Maritime Transport Network Optimisation with Respect to Environmental Footprint and Enhanced Resilience: A Case Study for the Aegean Sea

Abstract

Given the projection of the impact of climate change and the uncertainty caused by geopolitical volatility, minimising emissions has become an urgent priority for the shipping industry. In this context, the aim of the present study is the calculation and estimation of emissions generated by ship operations within a maritime transportation network, as well as the identification of the optimal route that minimises both emissions and travel time. Emission estimation is carried out using methodologies and assumptions from the Fourth IMO GHG Study. The decision-making, along with the optimisation process, is performed through backward dynamic programming, following a multi-objective optimisation framework. Specifically, the analysis is carried out on both a theoretical and a realistic network. In both cases, various scenarios are examined, including different approaches to vessel speed, some of which incorporate probabilistic speed distributions, as well as scenarios involving uncertainty regarding port availability. Additionally, the resilience of the network is examined, focusing on the additional burden in terms of emissions and travel time when a port is unexpectedly unavailable and a route adjustment is required. The calculations and optimisation are carried out using Excel and the @Risk software by Palisade, with the latter enabling the incorporation of probability distributions and the execution of Monte Carlo simulations.

1. Introduction

Maritime transport plays a vital role in the global economy, with ships having constituted for centuries the primary means of transporting goods across the globe. It is estimated that ships undertake about 90% of the transport volume arising from global trade [1]. In 2023, the volume of global seaborne trade was estimated to be almost 12.3 billion tonnes, showing a significant increase compared with 1990, when approximately only 4 billion tonnes of goods were loaded in ports worldwide [2]. In recent decades, however, emissions from shipping activities have risen significantly, contributing to global anthropogenic greenhouse gas emissions and, most importantly, impacting air pollution. These emissions have been linked to various environmental impacts, such as climate change, reduction of ozone layer thickness, and acid rain. Given the growing concern over ship-related air pollution, various legal frameworks have been established at the European and global levels. Among the most pressing global challenges today is global warming, driven primarily by the greenhouse gas (GHG) emissions resulting from increased fossil fuel consumption since industrialisation. There are two categories of ship air emissions: greenhouse gases, mainly carbon dioxide (CO₂), and atmospheric pollutants, mainly sulphur oxides (SO_x), nitrogen oxides (NO_x), and particulate matter (PM). Among these, the most important pollutants produced by ships in international routes and in port are PM, due to their impact on human health, SO_x and NO_x, due to their contribution to the formation of acid rain, as well as CO₂, nitrous oxide (N₂O), and methane (CH₄), due to their role in the greenhouse effect. Carbon dioxide is the most significant greenhouse gas released by ships, both in terms of the amount and the global warming potential. Moreover, maritime transportation is a major and growing contributor to global CO₂ emissions, with its environmental impact steadily increasing. The share of shipping in global anthropogenic CO₂ emissions increased from 2.76% in 2012 to 2.89% in 2018. According to the Fourth IMO Greenhouse Gas Study (2020), CO₂ emissions from ships increased from 0.962 billion tonnes in 2012 to 1.056 billion tonnes in 2018, representing a 9.3% rise. Projections for 2050 estimate that maritime CO₂ emissions could reach between 90% and 130% of the levels recorded in 2008 [3]. Therefore, even if strong efforts have been made so far, they should be further intensified to reduce and prevent the pollution caused by maritime ships. Shipowners are under increasing pressure to reduce GHG emissions from maritime transport. This shift toward greener shipping is driven by evolving regulations at the national, European Union (EU), and international (IMO) levels, alongside the growing use of alternative low-emission fuels, like methanol, ammonia, and hydrogen. Improving energy efficiency through technological advancements, like electrification and digitalisation systems, as well as implementing abatement technologies, such as scrubbers and carbon capture, are also key strategies in reducing the sector’s environmental impact.

Optimisation of ship energy efficiency is widely recognised as one of the most effective measures to reduce fuel consumption and emissions. In this context, the Genetic Algorithm–Long Short-Term Memory (GA–LSTM) model, representing one of the latest developments in ship energy consumption forecasting, has demonstrated significantly improved accuracy in predicting fuel usage under various operating conditions [4]. In addition, recent studies have highlighted that most current optimisation methods mainly focus on a single factor, such as navigation speed, route, or trim, without fully considering their coupling effects. To overcome this limitation, a collaborative optimisation framework has been proposed, combining a GA–LSTM-based forecast model with the Non-dominated Sorting Genetic Algorithm III (NSGA-III) for multi-objective optimisation, achieving fuel savings of up to 4.5% compared with conventional operations [5]. The accuracy of the prediction models depends strongly upon the modelling principles, influencing the parameters, and data quality, while intelligent algorithms and machine learning techniques have been shown to significantly improve their performance [6]. While the Genetic Algorithm represents a highly specialised and sophisticated optimisation technique, in this study, a dynamic programming approach is adopted as an efficient method for route and energy optimisation.

This study aims to analyse a maritime transportation network and to identify the optimal route that minimises both emissions and travel time, generally two conflicting goals. Within this framework, the present study seeks to address the issue of emissions and pollutants in the context of a maritime transportation network through a comprehensive quantitative assessment of ship emissions based on their operational profiles. Quantification of emissions is essential for optimising routes through dynamic programming, to identify paths that minimise both emissions and travel time. The insights obtained from this optimisation process aim to support the development of focused, evidence-based policies and strategies that effectively reduce emissions in maritime transport, thereby supporting broader environmental sustainability.

Section 2 provides the background, including a short literature review, a description of the methodology, as well as an overview of the case study, while Section 3 presents in detail this study’s results, followed by a thorough discussion of the results and next steps of this work in Section 4.

2. Materials and Methods

2.1. Background

This section presents the legislation and regulations concerning emissions, followed by a detailed description of the optimisation framework, focusing on dynamic programming, multi-objective optimisation, and voyage optimisation, after a thorough literature review. The section concludes with an overview of the resilience of transport networks.

2.1.1. Legislation and Regulations

Significant efforts have been undertaken to minimise and to prevent the pollution caused by maritime ships. The regulation of ship emissions is governed by both international and European legal frameworks, each playing a vital role in reducing environmental impacts from maritime transport. At an international level, the International Maritime Organisation (IMO) has implemented measures to limit ship air emissions through the adoption of the International Convention for the Prevention of Pollution from Ships (MARPOL) Annex VI [7,8]. Annex VI was introduced to the MARPOL 73/78 Convention in 1997 to regulate air pollution from ships by establishing limits on specific air pollutants, including sulphur and nitrogen oxides. Compliance with these limits is mandatory for all ships constructed from 2005 onwards. In addition to these defined limits, the MARPOL Convention sets stricter air emission standards in specific regions known as emission control areas (ECAs) [9]. In an effort to reduce GHG emissions from ships, the IMO has implemented two mandatory measures under the MARPOL Convention. The first measure is the obligatory implementation of the energy efficiency design index (EEDI) for new ships, which sets a minimum energy efficiency level per capacity mile (ton mile) across various ship types and size categories, along with the adoption of the Ship Energy Efficiency Management Plan (SEEMP). The SEEMP provides shipping companies with a framework to monitor ship efficiency by utilising the Energy Efficiency Operational Indicator (EEOI), which allows shipowners to measure a vessel’s fuel consumption during operation and to assess the impact of operational adjustments. The second measure is the Initial Strategy on Reduction of GHG Emissions, adopted in 2018 by the Marine Environment Protection Committee (MEPC) at its seventy-second session in London [10]. In addition to the regulations adopted by the IMO, the EU has implemented directives and established regulations aimed at reducing emissions from shipping within its territorial waters. There are various regulations and directives concerning the sulphur content of marine fuels, the Monitoring, Reporting, and Verification (MRV) of CO₂ emissions, as well as the European Climate Law. The EU, through the European Climate Law (Regulation (EU) 2021/1119), has established legally binding targets to reduce greenhouse gas emissions by 55% by 2030 compared to 1990 levels, and to achieve climate neutrality by 2050. To support this transition, the European Green Deal outlines the necessary changes to achieve these goals. A key component of this plan is the “Fit for 55” package, introduced in 2021, which aligns EU policies with the new climate targets. Within this package, two significant pieces of legislation are the revision of the EU Emissions Trading System (EU ETS) and the adoption of the FuelEU Maritime Regulation [11,12]. Together, these frameworks create a comprehensive approach for addressing ship emissions and ensuring a sustainable future for maritime transport [12].

2.1.2. Optimisation–Dynamic Programming

Optimisation plays a central role in decision-making by guiding the selection of the best possible solution from a set of feasible alternatives, often under multiple objectives and predefined limitations. In this study, dynamic programming (DP) is adopted as the primary methodology for addressing optimisation problems. DP is a powerful recursive technique used to solve complex problems by breaking them down into simpler subproblems. Depending on the presence or not of uncertainty, DP is classified into two main categories: deterministic dynamic programming (under certainty), where all parameters and transitions are known in advance, and stochastic dynamic programming, which accounts for uncertainty in the system’s evolution. DP is particularly well-suited for solving problems, where decisions are made sequentially and each decision influences subsequent ones [13,14,15]. First introduced by Richard Bellman in 1962, DP has become a foundational method for tackling sequential decision-making problems, including multi-stage Markov decision processes and path planning problems across various fields [16,17]. In deterministic problems, the state of the next stage is entirely determined by the current state and the decision at the current stage; whereas, in stochastic problems, the state of the next stage is governed by a probability distribution. In deterministic dynamic programming, both the transition cost and the next state are fully determined by the current state and the selected decision. By contrast, under uncertainty, the outcome of a decision also depends on a random variable that introduces stochastic behaviour into both the transition cost and the system’s next state. Specifically, the cost function and the transition to the next state are influenced by a random variable, whose outcome is beyond the control of the decision maker [13,18]. Due to uncertainty in both the transition cost and the system’s next state, stochastic dynamic programming does not allow for direct optimisation of the total return/cost across stages. Instead, the problem must be formulated as the optimisation of the expected total return, taking into account the order in which decisions are made and information becomes available.

Additionally, this study focuses on multi-objective dynamic programming (MODP), a methodological framework developed to address problems involving two or more conflicting objectives. In such problems, the concept of a single optimal solution is replaced by a set of Pareto optimal solutions, which represent efficient trade-offs among the objectives. Consequently, identifying the set of Pareto optimal solutions becomes essential for enabling the decision maker to select a compromise solution that satisfies all the objectives in the best possible way [19]. A solution $\overline{x}$ ∈ X is considered efficient (also known as the Pareto optimal or non-dominated) if there is no other point x ∈ X such that satisfies f(x) ≥ f($\overline{x}$) and f(x) ≠ f($\overline{x}$), meaning no other feasible solution performs at least as well as $\overline{x}$ in all of the criteria a problem and strictly better in at least one criterion [20]. In other words, Pareto optimal solutions are defined as those for which it is impossible to improve all objective values simultaneously; any improvement in one criterion would lead to deterioration in at least one other [19]. This field of multicriteria optimisation also encompasses the study of MODP. MODP, which is heavily based on the conventional dynamic programming technique, has been developed as a method for solving problems involving multiple, often conflicting, objectives that exhibit the characteristics of DP. Many of the techniques used are extensions of classical DP [15,20]. At each step of the process of MODP, only non-dominated (efficient) solutions are retained, eliminating inferior alternatives early.

Voyage optimisation has been the focus of extensive research, resulting in a variety of approaches. These can be generally categorised into traditional algorithms, heuristic and metaheuristic techniques, and modern AI-driven solutions. Voyage optimisation involves selecting the optimal route while considering several factors, such as weather conditions (e.g., wind, waves, currents), fuel consumption, departure and arrival windows, as well as safety and regulatory compliance [21]. Most voyage optimisation systems use speed as a control variable, with conventional Isochrone, Dijkstra’s algorithm, and 2D dynamic programming (2DDP) methods focusing mainly on course optimisation, while 3D dynamic programming (3DDP) and evolutionary algorithms, such as genetic algorithms, optimise both course and speed [22,23]. A 3DDP method is presented to optimise ship voyages by minimising fuel consumption, incorporating weather forecasts, ship motion constraints, comfort criteria, and estimated time of arrival, while also estimating added resistance and required engine power. The results reveal a Pareto frontier, highlighting the balance between minimising travel time and reducing fuel consumption across the different optimisation modes [14]. An improved 3DDP algorithm for ship route optimisation is proposed, which plans the optimal ship course and speed based on weather conditions and constraints. The case studies are carried out in a VLCC oil tanker, showing that the minimum fuel oil consumption (FOC) and estimated time of arrival (ETA) optimised route reduce fuel consumption by 1.39% and 1.13%, and save sailing time by 4.77% and 5.80%, respectively, while also supporting various optimisation objectives, such as minimum fuel cost and emissions [24]. Wang et al. [22] proposed a voyage optimisation method combining DP and genetic algorithms to optimise ship routes and engine power settings, aiming to minimise fuel consumption and GHG emissions while maintaining arrival punctuality. Kim and Lee proposed a DP-based method for optimal vessel speed adjustment, which generates more efficient speed plans compared to fixed-speed navigation, achieving up to 20% energy savings under high external force conditions [25].

2.1.3. Resilience and Disruption in Transport Systems

While maritime transport offers greater flexibility in routes and nodes compared to terrestrial transport, it remains vulnerable to various influences, including political (trade regulations and war), geographic (climatic conditions and tidal ranges), and technical factors (port accessibility and costs). Impacts might increase the risks of interrupting the shipping routes and ports. Shipping companies are often directly or indirectly responsible for delays and losses linked to route selection. To minimise the delays resulting from port disruptions, manufacturers and suppliers that depend on maritime transportation for the delivery or receipt of goods may request that shippers consider multiple alternative routing options. Consequently, ports must invest proactively in resilience measures to reduce their vulnerability and to safeguard or enhance their market position. Acknowledging the vital role of critical infrastructure in supporting essential services, the U.S. Presidential Policy Directive 21 explicitly emphasised the need to enhance the performance and resilience of the nation’s critical infrastructure, including transportation networks. Maritime transport is highly susceptible to disruptions caused by both natural and human-induced events, such as war, economic instability, natural disasters, and political transitions [26]. Different kinds of disruptions and threats to critical infrastructure, including the transport system, may require different tools of analysis and response strategies, effective for anticipation, prevention, mitigation, and restoration. Disruptive events in maritime transport can significantly affect scheduled operations, requiring timely and effective recovery strategies to restore service reliability. Speed adjustment, omitting port calls, or swapping port calls are three common recovery modes used to address scheduling disruptions [27].

2.2. Methodology

Several studies have examined the existing methodologies for estimating ship emissions, often supported by case studies, while the selection of an appropriate method typically depends on the availability of relevant data and technical parameters [28]. This section focuses on estimating fuel consumption and the associated emissions of CO₂, other greenhouse gases, and key air pollutants from shipping. The most commonly used methodologies for calculating ship emissions are the top–down (fuel based) approach and the bottom–up (activity based) approach. The emission calculation methodology used in this study is based on the bottom–up approach from the Fourth IMO GHG Study.

The methodology applied in the Fourth IMO GHG Study remains conceptually similar to that applied in the Third IMO GHG Study 2014. Depending on the pollutant, hourly emissions (EM) are the product of either power demand ($\dot{W}$) and energy-based emission factors (${EF}_e$) or fuel consumption (FC) and fuel-based emission factors (${EF}_f$), as described in Equations (1) and (2), for each of the three types of on-board machinery covered: the main engine (ME), auxiliary engine (AE), and auxiliary boiler (BO).

$${EM}_i={EF}_e·{\dot{W}}_i$$

(1)

$${EM}_i={FC}_i·{EF}_f$$

(2)

Additionally, it is possible to categorise emissions according to distinct operational phases of a ship’s activity. Ship operations are commonly categorised into five phases: at berth, at anchor, manoeuvring, slow transit, and at sea [29,30]. The classification of a ship’s operational phase is determined based on factors such as its proximity to land or port, its speed over ground, and its main engine load power. This categorisation allows for more detailed and spatially specific emissions estimates across different phases of operation. The model calculates the operational power demand ${\dot{W}}_i$ for the ME using the formula, as presented in Equation (3), which considers the vessel’s particular speed and draught. The power demand for AE and BO is determined using the default values provided in the Fourth IMO GHG Study, based on the ship’s type, size category, and operational phase.

$${\dot{W}}_i={\displaystyle \frac{{\dot{W}}_{ref}·{\delta }_w·{\left(\frac{t_i}{t_{ref}}\right)}^m·{\left(\frac{v_i}{v_{ref}}\right)}^n}{{\eta }_w·{\eta }_f}}$$

(3)

The hourly fuel consumption ${FC}_i$ for all engine types is given by the Equation (4), the specific fuel consumption ${SFC}_i$ for the main engine is given by Equation (5), and the specific fuel consumption ${SFC}_i$ for auxiliary engines and boilers is given by Equation (6).

$${FC}_i={SFC}_i·{\dot{W}}_i$$

(4)

$${SFC}_{ME,i}={SFC}_{base}·(0.455·{{LF}_i}^2-0.71·{LF}_i+1.28)$$

(5)

$${SFC}_{AE|BO,i}={SFC}_{base}$$

(6)

The parenthetic component of Equation (5) is known as the main engine load correction factor (CF_L). For auxiliary engines and boilers, it is assumed that they are not dependent on their load and, thus, are not corrected by the CF_L. For the model equations, the following parameters are defined and explained below, as they are essential for accurately estimating fuel consumption and emissions based on vessel characteristics and operational conditions:

• ${\dot{W}}_i$ is the calculated engine power for a single timestamp;
• ${\dot{W}}_{ref}$ is the reference power of the main engine, namely the installed power;
• $t_i$ is the instantaneous draught of the ship, as provided by AIS data;
• $t_{ref}$ is the design draught of the ship;
• $v_i$ is the instantaneous ship speed, as provided by AIS data;
• $v_{ref}$ is the design speed of the ship;
• m is the draught ratio exponent, and is equal to 0.66, for all ship types based on the Fourth IMO GHG Study [3];
• n is the speed ratio exponent, and is equal to 3, for all ship types based on the Fourth IMO GHG Study [3];
• ${\delta }_w$, a correction factor, is applied to certain ship types and sizes to adjust the speed–power relationship. It is equal to 0.75 for large containers, 0.7 for cruises, and 1 for the remaining ships [3];
• ${\eta }_w$ represents the weather modifier to the ship’s propulsive efficiency, and is equal to 0.909 for mainly small ships and 0.867 for all other ship types and sizes [3];
• ${\eta }_f$ represents the fouling modifier, and is equal to 0.917 for all ship types and sizes [3];
• ${SFC}_i$ is the specific fuel oil consumption;
• ${SFC}_{base}$ is the reference specific fuel consumption, and varies based on engine/system age, fuel type, engine type, and system, according to the Fourth IMO GHG Study [3];
• ${\mathrm{L}\mathrm{F}}_i$ is the hourly main engine loading given as a proportion, calculated by the calculated engine power and reference power.

The hourly emissions per engine type are summarised and presented as follows, in Equations (7)–(9) below:

$${EM}_{ME,i}={SFC}_{base}·(0.455·{{\mathrm{L}\mathrm{F}}_i}^2-0.71·{\mathrm{L}\mathrm{F}}_i+1.28)·{\dot{W}}_{ME,i}·{EF}_f$$

(7)

$${EM}_{AE,i}={SFC}_{base}·{\dot{W}}_{AE,i}·{EF}_f$$

(8)

$${EM}_{BO,i}={SFC}_{base}·{\dot{W}}_{BO,i}·{EF}_f$$

(9)

For each individual trip, the total emissions are calculated as follows, in Equation (10):

$$E_{trip}={\sum }_p{T}_p·{\sum }_e{EM}_{e,p}$$

(10)

where,

• E_trip is the emission over a complete trip (g);
• EM represents the hourly emissions (g);
• T is the duration of the operational phase p (h). The cruise time, if unknown, can be calculated as follows, in Equation (11):

$$T_{cruising}={\displaystyle \frac{Distancec ruised}{Average cruising speed}}$$

(11)

• e is the engine category;
• p represents the different phases of the trip

For the estimation of emissions, a pollutant emission factor is used, where each gas/pollutant has a different factor. Emission factors increase at varying rates when engine loads are below 20%, due to decreased combustion efficiency. To capture this behaviour, best-fit lines are employed to adjust pollutant emission factors under these conditions [3].

The GHG emissions are calculated as CO₂ equivalent, using the global warming potential over a 100-year time-horizon (GWP₁₀₀) to convert the emissions of other gases than CO₂, as given in the fifth IPCC Assessment Report, for CO₂, CH₄, and N₂O, through Equation (12) [31]:

$$\begin{gathered} g_{CO2eq\left(100y\right)}={GWP}_{CO2\left(100y\right)}·g_{CO2}+{GWP}_{CH4\left(100y\right)}·g_{CH4}+{GWP}_{N2O\left(100y\right)}·g_{N2O} \\ {g}_{CO2eq\left(100y\right)}=1·g_{CO2}+28·g_{CH4}+265·g_{N2O} \end{gathered}$$

(12)

2.3. Case Study Description

This section presents the case under investigation, which refers to the description of two distinct transportation networks, the theoretical model, and the realistic case. The goal is to identify the optimal route through the network, minimising key performance criteria, such as emissions and travel time. The problem is formulated and analysed using the methodologies presented in the previous section, which include emission estimation techniques and optimisation approaches based on dynamic programming. The purpose of this section is to establish the framework for the simulations and optimisations that follow, by presenting the structure of the network, the decision-making process, and the necessary input parameters, including vessel characteristics, such as size and type of ship, propulsion power, fuel type, and speed, which are essential for estimating accurately emissions and travel time in the evaluation of the optimal routing solution. For each network, a range of scenarios is examined regarding navigation speed and port availability (i.e., whether a port is open/accessible or not), incorporating both deterministic and stochastic approaches using probability distributions. The stochastic analysis is conducted with the aid of the @RISK 8.8 software by Palisade, making the modelling and simulation of uncertainty within the network feasible [32].

The transportation network shown in Figure 1, which represents the structure used for both the theoretical and the realistic case, is analysed to determine the optimal route for a vessel, starting from Point A and reaching to Point Z, with the goal of minimising both emissions and travel time. Most specifically, points A, B, C, …, Z represent ports, and the network is organised into four stages, with each stage requiring a decision on the next port to be visited. Only one port can be selected at each stage from a set of available alternatives. The variable n denotes the position of the decision to be made within the overall sequence of decisions. According to dynamic programming theory, in this network, there are four state variables Xn, that provide the necessary information for decision-making by indicating the vessel’s position within the network at each stage. For example, the state X₂ = 2 corresponds to the vessel being located at Port B during stage 2.

The vessel considered in the calculations of the theoretical model is a Suezmax oil tanker with a deadweight tonnage (DWT) between 120,000 and 199,999. The technical specifications of the Suezmax tanker that are essential for the quantification of emissions are summarised in

Table 1. Technical specifications of the Suezmax tanker used in the theoretical model.

Parameter	Specification
Ship Type	Suezmax Tanker
MCR (kW)	16,000
Vmax (kn)	15.85
Fuel Type	MDO
Main Engine Type	SSD

, while those of the Small tanker used in the realistic case are presented in

Table 2. Technical specifications and dimensions of the small tanker used in the realistic network.

Parameter	Specification
Ship Type	Oil Products Tanker
DWT	2454
Length overall (m)	89.95
Length between perpendiculars (m)	82.00
Breadth (m)	14.00
Depth (m)	6.50
Draught (m)	4.00
MCR (kW)	2040
Vmax (kn)	12.00
Fuel Type	MDO
Main Engine Type	MSD

.

In the realistic case, points A, B, C, …, Z represent actual ports and islands in the Cyclades region, as shown in

Table 3. Actual ports and islands corresponding to network points A–Z (realistic).

Cyclades
Node	Island
A	Elefsis
B	Serifos
C	Milos
K	Sifnos
E	Paros
D	Folegandros
Z	Amorgos (Katapola)

and Figure 2. In Figure 2, the ports and islands of interest are highlighted for emphasis. The distances between ports, for the theoretical and the realistic network, are presented in

Table 4. Overview of node-to-node distances in the networks.

Segment	S_theoretical [nm]	S_realistic [nm]
AB	65	78
BD	50	39
DZ	40	46
BE	30	32.6
EZ	50	52
AC	80	93
CD	40	36
CE	40	48.4
AK	70	87
KE	40	30.8
KD	45	30

, defining the geometry of each network.

In the theoretical case examined in this study, three distinct operational phases are considered for the vessel: navigation, manoeuvring, and at berth. With respect to navigation speeds, different assumptions are incorporated into the analysis to examine various operational scenarios. For example, in one scenario, a constant speed is applied for each route between two ports, with values corresponding to those shown in

Table 5. Overview of node-to-node speeds, expressing either constant or normal distributions (theoretical).

	Constant Speeds	Speed Distributions
Segment	V [kn]	V Mean [kn]	Standard Deviation
AB	15	15	0.1
BD	14	14	0.1
DZ	13.5	13.5	0.1
BE	15	15	0.1
EZ	14.5	14.5	0.1
AC	14	14	0.1
CD	13.5	13.5	0.1
CE	14	14	0.1
AK	14.5	14.5	0.1
KE	14	14	0.1
KD	14	14	0.1

. In an alternative scenario, navigation speed is modelled as a normally distributed random variable, where the mean values align with the speeds presented in the same table, and with a standard deviation equal to 0.1. For the manoeuvring phase, it is assumed that the vessel travels at a constant speed of 5 knots over a distance of 2 nautical miles. Additionally, it is assumed that the vessel remains at berth for a duration of 2 h at each port. Furthermore, the total emissions calculated for the entire route correspond to the emissions from the moment of departure from Port A until arrival at Port Z.

In the realistic case, the emission estimation method developed in this study is based on vessel activity, for which the data were retrieved from the Automatic Identification System (AIS). Originally designed to support radar augmentation and vessel traffic services (VTS), the AIS data have also proven valuable for monitoring maritime traffic. AIS data have been increasingly utilised in research to analyse various maritime issues, including ship surveillance, tracking, maritime security, collision risk assessment, shipping noise levels, and vessel emissions. The MarineTraffic website collects and processes AIS signals. Data, such as vessel position, speed, navigational status, as well as additional information including vessel characteristics, are available on the platform [33]. Leveraging the extensive functionalities of the Sea-web platform, which serves as a maritime database integrating the data for ship movements, port calls, and operational characteristics, the specific vessel of the case was identified approaching the ports of interest [34]. By examining the travel dates associated with these port calls, it becomes feasible to extract the corresponding AIS data for each route segment via MarineTraffic for further analysis. Utilising the AIS data obtained from MarineTraffic, instantaneous speeds and draughts were recorded for each individual route. It was observed that the draught values remain constant over time and do not vary; therefore, the analysis is focused on the instantaneous speeds. Particularly, for each route segment (i.e., Elefsis–Serifos), the AIS data were extracted from a voyage actually performed by the small tanker in 2025, as recorded on the MarineTraffic platform. This specific voyage was selected as it provided a complete set of AIS records without significant gaps, thereby ensuring the reliability of the subsequent analysis. Specifically, AIS data were collected for a two-month reference period (March–April 2025), encompassing all available voyages of the selected vessel across the studied network routes (i.e., Elefsis–Serifos, Elefsis–Sifnos, Paros–Amorgos). Given the limited frequency of these routes and the difficulty in obtaining sufficient AIS records for all possible route combinations, the analysis relied on the single available voyage within the reference period for each route segment, ensuring that the dataset adequately represents the operational conditions of each route segment. In order to obtain a consistent dataset for deriving speed distributions, instantaneous speed values are recorded at 5-min intervals for the navigation phase, while for the manoeuvring phase, they are recorded every 2 min. This approach ensures adequate representation of the frequency with which each speed value occurs. In addition, the classification of vessel operations into phases (i.e., navigation, manoeuvring, or at berth) was performed following the definitions provided by the Fourth IMO GHG Study, which determines a ship’s phase based on its speed over ground, proximity to land or port, and main engine load power. Accordingly, the AIS-derived speed data and the vessel’s proximity to port areas were used to distinguish between the navigation, manoeuvring, and at-berth phases in the present study. Based on the available AIS data, speed distributions are derived, allowing the speed of each segment to be approximated by a fitted probability distribution, with the assistance of @Risk. The selection of the appropriate probability distribution for speeds was determined using the Anderson–Darling (AD) statistic. The AD statistic is a commonly used goodness-of-fit measure that evaluates how well a theoretical distribution matches a given data set. Lower values of the statistic indicate a better fit between the model and the observed data [35]. This test accounts for deviations across the entire data range, as it places greater emphasis on the tails and does not rely on binning, while also being suitable for cases with limited sample sizes [32,36]. With respect to navigational speed, the ExtValueMin distribution was ultimately chosen as the best fit and, for the speeds recorded during the manoeuvring phase, a Weibull distribution was used. The RiskExtValueMin (alpha, beta) distribution, also known as the minimum extreme value distribution, is characterised by a location parameter (alpha) and a shape parameter (beta), while RiskWeibull (α,β) is defined by a shape parameter (α) and a scale parameter (β). For the manoeuvring phase, it was observed that the vessel typically manoeuvres over a distance of approximately 0.25 nautical miles upon departure from the origin port and an additional 0.25 nautical miles upon arrival at the destination port, resulting in a total manoeuvring distance of 0.5 nautical miles per route segment. As an example, Figure 3 illustrates the fitted probability distribution for the navigation speed and manoeuvring speed data on the Elefsis–Serifos route. The histogram of the data is shown in blue, overlaid with the selected theoretical distribution curve in red. Additionally, with respect to the duration at berth, the average of the various recorded berthing times at each port was considered, as presented in

Table 6. Time at berth per island for the realistic case.

Island	Mean Time at Berth (h)
Serifos	7.45
Sifnos	7.51
Milos	7.39
Paros	6.57
Folegandros	6.31

, due to significant variability observed across different trips for the same port.

The scenarios examined in this study for the theoretical model involve different approaches regarding the vessel speed, depending on whether the value is constant for each route segment or it is defined by a probability distribution. Additionally, both for the theoretical and the realistic models, to explore more realistic conditions and to introduce uncertainty into the network, scenarios in which access to Port B is not guaranteed are examined, reflecting possible unavailability due to factors such as adverse weather conditions, terrorist attacks, or other disruptions. Indicatively, in certain cases, it is assumed that Port B is accessible with a probability of 80%. If Port B is closed, then there is a 10% probability that the vessel will be redirected to Port C and another 10% that it will be redirected to Port K. The information regarding Port B’s inaccessibility becomes available during the transition from Port A to Port B. The point along the route at which this information is revealed, assuming that Port B is indeed closed, is modelled using a uniform distribution. A similar approach is also applied in other scenarios, where the probability of Port B being accessible is modelled using a normal distribution. As a result, choosing to move from A toward B does not guarantee that the vessel will actually reach Port B. This introduces uncertainty into the decision-making process, which necessitates the use of the stochastic dynamic programming methodology, described in Section 2.1. Figure 4 illustrates the structure of the transportation network under conditions of uncertainty, where K1 and C1 correspond to K and C, respectively. The decision variable y_n(Χ_n) specifies the set of decisions or actions to be taken at stage n, given that the system is in state Χ_n. For example, y_n = K corresponds, at stage n, to the decision to proceed to Port K. The scenarios considered for both the theoretical and realistic cases are summarised in

Table 7. Scenario definitions based on navigation speed, port availability, and location of information regarding port closure (theoretical).

Scenario	Navigation Speed (per Segment)	Port B Availability	Location of Information Received (Port B)
1	Constant	All ports available	-
2	Normal dist.	All ports available	-
3	Constant	80% open	Fixed
4	Normal dist.	80% open	Uniform dist.
5	Constant	80% open (normal dist.)	Uniform dist.
6	Normal dist.	80% open (normal dist.)	Uniform dist.
7	Normal dist.	Not available	Uniform dist.
8 (fleet)	Normal dist.	80% open	Uniform dist.

and

Table 8. Scenario definitions based on port availability and location of information regarding port closure (realistic).

Scenario	Port B Availability	Location of Information Received (Port B)
1	All ports available	-
2	80% open	Uniform dist.
3	80% open (normal dist.)	Uniform dist.
4	Not available	Uniform dist.

. These scenarios vary in terms of navigation speed, port availability, and the location of information received regarding Port B closure. In this study, resilience is defined as the additional burden, expressed as a percentage increase in key performance metrics, where the two primary parameters under investigation are emissions and travel time. A higher burden percentage indicates a lower resilience of the network. Therefore, it is desirable for the burden, namely the additional emissions and transit time caused by the disruption, to be as minimal as possible.

For Scenario 8 of the theoretical case, the analysis is conducted for a fleet rather than a single vessel. The fleet consists of eight sister ships with identical characteristics. The only varying factor between vessels in this scenario is the speed across the different route segments. The speeds are modelled using a normal distribution for each intermediate segment, with mean values equal to those presented in

Table 9. Mean speed values (kn) for each route segment and vessel, with standard deviation 0.1.

	V [kn]
Segment	Ship 1	Ship 2	Ship 3	Ship 4	Ship 5	Ship 6	Ship 7	Ship 8
AB	15	15	15	15	15	15	15	15
BD	14	14.5	14.5	14.5	14.5	14	14	14
DZ	13.5	14.5	14	14.5	14	14	14	14.5
BE	15	15	15	15	15	15	15	15
EZ	14.5	14.5	14.5	14.5	14.5	14.5	15	15
AC	14	14	14.5	14	14	13.5	13.5	13.5
CD	13.5	14	14	14	14	14	14	14
CE	14	14	14	14	13.5	14.5	14.5	14.5
AK	14.5	14.5	14.5	14.5	14.5	14.5	14	14.5
KE	14	14	14.5	14.5	14	14.5	14.5	14.5
KD	14	14	14.5	14.5	14.5	14.5	14.5	14.5
AK1	14	14	14	14.5	14.5	14	14	14
AC1	14	14	14	14.5	14.5	14	14	14

and standard deviation equal to 0.1. This case is studied in order to assess the extent to which variations in speed can influence emissions, travel time and, consequently, the optimal route selection. In this case, access to Port B is not guaranteed and may be closed (B is accessible with a probability of 0.8). Speeds that deviate from those assigned to Ship 1 are highlighted in bold for comparison purposes.

3. Results

Following the presentation of the case study and the implementation of the required calculations in Excel using the @Risk add-in software, this section focuses on the respective outputs and their analysis. Specifically, the emissions and travel times associated with each route are examined, along with the Pareto front and the corresponding optimal paths identified in each scenario. As already outlined, eight scenarios are developed for the theoretical model and four for the realistic case. However, instead of assessing all of them, specific scenarios are selected for evaluation and comparison. In particular, Scenarios 1 and 3 of the realistic case are examined to compare the simplest and most complex configurations. Additionally, Scenario 4 of the realistic case is analysed to investigate the network’s resilience. Finally, Scenario 8 of the theoretical model is presented to examine the impact of operating a fleet under the defined conditions.

3.1. Results of the Realistic Case

3.1.1. Scenario 1

Regarding the realistic network, in the scenario where all ports are assumed to be fully accessible, four routes are identified as Pareto optimal based on dynamic programming. These routes are ABEZ, ABDZ, AKEZ, and AKDZ. This implies that, starting from Elefsis, if the vessel proceeds either towards Serifos or Sifnos and subsequently towards Paros or Folegandros, the resulting route will be considered optimal in terms of the emissions–time trade-off. Consequently, the direction towards Milos is not selected, as it results in higher emissions and longer travel times. This is mainly due to the fact that the four routes, excluding Milos, have relatively small differences in total distance, leading to comparable emissions and travel times. On the other hand, the routes passing through Milos, namely Elefsis–Milos–Folegandros–Amorgos and Elefsis–Milos–Paros–Amorgos, are considerably longer, which results in a notable increase in both time and emissions. The presence of four Pareto optimal routes in this scenario can be attributed to the minimal differences in their total distances. Additionally, since the mean speeds across the individual route segments do not vary significantly, the combination of these input parameters results in non-dominated solutions, as no single route outperforms the others in both objectives simultaneously. The values reported in

Table 10. Summary of travel time and emissions for Scenario 1 (realistic), with optimal solutions shown in bold.

		CO2	CH4	N2O	CO2eq	SOx	NOx	PM10	PM2.5
Route	T [h]	E [Kg]	E [Kg]	E [Kg]	E [Kg]	E [Kg]	E [Kg]	E [Kg]	E [Kg]
ABEZ	29.20	27,086.01	0.36	1.37	27,458.00	1156.22	286.28	99.70	91.72
ACDZ	30.45	27,954.68	0.37	1.40	28,335.52	1176.44	295.70	101.50	93.38
ACEZ	32.67	29,591.02	0.40	1.49	29,996.75	1246.29	318.86	107.62	99.01
ABDZ	28.91	27,120.18	0.36	1.37	27,492.90	1157.68	289.84	99.83	91.85
AKEZ	30.59	27,032.47	0.36	1.37	27,404.44	1153.93	281.35	99.49	91.54
AKDZ	29.16	27,105.58	0.36	1.37	27,478.21	1157.05	288.34	99.75	91.77

represent the mean values of time and emissions for each route, derived from the fitted probability distributions of speeds. Furthermore, since the trends observed in the CO₂ and CO₂ equivalent emissions diagrams remain practically identical, given that CH₄ and N₂O values are significantly lower than those of CO₂, only the diagrams based on CO₂ equivalent emissions are presented. Figure 5 depicts the CO₂ equivalent emissions and the travel time for each route, with the four non-dominated solutions marked by diamonds with a black border.

3.1.2. Scenario 3

In this scenario, uncertainty is incorporated through the probability of Port B (Serifos) being accessible, which is now modelled using a normal distribution. In this case, three routes, AKEZ, AKDZ and ABDZ, AK1DZ, AC1DZ, are Pareto optimal. Figure 6 depicts the CO₂ equivalent emissions and the travel time for each route, with the three non-dominated solutions marked by diamonds with a black border. Introducing distributions with mean values equal to the constant values used in other scenarios, i.e., probabilities, does not influence the outcome when comparing the final mean values.

Table 11. Summary of travel time and emissions for Scenario 3 (realistic), with optimal solutions shown in bold.

		CO2	CH4	N2O	CO2eq	SOx	NOx	PM10	PM2.5
Route	T [h]	E [Kg]	E [Kg]	E [Kg]	E [Kg]	E [Kg]	E [Kg]	E [Kg]	E [Kg]
ΑΒΕΖ, AK1DZ, AC1DΖ	29.30	27,365.56	0.37	1.38	27,741.58	1168.15	291.42	100.73	92.68
ACDZ	30.45	27,954.68	0.37	1.40	28,335.52	1176.44	295.70	101.50	93.38
ΑCΕΖ	32.67	29,591.02	0.40	1.49	29,996.75	1246.29	318.86	107.62	99.01
ΑΒDΖ, AK1DZ, AC1DΖ	29.07	27,392.94	0.37	1.38	27,769.55	1169.32	294.28	100.84	92.77
ΑΚΕΖ	30.59	27,032.47	0.36	1.37	27,404.44	1153.93	281.35	99.49	91.54
ΑΚDΖ	29.16	27,105.58	0.36	1.37	27,478.21	1157.05	288.34	99.75	91.77
ΑΒΕΖ, AK1DZ, AC1ΕΖ	29.52	27,529.24	0.37	1.39	27,907.75	1175.14	293.74	101.35	93.24
ΑΒΕΖ, AK1ΕZ, AC1ΕΖ	29.67	27,521.93	0.37	1.39	27,900.38	1174.83	293.04	101.32	93.22
ΑΒΕΖ, AK1ΕZ, AC1DΖ	29.44	27,358.26	0.37	1.38	27,734.21	1167.84	290.72	100.71	92.65
ΑΒDΖ, AK1DZ, AC1ΕΖ	29.29	27,556.61	0.37	1.39	27,935.71	1176.31	296.59	101.45	93.34
ΑΒDΖ, AK1ΕZ, AC1ΕΖ	29.43	27,549.31	0.37	1.39	27,928.34	1175.99	295.90	101.43	93.31
ΑΒDΖ, AK1ΕZ, AC1DΖ	29.21	27,385.63	0.37	1.38	27,762.18	1169.01	293.58	100.82	92.75

represents the mean values of time and emissions for each route, derived from the probability distributions of speed and port accessibility.

When comparing Scenario 1, where all ports are available, and Scenario 3, where uncertainty is introduced regarding the availability of a specific port, there is a shift in the set of optimal routes. While Scenario 1 yields four optimal paths (ABEZ, ABDZ, AKEZ, and AKDZ), Scenario 3 identifies three optimal routes (AKEZ, AKDZ, and ABDZ, AK1DZ, AC1DZ). For the route ABDZ, AK1DZ, AC1DZ, which is influenced by whether Port B closes or not, there is a 1% increase in CO₂ equivalent emissions (from 27,493 kg to 27,770 kg) and a 0.6% increase in travel time (from 28.91 h to 29.07 h) when uncertainty is introduced into the network. Thus, with the introduction of uncertainty into dynamic programming, the routes that include Port B exhibit increased emissions and travel time compared to their counterparts in scenarios, where Port B was assumed to be certainly available, affecting the set of non-dominated solutions.

Additionally, the output distributions of CO₂ equivalent emissions and travel time are presented for one of these three optimal routes in Figure 7. These distributions result from the combination of input distributions for navigation speeds (ExtValueMin), manoeuvring speeds (Weibull), the position (Uniform) at which the closure of Port B is identified (if it is indeed closed), and the probability of Port B being accessible (Normal). As a result, combining the speed distributions of each segment produces final distributions with different characteristics and shapes, although the form of the input speed distribution types remains consistent.

A comparison is made regarding the emissions by engine type and operating phase, where the results are presented as percentages through bar charts. This comparison is conducted for one of the optimal routes, specifically the AKEZ route. As shown in Figure 8, the majority of emissions originate from the main engine. Specifically, depending on the type of emission, the main engine contributes between 52% and 58% of the total emissions, with the exception of NOx emissions, which are exclusively produced by the main engine. Similarly, regarding the comparison between operational phases, as shown in Figure 8, most emissions are produced during navigation. The relatively large difference between at-berth and manoeuvring emissions is due to the fact that ships remain at berth for several hours, whereas manoeuvring lasts only a few minutes, as shown in Figure 9.

3.1.3. Scenario 4

This scenario is implemented to assess network resilience, specifically by estimating the increase in emissions and travel time in case Port B is definitively closed. In this case, speeds continue to be modelled using the fitted probability distributions; however, it is now assumed that, upon confirmation of Serifos Port inaccessibility, the vessel has a 50% probability of being redirected to Port C (Milos) and a 50% probability of being redirected to Port K (Sifnos). In Scenario 3, the optimal routes identified were AKEZ, AKDZ and ABDZ, AK1DZ, and AC1DZ. Therefore, only the routes ABDZ, AK1DZ, and AC1DZ are affected by the unavailability of Port B. The mean, maximum, and minimum values of the output distributions for emissions and travel time are presented for Scenario 4 in

Table 12. CO₂, CO₂eq, and travel time and for route ΑΒDΖ, AK1DZ, AC1DΖ per the scenarios examined.

		Mean Values			Max Values			Min Values
Scenario	Route	CO2 [kg]	CO2eq [kg]	T [h]	CO2 [kg]	CO2eq [kg]	T [h]	CO2 [kg]	CO2eq [kg]	T [h]
Scen. 4	ΑΒDΖ, AK1DZ, AC1DΖ	28,484	28,876	29.71	29,256	29,658	31.29	26,506	26,869	29.08
Scen. 3	ΑΒDΖ, AK1DZ, AC1DΖ	27,393	27,770	29.07	28,227	28,614	30.82	25,401	25,751	28.50
Scen. 1	ΑΒDΖ	27,120	27,493	28.91	28,055	28,439	30.46	24,965	25,308	28.36
		CO2 [%]	CO2eq [%]	T [%]	CO2 [%]	CO2eq [%]	T [%]	CO2 [%]	CO2eq [%]	T [%]
Scen. 4 vs. 3	Increase %	3.98	3.99	2.22	3.64	3.65	1.51	4.35	4.34	2.01
Scen. 4 vs. 1		5.03	5.03	2.79	4.28	4.29	2.72	6.17	6.17	2.52

, along with those from Scenarios 3 and 1, in order to enable a comparison. Specifically, the results for the affected route are compared under three distinct conditions: when Serifos is entirely inaccessible (Scenario 4); when Serifos is accessible with a probability modelled by a normal distribution with a mean of 80% (Scenario 3); and when Serifos is fully accessible (Scenario 1). In the latter case, the comparison is made with route ABDZ, which was included among the Pareto optimal solutions. In all scenarios, navigation and manoeuvring speed are modelled using fitted distributions. Based on the mean values, when Serifos is fully inaccessible, emissions (both CO₂ and CO₂ eq.) increase by 5%, and travel time increases by 2.8%, compared to the case where Serifos Port’s potential closure is not considered. On the other hand, when the probability of Serifos Port being inaccessible is incorporated into the model, the eventual closure of the port results in a 4% increase in emissions and a 2.2% increase in travel time. A comparison of the maximum values indicates that, while the closure of Serifos Port results in an increase of 4.3% in emissions and 2.7% in travel time compared to Serifos Port being certainly open, incorporating uncertainty reduces the impact to 3.6% and 1.5%, respectively. For the minimum values, an increase of 6.2% in emissions and 2.5% in travel time decreases to 4.3% and 2%, respectively, when uncertainty is taken into account.

However, it is worth noting that, due to the difference in the order of magnitude of emissions between the theoretical and realistic networks, as well as the specific geometry shaped by the geographic location of the islands, the percentage increases in both emissions and travel time, when Port B is closed, are smaller. For these reasons, the reduction in the additional burden of emissions and travel time, achieved when uncertainty is incorporated into the analysis, is smaller in the realistic network compared to the theoretical one. Nevertheless, focusing on the mean values, it can be observed that, by incorporating uncertainty into the decision-making process, a strategy can be chosen that minimises the increase in emissions and travel time, in case Port B ultimately shuts down, as it takes into account multiple possible scenarios and helps avoid significant deviations from the expected outcomes.

3.2. Results of the Theoretical Model

Scenario 8

In this scenario, a fleet of eight identical vessels is examined, where certain mean speed values vary between the ships. The aim is to investigate how speed influences emissions and travel time, and consequently, the selection of the optimal route. Ship 1 corresponds to Scenario 4 of the theoretical case and has three non-dominated (Pareto optimal) solutions: ACDZ, AKDZ, and ABEZ, AK1DZ, AC1DZ. By contrast, for Ships 2, 3, 4, and 5, only one optimal solution is observed in each case. For instance, in the case of Ship 2, where there is an increase in the mean navigation speed by 1 kn in segment DZ, it is observed that the CO₂ equivalent emissions in AKDZ increased by 2.3% (from 123,733.39 to 126,612.23 kg), while the travel time did not decrease significantly enough for route AKDZ in order to remain non-dominated. A similar outcome is observed for route ACDZ, where, in addition to the 1 kn increase in mean speed in segment DZ, the speed in segment CD was also increased by 0.5 kn. As a result, the route ABEZ, AK1DZ, AC1DZ emerges as the only non-dominated solution, achieving both lower emissions and a shorter travel time. This analysis indicates that different combinations of speeds can affect both emissions and travel time, hence influencing the non-dominated solutions within the trade-off. Figure 10 presents the non-dominated solutions for each case, which are connected by dashed lines when their number exceeds two, thereby highlighting the number of optimal routing options for each vessel. For the given cases, the route ABEZ, AK1DZ, and AC1DZ consistently belongs to the set of optimal solutions, whereas ACDZ is optimal in four cases and AKDZ in only two.

Furthermore, considering these vessels as a fleet and assuming one voyage per ship, if the maximum-emission route is chosen for every ship, the total CO₂ equivalent emissions across the fleet increase by 7.82% compared to the scenario where the minimum-emission route is chosen for each ship. Additionally, comparing the average emissions across all possible routes for all ships, with the respective ones from the minimum-emission route for each ship, a 3.09% increase in total fleet emissions is observed.

4. Discussion

Within the context of a network where route distances vary, it becomes evident that total distance plays a critical role in determining emissions, travel duration, as well as the optimal solution. Therefore, the optimal route within such a network is influenced by multiple parameters, including the distances and the combination of speeds of each segment, which together define the total emissions and travel time for each possible path. Thus, by applying dynamic programming and multi-objective optimisation theory in combination with the concept of Pareto front optimality, it becomes feasible to identify and to visualise the set of optimal routes across different scenarios. Parameters, such as speed and port availability, are expressed as either a constant value or as a probability distribution, determining the problem as a static or dynamic one, respectively, with the latter to reflect better real-world conditions. Monte Carlo simulations can be performed to derive the output distributions of emissions and travel time, providing statistical information, such as mean, minimum, and maximum values, thus offering a more comprehensive view compared to the static/deterministic problem. However, the results indicate that introducing probability distributions with mean values equal to the constant values used in other scenarios (i.e., speeds or probabilities) does not significantly impact the final mean outcomes, as the absolute differences observed are minimal.

Dynamic programming under uncertainty provides a decision policy (strategy) rather than a fixed solution, since it is based on expected values. Actual outcomes may vary due to randomness, so the approach offers guidance for decision-making that accounts for variability rather than a single deterministic result. The realistic network adopts the same structure as the theoretical model, starting from Point A and reaching to Point Z, aiming at minimising both emissions and travel time. It is organised into four stages, each requiring a decision regarding the next port to be visited. However, in the realistic case study focusing on the Cyclades area, it was not feasible to consider a Suezmax tanker, as in the theoretical model, due to the region’s port operational constraints. Instead, the analysis was conducted for a smaller tanker identified through the MarineTraffic and Sea-web databases. The differences in vessel size and route-segment distances between the two networks substantially influence the results, leading to significantly higher emission values in the theoretical network due to the much greater energy demands of the Suezmax tanker. With the introduction of uncertainty into dynamic programming, namely with the probability that the ship may need to reroute due to the unavailability of Port B, the routes that include Port B exhibit increased emissions and travel time compared to their counterparts in scenarios where Port B was assumed to be certainly available. As a result, these parameters can be affected to such an extent that may also alter the set of non-dominated (Pareto optimal) solutions. Indicatively, in the theoretical network, while Scenario 2 yields a single optimal route, Scenario 6, which introduces uncertainty, identifies three optimal routes. For the optimal route ABEZ, AK1DZ, AC1DZ, which is affected by the potential closure of Port B, there is a 2.3% increase in CO₂ equivalent emissions and a 4.1% increase in travel time when uncertainty regarding the port’s availability is introduced into the network. Similarly, for the realistic network, while Scenario 1 yields four optimal paths, Scenario 3, which introduces uncertainty, identifies three optimal routes. For the route ABDZ, AK1DZ, AC1DZ, which is influenced by whether Port B closes or not, there is a 1% increase in CO₂ equivalent emissions and a 0.6% increase in travel time when uncertainty is introduced into the network. The percentage increases in both emissions and travel time are smaller in the realistic network compared to the theoretical one, due to differences in the order of magnitude of emissions and time, as well as the specific geography shaped by the location of the islands. In the theoretical network, emissions are significantly higher due to the substantially greater energy demands of a Suezmax tanker, compared to the small tanker used in the realistic network. In addition, it can be observed that uncertainty can either increase or decrease the number of Pareto optimal solutions, depending on the magnitude of its impact on the parameters (emissions, travel time) and the geographical/technical structure of the network. In the theoretical network, uncertainty increases the number of optimal solutions (from one to three) because the variations in emissions and travel time are significant, making different routes competitive. However, the variations in the realistic network are smaller due to the smaller vessel and the more geometrically constrained system (shorter distances between ports when Port B is closed), so uncertainty has a limited effect on the number of optimal solutions (from four to three).

Regarding the network’s resilience, it is specifically defined as the additional burden, expressed as a percentage increase in emissions and travel time, incurred in case Port B is definitively closed. In the case of the theoretical network, a comparison of the mean values of optimal route ABEZ, AK1DZ, and AC1DZ indicates that, while the closure of Port B results in an increase of 11.3% in emissions and 20.5% in travel time compared to Port B being certainly open, incorporating uncertainty reduces the impact to 8.9% and 15.7%, respectively. In the realistic network, an analysis of the mean values for optimal route ABDZ, AK1DZ, AC1DZ reveals that the closure of Port B results in a 5% increase in emissions and a 2.8% increase in travel time, compared to the case where Port B is certainly open. However, when uncertainty is incorporated, these impacts are reduced to 4% and 2.2%, respectively. Therefore, incorporating uncertainty into the modelling process enhances the resilience of the transport network, allowing for the development of strategies that remain effective despite unexpected disruptions, such as port closures. By considering a range of possible scenarios, the model identifies and selects the routes that minimise the impact of adverse events on emissions and travel time.

Regarding the real network, a comparison is made regarding the emissions by engine type and operating phase for a specific route and all emission types. The majority of CO₂ equivalent emissions are produced by the main engine and during the navigation phase. Approximately 57.7% of the total emissions originate from the main engine, with 27.8% and 14.6% coming from the boilers and auxiliary engines, respectively. Regarding operating phases, 64.9% of the CO₂ equivalent emissions are attributed to navigation, 34.4% to berthing, and 0.7% to manoeuvring. The significant difference between at-berth and manoeuvring emissions is due to the fact that ships remain at berth for several hours, whereas manoeuvring lasts only a few minutes.

Additionally, within the framework of the theoretical network, a fleet of eight identical vessels is examined, where certain mean speed values vary between the ships. The analysis indicates that different combinations of speeds across route segments, with relative changes of no more than 1 knot, can affect both the emissions and the travel time, thereby influencing the set of non-dominated solutions. Furthermore, choosing the maximum- or average-emission route per ship increases the total fleet CO₂ equivalent emissions by 7.8% and 3.1%, respectively, compared to the minimum-emission route, highlighting the value of optimal routing.

The validation of the proposed model and its corresponding results was conducted by comparing the estimated emissions per ship type with the values reported in the literature. Zhao et al. [37] quantified approximately 108 Mt of CO₂ emissions resulting from the operations of 2491 crude oil tankers worldwide in 2020. Specifically, considering 550 Suezmax tankers emitting 25 Mt of CO₂ annually, it can be estimated that a single Suezmax tanker emits approximately 124.5 t of CO₂ per day. In the present study, the CO₂ emissions of a Suezmax tanker within the theoretical model range from 122 to 125 t per voyage of 16 to 17.2 h, indicating that the order of magnitude of the calculated emissions aligns closely with the estimates of Zhao et al. [37]. Similarly, the CO₂ emissions of the small tanker in the realistic network are consistent with the literature values, with 27.4 t of CO₂ emitted over a 29-h voyage, as Zhao et al. [37] reported a daily emission of 34.5 t CO₂ for small tankers, based on 160 vessels producing 20 Mt CO₂ annually. Minor discrepancies arise because the present study considers the specific characteristics of individual vessels, whereas Zhao et al. [37] provide average emissions per ship type. Another point of comparison is the distribution of emissions across operational phases. In the realistic network of the present study, 64.9% of the total CO₂ emissions occur during the navigation phase and 34.5% at berth. Zhao et al. [37] reported similar shares of 67% and 33%, respectively, highlighting the close agreement between the two approaches. Finally, the emissions from an oil tanker can be contrasted with those of small container ships, as reported by Fadillah and Ayub, which are less than 300 t CO₂/day [38]. Given that container ships generally have higher annual emissions and emissions per distance than oil tankers, based on the European Commission, it is reasonable to conclude that small oil tankers emit significantly less than 300 t CO₂/day, consistent with the findings of the present study [39].

Therefore, the analysis of this study highlights the importance of selecting optimal routes, demonstrating that route optimisation contributes significantly to fuel savings and consequently to emission reductions, as well as minimising travel time. Although alternative fuel technologies, like methanol, ammonia, and hydrogen, offer the potential for substantial emission reductions and eventual elimination, their widespread implementation remains a long-term prospect. Thus, route optimisation emerges as an effective interim measure for reducing emissions, offering immediate environmental benefits while the industry gradually transitions to cleaner alternative fuels. From a practical perspective, shipping companies are encouraged to integrate advanced route optimisation algorithms into their voyage planning systems, coupled with real-time environmental data, to enhance operational efficiency and to reduce fuel consumption. Moreover, incorporating uncertainty modelling and accounting for factors such as weather variability, traffic conditions, and operational constraints into route optimisation can further improve the robustness and reliability of decision-making in real-world maritime operations. Policymakers, in turn, can facilitate this transition by incentivising the adoption of digital optimisation tools and incorporating them into regulatory frameworks to accelerate progress toward IMO decarbonisation goals.

While the present analysis focused on emissions and travel time on a per voyage basis, it is worth noting that the same framework could be adapted to longer time horizons, for example, covering a full operational year for a single vessel or even an entire fleet. Such an extension would require access to and processing of large volumes of operational data. In parallel, the methodology could be applied to different vessel types and sizes. By adjusting the parameters and factors that depend on ship type and size, according to the data provided in the Fourth IMO GHG Study, the model can be adapted to simulate different ship categories, such as container ships, bulk carriers, or Ro-Ro vessels. Thus, a comparative analysis can take place, enabling the assessment of both absolute and relative emissions (i.e., by engine type and operational phase), offering insight into how ship type and characteristics influence emissions. In addition to the methodology provided by the Fourth IMO GHG Study, alternative approaches for estimating emissions could be explored in future work. These may include models based on real operational data, emission factors from other international frameworks, or even machine learning techniques trained on historical voyage and fuel consumption data. In this study, and specifically for the realistic network, AIS data were used, with speed values recorded every 5 min during navigation and every 2 min during manoeuvring. However, if a database with more frequent speed recordings was available, the speed distributions could be modelled with greater accuracy, potentially enhancing the overall reliability of the emission estimates. Future research could extend the model to consider multiple simultaneous or sequential disruptions, as the current analysis is focused on the closure of a single port, providing a more comprehensive assessment of network robustness under varied failure scenarios. In cases where multiple ports may be closed, the same dynamic programming methodology can be applied, although the system will become more complex. Specifically, multiple random variables need to be incorporated to account for the uncertainty associated with each possible port closure. As a result, a larger number of possible routes would emerge, but by following the same methodological framework, the optimal solutions can still be identified. Additionally, the probability of a specific port being closed can be derived or estimated, based on statistical data or modelled as a function of specific parameters, such as weather conditions or geopolitical status. For instance, ports located in conflict zones are more likely to face closures compared to ports in neutral areas. Therefore, developing a method to model such probabilities dynamically, based on relevant factors, would significantly strengthen the model’s reliability and its ability to simulate real-world scenarios.