Operational Research in Engineering Sciences: Theory and Applications First online ISSN: 2620-1607 eISSN: 2620-1747 DOI: https://doi.org/10.31181/oresta241122166s * Corresponding author. sahamech90@gmail.com (S. Saha) saikatjumtech@gmail.com (S.R. Maity), infodrsudip@gmail.com (S. Dey) COMPARATIVE ANALYSIS OF METAHEURISTIC OPTIMIZERS IN THE PERFORMANCE OPTIMIZATION OF WIRE ELECTRIC DISCHARGE MACHINING PROCESSES Subhankar Saha 1, 2, Saikat Ranjan Maity 1*, Sudip Dey 1 1 Department of Mechanical Engineering, National Institute of Technology Silchar, India 2 Department of Mechanical Engineering, St. Martin’s Engineering College, Telangana, India Received: 25 July 2022 Accepted: 29 October 2022 First online: 24 November 2022 Research paper Abstract: WEDM is an intricate process whereby improper selection of machine parameters often leads to undesirable performances. Therefore, the extraction of optimal machining parameters is pivotal for achieving better performances in WEDM. Metaheuristic optimizers have gained immense popularity due to their capability of providing global optimal solutions. The application of recently reported metaheuristic optimizers in non-traditional machining processes is rarely being explored. In light of the above, the current paper examines the use of six recently reported metaheuristic optimizers, namely the ant lion optimization (ALO), chimp optimization algorithm (ChoA), moth flame optimization (MFO), spotted hyena optimization (SHO), Harris Hawk optimization algorithm (HHO), Marine predator algorithm (MPA) to optimize WEDM performances in three WEDM processes. Particle swarm optimization (PSO) and Teaching learning-based optimization (TLBO), two well-known existing optimization approaches, are also included in this study to enable a reasonable comparison of the algorithms' performance. The algorithms are compared with parameters such as the quality of optimal solutions, convergence behavior, and average computational time. HHO algorithm is found to be robust amongst the eight competitors in terms of culminating the global optimal solution and propensity to quickly converge to the global optimal solution which corroborates the high exploration and exploitation capability of the algorithm. Therefore, HHO optimizer can be exploited in future to determine the optimal operating conditions for other manufacturing processes. Key words: WEDM, optimization, metaheuristic algorithms, two sample t-test, sensitivity analysis Saha et al./Oper. Res. Eng. Sci. Theor. Appl. First online 1. Introduction In the era of technological advancements, there is a growing demand for advanced materials which are hard and difficult-to-machine. The machining of such advanced materials with high geometrical accuracy using traditional machining approaches is an impossible task. As a result, a number of non-traditional machining (NTM) techniques are available to meet the requirement for high geometrical accuracy. WEDM is a non-traditional machining approach that has garnered a lot of interest in the industry because of its ability to create intricate curves with high geometrical accuracy [Majumder & Maity, 2018]. The material removal in the WEDM process commences when an electric spark emerges amid the wire-workpiece interface. The spark liquidifies the material, and subsequently the molten debris is cleaned by the dielectric fluid injected from the top and bottom nozzles. The simplified view of the WEDM procedure is portrayed in Figure 1. Figure 1. Simplified view of WEDM process The performance attributes in WEDM are not always acquired at the envisioned level due to the process's intrinsic nature and a number of processing parameters (pulse duration, pulse interval, servo voltage, wire feed, wire speed, and so on), i.e., each process performance enhancement comes at the expense of another. As a result, machining under optimal operating conditions guarantees that a trade-off between the process performances is adequately maintained. In light of the preceding, researchers looked at several optimization techniques for selecting the best combination of process attributes. The next section provides a concise description of the optimization techniques that have been reported in WEDM operation, the importance of metaheursitic optimization algorithms and the advantages and limitations of different metaheuristic optimizers. 2. Literature Review Mandal et al. (2016) derived the optimal operating conditions by the desirability function while processing Nimonic C-263 superalloy through WEDM. In a recent investigation, the research group adopted a hybrid strategy i.e., signal-to-noise ratio and the Taguchi methodology to optimize the performance variables of the WEDM Comparative Analysis of Metaheuristic Optimizers in the Performance Optimization of Wire Electric Discharge Machining Processes process concurrently (Ramakrishnan & Karunamoorthy, 2008). A group of researchers optimized the performances using RSM in WEDM of Inconel 718 (Tonday & Tigga, 2019). Khan et al. (2014) implemented grey relational strategy to optimize microhardness and surface roughness simultaneously during the WEDM process. It is worth pointing that conventional techniques such as the Taguchi approach and grey relational analysis do not guarantee global optimum solutions as they commence the optimization with the specific level of process parameters. As a result, researchers are intrigued to adopt metaheuristic algorithms in WEDM process since they provide global optimum solutions or the best solutions. Metaheuristic algorithms are algorithms that are used in tackling a spectrum of complicated optimization problems without requiring to substantially adapt to each problem. The greek word "meta," which appears in the term, denotes that these algorithms are "higher level" heuristics, as contrary to problem-specific heuristics. Metaheuristic algorithms are commonly used to address problems for which no appropriate problem-specific algorithm exists. The following traits are shared by almost all metaheuristics: They are nature-inspired; they use stochastic components (involving random variables); they do not evaluate the gradient or Hessian matrix of the objective function. The research community has been using metaheuristic algorithms in WEDM for the past thirty years. One pioneer contribution is the proposition of the simulated annealing method in WEDM to discover the optimal operating condition for the cutting rate and surface roughness (Tarng et al. 1995). In a similar manner, Sadeghi et al. (2011) explored the Tabu-search algorithm for optimizing the performance parameters. In WEDM of Inconel-690, a modified version of cuckoo search algorithm is proposed to assess the optimal outcomes (Rao & Venkaiah, 2017). A group of two researcher performed optimization of the performance parameters employing bat algorithm in taper formation in Inconel 718 exploiting WEDM. In a research effort, NSGA methodology is executed to track the various optimal parametric combinations (Pareto set) for two performance parameters in WEDM of Ti6Al4V (Nayak & Mahapatra, 2016). In a similar manner, Garg et al. (2012) found a set of Pareto optimal solutions in WEDM of Ti6Al-4V alloy employing the NSGA-II algorithm. In view of the above, it is observed that there are limited research in WEDM which have documented the use of metaheuristic optimizers in WEDM processes. It is worth emphasizing that the No-Free-Lunch (NFL) theorem asserts that one specific algorithm cannot solve all sorts of optimization problems (Wolpert & Macready, 1997). Furthermore, previous studies have not documented the use of recently developed metaheuristic optimizers in WEDM. As a result, we plan to investigate the algorithmic performance of six recently reported metaheuristic optimizers, as well as two popular state-of-the-art metaheuristic optimizers, while optimizing WEDM performances either individually (single-objective) or collectively (multi-objective) for three WEDM processes. The six recently reported metaheuristic optimizers are ant lion optimization (ALO), chimp optimization algorithm (ChoA), moth flame optimization (MFO), spotted hyena optimization (SHO), Harris Hawk optimization (HHO), and Marine predator algorithm (MPA). Whereas, the two popular metaheuristic optimizers are particle swarm optimization and teaching learning- based optimization. We discussed the typical characteristics of each representative algorithm as follows to support the decisions made during the algorithm selection process. The PSO optimizer is simpler (Lee & Park, 2006). However, the main Saha et al./Oper. Res. Eng. Sci. Theor. Appl. First online loophole of this optimizer is the quick convergence of all solutions which undermines the diversity in the population (Juneja & Nagar, 2016). The TLBO optimizer has fewer tuning parameters than other optimizers, doesn't get stuck in local optima like other optimizers, and provides an accurate global optimal solution in minimum time (Uzlu et al., 2014). The limitation of this optimizer is that it ends with near-optimal solution in minimum iterative step (Sultana & Roy, 2014). SHO requires a low computational effort when tackling problems with high dimensions (Krishna et al., 2021). However, it is found that the problem space remains partially explored using SHO because of the concentrated search around the current optimal solution which might be a local optimum (Sabahno & Safara, 2021). ChoA has ample of advantages such as high exploration, a semi-deterministic feature of chaotic maps assists in high exploitation, local optima avoidance is very high, few parameters to tune, ease in implementation due to the parallel structure of independent groups (Khishe & Mosavi, 2020). In contrary, it has few limitations such as premature convergence, a slow rate of convergence, discovering local minima rather than global minima, and a low balance between exploitation and exploration (Kaur et al., 2021). The advantage of using MFO is that it is simple, and can be easily hybridized with other algorithms (Shehab et al., 2021). But, it may easily fall into the local optima because it emphasizes on exploitation more than exploration which causes premature convergence, and the search ability is insufficient (Shan et al. 2021). Population diversity, strong optimization ability, and fewer adjustment factors are the typical advantages of ALO algorithm (Yao et al. 2021). Due to the roulette wheel selection technique, ALO algorithm suffers from rapid convergence (Abualigah et al., 2021). MPA optimizer has limited number of algorithmic variables. Moreover, the procedures are simple and converge fast with the added benefits of flexibility, and robustness (Yakout et al. 2021). However, it exhibits premature convergence in complex and high dimensional problems, and falls in local optima (Houssein et al., 2021). HHO optimizer is simple and has a few exploratory and exploitative mechanisms (Mansoor et al., 2020). But it has the major limitation of displaying finite exploration behavior as the exploration behavior depends on the equal perching chance, and in the mid-flight, the escape energy gets limited within unity (Naik et al., 2021). In the present study, the goal is to compare the considered algorithms' performances based on several parameters such as the quality of optimal solutions, convergence behavior, and average computing time. The motive behind the comparative analysis is to find the most reliable optimization algorithm. Performance stability of the optimizers are retrieved exploiting the sensitivity analyses. Lastly, we tested the performance of the eight competing optimizers on benchmark test functions (i.e., the Sphere function and the Generalized Rastrigin's function) to determine the robust algorithm. The strategy adopted to accomplish the goal of the current research is delineated with a flowchart (see in Figure 2). Comparative Analysis of Metaheuristic Optimizers in the Performance Optimization of Wire Electric Discharge Machining Processes Figure 2. Flowchart showing the strategy adopted for the present work 3. Metaheuristic optimization algorithms 3.1. Teaching Learning Based Optimization (TLBO) The fundamental concept of TLBO is to simulate a two-stage learning process in a traditional classroom setting (Rao et al., 2011). The communication of knowledge between a teacher and students occurs in the first stage, known as the Teacher Phase. The amount of knowledge gained by students is proportional to the amount of teacher's knowledge. However, in practice, the likelihood of the teaching to become successful is distributed according to Gaussian law. Only a small percentage of students can comprehend everything indicated by the right end of the Gaussian distribution. However, the chances of learning new things aren't entirely eliminated. Saha et al./Oper. Res. Eng. Sci. Theor. Appl. First online A student can understand from the fellow students at the second stage, known as the Learner Phase. Overall, the amount of knowledge conveyed to a student is determined by his or her teacher and by peer learning exchanges. 3.1.1. Teacher Phase In this phase, a teacher intends to improve the average performance in the subject being taught. The teaching job is first assigned to the best individual in the population teacher X , after which the algorithm improves other individuals i X by adjusting their positions towards that of the teacher teacher X . The current mean value of the individuals mean X is used to create each individual's position, which symbolizes the traits of all learners in the current generation. The disparity amid the teacher's knowledge and the students’ knowledge is simulated in Eq. (1), which shows how the difference in student performance is affected by the difference in teacher’s knowledge and the students’ knowledge.   new i teacher F meanX X r X T X   (1) The F T in Eq. (1) refers to a teaching factor which depicts the altered mean value, and r refers a random number in [0,1]. 3.1.2. Learner Phase Increasing an individual's knowledge  iX is done in this phase through peer learning from any student ii X and interaction amid the individual and other learners. Two states can arise based on the relative knowledge levels of these two students: if ii X is better than i X , i X will move towards iiX (shown in Eq. (2)), and if iiX is worse than i X , iX will be moved away from iiX (shown in Eq. (3). Student will be allowed into the population if he or she performs better by using Eq. (2), and Eq. (3). The algorithm will iterate till the end condition is reached.  new i ii iX X r X X   (2)  new i i iiX X r X X   (3) 3.2. Particle Swarm Optimization PSO is led by swarm intelligence behavior which takes advantage of the social information sharing model. Individuals (i.e., particles) fly across a higher- dimensional search space in PSO (Poli et al. 2007). The individuals' tendency to imitate the success of others in population leads to changes in particle positions within the search space (called swarm). The knowledge, of a particle's surroundings thereby affects its modification within the swarm. The search characteristic of a particle is affected by the search characteristic of other particles in the swarm. The particle keeps track of its location in the problem space, which is related to its best solution so far, known as best p , and the overall best value is the best value recorded by the particle swarm optimizer when globally treated. Furthermore, its current Comparative Analysis of Metaheuristic Optimizers in the Performance Optimization of Wire Electric Discharge Machining Processes location, as determined by any particle in the population, is known as best g . Each particle's velocity is changed as it moves toward its best p and bestg positions in the particle swarm optimization process. Separate random values are created for acceleration towards the best p and bestg positions, which are weighted by random terms. The PSO method adjusts the particle's velocities and positions as shown in the equations below.          1 1 2 2 2 1 2 2 1 , , 2 4 where , 4 ii i best i best i vel t vel t c r p z t c r g z t c c                        (4)      1 1i i iz t z t vel t    (5) where, c1 and c2 are the positive constants which represent the cognitive learning factor, and social learning factor respectively, r1 and r2 are random numbers in range [0,1].  1 2 dim, ,..., T i i i i z z z z depicts the ith particle position in the search space of dimension dim, and,  1 2, ,..., T i i i in vel vel vel vel depicts the ith particle velocity. 3.3. Spotted Hyena optimizer Spotted Hyena Optimizer is a new bio-inspired optimization algorithm, which simulates the collaborative behavior of a group of spotted hyenas during encircling, hunting, and attacking the prey (Dhiman & Kumar, 2017). 3.3.1. Prey encircling During prey encircling, the target prey is assumed to be the best solution, and the other spotted hyenas change their positions by following the best solution. This behavior is mathematically modeled as follows.     Prey . h SH D A P t P t  (6)     Prey 1 . SH h P t P t B D   (7) where h D denotes the separation between the spotted hyena and prey, t denotes the present iteration, A and B are co-efficient vectors, Prey P represent prey’s position vector, and SH P represent spotted hyena’s position vector. 12 A r  (8) 22B s r s   (9) 5 5 MaxIter s t         (10) where s reduces linearly from 5 to 0, and 1r , and 2r randomly changes between 0 and 1. Saha et al./Oper. Res. Eng. Sci. Theor. Appl. First online 3.3.2. Prey hunting and Prey searching During prey hunting, the hunting strategy adopted by spotted hyenas in the SHO algorithm is modeled mathematically as follows: . h KbSH D A P P  (11) K bSH h P P B D   (12) 1 .... K K K Mh C P P P       (13) where, bSH P is the initial best position of spotted hyena, K P represents the position of other spotted hyenas, and M represents the number of spotted hyenas (shown in Eq.(13)).   1 2 ,, , .....bSH bSH bSH bSHM count P P P P R   (14) R randomly varies between 0.5 and 1, and h C is a cluster of M number of optimal solutions. During prey hunting, spotted Hyenas attack the prey in a way that is mathematically expressed below:  1 hSH C P t M   (15) where,  1SHP t  saves the best solutions, and the positions of the remaining spotted hyenas' changes relative to the best-spotted hyena’s position. During prey searching, the vector A in Eq. (11) provides random values during the iteration process, which aids in exploration. 3.4. Chimp Optimization algorithm The Chimp optimization algorithm, a metaheuristic optimizer, is motivated by the intelligence behavior exhibited by the chimps during hunting in their communities (Khishe & Mosavi, 2020). There are four categories of chimps, i.e., attacker, chaser, barrier, and driver, with different capabilities. The chaser, barrier, and driver lead the exploration process while the function of the attacker leads the exploitation process. The behavior of chimps modeled as follows:     prey chimp . . d c P t o P t  (16)     chimp prey 1 .P t P t a d   (17) where prey P and chimp P indicate the position vectors of the prey and the chimp respectively. The co-efficient vectors c, o, and a are determined as follows: 1 2. . a g r g  (18) 2 2. c r (19) o = chaotic_value (20) where g diminishes from 2.5 to 0 through the iteration process, o is a chaotic vector determined using chaotic maps. The generation of stochastic population of chimps is the initial step in the chimp optimization algorithm. Then in the next step, chimps are classified into four varying categories: driver, barrier, attacker, and chaser. The best Comparative Analysis of Metaheuristic Optimizers in the Performance Optimization of Wire Electric Discharge Machining Processes chimps are the initial attacker, barrier, driver, and chaser as they are aware of the prey's position. Therefore, amongst the entire set of best solutions, four best solutions are used to represent them. The rest of the chimps are compelled to change their locations on the basis of the best chimp locations. This behavior can be mathematically expressed as below: Attacker Attacker1 1 . ( ) . d c P t o P  (21)   Barrier Barrier2 2 . . d c P t o P  (22)   Chaser Chaser 3 3 . . d c P t o P  (23)   Driver Driver4 4 . . d c P t o P  (24)   Attacker Attacker1 1 . P P t a d  (25)   Barrier Barrier2 2 . P P t a d  (26)   Chaser3 3 . Chaser P P t a d  (27)  4 4. Driver DriverP P t a d  (28)    1 2 3 41 / 4P t P P P P     (29) where, Attacker d , Barrier d , Chaser d , and Driver d analogs with d in Eq. (16). For updating the location of the chimps during the searching period, a probability of 50% is chosen between two alternatives, i.e., the usual updating rule and the chaotic model, which is mathematically expressed below:     prey chimp . if 0.5 1 Chaotic_value if 0.5 P t a d P t          (30) 3.5. Moth Flame Optimization Transverse orientation for navigation of moths at night using moonlight forms the motivation of this MFO algorithm (Mirjalili, 2015). In the MFO algorithm, the candidate solutions are the moths and the problem variables refers to their positions in the search space. The set of moths is represented as a matrix with n moths and dim dimensions which is shown below: 1,1 1,2 1,dim 2,1 2,2 2,dim ,1 ,2 ,dim n n n m m m m m m M m m m                (31) We further suppose that the fitness values for all the moths are stored in an array, as follows: 1 2 = n OM OM OM OM             (32) Saha et al./Oper. Res. Eng. Sci. Theor. Appl. First online Another key part of the MFO algorithm is flames. The following is a matrix that is identical to the moth matrix: 1,1 1,2 1,dim 2,1 2,2 2,dim ,1 ,2 ,dim n n n F F F F F F F F F F                (33) where n is the number of moths and dim is the dimension. The dimension of the flame matrix is the same as the dimension of the moth matrix. Both the moth and the flame are solutions, but the moth is the search agent and the flame is the moth's best position. Flames are the flags that moths drop during the search process, and the moths travel around the flags and update accordingly. As a result of this, the moths never lose their best solution. According to the equation below, moths update their position in relation to flame.  = ,i i jM S M F (34) where i M represents the ith moth, j F represents the jth flame, and the spiral function is represented by S. The logarithmic spiral motion of the moth is given below:    , . .cos 2bti j i jS M F D e r F  (35) where, i j i D F M  (36) b is a constant that determines the form of spiral motion, r refers to random number within [-1, 1]. Flame gets updated over the course of iterations as follows: 1 flame no = round MaxIter N N t        (37) where N is the maximum number of flame. 3.6. Ant Lion Optimization Ant Lion Optimizer is a metaheuristic optimizer which is conceptualized based on the chasing strategy of antlions in catching their prey (Mirjalili, 2015). The ant lions hide underneath the base of the cone-shaped cavities in the sand and then wait for the ants to get captured in the hole. They throw sand at the tip of the trap so that the ants fail to escape and slide down to the bottom of the trap. In this manner, the ants get captured by the ant lions. The pits are rebuilded to capture other ants. The positions of the ants are stored in the matrix Ant M (shown in Eq. (38)) which is employed during the optimization. 11 1dim 1 dim Ant n n A A M A A            (38) where n refers to the quantity of ants, and dim refers to the dimension of the problem. Fitness function f is utilized for the evaluation of the fitness of each ant during optimization; the fitness values are stored in the matrix OAnt M as shown below: Comparative Analysis of Metaheuristic Optimizers in the Performance Optimization of Wire Electric Discharge Machining Processes       1,1 1,2 1,dim 2,1 2,2 2,dim ,1 ,2 ,dim f ,..., f ,..., f ,..., OAnt n n n A A A A A A M A A A                          (39) Apart from ants, the ant lions also have their hideouts in the search domain. The matrices Antlion M and OAntlion M save the positions and fitness values of the ant lions, respectively. 1,1 1, ,1 , m Antlion n n m A A M A A            (40)       1,1 1,2 1, 2,1 2,2 2, ,1 ,2 , m m OAntlion n n n m f AL AL AL f AL AL AL M f AL AL AL                          (41) While searching for food, ants move in a stochastic fashion; thus, a random walk is selected to simulate ants’ movement as below:           1 20, csum 2 1 , csum 2 1 ..., csum 2 1MaxIterX t r t r t r t      (42) where csum reveals the cumulative sum, t reveals the random walk steps, and  r t is a stochastic function which is enumerated as follows:   1 if rand 0.5 0 if rand 0.5 r t     (43) To limit the movement within the search space in a random fashion, Eq. (42) is normalized exploiting the Eq. (44).       t t i i i it i it i i X a b c X c d a       (44) Eq. (45), and Eq. (46) show how ants’ slide down into pits. t t c c I  (45) t t d d I  (46) Saha et al./Oper. Res. Eng. Sci. Theor. Appl. First online When the ant reaches the pit bottom, the antlion snatches it and consumes it. To improve its chances of obtaining new prey, an antlion must update its position to the most recent position of the chased ant. Eq. (47) represents this procedure.    t i t tj i i jAntlion Ant if f Ant > f Antlion  (47) In every step, elitism is employed to keep the best solutions. The fittest antlion is treated as elite, which is the best antlion achieved. In every step, the elite should have an impact on the antlion (random movement). For this, every ant is assumed to associate with an antlion by Roulette wheel and elite, which Eq. (48) gives. t t t A E i R R Ant 2   (48) t A R , and tER represent the random walk around the selected antlion and elite at t th iteration respectively. 3.7. Marine Predator Algorithm (MPA) The marine predator algorithm is a novel nature-inspired metaheuristic algorithm that replicates the biological interaction between marine predators and prey (Faramarzi et al., 2020). Predators are inspired in this algorithm to use the widespread foraging methods known as the Brownian and Levy random movement in the marine ecosystem. Predators utilize the Brownian approach if there exists a large concentration of prey in the hunting region, and the Levy method when there is a low concentration of prey. However, environmental factors namely eddy generation and the effects of fish aggregating devices (FADs) are among the elements that influence marine predator behavior. The steps of the algorithm are enumerated as follows: 3.7.1. Initialization Both the Prey (P) and Elite (E) matrices are formed during the initialization phase. In accordance with the survival of the fittest argument, the skilled foragers are the top predators in nature. Thus, in order to construct the Elite matrix, the fittest solution is designated as a top predator. Prey is another matrix with the same dimension as Elite, and predators use it to update their positions. 3.7.2. Phase 1 This phase commences during the one-third of iterations and is implemented by a large velocity ratio  10v  for an adequate exploration ability, wherein the movement of the prey is faster than the predator. The prey moves quickly to guard their food. Whereas the fittest predators are stationary during this stage. This stage is mathematically illustrated with the help of equations (Eq. 49 & Eq. 50). MaxIter While t 3     1, 2,....,ii B i BS R E R P i n     (49)  0.5i i iP P R S   (50) Comparative Analysis of Metaheuristic Optimizers in the Performance Optimization of Wire Electric Discharge Machining Processes where i S indicate the step size of the predator, B R is the random vector based on normally distributed Brownian motion, R indicate a uniformly distributed random variable, and n indicates the search agents per population. The notation  indicate entry-wise multiplications. 3.7.3. Phase 2 In this phase, there is a transient transformation from exploration to exploitation. Here, the velocity ratio of unity (v ≈ 1) indicates that both the predator and the prey moves at an identical speed. MaxIter 2 While < t < MaxIter 3 3 The first half population gets updated based on Levy strategy as follows:    1, 2,...., / 2ii L i LS R E R P i n     (51)  0.5i i iP P R S   (52) where L R is a uniformly distributed random vector based on Levy motion. On contrary, the second half population is updated using Brownian strategy as follows (shown in Eq. (53) & Eq. (54):    / 2,....,ii B B iS R R E P i n n     (53)  0.5i i ifP E X S   (54) where f X is a variable that monitor the predator’s step size and is evaluated by the following Eq. (55)   2 / 1 t MaxIter f t X MaxIter            (55) 3.7.4. Phase 3 This phase is usually marked with a high level of exploitation capacity. This phase is marked by a low velocity ratio (v = 0.1), in which the predator runs past the prey. This phase is based on Levy movement which is mathematically expressed as follows: 2 While t MaxIter 3     1, 2,....,ii L B iS R R E P i n     (56)  0.5i i ifP E X S   (57) 3.7.5. Finishing After each iteration, the best solutions gets stored in the Elite (E) matrix. The final solution is then achieved after the last iteration. Saha et al./Oper. Res. Eng. Sci. Theor. Appl. First online 3.8. Harris Hawk Optimization Harris Hawk optimization (HHO) is a new nature-inspired optimizer that imitates the chasing trait of Harris hawks in order to catch their prey (rabbit), which are the best solutions in the search space (Heidari et al., 2019). HHO goes through two stages: the first is looking for prey with a group of hawks, and this stage is referred to as the exploration phase in the algorithm. The second stage involves hunting the prey in order to catch it, which is depicted in the optimization algorithm as the exploitation phase. The balance between exploitation and exploration of search space is determined by the rabbit's energy escape, with hawks having the potential to explore for large energy and exploitation for small energy. In the exploration phase, the HHO algorithm employs two alternative search strategies. These strategies are chosen based on α; if α is greater than 0.5, the first strategy is employed to search near one of the other hawks at random, but if α is lesser than 0.5, the second strategy, stated in Eq. (59) is employed for the search operation.         1 21 2 0.5rand randX t X t s X t s X t      (58)           3 41 0.5rabbit mX t X t X t s LB s UB LB        (59)     1 1 where N m i i X t X t N    The mathematical model to demonstrate the mechanism which is exploited to get transformed from the exploration phase to the exploitation phase is shown in Eq. (60). 0 2 1 t E E T        (60) The algorithm arrives the exploration phase when 1E  whereas the algorithm arrives the exploitation phase when 1E  . E diminishes when the iteration count increases. In the exploitation phase, the HHO algorithm utilizes four different ways to conduct optimization operations. If E is greater than 0.5, two techniques are used: besiege and soft besiege with increasing quick dives. If E is less than 0.5, two techniques are used: besiege and hard besiege with progressive quick dives. The illustration of the strategies can be found in the literature. 4. WEDM performance optimization To assess the efficacy of the eight metaheuristic optimization techniques, single and multiple objective optimization is carried out for two WEDM processes (elaborated in case 1, and case 2). The codes for the eight optimizers are built in MATLAB R2018a and executed on Windows 10 OS, Intel(R) Core™ i5 processor, and 8.00 GB RAM. For unprejudiced comparison amid the performances of the considered optimizers, population size, and maximum generation is kept at 50 and 100 respectively for all the considered algorithms. Comparative Analysis of Metaheuristic Optimizers in the Performance Optimization of Wire Electric Discharge Machining Processes 4.1. Case 1 Performance optimization in WEDM of A286 superalloy We assessed the WEDM performances such as material removal rate (MRR, in mm2/min) and surface roughness (SR in µm) for an A286 Superalloy. Machining of the samples are accomplished using an Ultra cut F1 model, a variant of the WEDM machine tool. Twenty-seven sets of experiments are undergone under the L27 scheme. Five parameters are tuned in three subsequent levels (depicted in Table 1) within the stipulated bounds while machining. Finally, multiple performances are optimized simultaneously using a multi-objective evolutionary algorithm and a decision making tool (Saha et al., 2021). In this research, we use the eight different metaheuristic optimizers to optimize individual performances as well as two performances at the same time. The goal is to compare the performances of the optimizers. To accomplish the task, we intend to exploit the mathematical expressions used for devising the correlation between the response variables and the explanatory variables in the previous investigation (Saha et al., 2021). The mathematical models are shown below: 2 2 1 2 3 4 5 1 2 2 2 2 3 4 5 1 2 1 3 1 4 1 5 2 3 2 4 2 5 3 4 4 5 567 20.39 2.91 102.8 28.4 2.83 0.0497 0.0354 8.192 0.037 0.0302 0.0117 * 0.858 * 0.1511 * 0.0034 * 0.244 * 0.1140 * 0.0196 * 0.326 * 0.145 * MRR x x x x x x x x x x x x x x x x x x x x x x x x x x x x                     (61) 2 2 1 2 3 4 5 1 2 2 2 2 3 4 5 1 2 1 3 1 4 1 5 2 3 2 4 2 5 3 4 177.3 0.59 0.889 27.26 1.82 0.110 0.00276 0.00934 1.2656 0.0323 0.00642 0.00014 * 0.0177 * 0.0157 * 0.0034 * 0.244 * 0.1140 * 0.0196 * 0.326 * 0.14 SR x x x x x x x x x x x x x x x x x x x x x x x x x x                      3 5 5 *x x (62) Table 1. Process variables and levels Process variables Level 1 Level 2 Level 3  1x Pulse on period (μs) 120 125 130  2x Pulse off period (μs) 48 52 56  3x Peak current (A) 10 11 12  4x Wire feed rate (m/min) 5 7 9  5x Servo voltage (v) 30 35 40 4.1.1. Single-objective optimization The optimization of MRR and SR is accomplished under a set of constraints i.e., 1 120 130,x  2 48 56x  , 3 10 12,x  35 9,x  and 430 40x  . The results of the different optimizers for the two performance attributes is demonstrated in Table 2. It is evident that ChoA, MFO, HHO, MPA, and PSO are able to produce the optimized MRR of 37.527 mm2/min which is close to the maximum MRR present in the experimental dataset (Saha et al., 2021). However, ALO, SHO, and TLBO produces Saha et al./Oper. Res. Eng. Sci. Theor. Appl. First online optimized MRR of 35.701 mm2/min, 29.614 mm2/min and 6.718 mm2/min respectively, which is relatively poor. Figure 3. Convergence behavior of ALO, ChoA, HHO, MFO, MPA, PSO, SHO, and TLBO for MRR For SR, unlike the SHO, we detect that all the algorithms have produced similar results, but better than all the results reported in the previous investigation (Saha et al., 2021). To realize the convergence traits of the optimizers, we plotted the convergence history of the competing optimizers while optimizing the response MRR (shown in Figure 3). It is noted that HHO algorithm rapidly converges to the global optimal solution which exposes the algorithm’s outstanding exploitation capability. Table 2. Single-objective optimization outcomes. Optimizer Response Optimal Value Pulse on period Pulse off period Peak Current Wire feed rate Servo voltage ALO MRR 35.701 130 48 11.51 5.01 31.14 SR 0.4776 120 56 10 5 40 ChoA MRR 37.527 130 48 11.63 7 30 SR 0.4776 120 56 10 5 40 MFO MRR 37.527 130 48 11.59 7 30 SR 0.4776 120 56 10 5 40 SHO MRR 29.614 130 48 12 5 30 SR 0.5320 120 56 10 9 40 MPA MRR 37.527 130 48 11.59 7 30 SR 0.4776 120 56 10 5 40 HHO MRR 37.527 130 48 11.59 7 30 SR 0.4776 120 56 10 5 40 PSO MRR 37.527 130 48 11.59 5 30 SR 0.4776 120 56 10 5 40 TLBO MRR 6.718 120 56 10 9 40 SR 0.4776 120 56 10 5 40 Comparative Analysis of Metaheuristic Optimizers in the Performance Optimization of Wire Electric Discharge Machining Processes 4.1.2. Multiple objective optimization To perform optimization of the two performance attributes (MRR and SR) simultaneously, we formed the objective function using weighted-sum method as follows:       1 2 min max SR MRR Min SR MRR Y Y Y w w  (63) where 1 w and 2 w are the preference weights assigned to SR and MRR, respectively. Here, equal weights for all the responses are considered, i.e., 1 2 0.5w w  . min SR is the minimum surface roughness and max MRR is the maximum material removal rate, which are procured from the single-objective optimization outcomes. Table 3 shows the optimal MRR and SR values recommended using the eight competitor techniques (ALO, MFO, ChoA, MPA, SHO, HHO, PSO, and TLBO). HHO portrays the superior performance i.e., it produces the global optimal responses of MRR and SR at minimal value of the combined fitness function (Y = 0.306708). The corresponding optimal process parameters are pulse on period = 130 s, pulse off period = 52 s, peak current = 10 A, wire feed rate = 5 m/min, and servo voltage = 30 volt. Figure 4 analyzed the convergence traits of the eight competing algorithms. The objective function of the HHO algorithm approaches the least fitness value (global optimum) in the fewest generations possible, demonstrating the algorithm's excellent exploitation capability. Table 4 shows the average computing time (seconds) consumed by the optimizers while optimizing the multiple performances. As seen in Table 4, TLBO has the shortest average computation time. Table 3. Multiple objective optimization outcomes. Optimizer Response Optimal Value Y Pulse on period Pulse off period Curre nt Wire Speed Servo voltag e Saha et al. (2021) MRR 36.04 130 52 10 5 30 SR 3.49 ALO MRR 1.51 0.47 967 120 56 10 5 39 SR 0.96 MFO MRR 1.20 0.47 984 120 56 10 5 40 SR 0.79 SHO MRR 1.20 0.47 984 120 56 10 5 40 SR 0.79 PSO MRR 1.20 0.47 984 120 56 10 5 40 SR 0.79 TLBO MRR 1.20 0.47 984 120 56 10 5 40 SR 0.79 HHO MRR 36.04 0.30 670 130 52 10 5 30 SR 3.49 MPA MRR 1.20 0.47 984 120 56 10 5 40 SR 0.79 ChoA MRR 1.20 0.47 984 120 56 10 5 40 SR 0.79 Saha et al./Oper. Res. Eng. Sci. Theor. Appl. First online Figure 4. Convergence behavior of ALO, ChoA, MFO, SHO, HHO, MPA, PSO, and TLBO for multi-objective function. Table 4. Average computation time for the eight optimizers Optimizer Average Computational time (secs) ALO 2.20241 ChoA 1.83478 MFO 1.53465 SHO 1.85662 HHO 2.53421 MPA 1.25715 PSO 1.23427 TLBO 1.20124 4.2. Case 2 Performance optimization in WEDM of Ti-6Al-4V alloy Devarajaiah & Muthumari, (2018) conducted WEDM machining on Ti-6Al-4V employing wire EDM machine tool of Model: DK 7732. The machine tool used in this work is based on reusable wire technology and doesn't need air-conditioning below 40 degrees centigrade. Molybdenum wire electrode (diameter of 0.18 mm) is employed as wire electrode. Four parameters, i.e., pulse duration, pulse off period, applied current, and wire-speed, were selected as control variables. The levels considered for the four control variables are revealed in Table 5. Two vital process performance measures were considered as responses (i.e., material removal rate (MRR in g/min) and power consumption (PC in kW)). The experimental trials were carried out as per Taguchi L16 design, and each trial is repeated thrice to capture the variability in the WEDM responses. Furthermore, regression analysis was employed Comparative Analysis of Metaheuristic Optimizers in the Performance Optimization of Wire Electric Discharge Machining Processes by Devarajaiah & Muthumari, (2018) to correlate the performance measures with the control variables. The regression models for MRR and PC are shown below (Eq. (64) & Eq. (65)): 2 1 2 3 4 1 2 2 2 3 1 4 2 3 0.00249 0.0056 0.0151 0.000011 0.000065 0.00039 0.00008 0.000001 0.00088 MRR x x x x x x x x x x x          (64) 2 1 2 3 4 1 2 2 2 3 0.756 0.002 0.0569 0.0133 0.000045 0.000036 0.00273 0.002 PC x x x x x x x         (65) Table 5. Process variables with levels Process Variables Level 1 Level 2 Level 3 Level 4  1x Pulse on period (μs) 13 20 27 36  2x Pulse off period (μs) 4 6 8 10  3x Current (A) 1 2 4 5  4x Wire speed (rpm) 350 700 1050 1400 4.2.1. Single objective optimization In this case, two responses, i.e., MRR and PC, are optimized separately engaging the eight metaheuristic optimization algorithms. In other words, we intend to discover the optimal parametric condition for both the responses separately using the competing algorithms. The goal is to maximize the MRR and minimize the PC subjected to the imposed constraints as follows: 1 13 36,x  2 4 10x  , 3 1 5,x  and 4 350 1400x  . Table 6 exhibits the single objective optimization solutions derived by the eight metaheuristic optimizers. It is observed that all the competing optimizers furnished improved optimal MRR than the optimal MRR derived by (Devarajaiah & Muthumari, 2018). Unlike the SHO algorithm, all the algorithms have drastically improved the MRR from its initial value of 0.0647gm/min (Devarajaiah & Muthumari, 2018). From the optimal PC values as registered by the eight competitor algorithms (shown in Table 6), it is worth pointing that all the algorithms deliver almost similar performance. Besides, the optimal PC provided by the optimizers are found to be relatively better than the optimal PC endorsed by (Devarajaiah & Muthumari, 2018). When the convergence traits of the eight competitor algorithms are analyzed (shown in Figure 5), it is noted that HHO, and TLBO have accelerated tendency to converge faster to global optimal solution implying better exploitation potential of the two algorithms. Saha et al./Oper. Res. Eng. Sci. Theor. Appl. First online Table 6. Single objective optimization outcomes. Optimizer Response Optimal value Pulse duration Pulse off period Current Wire Speed (Devarajaiah & Muthumari, 2018). MRR 0.0647 27 4 4 1400 PC 0.589 27 8 1 700 ALO MRR 0.0825 29.92 4 5 1400 PC 0.523 13 10 1 350 ChoA MRR 0.0825 29.92 4 5 1400 PC 0.523 13 10 1 350 HHO MRR 0.0825 29.92 4 5 1400 PC 0.523 13 10 1 350 MPA MRR 0.0825 29.92 4 5 1400 PC 0.52313 13 10 1 350 SHO MRR 0.0669 21.17 4 5 350 PC 0.52313 13 10 1 350 MFO MRR 0.0825 29.92 4 5 1400 PC 0.52313 13 10 1 350 PSO MRR 0.0825 29.92 4 5 1400 PC 0.52313 13 10 1 350 TLBO MRR 0.0825 29.92 4 5 1400 PC 0.52313 13 10 1 350 Figure 5. Convergence behavior of ALO, ChoA, MFO, SHO, HHO, MPA, PSO, and TLBO for MRR. 4.2.2. Multiple objective optimization For multiple performance optimization, Eq. (66) is exploited as the objective function which is displayed below: Comparative Analysis of Metaheuristic Optimizers in the Performance Optimization of Wire Electric Discharge Machining Processes       1 2 min max PC MRR Y PC MRR Y Y Min w w  (66) where 1 w and 2w are the preference weights to PC and MRR respectively. In the present paper, we have assigned equal weights to PC and MRR, respectively. min PC is the minimum power consumption, and max MRR is the maximum material removal rate. The values are attained from the single-objective optimization results. Table 7 reported the findings of multiple performance optimization exploiting eight metaheuristic optimization algorithms. ALO, MFO, ChoA, HHO, TLBO, and MPA have been discovered to seek the best trade-off condition for both the performance attributes as corroborated when compared with the reported results by (Devarajaiah & Muthumari, 2018). Substantial improvement in MRR with a marginal decrement in the performance of PC is evident utilizing these algorithms. Conversely, SHO and PSO algorithms are found to deliver mediocre optimal performances. When the convergence traits of the eight competing algorithms are compared, it is discovered that the HHO algorithm rapidly converges to the minimal function value at minimal generation (as shown in Figure 6), confirming the HHO algorithm's exceptional exploitation capability. The comparison of average computational time for the eight algorithms for multiple performance optimization is exhibited in Table 8. It is noted from Table 8 that the TLBO algorithm requires the least computational time to reach the optimality condition. Table 7. Multiple-objective optimization outcomes Optimizer Response Optimal Value Y Pulse on time Pulse off time Current Wire Speed (Devarajaiah & Muthumari, 2018) MRR 0.0348 20 6 2 1050 PC 0.625 ALO MRR 0.049 0.53072 16.12 6.71 5 350 PC 0.670 MFO MRR 0.048 0.53228 16.09 6.77 5 350 PC 0.669 ChoA MRR 0.049 0.53072 16.12 6.71 5 350 PC 0.670 HHO MRR 0.049 0.53072 16.12 6.71 5 350 PC 0.670 SHO MRR 0.044 0.53089 15.75 6.40 5 350 PC 0.676 PSO MRR 0.0255 0.53199 16.12 10 2.88 350 PC 0.572 TLBO MRR 0.049 0.53072 16.12 6.71 5 350 PC 0.670 MPA MRR 0.049 0.53072 16.12 6.71 5 350 PC 0.670 Saha et al./Oper. Res. Eng. Sci. Theor. Appl. First online Figure 6. Convergence behavior of ALO, ChoA, MFO, SHO, HHO, MPA, PSO, and TLBO for multi-objective function. Table 8. Average computation time for the eight optimizers Optimizer Average computational time (secs) ALO 5.070851 ChoA 1.664359 MFO 1.119026 HHO 1.052527 MPA 0.720689 SHO 0.1390570 PSO 0.2506701 TLBO 0.3054067 5. Statistical and sensitivity Analysis To summarize the robustness and performance stability of the metaheuristic optimizers, we retrieved the statistical data evaluated for all the optimizers while dealing with multi-objective optimization problems in the two considered cases. The statistical metrics such as the mean and co-efficient of variation are evaluated after the execution of the optimizers for 30 number of runs. Comparative Analysis of Metaheuristic Optimizers in the Performance Optimization of Wire Electric Discharge Machining Processes Figure 7. Bar plots revealing mean value of objective function derived using optimizers for the two cases. Case-1 Performance optimization in WEDM of A286 superalloy; Case- 2 Performance optimization in WEDM of Ti-6Al-4V superalloy The mean value of objective function procured using the optimizers for the two cases are plotted in the form of 3D bar plots in Figure (7). From Figure (7), it is clear that HHO provides the least mean value of objective function for all the cases corroborating the robustness of HHO over the other competing algorithms in terms of tracking the global optimal solution. Figure (8) portrays the coefficient of variation procured using the optimizers for the two cases in the form of 3D bar plots. It is noted that HHO exhibits the least coefficient of variation for all the cases corroborating that HHO optimizer has the maximum stability over its other competitors. To summarize, it can also be inferred that HHO exhibits robustness in bolstering an adequate balance between two phases (i.e., exploration and exploitation). Figure 8. Bar plots revealing coefficient of variation derived using optimizers for the two cases. Case-1 Performance optimization in WEDM of A286 superalloy; Case-2 Performance optimization in WEDM of Ti-6Al-4V superalloy Saha et al./Oper. Res. Eng. Sci. Theor. Appl. First online 6. Performance of optimizers on benchmark test functions As corroborated from the different cases investigated in WEDM on the superiority of HHO over the other competing algorithms, we further intend to investigate the robustness of HHO by comparing its performance with the other competing optimizers on standard test functions. The details of the standard test functions can be found in the literature (Zhu & Kwong, 2010). The standard test functions considered in the present work are the Sphere function, and the Generalized Rastrigin’s function, respectively. Sphere function is a unimodal function which contains only one optimum point. Whereas Generalized Rastrigin’s function is a multimodal function which contains many local optima but only one global optimum. The mathematical description of the functions is illustrated below: Sphere function dim 30 2 1 ( ) -100 100 i i i F x x x      (67) where dim denotes the dimension of the solution space, and -100 100 i x  depicts the initial range of i x . Generalized Rastrigin’s function     dim 30 2 1 10 cos 2 10 -5.12 5.12 i i i i F x x x x          (68) where dim refers the dimension of the solution space, and -5.12 5.12 i x  depicts the initial range of i x . Table 9 depicts the results (Average ± standard deviation) of the eight optimizers on the optimization of benchmark test functions. HHO algorithm is found to be superior over the other optimizers in terms of robustness. Table 9. Performance comparison on benchmark test functions Optimizer ALO ChoA MFO HH O MPA SHO PSO TLBO Sphere Function 5.30E- 09 ±3.4035 E-09 2.60E- 06 ±6.43 E-06 3.38E -13 ±5.21 E-13 0 ± 0 2.76E -21 ±3.66 E-21 6.61 E- 100 ±1.5 E-99 3.39E-07 ±8.40871 E-07 2.83E -89 ±6.78 E-89 Generalize d Rastrigin’s Function 23.4809 8± 12.6612 3 10.63 647 ± 10.56 995 20.70 2215 ± 11.50 4771 2 0 ± 0 0 ± 0 0 ± 0 12.43697 ± 6.52438 2.45E +00 ± 2.154 3798 09 Comparative Analysis of Metaheuristic Optimizers in the Performance Optimization of Wire Electric Discharge Machining Processes 7. Conclusions WEDM is a complicated machining process. Machining in WEDM at any parametric combination does not result in enhanced performance outcomes. For improved performances in WEDM, machining must be carried out in compliance with the optimal parametric settings. Metaheuristic optimizers have grown in prominence due to their potential to provide global optimal solutions. However, the use of recently reported metaheuristic optimizers in non-traditional machining techniques has received little attention. The novelty of the present article is to explore the six recently reported metaheuristic optimizers namely the ant lion optimization (ALO), chimp optimization algorithm (ChoA), moth flame optimization (MFO), spotted hyena optimization (SHO), Harris Hawk optimization algorithm (HHO), and Marine predator algorithm (MPA) in the optimization of WEDM performances for two WEDM processes. Two well-known existing optimization approaches (i.e., PSO and TLBO), are also included in this study to allow a fair evaluation of the algorithms' performance. The comparison between the eight algorithms are carried out in terms of the optimal solutions, convergence rate, and average computational time. The goal of the comparative analyses is to select the robust optimizer. It is observed that the HHO algorithm is extremely robust in yielding global optimal solutions. 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Gbest-guided artificial bee colony algorithm for numerical function optimization. Applied Mathematics and Computation, 217, 3166- 3173. https://doi.org/10.1016/j.amc.2010.08.049 © 2022 by the authors. Submitted for possible open access publication under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/). https://doi.org/10.1016/j.measurement.2019.04.003 https://doi.org/10.1016/0890-6955(95)00019-T https://doi.org/10.1016/0890-6955(95)00019-T https://doi.org/10.1016/j.energy.2014.07.078 https://doi.org/10.1109/4235.585893 https://doi.org/10.1109/JSEN.2021.3091619 https://doi.org/10.1016/j.aej.2021.11.011 https://doi.org/10.1016/j.amc.2010.08.049 COMPARATIVE ANALYSIS OF METAHEURISTIC OPTIMIZERS IN THE PERFORMANCE OPTIMIZATION OF WIRE ELECTRIC DISCHARGE MACHINING PROCESSES Subhankar Saha 1, 2, Saikat Ranjan Maity 1*, Sudip Dey 1 1. Introduction 2. Literature Review 3. Metaheuristic optimization algorithms 3.1. Teaching Learning Based Optimization (TLBO) 3.1.1. Teacher Phase 3.1.2. Learner Phase 3.2. Particle Swarm Optimization 3.3. Spotted Hyena optimizer 3.3.1. Prey encircling 3.3.2. Prey hunting and Prey searching 3.4. Chimp Optimization algorithm 3.5. Moth Flame Optimization 3.6. Ant Lion Optimization 3.7. Marine Predator Algorithm (MPA) 3.7.1. Initialization 3.7.2. Phase 1 3.7.3. Phase 2 3.7.4. Phase 3 3.7.5. Finishing 3.8. Harris Hawk Optimization 4. WEDM performance optimization 4.1. Case 1 Performance optimization in WEDM of A286 superalloy 4.1.1. Single-objective optimization 4.1.2. Multiple objective optimization 4.2. Case 2 Performance optimization in WEDM of Ti-6Al-4V alloy 4.2.1. Single objective optimization 4.2.2. Multiple objective optimization 5. Statistical and sensitivity Analysis 6. Performance of optimizers on benchmark test functions 7. Conclusions