Enhanced Decision Support for Multi-Objective Factory Layout Optimization: Integrating Human Well-Being and System Performance Analysis

Abstract

This paper presents a decision support approach to enable decision-makers to identify no-preference solutions in multi-objective optimization for factory layout planning. Using a set of trade-off solutions for a battery production assembly station, a decision support method is introduced to select three solutions that balance all conflicting objectives, namely, the solution closest to the ideal point, the solution furthest from the nadir point, and the one that is best performing along the ideal nadir vector. To further support decision-making, additional analyses of system performance and worker well-being metrics are integrated. This approach emphasizes balancing operational efficiency with human-centric design, aligning with human factors and ergonomics (HFE) principles and Industry 4.0–5.0. The findings demonstrate that objective decision support based on Pareto front analysis can effectively guide stakeholders in selecting optimal solutions that enhance both system performance and worker well-being. Future work could explore applying this framework with alternative multi-objective optimization algorithms.

1. Introduction

In factory layout planning (FLP), the strategic arrangement of resources and equipment within manufacturing facilities is crucial to ensure efficient manufacturing processes. Systematic layout planning (SLP) is widely recognized as the state-of-the-art method for factory layout design and planning [1]. Traditionally, SLP focuses on how to plan and optimize positions of rectangular or quadratic resources and equipment in a 2D top-view environment; emphasizing the shortest distance between rectangular and quadratic resources as the primary optimization objective in such layouts, see, e.g., [2,3,4,5,6,7]. While effective for space management in two dimensions, these approaches neglect critical considerations of the vertical dimensions of layout planning, limiting their ability to fully account for human-centric factors, spatial aspects related to the height of resources and operational flexibility.

FLP involves multiple, often conflicting, objectives, such as minimizing space utilization, improving production throughput, optimizing material flow, reducing bottlenecks, and ensuring worker safety and well-being. These factors must be balanced alongside equipment requirements, production needs, capacity, logistics, and safety considerations. The inherent complexity of FLP makes it a classic example of a multi-objective optimization (MOO) problem, hence resulting in solutions that often represent trade-offs between competing objectives [8].

The increasing focus on human-centricity in design emphasizes the need to incorporate human factors into the process. Prioritizing worker well-being and safety alongside productivity is essential, particularly in the context of Industry 5.0, which motivates the creation of more human-centered, sustainable, and resilient industrial environments [9]. Unlike Industry 4.0, which focuses primarily on automation and data exchange, Industry 5.0 emphasizes collaboration between humans and machines, integrating social and environmental considerations to enhance worker well-being and system performance [10]. Integrating human factors early in FLP allows for proactive assessments of worker well-being, mitigating risks such as musculoskeletal disorders (MSDs).

Human simulation tools, known as digital human modeling (DHM) tools, can facilitate sophisticated simulations to evaluate, verify, and visualize potential worker well-being in virtual environments. DHM tools like Siemens Jack [11], RAMSIS [12], SAMMIE [13], SANTOS [14], and IPS IMMA [15], each offer unique simulation capabilities that contribute valuable insights into worker well-being assessments within virtual environments [16]. While such simulations provide foundational insights into human well-being optimization, they frequently focus on smaller environments, such as cockpits or workbenches, rather than expansive settings such as factory layouts. Although contemporary DHM tools primarily focus on simulating concerns related to human well-being, e.g., simulating humans performing work tasks and assessing consequential risks for MSDs, it is important to recognize that human factors and ergonomics (HFE) encompass both human well-being and system performance [17]. This broader scope underscores the commitment of the HFE discipline to enhancing both individual and systemic outcomes.

1.1. Related Work

While MOO techniques have been widely explored in various optimization fields, including FLP, much research has focused on generating Pareto front solutions.

General guidelines for selecting solutions in MOO typically focus on established metrics, such as the hypervolume indicator, spacing metrics, and diversity in objective values, which are commonly used to assess solution quality [18,19]. These metrics aid in evaluating convergence, diversity, and solution spread across the Pareto front, offering valuable insights. However, their application in FLP remains underexplored.

Recent studies have highlighted the importance of MOO in layout planning and design. For example, ref. [20] focuses on minimizing logistics costs and carbon emissions in 2D factory layout optimization, while ref. [21] employs a 2D grid method for construction site layout optimization in prefabricated buildings, aiming to balance safety, hoisting duration, and economic efficiency. These studies demonstrate how multi-objective approaches can be effectively applied in 2D environments to achieve more sustainable and operationally efficient layouts.

Optimizing 3D factory layouts, i.e., accounting also for vertical aspects of layout planning and design, introduces additional complexities but also opportunities. As an example, for workstation design, the added vertical dimension (commonly the z-axis) enables the consideration of reachability, visibility, and lifting heights for workers. Conventional 2D methods are insufficient for addressing these challenges, driving the rise in simulation-based optimization approaches that provide a more holistic analysis of multiple layout scenarios in 3D virtual environments [5,22]. By simulating various layout scenarios, it is possible to evaluate their performance across multiple objectives, considering real-world constraints and potential variations in operational conditions. Simulation-based optimization can be applied to both single-objective and multi-objective problems. In a multi-objective context, it provides greater flexibility by allowing decision-makers to evaluate trade-offs between conflicting goals, such as system performance and worker well-being.

Although there is a scarcity of studies exploring MOO in 3D simulations for workstation design, ref. [23] offers one example of such a study, particularly focusing on improving HFE objectives. The study presented in [23] lays an important foundation; however, further research is needed to develop and refine these approaches, especially in integrating worker well-being with system performance in a 3D context.

1.2. Objective of the Research

This paper addresses the gap between traditional 2D layout optimization and the integration of human-centric factors, such as worker well-being measurement, accessibility, and compliance with safety standards, into 3D factory layout planning by introducing a novel approach that advances Pareto front analysis and provides a systematic method for selecting the most suitable solution in multi-objective optimization for 3D factory layout planning. Even when decision-makers can express preferences, this approach offers valuable decision support by presenting new perspectives, uncovering insights, sparking discussions, or confirming decisions, leading to a more informed and well-rounded selection process.

By leveraging metrics such as the hypervolume indicator, spacing metrics over iterations, and having diversity in objective values, along with the consideration of ideal and nadir points, this approach enhances decision-making in MOO for FLP. The analysis integrates system performance and worker well-being metrics from generated layout proposals, helping stakeholders evaluate and select optimal solutions, thereby improving decision-making in alignment with HFE principles.

The primary objective of this research is to develop and demonstrate a comprehensive framework for selecting no-preference solutions from trade-offs generated by MOO in FLP. The approach evaluates algorithm convergence, diversity, and spacing, presenting objective metrics to support decision-making, particularly in human-centric design considerations.

2. Methods

This study builds on previous research that focused on the MOO of an assembly station in a battery production factory [23]. The assembly station layout is part of a battery production process, and the goal was to optimize its configuration considering three conflicting objectives: space utilization, worker well-being, and production efficiency.

2.1. Assembly Station Simulation

The software Industrial Path Solutions (IPS, Göteborg, Sweden, https://industrialpathsolutions.se/ (accessed on 15 September 2024)), together with its DHM module, Intelligently Moving Manikins (IMMA) [15], developed by IPS AB, and used in this research in its 2023 release (Version 1), was utilized to simulate all MOO generated layout proposals of the assembly station in a battery production factory [23]. The virtual environment consists of models of resource objects and parts, spatially positioned to form the layout of the assembly station (Figure 1). An operation sequence was established in the virtual environment, defining the tasks for the manikins (human models) to perform during the simulation. This sequence includes six tasks: positioning a busbar support, mounting it with a bolt, tightening the bolt, positioning busbars one and two, and mounting the busbars with four bolts.

Iteratively, new layout setups were sent to the virtual environment, triggering simulations to retrieve corresponding objective values. The objectives included the analysis of spaghetti diagrams (which measure simulated workers’ distance walked within the layout), the OWAS Lundqvist Index (assessing worker well-being scores derived from simulations of a Swedish population within a 95% confidence interval), and area utilization (the necessary space reservation for equipment within the layout) (further details given in [23]). All these metrics reflect HFE measurements, encompassing both system performance and detailed assessments of worker well-being.

This original start setup for the simulation sequence is presented online at https://doi.org/10.5878/gat9-m562 (accessed on 15 September 2024). For further details on the assembly station simulation process see [23].

In total there are eleven decision variables representing the spatial information of the rearrangeable objects for this MOO problem (

Table 1. The eleven decision variables of the rearrangeable objects for the MOO problem.

Decision Variable	Description	Datatype
Var. 1	Object 1: x coordinate	Float value with 3 decimals [m]
Var. 2	Object 1: y coordinate	Float value with 3 decimals [m]
Var. 3	Object 2: x coordinate	Float value with 3 decimals [m]
Var. 4	Object 2: y coordinate	Float value with 3 decimals [m]
Var. 5	Object 3: x coordinate	Float value with 3 decimals [m]
Var. 6	Object 3: y coordinate	Float value with 3 decimals [m]
Var. 7	Object 3, shelf 1: z coordinate	Float value with 3 decimals [m]
Var. 8	Object 3, shelf 2: z coordinate	Float value with 3 decimals [m]
Var. 9	Object 4: x coordinate	Float value with 3 decimals [m]
Var. 10	Object 4: y coordinate	Float value with 3 decimals [m]
Var. 11	Object 4: z coordinate	Float value with 3 decimals [m]

). This includes the spatial information in the xy-plane for the objects as well as z position of two shelves and the z position of a handheld tool used by the manikin in the simulation process of the sequence carried out at the station.

2.2. Optimization Process

This study builds upon and utilizes the same dataset as the previous research on MOO in FLP of an assembly station in a battery production factory [23]. The dataset, authorized for further study, consists of 3156 solutions generated through the MOO process.

As a brief recap of the optimization process presented in [23], the initial phase involved random sampling, where it took 1506 attempts to identify 110 feasible solutions. These 110 feasible solutions formed the initial population, referred to as population zero (P(0)). After establishing P(0), the Non-dominated Sorting Genetic Algorithm II (NSGA-II) [24] was applied, evolving the population over 10 generations (up to population ten, P(10)), to yield a set of solutions that represent various trade-offs among three conflicting objectives: space utilization, worker well-being, and distance walked by the worker (affecting production efficiency).

The NSGA-II is a genetic algorithm that simulates natural evolutionary processes. It leverages biologically inspired mechanisms such as crossover and mutation to explore the solution space. Here, genes are synonymous with decision variables.

In NSGA-II, the optimization process begins with an initial set of solutions that is often generated randomly. This initial set of solutions, known as population zero (P(0)), constitutes a defined population size. A general rule of thumb is to have a population size approximately ten times the number of decision variables [24]. These initial solutions are also called parents. During each generation of the NSGA-II algorithm, the parent population produces new candidate solutions, called offspring, through crossover and mutation, setting up new gene/decision variable constitutions.

The algorithm then combines the offspring and parent solutions to form a temporary set of solutions of double the initial population size. It applies non-dominated sorting and calculates the crowding distance to rank the temporary combined list of solutions. The top-ranked solutions that fit within the defined population size are then selected to form the next parent population for the subsequent generation. This process ensures that the best solutions survive and are carried forward to the next generation.

Given the iterative nature of NSGA-II, the terms population and generation are closely related but have distinct meanings: a generation represents one iteration of the algorithm, whereas a population refers to the set of solutions at each iteration.

In this extended study, the optimization process continues from population ten (P(10)). Over an additional five generations (up to population fifteen, P(15)), NSGA-II generated 110 offspring solutions, Q(t), in each generation. Following non-dominated sorting and selection, the top 110 solutions from each generation were carried forward as the subsequent parent population, P(t + 1). This process was repeated until the final population, P(15), resulting in a complete dataset of 3156 solutions, including those generated during both the previous and current stages of the study.

It is important to note that while the final set of 3156 solutions represents a diverse set of trade-off solutions among the objectives, it does not cover the entire feasible solution space. The hypervolume indicator confirms the convergence of the algorithm, indicating that a representative set of trade-off solutions has been obtained, where the Pareto front represents the best trade-off solutions of the investigated solution space.

2.3. Convergence Analysis

To provide a comprehensive evaluation process, this study employs additional metrics to assess both the convergence of the optimization results achieved using the NSGA-II algorithm and the diversity of the solutions.

• Hypervolume indicator: The hypervolume measure was calculated using the raw objective values, with a reference point established at [1000, 1000, 1000]. This reference point symbolizes an estimated worst-case solution in the objective space. By measuring the hypervolume, one can assess the extent to which the trade-off solutions dominate the objective space over time. This metric is well-regarded in MOO, serving to evaluate both the convergence and diversity of the obtained solutions [18].
• Spacing metric and diversity in objective values: The spacing metric was calculated to quantify the uniformity of the solutions along the Pareto front. Specifically, it was computed using the formula:

$$S={\displaystyle \frac{{\sum }_{i=1}^{N-1}\left|d_i-\overline{d}\right|}{\left(N-1\right)\times \overline{d}}},$$

(1)

where $d_i$ is the distance between consecutive solutions along the Pareto front, $\overline{d}$ is the average of these distances, and N is the number of solutions in the current generation. This metric provides an indication of how evenly the solutions are spread along the front [19]. The diversity measure was used to evaluate the distribution of solutions across the objective space. In this study, diversity was assessed using the Diversity Index (DI), which considers both the extreme solutions and the spread among solutions in the objective space. The DI is computed as:

$$DI={\displaystyle \frac{{\sum }_{i=1}^{N-1}\left|f_i-f_{i+1}\right|}{Range}},$$

(2)

where $f_i$ and $f_{i+1}$ represent objective values of consecutive solutions, and Range is the difference between the minimum and maximum objective values. This measure helps to ensure that the Pareto front remains well-distributed and does not become overcrowded in specific regions [19].

Notably, no normalization was applied to the objective values during the convergence analysis; instead, the raw values for each objective were utilized directly in the optimization metrics. This approach preserves the original scales of the objectives, ensuring that the analysis reflects true magnitudes and trade-offs without introducing artificial scaling [25]. Retaining the raw objective values provides a transparent view of how the solutions evolve over time and offers a more interpretable understanding of trade-offs between the objectives. This follows well-established guidelines in MOO, such as those described in evolutionary optimization techniques like the NSGA-II algorithm [26]. Using raw values ensures that the absolute differences in magnitude between the objectives are preserved, offering direct insights into the actual trade-offs in real-world scenarios. Understanding the convergence and diversity of solutions is essential for assessing the multi-objective optimization processes, particularly in complex scenarios such as factory layout planning.

2.4. Selecting a Solution from the Pareto Front

After evaluating the convergence and diversity metrics using raw objective values, the next step is selecting the most suitable solution from the Pareto front. To ensure fairness when selecting a final solution, normalization is applied to account for differences in objective scales, providing an equal footing for all objectives during the decision-making process. Three strategies guide this selection process:

• Distance to ideal point, $z_{ideal}$: The ideal point was calculated relative to each generated population, by determining the best-achieved values for each objective within the solution space for each population. However, for the objective OWAS Lundqvist Index, where a known theoretical range exists (100 to 400) [27], normalization was based on these theoretical limits. For the objectives without known ranges, such as distance walked and area utilization, normalization is performed using the observed minimum and maximum values from the corresponding population [26]. Once the ideal point is established in this normalized space, the Euclidean distance from this ideal point to all solutions on the Pareto front is calculated [18]. In the absence of any other preferences from the decision-maker, the solution closest to the ideal point in the normalized objective space represents a balanced trade-off among objectives and is hence often considered the most desirable solution. This method ensures a balanced selection, focusing on achieving the best possible outcomes across all objectives [27].
• Distance from nadir point, $z_{nadir}$: The nadir point, representing the worst-achieved values across all objectives on the Pareto front, is also calculated in the normalized space. As with the ideal point, normalization of objectives is handled according to known theoretical ranges (for the OWAS Lundqvist Index) or observed data ranges (for distance walked and area utilization). The Euclidean distance from the nadir point to each Pareto front solution is then computed [18]. The solution furthest from the nadir point may be preferable in scenarios where avoiding poor performance across any objective is critical. This method identifies robust and risk-averse solutions but may sacrifice some potential for optimal performance in favor of avoiding the worst-case outcomes [28].
• Distance along the vector joining ideal and nadir points, d: A third technique that combines aspects of the previous approaches involves projecting the obtained solutions onto the vector joining the ideal and nadir points. The projection that is closest to the ideal point, and hence furthest from the nadir point, is often selected as a compromise solution. The distance from the ideal point along this vector is given by the equation (where s represents the solution being evaluated on the Pareto front):

  $$d={\displaystyle \frac{\left(\left(s-z_{ideal}\right)\times \left(z_{nadir}-z_{ideal}\right)\right)}{‖z_{nadir}-z_{ideal}‖}},$$

  (3)

2.5. Knowledge Discovery in the Decision Space for Decision Support

In addition to the analysis of the objective space, the decision space (decision variables) was analyzed to further enhance the decision-making process. A comprehensive understanding of how decision variables influence the objective space, particularly the Pareto front, is essential for practical decision-making. In this study, the web-based multi-criteria decision support tool Mimer [29], available at https://assar.his.se/mimer/html/ (accessed on 22 September 2024), was used to extract knowledge from the variable space. Mimer leverages an interactive data mining algorithm called flexible pattern mining (FPM) to derive human-comprehensible rules.

2.6. Details of System Performance Metrics

With each simulated layout proposal, extended system performance metrics can be retrieved, e.g., the time spent performing tasks of an assembly sequence can be classified into value-adding tasks and non-value-adding tasks [30]. This study presents this categorization of task times within the assembly station into three types, Green, Yellow, and Red times:

• Green time: Represents value-adding activities, such as the actual assembly tasks performed on the battery.
• Yellow time: Encompasses necessary but non-value-adding activities, such as picking parts from racks.
• Red time: Represents non-value-adding activities, such as walking time.

These classifications enable a nuanced analysis of how different layout configurations impact the efficiency and effectiveness of the production process. They are designed to assist stakeholders in making informed decisions when selecting the most appropriate solution from the Pareto front. By considering these additional factors alongside worker well-being evaluations (e.g., the OWAS Lundqvist Index), the study enhances its analytical depth and contributes to a more comprehensive understanding of operational dynamics in factory layout design.

The Green, Yellow, and Red times serve as extended system performance metrics that complement the primary objectives of this study: worker well-being (assessed via the OWAS Lundqvist Index), area utilization (as a system performance index measuring the area used by the proposed layout setup), and distance walked (affecting cycle time and efficiency). While these primary metrics provide a direct evaluation of system performance and human-centered factors, the Green, Yellow, and Red times offer additional details by distinguishing between value-adding, necessary non-value-adding, and non-value-adding activities. This breakdown allows for a more nuanced analysis of how different layout configurations impact task efficiency and overall production effectiveness.

3. Results

In this section, the outcomes of the MOO process applied to FLP of the assembly station are presented. The results demonstrate the convergence of the NSGA-II algorithm and its ability to generate solutions that balance the conflicting objectives of area utilization, worker well-being (OWAS Lundqvist Index), and walking distance. Key solutions were selected for further analysis based on their proximity to the ideal point, distance from the nadir point, and distance along the vector joining the ideal and nadir points.

3.1. Hypervolume Convergence

The hypervolume indicator [18] is commonly used to measure the convergence of solutions towards the Pareto front. Figure 2 shows the progression of hypervolume across 15 generations of the NSGA-II algorithm with a population size of 110 solutions, computed based on the chosen reference point specified in Section 2.3.

While there are fluctuations between populations 9 and 11, the hypervolume stabilizes more consistently between populations 12 and 15, suggesting that the algorithm is unable to find a better set of trade-off solutions. Although convergence could theoretically be called when changes between consecutive populations become minimal, it is more robust to observe several generations of stability before confirming that the trade-off solutions have fully converged. In this case, the hypervolume across populations from 12 to 15 provides such evidence of convergence.

The initial generations show a low hypervolume, as expected due to the random initialization of solutions, indicating that they are far from the Pareto front. As the algorithm progresses, a significant increase in hypervolume is observed, particularly between generations 8 and 15, signaling effective convergence towards the Pareto-optimal frontier. By the final generation, the hypervolume has reached a stable point, showing that the NSGA-II algorithm has found a well-distributed set of solutions close to the Pareto front, representing optimal trade-offs among the objectives.

3.2. Spacing Metric

The spacing metric assesses the uniformity of solution distribution along the Pareto front (Figure 3). A consistently low spacing value suggests a well-spread set of solutions, while higher values indicate clustering or uneven distribution.

In the early generations, the spacing metric exhibits fluctuation, with a peak at generation 4, indicating uneven distribution as the algorithm explores the solution space. As the algorithm progresses, the spacing metric decreases, particularly around generations 9, 10, and 11, reflecting a more uniform distribution of solutions as NSGA-II converges toward a well-distributed Pareto front.

In later generations, a slight increase in the spacing metric is observed, specifically around generations 13 to 15. This fluctuation is likely influenced by genetic operations, such as crossover and mutation, that introduce variability to maintain diversity. The spacing metric is sensitive to the specific set of solutions in each generation, which can lead to minor fluctuations as the algorithm refines the Pareto front. Overall, the decreasing trend in spacing values suggests that the algorithm achieves a balanced distribution over time.

3.3. Diversity in Objective Values

The diversity in objective values refers to the range of the three objectives, area utilization, OWAS Lundqvist Index, and walking distance, across the solutions generated throughout the optimization process. Figure 4 illustrates the progression of diversity in these objectives over 15 generations of the NSGA-II algorithm, providing an insight into the trade-offs inherent in the optimization process.

Maintaining diversity is crucial for ensuring that the algorithm explores the search space adequately, avoiding premature convergence to suboptimal solutions. This is particularly important in multi-objective optimization, where the goal is to find a well-distributed set of Pareto-optimal solutions across conflicting objectives. Proper exploration ensures that the final solutions represent a wide range of trade-offs, reflecting the inherent diversity in the problem space [18,24].

It is also important to note that no duplicate solutions exist in the dataset, as the NSGA-II algorithm preserves only the best solutions (parents and offspring) for each generation, ensuring a diverse and well-distributed set of solutions across the Pareto front. Non-dominated solutions remain on the Pareto front if they continue to represent optimal trade-offs in the population’s solution space.

3.4. Selecting a Solution on the Pareto Front

To assist stakeholders in selecting the most appropriate solution from the Pareto front, the following strategies were applied to the final population:

• Solution closest to the ideal point: This solution offers the best trade-off between minimal area utilization, improved worker well-being, and the shortest walking distance. This method focuses on achieving optimal outcomes across all objectives while maintaining balance between the objectives [28].
• Solution furthest from the nadir point: This solution, although having the largest area utilization, was furthest from the nadir point. It maintained acceptable worker well-being and minimized walking distance. This solution may be preferable when prioritizing the avoidance of poor outcomes across objectives [28].
• Best solution along the ideal nadir vector: The solution selected is the one that lies closest to the ideal point along this vector, which signifies a strong compromise between competing objectives. This solution provides stakeholders with a balanced approach, where the trade-offs between objectives are well-managed, making it a suitable option when neither extreme (ideal or nadir) is perfectly attainable [28].

Figure 5 illustrates generation 15, showcasing the Pareto front along with all solutions of the solution search space. The gray points represent all the solutions tested by the NSGA-II algorithm, while the green points highlight the Pareto front solutions. The mesh binding these points visually represents the trade-offs among objectives.

The calculated nadir point is marked in purple, while the ideal point is indicated in blue. The solution furthest from the nadir point is shown in orange, and the solution closest to the ideal point is in red. The solution closest to the ideal nadir vector is marked in yellow. This visualization helps stakeholders identify which solutions optimize multiple objectives and which ones offer more robust, risk-averse trade-offs.

The solution closest to the ideal point, which was also the solution closest to the ideal nadir vector (solution 2808), and the solution furthest from the nadir point (solution 2466), were both further evaluated for system performance using additional metrics (Figure 6). However, it is important to note that this alignment between the solution closest to the ideal point and the one closest to the ideal nadir vector is not guaranteed in every case and may vary depending on the nature of the Pareto front in different optimization runs.

System Performance Measures

As shown in Figure 6, the system performance metrics for both solutions are evaluated based on Green time, Yellow time, and Red time:

• Solution 2466 exhibits the longest Green time at 23%, meaning it allocates slightly more time to value-adding tasks than Solution 2808, which has 21% Green time;
• Solution 2808 shows a slightly longer Red time (59%) compared with Solution 2466 (55%). Minimizing Red time is crucial as it corresponds to wasted time in the layout, particularly walking distance;
• Solution 2466 exhibits longer Yellow time (22%) compared with solution 2808 (20%).

Since solution 2808 has a slightly longer distance walked (37.80 m) compared with solution 2466 (36.3 m), it is expected the Red time is longer for solution 2808.

3.5. Using Knowledge Discovery of the Solutions Space for Decision Support

Using the Mimer tool, knowledge discovery provided valuable insights by identifying a human-comprehensible rule in the solution space that aligns with the found non-dominated trade-off solutions. The flexible pattern mining (FPM) algorithm extracted a key rule applied to 61.82% of the trade-off front and only 25.15% of dominated solutions:

var 8 < 1.094 ⋀ var 5 > 59.328 ⋀ var 6 > 29.646 ⋀ var 10 > 30.256 ⋀ var 10 < 30.331

(4)

The parallel coordinate plot in Figure 7 visualizes these patterns.

Table 2. The decision variables (var.) of the solutions 2466 and 2808.

Solution Number	Var. 1	Var. 2	Var. 3	Var. 4	Var. 5	Var. 6	Var. 7	Var. 8	Var. 9	Var. 10	Var. 11
2466	59.925	28.287	59.607	31.030	59.366	29.657	0.583	1.093	57.294	30.300	2.193
2808	59.897	28.287	59.925	31.030	59.329	29.657	0.569	1.087	57.298	30.300	1.989

presents the decision variables for Solutions 2466 and 2808, which both align with the extracted rule.

3.6. Summary of Results

In summary, the NSGA-II algorithm effectively generated a diverse set of well-distributed Pareto-optimal solutions for the FLP problem. The analysis focused on two particularly interesting solutions: the one closest to the ideal point and the one furthest from the nadir point. These solutions demonstrated significant improvements in area utilization, worker well-being, and walking distance compared with the initial layout and other generated solutions. By narrowing the selection process to these two solutions, a more focused comparison is achieved, highlighting the trade-offs involved in MOO in layout configurations.

The integration of system performance metrics, such as the Green, Yellow, and Red times, into the decision-making process provides a comprehensive view of production efficiency, further aiding stakeholders in making informed decisions.

While the NSGA-II algorithm was used in this study, the method of utilizing the ideal and nadir points for analyzing the Pareto front is applicable to other multi-objective optimization algorithms.

This approach offers a replicable and objective approach for evaluating trade-offs in MOO problems. In the context of the FLP problem in this study, it enabled the selection of suitable solutions for human-centric factory layout designs, considering both system performance and worker well-being.

4. Discussion

This study presents an objective decision approach to support decision-makers in selecting solutions generated through MOO in FLP. In this case, the NSGA-II algorithm was used to generate a set of Pareto-optimal solutions that balance system performance and worker well-being. The focus lies on how decision-makers can leverage these trade-off solutions to objectively select the most appropriate configurations for implementation, using decision support principles grounded in human-centric design and human factors and ergonomics (HFE). HFE encompasses both human well-being and system performance, making it a critical framework for evaluating and optimizing factory layout planning.

By integrating these principles, this study ensures that the selected solutions not only improve worker well-being but also contribute to overall production efficiency. This dual focus provides a comprehensive approach to decision-making in FLP, where achieving a balance between human and operational objectives is essential.

4.1. Decision Support to Select a Solution on the Pareto Front

A critical challenge when employing MOO in FLP is selecting an optimal solution from the diverse set of Pareto front solution configurations. This study proposes an objective decision-support approach based on the proximity of solutions to the ideal and the nadir points on the Pareto front. The dual strategy of leveraging the ideal point and nadir point allows decision-makers to address different organizational priorities: the ideal point focuses on achieving the best possible trade-offs among conflicting objectives, while the nadir point provides a more risk-averse perspective by avoiding poorly performing solutions.

The NSGA-II algorithm was chosen for this study because of its efficiency in handling complex multi-objective optimization problems and generating a diverse set of trade-off solutions. Genetic algorithms like NSGA-II are particularly well-suited for multi-objective optimization due to their ability to explore a large solution space and balance conflicting objectives effectively. While other algorithms can be used to produce a Pareto front, NSGA-II’s robust handling of solution diversity and convergence makes it ideal for this study’s objectives. Furthermore, the ideal point, nadir point, and ideal nadir vector used in our analysis can be applied to Pareto fronts generated by alternative algorithms, ensuring the generalizability of our decision-support approach.

The introduction of the ideal nadir vector further enriches the decision-making process by offering a compromise solution that balances the proximity to both the ideal and nadir points. This approach not only guides decision-makers in objectively selecting suitable configurations but also enhances transparency by providing evidence-based criteria.

One of the significant implications of this decision-support approach is its potential to reduce subjectivity in selecting solutions, a common challenge in multi-objective optimization. By grounding the selection process in well-defined reference points, the approach helps decision-makers align their choices with organizational goals, such as minimizing space utilization, improving worker well-being, and optimizing efficiency.

However, it is important to acknowledge some limitations of this approach. For instance, the selection of the ideal and nadir points is inherently dependent on the specific problem context and the chosen objective functions. In scenarios with more complex objectives or additional constraints, the definition of these reference points may require further refinement. Additionally, while this study demonstrated the effectiveness of using the ideal and nadir points, alternative methods such as preference-based selection or incorporating decision-makers’ subjective inputs could complement this approach in more nuanced scenarios.

Furthermore, knowledge discovery using the flexible pattern mining (FPM) algorithm and visualizations such as parallel coordinates plots can offer additional insights into the characteristic features of Pareto front solutions. The identified human-comprehensible rules align with non-dominated trade-off solutions, enhancing decision-making in complex multi-objective scenarios. By integrating FPM-based visualization techniques, stakeholders can better understand the underlying patterns and trade-offs, leading to more informed decisions.

In this study, we opted to maintain an objective approach by avoiding the application of specific weights to each objective. While weighting could be performed, it would introduce an element of subjectivity. Our goal was to allow each objective, i.e., worker well-being, area utilization, and distance walked, to be evaluated independently to reflect the inherent conflicts between them. For example, although a longer walking distance could potentially provide rest periods beneficial to worker well-being, it remains preferable to minimize walking distance to enhance productivity. By not applying weights, we avoid subjective assumptions and enable a balanced evaluation of trade-offs based on each objective’s intrinsic value. This approach, however, could be complemented in future studies by incorporating weighted preferences if organizational priorities suggest a need for it in specific contexts.

Overall, the proposed decision-support framework offers a replicable, objective method for evaluating trade-offs in MOO problems. This approach has practical relevance in scenarios where balancing human-centric and operational objectives is critical, such as factory layout planning. Future studies could explore the applicability of this framework to other domains or expand its capabilities to incorporate dynamic preferences or real-time decision-making scenarios.

4.2. System Performance and Worker Well-Being as Analysis Metrics

Once the set of solutions has been narrowed down to those closest to the ideal point and furthest from the nadir point, further evaluation is necessary to ensure the selected configurations meet broader operational goals, particularly under HFE principles. In this study, system performance was represented by the analysis of Green, Yellow, and Red times, which provides insights into how efficiently a layout supports production processes. Green time, which represents value-adding activities, was longer for the solution furthest from the nadir point (Solution 2466), and Red time, which represents non-value-adding activities such as walking, was also then shorter compared with solution 2808, demonstrating a favorable balance between system performance and spatial efficiency. However, the comparison is close and for solution 2808 the objective value of area utilization is lower than for solution 2466.

The inclusion of these assessments into the decision support framework aligns with Industry 5.0 principles, which emphasize human-centric design in industrial systems. Industry 5.0 aims to balance technological efficiency with human well-being, building on Lean principles that focus on minimizing waste and maximizing value-added activities. However, while Lean emphasizes efficiency and waste reduction, it often focuses primarily on operational metrics. HFE expands this by integrating worker well-being as a central aspect of production design. By evaluating Green, Yellow, and Red times and assessing task-specific strain on workers, this framework incorporates both Lean’s focus on efficient workflows and HFE’s emphasis on human well-being. This dual focus ensures that the selected configurations not only achieve high production efficiency but also enhance worker health and safety.

A potential enhancement to this decision support framework could involve the use of knee point detection methods. Knee points are solutions on the Pareto front that represent the most significant trade-offs between competing objectives. Identifying these points could provide decision-makers with an additional perspective, highlighting configurations that offer a balanced compromise without extreme sacrifices in any objective. Integrating knee point detection would enhance the robustness of the selection process and offer a nuanced understanding of trade-offs in complex multi-objective optimization scenarios.

5. Conclusions

This study presents a decision support approach to enhance the decision process for factory layout planning (FLP) by utilizing multi-objective optimization (MOO) and Pareto front analysis. Through the application of the NSGA-II algorithm, a set of Pareto-optimal solutions was generated, and an objective selection process was employed to identify the most appropriate configurations. By evaluating solutions based on their proximity to the ideal point (offering optimal trade-offs) and the nadir point (providing risk-averse choices), this method allows decision-makers to reduce the complexity of selecting from multiple solutions.

In addition, further evaluation of system performance according to lean classifications (using Green, Yellow, and Red times) provides an in-depth understanding and comparison between solutions. This objective evaluation method ensures that the selected solutions align with both human factors and ergonomics (HFE) principles and the human-centric approach of Industry 5.0 to production development.

The proposed method not only offers a structured and fair approach to decision-making in MOO problems but also contributes to the broader goal of creating more sustainable and human-centered industrial environments. Future work could explore the application of this decision support approach with other MOO algorithms to further enhance its generalizability and robustness across different industrial contexts.