AI-assisted bilevel optimization for sustainable maritime operations under the EU emissions trading system: model-implied carbon-cost signals and shipping network response

Abstract

Maritime decarbonization is increasingly urgent as the European Union Emissions Trading System (EU ETS) extends carbon pricing to liner shipping. This study develops an AI-assisted bilevel framework combining Bayesian Optimization (BO) and Mixed-Integer Linear Programming (MILP) to link an emission-indexed support path, a unit-consistent Marginal Abatement Cost–Total Number of Allowances in Circulation–Market Stability Reserve (MAC–TNAC–MSR) feedback mechanism, and discrete carrier network responses. In the Asia–Europe case, a representative best-evaluated policy reduces cumulative physical and ETS-covered emissions by 28.39% and 32.94%, respectively, relative to a matched nine-hub baseline, while increasing resource cost by only 0.213%. It captures 85.91% of the same-constraint short-run abatement potential. In an exact 248-policy micro-instance, BO achieves a 0.00% median optimality gap, a 4.35% mean gap, and an 80.0% exact-hit rate across 30 seeds under a common 20-evaluation budget. These results demonstrate the sample efficiency of BO–MILP for discontinuous maritime policy-response problems and provide practical guidance for designing support policies that promote lower-emission vessel deployment without substantial additional resource costs.

1 Introduction

Maritime transport carries more than 80% of world merchandise trade by volume and accounts for approximately 3% of global greenhouse gas emissions, making it indispensable to international trade but difficult to decarbonize (IMO, 2025). Ships monitored under the EU Monitoring, Reporting and Verification system emitted 135.5 Mt CO₂ in 2022 (European Commission, 2024). Since January 2024, maritime transport has been progressively incorporated into the European Union Emissions Trading System (EU ETS), with allowance-surrender obligations increasing from 40% of verified emissions in 2024 to 100% from 2026 onward (European Commission, 2025a). Early implementation monitoring covers approximately 12,000 large ships and has also examined potential avoidance behavior (European Commission, 2025b). More broadly, evidence from carbon-pricing programs shows that their effectiveness depends on policy credibility, available abatement channels, and complementary instruments (Döbbeling-Hildebrandt et al., 2024). Carbon regulation has therefore become an important determinant of shipping costs and operational decisions.

Recent methodological developments provide useful foundations for analyzing this transition. Data-driven studies have applied ensemble learning and machine-learning methods to ship energy-consumption prediction, emissions forecasting, and clean-energy shipbuilding (Xu et al., 2025b; Xu et al., 2025c; Xu et al., 2026). Research on ESG-constrained digital technology adoption further emphasizes the importance of coordination among shipping stakeholders (Qi et al., 2026), while graph-based assignment models demonstrate the value of preserving network topology and flow-conservation relationships (Xiao G. et al., 2026). Bilevel optimization offers a complementary approach for representing sequential decisions between policy designers and transport operators (Qi et al., 2021; Tan et al., 2023). Such models are particularly relevant to shipping, where regulatory signals interact with long-lived assets, discrete operational decisions, and sector-wide decarbonization pathways (Müller-Casseres et al., 2024).

The maritime literature has examined these interactions from several perspectives. Carbon-aware freight-network planning and maritime EU ETS assessments address route configuration and policy implementation (Kotzampasakis, 2025; Zhang et al., 2025), while other studies evaluate compliance costs, modal shifts, allowance allocation, route choice, and incentive alignment (Flodén et al., 2024; Sun Y. et al., 2024; Bucak et al., 2025). Hub-and-spoke network design and disruption analyses show that changes in connectivity can alter both operating costs and emissions (Xu et al., 2021; Peng et al., 2024). Related port-cluster research further indicates that inter-port connectivity and coordinated development can shape the functional differentiation and structural evolution of regional port systems (Li et al., 2026). Maritime allowance-allocation studies further demonstrate that regulatory design can influence carrier behavior and the distribution of abatement responsibilities (Zhu et al., 2023; Shangguan et al., 2025). At the market level, allowance scarcity, price corridors, and Market Stability Reserve rules affect carbon-price expectations and abatement incentives (Gerlagh et al., 2022; Pahle et al., 2025). Related research on shore-to-ship electricity also shows that the effectiveness of low-carbon support depends on the interaction between subsidy design and port competition (Xu et al., 2025a).

Despite this progress, four research gaps remain. First, existing studies generally examine carbon-market regulation, shipping-network design, or clean-technology support separately, although these mechanisms interact in practice. Second, maritime optimization models commonly treat the allowance price as an exogenous scenario input and therefore omit the feedback from solved carrier emissions to the carbon-cost signal. Third, carrier-level emissions and EU-wide TNAC and MSR quantities operate at different scales; without an explicit conversion, their direct coupling may produce mechanically driven rather than economically interpretable outcomes. Fourth, policy evaluation requires repeatedly solving a discontinuous mixed-integer carrier-response model, yet the reliability of surrogate or heuristic search is rarely assessed against an exact benchmark.

To address these gaps, this study develops an AI-assisted bilevel Bayesian Optimization–Mixed-Integer Linear Programming (BO–MILP) framework. The upper layer is interpreted as a normative policy-design and decision-support proxy rather than a literal representation of the European Commission. It parameterizes a monotone emission-indexed support path, while the lower-level MILP determines the carrier’s network, vessel deployment, voyages, cargo flows, and compliance decisions. Solved carrier emissions are converted to sector-equivalent quantities and returned to a unit-consistent MAC–TNAC–MSR controller, which generates a feedback-adjusted, corridor-constrained carbon-cost signal. Gaussian-process BO is then used to search the discontinuous policy-response surface under a limited number of MILP evaluations.

This study makes three contributions. First, it establishes a scale-consistent and auditable feedback mechanism between carrier emissions and the carbon-cost signal without interpreting that signal as a market-clearing EU allowance price. Second, it integrates the policy signal with discrete carrier decisions while separately measuring physical emissions, ETS-covered emissions, private carrier cost, and policy-inclusive resource cost. Third, it validates BO through exhaustive enumeration of a structurally matched micro-instance and compares alternative algorithms under an identical evaluation budget. The framework thereby provides a transparent tool for evaluating the short-run emissions, cost, and operational implications of maritime support policies.

The remainder of the paper is organized as follows. Section 2 reviews the literature, Sections 3 and 4 present the model and solution procedure, Section 5 reports the numerical results, and Sections 6 and 7 provide the discussion and conclusions. Figure 1 provides an overview of the research framework and paper organization.

Figure 1

2 Literature review

2.1 Evolution of carbon market mechanisms and price endogeneity

Carbon pricing mechanisms, particularly the EU ETS, have become central to maritime decarbonization research. Existing studies mainly examine allowance allocation, cost pass-through, and carbon leakage risks after shipping is brought within the ETS framework (Gerlagh et al., 2022; Pahle et al., 2025). This stream clarifies how carbon costs are redistributed across carriers, routes, and trading partners once maritime transport is incorporated into an emissions trading regime. A related strand moves beyond static policy assessment and discusses the dynamic features of cap-and-trade systems, especially the role of reserve mechanisms and allowance scarcity in shaping carbon-price signals (Tan et al., 2023). Viewed together, these studies indicate that the EU ETS is more than a regulatory surcharge: it is a market-based policy instrument that can influence strategic behavior across the shipping system.

Nevertheless, an important gap remains. Maritime applications commonly treat the allowance price as an external scenario input and therefore omit the computational feedback from carrier emissions to the allowance signal and back to network decisions. This study addresses that gap through a unit-consistent controller that maps solved carrier emissions to sector-equivalent quantities, updates the corridor-constrained MAC–TNAC–MSR signal, and resolves the carrier MILP until the annual emissions gap converges.

2.2 Shipping network topology and navigational decisions under carbon constraints

Under carbon constraints, strategic adjustment of shipping networks has become a central research theme. Existing studies can be grouped into three strands. The first examines route choice and speed optimization, emphasizing the trade-off between operating cost and emissions reduction (Ye and Yuan, 2025; Zhang et al., 2025). The second evaluates alternative fuels and their implications for network performance (Zhang et al., 2025). The third investigates network reconfiguration, including the addition of transshipment ports to diversify risk, even when such adjustments may increase system-wide emissions (Xu et al., 2021). These studies demonstrate that speed optimization significantly reduces fuel consumption, though it requires a trade-off with time-related costs (Peng et al., 2024).

Within the maritime EU ETS literature, operational studies often examine partial adjustments such as rerouting to reduce regulated voyage exposure (Sun L. et al., 2024). Fewer studies jointly determine hub selection, route opening, vessel deployment, and cargo-flow assignment. The lower-level MILP addresses this network-design gap, while the feedback controller returns the solved carrier response to the policy-signal module at a consistent sector scale (Huang et al., 2022; Lynce de Faria, 2024; Monferdini et al., 2025; Zhu et al., 2025).

2.3 Hierarchical decision modeling of policy regulation and shipping response

Hierarchical models are widely used to represent sequential policy and firm decisions (Uy et al., 2024). Recent studies examine incentive compatibility, maritime taxation, and multi-actor adjustment under carbon regulation (Wang and Zhu, 2026), while bilevel formulations represent enterprise responses to policy constraints (Zhu et al., 2023) and carbon-mitigation measures under explicit leader–follower structures (Caruso et al., 2025). Related game-theoretic research in digital mobility shows how platform rules affect participation and welfare (Guo and Xiao, 2026), providing cross-sector methodological evidence for rule-dependent operational response.

Such research commonly adopts a Stackelberg structure in which a policy designer precedes a carrier response (Zhao et al., 2025). The general formulation records cap, support, investment, and leakage-control instruments discussed in related studies (Liang et al., 2024; Xue et al., 2024; Li, 2025; Martínez-Moya et al., 2025). The implemented main experiment is deliberately narrower: after a matched screening shows that the cap perturbation does not change the carrier solution, the active search is restricted to the emission-indexed support path.

2.4 Literature synthesis and methodological positioning

The literature suggests three requirements for a maritime ETS decision model. Carbon-pricing effects depend on credible policy and available adjustment channels (Döbbeling-Hildebrandt et al., 2024), while shipping decarbonization requires coordinated technological and operational change (Müller-Casseres et al., 2024). Network studies further show that hub location, deployment, transshipment, and green-technology decisions interact (Zhen et al., 2020; Wang et al., 2022), consistent with operational evidence on fleet deployment and technology adoption (Shahriari et al., 2016; Kleinert et al., 2021). Cross-sector low-carbon scheduling research likewise demonstrates the value of integrating technology and operational decisions (Xiao Y. et al., 2026).

Accordingly, the paper develops an AI-assisted policy-response architecture in which the main experiment searches a monotone support path, the carrier MILP resolves discrete network and deployment decisions, and the feedback controller returns solved emissions to the MAC–TNAC–MSR signal at a consistent sector scale.

3 Model formulation

3.1 Problem description and model framework

Figure 2 summarizes the implemented policy-response architecture. The upper layer proposes an emission-indexed support path, and the lower layer returns a cost-minimizing carrier response under the associated carbon-cost signal.

Figure 2

Upper level: the normative policy-design layer parameterizes the start and end of the 2024–2026 support path. The general formulation contains additional policy instruments, but these are fixed rather than searched in the implemented main experiment.

Lower level (Follower): The shipping company selects hub ports, opens trunk routes, deploys vessels, and allocates cargo flows so as to minimize total discounted cost subject to carbon-compliance constraints.

The formulation operates through a closed policy-response sequence. A candidate support path enters the carbon-cost controller, the carrier MILP returns network, deployment, voyage, flow, and compliance decisions, and solved emissions are mapped back to sector scale before the carbon-cost signal is updated.

3.2 Sets and notation

The notation is grouped by decision level. Tables 1 and 2 report the policy and carrier variables, and Table 3 lists the exogenous parameters. Vessel- and arc-level emissions are expressed in tCO2; reported annual emissions are converted to ktCO2 or sector-equivalent MtCO2 as stated; policy expenditures and system costs are reported in million EUR.

Table 3 distinguishes observed and publicly reported inputs from author-constructed or calibrated values.

3.3 Upper-level model: EU ETS policy optimization

The upper-level model is a normative decision-support layer. It evaluates candidate policy settings through the carrier response generated by the lower-level MILP.

3.3.1 Upper-level objective function

The upper-level objective minimizes cumulative emissions (Equation 1):

where the symbol denotes the carrier emissions response generated by the lower-level MILP under the selected policy vector. Carrier-level quantities are reported in tCO2 or ktCO2, while the quantities entering the MAC–TNAC–MSR controller are converted to sector-equivalent MtCO2.

3.3.2 Carbon cap constraints

The carbon cap follows the EU ETS Phase 4 linear reduction rule (Equations 2, 3):

where MtCO₂ is the base-year cap, and is the annual reduction factor under EU ETS Phase 4.

3.3.3 Marginal abatement cost curve

The model-implied carbon-cost signal is computed from the following marginal abatement cost (MAC) mapping (Equation 4):

where c0 is the baseline MAC intercept, c1 is the first-order sensitivity, and c2 is the convexity coefficient. The mapping translates the sector-equivalent emissions gap into a structural carbon-cost signal that is subsequently restricted by the annual corridor.

Equation 4 supplies the MAC component of the controller. The final carbon-cost signal also depends on the TNAC–MSR adjustment and the annual corridor.

3.3.4 TNAC dynamics and market stability reserve

The Total Number of Allowances in Circulation (TNAC) follows (Equation 5):

The Market Stability Reserve (MSR) automatically adjusts allowance supply (Equation 6):

The MSR rule supplies the quantity-side adjustment. Because its thresholds are defined at EU scale, solved carrier emissions are converted to sector-equivalent MtCO2 before TNAC and MSR are updated. For each year t, the representative-carrier share is calibrated as s_t = E_car,ref,t/(10⁶E_sec,ref,t), and the feedback input is E_sec,t = E_car,t/(10⁶s_t). Using the matched zero-support reference emissions and the sector reference path of 287.1/274.8/263.0 MtCO2 gives s_t = 4.24765 × 10⁻⁵, 5.25410 × 10⁻⁵, and 4.54732 × 10⁻⁵ for 2024–2026. Only the sector-equivalent quantity enters the TNAC–MSR update; carrier compliance quantities remain in tCO2.

The same controller is applied to every main-case policy evaluation. Convergence is measured by the maximum annual relative gap between the emissions supplied to the controller and those returned by the solved carrier response.

3.3.5 Carbon price corridor

The carbon price is bounded by a price corridor and satisfies a monotonic non-decreasing condition (Equations 7, 8):

The monotonic condition in Equation 8 prevents the modeled signal from declining over the planning horizon. The annual scenario corridors are 60–75, 60–85, and 84–93 EUR/tCO2 for 2024, 2025, and 2026, respectively.

These bounds are author-defined scenario limits rather than a statutory EU ETS price corridor. Their levels are calibrated against the EEX primary-market auction record and the information set available at the calibration date (Eu Ets Auctions | Eex).

3.3.6 Technology-response descriptor

Equation 9 defines an auxiliary technology-response descriptor for interpreting policy signals. It is not used as an endogenous fleet-conversion decision in the implemented carrier MILP (Equations 9–11).

where alpha_j is the investment-efficiency coefficient, beta_j captures diminishing returns, gamma_j is the carbon-price sensitivity, and p-hat_j is the technology trigger price. In the controlled experiment, these parameters describe technology-response potential; the carrier MILP retains the available vessels’ fixed fuel types and optimizes their voyage deployment.

The carrier model instead fixes the fuel type of each available vessel and optimizes voyage deployment. Fuel-specific results therefore describe operating use of the available fleet.

3.3.7 Innovation fund budget constraint

where is the maximum proportion of carbon revenue allocated to the innovation fund.

3.3.8 IMO 2030 emission reduction target

3.4 Lower-level model: shipping network optimization

The lower-level model represents the shipping company’s operational optimization under the EU ETS policy constraints.

3.4.1 Lower-level objective function

The objective minimizes total discounted cost including fuel, port, transshipment, and carbon compliance costs (Equations 12–18):

The individual cost components are defined below.

(1) Hub facility cost

(2) Fuel cost (fixed-speed linearization)

where the fuel consumption function is a cubic polynomial (fitted to IMO 4th GHG Study data):

is the clean fuel indicator (1 for LNG/Methanol/Ammonia, 0 otherwise), and is the constant fuel consumption rate at fixed speed.

(3) Port and handling cost

(4) Transshipment cost

(5) EU ETS carbon compliance cost

3.4.2 Network topology constraints

Hub count bounds (Equations 19–24):

Minimum EU hubs (to ensure EU ETS analysis validity):

Constraint (20) maintains at least three EU gateways so that all policy cases share a comparable multi-gateway Asia–Europe topology.

Single assignment — each spoke is assigned to exactly one hub:

Hub-assignment consistency — a spoke can only be assigned to a selected hub:

Trunk route opening — a trunk arc requires both endpoints to be selected hubs:

Hub connectivity — every selected hub must participate in at least one trunk route:

3.4.3 Flow and capacity constraints

Node flow balance (Equations 25–28):

Hub transshipment conservation:

Arc capacity — flow must not exceed deployed vessel capacity:

Hub throughput capacity:

3.4.4 Vessel operation constraints

Vessel-route matching — voyage count linked to deployment (Equations 29–31):

Speed bounds:

Arc emission calculation:

3.4.5 Carbon emission compliance constraints

EU ETS covered emission aggregation (Equations 32–34):

The EU ETS coverage coefficient is defined as:

Allowance balance with phase-in of maritime EU ETS coverage:

All terms in Equation 34 are carrier-level quantities measured in tCO2. The phase-in coefficient denotes the statutory share of ETS-covered emissions that must be surrendered in year t, and A_t^car denotes carrier-specific free allocation, which is set to zero in the implemented 2024–2026 case. The resulting obligation is balanced through allowance purchase, sale, and banking decisions. The sector cap Cap_t^m does not enter this carrier balance; it enters only the sector-scale carbon-cost feedback controller.

Banking upper bound (Equation 35):

Carbon leakage avoidance ratio limit (Equation 36):

3.5 Model linearization strategy

The implemented lower level evaluates fuel burn at an economic-speed anchor, leaving voyage frequency as the principal operating variable. This local approximation keeps each policy evaluation within a MILP.

3.5.1 Fixed-speed linearization

For each vessel class, the sailing speed is fixed at an economic cruising anchor (Equation 37):

The fuel consumption rate is approximated as a constant (Equation 38):

Evaluating the cubic fuel-burn function at the economic-speed anchor converts the speed-dependent coefficient into a constant, so the remaining cost and emission expressions become linear in voyage frequency. This transformation is what allows the lower level to be solved repeatedly as a MILP inside the BO loop. The approximation is deliberately local: it is intended to represent the cruising regime around normal liner-service speeds, not the full continuous speed-control problem.

3.6 Feedback-coupled policy–response mapping

The model is summarized as a feedback-coupled policy-response mapping from an upper-level policy vector theta to a solved carrier response x*(theta). For each candidate, the controller produces a carbon-cost signal, the carrier MILP returns discrete network and compliance decisions, and the solved emissions are mapped back to sector scale. Binary hub, route, and vessel decisions make this mapping discontinuous.

The resulting upper-level response is discontinuous because hub, route, vessel, and voyage decisions change discretely. Bayesian optimization is therefore used as a sample-efficient outer search, and an exhaustive micro-instance provides an independent accuracy benchmark.

4 Solution algorithm

4.1 Bilevel solution framework

The bilevel model in Section 3 is computationally hard because the lower-level carrier response contains binary hub-selection, route-opening, and vessel-deployment variables, integer voyage-frequency decisions, multi-commodity flow constraints, and capacity coupling. Even without the upper-level policy search, this lower-level problem generalizes capacitated hub location and fixed-charge network design, both of which are NP-hard. The upper-level objective is therefore an expensive black-box value function generated by repeatedly solving a discrete network MILP.

A standard KKT reformulation is not applicable because the lower level contains binary and integer decisions. The implemented method therefore evaluates candidate policies through nested BO and MILP solves.

The proposed BO-MILP strategy treats the upper-level policy path as an expensive black-box design problem. Each candidate is scored only after the feedback-coupled carrier MILP has converged, so the surrogate learns the realized emissions and resource-cost response of the discrete network rather than a reduced-form proxy. Figure 3 summarizes the feedback-coupled BO-MILP procedure and its independent validation layer.

Figure 3

4.2 Policy-path parameterization

A preliminary matched screening shows that the calibrated cap perturbation leaves the carrier solution unchanged. Equation 39 is therefore retained only to define the cap-only diagnostic; cap parameters are fixed in every BO evaluation and are not active search variables. The main experiment searches only two interpretable controls: the start and end of a monotone emission-indexed support path (Equation 39).

Let s0 denote the 2024 support level and q denote support progress. The terminal support level is defined as s2 = s0 + (1 - s0)q, and the 2025 value is obtained by linear interpolation between s0 and s2 (Equation 40).

The investment controls in the general formulation are fixed in the implemented experiment. Operating support is weighted by tank-to-wake CO2 improvement relative to HFO: HFO and VLSFO receive zero weight, while LNG, methanol, and ammonia receive weights of 0.1169, 0.5584, and 1.0000, respectively (Equation 41).

The two controls preserve the timing and monotonicity of the annual support path while avoiding a separate free variable for each year. The 2025 value is obtained by linear interpolation.

Table 4 defines the matched cases used to isolate the cap and support channels while holding demand, topology, vessel availability, and solver settings constant.

4.3 Gaussian-process surrogate model

A Matérn-5/2 Gaussian process approximates the response over the normalized two-control policy domain. Five prespecified anchor paths initialize the surrogate (Equation 42).

where is the scaled Euclidean distance.

The kernel hyperparameters are estimated from the accumulated policy evaluations. Expected improvement is then evaluated over the admissible support domain to select the next unevaluated policy path.

4.4 Expected-improvement acquisition

The controlled main search uses expected improvement as its acquisition function. The acquisition balances the predicted emissions-resource-cost score against posterior uncertainty and proposes one new support path per BO iteration.

Expected improvement is computed relative to the best certified score observed so far (Equation 43).

The numerical search implements Equation 1 through the explicit score S(θ) = E_phys(θ)/E_phys,base + 100[max{0, C_res(θ)/C_res,base − 1.05}]². Thus, cumulative physical emissions remain the primary criterion, while the second term acts only as a soft resource-cost guardrail; policies at or below 105% of the matched baseline resource cost are ranked solely by emissions. The search uses 15 lower-level evaluations, including five prespecified anchors. The independent micro-instance comparison uses 20 evaluations per method and seed across 30 seeds.

4.5 MILP evaluation and optimality control

Each main-case candidate requires a feedback-coupled lower-level MILP solve. HiGHS is run with a 600 s limit, one thread, and a requested relative MIP gap of 0.1%; the exact micro-instance uses a requested gap of 0% (Equation 44).

In the exact micro-instance, every one of the 248 feasible upper-level policies is evaluated by a lower-level MILP solved through the HiGHS appsi interface with a 30 s limit, one thread, and a requested relative MIP gap of 0%. All 248 lower-level solves terminate optimally. The exhaustive outer enumeration takes 29.385 s of accumulated lower-solver time and establishes the reference objective used in every accuracy calculation.

4.6 Algorithm procedure and complexity

The total computational cost is dominated by MILP evaluations (Equation 45):

Table 5 reports the common initialization, operators, and evaluation budget used in the independent exact micro-instance comparison.

All five methods use the same feasible candidate set, seed-specific five-policy initialization, and 20-evaluation budget. Reported runtime includes method overhead and the stored solve time charged to each queried lower-level policy.

5 Numerical experiments

5.1 Case description and parameter settings

Port throughput is obtained from a Clarksons SIN annual time-series export generated on 2 February 2026, and coordinates are taken from the NGA World Port Index. The OD matrix is an author-constructed carrier scenario based on port scale and corridor weights rather than observed bilateral trade. Official EU MRV factors and maritime ETS rules anchor the regulatory inputs; remaining cost, fuel-path, MAC, and support parameters are calibrated structural assumptions. Table 6 summarizes the data sources and calibration procedures used in the numerical experiments.

5.2 Matched baseline and BO-selected support policy

Table 7 compares the matched baseline with a representative best-evaluated policy on the observed response plateau. Demand, topology, available fixed-fuel vessels, the cap reference, and solver settings are held constant.

Relative to the matched baseline, the selected policy reduces cumulative physical emissions from 36.887 to 26.414 ktCO2 and ETS-covered emissions from 9.501 to 6.371 ktCO2. Private carrier cost falls by 0.287%; after EUR 2.532 million of discounted support expenditure is included, resource cost increases by 0.213%.

The selected support path rises from 40% in 2024 to 60% in 2025 and 80% in 2026. The feedback-converged carbon-cost signal is 75.0, 85.0, and 86.7 EUR/tCO2. It reaches the upper corridor bound in 2024 and 2025 but remains below the 2026 ceiling. Figure 4 summarizes the emission-indexed support path and the feedback-converged carbon-cost signal. Figure 5 compares cumulative physical and ETS-covered emissions across the matched baseline, representative policy, and same-constraint operating bound.

Figure 4

Figure 5

5.3 Policy-channel diagnostic and cost decomposition

A matched cap-only diagnostic reproduces the baseline carrier solution, whereas the emission-indexed support path changes deployment among the available fixed-fuel vessels. Because demand is fixed, the interpretation concerns short-run deployment rather than fleet planning under demand uncertainty (Chua et al., 2023).

Within the calibrated lower-bound regime, cap tightening alone does not cross a discrete carrier-decision threshold. The modeled emissions change is therefore associated with the support-induced deployment response under fixed demand, fleet, and topology.

5.4 Annual feedback-converged outcomes

Table 8 reports annual physical emissions, ETS-covered emissions, allowance purchases, and the feedback-adjusted carbon-cost signal under the representative policy.

Physical emissions remain within 8.220–9.682 ktCO2 per year, while ETS-covered emissions increase from 1.832 to 2.562 ktCO2 as the statutory phase-in reaches 100%. The distinction between physical and covered emissions prevents changes in regulatory coverage from being interpreted as changes in combustion alone.

5.5 Network structure and feeder allocation

The controlled comparison uses the same nine-hub topology in all principal cases: Shanghai, Tokyo, Singapore, Xiamen, Rotterdam, Hamburg, Le Havre, Kuala Lumpur, and Hong Kong. The solved network contains 18 directed trunk arcs, eight regional spoke assignments, and four candidate-port assignments. Figure 6 shows the matched nine-hub topology and regional assignments.

Figure 6

The matched topology preserves the principal Asia–Europe trunk structure and its regional assignments. Differences between the baseline and policy cases therefore arise from voyage deployment and flow allocation rather than network replacement.

5.6 Fixed-fuel vessel deployment

Figure 7 compares the matched baseline and representative policy by vessel fuel and year. Baseline voyages are supplied by LNG vessels alongside 24 annual HFO voyages, whereas the policy case reallocates most of those LNG voyages to the available methanol vessel. This fixed-fleet substitution provides the principal operational mechanism behind the reported physical-emission reduction.

Figure 7

5.7 Computational validation of the proposed algorithm

The exact benchmark is a three-year micro-instance with four ordinal policy coordinates: carbon stringency, clean-fuel support, anti-avoidance, and infrastructure readiness. Each coordinate has five levels, and a policy-resource limit leaves 248 feasible combinations. Exhaustive enumeration solves every lower-level MILP to zero gap and establishes the reference objective of 110.400 normalized emission units. The five search methods are then compared over 30 common seeds and 20 evaluations per run. Table 9 reports the exactness, stability, and measured runtime of the upper-level search strategies on the micro-instance.

Under the common evaluation budget, BO attains a 0.00% median gap, a 4.35% mean gap, and an 80.0% exact-hit rate. The comparison methods have mean gaps of 27.20%–30.70%. BO incurs modest surrogate-fitting overhead but achieves substantially better solution quality per lower-level evaluation.

Figure 8 shows the gap to the independently known exhaustive optimum. BO separates from the comparison methods after the common five-policy initialization and reaches the lowest final mean gap under the same 20-evaluation budget.

Figure 8

5.8 BO search behavior and resource-cost guardrail

The 15 main-case evaluations reveal a threshold-shaped response: several support paths reach the same 28.39% physical-emission reduction, while higher nominal support does not produce further abatement after the deployment threshold is crossed.

Figure 9 displays the 15 evaluated policies over the two implemented controls. Surface height and color show physical-emission reduction, and base contours show resource-cost change. The 40%/80% path is a representative best-evaluated policy on the observed response plateau; several evaluated paths tie in emissions and resource cost, so the figure does not imply a unique optimum.

Figure 9

5.9 Physical operating bound under matched constraints

A separate physical-emission minimization is solved under the same demand, topology, available fixed-fuel vessels, and operational constraints as the main comparison. The solve requests a 0.1% MIP gap and terminates with a certified gap below that threshold.

The operating bound reduces cumulative physical emissions by 33.05% relative to the matched baseline. The BO-selected policy achieves a 28.39% reduction and therefore captures 85.91% of the attainable short-run abatement represented by the current model.

5.10 Price–emissions feedback convergence

For the representative support path, the price–emissions controller converges in two iterations after carrier emissions are converted from tCO2 to sector-equivalent MtCO2. The first iteration produces a 60.0/61.2/84.0 EUR/tCO2 signal. Feeding the solved emissions back into the controller yields the final 75.0/85.0/86.7 EUR/tCO2 path. All carrier MILP solves in the feedback history satisfy the requested 0.1% relative MIP-gap tolerance.

6 Discussion

The results demonstrate how an emission-indexed support policy can influence short-run carrier operations under a corridor-constrained carbon-cost signal. Relative to the matched nine-hub baseline, the representative best-evaluated policy reduces cumulative physical emissions from 36.887 to 26.414 ktCO₂, corresponding to a 28.39% reduction. ETS-covered emissions decrease from 9.501 to 6.371 ktCO₂, or 32.94%. Private carrier cost falls by 0.287%, whereas resource cost increases by only 0.213% after EUR 2.532 million of discounted support expenditure is included. The same-constraint physical-emission minimization yields a 33.05% attainable reduction, indicating that the representative policy captures 85.91% of the modeled short-run abatement potential.

The carbon-cost results clarify the role and limitations of the MAC–TNAC–MSR feedback mechanism. After solved carrier emissions are converted to sector-equivalent quantities and returned to the controller, the representative policy converges in two iterations to 75.0, 85.0, and 86.7 EUR/tCO₂ for 2024–2026. The first two values reach the imposed corridor ceilings, whereas the 2026 value lies within the corresponding corridor. The signal should therefore be interpreted as a feedback-adjusted and corridor-constrained carbon-cost signal rather than an empirically estimated or market-clearing EUA price. Its value lies in consistently transmitting changes in modeled emissions to the carrier’s compliance and operational decisions.

The matched cap-only diagnostic produces the same carrier solution as the baseline, indicating that the tested cap adjustment is inactive within the current operating range. The observed emission reduction is instead associated with the emission-indexed support path and the resulting redistribution of voyages among available fixed-fuel vessels. This distinction is important because Figure 7 represents solved voyage deployment shares rather than fleet conversion or technology penetration. The results consequently describe an operational response within the existing fleet and network, not rapid fleet-wide adoption of LNG, methanol, or other alternative fuels.

The exact micro-instance provides independent evidence on the computational value of BO. Exhaustive evaluation of all 248 feasible policies establishes the exact optimum, against which BO and four alternative search methods are assessed under the same initialization, lower-level oracle, 30 seeds, and 20-evaluation budget. BO achieves a 0.00% median optimality gap, a 4.35% mean gap, and an 80.0% exact-hit rate, whereas the competing methods produce substantially larger mean gaps. These findings support BO as a sample-efficient approach for searching a discontinuous policy-response surface when each policy evaluation requires solving a mixed-integer carrier model. They do not imply universal superiority over all deterministic or heuristic methods.

Several limitations define the interpretation of the findings. The case represents one normalized carrier over a short 2024–2026 horizon, with fixed demand, vessel fuel types, speed anchors, and network topology across the principal comparisons. Part of the OD matrix is constructed for the controlled case, and the carrier-to-sector conversion remains a calibrated representation rather than a complete multi-carrier allowance-market model. Moreover, exact outer-level optimality is established for the micro-instance, while the main Asia–Europe case reports the best policy identified within the available BO evaluation budget.

Future research could extend the framework to multiple interacting carriers and a sector-scale allowance market calibrated with observed carrier shares and TNAC–MSR data. Endogenous fuel choice, fleet renewal, slow steaming, fuel availability, methane slip, life-cycle emissions, demand response, and port congestion could also be incorporated. These extensions would allow the framework to evaluate longer-term technology transitions and policy robustness under market, fuel-price, and demand uncertainty.

7 Conclusion

This study develops an AI-assisted bilevel BO–MILP framework for analyzing the interaction between an emission-indexed support policy, a feedback-adjusted carbon-cost signal, and discrete carrier network responses under the maritime EU ETS. The framework distinguishes physical emissions, ETS-covered emissions, private carrier cost, policy expenditure, and resource cost, while feeding solved carrier emissions back to a unit-consistent MAC–TNAC–MSR controller.

In the Asia–Europe case, the representative best-evaluated policy reduces physical emissions by 28.39% and ETS-covered emissions by 32.94% relative to a matched nine-hub baseline, while increasing resource cost by 0.213%. It captures 85.91% of the same-constraint short-run abatement potential. The matched cap-only diagnostic produces no change, indicating that the modeled improvement is associated with support-induced voyage deployment among fixed-fuel vessels rather than cap tightening or fleet conversion. The resulting 75.0/85.0/86.7 EUR/tCO₂ path is interpreted as a corridor-constrained model signal rather than an EU-wide market-clearing allowance price.

Exact enumeration of the 248-policy micro-instance further shows that BO can identify high-quality policies with a limited number of lower-level MILP evaluations. Under a common 20-evaluation budget, BO achieves a 0.00% median gap and an 80.0% exact-hit rate across 30 seeds. The framework therefore provides a transparent and sample-efficient decision-support tool for assessing short-run maritime policy responses, while its conclusions remain conditional on the specified demand, fleet, network, and policy settings.

References

• 1

  BucakU.Ayazİ.S.EsmerS. (2025). The principal-agent problem in the inclusion of maritime transport into the EU ETS: Which kind of incentives are best? Res. Transportation Business Manage.62, 101459. doi: 10.1016/j.rtbm.2025.101459

  • Pubmed Abstract
  • CrossRef
  • Google Scholar

• 2

  CarusoF.CeparanoM. C.MorganJ. (2025). Lower Stackelberg equilibria: From bilevel optimization to Stackelberg games. Optimization74, 2857–2883. doi: 10.1080/02331934.2024.2422020

  • Pubmed Abstract
  • CrossRef
  • Google Scholar

• 3

  ChuaY. J.SoudagarI.NgS. H.MengQ. (2023). Impact analysis of environmental policies on shipping fleet planning under demand uncertainty. Transp. Res. Part D Transp. Environ.120, 103744. doi: 10.1016/j.trd.2023.103744

  • Pubmed Abstract
  • CrossRef
  • Google Scholar

• 4

  Döbbeling-HildebrandtN.MierschK.KhannaT. M.BacheletM.BrunsS. B.CallaghanM.et al. (2024). Systematic review and meta-analysis of ex-post evaluations on the effectiveness of carbon pricing. Nat. Commun.15, 4147. doi: 10.1038/s41467-024-48512-w

  • Pubmed Abstract
  • CrossRef
  • Google Scholar

• 5

  Eu Ets Auctions | Eex. Available online at: https://www.eex.com/en/markets/environmental-markets/eu-ets-auctions (Accessed July 16, 2026).

  • Google Scholar

• 6

  European Commission (2025a). FAQ - Maritime transport in EU Emissions Trading System (ETS). Available online at: https://climate.ec.europa.eu/eu-action/transport-decarbonisation/reducing-emissions-shipping-sector/faq-maritime-transport-eu-emissions-trading-system-ets_en (Accessed April 16, 2026).

  • Google Scholar

• 7

  European Commission (2025b). Report from the Commission on monitoring of the implementation of Directive 2003/87/EC in relation to maritime transport. Available online at: https://climate.ec.europa.eu/document/download/1bb8387b-bdc4-4489-9d76-7ff30239704d_en (Accessed May 4, 2026).

  • Google Scholar

• 8

  European Commission (2024). 2023 Report from the European Commission on CO2 Emissions from Maritime Transport. COM(2024) 151 final. Available online at: https://eur-lex.europa.eu/legal-content/EN/TXT/?uri=CELEX:52024DC0151 (Accessed May 4, 2026).

  • Google Scholar

• 9

  FlodénJ.ZetterbergL.ChristodoulouA.ParsmoR.FridellE.HanssonJ.et al. (2024). Shipping in the EU emissions trading system: Implications for mitigation, costs and modal split. Clim. Policy24, 969–987. doi: 10.1080/14693062.2024.2309167

  • Pubmed Abstract
  • CrossRef
  • Google Scholar

• 10

  GerlaghR.HeijmansR.RosendahlK. (2022). Shifting concerns for the EU ETS: Are carbon prices becoming too high? Environ. Res. Lett.17, 054018. doi: 10.1088/1748-9326/ac63d6

  • CrossRef
  • Google Scholar

• 11

  GuoX.XiaoG. (2026). Display slot competition and multi-homing in ride-hailing aggregator platforms: A game-theoretic analysis of profit and welfare implications. Sustainability18, 3625. doi: 10.3390/su18073625

  • Pubmed Abstract
  • CrossRef
  • Google Scholar

• 12

  HuangL.TanY.GuanX. (2022). Hub-and-spoke network design for container shipping considering disruption and congestion in the post COVID-19 era. Ocean Coast. Manage.225, 106230. doi: 10.1016/j.ocecoaman.2022.106230

  • Pubmed Abstract
  • CrossRef
  • Google Scholar

• 13

  International Maritime Organization (2025). IMO approves net-zero regulations for global shipping. Available online at: https://www.imo.org/en/mediacentre/pressbriefings/pages/imo-approves-netzero-regulations.aspx (Accessed May 4, 2026).

  • Google Scholar

• 14

  KleinertT.LabbéM.LjubićI.SchmidtM. (2021). A survey on mixed-integer programming techniques in bilevel optimization. EURO J. Comput. Optim.9, 100007. doi: 10.1016/j.ejco.2021.100007

  • Pubmed Abstract
  • CrossRef
  • Google Scholar

• 15

  KotzampasakisM. (2025). Maritime emissions trading in the EU: Systematic literature review and policy assessment. Transport Policy165, 28–41. doi: 10.1016/j.tranpol.2025.02.014

  • Pubmed Abstract
  • CrossRef
  • Google Scholar

• 16

  LiL. (2025). Decarbonizing China’s express freight market using high-speed rail services and carbon taxes: A bi-level optimization approach. Symmetry17, 1364. doi: 10.3390/sym17081364

  • Pubmed Abstract
  • CrossRef
  • Google Scholar

• 17

  LiX.XiaoG.LiA.LaiF.XueL. (2026). Network analysis of port clusters in the context of regional coordinated development: A case study of the Bohai Bay port cluster. Maritime Policy Manage., 1–22. doi: 10.1080/03088839.2026.2651210

  • Pubmed Abstract
  • CrossRef
  • Google Scholar

• 18

  LiangC.SunW.ShiJ.WangK.ZhangY.LimG. (2024). Decarbonizing maritime transport through green fuel-powered vessel retrofitting: A game-theoretic approach. J. Mar. Sci. Eng.12, 1174. doi: 10.3390/jmse12071174

  • Pubmed Abstract
  • CrossRef
  • Google Scholar

• 19

  Lynce de FariaD. (2024). The EU emission trading system tax regime and the issue of unfair maritime competition. Sustainability16, 9474. doi: 10.3390/su16219474

  • Pubmed Abstract
  • CrossRef
  • Google Scholar

• 20

  Maritime Safety Information. Available online at: https://msi.nga.mil/Publications/WPI (Accessed July 16, 2026).

  • Google Scholar

• 21

  Martínez-MoyaJ.Feo-ValeroM.VegaA. (2025). Policy-mix to decarbonise transport: Emission trading system and subsidies on short-sea-shipping demand. Transp. Res. Part D Transp. Environ.149, 105042. doi: 10.1016/j.trd.2025.105042

  • Pubmed Abstract
  • CrossRef
  • Google Scholar

• 22

  MonferdiniL.TebaldiL.BottaniE. (2025). From industry 4.0 to industry 5.0: Opportunities, challenges, and future perspectives in logistics. Proc. Comput. Sci.253, 2941–2950. doi: 10.1016/j.procs.2025.02.018

  • Pubmed Abstract
  • CrossRef
  • Google Scholar

• 23

  Müller-CasseresE.LeblancF.van den BergM.FragkosP.DessensO.NaghashH.et al. (2024). International shipping in a world below 2 °C. Nat. Clim. Change14, 600–607. doi: 10.1038/s41558-024-01997-1

  • Pubmed Abstract
  • CrossRef
  • Google Scholar

• 24

  PahleM.QueminS.OsorioS.GüntherC.PietzckerR. (2025). The emerging endgame: The EU ETS on the road towards climate neutrality. Resource Energy Econ81, 101476. doi: 10.1016/j.reseneeco.2024.101476

  • Pubmed Abstract
  • CrossRef
  • Google Scholar

• 25

  PengH.WangM.AnC. (2024). Implied threats of the red sea crisis to global maritime transport: Amplified carbon emissions and possible carbon pricing dysfunction. Environ. Res. Lett.19, 074053. doi: 10.1088/1748-9326/ad59b7

  • CrossRef
  • Google Scholar

• 26

  QiJ.WangS.PsaraftisH. (2021). Bi-level optimization model applications in managing air emissions from ships: A review. Commun. Transp. Res.1, 100020. doi: 10.1016/j.commtr.2021.100020

  • Pubmed Abstract
  • CrossRef
  • Google Scholar

• 27

  QiX.XiaoG.XuL. (2026). Collaborative governance mechanisms for digital technology adoption in the shipping industry under ESG constraint. Sustainability18, 5891. doi: 10.3390/su18125891

  • Pubmed Abstract
  • CrossRef
  • Google Scholar

• 28

  Regulation - 2015/757 - En - Eur-Lex. Available online at: https://eur-lex.europa.eu/eli/reg/2015/757/oj/eng (Accessed July 16, 2026).

  • Google Scholar

• 29

  ShahriariB.SwerskyK.WangZ.AdamsR. P.de FreitasN. (2016). Taking the human out of the loop: A review of Bayesian optimization. Proc. IEEE104, 148–175. doi: 10.1109/JPROC.2015.2494218

  • Pubmed Abstract
  • CrossRef
  • Google Scholar

• 30

  ShangguanY.TianX.JinY.WangS. (2025). Optimizing carbon emission allocation for liner shipping companies. Comput. Ind. Eng.208, 111348. doi: 10.1016/j.cie.2025.111348

  • Pubmed Abstract
  • CrossRef
  • Google Scholar

• 31

  SunL.WangX.HuZ.NingZ. (2024). Carbon and cost accounting for liner shipping under the European Union emission trading system. Front. Mar. Sci.11, 1291968. doi: 10.3389/fmars.2024.1291968

  • CrossRef
  • Google Scholar

• 32

  SunY.ZhengJ.YangL.LiX. (2024). Allocation and trading schemes of the maritime emissions trading system: Liner shipping route choice and carbon emissions. Transport Policy148, 60–78. doi: 10.1016/j.tranpol.2023.12.021

  • Pubmed Abstract
  • CrossRef
  • Google Scholar

• 33

  TanR. R.GueI. H. V.TapiaJ. F. D.AvisoK. B. (2023). Bilevel optimization model for maritime emissions reduction. J. Cleaner Prod.398, 136589. doi: 10.1016/j.jclepro.2023.136589

  • Pubmed Abstract
  • CrossRef
  • Google Scholar

• 34

  UyJ. G.GoR. K.AvisoK. B.San JuanJ. L. (2024). A bilevel integer programming approach for a Stackelberg game on sustainable aviation fuel production optimization. Chem. Eng. Trans.113, 205–210. doi: 10.3303/CET24113035

  • CrossRef
  • Google Scholar

• 35

  WangS.QiJ.LaporteG. (2022). Governmental subsidy plan modeling and optimization for liquefied natural gas as fuel for maritime transportation. Transp. Res. Part B Methodol.155, 304–321. doi: 10.1016/j.trb.2021.11.003

  • Pubmed Abstract
  • CrossRef
  • Google Scholar

• 36

  WangJ.ZhuW. (2026). Analysis of government taxation methods and maritime companies’ strategies under emission-reduction incentives. Res. Transportation Business Manage.64, 101559. doi: 10.1016/j.rtbm.2025.101559

  • Pubmed Abstract
  • CrossRef
  • Google Scholar

• 37

  XiaoG.XiaT.ChenX.NiA. (2026). Traffic assignment of urban road based on heterogeneous graph neural networks. Sustainability18, 5044. doi: 10.3390/su18105044

  • Pubmed Abstract
  • CrossRef
  • Google Scholar

• 38

  XiaoY.XiaoG.LiJ. (2026). Photovoltaic-energy storage systems empowered: Low-carbon and economic scheduling for electric buses. Transp. Res. Part D Transp. Environ.150, 105082. doi: 10.1016/j.trd.2025.105082

  • Pubmed Abstract
  • CrossRef
  • Google Scholar

• 39

  XuL.HouZ.SongY.ChenJ.ShuY. (2026). A feature selection-enhanced machine learning framework for clean-energy shipbuilding predictions. Ocean Coast. Manage.276, 108175. doi: 10.1016/j.ocecoaman.2026.108175

  • Pubmed Abstract
  • CrossRef
  • Google Scholar

• 40

  XuW.HuangJ.QiuY. (2021). Study on the optimization of hub-and-spoke logistics network regarding traffic congestion. J. Adv. Transp.2021, 8711964. doi: 10.1155/2021/8711964

  • CrossRef
  • Google Scholar

• 41

  XuL.LiX.YanR.ChenJ. (2025a). How to support shore-to-ship electricity constructions: Tradeoff between government subsidy and port competition. Transp. Res. Part E Logist. Transp. Rev.201, 104258. doi: 10.1016/j.tre.2025.104258

  • Pubmed Abstract
  • CrossRef
  • Google Scholar

• 42

  XuL.LinZ.MaW.HuZ.CaiL.LiJ. (2025b). A stacking ensemble learning approach for accurate and interpretable prediction of ship energy consumption. Front. Mar. Sci.12, 1679427. doi: 10.3389/fmars.2025.1679427

  • CrossRef
  • Google Scholar

• 43

  XuL.WuJ.YanR.ChenJ.FuS. (2025c). Who predicts better? A comparison of machine learning and econometrics in forecasting CO2 emissions from global shipping. Energy338, 138967. doi: 10.1016/j.energy.2025.138967

  • Pubmed Abstract
  • CrossRef
  • Google Scholar

• 44

  XueY.LaiK.WangC. (2024). How to invest decarbonization technology in shipping operations? Evidence from a game-theoretic investigation. Ocean Coast. Manage.251, 107076. doi: 10.1016/j.ocecoaman.2024.107076

  • Pubmed Abstract
  • CrossRef
  • Google Scholar

• 45

  YeZ.YuanX. (2025). Research on temporary shipping network model and optimization strategy under partial shipping network disruption. Sustain. Futures10, 101054. doi: 10.1016/j.sftr.2025.101054

  • Pubmed Abstract
  • CrossRef
  • Google Scholar

• 46

  ZhangY.ChenX.LiuJ.MaJ.BaoY.YuL. (2025). Planning and optimizing of a composite intermodal hub-and-spoke network for intercity freight considering carbon reduction policies and company attitudes. Comput. Ind. Eng.208, 111404. doi: 10.1016/j.cie.2025.111404

  • Pubmed Abstract
  • CrossRef
  • Google Scholar

• 47

  ZhaoF.DuL.ChuS. (2025). Optimized operation strategy for multi-regional integrated energy systems based on a bilevel Stackelberg game framework. Energies18, 4746. doi: 10.3390/en18174746

  • Pubmed Abstract
  • CrossRef
  • Google Scholar

• 48

  ZhenL.WuY.WangS.LaporteG. (2020). Green technology adoption for fleet deployment in a shipping network. Transp. Res. Part B Methodol.139, 388–410. doi: 10.1016/j.trb.2020.06.004

  • Pubmed Abstract
  • CrossRef
  • Google Scholar

• 49

  ZhuM.ShenS.ShiW. (2023). Carbon emission allowance allocation based on a bi-level multi-objective model in maritime shipping. Ocean Coast. Manage.241, 106665. doi: 10.1016/j.ocecoaman.2023.106665

  • Pubmed Abstract
  • CrossRef
  • Google Scholar

• 50

  ZhuL.ZhouR.LiX.LuS.LiuJ. (2025). Research on the optimization of collaborative decision making in shipping green fuel supply chains based on evolutionary game theory. Sustainability17, 5186. doi: 10.3390/su17115186

  • Pubmed Abstract
  • CrossRef
  • Google Scholar