Chicken swarm optimization and associated variants: a literature review

The difference between the CSO and other algorithms

Swarm optimization represents a metaheuristic approach drawing inspiration from diverse fields such as biological science (Tang et al., 2024; Wang, Yue & Cao, 2022), operations research, and physics (Qin et al., 2025). It stands as a vital methodology for addressing real-world optimization challenges. Metaheuristic techniques construct tailored models for particular problems by delving into the dynamics of biological, chemical, and social systems, subsequently engineering intelligent, iterative search and optimization algorithms (Abdel-Basset, Abdel-Fatah & Sangaiah, 2018). Some notable examples include Genetic Algorithm (GA) (Holland, 1992), Differential Evolution (DE) (Storn & Price, 1997), CSO (Meng et al., 2014), Particle Swarm Optimization (PSO) (Kennedy & Eberhart, 1995), Grey Wolf Optimizer (GWO) (Mirjalili, Mirjalili & Lewis, 2014), Harris Hawks Optimization (HHO) (Heidari et al., 2019), Reptile Search Algorithm (RSA) (Abualigah et al., 2022), Teaching–Learning–Based Optimization (TLBO) (Rao, Savsani & Vakharia, 2011), Artificial Ecosystem–Based Optimization (AEO) (Zhao, Wang & Zhang, 2020) Simulated Annealing (SA) (Kirkpatrick, Gelatt & Vecchi, 1983), Harmony Search (HS) (Geem, Kim & Loganathan, 2001), Marine Predators Algorithm (MPA) (Faramarzi et al., 2020), Arithmetic Optimization Algorithm (AOA) (Abualigah et al., 2021), Runge–Kutta Optimization (RUN) (Yıldız et al., 2022) etc., (see Fig. 1). With the continuous development of various metaheuristic methods, their unique advantages can be found in practical optimization problems (Zitar, 2022). CSO is a random optimization method based on the search behavior of chicken swarms, simulating the hierarchical structure and behavior of chicken swarms (Meng et al., 2014). Compared with other algorithms, it has the characteristics of fast convergence speed, high search accuracy, and strong robustness. With these advantages, CSO outperforms other classical algorithms in research and experiments and shows broad application prospects.

To further clarify the relative position and characteristics of CSO among existing metaheuristic algorithms presented in Fig. 1, a concise comparison with several representative algorithms is provided below. Among which PSO models the collective movement of bird flocks, where each particle updates its velocity and position by learning from its own best record and the swarm’s global (or neighborhood) best. This social-learning mechanism yields very fast convergence on continuous optimization but can suffer from premature stagnation without diversity control. Numerous variants add inertia weights, constriction factors, or topology changes to balance exploration and exploitation. Compared with CSO’s subgroup hierarchy, PSO uses a single-swarm structure and typically has fewer control rules, which simplifies implementation but may reduce robustness on complex multimodal landscapes (Kennedy & Eberhart, 1995).

GWO imitates the leadership hierarchy (alpha, beta, delta, omega) and encircling–attacking behaviors of grey wolves. Candidate solutions are guided by the weighted influence of the top wolves, gradually shrinking the search radius to intensify exploitation. Its rank-based leadership facilitates stable convergence with limited parameters, though it may converge early if elites are misleading. Relative to CSO, which uses role-specific motion rules (roosters, hens, chicks), GWO emphasizes elite guidance and distance-shrinking dynamics (Mirjalili, Mirjalili & Lewis, 2014).

HHO is inspired by the surprise pounce tactics of Harris hawks and alternates between exploration and exploitation via soft/hard besiege phases. A time-varying energy factor and randomized jumps help the algorithm switch strategies and avoid local traps. HHO has shown strong performance on complex functions and engineering tasks, but can still stagnate without diversification. Compared with CSO, HHO relies on an adaptive “attack” model rather than subgroup roles; hybrids often leverage CSO’s diversity and HHO’s aggressive exploitation (Heidari et al., 2019).

RSA mimics reptilian hunting—tracking, encircling, and ambushing—to design exploration and exploitation steps with adaptive control. It promotes wide coverage early and focuses on local refinement near promising regions later; several works report competitive results on high-dimensional benchmarks and feature selection. RSA, however, may require parameter tuning to maintain diversity. In comparison with CSO’s hierarchical interactions, RSA uses stage-driven movement policies; CSO–RSA hybrids are used to combine diversity (CSO) with RSA’s strong intensification (Abualigah et al., 2022).

DE is a classical, physics-inspired population heuristic that perturbs target vectors by scaled differences of randomly sampled individuals, followed by crossover and selection. Its few, interpretable parameters and powerful self-recombination make it a staple for continuous, nonconvex optimization. DE may stagnate if diversity vanishes, motivating adaptive and ensemble strategies. Relative to CSO, DE’s operators are vector arithmetic rather than behavior rules; CSO–DE hybrids often inject DE’s strong local refinement into CSO’s diverse search (Storn & Price, 1997).

GA formalizes Darwinian evolution—selection, crossover, and mutation—on encoded candidate solutions. Schema propagation enables broad exploration of the search space, while niche methods maintain diversity on multimodal problems. Drawbacks include parameter sensitivity and potentially slow convergence without elitism or adaptive rates. Compared with CSO’s role-based dynamics, GA focuses on genetic recombination; recent CSO hybrids using GA-style crossover can increase diversity between subgroups (Holland, 1992).

TLBO simulates a classroom where solutions learn from the best “teacher” (teacher phase) and from peers (learner phase). It needs few algorithm-specific parameters and often provides robust baseline performance across engineering designs. The absence of explicit mutation can slow escape from deep local minima; many improved TLBOs add adaptive or chaotic strategies. When paired with CSO, TLBO’s parameter-light exploitation complements CSO’s exploration from subgroup interactions (Rao, Savsani & Vakharia, 2011).

AEO abstracts producers, consumers, and decomposers in an ecosystem to construct sequential update modes for exploration (production/consumption) and exploitation (decomposition). The alternating roles encourage both diversity and intensified search near high-quality regions. Studies since 2020 show competitive results on engineering and energy scheduling problems; performance depends on properly balancing stage frequencies. Against CSO, AEO is also multi-role but emphasizes stage cycling of the entire population rather than fixed social hierarchies (Zhao, Wang & Zhang, 2020).

SA is a single-solution metaheuristic based on thermal annealing: at high “temperature,” worse moves may be accepted to escape local minima; as temperature cools, the search becomes greedy. Its simplicity and theoretical grounding are appealing, but performance hinges on a well-designed cooling schedule. SA is often embedded in hybrids for local polishing. In CSO contexts, SA can serve as a post-optimizer to refine CSO’s best subgroup solutions (Kirkpatrick, Gelatt & Vecchi, 1983).

HS emulates musical improvisation: new “harmonies” (solutions) are generated from a memory bank using memory consideration, pitch adjustment, and randomization. The memory structure maintains diversity while incremental pitch changes provide local improvement. Parameter settings (HMCR, PAR, bandwidth) influence the balance of exploration and exploitation. CSO–HS hybrids use HS’s memory to stabilize subgroup search while CSO supplies behavioral diversity (Geem, Kim & Loganathan, 2001).

MPA models predator–prey foraging with Brownian/Lévy motions and a fish aggregating devices (FADs) mechanism to alternate wide-range exploration and focused pursuit. It is noted for strong performance on complex multimodal tasks and has been widely applied since 2020 in imaging, energy, and scheduling. Parameter control remains important to avoid drift. Combining MPA with CSO often couples MPA’s long-jump exploration with CSO’s structured subgroup learning (Faramarzi et al., 2020).

AOA designs four arithmetic operators (addition, subtraction, multiplication, division) with a density factor and accelerator to shift from diversification to intensification. Its operator set yields flexible step sizes and competitive results with modest parameter burden. Reported issues include sensitivity to density scheduling on rugged landscapes; adaptive AOA variants address this. With CSO, AOA operators can replace or augment role updates to refine exploitation without sacrificing diversity (Abualigah et al., 2021).

RUN leverages the Runge–Kutta integration scheme to guide candidate updates via higher-order local approximations, improving exploitation accuracy while retaining randomized exploration. It has achieved strong results on engineering design and feature selection since 2022. Tuning exploration terms is key to avoid overfitting local regions. In CSO hybrids, RUN often acts as a local search layer to sharpen the best rooster/hen positions (Yıldız et al., 2022). To provide a clearer comparison between CSO and other representative metaheuristic algorithms, Table 1 summarizes their structural features, strengths, and limitations.

Research objective and motivation

The objective of this study is to provide a structural review of CSO and its variants, offering an organized synthesis of algorithmic mechanisms, improvement strategies, and application trends. Rather than performing a comprehensive empirical investigation, this review analyzes and classifies existing research according to algorithmic strategies, highlights the evolution of CSO, and identifies research gaps that remain unaddressed. This article specifically discusses the following aspects: Introduction to standard CSO; including inspiration, main components, flowchart, and algorithm steps. At the same time, it outlines the research methods used and the improvements of the CSO algorithm, as well as the research and application of CSO in different fields. Finally, it includes a discussion and conclusion. Although two prior CSO reviews exist, both leave clear gaps that justify the present study. The early survey “Recent Studies on Chicken Swarm Optimization Algorithm: A Review (2014–2018)” concentrates on the formative stage of CSO and mainly reports the standard model with a few preliminary variants, without a structured taxonomy of improvement strategies (e.g., hybridization, adaptive control, binary/discrete modeling, or multi-objective extensions), cross-domain performance comparisons, or quantitative trend analysis (Deb et al., 2020a). The more recent “A comprehensive survey on the chicken swarm optimization algorithm and its applications: state-of-the-art and research challenges” summarizes developments up to roughly 2023 and outlines application areas, but it does not provide (i) a year-wise publication distribution or an objective-oriented categorization (single vs. multi-objective; continuous vs. binary; hybrid vs. non-hybrid), (ii) a consolidated comparison of algorithmic mechanisms and parameter-adaptation schemes across CSO variants, or (iii) an explicit mapping from CSO strategy families to recurring problem archetypes (e.g., engineering design, energy scheduling, WSN routing, medical diagnosis) (Chen et al., 2024a).

In contrast, this review covers 2014–2025 and contributes: (i) an objective- and strategy-based taxonomy that organizes CSO variants (hybrid, adaptive, binary, multi-objective) together with their application domains; (ii) a quantitative, year-wise distribution to reveal research growth phases; and (iii) a synthesis of comparative insights on strengths/limitations and domain suitability of recent CSO advances (e.g., fuzzy/chaotic enhancements, dual-population/adaptive role updates, CSO–DE/PSO/GA/HHO hybrids). These additions complement and extend prior CSO surveys rather than duplicate them, clarifying where CSO variants are most effective and where open challenges remain.

Literature search and selection strategy

To ensure the reliability and academic quality of the reviewed materials, the majority of the studies analyzed in this article were sourced from peer-reviewed international journals indexed in Web of Science, Scopus, IEEE Xplore, and ScienceDirect. In cases where journal versions were not yet available, a limited number of high-impact conference publications were considered to capture recent advancements. This selection strategy ensures that the review is grounded in rigorously vetted research while preserving coverage of emerging CSO developments. To ensure transparency and reduce selection bias, the literature retrieval and filtering process was conducted in accordance with the PRISMA 2020 guidelines. The procedure consisted of four stages: identification, screening, eligibility assessment, and final inclusion.

A total of 198 studies related to CSO and its variants were initially identified. Among them, 180 records were retrieved from academic databases such as Google Scholar, while 18 additional studies were located through manual searches, including citation tracing and reference cross-checking. After the removal of 42 duplicates, 156 unique records were retained.

The titles and abstracts of the remaining 156 studies were screened to exclude irrelevant or clearly unsuitable publications. During this phase, 28 records were removed based on a lack of topic relevance, incomplete methodological description, or insufficient connection to CSO. A total of 130 articles remained for full-text assessment.

Full-text evaluation was performed on these 128 articles using predefined inclusion and exclusion criteria. During this stage, 10 publications were excluded due to one or more of the following reasons: absence of experimental validation (n = 4), insufficient algorithmic detail (n = 3), and lack of CSO relevance or comparative analysis (n = 3).

Finally, 138 eligible studies, These articles form the basis for examining the development, hybridization strategies, performance evaluation, and real-world applications of CSO and its variants. The entire identification and screening process is illustrated in the updated PRISMA 2020 flow diagram (Fig. 2).

Theoretical convergence considerations of hybrid CSO variants

Although metaheuristic algorithms are predominantly heuristic and stochastic in nature, their search behavior can still be theoretically justified under established convergence frameworks. Classical population-based optimizers, including CSO and its hybrid variants, can be interpreted as randomized iterative processes or Markov chains operating over a continuous or discrete search space (Marini & Walczak, 2015). Under the assumption of irreducibility and bounded stochastic perturbation, such processes are capable of converging in probability toward a global attractor set as the number of iterations tends to infinity.

In the context of hybrid CSO models, convergence is strengthened by the integration of mechanisms such as Lévy flight, differential evolution, adaptive parameter control, and elite retention. Lévy-based perturbations enlarge the global search radius and ensure a non-zero probability of escaping local minima (Verma, Sahu & Sahu, 2024), while DE-style mutation and crossover operators improve directional guidance in high-dimensional landscapes. The elite or memory-based retention of best-performing individuals preserves solution quality across generations, satisfying the conditions for asymptotic stability (Deng et al., 2021). Formally, if $x_t$ denotes the position of an individual (or best solution) at iteration $t$, and $x^{\ast }$ is a global optimum or element of the optimal set, then a common convergence statement in stochastic metaheuristics asserts:

(1) $$\underset{t\to \mathrm{\infty }}{lim}P\left(\Vert x_t-x^{\ast }\Vert \le \epsilon \right)=1,\mathrm{\forall }\epsilon >0.$$

This expression does not imply deterministic convergence to a single point, but rather convergence in probability to an $\epsilon$-neighborhood of the global optimum. The use of randomization, adaptive step sizes, and subgroup role updates in CSO satisfies two essential conditions typically cited in convergence theories: (i) a sufficiently diverse exploration phase with non-zero probability of sampling any region in the search space, and (ii) an exploitation mechanism that asymptotically narrows the search radius (Huang, Qiu & Riedl, 2023).

Moreover, hybrid CSO variants introduce multi-stage update rules that preserve ergodicity while reducing variance in the latter stages of search, thereby making the convergence bound tighter compared to the static version. Although a full formal proof for every variant is beyond the scope of this review, the combination of stochastic global reach, adaptive local search pressure, and elite solution retention aligns with the convergence models established for PSO, DE, and ACO-type metaheuristics. Consequently, the improved CSO frameworks surveyed in this article possess a reasonable theoretical foundation under stochastic convergence analysis, supporting their algorithmic stability and long-term search reliability.

Chicken swarm optimization algorithm

The CSO algorithm was initially introduced by Meng et al. (2014) in 2014. Their gregarious nature allows them to coordinate their foraging efforts (Yan et al., 2021). They posited that CSO functions as an intelligent optimization method, strategically orchestrating parameter adjustments by emulating the behavioural patterns observed in chicken flocks. The algorithm models the behaviour of chicken flocks by categorizing parameter variables into three distinct groups: roosters, hens, and chicks. It then establishes a parameter adjustment process that mirrors the foraging behaviour exhibited by these flocks (Carvalho et al., 2022). In this framework, the entire chicken flock is partitioned into multiple subgroups, each comprising a single rooster, several hens, and their respective chicks. Within each subgroup, these different categories of individuals adhere to their own unique movement rules and engage in diverse forms of interaction, such as learning and competition. At the same time, the hierarchical structure within the chicken flock will be continuously updated with multiple generations of evolution (Ishikawa et al., 2020).

Chicken swarm algorithm characteristics

Chickens are gregarious animals that usually forage in groups. The entire group can be divided into three types of individuals: roosters, hens, and chicks. Due to the difference in foraging ability, a clear hierarchy is formed within the group, as shown in Fig. 4. Hens have weaker foraging ability than roosters, so in the hierarchy, hens usually follow roosters to forage, while chicks with poorer foraging ability are ranked last. In a foraging group, the rooster is in the center of the group, the hens move around the roosters, and the chicks forage around the hens. Because of this, a pattern of mutual learning and competition is naturally formed between different members, and the positions of individuals are constantly updated according to their respective movement rules (Jonsson & Vahlne, 2023).

Within the entire population, the identities, mating relationships, and mother-child connections among individuals remain static over a span of G generations (where G represents the iteration cycle). Following this period, these relationships undergo an update (Singh & Kumar, 2023). Within each smaller subgroup of the population, hens typically adhere to their respective mates, the roosters, in their quest for sustenance, while concurrently engaging in random competitions for food with other members of the subgroup. Notably, individuals boasting superior fitness values are more likely to secure food resources (Wang et al., 2023a).

The entire flock can be categorized into three groups: RN, HN, and MN, which denote the respective counts of roosters, hens, and chicks. The rooster with the highest fitness value is designated as the best rooster (RN), while the chick with the lowest fitness value is classified as the worst chick (CN). All other individuals are considered hens. Thus, a total of N chickens are engaged in foraging activities within a D-dimensional space. Their position $x_{i,j}^t(i\in \{1,\dots ,N\},j\in \{1,\dots ,D)\}$, let $x_{i,j}^t$ represent the position of the represent the position of the chicken, chicken, represents the $t_i$ iteration, and $j$ represents the $j_i$ dimension of the search space, where $i\in \{1,\dots ,N\},j\in \{1,\dots ,D),t\in \{1,\dots ,T)$, and the maximum number of iterations is T.

Mathematical modeling

Parameter control within the CSO algorithm holds significant importance, and its key operational components are illustrated in Fig. 3 (Wu et al., 2015).

In the context of the CSO algorithm, the size of the flock is determined through empirical means. When the flock contains too few chickens, the quality of the solution to the optimization problem tends to be suboptimal. Conversely, an excessively large flock size can escalate the computational expense of running the algorithm. Furthermore, the ratio of hens, chicks, and roosters within the flock is crucial and exerts a notable influence on the algorithm’s search efficacy. A recommended ratio for the distribution of chicks, hens, and roosters is 2:6:2.

When the algorithm undergoes G iterations, it is essential to restructure the chicken hierarchy. Consequently, the choice of the dislike parameter G is of significant importance. If the value of G is set too high, it may hinder the CSO algorithm from effectively identifying the optimal solution to the problem. Conversely, if G is too low, the algorithm becomes susceptible to premature convergence. Therefore, selecting an appropriate value for G not only enhances the algorithm’s diversity but also provides opportunities for the algorithm to escape local optima during its execution. when $G\in [2,20]$, good optimization results can be achieved for most of the optimization problems.

In the CSO algorithm, the coefficient FL, which represents the learning factor for chicks following hens, plays a crucial role. When FL is set to a smaller value, it enhances the chicks’ ability to conduct preliminary local searches, thereby potentially improving their exploration in the early stages of the algorithm. However, if FL is too large, it may negatively affect the algorithm’s performance in later iterations. Typically, the FL value is maintained within the range of [0.4, 1] to balance exploration and exploitation effectively. When the CSO algorithm is updated after the maximum number of iterations, the algorithm will stop iterating down and output the final obtained optimal value, and when the convergence accuracy reaches a certain degree will also terminate the iteration.

The CSO algorithm begins by initializing the population and subsequently evaluates the fitness value of each individual within the population. Based on these fitness values, the chickens are categorized into roosters, hens, and chicks, establishing a hierarchical structure and a maternal-offspring relationship. Following the update of all individual positions, the current optimal position for each chicken is updated and potentially replaced. The overall best value of the entire chicken group is also updated, and the fitness value of each chicken is recalculated. The algorithm then checks whether the termination condition has been met to determine if the operation should be terminated. If the optimal position has been obtained, the iteration is halted, and the result is output. Otherwise, the algorithm reverts to the previous step, and this loop continues until the stopping criterion is satisfied. The standard CSO algorithm can be divided into three fundamental steps: parameter initialization, generation of new solutions, and updating the flock’s iterative positions. The flowchart for CSO is illustrated in Fig. 5, and the algorithm comprises the following nine steps.

(1) Initialize flock location.

In the CSO method, there are three distinct types of chickens: roosters, hens, and chicks. When the CSO algorithm is applied to solve an optimization problem, each chicken within the flock corresponds to a potential solution for that problem. Each type of chicken—roosters, hens, and chicks—employs a unique strategy in their pursuit of the optimal solution. Within the flock, the total number of chickens is denoted as N, which includes the combined count of roosters, hens, and chicks. Specifically, where $N_R$ denotes the number of roosters, the number of hens is denoted by $N_H$, The number of chicks is denoted by $N_C$, $X_i^j(t)$ denotes the value of the $j$ latitude of the $i$ individual at the t iteration, where $i\in \{1,2,\dots ,N\},j\in \{1,2,\dots ,D\},t\in \{1,2,\dots ,M\}$, represents the spatial dimension of the optimization problem, and Maximum iteration times at M. The initial positions $X_i=(x_{i1},x_{i2},\dots ,x_{iD})$ of the whole flock are randomly generated in the search space according to equation:

(2) $$x_{ij}=L_{\mathrm{m}\mathrm{i}\mathrm{n}}+\mathrm{R}\mathrm{a}\mathrm{n}\mathrm{d}(U_{\mathrm{m}\mathrm{a}\mathrm{x}j}-L_{\mathrm{m}\mathrm{i}\mathrm{n}j})$$where $U_{\mathrm{m}\mathrm{a}\mathrm{x}j}$ and $L_{\mathrm{m}\mathrm{i}\mathrm{n}j}$ denote the upper and lower bounds of everyone in the j dimension, respectively, and Rand denotes a random distribution between [0, 1].

In the chicken swarm optimization algorithm, the chickens are categorized into roosters, hens, and chicks based on their fitness values. Each category employs a distinct position update formula, which can be expressed as follows:

(2) Rooster location update

Roosters with superior fitness values possess a competitive edge in foraging compared to those with lower fitness, enabling them to explore a broader range of spaces in search of food. The equation governing the position update for roosters is presented as follows:

(3) $$x_i^j(t+1)=x_i^j(t)\cdot (1+\mathrm{R}\mathrm{a}\mathrm{n}\mathrm{d}\mathrm{n}(0,{\sigma }^2))$$

(4) $${\sigma }^2=\{\begin{array}{cc}1,\hfill & \mathrm{i}\mathrm{f}{f}_i\le f_k\hfill \\ \exp \left(\frac{f_k-f_i}{|f_i|+\epsilon }\right),\hfill & \mathrm{o}\mathrm{t}\mathrm{h}\mathrm{e}\mathrm{r}\mathrm{w}\mathrm{i}\mathrm{s}\mathrm{e}\hfill \end{array},k\in [1,N_R],k\ne i$$where $x_j^i$ is the selected rooster with index $i$, $\mathrm{R}\mathrm{a}\mathrm{n}\mathrm{d}(0,{\sigma }^2)$ represents a normal distribution with mean 0 and standard deviation ${\sigma }^2$, $\epsilon$ is a constant, and its meaning is such that the denominator is not 0. $f_i$ denotes the objective function of the current $i$ rooster, $f_k$ denotes the $k$ rooster’s evil objective function value.

(3) Hen location update

Hens not only followed the roosters of their subgroup in search of food but could also randomly steal food that other hens find. In feeding competitions, hens with higher fitness values have an advantage over hens with lower fitness values. The update formula for hen is as follows:

(5) $$x_{i,j}^{t+1}=x_{i,j}^t+S_1\cdot \mathrm{R}\mathrm{a}\mathrm{n}\mathrm{d}\cdot (x_{r1,j}^t-x_{i,j}^t)+S_2\cdot \mathrm{R}\mathrm{a}\mathrm{n}\mathrm{d}\cdot (x_{r2,j}^t-x_{i,j}^t)$$

(6) $$S_1=\exp \left(\frac{f_k-f_i}{|f_i|+\epsilon }\right)$$

(7) $$S_2=\exp (f_{r2}-f_i)$$where $I\in \{N_{R+1},N_{R+2},\dots ,N_H\},j\in \{1,2,\dots ,D\}$, Rand denotes the number of random distributions between [0, 1] again; $r_1$ corresponding to the $i$ hen, $r_2$ denotes either body inside the rooster and the hen, $r_1\ne r_2$.

(4) Chick position update

Within each subgroup, chicks forage in the vicinity of the hen and adjust their positions in accordance with Eq. (7).

(8) $$x_{i,j}^{t+1}=x_{i,j}^t+FL\cdot x_{m,j}^t-x_{i,j}^t$$where $x_{m,j}^t$ is the position of the $i^{t}h$ chick’s mother such that $m\in [1,N]$, FL is a parameter that signifies the extent to which a chick follows its mother in terms of speed. Taking into account the individual variations among chicks, FL is randomly selected within the range of [0.2].

Convergence and theoretical basis analysis

The convergence analysis of the swarm optimization algorithm is usually carried out with the help of Markov chain theory. Wang et al. (2019) first proposed a theoretical framework for using random processes to establish a Markov chain model for CSO convergence analysis. They constructed the following Markov chain model:

(9) $$P_r(T_s(x_i)=x_j)=\{\begin{array}{cc}\frac{1}{|x_1{R}_1|},\hfill & x_j\in [x_i,x_i+x_i{R}_1]\hfill \\ 0,\hfill & otherwise\hfill \end{array}.$$

Among them, the state transition probabilities of different roles (rooster, hen, chick) are defined respectively, and the conditions and limitations of CSO global convergence and local convergence are analyzed in detail. This theoretical analysis shows that although CSO has a strong global search capability, it is easy to fall into local optimality or premature convergence when the population size is unreasonable or the parameters are poorly set, which limits its performance in complex problems. To this end, it is necessary to improve its local search capability and parameter sensitivity through adaptive mechanisms or hybrid methods.

Comparative analysis of CSO-based and IEEE TEVC 2024–2025 algorithms

To provide a clearer performance perspective of CSO and its recent enhancements, a comparative analysis was conducted based on the original benchmark settings reported in the respective publications. Rather than forcing all methods into a single test environment, each algorithm is evaluated according to its native experimental protocol. This approach ensures that the reported improvements, dimensions, and baselines remain authentic and methodologically consistent.

First, representative CSO variants from 2017 to 2024 were summarized with respect to their originally adopted benchmark suites, performance highlights, and relative effectiveness. Tables 2 and 3 presents the comparative results. To avoid artificial ranking bias, relative performance is expressed using categories such as Top-1, Top-3, Top-5.

Following the analysis of CSO-based strategies, attention is extended to the latest evolutionary optimization research published in IEEE Transactions on Evolutionary Computation (TEVC) during 2024–2025. These algorithms frequently integrate concepts such as quantum-inspired operators, hybrid search strategies, orthogonal learning, or chaotic dynamics, and they often benchmark on CEC2017, CEC2020, CEC2022, BBOB, or real-world engineering problems. Table 3 provides an overview of these cutting-edge approaches using the same ranking categories to maintain comparability.

Overall, the CSO-derived algorithms demonstrate competitive performance among classical swarm intelligence approaches, particularly in unimodal and multimodal settings. Meanwhile, the emerging TEVC 2024–2025 methods highlight new research directions (e.g., quantum search mechanisms, hybrid exploitation–exploration strategies, and adaptive parameter control). The side-by-side comparison underscores a performance gap that motivates further integration of these advanced strategies into next-generation CSO variants.

Improvement of the swarm optimization algorithm

With the ongoing expansion of application domains and the escalating complexity of problems (Li et al., 2022), the CSO algorithm is simple and efficient. It can divide the parameters into three different populations. The search range and traversal range of different populations will also be different (Ding, Zhou & Bi, 2020). The CSO algorithm has progressively revealed certain limitations (Nadikattu, 2021), including sluggish convergence speed, inadequate local search capabilities, and a propensity to become trapped in local optima (Zouache et al., 2019). In the realm of population initialization, researchers have implemented a range of strategies, such as mutation mechanisms, chaos theory, and fuzzy theory, to augment population diversity (Ahmed, Hassanien & Bhattacharyya, 2017), thereby facilitating more efficient discovery of optimal solutions. Nevertheless, the majority of these enhancement methodologies predominantly concentrate on optimizing specific update procedures.

The most significant drawback of the CSO algorithm lies in its susceptibility to becoming ensnared in local optima, with improper parameter settings serving as the primary catalyst for this issue (Qi et al., 2022). When parameter configurations are ill-advised, the algorithm is unable to accurately forecast the optimal solution, leading to suboptimal final outcomes. Within the CSO algorithm, the hen individuals consistently adhere to the rooster for parameter substitution, while the chicks follow the hens, rendering the rooster a pivotal element in the positional updating of hens. Should the rooster operator succumb to local optimality, the hens and chicks will likewise become entrapped, prompting premature convergence of the algorithm (Ren & Long, 2021). This makes the algorithm form a closed loop, hindering the further optimization of subsequent operators. Compared with traditional swarm intelligence algorithms, CSO shows better efficiency and resilience, mainly because of the overall architecture, stability and portability of the algorithm (Singh et al., 2023). Key factors such as the number of hens in the CSO algorithm will affect the convergence of the algorithm (Wu, 2021). Therefore, in view of these limitations of the CSO algorithm, many researchers have improved it and derived multiple variants (Yang et al., 2023). This section will elaborate on the improvement methods of the CSO algorithm.

CSO algorithms classified by improvement principle

The improvement methods of the chicken swarm optimization algorithm can be summarized as follows (Box 1).

Application fields and performance analysis of CSO algorithm

The swarm optimization algorithm is widely used in multi-objective optimization, high-dimensional optimization, engineering design, energy management and medicine. The application characteristics of different fields are analyzed in depth as follows:

Deep learning–based methods and integration trends

In recent years, deep learning (DL) has rapidly emerged as a dominant paradigm across various fields, including computer vision, natural language processing, and increasingly, optimization and control systems. Its capacity for automatic feature extraction and nonlinear function approximation has made it a powerful tool for modeling complex relationships in high-dimensional and dynamic environments.

In the context of industrial control and cyber-physical systems (CPS), DL techniques have been widely adopted for anomaly detection, predictive maintenance, and energy management. A recent review by Aslam, Tufail & Irshad (2025) provides a comprehensive survey of deep learning frameworks tailored for securing industrial CPS, emphasizing their ability to process time-series data and detect real-time threats with minimal feature engineering.

Moreover, the integration of deep learning with swarm-based optimization has received increasing attention. For example, Azevedo, Rocha & Pereira (2024) proposed a novel framework that combines deep convolutional neural networks (CNNs) with swarm intelligence to improve the recognition accuracy of 3D human activity, demonstrating superior performance compared to traditional feature-engineered methods. This hybrid paradigm exemplifies how deep learning can benefit from population-based heuristics, enhancing global search while maintaining high adaptability.

In the energy and automation sectors, Abbassy et al. (2025) reviewed the application of data-driven and deep learning control strategies for CPS energy systems. They highlighted key challenges such as model generalization, real-time responsiveness, and robustness against noise—aspects that swarm intelligence algorithms like CSO can naturally complement through their exploration–exploitation balance and population diversity maintenance.

While these studies underline the strengths of DL in complex modeling, they often focus on purely data-driven learning pipelines and lack interpretability, controllability, or deployment readiness in real-time environments. Furthermore, few works explore the combination of deep learning with evolutionary algorithms under embedded platforms such as MATLAB/Simulink or Hardware-in-the-Loop (HIL) simulation.

Therefore, this study complements prior efforts by focusing on (your algorithmic framework or CSO variant) that retains the optimization interpretability of swarm-based models, while offering a potential bridge for future integration with DL frameworks. This also opens the door for adaptive hybrid approaches that leverage DL for feature representation and CSO variants for robust search and parameter adaptation.

Emerging integration with quantum-inspired and quantum-assisted optimization

Although CSO and its variants have demonstrated strong performance across engineering design, energy management, scheduling, healthcare analytics, and other real-world domains, recent research trends show a clear shift toward fusing swarm-based metaheuristics with quantum-inspired or quantum-assisted mechanisms. Foundational work on quantum-inspired evolutionary computation (QEA) (Han & Kim, 2002) and quantum-behaved particle swarm optimization (QPSO) (Sun, Feng & Xu, 2004) formalized q-bit representations, superposition-based search, and probabilistic state transitions that can strengthen exploration while controlling parameterization. Subsequent analyses and variants of QPSO (e.g., Gaussian-attractor QPSO) rigorously studied convergence behavior and reported performance gains on standard test suites (Sun et al., 2011, 2012).

Recent applied studies further validate quantum-inspired swarm ideas in practical settings. For instance, quantum-inspired PSO has been adopted for service placement and resource allocation under edge/cloud scenarios with competitive results on real workloads (Bey et al., 2024). Likewise, quantum-corrected Harris Hawks Optimization (QC-HHO) integrates quantum perturbation and chaos to improve robustness on CEC2014 and engineering problems (Zhu et al., 2022). At the landscape level, a 2024 scientometric review maps the growth and hotspots of quantum-inspired metaheuristics—including QEA/QPSO families—highlighting their acceleration in optimization and Industry 4.0 contexts (Pooja & Sood, 2024).

These advances indicate that similar hybridization strategies can be transferred to CSO-based frameworks, especially for high-dimensional, multimodal, or dynamic tasks. From a hardware perspective, demonstrated NISQ-era capabilities (Arute et al., 2019) suggest that variational hybrids (e.g., QAOA/VQE-style components) may become feasible partners for swarm heuristics in the near term. Consequently, evolving CSO toward quantum-inspired or quantum-assisted architectures is both credible and timely, aligning with current trends in high-impact venues and offering a principled path to scalability and efficiency.

Representation and analysis in multi-objective optimization problems

Liang, Wang & Ma (2022) introduced an enhanced CSO algorithm, referred to as RECSO, which incorporates replication and elimination diffusion operations inspired by the bacterial foraging algorithm (BFA) (Passino, 2002) and the particle swarm optimization algorithm (PSO) (Liu & Nishi, 2022). This enhancement was prompted by the tendency of the CSO algorithm to exhibit premature convergence when tackling multi-peak optimization problems. They identified chicks as the most vulnerable individuals within the CSO framework and, to bolster the algorithm’s search capabilities, integrated the replication operation and elimination mechanism from the BFA into the position updating process for chick foraging, thereby preventing chickens from settling into local. In terms of how to efficiently solve multi-objective optimization problems (MOPs), Huang et al. (2024) proposed a non-dominated sorting swarm optimization algorithm (NSCSO). This method uses fast non-dominated sorting to assign ranks to individuals in the population and uses the crowding distance strategy to sort operators within the same rank. In addition, the algorithm combines the elite adversarial learning strategy to improve the multi-objective optimization solution capability. Benchmark function evaluations and comparative analyses across various engineering design problems demonstrate that NSCSO exhibits superior performance. In scenarios where high-dimensional data in the stock market might lead to reduced prediction accuracy. Kiruthiga & Vennila (2020) proposed an innovative multi-objective chaotic Darwinian swarm optimization (CDCSO) system for energy-efficient QoS and load balance-aware task scheduling. The multi-objective CDCSO algorithm combines chaos theory and Darwinian theory into standard swarm optimization, thereby increasing the global search capability and accelerating the convergence process. CDCSO is able to effectively assign tasks to virtual machines (VMs) with optimal energy saving, cost and time minimization, while maintaining optimal load balance. Experimental results show that the proposed multi-objective CDCSO provides better performance in task scheduling, minimizing energy, cost and time while optimizing load balance. Tumor region classification in MRI brain imaging is a crucial task. Due to the diversity and complexity of cancer, MRI brain imaging has very high requirements for tumor classification and segmentation. Table 8 summarizes the optimization type, methods and characteristics, and application of all multi objective optimization of CSO.

Analysis of the advantages and disadvantages of CSO algorithm

The main advantages of the CSO algorithm are its simple and easy-to-implement structure, efficient global search capability, and hierarchical group collaboration mechanism, which makes it particularly suitable for multi-objective optimization and high-dimensional optimization scenarios. For example, in multi-objective problems, the group hierarchical structure can effectively balance the simultaneous optimization of multiple objectives and reduce the risk of the population falling into the local optimum; in high-dimensional optimization problems, the group update strategy of CSO promotes extensive exploration of the solution space. However, the CSO algorithm performs poorly in local search because the algorithm update mechanism is single and lacks a sophisticated local search strategy; in addition, the parameter sensitivity is high, and unreasonable parameter settings can easily cause the algorithm to fall into the local optimal solution. To solve these problems, the local optimization capability and robustness of the algorithm can be enhanced by integrating local search algorithms or introducing adaptive parameter control mechanisms.

Co-citation network and publication trend analysis

To deepen the bibliometric insights beyond descriptive statistics, two complementary perspectives were incorporated: (i) the temporal evolution of research output, and (ii) the structural connections among frequently cited works in the domain of CSO and its variants.

Figure 6 illustrates the annual growth of CSO-related publications over the period 2014–2025. The distribution follows a three-phase pattern: (1) Initial Stage (2014–2016): Only a limited number of studies appeared, reflecting the early adoption and conceptual exploration of swarm-based metaheuristics. (2) Growth Stage (2017–2020): A steady increase emerged as CSO and its modifications gained acceptance in engineering optimization, clustering, and scheduling problems. (3) Expansion Stage (2021–2025): A pronounced surge can be observed, driven by hybridization with differential evolution, feature selection tasks, biomedical data processing, and cross-domain optimization demands.

Co-citation network perspective

In addition to temporal trends, the intellectual structure of CSO research can be interpreted through co-citation patterns. Foundational works by Meng et al. (2014) on the original CSO model and subsequent enhancements (e.g., ICSO, ADPCCSO, MCSO) frequently co-occur with studies on PSO, DE, GWO, and hybrid swarm–evolutionary algorithms. Seminal references proposing benchmark comparisons (e.g., IEEE CEC test suites) also appear as central nodes in these co-citation clusters.

Recent expansions link CSO-related studies with broader metaheuristic families, including Harris Hawks Optimization, Grey Wolf Optimizer, and quantum-inspired swarm variants. This indicates an increasing convergence of algorithmic ideas and a shift from isolated swarm adaptations to integrative frameworks spanning feature selection, biomedical optimization, and resource allocation. Although a full visualization (e.g., via VOSviewer or CiteSpace) is not included here, the recurring co-citation of key authors and benchmark datasets reflects consolidated research communities and evolving collaborative networks. Together with the publication trend, these patterns underscore the maturation and diversification of CSO-based research in both theory and application.

Limitations and future probabilistic extensions of CSO

While CSO and its variants perform well on deterministic benchmarks, handling uncertainty (measurement noise, dynamic landscapes, stochastic constraints) remains under-explored. A practical route is to introduce stochastic or probabilistic mechanisms into CSO: e.g., distribution-driven role updates, randomized regrouping, variance-adaptive perturbations, or hybrids where CSO exploration is coupled with probabilistic search operators. Similar ideas have proved effective in related swarm frameworks. For example, the quantum-behaved PSO (QPSO) literature provides formal convergence analyses and improvement strategies under probabilistic sampling (e.g., $\delta$-potential well sampling and Gaussian local attractors), offering templates for uncertainty-aware design (Sun et al., 2012). Beyond PSO, stochastic-process viewpoints and random dynamical systems have been used to study stability or mean-square/consensus convergence of swarm dynamics (Erskine, Joyce & Herrmann, 2017).

Transferring these principles to CSO suggests two concrete steps: (i) define probabilistic transition rules for hen/rooster/chick updates and analyze convergence in expectation or mean-square under bounded-variance noise; (ii) study ergodicity via Markov-chain or RDS tools to obtain preliminary stability guarantees. Meanwhile, recent CSO improvements continue to appear in applications, reinforcing the need for uncertainty-aware formulations (Liang, Wang & Ma, 2022). Overall, stochastic CSO hybrids with preliminary stability discussion can close the current gap between empirical robustness and theoretical guarantees.