A new mathematical model for optimizing laser cutting parameters to improve fabric quality

3.3.1. ANN model for quality of cutting

The ANN model has three input neurons, which indicate power, speed, and fabric surface density. The output layer determines laser cutting quality. Eqs. (2) and (3) can be used to express the hidden layer output ( $v_l$) as well as the quality of the laser cutting result ($y$):

where $v_i$ represents the value of the hidden layer neuron, $x_{ij}$ denotes the input neuron, $\beta$ is the bias, $w_{ij}$ signifies the weight from the input to the hidden layer, $\phi$ is the activation function, and in this study, a sigmoid activation function is employed $y$ represents the weight of the output neuron. In this research, ANN architecture was created that incorporates three types of nodes into a single hidden layer. These variants were used to determine the best model, which was defined by minimal evaluation criteria. Fig. 2 shows the structure of the Perceptron used in this first model.

Fig. 2Neural network architecture model of perceptron

3.3.1.2. The feed forward ANNs with four nodes in one hidden layer

In the second ANN model architecture, a feed-forward propagation model was constructed, comprising four nodes within a single hidden layer. This architecture was evaluated under two different iteration scenarios: the first with 1,000 iterations and the second with 10,000 iterations. The effectiveness of both models was assessed using the correlation coefficient and mean absolute percentage error (MAPE). Fig. 3 depicts the architecture of this ANN model.

Fig. 3Architecture ANNs with four node one hidden layer

In Fig. 3, the arrows and circles represent the signal flow and neurons, respectively. The weight matrices of the first and second layers are denoted by the terms ($w_{ij}$) and ($w_{ij}^2$). The weighted sum can be determined using Eqs. (10) and (11):

10

${\mathrm{\Psi }}_i=\sum \sum w_{ij}{x}_j,$

11

${\left(\begin{array}{c}{\mathrm{\Psi }}_i\\ ⋮\\ {\mathrm{\Psi }}_4\end{array}\right)}_{4\times 1}={\left(\begin{array}{cc}\begin{array}{c}{w}_{11}\\ ⋮\end{array}& \begin{array}{c}\dots \\ w_{43}\end{array}\end{array}\right)}_{4\times 3}{\left(\begin{array}{c}{x}_1\\ ⋮\\ x_2\end{array}\right)}_{3\times 1}.$

The terms ${\psi }_i$ and $w_{ij}$ are the weighted sum of the first layer and the first layer weight matrix, respectively. The terms $x_1$, $x_2$, and $x_3$ are the input parameters. The neuron feeds the weighted sum into the sigmoid function and produces the result shown in Eq. (12):

12

${\left(\begin{array}{c}\begin{array}{c}{\varphi }_1\\ {\varphi }_2\end{array}\\ {\varphi }_3\\ {\varphi }_4\end{array}\right)}_{4\times 1}={\left(\begin{array}{c}\begin{array}{c}{\varsigma (\psi }_1)\\ {\varsigma (\psi }_2)\end{array}\\ {\varsigma (\psi }_3)\\ {\varsigma (\psi }_4)\end{array}\right)}_{4\times 1}.$

Eq. (12) is the input of the second layer, which results in a weighted sum, ${\pi }_i$, at the second layer, as seen in Eqs. (13) and (14):

13

${\pi }_i=\sum \sum {w_{ij}}^{\left(2\right)}{\varsigma (\psi }_j),$

14

${\pi }_i={\left(\begin{array}{ccc}{w_{11}}^{\left(2\right)}& \dots & {w_{14}}^{\left(2\right)}\end{array}\right)}_{1\times 4}{\left(\begin{array}{c}\begin{array}{c}{\varsigma (\psi }_1)\\ {\varsigma (\psi }_2)\end{array}\\ {\varsigma (\psi }_3)\\ {\varsigma (\psi }_4)\end{array}\right)}_{4\times 1}.$

The neuron inputs the weighted sum into a linear function, which generates the output. The neuron’s properties is characterized by this linear output function, as expressed in Eqs. (15) and (16):

15

$d=\lambda \left({\pi }_i\right)=\lambda \sum \sum {w_{ij}}^{\left(2\right)}{\varsigma (\psi }_j)={\left(\begin{array}{ccc}{w_{11}}^{\left(2\right)}& \dots & {w_{14}}^{\left(2\right)}\end{array}\right)}_{1\times 4}{\left(\begin{array}{c}\begin{array}{c}{\varsigma (\psi }_1)\\ {\varsigma (\psi }_2)\end{array}\\ {\varsigma (\psi }_3)\\ {\varsigma (\psi }_4)\end{array}\right)}_{4\times 1},$

16a

$d={\left(\begin{array}{ccc}{w_{11}}^{\left(2\right)}& \dots & {w_{14}}^{\left(2\right)}\end{array}\right)}_{1\times 4}{\left(\begin{array}{c}\begin{array}{c}{\varsigma (\psi }_1)\\ {\varsigma (\psi }_2)\end{array}\\ {\varsigma (\psi }_3)\\ {\varsigma (\psi }_4)\end{array}\right)}_{4\times 1},$

16b

$d=\lambda {\left(\begin{array}{ccc}{w_{11}}^{\left(2\right)}& \dots & {w_{14}}^{\left(2\right)}\end{array}\right)}_{1\times 4}\varsigma \left({\left(\begin{array}{cc}\begin{array}{c}{w}_{11}\\ ⋮\end{array}& \begin{array}{c}\dots \\ w_{43}\end{array}\end{array}\right)}_{\left(4\times 3\right)}{\left(\begin{array}{c}{x}_1\\ x_2\\ x_3\end{array}\right)}_{3\times 1}\right).$

The constants derived through the optimization process in Eqs. (15), (16a) and (16b), as well as the ANN architecture with four nodes in one hidden layer for 1,000 and 10,000 iterations, are detailed in Eq. (17a) and (17b):

17a

$d=\lambda {\left(\begin{array}{ccc}-1.8721& \dots & -3.5394\end{array}\right)}_{1\times 4}\varsigma \left({\left(\begin{array}{cc}\begin{array}{c}0.5302\\ ⋮\end{array}& \begin{array}{c}\dots \\ -1.8221\end{array}\end{array}\right)}_{\left(4\times 3\right)}{\left(\begin{array}{c}{x}_1\\ x_2\\ x_3\end{array}\right)}_{3\times 1}\right),$

17b

$d=\lambda {\left(\begin{array}{ccc}-14.9769\mathrm{}& \dots & 4.0604\end{array}\right)}_{1\times 4}\varsigma \left({\left(\begin{array}{cc}\begin{array}{c}0.9360\\ ⋮\end{array}& \begin{array}{c}\dots \\ 0.3716\end{array}\end{array}\right)}_{\left(4\times 3\right)}{\left(\begin{array}{c}{x}_1\\ x_2\\ x_3\end{array}\right)}_{3\times 1}\right).$

3.3.1.3. Feed-forward ANNs with a sixth node in one hidden layer

In the third ANN model architecture, a feed-forward propagation model was developed, comprising six nodes within a single hidden layer. This architecture was subsequently tested with two different iteration scenarios: the first with 1,000 iterations and the second with 10,000 iterations. The effectiveness of both models under these iteration variations was then assessed using the correlation coefficient and Mean Absolute Percentage Error (MAPE) metrics. Fig. 4 illustrates the architecture of this ANN model.

Fig. 4Architecture ANNs with sixth node one hidden layer

In Fig. 4, the arrows and circles represent the signal flow and neurons, respectively. The weight matrices of the first and second layers are denoted by the terms ($w_{ij}$) and ($w_{ij}^2$). The weighted sum can be determined using Eqs. (18) and (19):

18

${\mathrm{\Psi }}_i=\sum \sum w_{ij}{x}_j,$

19

${\left(\begin{array}{c}{\mathrm{\Psi }}_i\\ ⋮\\ {\mathrm{\Psi }}_6\end{array}\right)}_{6\times 1}={\left(\begin{array}{cc}\begin{array}{c}{w}_{11}\\ ⋮\end{array}& \begin{array}{c}\dots \\ w_{43}\end{array}\end{array}\right)}_{6\times 3}{\left(\begin{array}{c}{x}_1\\ ⋮\\ x_2\end{array}\right)}_{3\times 1}.$

The terms ${\psi }_i$ and $w_{ij}$ are the weighted sum of the first layer and the first layer weight matrix, respectively. The terms $x_1$, $x_2$, and $x_3$ are the input parameters. The neuron feeds the weighted sum into the sigmoid function and produces the result shown in Eq. (20):

20

${\left(\begin{array}{c}\begin{array}{c}{\varphi }_1\\ {\varphi }_2\end{array}\\ {\varphi }_3\\ \begin{array}{c}{\varphi }_4\\ {\varphi }_5\\ {\varphi }_6\end{array}\end{array}\right)}_{6\times 1}={\left(\begin{array}{c}\begin{array}{c}{\varsigma (\psi }_1)\\ {\varsigma (\psi }_2)\end{array}\\ {\varsigma (\psi }_3)\\ \begin{array}{c}{\varsigma (\psi }_4)\\ {\varsigma (\psi }_5)\\ {\varsigma (\psi }_6)\end{array}\end{array}\right)}_{6\times 1}.$

Eq. (20) is the input of the second layer, which results in a weighted sum, ${\pi }_i$, at the second layer, as seen in Eqs. (21) and (22):

21

${\pi }_i=\sum \sum {w_{ij}}^{\left(2\right)}{\varsigma (\psi }_j),$

22

${\pi }_i={\left(\begin{array}{ccc}{w_{11}}^{\left(2\right)}& \dots & {w_{16}}^{\left(2\right)}\end{array}\right)}_{1\times 6}{\left(\begin{array}{c}\begin{array}{c}{\varsigma (\psi }_1)\\ {\varsigma (\psi }_2)\end{array}\\ {\varsigma (\psi }_3)\\ \begin{array}{c}{\varsigma (\psi }_4)\\ \begin{array}{c}{\varsigma (\psi }_5)\\ {\varsigma (\psi }_6)\end{array}\end{array}\end{array}\right)}_{6\times 1}.$

The neuron passes the weighted sum to a linear function, which generates the output. The linear output function, as given in Eqs. (23) and (24), identifies the neurons function:

23

$d=\lambda \left({\pi }_i\right)=\lambda \sum \sum {w_{ij}}^{\left(2\right)}{\varsigma (\psi }_j)={\left(\begin{array}{ccc}{w_{11}}^{\left(2\right)}& \dots & {w_{16}}^{\left(2\right)}\end{array}\right)}_{1\times 6}{\left(\begin{array}{c}\begin{array}{c}{\varsigma (\psi }_1)\\ {\varsigma (\psi }_2)\end{array}\\ {\varsigma (\psi }_3)\\ \begin{array}{c}{\varsigma (\psi }_4)\\ \begin{array}{c}{\varsigma (\psi }_5)\\ {\varsigma (\psi }_6)\end{array}\end{array}\end{array}\right)}_{6\times 1},$

24a

$d={\left(\begin{array}{ccc}{w_{11}}^{\left(2\right)}& \dots & {w_{16}}^{\left(2\right)}\end{array}\right)}_{1\times 6}{\left(\begin{array}{c}\begin{array}{c}{\varsigma (\psi }_1)\\ {\varsigma (\psi }_2)\end{array}\\ {\varsigma (\psi }_3)\\ \begin{array}{c}{\varsigma (\psi }_4)\\ \begin{array}{c}{\varsigma (\psi }_5)\\ {\varsigma (\psi }_6)\end{array}\end{array}\end{array}\right)}_{6\times 1},$

24b

$d=\lambda {\left(\begin{array}{ccc}{w_{11}}^{\left(2\right)}& \dots & {w_{16}}^{\left(2\right)}\end{array}\right)}_{1\times 6}\varsigma \left({\left(\begin{array}{cc}\begin{array}{c}{w}_{11}\\ ⋮\end{array}& \begin{array}{c}\dots \\ w_{63}\end{array}\end{array}\right)}_{\left(6\times 3\right)}{\left(\begin{array}{c}{x}_1\\ x_2\\ x_3\end{array}\right)}_{3\times 1}\right).$

The constants derived through the optimization process in Eqs. (23), (24a) and (24b), as well as the ANN architecture with four nodes in one hidden layer for 1,000 and 10,000 iterations, are detailed in Eq. (25a) and (25b):

25a

$d=\lambda {\left(\begin{array}{ccc}-3.1459& \dots & -4.3038\end{array}\right)}_{1\times 6}\varsigma \left({\left(\begin{array}{cc}\begin{array}{c}-4.8379\\ ⋮\end{array}& \begin{array}{c}\dots \\ 1.4698\end{array}\end{array}\right)}_{\left(6\times 3\right)}{\left(\begin{array}{c}{x}_1\\ x_2\\ x_3\end{array}\right)}_{3\times 1}\right),$

25b

$d=\lambda {\left(\begin{array}{ccc}-5.8118\mathrm{}& \dots & 6.7229\end{array}\right)}_{1\times 6}\varsigma \left({\left(\begin{array}{cc}\begin{array}{c}-4.0728\\ ⋮\end{array}& \begin{array}{c}\dots \\ 0.6634\end{array}\end{array}\right)}_{\left(6\times 3\right)}{\left(\begin{array}{c}{x}_1\\ x_2\\ x_3\end{array}\right)}_{3\times 1}\right).$

The result of the mapping is measured based on three different ANN models, with different nodes in the hidden layer, and different iterations for each model. Table 4 shows the optimized results for the ANNs architecture model.

Table 4Result of ANNs model

Actual	Perceptron model	One hidden layer (4 nodes)		One hidden layer, (6 nodes)
		Iteration 1,000	Iteration 10,000	Iteration 1,000	Iteration 10,000
3	2.5	3.2836	3.0694	3.3619	2.9725
3	2.527	2.5426	2.9647	2.6119	3.0922
2	2.5537	1.855	1.8676	1.8586	1.8928
1	1.5807	1.4635	1.2769	1.4203	1.1776
1	1.6074	1.279	1.1026	1.2223	1.0375
3	2.7496	3.0604	3.1558	2.9011	3.0565
3	2.7757	3.1381	3.025	3.0214	2.9716
3	2.8015	3.2122	2.9053	3.1927	2.9074
3	2.8273	3.2734	3.064	3.3598	3.1069
4	3.8528	3.3169	3.85	3.481	3.8608

3.3.2. RSM model for quality of cutting

In this investigation, the RSM model is constructed based on three independent variables: Power ($x_1$), speed ($x_2$), and fabric surface density ($x_3$), which collectively influence the Quality of Fabric Cutting ($Q_t$). The core equation of RSM is represented as follows Eq. (26):

where, $x_1$, $x_2$, and $x_3$ denote the independent variables, specifically power, speed, and fabric surface density, respectively. The terms $a_0$, $a_1$, $a_2$, and $a_3$ represent the regression model constants for the respective variables, while ε represents the model error. Fig 5 illustrates the architecture of the RSM model used.

Fig. 5RSM architecture model

To obtain the constant value of Eq. (26), it can be determined by modeling it as illustrated in Eqs. (27) and (28):

The difference between the experimental data, $y$ and the model $Q_t$ is defined as the error $\left(ϵ\right)$, which is as follows:

Using Eqs. (29) and (30), we use to find the model:

Eq. (14) can be used to solve the linear equation produced by Eq. (26), resulting in the formulation shown in Eqs. (31) to (34):

Eq. (31) can be converted into matrix form as in Eq. (32):

The values of $a_0$, $a_1$, $a_2$ in Eq. (31) are obtained through the parameter optimization method, as in Eq. (33):

Then, Eq. (34) is derived to assess the quality of fabric cutting results using laser cutting, as follows:

In Table 5, Eq. (34) is used to calculate prediction calculations to assess the quality of fabric cutting after $a_0$, $a_1$, $a_2$, and $a_3$ have been determined.

3.4.1. MAPE (mean absolute percentage error)

A model's performance can be evaluated using the Mean Absolute Percentage Error (MAPE). Most commonly, it is used to determine the accuracy of predictive models, particularly in numerical or time series prediction. MAPE is calculated by comparing the actual and predicted values of a model. Eq. (35) demonstrates the calculation of MAPE:

MAPE is presented as the difference between the absolute amount predicted by the model (At) and the actual amount of experimental data ($F_t$) divided by the actual amount of experimental data (Ft) multiplied by 100 %. Therefore, using Eq. (35), the quality of fabric cutting predicted using both Artificial Neural Networks (ANN) and Response Surface Methodology (RSM) methods can be found in Table 6 as the absolute average of predictions.

3.4.2. Coefficient of determination (R²)

The coefficient of determination, also known as R-squared or $R^2$, is a statistical metric that measures how well a theoretical model matches actual observed data. In this context, $R^2$ indicates the quantity of variance in the dependent variable can be explained by the independent variables included in the model. The value of $R^2$ ranges from 0 to 1. A score of 1 shows that the model completely explains the variation in the data, whereas a value of 0 indicates that the model explains no variation and is simply a random error. Eq. (36) shows the $R^2$ calculations for the two model techniques developed:

The $R^2$ interpretation obtained from both the Artificial Neural Networks (ANNs) and Response Surface Methodology (RSM) models represents the developed model's ability to explain the complete variability in the dependent variable. Eq. (37) can be used to calculate the coefficient of determination in the prediction results for the total quantity of quality laser cutting performed with both the ANN and RSM techniques, as shown in Table 6.

This research requires neural computing because the process of laser-cutting fabric involves many complex and non-linear variables and interactions, which are difficult to model with deterministic rules or simple regression. Simple regression cannot effectively capture the complex relationships between process parameters and cutting quality, resulting in correctable prediction errors. Neural networks, with their ability to learn non-linear patterns and handle varying data, can provide more accurate and adaptive models, reducing prediction errors significantly.

3.4.3. Analyze evaluation result

The Response Surface Methodology (RSM) and Artificial Neural Networks (ANNs) can be used to optimize laser cutting settings for better fabric cutting quality. The ANN-based model, constructed through 10,000 iterations with a six-node architecture on a single hidden layer, demonstrated superior performance, with a mean absolute percentage error (MAPE) value of 0 % and a coefficient of determination ($R^2$) of 0.998. This implies that the model effectively accounts for 99.8 % of the variance in the actual data on fabric-cutting quality. The architecture type and complexity of the nodes and hidden layers used have a significant impact on the variability in validation results for ANN models. The RSM model performed well in predicting laser cutting parameters for cutting quality, with a MAPE score of 0 % and a $R^2$ of 0.952. Nonetheless, there are differences in the explained variance by the RSM method, which could be resolved by using more complex model optimization techniques such as ANNs. The gap in the R² value of the RSM model, which remains accessible to development, emphasizes the preference for ANN-based models for deeper insights. Fig. 6 shows the dispersion of actual cutting quality data received from garment industry professionals, compared with model outcomes, to clearly highlight the optimization model's variance. Furthermore, the findings highlight the significant impact of changing process factors like as power, speed, and fabric surface density on cutting quality. These findings may aid in the development of a standardized parameter index for optimizing laser cutting parameters in textile applications, hence improving fabric cut quality and process efficiency.

Fig. 6Pattern distribution of data