Mayfly optimization algorithm: a review

1. Introduction

Metaheuristic optimization is a method used to resolve the issues in optimization. Optimization basically involved in almost every application such as the design engineering, economical sector, internet of things applications as well [1]. In order to resolve the issues in optimization various algorithms are been used [2].

Artificial intelligence and machine learning offer solutions to a variety of real-world problems, ranging from discrete to continuous, and constrained to unconstrained challenges. However, it’s been shown that conventional techniques may struggle with large-scale multimodal situations that lack continuity or differentiability. In response, metaheuristic algorithms have emerged as effective alternatives due to their simplicity and ease of implementation. These strategies operate independently of gradient specifics or mathematical properties of the landscape, making them competitive options for a diverse array of problems [3, 4].

In a study, some algorithms from 2018 to 2020 were evaluated for the optimization [3]. They included the Atom Search Optimization (ASO) [5], Beetle Antennae Search (BAS) [5], Sunflower Optimization (SFO) [7], Mayfly Algorithm (MA) [8], Parasitism Predation Algorithm (PPA) [9], Slime Mould Algorithm (SMA) [10] and Tunicate Swarm Algorithm (TSA) [11]. In cuckoo algorithm [12] study is performed on egg laying pattern of cuckoo bird, in particle swarm optimization [13] research on swarm behaviour is observed, also behavioural pattern of firefly [14], grasshopper [15-18], salp swarm [19, 20] and bat [21] is studied.

On the basis of bio related inspirations, numerous types of developments in the bio-inspired algorithms [22]. These types of algorithms are basically categorized into the four categories. Namely are the evolutionary based [23], swarm intelligence [11], ecology based [24] and the multi objective algorithms [25].

The evolutionary based algorithms consist of the artificial neural networks, genetic algorithms, evolution strategies, differential evolution, paddy field algorithms. Further the second category which is the swarm intelligence consist of the ant colony optimization, artificial bee colony, fish swarm, firefly algorithms, group search optimizers and so on. The ecology-based algorithms include the invasive weed optimization, biogeography-based optimization, etc. The last category which is the multi-objective optimization basically consist of the non-dominated sorting genetic algorithm, population-based ant colony algorithm, etc.

The mayfly is a small, weak, and gentle bug that numbers way extra than out of 3,100 types in the globe. Nevertheless, this bug takes nearly a yearly to be born; passes away after a maximal of one period day of life. The major goal of their childbirth is coupling [26]. A number of them are not even bothered to eat [3]. Mayflies intended for coupling normally consist a number of males of a small number of tons of separate ones, that are roughly around meters of 1 to 4 more than bottom for roughly about 1.5 to 2 hourly periods early in forenoon. Forming pattern and taking adult female, male perform a nuptial dance to a characteristically upward along with downward motion orientation. This mayfly algorithm uses this concept for optimization.

In 2020, Zervoudakis and Tsafarakis developed the mayfly (MA) algorithm inspired by the mating behaviour of mayflies. Upon hatching, mayflies become adults and, although the lifespan may be short, are the healthiest mayflies that can survive. The solutions to an optimization problem are given by the position of the mayfly in a search space [8].

2. Mayfly algorithm (MA)

To serve men and women, the algorithm initially generates two sets of instants. It is assumed that all adult mayflies, including the best-adapted ones, survive after hatching from eggs, regardless of how long they live. The condition of each mayfly in the search room represents a potential fix for this problem. Here's how the algorithm operates: First, a random set-in number of twos of the mayflies are generated, designating the male and female populations separately. It indicates that each mayfly is positioned at a random room in form of contender solution defined as vector of dimension $d$, $x=\left(x_1,\dots ,x_d\right)$ and its performances are examined on the default functional objective $f\left(x\right)$ [8]. The change in its place and direction of flight of each one mayfly is a robust communication of the individual and communal flight experience which defines the speed of a mayfly. Specifically, each one mayfly adapts its flight path to theirs best one’s own position ($pbest$) until now, in addition the best distinction achieved by any mayfly flock until now ($gbest$) [27].

3. Algorithm modelling

The implementation of the mayfly optimization algorithm could well be summarized with the following steps [28]:

(1) Initialization: The population of corresponding masculine and feminine are firstly denoted by $x=[x_1,\dots ,x_d]$ and $y=[y_1,\dots ,y_d]$ respectively. Also, the velocity is $v=[v_1,\dots ,v_d]$ [28].

(2) Male movements: The male mayflies are not able to move at a higher speed when they dance over a certain height of water. The functional objective $f\left(x\right)$, which marks the $f\left(gbest\right)$ decides the best overall position for the current iteration.

Assume that the $x_i^t$ is the present position of the male mayfly denoted by $i$ in the search room at step time period $t$, then its positional value is modified with the addition of $v_i^{t+1}$ which denotes the velocity to the ongoing present position. It is represented by:

Therefore, the velocity of the male mayfly $i$ is calculated as follows [28]:

where $x_{ij}^t$ denotes mayfly $i$ in dimension $j$ at the current iteration of the space time step $t$. $a_1$and $a_2$ are the positive constants that represent the attraction. $pbest$ is the best position the mayfly $i$ has visited. The best position for the mayfly can be found out at step time $t+1$ as following:

where $f:Rn\to R$ is function to minimize ${gbest}_j$ is the best global reach during the iteration $t$. In the Eq. (2) $r_p$ is the distance of $x_i$ from ${pbest}_i$ and $r_g$ is the distance of$x_i$from ${gbest}_j$. $r_p$ and $r_g$ are calculated by the following:

where $x_{ij}$ is the $j$th element of mayfly $i$, while $X_i$ is corresponding to ${pbest}_i$ and ${gbest}_j$.

The velocity of best mayfly is given by $v^{t+1}=v^t+d*r$, where $d$is the nuptial dance and $r$ is the random variable in [1, –1] [28].

(3) Female movements: Compared to male mayflies, female mayflies do not gather in swarm. Instead, they fly towards male in order to breed. Let us consider $y_i^t$ is the current position of the female mayfly $i$ in space time step $t$. The location change can be given as:

The female mayflies’ velocities can be given as $y$:

where $y$ is the female mayfly, $a_3$ is learning coefficient, $\beta$ is distance coefficient which is constant, $r_{mf}$ is the interspace between the masculine and feminine mayflies, $f_l$ denotes the arbitrary walk value when the feminine is not being allured by the masculine, $r$ is random value in [1, –1].

(4) Mating of mayflies: The mate process within the two mayflies is represented with the help of a cross operator as following: selection of one parent is from the masculine population along with the other from the feminine population is done. The selection of the parents is done on the same basis of how the feminine is attracted to the masculine. Specifically, the selection can be based on either of the randomly nature or on their fitness function. In second of the two, the finest feminine mates with the finest masculine, the second-finest feminine with the second-finest male, and so forth. Two offspring are generated as the result of the cross operator as follows:

where $L$ is random number, initial velocity of two offsprings set 0 [28].

(5) Updating: The solution which is not suitable is replaced by the best suitable solution and the above procedure is repeated till the stopping criteria is satisfied. The upcoming next position could be acquired with the addition of $v_i^{t+1}$ which denotes the velocity. It could be represented by:

(6) Calculating fitness and updating $pbest$ and $gbest$.

(7) If the stopping criterion is encountered, do the exit and output the result. Else do repeat the steps (5)-(7).

The Mayfly Optimization Algorithm flowchart is depicted in Fig. 1.

Fig. 1Flowchart of algorithm [27]

4. Pseudo code of algorithm

The following pseudo code summarises the fundamental phases of the Mayfly Algorithm (MA) which is shown in Fig. 2.

Fig. 2Pseudo code of Mayfly algorithm

5. Different variants of Mayfly

6. Application areas

Effectiveness of the applications of Mayfly Optimization Algorithm have promisingly been shown in the optimization problems in engineering sector. In the literature, improved solutions than the current existing solutions have been achieved by the MA. Some of the MA applications are:

Yuhu Liu et al. [33] provided the Initial Center Frequency Guided Filter (ICFGF) which is a novel method of selecting the resonant demodulation frequency band to detect the bearing failure [23]. Through the initial step, a statistical index for variance is applied to estimate the distribution of the spectrum of energy, which can compatibly decide the central frequency of the faulty pulse and effectively quash the intervention of the arbitrary pulse. Based on the central frequency to find out the excellent resonant frequency from the initial step, a modified mayfly algorithm (MMA) is applied as it has a speedy convergence rate.

Mohamed A.M Shaheen et al. [27] presented a new optimization method “Chaotic MayFly Optimization Algorithm” (CMOA) to obtain the parameters of proton exchange membrane fuel cells (PEMFC) [34]. A nonlinear problem has been worked out as the optimization problem into this case. As the metaheuristic optimization approaches are eminently affected by the initialization problem, a newer hybridization within disorderly mapping also the mayfly optimization algorithm is used for solving the problems in estimating variables of the project called PEMFC and also achieve improved outcomes. In order of finding the better solution for the functional objective in satisfying the prefixed situations, a CMOA is put in application. Also, the application from the CMOA results in a precise development of the PEMFC model.

Aihua Hu et al. [35] used the mayfly optimization algorithm in finding the optimal station of the semi-passive positioning system of the multiple base station in flight of the unmanned aerial vehicle (UAV) [36]. Comparing to the similar terminal layout methods alike the genetic algorithm (GA), artificial bee colony (ABC), particle swarm optimization (PSO), a simulation and optimization analysis of different numbers of base terminals is offered [3]. Along the simulation, the implementation of the four base terminals and five base terminals in numerous circumstances, the MA could accomplish superior distribution effects.

Ahmed Fathy et al. [37] used the mayfly optimization algorithm along with other optimization algorithms as the strategy for energy management based on the recent metaheuristic optimizing of the hybrid origin parasitism-predation algorithm employed includes photovoltaics, fuel cells, batteries, supercapacitors to refuel aerodyne into urgent situation state throughout the time of landing [38, 39]. One of the most important objectives is the minimization of consumption in hydrogen, so makes it possible to improve the duration of the power supply of the aerodyne in the instances for the reduction of the important source of power. Achieved outcomes certified the prevalence for planned technique accomplishing 95.34 % efficiency along with the minimal utilization of 15.7559 gm of $H$ where MA came second in order of efficiency 91.83 % and $H$ utilization of 16.0579 gm.

Lian Cheng et al. [40] used an improved metaheuristic optimization called as balanced mayfly algorithm (BMA) to optimize the EVCS configuration taking into accounting of the Real Power Loss Reduction Index (PLRI), the Reactive Power Loss Reduction Index (QLRI), the Voltage Profile Improvement Index (VPII) and the cost of preparatory evolution in order to obtain the minimal value for installing cost also providing a better high-quality of boundaries in the electrical network. BMA used two changes, including pairs of elites and chaos mechanism to solve these problems as this is possible [9].

Ali Farki et al. [41] developed a classic setup in correcting detections of COVID19 established on (FCOM) Opti mized Fuzzy Wired Media also an enhanced sort like the (ECN) Enhanced Capsule Network [42]. ECN has been bettered based on the mayfly optimization algorithm (MOA). This method then was carried out on COVID19 x-ray images of chest from generally ready datasets [43]. The outcomes were evaluated by using the comparison methods that included the MID, FOMPA, and 4SDT, also the outcomes displayed highest efficiency for Mayfly optimization approach.

Mohamed Abd Elaziz et al. [44] used the mayfly-based (MO) optimization algorithm to implement with the (RVFL) Random Vector Functional Link Network to amplify the precision of prediction. The suggested AI model which is hybrid has been instructed and tried out by utilizing the experimental statistics. For the suggested PVTCEHP system, outside experiments were performed which works along the two distinct coolants named as the water and air into the weather conditions of India, their outcomes were then collated with the assumed outcomes of RVFLMO also the prevailing RVFL [45, 46].

Mahmoud Elsisi et al. [37] used the optimization method of mayfly algorithm (MA) for finding excellent boundaries of (PID) the proportional integral derivative controller in order to search the excellent data set in the instructing of controller and trying out of the Wind Energy Conversion Systems (WECS) based on the adaptive neuro-fuzzy interface system (ANFIS) [48]. In order to give a demonstration of the benefits of the suggested method, this was collated with three distinct algorithms. The suggested method assures the highly minimized of stabilization period of time and the maximized exceed of 0.4497 and 0.7989 %, individually, collated to the alike three methods. The suggested ANFIS based on MA could balance WECS also manage the variations in speed of the wind, variations in demand of loading, lagging issues of time, also uncertainty in boundaries for the structure.

Zhenkun Liu et al. [49] proposed a whole thing prediction system for the integration of data break down applied science, selection for the sub model, a new version of the multi aim of the mayfly algorithm and several assumptions to best perform the speculations and variation into the data from speed of wind [50]. Established from the three trials and four evaluations, the whole suggested structure is confirmed to be effectual in order to achieve timely and periodic accurate and firm forecast performance, thus helping to plan and distribute the power grid.

Ratna Patil et al. [51] used a vector machine which is modified with the help of mayfly support for the implementation of accuracy in the stage of prediabetes. To evaluate the efficacy from the suggested model, benchmarking research had been taken on and collated to assumption replicas which is T2DM formed by similar analysts during over the past five years. The proposed model has been validated on the data collected by the communal health centres also (PIMA) reference data set accessible on the depository of (UCI). The researches discover this the revised MayflySVM have a significant advantage onto the metaheuristic algorithms in the communal as well as worldwide search abilities also achieved a maximal correctness of the test to be 94.5 % when compared to the (PIMA).

Eunice Oluwabunmi Owoola et al. [52] proposed a mayfly algorithm to resolve the pattern array synthesis in antennas [53]. In this task, he put in an application to the linear antenna arrays [54] to obtain better outcomes. He performed it as follows: he took two cases for the optimization of the antenna. The two cases were to preserve unfluctuating positioning and other was unfluctuating excitation. After comparing the results with the other alike algorithms, he performed a simulation of electromagnetism onto the Ansys software to analyse the behaviour of the MA for the optimization of beam pattern. The outcomes showcased that the suggested algorithm had the ability for optimizing the antenna arrays with excellent outcomes. He also suggested to improve the MA for the reduction of computing time period along with the boundary counts.

Syed Kumayl Raza Moosavi et al. [55] suggested a mayfly algorithm to classify the artificial neural network [56] structure. This method was used for finding a global minimal costing for the lesser number of repetitions along with greater accurate outcomes. He used two inputs of database available from the University of California Irvine which were the Banknote Authentication and other one as the Cryotherapy. He then collated the MA along with the two algorithms named as grey wolf optimization neural network and the particle swarm optimization neural network [56]. The suggested MA showcased the outcomes as efficacy of around 1 %-2 % while the training database and an efficacy of around 2 % while the tested database [57].

Juan Zhao et al. [58] used the Chebyshev mapping technique for the further improvement of the mayfly algorithm. For the experimentation purpose he used a disordered MA along the Chebyshev map for the improvement of the algorithm. A simulation method called Monte Carlo was also taken into account for the outcomes. The outcomes of the suggested technique showcased the algorithm improved with the help of the Chebyshev map.

Manickam Amudha et al. [59] formed a mayfly algorithm that associates the benefits of alike algorithms such as PSO, GA and so on. On collating the suggested algorithm along with other such seven approaches, it was found out that the MA was the dominant one [60]. They used the algorithm for the prediction of sea level rise resulting climatic change over the Atlantic as well as the Pacific in northern hemisphere. The outcomes showed that the suggested algorithm produced superior outcomes than the combination of (PSO) and the (DE).

Muhammad Hamza Zafar et al. [61] introduces a technique named as the (MPPT) which is formed by combining the (MPA) and the (MA) to overcome the flaws occurring in the photovoltaic structures such as the partial shading situations, [62]. They created a mathematical representation of the PV structure for obtaining the outcomes. He connected the PV structure with a double diode and a loading and put into application the suggested technique [63]. They then collated the suggested algorithm with the alike algorithms namely (PSO), (GWO), (CS) [54], [55]. They found out that the suggested technique showcased efficacy of up to 99.99 %, speedy tracking of nearly 20 %-50 % and a reduction in the oscillation of 97 %.

Kingsuk Majumdar et al. [65] introduces the mayfly algorithm for obtaining superior outcomes for the scheduling in hydro thermal solar wind application. They combined the suggested algorithm with a number of schedulers say IEEE 9, IEEE 39, IEEE 118 buses for improving the efficacy. Later on, performing the experimental work it was found out that the suggested method showed superior outcomes by 19 % and an enhancement of about 56 % in (ATC) [16].

Mohamed A. M. Shaheen et al. [46] tried to improve modeling and simulation results by accurately extracting PEMFC models’ design factors. Under varied settings, the CMOA is used to minimize the sum of square errors between the measured and terminal voltages of fuel cells. The approach highlights the nonlinearity of the PEMFC model by predicting unknown design factors well and exhibiting great sensitivity to certain parameters. The simulation results demonstrate the convergence of the optimization methodologies and the comparison between estimated and empirically measured voltages for several PEMFC types, including the Ballard Mark V fuel cell.

7. Conclusions

The presented paper gives a description on optimization algorithm known as the Mayfly (MA) algorithm. The paper describes its operating principle and its various fields of application. The described method which is based on population unites the main benefits of current algorithms also provoked on behavioural by mayflies which are adult, incorporating the methods of crossing, evolution, flock aggregation, method of proposed walking in arbitrary form and dancing of nuptial, which improve exploration. Throughout this study, has been established that with the help of two distinct equations into every population (masculine and feminine), survey is enhanced.

Even though the described process does not look to be speedy when compared with the rest of the processes from whose outcomes are obtained, has a greater likeliness for detecting an overall excellence. The behaviour of combination from suggested algorithm looks likewise remarkable, because it most identically reaches excellent global solutions through first iteration carried out. The mayfly algorithm outcomes look sufficient good for problems that are discrete and for optimization with numerous objectives.