Improving Scheduling Efficiency: A Mathematical Approach to Multi-Operation Optimization in MSMEs

Abstract

Optimizing the use of resources is a key aspect of organizational management. Various methods have been developed and applied to optimize different variables, including sequencing methods that aim to minimize work time. This paper presents an integrated approach for optimizing the sequencing of operations, considering indicators such as usage time, completion time, waiting time, delivery delay, and flow time. A multi-criteria optimization method with weighted aggregation was used, employing either an exhaustive search or a heuristic algorithm with nested loops, in which multiple possible combinations of operational sequences were evaluated, considering several key indicators and their respective weights. The application of the methodology in a press validated its effectiveness, providing managers with key information to prioritize the indicators according to their needs, whether optimizing resource usage or minimizing waiting times and delays. The application resulted in a 95.3% improvement in the level of utilization; a 79.3% reduction in the average completion time; a 90.5% reduction in machine waiting time; and a 90.9% decrease in product delivery delay. The results show that prioritizing the objective function leads to a balanced optimization of all indicators, improving operational efficiency and reducing flow time. This study contributes to the body of knowledge on production scheduling by offering a novel multi-criteria optimization approach in manufacturing settings. The validated methodology can be adapted to a variety of industries and offers flexibility to align with the specific interests of each organization.

1. Introduction

Time, along with money, constitutes one of the most crucial resources managed by companies in their daily operations. In general, both variables are strategically managed together, as they are integral to the company’s core objectives. Indeed, the fundamental goal of any company can be summarized in its ability to generate revenue within the shortest possible time frame [1]. In a highly competitive business environment, the optimization of both time and financial resources is essential for maintaining efficiency and profitability.

The management of time has been a central focus since the early development of administrative sciences [2]. One of the earliest and most influential contributions came from Frederick Taylor, whose pioneering studies, conducted at the beginning of the 20th century, were dedicated to the analysis and improvement of work methods aimed at optimizing performance over time. Taylor introduced the concept of scientific management, which also involved the development of techniques for measuring the time of operations and actions to reduce inefficiencies. Additionally, he advanced the idea of specialization as a way to better organize labor and improve overall productivity.

Time, particularly the time spent on the shop floor, is a variable that directly affects business productivity and is closely linked to the management of product or service quality. In this context, delivery times and the ability to meet these deadlines are vital factors in determining the level of service quality a company provides. Adherence to delivery schedules is not just an operational concern but also a measure of the company’s reputation and reliability. Numerous investigations have been dedicated to evaluating the significant impact that time has on the quality of services provided to customers [3,4].

A critical indicator often used in business management to assess the efficiency of resource usage is the level of capacity utilization [5]. This indicator serves to quantify how effectively a company’s technological investments are being used to generate returns, ensuring that capital and equipment are not left idle but are instead employed in productive activities. As such, the level of capacity utilization should be considered as an additional key factor when addressing activity sequencing problems, as it directly impacts the optimization of operational processes.

In the 1940s and 1950s, the emergence of a mathematical school of thought within administrative sciences marked a significant turning point in operations management. This school of thought was primarily focused on applying mathematical methods to optimize the performance of various resources, including time, within the framework of existing constraints. During this period, several important methods were proposed to solve different types of operational problems. These include the simplex method [6], transportation problems [7], and task assignment problems [8], among others, all of which have had lasting influence on the field of operations research.

Activity sequencing problems, which fall within this broader domain, led to the development of practical, yet powerful proposals. One of the most famous contributions is Johnson’s well-known sequencing rule [9,10], which optimizes the sequencing of tasks in systems with two or three machines. This rule represents a simple approach to solving complex sequencing challenges. The focus of the current work is to extend these foundational approaches to address problems involving the sequencing of multiple tasks or activities across multiple machines or operations, with the aim of optimizing multiple performance indicators simultaneously.

In the broader context of operations management research, numerous studies have been conducted on the sequencing of operations. A search for the term “sequencing of operations” in the Scopus database, specifically in the TITLE-ABS-KEY field, reveals fewer than 100 publications on the topic. The general characterization and analysis of these publications can be effectively achieved using Bibliometrix software (Version 4.3.0). When applying this tool, the results displayed in Figure 1 provide a detailed overview of the scope and trends in the existing literature on this subject.

In Figure 1, it is possible to observe that the distribution of research is not homogeneous across different countries. Instead, it is predominantly concentrated in nations with the highest levels of industrialization and economic development.

This concentration of research output suggests that industrialized nations are more likely to have the infrastructure, resources, and expertise necessary to conduct advanced research in this field. Moreover, this distribution highlights an important gap, as it is not uniform across all countries. Additionally, studies on the topic show a low level of interaction with researchers from other countries, indicating that there may be a lack of global collaboration in this area. This could be an area for potential improvement, as international cooperation often leads to richer and more diverse research findings. Over the past few years, the themes and variables within the research on sequencing problems have diversified significantly. In the late decades of the 20th century, the focus was predominantly on three main areas: planning or scheduling, the development of heuristic algorithms, and process duration as a key variable to optimize. These themes have remained critical; however, in the early decades of the 21st century, there has been a noticeable expansion in the range of variables considered, while still maintaining a connection to the original focus areas. In the lower part of the figure, it is possible to discern the continued dominance of the initial themes, but with the addition of other variables, although they have a relatively lesser emphasis.

Johnson’s Rule stands out as one of the earliest and most widely recognized algorithms for solving sequencing problems. Its significance is reflected in the numerous applications of the algorithm, many of which continue to be reported in current research [11,12]. The rules associated with this algorithm are often attributed to Roger W. Johnson; however, it is also frequently credited to Selmer M. Johnson, who introduced algorithms for scheduling tasks on machines, taking into account their operating times [13]. The main objective of Johnson’s Rule is to optimize the sequencing of tasks with respect to machine operations. The rules themselves are directed toward achieving one of two goals: either minimizing the total operation completion times (case 1) or minimizing the starting times of the operations (case 2). These objectives are fundamental to improving efficiency in various industrial and manufacturing settings. The Johnson’s Rule methodology can be summarized in three key steps:

• Identify the shortest processing time among the remaining operations.
• Assign the operation to the machine with the shortest processing time (in the case of minimizing completion times) or to the machine with the longest processing time (in the case of minimizing starting times).
• Remove the assigned operation and repeat the process until all operations have been scheduled.

Despite being first proposed many years ago, the regulatory framework of Johnson’s Rule continues to be cited in contemporary research, with numerous studies showcasing its ongoing relevance and application in optimizing sequencing problems [14,15]. Although a variety of alternative algorithms have been developed in recent years to address sequencing issues, Johnson’s Rule remains one of the most widely used and referenced. Some of these alternative algorithms include the Neighborhood Search Algorithm [16,17], the Campbell Dudek Smith (CDS) Algorithm [18,19], the Migrating Bird Algorithm [20,21], Genetic Algorithms [22,23], Simulated Annealing Algorithms [24,25], Ant Colony Optimization [26,27], and the application of Fuzzy Logic approaches [28,29]. These diverse algorithms reflect the growing complexity and scope of sequencing problems in modern operations research.

In addition to the various algorithms developed for sequencing, there are multiple real-world environments where applications of these algorithms are being reported. For instance, sequencing problems have been successfully applied to address challenges in transportation systems [30], operations related to data management and Internet services [31,32], process automation, and the deployment of robots in industrial settings [33,34], as well as in e-commerce and retail operations [35]. Each of these environments presents unique challenges that require specialized algorithms to effectively optimize performance and minimize inefficiencies.

Furthermore, the conditions under which solutions to sequencing problems are developed have evolved over time, incorporating new variables and constraints. For example, many modern problems now consider the limitations imposed by machine availability and usage restrictions [36], constraints on internal transportation [37], the need to reduce electricity consumption and associated costs [38], and the incorporation of independent preparation times for tasks [39]. These evolving conditions underscore the increasing complexity of sequencing problems and the need for adaptable solutions that can accommodate a variety of operational constraints.

Finally, the objectives targeted by the various studies in this field have also diversified. While the reduction in total flow time remains the most commonly pursued optimization objective [40,41], other factors such as usage level [42] and waiting times [43] are also commonly optimized in the literature.

Interestingly, a review of the reference framework did not reveal any studies specifically focused on managing the sequence of activities for multiple products and operations, where the goal is to optimize an objective function incorporating multiple indicators [44]. These indicators might include flow duration, waiting times, delay times, and the level of capacity utilization.

This methodological gap identified in the literature is combined in this research with the limitations faced by micro, small, and medium-sized enterprises (MSMEs), which are characterized by having less complex processes compared to large industries and greater stability, as well as by facing constraints such as a lack of highly skilled professionals capable of applying advanced solution algorithms and limited financial resources to access the necessary computing tools.

The business landscape in most countries, and particularly in developing nations, shows that MSMEs represent the predominant organizational form. Despite this significant feature, these types of enterprises are not often the practical focus of in-depth research within this field of study.

The combination of the methodological gap and the practical study object represents the main limitations that this research aimed to address.

2. Materials and Methods

The development of the applied methodology is outlined below, with its graphical representation provided in Figure 2. The methodology begins by gathering operational data from the entity under study. This initial step is crucial, as the accuracy and reliability of the input data directly influence the quality of the results generated by the methodology. The required input variables are as follows, each contributing key information to the process of optimization:

• Number of different products to be developed (CPi): This variable represents the total count of distinct products that the company or entity plans to manufacture. Each product may have its own unique set of requirements, which need to be considered when scheduling operations and allocating resources. The diversity of products introduces a level of complexity, as different products may require different operations or varying levels of production capacity.
• Number of different operations that require each product (CO): For each product, there may be a series of distinct operations that must be performed. These operations could include processes such as assembly, quality control, packaging, or testing. Understanding the total number of operations required for each product is essential for determining the workflow and for optimizing the sequencing of tasks across different machines or workstations.
• Duration of each operation for each product (DOij): The duration of each operation is a critical piece of information. This variable indicates the time required to complete a specific operation for a given product. The duration will likely vary depending on the nature of the operation and the specific characteristics of the product being processed. Accurately estimating operation durations is vital for creating an efficient production schedule and minimizing idle time or bottlenecks in the workflow.
• Production volume required for each product (TLPi): This variable defines the total quantity of each product that needs to be produced within a certain timeframe. The production volume is a key input for resource planning, as it directly influences the amount of time and resources needed to meet production goals. Meeting production targets on time requires careful coordination of operations, resources, and scheduling to ensure that capacity is sufficient and production goals are met without delays.
• Delivery dates for each batch of products (FEPi): The delivery dates are essential for understanding the timeline within which each batch of products must be completed and shipped. These dates are usually driven by customer demand, contracts, or business commitments. Properly aligning production schedules with delivery dates is a critical aspect of customer satisfaction and overall operational efficiency. Meeting these deadlines often requires advanced planning and sequencing of operations, as well as the efficient management of resources to ensure timely delivery.

Each of these variables plays a pivotal role in the overall production planning and scheduling process. The successful application of the methodology depends on the careful collection and integration of these inputs, which will then guide the optimization process aimed at improving efficiency and meeting production goals.

Once the basic information is captured, the possible combination of sequencing alternatives to be applied is generated. The number of alternatives to be applied is determined through Equation (1), where the number of alternatives is the result of all possible permutations (the number of different ways in which the operations can be arranged) without restrictions, which mathematically corresponds to the factorial of the number of operations.

CAA = COǃ

(1)

Once the number of alternatives has been determined, the next critical step involves determining the start date (FIO) and completion date (FTO) for each operation associated with each product. This step is crucial, as it lays the foundation for scheduling the sequence of operations required for production. For each operation, determining the exact start and finish times helps to ensure that resources are allocated effectively and that there are no overlaps or bottlenecks in the process. These dates serve as the cornerstone for constructing a realistic and optimized production schedule.

The methodology assumes that for each alternative, two matrices must be established. These matrices are designed to organize and represent the flow of operations for each product, with the specific dates of start and completion for each task. The creation of these matrices is essential because they serve as the operational framework for the production planning process. The matrices are structured in such a way that they capture the temporal aspects of the workflow, ensuring that all operations for each product are completed in a timely manner while adhering to the overall production timeline.

To establish these matrices, certain key assumptions must be made regarding the closure times for each product and each operation. These closures are critical to ensuring that the sequence of tasks is properly aligned and that each operation begins only when the necessary prerequisites have been completed. The closure times for each operation in each alternative are defined by the relationships between the start and completion dates of the tasks. These relationships are mathematically represented in Equations (2) and (3), which serve to define the temporal constraints and dependencies between the operations.

By utilizing these two matrices and the corresponding equations, it is possible to create a comprehensive and optimized scheduling system that takes into account the various alternatives for operation sequencing. The proper definition and calculation of the start and completion dates for each operation are crucial in minimizing idle times, ensuring that resources are used efficiently, and meeting the required delivery dates for each product. This approach provides a clear roadmap for achieving production goals and improving overall operational efficiency.

$$\mathrm{FIO}=\left[\begin{array}{ccc}{FIO}_{11}& {FIO}_{1j}& \\ {FIO}_{i1}& & \\ & & {FIO}_{ij}\end{array}\right]$$

(2)

$$\mathrm{FTO}= \left[\begin{array}{ccc}{FTO}_{11}& {FTO}_{1j}& \\ & & \\ & & {FTO}_{ij}\end{array}\right]$$

(3)

where

i: product;

j: operation.

To configure these matrices, the following steps are applied. The values in both matrices are determined according to the following rules:

The initial operation is scheduled to start at time zero.

FIO_1,1 = 0

(4)

The completion time of the subsequent operations for the first product depends on the start time plus the duration of each operation.

FTO_1,j = FIO_1,j + DO_1,j

(5)

Each subsequent operation for the first product begins at the completion time of the preceding operation.

FIO_1,j = FTO_1,j−1

(6)

The start time of the first operation for subsequent products will be equal to the completion time of the corresponding operation in the previous product.

FIO_i,1 = FTO_i−1,j

(7)

The start time of subsequent operations in the following products will be equal to the greater value between the start time of operation j − 1 in product i and the completion time of operation j in product i − 1.

To i > 1

$${\mathrm{FIO}}_{\mathrm{i},\mathrm{j}}=\mathrm{maximum} \mathrm{of} \left\{{FIO}_{i,j-1}y{FTO}_{i-1,j}\right\}$$

(8)

Once the start and end points of each operation have been established, the calculation of the management indicators to be optimized is carried out. The indicators to optimize are the following:

2.1. Average Delay (AD)

To determine the average delay, Equation (8) applies, since a delay only occurs when the completion date exceeds the agreed delivery date, the delay is defined as the difference between the completion date and the delivery date.

$$\mathrm{R}(\mathrm{i})=\left\{\begin{array}{c}If OPF(F,\mathrm{j})>FE\left(\mathrm{i}\right) \mathrm{t}\mathrm{h}\mathrm{e}\mathrm{n} \mathrm{R}\left(\mathrm{i}\right)=\mathrm{O}\mathrm{P}\mathrm{F}(\mathrm{F},\mathrm{j})-\mathrm{F}\mathrm{E}\left(\mathrm{i}\right)\\ If OPF(F,j)\le FE\left(\mathrm{i}\right) \mathrm{t}\mathrm{h}\mathrm{e}\mathrm{n} \mathrm{R}\left(\mathrm{i}\right)=0\end{array}\right.$$

(9)

$$\mathrm{AD}={\displaystyle \frac{{\sum }_{i=1}^n{R}_i}{n}}$$

(10)

where

OPF(F,j): completion time of product i;

FE(i): product i delivery date;

n: number of product s.

2.2. Waiting Time (WT)

The waiting time is determined using Equation (11), as the total waiting time results from the sum of all differences between the start time of each operation and the completion time of the preceding operations.

$$\mathrm{WT}={\sum }_{i=1}^n{\sum }_{j=1}^{k}OPI\left(i,j\right)-OPF(i-1;j)$$

(11)

2.3. Flow Time (FT)

To calculate the flow time, Equation (12) must be used, where the flow time is defined as the completion time of the final operation in the last product.

FT = OPF(i,f)

(12)

OPF(i,f): final value of the OPF matrix—decides the final value of the last operation of the last product produced.

2.4. Average Completion Time (ACT)

The intermediate completion time is determined using Equation (13), considering TTP as the average of the completion times for each product.

$$\mathrm{ACT}={\displaystyle \frac{{\sum }_{i=1}^{n}OPF(i,f)}{n}}$$

(13)

where

OPF(i,f): value of the final operation (f) for product i.

2.5. Usage Time (UT)

To determine the level of utilization, use the application of two different options. Total usage time (UT) assumes that all equipment will be assigned to produce only the analyzed products from the start of the operation of the first product until the end of the operation of the last product. Relative usage (RU) considers that once all the products of an operation are finished, the equipment will be available for other products.

2.6. Total Usage Time (TUT)

For this indicator, the numerator includes the sum of all operation times, and it is assumed that all equipment remains available for use until the final minute of completion of the last product.

$$\mathrm{TUT}={\displaystyle \frac{{\sum }_{i=1}^n{\sum }_{j=1}^m{DO}_{ij}}{n\ast OPF(n,f)}}$$

(14)

where

OPF(n,f): value of the final operation (f) for the last product that is produced.

2.7. Relative Usage (RU)

In this case, the numerator includes the sum of all operation times, while the denominator only considers the completion times of the last operation on each piece of equipment. In other words, the time difference between the completion of the last operation on each machine and the completion of the final operation in the last product is not included in the total time base. This means that each piece of equipment is considered fully available for new tasks as soon as it completes its assigned operations for the analyzed products.

$$\mathrm{RU}={\displaystyle \frac{{\sum }_{i=1}^n{\sum }_{j=1}^m{DO}_{ij}}{{\sum }_{i=1}^{n}OPF(i,f)}}$$

(15)

These indicators must be determined for each of the possible sequences of operations. Likewise, it is possible to identify which combinations achieve the optimum value in each indicator.

2.8. Definition of Objective Function

Having determined the indicators, we must proceed to establish the objective function. This is the result of the weighted sum of previous indicators considering the weight of influence or importance of the indicators, and the meaning of each indicator. That is, there are indicators, such as profit, that add to the objective function that is defined based on a maximization criterion, but others, such as average delay, waiting time, or flow time, subtract from the objective function. The objective function can be denoted by Equation (16).

Max OF = (PIi × UT) − (PIi × RP) − (PIi × TE) − (PIi × TTP)

(16)

where

PIi: weight or importance coefficient of the indicator i.

The weight of each indicator must be defined based on the specific context and policies of the organization where the methodology is applied. For this purpose, the Delphi method is recommended. Even though indicators for the level of utilization of capabilities to gain accuracy in the analysis will be defined, considering that these are highly correlated, only the level of total utilization is introduced into the OF. For similar reasons, although the flow time and intermediate termination time are determined, the latter is included.

Likewise, considering that the indicators are not limited to the same scale, to be able to introduce these into the objective function, the indicator values for each alternative are standardized and account for the positive trend in each indicator. That is to say, the utility is better as it grows while others are better as it decreases. To standardize indicators, Equations (17) and (18) are used.

$$\mathrm{Standardization}-={\displaystyle \frac{MAX Indicador-Valor Indicador}{Max Indicador-Min Indicador}}$$

(17)

$$\mathrm{Standardization}+={\displaystyle \frac{Valor Indicador-Min Indicador}{Max Indicador-Min Indicador}}$$

(18)

The search for the best alternative is carried out using a heuristic algorithm with nested loops, which enable the efficient exploration of multiple combinations of operational sequences. This approach represents an application of the Weighted Sum Model within multi-criteria decision-making (MCDM) techniques, facilitating the identification of the solution that optimizes the overall performance of the system under study.

3. Results

The methodology described earlier was implemented in a printing workshop, where five basic operations were defined for processing five types of products, based on the orders received. These products varied in nature and required different operational steps, each with its own distinct characteristics and production needs. The information regarding the products and their corresponding operations is summarized in

Table 1. General operational data.

Products	Operations					Quantity	Delivery Date
	Design	Impression	Bookbinding	Cutting	Packaged
Invoices	1	8	3	4	2	1000	28
Cards	2	6	1	3	1	4000	30
Books	8	24	15	10	16	800	85
Advertising	6	10	1	7	4	750	42
Gigantography	12	14	1	8	8	1000	58

Note: The delivery date is expressed in hours, assuming eight hours of work per day.

. This table provides a comprehensive overview of the production process, including the number of operations each product undergoes, the quantity of each product to be produced, and the required delivery dates for each batch. These details are essential for understanding the complexity of the production scheduling task and for ensuring that the methodology is applied effectively.

In line with the objective function, the experts within the organization chose to assign equal weights (0.25) to each of the operational indicators considered in the decision-making process. By allocating equal importance to all the indicators, the methodology ensures a balanced approach to optimization. The goal was to find an optimal sequencing solution that minimizes delays, optimizes resource usage, and ensures that production targets are met within the stipulated timeframes.

Given that five different products were involved in the process, a total of 120 possible sequences could be generated. Each of these sequences needed to be evaluated based on the defined operational indicators, and the best sequence was selected according to the objective function. This approach is vital because it enables the company to analyze all potential sequencing options and select the one that optimizes key operational factors.

Table 2. General behavior of indicators.

Indicators	Minimum	Average	Maximum	Unit
Total usage	0.3043	0.3597	40.23	%
Partial usage	0.4310	0.4950	54.83	%
Average completion time	63.8	71.0	81.2	hours
Waiting time	24	107.52	172	hours
Average delay	11	24.1	36.2	hours
Flow time	87	98.033	115	hours
Objective function (OF)	−0.2562	0.08	0.3433

summarizes the general behavior of the indicators, providing a clear snapshot of the performance across all sequences and helping to identify trends and areas for improvement.

These indicators provide critical insights into the efficiency of the various sequencing alternatives. For instance, the total and partial usage rates reflect how well the available resources are being utilized, with higher values indicating more efficient use of production capacity. Similarly, the average completion time, waiting time, and average delay offer insights into the overall timeliness of the production process. The flow time, which indicates the total time required to complete a product from start to finish, also serves as a vital metric for assessing overall efficiency. Lastly, the objective function (OF) value, calculated from the combination of these indicators, provides an overall measure of the optimization achieved. A positive OF indicates a better performance, while a negative value suggests that the current sequencing may need adjustment.

By analyzing these indicators and comparing them across the various possible sequences, the methodology enables the selection of the most optimal sequencing solution for the production process. This approach not only improves operational efficiency but also helps in meeting customer demands and delivery timelines, ensuring that the company can operate smoothly and competitively in the marketplace.

The values of the indicators are prioritized according to each sequence, and the resulting performance of each alternative is evaluated.

Table 3. Indicator values prioritizing each indicator based on the corresponding sequence.

Prioritized Indicator	Alternative	Sequence	Total Usage	Average Completion Time	Waiting Time	Average Delay	Objective Function
Total usage	109	A-C-E-D-B	40.23	63.8	114	23.4	0.2968
Average completion time	4	E-D-B-A-C	30.43	81.2	68	21.4	−0.2349
Waiting time	115	A-B-E-C-D	0.3365	72.4	24	11	−0.1772
Average delay	115	A-B-E-C-D	0.3365	72.4	24	11	−0.1772
Objective function (OF)	50	C-E-D-A-B	0.9533	0.2069	0.0946	0.0794	0.3436

Note: The values for the objective function (OF) are normalized, while the remaining indicators are presented on their normal scale.

presents the prioritization of the indicators and their corresponding sequence assignments. Each row in the table represents a different sequence for the production process, with the key performance indicators (KPIs) calculated for each sequence. This allows for a direct comparison of how each sequencing alternative impacts the various operational metrics.

The table demonstrates how different sequencing strategies prioritize certain performance indicators. For example, when the total usage is the prioritized indicator, the sequence A-C-E-D-B results in the highest total usage value (40.23%), but also reflects certain trade-offs in other areas, such as a relatively higher waiting time (114 h) and average delay (23.4 h). On the other hand, when the objective function is prioritized, the C-E-D-A-B sequence achieves the highest normalized value for the objective function (0.3436), showing that it provides the best overall performance across the evaluated indicators, including lower waiting times and delays (Figure 3).

The objective function serves as a composite measure of performance, synthesizing the individual indicators into a single value that reflects the overall effectiveness of the sequence in optimizing the production process. While individual indicators such as total usage, average completion time, and waiting time are important, the objective function allows for a more holistic assessment of how well the chosen sequence balances the trade-offs between different operational goals (Figure 4).

Similarly, in Figure 5, the relationship between the indicators that negatively influence the objective function and the utilization indicator, which has a positive influence, can be observed. When comparing these indicators, a clear pattern emerges: the negative influence indicators tend to intersect with the utilization indicator in the central area of the alternatives. This central intersection is significant because it reflects a balance point where the trade-offs between the different indicators become apparent. The positioning of these intersections corresponds with the overall behavior observed in the other graphical representations.

This central area represents a critical zone within the optimization process, where adjustments to certain indicators can have a direct impact on others. Specifically, the negative influence indicators, such as waiting time, average delay, or flow time, show a tendency to converge with the utilization indicator, suggesting that higher resource utilization may be associated with higher waiting times or delays in the production process. This intersection highlights the inherent trade-offs in production scheduling, where maximizing one performance metric (such as utilization) can sometimes lead to negative consequences in others.

The central intersection in Figure 5 aligns with the findings in the other graphs, reinforcing the concept that achieving an optimal solution often involves balancing conflicting objectives. The intersection area thus serves as a visual representation of the decision-making space where different strategies and sequencing alternatives are evaluated. By understanding this relationship, it becomes easier to make informed decisions regarding the prioritization of certain indicators over others in pursuit of the best overall optimization of the production process.

Finally, Figure 6 presents the Gantt charts, which visually depict the sequencing of activities based on the prioritization of the fundamental indicators. Gantt charts are invaluable tools in project management as they provide a clear and concise overview of how different tasks are scheduled over time, helping to identify potential overlaps, delays, or underutilized periods. In this case, the Gantt charts illustrate the behavior of the various sequencing alternatives, with each chart reflecting the prioritization of a specific indicator.

When prioritizing the level of utilization, as shown in the first Gantt chart, the activities appear more compact, indicating that the available resources are being used efficiently with minimal idle time. The shorter waiting times between operations highlight the effective allocation of resources, resulting in a smoother and more continuous workflow. This is an optimal scenario for resource utilization, where the goal is to maximize the throughput of operations without unnecessary delays or idle periods. The compactness of the activities suggests that the production process is streamlined, and resources are being utilized as much as possible within the available time frame.

On the other hand, when the average delay is prioritized, the Gantt chart reveals a different scenario. In this case, an increase in waiting times becomes apparent, as well as a greater dispersion of activities. This suggests that prioritizing delay minimization can lead to a more fragmented production schedule, with gaps between operations that may be necessary to manage delays more effectively. Additionally, this approach results in an increase in flow time, as the overall time from start to finish of the production process is extended due to the added delays and less efficient activity sequencing.

The third Gantt chart, which prioritizes the objective function, demonstrates a more balanced outcome. Here, the optimization of flow time is achieved, leading to a decrease in the overall time required to complete the production process. Furthermore, the concentration of activities indicates that resources are being more effectively allocated, resulting in fewer delays and idle periods. This optimization ensures that production proceeds efficiently while meeting the goals of minimizing both waiting times and flow time. The Gantt chart shows that the objective function, by considering multiple factors simultaneously, leads to a more cohesive and efficient production schedule.

Overall, Figure 6 provides valuable insight into how the prioritization of different indicators impacts the sequencing of activities. The visual representation of the Gantt charts allows for a clear understanding of how resource utilization, waiting times, delays, and flow times interact, helping to inform decision-making and ultimately optimize the production process.

4. Discussion

The results obtained in this study were compared with the findings from several previous investigations in the field of operations management and sequencing problems. One of the main points of comparison is the utilization of Johnson’s Rule, which is a well-established method for optimizing sequencing tasks, particularly in two- or three-machine systems. Although Johnson’s Rule has been widely applied, our research extended this classical approach by incorporating multiple operations and products, considering several performance indicators such as resource utilization, average completion time, waiting time, average delay, and flow time. Previous studies like those of Johnson [13] and Agrawal et al. [34] have focused on relatively simpler setups, while our study addressed a more complex system with five operations and five products.

In addition to extending Johnson’s foundational approach, this study addresses gaps identified in recent literature. For instance, Wu et al. [9] and Meng et al. [10] validate Johnson’s Rule in two-stage systems but do not address multi-product, multi-operation contexts. Meanwhile, advances in genetic algorithms and fuzzy multicriteria methods [11,12,29] have shown strong results for complex problems, but their application in MSMEs remains limited due to computational demands. Heuristic methods like adaptive variable neighborhood search [17] and simulated annealing [25] require expert tuning and are rarely tailored for resource-constrained environments. Our model complements these approaches by offering a weighted aggregation strategy that can be adapted with expert judgment (e.g., Delphi method), aligning better with the operational context of MSMEs.

Additionally, the integration of usage time, waiting time, average delay, and flow time into a unified objective function addresses the challenge identified by Xiong et al. [11] and Ren and Meng [27], who emphasized the need to balance multiple performance criteria. By explicitly modeling trade-offs among these indicators and validating the approach in a real-world MSME setting—an aspect often absent in algorithm-focused literature—the study goes beyond theoretical experimentation and contributes to applied operations management. This balance between theory and practice positions the proposed method as both innovative and accessible to a broader range of industrial applications.

Our methodology also accounts for the trade-offs between different indicators, which is a more advanced approach compared to previous work. For example, optimizing total usage often leads to higher waiting times and delays, as seen in Table 3. This observation is consistent with the findings of Abdi and Labib [5], who discussed how increasing machine utilization can negatively impact waiting times and delays in certain scheduling models. In contrast, prioritizing waiting time results in increased flow time and a more dispersed scheduling arrangement, similar to what was observed in previous studies on scheduling with multiple objectives, such as those by Mangelsdorf et al. [2] and He et al. [30], where there was a clear indication of balancing trade-offs.

Furthermore, our study aligns with the work of Mashuri et al. [18], who used the CDS algorithm to optimize production time and minimize waiting and idle times. While their focus was on heuristic algorithms, our methodology integrates a comprehensive objective function that includes multiple performance indicators, a step further in optimizing production scheduling for multiple products. The objective function’s role as a composite measure reflects findings in the literature, particularly from studies like those of Pérez-Campdesuñer et al. [4], which emphasized multi-objective optimization in manufacturing scheduling.

The use of Gantt charts in our research provided clear visual insights into how different priorities influence scheduling, showing that prioritizing utilization leads to a more compact schedule with fewer idle times, as seen in Figure 6. This outcome is consistent with findings from previous studies, such as those of Xiong et al. [15], where the optimization of scheduling resulted in more efficient use of resources. However, prioritizing the objective function resulted in better overall performance by balancing different objectives, reflecting findings from Shan et al. [33], who similarly emphasized the importance of balancing conflicting goals to improve scheduling outcomes.

In summary, the study’s results corroborate and extend previous research by applying advanced mathematical methods to optimize sequencing across multiple operations and products. The objective function developed here offers a more holistic approach compared to traditional methods, allowing for more nuanced decision-making that balances performance metrics. This makes the methodology particularly suitable for practical adoption in small-scale manufacturing environments facing resource and capacity constraints.

5. Conclusions

This study presents a robust and practical methodology for optimizing production sequencing by integrating multiple operational indicators: total usage, waiting time, average delay, and flow time, into a single, adaptable objective function. This comprehensive approach to production planning allows managers to prioritize indicators based on their specific goals, whether maximizing resource utilization or minimizing delays and waiting times. Its application in a real-world press organization validated the model’s effectiveness and operational relevance.

The results demonstrate that prioritizing the objective function, which simultaneously considers all indicators, yields the most balanced and efficient scheduling solution. This leads to improved operational performance, optimized resource allocation, and timely product delivery, even in complex production environments. Importantly, the methodology offers a clear visualization of trade-offs between different performance metrics, aiding decision-making in practical settings.

The flexibility of the proposed approach makes it applicable across various industries and manufacturing systems, particularly those operating with limited resources. However, the present study is based on a specific organizational case, and future investigations are needed to test its generalizability under new conditions and more complex operational environments. Such exploration will help identify necessary adaptations for broader applicability.

Some limitations must be acknowledged. The methodology assumes that all data inputs are accurate, consistent, and readily available, conditions that may not always be present in real-world scenarios. Moreover, the computational complexity increases with the number of products and operations, which may pose constraints for large-scale systems. Future research should explore how the model behaves with larger datasets, and how it can incorporate real-time or variable data for dynamic scheduling adjustments.

In addition, while this study was applied to a relatively small number of products and operations, it provides a foundation for further research on scalability. This includes evaluating the model’s effectiveness in more complex production structures and exploring the integration of machine learning techniques to enhance the prediction of operation durations and resource availability. Future work should also compare the proposed methodology with other solution approaches, not only in terms of performance outcomes and processing efficiency, but also by considering technical and financial constraints, especially in MSMEs from developing economies.

It is important to emphasize that, although the proposed methodology does not introduce a new metaheuristic algorithm, it offers a novel integration of multiple operational indicators into a unified optimization function tailored to the realities of MSMEs, a segment often overlooked in sequencing research. Its successful implementation in a real-world context highlights both its feasibility and practical value. A promising line of future research is the formal benchmarking of this method against established algorithms such as Genetic Algorithms, Simulated Annealing, or Ant Colony Optimization, particularly with respect to scalability and performance in more demanding scenarios.

In conclusion, this research contributes to the body of knowledge on production scheduling by presenting a novel multi-criteria optimization approach that is both operationally efficient and accessible to resource-constrained environments. The proposed methodology not only enhances production efficiency but also equips decision-makers with a practical tool to balance competing operational goals, making it highly relevant for modern manufacturing contexts.