Reducing Electricity and Water Consumption in Textile Dyeing Industries

Abstract

The high electricity and water consumption in industrial textile dyeing processes represents an environmental and economic challenge, requiring optimization strategies to reduce costs and impacts toward cleaner production. This work proposes an optimization model to minimize costs associated with water and electricity consumption in industrial textile dyeing processes. The model has a Mixed Integer Linear Programming (MILP) formulation. The objective function to be minimized is the total process costs. The constraints consider production capacity, daily production limits, and specific costs per material. A case study was conducted in a real industrial process for three types of tissue: cotton, polyester, and polyamide. The model was coded in GAMS and the CPLEX solver was used to solve the problem. The results showed that water consumption accounted for 78.2% of the total cost in the optimal solution. Using the same model, an alternative simulation was performed, replacing four smaller-capacity machines with a single larger-capacity machine, resulting in a marginal reduction in total costs. Simulations were also performed to replace the current machines with highly efficient automated HT (High Temperature) machines, indicating a potential 71.39% reduction in water consumption costs. The conclusion is that the proposed model is effective for optimizing textile dyeing processes, balancing operational efficiency and sustainability, and is applicable in complex industrial scenarios.

1. Introduction

Drinking water is essential for sustaining life. Adequate water availability for drinking, sanitation, and industrial use is one of the most significant challenges of the 21st century. The rapid increase in the global population and industrialization puts significant pressure on society regarding the use of natural resources and poses enormous challenges to our ecosystem. This scenario underscores the urgency of adopting sustainable practices and effective measures to protect water resources from contamination and overexploitation, particularly in the industrial and agricultural sectors.

In addition to water scarcity in various parts of the world, industrial processes in general have been facing several problems, including declining availability of raw materials, large volumes of effluents, rising labor costs, environmental legislation governing effluents, and growing consumer awareness of environmental issues. These factors are forcing industries to review, restructure, and reduce their environmental impact [1]. Cleaner production approaches aim to avoid excessive resource consumption and the generation of pollutants resulting from inefficiency in production processes. Such approaches optimize production by considering environmental aspects, promoting sustainability, and reducing waste [2].

In industrial processes, environmental impacts and effluent generation can be significant, compromising environmental quality and human health. Furthermore, industries face an increasingly dynamic and competitive market, and those who can stay ahead of the curve in these areas will certainly secure a competitive advantage [3].

As one of the most traditional industries in the world, the textile industry is a significant market, primarily because of the various production links in its supply chain. The global textile market continues to grow, with both production volumes and foreign trade expanding. In 2022, global textile fiber consumption exceeded 120 million tons, highlighting the sector’s significant environmental impacts on climate change, energy consumption, land use, and water demand [4,5]. This sector contributes significantly to the pollution of more than a quarter of the planet’s water bodies [6].

Brazil is one of the world’s largest producers and consumers of textiles, producing approximately 2.1 million tons in 2022. In the finishing segment alone, 407,052 tons of tissues were finished [4].

With the increase in textile production over the past few decades, the need for modern, efficient designs has become evident. Lean manufacturing principles, such as the pursuit of quality from the outset, waste minimization, continuous improvement, and flexibility, have been applied to address the sector’s challenges [7]. However, processes such as textile dyeing continue to pose significant challenges due to high water and electricity consumption, as well as the generation of complex effluents that hinder adequate treatment [8]. The need for more sustainable dyeing has driven water conservation efforts across various segments of the textile industry, including wastewater treatment, recycling and reuse for industrial purposes, machine, process, and chemical innovations, advanced water analysis, and water-saving tools [1].

Chemical replacement and reduction practices in textile factories have shown promise for increasing production efficiency, meeting regulatory standards, minimizing environmental impacts, and protecting the health of workers and consumers [9]. Numerous techniques have also been adopted to treat textile dyeing wastewater, including physicochemical, biochemical, and biological treatments, combined treatment processes, and other technologies. Conventional biological treatment processes are commonly time-consuming, require large operational footprints, and are ineffective for effluents containing toxic elements. Advanced oxidation techniques entail high treatment costs and are typically used to produce high-purity water. Chemical coagulation is slow and generates large amounts of sludge [10].

One alternative to reducing water and electricity consumption and, consequently, the generation of industrial wastewater at the source is the application of optimization techniques. This alternative is an important and strategic step toward achieving sustainable development, particularly with potential environmental, economic, and social benefits [11]. According to Klemeš et al. [12], adopting these measures in industry is estimated to reduce consumption by 25–30% relative to current levels.

Minimizing the use of natural resources, such as water and energy, has gained prominence in modern industries, driven by the need to reduce costs and promote environmental sustainability. Batch sizing, a production planning and control technique, has been widely studied as a practical approach to minimizing resource consumption in industrial processes. According to Wagner and Whitin [13], batch sizing determines the optimal size of production batches, considering operational and environmental constraints, resulting in reduced waste and increased overall process efficiency. This technique is particularly relevant in sectors such as chemicals and manufacturing, where intensive water and energy use can be significantly optimized through appropriate planning strategies.

Recent studies have advanced the application of optimization techniques to textile dyeing. For example, Gao et al. [14] proposed a genetic algorithm (GA) to optimize dyeing scheduling in multi-machine systems, maximizing profits with constraints on water, machine capacity, and deadlines. The results demonstrated efficiency, validating the balance between productivity and sustainability. However, the GA presented scalability limitations for large order volumes, suggesting the use of reinforcement learning as an alternative.

Another significant advance was the mixed-integer programming (MIP) model proposed by Sung and Maravelias [15] for the general capacitated lot-sizing problem (GCLSP). This model relaxed traditional assumptions, allowing greater flexibility in production planning and demonstrating effectiveness in reducing costs and improving resource utilization. Villas Boas et al. [16] also proposed an MIP model to solve the integrated lot-sizing and production scheduling problem in complex industrial contexts. Their approach, based on heuristics such as Relax-and-Fix and Fix-and-Optimize, proved viable for large-scale problems, balancing solution quality and computational time.

The textile industry stands out among the most important industrial sectors globally [17]. However, its operations face significant challenges, such as high resource consumption, low operational efficiency, and pollutant emissions [18]. In particular, production processes require substantial water and energy [19], underscoring the urgent need to adopt sustainable, innovative practices.

Given this scenario, this work proposes an optimization-at-the-source approach to address a critical gap in the literature. While previous studies have applied heuristic methods that do not guarantee optimal solutions or have focused on more general lot-sizing problems, the novelty of this research lies in the formulation and application of a Mixed-Integer Linear Programming (MILP) model that robustly integrates batch sizing and production scheduling decisions, specifically for the textile dyeing process. The main advantage of this approach is its ability to find the mathematically optimal solution that simultaneously minimizes both water and electricity costs. The model serves as a practical management tool that not only optimizes daily operations but also enables the simulation of strategic investment scenarios in a complex industrial environment.

2. Materials and Methods

In this work, a quantitative production planning model was developed, formulated as a batch-sizing and sequencing optimization problem for the textile industry’s knitted fabric dyeing stage. The dyeing stage is critical, given its high water and electricity consumption. Dyeing takes place in machines called HT, which facilitate interaction between the textile meshes and the chemicals used, whose activation is mediated by mechanical and thermal processes. Another factor considered in the model was the composition of the tissues, as it affects processing times and, in turn, causes fluctuations in electricity and water consumption. Cotton, polyester, and polyamide are the most dominant compositions in the industrial and commercial markets and, therefore, were the subjects of this analysis. The model’s constraints consider initial inventory control and product availability for processing, maximum and minimum daily production limits for each product type, the maximum and minimum capacities of each machine, compatibility and balance between the quantities processed on the HT machines, restrictions to ensure that only one process is selected per machine and per cycle, and the calculation of costs for each cycle and process. These constraints ensure that quantities and availability are consistent with the system’s needs, while the binary variables represent allocation decisions. Other constraints, such as cost and quantity constraints, are essential to ensure the system’s economic and operational viability.

The indices, sets, parameters, continuous, and binary variables used in the proposed mathematical model are:

Indices
M	Pieces in kg (m = 1,…,M)
i	Product type (i = 1,…,I)
j	HT dyeing machines (j = 1,…,J)
C	Planning cycles (c = 1,…,C)
Sets
m ∈ M	Set of all tissues
i ∈ I	Set of all products
j ∈ J	Set of all HT dyeing machines
c ∈ C	Set of all cycles
Parameters
${ProdFcast}_m$	Production forecast for mesh m
${ElecCost}_{ji}$	Electricity cost to process product i on machine j
${WaterCost}_{ji}$	Water cost to process product i on machine j
${MaxCap}_{ji}$	Maximum capacity to process product i on machine j
${MinCap}_{ji}$	Minimum capacity to process product i on machine j
${MaxDaily}_i$	Maximum daily production limit for product i
${MinDaily}_i$	Minimum daily production limit for product i
${BatchTime}_i$	Processing time (in hours) per batch of product i
Continuous variables
$TotalCost$	Total dyeing cost (objective variable)
$x_{ijc}$	mass (kg) of product type i processed on machine j in cycle c
${AvailMesh}_{mc}$	mass (kg) of mesh m available at the beginning of cycle c
${ProcMesh}_{mc}$	mass (kg) of mesh m actually processed in cycle c
${CycleQty}_{ic}$	Total mass of product type i processed in cycle c
${TotProdQty}_i$	Total mass of product type i processed in all cycles
${TotMachCost}_j$	Total cost related to machine j in all cycles
$TotalElecCost$	Total cost of electricity
$TotalWaterCost$	Total cost of water
${NumBatches}_i$	Number of batches required to process product i
${TotalTime}_i$	Total time (h) required to process product i
Binary variable
$y_{ijc}$	It is equal to 1 if product i is processed on machine j in cycle c, and equal to 0 otherwise

The objective function to be minimized relates to the costs associated with the consumption of electricity and water during the dyeing process:

$$TotalCost= \sum _{i=1}^I\sum _{j=1}^J\sum _{c=1}^C{ElecCost}_{ji}·x_{ijc}+\sum _{i=1}^I\sum _{j=1}^J\sum _{c=1}^C{WaterCost}_{ji}·x_{ijc}$$

(1)

The assignment of start, progression, and end points for various processes depends on process constraints. Constraints (2a) and (2b) define the inventory balance for each loop m over cycles c. Equation (2a) establishes the initial boundary condition, in which the production forecast parametrically defines material availability (AvailMesh) for the first period. Equation (2b) in turn specifies the recursive relationship for subsequent periods. In each cycle, the available inventory is the remaining inventory from the previous cycle, connecting production decisions and ensuring that consumption does not exceed inventory.

$${AvailMesh}_{m,1}={ProdFcast}_m m=1,\dots M$$

(2a)

$${AvailMesh}_{mc}={AvailMesh}_{m,c-1}-{ProcMesh}_{m,c-1} m=1,\dots M; c=1,\dots C; c>1$$

(2b)

Constraint (3) ensures that the quantity of mesh to be processed does not exceed the mass available in stock.

$${ProcMesh}_{mc}\le {AvailMesh}_{mc} m=1,\dots M; c=1,\dots C$$

(3)

Constraint (4) imposes the balance of the material flow within each cycle c. This equality ensures that the total quantity of material entering the production area in a cycle is precisely equal to the sum of the quantities processed by all machines in that same cycle.

$${\sum }_{m=1}^M{ProcMesh}_{mc}={\sum }_{i=1}^I{\sum }_{j=1}^J{x}_{ijc} c=1,\dots C$$

(4)

Constraint (5) establishes a global consumption condition for the entire planning horizon. It requires the model to ensure that the total cumulative production across all machines and all cycles is exactly equal to the total inventory that entered the system through the initial forecast.

$$\sum _{i=1}^I\sum _{j=1}^J\sum _{c=1}^C{x}_{ijc}=\sum _{m=1}^M{ProdFcast}_m$$

(5)

Constraints (6) and (7) impose maximum and minimum daily production limits for each type of product.

$${\sum }_{j=1}^J{\sum }_{c=1}^C{x}_{ijc}\le {MaxDaily}_i i=1,\dots ,I$$

(6)

$${\sum }_{j=1}^J{\sum }_{c=1}^C{x}_{ijc}\ge {MinDaily}_i i=1,\dots ,I$$

(7)

Constraints (8) and (9) ensure that, if a machine is used, the quantity processed respects its capacity limits. If the machine is not used, production must be zero.

$$x_{ijc}\le {MaxCap}_{ji}·y_{ijc} i=1,\dots ,I;j=1,\dots J; c=1,\dots C$$

(8)

$$x_{ijc}\ge {MinCap}_{ji}·y_{ijc} i=1,\dots ,I;j=1,\dots J; c=1,\dots C$$

(9)

Constraint (10) imposes that only one type of product i is processed on each of the machines j. Thus, the constraint ensures that mixed production with more than one product type per machine is not considered.

$${\sum }_{i=1}^I{y}_{ijc}=1 y_{ijc}\in \left\{\mathrm{0,1}\right\};$$

(10)

Constraints (11) and (12) are definition equations that calculate the total mass processed by product type in each cycle and in the total planning horizon, respectively.

$${CycleQty}_{ic}={\sum }_{j=1}^J{x}_{ijc} i=1,\dots ,I; c=1,\dots C$$

(11)

$${TotProdQty}_i={\sum }_{c=1}^C{CycleQty}_{ic} i=1,\dots ,I$$

(12)

Equation (13) calculates the cost associated with each dyeing machine over the entire horizon.

$${TotMachCost}_j={\sum }_{i=1}^I{\sum }_{c=1}^C{ElecCost}_{ji}·x_{ijc}+{\sum }_{i=1}^I{\sum }_{c=1}^C{WaterCost}_{ji}·x_{ijc} j=1,\dots J$$

(13)

Equation (14) establishes the cost related to electricity consumption.

$$TotalElecCost=\sum _{i=1}^I\sum _{j=1}^J\sum _{c=1}^C{ElecCost}_{ji}·x_{ijc}$$

(14)

Equation (15) stipulates the total cost associated with water use.

$$TotalWaterCost=\sum _{i=1}^I\sum _{j=1}^J\sum _{c=1}^C{WaterCost}_{ji}·x_{ijc}$$

(15)

Equation (16) calculates the number of batches required to process the total quantity of a product type, and Equation (17) calculates the total time required to process these batches, rounding the number of batches to the nearest higher integer.

$${NumBatches}_i={\displaystyle \frac{{TotProdQty}_i}{\sum _{j=1}^J{MaxCap}_{ji}}} i=1,\dots ,I$$

(16)

$${TotalTime}_i={NumBatches}_i·{BatchTime}_i i=1,\dots ,I$$

(17)

The subsequent expressions establish the non-negativity conditions for all continuous variables in the model.

$${AvailMesh}_{mc}\ge 0;$$

(18)

$${ProcMesh}_{mc}\ge 0;$$

(19)

$$x_{ijc}\ge 0;$$

(20)

$${CycleQty}_{ic}\ge 0;$$

(21)

$${TotProdQty}_i\ge 0;$$

(22)

$${TotMachCost}_j\ge 0;$$

(23)

$$TotalElecCost\ge 0;$$

(24)

$$TotalWaterCost\ge 0$$

(25)

$${NumBatches}_i\ge 0$$

(26)

$${TotalTime}_i\ge 0$$

(27)

$$TotalCost\ge 0$$

(28)

Thus, the optimization problem consists of minimizing the objective function given by Equation (1) subject to constraints (2) to (28) and presents an MILP formulation. Batch sizing is addressed in the model by defining the optimal batch size, the number of batches required to process each product type, and the processing time for each batch. Production scheduling can be identified in the allocation of resources (how products will be allocated to the HT machines) throughout the production cycles, in the sequencing of tasks (considering the availability of products for processing in each cycle and defining the quantity of mesh to be processed on each machine), and in the capacity constraints, all of which minimize total production costs. This integration between batch sizing and production scheduling is essential to optimize production efficiency in complex environments, such as the textile dyeing plant described in the model.

3. Case Study

In this case study, the optimization model aims to minimize the total production cost, which is composed of electricity and water costs. It is essential to understand the parameters that influence each of these costs. The electricity cost is directly affected by the specific power of each HT machine, the processing time required for each fabric type, and the energy cost. The key optimization decision here is the allocation of production (the variable x_ijc) to the machines with the highest energy efficiency. On the other hand, water costs depend on the specific consumption (L/kg) of each product and the water price. Although this cost does not vary among machines, it is a decisive factor in optimizing the production mix to meet daily targets at the lowest total cost. Therefore, the model’s function is to determine an optimal production schedule by deciding which machines to use for these products, and in what quantities throughout the planning cycles, effectively balancing these competing cost factors.

The industrial-scale study presented in this article was based on a Brazilian textile dyeing unit in the northwestern region of Paraná, which is responsible for most of the region’s knitwear dyeing. The model’s input parameters were collected based on the unit’s industrial operations, focusing specifically on the dyeing process on HT machines, supplemented by technical information from machine suppliers and literature data.

The study considered 16 HT machines and a planning horizon of 24 operating cycles, corresponding to one production day. The objective function computes the total cost over the entire horizon. Three types of raw materials were analyzed: cotton, polyester, and polyamide, covering the primary fibers processed by the industry under study. Daily production targets, in the form of minimum and maximum processing limits, were established for each raw material type, as detailed in

Table 1. Daily production limits per raw material type.

Material Type	Minimum Daily Production (kg)	Maximum Daily Production (kg)
Cotton	35,000	46,000
Polyester	22,000	35,500
Polyamide	20,000	33,000

. The daily production target was estimated at 109,000 kg of mesh.

It is important to note that the model does not impose a predefined daily operating time limit per machine. Instead, the operating time for each machine is an output of the optimization, resulting from the production allocation required to meet the daily targets most economically within the available 24 cycles.

Regarding water consumption, there are specific demands for each type of raw material. Processing 1 kg of cotton mesh requires 90 L of water, while polyester and polyamide require 102 L and 74 L, respectively. In Brazil, a criterion adopted by SABESP [20] attributes a cost of R$13.79/m³. The values per kg processed are equivalent to R$1.24 (cotton), R$1.41 (polyester), and R$1.02 (polyamide). To determine electricity costs during the batch dyeing process on HT machines, processing times of 8 h for cotton, 5 h for polyester, and 6 h for polyamide were assumed. The kWh cost for the industry was obtained from the National Electric Energy Agency, considering an average value of R$684.77 per MWh [21]. For the power (kW) of each machine, the Dilmenler Makine Sanayi HT machine manufacturer catalog was used as a reference, based on the selected capacity.

Regardless of the type of product to be processed, the machines will be the same. Regarding the production volume of the HT dyeing machines, equipment with varying capacities was selected to achieve more efficient batch sizing and sequencing.

Table 2. Capacity and power of HT machines.

Machine	Mass (kg)		Power (kW)
	Minimum	Maximum
HT1–HT2	20	25	6.84
HT3–HT6	120	150	12.45
HT7–HT11	240	300	22.78
HT12–HT15	360	450	32.77
HT16	480	600	42.36

presents the maximum and minimum production capacities of the HT machines for processing product i, along with their respective power outputs (kW).

The costs associated with the consumption of electricity for processing product i on machine j are presented in

Table 3. Electricity cost per kilogram of mesh to process i on machine j.

Machine	Cotton	Polyester	Polyamide
HT1–HT2	R$1.50	R$0.94	R$1.12
HT3–HT6	R$0.46	R$0.28	R$0.34
HT7–HT11	R$0.42	R$0.26	R$0.31
HT12–HT15	R$0.40	R$0.25	R$0.30
HT16	R$0.39	R$0.24	R$0.29

. It is important to note that, unlike electricity costs, water costs do not vary by machine but rather by fabric type. As previously mentioned, the values are R$1.24/kg for cotton, R$1.41/kg for polyester, and R$1.02/kg for polyamide.

The configuration of this case study establishes a solid basis for the validation of the optimization model. The quantitative results and their respective implications will be discussed in detail in the following section.

4. Results and Discussion

The application of the optimization model to the industrial case study resulted in an optimal solution, the details of which are presented in this section. The analysis begins with the quantitative results, including the minimized total cost and the production distribution, and then proceeds to a broader discussion of the methodological choices and the contribution of this work in the context of the existing literature.

The optimization model was coded and solved in GAMS v49. As the problem has an MILP formulation, the CPLEX optimization solver v22.1.1 was used for this specific case. The program was executed on a Windows 11, Acer computer (Acer Inc., New Taipei City, Taiwan) with an 11th Gen Intel^® Core™ i5-1135G7 CPU @ 2.40/2.42 GHz, and 8 GB RAM. The optimal integer solution found for the problem has an objective function value of R$170,114.44. No branch-and-bound nodes were required, indicating that the entire solution was found directly from linear relaxation.

Optimality gap analysis revealed that the solution found is very close to the theoretical lower bound (best possible value), with an absolute gap of R$1.38 and a relative gap of 0.0008% of the best possible value. Since the relative gap is below 0.1%, it can be concluded that the CPLEX solver demonstrated effectiveness in solving the problem, providing an optimal solution within the established tolerance and with minimal computational time. The results meet the defined tolerances and validate the application of the proposed model to the scenario under study.

Regarding the results analysis,

Table 4. Distribution of production by fabric type.

Tissue	Total Mass (kg)	% of the Production	Number of Lots	Total Time (h)
Cotton	45,900	42.1%	11	88
Polyester	30,100	27.6%	7	35
Polyamide	33,000	30.3%	8	48
Total	109,000	100%	26	171

shows the production distribution by tissue type, indicating the best batch sizing and sequencing across 24 work cycles. The number of cycles was determined based on the expected production and the total capacity of the dyeing machines. In the table, one can also see the number of rounded batches and the total processing time.

The total mass (kg) processed for each raw material remained within the daily limits, complying with one of the modeling constraints. Figure 1 illustrates the production hierarchy of the dyeing machines, highlighting the disparity in volumes processed per unit over the 24 operating cycles. The graphical representation of accumulated production by machine clearly highlights the production hierarchy. Machine HT16 was the most productive unit in the industrial park, with a constant capacity of 600 kg per cycle, demonstrating high operational reliability and contributing significantly to the total production volume. Machines HT1 and HT2 were the only ones that showed reduced production, both with capacities of 20 to 25 kg per cycle.

Figure 1 shows the consolidated optimal production schedule, whose resulting strategy is clear: prioritize the larger-capacity machines (HT12-HT16), especially HT16, for cotton processing. This decision is a direct consequence of its lower energy cost per kilogram (Table 3). Concurrently, the schedule allocates the medium-sized machines (HT7-HT11) mainly to polyamide, and the smaller-capacity machines (HT1-HT6) to polyester. This distribution reflects an optimal balance: while the larger machines maximize energy efficiency, the medium-sized ones are better suited for the specific conditions of polyamide, and the smaller ones become advantageous for polyester due to its shorter processing time. Therefore, the analysis validates the model as a practical scheduling tool that balances energy efficiency, process suitability, and cycle time to achieve the lowest overall cost.

This batch distribution suggests higher costs for polyamide and polyester meshes, as smaller machine capacity is associated with higher electricity costs. However, polyamide offered economic advantages due to its shorter dyeing time and lower water consumption. Regarding the processing costs per kg of mesh per machine, Figure 2 shows the results achieved.

The HT16 machine was expected to have lower processing costs for cotton, since machines with larger capacities consume less electricity. However, dyeing cotton requires a significant amount of water—approximately 90 L/kg—and a longer dyeing time of 8 h. On the other hand, the HT7 and HT11 machines, designed for polyamide, stand out as the most economical, likely because they consume less water (approximately 74 L/kg) and have a shorter dyeing time, despite operating at higher temperatures (130 °C), which implies greater thermal energy consumption. Both machines have a maximum capacity of 300 kg of fabric per production cycle.

The HT3-HT6 and HT9 machines, both designed for polyester, have similar costs, ranging from R$1.69 to R$1.67 per kg. Processing this fiber requires approximately 102 L of water per kg of fabric, with cycles approximately 5 h long, but at high temperatures (120–130 °C), as mentioned by Clark [22].

The HT1 and HT2 (polyester) machines are the least efficient, costing R$2.34 per kg of dyed fabric. Therefore, it is recommended to prioritize HT3 to HT6 machines, which, despite being designed for larger batches, are more economical than lower-capacity machines.

To reduce total production costs, based on previous results, a new simulation was performed, replacing the lower-efficiency machines (HT1, HT2, HT3, HT4, and HT9) with a single, higher-capacity machine (HT17, with 750 kg of processing capacity). The results demonstrated a marginal reduction in electricity costs, which now totaled R$35,672.70, while total costs reached R$168,654.35. Although this reduction is not statistically significant in terms of energy and water consumption, it is important to note that the replacement eliminated four machines from the production process. Furthermore, if indirect labor and maintenance costs were considered, the optimization could have a more significant impact, suggesting the need for additional analyses that integrate operational and logistical variables for a more comprehensive assessment.

A complementary cost-reduction strategy involved applying real operational data from modern equipment. Current machines, developed with technological innovations, are more efficient in resource use and significantly reduce production costs. Recent data from the machine manufacturer (Dilmenler Makine Sanayi, Istambul, Turquia) show that the average water consumption in textile dyeing is 35.5 L/kg for cotton, 20 L/kg for polyester, and 16 L/kg for polyamide, values substantially lower than those observed with conventional equipment.

In a new simulation comparing the scenario initially studied, an optimized objective function value of R$75,182.22 was obtained, distributed as R$37,132.79 for electricity consumption and R$38,049.43 for water use. This result represents a 71.39% reduction in water costs compared to previous parameters, demonstrating that theoretical data in the literature on water consumption per kg of fabric are outdated and do not reflect the technological advances recently implemented in the sector, which have led to machines consuming less water.

The choice of a Mixed-Integer Linear Programming (MILP) model for this research was a deliberate methodological decision, as its structure is uniquely suited to the problem’s inherently hybrid nature, which involves both continuous decisions (production volume) and discrete/binary decisions (machine allocation). Crucially, this approach guarantees the achievement of the mathematically optimal solution, a fundamental advantage over heuristic methods that cannot ensure global optimality. Furthermore, the MILP structure offers flexibility for strategic scenario analyses, as demonstrated by the machine replacement simulations in this study, elevating the model from a purely operational tool to a robust investment planning instrument.

To contextualize the contribution of this work, it is instructive to compare the proposed methodology with existing approaches in the literature, which are mainly divided into heuristic and exact methods. A significant line of research uses heuristic methods to optimize dyeing scheduling. For example, Gao et al. [14] employed a Genetic Algorithm (GA) to maximize profits. At the same time, Demir [23] proposed a multi-objective Genetic Algorithm (Lex-MOGA) specifically to handle large-scale problems in an acceptable computational time. Although such approaches are practical for finding high-quality solutions, their fundamental limitation is the inability to guarantee mathematical optimality.

The other line of research, to which this work is aligned, uses exact mathematical programming. Demir [23] himself also presents an Integer Programming-based approach (Lex-ILPA), acknowledging the value of methods that seek the optimal solution, as do more general lot-sizing studies such as Sung and Maravelias [15]. It is at this point that the advantage and novelty of our work stand out. While the mentioned models focus on objectives such as meeting deadlines, the number of setups, or generic lot problems, our formulation is the first, within an exact optimization framework, to integrate and quantify the specific economic-environmental trade-off between water and electricity costs in the dyeing process. This particular formulation, which addresses a central sustainability and cost challenge in the sector, is our main contribution, uniting the rigor of exact optimization with a practical, detailed application to provide a guaranteed minimum-cost schedule.

5. Conclusions

The main contribution of this work was the development and validation of an optimization model for textile dyeing that, unlike heuristic-based approaches, guarantees the achievement of the minimum-cost solution. The methodological novelty lies not in the MILP model itself, but in its specific and integrated formulation, which captures the complex trade-offs among water consumption, electricity costs, and machine efficiency. The applicability of this holistic model was tested through an industrial case study coded in GAMS and solved with the CPLEX solver; the results are detailed below.

The results demonstrated the effectiveness of the model for optimizing the textile dyeing process, highlighting the achievement of an optimal solution of R$170,114.44, solved in a reduced computational time (0.19 s), for a daily production of 109,000 kg of fabric. The analysis revealed that water consumption represents the main cost component (78.2% of the total), significantly exceeding electricity costs. Production distribution showed that larger-capacity machines, such as the HT16, performed better, while smaller-capacity machines (HT1-HT2) accounted for a marginal share of the total processed volume. The simulation with high-efficiency dyeing machines indicated a potential 71.39% reduction in water costs, highlighting the obsolescence of conventional theoretical parameters. Although replacing four machines with a single high-capacity machine (HT17) generated limited direct savings (0.86%), the strategy demonstrated significant indirect benefits, including reduced physical space, simplified production flow, and reduced labor. The results reinforce the importance of integrating the developed model with up-to-date operational data for decision-making in the industrial environment.

Furthermore, the identified trade-offs—between production capacity, processing time, and specific costs per material—suggest that optimizing batch sizing and prioritizing equipment can generate additional gains. The small optimality gap and reduced processing time validated the efficiency of the proposed model. As future perspectives, it is recommended to evaluate the economic feasibility of replacing current machines with more water-efficient alternatives, incorporate the setup times required for switching between different fabric types into the model, and apply the method to other industrial equipment with similar demands. The developed approach proved promising for optimizing textile processes and can be adapted to other industrial contexts with similar resource-allocation and cost-management challenges.