Optimal Scheduling of the Wine-Bottling Process: A Multi-Dependency Model with Hydraulic Considerations

Abstract

This study addresses the optimization of the bottling process in the wine industry, focusing particularly on the filtration stage, which accounts for approximately 30% of the total costs in the value chain. In the context of significant global production, but with challenges arising from declining demand and increasing costs, precise planning becomes necessary to ensure product quality and operational efficiency in order to control and minimize the costs associated with resource usage. The work proposes an operational approach based on a mixed-integer programming model that optimizes the sequencing of batches on the filtration and bottling lines, integrating setup times that depend on the processing order. An analysis of prevailing practices in the Chilean wine industry is conducted, complemented by a literature review that identifies knowledge gaps, particularly regarding the influence of the wine’s physicochemical properties and the potential for filter blockage. The results of the model allow for reduced processing times and improved batch allocation, highlighting the importance of considering hydraulic and operational variables for more precise planning. Furthermore, the study discusses the limitations of the current approach and suggests future directions to continue optimizing production processes in the wine industry.

1. Introduction

The International Organization of Vine and Wine (OIV) estimates that global wine production reached approximately 221 million hectoliters in 2023 [1]. The wine industry is a highly relevant economic sector, contributing significantly to employment, tourism, and exports in numerous countries [2,3]. Wine production, due to its specific nature, involves the consideration of multiple critical variables to ensure the quality of the final product [4]. In this context, effective planning and scheduling are essential for wineries, as they directly influence the ability to meet customer demand, maintain high-quality standards, and optimize the use of available resources. The bottling stage, in particular, represents approximately 30% of the total costs of the wine value chain [5]. Poor planning in this phase can lead to production-line disruptions, especially during the filtration stage, which significantly impacts the overall production process [6].

This study aims to develop and apply an operational model based on mixed-integer programming that optimizes the sequencing of batches on the filtration and bottling lines. The proposed methodology integrates sequence-dependent setup times with the objective of minimizing the total processing time for orders and reducing the costs associated with resource usage.

Although the literature has explored aspects related to bottling planning, the specific constraints of the filtration process have received less attention. In particular, the clogging index which measures the potential of particles present in the wine to obstruct the filtering surface is a determining factor that has not been sufficiently addressed [7]. This index is influenced by the mechanical properties of the particles, such as their degree of deformation and disintegration [8]. The accumulation of solids in the filters reduces the flow and increases the processing time for each batch. If not properly managed, the clogging index can reach a critical threshold, requiring the production line to be stopped to replace the filter, resulting in significant losses of time and product [9]. A robust setup plan could mitigate these risks and ensure the feasibility of the production schedule.

Currently, many wineries perform planning manually, which is often inefficient and leads to adjustments and rescheduling based on unforeseen circumstances, generating additional costs due to delays in fulfilling planned orders. Existing research mainly addresses these issues from a strategic or tactical perspective by formulating the simultaneous lot-sizing and scheduling (SLS) problem and examining its implications for inventory management and order fulfillment. However, these studies do not exhaustively consider the constraints derived from the filtration process [10].

To address the challenges within the wine industry and contribute to operational decision-making in daily bottling planning, this document is structured as follows: first, it presents an analysis of the predominant practices in the Chilean wine industry, based on a survey conducted with key companies in the sector. This analysis identifies the mechanisms and tools used in planning, as well as the operational considerations and critical variables involved in production processes. Next, a thorough literature review is provided to establish the state-of-the-art and highlight existing knowledge gaps. The third section details the problem and the formulation of the proposed model, followed by its practical application in a real operational context. Finally, the document concludes with a discussion of the results obtained, the limitations of the study, and potential future research directions that could further optimize these processes in the wine industry.

1.1. Industry Practices in the Chilean Wine Industry

Through the application of a questionnaire to production leaders and managers from 17 wine-producing companies in Chile during September and October 2024, complemented with additional plant visits and interviews, a mixed linear programming model that considers sequence-dependent setup times was established regarding industrial practices in the planning of the wine-bottling process in Chile. The wineries were selected based on the criteria that they had a bottling process and an annual production exceeding one million liters, and that they represent 48% of Chile’s total wine production.

1.1.1. Standard Format Characteristics

Currently, wineries have reduced the total number of SKUs, managing an average of between 100 and 200 products. The 750 mL bottle format is the most commonly used, representing approximately 80% of bottled wine. Other formats include capacities of 187.5 mL, 350 mL, 500 mL, 700 mL, 1000 mL, 1500 mL, 3000 mL, and 5000 mL. Bottles are classified into three main types: Bordeaux, Burgundy, and Special, each with variations in diameter, height, and weight. The types of closures used are mainly divided into crown caps and corks (natural or synthetic), chosen according to preservation requirements and market preferences. Regarding labeling, two main adhesion methods are used: glue-based and self-adhesive labels, selected according to the type of bottle and the product’s marketing approach.

1.1.2. Setup Times and Changeovers

In the bottling and packaging process, four types of technical adjustments are identified. The first adjustment focuses on the bottle type, taking into account volume, height, and diameter. The second adjustment relates to the type of closure, which may be a crown cap or cork. The third adjustment is linked to labeling machines, which must adapt to different types of labels and bottle formats. Finally, the fourth adjustment corresponds to the packaging system, organizing bottles into cases of 6, 12, or 24 units, according to commercial and logistical specifications.

Due to the modernization of equipment, the frequency of format changes or adjustments between different bottle types has significantly decreased in recent years. The use of modern machines with automatic adjustments and standardized inputs has enabled most companies to reduce the time required for complex adjustments to as little as 60 or 90 min. Additionally, many wineries have two bottling lines and two filtration lines that can operate simultaneously, achieving a bottling capacity ranging from 3000 to 4000 L per hour. These systems can process batches of between 10,000 and 25,000 L in shifts of approximately 9 h. However, the need to extend work shifts to meet bottling schedules has increased costs, while the reduction in working hours imposed by current legislation has added extra pressure to meet objectives within the available time.

1.1.3. Bottling-Process Planning

The planning of the bottling process is defined in weekly meetings with the participation of the commercial, quality, operations, logistics, procurement, and production departments. During these meetings, both the weekly and daily bottling plans are established, determining the SKUs to be processed, required volumes, and specific quality and packaging requirements. The implementation of the daily plan and any necessary process adjustments fall under the responsibility of the production area, particularly the production supervisor. This is a key aspect of current operations: since the supervisor holds full responsibility for execution, the daily operational plan relies heavily on their experience, making it susceptible to human error and variability.

In this context, the marketing department maintains a strong interaction with the market, identifying trends, consumer needs, and characteristics for the development of new products and their positioning in different segments. Simultaneously, it gathers product demand, which materializes as purchase orders.

The basic information provided by the marketing department includes the SKU, which contains all specifications necessary for production—such as bottle type and format, closure system, specific label, and packaging configuration—along with the order content, volume (in number of bottles), and the required delivery date. This information is incorporated into the weekly planning meeting, where representatives from the aforementioned departments jointly determine the tactical and operational bottling plan. Based on the agreed plan, the operations department defines raw material requirements and coordinates with the supply department to ensure availability in the plant. The quality department, in turn, sets the product quality standards and operating conditions to be maintained during the bottling process.

Meanwhile, the production department receives the relevant information, requirements, and materials to plan daily execution. Using these, it is responsible for assigning wine batches to bottling lines, defining the production sequence, and coordinating the necessary adjustments and format changes. Finally, the logistics department is in charge of planning the delivery and dispatch routes of the finished products.

This planning dynamic presents particularities that significantly differentiate it from other industrial sectors. Unlike industries such as beer, dairy, water, or juice production, where bottling processes operate with high volumes and more homogeneous production standards, the wine industry deals with SKUs that represent unique products in both format and content. This diversity leads to a high frequency of setup and format changes on bottling lines, due to the wide variety of presentations and the relatively small size of production batches. Furthermore, wine is a product that is especially sensitive to microbiological contamination, which imposes strict sanitization and quality-control protocols. These conditions mean that the planning and execution of the bottling process in the wine industry face specific challenges that require differentiated approaches when compared to other beverage industries.

2. Literature Review

Scheduling problems in production systems have been widely studied, starting with the formulation of the Flow-Shop Problem (FSP), proposed by Johnson [11]. This model is particularly relevant to industries with structured production processes, such as bottling. Several extensions have been formulated to represent more complex production layouts. Rao [12] introduced the Hybrid Flow-Shop Problem (HFSP) which models a scheduling environment where n jobs must be processed through m stages, each containing k machines denoted by ($M_k$) with all jobs follow the same stage order, as opposed to the fixed machine sequence in the FSP.

Additionally, time-dependent scheduling models, for which an extensive review is found in Gawiejnowicz [13], classify processing times as either monotonic or non-monotonic, fluctuating based on job characteristics and machine performance.

To further address the limitations of traditional models, researchers have introduced family setups which group similar jobs and reduce transition times. Models for unrelated parallel machines, as explored by Graham [14], handle heterogeneous capacities. Finally, the inclusion of sequence-dependent setup times, as highlighted by Allahverdi et al. [15,16], underscores that setup times can vary significantly with the order of job processing, making sequencing a critical factor in optimizing production efficiency.

However, a significant limitation in the literature involves properly accounting for job ordering in multi-machine scenarios, especially when setup tasks require extensive reconfiguration processes. This is particularly significant in industries such as wine bottling, where changing from one wine type to another demands substantial cleaning and adjustment. Our work addresses this gap by proposing a continuous-time scheduling model that integrates hydraulic considerations and complex sequence-dependent setups.

2.1. Scheduling of Bottling Operations

There has been significant interest in researching the final stage of beverage production from a tactical perspective. The General Lot Sizing and Scheduling Problem (GLSP), introduced by Fleischmann [17], laid the foundation for addressing production quantities and scheduling decisions sequentially. Meyr (2000) [18] expanded this model to include sequence-dependent setup times (GLSPST), where setup costs vary with the processing order, and later extended it to parallel production lines (GLSPPL) for high-volume operations [19].

In a historical context, the Simultaneous Lot Sizing and Scheduling Problem (SLS), formulated later by Clark (2000) [20], integrates lot sizing and scheduling decisions into a single model. Additionally, Clark’s extension in 2011 [21] incorporates time-dependent setups and period overlapping, further adapting the model to real production systems where both time and sequence are critical.

These models have proven valuable in bottling operations across various industries, including the works of Ferreira et al. (2012) [22] and Toledo et al. (2015) [23] in the soft-drink and mineral-water sectors, as well as studies on beer by Baldo et al. (2017) [24] and Toscano et al. (2019) on fruit-based beverages [25]. Toscano et al. (2020) [26] emphasize that efficient synchronization between preparation and bottling stages is essential to minimizing idle time and ensuring continuous production flows.

In the specific case of wine-bottling operations, however, the available models are more limited. Berruto et al. (2006) [27] propose a simple scheduling model for a single production line with multiple products that primarily considers personnel capacity via normal and overtime hours, lacking flexibility in team allocation. Basso and Varas (2017) [28] introduced an MIP formulation to address bottling scheduling, while Basso (2020) [29] proposed a collaborative approach to scheduling in this context. Mac Cawley et al. (2022) [30] further contributed by developing a robust scheduling method for large-scale wine-bottling operations, employing decomposition techniques to efficiently handle the complexities of sequence-dependent setups and reconfiguration processes. These contributions underscore the need for an integrated, advanced scheduling model that can effectively address the challenges unique to wine-bottling operations.

2.1.1. Physicochemical Properties and Their Relationship with Filtration

The efficiency of wine filtration is significantly influenced by its physicochemical properties, including viscosity, density, and surface tension. De la Garza et al. noted that an increase in viscosity leads to greater flow resistance through porous media, which requires higher pressures to maintain an adequate filtration rate [31]. This phenomenon is particularly relevant in continuous filtration applications, where variations in wine composition directly impact both the duration and quality of the filtration process. Alsawaftah et al. further explain that surface tension and density influence the formation of filter cakes that accumulate on the membrane surface [32]. These layers of particles affect permeability and increase the frequency of required maintenance, highlighting the importance of controlling the physicochemical properties of wine to enhance process efficiency and minimize membrane fouling. Additionally, recent studies have explored in greater depth the relationship between fluid dynamics and filtration performance. Yanniotis et al. demonstrated that compounds such as ethanol, dry extract, and glycerol significantly affect wine viscosity, influencing its behavior when flowing through filtering media [33]. Moreover, Lee and Bush analyzed how surface tension and density determine bubble stability, a phenomenon relevant to the liquid–gas interface in filtration systems [34]. Finally, Venerus and Nieto Simavilla described how surface-tension gradients generate capillary flows in complex liquids such as wine, which can be extrapolated to the formation of deposits on membranes during filtration [35].

2.1.2. Impact of Particle Content on Membrane Fouling

The accumulation of suspended particles, including phenolic compounds, proteins, and sediments, plays a central role in membrane fouling during wine filtration. These particles tend to form deposits on the membrane surface, reducing permeability and increasing hydraulic resistance over time. Umiker and colleagues demonstrated that such particle accumulation decreases the effective filtration area and leads to a rise in the pressure required to maintain a constant flow rate, which in turn raises operational costs and prolongs filtration cycles [36]. This phenomenon is further intensified in fluids with high organic load or fine particulate matter, where pore blockage and surface layering become more frequent [7,9]. Mathematical models developed by researchers such as Sanaei and Song have provided tools to simulate these effects by incorporating particle transport, adsorption dynamics, and resistance accumulation into predictive frameworks [37,38]. These models highlight how particle content not only influences membrane performance but also directly impacts the frequency of maintenance, cleaning protocols, and overall process stability. Understanding the interaction between particle properties and membrane dynamics is therefore essential for improving filtration efficiency and designing more resilient wine-processing systems.

In Figure 1, the evolution of the hydraulic performance of a bottling line is presented. The figure shows how the progressive reduction in flow rate, due to the obstruction of pores on the filtration surface, affects the processing time of a batch.

2.1.3. Methods for Quantifying Clogging Potential

To quantify the fouling potential of wine, experimental methods are used to evaluate its hydraulic behavior under controlled conditions. These tests do not aim to replicate the industrial filtration process exactly—where both flow rate and pressure may vary dynamically—but rather employ a constant-pressure system to isolate the effect of filter clogging on flow capacity.

In the wine industry, the most commonly used indicators for this purpose are the Laurenty Index and Vmax [7,39]. Both indicators are calculated under the same operational conditions: a temperature of 20 °C, a constant pressure of 30 psi, and filter discs of 25 mm in diameter with a pore size of 0.65 µm, using a constant-pressure setup [40,41]. The test involves filtering a total volume of 400 mL using a sealed, pressurized system that maintains steady pressure throughout the procedure.

The Laurenty Indexis calculated by recording the time required to filter the first 200 mL and comparing it with the total time taken to filter all 400 mL. The difference between these two times reflects the rate at which the flow decreases due to progressive clogging of the filter. A larger time gap indicates higher fouling potential. In industrial practice, wines with a Laurenty Index below 35 are typically considered suitable for bottling, as values above this threshold are associated with significant filtration deviations and potential operational disruptions.

On the other hand, Vmax is defined as the maximum volume that can be filtered before reaching a critical operating threshold—such as a minimum acceptable flow rate or a critical pressure drop [8]. This metric is directly linked to the effective filtration area, providing a more direct and operationally useful interpretation of hydraulic performance. In this context, Vmax offers a clearer interpretation, as it allows the filtered volume to be associated with a known surface area [40,41]. The industry generally requires wines to exceed a minimum Vmax value of 4000 mL to ensure process efficiency and avoid excessive filter replacements.

These evaluations are carried out systematically once the wine has completed its stabilization and is ready for bottling, as part of quality-control procedures aimed at anticipating hydraulic behavior, optimizing lot allocation, and minimizing unplanned downtimes in bottling operations [42].

2.1.4. Application of Darcy’s Law for Calculating Flow Rate and Processing Time

Darcy’s law is fundamental in modeling fluid flow through porous media. Delagarza highlighted that this equation incorporates fluid properties, such as viscosity, as well as filtration-medium characteristics and operational conditions, allowing for a precise prediction of initial flow rates and pressure adjustments based on wine conditions [31]. By integrating these factors, Darcy’s law facilitates operational planning by allowing wineries to anticipate processing times and reduce the impact of fouling on membranes, optimizing the filtration system’s efficiency according to wine characteristics and operational demands.

In Equation (1), Darcy’s law is represented, establishing that viscosity has a negative association with flow rate and a positive association with the pressure differential during the filtration stage. According to the literature, this variable is influenced by factors such as temperature, sugar content, the use of additives (e.g., arabic gum and carboxymethyl cellulose), and alcohol content. Additionally, operational conditions are represented through the pressure differential, which is directly associated with the flow rate (m²). In this context, higher flow rates result in higher pressure differentials. Furthermore, fouling capacity impacts the flow rate through a progressive reduction in the filtration area as particles obstruct the filter surface.

In Figure 2, the interaction of variables affecting the filtration and bottling process is illustrated, highlighting that assigning a wine batch to a specific filtration and bottling line is a non-trivial decision. This is because the processing time associated with a batch, as presented in Equation (15), depends on both hydraulic properties and the characteristics of the filtration medium, as well as the fouling capacity. Therefore, selecting the optimal route for bottling must necessarily consider the actual flow rate and effective processing time, rather than relying solely on the nominal flow rate, as is commonly done in current industrial practice.

2.2. Resolution Algorithms

The scheduling of operations in the wine industry has been tackled using various resolution methods including heuristics, metaheuristics, and exact approaches, each offering distinct trade-offs. Heuristic methods, like those proposed by Berruto [27], provide fast, feasible solutions to reduce downtime but often lack precision and scalability in environments with high variability and frequent format changes. Metaheuristics (e.g., genetic algorithms as applied by Baldo [24]) can reduce transition times but may fall short when exact solutions are required. In contrast, mixed-integer programming (MIP) models, such as those developed by Basso [29], deliver detailed optimization by incorporating sequence-dependent setup times; however, they struggle with simultaneously allocating multiple batches across different production lines.

Mac Cawley et al. [30] have further contributed by employing a decomposition algorithm for large-scale wine bottling, achieving cost reductions of 15–30% through effective handling of sequence-dependent setups across parallel lines—though their approach can be computationally intensive and is often tied to specific operational data under the assumption of two setup types. In light of these limitations, our work proposes an exact and continuous scheduling approach that avoids time discretization, enabling a more precise modeling of processing and setup times. This method thoroughly evaluates all possible batch combinations for filtration and bottling, supports daily operational decision making in environments with a finite number of lines, and aligns with Meyr’s observations on the critical role of sequence-dependent setups in optimizing production flow. Overall, our framework offers robust, scalable, and adaptable support for precise and efficient daily planning in the wine-bottling industry.

2.3. Research Gaps and the Contribution of This Study

Despite advances in bottling operations modeling, the reviewed literature reveals significant limitations in the representation of critical physical conditions, particularly during the filtration stage. Most existing approaches assume constant nominal flow rates, overlook the dynamic effects of clogging potential, and fail to explicitly incorporate the physicochemical properties of wine, such as viscosity or particle composition. Additionally, although some studies have considered sequence-dependent setup times, their application has been mostly confined to generic contexts or based on simplified assumptions. In response to these gaps, this study proposes an exact mixed-integer programming model that incorporates a continuous-time processing approach based on Darcy’s law, allowing for a realistic representation of flow-rate reduction due to clogging. The model integrates both the hydraulic parameters of the wine and the technical configurations of filtration and bottling lines, including setup times associated with format changes. Through this formulation, the study captures the interaction between product properties, line configuration, and operational decisions, offering a robust daily planning tool applicable to real-world settings such as the Chilean wineries included in the case study.

3. Problem Description

The problem focuses on the efficient planning and sequencing of operations in a winery, with an emphasis on the filtration stage, an area that has been little explored in the literature. Filtration presents critical constraints due to the clogging potential of particles in the wine, which is measured using indicators that estimate filtration speed, flow reduction, and pressure differential increase. If not controlled, this process can cause interruptions, increase operational costs, and introduce variability in processing times. These factors directly affect production efficiency, compromising the ability to meet demand profitably.

The developed model aims to minimize the total production time by optimally assigning wine batches to filtration and bottling lines. Each filtration line has different pore sizes and filtering areas, which affect hydraulic performance depending on the physicochemical properties of the wine, such as viscosity. In addition, the assignment to the bottling line is limited by maximum flow rate and bottle type. The solid content in the wine, measured by the clogging index, impacts the flow and processing time. After each batch, it is possible to perform washes or adjustments that clean the filters and extend their lifespan, reducing uncertainty and allowing for better planning.

Based on the notation of [14] $\alpha /\beta /\gamma$, we propose a $FFR$/$S{T}_{sd,b}$/$C_{max}$ indicating a flexible (Hybrid Flow Shop) with unrelated parallel machines with a sequence-dependent batch or family setup time that minimizes makespan.

3.1. Transitions and States in the Bottling Process

The states of the bottling system refer to the different levels of operation, adjustment, preparation, and cleaning of the production lines, which are necessary to ensure compatibility between different types of wine and packaging formats. State $W_1$ represents the system during the bottling process, that is, when the line is operational and actively processing a batch of wine. In this state, the production flow is expected to be continuous, and all operational parameters are adjusted to maintain the efficiency of the process.

State $W_2$ corresponds to the adjustment of a format or parameter of the production line. This adjustment may include changes in the type of bottle, the closure system (such as cork or crown cap), labeling, or packaging, which are necessary to adapt to the SKU specifications. Additionally, it may include a cleanse or replacement of the filter based on the current condition and the filtration requirement of the following batch. This state implies a brief interruption in the production flow while these changes are made to ensure that the equipment is properly configured for the new format.

In contrast, states $W_3$, $W_4$, and $W_5$ correspond to known and fixed values that represent sequential cleaning and adjustment activities.

3.2. Batch-Processing Time

Our approach to the scheduling problem seeks to sensitize the batch-processing time considering a decrease in the flow over time due to clogging. Each production line comprises two stages: filtering and bottling. The nominal flow for both stages is determined by SKU-specific parameters. In the case of filtration, the admissible flow rate is determined by the filtering area available, calculated for the number of filters that a housing can contain and the pore diameter of the filter itself, which is specified in accordance with the filtering requirements of the batch. For the bottling stage, bottle size is the parameter that conditions the capacity of the line; furthermore, it is assumed that the flow will remain constant for this stage. With regard to the aggregated flow, the minimum of both is considered.

In the literature, authors have considered a processing-time calculation based on the nominal flow of the line, thereby overestimating the actual production capacity.

In order to model the reduction in flow rate resulting from filter clogging, a Darcy estimation is employed. This is a validated approach for describing the hydraulic performance of a fluid in a porous medium [43].

In principle, there are two possible scenarios: (1) The admitted flow rate through the filtration line (i.e., the Darcy flow rate, $Q_{Darcy}$) is greater than the flow rate through the bottling line (i.e., the operational flow rate, $Q_{op}$). In this case, the latter will be considered the initial flow rate. (2) Conversely, if the operational flow rate is greater than the Darcy flow rate, the initial flow considered is that from the filtration stage. In both scenarios, the objective is to model the decline of the flow rate, represented by the function $Q\left(t\right)$, to the minimum acceptable threshold, $Q_{min}$, as a fraction of the initial condition. At this point, the filters must be cleaned or replaced to restore efficient operation within tolerable margins. The reduction in flow rate is attributed to the accumulation of solids within the filters, which decreases the available filter area, $A\left(t\right)$, resulting in an exponentially distributed decline. The rate of filter clogging depends on the physicochemical characteristics of the wine batch being processed.

The variables and the modeling procedure will be defined in the following section, according to the cases described.

3.2.1. Variables

• $V_{\mathrm{total}}$: Total volume of the batch to be filtered.
• $Q_{\mathrm{op}}$: Operational flow rate.
• $Q_{\mathrm{Darcy}}$: Flow rate determined by Darcy’s law.
• $t_1$: Time when the flow rate starts to decrease due to clogging.
• $t_{\mathrm{total}}$: Total filtration time.
• $r_s$: Filter clogging rate.
• K: Combined permeability and thickness coefficient, $K={\displaystyle \frac{k}{L}}$.

  -

  k: Filter permeability coefficient.

  -

  L: Filter thickness.

• $A_0$: Initial filtration area.
• $\mathrm{\Delta }P$: Pressure differential.
• $\mu$: Fluid viscosity.
• $Q_{\min }$: Minimum allowable flow rate.

3.2.2. Development

• Part 1: Condition on Darcy’s flow rate

During the initial phase, the flow rate is given by:

$$Q_{\mathrm{Darcy}}={\displaystyle \frac{K·A_0·\mathrm{\Delta }P}{\mu }}$$

(1)

There are two possible cases:

Case 1: $Q_{\mathrm{Darcy}}\ge Q_{\mathrm{op}}$. In this case, the flow rate is forced to $Q_{\mathrm{op}}$ and the volume filtered during this phase is:

$$V_1=Q_{\mathrm{op}}·t_1$$

(2)

Case 2: $Q_{\min }<Q_{\mathrm{Darcy}}<Q_{\mathrm{op}}$. In this case, the flow rate is determined by Darcy’s law, i.e., $Q\left(t\right)=Q_{\mathrm{Darcy}}$, and the filtered volume is:

$$V_1=Q_{\mathrm{Darcy}}·t_1={\displaystyle \frac{K·A_0·\mathrm{\Delta }P}{\mu }}·t_1$$

(3)

If $Q_{\mathrm{Darcy}}\le Q_{\min }$, filtration does not occur.

• Part 2: Decreasing flow rate due to clogging

From $t_1$ onwards, the flow rate starts to decrease due to clogging. The effective filtration area decreases exponentially:

$$A\left(t\right)=A_0·e^{-r_s·(t-t_1)}$$

(4)

The flow rate during this phase follows Darcy’s law:

$$Q\left(t\right)={\displaystyle \frac{K·A_0·e^{-r_s·(t-t_1)}·\mathrm{\Delta }P}{\mu }}$$

(5)

The volume filtered during the second phase $V_2$ is:

$$V_2={\int }_{t_1}^{t_{\mathrm{total}}}Q\left(t\right) dt={\int }_{t_1}^{t_{\mathrm{total}}}{\displaystyle \frac{K·A_0·e^{-r_s·(t-t_1)}·\mathrm{\Delta }P}{\mu }} dt$$

(6)

Solving the integral:

$$V_2={\displaystyle \frac{K·A_0·\mathrm{\Delta }P}{\mu ·r_s}}\left(1-e^{-r_s·(t_{\mathrm{total}}-t_1)}\right)$$

(7)

Total filtered volume

The total filtered volume is the sum of the volumes filtered in both phases:

$$V_{\mathrm{total}}=V_1+V_2$$

(8)

Substituting the expressions for $V_1$ and $V_2$, we have two cases:

Case 1: $Q_{\mathrm{Darcy}}\ge Q_{\mathrm{op}}$:

$$V_{\mathrm{total}}=Q_{\mathrm{op}}·t_1+{\displaystyle \frac{K·A_0·\mathrm{\Delta }P}{\mu ·r_s}}\left(1-e^{-r_s·(t_{\mathrm{total}}-t_1)}\right)$$

(9)

Case 2: $Q_{\min }<Q_{\mathrm{Darcy}}<Q_{\mathrm{op}}$:

$$V_{\mathrm{total}}={\displaystyle \frac{K·A_0·\mathrm{\Delta }P}{\mu }}·t_1+{\displaystyle \frac{K·A_0·\mathrm{\Delta }P}{\mu ·r_s}}\left(1-e^{-r_s·(t_{\mathrm{total}}-t_1)}\right)$$

(10)

Solving for total filtration time $t_{\mathrm{total}}$

To solve for $t_{\mathrm{total}}$, we begin by the right-hand side of the equation:

$$V_{\mathrm{total}}-V_1={\displaystyle \frac{K·A_0·\mathrm{\Delta }P}{\mu ·r_s}}\left(1-e^{-r_s·(t_{\mathrm{total}}-t_1)}\right)$$

(11)

Multiplying both sides by ${\displaystyle \frac{\mu ·r_s}{K·A_0·\mathrm{\Delta }P}}$:

$$r_s·{\displaystyle \frac{V_{\mathrm{total}}-V_1}{K·A_0·\mathrm{\Delta }P/\mu }}=1-e^{-r_s·(t_{\mathrm{total}}-t_1)}$$

(12)

Solving for $e^{-r_s·(t_{\mathrm{total}}-t_1)}$:

$$e^{-r_s·(t_{\mathrm{total}}-t_1)}=1-r_s·{\displaystyle \frac{V_{\mathrm{total}}-V_1}{K·A_0·\mathrm{\Delta }P/\mu }}$$

(13)

Finally, taking the natural logarithm for both sides:

$$t_{\mathrm{total}}=t_1-{\displaystyle \frac{1}{r_s}}ln\left(1-r_s·{\displaystyle \frac{V_{\mathrm{total}}-V_1}{K·A_0·\mathrm{\Delta }P/\mu }}\right)$$

(14)

Final equation

$$t_{\mathrm{total}}=t_1-{\displaystyle \frac{1}{r_s}}ln\left(1-r_s·{\displaystyle \frac{V_{\mathrm{total}}-V_1}{K·A_0·\mathrm{\Delta }P/\mu }}\right)$$

(15)

where $V_1$ will depend on whether $Q_{\mathrm{Darcy}}\ge Q_{\mathrm{op}}$ or $Q_{\min }<Q_{\mathrm{Darcy}}<Q_{\mathrm{op}}$.

3.3. Sequence-Dependent Setup Transitions

A crucial aspect to address in the model is the sequence-dependent setup times. In the literature, this is typically classified into families of setups, including both major and minor operations. Major operations involve tasks such as bottle changes, while minor operations include cleaning the line or labelling [24,30].

In particular, we define one aspect of the sequence-dependent setup times for the desired SKU format within a specified batch, denoted as $W_2$. This refers to adjustments made to the bottling line to meet the specific requirements of the SKU. Factors to consider include the type of bottle (size, shape, weight), labeling, the type of closing (crown or corking with respective capsule), and packaging. It must be acknowledged that the operational constraints of each line determine the admissibility of specific formats.

Subsequently, a transition matrix will be defined for each setup task, as changes from one SKU to another may require one or more adjustments, with execution times varying accordingly. In general, large wineries handle between 100 to 200 SKUs. The more complex setup processes typically require approximately 60 to 90 min, reflecting enhanced operational efficiency enabled by novel machinery. Additionally, the setup transition matrix can be clustered in order to reduce its dimensionality. For instance, labeling can be defined as fixed with a lower-bound setup time, as it is necessary to change the label for every SKU transition.

Moreover, accounting for available workforce is essential for effective planning [30]. We will consider the maximum setup time across all tasks, allowing for parallel execution if there are sufficient workers. Otherwise, if the workforce is limited, we will sum the individual task-execution times. This approach allows for the adaptation of setup times based on the operational conditions of the winery.

4. Methodology

In this section, we present the developed model aimed at optimizing the production process. The model is built upon a set of foundational assumptions that ensure its coherence and effectiveness. Following these assumptions, the model is constructed by incorporating various parameters and variables that capture the essential aspects of the production workflow, such as processing times, flow rates, and SKU requirements. Additionally, an objective function is formulated to minimize the overall completion time of the production schedule. This comprehensive approach facilitates efficient and adaptable planning, ensuring that the model can effectively address the daily demands of production.

4.1. Assumptions

• A line can process only one job at a time.
• A task is fully completed before the next task is taken.
• A task cannot be stopped midway for another task.
• A job is assigned on only one line. There is no job splitting.
• Raw materials are always available.
• All tanks are filled beforehand, with no constraints on number of intermediate tanks.
• SKU and volumes are defined for the daily production plan.
• All batches must be completed on the daily schedule.
• If different clients place the same order, SKU is taken into consideration as the sum of both volumes.
• Every day starts and finishes with a sanitization $\Rightarrow W_5$.

4.2. Optimization Model Formulation

We denote by O the set or orders, (indexed by o). By F and B, the filtering and bottling units, respectively, are indexed by f and b. Finally the set of lines, denoted by L and indexed by l, is given by the machine assignation on the ordered pair of filtering and bottling units $(f,b)$.

4.2.1. Sets

• O: Set of orders, $o\in O$.
• F: Set of filtering machines, $\{1,2\} f\in F$.
• B: Set of bottling machines, $\{1,2\} b\in B$.
• L: Set of possible lines (machine pairs), where $L=\left\{\right(f,b\left)\right|f\in F,b\in B\}$/

4.2.2. Parameters

We have parameters indicating machine capacities, setup times, and batch-dependant processing times. We would consider the filtering-stage processing time as minimal; for modelling purposes, we treat them independently, but essentially, once assigned both machines, they operate as one. The ${\beta }_{ob}$ matrix or parameters, in particular, indicate SKU format admissibility on bottling lines, due to real production-environment constraints.

• $p_o^F$: Filtering time for order o.
• $p_o^B$: Bottling time for order o.
• $CT$: Time required to reconfigure line.
• M: A sufficiently large constant (Big M).
• $Q_f^F$: Flow rate of filtration unit f.
• $Q_b^B$: Flow rate of bottling unit b.
• ${\beta }_{ob}:\left\{\begin{array}{cc}1\hfill & \mathrm{if} \mathrm{bottling} \mathrm{unit} b \mathrm{can} \mathrm{process} \mathrm{order} o\hfill \\ 0\hfill & \mathrm{otherwise}\hfill \end{array}\right.$

Every order has a vector of attributes that will be considered for the calculation of the sequence-dependent setup times. Specifically, they include the volume of the order to be processed $V_o$ and the respective SKU requirements. We have also specified setup times for each task to be performed in the aggregated setup time.

• $V_o$: Volume of order o.
• Batch Attributes:
  ${\mathrm{BottleSize}}_o$; ${\mathrm{Color}}_o$; ${\mathrm{Complexity}}_o$; ${\mathrm{Height}}_o$; ${\mathrm{Diameter}}_o$; ${\mathrm{FilterReq}}_o$; ${\mathrm{Corking}}_o$; ${\mathrm{Packaging}}_o$.
• Setup Time Components:

  -

  $W_3$: Base wash time.

  -

  $W_4$: Wash time if wine color changes.

  -

  $W_5$: Sanitation time if complexity changes.

  -

  $s{t}_{\mathrm{bottle}}$: Time for bottle format changeover.

  -

  $s{t}_{\mathrm{height}}$: Time for height adjustment.

  -

  $s{t}_{\mathrm{diameter}}$: Time for diameter adjustment.

  -

  $s{t}_{\mathrm{filter}}$: Time for filter changeover.

  -

  $s{t}_{\mathrm{corking}}$: Time for corking system change.

  -

  $s{t}_{\mathrm{packaging}}$: Time for packaging format change.

  -

  $s{t}_{\mathrm{label}}$: Time for label change.

In the following expression, the bottleneck of the production setup is highlighted. Due to operational constraints, the production line operates at the minimal flow rate between both machines. So the processing time would initially consider the volume over the flow rate.

$$\begin{array}{cccc}\hfill & Q_l=min\left(Q_f,Q_b\right),\hfill & \hfill & \forall l\in L\hfill \\ \hfill & p_o^F=p_o^B={\displaystyle \frac{V_o}{Q_l}},\hfill & \hfill & \forall i\in I,\forall l\in L\hfill \end{array}$$

4.2.3. Variables

The following set of variables is meant to allow the sequencing of orders into their respective machines.

• $x_l^o:\left\{\begin{array}{cc}1\hfill & \mathrm{if}\mathrm{order}\mathrm{o}\mathrm{is}\mathrm{assigned}\mathrm{to}\mathrm{line}\mathit{l}\hfill \\ 0\hfill & \mathrm{otherwise}\hfill \end{array}\right.$
• $S_o^F$: Start time of filtering for order o.
• $C_o^F$: Completion time of filtering for order o.
• $S_o^B$: Start time of bottling for order o.
• $C_o^B$: Completion time of bottling for order o.
• ${\delta }_{op}^F:\left\{\begin{array}{cc}1\hfill & \mathrm{if}\mathrm{order}\mathit{o}\mathrm{precedes}\mathit{p}\mathrm{on}\mathrm{their}\mathrm{assigned}\mathrm{filtering}\mathrm{machine}\hfill \\ 0\hfill & \mathrm{otherwise}\hfill \end{array}\right.$
• ${\delta }_{op}^B:\left\{\begin{array}{cc}1\hfill & \mathrm{if}\mathrm{order}\mathit{o}\mathrm{precedes}\mathit{p}\mathrm{on}\mathrm{their}\mathrm{assigned}\mathrm{bottling}\mathrm{machine}\hfill \\ 0\hfill & \mathrm{otherwise}\hfill \end{array}\right.$

This set of variables include the assignation of orders to the machines at different stages, allowing for changeover in reassignation.

• $c_{op}^F=\left\{\begin{array}{cc}1\hfill & \mathrm{if} \mathrm{a} \mathrm{changeover} \mathrm{is} \mathrm{needed} \mathrm{between} \mathrm{orders} o \mathrm{and} p\hfill \\ & \mathrm{on} \mathrm{their} \mathrm{assigned} \mathrm{filtering} \mathrm{machine}\hfill \\ 0\hfill & \mathrm{otherwise}\hfill \end{array}\right.$
• $c_{op}^B=\left\{\begin{array}{cc}1\hfill & \mathrm{if} \mathrm{a} \mathrm{changeover} \mathrm{is} \mathrm{needed} \mathrm{between} \mathrm{orders} o \mathrm{and} p\hfill \\ & \mathrm{on} \mathrm{their} \mathrm{assigned} \mathrm{bottling} \mathrm{machine}\hfill \\ 0\hfill & \mathrm{otherwise}\hfill \end{array}\right.$
• $y_{of}=\left\{\begin{array}{cc}1\hfill & \mathrm{if} \mathrm{order} o \mathrm{uses} \mathrm{filtering} \mathrm{machine} f\hfill \\ 0\hfill & \mathrm{otherwise}\hfill \end{array}\right.$
• $z_{ob}=\left\{\begin{array}{cc}1\hfill & \mathrm{if} \mathrm{order} o \mathrm{uses} \mathrm{bottling} \mathrm{machine} b\hfill \\ 0\hfill & \mathrm{otherwise}\hfill \end{array}\right.$
• $m_{op}^F=\left\{\begin{array}{cc}1\hfill & \mathrm{if} \mathrm{a} \mathrm{machine} \mathrm{pairing} \mathrm{changeover} \mathrm{is} \mathrm{needed} \mathrm{from} \mathrm{order} o\hfill \\ & \mathrm{to} p \mathrm{on} \mathrm{filtering} \mathrm{machines}\hfill \\ 0\hfill & \mathrm{otherwise}\hfill \end{array}\right.$
• $m_{op}^B=\left\{\begin{array}{cc}1\hfill & \mathrm{if} \mathrm{a} \mathrm{machine} \mathrm{pairing} \mathrm{changeover} \mathrm{is} \mathrm{needed} \mathrm{from} \mathrm{order} o\hfill \\ & \mathrm{to} p \mathrm{on} \mathrm{bottling} \mathrm{machines}\hfill \\ 0\hfill & \mathrm{otherwise}\hfill \end{array}\right.$

Finally, the following is the set of variables necessary to address the sequence-dependent setup times, respectively, on the filtering and bottling machines. Also, for each one of the setup tasks needed for each possible SKU, the transition auxiliary decision variables are defined.

Setup Variables

• $S{U}_{op}^b$: Setup time from order o to p in bottling machine b.
• $S{U}_{op}^f$: Setup time from order o to p in filtering machine f.

  -

  ${\delta }_{color}^{op}:\left\{\begin{array}{cc}1\hfill & \mathrm{if}\mathrm{color}\mathrm{set}\mathrm{up}\mathrm{is}\mathrm{required}\mathrm{from}\mathrm{order}\mathit{o}\mathrm{to}\mathit{p}\hfill \\ 0\hfill & \mathrm{otherwise}\hfill \end{array}\right.$

  -

  ${\delta }_{complexity}^{op}:\left\{\begin{array}{cc}1\hfill & \mathrm{if}\mathrm{complexity}\mathrm{set}\mathrm{up}\mathrm{is}\mathrm{required}\mathrm{from}\mathrm{order}\mathit{o}\mathrm{to}\mathit{p}\hfill \\ 0\hfill & \mathrm{otherwise}\hfill \end{array}\right.$

  -

  ${\delta }_{bottle}^{op}:\left\{\begin{array}{cc}1\hfill & \mathrm{if}\mathrm{bottle}\mathrm{changeover}\mathrm{is}\mathrm{required}\mathrm{from}\mathrm{order}\mathit{o}\mathrm{to}\mathit{p}\hfill \\ 0\hfill & \mathrm{otherwise}\hfill \end{array}\right.$

  -

  ${\delta }_{height}^{op}:\left\{\begin{array}{cc}1\hfill & \mathrm{if}\mathrm{bottle}\mathrm{height}\mathrm{set}\mathrm{up}\mathrm{is}\mathrm{required}\mathrm{from}\mathrm{order}\mathit{o}\mathrm{to}\mathit{p}\hfill \\ 0\hfill & \mathrm{otherwise}\hfill \end{array}\right.$

  -

  ${\delta }_{diameter}^{op}:\left\{\begin{array}{cc}1\hfill & \mathrm{if}\mathrm{bottle}\mathrm{diameter}\mathrm{set}\mathrm{up}\mathrm{is}\mathrm{required}\mathrm{from}\mathrm{order}\mathit{o}\mathrm{to}\mathit{p}\hfill \\ 0\hfill & \mathrm{otherwise}\hfill \end{array}\right.$

  -

  ${\delta }_{filter}^{op}:\left\{\begin{array}{cc}1\hfill & \mathrm{if}\mathrm{filtering}\mathrm{requirement}\mathrm{set}\mathrm{up}\mathrm{is}\mathrm{required}\mathrm{from}\mathrm{order}\mathit{o}\mathrm{to}\mathit{p}\hfill \\ 0\hfill & \mathrm{otherwise}\hfill \end{array}\right.$

  -

  ${\delta }_{packaging}^{op}:\left\{\begin{array}{cc}1\hfill & \mathrm{if}\mathrm{packaging}\mathrm{set}\mathrm{up}\mathrm{is}\mathrm{required}\mathrm{from}\mathrm{order}\mathit{o}\mathrm{to}\mathit{p}\hfill \\ 0\hfill & \mathrm{otherwise}\hfill \end{array}\right.$

The objective function seeks to minimize the maximum completion time of the bottling machine i.e., the last stage of production process. This means it accounts for all possible combinations of batch assigning with setup and changeover and, while minimizing the makespan of the production plan, it also minimizes the necessary adjustments during the daily-planning horizon.

4.2.4. Objective Function

$$min ma{x}_{o\in O} C_o^B$$

(16)

4.2.5. Constraints

Constraint (1) is ensuring each order is assigned to exactly one production line. Constraints (2) and (3) link the assignment of orders to lines, specifying which machine is used at each stage, and verifying there is only one filtering and bottling machine, respectively. Constraint (4) looks for the sequencing of different orders when assigned to the same filtering machine and (5) considers the completion time of the former order to account for the setup and changeover time to process the latter. Similarly, (6) and (7) carry out the same for bottling machines. Constraints (8) and (9) determine if machine changeovers are necessary when processing different batches for filtering and bottling machines, respectively.

4.2.6. Assignment Constraints

Each product is assigned to exactly one line

$$\begin{array}{ccc}\hfill \sum _{l\in L}{x}_o^l& =1,\hfill & \hfill \forall o\in O\end{array}$$

(17)

4.2.7. Machine Usage Indicators

Linking line assignments to machine usage

$$\begin{array}{ccc}\hfill y_{of}& =\sum _{b\in B}{x}_o^l,\hfill & \hfill \forall o\in O,\forall f\in F\end{array}$$

(18)

$$\begin{array}{ccc}\hfill z_{ob}& =\sum _{f\in F}{x}_o^l,\hfill & \hfill \forall o\in O,\forall b\in B\end{array}$$

(19)

4.2.8. Sequencing on Filtering Machines

For products assigned to the same filtering machine, it ensures that when orders are assigned to the same filtering machine, they have to sequenced

$$\begin{array}{ccc}\hfill {\delta }_{op}^F+{\delta }_{po}^F+1& \ge y_{of}+y_{pf},\hfill & \hfill \forall o,p\in O,o\ne p,\forall f\in F\end{array}$$

(20)

No Overlap with Changeover on Filtering Machines.

$$\begin{array}{ccc}\hfill C_o^F+\mathrm{CT}·c_{op}^F+S{U}_{op}^F& \le S_p^F+M(1-{\delta }_{op}^F),\hfill & \hfill \forall o,p\in O,o\ne p,\forall f\in F\end{array}$$

(21)

4.2.9. Sequencing on Bottling Machines

Similarly, for bottling machines

$$\begin{array}{ccc}\hfill {\delta }_{op}^B+{\delta }_{po}^B+1& \ge z_{o,b}+z_{p,b},\hfill & \hfill \forall o,p\in O,o\ne p,\forall b\in B\end{array}$$

(22)

No Overlap with Changeover on Bottling Machines

$$\begin{array}{ccc}\hfill C_o^B+\mathrm{CT}·c_{op}^B+S{U}_{op}^B& \le S_p^B+M(1-{\delta }_{op}^B),\hfill & \hfill \forall o,p\in O,o\ne p,\forall b\in B\end{array}$$

(23)

4.2.10. Changeover Detection

Changeover Detection on Filtering Machines

$$\begin{array}{ccc}\hfill c_{op}^F& \ge {\delta }_{op}^F+m_{op}^F-1,\hfill & \hfill \forall o,p\in O,o\ne p\end{array}$$

(24)

Changeover Detection on Bottling Machines

$$\begin{array}{ccc}\hfill c_{op}^B& \ge {\delta }_{op}^B+m_{op}^B-1,\hfill & \hfill \forall o,p\in O,o\ne p\end{array}$$

(25)

4.2.11. Machine-Pair Difference Indicators

Machine-Pair Difference Indicators for Filtering Machines

$$\begin{array}{ccc}\hfill m_{op}^F& \ge z_{ob}+z_{p{b}^{\prime }}-1,\hfill & \hfill \forall o,p\in O,o\ne p;\forall b,b^{\prime }\in B,b\ne b^{\prime }\end{array}$$

(26)

Machine-Pair Difference Indicators for Bottling Machines

$$\begin{array}{ccc}\hfill m_{op}^B& \ge y_{of}+y_{p{f}^{\prime }}-1,\hfill & \hfill \forall o,p\in O,o\ne p;\forall f,f^{\prime }\in F,f\ne f^{\prime }\end{array}$$

(27)

4.2.12. Processing Times

$$\begin{array}{ccc}\hfill C_o^F& =S_o^F+{\mathrm{p}}_o^F,\hfill & \hfill \forall o\in O\end{array}$$

(28)

$$\begin{array}{ccc}\hfill C_o^B& =S_o^B+{\mathrm{p}}_o^B,\hfill & \hfill \forall o\in O\end{array}$$

(29)

4.2.13. Synchronization Constraints

$$\begin{array}{ccc}\hfill S_o^B& \ge C_o^F,\hfill & \hfill \forall o\in O\end{array}$$

(30)

4.2.14. Total Setup Time

$$\begin{array}{ccc} \hfill & S{U}_{op}^F=s_{\mathrm{wine}}^{op}+s_{\mathrm{filter}}^{op}\hfill & \hfill \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\forall o,p\in O,o\ne p\end{array}$$

(31)

$$\begin{array}{ccc} \hfill & S{U}_{op}^B=s_{\mathrm{bottle}}^{op}+s_{\mathrm{format}}^{op}+s_{\mathrm{corking}}^{op}+s_{\mathrm{packaging}}^{op}+s_{\mathrm{label}}\hfill & \forall o,p\in O,o\ne p\hfill \end{array}$$

(32)

4.2.15. Component Setup Times

$$\begin{array}{ccc}\hfill & s_{\mathrm{wine}}^{op}=(1-{\delta }_{color}^{op})·W_3+W_4·{\delta }_{color}^{op}+W_5·{\delta }_{\mathrm{complexity}}^{op}\hfill & \hfill \forall o,p\in O,o\ne p\end{array}$$

(33)

$$\begin{array}{ccc}\hfill & {\delta }_{\mathrm{color}}^{op}+{\delta }_{\mathrm{complexity}}^{op}=1\hfill & \hfill \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em} \forall o,p\in O,o\ne p\end{array}$$

(34)

$$\begin{array}{ccc}\hfill & s_{\mathrm{format}}^{op}=s_{\mathrm{height}}·{\delta }_{\mathrm{height}}^{op}+s_{\mathrm{diameter}}·{\delta }_{\mathrm{diameter}}^{op}\hfill & \hfill \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\forall o,p\in O,o\ne p\end{array}$$

(35)

$$\begin{array}{ccc}\hfill & 1-{\delta }_{bottle}^{op}\le {\delta }_{height}^{op}+{\delta }_{diameter}^{op}\hfill & \hfill \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em} \forall o,p\in O,o\ne p\end{array}$$

(36)

$$\begin{array}{ccc}\hfill & s_{bottle}^{op}={\delta }_{bottle}^{op}·s_{bottle}\hfill & \hfill \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em} \forall o,p\in O,o\ne p\end{array}$$

(37)

$$\begin{array}{ccc}\hfill & s_{\mathrm{filter}}^{op}=s_{\mathrm{filter}}·{\delta }_{\mathrm{filter}}^{op}+W_4·{\delta }_{\mathrm{filter}}^{i,j}\hfill & \hfill \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\forall o,p\in O,o\ne p\end{array}$$

(38)

$$\begin{array}{ccc}\hfill & s_{\mathrm{corking}}^{op}=s_{\mathrm{corking}}·{\delta }_{\mathrm{corking}}^{op}\hfill & \hfill \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\forall o,p\in O,o\ne p\end{array}$$

(39)

$$\begin{array}{ccc}\hfill & s_{\mathrm{packaging}}^{op}=s_{\mathrm{packaging}}·{\delta }_{\mathrm{packaging}}^{op}\hfill & \hfill \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\forall o,p\in O,o\ne p\end{array}$$

(40)

4.2.16. Format Admissibility

$${\beta }_{ob}=z_{ob} \forall o\in O, \forall b\in B$$

(41)

4.3. Additional Constraints

Filter Degradation Over Time: Introduce a variable to track filter condition over time, affecting subsequent batches. $A_0$

Non-linear Time Calculations: The current model uses a linear approximation for some non-linear time calculations. Consider piecewise linear approximations or non-linear programming techniques for more accuracy.

5. Case Study

This study focuses on a vineyard located in the southern region of Chile, which manages two filtration lines and two bottling lines. The characteristics of the bottling lines are detailed in

Table 1. Characteristics of bottling lines 1 and 2.

Line	Max. Flow Rate (m3/h)	Accepted Formats (mL)	Top Closure Type
Line 1	3.2	187, 330, 700, 750, 1000, 1500	Cork, Screw Cap
Line 2	2.8	187, 330, 700, 750, 1000, 1500	Cork, Screw Cap

, while the characteristics of the filtration lines are presented in

Table 2. Characteristics of filtration lines 1 and 2.

Line	Filtration Area (m2)	Max. Flow Rate (m3/h)	Accepted Formats (µm)
Line 1	12	4.8	0.45–0.65
Line 2	18	7.2	0.45–0.65

. These lines are assigned and combined based on production needs and quality and process standards. The production period considered spans three days, detailed in

Table 3. The general characteristics of production batches associated with the 3 packaging days.

Day	Batch	SKU	Format (mL)	Vmax	Volume (m3)	Sugar	Color
1	1	1001	750	4500	5000	High	White
1	2	1002	750	1800	6500	High	Red
1	3	1003	750	3600	4500	Low	White
1	4	1004	187	4900	3200	Low	Red
1	5	1005	750	2300	2500	Low	White
1	6	1006	750	3950	3500	Low	White
1	7	1007	750	4200	3500	Low	Red
1	8	1008	750	1980	2700	High	White
1	9	1009	750	5300	4300	Low	White
1	10	1010	1500	3200	3200	Low	White

. Daily production-lot assignments are established in weekly coordination meetings, while specific operational decisions regarding the order and assignment of lots are made by the production area.

5.1. General Characteristics of the Production Plan

The production plan considered, represented in Table 3, consists of 1 day of operation during which a total of 10 batches were processed. Regarding bottle format, 70% of the batches correspond to the 750 mL format, while the 187 mL and 1500 mL formats represent 10% and 20%, respectively. In terms of wine color, white wines constitute 50% of the processed batches, while red wines account for the other 50%.

The average Vmax for the batches is 3573, with a standard deviation of 1107.67, reflecting moderate variability in filtration capacity among them. The total volume processed over the 1 day amounts to 41,200 m³, with an average of 4120 m³ per batch. These data provide a detailed overview of the operational characteristics of the production plan and facilitate the analysis of workload distribution across the filtration and bottling lines.

It is important to note that the model parameters are categorical and must be coherent with the production plan. For instance, if the production schedule includes specific bottle formats (e.g., 187 mL, 750 mL, 1500 mL), the assigned lines must admit those combinations. These categories reflect standard configurations observed in Chilean bottling operations, as discussed in earlier sections.

5.2. Comparative Scenario Analysis

The analysis conducted evaluates the discrepancies between the estimated production time, based on the nominal capacities of the system, and the actual time observed, taking into account the hydraulic characteristics of the wine, its clogging potential, and the hydraulic performance of the filtration system. In response to these findings, an adjusted production plan is proposed that aims to minimize the total production time. This plan considers the optimal combination of assigning production lots to filtration and bottling lines, as well as the processing sequence and necessary adjustment activities.

With these results, a critical comparison is made between the three described scenarios, evaluating the operational implications and resource-allocation considerations that involve accounting for the filtration stage and the clogging potential of the wine in the compliance and optimization of the production plan.

5.3. Problem Resolution

The problem was solved by implementing a mixed-integer linear programming (MILP) model, using an exact approach to ensure the attainment of optimal solutions. The model was developed in Python 3.12 and solved using the Gurobi optimizer, selected for its well-established computational efficiency and numerical robustness in addressing highly combinatorial problems. Recent studies have shown that Gurobi consistently outperforms other commercial solvers in terms of solution speed, quality, and scalability, particularly in industrial contexts involving multiple binary variables and complex structural constraints [44]. Prior to solving the model, a data preprocessing phase was carried out, in which certain characteristics were grouped based on similarities in setup and processing times. This allowed for the construction of transition and combination matrices that represent feasible batch assignments and production sequences. This step contributed to simplifying the model while effectively incorporating the hydraulic characteristics of the wine, such as its clogging potential, thereby optimizing batch allocation across the filtration and bottling lines.

6. Results

This section presents the results obtained from comparing the actual production plan with the optimal allocation to the bottling lines. The results include the comparison of the duration times in the different states of the system efficiency achieved through the optimization of production batch assignments. These findings allow for the evaluation of the impact of the implemented strategies on improving the bottling process and reducing the total processing time.

6.1. Comparison Between Bottling Planning and Real Performance

The analysis of the 10 production batches and their initial assignment, based on the weekly coordination meeting, is represented in Table 3. The initial production plan considered a total time of 565 min for Line 1 and 507 min for Line 2. For Line 1, 431 min were allocated to the bottling process, with the remaining 134 min designated for format transitions, cleaning, and product changes. Similarly, for Line 2, 431 min were allocated to bottling and the remaining 76 min for transitions and adjustments. It is important to note that the times assigned for adjustments between batch changes do not vary, as only the difference between the planned and actual bottling times is recorded.

Due to the nature of the wine in terms of fouling capacity and hydraulic properties, deviations of 63 min were observed in the bottling plan for Line 1 and 37 min for Line 2. These deviations represent an approximate 9.3% increase in the daily production plan, resulting in additional costs for the company in terms of labor hours and energy consumption.

It is worth noting that these discrepancies between planned and actual observations are common, and the company has internalized the economic impact of this scenario, incorporating it into its projections and operational strategies.

The difference in filtration areas between the systems of lines 1 and 2 has a significant impact on their performance. Line 1 has a filtration area of 12 m², while Line 2 features a larger capacity with an area of 18 m². This difference in filtration capacity allows Line 2 to handle wines with higher fouling properties more efficiently, as the larger surface area reduces cumulative obstruction and helps maintain a more stable flow rate during the process.

However, despite the larger capacity of Line 2, a deviation of 63 min was observed in its bottling plan, compared to the 61-minute deviation in Line 1. This suggests that although the larger filtration area partially mitigates the effects of fouling capacity, it does not completely eliminate the hydraulic limitations caused by the wine’s properties. These observations are consistent with findings in the literature, which emphasize that larger filtration areas provide greater flow stability, but the cumulative impact of suspended particles can still affect process efficiency.

The comparison between the lines highlights the need to carefully consider the relationship between filtration area and specific wine properties, such as viscosity and particle content, when planning and optimizing bottling processes.

6.2. Time Spent in Different System States

Based on the states described in Figure 2, the changes in the duration of stay in the five states are analyzed. This analysis considers the real scenario obtained from the production plan for lines 1 and 2 (L1-R, L2-R) and evaluates the duration of stay between the states considering the optimal allocation of the bottling lines and their sequence (L1-O, L2-O).

The results presented in Figure 3 show a significant reduction in the duration of stay in state W2, corresponding to the bottle filling process. For line 1, the time in W2 is reduced from 492.35 to 427 min, representing a 13% decrease. In line 2, the time in W2 decreases from 378.94 to 291.65 min, equivalent to a 23% reduction. Additionally, a decrease is observed in the total processing time: from 627.35 min to 577 min in line 1 (a reduction of 8%) and from 543.94 min to 501.65 min in line 2 (a reduction of 7%) as shown in Figure 4.

These results suggest that the optimal allocation of the bottling lines allows for reducing the specific time of the filling process and also optimizing the total processing time, considering the transitions between formats and changes of wine batches. The efficient allocation of the wines, taking into account their hydraulic properties and the operational characteristics of the bottling and filtration lines, contributes to a comprehensive improvement in the efficiency of the production system.

6.3. Production Plan Based on Optimal Assignment

The production plan based on the proposed mathematical model established that the filtration stages and bottling lines will operate jointly, assigning filtration line F2 to work exclusively with bottling line 1 (L1), and filtration line F1 to work in conjunction with bottling line 2 (L2) (Figure 5). This decision is likely due to the fact that switching between filtration lines requires an additional 90 min, which would significantly impact the total production time.

Under this assignment, the model allocated batches 1, 2, 3, 5, 6, and 9 to the F2-L1 set, while batches 4, 7, 8, and 10 were assigned to the F1-L2 set, as shown in Figure 3. For the F2-L1 set, the total estimated time was 535 min, of which 457 min were allocated to the bottling process and 78 min to adjustments and transitions. In the case of F1-L2, the total time was 510 min, with 441 min for bottling and 69 min for transitions and adjustments.

Compared to the initial assignment, where batches were distributed without considering the specific properties of the wine or its fouling potential, this new assignment achieves an 8% reduction in total production time for Line 1 (L1-O) and a 7% reduction for Line 2 (L2-O). This demonstrates that the optimal allocation takes into account not only the hydraulic properties of the wine and their impact on processing time but also the filtration capacity of each line.

Figure 6 shows the result of the optimal allocation of production batches to Lines 1 and 2. This graph complements the analysis by providing an example of how the production lines are affected in the time of each stage when a change in planning is made. In gray, the transition between one wine batch and another is presented, which is determined by the system’s state, whether in cleaning processes or line adjustments. Light blue represents the bottling of wine, and the values contained in the graph indicate the duration time of each stage in minutes.

The model optimized the process by assigning batches with higher fouling properties to the filtration line with the larger area (F2, with 18 m²), thus reducing the impact of particle accumulation on processing time. Additionally, the joint assignment of filtration and bottling lines, combined with optimized transition times between stages, allowed for a reduction in total production time without compromising operational efficiency. These results highlight that the planning based on the mathematical model not only improves the overall efficiency of the system but also addresses technical limitations without incurring additional costs.

7. Discussion

The results reveal a complex interaction between the hydraulic properties of the wine, fouling capacity, and the physical characteristics of the filtration and bottling systems. While the deviations observed in lines 1 and 2 (61 and 63 min, respectively) represent an 11% increase in the total production plan time, these differences cannot be solely attributed to the physical filtration capacity. Line 2, with a larger filtration area (18 m² compared to 12 m² for line 1), showed a similar deviation, highlighting that increasing the filtration area does not guarantee a proportional reduction in operational deviations.

This behavior suggests that the wine’s fouling capacity and hydraulic properties, such as viscosity and suspended particle concentration, play a dominant role in reducing flow rate and system efficiency, regardless of the filtration area size. While a larger area may delay the impact of fouling, it does not completely eliminate the limitations imposed by the wine’s intrinsic properties.

Variations in wine viscosity or fouling capacity (e.g., increased solids content) lead to longer bottling times and may consequently necessitate recalculating the optimal solution to adjust both the processing sequence and batch assignment. Nonetheless, the statistical validity of the allocation obtained through the MILP model does not rely on hypothesis testing or random sampling, since the approach is deterministic: the optimal solution is grounded in the mathematical formulation itself, rather than in probability-based variability. Accordingly, the model remains applicable as long as the physicochemical parameters (e.g., viscosity) or fouling indicators are properly updated; this enables dynamic, continuously optimal scheduling in line with the wine’s actual conditions, rather than focusing solely on operational transitions. Instead of employing statistical tests, the robustness of the results is determined by directly comparing the proposed allocation with the company’s current policy and by conducting sensitivity analyses, thereby confirming the model’s effectiveness under reasonable parameter changes.

Additionally, the times allocated for batch transitions remained consistent as planned, indicating that the observed deviations are directly linked to the bottling process rather than peripheral operations. This reinforces the need for targeted strategies to mitigate hydraulic limitations, such as dynamically adjusting planned times based on the type of wine being processed. These results question the sustainability of the current production-planning model, which assumes homogeneity in batch performance and does not fully integrate hydraulic and fouling differences into its design. This limitation could be addressed through a more predictive approach, incorporating real-time data on flow rate and filtration resistance, enabling more accurate planning and reducing costs associated with operational deviations.

The development of optimization models that incorporate real variables significantly contributes to planning and scheduling in the wine industry. Throughout this work, it has been demonstrated that meticulously considering critical stages such as filtration, including factors like the clogging index, operational conditions of the process, and the physicochemical properties of the wine, allows for the generation of efficient production plans that better address the inherent uncertainty of this industry’s nature. By integrating these variables into the proposed model, the allocation and sequencing of batches in the filtration and bottling lines are optimized; similarly, the total processing times are minimized and interruptions in the production line are reduced. This operational approach, applicable to wineries of any size, underscores the importance of advancing optimization models that reflect practical production conditions and address the specific complexities of the wine sector. In this context, the results of the proposed mathematical model demonstrate the advantages of an optimized assignment between filtration and bottling lines, highlighting how the combination of filtration line F2 with bottling line L1, and F1 with L2, minimizes operational disruptions by avoiding line changes that require an additional 90 min. However, these results also reveal that the model’s efficiency heavily depends on the alignment between batch properties and the filtration-line capacities. While batches assigned to F2-L1 benefited from a larger filtration area (18 m²) that mitigated the impact of fouling, batches processed in F1-L2, with a smaller area (12 m²), required more precise coordination to avoid significant delays. This underscores the need to incorporate real-time data into production planning, enabling dynamic adjustments to further enhance operational efficiency. Additionally, considering filtration lines with greater capacity can significantly contribute to improving operational efficiency without increasing operational costs, as it reduces the time associated with hydraulic limitations and fouling effects.

Although the results indicate that increasing the filtration area does not guarantee a proportional reduction in operational deviations, the implementation of parallel filtration lines emerges as a promising strategy to significantly improve both productivity and system resilience. This configuration enhances operational flexibility by enabling workload redistribution, allowing for maintenance activities without production interruptions, and adapting more effectively to different filtration requirements based on the characteristics of each wine batch. While this approach requires a higher initial investment in infrastructure, the associated benefits—such as reduced downtime, greater continuity in production flow, and improved responsiveness to variations in viscosity, solid content, and fouling potential—can lead to a substantial increase in overall process efficiency. Therefore, the analysis suggests a favorable balance between the additional cost of incorporating parallel lines and the gains in productivity and robustness, particularly in settings where high product variability and demanding operational constraints are present. Although this strategy may not completely eliminate deviations, it supports the development of a more resilient system capable of dynamically adapting to changing process conditions.

In the context of production planning in the wine industry, it is feasible to establish a higher level of planning, such as optimal scheduling on a weekly basis or over other longer time intervals. This would allow for addressing resource allocation and batch sequencing from a strategic perspective, optimizing operational efficiency and the fulfillment of medium-term objectives. However, due to the complex nature of the problem and the considerable volume of data involved, the application of exact methods like MIP can become computationally impractical for longer planning horizons. Therefore, it is advisable to consider the use of heuristic or metaheuristic methods to tackle this optimization at a macro level. These approaches allow for obtaining high-quality approximate solutions in reasonable computational times, better adapting to long-term planning needs and facilitating a more agile response to variations in demand and operating conditions.

Nevertheless, in the present study, the model is conceived for an operational planning horizon restricted to a single workshift. Processing times are introduced as calculated parameters, obtained beforehand through calculation based on the wine’s physicochemical characteristics and the operational constraints of each possible line. This external calculation significantly reduces the dimensionality of the problem, enabling the effective use of exact methods such as mixed-integer programming. Consequently, under this operational approach, the incorporation of heuristics or metaheuristics is not required, as the computational complexity remains within a tractable range.

However, if future models were to incorporate the dynamic calculation of processing times within the optimization routine, by directly embedding hydraulic and physicochemical parameters into the decision variables or constraints, the resulting increase in model complexity could challenge the feasibility of exact methods. In such cases, the use of heuristic or metaheuristic approaches would be a promising alternative to handle the combinatorial explosion of configurations and ensure reasonable solution times for practical deployment.

The results obtained demonstrate that, although the proposed model achieves efficient batch allocation and reduces operational interruptions, its performance is conditioned by the computational complexity associated with problem size and variability in operating conditions. In this context, the value of complementing exact models with heuristic or metaheuristic approaches becomes evident, as these methods help reduce computation times without significantly compromising solution quality. The structure of the problem—with multiple lines, diverse products, and sensitive operational parameters—proves to be well suited for these techniques, which have shown strong performance in similar industrial scenarios. Additionally, the observed repetition of patterns in the behavior of certain batches and line combinations suggests that machine learning-based approaches could be integrated as predictive tools or decision-support mechanisms. Specifically, models trained on historical data could anticipate hydraulic behavior and fouling potential, allowing for dynamic adjustments in planning that further enhance operational efficiency. Therefore, the results not only validate the effectiveness of the proposed model but also support the integration of more computationally flexible methods to address larger-scale or higher-variability scenarios.

Limitations and Future Work

This work, by considering specific characteristics and calculations related to the hydraulic and physicochemical properties of wine, can be computationally intensive and require a substantial amount of detailed information. This complexity may limit the model’s applicability in environments with limited data resources or where exhaustive information collection is not feasible. Therefore, it is advisable to explore new approaches based on machine learning to estimate hydraulic performance and processing time using historical process information, which could simplify calculations and enhance the model’s adaptability to different operational conditions. Additionally, other models that incorporate probabilistic scenarios can be investigated, which would contribute to a more comprehensive consideration of processing time within the production plan, allowing for better management of the inherent uncertainty in viticultural processes. Another aspect to consider is the development of a new weekly planning method that optimizes resource allocation and batch sequencing over longer periods. Finally, the use of heuristics and metaheuristics as resolution methods could offer more scalable and computationally efficient solutions, especially in larger wineries with a wide variety of SKUs and multiple production lines. These future research directions would enhance operational efficiency and also increase the flexibility and robustness of optimization models in the wine sector.

Given the inherent variability in the physicochemical properties of wine and their impact on filtration performance, it is essential to develop mathematical optimization models under uncertainty to rigorously determine the optimal filtration-system configuration. These models could integrate variables such as projected demand, expected fouling frequency, average hydraulic properties, and the availability of technical resources, with the aim of identifying both the optimal batch-to-line assignment matrix and the total filtration area required. By incorporating uncertain scenarios and stochastic parameters, such models would enable the evaluation of different infrastructure and operational combinations, identifying those that achieve a balance between equipment investment and operational costs related to maintenance, downtime, and production efficiency. This approach would support strategic decision making aimed at optimizing system profitability in the medium and long term, offering a robust tool for capacity planning in complex production environments such as the wine industry.

Furthermore, we acknowledge that the current model estimates processing times externally based on predefined physicochemical parameters. While effective, this approach limits the exploration of how variations in critical parameters (such as viscosity, filter permeability, and clogging rate) impact not only processing time but also the sequence-dependent setup tasks required. As these three variables interact and jointly affect the filtration performance of a batch, conducting a structured sensitivity analysis would provide valuable insights into how this variability propagates through the scheduling model and influences its outputs. Although any variation in these parameters represents a different wine profile, such an analysis remains essential for understanding how to better manage operational risk within a production plan. In particular, it could support the classification and handling of high operational risk batches that require special scheduling strategies. Additionally, studying this interaction can inform future efforts to incorporate processing time calculations directly within the optimization model, especially considering heuristics-based methods, where dynamic parameter-driven modeling could enhance adaptability without overwhelming computational complexity.

In response to the computational challenges commonly faced by smaller-scale wineries, it is necessary to develop simplified and adaptable versions of the proposed model. In environments without access to advanced computing resources, an effective strategy involves reducing model complexity by clustering SKUs into product families with similar operational characteristics, significantly decreasing the dimensionality of the setup time matrix. Another practical approach is the implementation of heuristic methods or predefined assignment rules for sequencing, which, although sacrificing some degree of optimality, can provide sufficiently good solutions with lower computational effort. Moreover, instead of executing the model continuously or daily, wineries may apply weekly planning cycles or utilize preconfigured scenarios based on typical operating conditions, substantially reducing the processing load. Predictive models based on historical data can also be explored to estimate processing times or hydraulic behavior without fully solving the optimization model, thereby enabling faster and more feasible decision making in resource-limited contexts. Additional practical implementation strategies suitable for small-scale wineries include limiting the number of bottle formats to reduce line-adjustment complexity, grouping wine batches with similar hydraulic properties to minimize clogging effects, and adopting operational rules based on predefined changeover matrices, thereby decreasing the need to solve the full model daily. These approaches allow for the practical adaptation of the model to smaller-scale operations, maintaining its utility and contributing to enhanced production planning in the sector. Such strategies underscore the potential of integrating simplified heuristic methodologies and predictive models to support decision making in computationally constrained environments. Aligning model complexity with available infrastructure and incorporating data-driven predictive tools helps maintain high operational efficiency, ensuring wineries of varying scales can benefit from advanced optimization techniques. Future research could focus on integrating these alternatives into a unified decision-support framework, complemented by rigorous sensitivity analyses and validations based on historical data, to fine-tune model performance and further enhance production-planning processes in the wine industry.

In addition to the simplifications proposed for small-scale operations, it is also necessary to address the computational complexity introduced by the binary decision variables associated with sequence-dependent, particularly in larger or more intricate production environments. To this end, we suggest exploring structural decomposition techniques that separate configuration-related decisions (such as equipment compatibility and format changes) from operational sequencing. This layered approach can reduce the dimensionality of the optimization space and facilitate more efficient solving routines. Moreover, the integration of heuristic and metaheuristic methods, including Genetic Algorithms, Tabu Search, or GRASP (Greedy Randomized Adaptive Search Procedure), could offer robust approximations in significantly shorter computation times. These techniques are especially suitable when the number of SKUs and transitions becomes large, enabling planners to preserve solution quality while improving scalability and responsiveness. Future research could focus on benchmarking these alternative approaches against the exact method, identifying trade-offs between optimality and practicality across different planning scenarios.

Finally, the economic evaluation of the impact of a model such as the one proposed in this study represents a particularly relevant avenue for future research. Assessing the implications of the model in terms of tangible economic benefits would contribute significantly to its adoption in the industry, while also enhancing the efficiency and competitiveness of operations. In this regard, future studies could address this gap through the use of Data Envelopment Analysis (DEA), allowing for the assessment of the relative efficiency of different process lines or wineries by considering operational cost as the Decision-Making Unit (DMU). This approach would enable objective comparisons between production systems with varying levels of technological implementation and planning criteria, providing quantitative evidence of the added value of incorporating hydraulic variables into operational decision-making.

8. Conclusions

This work constitutes a significant contribution in the field of process optimization within the wine industry through a practical application that, for the first time, integrates the final filtration stage of wine and its impact on the planning of the bottling process. By considering this stage and its interaction with the rest of the production process, a realistic and optimal allocation and sequencing of batches has been achieved. The incorporation of the hydraulic properties, the filtration matrix, and the clogging potential of the wine into the planning model allows for reduced processing times and improved operational efficiency, avoiding scenarios of uncertainty and deviations from the expected processing times.

The optimization model presented in this study has the potential to become a key tool for improving operational efficiency in filtration and bottling processes. By integrating the hydraulic properties of wine, its fouling capacity, and the specific capabilities of production lines, the model provides optimal assignments that minimize total production times without increasing operational costs. However, to validate its contribution under diverse operational conditions, additional studies are needed that consider different types of wine and industrial scenarios. This will allow for an evaluation of its effectiveness in situations with greater variability in product properties and process requirements.

Exploring alternative solution methods, such as heuristics and metaheuristics, can provide more scalable and computationally efficient options compared to traditional mixed-integer programming (MIP) methods. These approaches would facilitate the management of larger wineries with a greater diversity of SKUs and multiple production lines, promoting more agile and robust planning. Additionally, the implementation of probabilistic models that consider the inherent variability in viticultural processes would allow for better uncertainty management, ensuring that production plans are more resilient to unforeseen events and fluctuations in the operational conditions or physicochemical parameters of the wine. Overall, these future directions would significantly contribute to operational efficiency in the production-planning process, increasing the flexibility and robustness of optimization models, fostering sustainable growth and competitiveness of wineries, and helping to overcome the challenges that this industry faces globally.