Flow-Balanced Scheduled Routing and Robust Refueling for Inland LNG-Fuelled Liner Shipping

Abstract

Inland LNG-fuelled liner shipping is emerging as a significant trend, yet limited refueling infrastructure presents operational challenges. The complexity of inland navigation requires frequent speed adjustments to meet scheduled arrivals, which directly affects fuel consumption and refueling strategies. Additionally, imbalances in domestic and foreign trade container flows further increase operating costs for liner shipping companies. Given estimated weekly demands, considering navigational restrictions such as water depth and bridge clearance, as well as streamflow velocity, port time windows, empty container repositioning, port selection, speed adjustment, and uncertain fuel consumption, two novel models based on empty container arc variables and node variables are formulated, aiming to maximize voyage profit. These models are extended from divisible demand to indivisible demand cases. The explicit expression for the maximum fuel consumption under the worst-case speed deviation is derived, and an external linear approximation algorithm is proposed to linearize the nonlinear models while controlling approximation errors. Furthermore, the NP-hardness of the problem, the strict equivalence of the two modeling approaches, and the solution properties are proved. A case study of LNG-fuelled liner shipping on the Yangtze River shows the following: (1) for divisible demand, both models achieve optimal solutions within seconds, while for indivisible demand, the node-variable model outperforms the arc-variable model; (2) tactical strategies should be flexibly adjusted based on seasonal water depth, fuel prices, carbon taxes, speed deviations, and expected lock passage times; and (3) increasing fuel prices and carbon taxes generally reduce port calls and sailing speeds, suggesting that stricter fuel price and carbon tax policies can support the transition to green shipping. This study provides both theoretical guidance and managerial insights, supporting shipping companies in optimizing operations and promoting the development of sustainable inland shipping.

1. Introduction

1.1. Research Background

Inland waterway transport plays a vital role in national economic and social development. By the end of 2023, the total length of navigable waterways in China had reached 128,200 km. In 2023, China’s annual inland waterway cargo volume totaled 4.791 billion tons [1], representing a year-on-year increase of 8.8%. The cargo throughput of inland ports reached 6.139 billion tons, up by 10.5%. Meanwhile, inland port container throughput amounted to 38 million TEUs, a year-on-year growth of 9.2%, maintaining the world’s leading position for 19 consecutive years [2]. Similarly, the European Union operates an interconnected network of approximately 37,000 km of navigable waterways across 13 member states, handling nearly 12 billion tons of inland cargo in 2023 [3].

As climate change intensifies, the transition to cleaner energy has become a priority for governments and industries worldwide. Inland waterway transport, as an essential mode of freight movement, is central to these decarbonization efforts. Under China’s “dual-carbon” goals and the “shift from land to water” strategy, the adoption of green ships—such as liquefied natural gas (LNG)-fuelled, electric, methanol-powered, hydrogen fuel cell, and hybrid ships—has accelerated [4]. By the end of 2024, more than 600 LNG-fuelled ships were operating in China, the majority on inland waterways [5]. LNG-fuelled ship technologies, including LNG fuel engines, storage systems, and supply infrastructure, have become increasingly mature, while safety technologies, regulatory standards, and port facilities are steadily improving to meet green development requirements. LNG fuel, with its storage convenience, high ignition point, and safe transportability, makes LNG-fuelled ships the most suitable ship type for inland waterway decarbonization in the medium to long term [6]. Compared with dual-fuel ships, single-fuel LNG-fuelled ships demonstrate clear advantages in emissions reduction, fuel economy, and overall operating costs, while fully meeting national emissions standards, making them a priority development direction.

Despite this progress, the expansion of LNG-fuelled inland shipping is constrained by limited refueling infrastructure. The scarcity and uneven distribution of inland LNG refueling stations hinder their ability to support high-density shipping routes. Currently, eight LNG refueling stations have been completed and are in operation along the main stem of the Yangtze River. Distributed across provinces such as Anhui, Hubei, and Jiangsu, these facilities form an initial refueling network that supports the use of LNG as a marine fuel in inland shipping. Nevertheless, difficulties in project approval, construction, and operation, together with the absence of a clear and viable business model, continue to constrain the expansion of LNG bunkering infrastructure [7]. The introduction of carbon tax and carbon trading policies has also begun to affect shipping decisions. Therefore, optimizing refueling and ensuring stable LNG fuel supply for ship operations is a critical issue in the organization of inland waterway transport.

Inland container liner shipping primarily carries industrial products and general consumer goods. These types of cargo typically involve a large number of shippers, small shipment volumes, high value, strong time sensitivity, and frequent transactions [8]. In this context, shipping companies must develop scheduled routes, which involve determining the sequence of port calls along the route and specifying the arrival and departure times at each port, in order to provide customers with regular and reliable liner shipping services. As illustrated in Figure 1, inland container liner often starts from a sea hub port, proceeds along inland waterways with sequential port calls, and is typically operated on a weekly basis. Unlike maritime liner shipping services, inland container liner routes usually consist of a round-trip structure: ships sail upstream from the starting port, sequentially calling at each port of the outbound route, reach the upstream terminal port, and then return downstream by calling at inbound ports in sequence before finally returning to the origin, forming a closed-loop route [9].

Inland waterways present additional navigational constraints, as water depths vary across different river sections and change seasonally, while bridge clearance restrictions on lower-grade channels limit ship loading capacity and influence shipping strategies [10]. Moreover, differences in regional economic structures and trade demands result in an imbalance between outbound and inbound container flows. Such imbalance necessitates the repositioning of empty containers, which increases shipping costs. To address this issue, Flow-Balanced scheduled routing and robust refueling (FSRR) for inland liner shipping must incorporate empty container repositioning, optimizing port call sequences to improve empty container circulation efficiency, reduce non-productive shipping, and enhance both economic viability and service stability. Furthermore, due to uncertainties such as weather conditions and navigational environments, actual sailing speeds often deviate from planned speeds, leading to variability in fuel consumption. Against the backdrop of limited LNG refueling infrastructure in inland waterways, FSRR must ensure fuel supply security under uncertain operating conditions [11].

This study aims to address the key tactical challenges faced by inland LNG-fuelled liner shipping companies. Focusing on the unique characteristics of inland waterway transport, this study integrates robust refueling strategies and empty container repositioning into the FSRR framework. This study provides important theoretical and practical insights for guiding the tactical decision-making of inland liner operators while promoting the deployment of LNG-fuelled ships in inland waterways and supporting the sustainable, green transition of the shipping sector.

1.2. Main Contributions

The main contributions are as follows:

(1)

We establish the functional relationship between fuel consumption and sailing speed under speed deviations, subject to port call time window constraints, and analyze its impact on overall fuel usage.

(2)

We propose two mixed-integer nonlinear programming models: (i) an arc-variable based model, where empty container flows are represented by arcs; and (ii) a node-variable based model, where flows are represented by a smaller set of node variables.

(3)

By analyzing the extreme case of speed deviations, we construct an explicit function for maximum fuel consumption. Subsequently, linearization techniques are applied to reformulate the model as a linear programming formulation. By using the minimum number of piecewise linear segments, approximation errors are kept within a reasonable range, enabling efficient and accurate solution by professional solvers. Furthermore, we demonstrate the correspondence between the upper and lower bounds of models.

(4)

As shown in the appendix, even after multiple constraint relaxations, the problem remains strongly NP-hard. We also demonstrate that the arc-variable-based and node-variable-based models exhibit equivalent relaxation bound strength, and we further investigate the solution properties.

(5)

Using Yangtze River liner shipping cases, we validate the proposed models and conduct sensitivity analyses. Both models quickly yield optimal solutions for divisible demand, while the node-variable based model performs better in complex and indivisible-demand cases. The sensitivity results reveal generalizable patterns and managerial insights that support green inland shipping.

The remainder of this paper is organized as follows: Section 2 reviews the related literature. Section 3 introduces the problem in detail and proves its NP-hardness. Section 4 formulates two FSRR models, establishes their mathematical equivalence and analyzes the properties of optimal solutions. Section 5 derives the explicit function for maximum fuel consumption under worst-case speed deviations and proposes a linearization approach. Section 6 extends the models to indivisible demand cases. Section 7 presents numerical experiments based on the Yangtze River case study. Section 8 concludes the paper.

2. Literature Review

The problem addressed in this paper involves speed optimization and route design, which were first explored in the maritime domain and later extended to the more complex context of inland waterways. Given the current state of research, this paper reviews the literature in four aspects: (1) sailing speed optimization for maritime liner shipping, (2) sailing speed optimization for inland liner shipping, (3) route design for maritime liner shipping, and (4) route design for inland liner shipping.

2.1. Sailing Speed Optimization for Maritime Liner Shipping

Significant progress has been made in maritime ship speed optimization. Early studies assumed constant sailing speeds. Dejax and Crainic [12] first identified the power-law relationship between fuel consumption and speed and optimized single-ship speeds. Ronen [13] extended this to joint speed and fleet deployment. Fagerholt [14] incorporated time windows and tardiness penalties, showing that suitable penalties improve reliability and reduce costs. Ting and Tzeng [15] minimized schedule deviations under time windows using dynamic programming. Within port–liner collaboration, Dulebenets [16] optimized ship speeds under multiple time windows and variable handling rates. Wang and Wang [17] jointly optimized heterogeneous fleet deployment, sequencing, and speed, while Wang et al. [18] considered cargo value depreciation in speed decisions. Qi and Song [19] incorporated stochastic port times and required arrival no later than announced schedules.

To address uncertain port service times, Wang and Meng [20] optimized speeds and proposed an exact cutting-plane algorithm, later developing a robust schedule design model to enhance reliability [21]. Reinhardt et al. [22] integrated extensive operational constraints to balance robustness and fuel consumption. Zhao et al. [23] introduced a conditional value-at-risk (CVaR) based two-stage model to manage disruption risks. With rising attention to green shipping, emission considerations have been integrated into speed decisions [24]. Doudnikoff and Lacoste [25] optimized speeds inside and outside emission control areas (ECAs) to reduce fuel use and quantify emissions. Kontovas [26] proposed a framework for green routing and scheduling. Fagerholt et al. [27] jointly optimized routes and speeds to minimize fuel consumption, and Zhen et al. [28] developed a bi-objective model for fuel cost and greenhouse gas emissions. Elmi et al. [29] examined deceleration incentives and ECAs in schedule recovery, while Hu et al. [30] studied liner disruption recovery under ECA constraints.

Recent work has expanded toward joint optimization of schedules, speeds, and refueling. Considering carbon emission costs, Wang and Chen [31] optimized speed, refueling port choice, and refueling amount. Wang and Meng [32] examined uncertainties in fuel consumption for integrated speed and refueling decisions. Aydin et al. [33] proposed a dynamic approach incorporating uncertain port times and time windows. Wang et al. [34] optimized speeds and refueling under uncertain fuel prices using second-order cone programming. De et al. [35] further analyzed dynamic refueling and speed decisions under fluctuating fuel prices and consumption. Wang et al. [36] incorporated container weight into a joint framework for speed, refueling, and container allocation. Considering alternative refueling ports, Wang et al. [37] evaluated how detours and refueling times influence voyage costs.

2.2. Sailing Speed Optimization for Inland Liner Shipping

Despite substantial progress in maritime speed optimization, inland waterways involve lock operations, streamflow directions, and variable velocities, requiring further methodological refinement. Addressing lock bottleneck capacity, Smith et al. [38] minimized total waiting times via a mixed-integer model and a two-stage heuristic. Passchyn et al. [39] studied scheduling in single-lock systems with parallel chambers and proposed a dynamic programming method to enhance throughput and reduce delays. However, these works generally assumed fixed ship speeds. To overcome this, Passchyn et al. [40] examined bi-directional lock passages and formulated two general speed-optimization models to reduce total waiting times. Zhang et al. [41] investigated coordinated scheduling of Yangtze River channels and locks, improving utilization, waiting times, and energy consumption. Golak et al. [42] optimized speeds in multi-reach inland waterways with multiple locks, aiming to minimize fuel use and introducing a heuristic to improve computing efficiency. Considering uncertain passage times, Buchem et al. [43] applied dynamic speed optimization for single-lock systems without explicit speed bounds. Tan et al. [44] proposed a bi-objective model incorporating probabilistic on-time arrivals and uncertainties in streamflow and lock durations, targeting reductions in fuel consumption and travel time. More recently, Li and Yang [45] studied speed optimization under streamflow and lockage uncertainties, while Zhang et al. [46] integrated multi-ship interactions for joint inland shipping scheduling.

2.3. Route Design for Maritime Liner Shipping

Route design is critical for meeting customer demand and improving economic performance. Existing studies mainly address uncertainties in demand, sailing conditions, and empty container repositioning. Rana and Vickson [47] first proposed a route design model without cargo transshipment, optimizing port sequences, container flows, and voyage frequencies. This model was later extended to heterogeneous fleets by incorporating ship–cargo compatibility and loading constraints, solved via a branch-and-bound algorithm [48]. Wang et al. [49] examined port selection and network redesign under fluctuating demand and freight rates through a two-stage optimization framework, emphasizing the impact of sailing environment uncertainties.

A key limitation of early studies is the neglect of empty container imbalances driven by trade asymmetry. Incorporating empty container repositioning is essential for reducing storage and leasing costs and sustaining system stability and profitability [12]. Shintani et al. [50] were the first to integrate empty container repositioning into route design, demonstrating its cost-saving potential. Dong and Song [51] extended this to multi-ship, multi-port, and multi-voyage operations under dynamic and unbalanced demand. Meng and Wang [52] introduced a hub-and-spoke structure within a mixed-integer linear programming model. Brouer et al. [53] jointly optimized empty repositioning, cargo allocation, and route design using Dantzig–Wolfe decomposition and column generation. Song and Dong [54] further advanced joint optimization of route design and fleet deployment for multi-cycle long-haul services with fixed port sequences. Akyüz and Lee [55] proved the NP-hardness of the joint fleet deployment and route design problem and proposed a branch-and-bound solution method.

2.4. Route Design for Inland Liner Shipping

Extensive studies on maritime liner route design offer useful insights but are not fully applicable to inland waterways due to unique operational characteristics. Inland liner shipping services typically operate on closed-loop routes where port sequence adjustments do not alter voyage distance, requiring different modeling structures. Moreover, inland shipping faces strict navigational constraints—including water depth and bridge clearance—and largely identical outbound and inbound port call sets. Although node-based modeling has been widely applied in maritime networks [56] and rail transport [57], its performance in inland liner shipping remains underexplored.

To address inland shipping operational challenges, Maraš [58] formulated a mixed-integer linear model for fleet deployment and route design, later extended with heuristic algorithms showing that variable neighborhood search achieves superior computational performance [59]. Alfandari et al. [60] incorporated port call selection to enhance fleet deployment flexibility and profitability. An et al. [61] jointly optimized empty container repositioning, route, and fleet deployment under indivisible demand, while Braekers et al. [62] proposed a network optimization model integrating capacity, service frequency, and route planning. Wang et al. [18] developed an integer programming model optimizing route design, port calls, fleet size, and ship type. Considering inland-specific constraints, Bian et al. [63] modeled Yangtze River container flows under bridge clearance and water depth limits using an improved particle swarm optimization method. Zhou et al. [8] examined hub-and-spoke network design for inland shipping. Feng et al. [64] analyzed cargo allocation in multimodal networks amid a bulk-to-container transition under navigational restrictions. More recently, Zhang et al. [65] evaluated foldable container applications under water depth and bridge constraints, though without integrating route design or empty container decisions. Li and Yang [66] optimized inland ship routes, loading strategies, and port call sequences under navigational restrictions.

A critical review of existing studies reveals several limitations: (1) Current models inadequately capture inland-specific constraints, such as water depth and bridge clearance restrictions. (2) The limited availability of LNG refueling facilities, combined with navigational uncertainties and carbon tax, underscores the need for green scheduled route design for inland LNG-fuelled liner shipping services with robust refueling strategy. (3) Strong interdependencies among laden container strategies, refueling, empty container repositioning, leasing, and storage remain underexplored, representing a critical gap in current modeling approaches. (4) Many container flow balance models overlook the fundamental asymmetry between laden containers, which typically travel one-way, and empty containers, which require repositioning in both directions. As a result, these models fail to coordinate outbound and inbound flows effectively, leading to increased operational costs for shipping companies.

These gaps highlight the need to integrate inland waterway characteristics with the distinct flow patterns of full and empty containers. Motivated by this, the present study investigates scheduled routing and robust refueling for inland LNG-fuelled liner shipping services, incorporating empty container repositioning to address the identified shortcomings.

3. Problem Definition and Notation

3.1. Problem Definition

3.1.1. Situation of Ships Arriving at the Port

Scheduled route design for inland LNG-fuelled liner shipping services falls within the scope of tactical planning and is typically adjusted every 3 to 6 months according to changes in shipping demand and navigational conditions. Shipping companies need to deploy an appropriate number of LNG-fuelled ships along the route to maintain service frequency, usually on a weekly basis. Once the scheduled route has been designed, the route and schedule are published on website to attract cargo. Due to limitations in port resources, channel characteristics, and variations in streamflow velocity, inland ports are subject to service time window constraints [67].

Figure 2 illustrates three possible arrival scenarios under such constraints: (1) The ship arrives before the start of the service time window, in which case it must wait at the anchorage (see Figure 2a). (2) The ship arrives within the service time window, enabling it to berth immediately and commence handling operations (see Figure 2b). The ship arrives after the service window ends, which constitutes a delayed arrival and may seriously disrupt port operations (see Figure 2c). Therefore, shipping companies must develop appropriate service plans to flexibly address challenges such as imbalances in container flows, variations in streamflow velocity, time window constraints at ports, ship capacity, and navigational restrictions. Proper planning of ship arrival and departure times, as well as port call sequences, is essential to reducing waiting time and improving voyage profit.

3.1.2. Relationship Between Port Calls and Service Time

When designing the scheduled route, multiple decisions are involved regarding port call selection and the determination of service times. Figure 3 illustrates that for each selected port, the ship’s service time consists of waiting time, potential delays, port time, arrival time, and departure time. For ports that are not called, no service time is assigned. It is important to note that a ship will incur waiting time only when its arrival precedes the beginning of the port service time window, even at the minimum sailing speed. Otherwise, the ship will adjust its speed to ensure arrival within the permissible service window while minimizing fuel costs. Given that anchorage capacity is relatively limited in inland waterways, port authorities may sometimes impose charges on ships waiting at anchor. In addition, delayed ship arrivals may interfere with port operations and result in extra costs.

3.1.3. LNG Refueling for Inland LNG-Fuelled Ships

Due to the high upfront investment required for LNG refueling infrastructure, coupled with the currently limited number of LNG-fuelled ships, the payback period for such facilities tends to be long, resulting in a restricted availability of LNG refueling services. Consequently, LNG-fuelled ships do not enjoy the same convenience in refueling as conventional oil-fuelled ships. Therefore, when designing a stable and environmentally friendly scheduled route for LNG-fuelled ships, refueling considerations must be incorporated. Moreover, inland navigation involves complex and uncertain conditions that significantly affect fuel consumption. For example, streamflow velocity directly influences resistance; ships may need to accelerate to overtake slower-moving ships to ensure navigational safety; and masters often adjust speed within a certain range based on real-time navigational conditions. These factors highlight the urgent need to adopt robust refueling strategies to ensure fuel safety during voyages.

Figure 4 illustrates the fuel inventory level and refueling situation of inland LNG-fuelled ships at ports. It is evident that if a ship does not choose to refuel at a given port, its fuel inventory level $o_i^2$ upon departure will be the same as its arrival level $o_i^1$. Conversely, if the ship decides to refuel, the departure level $o_i^2$ will exceed the arrival level $o_i^1$, and the refueling amount ${\vartheta }_i$ can be expressed as ${\vartheta }_i=o_i^2-o_i^1$. Due to the linear characteristics of inland port networks, ports offering refueling services do not conflict with those selected for cargo operations, meaning ships can refuel without altering sailing distance. It should also be noted that, in order to shorten the payback period of LNG refueling facilities and promote the adoption of LNG-fuelled ships, LNG prices may vary across ports, and refueling fees are often subject to amount-based discounts. Although LNG refueling methods include shore-based refueling, barge refueling, and tank swapping, the application of tank swapping remains limited. Thus, this study focuses solely on shore-based refueling and barge refueling, which allows shipowners to refuel according to their preferred amount. In addition, under the trend of green development, the introduction of carbon taxes has become a potential policy direction. Tactical level scheduled route design that incorporates refueling not only supports energy conservation and emission reduction but also improves cost efficiency. The interaction between refueling strategies and carbon tax policies may further influence ship speed, thereby affecting route design and shipping strategies.

3.1.4. Navigational Restrictions and Empty Container Repositioning

Ensuring that the total inflow and outflow of full and empty containers is balanced at each port is critical for reducing empty container leasing and storage costs [68]. Ship loading operations must also comply with various navigational restrictions, such as water depth, bridge clearance, capacity, and cockpit height, as shown in Figure 5.

Shipping too many full containers may violate water depth restrictions, whereas shipping too many empty containers may exceed bridge or cockpit height restrictions, as shown in Figure 5. Navigational restrictions and the need for empty container repositioning directly impact container throughput, port call decisions, ship speed, and port time, ultimately influencing the liner schedule. At the same time, LNG-fuelled ships incur higher construction and operating costs but are expected to benefit from lower LNG fuel prices, affecting overall voyage economics. Under government policies promoting green development and emission reduction, these ships may also benefit from preferential treatment in port operations or lock passages. Consequently, the scheduled route design must place greater emphasis on integrating empty container repositioning with detailed consideration of voyage costs and time structures.

3.1.5. Costs and Decisions

The objective of the FSRR is to maximize the overall voyage profit of inland liner services by optimizing revenue minus costs, including port charges, handling, empty container storage and leasing, waiting, ship operations, delay penalties, carbon emissions, and refueling. The decision-making scope involves:

(1)

Ship deployment numbers;

(2)

Port calling sequences in the outbound and inbound directions;

(3)

Refueling ports and fuel amounts;

(4)

Full and empty container shipments;

(5)

Empty container storage or leasing at ports;

(6)

Determining the voyage schedule.

For modeling, the following assumptions are made:

(1)

Ships are homogeneous [69,70,71];

(2)

Origin and destination ports are known [49,51,58];

(3)

Streamflow velocity is fixed per sailing stage [44,45,46];

(4)

Container flow imbalances can be managed via leasing, storage, or repositioning [51,52];

(5)

Container demand and freight rates are known [54,55];

(6)

Each container equally affects draft, hold height, and ship capacity [64,65].

3.2. Notation

Due to the nature of the problem, two models are developed, each with slightly different decision variables. For clarity and consistency in the subsequent analysis of FSRR, the relevant notation for the models is defined as follows:

(1) Sets

$N=\left\{1,\dots ,n\right\}$: set of callable ports, with port 1 as the origin and port $n$ as the destination in the outbound direction;

$N^{bk}=\left\{1,\dots ,n^{bk}\right\}$: ports where refueling is available;

$\Omega =\left\{1,\dots ,n-1\right\}$: set of legs.

(2) Parameters

$B_{ij}$, $i,j\in N,i\ne j$: weekly shipping demand of full containers from port $i$ to $j$ (TEU/week);

$A$: ship capacity (TEU);

$P_{ij}$, $i,j\in N,i\ne j$: unit full container freight rate (CNY/TEU);

$T_i^S$, $i\in N$: start time of service time window at port $i$ (h);

$T_i^E$, $i\in N$: end time of service time window at port $i$ (h);

$T_i^p$, $i\in N$: preparation time of port $i$ (h);

$T_i^f$, $i\in N$: refueling time of port $i$ (h);

$T_b^{dam}$, $b\in \Omega$: expected lock passage time of leg $b$ (h);

${\overline{T}}_b^{dam}$, $b\in \Omega$: average lock passage time of leg $b$ (h);

${\mathit{\xi }}_{T_b^{dam}}$, $b\in \Omega$: stochastic fluctuation of lock passage time (h);

$D_b$, $b\in \Omega$: distance of leg $b$ (km);

$V_b^{err}$, $b\in \Omega$: extreme speed deviation value of the ship on leg $b$ (km/h);

${\tilde{v}}_b$, $b\in \Omega$: streamflow velocity of leg $b$ (km/h);

$H_b$, $b\in \Omega$: bridge clearance restriction on leg $b$ (m);

$W_b$, $b\in \Omega$: water depth restriction on leg $b$ (m);

$C_i^{fp}$, $i\in N$: unit LNG fuel price of port $i$ for refueling amount below ${\xi }_i^1$ (CNY/ton);

$C_i^{fix}$, $i\in N$: port charge of port $i$ (CNY);

$C_i^{zl}$, $i\in N$: unit full container loading cost at port $i$ (CNY/TEU);

$C_i^{zu}$, $i\in N$: unit full container unloading cost at port $i$ (CNY/TEU);

$C_i^{el}$, $i\in N$: unit empty container loading cost at port $i$ (CNY/TEU);

$C_i^{eu}$, $i\in N$: unit empty container unloading cost at port $i$ (CNY/TEU);

$C_i^{lc}$, $i\in N$: unit empty container leasing cost at port $i$ (CNY/TEU);

$C_i^{sc}$, $i\in N$: unit empty container storage cost at port $i$ (CNY/TEU);

$C^{oc}$: fixed weekly operating cost of the ship (CNY/week);

$C_i^{py}$, $i\in N$: penalty cost for ship delays at port $i$ (CNY/h);

$C_i^{bk}$, $i\in N$: fixed LNG refueling cost of port $i$ (CNY);

$C_i^w$, $i\in N$: waiting cost of port $i$ (CNY/h);

$C^{c{o}_2}$: unit cost of CO₂ emissions (CNY/TEU);

$G_i$, $i\in N$: container handling rate of port $i$ (TEU/h);

${\xi }_i^1$, $i\in N$: refueling amount at port $i$ corresponding to the first discount threshold (ton);

${\xi }_i^2$, $i\in N$: refueling amount at port $i$ corresponding to the second discount threshold (ton);

${\iota }_i^1$, $i\in N$: first discount rate for refueling amount within $\left({\xi }_i^1,{\xi }_i^2\right]$ (%);

${\iota }_i^2$, $i\in N$: second discount rate for refueling amount above ${\xi }_i^2$ (%);

$\upsilon$: maximum capacity of LNG fuel tank (ton);

$\rho$: initial LNG inventory level (ton);

$\beta$: fuel consumption coefficient with respect to ship speed;

$\alpha$: hourly fuel consumption coefficient;

${\delta }^f$: increase value in water draft per full container (m/TEU);

${\delta }^e$: increase value in water draft per empty container (m/TEU);

${\lambda }^f$: weight contribution factor of per full container to ship capacity;

${\lambda }^e$: weight contribution factor of per empty container to ship capacity;

$\sigma$: increase value in cabin height per container (m/TEU);

$J$: relative height between cockpit and river surface under unladen condition (m);

$Y$: relative height between ship cockpit and cabin (m);

$R$: ship draft under unladen condition (m);

$L$: maximum refueling operations allowed per round trip (times);

$\theta$: CO₂ emission factor of LNG fuel;

$V^{\min }$: minimum ship sailing speed (km/h);

$V^{\max }$: maximum ship sailing speed (km/h);

${\varpi }^{\max }$: maximum allowable number of ships deployed on the route (vessels);

$\omega$, $\omega >0$: risk parameter, corresponding to $1-\omega$ is the confidence level;

$M$: a sufficiently large positive constant.

Next, ship turnaround time $T^{tot}$ is calculated as the sum of container handling, sailing, lockage, and port preparation and waiting times over all calling ports.

$$T^{tot}={\displaystyle \sum _{i\in N^{\prime }}{\displaystyle \sum _{j\in N^{\prime }}\frac{\left(z_{ij}+r_{ij}\right)}{G_i}+{\displaystyle \sum _{i\in N^{\prime }}{\displaystyle \sum _{j\in N^{\prime }}\frac{\left(z_{ij}+r_{ij}\right)}{G_j}}}}}+{\displaystyle \sum _{b\in {\Omega }^{out}}\left(\frac{D_b}{v_b+{\tilde{v}}_b}+T_b^{dam}\right)}+{\displaystyle \sum _{b\in {\Omega }^{in}}\left(\frac{D_b}{v_b+{\tilde{v}}_b}+T_b^{dam}\right)}+{\displaystyle \sum _{i\in N^{out}}\left(t_i^w+T_i^p\right)}+{\displaystyle \sum _{i\in N^{in}}\left(t_i^w+T_i^p\right)}$$

(1)

where $N^{out}\subseteq N$ and $N^{in}\subseteq N\backslash \left\{n\right\}$ is the sets of calling ports in the outbound and inbound directions, respectively; ${\Omega }^{out}$ and ${\Omega }^{in}$ are the corresponding sets of legs; $N^{\prime }=N^{out}\cup N^{in}$ is the all calling ports set; $T^{tot}$ is the ship turnaround time; $v_b$ is the ship speed of leg $b$; and $t_i^w$ is the waiting time at port $i$; $z_{ij}$ and $r_{ij}$ is the number of full and empty containers shipped from port $i$ to $j$, respectively.

Assume that the actual ship speed on leg $b$ at time $t$ is $v_{bt}$. Due to external uncertainties, a deviation occurs relative to the planned sailing speed ${\overline{v}}_b$, with the deviation range given as:

$${\overline{v}}_b+{\tilde{v}}_b-V_b^{err}\le v_{bt}\le {\overline{v}}_b+{\tilde{v}}_b+V_b^{err}$$

(2)

The ship hourly fuel consumption during navigation is assumed to follow a power-law relationship with its actual sailing speed [72]. Thus, the fuel consumption function $g_b\left(v_{bt}\right)$ on leg $b$ at time $t$ can be expressed as:

$$g_b\left(v_{bt}\right)=\alpha {v_{bt}}^{\beta },\alpha >0;\beta >2;b\in \Omega$$

(3)

When a ship encounters uncertainties during navigation along a leg, the captain may adjust the sailing speed to ensure arrival at the next port within the scheduled time window. As a result, the ship’s instantaneous fuel consumption varies over time. The total fuel consumption on leg $b$ as a function of sailing speed can thus be expressed as:

$$Q_b\left(v_{bt}\right)={{\displaystyle {\int }_0^{{\overline{t}}_b}g}}_b\left(v_{bt}\right)dt,b\in \Omega$$

(4)

where ${\overline{t}}_b$, $b\in \overline{\Omega }$ is the planned sailing time on leg $b$ (h).

Accordingly, the ship’s maximum fuel consumption $Q_b^{max}{\left({\overline{v}}_b\right)}^{im}$, accounting for the worst-case speed variation, can be calculated as:

$$Q_b^{max}{\left({\overline{v}}_b\right)}^{im}=\underset{v_{bt}}{\max }{{\displaystyle {\int }_0^{{\overline{t}}_b}g}}_b\left(v_{bt}\right)dt,b\in \Omega$$

(5)

To comply with the schedule and ensure timely service, the ship must arrive at the subsequent calling port on each leg within the specified time interval. Thus:

$${\displaystyle {\int }_0^{{\overline{t}}_b}\left(v_{bt}+{\tilde{v}}_{bt}\right)}dt=D_b,b\in \Omega$$

(6)

If the ship refuels at a port and two discount thresholds (${\iota }_i^1$ and ${\iota }_i^2$) are available, the corresponding refueling cost $f\left({\vartheta }_i\right)$ is given by:

$$f\left({\vartheta }_i\right)=\left\{\begin{array}{cc}{C}_i^{fp}{\vartheta }_i\hfill & \hfill ,0\le {\vartheta }_i\le {\xi }_i^1\\ C_i^{fp}\left[{\xi }_i^1+{\iota }_i^1\left({\vartheta }_i-{\xi }_i^1\right)\right]\hfill & \hfill ,{\xi }_i^1<{\vartheta }_i\le {\xi }_i^2\\ C_i^{fp}\left[{\xi }_i^1+{\iota }_i^1\left({\xi }_i^2-{\xi }_i^1\right)+{\iota }_i^2\left({\vartheta }_i-{\xi }_i^2\right)\right]\hfill & \hfill , {\xi }_i^2<{\vartheta }_i\le \upsilon \end{array}\right.; i\in N$$

(7)

where $0<{\iota }_i^2\le {\iota }_i^1\le 1$. Accordingly, the fuel cost of the ship for one round trip can be expressed as:

$$C^{fuel}={\displaystyle \sum _{i\in N^{\prime }}\left[C_i^{bk}{\chi }_i+f\left({\vartheta }_i\right)\right]}$$

(8)

The objective is to maximize the voyage profit of the shipping company, which is defined as the difference between container shipping revenue and the associated shipping costs. Accordingly, the liner voyage profit function can be expressed as:

$$\begin{array}{l}{\displaystyle \sum _{i\in N^{‘}}{\displaystyle \sum _{j\in N^{’}}{P}_{ij}}}{z}_{ij}-{\displaystyle \sum _{i\in N^{‘}}{\displaystyle \sum _{j\in N^{’}}\left(C_i^{zl}+C_j^{zu}\right)}}{z}_{ij}-{\displaystyle \sum _{i\in N^{‘}}{\displaystyle \sum _{j\in N^{’}}\left(C_i^{el}+C_j^{eu}\right)r_{ij}-\left({\displaystyle \sum _{i\in N^{out}}{C}_i^{fix}}+{\displaystyle \sum _{i\in N^{in}}{C}_i^{fix}}\right)}}-{\displaystyle \sum _{i\in N^{\prime }}\left(C_i^{sc}{s}_i+C_i^{lc}{l}_i\right)}\\ -C^{oc}m-{\displaystyle \sum _{i\in N^{\prime }}{C}_i^{py}{t}_i^l}-{\displaystyle \sum _{i\in N^{\prime }}{C}_i^w{t}_i^w}-C^{{co}_2}\left({\displaystyle \sum _{b\in {\Omega }^{out}}{Q}_b^{max}{\left({\overline{v}}_b\right)}^{im}}+{\displaystyle \sum _{b\in {\Omega }^{in}}{Q}_b^{max}{\left({\overline{v}}_b\right)}^{im}}\right)-{\displaystyle \sum _{i\in N^{\prime }}\left[C_i^{bk}{\chi }_i+g\left({\vartheta }_i\right)\right]}\end{array}$$

(9)

where $s_i$ and $l_i$ denote the numbers of empty containers stored and leased at port $i$, respectively; $t_i^l$ represents the ship delay time at port $i$; $m$ is the number of ships deployed on the route, satisfying $m=⌈{\displaystyle \frac{T^{tot}}{168}}⌉$, $m\le {\varpi }^{\max }$, and is equivalent to the ship turnaround time in weeks. In scheduled route design, $m$ should be aligned with ${\displaystyle \frac{T^{tot}}{168}}$, i.e., $168m=T^{tot}$.

3.3. Graph Transformation

The shipping path of each container is determined by its origin and destination ports, and any two ports can form a port pair. Along an inland shipping route, ports are arranged linearly, and container shipping demand may exist between any two ports. Figure 6 illustrates the characteristic of shipping demand, showing that container flows between port pairs involve not only outbound flows but also inbound flows.

From the analysis of Figure 6, it can be further observed that, in order to model the relationship between route design and cargo shipping demand more precisely, the container shipping demand between port pairs needs to be reasonably transformed. To this end, a new port set is introduced to represent the sequence of inbound ports. Through this transformation, the new set can systematically characterize the features of inbound shipping demand, thereby enhancing the representation of inbound shipping patterns at the modeling level. Figure 7 illustrates the specific implementation of this modeling approach.

In view of the above, and in order to better present the model while simplifying notation, two directed acyclic graphs (DAG), ${\Gamma }^f=\left(\overline{N},{\Psi }^f\right)$ and ${\Gamma }^e=\left(\overline{N},{\Psi }^e\right)$, are introduced to represent the shipping of full and empty containers, respectively. At the same time, the set of ports $\overline{N}=\left\{1,\dots ,2n-1\right\}$ that ships may call is redefined, which also serves as the same ordered node set for both graphs. Outbound ports and inbound ports are selected for each LNG-fuelled ship, where each outbound port $i$ has a mapped inbound port $\overline{i}$, $\overline{i}=2n-i$. Nodes $i\in \left\{1,\dots ,n\right\}$ and $i\in \left\{n+1,\dots ,2n-1\right\}$ represent the calling ports in the graphs. Since empty containers may continue shipping beyond the final port call in outbound direction, the sets of shipping paths for full and empty containers are defined as ${\Psi }^f=\left\{\left(i,j\right):i,j\in \overline{N},i<j\le n or n\le i<j\right\}$ and ${\Psi }^e=\left\{\left(i,j\right):i,j\in \overline{N},i<j\right\}$, forming the arcs in the respective DAGs. Additionally, the leg set $\overline{\Omega }=\left\{1,\dots ,2n-2\right\}$ and refueling port set ${\overline{N}}^{bk}=\left\{1,\dots ,2n^{bk}-1\right\}$ are defined to maintain correspondence with $\overline{N}=\left\{1,\dots ,2n-1\right\}$.

Based on the above description, the indexing of the following parameters is redefined:

For parameters $G_i,T_i^S,T_i^E,T_i^p,T_i^f,C_i^{fix},C_i^{py},C_i^w,C_i^{el},C_i^{zl},C_i^{eu},C_i^{zu},C_i^{bk},C_i^{fp},{\xi }_i^1,{\xi }_i^2,{\iota }_i^1,{\iota }_i^2$ previously indexed by $N$, the index set is now extended to $\overline{N}$. Similarly, parameters $D_b,T_b^{dam},{\tilde{v}}_b,H_b,W_b$ previously indexed by $\Omega$ are now defined over $\overline{\Omega }$. When $i\le n$, the index denotes an outbound port, consistent with the original definition $i$; when $i>n$, it denotes an inbound port, with updated to $\overline{i}$ corresponding to the inbound port.

Other redefined parameters include:

${\overline{B}}_{ij}$, $\left(i,j\right)\in {\Psi }^f$: weekly shipping demand of full containers from port $i$ to $j$, calculated as ${\overline{B}}_{ij}=\left\{\begin{array}{l}{B}_{ij},\hspace{1em}\mathrm{if} i<j\le n\\ B_{\overline{i}\overline{j}},\hspace{1em}\mathrm{if} n\le i<j\end{array}\right.$;

${\overline{P}}_{ij}$, $\left(i,j\right)\in {\Psi }^f$: net revenue for shipping unit full container from port $i$ to $j$, i.e., freight rate minus handling cost, calculated as ${\overline{P}}_{ij}=\left\{\begin{array}{l}{P}_{ij}-C_i^{zl}-C_j^{zu},\hspace{1em}\mathrm{if} i<j\le n\\ P_{\overline{i}\overline{j}}-C_{\overline{i}}^{zl}-C_{\overline{j}}^{zu},\hspace{1em}\mathrm{if} n\le i<j\end{array}\right.$;

${\overline{C}}_{ij}^e$, $\left(i,j\right)\in {\Psi }^e$: cost of shipping unit empty container from port $i$ to $j$, calculated as ${\overline{C}}_{ij}^e=C_i^{el}+C_j^{eu}$.

By selecting a subset of arcs ${\Psi }^{\prime }$, and associating each selected arc $\left(i,j\right)$ with full containers $z_{ij}$ and empty containers $r_{ij}$, a feasible solution can be constructed. This represents the shipment of containers from port $i$ to $j$. Simultaneously, the set of nodes associated with these arcs defines the port call sequences, forming a feasible route plan.

This is equivalent to shipping full and empty containers from port $i$ to $j$. At the same time, the set of nodes ${\Psi }^{\prime }$ associated with these arcs determines the sequence of port calls, thereby forming a feasible route plan.

Based on the above method, the transformation of relevant parameters is completed. Figure 8 provides an example illustrating the transformation process of key parameters, based on this, explains the container flow balance strategy in inland liner shipping. Assume The ship can call at 4 ports, with full container shipping demands of ${\overline{B}}_{12}=5$, ${\overline{B}}_{13}=5$, ${\overline{B}}_{34}=10$, ${\overline{B}}_{42}=5$, ${\overline{B}}_{57}=2$, ${\overline{B}}_{67}=15$, and a capacity of $A=12$ TEU. The water depth restrictions on the legs from port 1 to 2 and from port 2 to 3 are 1.2 m, while the water depth restriction from port 3 to 4 is 0.7 m. The ship draft increases by 0.1 m per full container and 0.05 m per empty container.

Figure 8 illustrates two alternative strategies for container flow balancing along the port call sequence 1→2→3→4→3→2→1. In the first strategy, storage at port 4 and leasing at port 3 are required to satisfy empty container demand. In the second strategy, empty containers are repositioned between ports, eliminating the need for storage or leasing, thus enhancing capacity utilization and reducing shipping service costs.

At this stage, the research problem has been fully formulated. To further elucidate its inherent complexity and to highlight the methodological contribution of this study, the NP-hardness of the problem is proved in Appendix A. This proof establishes the computational difficulty of the problem and motivates the development of a modeling approach based on empty container node variables.

4. MIP Models

This section presents two mixed-integer nonlinear programming (MINLP) models formulated to solve the FSRR problem. The models differ in their approaches to modeling empty containers. The common decision variables are defined as follows:

${\chi }_i$, $i\in \overline{N}$: binary variable, equal to 1 if the ship refueling at port $i$, 0 otherwise;

$x_i$, $i\in \overline{N}$: binary variable, equal to 1 if the ship calls at port $i$, 0 otherwise;

$z_{ij}$, $\left(i,j\right)\in {\Psi }^f$: number of full containers shipped from port $i$ to port $j$ (TEU);

$s_i$, $i\in N$: number of empty containers stored at port $i$ (TEU);

$l_i$, $i\in N$: number of empty containers leased at port $i$ (TEU);

$t_i^h$, $i\in \overline{N}$: container handling time at port $i$ (h);

$t_i^a$, $i\in \overline{N}$: ship arrival time at port $i$ (h);

$t_i^d$, $i\in \overline{N}$: ship departure time from port $i$ (h);

$t_i^l$, $i\in \overline{N}$: delay time of the ship at port $i$ (h);

$t_i^w$, $i\in \overline{N}$: ship waiting time at port $i$ (h);

${\vartheta }_i$, $i\in \overline{N}$: refueling amount at port $i$ (ton);

$o_i^1$, $i\in \overline{N}$: fuel inventory level upon arrival at port $i$ (ton);

$o_i^2$, $i\in \overline{N}$: fuel inventory level upon departure from port $i$ (ton);

${\overline{v}}_b$, $b\in \overline{\Omega }$: planned sailing speed on leg $b$ (km/h);

${\overline{t}}_b$, $b\in \overline{\Omega }$: planned sailing time on leg $b$ (h);

$m$: number of ships deployed on the route (vessels), equivalent to ship turnaround time in weeks.

In the directed acyclic graph ${\Gamma }^f$, for $\gamma \subset \overline{N}$, define ${\Phi }_f^{+}\left(\gamma \right)=\left\{\left(i,j\right)\in {\Psi }^f:i\in \gamma ,j\notin \gamma \right\}$ and ${\Phi }_f^{-}\left(\gamma \right)=\left\{\left(i,j\right)\in {\Psi }^f:i\notin \gamma ,j\in \gamma \right\}$ as the sets of outbound and inbound arcs for full containers, respectively. And, for $\gamma =\left\{i\right\}$, ${\Phi }_f^{+}\left(\gamma \right)$ and ${\Phi }_f^{-}\left(\gamma \right)$ are denoted as ${\Phi }_f^{+}\left(i\right)$ and ${\Phi }_f^{-}\left(i\right)$, respectively. Analogously, in ${\Psi }^e$, ${\Phi }_e^{+}\left(i\right)$ and ${\Phi }_e^{-}\left(i\right)$ correspond to empty containers. Moreover, for each port $i^{\prime }$, $i^{\prime }\in \overline{N}\backslash \left\{\overline{1}\right\}$, the sets of arcs connecting $i^{\prime }$ to its subsequent and preceding ports are defined as ${\Psi }_{i^{\prime }}^f=\left\{\left(i,j\right)\in {\Psi }^f:i\le i^{\prime },j>i^{\prime }\right\}$ and ${\Psi }_{i^{\prime }}^e=\left\{\left(i,j\right)\in {\Psi }^e:i\le i^{\prime },j>i^{\prime }\right\}$. Container summation over these arcs ${\Psi }_{i^{\prime }}^f$ and ${\Psi }_{i^{\prime }}^e$ determines loading levels along each leg $i^{\prime }$ (between port $i^{\prime }$ and $i^{\prime }+1$), forming the basis for ship capacity and navigational constraints.

4.1. First Model Based on Empty Container Arc Variables

In the first FSRR model, empty containers are modeled using arc-based decision variables. Specifically, the variable is defined as:

$r_{ij}$, $\left(i,j\right)\in {\Psi }^e$: number of empty containers shipped from port $i$ to $j$ (TEU).

For notational clarity and conciseness, the constraints are expressed in a simplified form:

$${\displaystyle \sum _j{z}_{ij}}+{\displaystyle \sum _j{r}_{ij}}\equiv {\displaystyle \sum _{\left(i,j\right)\in {\Phi }_f^{+}\left(i\right)}{z}_{ij}+{\displaystyle \sum _{\left(i,j\right)\in {\Phi }_e^{+}\left(i\right)}{r}_{ij}}}$$

(10)

$${\displaystyle \sum _j{z}_{ji}+{\displaystyle \sum _j{r}_{ji}}}\equiv {\displaystyle \sum _{\left(j,i\right)\in {\Phi }_f^{-}\left(i\right)}{z}_{ji}}+{\displaystyle \sum _{\left(j,i\right)\in {\Phi }_e^{-}\left(i\right)}{r}_{ji}}$$

(11)

Next, the model based on empty container arc variables, denoted as [MS1], is formulated. The objective function of this model, expressed in Equation (12), aims to maximize the total voyage profit of the liner container shipping service. Specifically, the first term represents the net revenue from shipping full containers; the second term corresponds to the cost of shipping empty containers; the third term accounts for the port call cost; the fourth term reflects the costs associated with empty container storage and leasing; the fifth term represents the operating cost of the service; the sixth term captures the penalty cost for ship delays at ports; the seventh term denotes the waiting cost; the eighth term corresponds to the emission cost associated with fuel consumption; and the ninth term represents the fuel cost of the ship along the entire route.

$$\begin{array}{l}\max F_1={\displaystyle \sum _{\left(i,j\right)\in {\Psi }^f}{\overline{P}}_{ij}{z}_{ij}}-{\displaystyle \sum _{\left(i,j\right)\in {\Psi }^e}{\overline{C}}_{ij}^e{r}_{ij}}-{\displaystyle \sum _{i\in \overline{N}}{C}_i^{fix}{x}_i}-{\displaystyle \sum _{i\in N}\left(C_i^{sc}{s}_i+C_i^{lc}{l}_i\right)}\\ -C^{oc}m-{\displaystyle \sum _{i\in \overline{N}}{C}_i^{py}{t}_i^l}-{\displaystyle \sum _{i\in \overline{N}}{C}_i^w{t}_i^w}-C^{c{o}_2}\theta {\displaystyle \sum _{b\in \overline{\Omega }}{Q}_b^{max}{\left({\overline{v}}_b\right)}^{im}}-{\displaystyle \sum _{i\in \overline{N}}\left[C_i^{bk}{\chi }_i+g\left({\vartheta }_i\right)\right]}\end{array}$$

(12)

Constraints (13)–(15) ensure that the ship calls at both the origin and destination ports.

$$x_1=1$$

(13)

$$x_{2n-1}=1$$

(14)

$$x_n=1$$

(15)

Constraints (16) and (17) restrict full container shipping to cases where both the port $i$ and $j$ are included in the ship’s calling sequence, while guaranteeing that the number of shipped full containers does not exceed the demand ${\overline{B}}_{ij}$.

$$z_{ij}\le {\overline{B}}_{ij}{x}_i,\left(i,j\right)\in {\Psi }^f$$

(16)

$$z_{ij}\le {\overline{B}}_{ij}{x}_j,\left(i,j\right)\in {\Psi }^f$$

(17)

Constraints (18) and (19) ensure that when a port $i$ is called, the total number of containers loaded at that port remains strictly below the ship capacity.

$${\lambda }^f{\displaystyle \sum _j{z}_{ij}+{\lambda }^e{\displaystyle \sum _j{r}_{ij}}}\le A{x}_i,i\in \overline{N}$$

(18)

$${\lambda }^f{\displaystyle \sum _j{z}_{ji}+{\lambda }^e{\displaystyle \sum _j{r}_{ji}}}\le A{x}_i,i\in \overline{N}$$

(19)

Constraint (20) ensures that the container shipping amount from port $i^{\prime }$ to $i^{\prime }+1$ (leg $i^{\prime }$) cannot exceed the ship capacity.

$${\lambda }^f{\displaystyle \sum _{\left(i,j\right)\in {\Psi }_{i^{\prime }}^f}{z}_{ij}}+{\lambda }^e{\displaystyle \sum _{\left(i,j\right)\in {\Psi }_{i^{\prime }}^e}{r}_{ij}}\le A,i^{\prime }\in \overline{N}\backslash \left\{\overline{1}\right\}$$

(20)

Constraint (21) ensures that the loading container height does not exceed the cockpit. Constraint (22) guarantees that the ship can safely pass beneath bridges along each leg. Constraint (23) enforces compliance with the water draft restriction on each leg.

$$\sigma \left({\displaystyle \sum _{\left(i,j\right)\in {\Psi }_{i^{\prime }}^f}{z}_{ij}}+{\displaystyle \sum _{\left(i,j\right)\in {\Psi }_{i^{\prime }}^e}{r}_{ij}}\right)\le Y,i^{\prime }\in \overline{N}\backslash \left\{\overline{1}\right\}$$

(21)

$$J-{\delta }^f{\displaystyle \sum _{\left(i,j\right)\in {\Psi }_{i^{\prime }}^f}{z}_{ij}}-{\delta }^e{\displaystyle \sum _{\left(i,j\right)\in {\Psi }_{i^{\prime }}^e}{r}_{ij}}\le H_{b=i^{\prime }},i^{\prime }\in \overline{N}\backslash \left\{\overline{1}\right\};b\in \overline{\Omega }$$

(22)

$$R+{\delta }^f{\displaystyle \sum _{\left(i,j\right)\in {\Psi }_{i^{\prime }}^f}{z}_{ij}}+{\delta }^e{\displaystyle \sum _{\left(i,j\right)\in {\Psi }_{i^{\prime }}^e}{r}_{ij}}\le W_{b=i^{\prime }},i^{\prime }\in \overline{N}\backslash \left\{\overline{1}\right\};b\in \overline{\Omega }$$

(23)

Constraint (24) represents the container flow balance at each port, ensuring that the difference between the numbers of loaded and unloaded containers is correctly accounted for. A positive difference indicates that empty containers must be leased at the corresponding port, whereas a negative difference implies that empty containers are stored there. Since the objective function maximizes total voyage profit and includes the cost term ${\displaystyle \sum _{i\in N}\left(C_i^{sc}{s}_i+C_i^{lc}{l}_i\right)}$, in the optimal solution, either $l_i\ge 0$, $s_i=0$ or $s_i\ge 0$, $l_i=0$ will occur, but not both simultaneously.

$${\displaystyle \sum _j\left(z_{ij}+r_{ij}\right)}-{\displaystyle \sum _j\left(z_{ji}+r_{ji}\right)}+{\displaystyle \sum _j\left(z_{\overline{i}j}+r_{\overline{i}j}\right)}-{\displaystyle \sum _j\left(z_{j\overline{i}}+r_{j\overline{i}}\right)}=l_i-s_i,i\in N$$

(24)

Constraint (25) defines the container handling time at each port. Constraint (26) specifies the ship departure time, and Constraint (27) represents the delay time at the port. Constraints (28) and (29) capture waiting time, while Constraints (30) and (31) denote arrival times at the ports. Constraint (32) ensures the total round-trip voyage duration is satisfied. Constraint (33) ensures that the starting time of the liner service is non-negative. Finally, Constraint (34) is a chance constraint, ensuring that the probability of the ship successfully passing through the lock within the reserved lock passage time is not less than $1-\omega$.

$$t_i^h={\displaystyle \frac{{\displaystyle \sum _{\left(i,j\right)\in {\Phi }_f^{+}\left(i\right)}{z}_{ij}}+{\displaystyle \sum _{\left(i,j\right)\in {\Phi }_e^{+}\left(i\right)}{r}_{ij}}+{\displaystyle \sum _{\left(j,i\right)\in {\Phi }_f^{-}\left(i\right)}{z}_{ji}}+{\displaystyle \sum _{\left(j,i\right)\in {\Phi }_e^{-}\left(i\right)}{r}_{ji}}}{G_i}},i\in \overline{N}\backslash \left\{\overline{1}\right\}$$

(25)

$$t_i^d=t_i^a+t_i^w+t_i^h+T_i^p{x}_i+T_i^f{\chi }_i,i\in \overline{N}\backslash \left\{\overline{1}\right\}$$

(26)

$$t_i^l\ge t_i^a-T_i^E\left(1+\left(1-x_i\right)M\right),i\in \overline{N}\backslash \left\{\overline{1}\right\}$$

(27)

$$t_{i+1}^w\ge T_{i+1}^S\left(1-\left(1-x_{i+1}\right)M\right)-t_i^a-t_i^h-t_i^w-T_i^p{x}_i-T_i^f{\chi }_i-T_{b=i}^{dam}-{\overline{t}}_{b=i},i\in \overline{N}\backslash \left\{\overline{1},\overline{2}\right\};b\in \overline{\Omega }\backslash \left\{2n-2\right\}$$

(28)

$$t_1^w\ge T_1^S\left(1-\left(1-x_1\right)M\right)-t_{\overline{2}}^a-t_{\overline{2}}^h-t_{\overline{2}}^w-T_{\overline{2}}^p{x}_{\overline{2}}-T_{\overline{2}}^f{\chi }_{\overline{2}}-T_{2n-2}^{dam}-{\overline{t}}_{2n-2}+168m$$

(29)

$$t_{i+1}^a=t_i^d+T_{b=i}^{dam}+{\overline{t}}_{b=i},i\in \overline{N}\backslash \left\{\overline{1},\overline{2}\right\};b\in \overline{\Omega }\backslash \left\{2n-2\right\}$$

(30)

$$t_1^a=t_{\overline{2}}^d+T_{2n-2}^{dam}-{\overline{t}}_{2n-2}-168m$$

(31)

$${\displaystyle \sum _{i\in \overline{N}\backslash \left\{\overline{1}\right\}}\left(t_i^d-t_i^a\right)+{\displaystyle \sum _{b\in \overline{\Omega }}\left(T_b^{dam}+{\overline{t}}_b\right)}}=168m$$

(32)

$$t_1^a\ge 0$$

(33)

$$\mathrm{Pr}\left\{{\overline{T}}_b^{dam}+{\mathit{\xi }}_{T_b^{dam}}\le T_b^{dam}\right\}\ge 1-\omega ,b\in \overline{\Omega }$$

(34)

Constraints (35) and (36) represent the refueling amount constraints at ports. Constraint (37) represents the ship initial fuel amount for the voyage. Constraints (38) and (39) represent the fuel inventory level constraints when the ship arriving and leaving a port, respectively. Constraints (40) and (41) represent the fuel consumption on each leg. Constraint (42) represents the sailing time on each leg. Constraint (43) represents the maximum number of refueling operations during a voyage.

$$0.1\upsilon {\chi }_i\le {\vartheta }_i\le \upsilon {\chi }_i,i\in \overline{N}\backslash \left\{\overline{1}\right\}$$

(35)

$${\vartheta }_i=o_i^2- o_i^1,i\in \overline{N}\backslash \left\{\overline{1}\right\}$$

(36)

$$o_1^1=\rho$$

(37)

$$o_i^1\ge 0.1\upsilon ,i\in \overline{N}\backslash \left\{\overline{1}\right\}$$

(38)

$$o_i^2\le \upsilon ,i\in \overline{N}\backslash \left\{\overline{1}\right\}$$

(39)

$$o_{i+1}^1=o_i^2-Q_{b=i}^{max}{\left({\overline{v}}_{b=i}\right)}^{im},i\in \overline{N}\backslash \left\{\overline{1},\overline{2}\right\};b\in \overline{\Omega }\backslash \left\{2n-2\right\}$$

(40)

$$o_1^1=o_{2n-2}^2-Q_{2n-2}^{max}{\left({\overline{v}}_{2n-2}\right)}^{im}$$

(41)

$${\overline{t}}_b={\displaystyle \frac{D_b}{{\overline{v}}_b+{\tilde{v}}_b}},b\in \overline{\Omega }$$

(42)

$${\displaystyle \sum _{i\in \overline{N}\backslash \left\{\overline{1}\right\}}{\chi }_i}\le L$$

(43)

Constraints (44)–(52) define the feasible ranges of the decision variables.

$$m\le {\varpi }^{\max }$$

(44)

$$s_i,l_i\ge 0,i\in N$$

(45)

$$x_i\in \left\{0,1\right\},i\in \overline{N}$$

(46)

$${\chi }_i=0,i\in \overline{N}\backslash {\overline{N}}^{bk}$$

(47)

$${\chi }_i\in \left\{0,1\right\},i\in {\overline{N}}^{bk}$$

(48)

$$m\in {\mathbb{Z}}_{+}$$

(49)

$$z_{ij}\in {\mathbb{Z}}_{+},\left(i,j\right)\in {\Psi }^f$$

(50)

$$r_{ij}\in {\mathbb{Z}}_{+},\left(i,j\right)\in {\Psi }^e$$

(51)

$$V^{min}+{\tilde{v}}_b+V_b^{err}\le {\overline{v}}_b\le V^{max}+{\tilde{v}}_b-V_b^{err},b\in \overline{\Omega }$$

(52)

4.2. Second Model Based on Empty Container Node Variables

In the FSRR model based on the empty container arc-variable, empty containers can be loaded or unloaded at any port of call, each with a fixed origin and destination. To reduce the number of decision variables while preserving model fidelity, the formulation is transformed. The transformed model [MS2] computes only the total number of empty containers loaded or unloaded at each port. Accordingly, variables $r_i^{in}$ and $r_i^{out}$ are introduced to replace $r_{ij}$, where

$r_i^{in}$, $i\in \overline{N}$: number of empty containers unloaded at port $i$ (TEU);

$r_i^{out}$, $i\in \overline{N}$: number of empty containers loaded at port $i$ (TEU).

The variables $r_{ij}$, $r_i^{in}$, and $r_i^{out}$ satisfy the following relationship:

$$r_i^{in}={\displaystyle \sum _{\left(j,i\right)\in {\Phi }_e^{-}\left(i\right)}{r}_{ji}}$$

(53)

$$r_i^{out}={\displaystyle \sum _{\left(i,j\right)\in {\Phi }_e^{+}\left(i\right)}{r}_{ij}}$$

(54)

Based on the above transformed variables, model [MS1] is adjusted to obtain model [MS2].

Objective function:

$$\begin{array}{l}\max F_2={\displaystyle \sum _{\left(i,j\right)\in {\Psi }^f}{\overline{P}}_{ij}{z}_{ij}}-{\displaystyle \sum _{i\in \overline{N}}\left(C_i^{el}{r}_i^{out}+C_i^{eu}{r}_i^{in}\right)-}{\displaystyle \sum _{i\in \overline{N}}{C}_i^{fix}{x}_i}-{\displaystyle \sum _{i\in N}\left(C_i^{sc}{s}_i+C_i^{lc}{l}_i\right)}\\ -C^{oc}m-{\displaystyle \sum _{i\in \overline{N}}{C}_i^{py}{t}_i^l}-{\displaystyle \sum _{i\in \overline{N}}{C}_i^w{t}_i^w}-C^{c{o}_2}\theta {\displaystyle \sum _{b\in \overline{\Omega }}{Q}_b^{max}{\left({\overline{v}}_b\right)}^{im}}-{\displaystyle \sum _{i\in \overline{N}}\left[C_i^{bk}{\chi }_i+g\left({\vartheta }_i\right)\right]}\end{array}$$

(55)

subject to constraints (13)–(17), (26)–(50), (52) and

$${\lambda }^f{\displaystyle \sum _{\left(i,j\right)\in {\Phi }_f^{+}\left(i\right)}{z}_{ij}}+{\lambda }^e{r}_i^{out}\le A{x}_i,i\in \overline{N}$$

(56)

$${\lambda }^f{\displaystyle \sum _{\left(j,i\right)\in {\Phi }_f^{-}\left(i\right)}{z}_{ji}}+{\lambda }^e{r}_i^{in}\le A{x}_i,i\in \overline{N}$$

(57)

$${\lambda }^f{\displaystyle \sum _{\left(i,j\right)\in {\Psi }_{i^{\prime }}^f}{z}_{ij}}+{\lambda }^e{\displaystyle \sum _{i\le i^{\prime }}\left(r_i^{out}-r_i^{in}\right)}\le A,i^{\prime }\in \overline{N}\backslash \left\{\overline{1}\right\}$$

(58)

$$\sigma \left({\displaystyle \sum _{\left(i,j\right)\in {\Psi }_{i^{\prime }}^f}{z}_{ij}}+{\displaystyle \sum _{i\le i^{\prime }}\left(r_i^{out}-r_i^{in}\right)}\right)\le Y,i^{\prime }\in \overline{N}\backslash \left\{\overline{1}\right\}$$

(59)

$$J-{\delta }^f{\displaystyle \sum _{\left(i,j\right)\in {\Psi }_{i^{\prime }}^f}{z}_{ij}}-{\delta }^e{\displaystyle \sum _{i\le i^{\prime }}\left(r_i^{out}-r_i^{in}\right)}\le H_{b=i^{\prime }},i^{\prime }\in \overline{N}\backslash \left\{\overline{1}\right\};b\in \overline{\Omega }$$

(60)

$$R+{\delta }^f{\displaystyle \sum _{\left(i,j\right)\in {\Psi }_{i^{\prime }}^f}{z}_{ij}}+{\delta }^e{\displaystyle \sum _{i\le i^{\prime }}\left(r_i^{out}-r_i^{in}\right)}\le W_{b=i^{\prime }},i^{\prime }\in \overline{N}\backslash \left\{\overline{1}\right\};b\in \overline{\Omega }$$

(61)

$${\displaystyle \sum _{\left(i,j\right)\in {\Phi }_f^{+}\left(i\right)}{z}_{ij}}+r_i^{out}-{\displaystyle \sum _{\left(j,i\right)\in {\Phi }_f^{-}\left(i\right)}{z}_{ji}}-r_i^{in}+{\displaystyle \sum _{\left(\overline{i},j\right)\in {\Phi }_f^{+}\left(\overline{i}\right)}{z}_{\overline{i}j}}+r_{\overline{i}}^{out}-{\displaystyle \sum _{\left(j,\overline{i}\right)\in {\Phi }_f^{-}\left(\overline{i}\right)}{z}_{j\overline{i}}}-r_{\overline{i}}^{in}=l_i-s_i,i\in N$$

(62)

$$t_i^h={\displaystyle \frac{{\displaystyle \sum _{\left(i,j\right)\in {\Phi }_f^{+}\left(i\right)}{z}_{ij}}+r_i^{out}+{\displaystyle \sum _{\left(j,i\right)\in {\Phi }_f^{-}\left(i\right)}{z}_{ji}}+r_i^{in}}{G_i}},i\in \overline{N}\backslash \left\{\overline{1}\right\}$$

(63)

$${\displaystyle \sum _{j<i}{r}_j^{out}-{\displaystyle \sum _{j<i}{r}_j^{in}}}\ge r_i^{in},i\in \overline{N}$$

(64)

$${\displaystyle \sum _{j>i}{r}_j^{in}-{\displaystyle \sum _{j>i}{r}_j^{out}}}\ge r_i^{out},i\in \overline{N}$$

(65)

$${\displaystyle \sum _{i\in \overline{N}}{r}_i^{out}}={\displaystyle \sum _{i\in \overline{N}}{r}_i^{in}}$$

(66)

$$r_i^{in},r_i^{out}\in {\mathbb{Z}}_{+},i\in \overline{N}$$

(67)

In model [MS2], the constraints (18)–(25) from [MS1], which originally involved $r_{ij}$, are replaced by $r_i^{in}$ and $r_i^{out}$, resulting in constraints (56)–(63). To preserve container flow balance, additional constraints (64)–(66) are introduced. Specifically, Constraint (64) ensures that the number of empty containers unloaded at each port does not exceed the remaining empty containers previously loaded by the ship. Constraint (65) imposes a similar condition for the number of empty containers loaded at each port. Constraint (66) enforces that the total number of loaded and unloaded empty containers across all ports is balanced, ensuring no empty containers remain at the end of the voyage. Constraint (67) ensures the integrality of the variables.

The two modeling methods have been fully introduced. Appendix B presents a proof of their mathematical equivalence, demonstrating that both models share identical LP-relaxation values. Furthermore, Appendix C provides an analysis of optimal solution properties, taking the empty container node-variable based model as an illustrative example.

5. Solution Method

5.1. Linearization of Chance Constraints

According to [73], if the random fluctuation ${\mathit{\xi }}_{T_b^{dam}}$ of lock passage time on the leg $b$ follows a normal distribution, then given its known distribution, the chance constraint can be linearized using stochastic programming methods, as follows:

$${\mathit{\mu }}_{T_b^{dam}}+{\mathit{\sigma }}_{T_b^{dam}}\left\{{\Phi }^{-1}\left(1-\omega \right)\right\}+{\overline{T}}_b^{dam}\le T_b^{dam},b\in \overline{\Omega }$$

(68)

where ${\mathit{\mu }}_{T_b^{dam}}$ denotes the expected value of the random fluctuation ${\mathit{\xi }}_{T_b^{dam}}$ of lock passage time on the leg $b$, and ${\mathit{\sigma }}_{T_b^{dam}}$ denotes its standard deviation.

5.2. Determination of the Maximum Fuel Consumption Function

In model [MS1], the objective function, as well as constraints (40) and (41), all involve the maximum fuel consumption function $Q_b^{max}{\left({\overline{v}}_b\right)}^{im}$ under the worst-case speed deviation. According to Equation (5), Under the condition that the actual speed $v_{bt}$ of the leg $b$ at time $t$ varies, $Q_b^{max}{\left({\overline{v}}_b\right)}^{im}$ is essentially implicit. To solve model [MS1], it is necessary to transform the implicit function $Q_b^{max}{\left({\overline{v}}_b\right)}^{im}$ into an explicit one $Q_b^{max}{\left({\overline{v}}_b\right)}^{ex}={\displaystyle \frac{{\overline{t}}_b}{2}}\alpha \left[{\left({\overline{v}}_b-V_b^{err}\right)}^{\beta }+{\left({\overline{v}}_b+V_b^{err}\right)}^{\beta }\right]$. The specific transformation method and the validation approach based on proof by contradiction are presented in Appendix D.

Accordingly, by applying Theorem 1, model [MS1] can be transformed into the FSRR model [MS3], which is formulated using empty container arc variables under the case of worst-case speed deviations as follows:

Objective function:

$$\begin{array}{l}\max F_3={\displaystyle \sum _{\left(i,j\right)\in {\Psi }^f}{\overline{P}}_{ij}{z}_{ij}}-{\displaystyle \sum _{\left(i,j\right)\in {\Psi }^e}{\overline{C}}_{ij}^e{r}_{ij}}-{\displaystyle \sum _{i\in \overline{N}}{C}_i^{fix}{x}_i}-{\displaystyle \sum _{i\in N}\left(C_i^{sc}{s}_i+C_i^{lc}{l}_i\right)}\\ -C^{oc}m-{\displaystyle \sum _{i\in \overline{N}}{C}_i^{py}{t}_i^l}-{\displaystyle \sum _{i\in \overline{N}}{C}_i^w{t}_i^w}-C^{c{o}_2}\theta {\displaystyle \sum _{b\in \overline{\Omega }}{Q}_b^{max}{\left({\overline{v}}_b\right)}^{ex}}-{\displaystyle \sum _{i\in \overline{N}}\left[C_i^{bk}{\chi }_i+g\left({\vartheta }_i\right)\right]}\end{array}$$

(69)

subject to constraints (13)–(33), (35)–(39), (42)–(52), (68) and

$$o_{i+1}^1=o_i^2-Q_{b=i}^{max}{\left({\overline{v}}_{b=i}\right)}^{ex},i\in \overline{N}\backslash \left\{\overline{1},\overline{2}\right\};b\in \overline{\Omega }\backslash \left\{2n-2\right\}$$

(70)

$$o_1^1=o_{2n-2}^2-Q_{2n-2}^{max}{\left({\overline{v}}_{2n-2}\right)}^{ex}$$

(71)

Similarly, model [MS2] can be transformed into the FSRR model [MS4] based on empty container node variables under the worst-case speed deviation, as follows:

Objective function:

$$\begin{array}{l}\max F_4={\displaystyle \sum _{\left(i,j\right)\in {\Psi }^f}{\overline{P}}_{ij}{z}_{ij}}-{\displaystyle \sum _{i\in \overline{N}}\left(C_i^{el}{r}_i^{out}+C_i^{eu}{r}_i^{in}\right)-}{\displaystyle \sum _{i\in \overline{N}}{C}_i^{fix}{x}_i}-{\displaystyle \sum _{i\in N}\left(C_i^{sc}{s}_i+C_i^{lc}{l}_i\right)}\\ -C^{oc}m-{\displaystyle \sum _{i\in \overline{N}}{C}_i^{py}{t}_i^l}-{\displaystyle \sum _{i\in \overline{N}}{C}_i^w{t}_i^w}-C^{c{o}_2}\theta {\displaystyle \sum _{b\in \overline{\Omega }}{Q}_b^{max}{\left({\overline{v}}_b\right)}^{ex}}-{\displaystyle \sum _{i\in \overline{N}}\left[C_i^{bk}{\chi }_i+g\left({\vartheta }_i\right)\right]}\end{array}$$

(72)

subject to constraints (13)–(17), (26)–(33), (35)–(39), (42)–(50), (52), (56)–(68), (70) and (71).

5.3. Linearization of Speed-Dependent Constraints

Both models feature the nonlinear constraint (42) and incorporate a nonlinear fuel consumption term in their objective functions. To linearize the nonlinear term ${\displaystyle \frac{1}{{\overline{v}}_b+{\tilde{v}}_b}}$ in constraint (42), a new auxiliary variable ${\overline{u}}_b={\displaystyle \frac{1}{{\overline{v}}_b+{\tilde{v}}_b}}$, $b\in \overline{\Omega }$ [69,70] is introduced as follows:

$${\overline{t}}_b=D_b{\overline{u}}_b,b\in \overline{\Omega }$$

(73)

Constraint (52) can be rewritten as:

$${\displaystyle \frac{1}{V^{max}+{\tilde{v}}_b-V_b^{err}}}={\overline{U}}_b^{\min }\le {\overline{u}}_b\le {\overline{U}}_b^{\max }={\displaystyle \frac{1}{V^{min}+{\tilde{v}}_b+V_b^{err}}},b\in \overline{\Omega }$$

(74)

The explicit fuel consumption function $Q_b^{max}{\left({\overline{v}}_b\right)}^{ex}$ of leg $b$ can be expressed as:

$$\begin{array}{ll}\hfill Q_b^{max}{\left({\overline{v}}_b\right)}^{ex}& ={\displaystyle \frac{{\overline{t}}_b}{2}}\alpha \left[{\left({\overline{v}}_b-V_b^{err}\right)}^{\beta }+{\left({\overline{v}}_b+V_b^{err}\right)}^{\beta }\right]\hfill \\ & ={\displaystyle \frac{D_b}{2\left({\overline{v}}_b+{\tilde{v}}_b\right)}}\alpha \left[{\left({\overline{v}}_b+{\tilde{v}}_b-{\tilde{v}}_b-V_b^{err}\right)}^{\beta }+{\left({\overline{v}}_b+{\tilde{v}}_b-{\tilde{v}}_b+V_b^{err}\right)}^{\beta }\right],b\in \overline{\Omega }\hfill \end{array}$$

(75)

$Q_b^{max}{\left({\overline{v}}_b\right)}^{ex}$ can be represented as the product of the leg distance $D_b$ and the fuel consumption rate per kilometer $Q_b\left({\overline{u}}_b\right)$:

$$Q_b\left({\overline{u}}_b\right)D_b={\displaystyle \frac{D_b{\overline{u}}_b}{2}}\alpha \left[{\left({\displaystyle \frac{1}{{\overline{u}}_b}}-{\tilde{v}}_b-V_b^{err}\right)}^{\beta }+{\left({\displaystyle \frac{1}{{\overline{u}}_b}}-{\tilde{v}}_b+V_b^{err}\right)}^{\beta }\right],b\in \overline{\Omega }$$

(76)

By substituting ${\displaystyle \sum _{b\in \overline{\Omega }}{Q}_b^{max}{\left({\overline{v}}_b\right)}^{im}}$ in model [MS3] with ${\displaystyle \sum _{b\in \overline{\Omega }}{Q}_b\left({\overline{u}}_b\right)D_b}$, an equivalent model [MS5] is obtained:

Objective function:

$$\begin{array}{l}\max F_5={\displaystyle \sum _{\left(i,j\right)\in {\Psi }^f}{\overline{P}}_{ij}{z}_{ij}}-{\displaystyle \sum _{\left(i,j\right)\in {\Psi }^e}{\overline{C}}_{ij}^e{r}_{ij}}-{\displaystyle \sum _{i\in \overline{N}}{C}_i^{fix}{x}_i}-{\displaystyle \sum _{i\in N}\left(C_i^{sc}{s}_i+C_i^{lc}{l}_i\right)}\\ -C^{oc}m-{\displaystyle \sum _{i\in \overline{N}}{C}_i^{py}{t}_i^l}-{\displaystyle \sum _{i\in \overline{N}}{C}_i^w{t}_i^w}-C^{c{o}_2}\theta {\displaystyle \sum _{b\in \overline{\Omega }}{Q}_b\left({\overline{u}}_b\right)D_b}-{\displaystyle \sum _{i\in \overline{N}}\left[C_i^{bk}{\chi }_i+g\left({\vartheta }_i\right)\right]}\end{array}$$

(77)

subject to constraints (13)–(33), (35)–(39), (43)–(51), (68), (73), (74) and

$$o_{i+1}^1=o_i^2-Q_{b=i}\left({\overline{u}}_{b=i}\right)D_{b=i},i\in \overline{N}\backslash \left\{\overline{1},\overline{2}\right\};b\in \overline{\Omega }\backslash \left\{2n-2\right\}$$

(78)

$$o_1^1=o_{2n-2}^2-Q_{2n-2}\left({\overline{u}}_{2n-2}\right)D_{2n-2}$$

(79)

Model [MS5] is a mixed-integer nonlinear programming model, with a nonlinear objective function (77) and refueling constraints (78) and (79). Due to the nature of model [MS5], establishing the convexity of the per kilometer fuel consumption function $Q_b\left({\overline{u}}_b\right)$, is sufficient to ensure that a global maximum value for model [MS5] exists.

$$Q_b\left({\overline{u}}_b\right)={\displaystyle \frac{{\overline{u}}_b}{2}}\alpha \left[{\left({\displaystyle \frac{1}{{\overline{u}}_b}}-{\tilde{v}}_b-V_b^{err}\right)}^{\beta }+{\left({\displaystyle \frac{1}{{\overline{u}}_b}}-{\tilde{v}}_b+V_b^{err}\right)}^{\beta }\right],b\in \overline{\Omega }$$

(80)

Based on regression analysis, the coefficients satisfy $\beta >2$ and $\alpha >0$. Obviously, $Q_b\left({\overline{u}}_b\right)$ is a convex function of speed over the interval $\left[{\overline{U}}_b^{\min },{\overline{U}}_b^{\max }\right]$.

By applying the same method, model [MS6] can be obtained based on model [MS4].

Objective function:

$$\begin{array}{l}\max F_6={\displaystyle \sum _{\left(i,j\right)\in {\Psi }^f}{\overline{P}}_{ij}{z}_{ij}}-{\displaystyle \sum _{i\in \overline{N}}\left(C_i^{el}{r}_i^{out}+C_i^{eu}{r}_i^{in}\right)-}{\displaystyle \sum _{i\in \overline{N}}{C}_i^{fix}{x}_i}-{\displaystyle \sum _{i\in N}\left(C_i^{sc}{s}_i+C_i^{lc}{l}_i\right)}\\ -C^{oc}m-{\displaystyle \sum _{i\in \overline{N}}{C}_i^{py}{t}_i^l}-{\displaystyle \sum _{i\in \overline{N}}{C}_i^w{t}_i^w}-C^{c{o}_2}\theta {\displaystyle \sum _{b\in \overline{\Omega }}{Q}_b\left({\overline{u}}_b\right)D_b}-{\displaystyle \sum _{i\in \overline{N}}\left[C_i^{bk}{\chi }_i+g\left({\vartheta }_i\right)\right]}\end{array}$$

(81)

subject to constraints (13)–(17), (26)–(33), (35)–(39), (43)–(50), (56)–(68), (73), (74), (78) and (79).

Currently, the model contains only one nonlinear component, namely the fuel cost function. To address this, a linear approximation technique for the nonlinear fuel consumption function is proposed in the following section.

5.4. Outer-Approximation Method

Increasing the number of tangents can improve approximation accuracy but also increases the computational burden [71]. To balance optimization precision and solution efficiency, the outer-approximation method is proposed. Firstly, an absolute objective tolerance $\epsilon$ (CNY) is specified to restrict the approximation error. The linear approximation method is required to produce a solution with an objective deviation no greater than $\epsilon$. The total tolerance is subsequently normalized by voyage distance to obtain a distance-based tolerance $\overline{\epsilon }$ (CNY·ton/km), calculated by Equation (82). When the local approximation error of $Q_b\left({\overline{u}}_b\right)$ is controlled within $\overline{\epsilon }$, the global objective error remains bounded by $\epsilon$ [69].

$$\overline{\epsilon }={\displaystyle \frac{\epsilon }{\max C_i^{fp}}}\times {\displaystyle \frac{1}{{\displaystyle \sum _{b\in \overline{\Omega }}{D}_b}}}$$

(82)

The outer-approximation method is designed to bound the approximation error of the nonlinear function $Q_b\left({\overline{u}}_b\right)$ at any point ${\overline{u}}_b$ within ${\overline{u}}_b\in \left[{\overline{U}}_b^{\min },{\overline{U}}_b^{\max }\right]$, while achieving the minimum number of linear segments required for accurate representation. Before formally presenting the method, the derivative of $Q_b\left({\overline{u}}_b\right)$ at ${\overline{u}}_b$, denoted by $Q_b^{\prime }\left({\overline{u}}_b\right)$, is defined as follows:

$$Q_b^{\prime }\left({\overline{u}}_b\right)={\displaystyle \frac{\alpha }{2}}{\left({\displaystyle \frac{1}{{\overline{u}}_b}}-{\tilde{v}}_b-V_b^{err}\right)}^{\beta }-{\displaystyle \frac{\alpha \beta }{2{\overline{u}}_b}}{\left({\displaystyle \frac{1}{{\overline{u}}_b}}-{\tilde{v}}_b+V_b^{err}\right)}^{\beta -1}$$

(83)

Based on Algorithm 1, a piecewise linear approximation function ${\overline{Q}}_b\left({\overline{u}}_b\right)$ of $Q_b\left({\overline{u}}_b\right)$ can be generated. The Algorithm 1 is as follows:

Algorithm 1. Construction of piecewise linear approximation function

Step 0: Initialize $\Lambda$ as the tangent line set, with ${\overline{U}}_b^{\min }={\displaystyle \frac{1}{V^{\max }+{\tilde{v}}_b-V_b^{err}}}$ and $U_b^{\max }={\displaystyle \frac{1}{V^{\min }+{\tilde{v}}_b+V_b^{err}}}$. Set $\Lambda :=\varnothing$, $k:=0$, ${\overline{u}}_{b1}={\overline{U}}_b^{\min }$, and $Q_{b1}=Q_b\left({\overline{u}}_{b1}\right)-\overline{\epsilon }$. Step 1: Increase the iteration index by one, i.e., $k:=k+1$. If the following inequality is satisfied:

$${\displaystyle \frac{Q_b\left({\overline{U}}_b^{\max }\right)-Q_{bk}}{{\overline{U}}_b^{\max }-{\overline{u}}_{bk}}}\ge Q_b^{\prime }\left({\overline{U}}_b^{\max }\right)$$ (84)

That is, the point $\left({\overline{u}}_{bk},Q_b\left({\overline{u}}_{bk}\right)\right)$ lies on or below the tangent line $Q_b\left({\overline{u}}_b\right)$ at ${\overline{U}}_b^{\max }$, as shown in Figure 9a. The tangent line $k$ is then added to the set of tangent lines $\Lambda$, and the set is updated as follows:

$$Q_b-Q_b\left({\overline{U}}_b^{\max }\right)=Q_b^{\prime }\left({\overline{U}}_b^{\max }\right)\left(u_b-{\overline{U}}_b^{\max }\right)$$ (85)

Next, proceed to Step 3. If the above condition is not satisfied, the tangent line $k$ passing through the point $\left({\overline{u}}_{bk},Q_{bk}\right)$ and supporting the function $Q_b\left({\overline{u}}_b\right)$ is added to the set of tangent lines $\Lambda$. Assum the tangent line $k$ supports the function $Q_b\left({\overline{u}}_b\right)$ at point $\left({\widehat{u}}_{bk},{\widehat{Q}}_{bk}\right)$, this point can be calculated using Equation (86).

$${\widehat{Q}}_{bk}={\displaystyle \frac{{\widehat{u}}_{bk}}{2}}\alpha \left[{\left({\displaystyle \frac{1}{{\widehat{u}}_{bk}}}-{\tilde{v}}_b-V_b^{err}\right)}^{\beta }+{\left({\displaystyle \frac{1}{{\widehat{u}}_{bk}}}-{\tilde{v}}_b+V_b^{err}\right)}^{\beta }\right]$$ (86)

According to the definition:

$$\begin{array}{ll}\hfill {\displaystyle \frac{{\widehat{Q}}_{bk}-Q_{bk}}{{\widehat{u}}_{bk}-{\overline{u}}_{bk}}}=Q_b^{\prime }\left({\widehat{u}}_{bk}\right)& ={\displaystyle \frac{\alpha }{2}}{\left({\displaystyle \frac{1}{{\widehat{u}}_{bk}}}-{\tilde{v}}_b-V_b^{err}\right)}^{\beta }-{\displaystyle \frac{\alpha \beta }{2{\widehat{u}}_{bk}}}{\left({\displaystyle \frac{1}{{\widehat{u}}_{bk}}}-{\tilde{v}}_b-V_b^{err}\right)}^{\beta -1}\hfill \\ & +{\displaystyle \frac{\alpha }{2}}{\left({\displaystyle \frac{1}{{\widehat{u}}_{bk}}}-{\tilde{v}}_b+V_b^{err}\right)}^{\beta }-{\displaystyle \frac{\alpha \beta }{2{\widehat{u}}_{bk}}}{\left({\displaystyle \frac{1}{{\widehat{u}}_{bk}}}-{\tilde{v}}_b+V_b^{err}\right)}^{\beta -1}\hfill \end{array}$$ (87)

By combining Equations (86) and (87), the value of ${\widehat{u}}_{bk}$ is obtained numerically via the bisection method. Therefore, according to Equation (88), the tangent line $k$ can be obtained.

$$Q_b-Q_{bk}={\displaystyle \frac{{\widehat{Q}}_{bk}-Q_{bk}}{{\widehat{u}}_{bk}-{\overline{u}}_{bk}}}\left({\overline{u}}_b-{\overline{u}}_{bk}\right)$$ (88)

Proceed to Step 2.

Step 2: For the tangent line $k$, assess whether the following inequality holds when ${\overline{u}}_b={\overline{U}}_b^{\max }$.

$$Q_{bk}+{\displaystyle \frac{{\widehat{Q}}_{bk}-Q_{bk}}{{\widehat{u}}_{bk}-{\overline{u}}_{bk}}}\left({\overline{U}}_b^{\max }-{\overline{u}}_{bk}\right)\ge Q_b\left({\overline{U}}_b^{\max }\right)-\overline{\epsilon }$$ (89)

When the inequality holds, the gap between the tangent line $k$ and $Q_b\left({\overline{u}}_b\right)$ at ${\overline{u}}_b$ is bounded by $\overline{\epsilon }$, even if ${\overline{u}}_b={\overline{U}}_b^{\max }$. This conclusion can be extended to any ${\overline{u}}_b$ between ${\overline{u}}_{bk}$ and ${\overline{U}}_b^{\max }$, as shown in Figure 9b, and then proceed to Step 3. Otherwise, there exists a unique point $\left({\overline{u}}_{b\left(k+1\right)},Q_{b\left(k+1\right)}\right)$ on the tangent line $k$ such that ${\overline{u}}_{bk}<{\overline{u}}_{b\left(k+1\right)}<{\overline{U}}_b^{\max }$ and $Q_{b\left(k+1\right)}=Q_b\left({\overline{u}}_{b\left(k+1\right)}\right)-\overline{\epsilon }$, as shown in Figure 9c. In this case, ${\overline{u}}_{b\left(k+1\right)}$ can be determined by the bisection method, followed by a return to Step 1. Step 3: Let $K_b$ be the number of tangent lines in the current set $\Lambda$. Each tangent line $k$, $k=1,2,\dots ,K_b$ in $\Lambda$ is expressed in the general form:

$$Q_b=slop{e}_{bk}\cdot {\overline{u}}_b+intercep{t}_{bk}$$ (90)

The piecewise linear approximation function ${\overline{Q}}_b\left({\overline{u}}_b\right)$(see Figure 9d) can be expressed as:

$${\overline{Q}}_b\left({\overline{u}}_b\right)=\max \left\{slop{e}_{bk}\cdot {\overline{u}}_b+intercep{t}_{bk},k=1,2,\dots ,K_b\right\}$$ (91)

Based on the above algorithm, by replacing $Q_b\left({\overline{u}}_b\right)$ in model [MS5] with the piecewise linear approximation function ${\overline{Q}}_b\left({\overline{u}}_b\right)$, the approximate model [MS7] is obtained:

Objective function:

$$\begin{array}{l}\max F_7={\displaystyle \sum _{\left(i,j\right)\in {\Psi }^f}{\overline{P}}_{ij}{z}_{ij}}-{\displaystyle \sum _{\left(i,j\right)\in {\Psi }^e}{\overline{C}}_{ij}^e{r}_{ij}}-{\displaystyle \sum _{i\in \overline{N}}{C}_i^{fix}{x}_i}-{\displaystyle \sum _{i\in N}\left(C_i^{sc}{s}_i+C_i^{lc}{l}_i\right)}\\ -C^{oc}m-{\displaystyle \sum _{i\in \overline{N}}{C}_i^{py}{t}_i^l}-{\displaystyle \sum _{i\in \overline{N}}{C}_i^w{t}_i^w}-C^{c{o}_2}{\displaystyle \sum _{b\in \overline{\Omega }}{\overline{Q}}_b\left({\overline{u}}_b\right)D_b}-{\displaystyle \sum _{i\in \overline{N}}\left[C_i^{bk}{\chi }_i+g\left({\vartheta }_i\right)\right]}\end{array}$$

(92)

subject to constraints (13)–(33), (35)–(39), (43)–(51), (68), (73), (74) and

$$o_{i+1}^1=o_i^2-{\overline{Q}}_{b=i}\left({\overline{u}}_{b=i}\right)D_{b=i},i\in \overline{N}\backslash \left\{\overline{1},\overline{2}\right\};b\in \overline{\Omega }\backslash \left\{2n-2\right\}$$

(93)

$$o_1^1=o_{2n-2}^2-{\overline{Q}}_{2n-2}\left({\overline{u}}_{2n-2}\right)D_{2n-2}$$

(94)

The principle of Algorithm 1 and the convexity of the function $Q_b\left({\overline{u}}_b\right)$ imply that:

$$Q_b\left({\overline{u}}_b\right)-\overline{\epsilon }\le {\overline{Q}}_b\left({\overline{u}}_b\right)\le Q_b\left({\overline{u}}_b\right),b\in \overline{\Omega };{\overline{u}}_b\in \left[{\overline{U}}_b^{\min },{\overline{U}}_b^{\max }\right]$$

(95)

Therefore, the total objective value error of model [MS7] with respect to the original model [MS1] can be controlled within the predefined tolerance $\epsilon$.

Furthermore, the convexity of the function $Q_b\left({\overline{u}}_b\right)$ ensures that its approximation $Q_b\left({\overline{u}}_b\right)$ remains convex. As a result, model [MS7] can be reformulated equivalently as model [MS8]:

Objective function:

$$\begin{array}{l}\max F_8={\displaystyle \sum _{\left(i,j\right)\in {\Psi }^f}{\overline{P}}_{ij}{z}_{ij}}-{\displaystyle \sum _{\left(i,j\right)\in {\Psi }^e}{\overline{C}}_{ij}^e{r}_{ij}}-{\displaystyle \sum _{i\in \overline{N}}{C}_i^{fix}{x}_i}-{\displaystyle \sum _{i\in N}\left(C_i^{sc}{s}_i+C_i^{lc}{l}_i\right)}-C^{oc}m\\ -{\displaystyle \sum _{i\in \overline{N}}{C}_i^{py}{t}_i^l}-{\displaystyle \sum _{i\in \overline{N}}{C}_i^w{t}_i^w}-C^{c{o}_2}\theta {\displaystyle \sum _{b\in \overline{\Omega }}{\overline{Q}}_b\left({\overline{u}}_b\right)D_b}-{\displaystyle \sum _{i\in \overline{N}}\left[C_i^{bk}{\chi }_i+g\left({\vartheta }_i\right)\right]}\end{array}$$

(96)

subject to constraints (13)–(33), (35)–(39), (43)–(51), (68), (73), (74), (93), (94) and

$$Q_b\left({\overline{u}}_b\right)\ge slop{e}_{bk}\cdot {\overline{u}}_b+intercep{t}_{bk},b\in \overline{\Omega };k=1,2,\dots ,K_b$$

(97)

Model [MS8] can be efficiently solved using a solver. Let the optimal objective values of the original model [MS1] and the approximate model [MS8] be $Opt$ and $MUB$, respectively, where $MUB$ is an upper bound of $Opt$, and let $\left(x_i^{*},z_{ij}^{*},s_i^{*},l_i^{*},{\overline{u}}_b^{*},t_i^{h*},t_i^{a*},t_i^{d*},t_i^{l*},t_i^{w*},m^{*},{\chi }_i^{*},{\vartheta }_i^{*},{\overline{t}}_b^{*},o_i^{1*},o_i^{2*},r_{ij}^{*},Q_b^{*}\right)$ be the optimal solution of model [MS8]. Obviously, $\begin{array}{l}(x_i=x_i^{*},z_{ij}=z_{ij}^{*},s_i=s_i^{*},l_i=l_i^{*},{\overline{v}}_b={\displaystyle \frac{1}{{\overline{u}}_b^{*}}},t_i^h=t_i^{h*},t_i^a=t_i^{a*},t_i^d=t_i^{d*},\\ t_i^l=t_i^{l*},t_i^w=t_i^{w*},m=m^{*},{\chi }_i={\chi }_i^{*},{\vartheta }_i={\vartheta }_i^{*},{\overline{t}}_b={\overline{t}}_b^{*},o_i^1=o_i^{1*},o_i^2=o_i^{2*},r_{ij}=r_{ij}^{*})\end{array}$ is a feasible solution for model [MS1], so a lower bound $MLB$ of $Opt$ can be determined as follows:

$$\begin{array}{l}MLB={\displaystyle \sum _{\left(i,j\right)\in {\Psi }^f}{\overline{P}}_{ij}{z}_{ij}^{*}}-{\displaystyle \sum _{\left(i,j\right)\in {\Psi }^e}{\overline{C}}_{ij}^e{r}_{ij}^{*}}-{\displaystyle \sum _{i\in \overline{N}}{C}_i^{fix}{x}_i^{*}}-{\displaystyle \sum _{i\in N}\left(C_i^{sc}{s}_i^{*}+C_i^{lc}{l}_i^{*}\right)}-C^{oc}{m}^{*}\\ -{\displaystyle \sum _{i\in \overline{N}}{C}_i^{py}{t}_i^{l*}}-{\displaystyle \sum _{i\in \overline{N}}{C}_i^w{t}_i^{w*}}-C^{c{o}_2}\theta {\displaystyle \sum _{b\in \overline{\Omega }}{Q}_b\left({\overline{u}}_b^{*}\right)D_b}-{\displaystyle \sum _{i\in \overline{N}}\left[C_i^{bk}{\chi }_i^{*}+g\left({\vartheta }_i^{*}\right)\right]}\end{array}$$

(98)

Based on the principle of the piecewise linear approximation method, the following conclusions can be drawn:

$$MUB-\epsilon \le MLB\le Opt\le MUB$$

(99)

By applying the same method, model [MS9] can be obtained from model [MS6]:

Objective function:

$$\begin{array}{l}\max F_7={\displaystyle \sum _{\left(i,j\right)\in {\Psi }^f}{\overline{P}}_{ij}{z}_{ij}}-{\displaystyle \sum _{i\in \overline{N}}\left(C_i^{el}{r}_i^{out}+C_i^{eu}{r}_i^{in}\right)-}{\displaystyle \sum _{i\in \overline{N}}{C}_i^{fix}{x}_i}-{\displaystyle \sum _{i\in N}\left(C_i^{sc}{s}_i+C_i^{lc}{l}_i\right)}\\ -C^{oc}m-{\displaystyle \sum _{i\in \overline{N}}{C}_i^{py}{t}_i^l}-C^{c{o}_2}\theta {\displaystyle \sum _{b\in \overline{\Omega }}{\overline{Q}}_b\left({\overline{u}}_b\right)D_b}-{\displaystyle \sum _{i\in \overline{N}}\left[C_i^{bk}{\chi }_i+g\left({\vartheta }_i\right)\right]}\end{array}$$

(100)

subject to constraints (13)–(17), (26)–(33), (35)–(39), (43)–(50), (56)–(68), (73), (74), (93), (94) and (97).

At this point, the linearization of the nonlinear models [MS1] and [MS2] has been completed, with the error of the approximation models effectively controlled.

6. Model Enhancement for Handling Indivisible Demand

Typically, shipping companies can select customers based on resource allocation and voyage planning (i.e., shipping container cargo selectively), meaning that demand is divisible. Nevertheless, in some cases, bookings are only available from freight forwarders, rendering container demand indivisible. Under this condition, the ship must either call at both ports and shipping all containers between the port pair, or not shipping any containers between them at all, i.e., all containers from port $i$ to $j$ are either fully shipped or not shipped at all. To effectively address this type of problem, a binary variable ${\gamma }_{ij}$ is introduced:

$${\gamma }_{ij}=\left\{\begin{array}{l}1 if \mathrm{shipping} demand from i to j is completely fulfilled by ship\\ 0 otherwise\end{array}\right. \left(i,j\right)\in {\Psi }^f$$

To address indivisible demand, the variables $z_{ij}$ in models [MS1] and [MS2] are replaced by ${\overline{B}}_{ij}{\gamma }_{ij}$, resulting in models [MU1] and [MU2], respectively. In this case, models [MU1] and [MU2] remain equivalent. The introduction of additional binary variables makes solving models [MU1] and [MU2] more challenging than solving their divisible demand counterparts. Studying the performance of these two models under indivisible demand provides valuable insight into the advantages of different modeling methods. Indivisible demand limits the flexibility of shipping operations and results in suboptimal vessel capacity utilization. In addition, due to the significant costs of empty container leasing and storage, shipping companies cannot rely solely on adjusting full container shipping strategies to balance container flows under indivisible demand; they must depend on empty container repositioning to maintain voyage profit. Using the same method, the models [MU1] and [MU2] for indivisible demand can be linearized, resulting in models [MU3] and [MU4].

Both models share the same lower bound when solving this problem, but their computational performance remains unclear. Compared with models [MS8]/[MU3], models [MS9]/[MU4] optimize model size by reducing decision variables while simultaneously introducing more constraints. Therefore, numerical experiments are needed to verify the solution efficiency and practical performance of both models.

7. Numerical Experiments

7.1. Benchmark Instances

As a case study, consider container liner shipping along the Yangtze River, see Figure 10. The ports where the ship can call from downstream to upstream include Shanghai, Nantong, Suzhou, Jiangyin, Taizhou, Yangzhou, Zhenjiang, Nanjing, Maanshan, Wuhu, Tongling, Anqing, Jiujiang, Wuhan, Yueyang, Jingzhou, Yichang, Fuling, Chongqing, Luzhou, and Yibin. Currently, LNG refueling ports along the Yangtze River have been established at Shanghai, Zhenjiang, Wuhu, Jiujiang, Wuhan, Yichang, and Chongqing.

Port preparation time $T_i^p$ is set to 2 h, and refueling time $T_i^f$ is set to 3 h. The ship minimum speed $V^{\min }$ and maximum speed $V^{\max }$ are set to 7 km/h and 31 km/h, respectively. The average streamflow velocity ${\widehat{v}}_b$ along the legs is set to 4 km/h, with an extreme speed deviation of 4 km/h. The maximum number of ships allowed on a route ${\varpi }^{\max }$ is 4. Anchorage fees apply while ships are waiting, but the cost is relatively low, so $C_i^w$ is set at 50 CNY/h. Ship delays significantly affect port operations, so the penalty cost $C_i^{py}$ is set at 5000 CNY/h. Port charges are related to ship capacity. $C_i^{fix}$ is set based on a charge of 10 CNY/TEU per unit container, but the cost shall not exceed 5000 CNY per calling port. Container handling rate $G_i$ is 23 TEU/h, and handling costs for both empty and full containers ($C_i^{el}$, $C_i^{eu}$, $C_i^{zl}$, $C_i^{zu}$) are all set at 100 CNY/TEU. Fixed LNG refueling cost $C_i^{bk}$ is set at 1000 CNY. The first LNG price discount rate ${\iota }_i^1$ and second LNG price discount rate ${\iota }_i^2$ are set at 90% and 80%, respectively, with LNG refueling amount at the first discount ${\xi }_i^1$ and second discount ${\xi }_i^2$ set at 10 ton and 20 ton, respectively. The maximum number of refueling operations $L$ for a round-trip voyage is 4. The average lock passage time $T_b^{dam}$ is set to 15 h, with stochastic lock passage time ${\mathit{\xi }}_{T_b^{dam}}$ having an expected value ${\mu }_{T_b^{dam}}$ of 5 and variance ${\sigma }_{T_b^{dam}}$ of 15 [74]. The risk parameter $\omega$ is set at 0.1. The absolute objective value tolerance $\overline{\epsilon }$ is set to 0.01$\alpha$ [70]. The weight of full containers is roughly 10 times that of empty containers, with parameter ${\sigma }^f$, ${\sigma }^e$ set to 1. According to calculations, the CO₂ emission factor of LNG fuel $\theta$ is 2.98. Most inland LNG-fuelled ships adopt fixed onboard fuel supply systems. Depending on ship type, these systems have capacities of 20 m³, 30 m³, 50 m³, which can be combined according to the ship operating route, deadweight tonnage, and budget. Based on LNG fuel properties, 1 m³ of LNG weighs approximately 0.43 ton. Accordingly, the typical inland LNG-fuelled container ship types on the Yangtze River are shown in

Table A1. Parameters of typical inland LNG-fuelled container ship types.

Ship Type	${\mathit{\lambda }}^{\mathit{f}}$ (m)	${\mathit{\delta }}^{\mathit{f}}$ (m)	$\mathit{A}$ (TEU)	Breadth (m)	$\mathit{Y}$ (m)	$\mathit{J}$ (m)	$\mathit{R}$ (m)	Designed Draft (m)	${\mathit{C}}^{\mathit{o}\mathit{c}}$ (CNY/Week)	$\mathit{\alpha }$	$\mathit{\beta }$	$\mathit{\upsilon }$ (ton)	$\mathit{\rho }$ (ton)
R1	0.07000	0.00500	100	9.0	7.0	7.5	2.0	2.5	30,000	0.000384	2.048	21.5	8.6
R2	0.05000	0.00267	150	10.5	7.5	8.0	2.6	3.0	50,000	0.000354	2.128	25.8	8.6
R3	0.04000	0.00250	200	10.8	8.0	8.5	3.0	4.0	60,000	0.000366	2.128	25.8	8.6
R4	0.03000	0.00167	300	13.6	9.0	9.5	3.5	4.0	80,000	0.000386	2.145	25.8	8.6
R5	0.02714	0.00143	350	16.4	9.5	10.0	3.5	4.0	90,000	0.000316	2.235	34.4	8.6
R6	0.02222	0.00133	450	17.5	10.0	10.5	3.6	4.2	100,000	0.000189	2.478	34.4	8.6
R7	0.02300	0.00160	500	19.0	11.5	12.0	3.7	4.5	120,000	0.000201	2.478	34.4	8.6
R8	0.02231	0.00154	650	21.5	14.5	15.0	5.0	6.0	130,000	0.000354	2.478	21.5	8.6
R9	0.01966	0.00106	941	25.6	18.5	19.0	5.0	6.0	150,000	0.000486	2.478	21.5	8.6
R10	0.01886	0.00088	1140	26.0	21.5	22.0	6.0	7.0	160,000	0.000521	2.478	21.5	8.6

.

The container shipping demand between port pairs $B_{ij}$, and the container freight rates $P_{ij}$, are commercial secrets. Therefore, estimates are made based on relevant studies and industry practice, as shown in

Table A2. Container shipping demands between port pairs (TEU).

No	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16	17	18	19	20	21
1	0	133	125	71	46	72	51	210	66	77	21	59	85	152	12	10	12	10	100	37	84
2	80	0	51	29	18	29	21	85	27	31	9	24	34	61	5	4	5	4	40	15	33
3	75	33	0	27	18	27	19	79	25	29	8	22	32	57	4	4	4	4	38	14	32
4	41	18	18	0	9	15	11	44	14	16	4	12	18	31	3	2	3	2	21	8	18
5	26	12	11	6	0	9	7	28	9	10	3	8	11	20	1	1	1	1	13	5	11
6	42	18	18	10	6	0	11	45	14	16	4	12	18	32	3	2	3	2	21	8	18
7	30	13	12	7	5	7	0	31	10	12	3	9	12	23	2	2	2	2	15	6	12
8	132	60	56	32	21	33	23	0	45	51	14	39	57	102	8	7	8	7	67	25	56
9	38	17	16	9	6	9	6	27	0	15	4	12	16	29	2	2	2	2	19	7	16
10	45	20	18	11	7	11	8	31	10	0	5	13	19	34	3	3	3	3	22	9	18
11	12	5	5	3	2	3	2	9	3	3	0	3	5	9	0	0	0	0	6	2	5
12	34	15	15	8	6	9	6	24	8	9	3	0	15	26	2	2	2	2	17	6	15
13	49	22	21	12	8	12	9	35	11	13	3	10	0	38	3	3	3	3	25	9	21
14	92	41	39	22	14	22	16	65	21	24	6	18	27	0	6	5	6	5	47	18	39
15	6	3	3	2	1	2	1	4	1	2	0	1	2	3	0	0	0	0	3	1	3
16	46	21	20	11	7	12	8	33	10	12	3	9	13	24	2	0	3	3	24	9	20
17	51	23	21	12	8	12	9	36	11	13	3	10	15	26	2	2	0	3	26	9	21
18	6	3	3	1	1	1	1	4	1	2	0	1	2	3	0	0	0	0	3	1	3
19	59	27	25	14	9	15	10	42	13	15	4	12	17	30	2	2	2	2	0	11	25
20	21	9	9	5	3	5	3	15	5	6	1	4	6	11	1	0	1	0	7	0	9
21	48	22	21	12	8	12	9	34	11	12	3	9	14	25	2	2	2	2	16	6	0

and

Table A3. Container freight rates between port pairs (CNY/TEU).

No	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16	17	18	19	20	21
1	0	1676	1726	1745	1791	1823	1831	1855	1880	1903	1947	1984	2023	2027	2304	2433	2677	3419	3512	3650	3748
2	1676	0	1600	1619	1676	1717	1728	1763	1791	1816	1865	1909	1958	1975	2235	2365	2601	3320	3413	3554	3652
3	1726	1600	0	1569	1630	1675	1687	1727	1755	1782	1833	1879	1932	1955	2208	2338	2570	3281	3374	3515	3613
4	1745	1619	1569	0	1612	1659	1671	1713	1742	1769	1821	1867	1923	1947	2198	2328	2559	3266	3359	3501	3599
5	1791	1676	1630	1612	0	1602	1615	1663	1694	1722	1777	1827	1888	1920	2162	2291	2518	3213	3306	3449	3547
6	1823	1717	1675	1659	1602	0	1565	1619	1650	1680	1738	1790	1856	1895	2129	2258	2481	3165	3258	3403	3500
7	1831	1728	1687	1671	1615	1565	0	1605	1637	1667	1726	1779	1847	1887	2118	2248	2470	3150	3243	3388	3486
8	1855	1763	1727	1713	1663	1619	1605	0	1583	1615	1677	1734	1808	1856	2077	2207	2424	3091	3184	3331	3428
9	1880	1791	1755	1742	1694	1650	1637	1583	0	1583	1646	1706	1784	1837	2052	2182	2395	3054	3147	3294	3392
10	1903	1816	1782	1769	1722	1680	1667	1615	1583	0	1616	1678	1759	1817	2026	2156	2366	3017	3110	3258	3356
11	1947	1865	1833	1821	1777	1738	1726	1677	1646	1616	0	1616	1707	1775	1971	2101	2305	2937	3029	3180	3278
12	1984	1909	1879	1867	1827	1790	1779	1734	1706	1678	1616	0	1649	1730	1911	2041	2237	2849	2942	3095	3193
13	2023	1958	1932	1923	1888	1856	1847	1808	1784	1759	1707	1649	0	1651	1807	1936	2121	2698	2791	2948	3046
14	2027	1975	1955	1947	1920	1895	1887	1856	1837	1817	1775	1730	1651	0	1673	1803	1971	2504	2597	2759	2857
15	2304	2235	2208	2198	2162	2129	2118	2077	2052	2026	1971	1911	1807	1673	0	1680	1833	2326	2418	2586	2683
16	2433	2365	2338	2328	2291	2258	2248	2207	2182	2156	2101	2041	1936	1803	1680	0	1688	2137	2230	2402	2500
17	2677	2601	2570	2559	2518	2481	2470	2424	2395	2366	2305	2237	2121	1971	1833	1688	0	1958	2051	2228	2326
18	3419	3320	3281	3266	3213	3165	3150	3091	3054	3017	2937	2849	2698	2504	2326	2137	1958	0	1643	1831	1929
19	3512	3413	3374	3359	3306	3258	3243	3184	3147	3110	3029	2942	2791	2597	2418	2230	2051	1643	0	1741	1839
20	3650	3554	3515	3501	3449	3403	3388	3331	3294	3258	3180	3095	2948	2759	2586	2402	2228	1831	1741	0	1648
21	3748	3652	3613	3599	3547	3500	3486	3428	3392	3356	3278	3193	3046	2857	2683	2500	2326	1929	1839	1648	0

[74]. The bridge clearance height $H_b$, water depth $W_b$, leg distance $D_b$, empty container leasing cost $C_i^{lc}$, empty container storage cost $C_i^{sc}$ for each leg, and LNG fuel price $C_i^{fp}$ at ports where refueling are shown in

Table A4. Parameters of legs and ports.

No	Port	Leg	${\mathit{D}}_{\mathit{b}}$ (km)	${\mathit{W}}_{\mathit{b}}$ (m)	${\mathit{H}}_{\mathit{b}}$ (m)	${\mathit{C}}_{\mathit{i}}^{\mathit{f}\mathit{p}}$ (CNY/ton)	${\mathit{C}}_{\mathit{i}}^{\mathit{s}}$ (CNY/TEU)	${\mathit{C}}_{\mathit{i}}^{\mathit{l}}$ (CNY/TEU)
1	Shanghai	Shanghai→Nantong	128	12.5	68	5600	200	550
2	Nantong	Nantong→Suzhou	51	12.5	60	–	200	500
3	Suzhou	Suzhou→Jiangyin	19	12.5	48	–	200	400
4	Jiangyin	Jiangyin→Taizhou	69	12.5	48	–	200	500
5	Taizhou	Taizhou→Yangzhou	62	12.5	48	–	200	300
6	Yangzhou	Yangzhou→Zhenjiang	19	12.5	48	–	200	200
7	Zhenjiang	Zhenjiang→Nanjing	77	12.5	48	6500	200	500
8	Nanjing	Nanjing→Maanshan	48	10.5	23	–	200	480
9	Maanshan	Maanshan→Wuhu	48	9.0	23	–	200	280
10	Wuhu	Wuhu→Tongling	104	6.0	23	6200	200	500
11	Tongling	Tongling→Anqing	113	6.0	23	–	200	500
12	Anqing	Anqing→Jiujiang	196	5.0	23	–	200	200
13	Jiujiang	Jiujiang→Wuhan	251	5.0	23	6000	200	500
14	Wuhan	Wuhan→Yueyang	231	4.2	17	5800	200	500
15	Yueyang	Yueyang→Jingzhou	244	3.8	17	–	200	200
16	Jingzhou	Jingzhou→Yichang	232	4.0	17	–	200	500
17	Yichang	Yichang→Fuling	528	4.5	17	6100	200	500
18	Fuling	Fuling→Chongqing	120	3.8	17	–	200	200
19	Chongqing	Chongqing→Luzhou	254	2.9	17	5700	200	500
20	Luzhou	Luzhou→Yibin	130	2.9	17	–	200	500
21	Yibin	–	–	–	–	–	200	500

. The time window start time $T_i^S$ of various typical ships at each port is shown in

Table A5. Time window start time for each port (h).

No	Port	R1	R2–R6	R7	R8–R10
1	Shanghai	0/672	0/672	0/504	0/504
2	Nantong	55/605	63/654	60/487	68/476
3	Suzhou	85/593	84/641	79/472	119/444
4	Jiangyin	100/584	103/640	98/462	147/434
5	Taizhou	113/580	122/628	117/451	172/423
6	Yangzhou	126/568	136/614	133/436	206/411
7	Zhenjiang	136/567	149/613	149/428	220/383
8	Nanjing	149/554	165/595	167/405	239/352
9	Maanshan	165/551	187/593	194/395	277/341
10	Wuhu	169/548	191/590	207/386	302
11	Tongling	184/542	199/584	224/379	–
12	Anqing	192/535	207/578	235/366	–
13	Jiujiang	206/523	221/567	260/343	–
14	Wuhan	223/507	241/543	294	–
15	Yueyang	239/493	270/528	–	–
16	Jingzhou	256/478	288/512	–	–
17	Yichang	273/463	306/497	–	–
18	Fuling	325/416	385/424	–	–
19	Chongqing	333/403	394	–	–
20	Luzhou	357/387	–	–	–
21	Yibin	366	–	–	–

, while the time window end time $T_i^E$ is calculated according to $T_i^E=T_i^S+10$. All tests are conducted on a computer equipped with an Intel (R) Core (TM) i5-12500H (2.50 GHz) processor and 16.0 GB of memory, under Windows 10 64-bit. All models were solved using IBM ILOG CPLEX 12.6.2, with all solver parameters set to their default values.

7.2. Reality Verification of the Method

Table 1. Comparison of the models in terms of decision variables and constraints.

Model	Indicator	R1	R2	R3	R4	R5	R6	R7	R8	R9	R10
Model [MS8] /Model [MU3]	constraint	3709	3555	3555	3627	4113	5535	3871	2610	2610	2610
	variable	1773	1525	1525	1525	1525	1525	976	607	607	607
Model [MS9] /Model [MU4]	constraint	3816	3652	3652	3724	4210	5632	3942	2662	2662	2662
	variable	1456	1276	1276	1276	1276	1276	861	565	565	565

provides a comparative summary of the two models in terms of the number of decision variables and constraints, thereby illustrating their structural complexity and computational characteristics. It is important to note that when the number of ports and the fuel consumption coefficient $\beta$ are fixed, the numbers of decision variables and constraints in the models remain unchanged.

From Table 1, it can be observed that the model [MS9]/[MU4], which is based on empty container node variables, contain fewer decision variables than the model [MS8]/[MU3], which is based on empty container arc variables, while [MS9]/[MU4] involves a larger number of constraints. However, the two models show no substantial difference in the number of constraints.

Table 2. Comparison of computational performance between models [MS8] and [MS9].

Ship Type	Model [MS8]					Model [MS9]
	Time (s)	Nodes	LP Gap (%)	$\mathit{L}\mathit{P}$	$\mathbf{Profit-}\mathit{L}\mathit{B}$	Time (s)	Nodes	LP Gap (%)	$\mathit{L}\mathit{P}$	$\mathbf{Profit-}\mathit{L}\mathit{B}$
R1	4.39	6439	0.00%	170,088.60	170,088.60	4.42	7456	0.00%	170,088.60	170,088.60
R2	1.78	2913	0.00%	318,827.37	318,827.37	2.05	5696	0.00%	318,827.37	318,827.37
R3	7.16	9387	0.00%	389,114.28	389,114.28	6.89	11,028	0.00%	389,114.28	389,114.28
R4	2.67	5406	0.00%	495,359.88	495,359.88	6.31	6845	0.00%	495,359.88	495,359.88
R5	4.02	5085	0.00%	593,178.48	593,178.48	5.06	6237	0.00%	593,178.48	593,178.48
R6	2.75	1432	0.00%	456,073.58	456,073.58	3.24	2779	0.00%	456,073.58	456,073.58
R7	0.20	0	0.00%	434,699.95	434,699.95	0.20	0	0.00%	434,699.95	434,699.95
R8	0.09	0	0.00%	200,613.27	200,613.27	0.11	0	0.00%	200,613.27	200,613.27
R9	0.16	0	0.00%	164,130.28	164,130.28	0.41	14	0.00%	164,130.28	164,130.28
R10	0.27	0	0.00%	123,117.51	123,117.51	0.14	0	0.00%	123,117.51	123,117.51

compares the models under divisible demand, constructed with different modeling methods, using indicators such as computation time, number of branch-and-bound nodes, LP gap, LP relaxation value, and the best-known lower bound (or optimal solution). The LP gap is computed as: $\mathrm{LP} \mathrm{gap} ={\displaystyle \frac{LP-LB}{LB}}\times 100\%$. where $LB$ is the best-known lower bound, and $LP$ is the LP relaxation value.

From Table 2, it can be seen that both models can solve the problem quickly and accurately. However, for R1, R2, R3–R6, R8, and R9, model [MS8] can obtain the same solution as model [MS9] in a shorter time.

Table 3. Comparison of computational performance between models [MU3] and [MU4].

Ship Type	Model [MU3]					Model [MU4]
	Time (s)	Nodes	Exit Gap (%)	$\mathit{U}\mathit{L}\mathit{P}$	$\mathbf{Profit-}\mathit{L}\mathit{B}$	Time (s)	Nodes	Exit Gap (%)	$\mathit{U}\mathit{L}\mathit{P}$	$\mathbf{Profit-}\mathit{L}\mathit{B}$
R1	5392.84	2,229,769	0.10%	118,125.54	118,009.78	2727.26	1,106,401	0.12%	118,151.28	118,009.78
R2	4656.72	1,546,588	0.12%	290,384.20	290,043.70	1243.16	644,610	0.06%	290,225.60	290,043.71
R3	3142.08	2,106,695	0.03%	361,276.52	361,151.74	2270.77	1,728,505	0.07%	361,414.82	361,151.74
R4	3187.97	1,253,782	0.01%	466,044.28	465,974.74	1866.27	1,214,510	0.06%	466,241.15	465,974.74
R5	6001.11	1,385,966	0.31%	567,637.69	565,905.90	3768.01	1,187,382	0.03%	566,071.95	565,905.90
R6	836.00	242,354	0.18%	434,005.91	433,207.91	397.38	199,464	0.09%	433,614.26	433,207.91
R7	31.56	76,014	0.00%	413,692.30	413,692.30	16.78	24,863	0.00%	413,692.30	413,692.30
R8	0.52	1275	0.00%	181,036.66	181,036.66	0.59	350	0.00%	181,036.66	181,036.66
R9	0.34	31	0.00%	155,003.59	155,003.59	0.31	56	0.00%	181,036.66	181,036.66
R10	0.27	0	0.00%	113,837.22	113,837.22	0.36	31	0.00%	113,837.22	113,837.22

summarizes the computational performance of models under indivisible demand scenarios. In comparison with Table 2, the LP gap is substituted with the Exit gap, while all other indicators remain unchanged. The Exit gap is defined as the relative difference between the LP relaxation value at algorithm termination and the best-known lower bound: $\mathrm{Exit} \mathrm{gap} ={\displaystyle \frac{ULP-LB}{LB}}\times 100\%$. Where $ULP$ is the LP relaxation value obtained at algorithm termination. The computation time for models [MU3] and [MU4] is set to 100 min.

From the results in Table 3, it is evident that all instances were solved satisfactorily within 100 min, with the solution gaps being very small relative to the optimal values. Under the same accuracy level, model [MU4] consistently requires less computation time than model [MU3], and its performance improvement is particularly pronounced for instances R1–R10. This suggests that model [MU4] is more suitable for problems with higher computational complexity. Furthermore, a comparison between Table 2 and Table 3 reveals that voyage profits decrease under indivisible demand. The results also indicate that, for indivisible demand, the modeling strategy based on empty container node variables offers distinct advantages.

7.3. Optimization Results Analysis

Taking the case of divisible demand as an example, Figure A3 illustrates the optimal port call sequence and container loading strategy of ships. As shown in Figure A3, water depth is a critical factor influencing ship deployment strategies. From R1, it can be observed that small LNG-fuelled container ships tend to prioritize upstream ports in order to transport high freight cargoes, while reducing calls at midstream ports to avoid low freight cargoes, thereby improving voyage profit. Fewer midstream port calls lead to reduced port time and extended sailing durations, which contribute to lower fuel consumption and increased voyage profit. Additionally, because downstream ports along the Yangtze River are closer and have deeper waterways, ships are more likely to call at multiple downstream ports to further optimize revenue.

A comparison of R2–R6 reveals that, under a fixed route length, the type of LNG-fuelled ship significantly influences the port call sequence. Compared with larger ships, smaller LNG-fuelled container ships typically call at fewer ports, spend less time in port, and allocate more time to sailing, thereby lowering fuel costs. In contrast, larger LNG-fuelled container ships, with greater carrying capacity, require optimized port call sequences and sailing speeds to ensure voyage profit. Notably, LNG-fuelled container ships with a capacity of 500 TEU or more primarily serve midstream and downstream ports along the Yangtze River. Owing to their larger capacity and the predominance of short-haul shipping tasks, these ships need to call at more ports to maximize voyage revenue.

Figure 11 presents the scheduled route of 10 typical ships, including arrival time, departure time, and port stay time. The slope between two consecutive ports reflects the sailing speed. An analysis of these slopes shows that, when the fleet size, origin, and destination ports are consistent, smaller ships with fewer port calls tend to operate at relatively lower speeds. Furthermore, both the port call sequence and sailing speeds vary across different ship types.

From the refueling strategy illustrated in Figure 12, it can be observed that all 10 ship types select Shanghai as the first refueling port to ensure fuel security for subsequent legs. Depending on the navigation conditions and fuel consumption levels, Chongqing and Yichang are generally selected as upstream refueling ports, Wuhan as the midstream port, and Wuhu as the downstream port. To guarantee fuel security, ships tend to minimize the number of refueling operations while maximizing the refueling amount in order to benefit from price discounts. Moreover, given the linear characteristics of inland waterways and the higher fuel consumption in the outbound direction, ships predominantly choose to refuel at outbound ports so as to secure fuel supply for the entire voyage and minimize overall fuel costs.

7.4. Sensitivity Analysis

This study selects R5 as the research object to analyze the impact of variable factors in practice on the optimization results, with the aim of revealing additional patterns and deriving managerial implications.

(1) Water depth

The water depth of the channel varies with the seasons, which affects the optimization results. Therefore, the impact of water depth on the optimization results is examined, as shown in Figure 13 and Figure 14.

From Figure 13, it can be observed that when the water depth is relatively shallow, the carrying capacity of ships is constrained. Consequently, shipping companies operate at lower sailing speeds and reduce port time in order to minimize fuel consumption. With increasing water depth, ships are able to access upstream ports and tend to prioritize the shipping of high freight rate and long-distance cargoes. To this end, ships appropriately increase sailing speed to shorten voyage time while also expanding the number of calling ports.

Figure 14 illustrates the relationship between various costs, the refueling strategy, and water depth. Figure 14a shows that deeper channels lead to higher voyage profits, while the fuel cost exhibits a trend of rising initially, then declining, and finally increasing again. As shown in Figure 14b, both the number of calling ports and container shipping increase with greater water depth. Notably, once water depth exceeds a certain threshold, ships gain more flexibility in serving high freight rates and long-distance cargoes, which reduce the container handling cost while maintaining the fuel cost within a reasonable range. Figure 14c indicates that shipping revenue increases with water depth, and the carbon emission cost follows a similar trend. Regarding the refueling strategy, when water depth is shallow, ships experience longer voyage times but lower fuel consumption and thus tend to refuel at Shanghai, Wuhu, and Chongqing. As water depth increases, ships shift to refueling at Shanghai and Yichang with larger refueling quantities. With further increases in depth, fuel consumption rises, and due to the limitation of LNG tank capacity, the frequency of refueling operations increases accordingly.

(2) Fuel price

Fuel prices fluctuate with international market dynamics, and Figure 15 and Figure 16 illustrate their impact on the optimization results.

As shown in Figure 15, when the fuel price increases, ships tend to reduce the number of calling ports, thereby shortening port time and extending sailing time. This strategy allows them to their lower sailing speed in order to minimize fuel costs.

Figure 16 presents the relationship between costs, refueling strategy, and fuel price. Figure 16a indicates that voyage profit is inversely related to fuel price, while fuel cost increases with rising prices. From Figure 16b, it can be observed that higher fuel prices lead to fewer containers handled at ports and a reduction in the number of calling ports. Furthermore, Figure 16c shows that carbon emission cost declines as fuel consumption decreases, while voyage profit decreases due to higher fuel prices and corresponding adjustments in shipping strategies. In terms of refueling strategy, as fuel prices rise, ships lower their sailing speed, and the preferred refueling ports shift from Shanghai, Wuhu, and Chongqing to Shanghai and Yichang. At the same time, refueling amount at selected ports increase to take advantage of discounts, with the LNG tank capacity becoming a key factor influencing routing and refueling decisions. However, as fuel prices continue to climb, the marginal benefit of refueling more fuel to exploit inter-port price differentials diminishes. To optimize total profit while aligning with shipping strategy, ships eventually return to refueling at three ports (see Figure 16d).

(3) Carbon tax

The intensity of government-imposed carbon tax may vary depending on the progress of green transitions in the shipping industry. Figure 17 and Figure 18 illustrate the impact of carbon tax variations on the optimization results.

As shown in Figure 17, when the carbon tax increases, ships reduce the number of calls at midstream ports, adjust their shipping strategies, and shorten port times while extending sailing durations. This adjustment aims to reduce both fuel cost and carbon tax expenditures.

Figure 18 further depicts the relationship between costs, refueling strategy, and carbon tax. Figure 18a demonstrates that voyage profit and fuel cost both exhibit an inverse relationship with the carbon tax. As revealed in Figure 18b, an increase in the carbon tax results in a decline in the number of containers handled and a reduction in the number of calling ports. Figure 18c indicates that carbon emission cost rises proportionally with the carbon tax, while overall voyage profit declines. With respect to refueling strategy, ships prefer to refuel at Shanghai and Yichang, keeping fuel amount constants at Shanghai while reducing the refueling amount at Yichang. Interestingly, the influence of carbon tax on optimization results differs slightly from that of fuel price changes. The underlying reason is that fuel price variations create nonlinear effects through port specific price differentials and amount-based discounts, whereas carbon tax exerts a linear influence across ports (see Figure 18d).

(4) Speed deviation value

The speed deviation value may vary due to factors such as the captain’s operational habits and seasonal changes in streamflow velocity. Therefore, an analysis of speed deviation was conducted, as illustrated in Figure 19 and Figure 20.

As shown in Figure 19, with an increase in speed deviation value, ship fuel consumption rises. To mitigate this effect, ships reduce port time and extend sailing time in order to lower fuel consumption. However, as the speed deviation value continues to increase, the flexibility of speed adjustments diminishes. By balancing port time and sailing time, ships ultimately increase port time to establish a more stable schedule.

Figure 20 presents the relationship between various costs, refueling strategy, and speed deviation value. Figure 20a indicates that voyage profit decreases with increasing speed deviation, while fuel cost rises. Figure 20b shows that the number of calling ports and handling cost initially decrease and then increase as the speed deviation value grows. This phenomenon occurs because ships must balance fuel cost against the flexibility of speed adjustment when determining routing strategy. Moreover, Figure 20c demonstrates that voyage revenue first decreases and then increases with higher speed deviation values, while fuel cost steadily increases, reflecting the adjustments in shipping strategy. Regarding refueling strategy, ships initially choose Shanghai and Chongqing as refueling ports, later adding Wuhan. As the speed deviation value rises further, the preferred refueling ports shift to Shanghai and Yichang, before eventually reverting to Shanghai, Wuhan, and Chongqing. This outcome results from the combined effects of fuel consumption patterns, inter-port fuel price differentials and discounts (see Figure 20d).

(5) Confidence level of lock passage time

The confidence level of lock passage time may vary depending on the preferences of decision-makers. Accordingly, Figure 21 and Figure 22 illustrate how optimization results change with different confidence levels.

As shown in Figure 21, ship speed increases as the confidence level of lock passage time rises, while port time initially increases and then decreases. This is because the confidence level of lock passage time significantly influences the overall time allocation, compelling shipowners to adjust both sailing and port times to achieve the optimal distribution between them.

Figure 22 presents the relationship between costs, refueling strategy, and the confidence level of lock passage time. Figure 22a indicates that fuel cost rises with increasing confidence levels, whereas voyage profit shows the opposite trend. Figure 22b reveals that both the number of calling ports and handling cost first increase and then decline. In Figure 22c, shipping revenues and carbon emission cost follow patterns consistent with handling cost and fuel cost. The underlying reason is that ships must balance fuel cost against shipping revenue to formulate rational shipping strategy. Regarding refueling strategy (see Figure 22d), ships initially select Shanghai, Wuhan, and Chongqing as refueling ports. As fuel consumption rises, they shift to refueling at Shanghai and Yichang, increasing fuel amount to benefit from discounts. However, due to the limited capacity of LNG tank, further increases in the confidence level eventually drive the refueling pattern back to Shanghai, Wuhan, and Chongqing.

8. Conclusions

This study investigates flow-balanced scheduled routing and robust refueling for inland LNG-fuelled liner shipping. Two scheduling models based on empty container arc variables and node variables are developed, alongside an external linear approximation algorithm to ensure computational efficiency while maintaining solution accuracy. The proposed modeling approach reduces the number of decision variables, thereby enhancing the quality and efficiency of problem solving. This research addresses the key challenges faced by inland LNG-fuelled ships, providing both methodological innovation and practical significance. Using the Yangtze River as a case study, the models yield optimal solutions for divisible demand instances within a few seconds and satisfactory solutions for indivisible demand cases within 100 min. The main managerial insights derived from this study are as follows:

(1)

Water depth is a critical determinant of voyage profit. Shipping companies should adjust tactical decisions and refueling strategy in line with seasonal variations in water depth. To maintain voyage profit and service reliability, companies may also consider seasonal freight rate adjustments. Ship upgrades can further enable efficient operations in midstream and upstream legs, thereby enhancing profitability.

(2)

Fuel price significantly influences tactical decisions and refueling strategy. Accurate forecasting of fuel prices is essential in scheduled route design. When fuel prices rise while ship allocated remains unchanged, shipping companies may reduce the number of calling ports, decrease sailing speeds, and maintain container flow balance to minimize fuel cost. For policy-makers, providing differentiated subsidies across routes can encourage the adoption of LNG-fuelled ships and ensure effective utilization of LNG refueling facilities.

(3)

The implementation of a carbon tax affects the operating costs of shipping companies, prompting adjustments in ship selection and operational strategies. To support the development of LNG-fuelled ships, governments should proactively adjust carbon tax intensity in anticipation of market responses.

(4)

Speed deviation value, influenced by seasonal channel conditions and captains’ operational habits, plays a vital role in ensuring the reliability of shipping plans. Properly managed speed deviation enhances schedule stability.

(5)

Confidence level of lock passage time has a significant effect on scheduled route design. Shipping companies should adjust their risk expectations based on market conditions and lock congestion, balancing safety in passage with the maximization of voyage profit.

While this study solves practical problems and provides methodological innovation, several limitations remain for future research: (1) The current study relies on certain assumptions, and future work may relax these assumptions to enhance realism. (2) The incorporation of multi-scheme carbon tax mechanisms into route design represents a promising research direction. (3) Considering the heterogeneity of containerized cargo types may yield additional managerial insights. (4) Investigating uncertainties in the shipping industry, such as freight rates, fuel prices, and streamflow velocities, as well as fleet heterogeneity, highlights the importance and interest of designing dynamic optimization methods. (5) Although the node-variable-based model provides high-quality solutions for indivisible demand instances within a reasonable timeframe, future research could explore exact algorithms to further accelerate solution efficiency.