1 Introduction

Meta-heuristic is one of the striking research areas accompanied by exceptionally significant progress with regard to generation of solution for numerous firm optimization problems. In the year 1976, the term “Meta-heuristic” was coined by Fred Glover [1] basically to exemplify heuristic method with no problem-specific characteristic. Over the last few decades, more attention is paid to the field of optimization using meta-heuristic and huge progress has been made from the time when the first meta-heuristic was anticipated and several novel algorithms are endorsed each day for resolving intricate and real-world predicaments. Appropriate trade-off among exploration and exploitation (chief functions of meta-heuristics) is the key to a proficient search process. Numerous ways of classifications of meta-heuristics have been offered based on utilization of exploration and exploitation mechanism, and the metaphor of the search procedures. In that regard, quite a few algorithms typically instigated by the natural phenomenon has been anticipated and exist in the literature and among those, meta-heuristic search algorithms with population-based outline [2] have revealed pleasing potential to crack high dimension optimization problems [3,4,5] appropriate for global searches due to global exploration and local exploitation capability. It involves the production of a set of assorted solutions at each run and the classification of population-based meta-heuristic algorithm into two main categories namely Evolutionary-Based and Nature-Inspired Algorithms [6, 7]. Further, the nature-inspired algorithms are categorized into five different classes i.e., Swarm-Based, Physics/Chemistry-Based, Human-Based [8], Plant-Based and Maths-Based Algorithms and the same is depicted in Fig. 1.

Fig. 1
figure 1

Classification of population-based meta-heuristic algorithms

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Evolutionary Algorithms (EA) is considered as the foremost class of population-based meta-heuristic optimization algorithms instigated from evolutionary phenomena of nature that make use of three main operators i.e., selection, recombination and mutation). A few popular EAs are: Genetic Algorithm [9], Differential Evolution [10], Evolutionary Programming [11], Evolution Strategies [12], Genetic Programming [13], Population-Based Incremental Learning [14], Biogeography-Based Optimizer [15], Memetic Algorithm [16] and Clonal Selection Algorithm [17]. The second class of population-based meta-heuristic optimization algorithms as depicted in Fig. 1 is Swarm-Based algorithm wherein the swarms (unsophisticated agents) tend to mimic the behavior of the social animals or agents available in our nature such as ants foraging, birds flocking, fish schooling, bacteria moulding, animals herding and many more. The algorithm basically emphasizes on direct and indirect interactions whereby the cooperative behavior of agent intermingling locally with the environment causes the comprehensible global pattern to arise. Some of the Swarm-Based algorithm that has been listed is: Ant Colony Optimization [18], Particle Swarm Optimization [19], Artificial Bee Colony Algorithm [20], Cuckoo Search [21], Firefly Algorithm [22], Bat Algorithm [23], Krill Herd Algorithm [24], Gray Wolf Optimization [25], Ant Lion Optimizer [26], Moth-Flame Optimization Algorithm [27], Dragonfly Algorithm [28], Whale Optimization Algorithm [29], Grasshopper Optimization Algorithm [30], Crow Search Algorithm [31], Salp Swarm Algorithm [32], Marine Predators Algorithm [2], Bee Algorithm [33], Harris Hawks Optimization [34], Social Spider Optimization [35], Intelligent Water Drop Algorithm [36], Glowworm Swarm Optimization [37], Manta Ray Foraging Optimization [38], Sail Fish Optimizer [39], Lion Swarm Optimization [40] and Emperor Penguin Optimizer [41].The third class of population-based meta-heuristic optimization algorithms as highlighted in Fig. 1 is Physics/Chemistry-Based algorithms wherein the main source of inspirations is the physical processes or nature of chemical reactions which are further formulated into solutions to resolve the problems. Few popular physics/chemistry-based algorithms are: Photosynthetic Algorithm [42], Galaxy-based Search Algorithm [43], Flow Direction Algorithm [44], Henry Gas Solubility Optimization [45], Nuclear Reaction Optimization [46], Chemical Reaction Optimization [47], Central Force Optimization [48], Big Bang-Big Crunch Algorithm [49], Magnetic Charged System Search [50], Multi-Verse Optimization [51], Thermal Exchange Optimization [52], Vibrating Particle System Algorithm [53], Artificial Physicomimetics Optimization [54], Ray Optimization [55], Atomic Orbital Search [56], Equilibrium Optimizer [57], Atom Search Optimization [58], Black Hole Algorithm [59], Artificial Chemical Reaction Optimization [60], Gravitational Search Algorithm [61], Electromagnetic Field Optimization [62], Water Evaporation Optimization [63], Optics Inspired Optimization [64], Electromagnetism-like Algorithm [65], Colliding Bodies Optimization [66], Charged System Search [67], Gravitational Local Search Optimization [68].

The fourth class of population-based meta-heuristic optimization algorithm, Human-Based algorithms imitates human behaviour, supremacy and intelligence. Few of the human-based algorithms as depicted in the figure are listed below: Cultural Algorithm [69], Imperialist Competitive Algorithm [70], Teaching Learning-Based Optimization [71], Brain Storm Optimization [72], Human Behavior-Based Optimization [73], Human Mental Search [74], Social Engineering Optimizer [75], Queuing Search Algorithm [76], Search and Rescue Optimization [77], Life Choice-Based Optimization [78], Social Ski-Driver Optimization [79], Gaining Sharing Knowledge-Based Algorithm [80], Future Search Algorithm [81], Forensic-Based Investigation Optimization [82], Political Optimizer [83], Heap-Based Optimizer [84], Human Urbanization Algorithm [85], Battle Royale Optimization [86], Corona virus Herd Immunity Optimization [87], Passing Vehicle Search [88], Jaya Algorithm [89], Seeker Optimization Algorithm [90], Interior Search Algorithm [91], Soccer League Competition Algorithm [92], Exchange Market Algorithm [93], Group Counseling Optimization Algorithm [94], Tug of War Optimization [95], Most Valuable Player Algorithm [96], Volleyball Premier League Algorithm [97], Dynastic Optimization Algorithm [98], Focus Group [99], Stock Exchange Trading Optimization [100], Anti Corona virus Optimization Algorithm [101], Socio Evolution and Learning Optimization [102], League Championship Algorithm [103], Ideology Algorithm [104], Cohort Intelligence [105], Social Group Optimization [106], Social Learning Optimization [107], Cultural Evolution Algorithm [108], Backtracking Search Optimization Algorithm [109], Football Game Algorithm [110], Class Topper Optimization [111], Ludo Game-based Swarm Intelligence [112], Team Game Algorithm [113], Election Algorithm [114], Election Campaign Optimization Algorithm [115], Anarchic Society Optimization [116], Society and Civilization [117] and Social Emotional Optimization Algorithm [118]. Plant-Based Algorithms has been categorized as the fifth class of population-based meta-heuristic optimization algorithm that mimics the intelligent behavior exhibited by plants. Some of the renowned plant-based algorithms are: Plant Growth Optimization [119], Root Growth Algorithm [120], Invasive Weed Optimization [121], Fertile Field Algorithm [122], Flower Pollination Algorithm [123], Paddy Field Algorithm [124], Root Mass Optimization Algorithm [125], Artificial Plant Optimization Algorithm [126], Sapling Growing up Algorithm [127], Photosynthetic Algorithm [42], Plant Propagation Algorithm [128], Rooted Tree Optimization [129], Path Planning inspired by Plant Growth [130] and Artificial Root Foraging Algorithm [131]. The last category that falls under the population-based meta-heuristic optimization algorithm is the Maths-Based Algorithms that basically tend to imitate the procedure of numerical techniques, mathematical programming and its orientation to resolve numerous constraints and optimization issues of the real environment. Some of the widely known maths -based algorithms are Hyper-Spherical Search Algorithm [132], Radial Movement Optimization [133], Stochastic Fractal Search [134], Golden Ratio Optimization Method [135], Sine Cosine Algorithm [136] [137] and Arithmetic Optimization Algorithm [138].

Marine Predators Algorithm (MPA) as highlighted in Fig. 1 is the algorithm that is considered among the list of algorithms available in this paper. MPA is one of the potential population-based meta-heuristic optimization algorithms that come under the class known as Swarm-Based Algorithms. This algorithm is employed to work out on abundant optimization problems specifically Mathematical and Engineering Optimization problems, Image processing, Photovoltaic Systems, Fog Computing, Wind-Solar Generation System and many other as mentioned earlier. MPA is formulated based on the different foraging strategy opted by the ocean predators and optimal encounter rates policy in biological interaction. The Levy and Brownian motions are dual strategies preferred by predators intended for the purpose of optimal foraging. MPA has time and again proved its capacity to present a good number of effectual designs and also spawned efficient statistical results when matched up with other well-regarded existing methods. The different strategies involved in terms of foraging and memories makes Marine Predators and overall MPA slightly different [2] and acceptable when compared with the other meta-heuristics algorithms presented in the literature:

  1. a.

    Marine Predators is well equipped with strategies for different scenarios. If the environment with less and sparse concentration of prey is encountered, MPA indulges in the usage of Levy strategy for foraging however, it navigates to Brownian movement on encountering of the environment [2] with higher and profuse concentration of prey.

  2. b.

    Marine Predators apart from quickly fluctuating the foraging strategy as well changes their actions with the objective [2] to discovery the areas with different concentrations of prey.

  3. c.

    In terms of memory, Marine predators are blessed with good memories [2] and predators takes the benefits of its skill to further track of the locations and additional help their subordinates to do the needful.

  4. d.

    The minimalism, easier to implement in conjunction with effectual and competent outcomes unquestionably put forth Marine Predators Algorithm, as an alternate optimization procedure to conventional techniques available in the literature.

This article hereby exemplifies a crisp survey of MPA, variants of MPA and further highlights the applications of MPA in diverse fields of research. Furthermore, to assemble the numerous published articles related to MPA, quite a few acclaimed publishers specifically IEEE, Elsevier, Springer, MDPI, AIMS press, Nature Portfolio, Taylor & Francis, Wiley, Hindawi and many more has been considered and in order to do so one of the liberally reachable web search engine that provides the full text of scholarly literature across the range of publishing disciplines i.e. Google Scholar is employed and the searching is done based on few of the terminologies (Not limited to though) as projected in Fig. 2. Number of recent variants of MPA (Revised and Hybridized) published by different publishers as per surveyed is depicted in Fig. 3. Figure 4 elaborates the top 10 Journals ranked based on publications of variants of MPA. Number of publications of research papers related to variants of MPA per year is depicted in Fig. 5.

Fig. 2
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Terminologies used to search the MPA research papers from google scholar

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Fig. 3
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Number of recent variants of MPA (Revised, Hybridized and Application based) published by different publishers (As per Surveyed)

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Fig. 4
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Top 10 Journals ranked based on publications of variants of MPA (Revised, Hybridized and Application based) (As per Surveyed)

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Fig. 5
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Number of publications of recent variants of MPA (Revised, Hybridized and Application based) per year (As per Surveyed)

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MPA is one of the prevalent swarm-based meta-heuristic and is considered as one of the exclusive members of NIOA family. The total number of citations of MPA (as per Google Scholar, dated: 19.10.2022) is 1040 (Only for the papers considered in this survey). As mentioned earlier, several variants of MPA have been proposed in the literature and applied in numerous application areas. However, as per the best of the knowledge there is no review or survey paper highlighting the different variants of MPA till date and this is the main objective behind the study i.e., basically reviews the existing work on MPA. This review article sheds lights on identifying, categorizing and further analyzing the different variants of MPA used in numerous application areas to resolve the real-world optimization issues. This review paper meticulously explores all research works linked with MPA thereby addressing five important pillars which is structured as follows:

  • The structure of the standard or original MPA is described in Sect. 2.

  • Section 3 highlights and further discusses the revised variants and hybrid version of MPA developed, introduced and applied so far.

  • The problem resolved by MPA that belongs to different applications areas are discussed in Sect. 4.

  • Section 5 highlights few representative articles of MPA used in the survey.

  • Finally, the paper is concluded and few potential future research directions for MPA is advocated in Sect. 6.

2 Original Marine Predators Algorithm (MPA)

Marine Predators Algorithm (MPA) is a popular nature-inspired swarm-based meta-heuristic optimization algorithm originally developed and introduced by Faramarzi and group in the year 2020 [2] based on foraging nature and meandering communications amongst the predators and prey in the oceanic ecological unit. The natural animals that tend to forage in groups basically employ the random walk strategy and one such exceptional variant of random walk strategy is Levy flight/move strategy that is typically grounded on the perception of optimal search. Several studies have clearly revealed and anticipated that many marine creatures including sharks, marlines, sunfish, tunas and swordfish make use of Levy strategy as the means to forage [139]. The other type of random walk strategy employed by the natural predators to traverse is popularly known as Brownian strategy and MPA during its life span thus uses both Levy as well as Brownian strategies to traverse or navigate diverse territories utilizing the first strategy i.e., Levy in the surroundings with inferior concentration of prey and the second strategy i.e., Brownian in the environment involving profuse number of preys. The pseudo-code for standard Marine Predators Algorithm is represented as Algorithm 1 and further Fig. 6 exemplifies the flowchart of the same. Like other population-based meta-heuristic algorithms, in MPA too, preliminary solution is unvaryingly disseminated over the search space and the same is depicted using Eq. 1. Two matrices of the same dimensions namely Elite and Prey is constructed [2] (as shown in Eqs. 2 and 3 respectively) that basically depicts the Predator’s and the Prey’s position that enables the predator to find its prey while the prey is in search for the food to survive as per the mechanism called “survival of the fittest”. Both the predator as well as prey is the searching agent in this scenario as mentioned earlier that predator is searching for prey and in turn prey searches for its food. The entire procedure of optimization revolves around these two matrices i.e., Elite and Prey matrices as depicted below.

$${X}_{0}={X}_{min}+rand ({X}_{max}-{X}_{min})$$
(1)
$$Elite={\left[\begin{array}{ccc}{X}_{\mathrm{1,1}}^{I}& \cdots & {X}_{1,d}^{I}\\ \vdots & \ddots & \vdots \\ {X}_{n,1}^{I}& \cdots & {X}_{n,d}^{I}\end{array}\right] }_{n\times d}$$
(2)
$$Prey={\left[\begin{array}{ccc}{X}_{\mathrm{1,1}}& \cdots & {X}_{1,d}\\ \vdots & \ddots & \vdots \\ {X}_{n,1}& \cdots & {X}_{n,d}\end{array}\right] }_{n\times d}$$
(3)

Here, \({X}_{min}\) denotes the lower variable bound, \({X}_{max}\) the upper variable bound and rand is a random vector that is uniform in nature ranging from 0 to 1[2]. Here, \(\overrightarrow{{X}^{I}}\) depicts the vector with regard to top predator that is simulated n times (reliant on the total search agent, n) to construct the Elite matrix with d as its dimension. In the entire process of searching the constructed Elite matrix keeps updating in search of the fittest predator. Further, \({X}_{i,j}\) corresponds to the location in the search space of the \({i}^{th}\) Prey in \({j}^{th}\) dimension.

Fig. 6
figure 6

Flowchart depicting the mechanism in Marine Predators Algorithm

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Subsequently with the MPA formulation, MPA optimization needs to be addressed and, in this regard, the entire optimization procedure is divided broadly into three major stages taking into account different velocity ratio however yet impersonating the life-cycle of predator as well as prey. Three different stages are highlighted as Stage I, II and III [2].

Stage I: “Whenever the predator navigates quicker than prey”. (High velocity ratio).

Stage II: “Whenever the predator and prey navigate almost with the same velocity”. (Unit velocity ratio).

Stage III: “Whenever the predator navigates slower than prey”. (Low velocity ratio).

In first stage, in a scenario wherein the predator is steering quicker than the prey, the velocity ratio is considered high, thereby the best strategy that can be adopted by the predator is to stop and not move at all. The mathematical formulation [2] of the same is depicted using Eq. 4.

\(While \,Iter< \frac{1}{3}\) Max_Iter then,

$$\overrightarrow {{Stepsize_{l} }} = \overrightarrow {{R_{B} }} \otimes \left( {\overrightarrow {{Elite_{l} }} - \overrightarrow {{R_{B} }} \otimes \overrightarrow {{Prey_{l} }} } \right)\,\,\,\,\,\,\,\,\,\,\,\,\,i = 1,........n$$
(4)

Here, \(\overrightarrow {{Prey_{l} }} = \left( {\overrightarrow {Prey} + P.\overrightarrow {R} \otimes \overrightarrow {{Stepsize_{l} }} } \right)\); \(\overrightarrow{{R}_{B}}\) is a vector comprising of random numbers built on Normal distribution symbolizing the Brownian motion; \(\otimes\) denotes entry-wise multiplication; P represents constant number initialized to 0.5 [2]; Iter is the current iteration; Mar_Iter symbolizes the maximum number of iteration and R designates vector of uniform random number in the range [0,1].

In Stage II, the scenario wherein the pace of predator as well as prey is almost the same, the velocity ratio is considered to be a unit velocity ratio, thereby the best strategy that can be rather need to be opted by Predator is the Brownian move and by the Prey is Levy move [2]. Herein, predator is accountable for exploration nevertheless the prey is responsible for the exploitation, depicted using Eqs. 5 and 6.

\(While \frac{1}{3}\) Max_Iter \(<Iter<\frac{2}{3}\) Max_Iter then,

For the Population (Prey)

$$\overrightarrow {{Stepsize_{l} }} = \overrightarrow {{R_{L} }} \otimes \left( {\overrightarrow {{Elite_{l} }} - \overrightarrow {{R_{L} }} \otimes \overrightarrow {{Prey_{l} }} } \right)\,\,\,\,\,\,\,\,\,\,\,\,\,i = 1,........n/2$$
(5)

Here, \(\overrightarrow{{Prey}_{i}}=\overrightarrow{{Prey}_{i}}+P.\overrightarrow{R} \otimes \overrightarrow{{Stepsize}_{i})}\); \(\overrightarrow{{R}_{L}}\) is a vector based on Levy distribution representing the Levy motion;

For the Population (Predator)

$$\overrightarrow {{Stepsize_{l} }} = \overrightarrow {{R_{B} }} \otimes \left( {\overrightarrow {{R_{B} }} \otimes \overrightarrow {{Elite_{l} }} - \overrightarrow {{Prey_{l} }} } \right)\,\,\,\,\,\,\,\,\,\,\,\,\,i = n/2,........n$$
(6)

Here,\(\overrightarrow{{Prey}_{i}}=\overrightarrow{Elite}+P.CF \otimes \overrightarrow{{Stepsize}_{i})}\); \(CF\) is used to control the step size and is given as \(CF={\left(1-\frac{Iter}{Ma{x}_{Iter}}\right)}^{{(}^{2}\frac{Iter}{Max\_Iter})}\); Multiplication \(\overrightarrow{{R}_{B}} \otimes \overrightarrow{{Elite}_{i}}\) denotes the Brownian move of the predator.

figure a

In the third stage, in the scenario wherein predator moves slower than that of the prey, the velocity ratio is considered to be a low velocity ratio, thereby the best strategy [2] that can be opted by the predator is Levy motion and the same is clearly depicted using Eq. 7.

\(While \,Iter> \frac{2}{3}\) Max_Iter then,

$$\overrightarrow {{Stepsize_{l} }} = \overrightarrow {{R_{L} }} \otimes \left( {\overrightarrow {{R_{L} }} \otimes \overrightarrow {{Elite_{l} }} - \overrightarrow {{Prey_{l} }} } \right)\,\,\,\,\,\,\,\,\,\,\,\,\,i = 1,........n$$
(7)

Here, \(\overrightarrow{{Prey}_{i}}=\overrightarrow{Elite}+P.CF \otimes \overrightarrow{{Stepsize}_{i})}\); Multiplication \(\overrightarrow{{R}_{L}} \otimes \overrightarrow{{Elite}_{i}}\) denotes the Levy move of the predator. Lastly, one important component that needs to be considered here in MPA is the cause of the behavioral change among the marine predators i.e., the environment concerns such as Eddy Formation or Fish Aggregating Devices commonly known as FADs effects [2]. The FADs effect is mathematically depicted as shown in Eq. 8.

$$\overrightarrow{{Prey}_{i}}=\left\{\begin{array}{c}\overrightarrow{{Prey}_{i}}+CF\left[\overrightarrow{{X}_{min}}+ \overrightarrow{R} \otimes \left(\overrightarrow{{X}_{max}} - \overrightarrow{{X}_{min}}\right)\right]\otimes \overrightarrow{U} if r \le FADs \\ \overrightarrow{{Prey}_{i}}+[FADs (1-r)+r] (\overrightarrow{{Prey}_{r1}}-\overrightarrow{{Prey}_{r2}}) if r> FADs\end{array}\right.$$
(8)

Here, FADs denote the probability for FADs effect on the optimization procedure initialized with value 0.2; \(\overrightarrow{\mathrm{U}}\) denotes the binary vectors with value 0 [2] (if array is < FADs) and 1 otherwise; \(\mathrm{r}\) depicts random number ranging between [0, 1] [2]; \(\overrightarrow{{\mathrm{X}}_{\mathrm{min}}}\) and \(\overrightarrow{{\mathrm{X}}_{\mathrm{max}}}\) are the vectors containing lower and upper bounds of the dimensions and \(\mathrm{r}1\) and \(\mathrm{r}2\) signifies the indexes of the prey matrix. (see Table 1).

Table 1 Abbreviations of different algorithms along with its full form for algorithms projected in Fig. 1
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3 Recent Variants of Marine Predators Algorithm

Since the inception of MPA, an extensive series of MPA’s variants have been developed and further introduced in the literature. Based on the same, recent variants of MPA are divided into two important categories namely: Revised variants of MPA and Hybridized variants of MPA and the same are illustrated in Table 2. The details in regard to the two categories as depicted in Table 2 are discussed in the subsequent sections. Furthermore, Abbreviations of different MPA variants along with its full form for algorithms projected in Fig. 4 is tabulated in Tables 3 and 4.

Table 2 Recent variants of marine predators algorithm (MPA)
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Table 3 Abbreviations of revised MPA variants along with its full form for algorithms projected in Table 2
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Table 4 Abbreviations of hybridized MPA variants along with its full form for algorithms projected in Table 2
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3.1 Revised Variants of Marine Predators Algorithm

The revised variants of MPA as per Fig. 7 has been categorized into different categories. The name of each category is devised as per the nomenclature provided by the different authors in their research papers and has not been altered. It is clear from given figure that around 26 different categories of the revised variants (with sub-variants) of MPA has been introduced since its development namely, Modified MPA, Improved MPA, Mutated MPA, Enhanced MPA, Binary MPA, Multi-Group, Multi-Objective, Extended MPA, IP Based MPA, Gold-Sine Dynamic MPA, Advanced MPA, Comprehensive MPA, Fuzzy MPA, Quantum MPA, Fusion MPA, Chaos MPA, Fractional MPA, Stochastic MPA, Non-Linear MPA, Heterogeneous MPA, Co-Evolutionary MPA, Comprehensive Learning Dynamic Multi-Swarm MPA, Adaptive MPA, Opposition-Based MPA, Multi-Strategy MPA, Fractional Order Comprehensive Learning MPA, Lambert MPA, Harmonic MPA, Ranking-Based MPA and Hybrid MPA. Further, depending on the different mechanism / operators / transfer functions used to resolve the problem under consideration, various sub-variants have been introduced under each category. There are 44 different revised version of MPA belonging to different categories that is IMPA-I, IMPA-II, IMPA-III, IMPOA, IMMPA, MMPA-I, MMPA-II, MMPO, MMPA-SA, BMPA-TVSinV, BMPA, MOMPA-I, MOMPA-II, MOMPA-III, MOEMPA, EMPA, LEO-EMPA, MSMPA-JRSSELM, MGMPA, QMPA, AMPA, FMPA, CMMPA, SMPA-MC, FMMPA, FO-MPA, CECMPA, CLDMMPA, H-MPA, IPMPA, MPA-OBL, GDMPA, MPAmu, N-MPA, EMPA, NMPA, FOCLMPA, CLMPA, ACMPA, OBL-MPA, HMPA, IMPARDR, SHE-MPA and MPALW. The same is depicted in Fig. 7. The citations as per Google scholar for different revised variants of MPA belonging to different categories is portrayed in Fig. 8. The total number of revised variants of MPA developed over years is highlighted in Fig. 9. The full-form of the same is depicted in Table 3. The details of each of the variants such as revised variants name, Methods / Mechanism used, application areas, results, citation (as per Google Scholar, dated: 19.10.2022) and publisher are tabulated in Table 5. Full form of the different terminologies used in Table 5 is projected in Table 7.

Fig. 7
figure 7

Proposed methods belonging to categories of revised MPA

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Fig. 8
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The citations as per Google scholar for different revised variants of MPA

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Fig. 9
figure 9

Total number of revised variants of MPA developed over years

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Table 5 Various revised MPA variants along with other related details
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3.2 Hybridized Variants of Marine Predators Algorithm

On the other hand, 16 research papers has been listed that focuses on the hybridization of MPA with numerous metaheuristic algorithms such as Salp Swarm Algorithm, Teaching–Learning mechanism, Golden Sine algorithm, Differential Evolution, Grey Wolf Optimizer, Sine–Cosine Algorithm, Slime Mould Algorithm, Mole Rat algorithm, Multi-Verse Optimization algorithm, Political Optimizers and Particle Swarm Optimization. The hybridization of MPA with all these algorithms has generated around 16 new algorithms that can applied to wide range of applications ranging from Image classification to Segmentation to Image Synthesis to Feature Selection to Optimization problems and many more. The hybridized algorithms are: MPASSA, EGMPA, ODMPA, MMPA-OLGWO, MPASCA, TLMPA, HMPA, MPA-PO, MPA-PSO, IMPAPSO, MPA-MVO, MpNMRA, MPAOA, DEMP, MMPA-TLBO and MPO-IPSO-OCR. The citations as per Google scholar for different hybridized variants of MPA belonging to different categories is depicted in Fig. 10. Various hybridized variants of MPA build up and projected over years since 2020 till date (as surveyed) is provided in Fig. 11. Full form of the same is highlighted in Table 4 and the hybridized variants of MPA and its related details are illustrated in Table 6. Full form of the different terminologies used in Table 6 is projected in Table 7.

Fig. 10
figure 10

The citations as per Google scholar for different hybridized variants of MPA

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Fig. 11
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Total number of revised variants of MPA developed over years

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Table 6 Various Hybridized MPA variants along with other related details
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Table 7 Full form of the different algorithms or terminologies as mentioned in Table 5 and 6
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4 Application Areas of Different Variants of Marine Predators Algorithm

Since its inception in the year 2020, Marine Predators Algorithm (MPA) has been employed to unravel assorted problems that belong to different application areas. The wide range of problems resolved by the algorithm and the details of the entire scenario are tabulated in Table 8. Full form of the different terminologies used in Table 8 is projected in Table 9. Also, kindly refer to Tables 3 and 4 for the remaining full forms of the different terminologies used in Table 8.

Table 8 The applications areas of Marine Predator Algorithm to solve various problems
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Table 9 Full Form of the different proposed methods as mentioned in Table 8
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5 Representative Articles of MPA Used in the Survey

This section briefly describes few articles used in the survey in the above sections that has been highlighted as a representative article in the manuscript. The choice of the articles is done on the basis of the highest number of citations done so far. The articles described in this section comprised of both the revised variant as well as the hybridized variant of Marine Predators algorithm developed so far and has been referred by many researchers to carry forth their research activities.

In the work of Basset et al. [165] a hybridized variant of MPA called as Improved Marine Predators Algorithm and a Ranking-Based Diversity Reduction Strategy (IMPARDR) to develop hybrid COVID-19 Detection model is proposed. The proposed method employs Improved Marine Predators Algorithm (IMPA) and a Ranking-Based Diversity Reduction Strategy (RDR). The RDR strategy is employed for the enhancement of the performance of IMPA to get better solution in lesser number of iterations. RDR basically identifies incompetent particles that cannot reach the better solutions within a specified number of iterations, thus moving those inept particles towards the best solutions generated so far. The proposed method is validated using the medical images i.e., nine chest X-Ray images with threshold levels amid 10 and 100 and is further equated with five state-of-art algorithms namely EO, WOA, SCA, HHA, and SSA. The experimental outcome clearly exposes that the hybrid model, IMPARDR outdoes other algorithms mentioned above in terms of fitness values, Std, and a range of threshold metrics. The paper further advocates a path to the interested researcher that the proposed method can be worked on with color image segmentation and other medical applications thus paving a way for more investigation.

Sahlol et al. [180] anticipated a revised variant of MPA called Fractional Order Marine Predators Algorithm (FO-MPA) for COVID-19 image classification. The proposed method uses CNN mechanism for feature extraction and Fractional Order (FO) with enhanced version of Marine Predators Algorithm (MPA) to choose the pertinent features. The proposed method’s performance is validated on two public COVID-19 X-ray datasets and the proposed FO-MPA method is compared with nine state-of-art algorithms namely SMA, HHO, HGSO, WOA, SCA, bGWO, SGA, BPSO, besides the classic MPA. The experimental result clearly highlights that the proposed method produces efficient result in terms of both classification as well as feature extraction when compared with the above-mentioned algorithms. The paper further suggests a direction to the researcher that the proposed method can be further applied for numerous image classification tasks and possibly will be decent alternative to other feature extractor and selector methods.

In the work of Basset et al. [181], Improved Modified Marine Predators Algorithm (IMMPA), a revised variants of MPA for the purpose of task scheduling in IoT based Fog Computing application is anticipated. The proposed method employs Modified MPA (MMPA) to improve the exploitation capability of the traditional MPA and ranking strategy-based initialization and mutation to get rid of the local optima and move towards the best so-far solution achieved. The IMMPA method is thereby compared with seven state-of-arts algorithms namely MPA, MMPA, WOA, SCA, SSA, GA and EOA and the experimental results clearly reveals the fact that the proposed method outperforms the above-mentioned algorithms. The evaluation has been performed based on five performance metrics such as energy consumed, make-span, cost, flow time and carbon dioxide emission rate. The paper further suggests a direction to the researcher that the proposed method can be further applied to schedule the dependent task in fog system and for answering multi-dimensional knapsack problems and DNA fragment assembly problem.

In the work of Elaziz et al. [217] a hybridized variant of MPA is proposed known as Random Vector Functional Link integrated with Marine Predators Algorithm (RVFL-MPA) for the tensile behavior prediction of dissimilar friction stir welded aluminum alloy joints. The proposed method employs MPA with RVFL to improve the prediction accuracy by employing the input parameters such as rotational speed, welding speed, tool axial force and pin profile with Tensile Elongation (TE) and Ultimate Tensile strength (UTS) as the output parameters. The RVFL-MPA method confirmed boundless promise amongst the experimental and projected outcomes which further indicates that it is not just precise but unfailing as well to expect the tensile behavior of welded aluminum joints.

In the work of Ridha [182] a hybridized variant of MPA is proposed known as Marine Predators Algorithm and Lambert W function (MPALW) for parameters extraction of single and double diodes photovoltaic models. The proposed method employs MPA with Lambert W function to tackle the parameter extraction optimization problem. The MPALW method is compared with six state-of-arts methods namely MPA, IEM, BHHO, DEAM, EO and SMA. The experimental result further clearly divulges the fact that the MPLW outperforms the other methods mentioned above in terms of accuracy as well as reliability. The paper further suggested the interested researchers that the proposed method can be applied to real engineering applications such as smart grids, energy sector, and fault error detection in future.

Ramezani et al [169]. in his work proposed a revised variant of MPA known as Modified Marine Predator Algorithm (MMPA) for the purpose of a real-world optimization problem based on PID control applied to a DC motor (PID controller tuning problem). The proposed method employs opposition-based learning method to improvise the initial population, population diversity as well as productivity; chaotic map function to discover the search space; self-adaptive population method to inevitably regulate the size of the population and adaptive method to switch amongst exploration and exploitation phases. The validation of the performance of MMPA is performed on the simulated MATLAB environment on standard test functions including CEC-06 2019 tests and is compared with five state-of-arts methods namely PSO, HHA, DS, JAYA, WOA, LCA, GOA and EO. The experimental result clearly reveals the fact that the MPLW outperforms the other methods mentioned above. The paper further suggest that the proposed method can be further applied to unravel discrete space, binary and multi-objective optimization problems, as well as for solving the problem of transmission sensitivity.

Eid et al. [167] in his work proposed a revised variant of MPA known as Improved Marine Predator Algorithm (IMPA) for the purpose of optimal allocation of active and reactive power resources in distribution networks. The proposed method employs Reactive power control strategy, Predator strategies to lessen the overall system losses and voltage deviations and make the most of the voltage stability further improving the distribution system’s total performance. The validation of the performance of IMPA is performed on two standard test systems, 69-bus and 118-bus distribution networks to prove the proposed algorithm’s efficiency as well as scalability. Further, the proposed IMPA method is compared with three state-of-arts methods namely such as MPA, AEO and PSO. The experimental result clearly reveals the fact that the IMPA methods is capable of finding optimal solution and outperforms the other methods mentioned above.

Houssein et al. [192] in his paper proposed a new revised variant of MPA known as Opposition-Based Marine Predators Algorithm (MPA-OBL) for the for global optimization and multilevel thresholding image segmentation. The proposed method employs Opposition-Based Learning (OBL) strategy to boost the performance of the traditional MPA basically by improving their search efficiency, enhancing the exploitation phase as well as convergence. The validation of the performance of MPA-OBL is performed to solve IEEE CEC’2020 benchmark problems. Further, the proposed method is compared with seven state-of-arts methods namely LSHADE-SPACMAOBL, CMA_ES-OBL, DE-OBL, HHO-OBL, SCA-OBL, SSA-OBL and MPA. The experimental result clearly reveals the fact that the MPA-OBL methods generates remarkably proficient outcomes in contrast with the other competitor algorithms as mentioned above. Additionally, the proposed method is used for image segmentation by means of two objective functions of Otsu and Kapur’s methods over a number of benchmark images at considering different threshold values using three evaluation matrices namely Peak signal-to-noise ratio (PSNR), Structural similarity (SSIM), and Feature similarity (FSIM) indices.

In the work of Dinh [210] a hybridized variant of MPA known as Three-Scale image Decomposition and Marine Predators Algorithm (TSD-MPA) for multi-modal image fusion is proposed. The proposed TSD-MPA employs Three-Scale Decomposition (TSD) technique to achieve the base and detail components; local energy function to preserve significant data and MPA for generating the optimal parameter. The validation of the performance of TSD-MPA is done with the help of the medical images and the proposed method is compared with five state-of-arts methods namely CSMCA, NSCT, CSR, NSST-PA-PCNN and NSST-MSMG-CNN. The experimental results clearly highlights that the TSD-MPA method meaningfully improves the quality of the fused image’s and preserves the information in regard to the edge.

6 Conclusion and Potential Future Research Directions

A number of studies projected using MPA has addressed and solved numerous optimization problems though, MPA was originally anticipated to deal with continuous optimization problems. Additionally, although MPA has vigorous parameters, still the issue of obtaining optimal or near optimal solution arises in some of the scenarios because of the local optima stagnation, low convergence speed and discrepancy between exploitation and exploration. Moreover, MPA has some crucial issue in terms of its structure i.e., the phases of algorithm wherein the number of iterations is inadequate to explore the search space and then find the optimal solutions thereby greatly affecting the searching mechanism. On the other hand, MPA suffers from few of the deficiency such as the incapability to yield a varied initial population with high productivity lack of quick escaping of the local optimization which needs to be taken care of.

This has led to the proposal and introduction of several variants of MPA to address the flaws and issues encountered in the standard MPA and convert MPA into a stronger, robust and effective algorithm that would be capable of managing diverse search spaces. In this paper, a comprehensive survey of MPA has been performed according to the revision, hybridizations and applications. The MPA variants i.e., both the revised and the hybridized are elaborated in the previous section. From the study so far, it is clear that MPA algorithm has gained enormous popularity and importance due to which39 revised or modified variants of MPA has been introduced, developed and implemented so far (in two years) to resolve problem from various application areas. Above all standard MPA as well as revised MPA has been integrated with the existing algorithms and strategies generating 35 new hybrid algorithms to resolve the numerous issues from different research domains. Since its development several research papers have been published by several researchers and academicians highlighting its effectiveness and such competitive performance of MPA are due to its effortlessness, superior convergence speed, realistic execution time and most importantly its high potential to blend and strive with new optimization techniques and strategies. No doubt, MPA and its variants have noticeably proved itself as a successful method to unravel vague real-world optimization problems; however, it can still be further investigated. Few of the prospective research directions have been anticipated below that shall expectantly turn out to be constructive for the researcher to exhume and discover MPA in other arena of research.

  1. 1.

    Numerous variants of MPA has been developed so far and all the variants have demonstrated the best of the results in different area of research, however, the mixed-integer variant of MPA (MIMPA), Constrained MPA (CMPA)or even parameter-less MPA (PMPA)could be an interesting area to explore in future. Furthermore, few revised variants of MPA had been explored in the field of robot path planning and navigation, however, more stringent variant of MPA could be devised such as Mobile MPA/Dynamic MPA that would have the capability to tackle and control dynamic trajectory, dynamic obstacles, dynamic goal etc., could definitely be a good work to work in future. The introduced revised variants of MPA utilized in the field of image processing could be utilized to work for color image classification, segmentation [221], enhancement [222] especially for medical images (MRI, CT etc.,) to extract regions containing clinical features.

  2. 2.

    Variety of swarm-based, math-based meta-heuristic algorithm has been hybridized with MPA and its variants to resolve different optimization problem however, in future one can even think of applying or integrating plant-based [119, 127], human-based [8] and even physics/chemistry [47, 68] based meta-heuristic algorithms to identify the potential of MPA and further progress the computational performance and generate quality solution.

  3. 3.

    Numerous problems belonging to wide range of applications areas has been explored using MPA and its variants, however, the researcher can focus on the devising the solution using MPA or its variants as an optimizer to optimize the existing classifier/mechanism and further apply to identify urban sprawl using the series of satellite images available. MPA technology can be further extended to solve different optimization problems in the power system applications, energy storage devices, smart grids, knowledge discovery, fog systems, DNA fragment assembly problem, signal denoising, work scheduling, parameter optimization and smart home applications.