Sustainable and Optimized Production in an Aluminum Extrusion Process

Abstract

In discussions on environmental policies, eco-efficiency is often underlined. Eco-efficiency is defined as delivering products and services with competitive value while simultaneously reducing the ecological impacts and meeting human needs. In highly competitive industrial environments, improvements in production processes are crucial for maintaining a strong differentiated position and competitive ability. Additionally, rationalizing energy consumption and optimizing the use of natural resources are essential for sustainability. This work presents an empirical study of a Portuguese industrial company focused on minimizing scrap production in extrusion processes. This is a common challenge in industrial extrusion processes worldwide, with significant economic and environmental implications. A literature review revealed strong relationships between key extrusion process parameters, including temperature, time, speed, pressure, and geometry. The main objective of this work is to model the aluminum extrusion process in a simple and replicable way, avoiding complex models such as nonlinear optimization or finite element methods, with a view toward potential machine learning applications for scrap reduction. Thus, simple multiple linear regression models enable the identification of the most influential variables involved in the process. The results identify key variables that impact scrap generation, aligning with findings from the literature. In this dataset, geometry-related factors are the parameters with notable scrap rates.

1. Introduction

Modern industrial environments are characterized by operational diversity and complexity, which demand a strong understanding of processes to ensure continuous improvement and effective performance. Sound organizational management is fundamental to navigating these challenges, especially as companies face growing pressure to enhance productivity while reducing material losses, such as scrap, which impacts both economic performance and sustainability [1].

In recent years, concerns related to environmental impacts and efficient resource use have led to broader integration of sustainable thinking into production systems. Although eco-efficiency—producing goods with a reduced environmental burden—has been widely promoted as a key principle for sustainable development [2], its application remains highly context-dependent and requires tailored operational strategies that balance cost, quality, and ecological responsibility [3].

Industrial competitiveness increasingly depends on streamlined and optimized operations. Efficient use of energy and raw materials is no longer a strategic advantage but rather a basic requirement. Achieving these goals involves a precise understanding of internal workflows and production variables, particularly in processes that are sensitive to fluctuations, such as those in the metal manufacturing sector [4].

The ongoing development of industrial technologies and the global push for responsible production have led many organizations to review and restructure their practices. The extrusion process, especially in the aluminum sector, serves as a prime example. It is a widely used method known for producing versatile profiles suited for various sectors, including construction and energy [5,6]. Its appeal lies in the potential for high recyclability and the production of lightweight, durable components. However, the process is influenced by numerous interdependent factors—such as temperature, speed, die geometry, and pressure—which, if not properly managed, can lead to increased scrap and reduced efficiency [7].

At the same time, manufacturing companies are facing tighter regulations and market demands, pushing them to align with global sustainability objectives such as carbon footprint reduction [8]. The challenge is to maintain high-quality outputs without escalating production costs or compromising environmental goals. Within this context, industries are expected not only to minimize resource consumption but also to take proactive roles in social responsibility and sustainable development [9].

In this study, we focus on the specific issue of scrap generation in aluminum extrusion. The objective is to explore how real industrial data can inform process improvements through optimization and data-driven decision-making, thereby contributing to both operational excellence and broader sustainability outcomes.

Taking the research literature into account, it was possible to identify emerging trends in the mitigation of waste, with a focus on the company’s sustainability, eco-efficiency, and competitiveness.

Taking into account this background about the growing importance of sustainability and waste reduction in aluminum extrusion industries as a critical problem with environmental and economic impact, the main motivation of the study is the challenge of minimizing scrap production. Therefore, the main objectives of this work are as follows:

• To present a model that minimizes the amount of scrap generated in the extrusion production sector, taking into account the example of a Portuguese company.
• To identify the most relevant parameters available in the data related to this waste, in order to support management decisions—such as acquiring new technologies or implementing more accurate information support systems, among other improvements.
• To improve the process and draw attention to environmental protection, improving behaviors, attitudes, and practices, with the potential to contribute to the development and competitiveness of the company in this important industrial sector, but also to their social responsibility and sustainability image.

In order to minimize the scrap created in the aluminum extrusion process, in this study, nonlinear and linear programming are considered. The process is modeled using six months of real industrial data from the company. By applying optimization and statistical techniques, it was possible to solve the problem and identify the importance of the variables under study, including temperature, time, speed, pressure, and geometry. Several other parameters can be considered in the extrusion process as factors influencing scrap generation, such as the composition, grade, and status of the aluminum alloy; mold aging and wear; and the mold maintenance cycle, among others, but the focus was on the data available.

The aluminum extrusion process is a well-known and extensively studied industrial process, already deeply described. The production process in the studied company follows the typical structure of an extrusion line, divided into three main phases, namely, pre-extrusion, extrusion, and post-extrusion. Moreover, the main contribution of this work is not the process description itself, but the application of a data-driven methodology based on real production data to identify the most relevant factors influencing scrap generation and to provide management insights for sustainable production.

The real database provided by the company contains information concerning the three phases of the production process under analysis, i.e., pre-extrusion, extrusion, and post-extrusion. These variables are expected to contribute to explaining the scrap variability during the process. The available database did not include detailed information regarding the specific composition, alloy grade, or metallurgical status of the billets used in each extrusion process. This limitation presents a possible direction for future research.

Nevertheless, it is important to emphasize that the variables analyzed (temperature, time, speed, pressure, and geometry) are parameters monitored and controlled daily by the company independently of the alloy, given the high standardization of their production processes. Additionally, the influence of these operational variables on scrap generation was validated both through statistical significance in the models and through alignment with findings from previous literature.

This paper is organized as follows: In Section 1, the subject is contextualized; Section 2 presents a brief literature review on the aluminum extrusion process, highlighting the importance of sustainable production and addressing the issue of wastefulness, which can be recycled at a high cost. The most commonly used methods for simulating the extrusion process are presented, followed by an optimization model. Section 3 introduces the empirical study and the proposed multiple regression models, as well as the validation and discussion of the industrial process problem model. This paper concludes with a discussion of the main findings and suggestions for future work in Section 4.

2. Literature Review

The aluminum extrusion industry has been around for more than 150 years and continues to play a critical role in various industrial applications such as construction, transportation, and renewable energy technologies [3,10]. Aluminum, although abundant in the Earth’s crust, is not found in pure form, but rather in combination with other elements. Its industrial use became feasible with the advent of the Bayer process, which enabled the extraction of aluminum from bauxite and other materials [11,12].

Aluminum extrusion technology is highly valued due to its ability to produce complex profiles with a high strength-to-weight ratio and excellent corrosion resistance [3,13]. The extrusion process involves forcing a billet through a die to produce continuous cross-sectional profiles. Depending on the alloy and desired properties, extrusion can be performed either hot or cold. However, direct hot extrusion remains the most widely used method for aluminum due to its adaptability and productivity [3,11].

A wide range of geometric shapes can be manufactured through extrusion, which offers flexibility in product design and industrial functionality [14,15]. Critical parameters such as temperature, time, extrusion speed, pressure, and billet geometry must be well-controlled to ensure high-quality output. Among these, the extrusion speed and the temperature of the billet are often cited as the most influential factors for productivity and defect minimization [3,16]. Research suggests that ideal billet temperatures typically range between 420 and 430 °C to minimize scrap while maintaining profile quality [17,18].

The optimization of these parameters is vital. Poor thermal control may result in surface imperfections, and excessive pressure can contribute to premature die wear and product inconsistency [19,20]. Furthermore, operational elements such as the timing of the sequence between the alloy changes and the condition of the mold also significantly affect the output [21,22].

Aluminum production generates significant scrap due to machining operations. The scrap amount depends on factors such as friction, billet temperature, and tool geometry [23]. Research has shown that adjusting the geometry and functionality of dies, particularly porthole dies, can improve die stability and reduce scrap production [24]. In terms of process parameters—die angle, load rate, and the total reduction in the cross-sectional area are key factors influencing extrusion pressures, as shown in studies that used empirical equations to assess the extrusion process [16]. Operating aluminum alloys at specific temperatures, typically between 420 and 430 °C, is crucial to avoid defects that lead to scrap. Exceeding these temperatures can cause issues with the profile quality [17]. The speed of extrusion also significantly affects the temperature and surface quality of the extruded product [25]. Some studies have found that extrusion yield is more dependent on temperature than on pressure. Recent advancements in extrusion modeling have provided more accurate predictions of effective strain, flow patterns, and extrusion pressure [18]. Additionally, there are efforts to modify alloy designs and improve material properties through better microstructure control. These modifications aim to optimize plastic deformation and recrystallization, both of which impact the final product’s texture and behavior [11]. The extrusion process requires careful management of both temperature and time to ensure proper material solubilization [12]. Many defects in extruded products are caused by die-related issues. Identifying and addressing these problems can help improve productivity and reduce the need for rework [23].

One study examined the causes of high scrap generation in aluminum extrusion by analyzing data from a Portuguese company. It was found that temperature, extrusion time, ram speed, pressure, and die geometry are critical factors for controlling scrap production [10]. Optimization methods have been shown to improve the yield of extrusion operations, particularly for complex extrudates, leading to significant savings for companies [25].

Modern studies highlight that extrusion pressure is influenced by factors such as billet and die geometry, deformation speed, extrusion ratio, and working temperature [5,26]. Critical variables identified in recent work include the extrusion ratio (ER), billet temperature ($T_B$), deformation speed ($V_R$), and alloy flow stress ($\sigma$). These factors directly impact metal flow, forming quality, and final product characteristics [27].

The die condition remains one of the main contributors to profile defects and subsequent scrap generation [7]. Effective maintenance and defect detection on dies are crucial for maintaining productivity. Material flow control, including friction management, not only reduces scrap but also enhances mechanical properties through better material orientation during deformation [28,29].

Aluminum’s unique combination of physical attributes, such as strength, ductility, corrosion resistance, and full recyclability, therefore offers a clear advantage in sustainable manufacturing [13,23]. In response to rising market demands, industries are simultaneously aiming to reduce costs and environmental impacts by improving energy efficiency and minimizing resource consumption during production [30,31].

Despite being infinitely recyclable, aluminum scrap recycling poses its own challenges, such as high energy use and metal loss during melting (up to 22% due to oxidation and slag formation) [25]. Thus, the most efficient and sustainable strategy is to minimize scrap generation at the source by controlling process variables effectively [10,23].

Sustainability efforts in the aluminum industry align with broader global initiatives like the Brundtland Report and Agenda 21, which emphasize social, economic, and environmental responsibility [32,33]. Organizations are increasingly evaluated using metrics like sustainable development indicators (SDIs), which track energy use, safety, and innovation capacity [13,34].

Circular economy (CE) practices have emerged as a complementary strategy, promoting recycling and reuse while encouraging efficient resource management. Several studies underline the CE’s role in reducing greenhouse gas emissions and enhancing operational performance in aluminum production [35,36,37,38]. For instance, solid-state recycling techniques have demonstrated up to 93% reductions in CO₂ emissions compared to traditional remelting methods [31].

Recent advancements in data-driven models and digital manufacturing have enabled companies to predict scrap more accurately and intervene earlier in the process. These approaches are especially beneficial to small- and medium-sized enterprises (SMEs), which often lack access to high-end simulation tools [31,39]. Compared to traditional finite element or finite volume models, data-driven solutions like multiple regression offer faster, more accessible alternatives.

Nevertheless, the success of such methodologies depends on quality data and internal expertise. Many firms still face challenges like limited personnel, insufficient digital infrastructure, or fragmented datasets [40]. Therefore, simplified models that provide actionable insights from real production data are critical for industrial practice.

Sustainable aluminum extrusion demands both technical and strategic approaches. It involves refining process parameters, embracing circular economy principles, and utilizing empirical data to guide improvements. Companies that integrate these elements can enhance both environmental performance and economic competitiveness, while also contributing to global sustainability goals [41,42].

Table 1. Dimensions according to the literature review.

Dimensions	Authors—Literature
Temperature	[5,10,12,15,17,22,25,26,35,40,43,44,45].
Time	[5,7,10,12,29,43].
Speed	[5,10,14,15,25,26,35,43,45].
Pressure	[5,10,16,18,40,43,44].
Geometry	[6,15,16,22,23,24,25,26,44].

summarizes the variables found in the literature as influencing scrap generation in aluminum extrusion processes.

Recently, research on digital simulation of the aluminum extrusion process has been tested in several studies [46,47,48], which underline the advantages of data-driven models compared to traditional physical models, highlighting the innovation of our methodology. The results obtained in these studies indicate that machine learning approaches are advantageous when large amounts of relevant data are available, allowing models to adapt to the specific conditions of a company. In contrast, physics-based models offer broader applicability and independence from historical data. The developed system supports decision-making by detecting hidden patterns in production data, identifying root causes of problems, and highlighting critical product parameters from a manufacturing perspective. In the context of additive manufacturing, especially 3D printing using material extrusion, machine learning has been increasingly applied to optimize process–structure–property relationships and to monitor the printing process in real time. Despite significant advances, future research should address challenges such as computational costs, appropriate sensor selection, and data sharing to create larger and more useful datasets for further improving manufacturing processes.

3. Methods and Models

The main objective of this study is to develop a simple, interpretable, and data-driven model to support decision-making in aluminum extrusion companies, aiming to minimize scrap generation and contribute to sustainable production practices. Previous works in the literature often rely on complex simulation methods (such as finite element models or non-linear optimization approaches), which are difficult to apply in small- and medium-sized companies due to their technical complexities, time requirements, and costs. Moreover, few studies explore the use of real industrial data to build practical optimization models that are directly applicable to the management of daily production processes. Therefore, the main research gap addressed in this study is the lack of simple and replicable modeling approaches using real production data that enable companies to identify the key variables influencing scrap generation and take action to improve operational performance and sustainability.

3.1. Methods

From previous research, it is evident that various parameters in the extrusion process have a direct impact on the amount of scrap produced. The issue at hand is considered a nonlinear optimization problem because most relationships between parameters and scrap generation are nonlinear.

There are several common methods for simulating the extrusion process, including the finite element method (FEM), which focuses on how material particle variables change based on coordinates and time. Another method is the finite volume method (FVM), which centers on particle velocity and acceleration. Additionally, the arbitrary Lagrangian–Eulerian (ALE) method combines FEM and FVM to model material flow and mesh movement [14].

However, modeling several nonlinear relationships is unnecessary when the initial objective is to identify the relative importance of data collected from the real process. Furthermore, it is time-consuming and requires sophisticated computational modeling methodologies and software tools, which are typically not available to small companies.

Optimization can play a significant role in process improvement [25]. Ref. described an optimization-based method used to analyze and improve the yield of aluminum extrusion operations at a local extrusion plant. This method resulted in significant savings, especially when the extrudate shape was complex.

3.2. Model

Traditional approaches that simulate the entire extrusion process—such as FEM, FVM, and ALE, as presented by [14] and briefly referred to in Section 3.1—are widely used in the literature. However, according to the same authors, these methods are often costly and time-consuming.

In fact, if we consider the dimensions or factors in Table 1, we easily have hundreds of variables with different relationships (linear and nonlinear) and weights related to the amount of waste, creating a very difficult nonlinear optimization problem to solve.

Table 2. Dimensions, including factors for scrap production.

Dimensions	Variables	Bounds
Temperature	$T_a,a=1,\dots ,k$	$l_a\le T_a\le u_a$
Time	$t_b,b=1,\dots ,p$	$l_b\le t_b\le u_b$
Speed	$S_c,c=1,\dots ,m$	$l_c\le S_c\le u_c$
Pressure	$P_d,d=1,\dots ,n$	$l_d\le P_d\le u_d$
Geometry	$G_e,e=1,\dots ,r$	$l_e\le G_e\le u_e$

presents a summary of the main dimensions monitored in the process and considered in this work, which are also identified in the literature (Table 1) as important factors for aluminum extrusion scrap generation. The parameters $k,p,m,n,$ and $r\in \mathrm{N}$, are described as follows:

• k—number of temperature parameters;
• p—number of time parameters;
• m—number of speed parameters;
• n—number of pressure parameters;
• r—number of geometric values in consideration.

• while the variables registered in the process are as follows:

• $T_a$—temperatures for $a=1,\dots ,k$;
• $t_b$—times for $b=1,\dots ,p$;
• $S_c$—speeds for $c=1,\dots ,m$;
• $P_d$—pressures for $d=1,\dots ,n$;
• $G_e$—geometric values for $e=1,\dots ,r$.

• and being the lower bounds for the extrusion parameters $l_a,l_b,l_c,l_d,l_e$, and their upper bounds, $u_a,u_b,u_c,u_d,u_e$.

According to the literature, the amount of scrap (S) depends on all these parameters in Table 2, thus, the amount of waste/scrap (S) from the aluminum extrusion process can be defined as follows:

$$\begin{array}{c}\begin{array}{cc}S =\hfill & {\beta }_0+{\beta }_1{f}_1\left(T_1\right)+\dots +{\beta }_k{f}_k\left(T_k\right)\hfill \\ & +{\beta }_{k+1}{f}_{k+1}\left(t_1\right)+\dots +{\beta }_{k+p}{f}_{k+p}\left(t_p\right)\hfill \\ & +{\beta }_{k+p+1}{f}_{k+p+1}\left(S_1\right)+\dots +{\beta }_{k+p+m}{f}_{k+p+m}\left(S_m\right)\hfill \\ & +{\beta }_{k+p+m+1}{f}_{k+p+m+1}\left(P_1\right)+\dots +{\beta }_{k+p+m+n}{f}_{k+p+m+n}\left(P_n\right)\hfill \\ & +{\beta }_{k+p+m+n+1}{f}_{k+p+m+n+1}\left(G_1\right)+\dots +{\beta }_q{f}_q\left(G_r\right)+\epsilon ,\hfill \end{array}\hfill \end{array}$$

(1)

i.e.,

$$S={\beta }_0+\sum _{i=1}^q{\beta }_i{f}_i\left(x_i\right)+\epsilon ,$$

(2)

with $\epsilon >0$, and $x_i,i=1,\dots ,q$, where $q=k+p+m+n+r$, represents all the variables considered. Also, ${\beta }_0$ is the constant (the scrap amount if all variables are null), and ${\beta }_i,i=1,\dots ,q$ correspond to the weights assigned to each variable in the model, representing the importance of each variable in the scrap amount. The functions representing the relation between each variable $x_i$ and the amount of waste created are denoted by $f_i$. Additionally, all variables are constrained, with $l_i$ and $u_i$ representing the non-negative lower and upper bounds for each variable $x_i,i=1,\dots ,q$.

Thus, given that the objective is to minimize the amount of scrap, the optimization problem to be addressed can be formulated as follows:

$$\begin{array}{cc}\underset{x\in R^q}{min}& S\left(x\right)\\ s.t.& l_i\le x_i\le u_i,\end{array}$$

(3)

being the objective function and scrap generation; the variables concerned with data about temperature, time, speed, pressure, and geometry-related variables; and the constraints defined by the lower and upper bounds of process variables, based on real data. The optimization formulation shown in Equation (3) can be classified as linear if all functions $f_i$ are linear. If at least one of them is nonlinear, then the problem becomes a nonlinear optimization problem. In either case, it remains an optimization problem with simple bounds.

Based on the literature, there is a strong connection between various parameters in the extrusion process and the amount of scrap produced. Since most of these relationships are nonlinear, the problem being addressed is a nonlinear optimization problem. There are many optimization algorithms available to solve problems like this, and the choice depends on factors such as the nature of the functions in the model (linear or nonlinear), the number of variables involved, and whether the problem is a global or local minimum. Nonlinear programming problems, such as the problems addressed in this work, are usually solved using simulation tools for the entire extrusion process. Nevertheless, there are many companies—namely small companies—that do not have trained employees, time, or money for this type of modeling, which is usually very specialized and time-consuming. Also, the cost of the sophisticated computational equipment required for these methodologies and simulation tools is high and often not feasible for such companies. However, modeling several nonlinear relationships is unnecessary when the initial objective is to identify the relative importance of data collected from the real process.

In this way, the model was simplified to a linear optimization model with simple bounds and incorporated statistical techniques—in particular, multiple linear regression models and the open-source software R, 4.1.2 version—to study the significance and relative relevance of the variables available in the data for waste generation. The simplification of the problem into its linear form has limitations and is not a perfect model, but it addresses the company’s main goal, namely, identifying the most influential variables in the available data that contribute to increased scrap production; thus, the model is a useful and easy-to-replicate model.

When all functions $f_i$, $i=1,\dots ,q$, are considered linear, i.e., assuming a linear relationship between the variables and scrap generation, the model $S\left(x\right)$ becomes a linear model. In this case, the amount of scrap is expressed as a combination of process variables. A multiple regression model can be used, where scrap is the dependent variable, and temperature, time, speed, pressure, and geometry are the independent variables. The next section will apply this approach. Since the minimum and maximum values (bounds) of the variables are known, minimizing S becomes a straightforward task. To reduce the amount of scrap, as predicted by the multiple linear regression model, it is necessary to decrease the variables that increase S, and increase the variables that reduce it. Therefore, there are several reasons for choosing a linear regression-based model for the data or similar data, namely, simplicity, interpretability, and suitability for companies without access to complex simulation tools.

4. Empirical Study

Production data from six months of continuous aluminum profile manufacturing were analyzed in previous research [10]. A stepwise method was applied to build a linear model for scrap, using different extrusion process variables. This method included all variables found to be statistically significant at the 5% level. The resulting model had an adjusted $R^2$ of around 46%, meaning that the model explains approximately 46% of the variation in scrap production based on the selected variables.

The present empirical study used the same database, but with a broader scope by including all stages of the aluminum extrusion process: pre-extrusion, extrusion, and post-extrusion. For comparison purposes, the data were standardized. The dataset contains 42,821 records; each record represents one extruded billet. It includes 65 variables—62 are quantitative (either discrete or continuous) and 3 are qualitative (nominal). The variables are grouped based on the stage of the process they refer to: pre-extrusion ($PR$), extrusion (E), and post-extrusion ($PO$), as detailed in

Table 3. Description of the variables included in the linear model.

Dimensions	Variables
Temperature	$E\text{\_}{T}_{CP}$: Post-Container Temperature (°C);
	$E\text{\_}{T}_{End}$: Max End Temperature (°C);
	$PR\text{\_}{T}_{{Z1}_{,Set}}$: Set Temperature Z1 (°C);
	$PR\text{\_}{T}_{BC}$: Billet Conical Temperature (°C);
Time	$E\text{\_}{t}_s$: Speed Time (s);
	$E\text{\_}{t}_D$: Dead Time (s);
	$E\text{\_}t$: Extrusion Time (s);
Speed	$E\text{\_}{S}_C$: Extrusion Speed (mm/s);
	$PO\text{\_}{S}_p$: Puller Speed (m/min);
	$PO\text{\_}{L}_p$: Pull Length (m);
	$PO\text{\_}{T}_{{rp}_{Dx}}$: DX Puller Traction
Pressure	$E\text{\_}{p}_{SL}$: Sealing Pressure (bar);
	$E\text{\_}{p}_{Max}$: Maximum Extrusion Pressure (bar)
Geometry	$E\text{\_}{L}_B$: Billet Length (mm);
	$E\text{\_}{L}_{Butt}$: Butt Length (mm);
	$E\text{\_}NH$: Number of Holes;
	$E\text{\_}SW$: Specific Weight;
	$PR\text{\_}NB$: Number of Billets;
	$PO\text{\_}LB$: Bar Length (mm);
	$PO\text{\_}NB$: Number of Bars (mm);
	$PO\text{\_}SS$: Saw Speed (%)

.

The amount of scrap in kilograms, $S{C}_{KG}$, is calculated by subtracting the bar weight ($PO\text{\_}{W}_B$) (kg) from the weight billet ($PR\text{\_}{W}_B$) (kg) [10], as shown in the following expression: $S{C}_{KG}=PR\text{\_}{W}_B-PO\text{\_}{W}_B$.

Optimizing the entire extrusion process and predicting aluminum behavior in all stages (pre-, during, and post-extrusion) helps identify key variables in advance. This makes it possible to adjust decisions to the company’s specific conditions and enhance process efficiency.

The goal is to find the best values for the extrusion variables that reduce the amount of scrap produced. This helps lower production costs and supports environmental sustainability. If a linear model is used to approximate S, then Problem (3) can be classified as a linear minimization problem with simple variable bounds. From the large database on the company’s six-month production, it was possible to extract the limits for the variables, which are presented in

Table 4. Descriptive statistics of the variables considered in the linear regression model (Equation (4)).

Variables	Minimum	Median	Mean	St. Deviation	Maximum
$E\text{\_}NH$	0	2	2.49	1.65	12
$E\text{\_}SW$	0	0.77	1.55	1.96	78.42
$E\text{\_}SC$	0.44	4.58	4.65	1.41	12.83
$E\text{\_}tS$	0	39	42.06	23.09	452
$E\text{\_}t$	56	167	173.93	49.09	1200
$E\text{\_}t\text{\_}D$	12	17	18.66	3.62	33
$PO\text{\_}LP$	0	43.93	43.62	8.65	58.19
$PR\text{\_}NB$	1	13	29.78	43.35	329
$E\text{\_}LB$	650	865	850.98	94.41	1113
$E\text{\_}L\text{\_}Butt$	12	22	23.62	4.47	150
$PO\text{\_}LB$	2050	6500	6542.21	1550.18	12,050
$PO\text{\_}NB$	0	14	17.24	17.19	228
$PO\text{\_}S\text{\_}p$	0	14.33	13.74	5.43	31.81
$PR\text{\_}T\text{\_}BC$	67	445	535.95	266.60	1372
$PR\text{\_}T\text{\_}Z1\text{\_},Set$	40	460	457.83	17.33	510
$E\text{\_}T\text{\_}End$	350	565	557.98	27.99	600
$PO\text{\_}Tr\text{\_}p,DX$	15	33	33.17	4	78
$PO\text{\_}T\text{\_}CH$	10	70	63.85	14.74	95
$E\text{\_}p\text{\_}Max$	133	240	236.47	18.43	275
$E\text{\_}pSL$	239	247	246.79	1.06	282
$E\text{\_}T\text{\_}CP$	407	435	435.43	4.06	451

.

If at least one of the functions in Problem (3) is not linear, the problem becomes a nonlinear one. In such cases, it is necessary to apply methods that are suitable for solving this nonlinear optimization problem type. To estimate the model for S, the stepwise method was applied using the R software. The analysis considered a 5% significance level, and all variables included in the model were found to be statistically significant. The resulting model achieved an adjusted R-squared value of approximately 53%. This means that 53% of the total variation in scrap production can be explained by the selected independent variables in the linear regression model. The model can be expressed as follows:

$$\begin{array}{lll}S{C}_{KG}\hfill & =& 60.447+1.097E\text{\_}{t}_D-0.620PO\text{\_}{S}_p\\ \hfill & & +0.146PO\text{\_}SS-0.063PR\text{\_}{T}_{{Z1}_{,Set}}\\ \hfill & & +1.110E\text{\_}SW-0.001PO\text{\_}LB\\ \hfill & & +0.141E\text{\_}{L}_{Butt}+1.442E\text{\_}NH\\ \hfill & & -0.028E\text{\_}{p}_{Max}-0.105PO\text{\_}NB\\ \hfill & & +0.010E\text{\_}t-0.117E\text{\_}{T}_{CP}\\ \hfill & & -0.121PO\text{\_}{Tr}_{p,DX}-0.012E\text{\_}tS\\ \hfill & & +0.032PO\text{\_}LP+0.003E\text{\_}LB\\ \hfill & & -0.216E\text{\_}SC+0.004PR\text{\_}NB\\ \hfill & & -0.004E\text{\_}{T}_{End}+0.061E\text{\_}{p}_{SL}\end{array}$$

(4)

The coefficients of model (4), shown in

Table 5. Regression coefficients and diagnosis of collinearity between variables.

	Unst	Std	St Coef	t	Sig.	Collinearity Stat
	Coef	Coef	Error			Tolerance	VIF
(Constant)	60.447	8.033		7.524	0.000
$E\text{\_}{t}_D$	1.097	0.01	0.428	112.873	0.000	0.763	1.31
$PO\text{\_}{S}_p$	−0.62	0.014	−0.363	−45.526	0.000	0.173	5.788
$PO\text{\_}SS$	0.146	0.003	0.231	45.017	0.000	0.415	2.41
$PR\text{\_}{T}_{{Z1}_{,Set}}$	−0.063	0.002	−0.117	−25.224	0.000	0.511	1.958
$E\text{\_}SW$	1.11	0.035	0.234	31.707	0.000	0.201	4.98
$PO\text{\_}LB$	−0.001	0.000	−0.115	−24.923	0.000	0.513	1,95
$E\text{\_}{L}_{Butt}$	0.141	0.008	0.068	18.755	0.000	0.842	1.188
$E\text{\_}NH$	1.442	0.049	0.257	29.181	0.000	0.141	7.069
$E\text{\_}{p}_{Max}$	−0.028	0.003	−0.055	−10.8	0.000	0.423	2.362
$PO\text{\_}NB$	−0.105	0.005	−0.194	−21.861	0.000	0.14	7.156
$E\text{\_}t$	0.01	0.001	0.052	7.677	0.000	0.237	4.226
$E\text{\_}{T}_{CP}$	−0.117	0.009	−0.051	−12.523	0.000	0.662	1.511
$PO\text{\_}{Tr}_{p,DX}$	−0.121	0.01	−0.052	−11.642	0.000	0.544	1.837
$E\text{\_}tS$	−0.012	0.002	−0.031	−7.842	0.000	0.722	1.386
$PO\text{\_}LP$	0.032	0.005	0.03	6.642	0.000	0.538	1.86
$E\text{\_}LB$	0.003	0.000	0.033	6.839	0.000	0.475	2.104
$E\text{\_}SC$	−0.216	0.041	−0.033	−5.267	0.000	0.283	3.537
$PR\text{\_}NB$	0.004	0.001	0.019	4.901	0.000	0.713	1.403
$E\text{\_}{T}_{End}$	−0.004	0.001	−0.014	−3.713	0.000	0.829	1.207
$E\text{\_}{p}_{SL}$	0.061	0.029	0.007	2.114	0.035	0.989	1.012

, indicate that all variables contribute significantly to explaining scrap production per billet. However, even though all variables are statistically significant, some have a greater influence on the model than others. The analysis of the standardized regression coefficients indicates that the variables $E\text{\_}NH$ (number of holes), $E\text{\_}{S}_C$ (extrusion speed), $E\text{\_}SW$ (specific weight), $E\text{\_}{p}_{SL}$ (sealing pressure), $PO\text{\_}{S}_p$ (puller speed), $E\text{\_}{t}_D$ (dead time), $PO\text{\_}{T}_{{rp}_{Dx}}$ (DX puller traction), and $E\text{\_}{T}_{CP}$ (post-container temperature) make the most significant relative contributions in explaining the variation in the dependent variable. The variables with positive coefficients are $E\text{\_}NH$ (number of holes), $E\text{\_}SW$ (specific weight), $E\text{\_}{p}_{SL}$ (sealing pressure), and $E\text{\_}{t}_D$ (dead time). These variables are those that most contribute to an increase in scrap production. The results suggest that reducing the values of these variables may lead to a decrease in scrap. Conversely, the variables $E\text{\_}{S}_C$ (extrusion speed), $PO\text{\_}{S}_p$ (puller speed), $PO\text{\_}{T}_{{rp}_{Dx}}$ (DX puller traction), and $E\text{\_}{T}_{CP}$ (post-container temperature) show negative coefficients, indicating that higher values of these variables are associated with reduced scrap generation.

The model presented in Equation (4) supports the findings from previous studies, as summarized in Table 1. The key variables that most influence scrap production during the aluminum extrusion process remain consistent across the two analyses. The linear regression model assumes that residuals follow specific conditions, namely, they must be independent, identically distributed, have a mean of zero, follow a normal distribution, and exhibit constant variance. These assumptions hold in this case. Given the large dataset, the central limit theorem [49] can be applied. This implies that the residuals should closely follow a normal distribution. To assess the independence of residuals, the Durbin–Watson (DW) statistic was calculated. The result of approximately 0.91 suggests some correlation among the residuals, which is a limitation of the model. This could be explained by the continuous nature of the billet extrusion process, where billets may fuse due to the high extrusion temperature and pressure [40]. Regarding multicollinearity, the tolerance values for all variables were found to be near zero, and the variance inflation factors (VIFs) were all less than 15 (as shown in Table 5). These results suggest that multicollinearity does not pose a significant issue for this model.

Thus, an optimal linear model can be formulated. By examining the descriptive statistics (Table 4) and identifying the bounds for each variable, scrap generation can be minimized. To achieve this, the maximum values of variables with positive coefficients and the minimum values of those with negative coefficients should be used, thereby minimizing waste and eliminating the need for recycling.

For additional analysis, the data were standardized to account for the company’s focus on sustainable production. Linear regression models were applied to the nine most common profiles in the production dataset, as well as to the nine profiles with an average scrap weight above 20 kg and production frequencies greater than 100 (see

Table 6. Standardized linear regressions for the most frequent profiles.

	A	B	C	D	E	F	G	H	I
(Constant)	−34.924	−1.5	−25.761	−8.202	−24.239	−51.198	−40.935	3.766	−8.292
$E\text{\_}{t}_D$						0.002		0.01
$PO\text{\_}{S}_p$	0.114	0.671	−0.013	−0.267	−0.471	−0.094	0.305	−0.007
$PO\text{\_}SS$	−0.071			0.022	0.089		0.031
$PO\text{\_}B/P$		0.187	0.313			−0.003
$PR\text{\_}{T}_{{Z1}_{,Set}}$					0.091
$PO\text{\_}LB$	−1.546	−0.721	−2.38	−1.309	−2.191	−2.51	−1.414	−1.439	−1.314
$E\text{\_}{L}_{Butt}$	−0.077	0.038			0.042		0.026	0.007	−0.005
$E\text{\_}SW$	−0.058	−2.163	−42.973	−10.639	−5.097	0.06	0.201		−14.299
$E\text{\_}t$		0.228		−0.236	−0.251	−0.047	0.061
$E\text{\_}NH$	10.07	1.098
$PO\text{\_}NB$	−52.309	−1.537	−2.603	−12.56	−34.033	−63.315	−50.194	−2.756	−10.024
$E\text{\_}{T}_{CP}$						−0.002
$E\text{\_}{p}_{Max}$	0.046				−0.066	−0.007	0.013	0.038
$PO\text{\_}{Tr}_{p,DX}$	−0.038	0.067		−0.059	0.097			0.016
$E\text{\_}tS$	−0.016	0.027	0.007	0.023				−0.01	−0.019
$E\text{\_}LB$	1.142		0.932	0.875	1.099	0.896	0.798	0.828	0.925
$E\text{\_}SC$		−0.251		−0.018	−0.076	−0.006
$PR\text{\_}NB$	−0.017	−0.049	−0.016	−0.013			−0.01	−0.006
$E\text{\_}{T}_{End}$		0.023	0.031		−0.147
$PO\text{\_}LP$	−0.064	0.024		−0.033	0.035
$E\text{\_}{PR}_{T_{BC}}$	−0.006	−0.012				0.01
$E\text{\_}{p}_{SL}$				−0.016	0.075		−0.02		−0.004
Frequency	2596	1331	1194	980	906	894	842	723	697
Scrap (KG)	11.415	9.627	6.801	8.159	10.594	11.294	10.461	8.485	8.81
$R_a^2$	0.804	0.606	0.96	0.966	0.925	0.996	0.962	0.998	0.986

and

Table 7. Standardized linear regressions of profiles with higher scrap rates.

	J	K	L	M	N	O	P	Q	R
(Constant)	−45.593	1.274	−49.869	−28.252	3.308	1.952	1.621	3.083	−2.039
$E\text{\_}{t}_D$									−0.028
$PO\text{\_}{S}_p$		0.102	−2.42			−0.121	−0.109
$PO\text{\_}SS$	−0.011			−0.102			0.346
$PO\text{\_}B/P$
$PR\text{\_}{T}_{{Z1}_{,Set}}$			0.648						0.036
$PO\text{\_}LB$	−1.254					−1.14	−0.56	−1.672	−0.312
$E\text{\_}{L}_{Butt}$	−0.013	0.018	−2.180	0.012	0.093	0.032	0.015		0.054
$E\text{\_}SW$	−71.739	−0.613	−3.702	−45.268	−1.191	−2.732	0.578
$E\text{\_}t$	−0.022	0.051	−1.472		0.05				0.098
$E\text{\_}NH$	−0.367								1.716
$PO\text{\_}NB$	−1.531	−5.427	−72.07	−0.669	−0.679	−1.363	−2.193	−2.312	−0.234
$E\text{\_}{T}_{CP}$			0.026	−0.041	0.021				0.035
$E\text{\_}{p}_{Max}$	−0.015					−0.068		0.009	0.078
$PO\text{\_}{Tr}_{p,DX}$				−0.033		0.033		0.016
$E\text{\_}tS$		0.039							0.07
$E\text{\_}LB$	1.015	0.526	1.152	0.654	0.512	0.942	0.728	0.86	0.096
$E\text{\_}SC$	0.033		−0.047			0.081
$PR\text{\_}NB$					0.041		−0.038
$E\text{\_}{T}_{End}$	−0.071	−0.093	0.063	0.043		0.246		−0.037
$PO\text{\_}LP$					0.876	−0.074	−0.417
$E\text{\_}{PR}_{T_{BC}}$	−0.017			0.044
$E\text{\_}{p}_{SL}$	0.021			0.004	0.037		0.02
Frequency	154	182	227	188	141	301	274	135	107
Scrap (KG)	43.55	43.518	43.236	41.574	35.273	25.476	23.291	22.147	20.4
$R_a^2$	0.986	0.983	0.993	0.951	0.975	0.995	0.999	1	0.999

). The regression analysis supports the expectation that frequently produced profiles tend to generate less scrap, while less frequently produced profiles result in higher scrap levels. This indicates a more stable production process for the higher frequency profiles.

The scrap quantity models ($S{C}_{KG}$: scrap (kg), dependent variable) for the 18 selected profiles, consisting of the 9 most frequent profiles and 9 profiles with more than 20 kg of scrap, are presented in Table 6 and Table 7.

The results from these tables indicate that the die and the number of holes in the die ($E_{NH}$) show minimal variation. In other words, these variables do not contribute significantly to the variability in scrap production. However, it is well established in the literature that die geometry and the materials used in the die are key factors in determining the amount of scrap produced in the extrusion process [6,15,23,24]. For instance, ref. [15] emphasized that the geometry must be specifically tailored to each type of alloy. According to [23], die geometry is one of the primary causes of defects in the extruded profiles. Furthermore, ref. [24] outlined guidelines for designing die matrices, which are crucial for producing high-quality profiles. Ref. [6] also argued that defects in dies and tooling are the leading causes of profile defects and scrap generation. The company has implemented various quality control systems [50,51], including a three-dimensional analysis of aluminum profiles, to address these issues.

The coefficients in the linear models, shown in Table 6 and Table 7, help identify which variables have the most influence on scrap production. Any missing coefficients correspond to variables that do not significantly contribute to the model.

The analysis of the most frequently produced profiles, as shown in Table 6, indicates that for these profiles, linear models were developed where geometric variables consistently presented significant coefficients, independently of the die matrices used. These findings highlight the importance of geometric factors in influencing scrap production. It can be concluded that, in addition to the die geometry and tools previously mentioned [6,15,23,24], other geometric aspects also play a substantial role in scrap generation. Notably, the variable “Length Bars” ($P{O}_{LB}$) consistently showed a negative contribution to scrap production across all frequently used dies. This suggests that reducing the length of the bars leads to an increase in scrap production. Similar patterns were observed for the “Number of Bars” ($P{O}_{NB}$) and “Number of Billets” ($P{R}_{NB}$). On the contrary, the “Billet Length” ($E_{L_B}$) variable showed a positive contribution, meaning that an increase in billet length leads to higher scrap generation.

In certain profiles, the “Butt Length” ($E_{L_{Butt}}$) and “Specific Weight” ($E_{SW}$) variables contributed positively to scrap production, as indicated by their positive coefficients. However, in other profiles, the contribution of these variables was negative. Due to the limited number of studies in the literature on billet length, a detailed analysis is challenging. Nevertheless, further investigation is warranted. The unexpected behavior observed could be attributed to the alloy type used [15] or the specific billet size calculated based on customer requirements.

As for temperature, the four variables analyzed—post-container temperature ($E_{T_{CP}}$), max end temperature ($E_{T_{End}}$), set temperature Z1 ($P{R}_{T_{{Z1}_{,Set}}}$), and billet conical temperature ($P{R}_{T_{BC}}$)—showed almost no impact on scrap production. Similar results were found for time, speed, and pressure. While the data did not indicate a clear influence of these parameters on scrap production for the profiles studied, it cannot be concluded that temperature, speed, and pressure do not affect scrap generation. Previous studies, as summarized in Table 1, clearly document that increasing temperature and pressure typically lead to more scrap, while time and speed are closely linked to these variables. However, all dependent variables are also influenced by the alloy type used to produce a specific profile [15].

The company’s empirical knowledge of how temperature, pressure, time, and speed affect scrap generation, based on experience, has led to heightened attention and greater control over these factors during production.

Table 7 presents the linear regression models developed for the profiles associated with the highest levels of scrap generation in the company’s extrusion operations. For each of these profiles, statistically significant models were obtained. As observed previously in Table 6, the geometric variables, namely butt length ($E_{L_{Butt}}$) and specific weight ($E_{SW}$), consistently exhibit a significant effect on scrap production, reinforcing their relevance in the process.

In particular, the variable length bars ($P{O}_{LB}$) demonstrate a negative association with scrap levels, indicating that shorter bar lengths are generally linked to increased waste. Similar trends are observed for the specific weight ($E_{SW}$) and number of bars ($P{O}_{NB}$), further substantiating the role of geometric characteristics. Notably, these geometric factors remain significant across all analyzed profiles, regardless of the die used, which highlights their widespread influence on extrusion outcomes.

Conversely, core process parameters such as extrusion time, speed, and pressure do not exhibit a statistically significant relationship with scrap generation for the profiles under consideration. While these findings do not negate the potential influence of these variables, they suggest that their impact may be less pronounced or more profile-dependent under current operating conditions.

In conclusion, the findings underscore the importance of geometric variables in minimizing scrap generation. Nonetheless, direct control over these parameters may be limited due to their dependence on customer specifications. In this context, enhanced oversight in die design, inspection, and correction procedures could represent a feasible strategy for process improvement. Moreover, the observed significance of geometric factors, despite limited attention in the existing literature, provides a valuable opportunity for further investigation and development in the context of sustainable aluminum extrusion.

5. Conclusions, Limitations, and Future Work

This work analyzed the aluminum extrusion process of a Portuguese company, aiming to minimize scrap generation through a multiple linear regression model based on real industrial data.

The findings highlight a strong interdependence among the various extrusion process parameters—including temperature, time, speed, pressure, and geometric characteristics—demonstrating their importance in reducing and managing scrap production. Among these, the geometric variables, notably the number of die holes, billet length, and specific weight, emerged as the most influential contributors to waste generation. Despite their relevance, geometric parameters are inherently challenging to manage due to their dependence on customer-specific product requirements. Nonetheless, the company may enhance control through improved production practices and more rigorous inspection and maintenance of extrusion dies. These insights underscore critical areas for process enhancement. Notably, the role of geometric factors, particularly those linked to die design and refurbishment, remains under-explored in the existing literature, indicating a valuable direction for future research and industrial practice. These findings suggest that while operational variables are well controlled by the company, further improvements may be achieved through better monitoring and management of geometric factors, whenever possible. It is important to note that the results may differ across companies, influenced by factors such as production scale, the diversity of profiles manufactured and sold, and the level of process knowledge and control over variables. As a direction for future research, one promising avenue is the application of nonlinear modeling techniques to better capture complex relationships among process variables. Additionally, a more detailed investigation into the die geometry is recommended, given its prominent role in scrap generation as evidenced in the current analysis. The adoption of machine learning methods also presents a valuable opportunity to enhance predictive capabilities and identify patterns not easily captured by traditional statistical approaches.

This work has the limitation of using a very simplified model of the real problem. Another limitation is that the empirical study was conducted with data from a single source, specifically in 2019. It is recommended that the model be applied to multiple databases to enable a more comprehensive evaluation through comparative analysis of the results. Some parameters that influence scrap generation were not considered in the present study. Unfortunately, the available database did not include detailed information regarding, for example, the specific composition, alloy grade, metallurgical status of the billets used in the extrusion process, mold aging and wear, and mold maintenance cycle, among others. Nevertheless, it is important to emphasize that the variables analyzed (temperature, time, speed, pressure, and geometry) are parameters monitored and controlled daily by the company, regardless of the alloy, given the high standardization of their production processes. Additionally, the influence of these operational variables on scrap generation was validated both through their statistical significance in the models and their alignment with findings from previous literature. Including alloy composition as a variable would further enrich the model and provide a more complete understanding of scrap variability. This suggestion represents a potential direction for future work, aiming to integrate metallurgical data whenever possible.

Several innovative points stand out. The modeling approach considered is simple but easily adaptable to similar processes or companies, including those in other sectors. It is easy to replicate the methodology with data from other companies since the main factors associated with scrap generation in the extrusion process are considered. The application of the suggested methodology does not increase company costs, as it can be easily implemented using an open-source tool, similar to what was done in this work, and may even contribute to reducing costs related to raw materials, energy, and time.

Thus, this work also serves as an incentive to focus on the sustainability and social responsibility of companies. It has enabled the optimization of the production process by identifying variables that need performance improvement, leading to greater profit, as well as a better image and environmental reputation for the company.