Maritime logistics is presently characterized by the increasing complexity of operations, which creates new challenges for ensuring the safety and reliability of port-ship interactions. The dynamic operating environment creates a new structure of risks that directly affect the efficiency and safety of port operations. In such conditions, traditional approaches to risk assessment based on static parameters and fixed scenarios are not sufficient to ensure adaptability and timely response.

There is a growing need to create models that can integrate dynamic risk reassessment, degradation of technical and organizational barriers, and optimization of management decisions in real time. Of particular importance are risk-informed planning approaches that combine quantitative modeling, multi-criteria assessment, and resource management under conditions of uncertainty. Such models should become the basis for the development of intelligent decision support systems (D).

High-profile incidents in recent years, such as the collision of the container ship DALI with a bridge pillar in the port of Baltimore (2024) and the accident of the Milano Bridge in the port of Busan (2020), have demonstrated systemic vulnerabilities in existing mooring procedures. The causes of the accidents go beyond technical errors and point to a lack of situational awareness, fragmented application of risk assessment methods, and insufficient integration of data into the decision-making process.

Research in the field of search and rescue operations outlines the main challenges for planning and demonstrates the potential of AI-oriented algorithms to reduce response times and improve the accuracy of incident localization.¹ These ideas correlate with the work on autonomous navigation, where adaptive multi-source quantitative risk assessment allows to take into account the dynamics of the environment and reduce the likelihood of collisions.² At the macro level of industry policy, the focus is shifting to environmental goals: research on achieving greenhouse gas emission reductions emphasizes the risks of not meeting international standards,³ while a review of ML applications in maritime risk management⁴ outlines methodological gaps and prospects for explanatory AI. Expanding the analytical framework through game theory and network approaches⁵ allows for the integration of reliability issues in complex transportation systems.

In the area of sensor networks, the use of LoRaWAN for decentralized monitoring, which increases the stability of data transmission channels in marine areas, is attracting attention.⁶ This approach is enhanced by the coordinated route planning of unmanned vehicles (UAVs and USVs), which can provide real-time monitoring despite the uncertainty in travel time.⁷ Security studies demonstrate the effectiveness of probabilistic networks in modeling the role of barriers,⁸ assessing the risks of transporting dangerous goods, including electric vehicles,⁹ and assessing the sustainability of green shipping policies.¹⁰ At the same time, deep learning models provide forecasting of offshore winds, which are critical for safe navigation.¹¹

An important area is the integration of cognitive maps with causal analysis methods and expert opinions, which allows to reproduce complex human factors.¹² Algorithms based on Transformer networks show potential for predicting “near-miss” events and explaining patterns.¹³ This correlates with the use of fuzzy logic and Bayesian networks in the study of reliability of inert gas operations¹⁴ and general overviews of BN applications in the maritime industry.¹⁵ The safe trajectory of autonomous vessels is also built through risk-based planning¹⁶ and multi-scale scenario analysis.¹⁷ In parallel, FRAM combined with AIS data shows how system interactions lead to incidents in port operations.¹⁸

Real-time monitoring of containers is realized through hybrid IoT networks,¹⁹ while the resilience of LNG chains is analyzed by methods ranging from static Bow-tie to dynamic epidemiological modeling.²⁰ Bayesian networks are increasingly being used to evaluate trade routes under uncertainty,²¹ optimize the placement of SAR equipment,²² plan AUV courses,²³ and enhance process security on a regional scale.²⁴

Against this background, it is worth noting the contribution of Ukrainian research. Papers^25,26,27 form the basis of a probabilistic approach to hydrometeorological risks and navigation safety assessment. The development of a methodology for expert assessment of the quality of risk analysis of operations²⁷ is directly integrated into modern DSS systems. The analysis of the efficiency of the Trumpet fleet²⁸ and the impact of hull geometry on maneuverability²⁹ add a practical engineering component, while the study of ship systems efficiency³⁰ emphasizes the reduction of energy consumption.

Special mention should be made of the work on improving the fuel efficiency of engines through additives³¹ and ensuring the safe operation of diesel plants.³² These are examples of applied technical solutions that are consistent with the general paradigm of sustainable development. Although the works on mathematical analysis³³ or optimization of port equipment³⁴ are more of a fundamental nature, they demonstrate the connection between accurate models and applied logistics problems. Finally, new methods based on Markov processes and fuzzy sets^35,36,37 bring reliability assessment to a level capable of taking into account uncertainty and multicriteria. Studies in the field of law and policy^38,39 introduce into the discussion the dimension of “gray areas” and cluster management, which is important for the institutional context of risk management.

Hydrodynamic aspects are revealed in the modeling of the impact of cold-water pipes and mooring systems on OTEC-type platforms,⁴⁰ which emphasizes the importance of physical models and numerical simulations in the overall safety loop.The increasing complexity of maritime transportation, combined with new environmental requirements and the transition to autonomous solutions, necessitates a systematic review of current approaches to risk management.

Works^41,42 analyze methods of managing the environmental safety of shipping and the parameters of marine diesel engines, emphasizing the need to balance technical efficiency and environmental constraints. The study⁴³ proposes an optimization model for energy flows and supply that is directly applicable to port networks operating in a mixed energy supply mode. Article⁴⁴ presents the application of wavelet transformation for diagnosing the technical condition of systems, which reinforces the direction of predictive monitoring in port infrastructure risk management. The works^45,46 address the issues of information security in shipping, AIS data manipulation, and the development of strategies to reduce nitrogen oxide emissions, forming an integrated basis for combining the environmental, information, and technical sustainability of port networks within the framework of risk-oriented management.

Despite the existence of numerous studies in the field of port security and risk management, most of them are static in nature and do not take into account the time degradation of technical barriers, the interdependence of subsystems, and the dynamics of risk in a changing maritime environment. The problem of combining multi-criteria risk assessment, time degradation of efficiency, and optimization of interventions in a single mathematical structure remains unresolved. This scientific gap limits the possibilities of real-time adaptive management of port networks.

The aim of the study is to develop an integrated dynamic model of risk-informed planning that combines risk assessment, degradation of security barriers, and optimization of interventions with regard to resource constraints. The scientific novelty of the work lies in the creation of a quantitative model that reconciles expert assessments (via TOPSIS) with the dynamics of risk and performance, introduces a network resilience index (Resilience Index) and allows minimizing the expected loss of throughput under conditions of uncertainty. The proposed approach forms an analytical basis for building intelligent decision support systems in seaports.

The purpose of this study is to develop a risk-informed decision-support model for port networks operating under uncertainty. The model integrates three sequential stages: (1) dynamic risk assessment, (2) optimization of resource allocation, and (3) decision-making under constraints. The problem is formulated as a multi-stage optimization task that minimizes the expected total loss due to cascading failures while respecting operational, budgetary, and temporal constraints:

subject to system dynamics

where R(t) - risk state vector, B(t) - available budget, U(t) - control/intervention vector, and ξ_t stochastic disturbances.

The above formulation serves as a logical bridge between risk evaluation, mitigation optimization, and adaptive decision-making, forming the foundation of the proposed DSS.

In today’s maritime logistics environment, port complexes function as complex cyber-physical systems that intertwine physical infrastructure nodes (terminals, mooring posts, control centers) with a network of information exchange, navigation monitoring, and management response. The key challenges for such a system are maintaining the functional availability of the nodes and ensuring resilience to risks caused by internal and external influences.

The availability of a node refers to its current ability to handle vessels, which depends on its technical condition, personnel, weather conditions, queues, and other factors. The availability value x_i(t) ∈ [0, 1] can be interpreted as the normalized fraction of available capacity from the nominal level c_i > 0, usually expressed in TEU/h or shipcalls/h.

Risk in this context is a generalized assessment of the potential loss of node functionality due to the impact of adverse events. The integral risk is denoted as r_i(t) ∈ [0, 1], and reflects the probability of disruption, threats, or degradation at a particular point in the network. High values of r_i(t) indicate the criticality of the situation and the need for prompt intervention.

To reduce risk, safety barriers are used - technical, procedural or administrative means that implement the protection function. The effectiveness of the barrier at a given time is denoted as EB_i(t) ∈ (0,1], and usually decreases over time due to degradation or wear and tear. A decrease in the effectiveness of barriers creates the need for managed intervention u_i(t) ≥ 0, which may include additional resources, changes in procedures, activation of backup systems, etc.

Table 1 summarizes examples of port network elements, their potential risks, and corresponding barriers.

Figure 1 shows a generalized diagram of the key elements of the port operational network, indicating potential risks and examples of security barriers for each. Within the three main nodes - terminal, tug service and traffic control - typical threats such as reduced visibility, tug failure, and human error are shown, as well as the corresponding risk management tools: backup lighting, redundant communication channels, implementation of SOPs (standard operating procedures), decision automation and tug redundancy.

Most traditional approaches to risk management look at events in isolation without considering the cascading effects of one node on others. Instead, the modern concept of network risk dynamics focuses on the interconnections between system components, taking into account that a disruption in a critical element can cause a domino effect throughout the network. Therefore, the key task is to develop a dynamic model that takes into account changes in risk and availability over time, describes the degradation of barriers and the effectiveness of interventions. In addition, it allows for the evaluation of integrated metrics (resilience, reliability, ETL) and serves as the basis for building an optimization problem of management in conditions of limited resources.

Seaports nowadays are complex cyber-physical logistics systems consisting of many interdependent elements: mooring terminals, towing stations, approach channels, control centers, energy and navigation systems. Ensuring the sustainability and functionality of the port in the face of dynamic changes in the external environment (weather conditions, traffic intensity, technical failures, cyber threats) requires the use of adaptive risk analysis and decision-making models.

The networked nature of the port is particularly challenging, where the failure of a single node or degradation of a particular subsystem can cause cascading effects that affect overall performance. Therefore, an approach that combines local risks with global impacts, also takes into account the degradation of security barriers over time and allows for optimized interventions (including tugs, slot transfers, routing) with limited resources, and is also focused on assessing the resilience index of the port network is relevant.

The port system is represented as a directed graph:

where: V - set of nodes (terminals, pilotage, towing service, traffic control); E - a set of arcs (logistics links, approaches, channels, interaction, etc.).

Each node i ∈ V - is characterized:

The diagram in Figure 2 shows the logic of risk assessment and efficiency of the port network based on the availability of its key elements.

The central component is Availability - an integrated assessment of the effective throughput of a network node (terminal, towing post, etc.). It is determined under the influence of risks r_i(t), barriers Z_i(t), and managed interventions u_i(t) that can compensate for infrastructure degradation or external threats.

From the accessibility indicator, four key system-level metrics are further calculated, namely:

Thus, resilience determines how quickly and efficiently the port network is able to adapt to risks and external changes, and the relevant metrics quantify this adaptation to support informed management decisions (Table 2).

Figure 3 shows a generalized structure of the interaction between risk, availability, intervention and network indicators. The diagram shows how local risks shape the overall dynamics of the system by affecting availability, which in turn affects performance, resilience, expected losses (ETL) and reliability (R_net).

The dynamic behavior of the port network resilience is represented by the time-dependent performance function Φ(t), which describes the normalized operational state of the system under uncertainty.

The aggregated network risk level R_net(t) is defined as a weighted combination of subsystem performance indicators:

where Φ_i(t) – normalized performance level of subsystem i at time t, w_i - – normalized risk weight of subsystem i (∑_iw_i = 1).

We define the Expected Throughput Loss (ETL) as a normalized time–integral measure of degraded performance that quantifies the cumulative degradation of system performance during the observation horizon T:

where R_net(t) ∈ [0, 1] denotes the instantaneous network reliability and T is the evaluation horizon. The normalization by T ensures that ETL is dimensionless. The resilience index (RI) is then defined as:

Thus, higher RI values indicate more stable and resilient port operation, while ETL reflects the total performance loss over time due to disturbances or cascading failures.

The proposed framework uses several resilienceoriented indicators:

All indicators are bounded in [0,1], allowing cross-scenario comparison.

The modeling results allowed us to quantify the impact of technical, organizational and climatic factors on the functioning of the port network under conditions of operational uncertainty. The simulation study covers three typical scenarios - Baseline, Adverse metocean and Adverse + Mitigation, which reflect different levels of risk load on port infrastructure and ship operations. The analysis of the dynamics of system performance, reliability and resilience showed that the implementation of controlled interventions can significantly reduce efficiency losses and stabilize the network functioning in real time.

This section summarizes the results of the calculations for the key indicators - relative performance, network reliability, resilience index, and expected capacity losses. For clarity, the system behavior is analyzed using graphical visualizations (Resilience curve, Network reliability, Risk heatmap, and degradation of security barriers) that reflect the evolution of the port network state in time and space. The obtained dependencies are interpreted from the perspective of risk management, intervention planning, and increasing the adaptability of port systems to changing operating conditions.

Simulation parameters:

The degradation rates and risk-transfer coefficients were estimated using a combination of expert elicitation and validation against port maintenance data collected over a six-month observation period. The parameters α and β were tuned within ±20% during calibration to ensure consistency with the observed failure frequencies and recovery dynamics, allows the model to remain interpretable while preserving empirical relevance without relying on confidential operational data.

The results obtained confirm that the application of a dynamic risk-based approach can increase the sustainability and reliability of the port network even in the face of external disturbances. The proposed model combines analytical rigor and practical adaptability, which makes it possible to formalize the relationships between technical, behavioral, and climatic factors. The modeling process revealed that control and power supply systems remain the most sensitive to degradation, while communication nodes demonstrate the greatest stability but accumulate latent risks under prolonged loads. This is in line with empirical data from maritime practice, where control errors or delays in crew interaction with port services are the main triggers of emergencies.

Comparison with previous studies shows that the developed model outperforms traditional risk assessment methods (FMEA, static risk matrices) by taking into account time degradation and the possibility of dynamic parameter updates. The inclusion of degradation parameters and adaptive interventions makes the model particularly relevant for real-time decision-support systems (DSS) that integrate operational and climatic data streams, bridging the gap between predictive analytics and port logistics. It also provides a quantitative integration of the results of multi-criteria analysis (TOPSIS) with real operational data, which creates the basis for using the model in digital port monitoring systems. The practical result is to prove the effectiveness of management interventions that not only reduce risks but also stabilize network capacity, increasing the overall resilience index by 2–3 percentage points.

Thus, the model can be considered as a basic element of a modern maritime transport safety and efficiency management system based on the principles of forecasting, adaptation, and continuous improvement. Its use will contribute to the formation of a new paradigm of port project management - from reactive to proactive, where decisions are made based on risk analytics, resource flexibility, and digital sustainability indicators.

Limitations of the present work include the reliance on expert-driven weighting and the assumption of exponential degradation for barriers, where future research may extend the model to cover multi-port interactions and cyber-physical dependencies within port clusters.

The developed model can be implemented within port Decision Support Systems (DSS) for maintenance scheduling, anomaly prioritization, and predictive failure management. It allows operators to dynamically allocate maintenance crews and energy resources under uncertainty, improving response times and reducing unplanned downtime. The clarified definitions of ETL (normalized, dimensionless) and RI, together with the explicit risk–reliability coupling, improve interpretability and comparability across scenarios. The approach also provides a methodological foundation for digital-twin integration and real-time resilience monitoring.

This paper presents an integrated mathematical model for port network risk management under conditions of operational uncertainty that combines dynamic risk assessment, degradation of safety barriers, and optimization of interventions within available resources. The model allows to quantify the relationships between technical, behavioral and climatic factors, determining their impact on the performance, reliability and resilience of the port system. The combination of the dynamic approach with the TOPSIS method ensures coherence between expert opinions and digital parameters of the decision support system (DSS). The study provides full computational reproducibility (Appendix A), including CVXPY specifications, parameter files, and fixed random seeds.

The simulation results confirmed that the use of Risk-Informed Planning can increase the network resilience index by 2–3 p.p. and reduce the expected performance losses by almost half compared to the scenario without interventions. The proposed approach can be used for adaptive management of port operations, prediction of critical node states, and optimization of maintenance. Thus, the model forms a scientifically sound basis for the creation of intelligent systems for managing the safety and efficiency of seaports in the context of the digital transformation of the industry. Future research should focus on extending the model to multi-port systems and integrating cyber–physical risks associated with digital twins and AI-based control frameworks.