Recent Developments in Equilibrium Optimizer Algorithm: Its Variants and Applications

Abstract

There have been many algorithms created and introduced in the literature inspired by various events observable in nature, such as evolutionary phenomena, the actions of social creatures or agents, broad principles based on physical processes, the nature of chemical reactions, human behavior, superiority, and intelligence, intelligent behavior of plants, numerical techniques and mathematics programming procedure and its orientation. Nature-inspired metaheuristic algorithms have dominated the scientific literature and have become a widely used computing paradigm over the past two decades. Equilibrium Optimizer, popularly known as EO, is a population-based, nature-inspired meta-heuristics that belongs to the class of Physics based optimization algorithms, enthused by dynamic source and sink models with a physics foundation that are used to make educated guesses about equilibrium states. EO has achieved massive recognition, and there are quite a few changes made to existing EOs. This article gives a thorough review of EO and its variations. We started with 175 research articles published by several major publishers. Additionally, we discuss the strengths and weaknesses of the algorithms to help researchers find the variant that best suits their needs. The core optimization problems from numerous application areas using EO are also covered in the study, including image classification, scheduling problems, and many others. Lastly, this work recommends a few potential areas for EO research in the future.

1 Introduction

Finding the ideal configuration of a set of decision variables to reduce or exploit precise measures, along with constraints and parameters, articulated in the objective function [1] is typically the focus of Optimization. The optimization procedure with the fundamental objective function explores a search space in which the optimal solution to a particular problem must be identified. The zone of optimization by means of meta-heuristics has gained massive acceptance with swelling curiosity from researchers from diverse arenas, which is the reason why quite a lot of meta-heuristics are being anticipated recurrently as a solution to composite and real-world problems. A metaheuristic algorithm, or population-based method as it is more frequently called, offers a universal optimization basis that is effortlessly adaptable to a wide range of optimization problems, thereby discovering and manipulating the search space through the use of operators and performing operations, namely parameter tuning and modification. Population-based nature-inspired metaheuristic are essentially categorized in numerous ways (6 categories or classes) based on the inspiration, such as “swarm-inspired”, “chemistry-inspired”, “human-inspired”, “plant-inspired”, “maths-inspired” and “physics-inspired”, and the same has been projected in Table 1 along with a few of the algorithms under each category.

Table 1 Classification of population-based nature-inspired meta-heuristic algorithms

Swarm-Inspired metaheuristics are the first class of population-based nature-inspired metaheuristics, wherein swarms imitate actions of social animals available in nature such as foraging behavior of ants, flocking behavior of birds, schooling of fish, moulding behavior of bacteria, herding behavior of animals, and many more. The following are a few of the swarm-inspired algorithms listed, Ant Colony Optimization [2], Cuckoo Search [3], Artificial Bee Colony Algorithm [4], Particle Swarm Optimization [5], Firefly Algorithm [6], Krill Herd Algorithm [7], Bat Algorithm [8], Manta Ray Foraging Optimization [9], Moth-Flame Optimization Algorithm [10], Dragonfly Algorithm [11], Marine Predators Algorithm [12], Gray Wolf Optimization [13] and Whale Optimization Algorithm [14]. The second class of Population-based Nature-inspired metaheuristic is Chemistry-Inspired where the nature of chemical processes serves as the primary foundation of inspiration for developing answers to challenges that arise in the present. Some of the algorithms under this category are Chemical Reaction Optimization [15], Henry Gas Solubility Optimization [16], and Artificial Chemical Reaction Optimization [17].

As highlighted in Table 1, a third class of population-based, nature-inspired metaheuristics is Human-Inspired that impersonate human behavior, dominance, and intelligence. The following is a list of some of the human-based algorithms shown in Table 1 i.e., Brain Storm Optimization [18], Teaching Learning-Based Optimization [19], Class Topper Optimization [20], Imperialist Competitive Algorithm [21], Human Mental Search [22], Search and Rescue Optimization [23], Election Algorithm [24], Gaining Sharing Knowledge-Based Algorithm [25], Forensic-Based Investigation Optimization [26], Football Game Algorithm [27], Life Choice-Based Optimization [28], Corona virus Herd Immunity Optimization [29] and Battle Royale Optimization [30]. The fifth class of metaheuristics, called Plant-Inspired population-based nature-inspired metaheuristics, essentially imitates the intelligent behavior displayed by plants. Some of the well-known plant-based algorithms include the ones listed below: Plant Growth Optimization [31], Photosynthetic Algorithm [32], Plant Propagation Algorithm [33], Artificial Plant Optimization Algorithm [34], Paddy Field Algorithm [35], Root Mass Optimization Algorithm [36], Fertile Field Algorithm [37], Flower Pollination Algorithm [38], Path Planning inspired by Plant Growth [39], Invasive Weed Optimization [40], Rooted Tree Optimization [41], Sapling Growing up Algorithm [42] and Root Growth Algorithm [43]. The Math-Inspired is sixth category under population-based, nature-inspired meta-heuristic optimization, which essentially tends to mimic the method of numerical techniques, mathematical programming, and its focus on resolving a variety of constraints and optimization problems in the real world. Several well-known math-inspired algorithms include Hyper-Spherical Search Algorithm [44], Radial Movement Optimization [45], Stochastic Fractal Search [46], Golden Ratio Optimization Method [47], Sine Cosine Algorithm [48] [49] and Arithmetic Optimization Algorithm [50]. Finally, the last class of Population-based Nature-inspired meta-heuristic optimization portrayed in Fig. 1 is Physics-Inspired in which the main source of inspiration is the physical processes, which are further formulated into solutions to resolve the problems. A few popular physics-inspired algorithms are: Equilibrium Optimizer [51], Multi-Verse Optimization [52], Bang-Big Big-Crunch Algorithm [53], Magnetic Charged System Search [54], Central Force Optimization [55], Thermal Exchange Optimization [56], Ray Optimization [57], Gravitational Search Algorithm [58], Artificial Physicomimetics Optimization [59], Optics Inspired Optimization [60], Electromagnetic Field Optimization [61], Gravitational Local Search Optimization [62] and Electromagnetism-like Algorithm [63]. As already mentioned, all these classes of algorithms are enthused by nature and its natural phenomena, and are both robust and flexible to adapt across scenarios and problems.

Fig. 1

Terminologies used for searching EO research papers from Google Scholar

Equilibrium Optimizer (EO) [51], is an example of a predominant physics-inspired population-based nature-inspired meta-heuristic optimization algorithms announced in the year 2020 by Afshin Faramarzi and group [51] that basically mimic the dynamic mass balance on a control volume. EO has seen a sharp increase in popularity in recent years, reaching 1842 total citations according to the survey performed on Google Scholar on February 5, 2023. Several EO’s variants have been developed by the researchers and used for various optimization problems, for instance, feature selection, photovoltaic models, medical data and image classification, image segmentation, scheduling, sentiment analysis, technical problems, and so on discussed and summarized in the subsequent section. To the best of our knowledge, no review or survey paper on EO has been published as of yet, and this is the main impetus behind this work. The purpose of this paper is therefore to present a thorough and understandable description of the pertinent work related to EO and its variations, as well as the proposed applications for further investigation of EO in order to fix various problems and find optimal solutions for the realm. This article further discusses the strengths and weaknesses of the aforementioned algorithms. This will undoubtedly help interested researchers and students in choosing the best option to address the issue at hand. In this review paper, all work related to the Equilibrium Optimizer (EO) will be rigorously examined, addressing five pillars:

• The evolution of EO-related articles in the literature related to amount of citations, publications, year of development, top 10 publishers, and top 10 journals is depicted in Sect. 2.

• The basic concept of EO is highlighted in terms of inspiration, mathematical model, original algorithm, procedural steps, and flowcharts in Sect. 3. Along with the other details, a summary of the various research works carried out using the basic EO and original EO is highlighted.

• An in-depth literature review of basic EO and its variants (Revised and Hybridized) is discussed in Sect. 4.

• All applications tackled by EO and its variants are also illustrated in Sect. 5.

• Finally, this paper concludes and endorses some of the possible future research directions for EO, which is available in Sect. 6.

2 The Evolution of Equilibrium Optimizer Algorithm in the Literature

No one algorithm can produce the best solutions for all optimization problems, or for all instances of the same issue, as the No Free Lunch (NFL) theorem correctly states. Hence, the basic form of any nature-inspired meta-heuristic algorithm can be revised or even hybridized in order to meet the requirement. Thus, numerous problem-dependent variants of EO have been proposed since 2020 to enhance not just its searching behavior, but also to generate the optimal solutions to the problem at hand. This section provides an overview of how EO has been used in academic articles to evaluate the evolution of EO in the literature. Numerous publications from leading publishers such as IEEE, Elsevier, Springer, MDPI, AIMS Press, Nature Portfolio, Taylor & Francis, Wiley and Hindawi were considered. To do this, the web search engines listed below are used, which offer the full text of academic literature across a variety of publication disciplines and search is conducted using some of the terms shown in Fig. 1 (albeit not exclusively).

• (a) Google Scholar (https://scholar.google.com),

• (b) IEEE Xplore (https://ieeexplore.ieee.org),

• (c) ScienceDirect (https://www.sciencedirect.com),

• (d) SpringerLink (https://www.springerlink.com),

• (e) ACM Digital Library (https://dl.acm.org) and

• (f) DBLP (https://dblp.uni-trier.de)

Furthermore, to scrutinize the huge number of research papers found in the web search engines, a few enclosure and omission norms were well-thought-out and used, as listed below:

Enclosure norms	Omission norms
Papers published in authentic journals/conferences	Predatory publication at conferences and journals
English language research papers	Papers written in languages other than English
If full papers are available for download	If the full paper cannot be downloaded

After utilizing the enclosure and omission norms, a collection of 175 research papers (3 papers are pre-print versions) related to EO, published in different conferences and journals, has been finalized as depicted in Fig. 2. Year-wise publications of the above-mentioned 175 EO related research papers are depicted in Fig. 3. Researchers from numerous domains considered EO to be the leading algorithm in the years mentioned, and this is also reflected in their publications. In 2020, when EO was first introduced, the number of EO-based publications was twenty, grew to 68 by the end of 2021, reaching 80 publications in 2022. EO gained much attention from the research community, leading to seven new publications in the first few months of 2023. As per the survey, as mentioned earlier, over the past few years (2020 till date), EO has received a total of 1842 citations according to Google Scholar and in this context, the year-wise citations for the different EO-related papers are highlighted in Fig. 4. Furthermore, as shown in Fig. 5, EO-related articles are published in reputable journals and conferences by well-established publishers, which depicts various EO-related articles published by the top 10 publishers, namely, Elsevier, IEEE, Springer, MDPI, Hindawi, Wiley, Taylor & Francis, Taiwan Ubiquitous Information, World Scientific, and Plos as per the survey. Last but not least, Fig. 6 lists the top 10 journals for EO-related article publishing.

Fig. 2

Different EO related papers published in different journals and conferences (As per the survey)

Fig. 3

Year-wise EO related research papers published (As per the survey)

Fig. 4

Year-wise citations for the EO related research papers published (As per the survey)

Fig. 5

Top 10 publishers publishing the EO-related research papers (As per the survey)

Fig. 6

Top 10 journals publishing the EO-related research papers (As per the survey)

3 Equilibrium Optimizer Algorithm: An Overview

In this section, an overview of Equilibrium Optimizer (EO) is presented that mostly focuses on the inspirational source, mathematical model, original algorithm, and procedural steps of EO. The Equilibrium Optimizer (EO) algorithm is the population-based nature-inspired physics-based meta-heuristic optimization algorithms introduced in 2020 by Afshin Faramarzi and his group [51] that mimics the simple dynamic mass balance on a control volume. Equations related to mass balance of EO provide a physical basis for conserving the arrived, departed, and generated masses in control volumes.EO starts the optimization process by creating an initial population, just like other metaheuristic algorithms. Similar to the PSO method, in EO, particles correspond to solutions and concentrations correspond to particle positions.

Few important terminologies are intensive in EO in regard to the updating rules of the particles and positions of particles namely, Equilibrium concentration, Equilibrium pool, Exponential term and Generation rate. Also defined below are the means by which each term influences the search pattern. Equation 1 illustrates how the starting population is made up of the quantity of particles with the least and maximum dimension given a uniform random initialization in the search space.

$$C_{i}^{initial} = C_{min} + rand_{i} \left( {C_{max} - C_{min } } \right)\,\,\,i = 1,2 \ldots .N$$

(1)

Here, $C_{i}^{initial}$ represents the initial concentration vector of the i-th particle; N is the number of particles; C indicates the position of the particle; $C_{max} and C_{min }$ are the maximum and minimum values for the dimensions; rand is a random vector with value between 0 and 1.

Once the initial population generation is complete, the EO algorithm proceeds to the next step. There, particles are evaluated using a fitness function and permuted to identify equilibrium candidates. Although knowledge of the equilibrium state is unknown in the early stages of the optimization process, equilibrium candidates must be identified in order to determine particle’s search pattern, and the identified suitable candidates are kept in the equilibrium pool. By identifying the full optimization process, 4 best performing particles to date from the pool are selected and used in order to update.

The selected candidates help explore the unexplored search space, enhancing EO's ability to achieve better searches. Further, averaging these candidates enhances the ability to exploit EO to achieve global optima. With this concept in mind, the Equilibrium vector pool wherein the promising candidates are stored during the entire process is constructed as shown in Eqs. 2 and 3. The particle placements are updated based on a random pick among the five candidates available from the pool at each iteration until the optimization process is complete.

$$C_{eq.pool} = \left\{ {C_{eq\left( 1 \right)} , C_{eq\left( 2 \right)} , C_{eq\left( 3 \right)} , C_{eq\left( 4 \right)} , C_{{eq\left( {ave} \right)}} } \right\}$$

(2)

$$C_{{eq\left( {ave} \right)}} = \frac{{ C_{eq\left( 1 \right)} + C_{eq\left( 2 \right)} + C_{eq\left( 3 \right)} + C_{eq\left( 4 \right)} }}{4}$$

(3)

Here, $C_{eq.pool}$ is the Equilibrium pool; $C_{eq\left( 1 \right)} , C_{eq\left( 2 \right)} , C_{eq\left( 3 \right)} , C_{eq\left( 4 \right)}$ are the four best-so-far candidates chosen for the pool; $C_{{eq\left( {ave} \right)}}$ is the average of the four best-so-far candidates.

To maintain adequate stability between global and local searches in the search space, or between exploration and exploitation, exponential terms that primarily contribute to the update process needs to be evaluated, as shown in Eq. 4.

$$F = {\text{exp}}\left( { - \lambda \left( {t - t_{0} } \right)} \right)$$

(4)

Here, F denotes the Exponential term; $\lambda$ is the random vector [1]; and t is the time and the same is computed as shown in Eq. 5.

$$t = \left( {1 - \frac{Iter}{{MaxIter}} } \right)^{{\left( {\alpha { }\frac{Iter}{{MaxIter}}} \right)}}$$

(5)

Here, Iter refers to current iteration; MaxIter denotes the maximum number of iterations; α is a constant basically employed to regulate the behavior of the local search mechanism and $t_{0}$ denotes the parameter used to carry out exploration and exploitation as shown in Eq. 6.

$$t_{0} = \frac{1}{\lambda }{\text{In}}\left( { - \beta sign\left( {r - 0.5} \right)\left[ {1 - \exp \left( { - \lambda } \right)} \right]} \right) - 1$$

(6)

Further, Eq. 7 is the revised variant of Eq. 4 along with the substitution of Eq. 6 in Eq. 4.

$$F = \beta sign\left( {r - 0.5} \right)\left[ {\exp \left( { - \lambda t} \right)} \right. - 1$$

(7)

Here, r is a random vector [0,1]; $\beta$ refers to the constant for handling the exploration ability; α and $\beta$ are assigned values 1 and 2 respectively as per [51]; sign (r $-$ 0.5) is employed to advance the direction of exploration and exploitation.

The generation rate needs to be expressed as stated in Eq. 7 in order to further explore the search spaces and enhance the exploitation phase to deliver the precise response to the problem.

$$G = G_{0} \exp \left( { - \lambda \left( {t - t_{0} } \right)} \right) = G_{0} F$$

(8)

$$G_{0} = GCP\left( {C_{eq} - \lambda X} \right)$$

(9)

$$GCP = \left\{ {\begin{array}{ll} {0.5, \,\,\,r_{1} , r_{2} \ge GP} \\ {0,\,\,\, r_{2} < GP} \\ \end{array} } \right.$$

(10)

Here, G denotes the Generate rate; $C_{eq}$ denotes the random candidate chosen from the Equilibrium pool; $r_{1}$ and $r_{2}$ are random vectors [0,1]; $G_{0}$ refers to the initial value and is generated using Eq. 8; GCP refers to the Generation rate Control Parameter evaluated using Eq. 9 and GP signifies the Generation Probability.

Finally, the rule for updating in EO is defined using Eq. 10, wherein F refers to the exponential term; G denotes the Generate rate; V denotes a constant initialized with a value 1 as per [2]; C denotes the position of the particle; and $C_{eq}$ represents a random candidate chosen from the Equilibrium pool.

$$C = C_{eq} + \left( {C - C_{eq} } \right)F + \frac{G}{ \lambda V}\left( {1 - F} \right)$$

(11)

Incorporating the particle's memory saving mechanism that benefits the algorithm's exploitation capacity would undoubtedly improve its performance, so it's worth noting that the EO algorithm does so. The procedural steps of EO is portrayed in Fig. 7, and its pseudocode [51] is presented in Algorithm 1. Lastly, the summary of the different research works carried out by different researchers using the basic or original EO to resolve different problems belonging to numerous applications areas is discussed and tabulated in Table 2.

Fig. 7

Flowchart of EO algorithm

Table 2 Summary of the different research work carried out using Basic/Original EO algorithm

Finally, Table 2 discusses and tabulates the different research works conducted by various researchers using the basic or original EO to solve various problems in a variety of application areas.

From the summarized details specified in Table 2, it is clear that EO is employed in numerous application areas to resolve several problems. The year-wise depiction of various research articles based on basic or original EO is highlighted in Fig. 8. Also, the full form of each acronym used in Table 2 is tabulated in Table 7.

Fig. 8

Year-wise depiction of the various research article based on Basic EO (As per the survey)

4 Recent Variants of Equilibrium Optimizer Algorithm

The Equilibrium Optimizer is one of the operational physics-inspired algorithms with an exploratory search mechanism plus a large-scale exploitation mechanism. Primarily, however, it has been proposed and expected to explore and exploit continuous search spaces to address continuous optimization problems. On the one hand, for a unimodal function, the stability of EO is comparatively decent in optimization and the main reason for this is the existence of a generation rate. On the other hand, standard EO lacks immature stability in exploration and utilization during the process of optimizing multimodal functions. EO has the tendency to effortlessly get trapped in local minima, thus leading towards meagre stability and ruining the overall performance of EO, which requires improving EO. Therefore, with time, though the EO algorithm is just three years old, several researchers paved the way towards revising the basic EO so that it could be adjusted to meet the requirements of different search spaces along with its application beyond the continuous optimization problems. In simple words, EO is revised to not just obtain a better/optimal solution but also to improve its searching rules. The alteration is proposed in two variants, namely, Revised and Hybridized, and the same is depicted in Fig. 9. The number of articles surveyed related to basic EO, revised EO, and hybridized EO is depicted in Fig. 10. The following subsections provide relevant work on the various variants of EO and a summary of each variant in tabular form.

Fig. 9

Revised and Hybridized variants of EO

Fig. 10

Year-wise depiction of the various research article based on Basic EO and variants of EO (As per the survey)

4.1 Revised Variants of Equilibrium Optimizer (EO)

Compared to basic or original EO, revised variants of EO are usually developed by layering specific additional strategies on top of existing EO, thus solving a wide range of complex optimization problems in the real world. Figure 11 shows the various revised variants of the EO and Fig. 12 shows the annual presentation of various published research papers on revised EO.

Fig. 11

Different revised variants of EO

Fig. 12

Year-wise depiction of the various research article based on revised variants of EO (As per the survey)

The description of the research articles that deals with the introduction of various categories of the revised variants, namely Dynamic, Adaptive, Fuzzy, Chaotic, Gaussian, Binary, Levy-Flight, Multi-Objective, Disturbance-Based, Opposition-Based Learning, Improved, Quantum and Communication is discussed further and the same has been highlighted in Fig. 13.

Fig. 13

Proportions of research articles addressing the various types of revised EO. (As per the survey)

4.1.1 Dynamic Equilibrium Optimizer

In order to increase the effectiveness of the ordinary EO, He et al. presented the Improved Equilibrium Optimizer (IEO) in 2022 [132]. The IEO combines the chaotic initialization technique with the dynamic sine and cosine components. Additionally, the experimental findings unmistakably demonstrate that the proposed IEO outperforms other current algorithms, opening the door for both financial and environmental advantages in the operation of the power system. Improved Equilibrium Optimizer (IEO), a revised version of EO proposed by Lan et al. in 2022 [133], uses a chaotic equilibrium pool to improve information interaction, a nonlinear dynamic generation strategy to balance global and local abilities, and a golden sine strategy to update and avoid local optimums. This revision of EO improves on standard EO's performance overall. Additionally, the experimental findings unmistakably show that the suggested IEO outperforms m-EO, AEO, OB-L-EO, and HEO for the optimal power flow problem. A new version of EO known as the United Equilibrium Optimizer (UEO), which uses combined parameters in the equilibrium pool (EOU) and dynamic parameters (EOD) to provide balance in standard EO, was proposed by Gui et al. in the year 2021 [134]. Furthermore, it is abundantly obvious from the experimental results that for multimodal image registration, the suggested UEO performs better than PSO, GWO, GA, GSA, SSA, and CMA-ES.

4.1.2 Adaptive Equilibrium Optimizer

Self-Adaptive Quantum Equilibrium Optimizer with Artificial Bee Colony (SQEOABC) is a new variant of revised and hybridized EO that Zhong. et al. proposed in 2023 [135]. It incorporates quantum theory and self-adaptive mechanisms into the EO's updating rule for convergence enhancement and further uses the updating mechanism of ABC to arrive at the right solution. Additionally, the experimental results clearly show that the proposed SQEOABC performs better than various state-of-the-art metaheuristics such as PSO, CMAES, GWO, SSA, HHO, SMA, WSO, TLSMA, DDSRPSO, EESHHO, and LSHADE-cnEpSin for COVID-19 feature selection problem. In order to ensure the proper balance and transition between exploration and exploitation and to further improve the convergence rate towards reaching the near-optimal region, Houssein et al. in the year 2022 [136] proposed a new variant of revised EO referred to as Self-Adaptive Equilibrium Optimizer (self-EO). This algorithm incorporates the strategies namely exploration/exploitation balancing, self-adaptive control parameters, and population reduction. Additionally, the experimental results clearly show that the suggested self-EO beats other state-of-the-art metaheuristics, such as EO, SCA, GSA, HHO, AGSK, IMODE, TLBO, and PSO, demonstrating its effectiveness in global, combinatorial, engineering, and multi-objective issues. Equilibrium Optimizer with Divided Population (DEO), which Li et al. proposed in 2022 [137], is a new variation of the revised EO that incorporates the adaptive population division strategy with beta distribution, distance factor, and disturbance factor to improve the standard EO and balance exploration and exploitation while enhancing convergence and breaking out of local optimum. Additionally, the experimental findings clearly reveal that the proposed LWMEO outperforms other current state-of-the-art metaheuristics, demonstrating its effectiveness in a variety of feature selection challenges. Hybrid Equilibrium Optimizer with Capsule Auto Encoder (HEOCAE), which incorporates skull stripping, Normalized Linear Smoothing and Median Joint (NLSMJ) filtering, an adaptive Fuzzy based Atom Search Optimizer, Adaptive Rain Optimization, and capsule auto encoder at different stages to enhance the standard EO, was proposed by Ansingkar et al. in the year 2022 [138]. Also, the experimental results clearly reveal that the proposed HEOCAE performs better than EO and other current state-of-the-art metaheuristics, demonstrating its effectiveness in the Multi class Alzheimer detection challenge. In 2022 [139], Das et al. developed a new version of EO called the Adaptive Equilibrium Optimizer (AEO), which uses a non-Entropic objective function and an adaptive decision-making mechanism to improve traditional EO. Furthermore, it is amply understandable from the experimental results that, for multilayer optimum threshold selection issues, the proposed AEO outperforms AEO-Tsallis and AEO-Otsu. The Automata-based Improved Equilibrium Optimizer with U-shaped Transfer Function (AIEOU), which Ahmed et al. suggested in 2021 [140], uses learning-based automata, a U-shaped transfer function, and adaptive hill climbing to improve standard EO. Furthermore, it is abundantly obvious from the experimental findings that the proposed AIEOU outperforms SSD + LAHC, RTHS, A BSF, HSGW, RSGW, ASGW, BGA, and PSO in terms of feature selection. An updated version of EO called the Adaptive Equilibrium Optimizer (AEO), which employs interdependence-based approaches and adaptive decision making to improve the standard EO, was proposed by Wunnava et al. in the year 2020 [141]. Additionally, experimental results show that the suggested AEO offers greater performance for multilevel thresholding when compared to EO, GWO, WOA, SSA, and WDO.

4.1.3 Fuzzy Equilibrium Optimizer

Hybrid Equilibrium Optimizer with Capsule Auto Encoder (HEOCAE), which incorporates skull stripping, Normalized Linear Smoothing and Median Joint (NLSMJ) filtering, an Adaptive Fuzzy based Atom Search Optimizer, Adaptive Rain Optimization, and capsule auto encoder at different stages to enhance the standard EO, was proposed by Ansingkar et al. in the year 2022 [138]. Also, the experimental results clearly reveal that the proposed HEOCAE performs better than EO and other current state-of-the-art metaheuristics, demonstrating its effectiveness in the Multi class Alzheimer detection problem. The Hybrid Fuzzy Equilibrium Optimizer (HFEO), which incorporates fuzzy-based fitness functions to improve the standard EO, was proposed by Yehia et al. in 2022 [142]. Also, it is profusely clear from the results of the experiment that proposed HFEO performs better than GWO, MFO, FPA, GOA, HFEO, and PSO when it comes to estimating the size and position of distributed generators. Modified Equilibrium Optimization Algorithm-based Interval Type-2 Fuzzy Proportional Integral Derivative Controller (MEO-IT2FPID), which combines IT2FPID to lessen the frequency control issue and some scaling factor into regular EO, was proposed by Khadanga et al. in 2021 [143]. Additionally, the experimental findings unmistakably show that the suggested CSED-OEO approach outperforms EO, WOA, and MWOA in terms of frequency control of an AC microgrid.

4.1.4 Chaotic Equilibrium Optimizer

Agarwal et al. in 2022 [144] proposed a new variant of revised EO termed as Normalized Mutual Information-based equilibrium optimizer (NMIEO) that incorporates the local search strategy on Normalized mutual information and chaotic maps to boost the exploitation ability and diversity of the problem as well as enhance the population initialization of the standard EO. Further, the experimental result evidently highlights that the proposed NMIEO outperforms other state-of-the-art metaheuristics showcasing its efficacy in feature selection domain. In order to increase the effectiveness of the ordinary EO, He et al. presented the Improved Equilibrium Optimizer (IEO) in the year 2022 [132]. The IEO combines the chaotic initialization technique with the dynamic sine and cosine parameters. Additionally, the experimental findings unambiguously illustrates that the proposed IEO surpasses other current algorithms, paving the way for the operation of the power system to be both economical and environmentally friendly. Improved Equilibrium Optimizer (IEO), which Lan et al. proposed in 2022 [133], is a revised version of the equilibrium optimizer that uses chaotic equilibrium pool to improve information interaction, nonlinear dynamic generation strategy to balance global and local ability, and golden sine strategy for updating and avoiding local optimum to improve standard EO's performance overall. Additionally, the experimental findings distinctly show that the suggested IEO outperforms m-EO, AEO, OB-L-EO, and HEO for the optimal power flow problem. Disturbance-Inspired Equilibrium Optimizer (DIEO), an updated version of EO proposed by Wang et al. in 2022 [145], uses chaotic sequence and excellent nodes set theory as deterministic and stochastic components to improve traditional EO. Furthermore, the experimental findings unequivocally illustrate that proposed DIEO outperforms EO, AOA, HGS, and AGPSO for the acquisition of hydrogeological parameters. In order to improve the standard EO, Sayed et al. introduced the Chaotic Equilibrium Optimizer Algorithm (CEO) in 2022 [146]. CEO uses eight alternative S-shaped and V-shaped transfer functions. Additionally, the experimental findings clearly highlight that the suggested CEO performs better than EO, PSO, QABC, CSO, ALO, SSA, ASO, and BOA for problems involving feature selection and global optimization. Mousa et al. in 2021 [147] proposed a revised variant of EO termed Chaotic Search based Constrained Equilibrium Optimizer Algorithm (CS-CEA) that incorporates Exhaustive local search, chaos-based search algorithm to enhance the standard EO. Further, experimental results clearly highlight that proposed CS-CEOA method provides superior performance when compared with GA and PSO, for Non-Linear Programming and Petrochemical application. The Enhanced Equilibrium Optimization (E²O) Algorithm, which uses Fractional Order Chaotic (FOC) system models and controller parameter optimization, was proposed by Ates et al. in the year 2021 [148]. Further, experimental results clearly highlight that proposed E²O method provides superior performance when compared with EO for Engineering problems. Improved Equilibrium Optimizer (IEO), a modified version of EO introduced by Lan et al. in 2021 [149], uses chaotic mechanisms for population range expansion, adaptive weights, and an adaptive convergence factor to prevent becoming caught in local optima, improving the power of standard EO. Additionally, experimental results demonstrate that the suggested IEO approach for the LSTM Neural Network outperforms PSO, GWO, SOA, WOA, CSA, MPA, COA, CPA, TSA, and GA. In order to improve the performance of the ordinary EO, Zheng et al. suggested the Chaotic Equilibrium Optimizer Algorithm (CEOA) in the year 2020 [150], CEOA uses tent maps and the chaos approach. Additionally, experimental results unmistakably show that as compared to original EO, the suggested CEOA offers greater performance for the optimization of traditional benchmark functions.

4.1.5 Gaussian Equilibrium Optimizer

Extreme Learning Machine Modified Equilibrium Optimizer (ELM-MEO), which Bardhan et al. suggested in the year 2022 [151], incorporates Gaussian mutation with the aid of an exploratory search mechanism and ELM to create and optimize the learning parameters of standard EO. Additionally, the experimental results clearly reveal that the proposed ELM-MEO outperforms other state-of-the-art metaheuristics like EO, HHO, SMA, and MPA, demonstrating its effectiveness in predicting soil compression index, particularly for roads, railroads, and airport runways. In order to improve the standard EO, Abdel et al. proposed the MOEO-EED-G, OMOEO-EED-G revision of EO in 2021 [152]. This revision incorporates the Crowding Distance Approach, Linear and Non-Linear Equations, Exploration–Exploitation Dominance Strategy, Gaussian-based Mutation (G) Strategy, and Opposition-Based Learning. Additionally, experimental results show that the suggested method outperforms MOEOL, MOEOC, MOEOLog, MOEOT, SMPSO, NS GAII, NSGAIII, PAES2, SMSEMOA, ASMSEMOA, ANSGAII, and ANSGAIII for the optimization of benchmark functions. An improved version of EO known as the Multiple Population Hybrid Equilibrium Optimizer (MHEO), which uses Lévy flight and the inferior solution shift technique to minimize stagnation, was developed by Tang et al. in the year 2021 [153]. Additionally, experimental results demonstrate that the proposed MHEO technique for UAV path planning outperforms EO, MPA, SMA, VCS, GEDGWO, and HFPSO. The modified Equilibrium Optimizer (m-EO), which Gupta et al. introduced in 2020 [154], uses Gaussian mutation based on population division and reconstruction to increase the standard EO's rate of convergence. Additionally, experimental results show that, when compared to GWO, SCA, SSA, OBSCA, OBGWO, PSO, and FA for numerical optimization, m-EO offers greater performance.

4.1.6 Binary Equilibrium Optimizer

In 2022 [155], Faramarzi et al. developed a new version of the improved EO known as the Binary Equilibrium Optimizer (BiEO), which includes the benefits of velocity-based approaches by redesigning discrete binary issues using a V-shape transfer function and tweaking the traditional EO. Additional evidence of the proposed BiEO's effectiveness in optimum control issues comes from the experimental results, which clearly show that it outperforms other state-of-the-art metaheuristics including BPSO/S, BPOS/V, BDA, and GA. ReliefF Binary Equilibrium Optimizer with Local Search (RBEO-LS), which was proposed by Quadfel et al. in the year 2022 [156], is a new variants of revised EO that incorporates the ReliefF filter method as a population strategy to reduce the search space and Binary EO with Local Search strategy as a wrapper mechanism to speed up EO's convergence speed. The experimental results also clearly show that the proposed BiEO beats various cutting-edge metaheuristics, such as EO, SSA, SCA, DE, BAT, BPSO, and HGSO, demonstrating its effectiveness in the feature selection area. In 2022 [157], Varzaneh et al. proposed a new revision of the EO called the Binary Improved Equilibrium Optimizer (BIMEO), which incorporates the strategies mRMR, Wrapper feature selection approach, Entropy-based and Levy-flight operators to prevent the standard EO from becoming stuck in a local optimum and thereby balancing the exploration and exploitation mechanism. Additionally, the experimental results evidently disclose that the proposed BIMEO performs better than other contemporary state-of-the-art metaheuristics, demonstrating its effectiveness in Feature Selection issues. With the purpose of improving the traditional EO, Minocha et al. in the year 2022 [158] devised the AV- Binary Modified Equilibrium Optimizer (AV-BMEO), which combines the AV-shape transfer function, opposition-based learning, and k-nearest neighbour classifier. The experimental results also clearly reveal that the proposed AV-BMEO performs better than other current state-of-the-art metaheuristics, demonstrating its effectiveness in Feature Selection. The ReliefF-guided Novel Binary Equilibrium Optimizer (RG-NBEO), which Zhang et al. proposed in 2022 [159], combines SSr and VVr transfer functions based on opposition learning with ReliefF bootstrapping strategy to achieve good stability between exploration and exploitation and to convert the continuous search space into a binary search space. Additionally, it is clear from the experimental results that the proposed RG-NBEO surpasses other current state-of-the-art metaheuristics when projecting its effectiveness in Feature selection problem. Binary Modified Equilibrium Optimizer (BMEO), which integrates Band selection and V-shape transfer function to maintain the balance between exploration and exploitation and further improves the exploration ability of conventional EO, was proposed by Minocha et al. in the year 2022 [160]. Also, it is clear from the experimental results that the proposed BMEO performs better than contemporary state-of-the-art metaheuristics when projecting its effectiveness in hyperspectral image classification. The Novel Discrete Equilibrium Optimizer algorithm (NDEOA), which was proposed by Haouassi et al. in the year 2022 [161], is a new variation of the revised EO that incorporates discrete particle encoding as well as new discrete operators for defining the classification rule and for updating the particle's position in a discrete search space. Also, the experimental data clearly shows that the suggested NDEOA attained good levels of accuracy, sensitivity, and specificity and may thus be successfully applied to the generation of classification rules. Mohamed et al. in the year 2022 [162] proposed a revised variant of EO termed as Binary version of the Equilibrium Optimization Algorithm (BEOA) that incorporates binary encoding mechanism. Further, the experimental results clearly highlight that proposed O-LEO has superior performance when compared with BGWO, BPOS, BWOA, BSMO, BGO and BAEO and BEHO proving its significance in domination metric dimension problem. In 2022 [163], Hu et al. introduced an updated version of EO called the Binary version of the Equilibrium Optimizer (BEO), which only improves the standard EO by changing the equilibrium concentration and position updation equation. Also, the experimental findings unmistakably show that proposed BEO outperforms BBA, BDE, BGWO, BPOS, and BPSOGSA for Feature Selection. Using a Discrete Equilibrium Optimization Algorithm (DEOA-CRM) that incorporates Sequential Covering Strategy for Classification Rule Mining and Discrete operators to improve the performance of standard EO, Malik et al. proposed a revised variant of EO called Classification Rule Mining in the year 2022 [164]. Furthermore, the experimental findings distinctly show that the proposed I-EO outperforms DEOA-CRM, ACO/PSO 2, cAnt-MinerPB, Ant-MinerPAE, and ILS-AntMiner in terms of generating classification rules. With the aim to improve upon the traditional EO, Guha et al. suggested Discrete Equilibrium Optimizer Simulated Annealing (DEOSA) in the year 2022 [165]. DEOSA integrates U-shaped transfer function and SA for local search technique. Additionally, it is amply understandable from the experimental findings that the proposed DEOSA outperforms DEO, GA, PSO, GSA, BBA, bALO-S, bALO-V, HGSA, CPBGSA, and DGA for Feature Selection. Rahab et al. in the year 2022 [166] proposed a revised variant of EO named as Binary Equilibrium Optimization Algorithm (BEOA) that permits BEOA to perform Classification rule extraction. Further, it is profusely comprehensible from the experimental results that proposed BEOA has superior performance when compared with PART, RIPPER, OneR, C4.5, REPTree, SVM, K-NN and NB for sentiment analysis. Improved Binary version of the Equilibrium Optimizer Algorithm (IBEO), which employs Opposition Based Learning and Local search algorithm to increase population variety and optimize the exploitation of conventional EO, was proposed by Elmanakhly et al. in the year 2021 [167]. Additionally, the experimental findings unambiguously illustrate that the proposed IBEO outperforms PSO, GOA, GWO, WOA, DA, and ISSA for the feature selection problem. With the intention to improve the performance of traditional EO, Abdel et al. created Binary version of Equilibrium Optimization (BEO) in the year 2021 [168]. This improved version of EO includes eight transfer functions, including V-shaped and S-shaped, a penalty function, and the Repair Algorithm (RA). Also, the experimental findings demonstrate that proposed BEO outperforms EOS2, SA, GA, IGA-SA, GSA, and BB for the 0–1 Knapsack Problem. The Binary Equilibrium Optimizer Algorithm based on S and V Transfer function (BEO-S and BEO-V) and Equilibrium Optimizer Algorithm based on Target (BEO-T) employ S and V Transfer function and target dimension values to improve the standard EO, respectively, and were proposed by Gao et al. in the year 2020 [169]. Additionally, experimental results show that, when compared to other metaheuristic algorithms like BGOA, BGWO, and BPSO in the largest number of datasets for feature selection problems, BEO-V2 offers greater performance among the suggested approaches.

4.1.7 Levy-Flight Equilibrium Optimizer

In the year 2022 [170], Liu et al. proposed a new variant of revised and hybridized EO called the Levy Whale Optimization Mutation Equilibrium Optimizer (LWMEO), which combines the random walk strategy based on Lévy flight, the spiral encirclement mechanism of WOA, and the adaptive proportional mutation strategy to increase search range and exploration ability, exploitation ability, and speed up the convergence speed of the standard EO. The experimental results also evidently show that the proposed LWMEO performs better than existing state-of-the-art metaheuristics including EO, AEO, IEO, SSA, MSSA, and JADE, demonstrating its effectiveness in many engineering fields. Varzaneh et al. in the year 2022 [157] proposed a new variant of the EO called the Binary Improved Equilibrium Optimizer (BIMEO), which incorporates the strategies namely, mRMR, Wrapper feature selection approach, Entropy-based and Levy-flight operators to prevent the standard EO from becoming stuck in a local optimum and thereby balancing the exploration and exploitation mechanism. Additionally, the experimental results clearly reveal that the proposed BIMEO performs better than other contemporary state-of-the-art metaheuristics, demonstrating its effectiveness in Feature Selection problem. Modified Equilibrium Optimizer (MEO), which incorporates Iterative Cosine Operator (ICO), Levy flight, and heavy-tailed non-uniform levy distribution to reduce diversification to intensification and improve the exploration and exploitation ability of standard EO, was proposed by Minocha et al. in the year 2022 [171]. The experimental results evidently highlights that the proposed MEO works better than EO and other state-of-the-art metaheuristics, demonstrating its effectiveness in engineering problems. Levy Equilibrium Optimizer (LEO), a new variation of revised EO that incorporates the Levy flight update strategy to improve standard EO, was proposed by Aval et al. in the year 2022 [172]. Additionally, the experimental results clearly confirms that the proposed LEO performs better than EO and other contemporary state-of-the-art metaheuristics, including PSO, GWO, GA, GSA, and SSA, demonstrating its effectiveness in DNA storage. Improved Equilibrium Optimizer based on Levy Flight (IEOLV), which incorporates Levy flight method to eliminate local minimum stagnation and hence improves convergence efficiency of the standard EO, was proposed by Balakrishnan et al. in the year 2022 [173]. Additionally, it is clear from the experimental results that the proposed IEOLV outperforms EO and other current state-of-the-art metaheuristics when projecting its effectiveness in Feature Selection problem. Mostafa et al. in the year 2021 [174] proposed a revised variant of EO termed as Levy Spiral Flight Equilibrium Optimizer (LSFEO) that employs Levy Flight distribution to avoid stagnation and spiral motion of the particles to boost exploitation in standard EO. Further, experimental outcome evidently disclose the proposed CSED-OEO method provides superior performance when compared with EO, KHA, SKHA, ARCBBO, ABC, TLBO, MTLBO, GOA and AGOA for Optimal Power Flow (OPF) problem. An improved version of EO known as the Multiple Population Hybrid Equilibrium Optimizer (MHEO), which uses Lévy flight and the inferior solution shift technique to minimize stagnation, was developed by Tang et al. in the year 2021 [153]. Additionally, experimental results demonstrate that the proposed MHEO technique for UAV path planning outperforms EO, MPA, SMA, VCS, GEDGWO, and HFPSO. The Enhanced Equilibrium Optimization Algorithm with Levy Flight (Levy EOA), which Zhao et al. proposed in 2020 [175], uses levy flight to improve the performance of standard EO. Additionally, the experimental findings clearly show that the proposed Levy EOA outperforms PSO, BA, and GWO for the optimization of unimodal, multimodal, and the bottom flat-like benchmark functions.

4.1.8 Multi-Objective Equilibrium Optimizer

Improved Multi-objective Equilibrium Optimizer (IMOEO), which integrates the techniques of optimal PV energy consumption approach, limited non-dominant mechanism, and TOPSIS to improve the performance of conventional EO, was proposed by Yi et al. in the year 2022 [176]. Also, the experimental results clearly reveal that the proposed IMOEO works better than other contemporary state-of-the-art metaheuristics, demonstrating its effectiveness in residential housing scheduling. Abou et al. in the year 2022 [177] proposed a new variant of revised EO termed as Multi-objective Equilibrium Optimization technique (MOEOT) that incorporates Pareto optimal front and fuzzy concept to extract the best solution to enhance the power of standard EO. Further, the experimental result noticeably highlights that the proposed MOEOT outperforms EO and other recent state-of-the-art metaheuristics projecting its efficacy in Radial distribution system. In order to improve the performance of standard EO, Yi et al. proposed a new variant of revised EO in 2022 [178] called Multi-Objective Equilibrium Optimizer algorithm integrated Competition Mechanism (CMOEO). This variant incorporates Competition mechanism integrating Differential Evolution by swapping out the constant factors of exploration and exploitation with dynamic factors. Also, it is clear from the experimental results that the proposed CMOEO beats EO and other state-of-the-art metaheuristics estimating its effectiveness in Photovoltaic Energy Storage Systems. R₂ index hybrid multi-objective equilibrium optimization algorithm (R₂HMEOA), which includes R₂ index and elite archiving method, was proposed by Yu et al. in the year 2022 [179] in order to improve the rules of Pareto solution, reference point strategy, and to increase the standard EO. Also, it is profusely obvious from the experimental findings that the suggested R₂HMEOA outperforms NSGA-II, NSGA-III, MOSHO, and MOEA for the Mine Ventilation system. Premkumar et al. in the year 2022 [180] proposed a revised variant of EO termed as Multi-Objective Equilibrium Optimizer Algorithm (MOEO) that incorporates crowding distance mechanism, non-dominated sorting strategy and an archive with an update function to balance exploration and exploitation phase, preserve population diversity and uphold and improve the coverage of Pareto with optimal solutions. Further, the experimental results clearly highlight that proposed MOEO has superior performance when compared with NSGAII, MOEA/D and MOPSO for Multi-Objective Optimization Problems. Constrained Multiobjective Equilibrium Optimizer Algorithm (CMEO), which employs Pareto border minimization and Repair method to improve conventional EO for Combined Economic Emission Dispatch Problem (CEEDP), was proposed by El et al. in the year 2021 [181]. Additionally, it is obvious from the experimental findings that the suggested CMEO outperforms DRN-PSO, NSGA, NPGA, FCPSO, MSFLA, RCGA, and SPEA. In order to improve the standard EO, Abdel et al. proposed the revised variant MOEO, MOEOS, MOEO-EED, MOEO-EED, and OMOEO-EED-G in the year 2021 [152]. This variant incorporates the Crowding Distance Approach, Linear and Non-Linear Equations, Exploration–Exploitation Dominance Strategy, Gaussian-based Mutation (G) Strategy, and Opposition-Based Learning. Additionally, experimental results show that the suggested method outperforms when compared to MOEOL, MOEOC, MOEOLog, MOEOT, SMPSO, NS GAII, NSGAIII, PAES2, SMSEMOA, ASMSEMOA, ANSGAII, and ANSGAIII considering many benchmark functions. The Multi-objective Equilibrium Optimizer Algorithm (MEOA), which uses the Improvement-Based Reference Points Method (IBRPM) to improve the standard EO by boosting population diversity and accelerating convergence speed, was proposed by Abdel et al. in the year 2021 [182]. Additionally, experimental results demonstrate that the proposed MEOA technique outperforms other multi-objective optimization methods such as CDG, LMOCSO, LMEA, CMOPSO, CTAEA, CAMOEA, CMOEAD, AGEMOED, and ARMOEA. Disruption-based Multi-Objective Equilibrium Optimization Algorithm (DMOEOA), which employs Layered Disruption technique and disruption operators to improve the exploration and exploitation capabilities of the standard EO, was proposed by Chen et al. in the year 2020 [183]. Additionally, experimental findings demonstrate that the suggested BEO technique outperforms MOPSO, MOALO, MOWOA, NSGAII, and MOGWO for structural optimization issues.

4.1.9 Disturbance-Based Equilibrium Optimizer

Disturbance Inspired Equilibrium Optimizer (DIEO), which Wang et al. proposed in the year 2023 [184], is a new variation of the revised EO that incorporates the strategies of disturbance-based hybrid initialization, boundary check, and an adaptive global position disturbance approach that materially enhances the exploration and exploitation capability of the standard EO. Additionally, the experimental results clearly reveal that the proposed DIEO performs better than other state-of-the-art metaheuristics including EO, ASO, HFPSO, LHHO, HGS, and IGWO, demonstrating its effectiveness in a variety of real-world engineering issues. Wang et al. in the year 2022 [145] proposed a revised variant of EO termed as Disturbance-Inspired Equilibrium Optimizer (DIEO) that employs good nodes set theory and chaotic sequence as deterministic and stochastic components to enhance standard EO. Further, the experimental results clearly highlight that proposed DIEO has superior performance when compared with EO, AOA, HGS and AGPSO for Acquisition of Hydrogeological parameters. In order to balance exploration and exploitation in standard EO, Shao et al. proposed the Interswarm Interactive Learning Strategy Equilibrium Optimizer (IILEO) in the year 2021 [185]. IILEO uses the softmax method, Interswarm Interactive Learning, a global optimal disturbance strategy, and linearly decreasing inertia weights. Also, the experimental findings unmistakably show that proposed IILEO outperforms EO, PSO, GWO, and WILEO 4 for the UAV path planning problem.

4.1.10 Opposition-Based Learning Equilibrium Optimizer

Ameliorated Equilibrium Optimizer (AEO), which Wu Zhong et al. proposed in 2023 [186], is a new variation of the revised EO that incorporates the strategies Opposition-Based Learning, Centroid Opposition-Based Learning, and Self-learning strategies to improve the optimization performance of the standard EO by using various information about the opposite and neighbourhood spaces as well as about the entire population present in the search space. The experimental result also demonstrates the effectiveness of the proposed AEO in path planning for the Unmanned Ground Vehicle (UGV) problem, outperforming other state-of-the-art metaheuristics like EO, PSO, GWO, and SSA. Information-utilization Strengthened Equilibrium Optimizer (IS-EO), which Zhang et al. proposed in the year 2022 [187], incorporates the strategies of information guiding using the local and global best opposition learning and differential mutation strategy to construct and strengthen the information sharing among individuals. Additionally, the experimental results clearly reveal that the proposed IS-EO performs better than other contemporary state-of-the-art metaheuristics, demonstrating its effectiveness in Engineering design issues. Minocha et al. in the year 2022 [158] proposed a new variant of revised EO termed as AV- Binary Modified Equilibrium Optimizer (AV-BMEO) that incorporates AV-shape transfer function, opposition-based learning, k-nearest neighbor classifier to enhance the standard EO. Further, the experimental result evidently highlights that the proposed AV-BMEO outperforms other recent state-of-arts metaheuristics showcasing its efficacy in Feature Selection. The ReliefF-guided Novel Binary Equilibrium Optimizer (RG-NBEO), which Zhang et al. proposed in 2022 [159], combines SSr and VVr transfer functions based on opposition learning with ReliefF bootstrapping strategy to achieve good stability between exploration and exploitation and to convert the continuous search space into a binary search space. Additionally, it is clear from the experimental results that the proposed RG-NBEO surpasses other current state-of-the-art metaheuristics when projecting its effectiveness in Feature Selection problem. The Improved Equilibrium Optimizer Regularized Random Vector Functional Link (IEO-RRVFL), which combines the Opposition-Based Learning technique for initial population generation for greater convergence impact, is an updated form of EO that Zhaou et al. proposed in the year 2022 [188]. The experimental findings further demonstrate that the suggested IEO-RRVFL generates high accuracy and stability, demonstrating its effectiveness in comparison to existing algorithms. In order to reduce the segmentation error and enhance the threshold selection approach, Das et al. suggested a new version of EO in the year 2022 [189] known as Opposition Equilibrium Optimizer (OEO). OEO incorporates the Segmentation Score (SS) using Opposition-Based Learning mechanism on EO. The experimental results also show that the suggested OEO works better than previous art-entropy based techniques, demonstrating the effectiveness of the method in the area of image segmentation. Aswini et al. in the year 2022 [190] proposed a revised variant of EO termed as Opposition-based Laplacian Equilibrium Optimizer Algorithm (O-LEO) that incorporates Opposition-Based Learning and Laplacian mechanism to enhance the ability of standard EO. Further, the experimental results clearly highlight that proposed O-LEO has superior performance and proves to be efficient in Healthcare Services. Li et al. in the year 2022 [191] proposed a revised variant of EO termed as Multi-strategy Improved Equilibrium Optimizer (IEO) that uses tent mapping to initialize the algorithm, non-linear parameters to update the position equation for balancing exploration and exploitation of standard EO and Lens Opposition-based Learning (LOBL) to enhance the population diversity. Further, the experimental results clearly highlight that proposed IEO has superior performance when compared with EO, EO, SSA, SCA, BOA, PSO and BA for Numerical Optimization and Engineering problems. Elmanakhly et al. in the year 2021 [167] proposed a revised variant of EO termed as Improved Binary version of the Equilibrium Optimizer Algorithm (IBEO) that employs Opposition Based Learning and Local search algorithm to enhance the population diversity and enhance the exploitation of standard EO. Further, the experimental results clearly highlight that proposed IBEO has superior performance when compared with PSO, GOA, GWO, WOA, DA and ISSA for feature selection problem. Elgamal et al. in the year 2021 [192] proposed a revised variant of EO termed as Improved Equilibrium Optimization Algorithm (IEOA) that employs Elite Opposition-Based Learning (EOBL) and three Local Search approaches such as mutation search, mutation–neighborhood search, and a backup strategy to enhance the diversity in population and for prevention of getting stuck in local optima. Further, the experimental results clearly highlight that proposed IEOA has superior performance when compared with PSO, GA, WOA, GOA, ALO, SMA and BOA for feature selection problem. Fan et al. in the year 2021 [193] proposed a revised variant of EO termed as modified EO (m-EO) that employs opposition-based learning (OBL) and update rules: nonlinear time control strategy, novel population update rules and a chaos-based strategy to enhance the standard EO. Further, the experimental results clearly highlight that proposed m-EO has superior performance when compared with PSO, SCA, PFA, EO, HGSO, OTSA, OBSCA and SOGWO for Engineering Problems. Opposition-based Laplacian Equilibrium Optimizer (OB-L-EO), which combines Laplace distribution and opposition-based learning to improve the performance of standard EO, was proposed by Dinkar et al. in the year 2021 [194]. Also, it is abundantly obvious from the experimental findings that the suggested OB-L-EO performs better for image segmentation than HS, SCA, WOA, MFO, SSO, FASSO, FA, and EO. With the intention to improve the standard EO, Abdel et al. in the year 2021 [152] created the OMOEO-EED-G, a revised version of the EO that integrates the Crowding Distance Approach, Linear and Non-Linear Equations, Exploration–Exploitation Dominance Strategy, Gaussian-based Mutation (G) Strategy, and Opposition-Based Learning. Additionally, experimental results show that the suggested method outperforms when compared to MOEOL, MOEOC, MOEOLog, MOEOT, SMPSO, NS GAII, NSGAIII, PAES2, SMSEMOA, ASMSEMOA, ANSGAII, and ANSGAIII considering several benchmark functions. Naik et al. in the year 2021 [195] proposed a revised variant of EO termed as Context-Sensitive Entropy Dependency based Opposition Equilibrium Optimizer (CSED-OEO) that employs CSED, opposition-based learning and escaping strategy. Further, experimental results clearly highlight that proposed CSED-OEO method provides superior performance when compared with EO, HHO, SFO, WOA, GWO, PSO and DE for multi-thresholding of remote sensing images. Opposition-Based Equilibrium Optimization (OBEO), which uses an opposition-based learning-based update technique to increase the power of regular EO, was proposed by et al. in the year 2021 [196]. Additionally, the experimental findings unmistakably show that the suggested E2O technique outperforms EO, ALO, GWO, WOA, PSOGWO, and GWOCS for parameter estimation in PV models.

4.1.11 Communication Equilibrium Optimizer

An improved version of EO called the Parallel Equilibrium Optimizer (PEO), which Gui et al. introduced in 2021 [197], uses two communication mechanisms to speed up convergence and look for better solutions in the search space. Additionally, the experimental findings demonstrate that the suggested PEO performs better than PSO, GWO, and PPSO for the Capacitated Vehicle Routing Issue (CVRP). Xu et al. in the year 2021 [198] proposed a revised variant of EO termed as compact Equilibrium Optimizer (cEO) algorithm that employs update interval method, parallel communication strategy to enhance the standard EO. Further, the experimental results clearly highlight that proposed cEO has superior performance when compared with PSO, GA, BOA, GSA and GWO for power System networks.

4.1.12 Quantum Equilibrium Optimizer

In 2021 [199], Pan et al. suggested a new version of EO called the Advanced Equilibrium Optimizer (AEO), which uses the multi-population approach, the quantum operator, and the pollination operator to improve the standard EO's global exploration capabilities. Additionally, the experimental findings demonstrate that the proposed AEO outperforms DE, EO, PSO, FPA, and GWO for the Electric Vehicle Routing Problem with Time Windows (EVRPTW).

4.1.13 Other Improved Equilibrium Optimizer

In 2023 [200], Altantawy et al. proposed a new version of the revised EO known as the Minkowski-based Equilibrium Optimizer (MEO), which combines the Sparsification, Pearson correlation, wrapper, and Minkowski-based scheme strategies to improve feature selection, seek the best dimensionality reduction, and improve local minimum avoidance to improve EO performance. The experimental findings also clearly reveal that the proposed MEO outperforms competing classifiers, demonstrating its effectiveness in predicting COVID-19 infection while requiring a small number of blood samples. Vommi et al. in the year 2023 [201] proposed a new variant of revised EO termed as Bi-phase Mutation based Equilibrium Optimizer (BMHEO) that incorporates Bi-phase Mutation (BM) scheme to enhance the exploitation phase of the EO algorithm and eight S-shaped and V-shaped transfer functions are integrated to convert the solutions to binary form. Further, the experimental result evidently highlights that the proposed BMHEO outperforms other methods namely GA, PSO, GWO, SCA, WOA, HHO, SMA, EO and HEO showcasing its efficacy in medical datasets classifications. The Improved Equilibrium Optimizer (EEO), which combines discretization processing to discretize the standard EO, was proposed by Sun et al. in the year 2022 [202]. Additional evidence of the suggested EEO's effectiveness in Work Shop Scheduling problems comes from the experimental results, which clearly show that it outperforms other cutting-edge metaheuristics like GWO, MVO, DE, and WOA.

The Enhanced Equilibrium Optimizer (EEO), which Eid et al. presented in the year 2022 [203], is a new variation of the improved EO that integrates the technique of optimizing the PV-BES size, position, and power factor to strengthen the current standard EO. The experimental results also clearly reveal that the proposed EEO works better than other current state-of-the-art metaheuristics, such as EO and SCA, demonstrating its effectiveness in Photovoltaic modules and Battery Energy System combination (PV-BES) unit optimization. Aval et al. in the year 2022 [204] proposed a new variant of revised EO termed as Residual Moment-Based Joint Damage Index with Equilibrium Optimizer (RMBJDI-EO) that incorporates Residual Moment-Based Joint Damage Index to enhance standard EO. Further, the experimental result evidently highlights that the proposed RMBJDI-EO outperforms EO and other recent state-of-arts metaheuristics showcasing its efficacy in Joint damage identification. Equilibrium Optimizer—Pattern Search (EO-PS), a novel variation of revised EO that integrates Pattern Search approaches to increase the exploitation capacity further increasing standard EO, was proposed by Rizk et al. in the year 2022 [205]. Additionally, the experimental results clearly reveal that the proposed EO-PS beats EO and other contemporary state-of-the-art metaheuristics, demonstrating its effectiveness in the challenge of optimizing the layout of wind farms. Duan et al. in the year 2022 [206] proposed a new variant of revised EO termed as Hybrid Equilibrium Optimizer based on the Crisscross Strategy (HEOC) that incorporates APBOP, t-distributed stochastic neighbor embedding, crisscross strategy to enhance local search and convergence efficiency of standard EO. Further, the experimental result evidently highlights that the proposed HEOC outperforms ANDE, AC-DPHS and AC-MeanABC and other recent state-of-arts metaheuristics projecting its efficacy in real world optimization problems. Modified Equilibrium Optimizer (MEO), a novel variation of the revised EO that combines several scaling factors on the standard EO to improve its performance, was proposed by Kumar et al. in the year 2022 [207]. Additionally, it is clear from the experimental results that the proposed MEO outperforms EO and other modern, state-of-the-art metaheuristics when predicting its effectiveness in hybrid distributed power systems. Dinh et al. in the year 2022 [208] proposed a revised variant of EO termed as Improved Equilibrium Optimizer (IEO) that works by modifying the equations updating solutions in the standard EO. Further, the experimental results clearly highlight that proposed IEO has superior performance when compared with EO, ABC, SSA, GA, PSO, GA/PSO, SA, HAS, BFOA and BSOA. Houssein et al. in the year 2022 [209] proposed a revised variant of EO termed as Improved version of Equilibrium Optimizer (I-EO) that incorporates Standard operators with the Dimension Learning Hunting (DLH) to enhance the performance of standard EO. Further, the experimental results clearly highlight that proposed I-EO has superior performance when compared with AGDE, GWO, MFO, SCA, HHO, TSA and EO for COVID-19 CT images segmentation. Improved Equilibrium Optimizer (IEO), an updated version of EO that especially focuses on population variety maintenance mechanisms to balance the exploration and exploitation in regular EO, was proposed by Yang et al. in the year 2022 [210]. Additionally, the experimental findings unmistakably show that the proposed IEO outperforms EO, GGSA, HGSA, and RGBSO for the problems of economic dispatch, spacecraft trajectory optimization, and artificial neural network model training. Zhu et al. in the year 2022 [211] proposed a revised variant of EO termed as Improved Equilibrium Optimizer (IEO) that make use of elite guidance and balance probability mechanism to speed up the convergence as well as exploitation ability and also to balance the exploration and exploitation ability of stand EO. Further, the experimental results clearly highlight that proposed IEO has superior performance when compared with EO, SMA, MPA, PFA and WOA for ORPD problems. Improved Equilibrium Optimizer Algorithm (IEO), which leverages function independent of the number of iterations to improve the performance of standard EO, was proposed by Nguyen et al. in the year 2022 [212]. Additionally, it is abundantly obvious from the experimental findings that the suggested IEO outperforms EO, AEO, CSA, TLBO, and ABC for the optimal power flow problem. Ates et al. in the year 2021 [213] proposed a revised variant of EO termed as Enhanced Equilibrium Optimizer (E2O) that employs random coefficients, K feedback gain vector parameters to enhance the standard EO. Further, the experimental results clearly highlight that proposed E2O has superior performance when compared with SMDO and DSO for Hover Flight System. A new version of EO called the Enhanced Equilibrium Optimizer (EEO), which incorporates variable selection probabilities based on its own fitness function to improve the standard EO, was proposed by Sun et al. in the year 2021 [214]. Furthermore, it is abundantly obvious from the experimental data that for optimal control designs, proposed EEO performs better than EO and GA. Global Learning Equilibrium Optimization (GLEO), which integrates a general learning method to avoid the local optima in regular EO and improve the exploration mechanism, was proposed by Too et al. in the year 2021 [215]. Also, it is abundantly obvious from the experimental findings that the suggested GLEO performs better than EO, BOA, GWO, PSO, and RF when it comes to classifying medical data. Wang et al. in the year 2021 [216] proposed a revised variant of EO termed as Improved Equilibrium Optimizer (IEO) that incorporates back propagation Neural Network to predict more output data of PV cell. Further, the experimental results clearly highlight that proposed IEO has superior performance when compared with ABC, BSA, GWO, MFO, PSO, WOA and EO for Photovoltaic cell parameter estimation. Zhao et al. in the year 2020 [217] proposed a revised variant of EO termed as Random Walk Equilibrium Optimizer Algorithm (RW-EOA) that employs random walk with Cauchy distribution to enhance the performance of standard EO. Further, experimental results clearly highlight that among the proposed RW-EOA provides superior performance when compared with original EO for optimization of classical benchmark functions. Abdel et al. in the year 2020 [218] proposed a revised variant of EO termed as Improved Equilibrium Optimizer (IEO) that employs Linear Reduction Diversity (LRD) and Local Minima Elimination method (MEM) to boost the standard EO. Further, experimental results clearly highlight that among the proposed IEO provides superior performance when compared with CPMPSO, MADE, ITLBO and MLBSA for multilevel thresholding.

The various methods proposed under each category of revised EO, along with other details, are tabulated in Table 3. The improvement strategies or methods along with other details, of the revised variants of EO proposed by various researchers that have been surveyed in this article, are tabulated in Table 4. Also, the full form of each acronym used in Table 4 is tabulated in Table 7.

Table 3 Various proposed methods under each category of revised EO, along with other details

Table 4 Summary of the different research work carried out using revised variants of EO algorithm

4.2 Hybridized variants of Equilibrium Optimizer (EO)

One of the prominent ways to improve the ability of any nature-inspire metaheuristic algorithm is by means of hybridization. Hybridization not only enables the balancing exploration–exploitation ability of algorithm, but it also enhances their ability to become acquainted with specific information in a given situation. This section discusses the hybridized variants of EO as depicted in Fig. 14, and the year-wise depiction of the research articles discussing the same is shown in Fig. 15.

Fig. 14

Hybridized variants of EO

Fig. 15

Year-wise depiction of the various research article based on hybridized variants of EO (As per the survey)

A number of metaheuristic algorithms have been hybridized with EO, proposing and introducing new hybridized variants of EO. A description of the research article dealing with the aforementioned method proposal is described below and shown in Fig. 16.

Fig. 16

Proportions of research articles addressing the various meta-heuristics algorithms hybridized with EO (As per the survey)

4.2.1 Hybridization with Artificial Bee Colony (ABC)

The Self-Adaptive Quantum Equilibrium Optimizer with Artificial Bee Colony (SQEOABC) is a new version of the revised and hybridized EO that Zhong et al. proposed in 2023 [135]. It uses the updating mechanism of the ABC to find the best solution and incorporates quantum theory and self-adaptive mechanisms into the EO's updating rule to improve convergence. The experimental results further reveal that the proposed SQEOABC performs better than existing state-of-the-art metaheuristics such as PSO, CMAES, GWO, SSA, HHO, SMA, WSO, TLSMA, DDSRPSO, EESHHO, and LSHADE-cnEpSin for the COVID-19 feature selection problem.

4.2.2 Hybridization with Teaching–Learning-Based Optimization (TLBO)

Hybrid Teaching Learning Based Optimization-Equilibrium Optimization (TLBO-EO), which combines the high exploration ability of TLBO with the high exploitation ability of EO, was proposed by Sayed et al. in the year 2023 [219]. The experimental result evidently highlights that, for estimating PV cell performance, the proposed TLBO-EO beats existing cutting-edge metaheuristics like PSO and PSO-CFA. The Modified Equilibrium Optimizer using Teaching–Learning-based Optimization and Opposition-Based Learning (OTLEO) is a hybridized EO that Wang et al. proposed in 2022 [220]. It combines EO with TLBO and OBL to preserve the diversity of the solutions, expand the search space, improve exploration and exploitation skills, and further avoid local optima. Additionally, it is clear from the experimental results that the proposed OTLEO outperforms EO and m-EO when tested against a variety of traditional benchmark functions.

4.2.3 Hybridization with Whale Optimization Algorithm (WOA)

In the year 2022 [170], Liu et al. proposed a new variant of revised and hybridized EO called the Levy Whale Optimization Mutation Equilibrium Optimizer (LWMEO), which combines the random walk strategy based on Lévy flight, the spiral encirclement mechanism of WOA, and the adaptive proportional mutation strategy to increase search range and exploration ability, exploitation ability, and speed up the convergence speed of the standard EO. The experimental results also clearly show that the proposed LWMEO performs better than existing state-of-the-art metaheuristics including EO, AEO, IEO, SSA, MSSA, and JADE, demonstrating its effectiveness in many engineering fields. Hybrid multi-objective Equilibrium Optimizer and Whale Optimization Algorithm (R2-HMEWO), which executes hybridization in the form of structure and operators, was proposed by Tahernezhad et al. in the year 2022 [221]. Furthermore, the experimental outcome clearly shows that for benchmark test issues, the proposed R2-HMEWO beats existing state-of-the-art metaheuristics like NSGA-III, NSGA-II, MOEA/D, MOMBI-II, MOEA/IGD-NS, and dMOPSO. Equilibrium Whale Optimization Algorithm (EWOA), which combines the weight balance technique of EO with the encircling and net-bubble attacking mechanisms of WOA, was proposed by Tan et al. in the year 2022 [222]. Additionally, it is clear from the experimental results that the proposed EWOA performs better than DMOA, POA, DOA, AOA, EO, and WOA for a number of benchmark test problems.

4.2.4 Hybridization with Adaptive Fuzzy based Atom Search Optimizer (AFASO) and Adaptive Rain Optimization (ARO)

Hybrid Equilibrium Optimizer with Capsule Auto Encoder (HEOCAE), which incorporates Skull stripping, Normalized Linear Smoothing and Median Joint (NLSMJ) filtering, Adaptive Fuzzy based Atom Search Optimizer, Adaptive rain optimization, and capsule auto encoder at different stages to enhance the standard EO, was proposed by Ansingkar et al. in the year 2022 [138]. The experimental results also clearly reveal that the proposed HEOCAE works better than EO and other contemporary state-of-the-art metaheuristics, demonstrating its effectiveness in the Multi class Alzheimer detection problem.

4.2.5 Hybridization with Genetic Algorithm (GA)

In 2022 [223], Heidari et al. introduced the Genetic Algorithm and Equilibrium Optimizer (GA-EO), a hybridized EO that combines EO and GA for the aim of choosing the path between the base station and cluster heads. Additionally, it is clear from the experimental results that the proposed GA-EO outperforms existing state-of-the-art metaheuristics for clustering and routing in WSN. In the year 2022 [224], Hemeida et al. presented a hybridized EO known as Genetic Algorithm-Equilibrium Optimization (GAEO), which combines the exploitative capabilities of EO with the explorative capabilities of GA to create a more effective technique. Also, the experimental results clearly show that for the reactive power management problem, the suggested GAEO performs better than SCA, CBA, and ISSA. Hybrid Genetic Algorithm Equilibrium Optimizer (GAEO), which combines EO and GA to generate a switch between exploration and exploitation, was proposed by Bakry et al. in the year 2022 [225]. Furthermore, the experimental data clearly shows that for distribution networks, the proposed GAEO outperforms GA, GASBO, SBO, TLBO, PSO, GA/PSO, BFOA, SA, QOTLBO, and IA.

4.2.6 Hybridization with Slime Mould Algorithm (SMA)

Equilibrium Optimizer Slime Mould Algorithm (EOSMA), a hybridized EO that includes the exploration and exploitation capabilities of SMA to improve the algorithm, was proposed by Yin et al. in the year 2022 [226]. Additionally, the experimental result clearly shows that the proposed EOSMA outperforms SMA, EO, MPA, IGWO, AGPSO, and MTDE employed for various cutting-edge metaheuristics for engineering design issues. A hybridized EO called the Equilibrium Optimizer Slime Mould Algorithm (EOSMA) and a Multi-Objective EOSMA (MOEOSMA) were proposed by Yin et al. in the year 2022 [227]. These algorithms use the concentration update operator of the EO to direct the SMA search mechanism, a greedy strategy to speed up convergence and escape from local optima, and a random difference mutation operator. Furthermore, the experimental data clearly shows that for inverse kinematics of the 7 DOF robotic manipulator, the proposed approaches perform better than MOMPA, MOGWO, MOALO, and MOSMA. Equilibrium Optimizer into Slime Mould Algorithm (EOSMA), which essentially combines SMA with EO and Centroid Opposition-based Computation (COBC) strategy, was proposed by Wei et al. in the year 2022 [228] in order to strengthen exploration and exploitation, accelerate convergence, and prevent stagnation. Additionally, the experimental results clearly show that for the Work Shop Scheduling problem, the suggested EOSMA outperforms EO, MPA, AO, BES, and SMA.

4.2.7 Hybridization with Cuckoo Search (CS)

Equilibrium-Cuckoo Search Optimizer (ECSO), a hybridized EO that uses CS and EO to reduce Differential Exponential Entropy (DEE), was proposed by Swain et al. in the year 2022 [229]. Furthermore, the experimental results clearly show that for multilevel threshold selection in colour satellite images, the proposed ECSO outperforms other state-of-the-art metaheuristics such as CSA and EO.

4.2.8 Hybridization with Atom Search Optimization (ASO)

In 2022 [230], Chattopadhyay et al. introduced the Clustering-based Equilibrium Optimizer and Atom Search Optimization algorithm (CEOAS), a hybridized EO that uses both EO and ASO to help reduce feature dimensions and hence increase classification accuracy. Additionally, it is clear from the experimental results that the proposed CEOAS outperforms other state-of-the-art metaheuristics for speech emotion identification.

4.2.9 Hybridization with White Shark Optimizer (WSO)

White Shark Equilibrium Optimizer (WSEO), a hybridized EO proposed by Makhadmeh et al. in the year 2022 [231], uses the searching components of WSO and EO to improve the performance of the WSO search agents in order to find the best schedules and maintain an ideal balance between exploration and exploitation in order to prevent stagnation in local optima. Furthermore, the experimental results clearly show that for power scheduling issues, the suggested WSEO outperforms DE, DMOA, GWO, SSA, and WSO.

4.2.10 Hybridization with Evaporation Rate Water Cycle Algorithm (ERWCA)

Equilibrium Optimizer- Evaporation Rate Water Cycle Algorithm (EO-ERWCA), a hybridized EO that uses EO over one subpopulation and ERWCA over another subpopulation, was proposed by Calasan et al. in the year 2022 [232]. Also, the experimental results clearly show that the suggested EO-ERWCA performs better for PID controller in AVR systems than CS, WOA, and YSGA.

4.2.11 Hybridization with Thermal Exchange Optimization (TEO)

High Equilibrium Optimizer (HEO), which combines EO and TEO to create a highly balanced optimizer without altering the memory mode and guiding strategy of the standard EO, was proposed by Jia et al. in the year 2021 [233]. The experimental results also clearly show that the proposed HEO performs better than existing state-of-the-art metaheuristics including EO, TEO, GSA, MVO, SHO, GWO, and SCA, demonstrating its effectiveness in a variety of engineering design domains.

4.2.12 Hybridization with Grasshopper Optimization Algorithm (GOA)

Hybrid Equilibrium Optimizer (HEO), a novel variation of hybridized EO that employs the benefits of GOA to improve the optimization capability of regular EO, was proposed by Qi et al. in the year 2021 [234]. Additionally, the experimental results clearly reveal that the proposed HEO performs better than other state-of-the-art metaheuristics including EOA, CSA, FPA, PSO, and BA, demonstrating its effectiveness in the multi-level picture segmentation domain.

4.2.13 Hybridization with Modified Particle Swarm Optimization (MPSO)

Modified Particle Swarm Optimization with Equilibrium Optimization (MPSO-EO), which combines MPSO with EO to boost the population variety of EO and improve PSO's capacity to escape local minima, was proposed by Sayed et al. in the year 2021 [235]. Additionally, the experimental results clearly show that for the Unit Commitment Problem, the proposed MPSO-EO beats existing metaheuristics including EO, PSO, GWO, and SCA.

4.2.14 Hybridization with Group Teaching Optimization Algorithm (GTA)

In 2021, Hemalatha et al. [236] introduced a novel hybridized EO known as the Group Teaching Optimization Algorithm- Adaptive Equilibrium Optimizer (GTA-AO), which uses the GTA for route discovery and the AO for effective packet transmission. Furthermore, the experimental results clearly show that the suggested GTA-AO performs better than other approaches like B-iHTRP, HyphaNet, QNewBee, and Crowwhale-ETR in MANET.

4.2.15 Hybridization with Golden Ratio Optimization (GRO)

In 2021, Dey et al. [237] proposed a new hybridized EO variant known as the Golden Ratio based Equilibrium Optimization algorithm (GREO), which uses the Sequential One Point Flipping (SOPF) technique to find the closest neighbour of the final candidate solution vector and the Average Weighted Combination Mean (AWCM) to combine EO with GRO. Also, the experimental data clearly shows that the suggested GREO outperforms competing approaches for performing feature selection for speech emotion recognition, including SOLO-EO, SOLO-GRBA, PSO, ASO, SFO, GBO, GA, GWO, and HSO.

4.2.16 Hybridization with Simulated Annealing (SA)

Binary Equilibrium Optimizer with Simulated Annealing (BEOSA), a novel hybridized EO variant developed by Ghosh et al. in 2020 [238], uses SA to change the exploitative properties of EO. Also, the experimental results clearly show that the proposed BEOSA outperforms existing feature selection approaches, including BEO, GSA based, ALO based, GWO, and WOA.

Details of various metaheuristic algorithms hybridized with EO are shown in Table 5 below and the summary of the various research articles related to hybridized variants of EO proposed by various researchers that have been surveyed is tabulated in Table 6. Also, the full form of each acronym used in Table 6 is tabulated in Table 7.

Table 5 Various meta-heuristic algorithms hybridized with EO

Table 6 Summary of the different research work carried out using Hybridized variants of EO algorithm

Table 7 The full form of each acronym used in Tables 2, 4 and 6

3.

5 Applications Tackled by EO and Its Variants

EO and its variants have been employed to unravel assorted problems that belong to different application areas. The application areas and other related details are presented in Table 8.

Table 8 Application areas covered by EO and its variants

6 Conclusion and Future research Directions

The goal of this study is to describe, discuss, and evaluate the most recent EO algorithms and their variations. From the study, it is evident that EO chooses a novel principle to handle optimization problems and has competitive advantages over other intelligent optimization algorithms, including ease of implementation, strong adaptability, fewer parameters, and the capacity to easily hybridize with other algorithms. Finding an optimal or nearly optimal solution is nevertheless difficult in some circumstances, especially when facing difficult multi-modal problems, for a variety of reasons:

• 1. The great propensity to become trapped in local optima (local optima stagnation).

• 2. Slow convergence speed and

• 3. The discrepancy points to an immature equilibrium between extraction and exploration.

Major reasons for the above-mentioned issues with EO are:

• 1. Low quality initial population particles are generated randomly.

• 2. An uneven exploration and exploitation process during iterative phase and

• 3. Most significantly, population diversity is reduced as a result of particle aggregation in the latter iteration phase.

This prompted the development and launch of various EO variations to address the shortcomings and problems with the original algorithm and make it into a more powerful, reliable, and efficient algorithm that could handle a variety of search areas. Therefore, the study represents an up-to-date survey on EO and its different variants, along with their applications in different optimization fields. This review paper includes 175 research articles related to the EO algorithm published in numerous reputed journals and conferences since its inception in the year 2020 until January 2023. According to the research so far, it is evident that the EO algorithm has become quite well-known and important, which has led to introduction of 87 different methods as the revised variants of EO since 2020 to resolve problems in various application areas. Above all, the basic and revised EO have been integrated with the existing metaheuristic algorithms, resulting in the development of 23 new hybrid algorithms to address a wide variety of issues from numerous research disciplines.

Additionally, the following limitations are based on the articles surveyed:

• 1. PF problems, PV systems, distribution system networks, feature selection, and engineering design problems have all seen extensive use of EO.

• 2. Levy flight and Opposition-based Learning are most frequently utilized strategies to boost the performance of EO.

• 3. Although chaotic sequence, fuzzy, and quantum mechanisms have been used, they have not been thoroughly investigated.

Without a doubt, EO and its variations have demonstrably proven to be successful methods for resolving ambiguous optimization challenges in the real world, but more research is still needed. Some of the potential research routes have been anticipated here, and maybe they will prove useful for the researcher as they dig up and uncover EO in other research fields.

• 1. To concentrate on the development of revised variants of EO capable of not only improving exploration and exploitation ability and avoiding local stagnation issues, but also focusing on diversity of solutions, expansion of algorithm’s search range, and so on, by introducing a multi-strategy optimization mechanism incorporating strategies that shall evenly distribute the particles in the search space rather than random initialization; one or more dynamic control parameter strategies to create exploration and exploitation stability; to include strategies in late iteration phases to avoid the algorithm from entering into local optima and particle aggregation.

• 2. EO has so far been created in a variety of forms (Table 4), and all of the variations have shown to produce the finest outcomes across many research fields; yet Mixed-integer EO (MEO), Parameter-less EO (PEO), Quantum-inspired EO(QEO), Fuzzy-controlled EO (FEO), etc., could be worthwhile to investigate this further in the future.

• 3. As listed in Table 6 of Sect. 4.2, many swarm-based [239] and human-based meta-heuristic algorithms [240] have been combined with EO for different types of optimization problems. Further applications or integrations of other NIOA algorithm [241] types are possible, such as Plant-based, Maths-based or even Physics-Chemistry-based meta-heuristic algorithms [242, 243], to determine the EO's potential, improve computational performance, and produce good solutions.

• 4. Table 8 of Sect. 5 brings to light the usage of EO in countless application areas; however, very little work has been proposed using EO for path forecasting and map reading. However, more rigorous versions of EO might be developed, such as Itinerant EO, which should be able to handle and control dynamic routes, barriers, goals, etc. This would undoubtedly be an excellent topic to work in the future.

• 5. In a similar line, EO technology could be additionally stretched to work on diverse optimization problems namely, Energy storage devices, Smart grids, Knowledge discovery systems, Fog systems, Abrupt motion tracking, DNA fragment assembly problems, Electric Car component design, Production planning, Rule mining, Image encryption and decryption, Vehicle routing problems, Shrimp freshness detection, Signal denoising, Pathology Image Clustering [244, 245], Work scheduling, Multi-objective feature selection, Smart home applications etc.

• 6. Also, the application of EO to resolving discrete problems could be a new area to ponder.