a black box is a double-edged weapon. However, this can be considered as a strength of the metaheuristic’s algorithms over other algorithms, but it also a challenge. Considering and integrating the domain knowledge of the problem with the designed algorithm will dramatically increase its performance. For example, a problem-orientated research direction can be obtained by designing the algorithm's operator and search mechanisms based on the characteristics of the problem which also can be benefit in reducing the complexity of the algorithm by considering the optimality conditions of the problem being considered [ 273 ].

In summary, the following observations from the experiment are:

Apply the MAs on parallel computing and combine the metaheuristic techniques with the modern parallel computing technologies to generate a powerful method matched with the future generation of computing.

Exploit the benefits of artificial intelligence and machine learning techniques to provide new algorithms that have the ability to automatically adjust the parameters and automatically analyze the algorithms.

Developing new methods directed towards strengthens the ability of MAs in addressing the large-scale global optimization (LSGO) problems.

A great effort must be paid for the hybridization process to allow the algorithms to use the Powers of many algorithms, also generating intelligent techniques that can provide the researcher with insights about what algorithms best suited to be hybridize together?

7.1 Emerging Technologies

After discussing the open issues and challenges, we see that there is much future work in the field of metaheuristics, a set of guidelines must be declared to help the future researcher in the field to address these challenges. In this section, the guidelines used to dive deeper into potential future research directions are introduced. Specifically, we will concentrate on two emerging technologies which are machine learning and quantum computing and how these technologies enhance the optimization process.

7.1.1 Quantum-Inspired Metaheuristics

Metaheuristics can be employed to obtain a global optimal solution for a wide range of different problems in different computational aspects. These methods can benefit from the concept of quantum computing (QC) to enhance the solutions obtained. Hybridizing the quantum computing with the metaheuristics will produce a quantum-inspired metaheuristic algorithm (QIMAs). QIMAs can be considered as an alternative approach to classical optimization methods for solving the optimization algorithm [ 274 ]. The main idea behind the QIMAs is to better use the quantum computing principles with the metaheuristics in order to boost the performance of the classical optimization algorithms by scoring a higher-performing results than traditional metaheuristic algorithms. Specifically, the use of QC in metaheuristics will accelerate convergence, enhance exploration, enhance exploitation, and provide a good balance between the two capabilities of the algorithm. The most promising merit that affects the performance of the algorithm is the parallel processing feature in QC [ 275 ]. Finally, QIMAs can be used in different disciplines such as engineering and science.

7.1.2 Intelligent Optimization

In this section we will introduce a new type of optimization that is considered as one of the most promising topics in the future of the metaheuristic field. Intelligent optimization (IO) is developed as a test to intelligently adjust the set of inputs and their values to achieve an optimal output(s). In other words, IO cost minimal consumption in determining and choosing the optimal solution among all possible solutions of the problem. The importance of using the IO is dramatically increased when solving complex and NP-hard problems in which the selection of the optimal solution through an exhaustive search is considered impossible or practically difficult. In addition, IO can be used as an important solution for the time-consuming problem of many optimization algorithms. IO can be used in all steps of the optimization process, such as defining the problem, handling, and formulating the objective function(s) and constraints.

7.1.3 Hybrid Metaheuristics and Mathematical Programming

In the last years, hybrid optimization algorithms have achieved promising results compared to classical optimization algorithms. The main aim behind the hybrid metaheuristics is to provide a reliable and high-performance solutions for the large and complex problems. One of the most widely used combinations is hybrid metaheuristics with mathematical programming approaches. This combination will increase the quality of the solution, as it benefits from the two methods in determining an exact solution in a reasonable amount of time. The following points define the mathematical programming approaches that can be used with metaheuristics to increase the quality of the solutions obtained [ 276 ].

Enumerative algorithms: in this approach we can use one of the well-known tree search methods such as dynamic programming and branch and bound. These methods follow the divide-and-conquer strategy where the search space can be divided into smaller search spaces, and then in each sub area we apply the optimization separately. By applying this strategy, the quality of the solution will increase, and the time consumed will decrease.

Decomposition and Relaxation methods: in this approach we can decompose the large problem using the Bender’s decomposition method or apply the Lagrangian relaxation method to convert the problem into smaller problems.

Pricing and Cutting plane algorithms: in this approach, we prune using polyhedral combinatorics.

8 Conclusion

In this review, a comprehensive study of metaheuristic algorithms is introduced that involves defining the concept of optimization. Studying the appearance of metaheuristic term. Introducing an explanation of the features of the MAs more than other techniques; Different taxonomies of the MAs according to different aspects such as inspiration source, number of search agents, population updating mechanisms, and number of parameters. Studying the metrics used in the Performance Evaluation of the algorithm. A great effort is paid to clarify the optimization problem in detail, concentrating on different classification techniques, and, moreover, the study reviews the use of metaheuristics in different application areas such as engineering design problems, NP hard problems, medical science, and robotics. Finally, we introduce some of the issues that exist in the MAs literature and the future directions of this important field.

Data Availability

Data sharing is not applicable to this article as no data sets were generated or analyzed during the current study.

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Essam H. Houssein, Mahmoud Khalaf Saeed & Mustafa M. Al-Sayed

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Essam H. Houssein: Supervision, Software, Methodology, Conceptualization, Formal analysis, Investigation, Visualization, Writing—review and editing. Mahmoud Khalaf Saeed: Conceptualization, Visualization, Software, Data curation, Resources, Writing—original draft. Gang Hu: Formal analysis, Resources, Validation, Writing—review and editing. Mustafa M. Al-Sayed: Conceptualization, Formal analysis, Visualization, Resources, Validation, Writing—review and editing. All authors read and approved the final paper.

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Houssein, E.H., Saeed, M.K., Hu, G. et al. Metaheuristics for Solving Global and Engineering Optimization Problems: Review, Applications, Open Issues and Challenges. Arch Computat Methods Eng (2024). https://doi.org/10.1007/s11831-024-10168-6

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Received : 19 December 2023

Accepted : 25 March 2024

Published : 21 August 2024

DOI : https://doi.org/10.1007/s11831-024-10168-6

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