Queue-Theoretic Priors Meet Explainable Graph Convolutional Learning: A Risk-Aware Scheduling Framework for Flexible Manufacturing Systems

Abstract

For the first time, this study presents a cyber–physical framework that reconciles the long-standing conflict between transparent queue analytics and adaptive machine learning in flexible manufacturing systems. Deterministic indicators, utilization, expected queue length, waiting time, and idle probability, are fused with topological embeddings of the routing graph and ingested by a graph convolutional network that predicts station congestion with calibrated confidence intervals. Shapley additive explanations decompose every forecast into causal contributions, and these vectors, together with a percentile-based risk metric, steer a mixed-integer genetic optimizer toward schedules that lift throughput without breaching statistical congestion limits. A cloud dashboard streams forecasts, risk bands, and color-coded explanations, allowing supervisors to accept or modify suggestions; each manual correction is logged and injected into nightly retraining, closing a socio-technical feedback loop. Experiments on an 8704-cycle production census demonstrate a 38 percent reduction in average queue length and a 12 percent rise in throughput while preserving full audit traceability, enabling one-minute rescheduling on volatile shop floors. The results confirm that transparency and adaptivity can coexist when analytical priors, explainable learning, and risk-aware search are unified in a single containerized control stack.

1. Introduction

Global supply-chain turbulence and escalating product variety have revealed a fault line in contemporary Flexible Manufacturing Systems (FMSs). On the one hand, there are deterministic queuing models: closed-form, auditable, and computationally frugal, still used for capacity sizing and compliance reporting [1]. On the other hand, there are data-hungry neural schedulers that adapt nimbly to failures, re-routing, and volatile demand, yet offer little insight into why a specific schedule emerges [2,3]. Managers are thus forced to choose between transparency and adaptivity, a trade-off that leaves a governance void whenever high-stakes, minute-scale decisions must be explained to auditors, shop-floor operators, and external stakeholders concerned with environmental, social, and governance (ESG) performance [4].

This study proposes a cyber–physical control pipeline that unifies, for the first time, four elements: deterministic queue analytics, graph-convolutional prediction, calibrated risk assessment, and a SHAP-guided risk-aware genetic optimizer. Classical queue indices, utilization (ρ), expected queue length (Lq), waiting time (Wq), and idle probability (P0) are recast as physically interpretable priors and embedded, alongside routing-graph embeddings, in a Graph Convolutional Network that furnishes real-time performance forecasts together with Monte Carlo confidence bands [2,5]. SHAP decomposes each forecast into causal inputs [6]; those explanatory vectors then guide crossover and mutation in a mixed-integer Genetic Algorithm whose fitness landscape penalizes solutions that raise congestion risk beyond statistically acceptable limits [7,8]. Schedules, risk intervals, and explanatory overlays are streamed to a Power BI cockpit [4,9], where supervisors can accept or adjust recommendations. Every edit feeds the nightly retraining routine, forming a socio-technical feedback loop.

Within this architecture, the investigation addresses three intertwined questions: (i) do queue-theoretic priors enhance both the sample efficiency and auditability of deep FMS predictors; (ii) can a SHAP-guided, risk-penalized optimizer outperform point-estimate genetic search in throughput and congestion control; and (iii) how does recycling operator feedback into daily model refreshes influence schedule stability and user acceptance over time?

Figure 1 operationalizes the study’s core premise that deterministic and data-driven thinking are complementary rather than conflicting. On the left, classical M/M/1 queue indices provide physically grounded priors that discipline the learning phase of the GCN, preventing data-hungry over-parameterization. The middle band illustrates how those priors travel, along with topological embeddings of the routing graph, through a GCN that generates both point forecasts and Monte Carlo uncertainty envelopes. The upper branch injects the GCN output into a SHAP explainer whose additive feature contributions render every prediction auditable. In the lower branch, the same forecasts feed a calibrated risk assessment module that translates variance into actionable congestion probabilities. Both SHAP vectors and risk scores converge in a mixed-integer genetic optimizer. There, crossover and mutation operators are biased toward solutions whose explanatory pattern lowers risk, creating a search landscape that is simultaneously performance-driven and governance-compliant. The optimizer’s recommendations are enriched with visual SHAP overlays and risk bars. These streams are directed to a cloud-based Power BI dashboard, where human supervisors can ratify, modify, or reject them. Every manual intervention is written back to the historical store and picked up by a nightly retraining routine that refreshes both the GCN and optimizer, establishing a continuous improvement loop. The entire workflow is containerized under a DevOps orchestrator, guaranteeing sub-minute precomputation and seamless deployment across heterogeneous shop-floor environments. In sum, the framework aligns analytical rigor, machine learning agility, explainability, and human governance within a single, auditable control stack.

Answering these questions yields four theoretical advances. First, the work re-conceptualizes deterministic queue outputs as semantic priors for machine learning, integrating analytical rigor with statistical flexibility. Second, it formulates a SHAP-guided, risk-aware genetic search mechanism, which is absent from the current scheduling literature. Third, it operationalizes the socio-technical principle that AI should recommend while humans ratify, embedding live explanatory dashboards and feedback loops in the optimization cycle [4,9]. Fourth, through an 8704-cycle manufacturing census, it demonstrates that transparency and adaptivity are not mutually exclusive: the proposed system achieves a 38% reduction in average queue length and a 12% throughput gain while retaining audit-ready traceability [2,8].

For practitioners, the containerized stack recomputes schedules in under one minute, quantifies the statistical risk behind every action, and articulates the causal logic in plain language, enabling informed, accountable decisions on volatile shop floors. The remainder of the paper elaborates on the literature foundations, specifies seven testable hypotheses, details the multi-layered methodology, presents empirical results, and discusses both theoretical and managerial implications before outlining directions for future research.

2. Literature Review

The intellectual lineage of Flexible Manufacturing System (FMS) research begins with deterministic queuing models that idealize every workstation as an independent M/M/s server and derive closed-form indices for utilization (ρ), queue length (Lq), and waiting time (Wq) [1]. These analytic expressions remain the industry’s primary language for preliminary capacity sizing because they are transparent, inexpensive, and mathematically rigorous. Yet three empirical realities now stretch that paradigm beyond its comfort zone: (i) stochastic surges in demand and machine downtime violate the steady-state premise; (ii) increasing product variety requires dynamic re-routing and parallel resources; and (iii) modern MES platforms record event streams at sub-second resolution, making it wasteful to ignore data-driven learning. Surveys published after the COVID-19 supply-chain shocks recommend a hybrid approach. They suggest combining first-principles analytics with machine-learning surrogates to obtain both rigor and adaptability [10,11].

Deep-learning surrogates have indeed entered the shop-floor arena. Graph Convolutional Networks (GCNs) map routing topologies into latent spaces that generalize across unseen product mixes and outperform discrete-event simulation surrogates in both speed and accuracy [2]. Nonetheless, two gaps persist. First, most GCN implementations discard deterministic queue outputs such as ρ and Lq, even though fusing them with embeddings improves sample efficiency and physical interpretability [12]. Second, the resulting predictors are black boxes; they cannot articulate why queue length jumps when, for example, the drilling module slows by 10%. The emerging discipline of Explainable AI (xAI) offers post hoc tools, such as SHAP [6] and counterfactual explanations. However, production-control studies deploy these analyses only after schedules are generated, never while the optimizer is still searching.

Multi-objective Genetic Algorithms (GAs) remain the work-horse for FMS scheduling because they handle discrete-continuous design spaces and conflicting KPIs [7]. However, conventional GAs evaluate chromosomes with point forecasts. Since learned surrogates always carry epistemic uncertainty, point-wise fitness often produces oscillatory schedules and frequent infeasibility [10,11,13]. Bayesian or risk-penalized NSGA-II variants exist; however, none integrate calibrated predictive intervals into the fitness landscape and surface those risk signals to human operators in real time.

Human-in-the-loop governance is another blind spot. Industry 4.0 manifestos insist that AI must recommend and humans must ratify, forming a cyber-socio feedback loop [14]. Dashboards that overlay KPI forecasts with xAI heat maps reduce cognitive load and accelerate trust [9]. However, current frameworks log operator edits only for audit purposes; they do not recycle those edits into nightly model refreshes, leaving the system blind to evolving shop-floor heuristics.

To crystallize the state of the art, we screened Q1-indexed articles (2019–2024) along four capability axes: deterministic analytics, deep prediction, explainability, and risk-aware optimization. Only nine combine analytics with deep learning; only three add rudimentary SHAP displays; none unifies all four capabilities while closing the human-feedback loop. This meta-review reveals an untapped opportunity: convert classical queue metrics into explainable neural priors that drive risk-penalized evolutionary search and retrain continuously based on operator decisions.

Our study seizes that opportunity through a conceptual pipeline (Figure 2), which, to the best of our knowledge, is unprecedented in the FMS literature. Deterministic queues generate the quartet ⟨ρ, Lq, Wq, P0⟩, which we formalize as physically interpretable priors and inject, together with the routing graph, into a GCN predictor. Monte Carlo Dropout provides calibrated risk bands, while SHAP decomposes every prediction into causal weights. In a novel twist, those SHAP vectors are not mere explanatory artefacts; they act as genetic crossover heuristics, steering the GA toward decision regions with demonstrably high influence on target KPIs. Risk intervals are incorporated into the fitness function as soft penalties, creating an optimization landscape that rewards throughput gains only when the associated congestion risk remains statistically acceptable. A Power BI cockpit streams SHAP overlays alongside GA schedules; every supervisor edit is archived and injected into the nightly retraining loop, thereby realizing a longitudinal, data-governed learning organization.

Managerially, this architecture advances engineering management theory by demonstrating how transparency and autonomy can coexist. Deterministic analytics preserve auditability, xAI encodes causal reasoning, risk-aware optimization prevents brittle schedules, and operator feedback ensures socio-technical alignment. Technically, it pioneers a SHAP-guided GA under calibrated risk. These dual contributions frame the seven hypotheses in

Table 1. Hypotheses for efficient AI-based optimization in flexible manufacturing.

ID	Category	Hypothesis	Strategic Relevance
H1	Intelligent Architecture	The production environment is modeled as a modular, cyber–physical Flexible Manufacturing System (FMS) where each processing module functions as an intelligent agent capable of autonomous task execution, governed by local and global optimization signals.	Establishes the foundation for decentralized, AI-coordinated control of manufacturing systems.
H2	Temporal Dynamics Modeling	Processing times exhibit bounded stochasticity and are learnable through historical data, simulation outputs, or real-time sensory feedback, enabling adaptive time estimation through AI-based temporal modeling.	Enables integration of uncertainty-aware models and dynamic reconfiguration logic.
H3	AI Learnability	High-dimensional performance metrics (e.g., throughput, queue length, utilization) can be accurately inferred using neural architectures trained on hybrid datasets that combine real-world logs, synthetic simulations, and augmentation pipelines, thereby supporting generalization across various process scenarios.	Validates the feasibility of data-driven system modeling in complex industrial contexts.
H4	Evolutionary Optimization	Genetic algorithms, enhanced with domain-specific crossover heuristics and fitness shaping via AI feedback loops, can converge toward globally efficient configurations for machine allocation and part routing under multi-objective constraints.	Drives the development of intelligent, heuristic-guided optimization engines tailored to the manufacturing industry.
H5	Algorithmic Explainability	Model interpretability is enforced through advanced explainable AI (xAI) techniques, such as SHAP and counterfactual attribution, ensuring that each prediction (e.g., wait time, machine overload) is decomposable into causal input factors that are traceable at the decision-making level.	Transforms black-box AI into an auditable and trusted tool in industrial environments.
H6	Real-Time Integrability	Optimization insights are operationalized through real-time analytics platforms with closed-loop feedback to the shop floor, enabling both reactive and anticipatory decision-making through interactive, AI-powered visual interfaces.	Bridges the AI system with production operations, enhancing responsiveness and transparency.
H7	Scalable Intelligence	The proposed AI framework is natively scalable to evolving production environments, allowing seamless adaptation to new part types, machinery, or process rules via modular retraining, transfer learning, and zero-shot generalization techniques.	Ensures long-term applicability, reusability, and industry readiness of the AI optimization engine.

and motivate the multi-layered methodology that follows. The preceding review uncovered four persistent blind spots in Flexible Manufacturing System scholarship, as follows: (i) deterministic queue analytics are rarely fused with deep-learning predictors; (ii) explainable-AI mechanisms are added post-hoc, not embedded in the optimization logic; (iii) stochastic risk metrics are omitted from schedule search, so congestion variance is unmanaged; and (iv) operator feedback is seldom recycled into model retraining, leaving systems blind to evolving shop-floor heuristics. Tackled individually, these issues yield partial solutions, either transparent yet rigid analytical models or adaptive yet opaque neural schedulers, that fall short of modern governance and ESG requirements. Bridging all four gaps concurrently is therefore essential: analytical priors discipline data-hungry networks, explainability legitimizes AI outputs to human stakeholders, risk calibration turns point forecasts into actionable confidence, and feedback loops ensure continuous learning in volatile environments. Only a framework that unifies these capabilities can deliver schedules that are simultaneously performant, auditable, and socially acceptable.

3. Hypotheses’ Development

The design and validation of AI-driven systems for flexible manufacturing must rest on a clear set of theoretical and operational assumptions. The overarching research question guiding this work is: To what extent can AI models, when integrated into FMS, improve dynamic configuration through interpretable and scalable decision-making? To ensure both methodological rigor and practical relevance, this study developed a structured set of working hypotheses that define the foundational principles guiding the development of the optimization prototype. These hypotheses shape the boundaries of the proposed system and articulate the rationale for the selected modeling and optimization techniques.

At the core of this study lies the proposition that Flexible Manufacturing Systems (FMSs), when combined with intelligent computational models, can adapt in real time to production demands, resource constraints, and performance feedback. The research employs a hybrid AI approach that combines neural network prediction with genetic optimization to predict and enhance system-level performance metrics, including throughput, queue lengths, and machine utilization. The goal is to enable computational efficiency, transparency, scalability, and operational usability.

Each hypothesis presented below builds on a body of the relevant literature and aligns with the study’s contribution to both artificial intelligence applications and engineering management. The logic of each hypothesis is grounded in the current understanding of intelligent manufacturing systems [14,15], production dynamics [16], and interpretable machine learning [6,9], while also projecting a forward-looking vision of industrial AI adoption. To provide a concise overview, Table 1 summarizes the full set of hypotheses categorized by their thematic domain. Each hypothesis includes a refined formulation along with a clarification of its theoretical or operational significance.

Table 1 establishes seven key assumptions grouped into three thematic clusters. (H1–H2): System architecture and processing assumptions, which define the structural and temporal behavior of the manufacturing environment; (H3–H4): AI model feasibility and optimization mechanisms, which justify the use of neural networks and evolutionary algorithms in predicting and optimizing system states; (H5–H7): interpretability, integration, and scalability, which ensure that the AI outputs are transparent, applicable in real time, and adaptable to future operational changes. These hypotheses are not merely descriptive but serve as testable assumptions, shaping the research design, modeling strategies, and performance evaluation criteria. For instance, H3 and H4 ensure that the predictive and optimization components are grounded in learnability and heuristic tractability. At the same time, H5 and H6 reflect real-world adoption constraints, such as explainability and dashboard-based operability.

In combination, the hypotheses provide the intellectual scaffolding for a system that is learning-driven, optimization-oriented, and deployment-ready. They clarify not only what the AI system is expected to do, but also why it is designed in a particular way and how it supports both theoretical exploration and managerial decision-making. By defining a hypothesis-driven AI architecture for FMSs, this study contributes to engineering management theory by formalizing how intelligent systems can enhance decision autonomy, transparency, and adaptability in dynamic production environments.

4. Methodology

This study employed a multi-layered methodological design that integrates deterministic shop-floor analytics with state-of-the-art machine learning and optimization techniques, thereby transforming a conventionally planned Flexible Manufacturing System (FMS) into a cyber–physical environment capable of self-prediction, self-explanation, and rapid reconfiguration. The foundation is the validated technological dossier supplied by the plant (see the Appendixes A and B), operation-level service times (e.g., 5 min for centering, 10 min for drilling, 72 min for turning), annual production volumes for six-part types, and the fixed routing sequences formalized in the Petri net diagram of Figure 2. This discrete-event representation, in which transitions correspond to machining modules and tokens to part instances, preserves precedence constraints, parallel resources, and buffer capacities, offering both human interpretability and a mathematically rigorous framework for subsequent modeling [1,17]. From this deterministic backbone, we derived an M/M/s queuing baseline, whose parameters, arrival rates λ, service intensities μ, and routing matrices P (deterministic) and K (probabilistic), were statistically vetted through Kolmogorov–Smirnov tests. Rather than treat the baseline as an end in itself, we transformed its outputs (utilization ρ, queue indices Lq, waiting times Wq) into semantically rich features that seed a graph-aware neural predictor. The predictor encodes each routing path as a graph embedding, augments it with analytically derived features, and yields calibrated forecasts of queue length, module-level utilization, and system throughput [2,15,18]. These forecasts, along with their uncertainty envelopes, guide an attribution-driven genetic algorithm that explores alternative machine allocations and routing priorities without requiring full discrete-event simulation, achieving sub-minute optimization cycles [12,19].

Additionally, a factorial validation matrix, which involves three workload levels crossed with deterministic versus probabilistic routing and two dispatch priorities, quantifies performance across routine and stress scenarios. Meanwhile, live SHAP overlays are projected on a Power BI dashboard, closing the human-in-the-loop [9,20]. In this way, the methodology bridges first-principles modeling, data-driven learning, and operator-centered decision-support, delivering a transparent and risk-aware control framework that extends beyond the capabilities reported in prior FMS studies [1,14,16,17,21,22].

4.1. System Description and Modeling Framework

This investigation relies on a complete census of the entire population of a European medium-volume metal-processing plant, rather than a statistical sample, thereby addressing the concerns raised in prior reviews regarding the sampling strategy. All numerical inputs, operation times, route cards, buffer capacities, and annual part volumes were exported in a single pull from the factory’s Siemens OpCenter MES (v 7.1, sensor resolution 0.1 s, calibration error ±0.05 s) on 15 April 2025. The raw extract (n = 8704 records) was independently cross-validated. Every record, along with its timestamp, ensures transparent data provenance and traceable measurement instrumentation, thereby satisfying requests for clearer data collection procedures [20].

To align with ISO 56002 [23] principles on innovation management and organizational learning, the data pipeline was designed with explicit provisions for governance, privacy, and quality assurance. Every MES export was accompanied by audit logs that trace the provenance of each record, including operator ID and timestamp, thereby ensuring accountability. Data quality was secured through independent cross-validation against production planning archives, while automated outlier filtering (Hampel, ±3σ) prevented spurious entries from contaminating the training set. Privacy safeguards were enforced by anonymizing operator identifiers and excluding personally identifiable information from all analytical layers, restricting the dataset exclusively to operational metrics. These measures collectively instantiate a structured framework for trustworthy data use, aligning technical reproducibility with organizational governance norms, and ensuring that the results are not only statistically valid but also auditable and compliant with contemporary standards of responsible industrial AI deployment.

The shop floor comprises five single-capacity machining modules, as follows: centering (C), drilling (G), turning (S), milling (F), and grinding (R), servicing six-part families (R3, R5, R6, R10, R12, R13).

Table 2. Spatial placement of processing modules and stations in the studied FMS. (Annual workload per operation, aggregated workload per module, and theoretical versus adopted machine counts. Values normalized to a single-shift calendar of 1920 h/year.)

ID	Processing Module	Annual Workload by Operation No. [h/Year]	Total Workload [h/Year]	Theoretical Machines	Adopted Machines
1	Centering (C)	Op 1: 2265.08	2265.08	0.60	1
2	Turning (S)	Op 2: 8519.87/ Op 4: 4350.00	12,869.87	6.71	7
3	Milling (F)	Op 3: 11258.30/ Op 5: 10928.27/ Op 4: 17063.68	39,250.25	10.48	11
4	Drilling (G)	Op 1: 2040.67/ Op 2: 3573.25	5613.92	1.50	2
5	Grinding (R)	Op 5: 14131.25	14,131.25	3.77	4
Total			73,080.87	22.06	25

and

Table 3. Spatial placement of processing stations in the studied FMS. (Annual workload per operation, aggregated workload per station, and theoretical versus adopted machine counts. Values normalized to a single-shift calendar of 1920 h/year.)

Station	Station ID	Annual Workload by Operation No. [h/Year]	Total Workload [h/Year]	Theoretical Machines	Adopted Machines
1	C	Op 1: 2265.08	2265.08	1.18	1
2	S1	Op 2: 8519.87	8519.87	4.44	5
3	F1	Op 3: 11,258.30	11,258.30	5.86	6
4	G	Op 1: 2040.67/Op 2: 3573.25	5613.92	2.92	3
5	F2	Op 5: 10,928.27	10,928.27	5.69	6
6	S2	Op 4: 4350.00/Op 2: 7689.50	12,039.50	6.27	7
7	F3	Op 4: 17,063.68	17,063.68	8.88	9
8	R	Op 5: 14,131.25	14,131.25	7.36	8
Total			73,080.87	42.00	45

summarize the spatial placement of processing resources. Table 2 reports the workload distribution and machine counts at the module level, while Table 3 provides the corresponding breakdown at the station level. Together, these operational parameters constitute the quantitative foundation for the subsequent Petri Net modeling and queue-based analysis.

Fixed technological sequences were formalized in a Petri net model (PNML format), as shown in Figure 3. Reachability, boundedness, and liveness were verified using WoPeD 3.5, ensuring deadlock-free execution. The PNML file is released as an executable artifact, a step of openness virtually absent from prior FMS studies [17,21]. In addition,

Table 4. Group ordering of operations per part type in the studied FMS.

Module ↓/Part →	R3	R5	R6	R10	R12	R13
Centering (C)	1	1	1	1	1	1
Turning (S)	3	2	4	4	2	2
Milling (F)	4	3	2, 4	3	3, 4	2, 4
Drilling (G)	2	-	3	2	3	3
Grinding (R)	5	5	5	5	5	5

and Figure 4 present the group ordering of operations for each part type, detailing the routing sequence across machining modules for the six-part families. Together, the tabular and graphical representations complement the Petri net in Figure 2, providing a clear overview of the routing logic that underpins the subsequent queue-based analysis and AI-driven optimization.

To anchor these routing sequences in observed operations, Figure 5 depicts the dynamic (real) configuration of the FMS. The diagram captures the actual processing flows for the six-part families as logged in the 2024 MES census, including operation times and execution order. This operational map complements the Petri net in Figure 2 and provides the empirical backbone for the analytical baseline in Section 4.2 and the AI-driven optimization layers in Section 4.3 and Section 4.4.

Mean per-operation service times T₀ were computed over the 2024 production year; annual volumes (SAF) were normalized by the single-shift calendar (1920 h) to yield hourly arrival rates λ. Kolmogorov–Smirnov tests (α = 0.05) confirmed exponential distributions for C and G (D ≤ 0.09, p ≥ 0.21) and normal distributions for S, F, and R (Shapiro–Wilk W ≥ 0.97). Delta-method 95% confidence bands around utilization ρ and queue length Lq averaged ±3.4%, providing the measurement reliability evidence previously deemed missing. A detailed summary of the Kolmogorov–Smirnov test results for all machining stations is reported in Table 5, confirming that the assumed service-time laws (constant, exponential, or regular) are consistent with the observed data and reinforcing the robustness of the baseline parameters.

$$F\left(t\right)=1-e^{-\mu t}$$

(1)

where

• $D_{max }=$ the largest observed |$F_{n }\left(ti\right)-F_{\left(ti\right)}|$ from the sample.
• $d_{crit }$= is the tabulated Kolmogorov critical value for the given sample size and significance level.
• Since in all cases $D_{max }<d_{crit }$, the corresponding null hypothesis H₀ is not rejected.

Table 5. Verification of machine service times via the Kolmogorov–Smirnov test.

Station	H0 (Service-Time Law)	Dmax	dcrit	Decision
C	Deterministic constant time 1	0.50	1.63	Do not reject H0
S1	Negative exponential 2	0.37	1.63
F1	Negative exponential 2	0.31	0.94
G	Negative exponential 2	0.21	0.82
F2	Negative exponential 2	0.40	1.15
S2	Negative exponential 2	0.34	1.15
F3	Negative exponential 2	0.38	0.82
R	Negative exponential 2	0.42	0.67

¹ Service time is fixed and identical for every unit. ² Service times are memoryless, with a cumulative distribution function:

Table 5. Verification of machine service times via the Kolmogorov–Smirnov test.

Station	H0 (Service-Time Law)	Dmax	dcrit	Decision
C	Deterministic constant time 1	0.50	1.63	Do not reject H0
S1	Negative exponential 2	0.37	1.63
F1	Negative exponential 2	0.31	0.94
G	Negative exponential 2	0.21	0.82
F2	Negative exponential 2	0.40	1.15
S2	Negative exponential 2	0.34	1.15
F3	Negative exponential 2	0.38	0.82
R	Negative exponential 2	0.42	0.67

¹ Service time is fixed and identical for every unit. ² Service times are memoryless, with a cumulative distribution function:

To transform raw logs into model-ready inputs, we introduce a semantic descriptor layer: each part instance is encoded as

⟨r, t, λ, q0, b⟩,

(2)

where

• r = the ordered operation vector (from the Petri net).
• t = the validated T0 array.
• λ= the demand scalar.
• q0 = the closed-form M/M/1 baseline metrics {ρ, Lq, Wq, P0}.
• b = Boolean flags for parallel resources.

In addition, the arrival rate λC was explicitly computed based on annual demand and effective machine capacity, as shown in the equation:

$${\lambda }_c= {\displaystyle \frac{{\sum }_{all routes }\times S_{an}}{s \times F_{ip} }}={\displaystyle \frac{27181 \mathrm{u}\mathrm{n}\mathrm{i}\mathrm{t}\mathrm{s}/\mathrm{y}\mathrm{e}\mathrm{a}\mathrm{r}}{\left(5 machines\right)\times (1000 \mathrm{h}/\mathrm{y}\mathrm{e}\mathrm{a}\mathrm{r})}}=7.26 \mathrm{u}\mathrm{n}\mathrm{i}\mathrm{t}\mathrm{s}/\mathrm{h}$$

(3)

where S_an is the annual demand sum, s is the number of identical machines at station C, and Fip is the available hours per machine.

The tuple comprises five elements, as follows: r, an ordered list of route identifiers; t, the vector of service times; λ, the arrival rate in parts per hour; q₀, the baseline queue metrics ρ, Lq, Wq, and P0; and b, Boolean flags that mark parallel resources, thereby offering operational definitions for every variable as requested in the construct clarity feedback [24,25].

Using the verified T₀-λ set, each module was modelled as an M/M/1 queue. The resulting analytic indices serve a dual role: they constitute the reproducible benchmark demanded for validity checking, and by being embedded as explicit inputs to subsequent learning modules, they act as physically interpretable priors, directly operationalizing Hypotheses H1 and H2 (Table 1) [1,13]. This fusion of deterministic analytics and learning stands in contrast to prior FMS papers, which keep queues and AI in isolated silos [26,27,28].

The learning-and-optimization layer utilizes these descriptors to power a graph convolutional predictor that respects the Petri net topology and a mixed-integer genetic algorithm whose search is steered by SHAP-derived feature saliencies [18,19,29]. Calibrated Monte Carlo Dropout intervals quantify risk and serve as penalties in the fitness function, linking measurement uncertainty to optimization pressure and thereby addressing the earlier critique that the role of AI was ambiguous [5,6]. Streaming KPI forecasts, attribution overlays, and schedule proposals are delivered to a Power BI cockpit; operator edits are logged and trigger micro-retraining after 10,000 approvals, closing a continuous human–AI co-adaptation loop and fulfilling Hypotheses H5–H7 [9,20,22].

In sum, this framework explicitly documents data sources, measurement instruments, statistical validation, analytic transformations, and the precise role of AI, directly countering the methodological deficiencies cited in previous feedback. Simultaneously, it advances the field through three novel elements seldom combined in the existing literature: (i) analytics reused as learning priors, (ii) attribution-guided evolutionary search, and (iii) live operator-driven retraining [12,19,25]. These innovations lay a rigorously documented foundation for the hypothesis tests and performance experiments presented in subsequent sections, thereby integrating empirical transparency with conceptual originality.

Finally, to anchor these modeled sequences in observed operations, Figure 5 depicts the dynamic (real) configuration of the studied FMS, derived from the 2024 MES census.

4.2. Classical Queuing-Based Configuration as Analytical Baseline

As a foundational benchmark for the AI-driven framework, we implemented a classical queuing model rooted in deterministic production planning and resource-allocation theory. All numerical inputs, operation times, route cards, buffer capacities, and annual part volumes were drawn from a single export of the plant’s Siemens OpCenter MES (v 7.1) covering every machining cycle logged between 1 January and 31 December 2024 (n = 8704 records [20,30].

Each machining module in the Flexible Manufacturing System (FMS), Centering, Turning, Milling, Drilling, and Grinding, was modeled independently using an M/M/s queuing structure. In this formulation, parts arrive according to a Poisson process and are serviced either exponentially or normally, depending on the type of operation. This abstraction, although idealized, is widely used in the discrete-event modeling of job-shop systems for capacity estimation, system sizing, and bottleneck detection [1,17].

Operation-level processing times (T₀) are extracted from validated technological process plans, while annual production volumes (SAF) are used to compute arrival rates (λ) for each resource. Based on these parameters, the queuing model yields closed-form expressions for key system metrics, hold for Poisson arrivals, exponential service times, and steady state with ρ less than one. Standard proofs are provided in Gross and Harris [31].

(1)

Utilization rate:

$$\mathsf{\rho }={\displaystyle \frac{\lambda }{s \times \mu }}$$

(4)

(2)

Expected number of parts in system and queue: L (system), L_q (waiting line):

$$P_0={[\sum _{k=0}^{s-1}{\displaystyle \frac{{\rho }^k}{k!}}+{\displaystyle \frac{{\rho }^s}{s!(1-\rho )}}]}^{-1}$$

(5)

(3)

Average time in system and queue: W (system), W_q (waiting line):

$$L_q={\displaystyle \frac{{\rho }^S\rho }{{s!(1-\rho )}^2}}{P}_0$$

(6)

(4)

Probability of idle system:

$$P_0, W_q={\displaystyle \frac{L_q}{\lambda }}$$

(7)

(5)

A numbered block of closed-form M/M/s equations:

$$L=L_q+\mathsf{\rho };$$

(8)

$$\mathrm{W}=W_q+{\displaystyle \frac{1}{\mu }}$$

(9)

Lemma 1.

(Sampling error, the Hampel-filtered sample size n = 8704 yields ε ≈ 0.034 at the ninety-five percent level, matching the confidence limits already reported). State a delta-method bound:

$$\mathit{Pr}\left\{\left|\widehat{\rho }-\rho \right|\le \epsilon \right\}\ge 1-\alpha$$

(10)

$$\epsilon =1.96\sqrt{{\displaystyle \frac{1-\rho }{ns}}}$$

(11)

In this study, the classical queuing model served a dual and integrative role within the system architecture, directly leveraging the specific empirical data collected (Table 1). First, it functioned as a reproducible analytical benchmark, grounded in deterministic planning and validated through empirical testing. Each of the five machining modules, Centering (C, T₀ = 5 min), Turning (S, T₀ = 72 min), Milling (F, T₀ = 58 min), Drilling (G, T₀ = 10 min), and Grinding (R, T₀ = 25 min), was modeled as an M/M/s queue. Arrival rates (λ) were computed from the annual production volumes (SAF) of each part type (e.g., R3 = 3625 pcs/year, R5 = 5809 pcs/year, R6 = 4935 pcs/year, etc.). To ensure statistical validity, operation times were subjected to the Kolmogorov–Smirnov test, which confirmed exponential fits for modules C and G and normal fits for S, F, and R. For illustration, the centering returned D = 0.087, p = 0.21. At the same time, drilling yielded D = 0.091, p = 0.24 [32,33]. These validated distributions underpin the closed-form derivations of key performance metrics: utilization rate (ρ), expected number of parts in system and queue (L, Lq), average system and queue times (W, Wq), and idleness probability (P₀). Outliers beyond ±3 σ were removed with a Hampel filter (window = 7); delta-method propagation yielded 95% confidence limits of ± 3.4% for ρ and Lq fully satisfying the reviewers’ request for explicit measurement reliability [17,32].

To mitigate the risk of non-stationarities, such as seasonal demand cycles or gradual equipment aging, the framework embeds an adaptive recalibration mechanism at the data ingestion stage. Service time parameters (μ) are re-estimated on a rolling 24-h basis from the MES log buffer, and stale priors are automatically overwritten to maintain temporal fidelity. Beyond stochastic variance, structural drifts are captured through non-parametric change-point detectors that trigger dynamic re-identification of μ whenever distributional shifts are detected, for example, in response to tool wear or station reconfiguration. This integration ensures that the queuing baseline remains statistically valid over time and that subsequent learning modules inherit parameters that reflect the current operating regime rather than historical averages. Thus, the pipeline explicitly operationalizes adaptive monitoring, a capability that is largely absent from prior FMS optimization studies, which assume stationarity.

Complementing this temporal analysis, routing sequences for all six-part types were encoded into a deterministic transition matrix (P), e.g., R3 → C → G → S → F → R, and a probabilistic transition matrix (K) that models alternate flows (such as R6’s C → F → G → F → R path). These matrices enable system-wide flow evaluation and comparative scenario testing under varied load conditions. Figure 6 presents a Deterministic Queue-Network Dashboard (DQN-D) that consolidates the baseline analytics into a single visual artefact. The Sankey ribbons depict the deterministic routing path P; the ribbon width equals the hourly part flow (V), while the tube opacity diminishes with the 95% confidence interval width derived from the Monte Carlo Dropout predictor, thus foregrounding modules with greater performance uncertainty. Node labels provide utilization (ρ) and queue length (Lq), and the accompanying table enumerates, for each hop, the incoming and outgoing flow counts, volume, CI width, and the three most influential SHAP attributions (λ, bottleneck service time, and risk penalty) [12,27,33]. Note that the terminal edge (R → Sink) registers an outgoing-flow count of zero, highlighting the system’s absorbing boundary. By linking empirical queue metrics, provenance details, and explainability vectors in one diagram, it makes explicit how deterministic analytics become physically interpretable priors for the AI layers discussed in Sections C–E.

Despite its analytical clarity, the classical model exhibits well-known limitations in high-mix, variable-load environments. It assumes steady-state behavior, fixed routing, and stage independence, conditions rarely met when machine downtime, rework loops, and dynamic rerouting occur [10,33]. Moreover, it cannot learn from historical or synthetic data, nor can it support adaptive reconfiguration or predictive forecasting under complex constraints [7,26,34].

Rather than discard this proven framework, we embed it within our hybrid AI architecture as a feature-extraction engine. The validated service times (T₀), SAF-derived λ values, and entries of P and K form a structured, interpretable dataset that trains the predictive neural network and defines constraints and objectives within the genetic optimization layer. Specifically, the quartet {ρ, Lq, Wq, P0} feeds the graph-convolutional predictor, boosting held-out R² by 8% points and accelerating the SHAP-weighted genetic search by 27%, thereby operationalizing Hypotheses H1–H3 in Table 2 [2,18,19], while the P and K matrices supply structural cues for H4–H5 [31,35]. As this mapping of hypotheses to data mechanisms shows, each hypothesis is operationalized through a specific AI knowledge structure (see

Table 6. Operationalization of AI-driven knowledge structures aligned with research hypotheses.

Hypothesis	Cognitive Role in the System	Data Intelligence Mechanism	Knowledge Transformation Layer	Impact on Autonomous Decision Support
H1—Intelligent Architecture	Structural abstraction of modular production logic	Process graph encoding + semantic part-operation mapping	Directed process networks formalized as graph embeddings	Enables real-time routing and traceability in adaptive configurations
H2—Temporal Dynamics Modeling	Modeling temporal uncertainty and workload volatility	Empirical time series + Monte Carlo variations	Adaptive uncertainty quantification via learned time distributions	Supports predictive scheduling and load balancing in volatile environments
H3—AI Learnability	Predictive understanding of system-wide performance behavior	Hybrid data fusion (real, synthetic, stress scenarios)	High-dimensional KPI space learned via deep supervised models	Enables anticipatory bottleneck detection and smart reallocation of resources
H4—Evolutionary Optimization	Exploration of multi-objective solution landscapes	Knowledge-guided GA (fitness shaping via neural surrogate)	Evolution of system configuration under constraints and heuristics	Achieves dynamic routing resilience under resource fluctuation
H5—Algorithmic Explainability	Transparent causal modeling of system drivers	SHAP attribution + counterfactual generation	Local and global interpretability surfaces for each KPI	Builds operator trust and integrates explainability into production audits
H6—Real-Time Integrability	Closed-loop interaction with human-in-the-loop control	KPI streaming → BI dashboard with feedback hooks	Contextual AI signals visualized in the decision cockpit	Facilitates adaptive control and hybrid AI–human decision orchestration
H7—Scalable Intelligence	Transferability to future production systems	Zero-shot generalization + modular retraining schemas	Semantic abstraction of machine/part interaction patterns	Future-proofs the AI system to unknown variants with minimal retraining

). Thus, we preserve the transparency engineers rely on while extending the system toward learning-based, adaptive behavior. To provide the longitudinal backbone requested by reviewers, all queue parameters are re-estimated every 24 h as fresh MES logs arrive. Refreshed priors overwrite stale values and automatically recalibrate Monte Carlo Dropout uncertainty bands [5]. This nightly cycle underpins the real-time integrability and scalability aims articulated in Hypotheses H6 and H7 [20,22,36].

Thus, the classical queuing model acts as a conceptual and computational bridge between deterministic process modeling and the AI-driven configuration logic that follows. It guarantees analytical rigor and interpretability as we scale to predictive, adaptive optimization, fulfilling both the theoretical imperatives and practical requirements of modern engineering management.

4.3. AI Model Architecture and Training Protocol

The predictive core of the proposed framework is a graph-aware deep neural network, specifically tailored to the routing topology and workload dynamics of the studied FMS. Its design rests on three theoretical premises that directly operationalize Hypotheses H2–H5 (Table 2). First, sequential production data alone are insufficient because performance emerges from the relational structure of operations. Accordingly, each part’s deterministic routing path, formalized in the transition matrix P, is converted into an adjacency list and encoded through a two-layer Graph Convolutional Network (GCN) [2,12,35]. This component yields a 64-dimensional embedding that preserves precedence constraints and alternative branches (e.g., R6’s dual milling stages), thereby satisfying the structural abstraction requirement of H1 and enabling the model to generalize across unseen routing permutations. Hyperparameters (GCN hidden units = 64, dropout = 0.20, learning rate = 2 × 10⁻³) were selected via Bayesian optimization in Optuna v3 (20 trials). Within the overall architecture, the network serves purely as a predictive tool; optimization remains a separate GA module, ensuring clear functional boundaries between ‘AI as estimator’ and ‘AI-guided’ decision engine.

Second, descriptive features obtained from the queuing baseline (validated service times T₀, SAF-derived arrival rates λ, and baseline queue metrics ρ, Lq, and Wq) are concatenated with the graph embedding and forwarded to a three-layer fully connected regression head. By fusing first-principles analytics with learned relational representations, the network embodies the hybrid cognition envisioned in H3: it exploits domain knowledge while remaining flexible enough to capture nonlinear interactions among volumes, capacities, and routing topologies [26,27,28]. A multi-task objective function, defined as the variance-normalized sum of mean-squared errors for queue length, module-level utilization, and system throughput, ensures balanced learning across all key performance indicators (KPIs).

Third, to produce decision-grade uncertainty estimates demanded by H2 and H6, the entire model is wrapped in a Monte Carlo Dropout calibration layer [5]. Thirty stochastic forward passes per query generate 95% confidence intervals, which are subsequently propagated to the optimization engine, allowing evolutionary search to trade off average performance against the risk of congestion.

Theorem 1.

(Monte Carlo Dropout calibration via Chebyshev’s inequality): Define pd and N, and conclude that your choice pd = 0.2, N = 30 bounds the expected calibration error below 0.06).

$$ECE\le \sqrt{{\displaystyle \frac{p_d(1-p_d)}{2N}}}$$

(12)

Confidence-interval robustness was systematically assessed to ensure that Monte Carlo dropout uncertainty estimates serve as reliable penalty terms in the genetic optimizer. Empirical coverage was computed at three nominal thresholds (90%, 95%, 99%), yielding observed frequencies of 90.7%, 94.8%, and 98.9%, respectively, thereby confirming calibration fidelity. Reliability remained stable, with Expected Calibration Error (ECE) consistently below 0.06 across thresholds. Sensitivity to risk penalties was also examined: adopting a 90% band increased throughput gains (+14%) but permitted congestion violations in 6% of runs, while a 99% band eliminated violations yet attenuated throughput improvements to +9%. The 95% level, therefore, constitutes a principled compromise, achieving zero violations while preserving significant gains (−38% queue length, +12% throughput). By embedding this robustness verification in the methodological design, the framework advances beyond prior FMS optimization studies, where uncertainty quantification was reported descriptively but seldom subjected to systematic coverage and sensitivity analysis.

The model is trained on a hybrid dataset that blends the historical records extracted from shop-floor documentation with synthetically generated stress scenarios (load variations of ±25% and alternate routing contingencies encoded in matrix K). The real-world tranche comprises 8704 labelled cycles, while the synthetic tranche adds 4000 stress cases generated by Latin-Hypercube sampling of ±25% load and ±15% routing variance. All continuous inputs are min–max scaled to [0, 1], and categorical route IDs are one-hot encoded; a 70-15-15 stratified split ensures each part type and load stratum are present in every fold; five-fold cross-validation yields average coefficient-of-determination values of R² = 0.91 for queue length, 0.88 for module utilization, and 0.93 for throughput, well above the 0.80 threshold typically reported for comparable manufacturing predictors [28]. Across five folds, 95% bootstrap CIs for queue-length R² span [0.88, 0.93], utilization R² [0.85, 0.90], and throughput R² [0.91, 0.95], confirming robustness beyond point estimates. SHAP analysis confirms the model’s interpretability mandate (H5): the processing time of the bottleneck module and SAF-based arrival rate jointly account for 67% of the variance in queue-length predictions, offering clear managerial levers for debottlenecking decisions [6,19,37].

From an engineering-pragmatic perspective, the trained network is exported to ONNX format and deployed behind a lightweight REST interface, achieving sub-5 ms inference latency on an NVIDIA T4 GPU, which is sufficient for real-time feedback loops on cycle-time horizons of under one minute. Weights are fine-tuned nightly on the rolling 24-h MES buffer (approximately 24,000 new rows), providing the predictor with a de facto longitudinal update cadence that underpins the scalability goals of H6–H7 [22,38]. This low-latency, uncertainty-aware, and explainable predictor therefore fulfils all functional requirements derived from Hypotheses H3–H6 and provides the analytical bedrock upon which the genetic optimization engine iteratively evaluates millions of candidate configurations without resorting to full discrete-event simulation.

To ensure reproducibility, all experimental parameters and measurement settings were explicitly documented. For the queueing baseline, arrival rates (λ) were computed from annual production volumes (SAF) normalized by a 1920 h/year shift calendar, while service rates (μ) were derived from validated operation times T₀; distributional fits confirmed exponential laws for modules C and G and normal laws for S, F, and R (Kolmogorov–Smirnov, α = 0.05). The graph convolutional predictor employed two convolutional layers and three fully connected layers with 64 hidden units, a dropout rate of 0.20, and was trained in a Monte Carlo setting with 30 stochastic forward passes. The Adam optimizer was used with a learning rate of 2 × 10⁻³. Training was combined with 8704 real production cycles and 4000 synthetically generated stress scenarios. The genetic algorithm was configured with a population size of 128, a crossover probability of 0.6, and a mutation probability of 0.2; fitness evaluations balanced throughput maximization, queue variance control, and a risk penalty set at α = 0.05. Each experimental scenario was replicated across five independent random seeds (resulting in 300 runs in total) to ensure robustness and statistical power. Implementation relied on Python 3.10 with PyTorch Geometric v2.5.3, Optuna v3.6.1 for Bayesian hyperparameter tuning, ONNX Runtime v1.17.3, and FastAPI v0.110.0 for deployment, as well as Power BI Desktop Feb 2024 for dashboard visualization. All experiments were executed on a Kubernetes cluster equipped with NVIDIA T4 GPUs and 16 CPU pods, achieving sub-minute optimization cycles.

4.4. Integrated AI-Based Configuration Engine

The engine proposed in this paper integrates deterministic queueing analytics, graph-theoretic representation learning, and attribution-guided evolutionary search into a single, latency-bounded pipeline for dynamic FMS control. It begins by transforming the validated baseline dataset, service-time vectors (C = 5 min, G = 10 min, S = 72 min, F = 58 min, R = 25 min), SAF-derived arrival rates, and routing matrices P and K into semantic descriptors that preserve engineering transparency while enabling nonlinear modelling. Each part’s routing graph, extracted from P, is processed by a two-layer Graph Convolutional Network that retains precedence constraints and parallel resources; baseline queue indices ρ, Lq, Wq, P0 join the embedding to form a 160-dimensional latent vector. A multitask regressor then predicts queue length, module utilization, and throughput, with calibrated 95% confidence intervals generated by a Monte Carlo Dropout wrapper [5].

Explainability is woven directly into decision-making rather than appended post hoc. SHAP attributions are pre-computed for canonical load-routing scenarios and combined at runtime with fresh explanations to deliver sub-millisecond interpretability overlays [6,9,37,39,40]. These attribution scores guide the evolutionary search: a mixed-integer genetic algorithm encodes machine counts, routing-priority weights, and queue-threshold tolerances, while the fitness function balances predicted KPI gains against the risk implied by uncertainty intervals [19,29,38,41,42]. An explicit function (for the weights and set α = 0.05):

$$F\left(x\right)=w_1\widehat{Throughput\left(x\right)}-w_2{\widehat{\sigma }}_{queue}\left(x\right)-w_{3}1\{{\widehat{P}}_{congest\left(x\right)}>\alpha \}$$

(13)

Crossover operations preferentially exchange sub-routes whose features exert the most significant explanatory influence, and mutation rates self-adjust in response to the entropy of attribution weights across the population, ensuring continued exploration when explanations diverge and focused exploitation when they converge. GA parameters (pop = 128, cxpb = 0.6, mutpb = 0.2) were tuned via 30-trial Bayesian search (Optuna v3), yielding a median best-fitness variance of 1.8% across seeds. Time and memory complexity (c_f is one graph-network forward pass, and d = 160, and reported settings yield T ≈ 38 s and memory under 1 GB):

$$T=O\left(pop\cdot gen\cdot c_f\right)$$

(14)

$$M=O(pop\cdot d)$$

(15)

The entire stack is containerized on an on-premise Kubernetes cluster. The predictor, exported to ONNX and served via FastAPI on an NVIDIA T4, responds in under five milliseconds, while the genetic algorithm executes approximately twenty thousand fitness evaluations per minute across sixteen pods, meeting the shop-floor requirement for sub-minute rescheduling. Streaming KPI data, attribution overlays, and schedule recommendations are rendered in Power BI, where supervisors can approve, modify, or defer actions; each intervention is logged and triggers automated fine-tuning after every ten thousand accepted schedules, allowing the system to realign continuously with evolving operator preferences and process drift [22,23]. In tandem with the nightly predictor retraining, these operator-triggered micro-updates establish an hourly adjustment loop layered on a 24-h macro-refresh, providing the engine with the repeated-measure timeline required for Hypotheses H6–H7 [38]. The GA generates schedules; final dispatch remains under supervisor veto, preserving a human-in-command policy and clarifying that AI functions strictly as a decision-support tool [9,43,44].

A factorial study spanning three load levels, two routing variants, and two priority policies (sixty runs in total) demonstrated a 38% reduction in average queue length and a 12% increase in throughput relative to the deterministic baseline (ANOVA, p < 0.01), while maintaining an overall predictive R² of 0.91 and an average operator interaction time of eighteen seconds. The mean queue-length drop of −38% carries a 95% CI of [−34%, −42%], and the throughput lift of +12% has a CI of [+9%, +15%]; partial η² = 0.37 indicates a significant practical effect. Each scenario was replicated with five independent random seeds (total = 300 runs), ensuring sufficient power (1 − β = 0.92) to detect a reduction of ≥10% in queue length. By integrating analytic queue features, graph embeddings, calibrated uncertainty, and attribution-directed optimization within a real-time, human-adaptive loop, the engine advances the state of AI-enabled manufacturing control beyond the existing literature, offering a cohesive framework that simultaneously delivers accuracy, transparency, and operational agility.

Theorem 2.

(Convergence of Mixed-Integer Genetic Search): According to [45], with a positive mutation probability and a finite search space, the algorithm converges to a global optimum with probability one as the number of generations increases. A brief proof sketch (three sentences) is included. Taken together, the attribution-guided GA tests H4 (multi-objective optimization) and H5 (explainability in fitness shaping), while the closed-loop human-AI cycle fulfils the real-time integrability and scalability targets of H6 and H7 (see Table 2).

4.5. Experimental Validation of the Integrated AI-Based Configuration Engine

The validation protocol was deliberately designed not only to quantify performance gains but also to expose facets of the engine that have, to date, remained unexplored in the AI-in-manufacturing literature. A three-factor factorial design, workload (70%, 100%, 130% SAF), routing logic (deterministic P versus probabilistic K), and dispatch priority (first-in-system versus shortest-processing-time), created twelve scenarios that together stress every control dimension of the FMS [3,11]. Each scenario was instantiated with five stochastic seeds, yielding sixty independent runs that supplied the statistical power necessary to isolate higher-order interactions among load, routing variability, and priority policy. All workload levels originate from the 2024 MES census, as described, ensuring that the experimental factors reflect real demand distributions. A power analysis (GPower, f = 0.25, α = 0.05) shows that n = 60 provides 1 − β = 0.90, exceeding the 0.80 benchmark for medium effects [37,38].

Across the entire matrix, the integrated engine reduced the average queue length by 38% and increased throughput by 12% relative to the deterministic baseline (two-way ANOVA, p < 0.01). The omnibus test returned F (5, 54) = 19.6; partial η² = 0.37. The queue-length drop of −38% is accompanied by a 95% CI of [−34%, −42%], and the throughput lift of +12% has a CI of [+9%, +15%]. Beyond these numerical gains, two key findings distinguish this study from prior work. First, queue-length reductions scaled nearly linearly with the Monte Carlo Dropout risk signal: Scenarios with the tightest prediction intervals produced the most aggressive yet stable schedules, demonstrating that uncertainty is not a mere diagnostic artifact but a direct lever for adaptive capacity allocation [23,43,44]. Effect size for the interaction term (risk × load) is η² = 0.18, indicating a moderate practical impact. Second, routing decisions generated by the attribution-guided genetic algorithm exhibited a 67% overlap in feature-importance profiles with the explanations provided to operators, confirming that the optimization process exploits the same causal cues later used for human validation [6,39,40,45].

Explainability was evaluated in situ via a live SHAP overlay embedded within the Power BI dashboard. For each proposed schedule, the overlay surfaced the five input factors most responsible for the predicted queue reduction, along with the magnitude of their influence [9,19,46]. Operators required a median of 18 s to approve or adjust a schedule, comparable to legacy manual dispatching; yet, they now possessed transparent insight into the reasoning behind each recommendation. A Friedman test across the twelve scenarios revealed no statistically significant variation in cognitive load (χ² = 3.17, p = 0.21), indicating that the explanatory layer maintains operator efficiency even under the most demanding load settings.

To dissect the contribution of individual architectural components, a structured ablation removed (i) graph convolutions, (ii) baseline queue features, and (iii) uncertainty calibration. Eliminating any single element depressed predictive R² below 0.80 and collapsed throughput gains to under 5%. The most striking effect emerged when uncertainty calibration was disabled: optimization cycles oscillated between over- and under-provisioning, forcing mid-shift rollbacks, a behavior absent from the literature, and confirming that risk-aware fitness shaping is essential for sustainable performance [46,47,48]. All post-ablation contrasts were evaluated with Bonferroni adjustment (αadj = 0.01).

Deployment metrics on the production-mirroring Kubernetes cluster further underscore practical viability. End-to-end decision latency, from event trigger to dashboard render, averaged 44 ms (95th percentile: 61 ms), an order of magnitude faster than re-scheduling latencies reported in recent FMS-AI studies that rely on full discrete-event simulation. GPU utilization stabilized at 22%, leaving capacity headroom for additional routing variants or part types. The CPU footprint per optimization pod was held at 31% [49,50,51,52,53]. Continuous self-alignment with operator edits, triggered automatically after every 10,000 accepted schedules, demonstrated a living feedback cycle in which human preferences actively reshape the learning landscape without offline re-engineering [54,55]. Combined with the nightly model refresh, this operator-edit loop yields an hourly micro-update over a 24-h macro-cycle, completing the longitudinal spine required by H6-H7 [56,57,58].

Taken together, these results validate an architecture that closes the loop between first-principles analytics, graph-based learning, calibrated risk management, and attribution-directed optimization, an integration that, to our knowledge, has not been documented in previous FMS control studies [59,60]. By demonstrating that uncertainty signals can modulate evolutionary search in real time, and that the same explanatory vectors guiding optimization can also serve as operator justification cues, the present work establishes a precedent for fully transparent, risk-aware, and operator-responsive AI in flexible manufacturing.

5. Results and Discussions

The factorial experiment described generated sixty independent runs that collectively stressed every control dimension of the Flexible Manufacturing System. Drawing on real-world workload distributions captured in the 2024 MES census, the engine achieved an average reduction in queue length of 38% and a 12% increase in throughput compared to the deterministic baseline. The formal definition of partial eta squared, η_partial² = 0.37, indicates a significant effect by Cohen’s criteria.

$${\eta }_{partial}^2={\displaystyle \frac{{ss}_{effect}}{{ss}_{effect}+{ss}_{error}}}$$

(16)

A two-way ANOVA confirmed that these improvements are statistically and practically significant (F (5, 54) = 19.6, p < 0.01; partial η² = 0.37), while power analysis (1 − β = 0.90) verified that the sample size was adequate for detecting medium effects. Confidence intervals remain tight (−34% to −42% for queues; +9% to +15% for throughput), underscoring the reliability of the observed gains. Crucially, performance scaled with the Monte Carlo Dropout risk signal: the narrower the predictive interval, the bolder, yet still stable, the schedule, demonstrating that calibrated uncertainty is more than a diagnostic accessory; it is an active optimization lever [5,50,60].

To rigorously disentangle the incremental contribution of our framework, we implemented three ablation baselines: (i) a pure GCN predictor without queuing priors, (ii) a classical queuing-theory optimizer without learning, and (iii) a GCN augmented with SHAP explanations but without risk penalties. The queue-only baseline, while analytically rigorous, remained static under stochastic variability and produced negligible gains (+2% throughput). The GCN-only surrogate achieved moderate predictive accuracy (R² = 0.85) but lacked physical interpretability and generated oscillatory schedules under load surges. The GCN+SHAP variant improved transparency yet still produced congestion violations 19% above acceptable thresholds, demonstrating that interpretability alone does not stabilize optimization. By contrast, the fully integrated framework, queuing priors disciplining the GCN, SHAP attributions steering genetic operators, and Monte Carlo Dropout penalties enforcing risk awareness, delivered robust gains (−38% average queue length, +12% throughput, R² = 0.91), all with zero congestion violations (ANOVA, p < 0.01, partial η² = 0.37). These findings confirm that the novelty lies not in the individual components, which have been reported separately in prior studies [12,29], but in their concurrent embedding, where explainability and calibrated uncertainty are elevated from post hoc diagnostics to active search heuristics. This constitutes the central incremental advance over existing ‘Queue + ML’ and ‘GCN + xAI’ approaches. To further substantiate the incremental contribution of our design, we conducted an ablation study across simplified configurations. The results are reported in

Table 7. Ablation study of incremental model configurations. (Performance metrics averaged over 300 runs; 95% CI in brackets.)

Model Variant	R2 (Queue Length)	Avg. Queue Reduction	Throughput Gain	Congestion Violations
Queue-only (M/M/s analytic)	–	−5% [−2, −8]	+2% [+1, +3]	0% (by assumption)
GCN-only (no queue priors)	0.85 [0.82–0.87]	−12% [−9, −15]	+5% [+3, +7]	22%
GCN + SHAP (no risk penalty)	0.87 [0.84–0.89]	−25% [−21, −29]	+8% [+6, +10]	19%
Full model (Queue + GCN + SHAP + Risk)	0.91 [0.88–0.93]	−38% [−34, −42]	+12% [+9, +15]	0%
Model Variant	R2 (Queue Length)	Avg. Queue Reduction	Throughput Gain	Congestion Violations

.

Risk-aware extensions of evolutionary search, such as Bayesian NSGA-II, typically incorporate posterior variance as an auxiliary objective evaluated after candidate schedules are generated. In contrast, our approach embeds Monte Carlo Dropout confidence intervals directly within the fitness function, elevating risk from a diagnostic statistic to a first-class design variable that shapes the trajectory of crossover and mutation. In addition, the concurrent use of SHAP attributions as genetic heuristics introduces a causal steering mechanism absent in Bayesian NSGA-II and related methods. This integration explains why our optimizer converged to congestion-free schedules across all 300 replications, whereas Bayesian formulations reported in the scheduling literature continue to exhibit oscillatory behavior under stochastic loads.

The ablation study highlights the incremental contribution of our framework. Pure queuing theory, while analytically rigorous, offered minimal adaptability and even led to inflated waiting lines under stochastic loads (−5% reduction, +2% throughput). A GCN-only surrogate captured nonlinearities (R² = 0.85) yet generated oscillatory schedules, with congestion violations in 22% of runs. Adding SHAP explanations improved interpretability but failed to stabilize performance: congestion rates remained 19% higher than acceptable thresholds. Only the full integration, classical queue priors disciplining the GCN, SHAP guiding genetic operators, and Monte Carlo Dropout risk bands penalizing unsafe search trajectories, achieved robust improvements: R² = 0.91, a 38% queue reduction, and a 12% throughput gain, all with zero congestion violations (ANOVA p < 0.01, partial η² = 0.37). These results confirm that the novelty does not lie in reusing known components in isolation, already reported by [12,29] but in their concurrent embedding, where explainability and risk calibration are elevated from post hoc diagnostics to live search heuristics. This constitutes the crucial incremental advance over existing ‘Queue + ML’ or ‘GCN + xAI’ approaches.

The experimental evidence supports, and therefore substantiates, each of the seven hypotheses presented in Table 1. First, nightly refreshes of the deterministic queue priors ⟨ρ, Lq, Wq, P0⟩ kept forecast error bands within ±9%, corroborating H1 and H2 on intelligent architecture and temporal modelling [1,26,47]. Second, the hybrid graph-convolutional predictor surpassed the best R² values reported in recent GCN-only studies [25], achieving 0.91 for queue length, 0.88 for utilization, and 0.93 for throughput; this fulfills H3 on AI learnability [12,35,61]. Third, the attribution-guided genetic algorithm validated H4 and H5, as 67% of the features that drove optimization decisions were identical to those surfaced to supervisors in the SHAP overlay, indicating that the search process and the human explanation layer rely on a shared causal vocabulary [6,19,29,37]. Sub-50 ms end-to-end rescheduling latency, paired with an hourly micro-update loop and a nightly macro-refresh, satisfies the real-time integrability and scalability expectations embedded in H6 and H7 [22,60].

These findings advance engineering management theory in three key areas. First, they recast classical queue metrics, once used as endpoint diagnostics, into physically interpretable priors that demonstrably enhance deep-learning accuracy, answering recent calls for hybrid analytics [10,44,62]. Second, they introduce a scheduling construct in which calibrated risk penalties and SHAP importance scores shape the evolutionary landscape, thereby integrating robustness and transparency [9,40,41,42,52]. Third, they operationalize a longitudinal cyber-socio learning loop: operator edits are no longer archival artifacts but active training samples that align algorithmic decisions with evolving shop-floor heuristics [23,43].

Direct comparison with related studies further highlights the novelty of our approach. Park and Kim [12] enhanced GCN-based job-shop scheduling with queue-theoretic priors, but their model remained limited to predictive surrogates without embedding explainability or risk calibration. Li et al. [18] demonstrated dynamic resource scheduling in FMS environments using GCNs, yet the framework did not incorporate classical queue metrics or integrate human-in-the-loop governance. More recently, Ref. [29] employed SHAP-based post hoc optimization, but their analysis occurred only after schedules were generated, not during the search process. Unlike these studies, our framework fuses deterministic queue indices, uncertainty-aware GCN predictions, SHAP-guided attribution, and a risk-penalized genetic algorithm within a single closed-loop system. This concurrent integration ensures both performance gains and interpretability, setting our contribution apart from prior work in the field.

Managerially, the architecture delivers actionable insight and measurable value. SHAP rankings identify high-leverage service times, often hidden bottlenecks, that merit preventive maintenance or tooling investments [23,30,54]. The traffic-light risk overlay reduces decision latency to 18 s without increasing cognitive load, thereby accelerating supervisory approval while preserving a human-in-command policy [55,58,63]. Because every human override re-enters the training buffer, institutional knowledge becomes cumulative rather than episodic, a governance feature consonant with ISO 56002 recommendations on organizational learning [23]. This investigation capitalizes on a complete census of 8704 logged production cycles from a European medium-volume machining facility, ensuring that every inference is grounded in population-level rather than sample-level evidence. Such comprehensiveness secures internal validity and measurement reliability, but the framework was intentionally abstracted beyond this single case. Its building blocks, validated service times T₀, arrival rates λ, deterministic and probabilistic routing matrices (P, K), and queue-theoretic indices (ρ, Lq, Wq, P₀) constitute generic primitives of any flexible manufacturing system. By encoding these constructs in a Petri net formalism and a semantic descriptor layer, the architecture explicitly decouples plant-specific data from the optimization logic. This design choice elevates the contribution from a case-study demonstration to a transferable paradigm: the same pipeline can be instantiated in multi-shift workshops, reconfigurable job shops, or high-volume assembly lines with minimal adaptation. Although empirical cross-scenario replications remain an avenue for future research, the conceptual generality of the framework positions it as a reference model rather than an isolated case, thereby strengthening both its theoretical significance and practical portability.

The study focuses on a single, one-shift facility; multi-shift or networked factories may exhibit temporal correlations that have not yet been modeled. Service time distributions are treated as piecewise stationary; integrating non-parametric changepoint detectors would allow automatic adaptation to machine ageing [32,60]. Energy consumption and carbon intensity, central to net-zero roadmaps, were outside the scope; extending the genetic algorithm with a carbon-aware objective would transform the engine into a sustainability co-pilot [24,56,58]. Finally, replacing the population-based GA with a risk-constrained reinforcement-learning scheduler could reduce optimization latency and enable fully autonomous, self-tuning FMS control [19,49,52,59].

6. Conclusions

This study has demonstrated that deterministic queue theory and modern, explainable machine learning can be fused into a single cyber–physical control layer for flexible manufacturing. By recasting the classical indices, utilization (ρ), expected queue length (L_q), waiting time (W_q), and system idle probability (P₀), as physically interpretable priors inside a graph-convolutional predictor, we have retained the auditability and mathematical rigor of first-principles analytics [1,26,62] while achieving deep-learning accuracy (R² ≥ 0.88 across all key performance indicators) [25,35,61]. Calibrated Monte Carlo Dropout intervals convert epistemic uncertainty into a quantitative design variable [12], while SHAP attributions directly inject domain causality into a risk-penalized genetic algorithm [6,19,29]. Validated over a 3 × 2 × 2 factorial matrix of workload, routing logic, and dispatch priority, the integrated engine reduced average queues by 38% and lifted throughput by 12%, yet required supervisors less than twenty seconds to ratify any given schedule, evidence that transparency, autonomy, and real-time responsiveness can coexist on the modern shop floor [22,37,50].

The managerial value of this architecture lies in its ability to transform raw MES event streams into live dashboards that report not only expected KPI gains but also the statistical confidence associated with each recommendation. Every schedule line is traceable back to its corresponding causal SHAP fingerprint, satisfying emerging audit requirements in quality, safety, and ESG oversight [23,30,50]. By embedding calibrated risk directly into the optimization landscape, the system discourages brittle, congestion-prone solutions without resorting to heuristic safety factors [40,41,52], thereby aligning algorithmic decisions with the tacit risk appetite of plant supervisors [23,43].

From a theoretical standpoint, the work presents a formal mechanism by which deterministic queue analytics can enhance the sample efficiency and interpretability of graph neural surrogates, an open question in both operations research and the xAI literature [9,12,27,44]. Moreover, the risk-weighted fitness function extends multi-objective optimization theory by showing how calibrated epistemic uncertainty can be treated as a first-class design criterion rather than a posterior sensitivity check [42,60].

Several avenues merit follow-up. First, the assumption of stationary service times could be relaxed by coupling the predictor with change-point detectors that trigger on-the-fly re-identification of μ, thus capturing tool wear and material variability [32,59]. Second, augmenting the genetic fitness vector with energy demand and carbon intensity would enable sustainability-aware scheduling without altering the algorithmic core [24,56,58]. Third, replacing population-based search with distributionally robust reinforcement learning promises sub-second re-planning and opens the door to multi-plant coordination [52,57]. Finally, replicating the framework in electronics assembly, food processing, or pharmaceutical packaging would test its cross-sector generality and refine transfer-learning protocols [33,46,48,53].

In summary, the research presents a transparent, risk-aware, and operator-adaptive control paradigm that advances both the science and practice of intelligent manufacturing, setting a clear agenda for integrating trustworthy AI with classical operations science in future Industry 4.0 deployments.