Abstract
Throughout the centuries, nature has been a source of inspiration, with much still to learn from and discover about. Among many others, Swarm Intelligence (SI), a substantial branch of Artificial Intelligence, is built on the intelligent collective behavior of social swarms in nature. One of the most popular SI paradigms, the Particle Swarm Optimization algorithm (PSO), is presented in this work. Many changes have been made to PSO since its inception in the mid 1990s. Since their learning about the technique, researchers and practitioners have developed new applications, derived new versions, and published theoretical studies on the potential influence of various parameters and aspects of the algorithm. Various perspectives are surveyed in this paper on existing and ongoing research, including algorithm methods, diverse application domains, open issues, and future perspectives, based on the Systematic Review (SR) process. More specifically, this paper analyzes the existing research on methods and applications published between 2017 and 2019 in a technical taxonomy of the picked content, including hybridization, improvement, and variants of PSO, as well as realworld applications of the algorithm categorized into: healthcare, environmental, industrial, commercial, smart city, and general aspects applications. Some technical characteristics, including accuracy, evaluation environments, and proposed case study are involved to investigate the effectiveness of different PSO methods and applications. Each addressed study has some valuable advantages and unavoidable drawbacks which are discussed and has accordingly yielded some hints presented for addressing the weaknesses of those studies and highlighting the open issues and future research perspectives on the algorithm.
Similar content being viewed by others
Avoid common mistakes on your manuscript.
1 Introduction
It has been proven in the literature beyond any doubt that metaheuristic optimization algorithms perform well by optimally handling several versatile realworld optimization tasks, ranging from robotics [31], wireless networks [195, 206], power systems [40, 76], job shop scheduling [109, 137], to classification [35] and training of artificial neural networks [83]. While questing the global best (nearoptimum) solution, numerous fitness evaluations are required by the metaheuristic algorithms. This typically poses a serious barrier against the application of metaheuristic algorithms to highcomputational optimization problems that extensively exist in computational fluid dynamic optimization [21] and structural optimization [79], among many others. For working out those problems, the performance of candidate solutions is usually evaluated with highfidelity numerical analysis approaches (e.g., computational fluid dynamics simulations or finite element analysis), which may deduct CPU time from minutes to hours, or even days [80, 222]. Therefore, within the paradigm of metaheuristics, many Swarm Intelligence (SI) techniques and variants were proposed to tackle complex/largescale optimization problems.
SI, a substantial branch of optimization techniques, is the characteristic of a system where agents interact locally with their environment so that their collective behaviors render the emergence of cohesive functional global patterns. Unlike Evolutionary Algorithms (EAs), SI techniques are inspired by agents’ plain behaviors and selforganizing interactions, such as fish schooling, honey bees, bacterial growth, animal herding, bird flocking, ant colonies foraging, and so on. Indeed, Beni [17] was the first to use the SI term in cellular robotic systems, in which simple agents interact with neighbors to organize themselves. SI was formally established in [18, 19]. The rife SI algorithms include Ant Colony Optimization (ACO) [37] and Particle Swarm Optimization (PSO) [91]. Less widespread SI algorithms are Bacterial Foraging Optimization (BFO) [141], Artificial Bee Colony (ABC) [87], Firefly Algorithm (FA) [201], and many others. SI algorithms were primarily developed for steady optimization problems. However, dynamic environments involve several realworld optimization problems [124].
Typically, a swarm is defined as a vast number of simple, homogeneous agents interacting locally with their environment, as well as themselves, with decentralized control to authorize the emergence of a global important behavior. Swarmbased techniques have recently arisen as a family of swarmbased, natureinspired algorithms that have the ability to produce robust, fast, and low cost solutions to numerous complex problems [50, 69]. Therefore, SI can be figured out as a major category of Artificial Intelligence (AI) that is utilized to model the collective behavior of natural social swarms, such as honey bees, bird flocks, and ant colonies. These agents (swarm individuals or insects) are relatively gullible with simple own capabilities. However, they perform cooperatively tasks substantial for their survival through interacting together in particular behavioral manners. Socially, swarm individuals can directly or indirectly interact among themselves [202]. Direct interaction can be through audio or visual contact (e.g., a waggle dance of honey bees), while indirect interaction is evident when the environment is changed by one individual and the other individuals react to the new environment (e.g., pheromone tracks of ants that look for food sources through depositing on their way). This indirect pattern of interaction is known as “stigmergy”, which denotes communication through the environment [42]. The research area presented in this thorough paper focuses on SI. More specifically, this review strives to explore one of the most popular models of SI, PSO, which is inspired by birds’ flocking behavior.
PSO is a swarmbased stochastic algorithm proposed originally by Kennedy and Eberhart [44, 91], which exploits the concepts of the social behavior of animals like fish schooling and bird flocking. In PSO, each potential solution to a given problem is viewed as a particle with a certain velocity flying through the space of the problem just like a flock of birds. Each particle then combines – with some random disturbances – some aspect of the record of its own historical best location and current location with those of one or more agents of the swarm to determine its next movement through the search space. After all particles have been moved, the next iteration occurs. The swarm as a whole (e.g., a flock of birds collectively searching for food) is probably to gradually approach the objective function optimum. PSO has eventually gained prevalent vogue amongst researchers and emerged to provide high performance in an assortment of application areas, with the potential to hybridize and specialize and demonstrate some appealing emergent behaviors. PSO has a main advantage of having fewer parameters to tune. PSO obtains the best solution from particles’ interaction, but through highdimensional search space, it converges at a very slow speed towards the global optimum. Moreover, in regard to complex and large datasets, it shows poorquality results. If there is a large number of dimensions in the problem at hand, PSO usually fails to discover the global optimum solution. This phenomenon is caused not only by the presence of local optima trap, but also the potential fluctuation of the velocities of particles such that the successive range of trials is bounded within a subplain of the whole search hyperplain [178].
The local optima issue in PSO has been discussed and diverse variants of PSO algorithm were developed to tackle that issue. For example, some of these variants have been developed by incorporating the capabilities of EAs (e.g., the adaptation of PSO parameters, hybrid versions of PSO, etc.), thereby producing adaptive PSO versions. Other researchers have taken on incorporating evolutionary operators, such as crossover, mutation, selection, as well as the Differential Evolution algorithm (DE) itself, into PSO. Consequently, hybrid versions of PSO have been tested and produced, including the hybrid evolutionary PSO [128], Genetic Algorithm (GA) and PSO [48, 116], genetic programmingbased adaptable evolutionary hybrid PSO [154], and many others [62]. Such improvements perform well with PSO and have the potential to avoid getting stuck at local optima. However, the problem of premature convergence in some highdimensional complex problems still exists, even if the local optima obstacle is absent. Hence, PSO does not always work properly for highdimensional models [54].
To this point, the performance of PSO has generally been improved by developing different variants of the algorithm. However, few review papers and technical surveys have systematically addressed the PSO literature [190, 216]. Consequently, the primary objective of this work is to present a systematic survey by reviewing the PSO algorithm and its different methods, as well as a wide variety of indicative application domains. Moreover, some of the addressed studies have highlighted the main approaches of PSO applications, including healthcare, environmental, industrial, commercial, smart city, and general aspects. The key solicitude of those PSO methods and applications is satisfying quality standards: services arising from PSO methods and applications should support user’s requirements that cover the quality metrics of accuracy and efficiency, such as convergence rate, computational cost, consistency, stability, diversity, etc. In other words, this paper aims to present a timely, compendious, systematic and an indepth overview of the PSO algorithm between 2017 and 2019 and the opportunities and challenges imposed during this period.
The structure of this study is organized as follows. Section 2 presents the related works. A brief foundation of the PSO is presented in Sect. 3. Section 4 exploits the Systematic Review (SR) procedure to provide the proposed research approach and motivation. Section 5 demonstrates the various methods of PSO based on the SR method. Section 6 systematically outlines and categorizes the application approaches in PSO. Also, a technical classification and differentiation of the approaches in the addressed papers is presented in this section. Section 7 provides a discussion on the PSO approaches that have not yet been thoroughly analyzed. Finally, Sect. 9 concludes the paper along with its limitations.
2 Related Work
A swift explanation is presented in this section for the general related studies in the PSO algorithm.
Poli et al. [143] presented an overview of the great efforts which have given impetus and direction to research in particle swarms, as well as some important new applications and directions. An analysis of IEEE Xplore and Google Scholar citations and publications from 1995 to 2006 were presented in this work, illuminating the sense meant by Kennedy and Eberhart [92]. The strength of this study was to present comprehensive challenges and open issues in the PSO algorithm. However, this study did not consider the compatibility of PSO application with each presented approach.
Banks et al. [13, 14] offered, in two parts, a timely and brief review of the field in general, alongside the opportunities and challenges emanating from the versatile application of PSO. On the one hand, part I [13] has considered the history and background of PSO and its position within the broader paradigm of natural computing. The review then continued to discuss different improvements to the native formulation of PSO both in discrete and continuous problems, swarm behavior analysis, and measures considered to address stagnation. Furthermore, the review focused on research regarding adaptations for parallel implementation, algorithm configuration, and dynamic environments. The achievement of this study was identifying two significant areas of challenge for future further development: swarm stagnation and dynamic environments. The shortcoming of this part is the insufficiency of the explanation the related work. On the other hand, part II [14] has discussed recent studies in some of the impressive areas of research: constrained and multiobjective optimization, combinatorial problems, and hybridization. In that review, a number of domains were briefly touched upon, including the optimization of neural fuzzy networks, artificial neural networks, computational biology, image processing and medical imaging, optimization of electricity generation, network routing, and financial forecasting. Nevertheless, the study has a main defect of not analyzing the selected studies in terms of evaluation parameters, such as convergence rate, diversity, accuracy, and processing time, as quality factors in this area.
Kulkarni et al. [96] outlined some issues regrading Wireless Sensor Networks (WSNs), introduced PSO, and discussed the suitability of the algorithm for WSN applications. This study also presented a compendious survey of how PSO is tailored to tackle the issues of node localization, node deployment, data aggregation, and energyaware clustering as optimization problems. The strong point of this review is to comprehensively present open issues in WSNs. However, the compatibility of highspeed realtime applications with each approach was not considered.
Kothari et al. [94] reviewed the studies related to the modifications of native PSO and its practical application in realworld problems. The rapid modification of PSO has been emerged in different examples, including the twostep PSO and the PSOSupport Vector Machine (PSOSVM). The integration of the PSO and the practical implementation of PSO with the industry standard algorithm have also generated stunning results. The advantage of this survey was to present recent diverse variations in PSO and analyze the accuracy of PSO in different domains. The main deficiency of this study is the lack of statistical information about the discussed standard PSO and its application in various specified contexts.
Imran et al. [71] discussed diverse variants of PSO in regards to swarm initialization, mutation operators, and inertia weight. The main advantage of this overview was to highlight the importance of introducing the different mutation operators and inertia weight parameter to improve the performance of PSO. However, other promising variants of PSO were not considered.
Alam et al. [7] surveyed systematically the evolution of clustering techniques based on PSO and presented the results of rapidly increasing trends in the literature of SI, PSO paradigm, and PSObased data clustering approaches, proving that there is a tremendous increase in the popularity of such approaches. This research affirmed that the approaches are novel and simple to implement and builds on communication and collaboration. This study outlined different application domains of PSO relevant to clustering. However, applications to more complex problems are absent. Furthermore, variants of PSO for clustering highdimensional data were surveyed by Esmin et al. [49]. The main advantage of this survey was to present the attempts of different publications in reducing data dimensions. The weakness is still the absence of discussing more clusteringrelated applications.
Marini and Walczak [123] described comprehensively the PSO algorithm to show its ability to solve different kinds of optimization problems in chemometrics. The importance of the appropriate choice of PSO metaparameters has been highlighted in this work by means of selecting practical examples in the subject areas of variable selection, estimating robust Principal Component Analysis (PCA) solutions, and signal warping. This tutorial contributed to presenting works subject to chemometrics impressively. However, it lacked splashing other stateoftheart fields.
Zhang et al. [216] investigated comprehensively the PSO algorithm. They provided advances in PSO, including its theoretical analysis (convergence analysis, parameter tuning, etc.), hybridization (with GA, DE, ABC, ACO, biogeographybased optimization, harmonic search, Tabu Search (TS), Artificial Immune System (AIS), and Simulated Annealing (SA)), modifications (including fuzzy PSO, chaotic PSO, barebones PSO, quantumbehaved PSO), population topologies (including star, ring, random, von Neumann, fully connected, etc.), extensions (to binary, discrete, constrained, and multiobjective optimization), and parallel implementation (in cloud computing, multiprocessor, multicore, and GPU forms). Moreover, they introduced a survey on the applications of PSO to the following eight fields: biology, chemistry, medicine, electrical and electronic engineering, fuel and energy, mechanical engineering, operation research, communication theory, and automation control systems. Despite all that, the publication analysis demonstrated in this survey lacks the presentation of the annual exponential fluctuation of publications for each variant and application field.
Wang et al. [190] presented the inception and background of the PSO algorithm, and carried out a theoretical analysis of the algorithm. Then, they analyzed its current situation of application and investigation in algorithm structure, topology structure, parameter selection, multiobjective optimization, discrete and parallel PSO algorithm, and engineering applications. This overview is characterized by suggesting distinctive future research directions. However, no analytic discussion is introduced in this work.
Summarizing, Table 1 outlines the recent review studies on PSO with respect to the general survey and review studies discussed in this section. This table lists the key topics, publication year, and covered years (if available) for each study. The deficiencies in the existing review papers propose that a comprehensive and systematic literature review should be provided to address some of such common weaknesses as:

In existing studies, the methods and applications of PSO are not organized in a clear taxonomy, and do not have analytical assessment from which one can learn something beneficial.

Important assessment factors are not involved in the methods and applications of PSO in some current studies.

The systematic arrangement is typically absent in the structure of literature studies, and the paper selection criterion is vague as well.
3 Particle Swarm Optimization: PSO Mechanism
3.1 SelfOrganization Features
SI system has a major feature, namely, selforganization, in which the components of an initially disordered system interact locally to produce a coordination or global order. This process is characterized by spontaneousness; that is, no agent inside or outside of the system dominates the interaction. The selforganization in swarms was interpreted by Bonabeau et al. [25] through three key components as follows:

Robust dynamical nonlinearity (always comprising positive and negative feedback) convenient structures are promotionally being created with the help of positive feedback, while this positive feedback is counterbalanced and the collective pattern is stabilized with the help of negative feedback.

Tradeoff between exploration and exploitation A valuable mean creativity artificial approach is provided through a suitable balance that is identified by SI.

Multiple interactions Information coming from neighbor agents in the swarm are used by individual agents, allowing information to be disseminated throughout the network.
3.2 SI Features
Millonas [127] proposed that SI must satisfy five basic principles: adaptability, diverse response, stability, quality, and proximity. Table 2 lists their meanings.
3.3 Standard PSO Algorithmic Structure
A swarm of particles updates their relative positions from iteration to another, boosting the PSO algorithm to duly perform the search process. To get the optimum solution, each particle moves towards its prior personal best position (\({\mathbf {p}}_{best}\)) and the global best position (\({\mathbf {g}}_{best}\)) in the swarm [215]. Assuming a minimization problem, one have
where \(i\in \{1,2,\dotsc ,N\}\), and
where i denotes particle’s index, t is the current iteration’s number, f is the objective function to be optimized (minimized), \({\mathbf {x}}\) is the position vector (or a potential solution), and N is the total number of particles in the swarm. The following equations update, at each current iteration \(t+1\), the velocity \({\mathbf {v}}\) and position \({\mathbf {x}}\) of of each particle i as:
where \({\mathbf {v}}\) represents the velocity vector, \(\omega\) is the inertia weight utilized to balance the local exploitation and global exploration, \(\mathbf {r_1}\) and \(\mathbf {r_2}\) are random vectors uniformly distributed within the range \([0, 1]^D\) (D being the search space dimensionality or the size of the problem at hand), and \(c_1\) and \(c_2\), called “acceleration coefficients”, are positive constants.
An upper bound is commonly set for the velocity vector. As a means to prevent particles from shaving off the search space and forcing them to take a proper step size to comb the entire search domain, the “velocity clamping” method was used [163]. The “constriction coefficient” strategy is another method, proposed by Clerc and Kennedy [36], in which the velocities can be also constricted by theoretically observing and analyzing the swarm dynamics.
By scrutinizing Eq. (3), we can interpret that the first part, known as “inertia component”, represents the prior velocity, which provides the particles with appropriate momentum to rove across the search space. The second part, the “cognitive component”, denotes the own positiveness for every particle. It motivates the particles to move towards their own best positions found so far over subsequent iterations. The third part, known as the “social component”, indicates the collective effect of the particles to reach the global optimum solution [218].
3.4 PSO Pseudocode
Let \(f : {\mathbf {R}}^N \rightarrow {\mathbf {R}}\) be an objective function which needs to be simplified. Then, the function takes a vector of N real numbers denoting an N candidate solutions and outputs a real number that indicates the value of the objective function. The f gradient is either hard to calculate or often unknown. Then, the global minimum \(\left( \mathbf{g}_{best}\right)\) is pursued as exhibited in Algorithm 1.
3.5 PSO Flowchart
The flowchart to demonstrate the flux of PSO is depicted in Fig. 1.
4 Proposed Research Approach
This section depends on the SR method to present, in a taxonomy, a review of the research studies related to the methods and applications of PSO [71, 123, 143, 190, 216]. Other synonyms and alternatives of the key essential components are also considered to commit the final exploration string as [49, 94, 96]:

(“Methods” OR “Variants” OR “Hybrid” OR “Improved” OR “Software” OR “Application” OR “Application layer” OR “Applicationbased” OR “App”) AND (“PSO”) OR (“Particle Swarm Optimization”)
Figure 2 demonstrates the distribution (from 2017 to 2019) of research studies published by the most popular venues regarding the review methods and articles, including Elsevier, Springer, IEEE, ACM, MDPI, Taylor & Francis, Wiley, Inderscience, MDPI, etc.
In relation to the objectives of this SR paper, comprehensive answers are provided to the following four Analytical Questions (AQs) [156]:

AQ1: Into which domains can PSO methods and applications be classified?

AQ2: Which main contexts can be considered for PSO methods and applications?

AQ3: What assessment environments are used for evaluating the PSO methods and applications?

AQ4: What are generally the PSO open perspectives and future research directions that can be drawn from the work presented in the paper?
After outlining the AQs, the ultimate research selection was conducted by applying the inclusion/exclusion criteria. With respect to the number of published research papers, only conference papers and journal articles indexed in Web of Science (WoS) or Scopus proceedings are selected and analyzed as peerreviewed publications for the methods and applications of PSO. Finally, I selected 2,140 peerreviewed papers. Due to the length limitation of the paper, I addressed curtly only 84 stateoftheart research papers from reputed journals to address the four AQs in detail in Sects. 5 and 6. Figure 3 demonstrates the selection criteria and evaluation method for the literature studies. The exclusion phase was carried out by omitting white papers, short papers, book chapters, nonpeerreviewed studies, and lowquality researches (published in rapacious journals) that did not provide any technical information or scientific discussion. The ultimately selected studies are recognized by considering the inclusion criteria as follows:

Papers published online from 2017 to 2019.

Papers on PSO topics, including methods and applications.

Papers prepared in a technical quality method in PSO methods and applications.

Papers that follow the quality standards.
The ultimately selected studies are recognized by considering the exclusion criteria as follows:

Papers that present survey and review studies.

Papers indexed in neither WoS nor Scopus.

Papers written in a language other than English.

Papers not subject to peerreview.
5 Methods of PSO
In fact, the literature is rich in many PSO techniques. To organize them, a proper classification criterion is suggested, and various PSO methods are classified as per Fig. 4.
5.1 Hybridization
Hybridization is a generic model of two or more algorithms, which exploits their advantages while decreasing their impediments. The combination of techniques does well for tacking a specified problem, given that the obtained results can be improved by these combined techniques on their own. The exploitation and exploration of an algorithm can completely be enhanced through hybridization of algorithms [53, 177, 180]. For example, an algorithm can cater for the lack of its preciseness and refine the results through synergy with a local search method. The hybrid approach in the field of optimization is growing in popularity and utilizes the purport of hybridizing the components from leading optimization techniques to improve the performance of traditional optimization algorithms.
5.1.1 Metaheuristic Algorithms
Many natureinspired hybrid approaches have been developed by numerous researchers/scholars for achieving significant performance in the exploitation and exploration of existing algorithms. To optimally design a Labyrinth Spillway (LS) having quarterround or halfround crest shape, an evolutionary Hybrid Algorithm (HA) combining the PSO algorithm with the Bat Algorithm (BA) was developed by Ferdowsi et al. [52]. This way, the best outcomes of one algorithm substitute the worst ones from the other. Moreover, the proposed algorithm has a high convergence rate regarding the optimal absolute outcome and does not get stuck into the local minima. Responses and computational time of this HA proved that LS with quarterround crest requires more concrete, compared to the halfround one.
In another work, Melton [126] examined a strategy for hybridizing two heuristic algorithms, PSO and DE, to better address the stagnation problem when solving slewmaneuver timeoptimal problems having constraints in solution path. The stagnation was handled very well by the combined PSODE method, with a reduction of 40% in the computational time compared to just applying DE. In [34], a Dynamic MultiSwarm Differential Learning Particle Swarm Optimizer (DMSDLPSO) was proposed. In this study, the DE operators were incorporated into each one of the DMSDLPSO’s subswarms to form a novel method. DMSDLPSO has a good capability of exploitation and exploration by employing a local searcher like QuasiNewton method to enhance the exploitation capability, making, at the same time, use of the capability of exploration existing in the differential mutation. On the other hand, the optimization performance of DE was improved by developing a novel selfadaptive mutation DE algorithm based on PSO (DEPSO) [194]. DEPSO can significantly utilize the fast convergence capability of PSO and the strong global exploration capability of an improved DE mutation strategy. As a result, the diversity of the swarm was managed well throughout the evolution, resulting in a higher convergence speed.
In [12], Aydilek proposed a Hybrid algorithm combining both FA and PSO (HFPSO). HFPSO checks the global historically best fitness values in order to properly determine the proper start of the local search. The high accuracy of convergence and runtime was statistically approved by evaluating the method using expensive benchmarks from the highdimensional CEC’15 and CEC’17 functions.
Typically, a strong classifier can be created based on the strengths of penalization and, say, SVM, proving the effectiveness of Penalized Support Vector Machine (PSVM). In [6], AlThanoon et al. proposed a new hybrid PSO and FA to find the tuned parameters of PSVM, showing a high ability to escape from being trapped into the local optima trap.
In [223], a hybrid method of PSO and GA was proposed by Zhu et al. to optimize an antenna array deployment for locating the sources of Partial Discharge (PD) in an entire substation. First, an algorithm called DirectionofArrival (DOA) estimation was presented for testing its applicability to arbitrary array configurations. In order to minimize the objective functions, the hybrid PSOGA algorithm was applied to optimize the array deployments for the localization of both DOA and coordinates.
Genetic Learning PSO (GLPSO) breeds eminent exemplars to steer the motion of particles for the purpose of improving the performance of PSO. However, the performance of GLPSO is not satisfactory on complex optimization problems, in which a global topology is depicted for exemplary generation and efficient diversity cannot be retained to boost exploration. For the betterment of the adaptability and performance of GLPSO, Lin et al. [106] modified two versions of the basic GLPSO algorithm. In exemplar generation, a ring topology is depicted to boost diversity and so exploration, while the algorithm’s adaptability is improved by adopting a Global Learning Component (GLC) with control parameters which are linearly adjusted.
Nenavath et al. [135] proposed a novel hybrid algorithm of Sine Cosine Algorithm (SCA) and PSO (SCAPSO) to overcome the SCA premature convergence at local minima for the two major goals of object tracking and solving optimization problems. Two parameters of PSO, \({\mathbf {p}}_{best}\) and \({\mathbf {g}}_{best}\), were embedded into the conventional SCA to lead the process of pursuing the possible candidate solutions, so that the \({\mathbf {p}}_{best}\) in SCA is then used to initialize PSO swarm to further exploit the feasible search spaces.
PSO usually experiences premature convergence, so it is easily to be caught into a local optimum solution. Also, it is ineffective in balancing exploration and exploitation, especially in complex multipeak search functions. To overcome these shortcomings, a Hybrid Particle Swarm Optimizer with Sine Cosine Acceleration Coefficients (HPSOSCAC) was proposed [32]. First, Sine Cosine Acceleration Coefficients (SCAC) were presented to efficiently control the convergence to the global optimum solution by guiding the local search. Second, the swarm is initialized by adopting oppositionbased learning method. Additionally, the inertia weight is adjusted by utilizing a sine chaotic map. Finally, a modified formula for updating position was proposed.
The SCA is mainly stigmatized by its limitation to exploit only emerging regions, depending on the parameter settings. However, the SCA is well capable of exploration. Taking advantage of the PSO capability of strong exploitation, Issa et al. [73] developed a new enhanced version of the SCA by hybridizing it with PSO. Two layers have formed the resulting technique: In the bottom layer, the search agents of SCA explore the search space, and in the top layer, the region around the best result found by the layer at bottom is exploited. Hence, there is a balance between exploitation and exploration in the proposed technique, which maintains fast convergence and improves the solution quality.
In another research, Sanchez et al. [159] used a column generation framework to solve the Binary Cutting Stock (BCS) problem by developing a benchmark of different approaches. Master and subproblem are the main components of this framework. Classical integer linear programming was used to solve the master problem, while metaheuristic algorithms (PSO, SA, and GA) were used to solve the subproblem. The aim of this benchmark analysis was to compare the results of the hybrid metaheuristics with an exact approach.
In [68], Huang et al. presented a methodology to utilize various loadingunloading curves to extract the mechanical properties of its bulk materials using an Instrumented Indentation Test (IIT). Simulated Annealing PSO (SAPSO) coupled with finite iterative element simulation were used to obtain the mechanical properties, including strainhardening rate (K) and strainhardening exponent (n), from the IIT data.
Ibrahim et al. [70] proposed a hybrid optimization approach for solving the feature selection problem. The Salp Swarm Algorithm (SSA) was combined with the PSO as SSAPSO, in which the performance of the exploitation and exploration stages in PSO was significantly amended. Based on 15 benchmark functions alongside different UCI datasets, the original datasets were refined by removing the confusing or redundant features while yielding or keeping a better efficiency, demonstrating the improvement in the SSAPSO with respect to the prediction accuracy without affecting the computational cost.
Laskar et al. [98] proposed a new hybrid algorithm named as the Hybrid Whale PSO (HWPSO) algorithm. Since Whale Optimization Algorithm (WOA) is known for its very good capability of exploration, it is hybridized with PSO in a novel way to overwhelm the limitations related to a PSO phase (i.e., stagnation impact). The HWPSO has employed two novel techniques during hybridization: “Forced” whale in the exploration phase so that the WOA can guide PSO to better avoid getting stuck into local optima, and “Capping” phenomenon in the exploitation phase for restricting the search mechanism of WOA in order to converge to the global optimum value more fast.
A new form of multiswarm BBPSO, in which its updating distributions are reoriented by adaptively learning the optimal alignments, is presented by Vafashoar and Meybodi [183]. The idea is that, along appropriate directions, the prospect of generating new particles is maximized. Due to the lack of a priori knowledge of these alignment directions, the proposed method adaptively learns suitable alignments by employing a learning mechanism. Cellular learning automata were adopted to build this learning mechanism. For each particle in the presented method, multiple alignment strategies are developed. Moreover, these strategies are adjusted so that, during the search process, the particles are attracted toward the most promising regions based on the cellular learning automata.
DuránRosal et al. [43] proposed novel approaches based on time series segmentation. The proposed methods include the PSO algorithm adaptation to this problem, as well as more advanced variants of PSO (e.g., BBPSO and its exploitation variant (BBePSO)). Furthermore, a new algorithm, called Dynamic exploitation BBPSO (DBBePSO), was derived, in which the importance of the cognitive and social components are updated throughout the successive generations. A final local search step is considered to further improve the solutions obtained from these algorithms based on the incorporation of two popular traditional segmentation algorithms (TopDown and BottomUp).
In [134], Nagra et al. proposed an enhanced SelfInertia Weight Adaptive PSO algorithm with a gradientbased Local Search strategy (SIWAPSOLS) to solve the problem of premature convergence in PSO. This proposed algorithm takes the advantages of the exploration ability of the adaptive PSO with enhanced inertia weight, and the exploitation of the gradientbased local search strategy.
In [30], Cao et al. proposed a Comprehensive Learning Particle Swarm Optimizer (CLPSO) embedded with a local search utilizing the CLPSO’s capability of global search and the fast convergence capability of local search to achieve higher optimization performance. This work takes advantage of the introduced so called quasientropy index for addressing its main issue (specifically, when to start the local search) by developing an adaptive local search starting scheme.
In most of the abovediscussed studies, PSO was implemented without coherence with other metaheuristics. For instance, a good initial solution can be produced by utilizing DE or GA as a starting point, and the PSO can take care of the rest of the search. In some cases, an initial solution can be produced by using PSO as a starting point, and the search can be conducted by others. Furthermore, the PSO, as a supplementary tool, may be hybridized with the deterministic local search to boost the exploitation capability and produce highquality solutions. Additionally, the search speed may be elevated by the PSO.
5.1.2 Artificial Neural Networks (ANNs)
Artificial Neural Networks (ANNs) can be, in a word, summarized as learning via training [187]. To achieve that, a series of input and output vectors is adopted to handle a set of data that needs to be trained. At the training time, the ANN is applied to the training data iterative, so that the desired inputoutput mapping emerges by balancing weights of the network several times in a while. After the training stage completes, a foreseeing model is ready and the corresponding output vector can be then created from input ones unrelated to the training pairs. An appropriate set of weights and transferring functions should be carefully chosen (via, say, metaheuristic optimization) to ensure that the ANNs can present a quantifiable smooth function that would properly link future input and output vectors. Various components, including prediction and system control and modelling, are inferred to express the broadly connected neural systems.
In this regard, Moayedi et al. [131] introduced a PSOoptimized ANN model to solve the prediction problem of Landslide Susceptibility Mapping (LSM). The focus of this study was the prediction of landslide hazardous susceptibility mapping by applying a hybrid model of PSO and ANN (PSOANN). Two statistical performance metrics, Root MeanSquared Error (RMSE) and coefficient of determination (\(R^2\)), were used to assess the predicted results from both the original ANN and the PSOANN. Comparing the two models, the PSOANN model was observed to be more reliable in LSM estimation than the ANN.
In another work, Junior and Yen [84] proposed a novel algorithm based on PSO and Convolutional Neural Network (CNN), namely, PSOCNN. In comparison with other evolutionary approaches, the proposed algorithm is capable of fast convergence. In an application to image classification, it was able to automatically find deep meaningful CNN architectures. To allow the use of PSO optimization with CNN, a novel strategy for direct encoding, as well as a velocity operator were built.
5.1.3 Support Vector Machine (SVM)
Support Vector Machine (SVM) [56] is an intelligence algorithm lying under supervised machine learning [2]. In this algorithm, the data is analyzed and particular visible or hidden patterns are quested with the major aim to perform two main tasks of classification and regression. SVM is known for its potential to perform nonlinear classification and linear separation of the data, to build another dimensional space from existing training datasets by investigating the respective categories. The separation process is working by increasing the chasm between too close points of various categories (i.e., support vectors) to better explore the optimal separating hyper plane. It should also point out that SVM can perform other tasks like linear classification [41].
In this framework, Hoang et al. [61] proposed a novel Differential PSObased SVM (DPSOSVM) classifier to monitor the conditions of a surge arrester. In the proposed method, the parameters of SVM classifiers were optimized by investigating the DPSOSVM technique configuration that gives the best results. Input vectors composed of three extracted features are used to assess five arrester conditions, including degradation (D), tracking (T), prefault (A), normal (N), and abnormal (U). Meanwhile, a DPSObased ANN classifier was also involved, to carry out a comparative study on fault diagnosis.
In another work, Tharwat and Hassanien [179] optimized the SVM parameters by employing quantumbehaved PSO (QPSO) for reducing the classification error. The proposed model, called QPSOSVM, was evaluated by adopting 7 traditional classification datasets from the UCI data repository. The experimental results revealed the capability of the proposed model to obtain the best values of the parameters of SVM. Moreover, the results showed lower classification error rates than those obtained by the standard versions of PSO and GA.
5.1.4 Quantum
40 years ago, quantum computer [20] was invented, and in the late 1980s, the quantum computer formal definition was given. Due to the potential of the quantum computer in various special problems [85], this field has witnessed many great efforts. Various popular algorithms are developed, and the most wellknown one of them is Shor’s quantum factoring algorithm [185]. In 2004, Sun et al. [170] firstly introduced the Quantumbehaved PSO (QPSO) for improving the convergence rate of the native PSO. In quantum space, particles often have the guarantee to reach the real global optimum via searching throughout the full solution space. Later in recent decades, QPSO has drawn great attention from many scholars. In what follows, some of them are presented.
In [81], Jmal et al. proposed a QPSO method for solving the KTraveling Repairman Problem (KTRP). The proposed approach incorporates a KTRPspecific repair operator to ensure that a feasible solution space will be always there to guide the search process and that the quality of solutions is significantly improved as much as possible.
In [147], an optimal path planning was proposed for freefloating twowheel pendulum robot system based on its selfbalance. First, the corner trajectory of this pendulum robot is parametrized by QPSO. The native attitude and the control precision of the robot’s terminal attitude and position are utilized to formulate the objective function. The addressed problem of optimal path planning was considered as a nonlinear optimization problem that is solved using the QPSO algorithm to achieve the demanded goal.
Motivated by the characteristics of PSO and quantum theory, Xu et al. [199] proposed a new method, namely, QPSO. The QPSO used fractional calculus to enhance its global search ability. The discrete expression of the GrünwaldLetnikov definition, one of the most popular differential fractional definitions, was incorporated for better updating of the particles’ positions in QPSO.
5.1.5 Rough Set
The theory of rough set [142] is an approach devised to deal with uncertainty and ambiguity. The philosophy of this approach was built on the fact that every object in the universe has its own characteristics. The rough set theory is a mathematical tool utilized to extract knowledge from information obtained from uncertain data [105]. In feature selection, rough set analysis is utilized because it only requires the supplied basic data without the need for any supplementary information. Also, it is characterized by the suitability for exploring the qualitative and quantitative properties of data.
Fan et al. [51] proposed an algorithm called RoughPSO for solving the convergence to a local optimum in PSO, depending on the rough set theory. The RoughPSO uses upper and lowerapproximation rough sets to get the membership values. Then, these values were employed to refine the velocity and position for each particle.
In another work, Bhattacharya et al. [23] proposed a technique of Feature Selection based on an impromptu PSO algorithm and Rough Set (PSORSFS), to select the most relevant features for detecting permissionbased Android malwares. This work mainly contributed to the recommendation of a new method of random key encoding that converts the conventional PSO algorithm in discrete space. It also reduced the issues related to the sigmoid function in binary PSO, as well as the particles’ maximum velocity.
5.2 Improved PSO
This section is competent of the improvements in PSO based on some thirdparty methods as learning strategy, fuzzy logic, mutation, Lévy flight, oppositionbased learning, and surrogate.
5.2.1 Learning Strategy
In reality, increasing complexity in many optimization problems calls for further research on the exploration and improvement of diverse optimization algorithms. It has been proved that the PSO algorithm is an efficient tool to solve different types of optimization tasks. However, for the base PSO, the main aim of the updating strategy is to learn the global optimum, and it always experiences too fast convergence in addition to poor performance on various complex optimization tasks, principally for multimodal problems. The mechanism of learning in classical PSO is simple and easy to develop, but some likely issues, such as the phenomena of “two steps forward, one step back” and “oscillation”, may disrupt it. Therefore, PSO researchers have managed to design new efficacious learning strategies for avoiding the two phenomena and improving the overall search performance. Xu et al. [198] presented a Dimensional Learning Strategy (DLS) following the best personal experience of each particle to eventually discover and integrate the propitious information extracted from the optimal solution of swarm. Thereafter, different learning strategies are used to provide a TwoSwarm Learning PSO algorithm (TSLPSO). One of the two subswarms used the DLS to create learning exemplars for guiding the particles’ local search, while the other subswarm used the comprehensive learning strategy to create learning exemplars for managing the global search of particles.
Wang and Liu [193] proposed a Heterogeneous Comprehensive Learning PSO (HCLPSO) algorithm to attain 3D spatial trajectory tracking to realize a new saturated approach to control a quadrotor. First, the quadrotor model was divided into: an inner position control loop, inside cascaded control structure, and an outer attitude control loop. Second, the saturated control is applied to limit the quadrotor’s thrust force in the outer attitude control loop. Next, the parameter adjustment difficulty for the quadrotor was alleviated by employing the HCLPSO algorithm to optimize the control parameters.
5.2.2 Fuzzy Logic (FL)
Since the performance of PSO is strongly dependent on the selection of its settings (i.e., minimum and maximum velocity, social and cognitive factors, and inertia weight), Fuzzy Logic (FL) can be first exploited to determine the best values of these parameters. So far, the PSO implementations based on FL aim at calculating distinct settings for the swarm as a whole. Nobile et al. [138] proposed a new algorithm called Fuzzy SelfTuning PSO (FSTPSO) which, for every particle, independently calculates the minimum and maximum velocity, social and cognitive factors, and inertia by exploiting FL, thereby creating a completely settingfree variant of PSO. The strength and novelty of FSTPSO was derived from the fact that there is no experience required to formulate PSO, since the optimization process automatically adaptively adjusts the behavior of each particle.
In [114], López et al. proposed a Fuzzy Logic Controller (FLC) modified by PSO, namely, FuzzyPSO to increment the lifetime of power electronics with a faster response of drive’s speed in a brushless DC electric motor. Furthermore, the reference temperature and the desired speed are proposed as parameters to formulate an objective function.
5.2.3 Mutation
Salajegheh and Salajegheh [157] combined PSO with gradient directions of first and second order, thereby achieving a great increasing in the approach robustness. The literature was thoroughly searched to choose an appropriate set of design problems, and the results of the standard PSO and the counterparts are compared with each other. In the proposed approach, the overall optimization cost was trivial as a result of decreasing the number of initial particles.
A modified variant, namely, Repository and Mutation based PSO (RMPSO) was proposed in [75]. In RMPSO, global and personal best solutions with the same fitness values are stored in two maintained extra repositories. And, another proposed Enhanced Leader PSO (ELPSO) indicated that the swarm optimum could be improved by applying five successive mutation strategies.
5.2.4 Lévy Flight (LF)
Lévy Flight (LF) is a particular category of random walk in which a heavy power tails law is applied for the distribution of step lengths. A global search is done or conducted by an algorithm with the occasional aid of larger steps. A better tradeoff between the two capabilities of exploration and exploitation in an algorithm could be gained by applying the LF trajectory [47], and the ability to avoid local optima gives plus points.
In [213], Zhang et al. established an optimization model for the task scheduling model in the MultipleInput and MultipleOutput (MIMO) radar, and proposed a hybrid Discrete PSO (DPSO) algorithm with LF for solving the problem. The internal structure of tasks, the features of the MIMO radar task scheduling, and the three main principles of task scheduling were all considered in the optimization model. In another work, Tang et al. [175] employed the signal level of Automatic Identification System (AIS) to build a new method for estimating the atmospheric profile refractivity, specifically the parameters of surfacebased duct in the maritime environment. Due to the complexity and nonlinearity of this optimization problem, LF with QPSO algorithm (LFQPSO) was proposed to find the optimal solution.
In [100], Li et al. developed a novel Unsupervised LF with PSO method (ULPSO) for image classification, giving a good explorationexploitation balance. ULPSO depends on a new searching mechanism that targets the worst particle in the swarm at each iteration to update its position with LF.
5.2.5 OppositionBased Learning (OBL)
Tizhoosh [181] originally introduced the basic idea of OppositionBased Learning (OBL). Based on the concept of OBL, typically, the fitness values of current agents and their opposites are evaluated [119]. Then, the suggested algorithm selects N agents with the highest/lowest fitness values.
In [86], Kang et al. proposed a hybrid PSO algorithm to improve the PSO performance by maximizing PSO diversity introduced by OBL. Probabilistic OBL was adapted for a swarm by the proposed hybrid algorithms. Unlike other fusions of PSO and OBL, the fitness of the entire swarm was improved by selecting the fittest particles from the current swarm and its opposite one.
In [189], Wang et al. addressed the learning strategy impact on a scheduling problem with twostage assembly flow shop. Three different machines were used to, in turn, process several different workpieces. The total completion time was minimized by developing a branchandbound technique combined with a lower bound procedure and several developed dominance rules to obtain the optimal scheduling solution. Besides, highquality feasible solutions were obtained by devising a new Dominance Rulebased Oppositionbased PSO algorithm (DROPSO).
5.2.6 Surrogate
Surrogateassisted optimization was established to handle expensive and complex problems in realworld applications. The methodology of this type of optimization depends on the maximal exhaustion of the available information for minimizing the number of expensive evaluations required for a given objective function, thereby reducing the related costs, resources, and time [169].
Related to this, Yu et al. [207] proposed a hierarchical surrogateassisted particle swarm optimizer comprising a Social Learningbased PSO algorithm (SLPSO) and the standard PSO algorithm itself with the goal of solving highdimensional problems. The cooperation between SLPSO and standard PSO was proposed so that the search space can be thoroughly exploited and explored, and the surrogate model global and local search performance is simultaneously improved.
In [203], Yang et al. proposed an Improved SurrogateAssisted PSO (ISAPSO) algorithm, in which global and local surrogates are integrated into a hybrid PSO. The global surrogate uses an efficient global optimization algorithm to work as a global searcher for speeding up the PSO global search process, and reduces the computational burden by predicting the fitness values. Meanwhile, the local surrogate explores the expected optimum solution so that it can later locally search through the neighborhood of this solution.
5.3 Variants of PSO
Different variants of the PSO algorithm are addressed in this section.
5.3.1 Binary
Continuous metaheuristics cannot handle a variety of optimization problems. Therefore, combinatorial optimization has emerged, which includes a type of optimization called binary optimization. The {0, 1} set elements are distinct decision variables in binary optimization problems. Thus, the binary optimization problem can have 0 and 1 as potential values for each decision variable. In other words, the decision variables can be digitally expressed as 0 or 1 with respect to the binary optimization problem at hand.
Sameer et al. [158] proposed a new Modified Binary PSO with a Kernel Fuzzy Clustering Method (MBPSOKFCM) for selecting relevant features and determining the number of clusters in fuzzy data clustering. To enhance the quality of cluster analysis, GustafsonKessel (GK) algorithm was established and improved. The datasets Taiwanese, German, and Australian credit were used to test the performance of the three proposed algorithms. The results showed that fuzzy partition (classification) is robust based on the fuzzy Rand validity metrics, which provides an opportunity to reduce the risk associated with loans.
In [89], Karbassi Yazdi et al. proposed a Binary PSO variant (BPSO) for optimally solving the problem of ship routing and scheduling in Liquefied Natural Gas (LNG) extraction, transportation, and regasification.
5.3.2 Chaotic
Generally, chaotic, adapted from the term “chaos”, formally refers to such a property that describes the unpredictable behavior of a complicated system. In chaotic systems, a function is used to “map” or associate algorithm chaotic behavior with some parameters, using two typical chaotic maps: logistic and sine. Chaos has distinctive properties of nonrepeatability and periodicity. So, it has the ability to do general searches at higher speeds compared to the probabilistic stochastic searches [72].
In [200], Xu et al. proposed a new Chaotic Search PSO algorithm (CSPSO) which enhances the PSO algorithm with the CS method for solving combinatorial optimization problems. In the initialization phase, the particles were positioned depending on the advance knowledge of the combinatorial optimization problems. Furthermore, in the chaos perturbing phase, the positions and velocities of particles are perturbed by introducing a brandnew range of rules for satisfying the adaptability and the capability of the ideal global search with the major aim to effectively avoid the fruitless early convergence that haunts the standard PSO algorithm.
In another work, Wang et al. [188] presented an efficient multiobjective optimization approach that combines an adaptive chaotic PSO and the IsoGeometric Analysis (IGA) to optimize the Ceramic Volume Fraction (CVF) distribution under eigenfrequencies in Functionally Graded Plates (FGPs). And, the Bspline basis function was used to represent the CVF distribution.
In [209], Yue et al. proposed a hybrid PSO algorithm based on the chaos theory for solving the mobile location estimation problem. In the proposed location estimation approach, it was crucial to consider the influence of measurement error arising from the nonlinear optimization problem and Angle Of Arrival (AOA)/Time Difference Of Arrival (TDOA) hybrid location method. The proposed algorithm recognized the AOA/TDOA hybrid location algorithm to track the object to significantly enhance the locating accuracy and the performance of the estimation process in general.
5.3.3 Multiobjective
MultiObjective Problems (MOPs) [108] is to be introduced in this section. Minimizing or maximizing multiple conflicting objective functions are two main tasks often considered in MOPs [148, 221]. Unlike single objective optimization ones, MOPs involve multiple contradictory objective functions. It is not easy to simultaneously optimize many functions. Hence, for obtaining an optimal solution set, there should be balance between the objective functions. Some constraints and at least two objectives must be included in a multiobjective optimization problem. In a solution space, the objectives and constraints are satisfied through finding mutually acceptable solutions.
In [214], Zhang et al. proposed a multiobjective particle swarm optimizer based on a competitive mechanism, in which the current swarm performs, at each iteration, pairwise competitions which in turn are used to update the position particles. On the other hand, a biobjective problem was formulated for the hand posture estimation problem [160]. To solve this problem, the concepts of Evolutionary Population Dynamics (EPD) were employed for developing an improved version of MOPSO. This problem has different types of parameters. Therefore, some of the parameters were significantly calibrated by the MOPSO algorithm.
In [219], Zheng et al. presented an adaptive neighborhood function for developing a MultiObjective Cellular PSO (MOCPSO). Three objective functions were taken for drilling a wellbore trajectory less costly, more quickly, and safely, compared to other potential trajectories. Then, a set of Pareto optimal solutions is gained by applying the MOCPSO algorithm to the three objective functions for designing a cheaper and safer wellbore trajectory.
Adhikari and Srirama [5] developed a new Containerbased EnergyEfficient Scheduling method (CEES) that responds effectively to different types of fast submitted Internet of Things (IoT) and nonIoT tasks. A MultiObjective Accelerated PSO (MOAPSO) technique was used in the proposed method to ensure a minimum delay by finding the most appropriate container for executing each task. To better utilize cloud resources, cloud environment involves other important missions of resource scheduling, which can be explored in [64].
6 PSO Applications
In accordance with the SR process adopted in this study, this section is to provide a technical review of the designated PSO applications in the actual literature. Figure 5 demonstrates a comprehensive taxonomy of the PSO applications in different domains, including healthcare, environmental, industrial, commercial, smart city, and general aspects. Each category of PSO applications is likely to face some issues that should be highlighted to come up with thriving solutions, enabling further efficient and viable PSO implementation in future realworld applications. Consequently, studies focusing on some pivotal issues are reviewed to prop PSO applications in a particular context associated with these issues. For example, in environmental applications, the main contexts, such as economic emission dispatch, parameter identification of PhotoVoltaics (PV), pollution forecasting, segmentation and classification of plants, flood control and routing, water quality monitoring, and many other issues, are floated in different aspects of environmental PSO applications. Thus, this paper presents a taxonomy based on diverse categories of PSO applications in selected research studies in which special subjunctives are addressed and discussed. Considering the concerns and challenges in various types of PSO applications, I firstly addressed different categories in the PSO applications and then reported the main subjects which have been paid special attention in each category. PSO applications have some general concerns, so a division in the taxonomy, namely “general aspects”, was adopted to refer to the studies that introduced a proposal to cope with a particular challenge in any general type of PSO applications. In other words, a new conceptual approach is introduced based on the studies shown as being of general aspects, thus promoting the development of any type of PSO application.
The upcoming subsections present diverse approaches in PSO applications. In addition, various researches will be compared from several sides, such as key subject, case studies, strengths, shortcomings, and special outputs.
6.1 HealthCare Applications
Table 3 illustrates a summary of some seminal researches and important information to assess healthcare approaches in PSO applications. Some of the major contexts in the healthcare domain are based on intelligent diagnosis, disease detection by medical robots, medical image segmentation, and disease classification.
6.2 Environmental Applications
Table 4 illustrates a summary of the recent researches and important information to assess the environmental approaches in PSO applications. The approaches include case studies, such as wild vegetation environmental monitoring, agriculture environmental monitoring, flood control and routing, water quality monitoring, and pollutant concentration monitoring systems.
6.3 Industrial Applications
Table 5 illustrates a summary of the recent researches and important information to assess the industrial approaches in PSO applications. Some of the major contexts in the industrial domain include economic dispatch problem in power systems, optimizing Phasor Measurement Units (PMUs) placement, allocating daily electrical loads, deploying WSNs, predicting the defection of products, and optimizing the design and operation of microgrids.
6.4 Commercial Applications
Table 6 illustrates a summary of some seminal researches and important information to assess the commercial approaches in PSO applications. Some of the major contexts in the commercial domain include prediction of cost and price, risk assessment, and profit calculation.
6.5 Smart City Applications
Table 7 illustrates a summary of some seminal researches and important information to assess the smart city approaches in PSO applications. The approaches include case studies, such as smart city, smart home, appliance scheduling, and vehicular monitoring systems.
6.6 General Aspects of PSO Applications
Table 8 describes a summary of some seminal researches and important information to assess the general approaches in PSO applications. Some of the major contexts in the general domain include service allocation, image segmentation, scheduling, prediction, and security management.
7 Discussion and Comparisons
Similar to many other metaheuristic algorithms, PSO has distinctive advantages, as well as some unavoidable shortcomings. Although there is no evidence of convergence for this algorithm, the results presented in this study indicate its competitiveness over, for example, EAs in terms of convergence rates and accuracy in many cases. Table 9 lists some of the impacting strengths and weaknesses of PSO.
Previous sections discussed the review process of the selected studies regarding PSO methods and applications. In turn, this section statistically analyzes those methods and application areas. Furthermore, the proposed AQs proposed in Sect. 4 are called to present some analytical reports as follows:

AQ1: Into which domains can PSO methods and applications be classified?
According to the taxonomy exhibited in Sect. 5, Fig. 6 depicts a comparison of the percentages of PSO methods as of the date of this study. I considered three PSO methods that span hybridization, improvement, and variants. As shown in Fig. 7, PSO variants have the largest share of PSO methods in the literature at 42%. Of course, hybridization techniques have 32%, and improved PSO approaches have 26% usage of the PSO methods.
Similarly, according to the taxonomy exhibited in Sect. 6, Fig. 8 depicts a comparison of the portions of PSO applications as of the date of this study. Six PSO application domains, including healthcare, environmental, industrial, commercial, smart city, and general aspects applications, are considered. If we look closely, the general approach has the largest percentage of application areas in the literature with a usage ratio of 27%. Of course, industrial applications have 20%, environmental applications have 17%, smart city applications have 16%, healthcare applications have 11%, and commercial applications have 9% usage out of the overall PSO applications domain.

AQ2: Which main contexts can be considered for PSO methods and applications?
Within the framework of this study, the main contexts of PSO methods and applications are shown in Fig. 9. It has been observed that the electrical engineering field received the most attention from scholars and practitioners with 13 studies, while prediction, image processing, and mechanical engineering approaches were 6 studies.

AQ3: What assessment environments are used for evaluating the PSO methods and applications?
According to Fig. 10, it has been observed that 47% of the research studies apply simulation tools to evaluate the presented case studies in the PSO community. In addition, it has been observed that 33% of the research papers have implemented approaches to develop PSO methods and applications. Moreover, 11% of the researches have not presented any simulation or implementation for the indicated methods and application domain. Finally, 9% of the existing studies use datasets to apply analysis approaches (e.g., testing and prediction) for assessing the involved case studies.
8 Challenges and Future Opportunities
While PSO has been effectively applied in versatile domains, challenges are required to be addressed and drawn as future research directions. In the past few years, PSO has gained an increasing interest from researchers and has been utilized in different areas of applications. However, critical problems and issues remain. Thus, more research efforts should be done by scholars and researchers to conquer the challenges and problems that may hinder the future application of PSO. Moreover, further inspirations and more effective techniques should be pursued to develop novel PSO approaches. For example, the research community needs to address new methodologies for complex problems. Based on the literature review, I discuss below different open issues and related topics for potential future research.
I should point out that the assortment of PSO methods and applications mentioned in this study applied the SR process so that the challenges discussed in the following two subsections are presented as the AQ4:

AQ4: What are generally the PSO open perspectives and future research directions that can be drawn from the work presented in the paper?
8.1 Open Issues
Premature convergence When applying the traditional PSO algorithm, if the search gets closer to the local optimal solution through random initial conditions, the optimal solution for a single particle (individual particle optimum) and the optimal solution for all particles (the group/global optimum) would both converge towards the local optimal solution, and a deceptive optimization result will therefore be obtained. Thus, it is not possible to guarantee the capability of finding the global optimal solution. As a result, the fast convergence capability will be ineffective. Over the past few years, the issue of stagnation (premature convergence) has been a subject of research in the PSO community [197, 208]. A roadmap for future research in this regard can be created based on some topics, including particles’ stability analysis [26], redistributing mechanism [146], and random sampling of control parameters [171].
Convergence speed controller Despite the typical fast convergence of PSO, it may be trapped into a local optimum due to premature converge. Therefore, a convergence speed controller was devised for the ultimate goal of solving this problem. Two adaptive approaches were proposed to adjust the convergence speed. First, when the particle prematurely converges, the convergence speed is slowed down. Second, when the particle cannot update its best solution in the present time, the convergence speed of PSO is accelerated. Thus, a key residual challenge is developing an effective convergence speed controller [110]. In the same context, for improving the convergence speed of PSObased algorithms, the social and cognitive components of the velocity update formula are modified based on the fitness of each particle [125]. In addition, the convergence speed is adaptively controlled to empower the PSO algorithm for solving largescale numerical optimization problems [66]. Hence, designing an appropriate convergence speed controller framework for PSO has become a major challenge in this topic.
Highdimensional search space Because of the curse of dimensionality problem, the effectiveness of PSO application to classification of highdimensional data has recently gained a major concern [9]. In a recent study, the effectiveness of PSO approach for feature selection has been demonstrated. However, due to the large search space, the challenge sill exists to apply PSObased feature selection to highdimensional datasets with tens of thousands of features [182]. In addition, the PSO algorithm can be adapted for the problem of highdimensional feature selection, providing thereby an efficient technique for achieving similar or even better classification accuracy by not using all features but instead selecting only a small set of relevant features from a wide range of features [58]. The key challenges in this area include novel approaches like Monte Carlo methods [22] that simultaneously minimize the number of chosen features and maximize the classification accuracy in PSO applications.
Memory requirement All living systems have memory as an essential feature. According to the evolutionism perspective, it is worthwhile examining the potential helpful role of historical memory in the process of EAs. That is mainly because the explicit or implicit historical memory in PSO improves the search process by storing promising solutions and reusing them in later stages [101]. Furthermore, a new inertia weight can be generated by using historical memory through a parameter adaptation mechanism. Therefore, improving PSO using historical memory is a pressing challenge [60, 103]. And how to set the memory size adaptively can be also considered as an interesting topic to be addressed in the future [117].
Parameter & topology selection In PSObased algorithms, the best performance can be achieved by elaborately determining the control parameters. However, choosing these parameters cannot be adequately guided. To address the job of parameter selection, future efforts in this topic should include choosing the best parameters based on simulations [38], parametric analysis in a computational environment with limited resources [162], and hyperparameter selection based on heuristics [115]. On the other hand, the performance of PSO in engineering applications is significantly affected by topology selection, and each problem has its appropriate optimal topology. However, this issue still lacks much study. As a future direction, topology selection for PSO can be better guided by taking into account factors affecting the optimality of algorithmic parameters (i.e., the topological degree and the number of particles) with the aim of selecting a proper class of deterministic regular topologies [112]. Other challenges including, studying and evaluating the performance of tree topology [155] and suggesting different topologies for the PSO optimizer in order to reach, for example, an optimal ProportionalIntegralDerivative (PID) controller design [3], are also pressing issues which calls for further research and audit.
8.2 Future Perspectives
PSO was practically applied for the first time to the domain of neural network training, which was indicated by the algorithm itself [93]. Many more areas of PSO methods and application have been explored ever since, including signal processing, power systems, combinatorial optimization, design, data mining, control, telecommunications, and many others. To date, applications of PSO algorithms are reported in hundreds of publications [46, 59]. Although the main use of PSO is to solve unconstrained, singleobjective optimization problems, a development has been made to PSO algorithms for solving constrained problems, multiobjective optimization problems, problems with dynamically changing landscapes, discrete problems, and for finding multiple solutions as well [16]. However, several PSO methods and applications can be sophisticated in some areas as future research directions in this topic as follows:
Nuero Fuzzy Network (NFN) Nuero Fuzzy Network (NFN) is an intelligent method for system identification, modelling, prediction, and control. In NFN, gradientbased algorithms are generally used for training. However, these algorithms have some disadvantages concerning, for example, getting stuck at local minima, which needs for complex gradient computations to be conquered. Accordingly, Karakuzu et al. [88] used improved PSO to introduce the first embedded highspeed, lowcost implementation of NFN hardware through online training. It has been observed that that the effectiveness of the proposed NFN implementation is similar to other approaches in the literature, thereby generating a novel idea for future research. In a different approach to NFNs, Vijay and Jena [186] utilized the PSO to minimize the quadratic performance indices for obtaining optimal sliding control parameters and PID parameters. In this study, Lyapunov stability theorem was used to improve the stability of the system by integrating sliding mode control into artificial neuro fuzzy inference system. Thus, it can be concluded that NFN can be applied to the robot manipulator for realtime control, where good control performance can be obtained by adaptively changing sliding control parameters under different input disturbances.
Color image processing In this regard, Gaussian PSO (GPSO) was used to develop a color image quantization algorithm [15]. In the proposed algorithm, each particle in the swarm is randomly initialized with k centroids (i.e. color triplets). The cluster with the closest centroid to a pixel in the image is recognized to assign this pixel to that cluster. The centroids obtained by Kmeans clustering are refined by applying the GPSO. Experimentation showed that using GPSO method for the color image quantization gives significantly better image quality than conventional PSObased approaches. In anther work, a multilevel color image thresholding algorithm was developed on the bases of GLLA histogram and TsallisHavrdaCharvát entropy of degree \(\alpha\) [27]. Compared with the existing models, the proposed model achieved better performance results by applying the PSO algorithm. However, so far only a limited number of studies reported how to apply together the quantization and multilevel thresholding techniques to a color image.
Image registration Image registration is defined as taking several 2D images from various sources, such as Computer Assisted Tomography (CAT) and Magnetic Resonance Imaging (MRI) scans, and combining them into a 3D image. Recently, a hybrid approach for registering medical images has been developed by employing a PSO method [1] and an adjusted mutual information as a similarity index, or general metaheuristics [184]. However, there are still trend applications for future research, including registering the images of a printed circuit board placed on a conveyor belt using an improved scale invariant feature transform, feature extraction technique combined with PSO [39]. Also, using PSO algorithm for remote sensing images registration less affected by the correction rate [196] is another major trend for future research.
Computational biology A long DNA chain first needs to be divided into subset fragments for determining its sequence. Therefore, combinatorial optimization researchers used the DNA Fragment Assembly method (DFA) to solve the NPhard problem of reassembling the fragment. The DFA problem is solved by applying the overlaplayoutconsensus model to maximize the overlapping score measurement using a memetic PSO algorithm based on two initialization operators, as well as the local search operator [67]. On 19 DNA fragment datasets, the results revealed that the PSO algorithm combining SAbased variable neighborhood local search and TS achieves the best overlap scores. Future works could address interesting issues, such as reducing the computational time by using DNA sequence compression, improving the initialization method of the algorithm, using an alternative search approach, and so on. In a similar biological application, some variants of the PSO algorithm have been utilized to extensively tackle the molecular docking problem and the problem of conformational search in proteinligands [55, 136]. Ultimately, the literature has reported significant results. Thus, it is natural to solve biological problems by applying the PSO algorithm.
Recommender systems PSO was utilized as a tuning mechanism in a further area in which software tools are created to develop recommendations to entrepreneurs or even end users. Explicit feedback data (i.e., votes or ratings) are usually used to build most of the existing recommender models. However, reallife scenarios do not always contain explicit feedback data. For example, a hybrid music recommender system was suggested based on implicit feedback data by utilizing graphbased algorithms for making songs recommendations based on the users preferences and behavior [90]. Furthermore, PSObased webpage recommendation system was developed on healthcare multimedia data to track user navigation behavior by utilizing semantic web mining [121]. In the future work, these systems can be further extended to include other Ehealth care application, and/or social networking sites like twitter and Facebook.
9 Conclusions and Recommendations
In this literature, the exploration query on 3,632 papers published between 2017 and 2019 was executed to fulfill the proposed SR process. 2,140 papers related to PSO methods and applications were selected. Amongst them, I briefly discussed and analyzed only 84 papers to retain the proper length of the paper. According to AQ1, on the one hand, the PSO variants have the most percentage of the PSO methods in the literature by 42% of quota. Of course, hybridization techniques have 32%, and improved PSO techniques have 26% of portions of PSO methods. On the other hand, the general approach has the most percentage of the application domains in the literature by 27% of quota, emphasizing the broad application of PSO in diverse subject areas. Also, industrial applications, environmental applications, smart city applications, healthcare applications, commercial applications have 20%, 17%, 16%, 11%, and 9% portions of all existing PSO applications, respectively. It has also been observed that electrical engineering approaches have the highest percentage, with 13 studies, and prediction, image processing, and mechanical engineering approaches are with 6 studies, based on AQ2. According to AQ3, 47% of the research studies applied simulation tools to evaluate the case studies presented in the PSO environment.
According to the adopted SR process, I must acknowledge that not all present studies may have been analyzed; nonpeer reviewed, nonEnglish book chapters, survey articles, and editorial papers were omitted. In this study, the findings of more than 120 authors and diverse studies were considered to perform a comprehensive research of the methods and application approaches in PSO. However, as the research ended in November 2019, it is not possible to guarantee that all relevant seminal works are covered herein, especially with the increasing number of studies in this field.
Finally, the SR method presented in this review focused primarily on PSO methods and applications. Through this study, the PSO methods and applications were comprehensively understood. In addition, open issues and future perspectives were considered. However, the PSO algorithm still needs further research in the following areas:

Studying the effects of heterogeneity in PSO algorithms.

Hybridizing PSO with novel optimization techniques, such as Salp Swarm Algorithm (SSA) [130], Whale Optimization Algorithm (WOA) [129], Lion Optimization Algorithm (LOA) [204], Elephant Herding Optimization (EHO) [191], and Jaya Algorithm (JA) [63, 153].

Implementation of innovated smart city applications, such as smart metering, smart farming, smart logistics, and smart buildings.

Further applications for addressing different issues in cloud computing.

Addressing more complex/largescale realworld problems, NPhard problems, and discrete optimization problems to discover new limitations of PSO.
Change history
30 May 2022
A Correction to this paper has been published: https://doi.org/10.1007/s11831022097623
References
AbdelBasset M, Fakhry AE, ElHenawy I, Qiu T, Sangaiah AK (2017) Feature and intensity based medical image registration using particle swarm optimization. J Med Syst 41(12):197
Abdelkader HE, Gad AG, Abohany AA, Sorour SE (2022) An efficient data mining technique for assessing satisfaction level of online learning for higher education students during the covid19. IEEE Access
Aberbour J, Graba M, Kheldoun A (2015) Effect of cost function and pso topology selection on the optimum design of pid parameters for the avr system. In: 2015 4th international conference on electrical engineering (ICEE). IEEE, pp 1–5
Abid S, Zafar A, Khalid R, Javaid S, Qasim U, Khan ZA, Javaid N (2017) Managing energy in smart homes using binary particle swarm optimization. In: Conference on complex, intelligent, and software intensive systems. Springer, pp 189–196
Adhikari M, Srirama SN (2019) Multiobjective accelerated particle swarm optimization with a containerbased scheduling for internetofthings in cloud environment. J Netw Comput Appl 137:35–61
AlThanoon NA, Qasim OS, Algamal ZY (2019) A new hybrid firefly algorithm and particle swarm optimization for tuning parameter estimation in penalized support vector machine with application in chemometrics. Chemom Intell Lab Syst 184:142–152
Alam S, Dobbie G, Koh YS, Riddle P, Rehman SU (2014) Research on particle swarm optimization based clustering: a systematic review of literature and techniques. Swarm Evol Comput 17:1–13
Ali Ghorbani M, Kazempour R, Chau KW, Shamshirband S, Taherei Ghazvinei P (2018) Forecasting pan evaporation with an integrated artificial neural network quantumbehaved particle swarm optimization model: A case study in talesh, northern iran. Eng Appl Comput Fluid Mech 12(1):724–737
Ali Yahya A (2018) Centroid particle swarm optimisation for highdimensional data classification. J Exp Theor Artif Intell 30(6):857–886
Alnaqi AA, Moayedi H, Shahsavar A, Nguyen TK (2019) Prediction of energetic performance of a building integrated photovoltaic/thermal system thorough artificial neural network and hybrid particle swarm optimization models. Energy Convers Manag 183:137–148
Alswaitti M, Albughdadi M, Isa NAM (2018) Densitybased particle swarm optimization algorithm for data clustering. Expert Syst Appl 91:170–186
Aydilek IB (2018) A hybrid firefly and particle swarm optimization algorithm for computationally expensive numerical problems. Appl Soft Comput 66:232–249
Banks A, Vincent J, Anyakoha C (2007) A review of particle swarm optimization. Part i: background and development. Nat Comput 6(4):467–484
Banks A, Vincent J, Anyakoha C (2008) A review of particle swarm optimization. Part ii: hybridisation, combinatorial, multicriteria and constrained optimization, and indicative applications. Nat Comput 7(1):109–124
Barman D, Hasnat A, Sarkar S, Murshidanad MAR (2016) Color image quantization using gaussian particle swarm optimization (ciqgpso). In: 2016 international conference on inventive computation technologies (ICICT). IEEE, vol 1, pp 1–4
Beheshti Z, Shamsuddin SM, Hasan S (2015) Memetic binary particle swarm optimization for discrete optimization problems. Inf Sci 299:58–84
Beni G (1988) The concept of cellular robotic system. In: Proceedings IEEE international symposium on intelligent control 1988. IEEE, pp 57–62
Beni G, Hackwood S (1992) Stationary waves in cyclic swarms. In: Proceedings of the 1992 IEEE international symposium on intelligent control. IEEE, pp 234–242
Beni G, Wang J (1993) Swarm intelligence in cellular robotic systems. In: Robots and biological systems: towards a new bionics? Springer, pp 703–712
Benioff P (1980) The computer as a physical system: a microscopic quantum mechanical Hamiltonian model of computers as represented by turing machines. J Stat Phys 22(5):563–591
Bernardino HS, Barbosa HJ, Fonseca LG (2011) Surrogateassisted clonal selection algorithms for expensive optimization problems. Evol Intel 4(2):81–97
Beskos A, Crisan D, Jasra A, Kamatani K, Zhou Y (2017) A stable particle filter for a class of highdimensional statespace models. Adv Appl Probab 49(1):24–48
Bhattacharya A, Goswami RT, Mukherjee K (2018) A feature selection technique based on rough set and improvised pso algorithm (psorsfs) for permission based detection of android malwares. Int J Mach Learn Cybern, pp 1–15
Bhattacharya A, Goswami RT, Mukherjee K (2019) A feature selection technique based on rough set and improvised pso algorithm (psorsfs) for permission based detection of android malwares. Int J Mach Learn Cybern 10(7):1893–1907
Bonabeau E, Marco DdRDF, Dorigo M, Théraulaz G, Theraulaz G et al (1999) Swarm intelligence: from natural to artificial systems, 1st edn. Oxford University Press, Oxford
Bonyadi MR, Michalewicz Z (2015) Stability analysis of the particle swarm optimization without stagnation assumption. IEEE Trans Evol Comput 20(5):814–819
Borjigin S, Sahoo PK (2019) Color image segmentation based on multilevel tsallishavrdacharvát entropy and 2d histogram using pso algorithms. Pattern Recogn 92:107–118
Camci E, Kripalani DR, Ma L, Kayacan E, Khanesar MA (2018) An aerial robot for rice farm quality inspection with type2 fuzzy neural networks tuned by particle swarm optimizationsliding mode control hybrid algorithm. Swarm Evol Comput 41:1–8
Cao Y, Ye Y, Zhao H, Jiang Y, Wang H, Shang Y, Wang J (2018) Remote sensing of water quality based on hj1a hsi imagery with modified discrete binary particle swarm optimizationpartial least squares (mdbpsopls) in inland waters: a case in weishan lake. Eco Inform 44:21–32
Cao Y, Zhang H, Li W, Zhou M, Zhang Y, Chaovalitwongse WA (2018) Comprehensive learning particle swarm optimization algorithm with local search for multimodal functions. IEEE Trans Evol Comput
Chen CH, Liu TK, Chou JH (2014) A novel crowding genetic algorithm and its applications to manufacturing robots. IEEE Trans Ind Inf 10(3):1705–1716
Chen K, Zhou F, Yin L, Wang S, Wang Y, Wan F (2018) A hybrid particle swarm optimizer with sine cosine acceleration coefficients. Inf Sci 422:218–241
Chen S, Jq Wang, Hy Zhang (2019) A hybrid psosvm model based on clustering algorithm for shortterm atmospheric pollutant concentration forecasting. Technol Forecast Soc Chang 146:41–54
Chen Y, Li L, Peng H, Xiao J, Wu Q (2018) Dynamic multiswarm differential learning particle swarm optimizer. Swarm Evol Comput 39:209–221
Chernbumroong S, Cang S, Yu H (2014) Genetic algorithmbased classifiers fusion for multisensor activity recognition of elderly people. IEEE J Biomed Health Inform 19(1):282–289
Clerc M, Kennedy J (2002) The particle swarmexplosion, stability, and convergence in a multidimensional complex space. IEEE Trans Evol Comput 6(1):58–73
Colorni A, Dorigo M, Maniezzo V et al (1992) Distributed optimization by ant colonies. In: Proceedings of the first European conference on artificial life, Cambridge, MA, vol 142, pp 134–142
Cui H, Shu M, Song M, Wang Y (2017) Parameter selection and performance comparison of particle swarm optimization in sensor networks localization. Sensors 17(3):487
Dai L, Guan Q, Liu H (2018) Robust image registration of printed circuit boards using improved siftpso algorithm. J Eng 16:1793–1797
Del Valle Y, Venayagamoorthy GK, Mohagheghi S, Hernandez JC, Harley RG (2008) Particle swarm optimization: basic concepts, variants and applications in power systems. IEEE Trans Evol Comput 12(2):171–195
Deng W, Yao R, Zhao H, Yang X, Li G (2019) A novel intelligent diagnosis method using optimal lssvm with improved pso algorithm. Soft Comput 23(7):2445–2462
Dorigo M, Bonabeau E, Theraulaz G (2000) Ant algorithms and stigmergy. Future Gen Comput Syst 16(8):851–871
DuránRosal AM, Gutiérrez PA, CarmonaPoyato Á, HervásMartínez C (2019) A hybrid dynamic exploitation barebones particle swarm optimisation algorithm for time series segmentation. Neurocomputing 353:45–55
Eberhart R, Kennedy J (1995) A new optimizer using particle swarm theory. In: MHS’95. Proceedings of the sixth international symposium on micro machine and human science. IEEE, pp 39–43
Ehteram M, Binti Othman F, Mundher Yaseen Z, Abdulmohsin Afan H, Falah Allawi M, Najah Ahmed A, Shahid S, Singh PV, ElShafie A (2018) Improving the muskingum flood routing method using a hybrid of particle swarm optimization and bat algorithm. Water 10(6):807
Elsheikh A, Elaziz MA (2019) Review on applications of particle swarm optimization in solar energy systems. Int J Environ Sci Technol 16(2):1159–1170
Emary E, Zawbaa HM, Sharawi M (2019) Impact of Lévy flight on modern metaheuristic optimizers. Appl Soft Comput 75:775–789
Esmin AA, LambertTorres G, De Souza AZ (2005) A hybrid particle swarm optimization applied to loss power minimization. IEEE Trans Power Syst 20(2):859–866
Esmin AA, Coelho RA, Matwin S (2015) A review on particle swarm optimization algorithm and its variants to clustering highdimensional data. Artif Intell Rev 44(1):23–45
Ewees AA, Elaziz MA, Houssein EH (2018) Improved grasshopper optimization algorithm using oppositionbased learning. Expert Syst Appl 112:156–172
Fan JC, Li Y, Tang LY, Wu GK (2018) Roughpso: rough setbased particle swarm optimisation. Int J Bio Inspir Comput 12(4):245–253
Ferdowsi A, Farzin S, Mousavi SF, Karami H (2019) Hybrid bat & particle swarm algorithm for optimization of labyrinth spillway based on half & quarter round crest shapes. Flow Measurement and Instrumentation
Fister I, Strnad D, Yang XS (2015) Adaptation and hybridization in natureinspired algorithms. In: Adaptation and hybridization in computational intelligence. Springer, pp 3–50
Gao H, Xu W (2011) Particle swarm algorithm with hybrid mutation strategy. Appl Soft Comput 11(8):5129–5142
GarcíaNieto J, LópezCamacho E, GarcíaGodoy MJ, Nebro AJ, AldanaMontes JF (2019) Multiobjective ligandprotein docking with particle swarm optimizers. Swarm Evol Comput 44:439–452
Ghaddar B, NaoumSawaya J (2018) High dimensional data classification and feature selection using support vector machines. Eur J Oper Res 265(3):993–1004
Ghorbani N, Kasaeian A, Toopshekan A, Bahrami L, Maghami A (2018) Optimizing a hybrid windpvbattery system using gapso and mopso for reducing cost and increasing reliability. Energy 154:581–591
Gu S, Cheng R, Jin Y (2018) Feature selection for highdimensional classification using a competitive swarm optimizer. Soft Comput 22(3):811–822
Hajihassani M, Armaghani DJ, Kalatehjari R (2018) Applications of particle swarm optimization in geotechnical engineering: a comprehensive review. Geotech Geol Eng 36(2):705–722
Hino T, Ito S, Liu T, Maeda M (2016) Setbased particle swarm optimization with status memory for knapsack problem. Artif Life Robot 21(1):98–105
Hoang TT, Cho MY, Alam MN, Vu QT (2018) A novel differential particle swarm optimization for parameter selection of support vector machines for monitoring metaloxide surge arrester conditions. Swarm Evol Comput 38:120–126
Houssein EH, Gad AG, Hussain K, Suganthan PN (2021) Major advances in particle swarm optimization: theory, analysis, and application. Swarm Evol Comput 63:100868
Houssein EH, Gad AG, Wazery YM (2021) Jaya algorithm and applications: a comprehensive review. Metaheuristics and Optimization in Computer and Electrical Engineering, pp 3–24
Houssein EH, Gad AG, Wazery YM, Suganthan PN (2021) Task scheduling in cloud computing based on metaheuristics: review, taxonomy, open challenges, and future trends. Swarm Evol Comput 62:100841
Hu W, Wang H, Qiu Z, Nie C, Yan L (2018) A quantum particle swarm optimization driven urban traffic light scheduling model. Neural Comput Appl 29(3):901–911
Huang H, Lv L, Ye S, Hao Z (2019) Particle swarm optimization with convergence speed controller for largescale numerical optimization. Soft Comput 23(12):4421–4437
Huang KW, Chen JL, Yang CS, Tsai CW (2015) A memetic particle swarm optimization algorithm for solving the dna fragment assembly problem. Neural Comput Appl 26(3):495–506
Huang L, Guan K, Xu T, Zhang J, Wang Q (2019) Investigation of the mechanical properties of steel using instrumented indentation test with simulated annealing particle swarm optimization. Theor Appl Fract Mech 102:116–121
Hussien AG, Houssein EH, Hassanien AE (2017) A binary whale optimization algorithm with hyperbolic tangent fitness function for feature selection. In: 2017 eighth international conference on intelligent computing and information systems (ICICIS). IEEE, pp 166–172
Ibrahim RA, Ewees AA, Oliva D, Elaziz MA, Lu S (2018) Improved salp swarm algorithm based on particle swarm optimization for feature selection. J Ambient Intelli Hum Comput, pp 1–15
Imran M, Hashim R, Khalid NEA (2013) An overview of particle swarm optimization variants. Procedia Eng 53:491–496
Ismail FH, Houssein EH, Hassanien AE (2018) Chaotic bird swarm optimization algorithm. In: International conference on advanced intelligent systems and informatics. Springer, pp 294–303
Issa M, Hassanien AE, Oliva D, Helmi A, Ziedan I, Alzohairy A (2018) Ascapso: adaptive sine cosine optimization algorithm integrated with particle swarm for pairwise local sequence alignment. Expert Syst Appl 99:56–70
Jain I, Jain VK, Jain R (2018) Correlation feature selection based improvedbinary particle swarm optimization for gene selection and cancer classification. Appl Soft Comput 62:203–215
Jana B, Mitra S, Acharyya S (2019) Repository and mutation based particle swarm optimization (rmpso): a new pso variant applied to reconstruction of gene regulatory network. Appl Soft Comput 74:330–355
Ji B, Song X, Sciberras E, Cao W, Hu Y, Pickert V (2014) Multiobjective design optimization of igbt power modules considering power cycling and thermal cycling. IEEE Trans Power Electron 30(5):2493–2504
Jiang H, Kwong C, Park W, Yu K (2018) A multiobjective pso approach of mining association rules for affective design based on online customer reviews. J Eng Des 29(7):381–403
Jiao R, Huang X, Ouyang H, Li G, Zheng Q, Jiang Z (2019) Optimal electric business centre location by centredecentre quantum particle swarm optimization. Syst Sci Control Eng 7(1):222–233
Jin Y, Sendhoff B (2009) A systems approach to evolutionary multiobjective structural optimization and beyond. IEEE Comput Intell Mag 4(3):62–76
Jin Y, Olhofer M, Sendhoff B (2002) A framework for evolutionary optimization with approximate fitness functions. IEEE Trans Evol Comput 6(5):481–494
Jmal S, Haddar B, Chabchoub H (2019) Apply the quantum particle swarm optimization for the ktraveling repairman problem. Soft Computing, pp 1–14
Jordehi AR (2019) Binary particle swarm optimisation with quadratic transfer function: a new binary optimisation algorithm for optimal scheduling of appliances in smart homes. Appl Soft Comput
Juang CF (2004) A hybrid of genetic algorithm and particle swarm optimization for recurrent network design. IEEE Trans Syst Man Cybernet Part B (Cybern) 34(2):997–1006
Junior FEF, Yen GG (2019) Particle swarm optimization of deep neural networks architectures for image classification. Swarm and Evolutionary Computation
Kane BE (1998) A siliconbased nuclear spin quantum computer. Nature 393(6681):133
Kang Q, Xiong C, Zhou M, Meng L (2018) Oppositionbased hybrid strategy for particle swarm optimization in noisy environments. IEEE Access 6:21888–21900
Karaboga D (2005) An idea based on honey bee swarm for numerical optimization. Technical report, Technical reporttr06, Erciyes university, engineering faculty, computer
Karakuzu C, Karakaya F, Çavuşlu MA (2016) Fpga implementation of neurofuzzy system with improved pso learning. Neural Netw 79:128–140
Karbassi Yazdi A, Kaviani MA, Emrouznejad A, Sahebi H (2019) A binary particle swarm optimization algorithm for ship routing and scheduling of liquefied natural gas transportation. Transp Lett, pp 1–10
Katarya R, Verma OP (2018) Efficient music recommender system using context graph and particle swarm. Multimed Tools Appl 77(2):2673–2687
Kennedy J, Eberhart R (1995) Particle swarm optimization (pso). In: Proceedings of IEEE international conference on neural networks, Perth, Australia, pp 1942–1948
Kennedy J, Eberhart R, Shi Y (2001) Swarm intelligence. Morgan Kaufmann, San Francisco
Kennedy J, Eberhart R (1995) Particle swarm optimization. In: Proceedings of IEEE international conference on neural networks IV, vol 1000, p 33
Kothari V, Anuradha J, Shah S, Mittal P (2011) A survey on particle swarm optimization in feature selection. In: International conference on computing and communication systems.. Springer, pp 192–201
Kour VP, Arora S (2019) Particle swarm optimization based support vector machine (psvm) for the segmentation and classification of plants. IEEE Access 7:29374–29385
Kulkarni RV, Venayagamoorthy GK (2011) Particle swarm optimization in wirelesssensor networks: a brief survey. IEEE Trans Syst Man Cybern Part C (Applications and Reviews) 41(2):262–267
Kumar S, Pal SK, Singh R (2019) A novel hybrid model based on particle swarm optimisation and extreme learning machine for shortterm temperature prediction using ambient sensors. Sustain Cities Soc 49:101601
Laskar NM, Guha K, Chatterjee I, Chanda S, Baishnab KL, Paul PK (2019) Hwpso:a new hybrid whaleparticle swarm optimization algorithm and its application in electronic design optimization problems. Appl Intell 49(1):265–291
Le LT, Nguyen H, Zhou J, Dou J, Moayedi H et al (2019) Estimating the heating load of buildings for smart city planning using a novel artificial intelligence technique psoxgboost. Appl Sci 9(13):2714
Li H, Zhang S, Zhang C, Li P, Cropp R (2017) A novel unsupervised Lévy flight particle swarm optimization (ulpso) method for multispectral remotesensing image classification. Int J Remote Sens 38(23):6970–6992
Li J, Zhang J, Jiang C, Zhou M (2015) Composite particle swarm optimizer with historical memory for function optimization. IEEE Trans Cybern 45(10):2350–2363
Li L, Qin L, Qu X, Zhang J, Wang Y, Ran B (2019) Dayahead traffic flow forecasting based on a deep belief network optimized by the multiobjective particle swarm algorithm. Knowl Based Syst 172:1–14
Li W (2018) Improving particle swarm optimization based on neighborhood and historical memory for training multilayer perceptron. Information 9(1):16
Li Y, Bai X, Jiao L, Xue Y (2017) Partitionedcooperative quantumbehaved particle swarm optimization based on multilevel thresholding applied to medical image segmentation. Appl Soft Comput 56:345–356
Li Z, Shi K, Dey N, Ashour AS, Wang D, Balas VE, McCauley P, Shi F (2017) Rulebased back propagation neural networks for various precision rough set presented kansei knowledge prediction: a case study on shoe product form features extraction. Neural Comput Appl 28(3):613–630
Lin A, Sun W, Yu H, Wu G, Tang H (2019) Global genetic learning particle swarm optimization with diversity enhancement by ring topology. Swarm Evol Comput 44:571–583
Lin G, Guan J, Li Z, Feng H (2019) A hybrid binary particle swarm optimization with tabu search for the setunion knapsack problem. Expert Syst Appl
Lin Q, Ma Y, Chen J, Zhu Q, Coello CAC, Wong KC, Chen F (2018) An adaptive immuneinspired multiobjective algorithm with multiple differential evolution strategies. Inf Sci 430:46–64
Lin TL, Horng SJ, Kao TW, Chen YH, Run RS, Chen RJ, Lai JL, Kuo IH (2010) An efficient jobshop scheduling algorithm based on particle swarm optimization. Expert Syst Appl 37(3):2629–2636
Liu F, Huang H, Li X, Hao Z (2019) Automated test data generation based on particle swarm optimisation with convergence speed controller. CAAI Trans Intell Technol 2(2):73–79
Liu G, Chen W, Chen H (2019) Quantum particle swarm with teamwork evolutionary strategy for multiobjective optimization on electrooptical platform. IEEE Access 7:41205–41219
Liu Q, Wei W, Yuan H, Zhan ZH, Li Y (2016) Topology selection for particle swarm optimization. Inf Sci 363:154–173
Lopes RF, Costa FF, Oliveira A, Lima ACdC (2018) Algorithm based on particle swarm applied to electrical load scheduling in an industrial setting. Energy 147:1007–1015
López MG, Ponce P, Soriano LA, Molina A, Rivas JJR (2019) A novel fuzzypso controller for increasing the lifetime in power electronics stage for brushless dc drives. IEEE Access 7:47841–47855
Lorenzo PR, Nalepa J, Ramos LS, Pastor JR (2017) Hyperparameter selection in deep neural networks using parallel particle swarm optimization. In: Proceedings of the genetic and evolutionary computation conference companion. ACM, pp 1864–1871
Løvbjerg M, Rasmussen TK, Krink T (2001) Hybrid particle swarm optimiser with breeding and subpopulations. In: Proceedings of the 3rd annual conference on genetic and evolutionary computation. Morgan Kaufmann Publishers Inc., pp 469–476
Luo W, Sun J, Bu C, Liang H (2016) Speciesbased particle swarm optimizer enhanced by memory for dynamic optimization. Appl Soft Comput 47:130–140
Ma K, Hu S, Yang J, Xu X, Guan X (2018) Appliances scheduling via cooperative multiswarm pso under dayahead prices and photovoltaic generation. Appl Soft Comput 62:504–513
Mahdavi S, Rahnamayan S, Deb K (2018) Opposition based learning: a literature review. Swarm Evol Comput 39:1–23
Maiyar LM, Thakkar JJ (2019) Environmentally conscious logistics planning for food grain industry considering wastages employing multi objective hybrid particle swarm optimization. Transp Res Part E Log Transp Rev 127:220–248
Manikandan R, Saravanan V (2019) A novel approach on particle agent swarm optimization (paso) in semantic mining for web page recommender system of multimedia data: a health care perspective. Multimedia Tools and Applications, pp 1–23
Mansouri N, Zade BMH, Javidi MM (2019) Hybrid task scheduling strategy for cloud computing by modified particle swarm optimization and fuzzy theory. Comput Ind Eng 130:597–633
Marini F, Walczak B (2015) Particle swarm optimization (pso). A tutorial. Chemom Intell Lab Syst 149:153–165
Mavrovouniotis M, Li C, Yang S (2017) A survey of swarm intelligence for dynamic optimization: algorithms and applications. Swarm Evol Comput 33:1–17
Mehmood Y, Sadiq M, Shahzad W, Amin F (2018) Fitnessbased acceleration coefficients to enhance the convergence speed of novel binary particle swarm optimization. In: 2018 international conference on frontiers of information technology (FIT). IEEE, pp 355–360
Melton RG (2018) Differential evolution/particle swarm optimizer for constrained slew maneuvers. Acta Astronaut 148:246–259
Millonas MM, et al (1993) Swarms, phase transitions, and collective intelligence (paper 1); and a nonequilibrium statistical field theory of swarms and other spatially extended complex systems (paper 2). Technical rep
Miranda V, Fonseca N (2002) Epsoevolutionary particle swarm optimization, a new algorithm with applications in power systems. In IEEE/PES transmission and distribution conference and exhibition, vol 2. IEEE, pp 745–750
Mirjalili S, Lewis A (2016) The whale optimization algorithm. Adv Eng Softw 95:51–67
Mirjalili S, Gandomi AH, Mirjalili SZ, Saremi S, Faris H, Mirjalili SM (2017) Salp swarm algorithm: a bioinspired optimizer for engineering design problems. Adv Eng Softw 114:163–191
Moayedi H, Mehrabi M, Mosallanezhad M, Rashid ASA, Pradhan B (2018) Modification of landslide susceptibility mapping using optimized psoann technique. Engineering with Computers, pp 1–18
Mohebbi A, Achiche S, Baron L (2019) Integrated and concurrent detailed design of a mechatronic quadrotor system using a fuzzybased particle swarm optimization. Eng Appl Artif Intell 82:192–206
Mokhtari H, Noroozi A (2018) An efficient chaotic based pso for earliness/tardiness optimization in a batch processing flow shop scheduling problem. J Intell Manuf 29(5):1063–1081
Nagra AA, Han F, Ling QH (2018) An improved hybrid selfinertia weight adaptive particle swarm optimization algorithm with local search. Engineering Optimization, pp 1–18
Nenavath H, Jatoth RK, Das S (2018) A synergy of the sinecosine algorithm and particle swarm optimizer for improved global optimization and object tracking. Swarm Evol Comput 43:1–30
Ng MC, Fong S, Siu SW (2015) Psovina: the hybrid particle swarm optimization algorithm for proteinligand docking. J Bioinform Comput Biol 13(03):1541007
Nguyen S, Zhang M, Johnston M, Tan KC (2014) Automatic programming via iterated local search for dynamic job shop scheduling. IEEE Trans Cybern 45(1):1–14
Nobile MS, Cazzaniga P, Besozzi D, Colombo R, Mauri G, Pasi G (2018) Fuzzy selftuning pso: a settingsfree algorithm for global optimization. Swarm Evol Comput 39:70–85
Nouiri M, Bekrar A, Jemai A, Niar S, Ammari AC (2018) An effective and distributed particle swarm optimization algorithm for flexible jobshop scheduling problem. J Intell Manuf 29(3):603–615
Pashaei E, Pashaei E, Aydin N (2019) Gene selection using hybrid binary black hole algorithm and modified binary particle swarm optimization. Genomics 111(4):669–686
Passino KM (2002) Biomimicry of bacterial foraging for distributed optimization and control. IEEE Control Syst Mag 22(3):52–67
Pawlak Z (1982) International of computer and information science. Rough Set 11:341–356
Poli R, Kennedy J, Blackwell T (2007) Particle swarm optimization. Swarm Intell 1(1):33–57
Pradeepkumar D, Ravi V (2017) Forecasting financial time series volatility using particle swarm optimization trained quantile regression neural network. Appl Soft Comput 58:35–52
Qi C, Fourie A, Chen Q (2018) Neural network and particle swarm optimization for predicting the unconfined compressive strength of cemented paste backfill. Constr Build Mater 159:473–478
Qi X, Ju G, Xu S (2018) Efficient solution to the stagnation problem of the particle swarm optimization algorithm for phase diversity. Appl Opt 57(11):2747–2757
Qian Q, Wu J, Wang Z (2019) Optimal path planning for twowheeled selfbalancing vehicle pendulum robot based on quantumbehaved particle swarm optimization algorithm. Personal and Ubiquitous Computing, pp 1–11
Qu B, Zhu Y, Jiao Y, Wu M, Suganthan PN, Liang J (2018) A survey on multiobjective evolutionary algorithms for the solution of the environmental/economic dispatch problems. Swarm Evol Comput 38:1–11
Rahgoshay M, Feiznia S, Arian M, Hashemi SAA (2019) Simulation of daily suspended sediment load using an improved model of support vector machine and genetic algorithms and particle swarm. Arab J Geosci 12(9):277
Rahman NHA, Zobaa AF (2017) Integrated mutation strategy with modified binary pso algorithm for optimal pmus placement. IEEE Trans Ind Inf 13(6):3124–3133
Raj S, Ray KC (2017) Ecg signal analysis using dctbased dost and pso optimized svm. IEEE Trans Instrum Meas 66(3):470–478
Ramya ST, Arunagiri B, Rangarajan P (2017) Novel effective xpath particle swarm optimization based deprived video data retrieval for smart city. Cluster Computing, pp 1–10
Rao R (2016) Jaya: a simple and new optimization algorithm for solving constrained and unconstrained optimization problems. Int J Ind Eng Comput 7(1):19–34
Rashid M, Baig AR (2010) Psogp: a genetic programming based adaptable evolutionary hybrid particle swarm optimization. Int J Innov Comput Inf Control 6(1):287–296
RojasGarcía ÁA, HernándezAguirre A, Valdez SI (2019) On the selection of the optimal topology for particle swarm optimization: a study of the tree as the universal topology. In: Proceedings of the genetic and evolutionary computation conference. ACM, pp 55–62
Rouff CA, Hinchey MG, Truszkowski WF, Rash JL (2006) Experiences applying formal approaches in the development of swarmbased space exploration systems. Int J Softw Tools Technol Transf 8(6):587–603
Salajegheh F, Salajegheh E (2019) Psog: enhanced particle swarm optimization by a unit vector of first and second order gradient directions. Swarm Evol Comput 46:28–51
Sameer F, Bakar MA, Zaidan A, Zaidan B (2019) A new algorithm of modified binary particle swarm optimization based on the GustafsonKessel for credit risk assessment. Neural Comput Appl 31(2):337–346
Sanchez IAL, Vargas JM, Santos CA, Mendoza MG, Moctezuma CJM (2018) Solving binary cutting stock with matheuristics using particle swarm optimization and simulated annealing. Soft Comput 22(18):6111–6119
Saremi S, Mirjalili S, Lewis A, Liew AWC, Dong JS (2018) Enhanced multiobjective particle swarm optimisation for estimating hand postures. Knowl Based Syst 158:175–195
Sato M, Fukuyama Y, Iizaka T, Matsui T (2018) Total optimization of energy networks in a smart city by multiswarm differential evolutionary particle swarm optimization. IEEE Trans Sustain Energy
Serani A, Leotardi C, Iemma U, Campana EF, Fasano G, Diez M (2016) Parameter selection in synchronous and asynchronous deterministic particle swarm optimization for ship hydrodynamics problems. Appl Soft Comput 49:313–334
Shahzad F, Masood S, Khan NK (2014) Probabilistic oppositionbased particle swarm optimization with velocity clamping. Knowl Inf Syst 39(3):703–737
Sheikholeslami F, Navimipour NJ (2017) Service allocation in the cloud environments using multiobjective particle swarm optimization algorithm based on crowding distance. Swarm Evol Comput 35:53–64
Shen J, Han L (2019) Design process optimization and profit calculation module development simulation analysis of financial accounting information system based on particle swarm optimization (pso). Information Systems and eBusiness Management, pp 1–14
Sivaranjani R, Roomi SMM, Senthilarasi M (2019) Speckle noise removal in sar images using multiobjective pso (mopso) algorithm. Appl Soft Comput 76:671–681
Song M, Chen K, Wang J (2018) Threedimensional wind turbine positioning using gaussian particle swarm optimization with differential evolution. J Wind Eng Ind Aerodyn 172:317–324
Srisukkham W, Zhang L, Neoh SC, Todryk S, Lim CP (2017) Intelligent leukaemia diagnosis with barebones pso based feature optimization. Appl Soft Comput 56:405–419
Stork J, Friese M, Zaefferer M, BartzBeielstein T, Fischbach A, Breiderhoff B, Naujoks B, Tušar T (2020) Open issues in surrogateassisted optimization. In: Highperformance simulationbased optimization. Springer, pp 225–244
Sun J, Feng B, Xu W (2004) Particle swarm optimization with particles having quantum behavior. In: Proceedings of the 2004 congress on evolutionary computation (IEEE Cat. No. 04TH8753), vol 1. IEEE, pp 325–331
Sun L, Song X, Chen T (2019) An improved convergence particle swarm optimization algorithm with random sampling of control parameters. J Control Sci Eng 2019
Sun Z, Liu Y, Tao L (2018) Attack localization task allocation in wireless sensor networks based on multiobjective binary particle swarm optimization. J Netw Comput Appl 112:29–40
Suresh S, Lal S (2017) Multilevel thresholding based on chaotic Darwinian particle swarm optimization for segmentation of satellite images. Appl Soft Comput 55:503–522
Tang B, Han J, Guo G, Chen Y, Zhang S (2019) Building material prices forecasting based on least square support vector machine and improved particle swarm optimization. Archit Eng Des Manag 15(3):196–212
Tang W, Cha H, Wei M, Tian B (2019) Estimation of surfacebased duct parameters from automatic identification system using the Lévy flight quantumbehaved particle swarm optimization algorithm. J Electromagn Waves Appl, 1–11
Thabit S, Mohades A (2018) Multirobot path planning based on multiobjective particle swarm optimization. IEEE Access 7:2138–2147
Thangaraj R, Pant M, Abraham A, Bouvry P (2011) Particle swarm optimization: hybridization perspectives and experimental illustrations. Appl Math Comput 217(12):5208–5226
Thangaraj R, Pant M, Abraham A, Snasel V (2012) Modified particle swarm optimization with time varying velocity vector. Int J Innov Comput Inf Control 8(1):201–218
Tharwat A, Hassanien AE (2019) Quantumbehaved particle swarm optimization for parameter optimization of support vector machine. J Classif, pp 1–23
Ting T, Yang XS, Cheng S, Huang K (2015) Hybrid metaheuristic algorithms: past, present, and future. In: Recent advances in swarm intelligence and evolutionary computation. Springer, pp 71–83
Tizhoosh HR (2005) Oppositionbased learning: a new scheme for machine intelligence. In: International conference on computational intelligence for modelling, control and automation and international conference on intelligent agents, web technologies and internet commerce (CIMCAIAWTIC’06), vol 1. IEEE, pp 695–701
Tran B, Zhang M, Xue B (2016) A pso based hybrid feature selection algorithm for highdimensional classification. In: 2016 IEEE congress on evolutionary computation (CEC). IEEE, pp 3801–3808
Vafashoar R, Meybodi MR (2019) Cellular learning automata based bare bones pso with maximum likelihood rotated mutations. Swarm Evol Comput 44:680–694
Valsecchi A, Bermejo E, Damas S, Cordón O (2018) Metaheuristics for medical image registration. Handbook of Heuristics, pp 1079–1101
Vandersypen LM, Steffen M, Breyta G, Yannoni CS, Sherwood MH, Chuang IL (2001) Experimental realization of Shor’s quantum factoring algorithm using nuclear magnetic resonance. Nature 414(6866):883
Vijay M, Jena D (2017) Pso based neuro fuzzy sliding mode control for a robot manipulator. J Electr Syst Inf Technol 4(1):243–256
Villarrubia G, De Paz JF, Chamoso P, De la Prieta F (2018) Artificial neural networks used in optimization problems. Neurocomputing 272:10–16
Wang C, Yu T, CurielSosa JL, Xie N, Bui TQ (2019) Adaptive chaotic particle swarm algorithm for isogeometric multiobjective size optimization of fg plates. Structural and Multidisciplinary Optimization, pp 1–22
Wang D, Qiu H, Wu CC, Lin WC, Lai K, Cheng SR (2018) Dominance rule and oppositionbased particle swarm optimization for twostage assembly scheduling with time cumulated learning effect. Soft Computing, pp 1–12
Wang D, Tan D, Liu L (2018) Particle swarm optimization algorithm: an overview. Soft Comput 22(2):387–408
Wang GG, Deb S, Coelho LdS (2015) Elephant herding optimization. In: 2015 3rd international symposium on computational and business intelligence (ISCBI). IEEE, pp 1–5
Wang H, Peng Mj, Hines JW, Zheng Gy, Liu Yk, Upadhyaya BR (2019) A hybrid fault diagnosis methodology with support vector machine and improved particle swarm optimization for nuclear power plants. ISA Trans
Wang JJ, Liu GY (2019) Saturated control design of a quadrotor with heterogeneous comprehensive learning particle swarm optimization. Swarm Evol Comput 46:84–96
Wang S, Li Y, Yang H (2019) Selfadaptive mutation differential evolution algorithm based on particle swarm optimization. Appl Soft Comput 81:105496
Wu TY, Lin CH (2014) Lowsar path discovery by particle swarm optimization algorithm in wireless body area networks. IEEE Sens J 15(2):928–936
Wu Y, Miao Q, Ma W, Gong M, Wang S (2017) Psosac: particle swarm optimization sample consensus algorithm for remote sensing image registration. IEEE Geosci Remote Sens Lett 15(2):242–246
Xu G, Wu ZH, Jiang MZ (2015) Premature convergence of standard particle swarm optimisation algorithm based on Markov chain analysis. Int J Wirel Mobile Comput 9(4):377–382
Xu G, Cui Q, Shi X, Ge H, Zhan ZH, Lee HP, Liang Y, Tai R, Wu C (2019) Particle swarm optimization based on dimensional learning strategy. Swarm Evol Comput 45:33–51
Xu L, Muhammad A, Pu Y, Zhou J, Zhang Y (2019) Fractionalorder quantum particle swarm optimization. PLoS ONE 14(6):e0218285
Xu X, Rong H, Trovati M, Liptrott M, Bessis N (2018) Cspso: chaotic particle swarm optimization algorithm for solving combinatorial optimization problems. Soft Comput 22(3):783–795
Yang XS (2009) Firefly algorithms for multimodal optimization. In: International symposium on stochastic algorithms. Springer, pp 169–178
Yang XS, Cui Z, Xiao R, Gandomi AH, Karamanoglu M (2013) Swarm intelligence and bioinspired computation: theory and applications. Newnes, London
Yang Z, Qiu H, Gao L, Cai X, Jiang C, Chen L (2019) A surrogateassisted particle swarm optimization algorithm based on efficient global optimization for expensive blackbox problems. Eng Optim 51(4):549–566
Yazdani M, Jolai F (2016) Lion optimization algorithm (loa): a natureinspired metaheuristic algorithm. J Comput Des Eng 3(1):24–36
Yi T, Zheng H, Tian Y, Liu Jp (2018) Intelligent prediction of transmission line project cost based on least squares support vector machine optimized by particle swarm optimization. Mathematical Problems in Engineering 2018
Yoon Y, Kim YH (2013) An efficient genetic algorithm for maximum coverage deployment in wireless sensor networks. IEEE Trans Cybern 43(5):1473–1483
Yu H, Tan Y, Zeng J, Sun C, Jin Y (2018) Surrogateassisted hierarchical particle swarm optimization. Inf Sci 454:59–72
Yuan Q, Yin G (2014) Analyzing convergence and rates of convergence of particle swarm optimization algorithms using stochastic approximation methods. IEEE Trans Autom Control 60(7):1760–1773
Yue Y, Cao L, Hu J, Cai S, Hang B, Wu H (2019) A novel hybrid location algorithm based on chaotic particle swarm optimization for mobile position estimation. IEEE Access 7:58541–58552
Zarei A, Mousavi SF, Gordji ME, Karami H (2019) Optimal reservoir operation using bat and particle swarm algorithm and game theory based on optimal water allocation among consumers. Water Resources Management, pp 1–23
Zarrouk R, Bennour IE, Jemai A (2019) A twolevel particle swarm optimization algorithm for the flexible job shop scheduling problem. Swarm Intelligence, pp 1–24
Zeng N, Qiu H, Wang Z, Liu W, Zhang H, Li Y (2018) A new switchingdelayedpsobased optimized svm algorithm for diagnosis of Alzheimer’s disease. Neurocomputing 320:195–202
Zhang H, Xie J, Hu Q, Shao L, Chen T (2018) A hybrid dpso with Lévy flight for scheduling mimo radar tasks. Appl Soft Comput 71:242–254
Zhang X, Zheng X, Cheng R, Qiu J, Jin Y (2018) A competitive mechanism based multiobjective particle swarm optimizer with fast convergence. Inf Sci 427:63–76
Zhang Y, Balochian S, Agarwal P, Bhatnagar V, Housheya OJ (2014) Artificial intelligence and its applications. Mathematical Problems in Engineering 2014
Zhang Y, Wang S, Ji G (2015) A comprehensive survey on particle swarm optimization algorithm and its applications. Mathematical Problems in Engineering 2015
Zhang Y, Zhang Q, Farnoosh A, Chen S, Li Y (2019) Gisbased multiobjective particle swarm optimization of charging stations for electric vehicles. Energy 169:844–853
Zhang YD, Wang S, Dong Z (2014) Classification of Alzheimer disease based on structural magnetic resonance imaging by kernel support vector machine decision tree. Prog Electromagn Res 144:171–184
Zheng J, Lu C, Gao L (2019) Multiobjective cellular particle swarm optimization for wellbore trajectory design. Appl Soft Comput 77:106–117
Zhong Y, Lin J, Wang L, Zhang H (2018) Discrete comprehensive learning particle swarm optimization algorithm with metropolis acceptance criterion for traveling salesman problem. Swarm Evol Comput 42:77–88
Zhou A, Qu BY, Li H, Zhao SZ, Suganthan PN, Zhang Q (2011) Multiobjective evolutionary algorithms: a survey of the state of the art. Swarm Evol Comput 1(1):32–49
Zhou Z, Ong YS, Lim MH, Lee BS (2007) Memetic algorithm using multisurrogates for computationally expensive optimization problems. Soft Comput 11(10):957–971
Zhu M, Li J, Chang D, Zhang G, Chen J (2018) Optimization of antenna array deployment for partial discharge localization in substations by hybrid particle swarm optimization and genetic algorithm method. Energies 1(11):1813
Funding
Open access funding provided by The Science, Technology & Innovation Funding Authority (STDF) in cooperation with The Egyptian Knowledge Bank (EKB).
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Conflict of interest
The single/corresonding author declares no conflict of interest.
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.
About this article
Cite this article
Gad, A.G. Particle Swarm Optimization Algorithm and Its Applications: A Systematic Review. Arch Computat Methods Eng 29, 2531–2561 (2022). https://doi.org/10.1007/s11831021096944
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11831021096944