Optimization of Different Metal Casting Processes Using Three Simple and Efficient Advanced Algorithms

Abstract

This paper presents three simple and efficient advanced optimization algorithms, namely the best–worst–random (BWR), best–mean–random (BMR), and best–mean–worst–random (BMWR) algorithms designed to address unconstrained and constrained single- and multi-objective optimization tasks of the metal casting processes. The effectiveness of the algorithms is demonstrated through real case studies, including (i) optimization of a lost foam casting process for producing a fifth wheel coupling shell from EN-GJS-400-18 ductile iron, (ii) optimization of process parameters of die casting of A360 Al-alloy, (iii) optimization of wear rate in AA7178 alloy reinforced with nano-SiC particles fabricated via the stir-casting process, (iv) two-objectives optimization of a low-pressure casting process using a sand mold for producing A356 engine block, and (v) four-objectives optimization of a squeeze casting process for LM20 material. Results demonstrate that the proposed algorithms consistently achieve faster convergence, superior solution quality, and reduced function evaluations compared to simulation software (ProCAST, CAE, and FEA) and established metaheuristics (ABC, Rao-1, PSO, NSGA-II, and GA). For single-objective problems, BWR, BMR, and BMWR yield nearly identical solutions, whereas in multi-objective tasks, their behaviors diverge, offering well-distributed Pareto fronts and improved convergence. These findings establish BWR, BMR, and BMWR as efficient and robust optimizers, positioning them as promising decision support tools for industrial metal casting.

1. Introduction

Metal casting is one of the most versatile and widely employed manufacturing processes for producing complex shapes with high dimensional accuracy and economic viability. It serves as a foundational technique across industries such as automotive, aerospace, railways, and defense, where intricate geometries, weight reduction, and mechanical performance are critical. Despite its advantages, casting processes are often susceptible to a range of defects such as porosity, shrinkage, cold shuts, and incomplete filling. These defects arise due to the complex thermal–fluid–solid interactions, process parameter uncertainties, and material behaviors involved in casting. Consequently, there is an increasing demand for optimizing casting parameters to ensure high-quality cast components with minimal rework and reduced material waste.

In recent years, researchers have utilized various computational, statistical, and artificial intelligence-based approaches to model, simulate, and optimize casting processes. Xu et al. [1] developed a coupled response surface methodology (RSM)–Monte Carlo framework to predict and optimize the porosity and grain size of A360 aluminum alloy die castings. Their study demonstrated high prediction accuracy and validated the results. In the context of squeeze casting, Li et al. [2] combined finite element method (FEM) simulations with RSM to optimize mold temperature, alloy temperature, and pressure parameters for large aluminum alloy wheel hubs. Their work achieved substantial weight reduction and porosity control, showcasing the potential of simulation-based optimization.

The role of intelligent design integration was demonstrated by Triller et al. [3], who incorporated topology optimization, crashworthiness simulations, and response surface models enhanced by machine learning clustering to optimize automotive mega-castings produced via high-pressure die casting (HPDC). Their methodology offered a scalable, efficient, and physically interpretable workflow for complex structural components. Deng et al. [4] used non-dominated sorting genetic algorithm-II (NSGA-II) and RSM to simultaneously optimize solidification time and shrinkage porosity in A356 engine blocks, resulting in improvements in thermal behavior and casting integrity.

In the domain of composite casting, Bharat et al. [5] utilized multiple metaheuristic algorithms such as artificial bee colony (ABC), particle swarm optimization (PSO), and Rao-1 algorithms to optimize wear rate in nano-SiC-reinforced AA7178 matrix composites fabricated via stir casting. Their results emphasized the significance of algorithm selection and demonstrated Rao-1’s effectiveness. Similarly, He et al. [6] developed a two-stage evolutionary strategy using NSGA-II for gating and genetic algorithm (GA) for riser optimization in ZL104 sand castings, effectively improving yield and minimizing shrinkage porosity.

Optimization of TiB₂-reinforced AA6061 composites fabricated via stir casting was undertaken by Deshmukh et al. [7], who employed RSM to tune melt temperature, reinforcement level, and stirring speed, yielding significant mechanical improvements validated by SEM. In another advancement, Kavitha and Raja [8] optimized insert surface roughness and pouring conditions to maximize bond strength in (Cp)Al–SS304 bimetallic castings using an RSM–GA hybrid approach, resulting in enhanced metallurgical bonding.

In terms of innovative mold design, Patel and Nayak [9] proposed and tested the use of a copper slag mold for casting A356 aluminum alloy. Their study, based on thermal analysis and experimental trials, demonstrated improved cooling rates and casting integrity. The copper slag mold showed potential as a sustainable and cost-effective alternative to traditional mold materials, with benefits in microstructure refinement and reduced porosity formation.

Hybrid modeling using RSM and artificial neural networks (ANNs) was applied by Panicker and Kuriakose [10] to enhance prediction accuracy and optimize squeeze casting of LM20 alloy. Their studies highlighted the importance of pressurization delay and insert parameters in defect control, mechanical property enhancement, and interface bonding.

Modeling and optimization efforts in squeeze casting were also advanced by Patel et al. [11], who developed regression-based models for multiple quality responses including hardness, impact strength, and density. They used multi-objective optimization via genetic algorithms to determine optimal process settings, highlighting the critical influence of squeeze pressure, pouring temperature, and punch velocity. Their results provided a structured framework for enhancing mechanical performance and consistency in squeeze cast components.

Defect reduction in lost foam casting has also been a key focus, with Li et al. [12] employing ProCAST simulations and Box–Behnken RSM to optimize pouring temperature, pouring speed, and ferro-static head pressure for a ductile iron fifth wheel coupling shell. Their work significantly reduced shrinkage defects and validated results through industrial trials.

Taken together, these studies underscore the critical role of optimization techniques—ranging from statistical design of experiments to machine learning and metaheuristics—in advancing casting technology. They also highlight the diverse casting methods (sand casting, squeeze casting, die casting, stir casting, investment casting, and bimetallic casting) where such approaches have demonstrated substantial improvements in product quality, performance, and manufacturability.

The present study applies metaheuristic algorithms—advanced optimization techniques—to optimize various metal casting processes. Metaheuristics are particularly effective for tackling complex, nonlinear, and high-dimensional search spaces where conventional optimization methods often underperform. These algorithms iteratively refine a population of candidate solutions, balancing exploration (global search) and exploitation (local refinement) to progressively approach global optimality.

Metaheuristics have gained widespread popularity owing to the following inherent advantages: they are problem-independent, do not rely on gradient information, are well-suited for both constrained and unconstrained optimization, and offer effective strategies for handling multi-objective formulations. Moreover, their flexibility and adaptability make them suitable for a wide range of engineering problems [13,14,15,16]. However, despite their advantages, metaheuristics also have notable limitations that warrant careful consideration. A major challenge lies in the lack of formal theoretical guarantees for convergence to the global optimum—most metaheuristics are empirically driven and problem-specific. Furthermore, their success heavily depends on the fine-tuning of algorithm-specific control parameters, which can significantly impact convergence behavior, solution quality, and computational cost. For example, genetic algorithms (GAs) require calibration of parameters such as population size, selection strategy (e.g., tournament, roulette wheel), crossover probability, and mutation rate; PSO is sensitive to its inertia weight and the cognitive and social coefficients; ACO demands careful control of pheromone evaporation rates, as well as the relative influence of heuristic desirability versus accumulated pheromone; and artificial bee colony (ABC) algorithm tends to have slower convergence near the global optimum and requires proper setting of colony size and employed/onlooker bee ratios.

The sensitivity of these algorithms to parameter settings can pose a barrier to practitioners, especially in complex industrial applications where trial-and-error calibration is time-consuming and computationally expensive. Although a few parameter-free methods—such as the Rao and Jaya algorithms—have been proposed to mitigate this issue, the majority of metaheuristics still demand meticulous parameter tuning for satisfactory performance [17]. When poorly tuned, these algorithms may produce suboptimal solutions or incur excessive computational overhead.

Compounding this challenge is the recent proliferation of metaphor-based metaheuristics, inspired by natural, physical, biological, or sociocultural processes. While innovation in algorithm design is commendable, many of these approaches are constructed on loosely justified metaphors with minimal scientific grounding. Often, the metaphor serves more as an aesthetic or thematic overlay rather than a meaningful contributor to algorithmic structure or performance [18,19,20,21]. This trend risks diluting scientific rigor and may obscure reproducibility or comparability across methods.

In light of these observations, very recently, Rao and Davim [22] proposed two metaphor-free and parameter-free algorithms named as best–worst–random (BWR) and best–mean–random (BMR) algorithms for optimizing various manufacturing processes, including ultra-precision turning, friction stir processing, wire arc additive manufacturing, and laser powder bed fusion.

The present work pursues the following key objectives:

• To extend the application of the very recently developed BWR and BMR algorithms to the single- and multi-objective optimization of different metal casting processes.
• To introduce a new simple, metaphor-independent, and parameter-free algorithm, the best–mean–worst–random (BMWR) algorithm, along with its multi-objective variant and to apply the same to the single- and multi-objective optimization of different metal casting processes.
• To evaluate the convergence behavior, robustness, and solution quality of the proposed algorithms in solving the optimization problems.
• To apply a simple decision-making method as a post-optimization decision-making tool for selecting the most balanced compromise solution from the Pareto front in multi-objective contexts of metal casting processes.

The following section outlines the proposed optimization algorithms and their working principles.

2. Materials and Methods: Proposed Algorithms

2.1. Proposed Algorithms for Single-Objective Constrained Optimization Problems

This section presents three simple yet efficient optimization algorithms: best–worst–random (BWR), best–mean–random (BMR), and best–mean–worst–random (BMWR). These methods are metaphor-free and do not require tuning of algorithm-specific control parameters. They rely on basic arithmetic operations to explore the search space within a population of candidate solutions and are designed to handle constrained single-objective optimization problems.

2.1.1. BWR Algorithm

Let f(x) denotes the objective function to be minimized or maximized. Assume the problem involves m design variables and a population of c candidate solutions. Each variable x is bounded within a lower limit Lx and an upper limit Hx. The algorithm starts by randomly initializing a population of candidate solutions. If any variable exceeds its bounds, it is clipped to remain within the permissible range. Each candidate is then evaluated for feasibility. If a solution violates a constraint c_j(x), a penalty is imposed, typically as p_j(x) = c_j(x)² or │c_j(x)│ or any higher value as decided by the user. The penalized objective function, M(x), is computed as M(x) = f(x) + ∑ p_j(x) for minimization and M(x) = f(x) − ∑ p_j(x) for maximization.

During each iteration i for a given candidate k let X{x, k, } be the value of the x-th variable. Also, let the best, worst, mean, and random values of variable v be denoted by X{x,b,i}, X{x,w,i}, X{x,m,i}, and X{x,r,i}, respectively. Let r1, r2, r3, and r4 be random numbers uniformly drawn from [0, 1], and F be a random factor, taking a value of either 1 or 2 (either 1 or 2). The update rules are as follows:

If r4 > 0.5: X′{x, k, i} = X{x, k, i} + r1{x, i}(X{x, b, i} − F ∗ X{x, r, i}) − r2{x, i}(X{x, w, i} − X{x, r, i})

(1)

Else: X′{x,k,i} = Hx − (Hx − Lx) ∗ r3

(2)

After generating the new solution, constraints c_j(x) are checked again, penalties are applied as needed, and the penalized objective M(x) is calculated. If the new solution yields a better M(x) than the current one, it is retained for the next iteration. This process repeats until the termination condition—typically a predefined maximum number of iterations—is met.

2.1.2. BMR Algorithm

The BMR algorithm follows the same structure as BWR but uses the mean value instead of the worst solution in the update equation. This can be seen in the following:

If r4 > 0.5: X′{x, k, i} = X{x, k, i} + r1{x, i}(X{x, b, i} − F ∗ X{x, m, i}) + r2{x, i}(X{x, b, i} − X{x, r, i})

(3)

else, use the same Equation (2) as in BWR for r4 ≤ 0.5.

The subsequent steps—constraint evaluation, penalized objective computation, and population updating—are identical to those in the BWR algorithm.

Based on the BWR and BMR algorithms, a new algorithm named best–mean–worst–random (BMWR) is proposed now in this paper.

2.1.3. Best–Mean–Worst–Random (BMWR) Algorithm

The BMWR algorithm combines ideas from both BWR and BMR by incorporating best, mean, worst, and random values in its update step, which is as follows:

If r4 > 0.5: X′{x, k, i} = X{x, k, i} + r1{x, i}(X{x, b, i} − F ∗ X{x, m, i}) − r2{x, i}(X{x, w, i} − X{x, r, i})

(4)

else, use the same Equation (2) as in BWR for r4 ≤ 0.5.

All other operations follow the same approach as the BWR algorithm. Figure 1 presents the flowchart of the BWR, BMR, and BMWR algorithms.

The unified pseudocode for the BWR, BMR, and the newly proposed BMWR algorithms is given in Figure 2.

2.1.4. Demonstration of the BWR, BMR, and BMWR Algorithms on a Constrained Benchmark Problem

Demonstration of the BWR Algorithm

To demonstrate the effectiveness and underlying mechanisms of the proposed BWR, BMR, and BMWR algorithms, a widely recognized benchmark problem is utilized. The constrained objective function is defined as Equation (5):

f(x): Minimize x1² + x2²

(5)

Subject to the constraints given by Equations (6)–(8):

c₁(x)= x1 + x2 − 1 ≤ 0

(6)

c₂(x)= x1 − 0.2 ≥ 0

(7)

c₃(x)= x2 − 0.3 ≥ 0

(8)

The variable bounds are given by Equation (9):

0 ≤ x1, x2 ≤1

(9)

The global minimum value for this constrained problem is known to be f(x) = 0.13 for x1 = 0.2 and x2 = 0.3.

Step 1: The objective function involves two design variables, x1 and x2. For demonstration purposes, a population size of 5 is used, with the termination criterion set at 1000 iterations. The randomly generated initial solutions are presented in

Table 1. Initially generated random population.

Solution	x1	x2	f(x)	c1(x)	p1	c2(x)	p2	c3(x)	p3	M(x)
1	0.37454	0.950714	1.044138	0.325254	0.10579	−0.17454	0	−0.65071	0	1.149928 (worst)
2	0.731994	0.598658	0.894207	0.330652	0.109331	−0.53199	0	−0.29866	0	1.003538
3	0.156019	0.155995	0.048676	−0.68799	0	0.043981	0.001934	0.144005	0.020738	0.071348 (best)
4	0.058084	0.866176	0.753635	−0.07574	0	0.141916	0.02014	−0.56618	0	0.773775
5	0.601115	0.708073	0.862706	0.309188	0.095597	−0.40112	0	−0.40807	0	0.958303

.

Step 2: The optimal M(x) value of the objective function, considering the constraints, is determined and presented in Table 1. The corresponding best values of x1, x2, and M(x) are highlighted in bold, while the worst values of x1, x2, and M(x) are shown in italics.

Step 3: The optimal M(x) value of the objective function, subject to the constraints, corresponds to Solution 3, since the problem is formulated as a minimization task. Assume that the randomly generated numbers for x1 and x2 are r1 = 0.2, r2 = 0.4, and r3 = 0.2. These values are chosen solely for demonstration purposes; in actual implementation, the random numbers for x1 and x2 would be generated dynamically. The random number n4, generated during each iteration, determines whether Equation (1) or Equation (2) is used to update the values of x1 and x2. Assume that in the first iteration, r4 = 0.8, which is greater than 0.5. Therefore, Equations (1) and (2) are applied to modify x1 and x2. For example, in the first iteration, when Candidate Solution 1 randomly interacts with Candidate Solution 4, the updated values of x1 and x2 are calculated as follows:

X′_x1,1,1 = 0.37454 + 0.2 ∗ (0.156019 − 1 ∗ 0.058084) − 0.4 ∗ (0.37454 − 0.058084) = 0.267545

X′_x2,1,1 = 0.950714 + 0.2 ∗ (0.155995 − 1 ∗ 0.866176) − 0.4 ∗ (0.950714 − 0.866176) = 0.774863

The modified values of x1 and x2, along with the corresponding f(x), c1(x), p1, c2(x), p2, c3(x), p3, and M(x), are computed and presented in

Table 2. Modified variable values, constraint evaluations, penalty terms, and objective function results obtained using the BWR algorithm.

Solution	x1	x2	f(x)	c1(x)	p1	c2(x)	p2	c3(x)	p3	M(x)
1	0.267545	0.774863	0.671992	0.042407	0.001798	−0.067545	0	−0.47486	0	0.673791
2	0.733605	0.391178	0.691196	0.124783	0.015571	−0.533605	0	−0.091178	0	0.706767
3	0.183806	0	0.033788	−0.816194	0	0.016194	0.000262	0.3	0.09	0.124047
4	0	0.548288	0.30062	−0.451712	0	0.2	0.04	−0.248288	0	0.34062
5	0.557411	0.549129	0.61225	0.10654	0.011351	−0.357411	0	−0.249129	0	0.6236

. In this iteration, the new x1 and x2 values for Candidate Solutions 1, 2, 3, 4, and 5 are obtained by considering Candidate Solutions 4, 5, 2, 3, and 1, respectively, as the random interaction partners.

Step 4: The updated M(x) values in Table 2 are compared with their corresponding previous M(x) values from Table 1 for the first iteration. The best M(x) values are retained, and the resulting population is presented in

Table 3. Updated variable values, evaluated constraints, corresponding penalties, and objective function results obtained using the BWR algorithm.

Solution	x1	x2	f(x)	c1(x)	p1	c2(x)	p2	c3(x)	p3	M(x)
1	0.267545	0.774863	0.671992	0.042407	0.001798	−0.067545	0	−0.47486	0	0.673791
2	0.733605	0.391178	0.691196	0.124783	0.015571	−0.533605	0	−0.091178	0	0.706767 (worst)
3	0.156019	0.155995	0.048676	−0.68799	0	0.043981	0.001934	0.144005	0.020738	0.071348 (best)
4	0	0.548288	0.30062	−0.451712	0	0.2	0.04	−0.248288	0	0.34062
5	0.557411	0.549129	0.61225	0.10654	0.011351	−0.357411	0	−0.249129	0	0.6236

.

It is observed that, at the end of the first iteration, the best value is 0.071348 and the worst value is 0.706767, indicating an improvement in the worst solution compared to the randomly generated initial population.

Step 5: Based on the input from Table 3, the second iteration can be performed since the termination criterion has not been satisfied. However, the results of the second iteration are omitted here due to space limitations.

Step 6: Assuming the termination criterion of 1000 iterations is met, the final results at the end of the 1000th iteration are presented in

Table 4. Final (1000th iteration) values of the variables, constraints, penalties, and objective function results obtained using the BWR algorithm.

Solution	x1	x2	f(x)	c1(x)	p1	c2(x)	p2	c3(x)	p3	M(x)
1	0.2	0.3	0.13	−0.5	0	0	0	0	0	0.13
2	0.2	0.3	0.13	−0.5	0	0	0	0	0	0.13
3	0.2	0.3	0.13	−0.5	0	0	0	0	0	0.13
4	0.2	0.3	0.13	−0.5	0	0	0	0	0	0.13
5	0.2	0.3	0.13	−0.5	0	0	0	0	0	0.13

.

It is evident that the proposed BWR algorithm achieves an optimum solution of 0.13, corresponding to the optimal values x1 = 0.2 and x2 = 0.3.

Demonstration of BMR Algorithm

To illustrate the proposed BMR optimization algorithm, the same constrained objective function is used. For performance comparison with the BWR algorithm, the initial population from Table 1 is retained, and the updated values of x1 and x2 are computed using Equation (3) (or Equation (2) if the updated values exceed the bounds). The mean values obtained from Table 1 are x1 = 0.3843504 and x2 = 0.6559216. Using the same random interactions as in the BWR algorithm—where candidate solutions 1–5 interact with 4, 5, 2, 3, and 1, respectively—the updated population is presented in

Table 5. Modified variable values, constraint evaluations, penalty terms, and objective function results obtained using the BMR algorithm.

Solution	x1	x2	f(x)	c1(x)	p1	c2(x)	p2	c3(x)	p3	M(x)
1	0.368048	0.566656	0.456558	−0.065296	0	−0.168048	0	−0.266656	0	0.456558
2	0.508289	0.277833	0.335549	−0.213877	0	−0.308289	0	0.22167	0.000491	0.336041
3	0	0	0	−1	0	0.2	0.04	0.3	0.09	0.13
4	0.012418	0.766191	0.587202	−0.221392	0	0.187582	0.035187	−0.466191	0	0.622389
5	0.46804	0.2902	0.303278	−0.24176	0	−0.26804	0	0.0098	0.000096	0.303374

. For instance, in the first iteration, when candidate solution 1 interacts with candidate solution 4 the modified values of x1 and x2 are computed using the BMR algorithm as follows.

X′_x1,1,1 = 0.37454 + 0.2 ∗ (0.156019 − 1 ∗ 0.3843504) + 0.4 ∗ (0.156019 − 0.058084) = 0.368048

X′_x2,1,1 = 0.950714 + 0.2 ∗ (0.155995 − 1 ∗ 0.6559216) + 0.4 ∗ (0.155995 − 0.866176) = 0.566656

The modified values are computed and are given in Table 5.

The modified M(x) values in Table 5 are compared with the corresponding M(x) values from Table 1 for the first iteration. The best M(x) values are retained, and the resulting population is presented in

Table 6. Updated variable values, evaluated constraints, corresponding penalties, and objective function results obtained using the BMR algorithm.

Solution	x1	x2	f(x)	c1(x)	p1	c2(x)	p2	c3(x)	p3	M(x)
1	0.368048	0.566656	0.456558	−0.065296	0	−0.168048	0	−0.266656	0	0.456558
2	0.508289	0.277833	0.335549	−0.213877	0	−0.308289	0	0.22167	0.000491	0.336041
3	0.156019	0.155995	0.048676	−0.68799	0	0.043981	0.001934	0.144005	0.020738	0.071348
4	0.012418	0.766191	0.587202	−0.221392	0	0.187582	0.035187	−0.466191	0	0.622389
5	0.46804	0.2902	0.303278	−0.24176	0	−0.26804	0	0.0098	0.000096	0.303374

.

It can be seen that the best value at the end of the first iteration using the BMR algorithm is 0.071348 and the worst value is 0.622389. This shows the improvement in the worst solution compared to the randomly generated worst solution of 1.149928 given in Table 1.

Demonstration of BMWR Algorithm

Using Equations (2) and (4), the updated values of the variables, constraints, penalties, and objective function results obtained with the BMWR algorithm are presented in

Table 7. Modified variable values, constraint evaluations, penalty terms, and objective function results obtained using the BMWR algorithm.

Solution	x1	x2	f(x)	c1(x)	p1	c2(x)	p2	c3(x)	p3	M(x)
1	0.202291	0.816913	0.708269	0.019205	0.000369	−0.002291	0	−0.516913	0	0.708638
2	0.776958	0.401608	0.764953	0.178566	0.031886	−0.576958	0	−0.101608	0	0.796838
3	0.253334	0	0.064178	−0.746666	0	−0.053334	0	0.3	0.09	0.154178
4	0	0.448303	0.200976	−0.551697	0	0.2	0.04	−0.148303	0	0.240976
5	0.555449	0608088	0.678294	0.163536	0.026744	−0.355449	0	−0.308088	0	0.705038

. For instance, in the first iteration, when candidate solution 1 interacts randomly with candidate solution 4 the modified values of x1 and x2 are computed using the BMWR algorithm as follows:

X′_x1,1,1 = 0.37454 + 0.2 ∗ (0.156019 − 1 ∗ 0.3843504) − 0.4 ∗ (0.37454 − 0.058084) = 0.202291

X′_x2,1,1 = 0.950714 + 0.2 ∗ (0.155995 − 1 ∗ 0.6559216) − 0.4 ∗ (0.950714 − 0.866176) = 0.816913

The modified values are computed and are given in Table 7.

Now the values of M(x) in Table 7 are compared with those in Table 1 and the best solutions are taken to form

Table 8. Updated values of the variables, evaluated constraints, associated penalties, and the objective function results obtained using the BMWR algorithm.

Solution	x1	x2	f(x)	c1(x)	p1	c2(x)	p2	c3(x)	p3	M(x)
1	0.202291	0.816913	0.708269	0.019205	0.000369	−0.002291	0	−0.516913	0	0.708638
2	0.776958	0.401608	0.764953	0.178566	0.031886	−0.576958	0	−0.101608	0	0.796838
3	0.156019	0.155995	0.048676	−0.68799	0	0.043981	0.001934	0.144005	0.020738	0.071348
4	0	0.448303	0.200976	−0.551697	0	0.2	0.04	−0.148303	0	0.240976
5	0.555449	0608088	0.678294	0.163536	0.026744	−0.355449	0	−0.308088	0	0.705038

.

It is observed that, after the first iteration using the BMWR algorithm, the best value is 0.071348 and the worst value is 0.708638. This indicates an improvement in the worst solution compared to the initially generated worst value of 1.149928 presented in Table 1.

If the BWR, BMR, and BMWR algorithms are continued for 1000 iterations, then we can find that the global optimum value of f(x) = 0.13 for x1 = 0.2 and x2 = 0.3 is obtained. All the three algorithms converged to the same, as shown in Table 4. The convergence behaviors of the BWR, BMR, and BMWR algorithms are shown in Figure 3.

The trend for BWR for this constrained objective function is slower to converge compared to the other two, though it reaches the same final optimal value. The BMR algorithm shows efficient early performance and faster convergence. The BMWR algorithm showed best performance overall in terms of solution quality and convergence speed—reaching the feasible optimum with minimal fluctuation. The convergence occurred after 228, 312, and 207 iterations for the BWR, BMR, and BMWR algorithms, respectively.

Table 9. Salient features of the BWR, BMR, and BMWR algorithms.

Feature	BWR	BMR	BMWR
Search mechanism type	Hybrid arithmetic with best–worst–random guidance	Hybrid arithmetic with best–mean–random guidance	Hybrid arithmetic combining best–mean–worst–random
Novel equation logic	Combines (best–F × random) and (worst–random) differences; fallback to uniform sampling	Combines (best–F × mean) and (best–random); fallback to uniform sampling	Combines (best–F × mean) and (worst–random); fallback to uniform sampling
Handling of random influence	Direct—random solution explicitly part of update	Same as BWR	Same as BWR and BMR
Fallback sampling	Yes: if r4 ≤ 0.5, fallback to uniform random within bounds	Same as BWR	Same as BWR
Diversity mechanism	Very strong, due to combination of best, worst, and random solutions	Strong, but more conservative due to mean guidance	Very strong, integrates best, mean, worst, and random
Exploration–exploitation balance	Highly exploratory, still exploits best	More exploitative, smoother convergence via mean-based pull	Balanced—exploration from worst/random, exploitation from best/mean
Design simplicity	Slightly complex (two difference terms + fallback), but parameter-free and metaphor-free	Similarly to BWR and still parameter-free	Slightly more complex (all four references), but parameter-free
Robustness to local optima	Strong—fallback randomness + worst guidance help escape	Good, but conservative due to mean-based bias	Strongest—mixed forces reduce trapping in local optima
Risk of premature convergence	Low (due to worst + random)	Low–moderate (mean pulls population together)	Low (conflicting forces maintain diversity)
Parameter dependence	None beyond population size and iterations	None beyond population size and iterations	None beyond population size and iterations
Convergence speed	Slower—broad exploration before exploitation	Faster—mean accelerates convergence	Intermediate—not as fast as BMR, but more diverse
Constraint handling compatibility	Works well with penalty methods, strong exploration helps recover feasibility	May converge prematurely in tight feasible spaces	More robust than both, balances feasibility search and convergence
Multi-objective performance	Produces well-spread Pareto sets due to strong diversity	Generates smoother but sometimes narrower Pareto sets	Best compromise—well-distributed and convergent Pareto fronts
Scalability with dimensionality	May lose efficiency in very high dimensions (overexplores)	Stable, but risks local trapping as dimensions grow	Better scalability—multiple guiding forces adapt well
Computational cost per iteration	Light—simple difference calculations	Same as BWR	Slightly higher (two guiding forces), but still lighter than GA/PSO/DE

is presented below to present the novel features of the proposed BWR, BMR, and BMWR algorithms.

2.2. Proposed Algorithms for Multi-Objective Constrained Optimization Problems

This section outlines the multi-objective variants of the BWR, BMR, and BMWR algorithms. It also focuses on the multi-attribute decision-making (MADM) process used to select the most suitable compromise solution from the set of non-dominated Pareto-optimal solutions.

2.2.1. Multi-Objective Optimization with the BWR, BMR, and BMWR Algorithms

The extended versions of the BWR, BMR, and BMWR algorithms for handling multi-objective problems are referred to as MO-BWR, MO-BMR, and MO-BMWR, respectively. These algorithms incorporate several key features, including elite seeding, fast non-dominated sorting, constraint repair mechanisms, penalty application (when repair fails), local search enhancement, and edge-boosting strategies. A summary of the algorithmic procedure is provided in

Table 10. Procedural steps of the MO-BWR, MO-BMR, and MO-BMWR algorithms.

Step No.	Step Title	Description
1	Start	Initiate the algorithm.
2	Initialize population	Generate the initial set of solutions within defined variable bounds.
3	Elite seeding	Introduce high-quality or historically good solutions into the initial population.
4	Fast non-dominated sorting	Rank solutions based on Pareto dominance and compute crowding distance.
5	Constraint repair	Try to rectify any constraint violations in the solutions as the primary strategy.
6	Penalty application	Impose penalties on the objective functions if constraint violations persist (fallback strategy).
7	Objective function evaluation	Assess all objective functions for every solution in the population.
8	Edge boosting	Promote exploration in regions close to the extreme ends of the Pareto front.
9	Local exploration	Improve elite or high-potential solutions by conducting a search in their local neighborhood.
10	Population update (BWR/BMR/BMWR)	Create new candidate solutions using the specific update mechanism of BWR, BMR, or BMWR.
11	Check termination criterion	Check if the stopping condition (e.g., maximum iterations or evaluations) has been reached.
12	If not terminated, repeat (go to step 5)	If termination condition is not met, return to Step 5 and continue the process.
13	If terminated, output solutions	Terminate the algorithm and present the final set of non-dominated Pareto-optimal solutions.

.

The total computational complexities of the MO-BWR, MO-BMR, and MO-BMWR algorithms are identical and given by O(I·(M.c² + c·(m + t_f + t_p))), where M is the number of objectives, I is the number of iterations, c is the population size, m is the number of design variables, t_f represents the time required to evaluate the objective functions and constraints per individual, and t_p is the time needed for penalty computation (assumed proportional to the number of constraints). This complexity accounts for the quadratic cost associated with Pareto-based non-dominated sorting, and the linear cost associated with variable updates, evaluations, and penalties. The algorithms are well-suited for small to medium population sizes and can be made more scalable by incorporating faster sorting mechanisms such as those in NSGA-III. For the single-objective BWR, BMR, and BMWR algorithms, the total computational complexity is O(I·c·(m + t_f + t_p) as they do not involve non-dominated sorting. The only distinction between BWR, BMR, and BMWR lies in their update strategies, which does not impact their asymptotic complexity.

2.2.2. Validation of the MO-BWR, MO-BMR, MO-BMWR Algorithms for ZDT Functions

The standard Zitzler–Deb–Thiele (ZDT) benchmark functions are used to assess the performance of the proposed algorithms. A brief overview of these functions is presented below.

ZDT1:

f1(x) = x1

(10)

g(x) = 1 + 9 ∗ (1/(n − 1)) ∗ Σ xi for i = 2 to n

(11)

f2(x) = g(x) ∗ [1 − sqrt(f1(x)/g(x))]

(12)

ZDT2:

f1(x) = x1

(13)

g(x) = 1 + 9 ∗ (1/(n − 1)) ∗ Σ xi for i = 2 to n

(14)

f2(x) = g(x) ∗ [1 − (f1(x)/g(x))^2]

(15)

ZDT3:

f1(x) = x1

(16)

g(x) = 1 + 9 ∗ (1/(n − 1)) ∗ Σ xi for i = 2 to n

(17)

f2(x) = g(x) ∗ [1 − sqrt(f1(x)/g(x)) − (f1(x)/g(x)) ∗ sin(10π ∗ f1(x))]

(18)

ZDT4:

f1(x) = x1

(19)

g(x) = 1 + 10 ∗ (n − 1) + Σ [xi^2 − 10 ∗ cos(4π ∗ xi)] for i = 2 to n

(20)

f2(x) = g(x) ∗ [1 − sqrt(f1(x)/g(x))]

(21)

ZDT6:

f1(x) = 1 − exp(−4 ∗ x1) ∗ sin^6(6π ∗ x1)

(22)

g(x) = 1 + 9 ∗ (Σ xi for i = 2 to n)^(0.25)/(n − 1)

(23)

f2(x) = g(x) ∗ [1 − (f1(x)/g(x))^2]

(24)

The MO-BWR, MO-BMR, and MO-BMWR algorithms are applied to the ZDT benchmark problems using 40,000 function evaluations. An equal number of function evaluations is used for performance comparison with the non-dominated sorting genetic algorithm III (NSGA-III) for each problem. Each experiment is executed 30 times. The implementations of MO-BWR, MO-BMR, and MO-BMWR are developed in Python 3.9 on a laptop equipped with an AMD Ryzen 5 5600H processor and 16 GB DDR4 RAM. The Pymoo library is utilized for benchmark problem definitions, their true Pareto fronts, and the NSGA-III algorithm. The results of algorithms on ZDT functions and the convergence graphs are given in Appendix A.

The MO-BWR, MO-BMR, and MO-BMWR algorithms have successfully addressed the standard Zitzler–Deb–Thiele (ZDT) multi-objective benchmark problems. Likewise, these algorithms can be extended to solve other multi-objective benchmark functions as well. However, prior studies have already demonstrated that the base versions have been effectively applied to a variety of standard benchmark problems [22] and complex engineering design tasks. The primary focus of this paper is to apply these algorithms to solve single- and multi-objective optimization problems encountered in metal casting processes.

It is important to note that Pareto-optimal solutions are non-dominated—meaning no solution is strictly better than another across all objectives—and each is considered acceptable. However, if a decision-maker wishes to prioritize certain objectives over others, a multi-attribute decision-making (MADM) approach can be employed. Numerous MADM methods have been proposed in the literature. Among them, a recent method called BHARAT, developed by Rao [23], offers a simple and logical approach to assigning relative importance to objectives and ranking the solutions accordingly.

The following sections present the application of the BWR, BMR, and BMWR algorithms for single-objective optimization of metal casting processes, followed by the use of MO-BWR, MO-BMR, and MO-BMWR for multi-objective optimization. Subsequently, the BHARAT method is applied to select the most preferred solution from the obtained set of Pareto-optimal alternatives.

3. Results and Discussion on the Application of the Proposed Algorithms for Optimizing Various Metal Casting Processes

Optimization of metal casting processes is critical due to the complex nature of casting operations, the variability of input parameters, and the cost-sensitive environment in which foundries operate. Casting involves numerous interdependent variables, such as mold design, gating and riser dimensions, pouring temperature, and cooling rate. Each of these factors significantly influences the quality, efficiency, and cost of the final product. Without optimization, achieving the desired quality consistently becomes a trial-and-error process, leading to high rates of scrap, rework, and energy consumption. Optimization of metal casting processes is essential for improving product quality, reducing costs, increasing efficiency, and ensuring consistency in production. It enables data-driven decision-making, minimizes reliance on empirical adjustments, and is an integral part of modern, sustainable manufacturing strategies. Now, optimization of different metal casting processes is presented in the following sub-sections.

3.1. Single-Objective Optimization of Metal Casting Processes

To demonstrate the applicability of the proposed BWR, BMR, and BMWR algorithms for single-objective optimization of metal casting processes, three case studies are examined.

3.1.1. Optimization of the Lost Foam Casting Process for Manufacturing a Fifth Wheel Coupling Shell from EN-GJS-400-18 Ductile Iron

Li et al. [12] optimized defect reduction in the lost foam casting process for producing a fifth wheel coupling shell. The semi-truck fifth wheel coupling, also known as the traction seat, is a vital chassis component that links the trailer to the truck. It locks onto the truck’s kingpin, ensuring a secure connection and enabling the transfer of horizontal and vertical forces between the truck and trailer. This component must also withstand dynamic loads and impacts encountered during various driving conditions, including starting, accelerating, braking, steering, and decelerating. The casting of the fifth wheel coupling considered by Li et al. [12] had dimensions of 916 mm × 760 mm × 140 mm, with a weight of approximately 126 kg. The shell featured an average wall thickness of 16 mm, reaching up to 20 mm at its thickest sections.

Manufactured from EN-GJS-400-18 ductile iron, the coupling benefits from this alloy’s notable ductility and toughness, setting it apart from other cast iron types. These properties make it ideal for applications that demand a balance between hardness and abrasion resistance. Consequently, it sees widespread use in industries such as marine, railways, transportation, wind energy, and various industrial machinery, especially in structural and safety-critical components. The casting process employed natural silica sand as the mold material and expanded polystyrene (EPS) foam as the pattern. EPS foam begins to decompose at approximately 290 °C; depending on the surface heat flux, it may fully vaporize into a gas or partially melt into a liquid phase.

Li et al. [12] designed a pouring system for the EN-GJS-400-18 ductile iron fifth wheel coupling shell, and ProCAST 2021 simulation software was used to simulate filling and solidification. Based on these simulations, risers were added in suitable positions to reduce casting defects. The Box–Behnken response surface methodology (RSM) using Design Expert 13 data processing software was applied to optimize three key process parameters: pouring temperature (A) in °C, pouring speed (B) in m/s, and ferro-static head pressure (C) in Pa. The process output parameter was the porosity (P) of the produced casting. The statistical tests of ANOVA and F-tests confirmed the best fitness and accuracy of the regression equation developed for P in terms of the process input parameters.

The optimum conditions were found to be a pouring temperature of 1386 °C, a pouring speed of 6.42 kg/s, and a ferrostatic head pressure of 109,751 Pa. Under these conditions, shrinkage was reduced, cracks were eliminated, and porosity decreased. With the optimized parameters, ProCAST predicted the locations of possible casting defects. Test castings were then produced, and samples were examined for metallographic structure and tensile properties. The results showed no major defects, and the casting quality met production and performance requirements.

The mathematical regression model for P in terms of the input parameters is given by Equation (25). The bounds of the input parameters are given in Equation (26).

Porosity (P) = 0.011 ∗ A + 0.746 ∗ B + 6.758 ∗ 10⁻⁶ ∗ C + 0.4 ∗ 10⁻⁴ ∗ A ∗ B + 8.520 ∗ 10⁻⁸ ∗ A ∗ C − 6.130 ∗ 10⁻⁶ ∗ B ∗ C − 7.308 ∗ 10⁻⁶ ∗ A² − 0.014 ∗ B² − 4.512 ∗ 10⁻¹⁰ ∗ C²

(25)

1350 °C ≤ A ≤ 1450 °C; 4.5 m/s ≤ B ≤ 6.5 m/s; 105,000 Pa ≤ C ≤ 110,000 Pa

(26)

Substituting the optimal values of A, B, and C suggested by Li et al. [12] into Equation (25) gives a porosity value of 9.82%!! Hence, the data from the 17 experiments given in Table 4 of the paper of Li et al. [12] is revisited and a correct regression model is formed. It is almost the same as Equation (25), but an important constant term of −8.856569 is added. The corrected regression model for the porosity P is written now as Equation (27).

Porosity (P) = −8.856569 + 0.010943 ∗ A + 0.745645 ∗ B + 6.758 ∗ 10⁻⁶ ∗ C + 0.4 ∗ 10⁻⁴ ∗ A ∗ B + 8.520 ∗ 10⁻⁸ ∗ A ∗ C − 6.130 ∗ 10⁻⁶ ∗ B ∗ C − 7.308 ∗ 10⁻⁶ ∗ A² − 0.013845 ∗ B² − 4.512 ∗ 10⁻¹⁰ ∗ C²

(27)

Substituting the optimal parameters of pouring temperature of 1386 °C, pouring speed of 6.42 kg/s, and ferro-static head pressure of 109,751 Pa, as suggested by Li et al. [12], into the corrected equation for porosity (i.e., Equation (27)) leads to a porosity of 0.792% (not 0.82%, as reported by Li et al. [12]).

Now to check if there can be any further improvement in the optimal values of A, B, and C, and the corresponding P, the proposed BWR, BMR, and BMWR algorithms are applied to the same regression model with the same bounds of the input parameters. A population size of 10 and 50 iterations are used, resulting in a total of 500 function evaluations. The results are presented in

Table 11. Comparison of optimal porosity obtained in a lost foam casting process for producing a fifth wheel coupling shell from EN-GJS-400-18 ductile iron.

Method	Optimal Porosity (%)	Process Input Parameters
		Pouring Temperature A (°C)	Pouring Speed B (kg/s)	Ferro-Static Head Pressure C (Pa)
Simulation by RSM and ProCAST 2021 [12]	0.7920	1386	6.42	109751
BWR	0.7635	1350	6.5	110000
BMR	0.7635	1350	6.5	110000
BMWR	0.7635	1350	6.5	110000

, with the best optimal porosity value highlighted in bold.

From Table 11, it can be understood that the BWR, BMR, and BMWR algorithms have produced a minimum porosity of 0.7635% compared to the 0.792% suggested by Li et al. [12] using RSM and ProCAST 2021 simulation software. It can also be seen that the BWR, BMR, and BMWR algorithms required only a few iterations to reach the optimal solution.

Figure 4 presents the convergence graphs of the BWR, BMR, and BMWR algorithms. The BMWR algorithm exhibits faster convergence, and all three algorithms achieve the global optimum value of f(x) = 0.7635 cm³. The convergence of the BMWR algorithm is faster and the BMR lagged slightly. BWR converged after the 7th iteration, BMR after the 8th iteration, and BMWR after the 5th iteration.

The effects of the input parameters A, B, and C on porosity P are shown in the surface plots shown in Figure 5.

In the above plot of P vs. A and B, the value of C is fixed at its mid-point of 107,500 Pa; in the plot of P vs. A and C, the value of B is fixed at its mid-point of 5.5 m/s; and in the plot of P vs. B and C, the value of A is fixed at its mid-point of 1400 °C. The surface plots reveal that higher values of both A and B reduce the porosity (when C is fixed at its mid-point); higher values of A and C reduce the porosity (when B is fixed at its mid-point); and higher values of B and C reduce the porosity (when A is fixed at its mid-point). Based on the regression equation of P, and taking into account the linear, quadratic, and interaction effects of the parameters, it can be concluded that parameter C is the most dominant, followed by B, and then A.

3.1.2. Optimization of Process Parameters of Die Casting of A360 Al-Alloy

Die casting is widely used for its high surface finish, dimensional accuracy, and cost-effectiveness, but it often suffers from a high defect rate, largely influenced by the gating system design. A360 Al-Alloy is commonly used and is considered a viable alternative to A380 Al-Alloy. Xu et al. [1] applied response surface methodology (RSM) based on the ordinary least squares (OLS) method to reduce severe defects in A360 Al-alloy cover plates arising from excessive porosity during die casting. The effects of three key process parameters—casting temperature (TC), mold preheating temperature (TM), and fast injection speed (V)—on volumetric porosity were systematically evaluated using 17 distinct combinations of injection molding settings. Optimal process parameters were determined and experimentally validated through RSM combined with computer-aided engineering (CAE) simulation. A 3D model of the cover plate casting, including the gating system, was developed using 3D modeling [1], and a chill vent was added and connected to the vacuum system.

The mathematical model developed for the volumetric porosity P is given by Equation (28). The statistical tests of ANOVA and F-tests confirmed the best fitness and accuracy of the regression equation developed for P in terms of the process input parameters. The bounds of the three critical process parameters—casting temperature (TC), mold preheating temperature (TM), and fast injection speed (V)—are given by Equation (29).

Volumetric porosity P (cm³) = −1.092681 + 0.012221 ∗ TC − 0.013113 ∗ TM − 0.919275 ∗ V −0.000012 ∗ TC² + 0.00001 ∗ TC ∗ TM + 0.001485 ∗ TC*V + 0.000015 ∗ TM² − 0.000462 ∗ TM ∗ V + 0.0104 ∗ V²

(28)

630 °C ≤ TC ≤ 670 °C; 180 °C ≤ TM ≤ 220 °C; 2.5 m/s ≤ V ≤ 3.5 m/s

(29)

Through RSM and CAE simulations, Xu et al. [1] reported the optimal values of TC, TM, and V as 630 °C, 220 °C, and 2.5 m/s. The volumetric porosity P obtained by substituting the optimal parameter values of TC, TM, and V in Equation (28) gives P as 0.9203 cm³ (not 0.8745 cm³ as incorrectly reported by Xu et al. [1]).

To explore potential improvements in the optimal values of TC, TM, and V, as well as the corresponding P, the proposed BWR, BMR, and BMWR algorithms are applied to the same regression model using identical input parameter bounds. A population size of 10 and 100 iterations are employed, resulting in 1000 function evaluations.

Table 12. Comparison of optimal volumetric porosity obtained in the die casting process for producing A360 Al-alloy cover plate.

Method	Optimal Volumetric Porosity (cm3)	Process Input Parameters
		Casting Temperature TC (°C)	Mold Preheating Temperature TM (°C)	Fast Injection Speed V (m/s)
Simulation by RSM and CAE software [1]	0.9203 (corrected value)	630	220	2.5
BWR	0.8995	630	220	3.5
BMR	0.8995	630	220	3.5
BMWR	0.8995	630	220	3.5

gives the comparison of optimal volumetric porosity P obtained in the die casting process for producing A360 Al-alloy cover plate. The best value of P is indicated in bold.

From Table 12, it can be understood that the BWR, BMR, and BMWR algorithms have produced a minimum volumetric porosity of 0.8995 cm³ compared to the 0.9203 cm³ produced by the RSM and CAE used by Xu et al. [1]. Xu et al. [1] reported the optimal porosity as 0.8745 cm³, corresponding to a casting temperature TC of 630 °C, a mold preheating temperature TM of 220 °C, and a fast injection speed V of 2.5 m/s. However, substituting those values of TC, TM, and V lead to a porosity value of 0.9203 cm³ (instead of 0.8745 cm³, as reported by Li et al. [12]).

It can also be seen that the BWR, BMR, and BMWR algorithms required lower numbers of function evaluations to reach the optimal solution. Figure 6 shows the convergence graph of the BWR, BMR, and BMWR algorithms. The BWR algorithm showed best early convergence after 10 iterations, BMWR showed excellent stability and converged after 20 iterations, and BWR showed moderate stability and converged after the 25th iteration. All three algorithms (BWR, BMR, and BMWR) converged to the same optimum value of volumetric porosity P, which was 0.8995 cm³.

Figure 7 presents three surface plots illustrating the effects of casting temperature (TC), mold temperature (TM), and injection speed (V) on porosity (P).

In the above plot of P vs. TC and TM, the value of V is fixed at its mid-point of 3.0 m/s; in the plot of P vs. TC and V, the value of TM is fixed at its mid-point of 200 °C; and in the plot of P vs. TM and V, the value of TC is fixed at its mid-point of 650 °C. The surface plots reveal that lower TC and lower TM reduces the porosity (when V is fixed at its mid-point); lower TC and lower V reduces the porosity (when TM is fixed at its mid-point); and lower TM and lower V reduces the porosity (when TC is fixed at its mid-point). Based on the regression equation of P, and taking into account the linear, quadratic, and interaction effects of the parameters, it can be concluded that parameter TC is the most dominant, followed by TM, and then V.

3.1.3. Optimization of Wear Rate of an AA7178 Alloy Reinforced with Nano-SiC Particles Produced Using a Stir-Casting Process

Metal matrix composites (MMCs) are increasingly replacing traditional materials due to their superior mechanical and tribological properties. Ceramic nanoparticle reinforcements like SiC, TiO₂, Al₂O₃, ZrB₂, and TiC have shown great potential in aerospace, automotive, and biomedical applications. However, optimizing their wear behavior is challenging due to complex interactions among process parameters. While methods like response surface methodology (RSM) and Taguchi have been used, they struggle with nonlinear relationships. Metaheuristic optimization techniques offer a more effective alternative by efficiently handling complex, multi-parameter problems and identifying global optima. This has positioned metaheuristic approaches as valuable tools for advancing the performance and reliability of these advanced composite materials.

Bharat et al. [5] investigated the microstructure and wear behavior of AA7178 alloy reinforced with varying weight percentages (0, 1, 2, and 3%) of nano-SiC particles, fabricated via stir casting, to improve material performance under wear conditions. It addressed limitations in current aerospace materials, which often exhibit poor wear resistance, surface degradation, and reduced durability under abrasive environments—factors that lead to higher maintenance costs and decreased operational efficiency. AA7178 alloy was first dried in a muffle furnace at 450 °C for 45 min to remove moisture and impurities, then melted in a graphite crucible. Nano-SiC particles were added to the melt and stirred at 625 rpm for 10–15 min for uniform dispersion. Magnesium powder was introduced to improve wettability and reduce surface tension. The final slurry was poured into a preheated permanent iron mold.

The research hypothesized that incorporating nano-SiC into the AA7178 matrix would enhance wear resistance and contribute to the development of high-performance metal matrix composites. To evaluate this, dry sliding wear tests were performed using a pin-on-disk apparatus. Experiments were systematically varied across sliding velocity A (1–4 m/s), sliding wear distance B (500–2000 m), applied load C (9.81–39.24 N), and weight % of nano-SiC (0–3%). The L16 Taguchi design was used to plan and analyze the experiments in terms of the signal-to-noise ratio. To analyze the influence of process factors, Minitab 18 software was used, and ANOVA was performed to evaluate the relative importance of each parameter. The RSM model was used to predict the wear rate, given by Equation (30). The coefficient of determination (i.e., R²) was 99.88%, which indicates the best fit with highest accuracy. The bounds of the process input parameters are given by Equation (31):

Wear rate ((mm³/m) ∗ 10⁻³) = 2.381 − 0.106 ∗ A + 0.000657 ∗ B + 0.0357 ∗ C −0.353 ∗ D + 0.01004 ∗ A ∗ C + 0.00418 ∗ C ∗ D − 0.000012 ∗ C ∗ B − 0.0160 ∗ A ∗ D − 0.000061 ∗ A ∗ B − 0.000008 ∗ D ∗ B

(30)

1 m/s ≤ A ≤ 4 m/s, 500 m ≤ B ≤ 2000 m, 9.81 N ≤ C ≤ 39.24 N, and 0% ≤ D ≤ 3%

(31)

Bharat et al. [5] employed three metaheuristic algorithms—particle swarm optimization (PSO), artificial bee colony (ABC), and Rao-1—to determine the optimal values of AAA, BBB, CCC, and DDD. Using a population size of 50 and 50 iterations, they performed a total of 2500 function evaluations. To demonstrate the efficiency of the proposed BWR, BMR, and BMWR algorithms, a smaller population size of 10 and 50 iterations was used, resulting in only 500 function evaluations. The wear rate values obtained by the different algorithms are presented in

Table 13. Comparison of optimal wear rate obtained in stir casting by different algorithms.

Algorithm	Optimal Wear Rate ((mm3/m) ∗ 10−3)	Optimum Values of Input Variables
		A (m/s)	B (m)	C (N)	D (%)
ABC [5]	1.7093	Not given	Not given	Not given	Not given
Rao-1 [5]	1.7088	4	500	9.81	3
PSO [5]	2.1809	Not given	Not given	Not given	Not given
BWR	1.7088	4	500	9.81	3
BMR	1.7088	4	500	9.81	3
BMWR	1.7088	4	500	9.81	3

, with the best optimal wear rate highlighted in bold.

The optimum value of wear loss (i.e., f(x)) obtained by the BMR, BWR, and BMWR algorithms is f(x) = 1.7088 $\times$ 10⁻³ mm³/m, with the optimal values of A = 4 m/s, B = 500 m, C = 9.81 N, and D = 3%. The value of wear loss obtained by the Rao-1 algorithm used by Bharat et al. [5] was also 1.7088 ∗ 10⁻³ mm³/m. The optimum values obtained by Bharat et al. [5] using the ABC and PSO algorithms are not optimal. Unlike the ABC and PSO algorithms, the BMR, BWR, BMWR, and Rao-1 algorithms do not need to tune any algorithm-specific parameters. Furthermore, as can be seen in this example, compared to the Rao-1 algorithm, the BWR, BMR, and BMWR algorithms required a smaller number of function evaluations to reach the optimal solution.

Figure 8 presents the convergence graph of the BWR, BMR, and BMWR algorithms. The BWR algorithm demonstrates faster convergence, and all three algorithms achieve a global optimum value of f(x) = 1.7088 $\times$ 10⁻³ mm³/m.

The BWR and BMR are faster and more efficient for this problem, converging quickly. The convergence occurred from the 16th iteration. The BMWR offers a balance of exploration and exploitation, but at the cost of slower convergence. The convergence occurred from the 29th iteration. If robustness in avoiding local optima is important, BMWR may be better suited despite slower convergence. The surface plots showing the effect of the input variables A, B, C, and D on wear rate are shown in Figure 9.

In the above plot of wear rate vs. A and B, the values of C and D are fixed at their mid-point values of 24.53 N and 1.5%, respectively; in the plot of wear rate vs. A and C, the values of B and D are fixed at their mid-point values of 1250 m and 1.5%, respectively; in the plot of wear rate vs. A and D, the values of B and C are fixed at their mid-point values of 1250 m and 24.53 N, respectively; in the plot of wear rate vs. B and C, the values of A and D are fixed at their mid-point values of 2.5 m/s and 1.5%, respectively; in the plot of wear rate vs. B and D, the values of A and C are fixed at their mid-point values of 2.5 m/s and 24.53 N, respectively; and in the plot of wear rate vs. C and D, the values of A and D are fixed at their mid-point values of 2.5 m/s and 1250 m, respectively.

It is observed that the wear rate decreases with higher speed and shorter distance; increases with rising load (C) and decreases with increasing speed (A); and drops significantly with higher nano-SiC content (%) combined with increased speed. Furthermore, wear rate increases with load; wear rate decreases with higher D (nano-SiC) and lower B (distance); and wear rate increases with load (C) but decreases with nano-SiC (D). The speed (A) and nano-SiC (D) are the most beneficial parameters for reducing wear. The load (C) and distance (B) are the most detrimental if increased. Higher values of A and D help reduce wear rate.

3.2. Multi-Objective Optimization of Metal Casting Processes

Multi-objective optimization (MOO) refers to the simultaneous optimization of two or more conflicting objectives. In metal casting, these objectives may include:

• Minimizing porosity;
• Minimizing shrinkage defects;
• Minimizing solidification time;
• Maximizing mechanical properties (e.g., hardness, tensile strength);
• Reducing production costs.

MOO uses concepts such as Pareto-optimality to identify a set of optimal solutions, each representing a different trade-off scenario. This allows process engineers to select the most suitable solution based on specific constraints or priorities. The MOO allows balancing between competing objectives, provides a Pareto front offering flexibility in decision-making, helps in understanding how process parameters affect multiple outputs simultaneously, and the solutions are typically more resilient to parameter variations and uncertainties.

To demonstrate the applicability of the proposed BWR, BMR, and BMWR algorithms for multi-objective optimization of metal casting processes, two case studies are examined.

3.2.1. Multi-Objective Optimization of a Low-Pressure Casting Process Using a Sand Mold for Producing A356 Engine Block

Deng et al. [4] optimized the process parameters of a low-pressure casting process using a sand mold that was used for producing a four-cylinder A356 alloy engine block weighing 18.83 kg with the dimensions of 371.5 × 348.7 × 282.6 mm. The casting must be free from cold segregation, cracks, underpouring, sand burn-on, and shrinkage defects. Deng et al. [4] carried out multi-objective optimization of A356 engine block casting parameters using response surface methodology (RSM) in combination with the NSGA-II algorithm. Their study optimized the low-pressure casting process of A356 alloy engine blocks by analyzing four key parameters: pouring temperature A (°C), mold preheating temperature B (°C), filling time C (s), and holding pressure D (kPa). Using RSM and the NSGA-II algorithm through 29 test runs, two objectives were optimized—solidification time Y1 (s) and shrinkage volume Y2 (cm³). Both Y1 and Y2 are to be minimized. The mathematical regression models for Y1 and Y2 are given by Equations (32) and (33). The statistical tests of ANOVA and F-tests confirmed the best fitness and accuracy of the regression equations developed for Y1 and Y2 in terms of the process input parameters. The bounds of the process parameters are given by Equation (34).

Solidification time Y1 (s) = 3227.71435 − 10.52667 ∗ A + 0.519167 ∗ B + 66.25 ∗ C
+ 0.300926 ∗ D − 0.001438 ∗ A ∗ B − 0.07175 ∗ A ∗ C − 0.0005 ∗ A ∗ D + 0.0235 ∗ B ∗ C + 0.00825 ∗ B ∗ D + 0.004667 ∗ C ∗ D + 0.009823 ∗ A² − 0.00049 ∗ B² − 0.474333 ∗ C^{2 −} 0.003259 ∗ D²

(32)

Shrinkage volume Y2 (cm³) = 246.25699 − 0.488750 ∗ A − 0.243250 ∗ B − 1.29817 ∗ C + 0.487481 ∗ D + 0.0004 ∗ A ∗ B + 0.00135 ∗ A ∗ C − 0.000742 ∗ A ∗ D + 0.00145 ∗ B ∗ C − 0.0007 ∗ B ∗ D − 0.002333 ∗ C ∗ D + 0.000354 ∗ A² − 0.000049B² + 0.015417 ∗ C² + 0.000674D²

(33)

680 °C ≤ A ≤ 720 °C, 20 °C ≤ B ≤ 60 °C, 10 s ≤ C ≤ 20 s, and 50 kPa ≤ D ≤ 80 kPa

(34)

Deng et al. [4] used the NSGA-II algorithm and proposed the optimal values of parameters as pouring temperature A = 680 °C, mold preheat B = 20 °C, filling time C = 12 s, and pressure D = 50 kPa. Furthermore, using finite element analysis (FEA) simulation, Deng et al. [4] proposed the optimal values of Y1 as 745 s and Y2 as 74.25 cm³. The corresponding process input parameters obtained using FEA were not reported by Deng et al. [4].

Now, to explore potential improvements in the optimal sets of values of the parameters A–D and the corresponding non-dominated Pareto solutions the proposed MO-BWR, MO-BMR, and MO-BMWR algorithms are applied to the same regression models given by Equations (32) and (33), using identical input parameter bounds given by Equation (34). A population size of 50 and 200 iterations are employed, resulting in 10,000 function evaluations (same as those used by NSGA-II in [4]).

Before applying the MOO versions, BWR, BMR, and BMWR are applied to the individual objective functions individually to discover the best optimal values of solidification time Y1 and shrinkage volume Y2.

• All three algorithms have given the best minimum value of Y1 as 727.18127 s, corresponding to A = 680 °C, B = 20 °C, C = 10 s, and D = 80 kPa. The convergence graphs are shown in Figure 10.
• Similarly, all three algorithms have given the best minimum value of Y2 as 74.3563 cm³, corresponding to A = 720 °C, B = 20 °C, C = 15.01 s, and D = 71.057 kPa. The convergence graphs are shown in Figure 11.

However, in actual practice, both the objective functions Y1 and Y2 have to be attempted simultaneously to find the real optimum values of variables. For this purpose, the MO-BWR, MO-BMR, and MO-BMWR algorithms are applied to find out the optimum values of parameters A–D. Deng et al. [4] used the NSGA-II algorithm with a population size of 50 and number of iterations of 200 (i.e., function evaluations = 50 ∗ 200 = 10,000).

Now the proposed three algorithms are applied with the same number of function evaluations. The non-dominated Pareto-optimal solutions obtained are combined to form the composite front. The composite front contains 40 unique, non-repeating, non-dominated solutions. These 40 Pareto-optimal solutions are formed with MO-BMR’s 15 solutions, MO-BMWR’s 10 solutions, and MO-BWR’s 10 solutions.

Table 14. Performance metrics for the three algorithms for the low-pressure casting process.

Algorithm	GD	IGD	Spacing	Spread	Hypervolume
MO-BWR	0	0.162757	0.267771	0.836069	0.516014
MO-BMR	0	0.065284	0.021535	0.845045	0.623620
MO-BMWR	0	0.032764	0.045740	0.506841	0.731088
Composite	0	0	0.042701	0.811891	0.762125

shows the performance metrics of the three algorithms with reference to the composite front.

The MO-BMWR outperforms in IGD (0.032764) and has the highest hypervolume (0.731088) among the individual algorithms, indicating high-quality and well-spread solutions. The MO-BMR achieves the lowest spacing (0.021535), meaning its solutions are very evenly distributed. The MO-BWR lags behind in IGD and hypervolume, suggesting less extensive and less representative coverage. The composite front shows the best hypervolume and perfect IGD, as expected since it is the reference.

The composite Pareto front represents a non-dominated frontier obtained by merging the elite-boosted runs, providing the best trade-off solutions derived from the strategies of all three algorithms.

Table 15. Forty unique non-dominated optimal solutions from the composite front (MO-BWR + MO-BMR + MO-BMWR) for the low-pressure casting process.

S.No.	A (°C)	B (°C)	C (s)	D (kPa)	Y1 (s)	Y2 (cm3)	Algorithm	Normalized Y1	Normalized Y2	Score
1	680	20	10	50	734.7135	75.14654	MO-BWR	0.989748	0.992843	0.991296
2	685.4846	25.48459	11.37115	54.11345	757.3426	74.92711	MO-BWR	0.960175	0.995751	0.977963
3	708.8844	36.02023	15.90399	71.66332	827.3466	74.60873	MO-BWR	0.878932	1	0.939466
4	684.9727	24.97272	11.24318	53.72954	755.3044	74.94447	MO-BWR	0.962766	0.99552	0.979143
5	685.4491	25.44909	11.36227	54.08682	757.2017	74.9283	MO-BWR	0.960353	0.995735	0.978044
6	686.138	26.13801	11.5345	54.60351	759.9224	74.9059	MO-BWR	0.956915	0.996033	0.976474
7	682.0422	20	10.51054	51.53162	742.955	75.04257	MO-BWR	0.978769	0.994219	0.986494
8	688.066	28.06604	12.01651	56.04953	767.3905	74.84942	MO-BWR	0.947603	0.996784	0.972193
9	688.0047	28.00472	12.00118	56.00354	767.1563	74.85108	MO-BWR	0.947892	0.996762	0.972327
10	694.0565	29.85468	13.51413	60.54238	787.9312	74.69669	MO-BWR	0.922899	0.998822	0.960861
11	682.2537	20	10.56343	51.69028	743.7901	75.03229	MO-BWR	0.97767	0.994355	0.986012
12	691.7727	31.77268	12.94317	58.82951	781.1432	74.76651	MO-BWR	0.930919	0.99789	0.964405
13	699.3755	36.86892	14.84387	64.53161	805.9659	74.67923	MO-BWR	0.902248	0.999056	0.950652
14	684.9284	24.92842	11.2321	53.69631	755.1272	74.946	MO-BWR	0.962992	0.9955	0.979246
15	689.0101	29.01013	12.25253	56.75759	770.9689	74.8251	MO-BWR	0.943204	0.997108	0.970156
16	693.636	33.63601	13.409	60.22701	787.756	74.73758	MO-BMR	0.923105	0.998276	0.96069
17	690.4862	30.48621	12.62155	57.86466	776.4602	74.79146	MO-BMR	0.936534	0.997557	0.967045
18	680	20	10	80	727.1813	76.14287	MO-BMR	1	0.979852	0.989926
19	680	20	10	69.99374	730.3456	75.67571	MO-BMR	0.995667	0.985901	0.990784
20	693.3065	33.3065	13.32663	59.97988	786.6013	74.74207	MO-BMR	0.92446	0.998216	0.961338
21	680.6979	20.69788	10.17447	50.52341	737.6896	75.11451	MO-BMR	0.985755	0.993267	0.989511
22	680	20	10	63.60292	732.0251	75.44797	MO-BMR	0.993383	0.988877	0.99113
23	686.7382	26.73821	11.68455	55.05365	762.2703	74.88734	MO-BMR	0.953968	0.99628	0.975124
24	680	20	10	67.52154	731.0268	75.58108	MO-BMR	0.994739	0.987135	0.990937
25	685.892	25.89203	11.47301	54.41903	758.9542	74.91376	MO-BMR	0.958136	0.995928	0.977032
26	680	20	10	51.89557	734.4112	75.17359	MO-BMR	0.990156	0.992486	0.991321
27	680	20	10	68.7771	730.6858	75.62811	MO-BMR	0.995204	0.986521	0.990862
28	680	20	10	54.11347	734.0277	75.21138	MO-BMR	0.990673	0.991987	0.99133
29	690.0543	30.0543	12.51358	57.54073	774.8665	74.80075	MO-BMR	0.93846	0.997433	0.967947
30	680	20	10	74.86169	728.8877	75.88612	MO-BMR	0.997659	0.983167	0.990413
31	689.6883	29.68833	12.42208	57.26624	773.5076	74.80898	MO-BMWR	0.940109	0.997323	0.968716
32	684.2734	24.27337	11.06834	53.20503	752.4951	74.96921	MO-BMWR	0.96636	0.995192	0.980776
33	691.1338	31.13376	12.78344	58.35032	778.8293	74.77839	MO-BMWR	0.933685	0.997731	0.965708
34	688.7624	28.76243	12.19061	56.57182	770.035	74.83127	MO-BMWR	0.944348	0.997026	0.970687
35	684.201	24.20103	11.05026	53.15078	752.2029	74.97184	MO-BMWR	0.966736	0.995157	0.980946
36	690.0159	30.03474	12.50854	57.52065	774.7549	74.80144	MO-BMWR	0.938595	0.997424	0.96801
37	693.1592	33.15919	13.2898	59.86939	786.083	74.74417	MO-BMWR	0.925069	0.998188	0.961629
38	689.6283	29.58275	12.40043	57.2085	773.2337	74.81047	MO-BMWR	0.940442	0.997303	0.968873
39	687.2027	27.20275	11.80069	55.40206	764.0732	74.87358	MO-BMWR	0.951717	0.996463	0.97409
40	681.3968	21.39681	10.3492	51.04761	740.642	75.08364	MO-BMWR	0.981826	0.993675	0.98775

presents the set of 40 unique non-dominated optimal solutions produced by the three algorithms, after eliminating all duplicate solutions.

The metal casting process planners or decision-makers can select a point from the Pareto front based on their preference or constraints, e.g., if cycle time is critical, they can pick a point with lower Y₁ even if Y₂ is slightly higher. If defect minimization is more important, choose a solution with lower Y₂ even if Y₁ increases.

Figure 12 presents the non-dominated Pareto solutions of the composite front.

Now, the BHARAT method [23] is applied to find the best compromise Pareto-optimal solution out of the 29 non-dominated solutions of the composite front obtained by the combination of MO-BWR, MO-BMR, and MO-BMWR. The objectives values of Y1 and Y2 are normalized and shown in Table 15. As lower values are desired for both Y1 and Y2, the best values of Y1 and Y2 (i.e., Y₁^best = 727.1813 and Y₂^best = 74.60873, respectively) are taken from Table 15. The other values are normalized as Yj^normalized = Yi^best/Yj. For example, the normalized values of Y1 and Y2 for solution no. 1 are 0.989748 (=727.1813/734.7135) and 0.992843 (=74.60873/75.14654), respectively. Similarly, the values of Y1 and Y2 are normalized for the other solutions.

If the decision-maker gives equal importance to both objectives Y₁ and Y₂ (i.e., wᵧ₁ = wᵧ₂), then each objective is assigned an average rank of 1.5 (the mean of ranks 1 and 2). Based on the reference table provided in [23], the BHARAT method assigns an equal weight of 0.5 to each objective for the rank of 1.5. The corresponding scores (weighted sums) are presented in the second-last column in Table 15. For example, the score for solution no. 1 is calculated as 0.5 ∗ 0.989748 + 0.5 ∗ 0.992843 = 0.991296. It is clear from the scores of the solution given in Table 15 that solution 1 is the best Pareto solution.

Table 16. Comparison of optimization results for the low-pressure casting process.

Optimization Method	Optimum Pouring Temperature A (°C)	Optimum Mold Preheating Temperature B (°C)	Optimum Filling Time C (s)	Optimum Holding Pressure D (kPa)	Optimum Solidification Time Y1 (s)	Optimum Shrinkage Volume Y2 (cm3)
NSGA-II [4]	680	20	12	50	750.17	74.89
Finite Element Simulation [4]	Not given	Not given	Not given	Not given	745	74.25
Composite front of the present work	680	20	10	54.11	734.02	75.21 (MO-BMR)
	680	20	10	51.89	734.41	75.17 (MO-BMR)
	680	20	10	50	734.71	75.14 (MO-BWR)
	680	20	10	63.60	732.02	75.44 (MO-BMR)
	680	20	10	67.52	731.02	75.58 (MO-BMR)

makes a comparison of the optimization results. The best values of Y1 and Y2 are indicated in bold. The composite front has given a number of trade-off solutions and five of the such top solutions are included in Table 16.

It may be observed that the proposed algorithms have given trade-off solutions, and five of the top solutions given in Table 16 give a much better minimum solidification time Y1 compared to the NSGA-II and finite element simulation of Deng et al. [4]. The optimum shrinkage volume given by the proposed algorithms is also reasonable compared to the solutions given by Deng et al. [4]. In fact, Deng et al. [4] did not provide the values of the input parameters to check the values of Y1 and Y2 reported by them.

This example validates the proposed MO-BWR, MO-BMR, and MO-BMWR algorithms for the multi-objective optimization of the low-pressure casting process.

The surface plots showing the effects of the process parameters A, B, C, and D on Y1 and Y2 are shown in Figure 13. The 12 surface plots are arranged in four rows, with 3 plots per row, showing the effect of all pairwise combinations of A, B, C, and D on Y₁ (solidification time) and Y₂ (shrinkage volume).

In the above plot of solidification time Y1 vs. A and B, the values of C and D are fixed at their mid-point values of 15 s and 65 kPa, respectively. In the plot of Y1 vs. A and C, the values of B and D are fixed at their mid-point values of 40 °C and 65 kPa, respectively. In the plot of Y1 vs. A and D, the values of B and C are fixed at their mid-point values of 40 °C and 15 s, respectively. In the plot of Y1 vs. B and C, the values of A and D are fixed at their mid-point values of 700 °C and 65 kPa, respectively. In the plot of Y1 vs. B and D, the values of A and C are fixed at their mid-point values of 700 °C and 15 s, respectively. In the plot of Y1 vs. C and D, and the values of A and D are fixed at their mid-point values of 700 °C and 65 kPa, respectively. Similar fixing of the variables A, B, C, and D is performed for the plots of shrinkage volume Y2 versus the variables.

The surface plots reveal distinct and consistent trends in how the input parameters A–D influence the outputs Y₁ (solidification time) and Y₂ (shrinkage volume). For Y₁, solidification time generally increases with higher values of C (filling time) and A (pouring temperature), indicating their dominant influence. The roles of B and D are more moderate but still positive, suggesting a cumulative effect when all variables are high. Conversely, Y₂ (shrinkage volume) exhibits a more complex behavior. It tends to decrease with increasing C and D, indicating their effectiveness in minimizing shrinkage. There is a subtle curvature with respect to A and B, suggesting optimal intermediate values rather than extreme high or low settings. Overall, reducing Y₂ while keeping Y₁ low appears to require careful balancing—especially increasing C and D while avoiding excessively high A and B. The nonlinearity and interaction effects across the plots highlight the importance of multi-objective optimization for effective process parameter tuning.

3.2.2. Multi-Objective Optimization of a Squeeze Casting Process for LM20 Material

Optimizing squeeze casting parameters is vital for producing high-quality, low-cost castings, though limited studies exist. Patel et al. [11] carried out squeeze casting of LM20 aluminum alloy. LM20 alloy was selected for its excellent fluidity, pressure tightness, and resistance to hot tearing, wear, and corrosion, making it ideal for marine, automotive, engineering, and domestic applications. H13-grade chromium–molybdenum heat-treated steel was used for making the die. After cleaning and degassing, the molten metal was poured into a preheated die, pressed until solidification, then released and removed from the die. The squeeze casting process parameters considered were as follows: time delay (Td) in seconds, pressure duration (Dp) in seconds, squeeze pressure (Sp) in MPa, pouring temperature (Pt) in °C, and die temperature (Dt) in °C.

Patel et al. [11] employed statistical tools to develop nonlinear regression models and identify the significant effects of squeeze casting process parameters on surface roughness (Ra), hardness (H), yield strength (YS), and ultimate tensile strength (UTS) of the produced parts. The accuracy of these models was validated using ten test cases, showing that they could predict the responses with high reliability. The resulting mathematical input–output relationships offered practical guidance for foundry operators to make more accurate process predictions. This study addressed the four inherently conflicting objectives of Ra, H, YS, and UTS, which were mathematically formulated into a single objective function. The widely used evolutionary genetic algorithm (GA) was then applied to determine the optimal process parameters. Multi-objective optimization was performed using the evolutionary genetic algorithm (GA). The optimization was conducted by assigning different weights to the responses, resulting in a set of optimal process parameters determined according to the assigned priorities.

Now to demonstrate and validate the proposed BWR, BMR, and BMWR algorithms, the squeeze casting process parameters case study presented by Patel et al. [11] is considered to see if there can be any further improvement in the optimal solutions. The four objective functions (i.e., output responses) are described by Equations (35)–(38). The bounds of the process input parameters are given by Equation (39). These are all as follows:

Surface roughness Ra (μm) = 19.4116 − 0.0933924 ∗ Td − 0.0170929 ∗ Dp− 0.0124204 ∗ Sp − 0.0437485 ∗ Pt − 0.00421634 ∗ Dt+ 0.0109566 ∗ Td² + 0.000107382 ∗ Dp²+ 3.40018E − 05 ∗ Sp² + 2.85979E − 05 ∗ Pt²+ 6.76585E − 06 ∗ Dt²

(35)

Yield strength YS (MPa) = − 879.007 + 0.100644 ∗ Td+ 0.616749 ∗ Dp + 0.627314 ∗ Sp+ 2.53574 ∗ Pt + 0.610584 ∗ Dt− 0.189332 ∗ Td² − 0.00727915 ∗ Dp²− 0.0020318 ∗ Sp² − 0.00178919 ∗ Pt²− 0.00132646 ∗ Dt²

(36)

Ultimate tensile strength UTS (MPa) = −1414.09 − 5.86888 ∗ Td + 1.31391 ∗ Dp+ 0.815477 ∗ Sp + 4.15593 ∗ Pt+ 0.887826 ∗ Dt + 0.119206 ∗ Td²− 0.0152318 ∗ Dp² − 0.00243251 ∗ Sp²− 0.00293425 ∗ Pt² − 0.00184456 ∗ Dt²

(37)

Hardness H (BHN) = − 283.664 − 1.09353 ∗ Td + 0.214051 ∗ Dp+ 0.307208 ∗ Sp + 0.87264 ∗ Pt+ 0.208767 ∗ Dt − 0.0254625 ∗ Td² − 0.00190085 ∗ Dp² − 9.35689 ∗ E − 04 ∗ Sp²− 6.03363E − 04 ∗ Pt² − 3.81916E − 04 ∗ Dt²

(38)

3 s ≤ Td ≤ 11 s, 10 s ≤ Dp ≤ 50 s, 0.1 MPa ≤ Sp ≤ 200 MPa, 630 °C ≤ Pt ≤ 750 °C, 100 °C ≤ Dt ≤ 300 °C

(39)

The Ra is to be minimized and the YS, UTS, and H are to be maximized.

Patel et al. [11] used a genetic algorithm (GA) to solve this multi-objective problem using a priori approach (i.e., assigning different weightages to the objective functions). The population size and the iterations used in the GA were 120 and 100, respectively (i.e., the function evaluations were 120 ∗ 100 = 12,000). For the sake of comparison, the same are used in the present work using the BWR, BMR, and BMWR algorithms. At first, just to obtain the ideal optimal values of Ra, YS, UTS, and H, the three algorithms are run on the individual objective functions. The results of individual optimization are given in

Table 17. Results of optimization of individual objective functions of squeeze casting process.

Algorithm	Process Parameters					Objectives (i.e., Process Responses)
	Td (s)	Dp (s)	Sp (MPa)	Pt (°C)	Dt (°C)	Ra (µm)	YS (MPa)	UTS (MPa)	H (BHN)
BWR	4.261923	50	182.6434	750	300	0.111111
BMR	4.261923	50	182.6434	750	300	0.111111
BMWR	4.261923	50	182.6434	750	300	0.111111
BWR	3	42.41497	154.1347	708.4726	230.3095		149.7879
BMR	3	42.41497	154.1347	708.4726	230.3095		149.7879
BMWR	3	42.41497	154.1347	708.4726	230.3095		149.7879
BWR	3	43.12943	167.647	708.146	240.6784			244.4533
BMR	3	43.12943	167.647	708.146	240.6784			244.4533
BMWR	3	43.12943	167.647	708.146	240.6784			244.4533
BWR	3	50	164.1633	723.1391	273.3112				88.0455
BMR	3	50	164.1633	723.1391	273.3112				88.0455
BMWR	3	50	164.1633	723.1391	273.3112				88.0455

. The best values of the objectives obtained by the proposed algorithms are given in Table 17. It can be observed that all the three algorithms converged to the same values of respective objective functions.

Figure 14 shows the convergence graphs of the BWR, BMR, and BMWR algorithms for the four objectives Ra, YS, UTS, and H when solved for each objective function individually. All these algorithms converged to the same results.

Now, to deal with the task of simultaneously optimizing all four objectives, the proposed MO-BWR, MO-BMR, and MO-BMWR algorithms are applied each with a population size of 120 and iterations of 100 (same as those used by Patel et al. [11]). The composite front containing 94 non-repeating and Pareto-optimal non-dominated solutions (with 23 solutions from BWR, 52 from BMR, and 19 from BMWR) is formed and shown in

Table 18. Ninety-four Pareto-optimal solutions of the composite front for the squeeze casting process.

Solution	Squeeze Casting Process Parameters					Process Responses (i.e., Objectives)				Algorithm
	Td (s)	Dp (s)	Sp (MPa)	Pt (°C)	Dt (°C)	Ra (µm)	YS (MPa)	UTS (MPa)	H (BHN)	Algorithm
1	3	44.18163	161.653	711.2367	240.5869	0.293334	149.4999	244.3224	87.34117	MO-BMWR
2	3	44.54666	160.6631	711.2168	243.4631	0.289372	149.4262	244.2634	87.42063	MO-BWR
3	3	44.67159	159.4151	714.4595	239.9935	0.283978	149.5085	244.1367	87.37318	MO-BMR
4	3	44.83501	163.0088	712.1447	246.8998	0.27798	149.1982	244.2393	87.53026	MO-BWR
5	3	45.15195	166.6539	713.2368	242.3581	0.272064	149.1896	244.3084	87.45358	MO-BMWR
6	3	44.14191	157.9521	708.5658	234.6453	0.313732	149.7124	244.1432	87.10444	MO-BMWR
7	3	45.26108	161.1808	716.6505	241.7855	0.269015	149.3383	244.0703	87.47581	MO-BWR
8	3	45.10441	163.5533	714.0757	246.921	0.269522	149.1363	244.1793	87.56658	MO-BMR
9	3	44.17161	155.1055	708.3735	237.4219	0.316343	149.6931	244.0364	87.14081	MO-BWR
10	3	45.04359	163.7003	714.3668	249.1897	0.266989	149.0196	244.1136	87.61104	MO-BWR
11	3	44.09115	152.9356	711.4833	238.185	0.310642	149.6621	243.8713	87.16622	MO-BWR
12	3	47.68317	159.557	714.556	236.1015	0.26574	149.4178	243.8217	87.38652	MO-BMR
13	3	45.52768	158.0346	719.1317	245.1684	0.262198	149.1917	243.7526	87.5529	MO-BWR
14	3	44.99959	165.8949	715.562	251.3382	0.259457	148.7867	244.0225	87.65617	MO-BMWR
15	3	46.57805	159.0652	719.7985	246.6745	0.249873	149.0289	243.6312	87.63913	MO-BMR
16	3	44.73873	152.6987	714.4641	232.0572	0.30372	149.6756	243.6198	87.04827	MO-BMR
17	3	47.40826	166.9209	714.6235	251.2354	0.243811	148.6293	243.8452	87.73349	MO-BMWR
18	3	46.42214	159.6335	714.8783	253.8669	0.257495	148.7964	243.6796	87.73055	MO-BWR
19	3	47.67281	158.9126	719.9781	246.2668	0.242383	148.9663	243.4879	87.6682	MO-BMR
20	3	45.96921	160.8699	723.0505	236.6517	0.252846	149.1796	243.5409	87.39452	MO-BMWR
21	3	46.35167	161.9085	723.0113	237.7331	0.247604	149.1107	243.5543	87.44449	MO-BWR
22	3	49.3623	159.9883	720.2755	242.8822	0.23172	148.91	243.2814	87.65448	MO-BWR
23	3	45.59133	167.3896	719.3487	254.638	0.240582	148.3674	243.6343	87.75124	MO-BMWR
24	3	45.89205	162.5846	722.1216	253.2663	0.238232	148.5263	243.4117	87.75852	MO-BWR
25	3	47.35981	167.9442	721.7841	250.5326	0.224544	148.3718	243.4575	87.75625	MO-BMR
26	3	48.11982	158.2283	716.7499	257.6829	0.239842	148.3936	243.1094	87.84279	MO-BMR
27	3	47.6624	162.7167	720.2609	256.8525	0.227484	148.2548	243.1699	87.86863	MO-BMR
28	3	47.4309	156.3822	722.9969	252.775	0.235388	148.545	242.9492	87.75364	MO-BMR
29	3	46.76511	161.2831	728.2398	244.7107	0.22718	148.581	242.943	87.61221	MO-BMWR
30	3	47.84515	168.0365	722.0871	255.4358	0.216485	148.0183	243.1438	87.84824	MO-BMWR
31	3	49.658	165.1301	715.4336	259.0886	0.222522	147.9725	243.0083	87.92303	MO-BMR
32	3	49.26079	176.1734	721.0641	244.8636	0.212542	147.9127	243.183	87.57998	MO-BMWR
33	3	48.20985	161.7354	730.0359	243.004	0.214461	148.3903	242.5638	87.61149	MO-BMR
34	3	49.43683	170.3026	717.9203	258.9769	0.211921	147.6521	242.9326	87.90114	MO-BMR
35	3	48.71912	161.6153	719.7256	262.7071	0.219162	147.7617	242.6019	87.9556	MO-BMR
36	3	47.04752	167.5148	729.3287	255.0058	0.206618	147.6918	242.5271	87.79658	MO-BWR
37	3	49.71336	177.3987	723.7574	247.6101	0.200155	147.5042	242.7592	87.62196	MO-BMWR
38	3	46.90786	160.7199	732.3403	252.8415	0.211979	147.8673	242.1331	87.73107	MO-BWR
39	3	49.18771	163.0856	732.9972	241.0105	0.202341	148.0762	242.0365	87.56661	MO-BMR
40	3	47.92386	169.7216	732.1216	249.4968	0.197437	147.6008	242.2661	87.69338	MO-BWR
41	3	47.12514	168.6922	725.2975	263.2747	0.207707	147.2544	242.404	87.90041	MO-BWR
42	3	50	168.4982	725.7775	258.6422	0.19083	147.3557	242.2272	87.94152	MO-BMR
43	3	48.93892	165.129	733.511	244.8578	0.196892	147.8439	242.0085	87.64298	MO-BMR
44	3	49.72561	161.2063	721.2148	265.0324	0.207904	147.4018	242.0963	88.00218	MO-BMR
45	3	49.50244	159.7381	728.6085	258.757	0.198868	147.5593	241.8547	87.91588	MO-BMR
46	3	48.41383	167.4577	722.2672	270.1418	0.203078	146.7202	241.8423	87.98825	MO-BWR
47	3	50	163.571	724.5179	268.0383	0.193123	146.8365	241.5285	88.03343	MO-BMR
48	3	49.10861	164.1719	737.4422	250.7196	0.185275	147.2155	241.1802	87.70436	MO-BMR
49	3	47.96363	174.5148	730.1604	262.1093	0.188	146.5517	241.7152	87.8109	MO-BMWR
50	3	49.52507	168.6063	737.6373	249.0467	0.178757	147.0242	241.1516	87.6636	MO-BWR
51	3	50	169.9449	736.4611	250.7107	0.174982	146.9246	241.1875	87.71212	MO-BMR
52	3	48.29039	176.5516	729.8205	262.4282	0.185274	146.348	241.6051	87.78321	MO-BMWR
53	3	49.40561	172.8693	731.537	262.6046	0.176421	146.3965	241.297	87.87337	MO-BMR
54	3	50	166.9879	735.5043	258.3186	0.173539	146.6959	240.967	87.86002	MO-BMR
55	3	48.75456	161.8118	729.9932	271.2853	0.189963	146.3178	240.7628	87.97771	MO-BMR
56	3	50	166.186	740.2653	253.1963	0.17093	146.5852	240.4182	87.71029	MO-BMR
57	3	50	170.2129	725.9818	274.6315	0.179098	145.6913	240.6593	88.00575	MO-BMR
58	3	49.12464	160.7683	730.1268	274.9778	0.186854	145.8805	240.2057	87.98186	MO-BMR
59	3	50	166.504	741.4914	256.4905	0.166364	146.2125	240.0125	87.72923	MO-BMR
60	3	48.84152	179.2203	737.3762	261.4312	0.167885	145.4522	240.3316	87.62692	MO-BMR
61	3	49.3342	170.4196	742.972	258.7064	0.163329	145.7197	239.6947	87.67342	MO-BWR
62	3	50	168.0049	744.2395	255.4344	0.162127	145.8695	239.5153	87.64115	MO-BWR
63	3	49.55151	163.2018	742.4222	263.659	0.167013	145.7215	239.3609	87.77374	MO-BMR
64	3	50	178.4454	741.7557	255.6128	0.158415	145.3632	239.7285	87.52599	MO-BMR
65	3	50	169.0259	742.0731	265.8839	0.156637	145.2329	239.1847	87.78616	MO-BMR
66	3	50	176.5496	726.7183	280.2386	0.170892	144.4518	239.6424	87.87592	MO-BMR
67	3	50	161.689	732.8145	280.1212	0.172362	144.8967	238.9955	87.96572	MO-BMR
68	3	48.58098	163.7929	739.8322	274.5845	0.169847	144.966	238.902	87.83897	MO-BMR
69	3	50	173.458	744.9693	265.3419	0.149999	144.6185	238.5558	87.65304	MO-BWR
70	3	49.04694	163.6946	737.7385	279.922	0.16754	144.4851	238.475	87.87562	MO-BMR
71	3	50	171.1127	747.9459	266.6533	0.147703	144.2615	237.8177	87.61229	MO-BWR
72	3	49.63502	163.8374	740.196	282.707	0.158752	143.7751	237.5047	87.82736	MO-BMR
73	3	50	173.3144	750	269.6785	0.142494	143.5003	236.9697	87.527	MO-BWR
74	3	50	164.5898	741.8089	284.6633	0.152533	143.2407	236.8215	87.78603	MO-BMR
75	3.818123	49.32544	168.2566	743.2671	256.8835	0.151013	144.9756	235.6326	86.62875	MO-BMWR
76	3.763176	49.29088	170.8912	743.7041	259.0603	0.147349	144.6746	235.6573	86.68662	MO-BMWR
77	3.528248	49.61432	167.588	743.1029	273.3304	0.143412	143.8518	235.5756	87.11926	MO-BMR
78	3	50	167.3271	741.5766	289.3435	0.148178	142.4335	236.0892	87.73309	MO-BMR
79	3	50	167.8223	750	280.6006	0.141617	142.5583	235.6593	87.57763	MO-BMR
80	3.780539	49.46084	171.2914	745.534	261.7692	0.141748	144.1539	234.9432	86.64287	MO-BMWR
81	3	50	183.1604	750	272.721	0.137881	142.2142	236.1184	87.27256	MO-BMR
82	3	50	180.9857	750	276.1779	0.136228	142.0528	235.8404	87.34246	MO-BMR
83	3.530566	50	168.8275	744.5875	276.5317	0.136316	143.1702	234.7665	87.07542	MO-BMR
84	3.884643	49.69957	172.2824	746.8306	263.5071	0.13596	143.594	233.9302	86.4788	MO-BMWR
85	3.944815	50	178.8443	741.1399	270.7461	0.132873	142.9231	233.8073	86.44568	MO-BMR
86	3.918808	50	173.1911	748.3075	265.0832	0.130687	143.0979	233.1985	86.39477	MO-BMWR
87	3	49.62956	175.2274	745.1802	295.2004	0.138476	140.5178	234.1645	87.44597	MO-BMR
88	3.598671	50	169.1933	747.9599	283.8465	0.128237	141.6381	232.6015	86.85274	MO-BMR
89	3.956939	49.36912	169.062	744.4129	282.4436	0.132944	141.9107	231.9596	86.48653	MO-BMR
90	3	50	176.2479	747.7101	298.4581	0.132308	139.4704	232.8056	87.30336	MO-BMR
91	3.617944	49.90341	170.8667	748.8612	287.9335	0.12487	140.7853	231.6107	86.74068	MO-BMR
92	3	49.81773	180.8679	750	299.8957	0.129845	138.4436	231.7404	87.07502	MO-BWR
93	3.974298	50	171.5308	748.3405	287.3691	0.120769	140.4134	230.0316	86.29786	MO-BMR
94	4.761999	50	186.6837	747.9199	300	0.116302	135.5975	223.0101	84.65361	MO-BMWR

.

As this problem contains four objectives, the best compromise solution is to be selected from the 94 Pareto-optimal solutions. BHARAT method [24] is used for normalizing the data of the objectives. In the case of the minimization of Ra, the best value is taken as 0.116302 and it is kept in the numerator and the remaining values of Ra in that column are put in the denominator to obtain the normalized data of Ra. For example, for solution 1, the normalized Ra value is 0.396482 (=0.116302/0.293334). In the case of maximization of YS, the best value is taken as 149.7124 and it is kept in the denominator and the remaining values of YS in that column are put in the numerator to obtain the normalized data of YS. A similar procedure is carried out for normalizing the data of UTS and H. The normalized data of the Ra, YS, UTS, and H are given in

Table 19. Normalized objective data and composite scores of the solutions for various weight assignments to the objectives in the squeeze casting process.

Solution	Normalized Ra	Normalized YS	Normalized UTS	Normalized H	Algorithm	Composite Scores of the Solutions for Different Weightages
						Equation Wts. (Case 1)	WRa = 0.7 (Case 2)	WYS = 0.7 (Case 3)	WUTS = 0.7 (Case 4)	WH= 0.7 (Case 5)
1	0.396482	0.998581	1	0.992136	MO-BMWR	0.8468	0.576609	0.937869	0.93872	0.934002
2	0.40191	0.998089	0.999759	0.993039	MO-BWR	0.848199	0.580426	0.938133	0.939135	0.935103
3	0.409545	0.998638	0.99924	0.9925	MO-BMR	0.849981	0.585719	0.939175	0.939536	0.935492
4	0.418382	0.996565	0.99966	0.994284	MO-BWR	0.852223	0.591918	0.938828	0.940685	0.93746
5	0.427478	0.996509	0.999943	0.993413	MO-BMWR	0.854336	0.598221	0.939639	0.9417	0.937782
6	0.370704	1	0.999267	0.989447	MO-BMWR	0.839854	0.558364	0.935942	0.935502	0.92961
7	0.432324	0.997502	0.998968	0.993666	MO-BWR	0.855615	0.60164	0.940747	0.941627	0.938445
8	0.43151	0.996152	0.999414	0.994697	MO-BMR	0.855443	0.601084	0.939869	0.941826	0.938996
9	0.367644	0.999871	0.998829	0.98986	MO-BWR	0.839051	0.556207	0.935543	0.934918	0.929537
10	0.435605	0.995373	0.999145	0.995202	MO-BWR	0.856331	0.603895	0.939756	0.94202	0.939654
11	0.374392	0.999664	0.998154	0.990149	MO-BWR	0.84059	0.560871	0.936034	0.935128	0.930325
12	0.437652	0.998033	0.997951	0.992652	MO-BMR	0.856572	0.60522	0.941448	0.941399	0.93822
13	0.443565	0.996522	0.997668	0.994541	MO-BWR	0.858074	0.609368	0.941143	0.94183	0.939955
14	0.448251	0.993817	0.998773	0.995715	MO-BMWR	0.859139	0.612606	0.939946	0.942919	0.941084
15	0.465443	0.995435	0.997171	0.995521	MO-BMR	0.863392	0.624623	0.942618	0.943659	0.94267
16	0.382924	0.999755	0.997124	0.988809	MO-BMR	0.842153	0.566616	0.936714	0.935136	0.930147
17	0.477016	0.992766	0.998047	0.996593	MO-BMWR	0.866105	0.632652	0.942102	0.94527	0.944398
18	0.451666	0.993882	0.997369	0.99656	MO-BWR	0.859869	0.614948	0.940277	0.942369	0.941883
19	0.479826	0.995017	0.996585	0.995851	MO-BMR	0.86682	0.634623	0.943738	0.944679	0.944239
20	0.459971	0.996442	0.996801	0.992742	MO-BMWR	0.861489	0.620578	0.942461	0.942676	0.940241
21	0.469708	0.995981	0.996856	0.99331	MO-BWR	0.863964	0.627411	0.943174	0.943699	0.941572
22	0.501906	0.994641	0.995739	0.995695	MO-BWR	0.871995	0.649942	0.945583	0.946242	0.946215
23	0.483419	0.991016	0.997184	0.996794	MO-BMWR	0.867103	0.636892	0.941451	0.945152	0.944918
24	0.488186	0.992078	0.996273	0.996877	MO-BWR	0.868354	0.640253	0.942588	0.945105	0.945468
25	0.517946	0.991046	0.99646	0.996851	MO-BMR	0.875576	0.660998	0.944858	0.948106	0.948341
26	0.48491	0.991192	0.995035	0.997834	MO-BMR	0.867243	0.637843	0.941612	0.943918	0.945598
27	0.511251	0.990264	0.995283	0.998128	MO-BMR	0.873732	0.656243	0.943651	0.946662	0.948369
28	0.494084	0.992202	0.99438	0.996822	MO-BMR	0.869372	0.644199	0.94307	0.944377	0.945842
29	0.511936	0.992443	0.994354	0.995215	MO-BMWR	0.873487	0.656556	0.94486	0.946007	0.946524
30	0.537227	0.988685	0.995176	0.997896	MO-BMWR	0.879746	0.674235	0.945109	0.949004	0.950636
31	0.522651	0.988379	0.994621	0.998746	MO-BMR	0.876099	0.664031	0.943467	0.947213	0.949687
32	0.547194	0.987979	0.995337	0.994849	MO-BMWR	0.88134	0.680852	0.945323	0.949738	0.949445
33	0.542297	0.991169	0.992802	0.995207	MO-BMR	0.880369	0.677526	0.946849	0.947829	0.949272
34	0.548797	0.986239	0.994312	0.998497	MO-BMR	0.881961	0.682063	0.944528	0.949372	0.951883
35	0.530664	0.986971	0.992958	0.999116	MO-BMR	0.877427	0.66937	0.943153	0.946746	0.95044
36	0.562883	0.986503	0.992652	0.99731	MO-BWR	0.884837	0.691664	0.945837	0.949526	0.952321
37	0.581056	0.98525	0.993602	0.995326	MO-BMWR	0.888809	0.704157	0.946674	0.951685	0.952719
38	0.548647	0.987676	0.991039	0.996565	MO-BWR	0.880982	0.681581	0.944998	0.947016	0.950332
39	0.574782	0.989071	0.990644	0.994697	MO-BMR	0.887298	0.699788	0.948362	0.949306	0.951738
40	0.589057	0.985896	0.991584	0.996137	MO-BWR	0.890668	0.709702	0.947805	0.951218	0.95395
41	0.559931	0.983582	0.992148	0.998489	MO-BWR	0.883538	0.689374	0.943564	0.948704	0.952508
42	0.609453	0.984259	0.991424	0.998956	MO-BMR	0.896023	0.724081	0.948965	0.953264	0.957783
43	0.590688	0.98752	0.990529	0.995565	MO-BMR	0.891075	0.710843	0.948942	0.950748	0.953769
44	0.559401	0.984567	0.990888	0.999645	MO-BMR	0.883625	0.689091	0.94419	0.947983	0.953237
45	0.584819	0.985619	0.9899	0.998665	MO-BMR	0.889751	0.706792	0.947272	0.94984	0.955099
46	0.572695	0.980014	0.989849	0.999487	MO-BWR	0.885511	0.697821	0.942213	0.948114	0.953896
47	0.602215	0.980791	0.988565	1	MO-BMR	0.892892	0.718486	0.945631	0.950296	0.957157
48	0.627725	0.983322	0.987139	0.996262	MO-BMR	0.898612	0.73608	0.949438	0.951728	0.957202
49	0.618624	0.978889	0.989329	0.997472	MO-BMWR	0.896078	0.729606	0.945765	0.952029	0.956915
50	0.650612	0.982045	0.987022	0.995799	MO-BWR	0.903869	0.751915	0.950775	0.953761	0.959027
51	0.664651	0.981379	0.987169	0.99635	MO-BMR	0.907387	0.761745	0.951782	0.955256	0.960765
52	0.627729	0.977528	0.988878	0.997158	MO-BMWR	0.897823	0.735767	0.945646	0.952456	0.957424
53	0.659227	0.977852	0.987617	0.998182	MO-BMR	0.905719	0.757824	0.948999	0.954858	0.961197
54	0.670178	0.979852	0.986267	0.99803	MO-BMR	0.908581	0.765539	0.951344	0.955193	0.962251
55	0.612233	0.977326	0.985431	0.999367	MO-BMR	0.893589	0.724776	0.943831	0.948694	0.957056
56	0.680404	0.979112	0.98402	0.996329	MO-BMR	0.909966	0.772229	0.951454	0.954399	0.961784
57	0.649374	0.973141	0.985007	0.999686	MO-BMR	0.901802	0.750345	0.944605	0.951725	0.960532
58	0.622418	0.974405	0.983151	0.999414	MO-BMR	0.894847	0.73139	0.942582	0.947829	0.957587
59	0.699077	0.976622	0.98236	0.996545	MO-BMR	0.913651	0.784907	0.951434	0.954876	0.963387
60	0.692745	0.971544	0.983666	0.995382	MO-BMR	0.910834	0.77998	0.94726	0.954533	0.961563
61	0.712069	0.973331	0.981059	0.995911	MO-BWR	0.915593	0.793478	0.950236	0.954872	0.963783
62	0.71735	0.974332	0.980325	0.995544	MO-BWR	0.916888	0.797165	0.951354	0.95495	0.964081
63	0.696362	0.973343	0.979693	0.99705	MO-BMR	0.911612	0.782462	0.948651	0.95246	0.962875
64	0.734158	0.97095	0.981197	0.994236	MO-BMR	0.920135	0.808549	0.950624	0.956772	0.964596
65	0.742489	0.97008	0.978972	0.997191	MO-BMR	0.922183	0.814367	0.950921	0.956256	0.967188
66	0.680556	0.964862	0.980845	0.998211	MO-BMR	0.906119	0.770781	0.941365	0.950954	0.961374
67	0.674751	0.967834	0.978197	0.999231	MO-BMR	0.905003	0.766852	0.942702	0.94892	0.96154
68	0.684743	0.968297	0.977814	0.997791	MO-BMR	0.907161	0.77371	0.943843	0.949553	0.961539
69	0.775352	0.965975	0.976397	0.995679	MO-BWR	0.928351	0.836552	0.950926	0.957179	0.968748
70	0.694173	0.965084	0.976067	0.998207	MO-BMR	0.908383	0.779857	0.942404	0.948993	0.962278
71	0.7874	0.963591	0.973377	0.995216	MO-BWR	0.929896	0.844399	0.950113	0.955984	0.969088
72	0.732601	0.960342	0.972095	0.997659	MO-BMR	0.915674	0.80583	0.942475	0.949527	0.964865
73	0.816186	0.958507	0.969906	0.994247	MO-BWR	0.934711	0.863596	0.948989	0.955828	0.970433
74	0.76247	0.956773	0.969299	0.99719	MO-BMR	0.921433	0.826055	0.942637	0.950153	0.966887
75	0.770145	0.968361	0.964433	0.984044	MO-BMWR	0.921746	0.830785	0.949715	0.947358	0.959125
76	0.789294	0.96635	0.964534	0.984701	MO-BMWR	0.92622	0.844064	0.950298	0.949208	0.961309
77	0.810962	0.960855	0.9642	0.989616	MO-BMR	0.931408	0.85914	0.949076	0.951083	0.966333
78	0.784878	0.951381	0.966302	0.996588	MO-BMR	0.924787	0.840842	0.940744	0.949696	0.967868
79	0.821243	0.952215	0.964542	0.994822	MO-BMR	0.933205	0.866028	0.944611	0.952007	0.970176
80	0.820482	0.962872	0.961611	0.984204	MO-BMWR	0.932292	0.865206	0.95064	0.949884	0.963439
81	0.843492	0.949917	0.966421	0.991357	MO-BMR	0.937797	0.881214	0.945069	0.954972	0.969933
82	0.853726	0.948838	0.965284	0.992151	MO-BMR	0.94	0.888236	0.945303	0.95517	0.971291
83	0.853175	0.956302	0.960888	0.989118	MO-BMR	0.939871	0.887853	0.949729	0.952481	0.969419
84	0.855413	0.959133	0.957465	0.98234	MO-BMWR	0.938588	0.888683	0.950915	0.949914	0.964839
85	0.875287	0.954652	0.956962	0.981964	MO-BMR	0.942216	0.902059	0.949677	0.951064	0.966065
86	0.889924	0.955819	0.954471	0.981386	MO-BMWR	0.9454	0.912114	0.951651	0.950842	0.966991
87	0.839871	0.938585	0.958424	0.993327	MO-BMR	0.932552	0.876943	0.936172	0.948075	0.969017
88	0.906925	0.946068	0.952027	0.986588	MO-BMR	0.947902	0.923316	0.946802	0.950377	0.971114
89	0.874819	0.947889	0.9494	0.982428	MO-BMR	0.938634	0.900345	0.944187	0.945093	0.964911
90	0.879022	0.931589	0.952862	0.991707	MO-BMR	0.938795	0.902931	0.934472	0.947235	0.970542
91	0.931382	0.940372	0.947972	0.985315	MO-BMR	0.95126	0.939333	0.944727	0.949287	0.971693
92	0.895699	0.92473	0.948502	0.989113	MO-BWR	0.939511	0.913224	0.930643	0.944906	0.969272
93	0.96301	0.937888	0.941508	0.980285	MO-BMR	0.955673	0.960075	0.945002	0.947174	0.97044
94	1	0.90572	0.91277	0.961608	MO-BMWR	0.945024	0.97801	0.921442	0.925671	0.954974

.

The weights of importance of the objectives can be assigned following the BHARAT method. However, to make a fair comparison with the a priori GA approach followed by Patel et al. [11], the following steps are taken.

Patel et al. [11] considered following different situations of assigning different priorities (i.e., weightages) to the objectives.

• Case 1: Assigning equal weightages to all the objectives Ra, YS, UTS, and H, (i.e., W_Ra = W_YS = W_UTS = W_H = 0.25).
• Case 2: Assigning 70% weightage to Ra and assigning 10% weightage each to YS, UTS, and H (i.e., W_Ra = 0.7, W_YS = W_UTS = W_H = 0.10).
• Case 3: Assigning 70% weightage to YS and assigning 10% weightage each to Ra, UTS, and H (i.e., W_YS = 0.7, W_Ra = W_UTS = W_H = 0.10).
• Case 4: Assigning 70% weightage to UTS and assigning 10% weightage each to Ra, YS, and H (i.e., W_UTS = 0.7, W_Ra = W_YS = W_H = 0.10).
• Case 5: Assigning 70% weightage to H and assigning 10% weightage each to Ra, YS, and UTS (i.e., W_H = 0.7, W_Ra = W_YS = W_UTS = 0.10).

For demonstration, if case 1 is considered with equal weightages of 0.25 for each of the four objectives, the composite score for solution 1 is computed as: 0.25 ∗ 0.396482 + 0.25 ∗ 0.998581 + 0.25 ∗ 1 +0.25 ∗ 0.992136 = 0.8468. Similarly, the composite scores of the 94 solutions are computed for all five cases, and the scores are listed in Table 19. It may be remembered that the solution having the highest composite score will be considered as the best compromise solution (considering all the objectives simultaneously in multi-objective optimization). The best compromise solutions are indicated in bold in Table 19 for each of the five cases.

Patel et al. [11] used GA and obtained 100 solutions and presented the compromise solutions for five cases. Now the results of the application of the proposed algorithms presented in this paper are compared with those of Patel et al. [11].

Table 20. Comparison of multi-objective optimization results for the squeeze casting process.

Case No.	Td (s)	Dp (s)	Sp (MPa)	Pt (°C)	Dt (°C)	Ra (µm)	YS (MPa)	UTS (MPa)	H (BHN)
Case 1: Equal weights (i.e., WRa = WYS = WUTS = WH = 0.25)
GA [11]	4.013	49.99	179.42	749.62	293.95	0.1142	138.79	228.44	85.98
Present work	3.974298	50	171.5308	748.3405	287.3691	0.120769	140.4134	230.0316	86.29786
Case 2: WRa = 0.7, WYS = WUTS = WH = 0.10)
GA [11]	3.983	49.79	180.75	749.89	295.04	0.1139	138.03	227.27	85.76
Present work	4.761999	50	186.6837	747.9199	300	0.116302	135.5975	223.0101	84.65361
Case 3: WYS = 0.7, WRa = WUTS = WH = 0.10
GA [11]	3.102	49.87	177.15	748.52	279.96	0.1349	142.15	235.40	87.34
Present work	3	50	169.9449	736.4611	250.7107	0.174982	146.9246	241.1875	87.71212
Case 4: WUTS = 0.7, WRa = WYS = WH = 0.10
GA [11]	3.207	49.42	173.63	746.06	271.26	0.1437	143.69	236.76	87.33
Present work	3	50	173.458	744.9693	265.3419	0.149999	144.6185	238.5558	87.65304
Case 5: WH = 0.7, WRa = WYS = WUTS = 0.10
GA [11]	3.521	49.89	176.33	745.61	266.16	0.1365	143.64	235.63	86.95
Present work	3.617944	49.90341	170.8667	748.8612	287.9335	0.12487	140.7853	231.6107	86.74068

compares the results of multi-objective optimization for the squeeze casting process for five cases. The best values of the objectives for the five cases are indicated in bold.

Even though Patel et al. [11] presented five cases, finally they recommended to use the compromise solution of case 1 with Ra = 0.1142 (µm), YS = 138.79 MPa, UTS = 228.44 MPa, and H = 85.98 BHN. However, for the same case 1 and with the same weightages of the objectives, the composite front of the present work has recommended Ra = 0.120769 (µm), YS = 140.4134 MPa, UTS = 230.0316 MPa, and H = 86.29786 BHN. The solution recommended by the present work is much better in three out of the four objectives of case 1. Similarly, it can be observed that the results of the present work are much better for three out of five cases. Thus, the composite front formed by the proposed metaphor-less and algorithm-specific control parameter-less MO-BWR, BMR, and BMWR algorithms can be understood as a viable and convenient approach for the multi-objective optimization problems.

In fact, Patel et al. [11] carried out single-objective optimization of the objectives individually (ignoring the other objectives) and then formed a combined objective function to optimize (treating that combined objective as a single objective function) using an a priori approach of GA. However, the present work considered all objectives simultaneously following an a posteriori approach. In multi-objective optimization, an a posteriori approach offers significant advantages over the a priori approach by generating the complete Pareto-optimal set before any decision-making. This allows decision-makers to visualize trade-offs between objectives, explore unexpected high-quality solutions, and avoid bias from predefined or inaccurate preferences. It is particularly valuable when preferences are uncertain or may change, as choices can be made after analyzing actual results. Additionally, it supports multi-stakeholder scenarios by enabling different decision-makers to select different solutions from the same Pareto set, ensuring flexibility, robustness, and a more informed selection process.

The Python codes for the BWR, BMR, BMWR, MO-BWR, MO-BMR, and MO-BMWR algorithms are available at: https://sites.google.com/d/1qNsqo0kHkQD9Bhi4-prZlxo5K7QuntgI/p/1OeRlPp5dhwkqfeNvLP4CtHqFhwk74DqB/edit (Accessed on 10 August 2025).

It should be emphasized that the proposed BWR, BMR, and BMWR algorithms are capable of optimizing process parameters based on the previously established empirical or regression relationships developed by the previous researchers. It should be kept in mind that the reliability of the optimization results is inherently tied to the accuracy and robustness of the regression equations employed in the optimization framework. The previous researchers reported very good statistical values corresponding to ANOVA, F-ratios, p values, and very high values of the coefficient of determination (i.e., R²) for the regression equations developed by them. Hence, the proposed algorithms are applied to those regression equations, treating them as the objective functions and the optimum values of parameters are suggested. However, the outcomes may require further validation through dedicated experiments which is not in the purview of the present work (as the aim is to prove the effective optimization of process parameters of the casting processes based on the regression equations and the simulation results reported by the researchers during the last 5 years.

The proposed algorithms (BWR, BMR, BMWR, and their multi-objective variants) differ fundamentally from machine learning (ML), deep learning (DL), and artificial neural networks (ANNs) in that they are direct metaheuristic optimizers which require only the objective functions, constraints, and variable bounds, without relying on prior datasets. They are designed to explore the solution space stochastically and provide optimal or near-optimal parameter values, often generating Pareto fronts in multi-objective scenarios. In contrast, ML/DL/ANNs are data-driven approaches that excel in regression, classification, and prediction once trained on sufficiently large and high-quality datasets, but they do not inherently perform optimization unless coupled with an optimizer. While the proposed algorithms are generally transparent, robust, and computationally efficient for parameter tuning, ML/DL/ANNs often function as black boxes requiring significant training effort, though they enable rapid predictions after training. Importantly, the two approaches can complement each other as ANNs or DL models can act as surrogate predictors of expensive simulations or experiments, while the proposed algorithms can then be used to optimize over these surrogates, resulting in a powerful hybrid strategy for complex engineering problems.

Table 21. Comparison of the proposed algorithms with machine learning/deep learning/ANNs.

Aspect	BWR, BMR, BMWR	Machine Learning/Deep Learning/ANNs
Principle	Population-based, stochastic, metaphor-free metaheuristics.	Data-driven models that learn patterns and decision boundaries from datasets.
Search strategy	Guided by best, mean, worst, and random solutions; exploration–exploitation balance.	Learns mappings x → f(x); optimization via gradient descent, reinforcement learning, or surrogate-assisted search.
Parameter dependence	Parameter-free (except population size and iterations).	High dependence on hyperparameters (layers, neurons, learning rate, batch size, epochs, etc.).
Computational cost	Moderate—requires multiple evaluations of objective function.	High training cost (especially for deep models); once trained, predictions are cheap.
Scalability	Performs well up to medium–high dimensions (50–100+ variables), but may degrade in very large-scale spaces.	Scales well if sufficient data is available; dimensionality handled via feature engineering and network depth.
Constraint handling	Penalty functions, feasibility rules, ε-constraints, etc.	Constraints are often embedded into model architecture or handled via constrained optimization layers.
Exploration vs. exploitation	Explicitly balanced (BWR → exploratory, BMR → exploitative, BMWR → hybrid).	Exploitation-oriented; exploration requires reinforcement learning or hybridization.
Robustness to local optima	Strong—random + worst interactions help escape local traps.	Vulnerable to local minima in gradient descent; mitigated by advanced training strategies (Adam, momentum, etc.).
Data requirements	No data required; works directly on objective evaluations.	Requires large, high-quality datasets; poor generalization if training data is limited.
Multi-objective optimization	Natural extension with Pareto archives, crowding distance, and ε-dominance.	Multi-objectives handled via multi-task learning or scalarization; less transparent in decision diversity.
Interpretability	Transparent update logic, easy to trace search behavior.	Often a black box; DL/ANNs lack interpretability.
Best use cases	Problems without prior data, black-box functions, and engineering design optimization.	Problems with abundant historical/simulation data; surrogate-assisted optimization for expensive evaluations.

compares the proposed BWR, BMR, and BMWR algorithms with the AI8-based optimization tools such as machine learning (ML)/deep learning (DL)/artificial neural networks (ANNs).

4. Conclusions

In this work, three novel optimization algorithms—best–worst–random (BWR), best–mean–random (BMR), and best–mean–worst–random (BMWR)—are presented and validated on a wide range of parameter optimization problems of different metal casting processes.

• Applications to single-objective metal casting process: The BWR, BMR, and BMWR algorithms are applied to the real case studies of (i) optimization of a lost foam casting process for producing a fifth wheel coupling shell from EN-GJS-400-18 ductile iron, (ii) optimization of process parameters of die casting of A360 Al-alloy, and (iii) optimization of wear rate of an AA7178 alloy reinforced with nano-SiC particles produced using a stir-casting process. Comparisons of results are made with RSM and ProCAST 2021 simulations in case study (i), RSM and CAE simulation software in case study (ii), and the metaheuristics such as ABC, Rao-1, and PSO algorithms in case study (iii). The three algorithms achieve better results compared to the simulation software and the metaheuristics. The proposed algorithms showed faster convergence and superior solution quality with fewer function evaluations, highlighting their computational efficiency. For single-objective optimization, BWR, BMR, and BMWR provided the same results for the case studies considered. In general, these three algorithms provide almost the same results with minor variations in the case of single-objective optimization problems. But it is not so in the case of multi-objective optimization problems. Their performances may give different Pareto fronts (and some Pareto front solutions may match), and hence a composite front concept that takes into account all non-duplicated and unique non-dominated solutions is suggested in this paper for the multi-objective optimization of metal casting processes.
• Applications to multi-objective metal casting processes: The multi-objective variants (MO-BWR, MO-BMR, and MO-BMWR) are tested on (iv) two-objectives optimization of a low-pressure casting process using a sand mold for producing A356 engine block and (v) four-objective optimization of a squeeze casting process for LM20 material. These are compared with established methods such as NSGA-II and FEA simulation in case study (iv), and the a priori approach of GA in case study (v). The results show that competitive performance is achievable without added complexity, making the proposed algorithms transparent, interpretable, robust, and versatile for optimizing metal casting processes.
• Decision-making integration: This study employs the BHARAT method—a structured MADM approach—to select the most suitable compromise solutions from Pareto-optimal sets, especially when visual interpretation is impractical in multi-objective problems. This leads to better decision-making for choosing the optimal process parameters in the case of multi-objective optimization problems.
• Broader applicability: The findings suggest that these algorithms are well suited for a wide range of real-world manufacturing challenges across high-dimensional settings and constrained/unconstrained problems alike. Applications include additive manufacturing process optimization, toolpath planning, welding parameter tuning, multi-axis machining, and thermal processing, with further potential through coupling with ML, DL, and ANN models for data-driven workflows. The versatility, scalability, and robustness of the BWR, BMR, and BMWR algorithms make them promising tools for the future of intelligent manufacturing optimization.