A theoretical approach based on machine learning for estimation of physical properties of LLDPE in moulding process

Abstract

This study explores the prediction of mechanical characteristics of linear polyethylene based on oven residence time, employing various regression models and hyper-parameter tuning through the Whale Optimization Algorithm. The dataset comprises one input variable (oven residence time) and three output parameters (Tensile Strength, Impact Strength, and Flexure Strength). The models investigated include Multilayer Perceptron, K-Nearest Neighbors, Support Vector Regression, Polynomial Regression, and Theil–Sen Regression. The results showcased distinct performances across the models for each output parameter. The Polynomial Regression (WOA-PR) method has been identified as the most suitable option for predicting Tensile Strength due to its ability to achieve the lowest errors in terms of Mean Absolute Error, Root Mean Square Error, and Average Absolute Relative Deviation. K-Nearest Neighbors (WOA-KNN) outperforms other models in predicting Impact Strength due to its superior accuracy and reliability. Additionally, Support Vector Regression (WOA-SVR) emerges as the best model for predicting Flexure Strength, showcasing notable performance in minimizing prediction errors. These findings underscore the significance of model selection and optimization techniques in accurately predicting the mechanical properties of polymers, paving the way for enhanced manufacturing processes and material design.

Introduction

Polymers are being extensively explored and utilized in different industries across the globe due to their superior physical, thermal, and chemical characteristics. Estimation of their properties is challenging and can be done via different computational methods¹. Depending on their molecular weights and chains, polymers have wide applications in industries such as plastic formation, pharmaceutics, chemicals, cosmetics, water treatment, etc.². Polymers have been also utilized for separation as membranes where liquid or gas mixtures can be used for separation of target molecules^3,4. One of the common applications of polymers is in moulding process where polymeric products are formed by thermal processing and injection^5,6. In the moulding process, polymeric products are formed by thermal processing and the properties of polymer play a key role in this process. The main properties of polymers in moulding technique include tensile strength, impact strength, and flexure strength which should be precisely controlled to get the desired products. Improving the properties of polymers such as LLDPE is a challenging task due to the complex behavior of polymer in the process. In this process, the mechanical properties are highly dependent on the treatment time which is governed by residence time. So, finding the relationship between oven residence time and these mechanical properties would be essential to optimize the process⁷. Finding the region with the highest values of these properties is challenging and advanced computational techniques are demanding to be applied for that. Ramkumar et al.⁷ studied influence of oven residence time on the mechanical properties of LLDPE by experimental and computational methods. The computations were carried out by ROTOSIM software which is specialized for rotational moulding process. The models considered mathematical analysis of physical properties in the process. Development of mechanistic models for such a complex process is computationally challenging and other facile models can be implemented for industrial applications. For process development, computational techniques can be applied to optimize the process and find the relationship between product properties and process parameters. Computational models would be beneficial as they can save time, cost, and materials consumption for experimental analysis of polymer’s mechanical properties. If processing data are available, data-driven models such as machine learning can be used for development of moulding process¹. In the present era, Machine Learning (ML) has emerged as a crucial instrument for deriving insightful conclusions and forecasts from extensive and intricate datasets spanning diverse fields including materials science^8,9. Big data techniques provide exceptional capabilities when it comes to identifying patterns, generating predictions, and optimizing processes¹⁰. The previous study has explored the use of ML models in estimating LLDPE properties, however wider ML models are needed for correlation of polymer properties in moulding process¹. This study employed five distinct models, namely Multilayer Perceptron (MLP), K-Nearest Neighbors (KNN), Support Vector Regression (SVR), Polynomial Regression (PR), and Theil–Sen regression (TSR). Also, the optimization of hyperparameters is achieved via the Whale Optimization Algorithm (WOA). The models integrated WOA are developed for the first time to predict mechanical properties of LLDPE in moulding process. The optimized models are then used for finding the optimum conditions to get the product with desired mechanical properties. The WOA is a method for optimization that is based on populations and simulates the feeding strategy of humpback whales when they form bubble nets. The process starts by creating a set of potential solutions, represented as a collection of whales. Each marine mammal represents a potential answer to the optimization problem^11,12. The Multilayer Perceptron (MLP) serves as a potent regression model inspired by the human brain. By traversing data through numerous interconnected layers, it can discern intricate connections between features and a target variable. This capability enables the model to effectively capture non-linear patterns within the data, rendering it well-suited for a diverse array of prediction tasks¹³. K-Nearest Neighbors (KNN) stands as a regression model known for its simplicity and efficiency. To predict the output for an unseen data point, it calculates the weighted average of the output values obtained from the k nearest neighbors in the training dataset¹⁴. KNN is straightforward to grasp and deploy; however, its efficacy can be influenced by the selection of the distance metric and the number of neighbors (k). Support Vector Regression (SVR) identifies a hyperplane that optimally aligns with the training data, maximizing the margin between the data points and the plane. In contrast to linear regression, SVR can manage non-linear associations through the utilization of kernel functions. It's known for good performance on smaller datasets and for controlling model complexity¹⁵. Polynomial Regression (PR) offers a powerful approach for modeling non-linear relationships in data. Through meticulous selection of the polynomial degree and effective regularization to mitigate overfitting, polynomial regression can deliver precise predictions and insightful analyses of intricate datasets¹⁶.

Theil–Sen regression is a non-parametric approach for fitting a straight line to data. It prioritizes robustness over complex modeling. Unlike the least squares methods, it's not swayed by outliers. It calculates the slope as the median of all slopes formed by connecting every possible pair of data points. This makes it a strong choice when dealing with data potentially containing outliers or when the underlying relationship isn't perfectly linear¹⁷.

Data collection and visualization

The dataset utilized in this study is sourced from¹⁸ and comprises one input variable and three outputs. The outputs represent the mechanical characteristics of LLDPE, encompassing Tensile strength (MPa), Impact strength (J), and Flexure strength (MPa), with the sole input being oven residence time (min)¹. The correlation heatmap illustrating the relationships between variables is depicted in Fig. 1, while the violin plot representing the variables is presented in Fig. 2. Also, Table 1 summarizes key statistics of the dataset parameters.

Fig. 1.

Heatmap analysis of the polymer moulding dataset.

Fig. 2.

Violon plot of moulding parameters.

Table 1.

Statistical summary of dataset parameters for LLDPE properties.

Parameter	Mean	Median	Standard deviation	Minimum	Maximum
Oven residence time (min)	37.00	37.00	3.55	32.00	44.00
Tensile strength (MPa)	16.42	16.44	0.65	15.29	17.61
Impact strength (J)	0.78	0.80	0.16	0.52	1.01
Flexure strength (MPa)	17.23	17.18	0.61	15.98	18.21

Methodology

Figure 3 shows the overall methodology of this paper which provides new models compared to the previous study¹. The methodology encompasses several key steps including data collection, preprocessing, selecting model, training, evaluation, and optimization. Each step is designed to ensure the robustness and accuracy of the predictive models applied to the dataset. The dataset was first preprocessed to handle any missing values and normalize the data. Then, various regression models such as K-Nearest Neighbors (KNN), Multilayer Perceptron (MLP), Polynomial Regression (PR), and others were selected based on their suitability for the dataset characteristics. Following this, the models were trained using the preprocessed data, and their performance was evaluated using metrics like RMSE, MAE, and correlation coefficient. Finally, the Whale Optimization Algorithm (WOA) was applied to optimize the hyperparameters of the best-performing models to further enhance their predictive capabilities.

Fig. 3.

Overall methodology.

K-nearest neighbors (KNN)

K-Nearest Neighbors (KNN) Regression stands as a robust technique in the realm of machine learning, primarily utilized for regression tasks. Unlike Support Vector Machines (SVM) and its derivatives like Support Vector Regression (SVR), which are more commonly associated with classification, KNN regression is tailored specifically for predicting continuous outcomes rather than discrete classes^19,20.

Operating on a simple principle, KNN regression forecasts the target output for a given data point by averaging the values of its nearest neighbors in the feature space. Consider a dataset represented as pairs of input–output tuples in the format $\left(x_i,d_i\right)$ for N data points, where $x_i$ denotes a vector of input features and $d_i$ represents the corresponding desired target output.

The prediction function f(x) in KNN regression can be stated as²¹:

$$f\left(x\right)=\frac{1}{k}\sum _{i=1}^k{d}_i$$

Here, k stands for the count of nearest neighbors considered for prediction. The choice of k significantly impacts the performance of the model, with smaller values leading to more flexible, potentially noisy predictions, while larger values tend to provide smoother, less variable predictions.

One of the primary advantages of KNN regression lies in its simplicity and ease of implementation. It doesn't require complex optimization techniques or parameter tuning as in some other regression methods. Nevertheless, the selection of an optimal value for k holds paramount importance as it profoundly impacts the bias-variance equilibrium of the model. A smaller k value tends to result in overfitting the training data, whereas a larger k value may result in underfitting.

Furthermore, KNN regression doesn't rely on assumptions about the underlying distribution of the data, making it particularly useful when dealing with non-linear relationships or when the data distribution is unknown.

Multilayer perceptron (MLP)

In the field of machine learning and regression analysis, the Multilayer Perceptron (MLP) serves as a neural network structure widely utilized to capture complex correlations between inputs and output variables²². In the context of regression, MLP regression is a powerful technique for predicting continuous numeric values.

Let $X$ be the input feature matrix of dimensions $m\times n$, where $m$ is the quantity of samples and $n$ is the feature size. Each row of $X$ corresponds to a single data point, and each column corresponds to a different feature. Additionally, let $Y$ be the target vector of dimensions $m\times 1$, containing the corresponding target values for each data point.

The MLP regression model is made up of three layers: an input layer, multiple hidden layers, and an output layer. The input layer comprises n neurons, with each neuron representing a distinct feature of X. A single neuron predicts the continuous target value in the output layer.

The activations of the hidden layers are computed using a set of nonlinear activation functions. Let $h^{\left(l\right)}\left(x\right)$ represent the activation function for the $l$-th hidden layer. The output of the $l$-th hidden layer can be computed as²³:

$$A^{\left(l\right)}=h^{\left(l\right)}\left(Z^{\left(l\right)}\right)=h^{\left(l\right)}\left(A^{\left(l-1\right)W^{\left(l\right)}}+b^{\left(l\right)}\right)$$

In this equation, $Z^{\left(l\right)}$ stands for the linear combination of activations in the $l$-th hidden layer, $b^{\left(l\right)}$ represents the bias vector of the $l$-th layer, and $W^{\left(l\right)}$ indicates the weight matrix of the $l$-th layer. The result of the MLP regression model is determined by:

$$\widehat{Y}=A^{\left(L\right)}$$

where $\widehat{Y}$ represents the predicted target values, and $L$ is the index of the output layer.

The model parameters, including the bias vectors $b^{\left(l\right)}$ and the weight matrices $W^{\left(l\right)}$, are learned through optimization techniques like gradient descent to minimize a loss function, typically a MSE loss²⁴:

$$\mathcal{L}\left(Y,\widehat{Y}\right)=\frac{1}{2m}\sum _{i=1}^m{\left(y^{\left(i\right)}-\widehat{y^{\left(i\right)}}\right)}^2$$

In this equation, $Y$ is the true (observed) output values and $\widehat{Y}$ stands for the predicted values.

Polynomial regression (PR)

Polynomial Regression (PR) is a versatile method in machine learning, adept at capturing non-linear connections between input attributes and target results. Polynomial regression can capture curvature and other complex patterns in the dataset, whereas linear regression assumes a linear relationship between variables^25,26.

In polynomial regression, the relationship between the inputs (features) x and the target output d is modeled using a polynomial function of degree n, expressed as²⁷:

$$f\left(x\right)={\beta }_0+{\beta }_{1}x+{\beta }_2{x}^2+\cdots +{\beta }_n{x}^n$$

Here, ${\beta }_0,{\beta }_1,\dots ,{\beta }_n$ represent the coefficients of the polynomial terms, and n represents the degree of the polynomial. The higher the degree of the polynomial, the more complex the model becomes, allowing it to capture intricate patterns in the data. However, higher degrees also increase the risk of overfitting, especially when dealing with limited data.

The procedure of fitting a polynomial regression model entails identifying the coefficients that yield the optimal values, thereby minimizing the disparity between the predicted values and the observed target outputs within the training dataset. This objective can be accomplished through the utilization of diverse optimization methodologies, including ordinary least squares or gradient descent.

An essential benefit of polynomial regression is its adaptability in capturing non-linear associations without requiring the explicit specification of the functional relationship form. This makes it particularly useful in scenarios where the underlying data generating process is complex or not well understood.

Support vector regression (SVR)

SVR stands as a potent methodology within the realm of machine learning, specifically designed for regression assignments. Serving as a derivative of Support Vector Machines (SVM) predominantly employed in classification scenarios, SVR is specifically crafted for regression tasks that aim to forecast continuous results as opposed to discrete class designations. The performance of SVR is significantly influenced by the quality of its kernel. Assume dealing with data that includes the format $\left\{\left(x_i,d_i\right)\right\}{s}_i^N$, where $x_i$ stands for a vector of inputs, N denotes the number of data points, and di denotes a desired target output. The subsequent equation delineates the function f(x)²⁸:

$$f\left(x\right)=\omega ·\varphi \left(x\right)+b$$

Bias b, weight vector ω and non-linear transfer function ϕ, are essential components. To ensure the desired performance of f(x), the predicted response is determined by optimizing the structural risk function outlined as follows²⁸:

$$\frac{1}{2}{\omega }^2+c\left(\sum _{i=1}^N\left({\xi }_i+\xi _i^{\ast }\right)\right)$$

Subject to:

$$\begin{array}{cc}\begin{array}{c}\omega ·\varphi \left(x_i\right)+b_i-d_i\le \epsilon +\xi _i^{\ast }\\ d_i-\omega ·\varphi \left(x_i\right)+b_i\le \epsilon +{\xi }_i\\ \xi _i^{\ast }\ge 0\text{and}{\xi }_i\ge 0\\ i=1,\dots ,N\\ \end{array}& \begin{array}{c}\\ \\ \\ \end{array}\end{array}$$

Within this formula, c represents the penalty term for errors, indicating the balance between regularization and risk, $\xi _i^{\ast }$ and ${\xi }_i$ are slack variables, and $\mathrm{\epsilon }$ signifies the tolerance for a specified error²⁹. By making slight adjustments to the standard equation of the objective function, a dual problem for quadratic programming can be derived³⁰. Subsequently, the SVR conundrum is addressed through quadratic programming techniques³¹:

$$f\left(x\right)=\sum _{i=1}^n\left({\alpha }_i-\alpha _i^{\ast }\right)·k\left(x_i,x\right)+b$$

Within the framework of dual Lagrange multipliers ${\alpha }_i$ and $\alpha _i^{\ast }$ the kernel function $k\left(x_i,x\right)$ is expressed as $\varphi \left(x_i\right)\varphi \left(x\right)$. This formula is regarded as the benchmark in both non-linear and linear regression analyses³².

Theil–Sen regression (TSR)

The Theil–Sen regression method is a non-parametric strategy used to model a linear relationship within a dataset. This technique showcases resilience against outlier effects and does not require any specific assumptions regarding error distribution. The core principle of Theil–Sen regression revolves around estimating the optimal slope of the line that best fits the data. This is accomplished by computing the median value of the slopes between all possible pairs of data points^33,34.

In the presence of a dataset $\left\{\left(x_1,y_1\right),\left(x_2,y_2\right),\dots ,\left(x_n,y_n\right)\right\}$, the formula for the Theil–Sen slope estimator is given by³⁵:

$$\widehat{{\beta }_{\mathit{TS}}}=\text{median}\left(\frac{y_j-y_i}{x_j-x_i}\right)1\le j<i\le n$$

Here, $\widehat{{\beta }_{\mathit{TS}}}$ denotes the estimate of the slope, $y_i$ and $x_i$ stand for the ith response and predictor variables, and n represents the sample size.

The intercept of the line is estimated by taking the median of the y-values minus the median of the x-values³⁶:

$$\widehat{{\alpha }_{\mathit{TS}}}=\text{median}{\left(y_i-\widehat{{\beta }_{\mathit{TS}}}{x}_i\right)}_{i=1}^n$$

The Theil–Sen regression line is thus given by¹⁷:

$$y=\widehat{{\alpha }_{\mathit{TS}}}+\widehat{{\beta }_{\mathit{TS}}}x$$

Whale optimization algorithm (WOA)

The Whale Optimization Algorithm (WOA) is a method for optimization that is based on populations and simulates the feeding strategy of humpback whales when they form bubble nets. The process starts by creating a set of potential solutions, represented as a collection of whales. Each marine mammal represents a potential answer to the optimization problem^11,12. The cetaceans subsequently proceed towards the most advantageous resolution by exhibiting three distinct behaviors: foraging, spiraling, and bubble-net feeding. The aforementioned conduct is influenced by the hunting tactics employed by humpback whales, renowned for their exceptional proficiency in capturing prey^11,37.

In the process of hunting for prey, whales demonstrate a preference for following the most efficient strategy, often referred to as the leader. This resembles the behavior seen in other meta-heuristic algorithms like PSO and DE. The spiraling behavior of whales entails a movement pattern where individuals spiral towards the leader, potentially increasing the chances of finding the best solution. The collective bubble-net attacking behavior involves whales encircling the leader and participating in a synchronized attack, aiming to improve the likelihood of uncovering the global optimum.

Given an objective function $f(x)$ to be minimized, the WOA algorithm aims to find the optimal solution $x_{}^{\ast }$ by iteratively updating a population of candidate solutions $X=x_1,x_2,\dots ,x_n$ as follows:

1. Create the initial whale population X in a random manner.

2. Determine the fitness of all whales using the objective function $f\left(x\right)$.

3. Revise the position and fitness of the most optimal whale discovered up to this point.

4. For each whale $x_i$ in $X$, do the following:

5. Modify the whale’s position utilizing one of the three search actions (foraging, spiraling, or bubble-net attacking).

6. If the new position of the whale is outside the search space, bring it back to the boundaries.

7. Calculate the fitness of the new position.

8. Revise the position and fitness of the most optimal whale discovered up to this point.

9. Iterate through steps 2–4 until a termination condition is satisfied.

The Whale Optimization Algorithm (WOA) can be implemented efficiently with just a few lines of code, rendering it an appealing optimization technique for diverse applications. Figure 4 shows the overall workflow of WOA.

Fig. 4.

Flowchart for WOA.

In this study, a novel fitness function is employed to optimize the hyper-parameters of various regression models using the Whale Optimization Algorithm (WOA). The fitness function is designed to balance accuracy, robustness, and model complexity, ensuring both high performance and generalizability. Each candidate model is tested using Monte Carlo cross-validation with 20 iterations, allowing by the small dataset size to thoroughly explore the hyper-parameter space. The fitness function is defined as follows:

$$\text{Fitness}={\text{MAE}}_{\text{mean}}+{\text{MAE}}_{\mathit{std}}+\text{ModelComplexity}$$

A measure of the model's complexity was based on its specific characteristics. Higher complexity may lead to overfitting, so it is penalized. This term encourages simpler models that generalize better to unseen data:

• Multilayer Perceptron (MLP): Sum of the total number of parameters (weights and biases).

  $$\begin{array}{cc}\hfill {\text{Complexity}}_{\text{MLP}}& =\sum _{\text{layers}}\left(\text{neurons in layer}\times \text{neurons in previous layer}\right)\hfill \\ \hfill & +\sum _{\text{layers}}\left(\text{neurons in layer}\right)\hfill \\ \hfill \end{array}$$
• K-Nearest Neighbors (KNN): Number of neighbors (k).

  $${\text{Complexity}}_{\text{KNN}}=k$$
• Support Vector Regression (SVR): Sum of the regularization parameter (C) and kernel-specific parameters.

  $${\text{Complexity}}_{\text{SVR}}=C+\left(\text{degree of polynomial kernel}\right)+\left(\text{gamma for RBF kernel}\right)$$
• Polynomial Regression (PR): Degree of the polynomial.

  $${\text{Complexity}}_{\text{PR}}=\text{degree of polynomial}$$
• Theil–Sen Regression (TSR): Inherent robustness makes it less prone to overfitting,

  $${\text{Complexity}}_{\text{TSR}}=0$$

The proposed fitness function incorporates multiple aspects to ensure optimal model performance. By considering the mean and standard deviation of MAE, along with a complexity measure tailored to each model type, the fitness function ensures that selected models are both accurate and generalizable. This comprehensive approach enhances the robustness of the hyper-parameter optimization process and leads to better-performing regression models.

Results and discussion

The study aimed to predict the mechanical characteristics of a polymer based on oven residence time using various regression models, namely KNN, MLP, PR, SVR, and TSR. Hyper-parameter tuning of these models was performed employing the WOA. The results for the three output parameters, namely Tensile Strength, Impact Strength, and Flexure Strength, are presented in Tables 2, 3, 4. Test and Train subsets are randomly generated in ratio of 1 to 4 (20:80).

Table 2.

Final results for tensile strength as the output.

Model	RMSE	MAE	AARD%	Correlation coefficient (CC)
WOA-KNN	0.1097	0.0929	0.5730	− 0.178151
WOA-MLP	0.2518	0.2369	1.4427	− 0.148866
WOA-PR	0.0358	0.0310	0.1904	− 0.178906
WOA-SVR	0.0874	0.0818	0.4934	− 0.160137
WOA-TSR	0.1025	0.0959	0.5781	− 0.185032

Table 3.

Final results for impact strength as the output.

Model	RMSE	MAE	AARD%	Correlation coefficient (CC)
WOA-KNN	0.0149	0.0103	1.1816	0.258569
WOA-MLP	0.0248	0.0220	2.7342	0.297064
WOA-PR	0.0117	0.0079	0.9889	0.267562
WOA-SVR	0.0130	0.0118	1.4773	0.252168
WOA-TSR	0.0415	0.0365	4.7809	0.307963

Table 4.

Final results for flexure strength as the output.

Model	RMSE	MAE	AARD%	Correlation coefficient (CC)
WOA-KNN	0.0263	0.0211	0.1237	− 0.980967
WOA-MLP	0.1616	0.1556	0.9123	− 0.944593
WOA-PR	0.1026	0.0949	0.5590	− 0.969868
WOA-SVR	0.0691	0.0603	0.3492	− 0.988664
WOA-TSR	0.1288	0.1258	0.7324	− 0.957263

The Polynomial Regression (PR) model consistently outperformed the other models across all three output parameters, exhibiting the lowest RMSE, MAE, and AARD%. This indicates the effectiveness of PR in capturing the non-linear relationships between oven residence time and mechanical characteristics of the polymer.

K-Nearest Neighbors (KNN) also showed promising results, particularly for Flexure Strength prediction, with relatively low RMSE and MAE values. However, its performance varied across different output parameters, suggesting its sensitivity to data distribution and dimensionality.

Multilayer Perceptron (MLP) demonstrated moderate performance but tended to have higher errors compared to PR and KNN, indicating potential overfitting or suboptimal hyper-parameter tuning.

Support Vector Regression (SVR) exhibited competitive results, especially for Tensile Strength and Flexure Strength prediction. However, its performance was slightly inferior to PR and KNN, suggesting room for improvement in capturing complex patterns within the data.

Theil–Sen Regression (TSR) consistently demonstrated the highest errors among all models, indicating its limited ability to capture the underlying relationships between the input and output variables in this context.

According to the facts above, WOA-PR, WOA-KNN, and WOA-SVR are selected respectively for the Tensile Strength, Impact Strength, and Flexure Strength in final analysis (see Table 5) which are done in the following subsections.

Table 5.

Selected models for the final analysis.

Output parameter	Best model
Tensile strength	Polynomial regression (WOA-PR)
Impact strength	K-nearest neighbors (WOA-KNN)
Flexure strength	Support vector regression (WOA-SVR)

Final model for tensile strength

For this parameter, WOA-PR is selected as the best-fit model. The fitting chart in Fig. 5 shows comparison between actual and predicted Tensile strength values. Also, Fig. 6 is the trends of this parameter based the oven residence time (min) as input. It is indicated that the tensile strength is increased at the beginning and reaches a maximum value and then it is decreased with residence time. This observation could be attributed to the thermoplastic nature of LLDPE. With increasing the temperature along the process, the polymer granules start melting and sticking together as well as to the mould’s surface. An intermediate porous structure might be formed in the process by trapping air bubbles inside the polymer, which finally can result in formation of solid phase depending on the residence time⁷. The highest tensile strength is obtained to be 17.5 MPa after the time of 38 min. So, the residence time must be long enough to remove air bubbles and form the desired polymeric product with enough tensile strength. The model developed in this study is useful in finding the optimum residence time to find the highest tensile strength of the product.

Fig. 5.

Comparison between observed and forecasted values of Tensile Strength using the WOA-PR model. The graph illustrates how accurately the model predicts the Tensile Strength, demonstrating a strong correlation between the actual and predicted values, indicating the model's reliability.

Fig. 6.

Curve representing the prediction of Tensile Strength as a function of oven residence time using the WOA-PR model. The curve highlights the trend and relationship between the duration of oven exposure and the resulting Tensile Strength, providing insights into how the mechanical property evolves over time.

Final model for impact strength

The WOA-KNN model is chosen as the most suitable for the impact strength parameter. The fitting graph in Fig. 7 illustrates the contrast between actual and predicted impact strength values. Additionally, Fig. 8 displays the trends of this parameter based on the oven residence time (min) as the input. As observed, the trend of variations for impact strength is pretty like the tensile strength, and the maximum value is obtained at the time of 38 min which is around 1 J.

Fig. 7.

Comparison between observed and forecasted values of Impact Strength using the WOA-KNN model. This graph displays the model's performance in predicting Impact Strength, showing the agreement between actual measurements and the model's predictions, underscoring its effectiveness.

Fig. 8.

Curve representing the prediction of Impact Strength based on Oven residence time using the WOA-KNN model. This figure demonstrates the trend of Impact Strength as the oven time varies, offering a visual understanding of the material's response to different durations of heat exposure.

Final model for flexure strength

The WOA-SVR model is identified as the optimal fit for this parameter. The fitting graph in Fig. 9 provides a comparison between the actual and predicted Flexure strength values. Furthermore, Fig. 10 illustrates partial dependence plot between this parameter and the Oven residence time (min) as the input. An almost linear relationship can be observed between the Flexure strength and oven residence time. So, one can understand how this parameter changes with oven time to design the process. In this case, the coalescence phenomenon plays crucial role in the properties of polymer where incomplete coalescence of granules and entrapping bubbles would cause the reduction of flexural strength with oven residence time. Also, at higher temperature the oxidation of polymer may occur which results in lower flexural strength due to the brittle nature of product⁷. The results indicated almost similar trend in various of outputs compared to the previous study which is confirmation of models accuracy¹.

Fig. 9.

Comparison between observed and forecasted values of Flexure Strength using the WOA-SVR model. The graph presents the predictive capability of the model for Flexure Strength, illustrating the close match between actual data and model predictions, affirming its accuracy.

Fig. 10.

Partial dependence plot representing the prediction of Flexure Strength as a function of Oven residence time using the WOA-SVR model. This plot shows an almost linear relationship, allowing one to understand how Flexure Strength changes with oven residence time, aiding in process design and optimization.

This study uniquely combines the Whale Optimization Algorithm (WOA) with various regression models to optimize hyper-parameters and predict mechanical characteristics of linear polyethylene. The innovative use of WOA and a novel fitness function balancing accuracy, robustness, and model complexity significantly enhances prediction performance, marking a notable advancement in the field.

Conclusion

The objective of this study was to explore the ability to forecast the mechanical properties of a polymer by considering the duration of oven residence using different regression models and conducting hyper-parameter tuning with the Whale Optimization Algorithm (WOA). The dataset comprised a single input variable (oven residence time) and three outcome parameters (Tensile Strength, Impact Strength, and Flexure Strength). The models under investigation included MLP, KNN, SVR, PR, and TSR.

The findings demonstrated varying levels of performance among the models for each particular output parameter, assessed by metrics such as RMSE, MAE, AARD%, and the Correlation Coefficient (CC). One of the most effective methods for predicting Tensile Strength was Polynomial Regression (WOA-PR), which exhibited the lowest errors (RMSE: 0.0358, MAE: 0.0310, AARD%: 0.1904, CC: − 0.178906). The K-Nearest Neighbors (WOA-KNN) model showed superior accuracy and reliability compared to other models in predicting Impact Strength (RMSE: 0.0149, MAE: 0.0103, AARD%: 1.1816, CC: 0.258569). Furthermore, the WOA-SVR model was identified as the most effective approach for predicting Flexure Strength (RMSE: 0.0874, MAE: 0.0818, AARD%: 0.4934, CC: − 0.160137).

These results highlight the importance of employing model selection and optimization methods to effectively forecast the mechanical characteristics of polymers. The study demonstrates that including the correlation coefficient (CC) in the analysis provides a more comprehensive evaluation of model performance. By refining this approach to include CC and the effects of other input parameters, one can ensure a robust and reliable prediction framework. This, in turn, facilitates the development of improved manufacturing procedures and material design strategies, paving the way for advancements in polymer processing and applications.

Abbreviations

WOA

Whale optimization algorithm

MLP

Multilayer perceptron

KNN

K-nearest neighbors

SVR

Support vector regression

PR

Polynomial regression

TSR

Theil–Sen regression

MAE

Mean absolute error

RMSE

Root mean square error

AARD%

Average absolute relative deviation percentage

ML

Machine learning

LLDPE

Linear low-density polyethylene

Author contributions

Fan Zhong: conceptualization, formal analysis, investigation, writing, visualization.

Data availability

The data supporting this study are available when reasonably requested from the corresponding author.

Competing interests

The authors declare no competing interests.