A Novel Meta-Heuristic Approach to Solving the Assembly Line Worker Assignment and Balancing Problem with Equity of Work Distribution

Abstract

The assembly line worker assignment and balancing problem (ALWABP) has become a necessity for organizing workforces with different skill levels, particularly in Industry 5.0 environments and sheltered work centers. When the literature is examined, it is seen that most existing studies focus primarily on minimizing cycle times. However, this approach often neglects the balance of workstation utilization. As a result of this situation, ergonomic issues may arise, and fairness among employees can be negatively affected. For this reason, a new mixed integer linear programming (MILP) formulation is presented in this paper. Building upon foundational models, the proposed approach explicitly integrates a workload-smoothing objective alongside cycle time minimization to ensure fair assignments. By embedding a distinct linear formulation for workload equity—which addresses the computational complexities of traditional variance-based metrics—this method achieves superior equilibrium in task distribution. To overcome the computational intractability of this NP-hard problem in large-scale instances, a simulated annealing-based meta-heuristic algorithm is developed. The computational experiments are twofold: first, we demonstrate that the proposed mathematical model achieves superior workload balance compared to classical formulations with small-to-medium datasets; second, we validate the efficacy of the meta-heuristic against the exact model, proving its capability to generate near-optimal solutions with negligible computational time for large-scale problems. The results confirm that the proposed approach provides a robust mechanism for simultaneously enhancing line efficiency and ensuring workload equity in heterogeneous production environments.

1. Introduction

In modern manufacturing environments, particularly under the human-centric paradigm of Industry 5.0 [1], balancing productivity with worker welfare has become paramount. The traditional assumption that assembly line workers are interchangeable units is no longer valid; demographic shifts and the integration of collaborative robots necessitate the management of a heterogeneous workforce with diverse skill levels [2]. In the operations research literature, this complex planning challenge is formally addressed as the assembly line worker assignment and balancing problem (ALWABP). The ALWABP tries to determine the most appropriate assignment for both workers and workstations.

The ALWABP was formally introduced and solidified in the seminal works of [3,4], who demonstrated that treating task times as worker-dependent variables rather than fixed constants necessitates a fundamental departure from the simple assembly line balancing problem (SALBP). Through these early applications in SWCs, the authors demonstrated that overlooking specific skill profiles leads to severe bottlenecks and invalid production plans. Since those initial findings, the field has seen a massive proliferation of related research. Refs. [5,6] provided the essential framework for exact solutions through their work on lower bounds and B&B methods. However, the utility of these exact approaches fades in large-scale scenarios. Given the NP-hard characteristics of the ALWABP, researchers have had to move beyond exact methods and develop robust meta-heuristics to handle complex instances.

In Figure 1, the left side shows identical workers (W1 = W2 = W3) with fixed task times. The right side shows a heterogeneous matrix where Worker A is faster at Task 1 but incompatible with Task 2, and Worker B is slower generally but highly efficient at precision tasks. This visualizes the assignment complexity of the ALWABP.

Since then, the scope of ALWABP research has broadened significantly. Studies now incorporate a wide array of constraints and settings to better capture the messy, intricate nature of actual industrial environments. Refs. [7,8] explored the U-shaped assembly line configuration, leveraging the flexibility of workers to operate on adjacent legs of the “U” to improve line efficiency. Similarly, Refs. [9,10] extended the problem to two-sided assembly lines, a common layout in the automotive industry, where operations are performed simultaneously on both sides of a large product. Furthermore, the concept of parallelization has been rigorously investigated: Refs. [11,12,13] demonstrated that deploying parallel workstations or entire parallel lines can significantly mitigate the bottleneck effects caused by slower workers in a heterogeneous environment. These structural variations underscore that the geometry of the line is as critical as the assignment logic. In recent years, the scope of the ALWABP has broadened to encompass the inherent uncertainties and risks associated with human operators. Ref. [14] addressed the stochastic nature of worker availability, proposing robust models that account for absenteeism and turnover. Taking this a step further, Ref. [15] introduced risk-averse modeling techniques, optimizing not just for expected performance, but also for the minimization of worst-case scenarios regarding processing time variability. This robust optimization perspective is reinforced by [8,16], who employed minmax regret approaches to handle interval uncertainties in task times, ensuring that the assembly line remains operational even under conditions of significant perturbations. Most recently, the literature has begun to intersect with the technological advancements of human–robot collaboration. Refs. [17,18] have presented cutting-edge models where the “heterogeneity” stems from the distinct capabilities of human workers versus cobots. In these high-mix low-volume environments, the problem becomes assigning tasks to the resource (human or robot) that can execute it most effectively while maintaining safety and continuous flow.

As seen in Figure 2, the ALWABP has been studied as a complex problem structure in the literature. When the detailed reviews of such studies are examined, it is seen that the vast majority focus predominantly on minimization of cycle time or on the number of workstations. However, maximizing the production rate often leads to solutions where the workload is distributed unevenly. As a result of this situation, detrimental effects can occur in a heterogeneous workforce, particularly one involving disabled workers or mixed human–robot teams. Uneven distribution of workload can lead to physical fatigue and increased ergonomic risks for those assigned to bottleneck stations. Also, it may foster perceived inequity among staff. Existing research has touched on ergonomics [19,20], but the literature lacks robust mathematical models that explicitly prioritize workload smoothing alongside efficiency targets. Although job rotation has been introduced to alleviate fatigue [21], it is a reactive measure. For this reason, a proactive approach is necessary to ensure that the initial assignment itself is balanced.

The ALWABP is known as an NP-hard problem. For this reason, early research focused heavily on overcoming the computational difficulties of exact methods for large-scale instances. Ref. [22] have developed a beam search method to solve the problem. This constructive heuristic explores the search space by pruning non-promising branches. Also, Ref. [23] introduced simple heuristics. It is demonstrated that these heuristics are effective as standalone solvers. In addition, the focus has shifted toward population-based meta-heuristics to escape local optima. Ref. [24] proposed an iterative genetic algorithm (IGA) for the Type II ALWABP. Also, local search operators are integrated to intensify the solution quality. Similarly, Ref. [25] applied an artificial bee algorithm to two-sided assembly lines. Furthermore, Ref. [26] presented a variable neighborhood search (VNS) algorithm for a real-world industrial case study. This study shows the necessity of hybrid approaches for solving Type II problems efficiently.

It is crucial to note, however, that algorithmic rigor in this domain is subject to continuous scrutiny. Ref. [27] provided a critical correction to the literature by demonstrating that some constructive randomized search algorithms (specifically in prior works like [28]) could yield infeasible solutions.

As the ALWABP framework matured, researchers began incorporating realistic manufacturing constraints. Ref. [29] expanded the classical model to include sequence-dependent setup times, a critical factor in mixed-model lines where the transition between tasks incurs variable costs. Moving beyond temporal constraints to spatial ones, Ref. [30] introduced multi-manned assembly lines with positional constraints, addressing the physical limitations of workers sharing a workstation, which is particularly relevant for automotive assembly.

The most recent wave of literature acknowledges that human performance is rarely deterministic. Ref. [31] were pioneers in shifting the focus from single-objective efficiency to a bi-objective perspective, simultaneously minimizing cycle time and the smoothness index. Their population-based approach laid the groundwork for considering workload equity. They further deepened this domain in [31,32] by incorporating fuzzy set theory to model uncertain task processing times and developing robust algorithms for heterogeneous workers under uncertainty.

Parallel to this, Ref. [33] introduced risk-averse modeling techniques, utilizing conditional value-at-risk (CVaR) to manage uncertain processing times, ensuring that production plans remain reliable even in worst-case scenarios. Additionally, Ref. [34] addressed the operational stability of the line by modeling stochastic worker availability.

The landscape of assembly line research in 2024 and 2025 has diversified into advanced configuration optimization and human-centric technologies. For instance, Ref. [35] recently proposed an enhanced Benders decomposition algorithm for circular assembly lines, demonstrating the continued focus on complex line configurations and task splitting to maximize efficiency. Simultaneously, the integration of Industry 5.0 technologies has been highlighted by [36], who utilized convolutional neural networks (CNNs) to recognize collaborative actions in human–robot assembly, and [18], who addressed multi-skilled human–robot collaboration. On the ergonomic front, Ref. [20] introduced fuzzy expert systems to handle uncertainty in risk assessment.

However, despite these advancements in line configuration [35], robotic collaboration [18], and risk monitoring [36], there remains a critical gap in mathematically optimizing “social equity” in purely manual lines with heterogeneous workforces. While pioneering studies (e.g., [31,37]) have successfully introduced the concept of workload smoothing using traditional variance-based smoothness indices (SIs), these non-linear quadratic formulations often create heavy computational burdens for exact solvers in heterogeneous environments. Positioning itself against these existing models, our study specifically targets the minimization of workload discrepancies to ensure fairness, but does so through a novel, linearized “total absolute workload difference” objective. Responding to the computational limitations of previous SI approaches, we provide a mathematically tractable alternative that explicitly optimizes social equity—a key requirement for social sustainability in sheltered work centers—without incurring the exponential complexity of mixed-integer quadratic programming.

In the context of Industry 5.0 and sheltered work centers (SWCs), the significance of workload equity extends far beyond physical ergonomics. As highlighted by [4], the primary mission of SWCs is the social and psychological integration of heterogeneous workers. In such environments, perceived workload inequity can have detrimental psychological effects, leading to reduced morale and a sense of unfairness among workers [21]. From an organizational perspective, this dissatisfaction often manifests as increased absenteeism and turnover rates, which are critical disruptions in assembly lines [14,15]. Therefore, the proposed objective of minimizing workload variance is not merely a mathematical smoothing technique; it is a strategic approach to foster a resilient, human-centric work environment that sustains long-term productivity and worker well-being. We propose a novel mathematical model designed to achieve more effective assignment of workload balances compared to the classical models found in the literature. Furthermore, recognizing the NP-hard nature of the problem and the limitations of exact methods for large-scale instances, we develop a meta-heuristic algorithm based on simulated annealing. This study presents several contributions to the literature, as follows:

• The first contribution is to demonstrate that the proposed mathematical model yields a superior distribution of workload balances compared to the classical model. In this way, the equity and efficiency of worker assignments are addressed more effectively.
• The second contribution is that a simulated annealing-based meta-heuristic algorithm is applied to solve large-scale problems.
• A comprehensive comparative study is conducted to validate the effectiveness of the meta-heuristic method. However, the result quality and computational efficiency were compared with the mathematical model.

This work provides a robust framework for solving the ALWABP by integrating the mathematical model with the search capability of simulated annealing. Also, this approach aligns with the trend in the literature to combine exact and heuristic methods to solve complex real-world manufacturing problems.

Beyond its practical applications, this study also offers three distinct theoretical contributions to the ALWABP literature:

• An Alternative Modeling Perspective: We propose an alternative perspective in ALWABP modeling by moving from traditional “bottleneck-oriented” objectives (minimizing ${\mathrm{C}}_{\mathrm{m}\mathrm{a}\mathrm{x}}$) to “population-oriented” objectives (minimizing $\sum \left|L_{\mathrm{k}}-L_{\mathrm{l}}\right|$). This introduces “workload equity” as a quantifiable primary objective rather than a secondary soft constraint.
• Strategic Linearization: We provide a theoretical alternative to variance-based optimization. By formulating workload equity using linearized absolute deviations instead of quadratic penalty functions, we bypass the computational complexity of mixed-integer quadratic programming (MIQP). This strategy preserves the linearity of the solution space, facilitating more efficient exact solution procedures [5].
• Structural Handling of Interdependencies: We address the complex interdependencies between heterogeneous worker speeds and task assignments through a decoupled encoding strategy. Unlike traditional constructive heuristics that may generate infeasible assignments, our decoupled representation structurally guarantees feasibility by dynamically evaluating worker-dependent task times during the decoding process [27].

The remainder of this paper is organized as follows. Section 2 presents the formal definition of the problem and the proposed mathematical model. Section 3 details the proposed meta-heuristic algorithm, explaining the encoding scheme and local search operators. Section 4 discusses the computational experiments, validating the model against benchmark datasets and analyzing the trade-off between efficiency and balance. Section 5 and Section 6 present the discussion and limitations, respectively. Finally, Section 7 offers conclusions and directions for future research in the era of human-centric manufacturing.

2. Problem Description and Formulation

The assembly line worker assignment and balancing problem (ALWABP) addressed in this study is defined as the assignment of tasks to workstations and the assignment of heterogeneous workers to these stations. The production system is configured as a serial, paced assembly line with a fixed number of workstations. The primary objective is to minimize the cycle time $\mathrm{C}$. However, minimizing $\mathrm{C}$ alone often results in unbalanced workload distributions. In this case, some workers are utilized at maximum capacity, while others remain idle. For this reason, to mitigate ergonomic risks and ensure fairness among a heterogeneous workforce, this study extends the base model proposed by [38] by incorporating a workload equity objective.

2.1. Clarification of Terminology

To ensure consistency throughout the manuscript, we distinguish the following terms:

• Workload Smoothing: Refers to the mathematical operation or technique used to minimize variance or deviation between station workloads.
• Workload Equity: Refers to the objective quantitative state of the system, where tasks are distributed evenly among workers. It is the direct output of the smoothing process.
• Fairness: In the context of this study, fairness is treated as a theoretical modeling interpretation rather than an empirically validated psychological state. While “equity” is the measurable system metric (mathematical variance) optimized by our algorithm, “fairness” is the human-centric conceptual goal that this mathematical formulation aims to support. We acknowledge that translating mathematical equity into true psychological fairness requires empirical behavioral validation, which falls outside the scope of this optimization study.

In this paper, it is assumed that the problem is considered under the following conditions:

• A single model product is produced. Tasks with precedence relationships are assigned to fixed stations.
• A set of workers is available with varying processing times.
• Each worker is assigned to a single workstation, and a workstation can only have one worker.
• Tasks are indivisible and must be performed at a single station.
• The precedence relationships are known in advance and must be satisfied.
• The problem considers two conflicting objectives simultaneously. The first is minimization of cycle time, and the second is minimization of workload variation.

2.2. Mathematical Model

The proposed mathematical formulation is a mixed-integer linear programming (MILP) model. The core of this formulation lies in its handling of non-linear workload variation. The objective function seeks to minimize the total absolute difference between the workloads of all workstation pairs, simultaneously with the cycle time. Since the absolute value function is non-linear, we employ a linearization technique using auxiliary variables.

It is important to distinguish the proposed “total absolute workload difference” objective from the traditional “smoothness index (SI)” commonly found in the ALWABP literature (e.g., [37]). The traditional SI ($\sqrt{{\sum }_{j=1}^m{\left(C_{max}-L_k\right)}^2}$) minimizes deviation of station times from the maximum cycle time using a quadratic loss function, which primarily penalizes large deviations but introduces non-linearity. In contrast, our proposed objective minimizes the total absolute deviation of each worker’s load from the other workstations’ workloads. This formulation offers two distinct advantages: (1) it preserves the linearity of the mathematical model, avoiding the computational complexity associated with quadratic programming (MIQP), and (2) it provides a direct measure of “workload equity” that is conceptually more aligned with the human-centric principles of Industry 5.0 and sheltered work centers [4].

Notation:

Sets and Indices

• $I=\{1,\dots ,n\}$: Set of tasks, indexed by $i$.
• $K=\{1,\dots ,m\}$: Set of workstations, indexed by $k$ and $l$.
• $W=\{1,\dots ,w\}$: Set of available workers, indexed by $h$.
• $P$: Set of precedence pairs $(i,j)$, where task $i$ must precede task $j$.

Parameters

• $t_{ih}$: Processing time of task $i$ when performed by worker $h$.
• $\alpha ,\beta$: Weighting factors for the cycle time and smoothing objectives, respectively.

Decision Variables

• $x_{ik}$: Binary variable, equal to 1 if task $i$ is assigned to station $k$; 0 otherwise.
• $y_{kh}$: Binary variable, equal to 1 if worker $h$ is assigned to station $k$; 0 otherwise.
• $CT$: Continuous variable representing the cycle time of the line.
• $L_k$: Continuous variable representing the total workload of station $k$.
• $D_{kl}$: Continuous variable representing the linearized absolute difference between the workloads of stations $k$ and $l$.

2.2.1. Objective Function and Constraints

The objective function seeks to minimize the weighted sum of the cycle time and the total workload imbalance across the assembly line.

$$\mathrm{Minimize} Z=\alpha \cdot CT+\beta \cdot \sum _{k=1}^{m-1}\sum _{l=k+1}^m{D}_{kl}$$

(1)

The weighting parameters $\alpha$ and $\beta$ are introduced to strategically quantify the managerial trade-off between traditional line efficiency and social equity, respectively. Rather than being rigidly fixed constants, these parameters are designed to be context-dependent, allowing decision-makers to prioritize either production throughput or worker fairness based on the current operational needs of the sheltered work center. A comprehensive justification, Pareto analysis, and calibration guide for selecting these specific weights are detailed later in Section 5.4.

Subject to

$$\sum _{k\in K}{x}_{ik}=1,\forall i\in I$$

(2)

Constraint (2) ensures that each task is assigned to exactly one workstation.

$$\sum _{h\in W}{y}_{kh}=1,\forall k\in K$$

(3)

$$\sum _{k\in K}{y}_{kh}\le 1,\forall h\in W$$

(4)

Constraints (3) and (4) manage the worker assignments. Constraint (3) guarantees that every station is operated by exactly one worker, while Constraint (4) ensures that a worker is assigned to, at most, one station.

$$\sum _{k\in K}k\cdot x_{ik}\le \sum _{k\in K}k\cdot x_{jk},\forall \left(i,j\right)\in P$$

(5)

Constraint (5) enforces the precedence relationships. If task $i$ precedes task $j$, the station index of $i$ must be less than or equal to the station index of $j$.

2.2.2. Workload Calculation and Linearization

The workload of station $k \left(L_k\right)$ is determined by the sum of processing times of the assigned tasks, calculated according to the specific worker assigned to that station.

$$L_k=\sum _{i\in I}\sum _{h\in W}{t}_{ih}\cdot x_{ik}\cdot y_{kh},\forall k\in K$$

(6)

The choice of minimizing the total absolute workload difference ($\sum \left|L_k - L_l\right|$), rather than the traditional variance-based or quadratic smoothness index ($\sum {\left(L_k - L_l\right)}^2$), is driven by computational efficiency. Quadratic objectives classify the problem as mixed-integer quadratic programming (MIQP), which generally presents higher computational complexity regarding the relaxation mechanisms in branch-and-bound algorithms compared to linear models. By employing the proposed linearization with auxiliary variables, the problem remains a mixed-integer linear programming (MILP) model. This structure allows for the utilization of standard, highly optimized simplex-based solvers and typically yields tighter linear relaxation bounds, a critical factor for exact solution procedures in the ALWABP [5].

Since Equation (6) contains a non-linear term, standard linearization techniques are applied to maintain the linearity of the model.

The cycle time is defined by the station with the maximum workload:

$$L_k\le CT,\forall k\in K$$

(7)

To achieve workload equity, the model minimizes workload variation between stations. The absolute difference between any pair of stations $k$ and $l$ is represented by $|L_k-L_l|$. Since absolute values are non-linear, we introduce the auxiliary continuous variable $D_{kl}$ and apply the following linearization constraints:

$$L_k-L_l\le D_{kl},\forall k,l\in K,k<l$$

(8)

$$L_l-L_k\le D_{kl},\forall k,l\in K,k<l$$

(9)

To maintain the linearity of the model suitable for standard MILP solvers, the absolute workload difference is handled via the auxiliary variable $D_{kl}$ and Constraints (8) and (9), avoiding the non-linearity of quadratic formulations. This formulation allows precise control of workload distribution without compromising the model’s solvability in standard MILP solvers.

Finally, the definitions and domains of the decision variables are given as follows:

$$x_{ik}\in \left\{0,1\right\},\forall i\in I,\forall k\in K$$

(10)

$$y_{kh}\in \left\{0,1\right\},\forall k\in K,\forall h\in W$$

(11)

$$CT\ge 0$$

(12)

$$L_k\ge 0,\forall k\in K$$

(13)

$$D_{kl}\ge 0,\forall k,l\in K,k<l$$

(14)

Constraints (10) and (11) define the binary nature of the assignment variables for tasks and workers, respectively. Constraints (12)–(14) ensure the non-negativity of the continuous variables representing cycle time, station workload, and workload deviation.

2.3. Model Complexity and Formulation Justification

To clarify the computational size of the proposed MILP model, the number of decision variables and constraints can be expressed as a function of the problem parameters: number of tasks ($\mathrm{n}$), number of stations ($\mathrm{m}$), number of workers ($\mathrm{w}$), and precedence relations ($\left|\mathrm{P}\right|$).

• Binary Variables: $\mathrm{n}\cdot \mathrm{m}+\mathrm{m}\cdot \mathrm{w}$.
• Continuous Variables: $1+\mathrm{m}+{\displaystyle \frac{\mathrm{m}(\mathrm{m}-1)}{2}}$ (where the last term represents the $\mathcal{O}\left({\mathrm{m}}^2\right)$ auxiliary variables ${\mathrm{D}}_{\mathrm{k}\mathrm{l}}$).
• Constraints: $\mathrm{n}+2\mathrm{m}+\mathrm{w}+\left|\mathrm{P}\right|+\mathrm{m}(\mathrm{m}-1)$ (where the last term represents the linearization constraints).

We acknowledge that minimizing absolute deviation from the mean workload (e.g., $\sum \left|{\mathrm{L}}_{\mathrm{k}}-{\mathrm{L}}_{\mathrm{a}\mathrm{v}\mathrm{g}}\right|$) is an alternative linear formulation that only requires $\mathcal{O}\left(\mathrm{m}\right)$ auxiliary variables. However, in the heterogeneous ALWABP, task-processing times depend strictly on the assigned worker. Therefore, the “mean workload” (${\mathrm{L}}_{\mathrm{a}\mathrm{v}\mathrm{g}}$) is not a static parameter, but a dynamic variable depending on the assignment matrix. Calculating a dynamic mean introduces additional complexity. More importantly, the selected pairwise comparison approach ($\sum _{\mathrm{k}}\sum _{\mathrm{l}}\left|{\mathrm{L}}_{\mathrm{k}}-{\mathrm{L}}_{\mathrm{l}}\right|$) is mathematically stricter; it directly and heavily penalizes disparity between the most overloaded and most underloaded specific workers. This strict pairwise penalty serves as a mathematical proxy for the concept of “fairness” in Industry 5.0, interpreting it under the theoretical assumption that workers evaluate their specific load by directly comparing it against their peers’ loads.

While the $\mathcal{O}\left({\mathrm{m}}^2\right)$ growth of auxiliary variables and constraints remains manageable for small-to-medium datasets (Roszieg, Heskia), it contributes to thickening of the LP relaxation matrix. As the number of stations ($\mathrm{m}$) increases in large-scale datasets (Tonge, Wee-Mag), this $\mathcal{O}\left({\mathrm{m}}^2\right)$ growth, combined with the NP-hard binary assignment complexity, severely limits the scalability of exact solvers. This structural bottleneck is the precise motivation for developing the hybrid simulated annealing (HSA) algorithm presented in Section 3.

To visually illustrate this computational bottleneck, Figure 3 plots the growth of the auxiliary continuous variables and linearization constraints as a function of the number of workstations ($\mathrm{m}$). As observed, the quadratic $\mathcal{O}\left({\mathrm{m}}^2\right)$ expansion rapidly inflates the model size. While tractable for $\mathrm{m}<10$, the steep trajectory quickly renders the exact MILP approach intractable for large-scale industrial instances, visually underscoring the absolute necessity of the proposed meta-heuristic.

3. Solution Methodology

The ALWABP is known as an NP-hard problem. Exact methods, such as the MILP model presented in the previous section, are effective for small-sized instances. However, they struggle to converge within reasonable CPU times as the problem size increases. For this reason, a hybrid simulated annealing (HSA) algorithm is proposed in this study to address large-scale instances effectively.

Unlike traditional meta-heuristics that encode the entire problem into a single complex chromosome, a “decoupled encoding strategy” is adopted. The HSA focuses on optimizing the topological ordering of tasks and the segmentation of stations. In addition, the assignment of workers is handled by an embedded greedy constructive heuristic during the decoding phase. This hybridization significantly reduces the search space and ensures high-quality assignments.

To provide a clear roadmap of the proposed approach, Figure 4 illustrates the overall solution methodology flowchart, encompassing the MILP model development, the hybrid simulated annealing framework, the integrated encoding-decoding mechanism, and the post-optimization phase.

3.1. Encoding and Decoding Scheme

A critical advantage of the proposed decoupled encoding strategy is its inherent ability to prevent infeasible solutions, a common pitfall in prior constructive heuristics. As highlighted by [27], algorithms that rely on simultaneous task-worker assignment rules [28] may inadvertently violate cycle time constraints due to the complexity of worker-dependent processing times. Our approach avoids this by decoupling the decision variables into two distinct permutations: (1) a precedence-feasible task sequence and (2) a worker priority sequence. The decoding procedure operates constructively: it assigns tasks to the current worker strictly based on their specific processing times ($t_{ik}$) and only proceeds as long as the cycle time constraint is satisfied. If the capacity is reached, a new station is opened for the next worker. This mechanism structurally guarantees that every solution generated and evaluated by the algorithm is feasible, eliminating the need for complex penalty functions or repair mechanisms.

A robust representation scheme is vital for the efficiency of the algorithm. We employ a dual-vector representation:

• Task Sequence Vector $\left(S\right)$: A permutation of tasks $\{j_1, j_2, j_3, \dots ,j_n\}$ satisfying the precedence constraints.
• Partition Vector $\left(P\right)$: An integer array of size $m$ (number of stations), where $P_k$ indicates the number of tasks assigned to station $k$.

The Decoding Procedure (Fitness Evaluation): The core innovation of our methodology lies in the decoding mechanism. Converting a solution representation into a fitness value involves two steps:

Step 1: Task-to-Station Mapping: Tasks are distributed to stations sequentially based on the partition vector. For instance, the first $P_1$ tasks in sequence $\left(S\right)$ are assigned to Station 1, the next $P_2$ to Station 2, and so on.

Step 2: Greedy Worker Assignment (GWA): Once the workload (task bundle) of each station is determined, the assignment of heterogeneous workers is performed using a constructive heuristic rather than random assignment.

• Difficulty Assessment: For each station $k$, a “difficulty score” is calculated based on the minimum processing time required by the most capable available worker.
• Prioritization: Stations are sorted in descending order of difficulty.
• Assignment: The algorithm iterates through the sorted stations and assigns the most efficient available worker (the one who minimizes the processing time for that specific bundle) to the current station. This decoding strategy ensures that, for any given configuration of tasks, the worker assignment is locally optimal, minimizing potential bottlenecks dynamically.

3.2. Initial Solution Construction

In order to ensure the feasibility of the search process, the initial solution is generated using a topological randomized sort. The algorithm compiles a list of available tasks. Then, a task is randomly selected from this list and added to the sequence. The list is updated with new available tasks. This process repeats until all tasks are sequenced. Thus, it is guaranteed that no precedence constraints are violated. In addition, the partition vector is initialized by randomly distributing the total number of tasks across stations. In this case, it is ensured that at least one task is assigned to each station.

3.3. Neighborhood Structures

To explore the solution space, the proposed HSA employs two distinct perturbation operators chosen probabilistically:

• Feasible Swap Operator ($N_1$): Two tasks in the task sequence vector are selected. They are swapped only if the move does not violate precedence constraints. This operator explores the sequencing search space.
• Boundary Shift Operator ($N_2$): This operator modifies the partition vector to balance workloads. A station boundary is shifted by moving the last task of station $k$ to station $k+1$, or vice versa. This effectively resizes the stations without changing the relative order of tasks.

3.4. The Hybrid Simulated Annealing Algorithm

The overall framework follows standard SA logic, simulating the physical annealing process of solids. Starting with a high initial temperature ($T_0$), the algorithm accepts worse solutions with a probability $\mathrm{P}=\mathrm{e}\mathrm{x}\mathrm{p}(-∆/\mathrm{T})$ to escape local optima, where $∆$ is the deterioration in the objective function.

Post-Optimization (Memetic Component): Upon termination of the annealing schedule, a deterministic local search procedure, named boundary optimization, is triggered. This procedure iteratively tests all possible single-task shifts between adjacent stations on the best-found solution. It acts as an intensification phase, fine-tuning the station loads to minimize the “smoothing” objective $D_{kl}$ further. The pseudocode of the proposed HSA is presented in Algorithm 1.

Algorithm 1. Hybrid simulated annealing (HSA) for the ALWABP

Inputs: Set of tasks N, Set of workers W, Processing times t_iw, Precedence graph P, Initial Temp. T0, Final Temp. T_final, Cooling rate α, Markov chain length L Outputs: Best-found assignment S_best and its bi-objective value F(S_best) / Phase 1: Initialization / T ← T0 Generate an initial topologically sorted task sequence π_init respecting P S_current ← GreedyWorkerAssignment(π_init, t_iw) // Decoupled decoding for feasibility S_best ← S_current π_current ← π_init / Phase 2: Simulated Annealing Main Loop / While T > T_final Do For iter = 1 to L Do π_new ← FeasibleSwap(π_current, P) // Generate neighbor task sequence S_new ← GreedyWorkerAssignment(π_new, t_iw) // Decode new sequence ΔF ← F(S_new) − F(S_current) // Evaluate objective difference If ΔF ≤ 0 or Random(0,1) < exp(−ΔF/T) Then S_current ← S_new // Accept new solution π_current ← π_new If F(S_current) < F(S_best) Then S_best ← S_current // Update global best End If End If End For T ← α × T // Apply geometric cooling schedule End While / Phase 3: Memetic Intensification / S_best ← BoundaryOptimization(S_best) // Local search to improve workload equity Return S_best

3.5. Parameter Tuning and Sensitivity Analysis

To ensure the robustness of the proposed HSA algorithm, a sensitivity analysis was conducted to identify the optimal parameter settings. Following standard procedures in the literature [17], we focused on three critical parameters: initial temperature ($T_0$), cooling rate ($\alpha$), and epoch length ($L$). We employed a full factorial design with three levels for each parameter, as summarized in

Table 1. Parameter levels and selected configuration for HSA.

Parameter	Level 1 (Low)	Level 2 (Medium)	Level 3 (High)	Selected Value
$\mathrm{Initial} \mathrm{Temp}. \left(T_0\right)$	50	100	200	100
$\mathrm{Cooling} \mathrm{Rate} \left(\alpha \right)$	0.90	0.95	0.99	0.98
$\mathrm{Epoch} \mathrm{Length} \left(L\right)$	20	50	100	50

.

This calibration procedure was performed using a representative subset of medium-scale instances from the Heskia dataset to ensure generalized robustness across varying complexities. Furthermore, to account for the stochastic nature of the meta-heuristic, each parameter combination in the factorial design was executed for five independent runs, and the average objective performance was used to determine the final configuration.

The preliminary experiments indicated that the cooling rate ($\alpha$) has the most significant impact on solution quality. Lower values ($\alpha <0.90$) led to premature convergence (trapped in local optima), while extremely high values ($\alpha >0.99$) resulted in excessive computational time without significant quality improvement. The selected configuration provided the best trade-off between solution quality (workload equity) and computational efficiency.

4. Computational Study

In this section, the performance of the proposed mathematical model and the hybrid simulated annealing (HSA) algorithm is evaluated in a comprehensive computational experiment. The primary objectives of this study are twofold: (1) to validate the effectiveness of the proposed workload smoothing formulation in small-scale benchmark instances by comparing it with existing classical models, and (2) to assess the scalability and efficiency of the proposed HSA in large-scale instances where exact methods typically struggle.

The proposed simulated annealing algorithm is computationally efficient and capable of identifying high-quality solutions in a short timeframe. Since this paper is the first to introduce the ALWABP with a focus on work distribution equity, there are no existing studies available for comparison. Consequently, we validate the algorithm against the mathematical model using small-scale instances and provide detailed results for large-scale datasets to serve as a baseline for future research.

All experiments were executed on a PC equipped with an Intel(R) Core(TM) i7, 2.20 GHz processor and 8 GB of RAM. The mathematical models were implemented in C# 10 and solved using Gurobi Optimizer version 12, a state-of-the-art solver for mixed-integer programming. The meta-heuristic algorithm was also coded in C#.

To ensure full computational reproducibility, the exact configuration parameters applied to the Gurobi solver are explicitly defined in a consolidated manner. A strict termination criterion of 3600 s was imposed as the time limit for each exact problem instance group. The relative optimality gap tolerance (MIPGap) was kept at the solver’s default value of ${10}^{-4}$ (0.01%), ensuring strict convergence criteria. Furthermore, the presolve parameter was set to “automatic”, allowing the solver to dynamically apply bound tightening and coefficient reduction techniques prior to the root node relaxation. Finally, the concurrent thread parameter was also set to “automatic”, enabling Gurobi to fully utilize all available logical cores of the processor to parallelize the branch-and-cut tree exploration.

Benchmark data are considered from the ALWABP benchmark dataset [3]. The test bed consists of four well-known dataset families from the ALWABP literature: Roszieg and Heskia (representing small-to-medium complexity) and Tonge and We-MAG (representing large-scale complexity). The test instances were generated based on a design involving five experimental factors, each evaluated at two distinct levels (low and high). These parameters include the workforce size, number of tasks, processing time variability, order strength, and degree of task–worker incompatibility.

Table 2. Summary of instance characteristics for the benchmark datasets.

Dataset Family	$\mathbf{Number} \mathbf{of} \mathbf{Tasks} \mathbf{\left(}\mathit{n}\mathbf{\right)}$	$\mathbf{Number} \mathbf{of} \mathbf{Stations} \mathbf{\left(}\mathit{m}\mathbf{\right)}$	$\mathbf{Number} \mathbf{of} \mathbf{Workers} \mathbf{\left(}\mathit{w}\mathbf{\right)}$	Order Strength
Roszieg (Low)	25	4 (Groups 1–4)	4 (Groups 1–4)	71.67 (High)
		6 (Groups 5–8)	6 (Groups 5–8)
Heskia (Low)	28	4 (Groups 1–4)	4 (Groups 1–4)	22.49 (Low)
		7 (Groups 5–8)	7 (Groups 5–8)
Tonge (High)	70	10 (Groups 1–4)	10 (Groups 1–4)	59.42 (High)
		17 (Groups 5–8)	17 (Groups 5–8)
Wee-Mag (High)	75	11 (Groups 1–4)	11 (Groups 1–4)	22.67 (Low)
		19 (Groups 5–8)	19 (Groups 5–8)

provides a detailed summary of these characteristics.

As seen in

Table 3. Detailed results obtained by mathematical models for the Roszieg family.

Family (Roszieg)		Existing Model		Proposed Model		Imp (%)
Group	Test Problem	Best C	Total Workload Diff.	Best C	Total Workload Diff.
Group 1	1	20	15	20	9	40.00
	2	22	3	22	3	0.00
	3	18	6	18	6	0.00
	4	18	8	18	6	25.00
	5	17	9	17	0	100.00
	6	24	10	24	9	10.00
	7	21	7	21	3	57.14
	8	20	15	20	10	33.33
	9	22	16	22	9	43.75
	10	19	10	19	3	70.00
Group 2	11	30	89	30	89	0.00
	12	27	41	27	41	0.00
	13	76	253	76	210	17.00
	14	25	34	25	34	0.00
	15	26	50	26	47	6.00
	16	22	10	22	10	0.00
	17	22	14	22	14	0.00
	18	20	6	20	0	100.00
	19	27	35	27	31	11.43
	20	40	110	40	110	0.00
Group 3	21	28	10	28	10	0.00
	22	30	30	30	18	40.00
	23	26	22	26	3	86.36
	24	33	10	33	9	10.00
	25	28	3	28	3	0.00
	26	27	7	27	3	57.14
	27	21	18	21	18	0.00
	28	28	26	28	23	11.54
	29	27	27	27	27	0.00
	30	33	16	33	0	100.00
Group 4	31	31	33	31	9	72.73
	32	29	18	29	0	100.00
	33	32	11	32	10	9.09
	34	27	3	27	0	100.00
	35	27	13	27	13	0.00
	36	29	13	29	13	0.00
	37	27	19	27	0	100.00
	38	28	4	28	4	0.00
	39	21	6	21	3	50.00
	40	29	7	29	4	42.86
Group 5	41	10	28	10	0	100.00
	42	10	24	10	13	45.83
	43	10	29	10	15	48.28
	44	9	25	9	15	40.00
	45	12	28	12	5	82.14
	46	9	14	9	8	42.86
	47	10	17	10	10	41.18
	48	8	14	8	0	100.00
	49	10	17	10	8	52.94
	50	9	10	9	10	0.00
Group 6	51	11	53	11	23	56.60
	52	10	10	10	8	20.00
	53	10	18	10	18	0.00
	54	10	18	10	15	16.67
	55	11	46	11	20	56.52
	56	13	13	13	13	0.00
	57	13	33	13	5	84.85
	58	11	27	11	20	25.93
	59	12	28	12	13	53.57
	60	9	8	9	8	0.00
Group 7	61	16	30	16	20	33.33
	62	13	23	13	0	100.00
	63	19	18	19	9	50.00
	64	16	61	16	28	54.10
	65	14	33	14	8	75.76
	66	17	8	17	5	37.50
	67	17	54	17	8	85.19
	68	16	49	16	43	12.24
	69	15	55	15	40	27.27
	70	17	29	17	13	55.17
Group 8	71	15	43	15	27	37.21
	72	16	26	16	0	100.00
	73	16	22	16	16	27.27
	74	16	64	16	18	71.88
	75	16	18	16	8	55.56
	76	17	39	17	5	87.18
	77	13	18	13	13	27.78
	78	14	48	14	37	22.92
	79	14	24	14	5	79.17
	80	14	19	14	9	52.63
					Improvement Mean	40.66

Numbers written in bold indicate better workload results.

, the proposed model demonstrated effective performance in terms of workload distribution. For the Roszieg dataset, the proposed formulation achieved an average reduction of 40.66% in the total workload difference compared to the classical model. This result indicates that the proposed model successfully identifies solutions in which the line is equally efficient but significantly more balanced. In addition, for the Heskia instances, the model maintained the optimal cycle time while reducing the workload imbalance. Thus, the experimental results show that efficiency and equity can be optimized simultaneously in small-to-medium search spaces.

The second phase evaluates the proposed HSA algorithm with the Tonge and Wee-Mag datasets. These instances involve a larger number of tasks and workers, making them computationally intractable for exact solvers within reasonable time limits. Therefore, we compared the solutions found by the HSA against the best integer solutions obtained by the solver.

Summary of Computational Performance: To facilitate readability, we summarize the key aggregate indicators from the detailed data shown in Table 3,

Table 4. Detailed results obtained by mathematical models for the Heskia family.

Family (Heskia)		Existing Model		Proposed Model		Imp (%)
Group	Test Problem	Best C	Total Workload Diff.	Best C	Total Workload Diff.
Group 1	1	94	7	94	7	0
	2	95	3	95	0	100
	3	102	3	102	3	0
	4	103	47	103	0	100
	5	92	6	92	3	50
	6	98	22	98	6	72.7273
	7	116	7	116	3	57.1429
	8	86	10	86	3	70
	9	95	3	95	0	100
	10	142	36	142	31	13.8889
Group 2	11	169	3	169	3	0
	12	107	31	107	29	6.45161
	13	108	51	108	6	88.2353
	14	96	0	96	0	0
	15	130	7	130	0	100
	16	117	11	117	0	100
	17	146	32	146	27	15.625
	18	132	27	132	3	88.8889
	19	101	0	101	0	0
	20	120	24	120	0	100
Group 3	21	200	3	200	3	0
	22	147	69	147	3	95.6522
	23	186	75	186	13	82.6667
	24	181	29	181	15	48.2759
	25	142	19	142	4	78.9474
	26	194	74	194	0	100
	27	149	36	149	12	66.6667
	28	191	29	191	15	48.2759
	29	170	119	170	6	94.958
	30	165	112	165	3	97.3214
Group 4	31	204	3	204	3	0
	32	147	20	147	20	0
	33	211	21	211	7	66.6667
	34	127	6	127	3	50
	35	181	21	181	12	42.8571
	36	179	30	179	8	73.3333
	37	191	8	191	3	62.5
	38	152	19	152	3	84.2105
	39	167	35	167	12	65.7143
	40	153	3	153	3	0
Group 5	41	35	22	35	6	72.7273
	42	40	82	40	0	100
	43	35	68	35	10	85.2941
	44	30	18	30	6	66.6667
	45	40	134	40	12	91.0448
	46	29	36	29	6	83.3333
	47	25	28	25	16	42.8571
	48	43	112	43	22	80.3571
	49	38	66	38	0	100
	50	34	58	34	12	79.3103
Group 6	51	51	278	51	6	97.8417
	52	50	138	50	18	86.9565
	53	52	312	52	0	100
	54	33	78	33	48	38.4615
	55	38	22	38	18	18.1818
	56	34	72	34	6	91.6667
	57	42	58	42	10	82.7586
	58	39	72	39	40	44.4444
	59	59	186	59	6	96.7742
	60	28	90	28	0	100
Group 7	61	66	160	66	60	62.5
	62	56	72	56	10	86.1111
	63	69	324	69	48	85.1852
	64	126	792	126	10	98.7374
	65	107	982	107	0	100
	66	39	96	39	40	58.3333
	67	87	456	87	0	100
	68	62	100	62	42	58
	69	61	48	61	6	87.5
	70	79	174	79	6	96.5517
Group 8	71	91	366	91	6	98.3607
	72	65	86	65	48	44.186
	73	73	76	73	10	86.8421
	74	57	34	57	0	100
	75	65	72	65	22	69.4444
	76	70	74	70	6	91.8919
	77	71	54	71	24	55.5556
	78	58	58	58	28	51.7241
	79	46	42	46	10	76.1905
	80	76	132	76	30	77.2727
					Improvement Mean	67.0509

Numbers written in bold indicate better workload results.

,

Table 5. Detailed results obtained by mathematical models for the Tonge family.

Family (Tonge)		Proposed Model		HSA Solutions
Group	Test Problem	Best C	Total Workload Diff.	Best C	Total Workload Diff.
Group 1	1	92	46	87	82
	2	87	56	87	30
	3	102	73	100	88
	4	117	127	108	197
	5	92	97	89	106
	6	90	77	88	69
	7	95	81	94	96
	8	104	86	104	55
	9	71	127	69	57
	10	85	76	83	37
Group 2	11	118	187	110	221
	12	110	339	108	375
	13	107	154	103	466
	14	94	85	93	278
	15	97	75	89	88
	16	102	97	100	152
	17	124	151	120	265
	18	115	70	112	239
	19	132	55	119	111
	20	114	107	113	140
Group 3	21	158	466	158	281
	22	169	189	160	299
	23	148	70	144	429
	24	149	288	147	229
	25	169	255	169	189
	26	174	214	161	318
	27	146	62	146	349
	28	182	552	179	590
	29	190	478	182	251
	30	154	165	153	374
Group 4	31	179	412	167	226
	32	192	172	177	428
	33	179	182	157	440
	34	178	214	172	176
	35	149	121	146	228
	36	174	135	163	304
	37	167	222	166	163
	38	169	265	152	262
	39	211	325	180	304
	40	159	315	159	450
Group 5	41	32	810	28	754
	42	39	176	33	390
	43	42	302	32	228
	44	44	432	34	292
	45	40	654	32	540
	46	41	278	35	418
	47	36	366	30	296
	48	39	296	35	600
	49	36	316	33	324
	50	38	576	33	534
Group 6	51	40	292	37	372
	52	45	416	43	640
	53	43	474	32	270
	54	42	372	36	276
	55	57	948	40	712
	56	43	702	40	714
	57	40	408	39	536
	58	46	384	40	726
	59	40	194	35	480
	60	40	194	38	508
Group 7	61	74	982	63	766
	62	75	420	68	900
	63	69	908	64	544
	64	99	1798	97	1216
	65	72	914	60	910
	66	84	1684	67	606
	67	127	2968	52	840
	68	77	568	61	880
	69	85	1480	57	920
	70	75	706	60	776
Group 8	71	57	854	56	468
	72	76	560	57	808
	73	79	600	72	742
	74	79	236	69	942
	75	81	1256	64	788
	76	74	586	59	1516
	77	73	856	60	612
	78	75	1234	68	728
	79	73	498	61	512
	80	72	1404	62	688

and

Table 6. Detailed results obtained by mathematical models for the Wee-Mag family.

Family (Wee-Mag)		Proposed Model		HSA Solutions
Group	Test Problem	Best C	Total Workload Diff.	Best C	Total Workload Diff.
Group 1	1	29	10	26	82
	2	31	36	26	10
	3	31	18	26	68
	4	35	58	30	86
	5	34	48	30	88
	6	26	30	26	76
	7	29	10	24	28
	8	32	60	28	58
	9	29	28	26	58
	10	30	36	27	62
Group 2	11	35	10	30	54
	12	35	28	33	58
	13	36	106	31	90
	14	37	24	34	38
	15	36	18	33	58
	16	34	18	30	96
	17	36	10	32	62
	18	40	18	35	82
	19	35	24	33	18
	20	38	20	36	104
Group 3	21	57	46	49	68
	22	58	58	46	58
	23	52	72	47	146
	24	53	52	48	96
	25	51	64	46	84
	26	62	46	46	142
	27	57	62	51	88
	28	52	70	47	104
	29	44	34	44	76
	30	57	28	50	98
Group 4	31	56	78	49	298
	32	52	78	47	102
	33	61	34	48	52
	34	60	48	51	126
	35	50	90	42	38
	36	54	72	47	66
	37	51	46	40	106
	38	52	98	47	140
	39	47	46	47	104
	40	47	58	39	70
Group 5	41	14	226	10	198
	42	13	0	12	176
	43	15	88	11	184
	44	14	148	12	214
	45	15	172	12	272
	46	12	230	11	360
	47	16	120	13	88
	48	14	130	11	140
	49	11	238	10	78
	50	13	158	12	234
Group 6	51	17	138	14	136
	52	13	316	11	148
	53	15	166	12	174
	54	17	104	13	160
	55	13	144	12	128
	56	22	468	13	214
	57	14	232	13	210
	58	15	104	13	290
	59	20	208	16	300
	60	14	160	13	138
Group 7	61	19	412	18	300
	62	23	546	19	248
	63	26	574	22	366
	64	28	136	19	362
	65	29	308	18	288
	66	35	896	20	206
	67	31	264	21	214
	68	24	422	18	310
	69	27	286	21	322
	70	25	538	24	540
Group 8	71	27	148	21	276
	72	26	178	19	272
	73	31	680	18	122
	74	31	214	23	246
	75	24	510	19	230
	76	25	816	16	192
	77	24	280	19	366
	78	28	390	19	192
	79	26	220	21	272
	80	22	264	20	150

.

Small-to-Medium Instances: The proposed MILP model achieved an average reduction in workload difference of 40.66% for the Roszieg dataset and 67.05% for the Heskia dataset compared to the classical model, without compromising cycle time efficiency.

Large-Scale Instances: The proposed HSA algorithm demonstrated superior scalability. For the Tonge dataset, HSA improved the cycle time by an average of 9.79% compared to the best integer solution found by Gurobi within the 3600 s limit. A similar trend is observed in the Wee-Mag dataset, where HSA consistently identified lower cycle times in high-complexity instances.

In practical, real-world terms, this 40% to 67% reduction in absolute workload variance translates directly to a more synchronized and human-centric assembly line. For a plant manager or a supervisor in a sheltered work center, these numerical improvements mean that the physical fatigue and mental stress are distributed equitably among heterogeneous workers. It prevents the common operational pitfall of overworking highly capable individuals (bottleneck stations) while others experience excessive idle time. Consequently, achieving these specific equity metrics mathematically supports the social sustainability and workforce retention goals inherent to Industry 5.0.

5. Discussion

5.1. Comparative Analysis: MILP vs. Traditional MIQP

To empirically validate the methodological superiority of the proposed linearized $L_1 - norm$ formulation over traditional quadratic ($L_2 - norm$) approaches, we conducted a direct computational comparison. We implemented the workload equity objective, which relies on a quadratic loss function, as a mixed-integer quadratic programming (MIQP) model. Both formulations were then used to solve the identical small-to-medium instances (Roszieg and Heskia datasets) using the Gurobi solver under the same hardware conditions. The computational results, summarized in

Table 7. Computational comparison between MIQP and MILP formulations.

Dataset	Metric	Traditional MIQP (L2-Norm)	Proposed MILP (L1-Norm)	Improvement (%)
Roszieg	Avg. CPU Time (s)	455.51	326.68	28.28%
	Avg. Gap	0.0%	0.0%	-
Heskia	Avg. CPU Time (s)	386.07	282.64	26.79%
	Avg. Gap	0.0%	0.0%	-

, demonstrate a clear advantage for the proposed linear approach. While both models successfully proved optimality (achieving a 0.0% average optimality gap for both datasets), the proposed MILP formulation reached these optimal solutions significantly faster. For the Roszieg dataset, the average CPU time decreased from 455.51 s in the traditional MIQP model to 326.68 s in the proposed MILP model, representing a 28.28% improvement. Similarly, for the Heskia dataset, the average CPU time was reduced from 386.07 s to 282.64 s, yielding a 26.79% improvement in solution speed. This empirical evidence confirms that minimizing the “total absolute workload difference” via linearization avoids the computational burden associated with the quadratic relaxations in branch-and-bound algorithms. Consequently, the proposed approach is not only conceptually aligned with social equity, but is also methodologically and computationally superior to traditional quadratic smoothing methods.

5.2. Performance Analysis

The computational results demonstrate a clear divergence in performance as problem size increases. While the MILP model guarantees global optimality for small instances, it struggles with large-scale datasets like Tonge and Wee-Mag, as seen in Table 5 and Table 6. This behavior is consistent with the findings of [5,6], who established that the exact solution of the ALWABP via mathematical programming is limited by the exponential growth of binary variables. While the solver reached the 3600 s time limit for 100% of the large-scale instances without proving optimality, the HSA algorithm converged to high-quality solutions less than 600 s. This drastic reduction in CPU time confirms the algorithm’s suitability for real-time industrial applications. Also, the proposed HSA achieved an average improvement of 9.79% in cycle time for the Tonge instances compared to the best integer solution found by solver. This superior performance is attributed to the decoupled encoding strategy, which allows the meta-heuristic to efficiently explore the solution space without being hindered by the rigid structural constraints that slow down the branch-and-cut process. By decoupling task sequencing from worker assignment, the HSA avoids generating infeasible solutions, dedicating its entire computational power to improving solution quality.

To rigorously verify that the reported improvements in the Tonge datasets are statistically significant and to document the variability inherent in the stochastic hybrid simulated annealing (HSA) algorithm, we conducted an extended statistical analysis. Each large-scale instance was executed for 10 independent runs using varying random seeds to ensure a genuinely stochastic evaluation of the search space and to prevent biased convergence.

Firstly, to address the variability across independent runs, the performance of the HSA is evaluated not only by the best-found cycle time, but also by its stability. Across the Tonge dataset, the algorithm demonstrated high reliability, with an average performance reported as mean $\pm$ standard deviation (achieving an average standard deviation of 1.45 units).

Secondly, we applied the non-parametric Wilcoxon signed-rank test, considering a 95% confidence level ($\mathsf{\alpha }=0.05$), to compare the mean cycle time values obtained by the HSA against the best integer solutions found by the solver. The test yielded a p-value of $0.00014$, which is significantly lower than $0.05$, allowing us to reject the null hypothesis.

To determine the magnitude of this difference, we calculated the effect size using the $\mathrm{r}$ statistic ($\mathrm{r}={\displaystyle \frac{\mathrm{Z}}{\sqrt{\mathrm{N}}}}$). The analysis yielded an effect size of 0.82, which is conventionally considered a “large” effect in statistical literature. This confirms that the proposed HSA algorithm provides a practically substantial, not just statistically significant, improvement.

Finally, providing a transparent range for this performance gap, the 95% confidence interval (CI) for the mean percentage improvement in cycle time for the large-scale instances was calculated to be 8.5% to 11.1%. These comprehensive statistical indicators confirm that the proposed HSA is a robust, stable, and highly effective methodology for the ALWABP.

5.3. Implications for Industry 5.0

From a managerial perspective, the ability to solve large-scale instances with high “workload equity” validates the applicability of our approach in real-world Industry 5.0 environments [17]. The results confirm that achieving social fairness (minimized workload variance) does not necessarily compromise production efficiency (cycle time), provided that an effective meta-heuristic is employed.

5.4. Pareto Trade-Off Analysis and Parameter Calibration

The weighting parameters $\alpha$ and $\mathsf{\beta }$ in the objective function (Equation (1)) fundamentally define the managerial trade-off between line efficiency (cycle time minimization) and social equity (workload difference minimization). To evaluate the robustness of this balance, a sensitivity analysis was conducted on a representative medium-scale instance from the Heskia dataset. We systematically varied the weights across five configurations, scaling from a pure efficiency focus ($\mathsf{\alpha }=1.0,\mathsf{\beta }=0.0$) to a pure equity focus ($\mathsf{\alpha }=0.0,\mathsf{\beta }=1.0$).

As shown in

Table 8. Pareto trade-off analysis under multiple $\mathsf{\alpha }$ and $\mathsf{\beta }$ configurations (representative Heskia instance).

Scenario	$\mathbf{Weights} \mathbf{(}\mathsf{\alpha },\mathsf{\beta }\mathbf{)}$	Strategic Focus	Cycle Time (CT)	Total Workload Diff.
1	(1.00, 0.00)	Pure Efficiency (Traditional)	92	82
2	(0.75, 0.25)	Efficiency-Biased	94	34
3	(0.50, 0.50)	Balanced Approach	95	12
4	(0.25, 0.75)	Equity-Biased	102	3
5	(0.00, 1.00)	Pure Equity	116	0

, the relationship between efficiency and equity forms a clear Pareto frontier. When the model relies solely on traditional efficiency ($\mathsf{\alpha }=1.0$), it achieves the absolute minimum cycle time but results in a highly imbalanced line (workload difference = 82 units). Conversely, prioritizing only equity ($\mathsf{\beta }=1.0$) perfectly balances the line (difference = 0) but severely deteriorates the cycle time.

The most critical insight from this Pareto analysis is the non-linear nature of the trade-off. Moving from Scenario 1 to Scenario 2 requires a marginal sacrifice in cycle time (approx. $2.1\%$ increase), yet it yields a massive $58.5\%$ reduction in workload difference. This non-linearity highlights the core value of the proposed formulation.

Practical Calibration for Decision-Makers: In practice, plant managers can utilize this formulation as a dynamic decision support system (DSS) rather than a static plan. Calibration of $\mathsf{\alpha }$ and $\mathsf{\beta }$ should be context-dependent:

• During peak seasonal demand where throughput is critical, managers should calibrate the system towards Scenario 2 ($\mathsf{\alpha }>\mathsf{\beta }$).
• In sheltered work centers or during periods of high worker turnover/fatigue, prioritizing worker retention requires calibrating towards Scenario 4 ($\mathsf{\beta }>\mathsf{\alpha }$).
• Scenario 3 provides a robust default state for standard operations, proving that high equity can be achieved without catastrophic losses in production rates.

5.5. Managerial Implications

For plant managers and production planners, the proposed model serves as a strategic decision support system (DSS) rather than a rigid optimization tool. In practice, managers constantly face a trade-off between maximizing throughput (cycle time) and maintaining workforce stability. Our results demonstrate that the relationship between these two objectives is often non-linear; a marginal relaxation in cycle time constraints can lead to significant improvements in workload equity.

Practitioners can use this model to

• Visualize Trade-offs: Identify scenarios where a negligible loss in production rate (e.g., 0.5%) results in a drastic reduction in workload inequality, thereby choosing a more sustainable operating point.
• Enhance Retention: By implementing the “workload equity” objective, managers can mitigate the feelings of unfairness that drive high turnover and absenteeism [15]. This is particularly critical in Industry 5.0 environments, where retaining skilled, heterogeneous human labor is a competitive advantage [17].
• Operationalize Social Responsibility: For managers in sheltered work centers, this model transforms the abstract concept of “social integration” into a concrete scheduling plan, ensuring that tasks are distributed in a way that respects the diverse capabilities of disabled workers without compromising the overall line flow.
• Prioritize Equity over Efficiency in High-Risk Contexts: Plant managers should actively shift the operational focus towards workload equity under specific real-world scenarios. Efficiency should be treated as a secondary goal when integrating highly vulnerable populations (such as in SWCs), where excessive physical or cognitive stress can lead to severe health consequences. Furthermore, equity must be prioritized during periods of widespread workforce fatigue, high staff turnover, or in high-precision manufacturing environments where overworked employees at bottleneck stations are significantly more prone to making costly quality errors. In these contexts, the long-term sustainability and quality of the workforce heavily outweigh the short-term gains of a maximized production rate.

5.6. Algorithmic Ablation Study

To rigorously evaluate the algorithmic design and clarify the marginal contribution of each hybridized component, an ablation study was conducted. While simulated annealing and constructive heuristics are established techniques individually, their specific integration addresses distinct challenges of the bi-objective ALWABP. We compared three algorithmic variants using a representative subset of the large-scale Tonge dataset:

• Variant 1 (Base SA): A standard simulated annealing algorithm relying on the decoupled task sequence, but using a naive/random worker assignment mechanism and lacking any post-optimization.
• Variant 2 (SA + GWA): The simulated annealing algorithm integrated with the greedy worker assignment (GWA) mechanism during the decoding phase.
• Variant 3 (Full HSA): The complete proposed algorithm, integrating SA, GWA, and the memetic intensification phase (boundary optimization) applied at the end of the search.

The computational results of the ablation study are summarized in

Table 9. Ablation study results isolating the marginal contribution of algorithmic components (Tonge dataset).

Algorithmic Variant	Included Components	Avg. Cycle Time	Avg. Workload Diff.	Contribution Highlight
Variant 1 (Base SA)	Sequencing only	182.4	510.6	Baseline Performance
Variant 2 (SA + GWA)	Sequencing + GWA	161.5	345.2	Significant Cycle Time reduction (−11.4%)
Variant 3 (Full HSA)	Sequencing + GWA + Boundary Opt.	158.8	212.4	Massive Workload Smoothing (−38.4%)

.

The ablation study provides strong empirical evidence for the algorithmic design:

• The Role of GWA: Comparing Variant 1 and Variant 2 reveals that the greedy worker assignment is the primary driver for production efficiency. By dynamically matching task bundles to the most capable heterogeneous workers during decoding, the GWA drastically reduces the average cycle time (an 11.4% improvement) and prevents the generation of bottleneck-heavy infeasible solutions.
• The Role of Boundary Optimization: Comparing variant 2 and the full HSA demonstrates the absolute necessity of the memetic intensification phase for social equity. While boundary optimization provides only a marginal secondary improvement to cycle time, it systematically redistributes boundary tasks, resulting in a dramatic 38.4% reduction in total workload difference.

Consequently, the ablation study confirms that the full HSA is not merely a collection of standard techniques, but a structurally justified hybridization in which each component is strictly necessary to satisfy the conflicting objectives of efficiency and equity.

6. Limitations

Recent studies have increasingly focused on multi-objective optimization and contemporary applications of assembly line balancing [39,40]. Despite the significant contributions regarding workload equity, this study has certain limitations that should be acknowledged to guide future research.

From a methodological perspective, the proposed HSA is a meta-heuristic approach. While it outperforms MILP in large-scale instances, it does not guarantee global optimality and carries the inherent risk of becoming trapped in local optima, a trade-off that is common in NP-hard problems [5]. Furthermore, the performance of HSA is sensitive to calibration of control parameters, requiring a rigorous tuning process that adds to the computational effort.

Regarding the modeling scope, this study assumes deterministic task times. While this assumption allows for a clear definition of workload equity, it does not fully capture the stochastic variability of human workers or absenteeism, which are addressed in other robust optimization studies [15].

The proposed model is tailored for single-model assembly lines. Its applicability to high-mix, low-volume environments or mixed-model lines remains limited. Expanding the “workload equity” concept to such dynamic environments, potentially utilizing real-time data from Industry 5.0 technologies, represents a necessary evolution of this work.

The present study demonstrates the benefits of workload equity within a theoretical and mathematical framework. However, empirical validation of these theoretical improvements in real-world production environments (e.g., wearable sensors, IoT) lies beyond the scope of this work and remains a necessary avenue for future research.

7. Conclusions

In this paper, the assembly line worker assignment and balancing problem with a specific workload smoothing objective is introduced. Unlike traditional approaches that prioritize solely cycle time minimization, this study mathematically models the distribution of tasks to promote theoretical workload equity in heterogeneous workforces. While our computational experiments confirm significant improvements in mathematical balance, we acknowledge that empirical field studies are necessary to validate the translation of these theoretical metrics into actual ergonomic and psychological benefits. For this purpose, a mixed integer linear programming (MILP) formulation is developed. The proposed model is capable of solving some small-scale instances optimally using a Gurobi solver. As the mathematical model has difficulties finding optimal solutions for real-life environment problems with a reasonable CPU time, a hybrid simulated annealing (HSA) algorithm is also proposed. To demonstrate the efficiency of the proposed approaches, a comprehensive computational experiment is conducted. The experimental results show that the proposed mathematical model is effective in terms of workload distribution and achieves significant improvement in balance. Also, for large-scale instances, the proposed HSA algorithm proves to be robust and effective.

This study is a step toward “human-centric” manufacturing. Future studies might extend the presented problem, like incorporating stochastic task times. The current model assumes deterministic processing times. However, in realistic manufacturing environments, processing times are often subject to variability. Future studies should extend the “workload equity” objective to stochastic or fuzzy environments to test the robustness of the fairness solutions under conditions of uncertainty. And, the proposed formulation addresses the single-model ALWABP and can be extended to multi- or mixed-model assembly lines. Also, applying the workload-smoothing objective to U-shaped or two-sided assembly lines might be a valuable direction. Finally, exact decomposition algorithms might be investigated to solve larger instances.