Risk-Based Planning of Port Network Sustainability Under Conditions of Operational Uncertainty

Oleksiy Melnyk¹ , Hennady Shcheniavskyi¹, Sergiy Volyanskyy², Kostyantin Koryakin¹ and Volodymyr Kucherenko¹
1. Odesa National Maritime University, Odesa, Ukraine
2. Admiral Makarov National University of Shipbuilding, Mykolaiv, Ukraine
Correspondence to: Oleksiy Melnyk, m.onmu@ukr.net

DOI: https://doi.org/10.70389/PJS.100171

Additional information

• Ethical approval: This study did not involve human participants or animals, and therefore ethical approval was not required.
• Consent: N/a
• Funding: No industry funding
• Conflicts of interest: N/a
• Author contribution: Oleksiy Melnyk – Supervision, Project administration, Methodology, Conceptualization. Hennady Shcheniavskyi – Writing – review & editing, Resources, Methodology, Formal analysis. Sergiy Volyanskyy – Writing – review & editing, Writing – original draft, Visualization, Validation. Kostyantin Koryakin – Investigation, Formal analysis, Data curation. Volodymyr Kucherenko – Writing – review & editing, Writing – original draft, Visualization.
• Guarantor: Oleksiy Melnyk
• Provenance and peer-review: Unsolicited and externally peer-reviewed
• Data availability statement: Simulation datasets, parameter files, and scripts are available at the project repository, and from the corresponding author upon reasonable request.

Keywords: Dynamic risk-informed planning, Safety barrier degradation modeling, Port network resilience index, Topsis-based multicriteria risk assessment, Real-time intervention optimization.

Peer Review
Received: 16 October 2025
Last revised: 28 October 2025
Accepted: 28 October 2025
Version accepted: 4
Published: 17 December 2025

Plain Language Summary Infographic

Introduction

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^25–27 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^35–37 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.

Problem Formulation

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: (1)

subject to system dynamics: (2)

where 𝑅(𝑡) – risk state vector, 𝐵(𝑡) – available budget, 𝑈(𝑡) – control/intervention vector, and 𝜉𝑡 – 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.

Materials

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 𝑥𝑖(𝑡) ∈ [0, 1] can be interpreted as the normalized fraction of available capacity from the nominal level 𝑐𝑖 > 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 𝑟𝑖(𝑡) ∈ [0, 1], and reflects the probability of disruption, threats, or degradation at a particular point in the network. High values of 𝑟𝑖(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 𝐸𝐵𝑖(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 𝑢𝑖(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.

Table 1: Examples of network elements, risks and barriers.
Network Node	Potential Risk	Safety Barrier Example
Cargo terminal	Visibility reduction, crane breakdown	Backup lighting, double check
Towing service	Failure of the tug, storm waves	Emergency reserve of tugs, SOP
Port Traffic Management	Communication failure, human factor	Automation of decision-making, backup channels

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: (3)

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:

• availability 𝑥𝑖(𝑡) ∈ [0, 1] – which reflects the effective share of the node’s capacity;
• integral risk 𝑟𝑖(𝑡) ∈ [0, 1] – from low to critical;
• nominal capacity 𝑐𝑖 – for example, in teu/h;
• controlled intervention 𝑢𝑖(𝑡) ≥ 0, i.e., resources or actions that reduce the risk;
• the effectiveness of the safety barrier 𝐸𝐵𝑖(𝑡) ∈ [0, 1], which decreases over time according to the exponential law.

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 𝑟𝑖(𝑡), barriers 𝑧𝑖(𝑡), and managed interventions 𝑢𝑖(𝑡) that can compensate for infrastructure degradation or external threats. From the accessibility indicator, four key system-level metrics are further calculated, namely:

1. Performance: total effective current network performance Φ(𝑡);
2. Network Reliability (Rnet): the probability of network uptime at the current time;
3. Resilience Index (RI): network resilience index – the share of restored or preserved bandwidth during a period of stress;
4. Expected Throughput Loss (ETL): expected bandwidth losses averaged over a time horizon.

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 (Rnet).

Table 2: Key performance and risk metrics for port network assessment.
Metric	Notation	Description
Network Performance	Φ(t)	Total effective throughput capacity of the port network at time ttt, calculated as the sum of node capacities weighted by their availability.
Network Reliability	Rnet(t)	Probability that the port network remains operational at time ttt, considering the aggregated risk levels at each node (e.g., berths, tug stations).
Resilience Index	RI	Fraction of the maximum possible throughput preserved over a time horizon T; reflects the system’s ability to recover from disruptions.
Expected Throughput Loss (ETL)	ETL	Average relative productivity loss over a given time horizon; quantifies cumulative inefficiencies due to risk, failures, or degraded operations.

Mathematical Formulation

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 𝑅𝑛𝑒𝑡(𝑡) is defined as a weighted combination of subsystem performance indicators: (4)

where Φ𝑖(𝑡) – normalized performance level of subsystem i at time t, 𝑤𝑖 – normalized risk weight of subsystem i (∑𝑖𝑤𝑖 = 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: (5)

where 𝑅net(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: (6)

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 resilience-oriented indicators:

• Φ(𝑡): normalized operational performance function (0–1);
• 𝑅𝑛𝑒𝑡(𝑡): network-wide risk level, obtained by weighted aggregation of node risks;
• RI: resilience index, representing mean normalized availability;
• ETL: expected throughput loss, reflecting cumulative performance degradation;

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

Methods

Port network, Conditions and Risk

As indicated, a port system is considered as (1) where the set of nodes 𝑉 represents moorings, towing facilities, control centers and supporting technical systems, and the set of arcs 𝐸 represents logistical or technological links between them. Each node 𝑖 ∈ 𝑉 is characterized by the current availability state 𝑥𝑖(𝑡) (from 0 to 1), which reflects the share of preserved functionality or throughput. Node performance depends on the level of risk 𝑟𝑖(𝑡), which integrates technical, behavioral, and climatic factors, as well as the reliability of security barriers 𝐸𝐵𝑖(𝑡), which degrade over time. Thus, risk and accessibility are interrelated – an increase in 𝑟𝑖(𝑡) leads to a decrease in 𝑥𝑖(𝑡), and vice versa. To describe the behavior of the system, a dynamic pair of equations is introduced: (7)

where: 𝜎(⋅) – a logistic function that normalizes risk within [0,1]; 𝛾𝑖𝑗 – risk transfer coefficients between nodes (cascade effect); 𝑏𝑖·𝑧𝑖(𝑡) – Vector of external disturbances (weather, visibility, ship congestion); 𝑢𝑖(𝑡) – managed intervention (prevention, reserves, additional towing); 𝜉𝑖(𝑡) and 𝜔𝑖(𝑡) – stochastic noise components. This system of equations describes the bi-directional dynamics of risk and performance, which is the foundation of the port network risk management model. It is consistent with the concept of dynamic risk reassessment, which the authors of the article define as a transition from static standards to adaptive management. The effectiveness of each security barrier 𝐸𝐵𝑖(𝑡) decreases exponentially: (8)

where 𝐸0, = 1 – initial efficiency, and 𝜆𝑖 – degradation rate (h–1). The calibration of 𝜆𝑖 is performed by the (9):

The barrier degradation model allows to quantify the time aspect of safety degradation, which directly affects the operational risk 𝑟𝑖(𝑡). The largest value of 𝜆 (e.g., 0.010 for a control system) means a rapid decline in efficiency, and thus a higher priority in maintenance planning. For node 𝑖 at step 𝑡: criteria {𝑘} (equipment reliability, human factor, weather, tugboat traction margin, visibility etc.) with weights 𝑤𝑘. Let 𝐶𝑖(𝑡) ∈ [0, 1] be an indicator of proximity to the ideal solution – the closeness with TOPSIS; then 𝑟𝑖(𝑡) = 1 – 𝐶𝑖(𝑡). Thus, if the mooring scenario receives a high value of 𝐶𝑖 (for example, 0.78), the risk 𝑟𝑖(𝑡) is equal to 0.22, which automatically reduces the negative impact in the first equation of the system. This combination of methods allows us to move from a qualitative expert risk assessment (TOPSIS) to a dynamic model with quantitative updates. This ensures consistency between DSS decisions and the system’s actual state.

Supersystem Network Indicators

The analysis of the port system at the level of individual nodes (terminals, mooring posts, towing points) allows you to track local changes in the state, but for strategic management it is important to obtain integrated metrics that characterize the behavior of the entire network. Such metrics include:

• instantaneous network performance Φ(𝑡);
• probability of failure-free operation 𝑅net(𝑡);
• resilience Index (𝑅𝐼);
• expected Throughput Loss (ETL).

The total efficiency of the port network at time 𝑡 is defined as the aggregate throughput of all nodes, taking into account their availability: (10)

where 𝑐𝑖 – nominal (design) capacity of node 𝑖; 𝑥𝑖(𝑡) – current availability (0–1). The function Φ(𝑡) describes the actual efficiency of the system. If all elements are fully operational, 𝑥𝑖(𝑡) = 1, we obtain Φ(𝑡) = Φmax. A decrease in 𝑥𝑖(𝑡) due to risks or degradation of security barriers directly affects performance. (11)

A stochastic approximation based on node failure probabilities is used to estimate the integral reliability: (12)

or in a discrete form: (13)

If 𝑅𝐼 ≈ 1, the system operates stably even under the influence of external risks; if 𝑅𝐼 < 0.9, there is a significant loss of efficiency that requires corrective action. Indicator 𝐸𝑇𝐿 characterizes the average performance loss during the operation period: (14)

𝐸𝑇𝐿 measures the fraction of time that the system operates below its optimal level. This is a key criterion for evaluating the effectiveness of intervention strategies 𝑢𝑖(𝑡): the lower the 𝐸𝑇𝐿, the better the system performs under real-world uncertainty. The combination of indicators Φ(𝑡), 𝑅net(𝑡), 𝑅𝐼 and 𝐸𝑇𝐿 forms a supersystem monitoring level that allows assessing the operational state of the port network in real time, the impact of external disturbances on overall efficiency, the effectiveness of stabilization measures, and the comparative sustainability of alternative operating scenarios. Thus, these metrics are the basis for the subsequent optimization block – the intervention management problem, which determines how to minimize risks and productivity losses with limited resources. In other words, a risk-based management planning problem, where local dynamics 𝑟𝑖(𝑡), 𝑥𝑖(𝑡) and barrier degradation 𝐸𝐵𝑖(𝑡) are integrated into a strategic optimization model and the goal is to determine how, when and to which nodes it is advisable to direct interventions 𝑢𝑖(𝑡) to minimize the total loss of efficiency with limited resources.

Risk–Reliability Coupling

We explicitly distinguish reliability and risk. The network reliability aggregates node-level failure probabilities 𝑝𝑖(𝑡). Under conditional independence: (15)

When dependencies are non-negligible, we include pairwise correlation terms: (16)

where 𝜌𝑖𝑗 – dependence between nodes 𝑖 and 𝑗. Risk is defined as the expected loss weighted by consequence 𝐶(𝑡): (17)

Assumption of conditional independence is justified for decentralized subsystems; otherwise, 𝜌𝑖𝑗 ≠ 0 is calibrated from historical or simulated cascading events. This coupling clarifies the role of 𝑝𝑖(𝑡) in forming 𝑅net(𝑡) and its linkage to RI via ETL.

The task of Planning Risk-Based Management

Under conditions of environmental uncertainty (weather changes, queues, equipment failures), the port network must maintain stable performance while limiting the level of risk. Therefore, a multi-criteria optimization problem is formed that takes into account three key objectives:

• minimizing bandwidth losses (via 𝐸𝑇𝐿);
• minimizing the total risk of nodes 𝑟𝑖(𝑡);
• minimizing the cost of intervention 𝑢𝑖(𝑡).

 To formulate the objective function, the problem is written mathematically as follows: (18)

where: ϰ – weighting factor of productivity losses; 𝜇𝑖 – risk weight of a node 𝑖; 𝜈𝑖 – cost-effectiveness of the intervention; 𝐵(𝑡) – available resource or budget at the moment 𝑡; 𝐸 [⋅] – mathematical expectation under stochastic perturbations.

The logic of the objective function

1. The first term (ϰ𝐸𝑇𝐿(𝑡)) estimates the system performance loss – its minimization ensures that a stable flow of cargo is maintained.
2. The second term (∑𝜇𝑖𝑟𝑖(𝑡)) minimizes local risks – it prevents critical values of 𝑟𝑖(𝑡) from exceeding the nodes.
3. The third term (∑𝜈𝑖∥𝑢𝑖(𝑡)∥1) imposes a penalty for excessive or irrational interventions – it stimulates the optimal use of limited resources.

Together, these elements form a compromise multi-criteria task that balances safety, efficiency, and economic feasibility.

Implementation Details

• The optimization problem was solved using a discrete-time MPC approach with a prediction horizon of 10 steps (24 h). The solver CVXPY (Python) with an interior-point method was used.
• Weights were normalized as 𝑤𝑅 = 0.4, 𝑤𝐵 = 0.3, 𝑤𝑇 = 0.3.
• Constraints included: 𝑅(𝑡) ≤ 0.8, 𝐵(𝑡) ≥ 0, 𝑈(𝑡) ≤ 𝑈𝑚𝑎𝑥 = 1.0.
• The algorithm converged within 15 iterations on a 2.9 GHz CPU.
• Degradation and transfer parameters were estimated through a hybrid approach combining expert elicitation and analysis of maintenance event logs from a port terminal dataset. Rates 𝛼 and 𝛾 were adjusted within ±20% during calibration to achieve convergence with observed operational failure frequencies over a 6-month monitoring period, Table 3.

Table 3: Degradation and transfer parameters.
Parameter	Symbol	Value	Method of Estimation	Description
Barrier degradation rate	𝛼	0.20	Empirical calibration	Rate of exponential decay in subsystem reliability
Risk transfer coefficient	𝛾	0.15	Expert elicitation	Coupling intensity between interconnected nodes
Recovery coefficient	𝛽	0.10	Derived from maintenance data	Speed of recovery after mitigation
Disturbance factor	𝜇	0.05	Monte Carlo initialization	External stochastic influence
Budget constraint	𝐵(𝑡)	≤1.0	Normalized constant	Fraction of available mitigation resources

Resource Constraints and Managed Interventions

Interventions 𝑢𝑖(𝑡) reflect any actions aimed at reducing risk or strengthening barriers such as an additional tug during mooring, a backup power line, increased monitoring or manual control, or temporary relocation of the vessel to another berth. The total resource at time 𝑡 is limited by: (19)

where 𝐵(𝑡) can be interpreted as the number of tugs, energy reserves or man-hours available in a particular shift. Link to risk dynamics because interventions directly reduce risk in the dynamic model (eq. 7): (20)

where 𝜂𝑖𝐸𝐵𝑖(𝑡)𝑢𝑖(𝑡) reflects the risk reduction due to active action, adjusted for the current effectiveness of the barrier 𝐸𝐵𝑖(𝑡), which means that even an effective intervention 𝑢𝑖(𝑡) will be less effective if the barrier has already degraded (small 𝐸𝐵𝑖(𝑡)). The problem belongs to the class of stochastic dynamic optimizations, where we can use approaches such as Model Predictive Control (MPC): optimization on a rolling horizon with state updates every Δ𝑡 or Cut-and-Column Generation (C&CG): two-stage solution with uncertainty scenarios. In addition, Heuristic strategies: intervention priorities ∝𝜅𝑖𝑟𝑖(𝑡)𝑐𝑖, i.e. the greater the node’s contribution to performance and the higher the risk, the higher the priority of the action.

The proposed approach enables a systematic integration of risk management and logistics planning. Interventions are not distributed statically, but in accordance with the current risk dynamics and the state of the network. To implement adaptive control because the model can update decisions after each evaluation cycle 𝑟𝑖(𝑡) and 𝑥𝑖(𝑡), which ensures that the DSS responds to real changes in the environment. As well as assess the trade-off between efficiency and cost, the coefficients 𝜇𝑖, 𝜈𝑖, ϰ allow you to adjust priorities – from “maximum security” to “minimum cost”. The optimization module is implemented using a Model Predictive Control (MPC) framework, where the system state 𝑥(𝑡) evolves according to a discrete-time stochastic process. The planning horizon 𝐻 = 10 iterations corresponds to a 24-hour operational window. Constraints include: . Optimization is solved iteratively using a quadratic programming solver (Python–cvxpy, interior-point method) with a convergence tolerance of 10–5. Computational complexity grows linearly with the number of monitored subsystems, ensuring scalability for networks up to 50 nodes.

Results

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:

• Horizon: 𝑇 = 24 hrs (step 1 h).
• Subsystems and barrier degradation: 𝜆ctrl = 0.010, 𝜆eng = 0.007, 𝜆nav = 0.006, 𝜆com = 0.004.
• Scenarios: Baseline, Adverse metocean, Adverse + Mitigation (in mitigation, the 𝑢(𝑡), that strengthens barriers at the highest risk nodes and reduces 𝑟𝑖(𝑡)).
• Alignment with TOPSIS: local risks are taken as 𝑟𝑖(𝑡) = 1−𝐶𝑖(𝑡), where 𝐶𝑖(𝑡) ∈ [0,1] – proximity coefficient obtained from TOPSIS.

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.

Port Network Simulation Algorithm

Stage 1: Initialization of parameters. At the beginning of the simulation, the initial values of key variables are determined. For each node of the port network, full availability 𝑥𝑖(0) = 1, initial risk 𝑟𝑖(0) = 0.05, and the effectiveness of barrier mechanisms 𝐸𝐵𝑖(0) = 1 are set. In parallel, the numerical parameters of the model are set – the weights of self-risk 𝛼𝑖, inter-node impact 𝛾𝑖𝑗, barrier degradation 𝜆𝑖, risk penalties 𝜇𝑖, intervention costs 𝜈𝑖, and response efficiency 𝜂𝑖. The time horizon of the simulation is determined (e.g., 24 hours) and three scenarios are generated: normal (Baseline), adverse (Adverse metocean), and a scenario with compensatory measures (Mitigation).

Stage 2: Generation of external disturbances. At each time step, a stochastic vector of external influences 𝑧𝑖(𝑡) is generated, taking into account meteorological conditions (wind, wave, visibility), tug loads, and other operational factors. These disturbances affect the risk through the corresponding sensitivity coefficients 𝑏𝑖, reflecting the dynamics of disturbances or complications in the system.

Stage 3: Risk assessment using TOPSIS. The integral risk in a node at time step 𝑡 is calculated based on the TOPSIS method, a multi-criteria ranking method. Technical, human, natural and navigational factors are taken into account. For each node, an indicator of proximity to the ideal solution 𝐶𝑖(𝑡) ∈ [0, 1] is determined, after which the risk is calculated as 𝑟𝑖(𝑡) = 1 – 𝐶𝑖(𝑡), which allows for adaptive display of the node’s state change under conditions of uncertainty.

Stage 4: Updating the dynamics of states. At this stage, two key model equations are applied: for risk evolution and availability. The risk in a node is updated taking into account the impact of its own previous state, the risks of neighboring nodes (cascading effect), external disturbances, barriers, and interventions. The availability is updated based on the balance between the damage from the risk and the effect of the intervention, taking into account the degradation of barriers. The logistic smoothing function 𝜎(⋅) is used and projected onto the interval [0, 1].

Stage 5: Optimization of interventions. At each time step, the allocation of intervention resources is optimized. The minimization problem includes three components: performance loss (ETL), risk penalty, and cost of interventions. The resource constraint of the intervention budget ∑𝑖𝑐𝑜𝑠𝑡𝑖𝑢𝑖(𝑡) ≤ 𝐵(𝑡) is taken into account. Priority is given to nodes with high risk and high criticality (throughput), which meets the criterion ∝𝜅𝑖𝑟𝑖(𝑡)𝑐𝑖.

Stage 6: Calculation of network metrics. After updating the states, the system performance indicators Performance Φ(𝑡), Network Reliability 𝑅𝑛𝑒𝑡(𝑡), Resilience Index (RI) and Expected Throughput Loss (ETL) are calculated, which allow to quantitatively compare scenarios and evaluate the effectiveness of interventions.

Stage 7: Analysis and visualization. The modeling results are visualized as a series of graphs that show the dynamics of performance, reliability, risk, and barrier degradation over time. Key indicators are summarized in a comparative table. This makes it possible to draw conclusions about the system’s sensitivity to external factors and the benefits of risk-informed port network management (Figure 4).

Figure 5 shows the share of available network performance Φ(𝑡)/Φmax over time for the three scenarios. In Baseline, the curve is almost “flat” at ≈ 0.98; in Adverse, there is a systematic drop to ≈ 0.90–0.93 during periods of deteriorating conditions; in Mitigation (using targeted interventions 𝑢(𝑡)), the curve rises between these two, stabilizing at ≈ 0.94–0.96. The difference between Adverse and Mitigation reflects the effect of risk-informed management: the use of 𝑢(𝑡) can return a significant portion of the lost throughput by strengthening barriers in bottlenecks.

Figure 6 shows the evolution of network reliability 𝑅net(𝑡) with confidence bands (e.g., 5–95th percentile) based on the results of Monte Carlo simulations of random weather/load disturbances. In Baseline, 𝑅net remains at ≈ 0.98–0.99; in Adverse, it decreases to ≈ 0.95–0.97, and the bands become wider (greater variability); in Mitigation, the median shifts upward (≈ 0.97) and the spread decreases. The narrower bands for Mitigation indicate not only an increase in the average reliability level, but also a decrease in uncertainty – the grid becomes more predictable under the intervention.

Sensitivity and Robustness Analysis

To evaluate model stability, we performed one-factor sensitivity tests and Monte Carlo simulations. Variations of the key degradation parameter 𝛼 within the range [0.1–0.4] produced changes in the Expected Throughput Loss (ETL) of approximately ±18 %. Adjustments of the transfer coefficient 𝛾 within ±20 % resulted in less than 6 % variation in the Resilience Index (RI). A Monte Carlo procedure with 500 random draws from uniform parameter distributions confirmed the robustness of the optimization output, with 95 % confidence intervals for RI within ±0.05 of the mean. These results indicate that the proposed adaptive optimization framework maintains consistent performance under parameter uncertainty and limited data variability (Table 4).

Table 4: Sensitivity test summary.
Parameter Varied	Range Tested	Metric Affected	Change (%)	Interpretation
α (degradation rate)	0.1 – 0.4	ETL	±18	Moderate sensitivity
γ (risk transfer factor)	± 20 %	RI	±6	Low sensitivity
Budget 𝐵(𝑡)	± 15 %	Rnet	±9	Stable behavior

The heat map in Figure 7 (nodes on the vertical, time on the horizontal; shades – risk level 𝑟𝑖(𝑡)) shows the “hot zones”. In Adverse, temporary high-risk clusters appear at weather/visibility-sensitive nodes (e.g., approach channel, mooring terminal). In Mitigation, the intensity of the hot spots is significantly reduced, and the risk front is localized (smaller spatial and temporal scope). The heatmap allows for spatial and temporal localization of interventions: first, to stabilize the control and traction of tugs on the approach, then to strengthen navigation/communication barriers during periods of poor visibility.

Figure 8 shows four exponential curves 𝐸𝐵(𝑡) for the subsystems: Control (the fastest decline, 𝜆 = 0.010), Energy (0.007), Navigation (0.006), and Communication (0.004). At about the 100th hour, Control’s efficiency drops to ≈37% of the initial one, while Communication maintains the highest level. Such a difference in 𝜆 determines the priority of maintenance: short inspection/reinforcement intervals for Control and Energy; longer ones for Communication with regular monitoring.

Table 5 shows a comparison of the integrated performance indicators of the port network under three scenarios: baseline, adverse metocean, and mitigation. The metrics used provide a comprehensive picture of the system’s state over the daily observation horizon.

Table 5: Comparative results for maritime network scenarios (24 h horizon).
Scenario	Φ(𝑡)/Φmax	Rnet	RI	ETL
Baseline	0.98	0.985	0.975	0.020
Adverse metocean	0.92	0.961	0.926	0.080
Adverse + Mitigation	0.95	0.971	0.949	0.050

Thus, in the baseline scenario, the network exhibits a stable state (RI = 0.975, ETL = 0.02), corresponding to nearly full operability. When hydrometeorological conditions deteriorate, performance and reliability decline: the average throughput drops to 0.92, and RI decreases to 0.926, indicating a loss of about 8% of efficiency. The introduction of managed interventions – strengthening technical barriers, using backup tugs, flexible slot scheduling – increases RI to 0.949 and reduces ETL to 0.05. This means that approximately half of the performance loss was compensated.

Case Study Example

To demonstrate the model’s applicability, a simplified case was implemented using anonymized data from the Odesa port region. Three critical subsystems were modeled (power supply, IT control, and navigation aids). Baseline simulation without adaptive control yielded an average Expected Throughput Loss (ETL) of 0.082. Under the proposed adaptive DSS, ETL decreased to 0.054, indicating a 34% improvement in operational resilience.

Validation and Sensitivity Analysis

The model was tested on anonymized operational data from a port network with three interdependent subsystems. A Monte Carlo experiment with 1000 iterations was performed to estimate variability of RI and ETL. The resulting 95% confidence intervals were: 𝑅𝐼 = 0.93 ± 0.04, 𝐸𝑇𝐿 = 0.056 ± 0.008. The sensitivity analysis showed that varying the degradation parameter 𝛼 from 0.1 to 0.3 resulted in ±12% change in RI.

Ablation Study

To quantify the marginal contribution of each module, we evaluate three ablations: (i) no degradation model, (ii) no cascading, (iii) no MPC. The impact is reported as the change in resilience index (ΔRI) relative to the full model. The results, summarized in Table 6, indicate the relative drop in resilience index (ΔRI, in percentage points) compared to the full implementation.

Table 6: Impact of individual modules based on ablation scenarios.
Configuration	ΔRI (pp)
No degradation model	−7.5
No cascading	−4.8
No MPC	−9.2
Negative values indicate a reduction in resilience versus the full integrated framework.

Discussion

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.

Conclusion

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.

References

1. Kilic KI, Maity S, Sung I, Nielsen P. Challenges and AI-driven solutions in maritime search and rescue planning: A comprehensive literature review. Mar Policy. 2025;178:106692. https://doi.org/10.1016/j.marpol.2025.106692
2. Yang L, Liu J, Zhou Q, Liu Z, Chen Y, Wang Y, Liu Y. Enabling autonomous navigation: Adaptive multi-source risk quantification in maritime transportation. Reliab Eng Syst Saf. 2025;261:111118. https://doi.org/10.1016/j.ress.2025.111118
3. Li Y, Yuen KF, Zhou Y. Risk assessment of achieving greenhouse gas emission reduction target in the maritime industry. Transp Policy. 2024;155:29–46. https://doi.org/10.1016/j.tranpol.2024.06.001
4. Lin Y, Li X, Yuen KF. Machine learning applications for risk assessment in maritime transport: Current status and future directions. Eng Appl Artif Intell. 2025;155:110959. https://doi.org/10.1016/j.engappai.2025.110959
5. Hausken K, Welburn JW, Zhuang J. A review of game theory and risk and reliability analysis in infrastructures and networks. Reliab Eng Syst Saf. 2025;261:111123. https://doi.org/10.1016/j.ress.2025.111123
6. Elgharbi SE, Iturralde M, Dupuis Y, Gaugue A. Maritime monitoring through LoRaWAN: Resilient decentralised mesh networks for enhanced data transmission. Comput Commun. 2025;241:108276. https://doi.org/10.1016/j.comcom.2025.108276
7. He Q, Liu W, Liu T, Tian Q. Robust coordinated path planning for unmanned aerial vehicles and unmanned surface vehicles in maritime monitoring with travel time uncertainty. Transp Res Part B Methodol. 2025;199:103284. https://doi.org/ 10.1016/j.trb.2025.103284
8. Deng W, Ma X, Qiao W. A novel methodology to quantify the impact of safety barriers on maritime operational risk based on a probabilistic network. Reliab Eng Syst Saf. 2024;243:109884. https://doi.org/10.1016/j.ress.2023.109884
9. Waskito DH, Kurt RE, Gusti AP, Bowo LP. Assessing the risk of transporting battery electric vehicles through water transportation modes by integrating system theory and Bayesian Network approach. Ocean Eng. 2025;341:122606. https://doi.org/ 10.1016/j.oceaneng.2025.122606
10. Dong G, Zhang J, Dai M. Evaluating the resilience impacts of emission reduction policies in green liner shipping network along the New Maritime Silk Road. Reg Stud Mar Sci. 2025;86:104209. https://doi.org/ 10.1016/j.rsma.2025.104209
11. Liu Z, Deng J, Shu Y, Gan L, Song L, Li H, Yang Z. Spatiotemporal prediction of offshore wind fields based on a hybrid deep learning model for maritime navigation. Ocean Coast Manag. 2025;269:107841. https://doi.org/ 10.1016/j.ocecoaman.2025.107841
12. Liu P, Dong X, Wang P. An enhanced fuzzy cognitive map for human risk assessment in maritime transportation: Integrating causal mining and expert elicitation. Adv Eng Inform. 2025;68:103624. https://doi.org/ 10.1016/j.aei.2025.103624
13. Chen P, Zhang A, Zhang S, Dong T, Zeng X, Chen S, et al. Maritime Near-Miss prediction framework and model interpretation analysis method based on Transformer neural network model with multi-task classification variables. Reliab Eng Syst Saf. 2025;257:110845. https://doi.org/ 10.1016/j.ress.2025.110845
14. Ay C. Human reliability analysis in inert gas operations with fuzzy CREAM-based Bayesian networks. Ocean Eng. 2025;339:122196. https://doi.org/ 10.1016/j.oceaneng.2025.122196
15. Animah I. Application of bayesian network in the maritime industry: Comprehensive literature review. Ocean Eng. 2024;302:117610. https://doi.org/ 10.1016/j.oceaneng.2024.117610
16. Maidana RG, Kristensen SD, Utne IB, Sørensen AJ. Risk-based path planning for preventing collisions and groundings of maritime autonomous surface ships. Ocean Eng. 2023;290:116417. https://doi.org/10.1016/j.oceaneng.2023.116417
17. Kong D, Lin Z, Li W, He W. Development of an improved Bayesian network method for maritime accident safety assessment based on multiscale scenario analysis theory. Reliab Eng Syst Saf. 2024;251:110344. https://doi.org/10.1016/j.ress.2024.110344
18. Bokau JRK, Samad R, Park Y, Kim D. From managing risk to reality: A case of maritime safety in Makassar Port, Indonesia using FRAM and AIS data analysis. Ocean Eng. 2025;339:122012. https://doi.org/10.1016/j.oceaneng.2025.122012
19. Ledesma O, Lamo P. Hybrid IoT network for real-time monitoring of maritime containers. Comput Netw. 2025;271:111627. https://doi.org/10.1016/j.comnet.2025.111627
20. Xu Y, Peng P, Claramunt C, Qian J, Lu F. Resilience of the global maritime LNG network: From static Bow-tie to dynamic SIR modeling. Reliab Eng Syst Saf. 2025;265:111605. https://doi.org/10.1016/j.ress.2025.111605
21. Fan H, Jia H, He X, Lyu J. Navigating uncertainty: A dynamic Bayesian network-based risk assessment framework for maritime trade routes. Reliab Eng Syst Saf. 2024;250:110311. https://doi.org/10.1016/j.ress.2024.110311
22. Dong Y, Ren H, Tao R, Sun J, Zhou Y. Joint multi-objective optimization of maritime SAR equipment deployment using high-risk area recognition and SA-APSO. Comput Ind Eng. 2025;208:111399. https://doi.org/10.1016/j.cie.2025.111399
23. Cooper-Baldock Z, Turnock SR, Sammut K. Wake-informed 3D path planning for Autonomous Underwater Vehicles using A* and neural network approximations. Ocean Eng. 2025;332:121353. https://doi.org/10.1016/j.oceaneng.2025.121353
24. Wang W, Zhao J, Chen Y, Shao P, Jia P. Process safety enhancement in maritime operations: A Bayesian network-based risk assessment framework for the RCEP region. Process Saf Environ Prot. 2025;194:1235–56. https://doi.org/10.1016/j.psep.2024.12.072
25. Onyshchenko S, Melnyk O. Probabilistic assessment method of hydrometeorological conditions and their impact on the efficiency of ship operation. J Eng Sci Technol Rev. 2021;14(6):132–6. https://doi.org/10.25103/jestr.146.15
26. Melnyk O, Onyshchenko S, Koryakin K. Nature and origin of major security concerns and potential threats to the shipping industry. Sci J Silesian Univ Technol Ser Transp. 2021;113:145–53. https://doi.org/10.20858/SJSUTST.2021.113.11
27. Melnyk O, Bychkovsky Y, Onishchenko O, Onyshchenko S, Volianska Y. Development the method of shipboard operations risk assessment quality evaluation based on experts review. In: Studies in Systems, Decision and Control, vol. 481. Springer; 2023. p. 695–710. https://doi.org/10.1007/978-3-031-35088-7_40
28. Koskina Y, Onyshenko S, Drozhzhyn O, Melnyk O. Efficiency of tramp fleet operating under the contracts of affreightment. Sci J Silesian Univ Technol Ser Transp. 2023;120:137–49. https://doi.org/10.20858/sjsutst.2023.120.9
29. Shumylo O, Yarovenko V, Malaksiano M, Melnyk O. Comprehensive assessment of hull geometry influence of a modernized ship on maneuvering performance and propulsion system parameters. Pomorstvo. 2023;37(2):314–25. https://doi.org/10.31217/p.37.2.13
30. Onishchenko O, Bukaros A, Melnyk O, Yarovenko V, Voloshyn A, et al. Ship refrigeration system operating cycle efficiency assessment and identification of ways to reduce energy consumption of maritime transport. In: Studies in Systems, Decision and Control, vol. 481. Cham: Springer; 2023. p. 641–52. https://doi.org/10.1007/978-3-031-35088-7_36
31. Zablotsky YV, Sagin SV. Enhancing fuel efficiency and environmental specifications of a marine diesel when using fuel additives. Indian J Sci Technol. 2016;9(46):Article 107516. https://doi.org/10.17485/ijst/2016/v9i46/107516
32. Sagin S, Madey V, Sagin A, Stoliaryk T, Fomin O, et al. Ensuring reliable and safe operation of trunk diesel engines of marine transport vessels. J Mar Sci Eng. 2022;10(10):Article 1373. https://doi.org/10.3390/jmse10101373
33. Malaksiano NA. Exact inclusions of Gehring classes in Muckenhoupt classes. Mathematical Notes, 2001,70(5-6), 673–681. https://doi.org/10.1023/a:1012983028054
34. Lapkina IO, Malaksiano MO, Malaksiano MO. Optimization of the structure of sea port equipment fleet under unbalanced load. Actual Probl Econ. 2016;183(9):364–71.
35. Chachra A, Kumar A, Ram M. A Markovian approach to reliability estimation of series-parallel system with Fermatean fuzzy sets. Comput Ind Eng. 2024;190:Article 110081. https://doi.org/10.1016/j.cie.2024.110081
36. Singh KU, Kumar A, Singh T, Ram M. Image-based decision making for reliable and proper diagnosing in NIFTI format using watermarking. Multimed Tools Appl. 2022;81(27):39577–603. https://doi.org/10.1007/s11042-022-12192-9
37. Kumar A, Bisht S, Goyal N, Ram M. Fuzzy reliability based on hesitant and dual hesitant fuzzy set evaluation. Int J Math Eng Manag Sci. 2021;6(1):166–79. https://doi.org/10.33889/IJMEMS.2021.6.1.010
38. Kormych B, Malyarenko T, Wittke C. Rescaling the legal dimensions of grey zones: Evidence from Ukraine. Glob Policy. 2023;14(3):516–30. https://doi.org/ 10.1111/1758-5899.13233
39. Kormych B, Averochkina T, Demin S, Savych O. Management in the field of innovative and import-substituting cluster in the modern Asian economy. Int J Manag Bus Res. 2019;9(4):204–14.
40. Faizatama AM, Firdaus N, Adiputra R, Prabowo AR, Ghanbari-Ghazijahani T, Fazeres-Ferradosa T, et al. Hydrodynamic impact of cold-water pipes and mooring systems on KVLCC2 for floating OTEC platforms. Ocean Eng. 2025;341:Article 122491. https://doi.org/ 10.1016/j.oceaneng.2025.122491
41. Sagin SV, Sagin SS, Madey V. Analysis of methods of managing the environmental safety of the navigation passage of ships of maritime transport. Technol Audit Prod Res. 2023;4(3):33–42. https://doi.org/ 10.15587/2706-5448.2023.286039
42. Varbanets R, Minchev D, Kucherenko Y, Zalozh V, Kyrylash O, Tarasenko T. Methods of real-time parametric diagnostics for marine diesel engines. Pol Marit Res. 2024;31(3):71–84. https://doi.org/10.2478/pomr-2024-0037
43. Bilan T, Kaplin M, Makarov V, Perov M, Novitskii I, Zaporozhets A, et al. The balance and optimization model of coal supply in the flow representation of domestic production and imports: The Ukrainian case study. Energies. 2022;15(21):Article 8103. https://doi.org/10.3390/en15218103
44. Eremenko V, Zaporozhets A, Isaenko V, Babikova K. Application of wavelet transform for determining diagnostic signs. CEUR Workshop Proc. 2019;2387:202–14.
45. Melnyk O, Kuznichenko S, Onishchenko O. Impact of AIS manipulation on shipping safety and strategic countermeasures. Lex Portus. 2024;10(4):31–9. https://doi.org/10.62821/lp10403
46. Melnyk OM, Onishchenko OA, Shibaev OG, Kuznichenko SO, Bulgakov MP, Shcherbina OV, et al. Development of strategies for reducing nitrous oxide emissions from marine diesel engines. J Chem Technol. 2024;32(2):465–79. https://doi.org/10.15421/jchemtech.v32i2.297410

Glossary of Symbols

Symbol	Definition	Units / Range	Description
Φ(𝑡)	System performance function	[0–1]	Normalized operational state of the port network at time t.
𝑅(𝑡)	Instantaneous risk level	[0–1]	Probability-weighted measure of operational vulnerability.
𝑅𝑛𝑒𝑡(𝑡)	Network-level risk	[0–1]	Aggregated risk across all subsystems considering interdependencies.
𝑅𝐼	Resilience Index	[0–1]	Mean normalized availability; the higher the RI, the more resilient the system.
𝐸𝑇𝐿	Expected Throughput Loss	hours / normalized	Cumulative loss of functionality over time due to disruptions.
𝛼	Degradation rate	0.1–0.3	Parameter describing barrier performance decay rate.
𝛽	Recovery rate	0.05–0.25	Rate of performance restoration during mitigation.
𝛾	Risk transfer coefficient	0–1	Coupling intensity between connected nodes (risk propagation).
𝑤𝑖	Weighting factor	Σw_i = 1	Relative importance of each subsystem or criterion.
​𝐵(𝑡)	Budget / resource constraint	[0–1]	Available fraction of total mitigation resources at time t.
𝑈(𝑡)	Control input	[0–1]	Decision variable defining mitigation or maintenance action.
𝐻	Prediction horizon	steps (e.g., 10)	Time window for MPC optimization.
𝜇	Disturbance factor	–	Represents stochastic external effects (e.g., weather, cyber event).
𝑅𝑇𝑂	Recovery Time Objective	hours	Maximum acceptable downtime for restoration.
𝑅𝑅𝑅	Resilience Recovery Rate	[0–1]	Rate of recovery of system performance after disruption.
𝑓𝑖(𝑡)	Risk impact function	arbitrary	Quantifies the impact of risk event i over time.
𝐶𝑖	Cost coefficient	monetary units	Cost associated with mitigation or maintenance action.

Appendix

Cite this article as:
Melnyk O, Shcheniavskyi H, Volyanskyy S, Koryakin K and Kucherenko V. Risk-Based Planning of Port Network Sustainability Under Conditions of Operational Uncertainty. Premier Journal of Science 2025;14:100171

Export to EndNote Export to Mendeley