A conceptual comparison of six nature-inspired metaheuristic algorithms in process optimization

  Citation: Rajendran, S.; N., G.; ˇ Cep, R.; R. C., N.; Pal, S.; Kalita, K. A Conceptual Comparison of Six Nature-Inspired Metaheuristic Algorithms in Process Optimization. Processes 2022,10, 197. https:// doi.org/10.3390/pr10020197 Academic Editor: Blaž Likozar Received: 23 December 2021 Accepted: 13 January 2022 Published: 20 January 2022 Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. Copyright: © 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/). processes Article A Conceptual Comparison of Six Nature-Inspired Metaheuristic Algorithms in Process Optimization Shankar Rajendran 1, Ganesh N. 2,* , Robert ˇ Cep 3, Narayanan R. C. 4, Subham Pal 5 and Kanak Kalita 6,* 1Department of Computer Science and Engineering, Koneru Lakshmaiah Education Foundation, Guntur 522 502, India; [email protected] 2Department of Computer Science and Engineering, Vel Tech Multi Tech Dr. Rangarajan Dr. Sakunthala Engineering College, Chennai 600 062, India 3Department of Machining, Assembly and Engineering Metrology, Faculty of Mechanical Engineering, VSB-Technical University of Ostrava, 17. Listopadu 2172/15, 708 00 Ostrava, Czech Republic; [email protected] 4Department of Computer Science and Engineering, Sona College of Technology, Salem 636 005, India; [email protected] 5Department of Aerospace Engineering and Applied Mechanics, Indian Institute of Engineering Science and Technology, Howrah 711 103, India; [email protected] 6Department of Mechanical Engineering, Vel Tech Rangarajan Dr. Sagunthala R&D Institute of Science and Technology, Avadi 600 062, India *Correspondence: [email protected] (G.N.); [email protected] (K.K.) Abstract: In recent years, several high-performance nature-inspired metaheuristic algorithms have been proposed. It is important to study and compare the convergence, computational burden and statistical significance of these metaheuristics to aid future developments. This study focuses on six recent metaheuristics, namely, ant lion optimization (ALO), arithmetic optimization algorithm (AOA), dragonfly algorithm (DA), grey wolf optimizer (GWO), salp swarm algorithm (SSA) and whale optimization algorithm (WOA). Optimization of an industrial machining application is tackled in this paper. The optimal machining parameters (peak current, duty factor, wire tension and water pressure) of WEDM are predicted using the six aforementioned metaheuristics. The objective functions of the optimization study are to maximize the material removal rate (MRR) and minimize the wear ratio (WR) and surface roughness (SR). All of the current algorithms have been seen to surpass existing results, thereby indicating their superiority over conventional optimization algorithms. Keywords: optimization; non-traditional algorithms; process optimization; process parameters; algorithms 1. Introduction Nowadays, non-traditional machining processes are mostly used in electronic and aerospace industries, where machining with high accuracy is required. Non-traditional machining processes are defined as processes that remove material by various techniques like mechanical, thermal, electrical, chemical energy or combinations of these energies. Unlike traditional machining processes, non-traditional machining processes do not require any sharp cutting tools. Generally, materials that are difficult to machine using conventional machining processes are machined using non-traditional machining processes. Electrical discharge machining (EDM) is one of the most widely used non-traditional machining processes. In EDM, the material is removed by the thermoelectric process. A series of discrete electrical sparks between the workpiece and the electrode is generated to erode the undesired materials from the workpiece. The Wire Electrical Discharge Machining (WEDM) process is widely used to pattern tool steel for die making. WEDM is mostly used to make dies and punches in the aerospace and automotive industries [ 1 ]. In this process, Processes 2022,10, 197. https://doi.org/10.3390/pr10020197 https://www.mdpi.com/journal/processes Processes 2022,10, 197 2 of 20 a slowly moving wire travels along a pre-defined path and removes material from the workpiece. Wires in WEDM are mostly made with brass, copper, tungsten or molybdenum. Sometimes zincor brass-coated wires are also used. The wire in WEDM should have high tensile strength and good electrical conductivity. The surface roughness (SR) and material removal rate (MRR) of the machined surface by WEDM depends on different machining parameters, such as peak current (A), duty factor, wire tension (N), and water pressure (MPa). Thus, optimizing such process parameters is very important to maximize or minimize the response parameters and reduce machining time. In the last few decades, many researchers have proposed several optimization techniques, such as genetic algorithm (GA) [ 2 ], particle swarm optimization (PSO) [ 3 ], ant colony optimization (ACO), artificial bee colony (ABC) [ 4 ], cuckoo search optimization (CO), grey wolf optimizer (GWO) [ 5 ], arithmetic optimization algorithm (AOA) [ 6 ], salp swarm algorithm (SSA) [ 7 ], ant lion optimization (ALO) [ 8 ], whale optimization algorithm (WOA) [ 9 ], multi-verse optimization (MVO), bat algorithm [ 10 ], dragonfly algorithm (DA) [ 11 ]. These optimization techniques are called non-traditional optimization techniques or metaheuristic techniques. These optimizers have been used in many fields, like the production, project scheduling, management field, manufacturing field and design field. Hewidy et al. [ 12 ] used an RSM-based metamodel of the WEDM process to optimize the process parameters. The objective of their work was to find the maximum MRR and minimize SR and wear ratio (WR). The experiment was conducted on Inconel 601. Zhang et al. [ 13 ] performed optimization of process parameters for machining SKD11. They used a back-propagation neural network in link with a genetic algorithm (BPNN-GA) to maximize MRR and minimize SR. Shihab [ 14 ] examined the optimal conditions for machining parameters using a Box-Behnken design (BBD). Chaudhary et al. [ 15 ] optimized WEDM process parameters for machining of ASSAB’88 tool steel with RSM. The main objective was to calculate the optimal condition of process parameters to maximize MRR and minimize SR. Mahapatra and Patnaik [ 16 ] examined optimum process parameters of WEDM using the Taguchi method. They considered discharge current, pulse duration, pulse frequency, wire-speed, wire tension and dielectric flow rate as process parameters and MRR,SR and cutting width (kerf) response parameters. Nayak and Mahapatra [ 17 ] performed optimization of machining parameters of WEDM process parameters for a deep cryo-treated Inconel 718 material. Mukherjee et al. [ 18 ] performed a comparative study of six different non-conventional optimization techniques (i.e., genetic algorithm (GA), particle swarm optimization (PSO), sheep flock algorithm (SF), ant colony optimization (ACO), artificial bee colony (ABC) and biogeography-based optimization (BBO)). The objective of their study was to maximize the MRR and minimize the SR and WR values for the WEDM process. A literature survey revealed that many researchers have used varied techniques to optimize the process parameters of machining processes. However, most researchers have been limited to desirability analyses through RSM and the use of traditional metaheuristics like GA and PSO. Furthermore, among the metaheuristic techniques, no comparative study has been carried out in comparison of recently-proposed optimization techniques. There is very little literature available where recent nature-inspired optimization techniques are used to optimize the process parameters of machining processes. Thus, in this paper, a comparison of six newly proposed metaheuristic techniques, namely, ant lion optimization (ALO), arithmetic optimization algorithm (AOA), dragonfly algorithm (DA), grey wolf optimizer (GWO), salp swarm algorithm (SSA) and whale optimization algorithm (WOA), is made, and the results are compared with previously published results. The rest of the article is presented as follows: in the second section, the theoretical background of the six metaheuristic algorithms with their pseudo code are shown; in the third section, the problem description is shown; in the fourth section, the results and discussions are discussed, and at last, the conclusions are made. Processes 2022,10, 197 3 of 20 2. Nature-Inspired Metaheuristics 2.1. Ant Lion Optimization Ant lion optimization (ALO) is a newly proposed nature-inspired optimization technique proposed by Mirjalili [ 19 ]. The ant lion algorithm is inspired by the hunting mechanism of ant lions in nature [ 8 ]. The life cycle of the ant lion involves two stages: one is the larvae stage with the ant lion hunting prey (ants), while the other stage is the adult stage with the reproduction of the ant lion [ 20 ]. In nature, an ant lion digs a trap with a cone shape. The size of the trap defines the hunger level of an ant lion. A big cone trap represents an ant lion that is hungrier than one with a small cone trap. Generally, a big hole is dug by an elite ant lion, which has a better probability of catching prey. The ant lion hides at the bottom of the trap waiting for an ant, which moves randomly around the trap to fall. Once the ant lion realizes that there is an ant in the trap, it catches it. Once an insect is caught, the ant lions pull it under the cone trap and throw the sand towards the outer edges of the hole, using its big jaw so that the prey cannot escape. Then, the ant lion consumes the prey and digs another hole for the next hunt [ 21 ]. The fitness of ant lions and the quality of the traps improve in every hunt. Considering this hunting mechanism, the ALO algorithm follows the main processes. Such as random walks of ants, trapping in an ant lion’s hole, establishing a trap, sliding ants towards an ant lion, catching prey and reconstructing the hole, and elitism [22]. Random walks of ants: X(k)=[0, cs(2r(k1)−1),cs(2r(k2)−1), . . . . . . ·,cs(2r(km)−1)] (1) where cs is the cumulative sum, m is the maximum number of iterations, k is the stop of random walk (iteration), r(k)is a stochastic function and is given by: r(k)=1i f α>0.5 0i f α≤0.5 ,C∈[0, 1](2) where αis the random number generated with uniform distribution. To make the random movement inside the search space. The following equation is used [23]: Xk t=Xk t−atdk t−ck t (bt−at)+ct(3) where at and bt indicate the minimum and maximum of the random movement of the t th variables, respectively; ck t and dk t are the minimum and maximum of the t th variable at k th iteration, respectively. Trapping in an ant lion’s hole: The following equation is used to mathematically model the random walks of ants affected by the ant lion’s traps. ck t=Antlionk t+ckdk t=Antlionk t+dk(4) where ck and dk are the minimum and maximum of all variables at the k th iteration, ck t and dk t are the minimum and maximum of all variables for the t th ant and Antlionk t represents the position of the selected tth ant lion and kth iteration. Establishing trap: The largest trap belongs to the fittest ant lions. Thus, catching an ant by an ant lion is proportional to the fitness of that ant lion (i.e., the ant lion with the higher fitness has a higher chance to catch an ant) and the fittest ant lion is selected by applying the roulette wheel selection. Sliding ants towards an ant lion: Processes 2022,10, 197 4 of 20 To model the sliding ants towards ant lions, the scope of the random movement should be decreased adaptively. ck=ck Idk=dk I(5) where Iis a ratio that controls the exploration/exploitation rate in the ALO algorithm by limiting the random walk range of the ants and prey. The parameter I in the above equation is equal to 10 wk K , k indicates the current iteration, K is the iteration max and w represents a constant and is related to the accuracy of the exploration. The wvalues are presented as: w=           2i f k >0.1K 3i f k >0.5K 4i f k >0.75K 5i f k >0.9K 6i f k >0.95K            (6) Catching prey and reconstructing the hole: If an ant reaches the bottom of the pit, it will be caught and consumed by the ant lions. After this, the ant lions wait to catch new prey by updating their position. The following equation can express the above-mentioned process. Antlionk j=Antk ii f f Antk i>fAntlionk j(7) where krepresents the current iteration, Antlionk j and Antk i represent the position of the j th ant lion and the ith ant at the kth iteration. Elitism: As a key trait of evolutionary algorithms, elitism is utilized to store the best solutions during the optimization process. In this algorithm, the best ant lion obtained is regarded as an elite, which should have an impact on the all the ants in every stage. Thus, affected by the roulette wheel and the elite at the same time, every ant moves towards a selected ant lion, which can be expressed as follows: Antk i=Rk A+Rk E 2(8) where RA and RE indicate the random moves towards the ant lion selected by the roulette wheel and the elite, respectively. The entire evolutionary process of ant lion optimization is given in Algorithm 1. Algorithm 1 Pseudocode of Ant Lion Optimization Inputs:Population size (N), tmax (Max. iteration number) Initialize the first population of Nant and ant lions randomly Calculate the fitness of ant and ant lion Find the best ant lions and assume it is the elite. while (t < tmax) for every ant (i = 1, 2, 3 . . . . . . , N) Find an ant lion using Roulette wheel Update c and d using Equation (5) Create a random walk using Equation (1) and normalized walk by Equation (3) Update the position of ant by Equation (8) end Calculate fitness values for all updated ants Replace ant lion by ant if f(ant) > f(antlion) Replace the elite with the fittest solution t=t+1 end Return the elite solution as best solution Processes 2022,10, 197 5 of 20 2.2. Arithmetic Optimization Algorithm Arithmetic optimization algorithm (AOA) is a population-based meta-heuristic capable of solving optimization problems without calculating their derivatives [ 6 ]. The AOA algorithm follows basic arithmetic operators in math (i.e., multiplication ( × ), division ( ÷ ), subtraction ( − ), and addition (+)). Metaheuristic optimization techniques have two main parts, exploration and exploitation. In the exploration process, the search agents search for the optimal values around the search space so that the solution does not get trapped in local optima. In the exploitation phase, the optimizer takes advantage of exploration and finds global optimal values among the local optimal values. In this AOA optimization technique, the exploration and exploitation process is achieved by those arithmetic operations. In AOA, the optimization operation begins with randomly generated solutions known as candidate solutions ( × ), as shown in Equation (9). The best candidate solution in each iteration is known as an optimal or near-optimal solution. X=           x1,1 · · · · · · x1,jx1,n−1x1,n x2,1 · · · · · · x2,j· · · x2,n · · · · · · · · · · · · · · · · · · . . .. . .. . .. . .. . .. . . xN−1,1 · · · · · · xN−1,j· · · xN−1,n xN,1 · · · · · · xN,jxN,n−1xN,n           (9) Before it gets started, it selects the search phase (i.e., exploration and exploitation) by calculating a coefficient, math optimizer accelerated (MOA) function using Equation (10). MOA(t)=Min +t×Max – Min tmax (10) where, MOA(t) is the function value at the t th iteration, t is the current iteration, tmax is the maximum iteration, and Max and Min are the maximum and minimum values of the accelerated function, respectively. Exploration phase: The search area is explored randomly in several regions to find a better solution in the exploration phase. Two main search strategies, such as the division (D) search strategy and the multiplication (M) search strategy, as given in Equation (10). If the random number (r1) is greater than the math optimizer accelerated (MOA) function (r1 > MOA), then this exploration phase will take place. The first operator (D) of Equation (11) is applicable when r2 < 0.5; at that time, the second operator (M) is neglected. The multiplication search strategy is done after the complication of the division search strategy [6]. xi,j(t+1) =best xj÷(MOP +e)×UBj−LBj×µ+LBj,r2<0.5 best xj×MOP ×UBj−LBj×µ+LBj, otherwise (11) where xi,j(t+1) denotes the j th position of the i th solution at the next iteration and best (xj) is the j th position in the best-obtained solution so far, e is a small integer number, UBj and LBj denote the upper limit and lower limit value of the j th position, respectively. µ is a control parameter used to adjust the search process. A µof 0.5 is taken in this algorithm. MOP(k)=1−t1/α tmax 1/α(12) where the math optimizer probability (MOP) is a probability value in the range between 0 and 1. It is calculated using Equation (12). MOP(k) denotes the probability value at the k th iteration, t represents the current iteration and ( tmax ) denotes the maximum number of Processes 2022,10, 197 6 of 20 iterations. α is a sensitivity parameter and is responsible for exploiting accuracy over the iterations. An αvalue of 5 is taken in the present algorithm. Exploitation phase: The exploitation phase takes place when the randomly generated number (r1) is less than the math optimizer accelerated (MOA) function (r1 > MOA). In the exploitation of AOA optimization, the better solution search is based on two main strategies: the subtraction (S) search strategy and the addition (A) search strategy. The subtraction (S) search strategy and addition (A) search strategy are modelled as follows: xi,j(t+1)=best xj−MOP ×UBj−LBj×µ+LBj,r3<0.5 best xj+MOP ×UBj−LBj×µ+LBj, otherwise (13) The subtraction (S) search strategy takes place when r3 < 0.5 at that time, the other operator (A) is neglected. The procedures in this phase are similar to exploration, but operator S and A is used instead of operator D and M. The pseudocode of the AOA algorithm is given below in Algorithm 2. Algorithm 2 Pseudocode of Arithmetic Optimization Algorithm Inputs: Population size (N), tmax (Max. iteration number) Initialize the arithmetic optimization parameters α,µ. Initialize the solutions positions randomly. while(t < tmax) Evaluate the fitness value of the given solution Xbest = Select the best solution obtained so far. Update the MOA using Equation (10) Update the MOP using Equation (12) for i =1to Solutions for j =1to Positions Generate a random value between [0,1] for r1, r2, r3. ifr1> MOA Exploration phase ifr2> 0.5 Apply the division math operator (D) Update the ith solution position using the first equation of Equation (11) else Apply the multiplication math operator (M) Update the ith solution position using the second equation of Equation (11) end else Exploitation phase ifr3> 0.5 Apply the subtraction math operator (S) Update the ith solution position using the first equation of Equation (13) Else Apply the addition math operator (A) Update the ith solution position using the second equation of Equation (13) End end end end t = t+1 end Return the optimal solutions Processes 2022,10, 197 7 of 20 2.3. Dragonfly Algorithm The dragonfly algorithm (DA) was proposed by Mirjalili [ 24 ] in 2015. The dragonfly algorithm is inspired by the swarming behaviour of dragonflies—hunting and migration. The hunting process is known as a static swarm, and migration is known as a dynamic swarm. In a static swarm, dragonflies make small groups and move back and forth over a small area to hunt other flying prey, such as butterflies and mosquitoes. Local movements and abrupt changes in the flying path are the main characteristics of a static swarm. However, in dynamic swarms, many dragonflies swarm in one direction over long distances for migration. These two swarming behaviours are very similar to the exploration and exploitation techniques in metaheuristics. The main objective of any swarm is survival, so all individuals should be attracted towards food sources and distracting outward enemies. The mathematical model of swarming behaviour is given as follows [ 24 ]. The separation is formulated as follows: Si=− N ∑ j=1 X−Xj(14) where Xis the position of the current individual, Xj shows the position of the j th neighbouring individual and Nis the number of neighbouring individuals. Alignment is formulated as follows: Ai=∑N j=1Vj N(15) here, Vjis the velocity of the ith neighbouring individual. The cohesion is formulated as follows: Ci=∑N j=1Xj N−X(16) Attraction towards a food source is formulated as follows: Fi=X+−X(17) where X+is the position of the food source. Distraction outwards towards an enemy is formulated as follows: Ei=X−+X(18) where X−is the position of the enemy. These five techniques are combined to represent the behaviour of dragonflies mathematically. To update the position of artificial dragonflies in a search space and simulate their movements, two vectors are considered: step ( ∆X ) and position ( X ). The step vector shows the direction of the movement of the dragonflies. The mathematical model of the step vector is shown as follows: ∆Xt+1=(sSi+aAi+cCi+f Fi+eEi)+w∆Xt(19) where s , a , c , f , e and w are the separation weight, alignment weight, cohesion weight, food factor, enemy factor and inertia weight, respectively. Si , Ai , Ci , Fi and Ei are the separation, alignment, cohesion, food source and position of an enemy of the ith individual. ∆Xt is the step vector of the tth iteration. The position vectors are calculated as follows: Xt+1=Xt+∆Xt+1(20) Processes 2022,10, 197 8 of 20 where Xt+1 is the position vector for the (t+1)th iteration, ∆Xt+1 is the step vector for the (t+1)th iteration, Xt is the position vector of the tth iteration. The dragonfly algorithm is realized by using the pseudocode in Algorithm 3. Algorithm 3 Pseudocode of Dragonfly Algorithm Inputs: Population size (N), tmax (Max. iteration number) Initialize the dragonflies population Xi(i = 1, 2, . . . ., n) Initialize the step vectors ∆Xi(i = 1, 2, . . . ., n) while (t < tmax) Evaluate the fitness value of all dragonflies. Update the food source and enemy. Update w,s,a,c,f and e. Calculate S, A, C, F and Eusing Equations (14)–(18). Update neighbouring radius. if a dragonfly has at least one neighbouring dragonfly Update velocity vector using Equation (19). Update position vector using Equation (20). else Update position vector using Equation (20). end Check and correct the new positions based on the boundaries of variables. t = t+1 end 2.4. Grey Wolf Optimizer According to Mirjalili et al. [ 25 ], grey wolves live together and hunt in a group. Therefore, in the GWO algorithm, the social leadership and hunting mechanisms of grey wolves are mimicked. In all population-based optimization techniques, good exploration and exploitation capabilities are required. The computational time of any metaheuristic depends on how much time it spends in exploration and how fast it finds the global optimal that is exploitation. Therefore, the right balance of exploration and exploitation is essential for the fasted convergence to the global optimal. In the original GWO algorithm, half of the iterations are set for exploration, and the other half of the iterations are set for exploitation. In GWO, the solutions are divided into four groups, the fittest solutions are named as alpha ( α ), the second-best solutions are called beta ( β ) and the third-best solutions are quoted as delta ( δ ). All remaining solutions are termed as omega ( ω ). In the hierarchy of GWO, the omega wolf is controlled by the alpha, beta and delta wolves. The hunting technique of grey wolves is divided mainly into five parts [25]. Social hierarchy. Tracking, chasing and approaching the prey. Pursuing, encircling and harassing the prey until it stops moving. Attacking the prey (Exploration). Searching for prey (Exploitation). The mathematical model of encircling the prey is written as: → D=|→ C·→ Wp(t)−→ W(t)|(21) → W(t+1)=→ Wp(t)−→ A·→ D(22) where trepresents the current iteration, → Wp(t) represents the position vector of prey at the t th iteration, → W(t) represents the position vector of a grey wolf at the t th iteration and Processes 2022,10, 197 9 of 20 → W(t+1) is the updated position of the grey wolf. → A and → C are the coefficient vectors. These coefficient vectors are expressed as follows: → A=2→ a·→ r1−→ a(23) → C=2→ r2(24) where → r1 and → r2 are random vectors in [0,1]. Values of the → a vector linearly decreased from 2 to 0 with the progress in iteration, and it is given as: → a(t)=2−2t tmax (25) where tis the current iteration and tmax represents the maximum iteration number. After the encircling operation, a grey wolf starts to hunt the prey (i.e., best solutions). The best candidates (i.e., α , β , δ -wolves) have better information about the location of the prey. Other candidates ( ω -wolves) change their positions to the position of the three best candidates. The hunting mechanisms of the grey wolf are mathematically represented as follows: → Dα=|→ C1·→ Wα−→ W|→ Dβ=|→ C2·→ Wβ−→ W|→ Dδ=|→ C3·→ Wδ−→ W|(26) → W1=→ Wα−→ A1·( → Dα)→ Dβ=|→ C2·→ Wβ−→ W|→ W3=→ Wδ−→ A3·( → Dδ)(27) → W(t+1)= → W1(t)+→ W2(t)+→ W3(t) 3(28) The grey wolves end the hunting by attacking at last when the prey is not moving. To represent this mathematically, we decreased the value of → a . The fluctuation range of → A is also decreased by → a . → A is a random value in the interval [ − a,a]. When random values of → A are in [−1,1], the next position of the search agent can be any position between its current position and the position of the prey. The values of |A|< 1 forced the wolves to attack the prey and |A|> 1 forced the grey wolves to diverge from the prey to hopefully find fitter prey. After the hunting of prey, the grey wolves search for the prey in the next iteration. This process will continue until the termination criterion is satisfied. The pseudocode of the GWO algorithm is given in Algorithm 4. Algorithm 4 Pseudocode of Grey Wolf Optimization Inputs:Fitness function, lower bound, upper bounds, number of search agents, maximum iteration. Initialize a random population of grey wolves (Wi) (i=1,2,3. . . ., n). Initialize A, C and a; set t=0. Calculate the fitness values of all search agents. Select α,β,δ-wolves. (Considering fittest solution as α-wolves, second-best solution as β–wolves and third-best solution as δ-wolves). while(t<tmax) foreach search agent Update the position of the current search agent using Equation (28) end Update A, C and a. Calculate the fitness values of all search agents. Update the position of α,β,δ-wolves. t=t+1 end while(t=tmax) report the best individual. End. Processes 2022,10, 197 16 of 20 Table 2. Comparison of current results with solutions from the literature for MRR optimization. Algorithm x1x2x3x4Optimum % Improvement Hewidy et al. [12] 6 0.5 7 0.5 6.57 - ALO 3 0.3288 9 0.418 8.765 33.41% AOA 3.16 0.4349 8.835 0.477 8.188 24.63% DA 3 0.3288 9 0.417 8.765 33.41% GWO 3 0.3288 9 0.417 8.765 33.41% SSA 3 0.3288 9 0.417 8.765 33.41% WOA 3 0.3288 9 0.417 8.765 33.41% Figure 3shows the convergence of the algorithms while optimizing the WR. It was observed that except for AOA, all other algorithms have a similar convergence trend. Table 3contains the statistical summary of the 10 independent trials of WR optimization. Like MRR optimization, in the case of WR, AOA was unable to locate the best-known optima. All the other five algorithms reported the best-known optima to be 1.216, an improvement of approximately 25% over the AOA results. As seen from Equation (42), the WR model was linear with only two process parameters. This may be the reason behind the 100% success rate reported by all the other five algorithms except AOA. Processes 2022, 10, x FOR PEER REVIEW 16 of 20 a lower number of function evaluations with lower values, indicating that they had more rapid convergence towards the optimal zone. The optimal process parameters and the MRR, as reported by the various metaheuristics, are presented in Table 2. With respect to the Hewidy et al. [12] solutions, AOA showed a 24.63% improvement. All the other algorithms reported a 33.41% improvement over the existing solutions in the literature. Table 2. Comparison of current results with solutions from the literature for MRR optimization. Algorithm 𝒙𝟏 𝒙𝟐 𝒙𝟑 𝒙𝟒 Optimum % Improvement Hewidy et al. [12] 6 0.5 7 0.5 6.57 - ALO 3 0.3288 9 0.418 8.765 33.41% AOA 3.16 0.4349 8.835 0.477 8.188 24.63% DA 3 0.3288 9 0.417 8.765 33.41% GWO 3 0.3288 9 0.417 8.765 33.41% SSA 3 0.3288 9 0.417 8.765 33.41% WOA 3 0.3288 9 0.417 8.765 33.41% Figure 3 shows the convergence of the algorithms while optimizing the WR. It was observed that except for AOA, all other algorithms have a similar convergence trend. Table 3 contains the statistical summary of the 10 independent trials of WR optimization. Like MRR optimization, in the case of WR, AOA was unable to locate the best-known optima. All the other five algorithms reported the best-known optima to be 1.216, an improvement of approximately 25% over the AOA results. As seen from Equation (42), the WR model was linear with only two process parameters. This may be the reason behind the 100% success rate reported by all the other five algorithms except AOA. Figure 3. Convergence of optimum WR with respect to iterations. Table 3. Statistical summary of 10 trials in optimizing WR. Algorithm Mean Standard Deviation Median Best Worst Success Rate ALO 1.216 0 1.216 1.216 1.216 100% AOA 1.433 0.108 1.423 1.268 1.611 0% DA 1.216 0 1.216 1.216 1.216 100% GWO 1.216 0 1.216 1.216 1.216 100% SSA 1.216 0 1.216 1.216 1.216 100% WOA 1.216 0 1.216 1.216 1.216 100% Figure 3. Convergence of optimum WR with respect to iterations. Table 3. Statistical summary of 10 trials in optimizing WR. Algorithm Mean Standard Deviation Median Best Worst Success Rate ALO 1.216 0 1.216 1.216 1.216 100% AOA 1.433 0.108 1.423 1.268 1.611 0% DA 1.216 0 1.216 1.216 1.216 100% GWO 1.216 0 1.216 1.216 1.216 100% SSA 1.216 0 1.216 1.216 1.216 100% WOA 1.216 0 1.216 1.216 1.216 100% Figure 4shows the spread of the total function evaluations during a typical trial while optimizing WR. The overall pattern of the spread and distribution of the function evaluations in Figure 4is very similar to that in Figure 2. This indicates that the algorithms are unaffected by whether the optimization problem is a minimization or maximization type. GWO was seen to have the highest median function evaluation value among the algorithms. Moreover, the mean function evaluation value for GWO was very close to its median. This indicates that a very high percentage of GWO’s Processes 2022,10, 197 17 of 20 evaluated functions were in the optimal zone. This is generally preferred as it may be indicative of a high convergence rate. Processes 2022, 10, x FOR PEER REVIEW 17 of 20 Figure 4 shows the spread of the total function evaluations during a typical trial while optimizing WR. The overall pattern of the spread and distribution of the function evaluations in Figure 4 is very similar to that in Figure 2. This indicates that the algorithms are unaffected by whether the optimization problem is a minimization or maximization type. GWO was seen to have the highest median function evaluation value among the algorithms. Moreover, the mean function evaluation value for GWO was very close to its median. This indicates that a very high percentage of GWO’s evaluated functions were in the optimal zone. This is generally preferred as it may be indicative of a high convergence rate. Figure 4. Box plots denoting the spread of all the function evaluations during WR optimization in a typical independent run. The horizontal blue line (and numeric value) and the red line indicate the median and mean of all the function evaluations. The optimum process parameters for minimizing WR are presented in Table 4. With respect to Hewidy et al. [12], the current results are 70% better for AOA and 71.32% better for ALO, DA, GWO, SSA and WOA. Similarly, the optimum process parameters for minimizing SR are presented in Table 5. In this case, the current ALO and AOA results are observed to be 47.68% better than Hewidy et al. [12]. On the other hand, DA and GWO results were 47.82% better, while SSA and WOA were 48.05% better than Hewidy et al. [12]. However, it is important to point out that all six algorithms reported varied in optimized values of the process parameters. This indicates that the objective function search space was, perhaps, multimodal. Therefore, it is worth mentioning that Hewidy et al.[12] used an RSM-based model to calculate the optimum values. The optimum values of responses obtained by Hewidy et al.[12] were MRR = 6.57 mm3/min, WR = 4.24 and SR = 2.20µm. The best-known optimum values obtained in this study were MRR = 8.765 mm3/min, WR = 1.216 and SR = 1.143µm. Table 4. Comparison of current results with solutions from literature for WR optimization. Algorithm 𝒙𝟏 𝒙𝟒 Optimum % Improvement Hewidy et al. [12] 7 0.7 4.24 - ALO 3 0.3 1.216 71.32% AOA 3 0.321 1.268 70.09% DA 3 0.3 1.216 71.32% GWO 3 0.3 1.216 71.32% SSA 3 0.3 1.216 71.32% WOA 3 0.3 1.216 71.32% Figure 4. Box plots denoting the spread of all the function evaluations during WR optimization in a typical independent run. The horizontal blue line (and numeric value) and the red line indicate the median and mean of all the function evaluations. The optimum process parameters for minimizing WR are presented in Table 4. With respect to Hewidy et al. [ 12 ], the current results are 70% better for AOA and 71.32% better for ALO, DA, GWO, SSA and WOA. Similarly, the optimum process parameters for minimizing SR are presented in Table 5. In this case, the current ALO and AOA results are observed to be 47.68% better than Hewidy et al. [ 12 ]. On the other hand, DA and GWO results were 47.82% better, while SSA and WOA were 48.05% better than Hewidy et al. [ 12 ]. However, it is important to point out that all six algorithms reported varied in optimized values of the process parameters. This indicates that the objective function search space was, perhaps, multimodal. Therefore, it is worth mentioning that Hewidy et al. [ 12 ] used an RSM-based model to calculate the optimum values. The optimum values of responses obtained by Hewidy et al. [ 12 ] were MRR = 6.57 mm 3 /min, WR = 4.24 and SR = 2.20 µ m. The best-known optimum values obtained in this study were MRR = 8.765 mm 3 /min, WR = 1.216 and SR = 1.143 µm. Table 4. Comparison of current results with solutions from literature for WR optimization. Algorithm x1x4Optimum % Improvement Hewidy et al. [12] 7 0.7 4.24 - ALO 3 0.3 1.216 71.32% AOA 3 0.321 1.268 70.09% DA 3 0.3 1.216 71.32% GWO 3 0.3 1.216 71.32% SSA 3 0.3 1.216 71.32% WOA 3 0.3 1.216 71.32% Though the same number of function evaluations (i.e., 30 × 100 = 3000) were carried out by each algorithm, some algorithms were expected to be faster than others. Thus, the CPU time of each algorithm was noted and averaged for 10 independent trials for each response. From Figure 5, it is observed that ALO, DA and WOA were the three most expensive algorithms, whereas SSA, GWO and AOA were the three least expensive. Nevertheless, the standard deviation of computational time for WOA was very high, indicating that in some instances, it may take a very short computation time. Processes 2022,10, 197 18 of 20 Perhaps this is because optimizing WR was very fast in optimizing the linear problem with only two variables. Table 5. Comparison of current results with solutions from the literature for SR optimization. Algorithm x1x2x3x4Optimum % Improvement Hewidy et al. [12] 5 0.75 9 0.5 2.2 - ALO 4 0.623 8.2 0.62 1.151 47.68% AOA 3.8 0.605 7.2 0.5 1.151 47.68% DA 3.4 0.623 7.1 0.38 1.148 47.82% GWO 4.6 0.678 7.3 0.54 1.148 47.82% SSA 4.2 0.623 8.5 0.68 1.143 48.05% WOA 5.2 0.568 9 0.6 1.143 48.05% Processes 2022, 10, x FOR PEER REVIEW 18 of 20 Table 5. Comparison of current results with solutions from the literature for SR optimization. Algorithm 𝒙𝟏 𝒙𝟐 𝒙𝟑 𝒙𝟒 Optimum % Improvement Hewidy et al. [12] 5 0.75 9 0.5 2.2 - ALO 4 0.623 8.2 0.62 1.151 47.68% AOA 3.8 0.605 7.2 0.5 1.151 47.68% DA 3.4 0.623 7.1 0.38 1.148 47.82% GWO 4.6 0.678 7.3 0.54 1.148 47.82% SSA 4.2 0.623 8.5 0.68 1.143 48.05% WOA 5.2 0.568 9 0.6 1.143 48.05% Though the same number of function evaluations (i.e., 30 × 100 = 3000) were carried out by each algorithm, some algorithms were expected to be faster than others. Thus, the CPU time of each algorithm was noted and averaged for 10 independent trials for each response. From Figure 5, it is observed that ALO, DA and WOA were the three most expensive algorithms, whereas SSA, GWO and AOA were the three least expensive. Nevertheless, the standard deviation of computational time for WOA was very high, indicating that in some instances, it may take a very short computation time. Perhaps this is because optimizing WR was very fast in optimizing the linear problem with only two variables. Figure 5. Average time taken by the algorithms in the optimization of the responses. The bars and whiskers denote the mean and standard deviation of 10 runs (3 responses X 10 independent runs/response). All six recent algorithms performed better than the existing solutions by Hewidy et al. [12]. Hewidy et al. [12] used a desirability function-based approach, which is, in general, incapable of locating the global optima. The recent metaheuristics tested in this paper recorded at least 25%, 70% and 47% better solutions than Hewidy et al. [12] for MRR, WR and SR results. This improvement is perhaps due to the fact that the metaheuristics initiate a random population and algorithmically improve it over generations by continuously evolving the solutions. Based on the comprehensive evaluations of the algorithms on the three machining-related test functions, it can be summarized that despite AOA being the most recent algorithm among the six tested metaheuristics, it is not necessarily the best in locating the best-known optima. It is evident that the AOA is trapped in the local optima region, and for both the minimization type test functions, the solution of AOA was 1–2% poorer than the best-known optima. For the maximization-type test function, the bestknown optima were observed to be roughly 9% better than the AOA’s solution. However, the computational time requirement of AOA was quite low, nearly on par with SSA and Figure 5. Average time taken by the algorithms in the optimization of the responses. The bars and whiskers denote the mean and standard deviation of 10 runs (3 responses X 10 independent runs/response). All six recent algorithms performed better than the existing solutions by Hewidy et al. [ 12 ]. Hewidy et al. [ 12 ] used a desirability function-based approach, which is, in general, incapable of locating the global optima. The recent metaheuristics tested in this paper recorded at least 25%, 70% and 47% better solutions than Hewidy et al. [ 12 ] for MRR, WR and SR results. This improvement is perhaps due to the fact that the metaheuristics initiate a random population and algorithmically improve it over generations by continuously evolving the solutions. Based on the comprehensive evaluations of the algorithms on the three machining-related test functions, it can be summarized that despite AOA being the most recent algorithm among the six tested metaheuristics, it is not necessarily the best in locating the best-known optima. It is evident that the AOA is trapped in the local optima region, and for both the minimization type test functions, the solution of AOA was 1–2% poorer than the best-known optima. For the maximization-type test function, the best-known optima were observed to be roughly 9% better than the AOA’s solution. However, the computational time requirement of AOA was quite low, nearly on par with SSA and GWO. In terms of convergence, the SSA was observed to be the fastest. The median evaluated function value of SSA was seen to be quite a bit lower (for minimization problems) than its mean evaluated function value, indicating faster navigation to the optimal solution zone. Processes 2022,10, 197 19 of 20 5. Conclusions Machining process optimization is a necessary task for manufacturing industries and can lead to significant savings in material wastage, power consumption and tool wear and can improve productivity and efficiency of the process. Since a plethora of novel algorithms have been proposed in the recent past, with each having demonstrated capabilities in the literature, it is important to comprehensively compare them for their potential use in machining process optimization. In this article, six recently proposed nature-inspired algorithms, namely, ALO, AOA, DA, GWO, SSA and WOA, are comprehensively assessed, and the following conclusions were drawn. • Based on the ability to navigate and find the optimal solution, the tested algorithms may be ranked as SSA > WOA > GWO > DA > ALO > AOA. Both SSA and WOA were able to locate the best solution for all three responses. However, SSA’s success rate was 100% as opposed to 83% of WOA. • Based on the computational time, the tested algorithms may be ranked as SSA > GWO > AOA > WOA > DA > ALO. Both SSA and GWO had very minimal and similar computational requirements. However, SSA had a marginally lower standard deviation than GWO. As compared to ALO, SSA was observed to be about 25 times faster. • The convergence of SSA was observed to be slightly better than its counterparts. GWO also showed fast convergence. AOA was prone to be trapped in local optima. • As compared to the previous known best solutions, an average (on three responses) improvement of 50.8% and 47.47% was observed for ALO and AOA, respectively. DA and GWO showed a 50.85% improvement, whereas SSA and WOA recorded a 50.93% improvement. The limitations of this study are that there are several advanced and hybrid variants of the algorithms, which were not considered in this paper. The study is also limited to one class of optimization problems. Nevertheless, the current optimization problems are of immense importance to industries. In the future, this study can be extended to incorporate an in-depth analysis of hybrid metaheuristics and the use of advanced quantum and chaotic enhancements to these algorithms. Optimization under uncertainty could also be an interesting area for their application. Author Contributions: Conceptualization, S.R., G.N., R. ˇ C., N.R.C. and K.K.; data curation, S.P.; formal analysis, S.P. and K.K.; methodology, S.R., G.N., R. ˇ C. and N.R.C.; project administration, R. ˇ C.; resources, S.R., G.N., R. ˇ C. and N.R.C.; software, S.R., G.N. and N.R.C.; supervision, R. ˇ C.; validation, S.P. and K.K.; visualization, S.P. and K.K.; writing—original draft, S.R., G.N., N.R.C. and S.P.; writing—review and editing, R. ˇ C. and K.K. All authors have read and agreed to the published version of the manuscript. Funding: This research received no external funding. Institutional Review Board Statement: Not applicable. Informed Consent Statement: Not applicable. Data Availability Statement: The data presented in this study are available through email upon request to the corresponding author. Conflicts of Interest: The authors declare no conflict of interest. References 1. Ramaswamy, A.; Perumal, A.V.; Jagadeesan, J.; Nagarajan, H.V. Optimization of WEDM process parameters for D3 die steel using RSM. Mater. Today Proc. 2021,37, 2063–2069. [CrossRef] 2. Yong-Jie, M.; Wen-Xia, Y. Research progress of genetic algorithm. Appl. Res. Comput. 2012,29, 1201–1206. 3. Zhou, C.; Gao, H.B.; Gao, L.; Zhang, W.G. Particle swarm optimization (PSO) algorithm. Appl. Res. Comput. 2003,12, 7–11. 4. Gao, W.-F.; Liu, S.-Y. A modified artificial bee colony algorithm. Comput. Oper. Res. 2012,39, 687–697. [CrossRef] 5. Faris, H.; Aljarah, I.; Al-Betar, M.A.; Mirjalili, S. Grey wolf optimizer: A review of recent variants and applications. Neural Comput. Appl. 2018,30, 413–435. [CrossRef] Processes 2022,10, 197 20 of 20 6. Abualigah, L.; Diabat, A.; Mirjalili, S.; Elaziz, M.A.; Gandomi, A.H. The arithmetic optimization algorithm. Comput. Methods Appl. Mech. Eng. 2021,376, 113609. [CrossRef] 7. Abualigah, L.; Shehab, M.; Alshinwan, M.; Alabool, H. Salp swarm algorithm: A comprehensive survey. Neural Comput. Appl. 2020,32, 11195–11215. [CrossRef] 8. Abualigah, L.; Shehab, M.; Alshinwan, M.; Mirjalili, S.; Elaziz, M.A. Ant Lion Optimizer: A Comprehensive Survey of Its Variants and Applications. Arch. Comput. Methods Eng. 2021,28, 1397–1416. [CrossRef] 9. Gharehchopogh, F.S.; Gholizadeh, H. A comprehensive survey: Whale Optimization Algorithm and its applications. Swarm Evol. Comput. 2019,48, 1–24. [CrossRef] 10. Gandomi, A.H.; Yang, X.-S.; Alavi, A.H.; Talatahari, S. Bat algorithm for constrained optimization tasks. Neural Comput. Appl. 2013,22, 1239–1255. [CrossRef] 11. Meraihi, Y.; Ramdane-Cherif, A.; Acheli, D.; Mahseur, M. Dragonfly algorithm: A comprehensive review and applications. Neural Comput. Appl. 2020,32, 16625–16646. [CrossRef] 12. Hewidy, M.S.; El-Taweel, T.A.; El-Safty, M.F. Modelling the machining parameters of wire electrical discharge machining of Inconel 601 using RSM. J. Mater. Processing Technol. 2005,169, 328–336. [CrossRef] 13. Zhang, G.; Zhang, Z.; Guo, J.; Ming, W.; Li, M.; Huang, Y. Modeling and optimization of medium-speed WEDM process parameters for machining SKD11. Mater. Manuf. Processes 2013,28, 1124–1132. [CrossRef] 14. Shihab, S.K. Optimization of WEDM process parameters for machining of friction-stir-welded 5754 aluminum alloy using Box–Behnken design of RSM. Arab. J. Sci. Eng. 2018,43, 5017–5027. [CrossRef] 15. Chaudhary, A.; Sharma, S.; Verma, A. Optimization of WEDM process parameters for machining of heat treated ASSAB’88 tool steel using Response surface methodology (RSM). Mater. Today Proc. 2021. [CrossRef] 16. Mahapatra, S.S.; Patnaik, A. Optimization of wire electrical discharge machining (WEDM) process parameters using Taguchi method. Int. J. Adv. Manuf. Technol. 2007,34, 911–925. [CrossRef] 17. Nayak, B.B.; Mahapatra, S.S. Optimization of WEDM process parameters using deep cryo-treated Inconel 718 as work material. Eng. Sci. Technol. Int. J. 2016,19, 161–170. [CrossRef] 18. Mukherjee, R.; Chakraborty, S.; Samanta, S. Selection of wire electrical discharge machining process parameters using nontraditional optimization algorithms. Appl. Soft Comput. 2012,12, 2506–2516. [CrossRef] 19. Mirjalili, S. The ant lion optimizer. Adv. Eng. Softw. 2015,83, 80–98. [CrossRef] 20. Wang, J.; Du, P.; Niu, T.; Yang, W. A novel hybrid system based on a new proposed algorithm—Multi-Objective Whale Optimization Algorithm for wind speed forecasting. Appl. Energy 2017,208, 344–360. [CrossRef] 21. Zawbaa, H.M.; Emary, E.; Grosan, C. Feature selection via chaotic antlion optimization. PLoS ONE 2016 ,11, e0150652. [CrossRef] 22. Mafarja, M.; Eleyan, D.; Abdullah, S.; Mirjalili, S. S-shaped vs. V-shaped transfer functions for ant lion optimization algorithm in feature selection problem. In Proceedings of the International Conference on Future Networks and Distributed Systems, Cambridge, UK, 19 July 2017. 23. Mirjalili, S.; Jangir, P.; Saremi, S. Multi-objective ant lion optimizer: A multi-objective optimization algorithm for solving engineering problems. Appl. Intell. 2017,46, 79–95. [CrossRef] 24. Mirjalili, S. Dragonfly algorithm: A new meta-heuristic optimization technique for solving single-objective, discrete, and multi-objective problems. Neural Comput. Appl. 2016,27, 1053–1073. [CrossRef] 25. Mirjalili, S.; Mirjalili, S.M.; Lewis, A. Grey wolf optimizer. Adv. Eng. Softw. 2014,69, 46–61. [CrossRef] 26. Mirjalili, S.; Gandomi, A.H.; Mirjalili, S.Z.; Saremi, S.; Faris, H.; Mirjalili, S.M. Salp Swarm Algorithm: A bio-inspired optimizer for engineering design problems. Adv. Eng. Softw. 2017,114, 163–191. [CrossRef] 27. Mirjalili, S.; Lewis, A. The whale optimization algorithm. Adv. Eng. Softw. 2016,95, 51–67. [CrossRef] 28. Hof, P.R.; van der Gucht, E. Structure of the cerebral cortex of the humpback whale, Megaptera novaeangliae (Cetacea, Mysticeti, Balaenopteridae). Anat. Rec. Adv. Integr. Anat. Evol. Biol. 2007,290, 1–31. [CrossRef] [PubMed] 29. Hemeida, A.M.; Alkhalaf, S.; Mady, A.; Mahmoud, E.A.; Hussein, M.E.; Eldin, A.M.B. Implementation of nature-inspired optimization algorithms in some data mining tasks. Ain Shams Eng. J. 2020,11, 309–318. [CrossRef] 30. Nasiri, J.; Khiyabani, F.M. A whale optimization algorithm (WOA) approach for clustering. Cogent Math. Stat. 2018 ,5, 1483565. [CrossRef] 31. Oliva, D.; el Aziz, M.A.; Hassanien, A.E. Parameter estimation of photovoltaic cells using an improved chaotic whale optimization algorithm. Appl. Energy 2017,200, 141–154. [CrossRef] 32. Watkins, W.A.; Schevill, W.E. Aerial observation of feeding behavior in four baleen whales: Eubalaena glacialis, Balaenoptera borealis, Megaptera novaeangliae, and Balaenoptera physalus. J. Mammal. 1979,60, 155–163. [CrossRef] 33. Aljarah, I.; Faris, H.; Mirjalili, S. Optimizing connection weights in neural networks using the whale optimization algorithm. Soft Comput. 2018,22, 1–15. [CrossRef]