A Declarative Framework for Production Line Balancing with Disruption-Resilient and Sustainability-Focused Improvements

Abstract

This paper presents a declarative framework for resilient machining line planning, integrating line balancing and disruption handling within a unified, interactive decision-support environment. Building upon earlier constraint-based models, the proposed approach incorporates sustainability-oriented improvements through Pareto-based multi-criteria optimization. The model supports trade-off analysis across cost, energy consumption, tool wear, and schedule continuity, enabling predictive planning and adaptive dispatching under operational uncertainty. By combining proactive balancing with reactive disruption handling in a single declarative formulation, the framework addresses a key gap in the current production engineering methodologies. A case study employing real data and real-world-inspired disruption scenarios demonstrates the effectiveness of the approach. Compared to traditional sequential strategies, the framework yields superior performance in terms of solution diversity, responsiveness, and sustainability alignment, confirming its value for next-generation, resilient manufacturing systems.

1. Introduction

The increasing need for frequent replanning in production—covering aspects such as reconfiguration, rescheduling, rebalancing, reallocation of human resources, and reprioritization of orders—presents considerable challenges for modern production management [1]. The increasing complexity of modern production systems calls for intelligent, interactive decision-support tools capable of dynamically handling unexpected disruptions and operational variations [2,3]. Recent studies further reinforce this need by proposing hybrid and resilient decision-making frameworks that support real-time reconfiguration and adaptive planning in manufacturing environments [4,5]. This need is addressed through the design and validation of a prototype framework that combines line balancing, resilience, and optimization objectives within a Pareto-based multi-criteria model [6,7].

The proposed approach uses a declarative framework [8,9] to support interactive planning and dispatching, allowing for real-time scenario analysis, adaptive response, and control based on continuous feedback. The possibilities of using declarative modeling initiated in our earlier works [10,11] were focused on the problems of balancing machining lines, in which an integrated, single-argument objective function was optimized. The obtained results were extended in this work to include the possibility of robust balancing of production flows resistant to disruptions related to changes in working conditions (e.g., energy and/or tool costs), as well as machine malfunctions. Unlike most existing studies, which focus on individual elements, such as rescheduling, line balancing, or resource reconfiguration, this framework highlights the interconnected nature of these decision areas and addresses a significant gap in the current body of research [12].

By focusing decision-making efforts on equipment rebalancing and reconfiguration, the proposed framework facilitates interactive replanning, which is particularly well-suited to machining operations in iron casting. Moreover, the incorporation of production line characteristics into a nominal reference model enhances the system’s ability to dynamically adapt to evolving operational conditions, including changes in machine layout, tool wear, energy consumption, and variability in production orders.

The proposed approach develops a declarative, constraint-based decision support framework for dynamic production line balancing that integrates resilience, sustainability, and optimization goals. It models machine-level operational states and their impact on cycle time, tool wear, and energy consumption, enabling the generation of Pareto-optimal configurations under changing production conditions. The framework supports real-time, interactive replanning through scenario-based analysis and is validated via a case study in a machining context. At the core of this approach lies the model’s open architecture, which facilitates the integration of heterogeneous data sources and supports the flexible configuration of operational scenarios. This adaptability empowers dispatchers in determining viable corrective actions in real time, ensuring continuous alignment with operational objectives. In particular, it highlights the opportunities to interactively balance production flows by monitoring to detect deviations from expected performance and using adaptive control to implement corrective strategies to minimize downtime, reduce risk, and maximize throughput. Within this context, several critical research questions arise, which include the following:

• Can user-defined scenarios based on machining takt times, tool wear, energy consumption, and layout changes effectively support interactive rebalancing and dispatching?
• Does the declarative, computer-implemented model operate efficiently in real-world conditions, supporting timely decisions at an industrial scale?
• Which combinations of line configurations and machine parameters deliver optimal trade-offs between cost, efficiency, and resource utilization?
• Is there a viable set of Pareto-optimal scenarios that ensures that production remains within defined constraints while maintaining system responsiveness?

Seeking answers to questions such as these justifies the scope and importance of our contribution, which includes the following:

• A novel declarative modeling framework, enabling holistic decision-making in dynamic manufacturing environments that unifies proactive line balancing, reactive disruption management, and sustainability-aware multi-criteria optimization.
• Dynamic and transparent planning with constraint-based modeling that seamlessly incorporates changing decision-maker objectives, increasing adaptability and transparency in response to changing operational and economic conditions.
• A case study demonstrating superiority in a realistic machining context showing the improved responsiveness, robustness, and sustainability of the proposed approach compared to conventional methods.

The implementation of the proposed solution within the Decision Support System (DSS), providing a solid foundation for polyoptimization and trade-off decision-making, integrating domains, process planning, workforce scheduling, and resource allocation, seems to be competitive with the currently available approaches.

The remainder of this paper is organized as follows. Section 2 reviews the related work. Section 3 outlines the methodological foundations, including the problem formulation. Section 4 introduces the modeling framework, which is then applied in Section 5 to describe the concept of an interactive, follow-up control-driven decision support procedure. Section 6 discusses the results of computational experiments, demonstrating the applicability of the proposed approach using a case study example. Finally, Section 7 summarizes the key findings and suggests directions for future research.

2. Related Work

The increasing complexity of production systems, combined with sustainability imperatives and operational uncertainty, has led to intensified research into production line balancing (PLB), resilience strategies, and multi-criteria decision-making. This section surveys key developments in these areas and identifies the need for a unified framework that supports interactive, sustainable, and disruption-resilient planning.

2.1. Production Line Balancing and Disruption Handling

Production Line Balancing (PLB) traditionally focuses on static optimization problems aimed at minimizing idle time and balancing workloads across workstations [13,14,15]. These methods are typically heuristic or exact, with applications in repetitive and rhythmic production systems [11,13]. However, conventional approaches often fall short in dynamic environments where disruptions—such as machine breakdowns or order changes—necessitate continuous rebalancing [16].

To address these challenges, follow-up control strategies have emerged, enabling real-time adaptation through feedback-driven replanning [6,10]. Declarative models in this context allow operators to iteratively generate updated production scenarios based on evolving operational conditions and constraints [10,11]. The follow-up production scheduling algorithm (FPSA), for instance, enables cyclic reconfiguration of machining tasks while preserving production continuity [16].

To summarize, existing frameworks typically address line balancing under deterministic or mildly stochastic assumptions [17,18]. However, they often fail to incorporate real-time responsiveness or structured mechanisms to absorb or reconfigure under unplanned disruptions (e.g., sudden machine breakdowns or supply delays) [19]. For this reason, the goal of the proposed framework is the integration of disruption-resilient mechanisms directly into the decision logic using declarative constraints, which allow for adaptive rebalancing.

2.2. Multi-Criteria Optimization in Sustainable Manufacturing

Modern production systems increasingly require balancing multiple, often conflicting objectives—such as energy consumption, labor cost, and tool wear—under sustainability constraints [7,20,21]. Multi-criteria optimization methods, particularly Pareto-based approaches, are used to explore trade-offs among these objectives, yielding sets of non-dominated solutions that provide flexibility for decision-makers [7,20].

The Pareto frontier serves as a crucial decision-support tool in identifying optimal configurations where no single criterion can be improved without degrading another [20]. For instance, the authors of [20] focus on optimizing process parameters for sustainable machining, highlighting the trade-offs between tool wear and energy consumption. In addition, the authors of [7] review hybrid Pareto-based formulations for production line balancing, showing their effectiveness in dynamic environments. Additionally, interactive decision support systems, combining Pareto frontiers with operator input, are addressed in [6]. Nonetheless, in many production systems, the Pareto analysis is performed post hoc and is not integrated into real-time decision-making workflows. Recent efforts aim to embed such trade-off navigation directly into the modeling phase through constraint optimization techniques [10,11,16].

Metaheuristics, especially those enhanced by reinforcement learning, are increasingly used to address the computational challenges of high-dimensional optimization in dynamic environments [21,22,23]. These methods align with Industry 4.0 principles, allowing integration with data from MES and IoT systems to support predictive maintenance and dynamic load balancing [24].

In turn, the proposed approach in this paper extends the balancing problem by embedding Pareto-based multi-objective optimization that includes sustainability metrics, such as energy consumption and tool wear cost, enabling more comprehensive trade-off analyses.

2.3. Integrated, Declarative Approaches: Identified Gap

Despite the progress in each domain—balancing, disruption management, and multi-criteria optimization—a critical research gap remains. Few approaches offer a unified, declarative modeling framework that combines proactive balancing, reactive disruption handling, and sustainability-aware optimization [10,25]. Such integration is essential for modern manufacturing environments that demand transparency, adaptability, and responsiveness.

The use of declarative constraint-based models supports the formulation of both “what-if” and prescriptive planning scenarios without redesigning the underlying logic [10,15,21]. Moreover, this paradigm allows for a dynamic incorporation of decision-makers’ evolving goals, ensuring the model remains relevant across changing economic and operational contexts.

While some recent works have explored follow-up control or Pareto-based optimization, they often do so in isolation and lack architectural flexibility for real-time integration across multiple planning domains. For example, existing solutions typically handle rebalancing as a static pre-processing step, rely on sequential execution of reconfiguration and optimization modules, or address sustainability metrics only at a post hoc evaluation stage. This fragmented handling reduces responsiveness and limits practical deployment in fast-changing industrial environments. To our knowledge, no existing method offers a declarative, constraint-based framework that simultaneously

• Supports proactive and reactive decisions across multiple objectives;
• Enables integrated, on-the-fly scenario adaptation in response to disruptions;
• Handles sustainability-aware trade-offs during the planning—not just evaluation—phase.

We address this gap by proposing a unified framework in which interactive rebalancing, disruption resilience, and multi-criteria optimization are treated as co-dependent components of one declarative formulation. The system allows for continuous monitoring and responsive reconfiguration, making it well suited for machining operations where tool wear, energy prices, or task durations can shift dynamically.

In Section 6, we further support this claim through a practice case study that benchmarks our approach. The results demonstrate system responsiveness, solution diversity, and sustainability alignment. These outcomes validate the practical benefits of the proposed integration.

3. Problem Formulation

The line balancing problem under consideration pertains to a machining line $L$ (as illustrated in Figure 1), composed of $n$ machines, denoted as $m_i$, where $M =\left\{m_1,\dots ,m_i,\dots ,m_n\right\}$ represents the set of all machines. The machining line $L$ is designated for the serial production of a predetermined number of products, denoted by $TN$.

Each product corresponds to a task $J$, which involves a sequence of operations $o_1$,…,$o_i$,…,$o_n$, executed in a fixed order across the machines in set $M$. Each operation $o_i$ is characterized by

• its processing time $t_i$ (the cycle time of machine $m_i$),
• the tool lifetime $z_i$ (specific to the machine) used during its execution,
• the energy consumption $e_i$.

The aforementioned parameters are contingent upon the assumed states $x_i\in S{X}_i\subseteq \mathbb{N}$ of the machines within the set $M$, where $S{X}_i$ represents the set of possible states of machine $m_i$. Formally, the state $x_i$ of machine $m_i$ determines the sequence $P_i$ of its production parameters: $P_i=(t_i,z_i{,e}_i$): $S{X}_i⟶{\mathbb{Q}}_i$ (where ${\mathbb{Q}}_i$ denotes the set of admissible production parameters of machine $m_i$). Within this framework, the configuration of the machining line $L$ is defined as a sequence of states $x_i$ of the constituent machines, expressed as $X=\left(x_1,\dots ,x_i,\dots ,x_n\right)$.

The set of admissible configurations of line $L$ is denoted by $\mathbb{X}$. Modifications to configuration $X$ have a direct impact on the production cycle time $T$, total tool cost $CZ$, and total energy costs $CE$.

• The overall production cycle time, defined as $T= max \left\{t_1, \dots , t_i, \dots , t_n\right\},$ determines the production completion date ($PC$). This completion date is directly proportional to the available working time ($WT$)—which depends on factors such as the number of shifts, overtime, and weekend operations—and inversely proportional to the cycle time $T$ of the production line.
• The tool consumption costs of the machines constituting line $L$, defined as $CZ=\sum _{i=1}^{n}c{z}_i$ (where $c{z}_i$ denotes the tool cost associated with machine $m_i$), depend on the tool lifetime $z_i$, which in turn is a function of the adopted operational states $x_i$ of the constituent machines. Typically, as the production rate increases (i.e., the processing time titi decreases), tool wear accelerates significantly, leading to a substantial rise in tool-related costs.
• The energy consumption costs of the machines comprising line $L$, defined as $CE=\sum _{i=1}^n{ce}_i$ (where ${ce}_i$ denotes the energy cost incurred by machine mi), depend on the energy consumption $e_i$, which is also determined by the adopted operational states $x_i$ of the respective machines. As in the case of tool consumption, a higher production rate (reflected by a lower value of $t_i$) typically results in a sharp increase in energy usage, thereby escalating the associated energy costs.

The aforementioned criteria (hereafter referred to as partial criteria) enable the evaluation of the configuration of production line $L$ along three primary dimensions: production time, tool consumption costs, and energy costs. In general, additional criteria may be considered depending on the specific characteristics of the line under analysis. Nevertheless, in most practical cases, the considered criteria are mutually conflicting. From this perspective, line balancing becomes a problem of identifying such configurations $X$ that minimize the values of the criteria $T$, $CZ$, and $CE$ in the Pareto sense—that is, seeking solutions in which improving the value of one criterion is not possible without simultaneously worsening the value of at least one of the remaining two.

The utilization of Pareto-optimal solutions, forming the so-called Pareto front, in production planning is often challenging due to the potentially large number of solutions, which increases sharply with the number of considered dimensions (criteria). To address this issue, an additional criterion—referred to as the aggregated criterion—has been introduced to reduce the number of solutions under consideration in accordance with the current decision-maker’s preferences. This aggregated criterion corresponds to the total operating cost ($TOC$). The $TOC$ encompasses multiple components, including energy consumption $e_i$, tool wear $z_i$, and individual cycle time $t_i$ associated with each machine $m_i$. These factors collectively affect the labor cost $c{w}_i$ attributed to each machine and hence the overall operational cost of the production line.

Figure 2a presents an example of a production line consisting of three machines, for which the configuration $X=\left(\mathrm{33,78,67}\right)$ has been adopted. This configuration ensures Pareto optimality while simultaneously yielding the lowest value of the aggregated criterion, $TOC$. In practical terms, this means that among all feasible configurations of line $L$, the selected solution provides the minimum production cost, with a $TOC$ value of $3.68$ [m.u.] (m.u.—monetary units). For this configuration, the energy consumption cost amounts to $CE=0.23$ [m.u.], the tool wear cost is $CZ=1.76$ [m.u.], and the production cycle time is $T=52$ [s].

The considered (see Figure 1) configuration $X$ is one of the admissible configurations that make up the set $\mathbb{X}$. Each configuration $X\in \mathbb{X}$ corresponds to a criteria vector $Z=(T,CZ,CE)$, which forms the objective space $\mathbb{Z}$. This is illustrated in Figure 2b, which shows the space $\mathbb{Z}$ for the considered line $L$. The Pareto-optimal solutions have been distinguished by points marked in red and yellow. These points corresponding with the set $\mathbb{P}\subseteq \mathbb{X}$ containing such configurations $X$ for which the improvement of one partial criterion (e.g., $T$) cannot occur without a reduction of at least one of the others (e.g., $CE$ or $CZ$).

This means that the set $\mathbb{P}$ guarantees a choice from among the Pareto-optimal solutions, i.e., $X\in \mathbb{P}$. Among the remaining feasible solutions, there are alternative variants characterized by lower values of all the partial criteria. It is worth noting (Figure 2b) that the Pareto solutions create the so-called Pareto front, which is the edge of the point cloud of feasible solutions $\mathbb{X}$. The considered Pareto front is better visible in Figure 2c, which shows the projections of the space $\mathbb{X}$ onto the planes: $CZ/T$, $T/CE$ and $CE/CZ$. The selected sample configuration of line $X$ has been highlighted in green, which makes it easier to assess how a change in the line configuration within one criterion results in a change in the others.

For example, it may occur that the values of the considered criteria do not align with the decision-maker’s expectations—for instance, when the decision-maker seeks a solution that minimizes energy costs $CE$. In such a case, a decision may be made to modify the configuration of the line to an alternative arrangement that yields the following criterion values: $CE=0.22$ [m.u], $CZ=1.83$ [m.u], $T=47$ [s], and $TOC=3.72$ [m.u.]—the new configuration $X’$ is marked in orange. It is important to note, however, that such a change occurs along the Pareto front, meaning that the improvement of one criterion (e.g., $CE$) necessarily entails the deterioration of others (e.g., $CZ$), including the aggregated criterion $TOC$. In general, other configurations of the solution space can be analyzed. For example, the yellow color indicates those configurations that guarantee $TOC\le 3.80$ [m.u.]. In a similar way, depending on the decision-maker’s requirements, other subsets can be established. Representations of this type support the decision-maker in the process of interactive (supported by DSS class dialog solutions) search for configuration $X$ that provides both the expected values of partial criteria and the aggregate criterion. In practice, in situations related to the occurrence of multidimensional spaces $X$, such searches becomes very time-consuming. In response to such needs, a representation in the form of polar charts was introduced—see Figure 3.

In this representation, the values of individual criteria are presented on separate charts, in which the values of a given criterion increase with the angle α (creating spiral plots). Such a solution allows for an easy, intuitive way of selecting the sought solutions. For example, each of the charts in Figure 3 shows a ring zone illustrating the range of requirements specified by the decision maker: $T\in \left[\mathrm{51,53}\right]$ (Figure 3a), $CE\in \left[0.225, 0.235\right]$ (Figure 3b), and $CZ\in \left[1.75, 1.8\right]$ (Figure 3c). The Pareto solutions within these rings correspond to the configurations of lines that satisfy the selected requirements. By changing the range of rings, the decision-maker can easily identify the desired configurations and choose the one that satisfies his requirements.

Thus, the process of identifying a solution that satisfies the decision-maker’s preferences requires continuous analysis within a high-dimensional solution space. For this reason, it is essential to implement mechanisms that enable the selection of admissible solutions, i.e., the reduction of the solution space, to such subsets that effectively correspond to the decision-maker’s priorities.

The complexity of the problem increases further when considering the fact that the values of the aforementioned criteria are also influenced by the adopted structure $SL\in \mathbb{S}\mathbb{L}$ of the machining line $L$. This structure depends on the types of disruptions affecting line operation, such as machine failures, scheduled maintenance downtimes, delays due to staff absenteeism, and other similar factors. The occurrence of a disruption, therefore, affects the effectiveness of the adopted configuration $X$ of line $L$, i.e., it alters the values of the considered criteria. In other words, each disruption impacts the set Pareto solutions and requires a re-evaluation of the permissible line configuration. Figure 3 shows a case where disturbances changing the values of all partial criteria take them outside the required range—the adopted configuration (green point) has moved outside the range area limited by rings (moved to point ⊗). In such a situation, it is necessary to either correct the configuration $X$ or change the $SL$ structure of the $L$ line.

During the execution of a production order, the structure frequently changes and must be adapted to the current conditions. This continuous adaptation of the line has a significant impact on the considered partial criteria ($T$, $CZ$, and $CE$) determining production cost ($TOC$) and the completion time ($PC$). The structures considered in this study include the following cases: $\mathbb{S}\mathbb{L}=\left\{s{l}_1,s{l}_2,s{l}_3,s{l}_4\right\}$, where the structure $s{l}_1$ corresponds to uninterrupted production flow, while the remaining structures, $s{l}_2$, $s{l}_3$, $s{l}_4$, represent scenarios involving the failure of a single machine. These structures, in particular, reflect the following modes of operation:

• Nominal operating mode: All machines from the set $M$ are available, and production is carried out during a single shift—see structure $s{l}_1$ in Figure 1.
• Machine failure—two-shift operation mode: In the event of a failure of machine $m_i$, the operations originally assigned to $m_i$ are transferred to machine $m_{i-1}$, which performs them during the second shift—see structure $s{l}_2$ in Figure 4a. The resulting semi-finished products are subsequently processed on machines $m_{i+1},m_{i+2}\dots ,m_n$ during the first shift of the following day. This solution is admissible under the assumption that two consecutive machines, $m_{i-1}$ and $m_i$, are capable of substituting for one another.
• Machine failure—single-shift operation mode with operation merging: In the event of a failure of the $m_i$ machine, its operations are transferred to machine $m_{i-1}$, which simultaneously performs both (merged) operations—see structure $s{l}_3$ in Figure 4b. This approach is admissible under the assumption that machines $m_{i-1}$ and $m_i$, being consecutive in the line, are interchangeable.
• Machine failure—single-shift operation mode with the use of an alternative machine: If machine $m_i$ fails, its operations are redirected to a backup machine $m_i^{\prime }$—see structure $s{l}_4$ in Figure 4c. This solution is acceptable under the condition that machine $m_i$ has a designated substitute $m_i^{\prime }$, whose utilization does not cause additional losses (e.g., due to the delay of operations in the line in which this machine is deployed).

The above-presented arrangements of the machining line structure do not, of course, exhaust all possible disturbances and related regroupings of the machining line. The above-presented variants correspond to the situations most frequently occurring in the considered bearing production process.

In that context, the primary objective of line balancing is to identify the structure $SL$ of the machining line $L$ and its configuration $X=\left(x_1,\dots ,x_i,\dots ,x_n\right)$ that satisfies the following condition:

achievement of considered partial criteria: $T$, $CZ$, $CE$, and/or $TOC$ at or below a predefined threshold, i.e., $TOC \le TOR$ ($T\le TR$,  $CZ\le CZR$,  $CE\le CER$) and/or $TOC\to min$ ($T\to min$, $CZ\to min$, $CE\to min).$

Therefore, the set of admissible solutions (admissible production flow scenarios) is the set $AS=\left\{\left(SL,X\right)|SL\in \mathbb{S}\mathbb{L},X\in \mathbb{X}\right\}$ (where $\mathbb{S}\mathbb{L}$ is the set of admissible structures of machining line $L$; $\mathbb{X}$ is the set of admissible configurations of $L$). The ultimate objective is to determine the scenario of production configuration $\left(SL,X\right)$ that minimizes one from considered criteria ($T, CZ, CE$ or $TOC$) while ensuring compliance with the given constraints.

In the considered problem, the continuous adjustment of machining line operations in response to environmental changes (e.g., fluctuations in energy prices) and unforeseen disruptions (e.g., equipment failures) is assumed. In this context, balancing the machining line becomes a process of continuous follow-up control-driven decision-making guided by the following key questions:

I.

What production scenario $(SL,X)$ of machining line $L$ ensures an assumed level of partial $(T,CZ,CE)$ and/or aggregated $\left(TOC\right)$ criteria?

II.

Can the current production scenario $\left(SL,X\right)$ be modified to maintain the given $T, CZ, CE$ and/or $TOC$ level in response to changes in production parameters (e.g., energy costs)?

The questions presented justify the need for an interactive approach to production balancing, enabling both proactive planning of production scenarios and their reactive adjustment to changing operational conditions. The problem under consideration thus reduces to the following key question: Does the computer-based implementation of the proposed model support the planning of restructuring and balancing operations within the machining line in a way that facilitates online adaptation of production plans to emerging disturbances? A proposal for such a method is outlined in the subsequent section.

4. Framework and Modeling Methodology

To address the question of whether a computer-based implementation of the proposed model can support the dynamic restructuring and balancing of machining operations, the problem was formally defined using declarative modeling techniques. This methodological approach enables a precise and compact representation of system behavior and constraints, thereby establishing a robust foundation for real-time simulation and adaptive adjustment of production plans. The proposed framework is grounded in the following assumptions and methodological constructs:

• The operations, $o_1$,…,$o_i$,…,$o_n$, necessary for the production of a single unit of product (task $J$) are performed sequentially on machines, $m_1,\dots ,m_i,\dots ,m_n$, which constitute the machining line $L$.
• At any given time, a machine can perform only one operation.
• The product (task $J$) releases $m_i$ upon completing the operation $o_i$ and subsequently initiates operation $o_{i+1}$ on machine $m_{i+1}$, contingent on its availability.
• If the subsequent machine is occupied, the product remains in a waiting state until it becomes available. The machine buffer, which temporarily holds tasks awaiting machine availability, is assumed to have unlimited capacity.
• To ensure minimal buffer utilization, machines are assumed to be synchronized, meaning the difference between their cycle times must not exceed a predefined threshold $TD$, i.e., $\left|t_i-t_j\right|\le TD$, $\forall \left\{m_i,m_j\right\}\subseteq M$.
• It is assumed that within a single working day ($WD$), at least $DP$ units must be produced. If the production cycle time $T = max \{t_1, \dots , t_i, \dots , t_n\}$ does not allow for this (i.e., production is too slow), overtime is considered.
• Total production cost is calculated as the sum of tool wear costs, labor expenses (covering standard shifts and overtime), and energy usage costs incurred throughout the production process.

The mathematical statement of the MLB problem involves the following descriptions.

• Parameters:
• $M$: set of available machines in line $L$: $M=\left\{m_1,\dots ,m_i,\dots ,m_n\right\}$;
• $n$: number of available machines: $n=\left|M\right|$;
• $LK:$ number of machine operation states, defined as a sequence $LK=\left(l{k}_1,\dots ,l{k}_i,\dots ,{ln}_n\right)$, where $l{k}_i$ is the number of states for machine $m_i$;
• ${TA}_i$: sequence of admissible cycle times for machine $m_i$ for subsequent states given by ${TA}_i=\left(t{a}_{i,1},\dots ,t{a}_{i,q},\dots ,t{a}_{i,l{k}_i}\right)$, where $t{a}_{i,q}$ represents the cycle time ([u.t.]—unit of time) for state $q$ ($x_i=q$), determining the production time of a single unit of product;
• $TD$: predefined synchronization margin for machine cycle times;
• ${ZA}_i$: sequence of tool lifespan values for machine $m_i$ for subsequent states, given by ${ZA}_i=\left(z{a}_{i,1},\dots ,z{a}_{i,q},\dots ,z{a}_{i,l{k}_i}\right)$, where $z{a}_{i,q}$ denotes the number of units produced before tool replacement is required;
• ${EA}_i$: sequence of energy consumption values for machine $m_i$ for subsequent states given by ${EA}_i=\left(e{a}_{i,1},\dots ,e{a}_{i,q},\dots ,e{a}_{i,l{k}_i}\right)$, where $e{a}_{i,q}$ represents the energy consumption in state $q$ ($x_i=q$);
• $c{p}_i$: cost of purchasing a tool for machine $m_i$;
• $c{o}_i$: labor cost per unit time for an operator working at machine $m_i$;
• $c{n}_i$: overtime labor cost per unit time for machine $m_i$;
• $c{r}_i$: energy cost of machine $m_i$;
• $c{s}_i$: fixed cost of machine $m_i$;
• $DP$: minimum daily production requirement;
• $WD$: working time per workday [u.t].
• Decision variables:
• $x_i$: state of machine $m_i$, where $x_i\in \left\{1,\dots ,l{k}_i\right\}$, for example, $x_1=2$ indicates that machine $m_2$ operates in state number 2;
• $t_i$: machine cycle time of $m_i$, dependent on the operating state $x_i$: $t_i\in \mathbb{N}$;
• $z_i$: tool lifespan (e.g., lathe knife durability) for machine $m_i$, dependent on the operating state $x_i$, where $z_i\in \mathbb{N}$;
• $e_i$: energy consumption of machine $m_i$, dependent on the operating state $x_i$;
• $c{z}_i$: cost of tools (e.g., lathe knifes) for $m_i$, calculated per unit of product;
• $LP$: daily production output (number of units), excluding overtime;
• $c{w}_i$: operating cost of machine $m_i$ during the nominal working period (regular workday) for a given state $x_i$;
• $c{v}_i$: overtime cost of machine $m_i$ for a given state $x_i$;
• $c{e}_i$: energy cost of machine $m_i$ for a given state $x_i$;
• $CE$: total energy cost for the entire line $L$, calculated per unit of product;
• $T$: production cycle time of line $L$, composed of $n$ machines (see Figure 1);
• $CZ$: total tool cost for the entire machining line $L$, calculated per unit of product;
• $CW$: total labor cost for the entire line $L$, calculated per unit of product;
• $TOC$: total operating cost for the machining line $L$.

• Constraints:
  • Constraints that specify the machine cycle time $t_i$, tool lifespan $z_i$, and energy consumption depending on the adopted state $x_i$:

    $$t_i\left(x_i\right)=t{a}_{i,\left(x_i\right)}, \forall m_i\in M,$$

    (1)

    $$z_i\left(x_i\right)=z{a}_{i,\left(x_i\right)}, \forall m_i\in M,$$

    (2)

    $$e_i\left(x_i\right)=e{a}_{i,\left(x_i\right)}, \forall m_i\in M,$$

    (3)
• • Constraint enforcing the synchronization of machine cycle times:

    $$\left|t_i\left(x_i\right)-t_j\left(x_i\right)\right|\le TD, \forall \left\{m_i,m_j\right\}\subset M,$$

    (4)
• • Production cycle time of line $L$:

    $$T\left(X\right)=\underset{\forall m_i\in M}{\max }\left\{t_i\left(x_i\right)\right\},$$

    (5)
• • Constraint defining the value of $c{z}_i\left(x_i\right)$ (tool cost per unit of produced product):

    $$c{z}_i(x_i)={\displaystyle \frac{c{p}_i}{z_i(x_i)}},$$

    (6)
• • Aggregate cost of tools used across the entire line $L$, given the configuration $X=(x_1,\dots ,x_i,\dots ,x_n)$:

    $$CZ(X)={\sum }_{i=1}^{n}c{z}_i(x_i),$$

    (7)
• • Unit output constrained to regular shift duration (excluding overtime):

    $$LP(X)=⌊{\displaystyle \frac{WD}{T\left(X\right)}}⌋,$$

    (8)
• • Labor costs during the nominal working time for an individual machine.

    $$c{w}_i\left(x_i\right)=t_i\left(x_i\right)\times c{o}_i\times LP, \forall m_i\in M,$$

    (9)
• • Overtime costs for an individual machine:

    $$c{v}_i=c{n}_i\times T\left(X\right)\times max\left\{0,\left(LP\left(X\right)-DP\right)\right\}, \forall m_i\in M,$$

    (10)
• • Aggregate cost of labor for the entire line $L$, given the configuration $X$ (per unit of produced product):

    $$CW(X)={\displaystyle \frac{\sum _{i=1}^n\left(c{w}_i\left(x_i\right)+c{v}_i\right)}{\max \left\{LP,DP\right\}}},$$

    (11)
• • Energy costs for an individual machine:

    $${ce}_i\left(x_i\right)=c{s}_i+c{r}_i\times t_i\left(x_i\right)\times e_i\left(x_i\right)\times \mathit{max}\left\{LP\left(X\right),DP\right\}, \forall m_i\in M$$

    (12)
• • Aggregate cost of energy used across the entire line $L$, given the configuration $X=(x_1,\dots ,x_i,\dots ,x_n)$:

    $$CE(X)={\sum }_{i=1}^n{ce}_i\left(x_i\right),$$

    (13)
• • Each admissible configuration $X=(x_1,\dots ,x_i,\dots ,x_n)\in \mathbb{X}$ must satisfy the condition $TOC\left(X\right)\le TR$, where $\mathbb{X}$ is the set of feasible configurations and $TR$ is the cost limit.

    $$TOC\left(X\right)=CZ\left(X\right)+CW\left(X\right)+CE\left(X\right)\le TR,$$

    (14)
• • If $\mathbb{X}\ne \mathbb{\varnothing }$, the problem reduces to identifying a configuration $X=(x_1,\dots ,x_i,\dots ,x_n)$ for the machining line $L$ that minimizes the objective function:

    $$minimize TOC\left(X\right)=CZ\left(X\right)+CW\left(X\right)+CE\left(X\right), X\in \mathbb{X}.$$

    (15)

An extension of the presented production line balancing model may involve the searching for solutions optimal in the Pareto sense. Within this multi-objective framework, the goal is to simultaneously minimize three key parameters for the entire production line $L$: the production cycle time $T\left(X\right)$(5), the aggregate tool used cost $CZ\left(X\right)$ (7), and the aggregate energy used cost $CE\left(X\right)$ (13).

Formally the problem is defined over a decision vector $X=\left(x_1,x_2,\dots ,x_n\right)\in \mathbb{X}$, where $\mathbb{X}$ denotes the set of admissible configurations of the production line. The corresponding vector-valued objective function is defined as

$$F\left(X\right)=\left[CE\left(X\right), CZ\left(X\right), T\left(X\right)\right],$$

(16)

with $CE\left(X\right)$—the total production cost (13), $CZ\left(X\right)$—the total tool usage costs (7), $T\left(X\right)$—the total time cycle for configuration $X$ (5).

A configuration $X^{*}\in \mathbb{X}$ is called Pareto-optimal if there is no other configuration $X\in \mathbb{X}$ such that

$$F\left(X\right)\prec F\left(X^{*}\right)\iff \left\{\begin{array}{c}\left\{F_i\right.\left(X\right)\le F_i\left(X^{*}\right) \forall i\in \{1,\dots ,k\}\\ F_j\left(X\right)<F_j\left(X^{*}\right) \exists j\in \left\{1,\dots ,k\right\} \end{array}\right.$$

(17)

where $F\left(A\right)=\left[a_1,a_2,\dots {,a}_n\right]\prec F\left(B\right)=\left[b_1,b_2,\dots ,b_n\right]$ means that $a_i\le b_i$ for all $i\in \left\{1,\dots ,n\right\}$ and $a_j<b_j$ for at least one $j\in \left\{1,\dots ,n\right\}$.

In the considered case this means $CE\left(X\right)\le CE\left(X^{\mathrm{*}}\right),CZ\left(X\right)\le CZ\left(X^{\mathrm{*}}\right),T\left(X\right)\le T\left(X^{\mathrm{*}}\right)$ with at least one strict inequality. The set of all Pareto-optimal solutions forms the Pareto-front:

$$\mathbb{P}=\left\{X^{*}\in \mathbb{X}:\nexists X\in \mathbb{X},F\left(X\right)\prec F\left(X^{*}\right)\right\}$$

(18)

which represents the trade-offs between conflicting objectives and offers a spectrum of optimal alternatives to decision-makers. The set $\mathbb{D}=\mathbb{X}\backslash \mathbb{P}$ configurations that are not Pareto-optimal is hereinafter called dominated configurations.

Among the constraints introduced, relations (1)–(3) determine the parameters ($t_i\left(x_i\right)$, $z_i\left(x_i\right),$ $e_i\left(x_i\right)$) of operation executed on machine $m_i$ depending on configuration $x_i$. Constraints (4) and (5) determine the working cycle $t_i$ of the machines $M$ and the entire $L$ line. Constraints (6)–(13) relate to production costs ($CZ\left(X\right),CW\left(X\right),CE\left(X\right)$) depending on the configuration $X$. Constraint (14) defines the set of admissible solutions $\mathbb{X}$. Constraints (15) and (16) represent the aggregate and multi-criterial objective function. Related problem can be formulated in terms of the following constraint optimization problem (COP):

$$COP=\left(\left(\mathcal{V},\mathcal{D}\right),\mathcal{C},{\mathcal{C}}_{OPT}\right)$$

(19)

where $\mathcal{V}$ is a set of decision variables representing machines states ($x_i)$, operations parameters ($t_i\left(x_i\right)$, $z_i\left(x_i\right),$ $e_i\left(x_i\right)$), cycle time of line $L$ ($T$), and production costs ($c{z}_i$, $c{w}_i$, $c{v}_i$, $CZ\left(X\right),CW\left(X\right),CE\left(X\right)$, $TOC\left(X\right)$); $\mathcal{D}$ is a set of domains of decision variables: $x_i\in \left\{1,\dots ,l{k}_i\right\}$, $t_i\left(x_i\right)$, $z_i\left(x_i\right)$, $\dots$, $TOC\left(X\right)$, $TOC\in \mathbb{N}$; $\mathcal{C}$ is a set of constraints specified in interrelations (1)–(14); ${\mathcal{C}}_{OPT}$ is an objective function (15) or (16).

To solve the $COP$ (19), the decision variables $\mathcal{V}$ must be assigned values that satisfy all constraints in the set $\mathcal{C}$, thus defining the set of feasible solutions. The problem formulation can involve either a single-objective or a multi-objective function. In the single-objective case, the objective function $COP$ is minimized to identify the optimal configuration. In the multi-objective scenario, multiple criteria are considered simultaneously, and the solution seeks to satisfy constraints while optimizing a vector-valued objective function. Consequently, solving the $COP$ entails finding a configuration $X$ that meets production requirements and achieves the best trade-offs among competing objectives.

5. Interactive Production Line Balancing

As previously outlined, the production line balancing problem under consideration focuses on two main aspects. The first concerns the selection of a configuration $X$ that aligns with the decision-maker’s expectations with respect to partial criteria, such as production takt time, tool wear, and energy consumption, as well as the overall production cost. The second aspect pertains to the need for adapting the configuration $X$ in response to changing operating conditions (e.g., due to disruptions). On the one hand, the objective is to maximize the overall efficiency of the production line; on the other hand, it is essential to ensure its operational continuity and safety. Thus, production line balancing involves the continuous adaptation of its configuration to maximize performance within a dynamically changing environment.

Given the inherently interactive nature of production line planning, the proposed approach is based on a declarative representation of the underlying problem ($COP$ (19)). This representation features an open structure, allowing for the incorporation of diverse and evolving user-specific requirements without necessitating modifications to the underlying solution-search algorithms.

The conceptual framework of the proposed method was illustrated in Figure 5. It presumes continuous adaptation of line configuration to prevailing operational conditions (such as changes in production orders, disruptions, or cost fluctuations) and comprises the following key stages:

Figure 5. Relationships occurring in the interactive production line balancing process between the operator’s dashboard and the machining line.

Figure 5. Relationships occurring in the interactive production line balancing process between the operator’s dashboard and the machining line.

6. Case Study

To evaluate the practical applicability of the proposed approach, consider the following case study. A sequential machining line $L$ comprises four machines, $M=\left\{m_1,m_2,m_3,m_4\right\}$. Each machine $m\in M$ can operate in one of 100 distinct configurations or states $\left(l{k}_1=\dots =l{k}_4=100\right)$. Each state is characterized by specific operational parameters, including cycle time $t_i$ [s], tool lifespan $z_i$ [units], and energy consumption $e_i$ [kWh]. These parameter values are provided in

Table 1. Machines distinct states ($t_i$ in seconds, $z_i$ in cycles, $e_i$ in $\mathrm{k}\mathrm{W}\mathrm{h}$).

${\mathit{m}}_1$				${\mathit{m}}_2$				${\mathit{m}}_3$				${\mathit{m}}_4$
${\mathit{x}}_1$	${\mathit{t}}_1$	${\mathit{z}}_1$	${\mathit{e}}_1$	${\mathit{x}}_2$	${\mathit{t}}_2$	${\mathit{z}}_2$	${\mathit{e}}_2$	${\mathit{x}}_3$	${\mathit{t}}_3$	${\mathit{z}}_3$	${\mathit{e}}_3$	${\mathit{x}}_4$	${\mathit{t}}_4$	${\mathit{z}}_4$	${\mathit{e}}_4$
1	120	52	5.5	1	111	59	4.6	1	138	47	4.6	1	113	59	4.6
2	123	48	5.6	2	109	48	4.5	2	134	46	4.5	2	127	38	4.5
3	126	62	5.7	3	111	64	4.5	3	140	61	4.5	3	117	55	4.5
…	…	…	…	…	…	…	…	…	…	…	…	…	…	…	…
100	43	69	5.1	100	47	47	4.2	100	64	40	4.2	100	56	69	4.2

. The system must maintain operational synchronization, such that the maximum allowable deviation in cycle time between adjacent machines does not exceed $TD=10$ [s].

With these assumptions, a plan for the execution of an order for the production of $TN = 8000$ [units] of product pieces is sought. The production period is set at 21 working days between 1 June 2025 and 30 June 2025. Assuming a daily working time of $WD = 8$ h, the required level of daily production is $DP = 381$ [units/day]. In addition, it is required that the unit production cost does not exceed $TOC\le 10$ [m.u.]. The total production cost includes both machine and labor expenses, as described in

Table 2. Production costs.

Costs	${\mathit{m}}_1$	${\mathit{m}}_2$	${\mathit{m}}_3$	${\mathit{m}}_4$
cost of purchasing a tool $c{p}_i$ [m.u./unit]	55	45	50	65
labor cost $c{o}_i$ [m.u./h]	45	25	40	50
overtime labor cost $c{n}_i$ [m.u./h]	55	35	50	60
energy cost $c{r}_i$ [m.u./kWh]	1.11	1.11	1.11	1.11
fixed cost $c{s}_i$ [m.u.]	30	15	35	25

. The presented example illustrates a manufacturing cell from a real-world production process for rolling bearing components. The production line consists of four lathes. The absence of intermediate buffers necessitates the synchronization of machine operations to prevent the accumulation of workpieces in front of the slowest workstation.

The proposed experimental plan assumes the application of the developed approach (Section 5) to configure the considered machining line $L$ in the following three planning scenarios (Figure 6):

• Proactive planning—enabling the specification of operating conditions for line $L$ under nominal conditions. This is carried out to determine a configuration that guarantees the completion of a given order within a specified deadline. In other words, it allows for the identification of fixed line parameters for each working day (Section 6.1).
• Reactive planning triggered by changes in operating conditions, such as fluctuations in cycle times $t_i$. This allows for reconfiguration of the line, starting from the day the disturbance occurs (day 12), to a new configuration that still ensures on-time completion of the order (Section 6.2).
• Reactive planning triggered by changes in line configuration, such as a breakdown of machine $m_2$. This allows for restructuring the line, starting from the day the disruption occurs (day 23), to a configuration that maintains the timely completion of the specified order (Section 6.3).

The sequence of these stages is illustrated in the schedule presented in Figure 6.

6.1. Proactive Planning

It is assumed that the configuration of the considered production line $L$ corresponds to the structure $SL=s{l}_1$, as illustrated in Figure 1. Referring to the developed algorithm (Figure 3), a scenario ($SL,X$) can be sought that guarantees the minimum value of a single-objective function, which is minimizing the total operating cost ($TOC$), or a multi-objective Pareto function that aims to minimize three key performance indicators: total cycle time ($T$), tool consumption cost ($CZ$), and energy cost ($CE$). Scenario ($SL,X$) should additionally meet the requirements set by the decision maker, i.e., the values of partial criteria of the obtained solution should fall within the given ranges:

• $T\le 75$ [s]—achieving these values guarantees daily production at the level of $DP\ge 384$ [units/day] (which allows the order to be completed within the given deadline),
• $CE\le 0.5$ [m.u.]—the adopted limit results from economic requirements and the sustainable development strategy,
• $CZ\in [3.5;5.0]$ [m.u.]—the adopted limit is a consequence of the cutting knives available on the market that ensure the given production quality.

To address this problem, the developed declarative model was implemented using the Gurobi solver (version 11.0.3, Gurobi, version 3.13.5, Python API [26]), executed on a system equipped with an Intel Core i7-M4800MQ (2.7–4.82 GHz) and 32 GB RAM. The total computation time was less than 20 s. Among the evaluated scenarios, the number of admissible configurations satisfying the cost constraint $TO{C}_i\le 10$ [m.u.] was $\left|\mathbb{X}\right|=985$. The optimal scenario minimizing $TOC$ was identified as $X^{\left(1\right)}=\left(\mathrm{46,38,81,51}\right)$, resulting in $TOC=8.91$ [m.u.]. The obtained solution determines the following machine cycles: $t_1=63$ [s], $t_2=69$ [s],$t_3=63$ [s], $t_4=73$ [s], and the production cycle time: $T= 73$ [s]—see Figure 7.

The adoption of the configuration $X^{\left(1\right)}=\left(\mathrm{46,38,81,51}\right)$ for the considered line $L$ ensures obtaining the expected values of partial criteria: $T=73$ [s], $CE=0.46$ [m.u.], $CZ=3.53$ [m.u.] (see ring zones on the Figure 7b), and completion of the order within the specified time of 21 working days (see schedule on the Figure 7c).

Ensuring the above parameters is possible assuming no disruptions to the production process. In the event of a disruption, the line should be reconfigured (if possible)—as illustrated in the next section.

6.2. Reactive Planning—Disability of One Machine

Let us consider a situation in which, on the 12th day (a working day W9) of executing the considered order (proceeding according to the adopted proactive plan $X^{\left(1\right)}$), a partial failure of machine $m_2$ occurs. The partially malfunctioning machine may continue to operate on line $L$, albeit with reduced efficiency, which may be caused, for example, by a slowdown in the execution of scheduled activities deviating from the nominal durations specified in Table 1. The states of machine $m_2$ available after the occurrence of the disruption are presented in

Table 3. New states of machine $m_2$ ($t_i$ in seconds, $z_i$ in cycles, $e_i$ in $\mathrm{k}\mathrm{W}\mathrm{h}$).

${\mathit{m}}_2$				${\mathit{m}}_2$				${\mathit{m}}_2$				${\mathit{m}}_2$
${\mathit{x}}_1$	${\mathit{t}}_1$	${\mathit{z}}_1$	${\mathit{e}}_1$	${\mathit{x}}_1$	${\mathit{t}}_1$	${\mathit{z}}_1$	${\mathit{e}}_1$	${\mathit{x}}_1$	${\mathit{t}}_1$	${\mathit{z}}_1$	${\mathit{e}}_1$	${\mathit{x}}_1$	${\mathit{t}}_1$	${\mathit{z}}_1$	${\mathit{e}}_1$
1	136	102	5.5	6	132	78	4.6	11	175	55	4.6	16	102	73	4.6
2	131	131	5.6	7	158	79	4.5	12	96	45	4.5	17	131	73	4.5
3	128	114	5.7	8	161	91	4.5	13	108	39	4.5	18	114	75	4.5
4	138	106	4.5	9	158	92	4.5	14	117	72	4.5	…	…	…	…
5	155	102	5.1	10	166	87	4.2	15	109	69	4.2	100	66	47	4.2

.

An analysis of the current course of the process indicates that the adopted configuration $X^{\left(1\right)}=\left(\mathrm{46,38,81,51}\right)$ does not meet the predefined expectations due to desynchronization of the machines (the difference between the cycle times $t_i$ exceeds 10 s), as evidenced by the cycle time values: $t_1=63$ [s], $t_2=77$ [s], $t_3=63$ [s], and $t_4=73$ [s]. This implies that the synchronization condition of the machines, described by constraint (4), is not satisfied. Additionally, the values of two partial criteria have exceeded their acceptable ranges: the production cycle time, $T=77$ [s] $\nleqq 75$ [s], does not meet the specified requirements; in turn, the energy cost: $CE=0.48\le 0.5$, meets the specified requirements, whereas $CZ=5.1$ [m.u.] $\notin [3.5;5.0]$ does not meet the requirements. The change of point operating position of line $L$ (i.e., the values of the criteria for configuration $X^{\left(1\right)}$) resulting from the occurrence of the disruption is illustrated in Figure 8.

This situation raises the following question: For the given machine states from set $M$, does a configuration $X^{\left(2\right)}$ exist that allows for the execution of an order that minimizes unit production costs during the remaining 13 working days? The answer obtained in the Gurobi environment, in time $<10$ s, is a new configuration $X^{\left(2\right)}=\left(\mathrm{73,69,94,85}\right)$, resulting in new time values: $t_1=58$ [s], $t_2=48$ [s], $t_3=57$ [s], $t_4=55$ [s] and $TOC=9.27$ [m.u.]—see Figure 9.

The designated configuration $X^{\left(2\right)}$ results in the following values of the partial criteria: $T=58$ [s], $CE=0.378$ [m.u.] and $CZ = 4.81$ [m.u.]. These values fall within the specified ranges and ensure the completion of production within the required timeframe (production level $DP = 496$ units/day). In contrast to the nominal plan, configuration $X^{\left(2\right)}$ leads, however, to higher production costs, $TOC = 9.27$ [m.u.] (previously $TOC=8.91$ [m.u.]), which still remain within the acceptable limit of $TOC\le 10$ [m.u.].

It is worth noting that the replanning of the line configuration was completed in less than 10 s, i.e., in online mode. The newly adopted line configuration guarantees timely execution of the production order, although it may naturally be disrupted by subsequent changes in operating conditions—see next section.

6.3. Reactive Planning—Failure of One Machine

Let us consider a case in which machine $m_2$ experiences a complete failure (day 23, production state $NUP = 7865$ [units]), rendering the continuation of production impossible. In such a situation, it becomes essential to seek alternative solutions that enable the fulfillment of the order within the designated deadline while maintaining cost levels within the predefined limits. This need is addressed by implementing the line structure $SL=s{l}_3$—see Figure 4c—in which the execution of operation $o_2$ is allowed on a different machine $m_i\in M\backslash \left\{m_2\right\}$. In the considered scenario, operation $o_2$ can be performed on machine $m_1$, which results in an extension of the processing time per unit on machine $m_1$ and an increase in unit costs. The new states of machine $m_1$ in the reconfigured line $L$ are presented in

Table 4. New states of machine $m_1$ ($t_i$ in seconds, $z_i$ in cycles, $e_i$ in $\mathrm{k}\mathrm{W}\mathrm{h}$).

${\mathit{m}}_1$				${\mathit{m}}_1$				${\mathit{m}}_1$				${\mathit{m}}_1$
${\mathit{x}}_1$	${\mathit{t}}_1$	${\mathit{z}}_1$	${\mathit{e}}_1$	${\mathit{x}}_1$	${\mathit{t}}_1$	${\mathit{z}}_1$	${\mathit{e}}_1$	${\mathit{x}}_1$	${\mathit{t}}_1$	${\mathit{z}}_1$	${\mathit{e}}_1$	${\mathit{x}}_1$	${\mathit{t}}_1$	${\mathit{z}}_1$	${\mathit{e}}_1$
1	281	27	5.5	6	295	39	4.6	11	368	27	4.6	16	229	36	4.6
2	279	24	5.6	7	328	39	4.5	12	211	22	4.5	17	265	36	4.5
3	279	32	5.7	8	336	45	4.5	13	226	19	4.5	18	249	37	4.5
4	294	30	4.5	9	338	46	4.5	14	239	36	4.5	…	…	…	…
5	316	41	5.1	10	353	43	4.2	15	234	34	4.2	100	130	23	4.2

.

The adopted solution, however, implies the question: For the given machine states from the set $M\left\{m_2\right\}$, is there a configuration $X^{\left(3\right)}$ that allows the execution of an order that minimizes unit production costs within the remaining 5 working days? The answer obtained in the Gurobi environment, in time $<10$ [s], is a new configuration $X^{\left(3\right)}=\left(\mathrm{92,38,40}\right)$, giving new time values: $t_1=101$ [s], $t_2=91$ [s], $t_3=92$ [s] and $TOC=9.48$ [m.u.]—see Figure 10.

The designated configuration $X^{\left(3\right)}$ implies the following values of partial criteria: $T=101$ [s], $CE=0.493$ [m.u.], and $CZ = 4.05$ [m.u.]. Despite exceeding the value of $T>75$ [s] (for failure of machine $m_2$, the decision-maker changed the requirement for the production cycle time: $T \le 110$), the unit costs are within the assumed range, i.e., $TOC\le 10$ [m.u.], which allows for the introduction of an appropriate reconfiguration of the line structure and replanning of the related production flow.

The reported results correspond to the scale of machining lines typically found in the manufacturing cells of small- and medium-sized enterprises. This implies that the proposed approach can be effectively employed for the evaluation and synthesis of production scenarios in typical manufacturing cells, where the number of machines usually does not exceed ten.

To sum up, the implementation of the proposed approach to proactive–reactive planning of the balanced production flow allowed the order to be completed before the assumed deadline (production was completed on the 25th day). The calculation/reaction time for each of the discussed cases did not exceed 10 [s], which corresponds to the real needs of manufacturing companies.

6.4. Scalability

The presented line balancing examples have demonstrated that the developed approach can be effectively applied in both proactive (offline) and reactive (online) planning, aimed at identifying configurations that ensure desired values of multiple partial criteria ($T, CZ, CE$) as well as the aggregated criterion ($TOC$). This approach falls within the domain of multi-criteria optimization. In contrast to the existing approaches available in the literature [27,28,29], emphasis is placed here on a declarative formulation of the modeled problem. The open structure of the developed model, implemented within declarative programming environments, allows for unrestricted modification or updating of the planner’s/operator’s requirements. The scope of proposed changes may include previously unconsidered operating conditions (e.g., a new line structure) as well as new target values for both the partial criteria and the aggregated criterion (e.g., due to the occurrence of a disturbance).

The adopted representation using polar plots significantly facilitates the evaluation and selection of admissible solutions (the space $\mathbb{X}$), notably through the application of rings on the plots—resembling shooting targets—thus enhancing visual assessment.

The presented examples were limited to a line consisting of four machines; subsequent experiments were aimed at evaluating the scalability of the proposed approach. It should be noted that the computational complexity of the considered problem is significantly dependent on the number of assumed potential configuration options and on the complexity of the assumed production flow scenarios satisfying arbitrarily chosen synchronization constraints. The obtained results are summarized in Figure 11. They show that the computational feasibility of the considered problem (i.e., finding a solution within 10 min) is limited to

• $n=8$ machines, assuming that the number of states does not exceed 100;
• $n=6$ machines, assuming that the number of states does not exceed 150;
• n = 4 machines, assuming that the number of states does not exceed 200;
• $n=3$ machines, assuming that the number of states does not exceed 250.

These parameters are representative of a typical single production line, which rarely includes more than three machines.

In summary, it is worth emphasizing that the proposed approach, aligning with the current development trends in DSS systems [27,28,30], finds practical application in the planning of balanced production flow within machining lines for short- and medium-series manufacturing. Its main advantage lies in its potential for use in scenarios requiring dynamic scheduling disruption management [27]. While this paper limits itself to a qualitative assessment of the proposed approach, we note that the absence of a quantitative evaluation stems from a lack of literature sources presenting numerical research findings in the considered domain. This limitation hinders the ability to compare its competitiveness in terms of computational efficiency, the most frequently occurring disruptions and failures, operator training requirements, and other relevant factors.

7. Conclusions

This paper has proposed a novel declarative framework that unifies proactive production line balancing and reactive disruption management with sustainability-aware optimization in machining environments. By adopting a constraint-based, multi-criteria formulation, the approach addresses critical limitations of traditional sequential planning methods, particularly their inability to cope with dynamic disruptions and conflicting operational objectives. The integration of predictive planning and adaptive dispatching within a single decision-support system represents a significant advancement in the design of intelligent, interactive tools for modern manufacturing.

The framework supports real-time reconfiguration of resources through an open architecture, enabling seamless incorporation of user-defined objectives and operational constraints. The use of Pareto-based optimization facilitates informed trade-offs between key performance indicators, such as takt time, tool wear, energy consumption, and total operating cost. This structure supports both exploratory “what-if” analyses and prescriptive planning strategies without requiring modifications to the model’s core logic. As demonstrated through the case study, the proposed approach not only ensures production continuity under uncertain conditions but also improves system responsiveness, enhances sustainability alignment, and supports operator engagement through interactive visualizations.

Results of this work naturally intersect with the broader concept of sustainable development, going beyond the direct performance indicators of the considered case. For example, effective line balancing supports limiting overproduction, which reduces storage costs and energy usage. Rapid response to disruptions and failures minimizes the risk of delays, mitigates cascading failure effects (the so-called “domino effect”), and improves flow control. An open structure of the proposed model allows easy integration of additional sustainability-related criteria, such as waste generation, carbon footprint, and material efficiency.

In comparison to conventional methods that treat rescheduling, rebalancing, or reallocation as separate and sequential tasks, the presented solution addresses these interrelated challenges in a holistic and flexible manner. This positions the framework as a practical tool for short- and medium-series production, particularly in machining lines with frequent disruptions and high reconfiguration demands.

While the qualitative evaluation confirms the utility and feasibility of the approach, upcoming work will focus on developing quantitative performance benchmarks—particularly in terms of computational efficiency, failure handling patterns, and operator training requirements—while also extending the evaluation scope to include robustness measures, adaptability indices, and sustainability-related metrics.

In sum, this work lays a foundation for next-generation decision support systems that integrate resilience, sustainability, and adaptability as core design principles—bridging the gap between theoretical advances in declarative modeling and their practical application in industrial production systems.

The presented implementation has been validated using only a single case study, which may limit its generalizability to other manufacturing sectors. It is worth noting that the model’s effectiveness depends on the availability and accuracy of input data, which can vary across different industrial environments. This suggests that future research should explore the scalability of the approach in more complex systems $(n > 10)$, as well as its potential integration with real-time data acquisition systems and adaptive mechanisms based on machine learning.