Tensile Test Optimization Using the Design of Experiment and Soft Computing

Abstract

The tensile testing of various materials to evaluate the influence of different machining parameters is a fundamental requirement in every industry. The objective of this study is to investigate the effects of temperature, the area of the contact point, and the operator on the tensile test of brass samples. This study employs a hybrid soft computing approach, integrating an adaptive network-based fuzzy inference system (ANFIS), genetic algorithm (GA) optimization, and design of experiments (DOE). By combining these techniques, the study aims to leverage their individual strengths and achieve superior results. The results reveal that the area of the contact point exerts the most significant influence on the tensile test, followed by the operator and temperature. The optimal levels of these parameters are determined to be a level of two for the operator, a level of three for the area of the contact point, and a level of one for the temperature. The study demonstrated that the hybrid soft computing method outperformed the traditional DOE method, achieving a substantial improvement in elongation of 32.9%. The optimized combination of machining parameters led to a notable enhancement in the brass samples’ tensile properties, highlighting the effectiveness of the applied methodology. The marginal error of only 0.72% in the hybrid approach showcases its high precision and reliability in determining the optimal levels of machining parameters. These findings underscore the potential of the Taguchi optimization method, ANFIS, and GA in achieving superior results in the tensile testing of materials, particularly in cases where multiple parameters are involved. The research results provide valuable insights for industries relying on precise material characterization, offering a robust methodology for optimizing tensile testing procedures. The study’s success in leveraging a hybrid soft computing approach serves as a promising avenue for future research in the field of material testing and optimization techniques.

1. Introduction

1.1. Tensile Test

The tensile test serves as a fundamental method for characterizing the mechanical properties of materials. Through laboratory experiments meticulously designed to mimic real-world service conditions, researchers can accurately determine the mechanical behavior of materials. In this pursuit, factors such as the applied load’s magnitude, duration, and the surrounding environmental conditions play pivotal roles. Tensile testing, specifically, involves subjecting specimens to tensile forces to evaluate a diverse range of mechanical properties, including but not limited to, ultimate tensile strength, yield strength, strain, ductility, Poisson’s ratio, and Young’s modulus [1]. This entails the gradual application of a uniaxial tensile load along the specimen’s longitudinal axis, typically characterized by a circular cross-section, although rectangular geometries are also admissible. By harnessing the power of tensile testing, researchers gain invaluable insights into the material properties of newly developed substances. Notably, the laser powder bed fusion technique emerged as a critical tool in the realm of additive manufacturing. In a compelling illustration, an AlSi10Mg alloy underwent rigorous tensile testing to probe for any macroscopic anisotropy inherent to its composition. To comprehensively assess the material’s response, both tensile and compression tests were meticulously carried out, culminating in the derivation of Young’s modulus values soaring to an impressive 79.8 GPa. To further ascertain the material’s integrity, a complete tensile test until failure was conducted in strict adherence to the DIN EN ISO 6892-1: 2017-02 standard [2], enabling precise determinations of crucial metrics such as tensile strength (Rm), yield strength (Rp0.2), and elongation at failure [3]. The tensile strength of dry glass and glass fiber straps coated with epoxy was tested in the study. Significant differences in strain strength were revealed between the epoxy-coated and uncoated fiber straps. The researchers conducted four different types of tests to investigate the impact of various factors on the tensile strength and modulus of elasticity of glass fiber straps. These factors included specimen width, deformation speed, reinforced materials, and epoxy glue. During the tensile strength tests, an interesting observation was made: certain fibers within the glass fiber straps only became active after the most stretched fibers had already broken. This observation indicated that achieving equal stretching across all fibers was not possible. As a result, the tensile strength of individual fibers cannot be used as a determinant for the overall tensile strength of glass fiber straps or fabrics [4].

To assess the tensile strength of FRP composites and lamellas, modifications were made to the existing standards. These modifications were necessary as the prescribed standards were not originally designed for fabric and lacked rigid specimen requirements [5,6]. Another study focused on additive manufacturing and involved the 3D printing of mechanical testing samples using PLA (polylactic acid) and PLA/wood fiber composites. The investigation examined various 3D printing parameters, such as layer height, number of shells, and infill density, employing a design of experiments (DoE) approach. Among these parameters, it was discovered that the number of shells had the most significant impact on maximizing the tensile strengths of the PLA samples. Tensile tests were conducted on dog-bone-shaped samples created using the 3D printing technique. The samples were stretched until they fractured, with the tests performed at a crosshead speed of 5 mm/min in accordance with the guidelines outlined in the ASTM D638 standard [5]. Notably, the mechanical strengths of the PLA samples significantly surpassed those of their composite counterparts. Furthermore, increasing the number of shells during the 3D printing process enhanced the mechanical strengths of the printed samples [7].

The process parameters for three-dimensional (3D) printing using fused deposition modeling (FDM) significantly impacted the printed item. Consequently, careful selection of these parameters was necessary to improve the properties of the finished product. In this research paper, the influence of various printing parameters on the tensile strength of Polylactic acid (PLA) filament was investigated. The parameters examined included raster orientation, build orientation, extruder temperature, nozzle diameter, shell number, infill density, and extrusion speed. The investigation was carried out through a combination of experimental and statistical analyses.

To conduct the study, an FDM 3D printer was employed to manufacture PLA samples for 18 trials, following Taguchi’s mixed model fractional factorial design. The tensile strength of the PLA samples was evaluated using universal testing equipment. The optimal combination of parameters was determined using the S/N ratio. The primary objective was to establish a resilient system that would be less affected by variations in noise factors.

The results revealed that raster orientation had a significant impact of 0.46% on tensile strength, while construct orientation had a much more substantial influence, accounting for 44.68% of the variation. Similarly, only three parameters—nozzle diameter, infill rate, and build direction—were statistically significant, with a p-value of less than 0.05 [8].

The increasing demand for sustainable materials in various structural applications emphasized the need to explore alternative and eco-friendly sources for these materials. In this study, the impact of different production parameters on the performance of hybrid epoxy composites reinforced with chicken feathers was investigated. By implementing the robust Taguchi optimization technique, the study aimed to optimize the composite’s flexural strength, tensile strength, and impact energy by considering factors such as the content of chicken feather fiber, particle size, eggshell ash, and kaolin. The ultimate goal was to develop reliable and structurally sustainable materials.

The optimal values for the composite’s tensile strength, flexural strength, and impact energy were 64.66 MPa, 74.84 MPa, and 1.3659 J, respectively. Statistical analysis, conducted at a 95% confidence level with a p-value of 0.029, demonstrated the significance of the interaction between the fiber content of chicken feathers and the ash content of eggshells on the impact energy of the composite material.

Moreover, predictive equations were modeled to determine the tensile strength, flexural strength, and impact energy of the materials, achieving a high reliability of 93.43%, 59.57%, and 75.25%, respectively, for trend prediction. The results also indicated good matrix adhesion to the fibers, as no fiber pull-outs were observed. However, in composites with higher compositions of powder reinforcements, the formation of pores was observed [9].

Acquiring knowledge about the mechanical properties of various industrial products holds significant importance for all companies. Tube hydroforming has emerged as a widely adopted method in the automotive industry to achieve precise shaping of tubular metal parts using pressurized fluid. A comprehensive understanding of the mechanical properties of these parts plays a vital role in ensuring effective process control. In this study, a modified ring test, originally employed in nuclear industry applications, was introduced as a means to determine the hoop stress–strain curve of tubular materials. This innovative test method facilitated the measurement of tensile properties without causing any alterations to the material’s characteristics during specimen preparation. The paper details the newly proposed test method, outlines the data analysis procedure, and presents the results obtained from testing a commercial steel hydroforming tube [10].

A recent study has introduced an innovative method for conducting high-precision tensile tests on small-scale specimens. In this method, the specimens were fabricated using a water-cooled circular grinding process to ensure precise geometry and minimal alterations to the material. The test setup incorporated a mechanical polishing unit and an image-based system for displacement measurement, enabling the evaluation of true stress, true strain, and reduction in area. Successful demonstrations conducted on various metals have demonstrated improved quality and reduced fabrication and testing time through the utilization of this novel approach [11].

Rutting, fatigue, moisture susceptibility, and thermal cracking pose significant challenges in asphalt pavement. The objective of this study was to investigate the use of the indirect tensile (IDT) test in evaluating rutting and fatigue performance. Various blends of recycled concrete aggregate (RCA) and virgin aggregate were examined, and the IDT test demonstrated strong correlations with other tests, indicating its capability to predict rutting behavior. Furthermore, the IDT test proved effective in assessing fatigue behavior through cyclic testing. These findings underscore the potential of the IDT test as a comprehensive performance assessment tool for fatigue, rutting, thermal cracking, and moisture damage. Further validation through additional materials and field studies is recommended for a more comprehensive understanding [12].

Another study proposes a method to evaluate fracture toughness in a J-R curve format without automatic crack length measurement devices. By employing load and displacement pairs, a calibration curve based on material flow properties was utilized to evaluate crack growth. For each specimen, individual calibration curves were developed, considering both elastic and plastic displacement components. The study proposed three methods to determine the power term exponent and coefficient. Experimental testing on various materials revealed the potential of this method as a viable alternative to automatic crack length measurement techniques for evaluating J-R curves [13].

1.2. Taguchi Application

The Taguchi technique offers a valuable approach to attaining the optimal process parameters across various fields. By utilizing this technique, significant time and cost savings can be achieved, leading to improved performance and the production of higher-quality products. The implementation of the Taguchi method yields substantial enhancements in engineering output. It considers noise factors such as environmental changes, production variations, and component deterioration, ensuring robust design and mitigating costs associated with breakdowns [14,15,16].

The application of the Taguchi methodology, integrating experimental design theory and the concept of the quality loss function, has proven effective in addressing complex manufacturing challenges related to product and process design. Apart from identifying crucial input process parameters, the Taguchi technique also determines the key factors influencing the output response. Optimal management of variation can be achieved by selecting a controlling factor based on the behavior of the output. Within the Taguchi technique, repetition data can be transformed into a different value to assess output variation, with the S/N ratio serving as an indicator of how the signal has been transformed. The Taguchi technique employs the Signal-to-Noise (S/N) ratio to evaluate the quality characteristics of the output, distinguishing between lower-the-better, larger-the-better, and nominal-the-better characteristics. Through S/N ratio analysis, each process parameter is assessed, with a higher S/N ratio indicating superior quality characteristics, irrespective of their type. Consequently, the maximum S/N ratio signifies the optimal process parameter. Additionally, the ANOVA test can determine the significance of statistically significant output quality characteristics [17,18].

1.3. Soft Computing

Gibson et al. [19] employed Gaussian process regression to predict strain and the distribution of accumulated fatigue damage in complex environments. By considering model uncertainty, it provided valuable probabilistic information for robust risk assessment. The approach was demonstrated in aircraft wing fatigue damage prediction, showcasing its effectiveness in estimating damage accumulation and uncertainty. Tanhadoust et al. [20] investigated the mechanical performance of lightweight aggregate concrete (LWAC) exposed to high temperatures using an LSTM recurrent neural network. Hiew et al. [21] proposed a data-driven approach using sequential artificial neural networks to predict the stress–strain behavior and ultimate conditions of steel-confined ultra-high-performance concrete (UHPC). Milad et al. [22] introduced ensemble machine learning models (MARS, RF, and XGBoost) for forecasting fiber-reinforced polymer (FRP) composite strain, aiding in FRP composite design. Iravanian et al. [23] explored the use of sodium hydroxide (NaOH) to increase soil stability and compare various prediction models for the stress–strain behavior of treated soil, with the quadratic model (QM) showing good performance.

This study utilized a hybrid soft computing method, combining an adaptive network-based fuzzy inference system (ANFIS) and genetic algorithm (GA) optimization. The tensile test data were initially employed in the ANFIS to develop a surrogate model for the application. Subsequently, the surrogate model was integrated into the GA optimization algorithm to obtain the optimal solution that maximizes elongation.

While these studies have made significant contributions to understanding and optimizing material properties through various testing methods, there are some limitations to be considered:

Narrow Range of Process Parameters: In certain studies, the range of process parameters examined may have been somewhat limited. A broader exploration of parameters could provide a more comprehensive understanding of the material’s behavior under different conditions.

Dependency on Specific Testing Standards: Some studies relied heavily on established testing standards. While these standards provide a valuable framework, they may not always capture all relevant aspects of a material’s behavior in real-world applications.

Lack of Consideration for Environmental Factors: Environmental conditions, such as temperature and humidity, can significantly influence material properties. It would be beneficial for future studies to delve deeper into how these factors impact test outcomes.

Focus on Specific Applications: Certain studies were tailored towards specific applications, like additive manufacturing or asphalt pavement. While this specialization is valuable, it may not fully capture the broader implications of the material’s behavior.

Assumptions in Soft Computing Models: Soft computing methods, while powerful, rely on assumptions and training data. It is important to acknowledge potential limitations in the accuracy of predictions, especially when applied to novel or complex materials.

Possible Need for Validation in Real-World Applications: while laboratory tests provide crucial insights, it is essential to validate findings in real-world scenarios to ensure that the material’s behavior aligns with practical applications.

2. Methodology

The experimental configuration for this study involves several components, including hardware, software, and specific parameters. The tensile testing machine is the primary hardware used for conducting the tensile tests on the brass samples. It includes components like grips, load cells, and actuators for applying tension to the samples until they fracture. Also, the temperature control system ensures that the temperature during testing is controlled and maintained at the desired levels. It may include heating elements, thermocouples, and a control unit.

For the software configuration, MATLAB is a high-level programming language and an environment commonly used for data analysis, modeling, and simulation. In this study, it is utilized for designing and developing the hybrid soft computing model. ANFIS Toolbox is a specific toolbox within MATLAB for implementing Adaptive Network-Based Fuzzy Inference Systems. It allows for the creation, training, and evaluation of ANFIS models. MATLAB also provides a toolbox for implementing genetic algorithms. This tool assists in optimization processes, helping to find optimal solutions for complex problems. Also, DOE is a specialized software used for planning, conducting, and analyzing experiments. It aids in efficiently exploring the effects of different factors and their interactions.

For the parameter configuration, 3 significant parameters are selected, which are temperature, contact point area level, and operator level. Temperature levels are the different temperature settings at which the tensile tests are conducted. Each level represents a specific temperature condition (e.g., level 1 for low temperature, level 3 for high temperature). Contact point area levels represent the different sizes or areas of the contact point between the testing apparatus and the brass sample. Finally, operator levels refer to different operators involved in conducting the tests. Each level could represent a different operator or level of expertise.

2.1. Material Selection and Setup

The test specimen utilized in this study is composed of brass, as depicted in Figure 1, with cross-sectional dimensions measuring 12.2 mm × 1.5 mm.

The tensile test is performed using the Tinius Olsen 50ST materials testing machine (Tinius Olsen, Horsham, PA, USA) as shown in Figure 2. The initial velocity ranges from 3.1 to 3.2 mm per minute during the initial 15 s of the experiment, coinciding with a linear and consistent trend in the graph. Subsequently, as the graph transitions into a curved trajectory, the velocity is elevated to 26 mm per minute, maintaining this rate for the remainder of the experiment. As the specimen length increases, the grip surface between the machine’s calipers and the specimen decreases (e.g., 75 mm, 85 mm, and 95 mm correspond to 800 mm², 600 mm², and 400 mm² grip surfaces, respectively).

2.2. Experimental Setup

Parameter Selection: The elongation and ultimate stress of the selected samples are influenced by various parameters. Therefore, it is crucial to comprehend the significant process parameters in the tensile test. The careful consideration and selection of these process parameters are based on an extensive review of the existing literature. The chosen parameters are as follows: A = Operator, B = Specimen length (grip surface) (mm²), and C = Temperature (°C), as tabulated in

Table 1. Parameters and levels.

Parameters	Level 1	Level 2	Level 3
Temperature (°C), C	12	22	32
Specimen length (grip surface) (mm²) B	75	85	95
Operator, A	1	2	3

. Based on the available information, the author notes that there are limited studies investigating the impact of temperature and grip surface on tensile tests.

Selection of orthogonal array: The study utilizes an L9 orthogonal array of Taguchi based on the selected number of parameters and their respective levels.

Table 2. The L9 orthogonal array of the Taguchi illustrates the configuration of four chosen parameters.

Trials	A	B	C
1	1	1	1
2	1	2	2
3	1	3	3
4	2	1	2
5	2	2	3
6	2	3	1
7	3	1	3
8	3	2	1
9	3	3	2

displays the tabulated arrangement of this array, where each parameter is assigned three levels, resulting in a total of nine trials.

S/N Ratio Approach

The signal-to-noise ratio (S/N) quantifies noise reduction for various quality characteristic definitions. Typically, three quality characteristics are employed to calculate S/N: smaller S/N values signify better performance, the nominal value indicates the best performance, and larger S/N values also indicate improved performance. This metric facilitates the assessment of noise reduction effectiveness and offers valuable insights into the outcome’s quality based on diverse quality definitions.

In line with the study’s main objective of determining the strain value in tensile tests, Equations (1) and (2) demonstrate that an increase in strain value corresponds to an enhancement in the applied quality characteristic. Put differently, a higher strain value signifies a superior-quality outcome [24]:

$$S/N=-10 log \left(MSD\right)$$

(1)

Equation (2) can be employed to calculate the mean squared deviation (MSD) for quality characteristics that exhibit a larger-is-better behavior [24]:

$$MSD=\frac{1}{N}\left(\sum _{i=1}^n\frac{1}{{y_i}^2}\right)$$

(2)

In this context, the symbol N represents the total number of data points obtained for each experiment, while $y_i$ represents the strain value corresponding to different tests. It is imperative to conduct multiple iterations within each trial to ascertain the consistency of the attained target value. Here, ‘N’ denotes the quantity of iterations per trial. In cases where the target value remains invariant across multiple iterations, ‘N’ may assume a value of one.

2.3. Soft Computing

ANFIS

ANFIS, a hybrid system combining an artificial neural network and a fuzzy inference system (FIS), is a Takagi–Sugeno-type artificial intelligence model for tackling complex and nonlinear problems. ANFIS possesses self-learning and reasoning capabilities, making it a powerful tool. The fuzzy rules utilized by ANFIS are outlined as follows:

Rule 1: if $t$ is $O_1$, and $x$ is $P_1$ then $z_1=a_{1}t+a_{2}x+a_3$.

Rule 2: if $t$ is $O_2$, and x is $P_2$ then $z_2=b_{1}t+b_{2}x+b_3$.

Rules 1 and 2 in the system involve the input variables t and x. These rules determine the outputs $z_1$ and $z_2$. During the learning process, the parameters $a_1$, $a_2$, $a_3$, $b_1$, $b_2$, and $b_3$ are obtained. ANFIS consists of five levels. Each level consists of one output and two inputs. The purpose of each layer is defined as follows:

Layer 1: The fuzzification layer utilizes membership functions to compute fuzzy clusters based on the training input data. The indexes ai and bi are used to define the membership function and determine the degree of membership, as follows:

$$Q_i^1={\mu }_{A_i}\left(x\right)=\frac{1}{1+{\left|\frac{x-c}{a}\right|}^{2b}}; i=1,2$$

(3)

$$Q_i^1={\mu }_{B_i}\left(x\right); i=1,2$$

(4)

where ${\mu }_x$ and ${\mu }_y$ represent the membership degrees, $A_i$ and $B_i$ are the fuzzy sets, and x and $Q_i^1$ denote the input and the output of the ith node in the jth layer, respectively.

Layer 2: It is known as the rule layer, which is responsible for amplifying the firing power. The output is computed as follows:

$$Q_i^2=w_i={\mu }_{A_i}\left(x\right).{\mu }_{B_i}\left(y\right), i=1,2$$

(5)

Layer 3: It performs the task of normalizing the firing intensity from the preceding layer. The normalized values are obtained by dividing the firing strength of the ith rule by the sum of all firing strengths, ensuring relative proportions are maintained:

$$Q_i^3={\overline{w}}_i=\frac{w_i}{w_1+w_2+w_3+w_4}, i=1,\cdots ,4$$

(6)

Layer 4: This layer handles the process of defuzzification. The outcome is calculated by taking the dot product of the normalized firing strength and the parameter set ($p_i$, $q_i$ and $r_i$), resulting in the following:

$$Q_i^4={\overline{w}}_i{f}_i={\overline{w}}_i\left(p_{i}x+q_{i}y+r_i\right)$$

(7)

Layer 5: The outcome is obtained by combining the outputs of defuzzification from each rule:

$$Q_i^5=\sum _i{\overline{w}}_i{f}_i=\frac{\sum _i{w}_i{f}_i}{\sum _i{w}_i}$$

(8)

The computation of error signals, which flow from the output layer to the input layer, is carried out through the backpropagation algorithm based on the gradient descent method. The algorithm aims to minimize the training error by adjusting adjustable parameters.

GA Optimization: The GA is an optimization method that utilizes natural selection principles. It has gained popularity in solving various problems. The GA employs mutation, crossover, and selection operators inspired by biological processes. Mitchell (1998) developed this approach based on natural selection. A population of chromosomes is created and evaluated using an objective function. Chromosomes approximating the desired solution have a higher chance of reproducing. Careful parameter selection is essential for accurate convergence. The mutation and crossover parameters must be adjusted for success and quality output. A high mutation rate risks overlooking solutions, while a low mutation parameter may trap the algorithm. Crossover prevents exact replicas in offspring. Following Mitchell’s recommendation, suitable values are chosen for the mutation and crossover parameters [25]. A population size of 80 is employed, balancing capabilities and optimization length. The parameter values align with the research objectives.

In this study, GA is employed to extract the optimal inputs (P1, P2, P3) reaching that result in the highest strain of the samples. The extracted surrogate model via ANFIS is used within the cost function to calculate the objective function as follows:

$$J=100-\widehat{Strain}$$

(9)

where $\widehat{Strain}$ is the predicted strain via the ANFIS model. It has been substituted from 100 because the highest strain is our optimal situation. Trial and error methods are used to extract the optimal hyperparameters of the investigated ANFIS model in this study, and these parameters are reported in

Table 3. The optimal extracted hyperparameters of the proposed ANFIS using trial and error.

Hyperparameter	Value
Epoch Number	200
Membership functions for each input variable	3
Step-size decrease rate	0.5128
Step-size increase rate	1.0033
Initial training step size	0.0985

.

3. Results and Discussions

Due to the study’s numerous parameters and levels, an experimental design utilizing the Taguchi orthogonal array (Table 2) is selected. To evaluate the strain value, nine experiments are conducted, each requiring a change in the levels of the selected parameters to measure the strain value accurately. The tensile test is performed using the Tinius Olsen 50ST materials testing machine, employing brass specimens with cross-sectional dimensions of 12.2 mm × 1.5 mm. Before each test, the specimen temperature is checked using a K-type thermometer (Figure 3). To achieve the desired temperature of 12 °C, plastic reusable ice boxes are employed, while an ordinary fan is used to heat the specimen to 32 °C just before the start of the test.

Subsequently, the specimen is positioned on the testing machine, and the test is initiated (Figure 4). The HORIZON software (Version 10.2.4.22) is utilized to obtain the test results. Following each test, the final length of the specimen is measured using a Vernier caliper, enabling the calculation of elongation (Figure 5).

Referring to Table 2, a total of nine trials are conducted to assess the strain value, as presented in

Table 4. The strain measurement in the test cases using the L9 orthogonal array, considering the larger-the-better quality characteristic.

Trial	P1	P2 (mm²)	P3 (°C)	Strain	MSD	S/N
1	Operator 1	400	12	24.1	0.0017	27.64
2	Operator 1	600	22	26.4	0.0014	28.43
3	Operator 1	800	32	29.6	0.0011	29.43
4	Operator 2	400	22	25.3	0.0021	26.73
5	Operator 2	600	32	25.2	0.0016	28.03
6	Operator 2	800	12	31.3	0.0012	29.19
7	Operator 1 and 2	400	32	23.4	0.0018	27.38
8	Operator 1 and 2	600	12	25.6	0.0015	28.16
9	Operator 1 and 2	800	22	32.8	0.0011	29.77

. The S/N ratio for the larger-the-better quality characteristic is then calculated for these nine trials using Equations (1) and (2). The fourth trial corresponds to the minimum strain value, while the ninth trial corresponds to the maximum strain value.

3.1. Measurement of Specimen Length and Grip Size

Prior to measuring the specimen’s length, the positioning of the indication line is crucial for determining the grip size accurately. To accomplish this, a ruler is employed to identify the appropriate placement of the indication line relative to the specimen’s edge. For instance, when considering a grip size of 600 mm², the indication line is positioned 30 mm from the edge, considering the width of the specimen, which measures 20 mm. This procedure is performed on both the left and right sides of the specimen, ensuring consistency and reliability in subsequent measurements.

Following the determination of the indication line’s placement, the distance between the two lines is measured using Vernier calipers. The specimen is securely positioned on a table, and the Vernier caliper jaws are aligned with the indication lines, ensuring an accurate measurement. By employing this method, the length of the specimen and the grip size are precisely determined, contributing to the overall reliability of the experimental results.

3.2. Measurement of Broken Specimens

Upon completion of the experiment, the broken specimen is carefully removed from the testing apparatus and placed on a table for further analysis. To assess the length and grip size of the broken specimen, an identical measurement procedure to that employed during the initial assessment is followed. This ensures consistency and comparability of results between the intact and broken specimens, facilitating accurate data analysis and interpretation.

To ascertain the optimal levels of the three chosen parameters, the average signal-to-noise (S/N) ratio of responses needs to be computed, as demonstrated in

Table 5. The response table of the S/N ratio.

	P1	P2	P3
L1	28.49	27.69	28.57
L2	29.91	28.20	28.93
L3	28.62	29.88	28.27
|ΔT|	1.28	2.18	0.29
Rate	2	1	3

. The significance of a parameter increases with a larger value of |ΔT|. Referring to

Table 6. Elongation percentage for the optimum levels.

Trial	Temperature	Gripping Area	Operator	Modulus [MPa]	Ultimate Force [N]	Ultimate Stress [MPa]	Elongation [%]
1	22 °C	(800 mm2)	Operator 2	9870	7040	385	33.5

, it is evident that the most favorable outcome is achieved by selecting Level 2 for parameter P1, Level 3 for parameter P2, and Level 1 for parameter P3.

Regarding Table 4, the test results for the minimum and maximum values of strain are presented in Figure 6 and Figure 7, respectively.

To validate the optimal design, the experiment is conducted with P1 at level 2, P2 at level 3, and P3 at level 2, resulting in the best results. Figure 8 illustrates the experimental setup used to find the optimum strain value. As the specimen length increases, the grip surface between the calipers and specimen decreases. (Specimen lengths of 75 mm, 85 mm, and 95 mm correspond to grip surfaces of 800 mm², 600 mm², and 400 mm², respectively).

Figure 9 illustrates the stress and strain diagram for the optimum levels. The initial length of the specimen is 75 mm, and the final length is 98.5 mm, resulting in an elongation percentage of 33.5%, as presented in Table 6. Additionally, Figure 10 displays the strain values for nine experiments, along with the optimal level.

3.3. Soft Computing Method

MATLAB’s “ANFIS” function is used to design the ANFIS model. Furthermore, the GA optimization methods are implemented using MATLAB’s “ga” function. The algorithms presented in Section 3 are created within the MATLAB coding environment, and the plot function is utilized to visualize all the outcomes. The models are trained and tested using 80% and 20% of the extracted dataset, respectively.

Figure 11 shows the actual and estimated strain of the samples during the training and testing stages of the ANFIS. Based on the represented results, the mean squared error between the actual and predicted results using ANFIS is 8.4261 × 10⁻⁴. Also, the root-mean-square error reaches the acceptable value of 2.9 × 10⁻². It proves the efficiency of the proposed ANFIS model in extracting the surrogate model of the application.

Figure 12 presents the regression between the actual and estimated strain of the different samples based on the different distribution of input parameters, including P1, P2, and P3. Based on the represented results, the correlation coefficient and coefficient of determination (R2) between the actual and predicted strains are 0.99879 and 0.99997, respectively.

The extracted ANFIS model is employed within the objective function (Equation (9)) to calculate the optimal input parameters (P1, P2, and P3) that yield the highest strain. Based on the presented results, GA converges to the optimal solution after nine generations. The optimal solution extracted via GA is P1 = 3, P2 = 800, and P3 = 23.8. The ANFIS model predicts a strain of 33.14% for the given inputs. The extracted optimal solution obtained via ANFIS and GA is evaluated practically to assess the efficiency of the proposed method. The practical evaluation shows a strain of 32.9%. The margin of error between the actual and predicted strain is 0.72%, which falls within an acceptable range. The plot in Figure 13 illustrates the convergence of the GA optimization process used to determine the optimal values for the selected parameters.

The study incorporates the Taguchi optimization method, which is widely used for its efficiency in conducting experiments with multiple factors at various levels. This method is established and well-regarded in the field of experimental design. The use of S/N ratio calculation is a robust statistical method for identifying the optimal level of each parameter. It helps to filter out noise and to focus on the critical factors influencing the outcome. The inclusion of ANFIS is significant. ANFIS is known for its capability of modelling complex systems and making accurate predictions. Its integration in this research showcases a comprehensive approach to optimization. The integration of GA is another powerful optimization technique. GA is known for its ability to find global optima in complex, multi-dimensional search spaces. This complements the other methods used in this study. The combination of ANFIS, GA, and Taguchi optimization with DOE is innovative and represents a state-of-the-art approach. This hybrid method leverages the strengths of each technique, resulting in superior results.

The hybrid soft computing approach integrates multiple optimization techniques, providing a comprehensive strategy for determining the optimal levels of machining parameters. This allows for a more thorough evaluation compared to using individual methods in isolation. The proposed method achieves an elongation of 32.9% with a marginal error of 0.72%, demonstrating higher accuracy and precision compared to traditional methods. By combining ANFIS, GA, and Taguchi optimization, the approach can be tailored to specific scenarios, allowing for a more customized optimization process. The use of multiple optimization methods allows for cross-validation of results, enhancing the robustness and reliability of the findings. The hybrid approach is particularly beneficial in situations with multiple interacting factors, where a single optimization method may be insufficient.

In summary, the proposed hybrid soft computing approach offers a cutting-edge and effective means of optimizing machining parameters in tensile testing. Its integration of various techniques provides a more holistic and accurate evaluation, ultimately leading to superior results.

4. Conclusions

The study conducted a comprehensive examination of the effects of temperature, contact point area, and operator on the tensile behavior of brass. This multi-parameter analysis provided valuable insights into how various factors interacted and influenced the material’s mechanical properties. By employing the L9 orthogonal array of the Taguchi method, the research efficiently organized and conducted experiments with multiple parameters and levels. This demonstrated a structured and systematic approach to experimental design, which was crucial for obtaining reliable and meaningful results. Through the use of signal-to-noise (S/N) ratio calculations, the study identified the optimal levels for each parameter. This step was pivotal in achieving the highest quality output in the tensile test. The determination of the optimal settings was a key value proposition of this research. The adoption of the “larger-the-better” quality characteristic for strain analysis was a sound choice, particularly in materials testing. This indicated a focus on maximizing the desired outcome, which in this case was strain, for enhanced material performance. The validation of findings through the use of ANFIS as an additional optimization method added credibility to the results of the current study. This step confirmed the robustness of the conclusions drawn from the Taguchi optimization method. The development of a hybrid soft computing method combining ANFIS and GA to extract the optimal solution for achieving the highest strain was a noteworthy contribution. This approach showcased innovation in methodology and represented a potential advancement in materials testing techniques. The utilization of MATLAB 2019b software for designing and developing the model demonstrated proficiency in utilizing advanced computational tools. This approach not only enhanced the accuracy of the results but also allowed for a reproducible methodology. The extracted results, revealing an elongation of 32.9% with a marginal error of 0.72% compared to the Taguchi method, underscored the substantial improvement achieved through the proposed approach. This improvement in material behavior was a key value proposition of the research.