Machinability analysis in wire-EDM of cryogenically treated Ti6Al4V alloy and multi-objective optimization using MOAVOA and MOGA

Single objective optimization using GA and AVOA

This section presents a single objective optimization of MRR_CT and Ra_CT using GA and AVOA algorithms. GA is a well-known optimization technique inspired by the principles of natural selection and genetics. It simulates the process of evolution to find optimal or near-optimal solutions to complex real-life problems and works with a population of potential solutions, evolving them over generations through operations such as selection, crossover, and mutation. By iteratively improving the solutions based on a fitness function, GA effectively explores the search space to discover high-quality solutions⁵⁵. On the other hand, the African vultures optimization algorithm (AVOA) is a metaheuristic inspired by the scavenging behavior of African vultures. It emulates how vultures search for food using visual and sensory cues, striking a balance between exploring new areas and refining known good solutions. The algorithm starts with a population of potential solutions (vultures), evaluates their effectiveness, and adjusts their positions to enhance the quality of solutions⁴⁴. The convergence plot in Fig. 12 shows that AVOA converges faster than GA and outperforms it in finding minimum Ra_CT and maximum MRR_CT values. The minimum Ra_CT values obtained using GA and AVOA are 4.85 and 4.81 respectively while for MRR_CT the values are 6.77 and 6.84 respectively.

MOGA (multi objective genetic algorithm)

In 1993, a groundbreaking advancement emerged in genetic algorithms with the introduction of the multi objective genetic algorithm (MOGA)⁵⁶. This innovative approach prioritized identifying and preserving non-dominated solutions within a GA population, enhancing both solution optimality and diversity. Unlike traditional GA, MOGA redefined the fitness assignment process, focusing explicitly on non-dominated solutions while fostering a diverse array of solutions. MOGA stands out as a potent direct search technique tailored for tackling complex multi-objective optimization problems. Leveraging genetic algorithm principles, MOGA’s utilize their population-centric nature to explore multiple Pareto-optimal solutions simultaneously, contrasting with methods fixed on singular Pareto optimality. Successfully navigating multi-objective optimization via genetic algorithms requires a nuanced balance of elitism and solution diversity throughout the evolutionary process.

Procedure of MOGA

A straightforward method for combining several goal variables into an integer performance function is the weighted sum technique.

Here, n is the number of objective functions, x is an integer (i.e., an individual), f(x) is the sum of the performance function, and f_i(x) is the i-th objective function. W_i is a constant weight for f_i(x). The search direction in genetic algorithms becomes fixed if we use constant weights wi in Eq. (4). Therefore, using a variety of search strategies, we provide a selection process that includes random weights to investigate Pareto optimum solutions. When selecting a couple of strings for a crossover operation, we assign an arbitrary real number to each weight as follows:

A positive random integer is shown here by random j(·). Equation (5) indicates that w_i assumes a real value inside the range [0, 1]. Step by step procedures of MOGA are as follows:

Step 1: Initialization

Generate an initial population comprising N_pop strings, where N_pop represents the number of strings in each population.

Step 2: Evaluation

Compute the objective function values for the generated strings. Update a preliminary set of Pareto optimal solutions.

Step 3: Selection

Determine the fitness value of each string using random weights. Selecting pairs of strings from the current population based on a selection probability.

Repeat this process to select N_pop/2 pairs of strings from the current population.

Step 4: Crossover

For each selected pair, perform a crossover operation to create two new strings. Generate N pop new strings through the crossover operation. (It’s a genetic operation where two parent solutions are combined to create one or more offspring solutions. This is typically done by exchanging information between the parents to explore new areas of the search space.)

Step 5: Mutation

Apply a mutation operation with a specified mutation probability for each bit value of the strings generated by the crossover operation. (It’s a genetic operation where a small random change is applied to a single solution in order to introduce diversity and prevent premature convergence. Mutation helps explore new regions of the search space that might lead to better solutions).

Step 6: Elitist strategy

Randomly remove N_elite strings from the set of N_pop strings generated in the previous steps. Replace them with N_elite strings randomly chosen from a preliminary set of Pareto optimal solutions.

Step 7: Termination test

If a predetermined stopping condition is not met, return to Step 1.

Step 8: User selection

Present the final set of Pareto optimal solutions to the decision maker. The best solution is then chosen according to the decision maker’s preference. The flowchart is shown in Fig. 13.

Multi objective African vulture optimization algorithm (MOAVOA)

The multi objective African vultures optimization algorithm (MOAVOA) was introduced by Khodadadi et al.⁵⁰ for solving different real life non-trivial multi-objective problems. The MOAVOA involves integration of three different mechanisms such as archive, grid mechanism and leader selection mechanism. African vultures’ hunting and navigational habits served as the model for the innovative metaheuristic technique known as MOAVOA. It comprises exploration and exploitation phases, with the latter further divided into cooperative and competitive phases. MOAVOA determines its current phase primarily based on the hunger level F of the vulture, transitioning to exploration if hunger is below a threshold (F < r₁), cooperative if above another threshold (F > r₂), and otherwise enters the competition phase. MOAVOA appoints two vulture leaders to guide the rest of the vulture population. Unlike traditional methods, MOAVOA recognizes that there isn’t just one best solution, but rather a range of good solutions. Therefore, it designates two sets of Socially dominant vultures. The first batch includes all non-dominated solutions within the group. Each vulture’s first Socially dominant is selected from this set based on diversity and convergence metrics. The tournament selection process ensures that the optimal solutions are selected, with priority given to lower density values and convergence measures. If there are ties, the solution with the preferable diversity measure is selected. The second set comprises the best solutions for each objective function across the entire vulture population. This selection process aims to identify solutions that are closest to the Pareto Front, guiding the convergence of each iteration’s hyperplane towards it. Each vulture in this set is tasked with guiding new vultures towards the ideal point of the Pareto Front. The selection of the second Socially dominant for each vulture involves a random assignment from this set. The selection processes for both sets are detailed in the algorithms provided. MOAVOA employs a set of reference points to locate evenly distributed solutions and those in close proximity to the Pareto front.

Procedure of MOAVOA

In the iterative process of MOAVOA, several crucial steps are taken. Firstly, socially dominant are identified to facilitate the movement of solutions within the decision space. Secondly, to promote diversity and prevent premature convergence, mutation is applied to a portion of the vulture positions. Thirdly, environmental selection is conducted to choose the top vultures for the subsequent generation. Lastly, the external Arc (Archive) is updated to harbor only non-dominated solutions. These steps are conducted until the maximum iteration limit is reached.

In Eq. (6), the probability of selecting certain vultures to relocate others towards optimal solutions is calculated, with parameters \(L_ troy electrical company\) and \(L_ electrical installation\) ranging from 0 to 1 and the sum of both parameters 1.

The chance that each group will select the optimal solution using the Roulette wheel technique is found using Eq. (7).

Equation (8) simulates the shift in vulture behavior from the stages of exploration to exploitation, based on the pace at which vultures get satisfied or their appetite declines. This formula represents the declining trend in satiation and directs behavior in that direction. \(F\) indicates that the vultures are full. \(z\) ∈ (− 1, 1) is a random number. \(H\) ∈ (− 2, 2) is a random number, while \(rand_ troy electrical company\), \(rand_{p1}\) varies ∈ (0, 1). If \(z\) falls below 0, the vulture is considered starved, while reaching 0 indicates satiation.

Before the optimization process begins, the parameter w is first given a fixed value. It serves as a barometer for how much the optimization procedure influences the phases of discovery and development. As w increases in value, there is a greater chance of going into the exploration phase during the last phases of optimization. Conversely, when the parameter w is reduced, the probability of entering the exploration phase falls. The values given to the parameter w determine how the objective function, F, and the iteration number, t, behave during the optimization process.

To determine which strategy to employ during the exploration phase, rand_P1 ∈ (0, 1) is generated. If P₁ ≥ rand_P1, Eq. (12) is utilized. However, if P₁ < rand_P1 Eq. (13) is applied.

In Eq. (11), vultures haphazardly search for food in the vicinity of the best civilizations of either of the two tribes. Here, F represents the rate of satiation, which was determined by applying Eq. (9) in the current iteration, and \(P\left( {i + 1} \right)\) represents the position coordinate of the vulture in the subsequent iteration. In this version, one of the best vultures chosen by Eq. (6) is shown in Eq. (12) as R(i). Moreover, X indicates the area where vultures sporadically relocate to protect food sources from other predators. X is a coefficient vector that increases random movement; it is different for every iteration and can be computed as X = 2 × rand, where rand ∈ (0, 1).

R(i) in Eq. (13) represents a single of the finest vultures chosen in the present iteration using Eq. (6). The vulture hunger rate, F, in this iteration is defined by Eq. (9) and rand2 ∈ (0, 1). The variables’ lower and upper bounds are represented by the variables lb and ub, respectively. When Eq. (13) is employed, the MOAVOA generates a basic model for randomly generating solutions in the range specified by lb and ub. The randomness coefficient is amplified by rand₃, which affects the distribution of solutions. Random motion is added to lb by rand₃, which favors solutions with comparable patterns as it gets closer to 1. This increases the degree of unpredictability in the search environment, encouraging variety and travel across different areas of the search field. Simple models are utilized in MOAVOA to represent vulture motion. At this stage, MOAVOA efficiency is examined. When |F|< 1, the MOAVOA transitions into the exploitation phase, comprising two sub-phases, each employing distinct strategies. The selection of strategies within each sub-phase is controlled by parameters P₂ and P₃, with P₂ determining the strategies in the first sub-phase and P₃ in the second. Both parameters ∈ (0, 1) and are set prior to the search operation. The vultures’ foraging behaviors are mathematically formulated for problem-solving purposes. The Initial sub phase of the exploitation phase is initiated by the MOAVOA when |F|∈ (0.5, 1). In this sub-phase, two distinct strategies are employed: siege-fighting and rotational flight. The selected and preset procedure is determined by P₂ ∈ (0, 1). The first step in this sub-phase is to generate a random number, rand_P2 ∈ (0, 1). Gradually, the siege-fight strategy is applied if P₂ ≥ rand_P2. On the other hand, the rotational flying method is used if P₂ < rand_P2. Equation (14) provides specifics on this selection procedure.

When the absolute value of |F|≥ 0.5, indicating that vultures are relatively satiated, competition for food intensifies. In such scenarios, where multiple vultures converge on a single food source, conflicts over food acquisition escalate. Physically dominant vultures typically prefer not to share food, exhibiting a reluctance to cooperate with others. On the other hand, less powerful vultures use a different approach, congregating nearer to more powerful ones in an effort to wear them out and steal food. Equations (15) and (16) are used to represent this behavior.

D(i) is obtained by applying Eq. (12) to the vulture hunger rate F, which is provided by Eq. (9). The purpose of rand₄ ∈ (0, 1) is to introduce unpredictability. The distance that exists between the vulture and a single of the most desirable vultures in each group is calculated using Eq. (16), where P(i) represents the vulture’s present vector position. In the current iteration, R(i) is selected using Eq. (6) as one of the best vultures from both groups.

Vultures’ rotational waving simulates spiral motion by using repeated spiraling motions. Spiral equations between all vultures and one of the top two vultures are used to numerically simulate this motion.

Equations (17) and (18) describe circular flight, while Eq. (6) yields the displacement vector of a top vulture, represented by R(i). Furthermore, random numbers \(rand_ industrial electrician\) and \(rand_ electrical maintenance\) ∈ (0, 1) are used. Equation (17) is used to calculate S₁ and S₂, and Eq. (18) is used to update the locations of the vultures. Different vulture species congregate around the food supply during the second stage of exploitation, engaging in violent conflict and besieging each other in an effort to get food. When |F|< 0.5, this phase begins. rand_P3 ∈ (0, 1) at first. The tactic is to amass several kinds of vultures over the food supply if P₃ ≥ rand_P3. On the other hand, aggressive tactics for siege and warfare are used if P₃ < rand_P3. Equation (19) provides an explanation of this procedure.

Vulture species congregating at a particular food source occurs when vultures are starving and face intense competition for food. This phenomenon is modeled using Eqs. (20) and (21).

\(BestVulture_ troy electrical company \left( i \right)\) in Eq. (21) represents the most effective vulture from the first set in the present iteration, and \(BestVulture_ electrical installation \left( i \right)\) represents the most effective vulture from the second group. F is the vulture hunger rate as calculated by Eq. (9), and P(i) is the vector position of a vulture at any given moment.

Equation (22), in which \(A_ troy electrical company\) and \(A_ electrical installation\) are obtained from Eqs. (20) and (21), is then used to calculate the aggregate of all vultures. The vulture’s position vector in the subsequent iteration is \(P\left( {i + 1} \right)\).

When there is intense rivalry for food, as shown when |F|< 0.5, dominant vultures lose energy and become weak, making it difficult for them to successfully compete with other vultures. Analysis is done on how every vulture moves in the direction of a food source. Occasionally, vultures are hungry. Because of their intense competition for food, many vulture breeds may congregate at just one food source. Equations (21) and (22) are used to explain this vulture behavior.

As |F|< 0.5, dominant vultures lose their ability to battle other vultures and become weak and famished. However, other vultures approach the dominant vulture from different directions, acting aggressively as they search for food. Equation (23) provides a description of this movement pattern.

Equation (16) is used to calculate the distance, d
(24)