Scheduling of Mixed Fleet Passing Through River Bottleneck in Multiple Ways

Abstract

This paper addresses the scheduling problem of a mixed fleet passing through a river bottleneck in multiple ways, considering the impact of streamflow velocity, the fuel cost with different sailing speeds, and the potential opportunity cost of various types and sizes of vessels. From the perspective of centralized management by river bottleneck authorities, a unified scheduling approach is proposed, and a nonlinear model is constructed, where the total fuel cost and potential opportunity cost of the fleet are minimized. To handle the nonlinear terms in the model, an outer approximation technique is applied to linearize the model while ensuring the approximation error remains controlled. The optimal value range of the nonlinear variables is also proven to ensure solution speed. Furthermore, the applicability and effectiveness of the model and solution method are validated through a real-world case study on the Yangtze River. The results show the following: (1) Unified collaborative scheduling by bottleneck authorities can ensure the optimal total cost of the fleet is effectively met and that the vessels passing through the river bottleneck are arranged under rational ways. (2) When fuel consumption is the same as that of traditional oil-fuelled vessels, giving priority to liquefied natural gas (LNG)-fuelled vessels to pass through the river bottleneck can reduce the potential opportunity cost and the total cost of the fleet reasonably. (3) In accordance with changes in the fuel price, streamflow velocity, and proportion of LNG-fuelled vessels, timely adjusting the opportunity cost expectations, vessel arrival time, and service times of bottleneck passing ways is crucial for shipowners and authorities to reduce fleet waiting times at the bottleneck, delay time, and the total cost.

1. Introduction

1.1. Research Background

Inland waterways are important channels connecting inland cities with coastal areas and play a crucial role in regional economic development and international trade. With the rapid growth of freight traffic, congestion at locks has become a key factor hindering the development of inland waterway transportation. In view of the mismatch between the capacity of locks and the number of vessels passing through, locks have become typical bottlenecks in inland waterway transportation. For example, the average waiting time for vessels at the Three Gorges locks on the Yangtze River increased from 44 h per vessel in 2016 to 110 h in 2020, and with the growth in cargo volume and the number of vessels, the average waiting time is expected to continue to rise [1]. The long waiting time at bottlenecks leads to delays in vessel arrivals, increasing the potential opportunity cost for vessels during transportation [2]. The primary cause of congestion at river bottlenecks is the demand for passage exceeding the handling capacity of the bottlenecks. Enhancing the capacity of bottlenecks is an effective way to alleviate congestion, prompting authorities to explore ways to expand bottleneck capacity, such as constructing new lock channels, ship lifts, and trans-shipment facilities to bypass bottlenecks.

Because vessels passing through the bottleneck belong to different shipowners and lack a coordinated mechanism, some vessels may select to arrive early and wait at anchor, aiming to pass through the bottleneck sooner. On the contrary, other vessels may arrive later to avoid long waiting times at the bottleneck. It can be seen that in the absence of unified scheduling, shipowners decide how and when to pass through the bottleneck by weighing their own interests. Therefore, although capacity expansion at bottlenecks fundamentally increases passage capacity, congestion can still occur at river bottlenecks during certain periods due to the uneven distribution of demand over time or the disordered arrival of vessels, even when the handling capacity of the bottleneck is greater than the overall demand. To address the issue of congestion at river bottlenecks, even after capacity expansion, the problem of time allocation across different bottleneck passing ways requires a more detailed and structured approach. Firstly, an optimal time allocation approach can be introduced to manage the staggered passage of vessels through the river bottleneck. Secondly, this time allocation approach should consider specific lock operations and personnel arrangements [3]. Consequently, a scheduling model for a specific period can be developed, in which the service times allocated to vessels for multiple passing ways are optimized and provided by the authorities. This would ensure that vessels begin their passage at precisely designated times, maximizing bottleneck efficiency.

When vessels arrive at river bottlenecks in an uncoordinated manner, each shipowner formulates their sailing plan based on their own preferences, which inevitably leads to long waiting times at the bottleneck and also affects the fuel cost. It is important to note that fuel cost is proportional to the power of the sailing speed and accounts for a significant portion of the total operating cost, ranging from 50% to 75% [4]. In addition, green-powered vessels are increasingly being developed for inland waterways, with liquefied natural gas (LNG)-fuelled vessels being the most widespread [5]. This results in differences in the types of fuel used by different vessels, variations in fuel consumption across vessels of different sizes, and differences in the opportunity cost of various types and sizes of vessels. Therefore, based on all these factors, optimizing the collaborative scheduling of a mixed fleet passing through a river bottleneck, while taking into account the total fuel cost and potential opportunity cost of all vessels, is of great practical significance.

In summary, this paper addresses the congestion problem faced by vessels from a mixed fleet as they pass a river bottleneck with multiple passing ways. This study focuses on minimizing both total fuel cost and potential opportunity cost, crucial factors that influence the efficiency and sustainability of inland waterway transportation. Given the unique time constraints associated with each passing way, along with the varying cost structures, speed ranges, and fuel consumption characteristics of different vessel types, this paper seeks to solve several key problems: (I). How should authorities strategically assign passing ways to vessels at the bottleneck to optimize traffic flow? (II). In what order and at what specific times should vessels arrive at their designated passing ways to ensure smooth passage and reduce congestion? (III). What optimal speed should each vessel maintain while sailing the legs to balance cost and efficiency? (IV). How can an effective operational plan be developed for the various passing ways to maximize throughput? The practical implications of this research are significant. Efficiently managing vessel traffic at river bottlenecks not only enhances operational efficiency but also reduces operational cost and environmental impact and ensures timely logistics, benefitting both the river shipping industry and authorities. By addressing these challenges, the findings of this study can inform policy decisions and operational strategies, ultimately contributing to a more sustainable and effective management of inland waterways.

1.2. Main Contributions

This paper studies the problem of scheduling a mixed fleet to pass through a river bottleneck selecting different passing ways. This can assist bottleneck authorities in developing effective scheduling plans for both green and traditional vessels, as well as operational schedules for various bottleneck passing ways. In addition, it can also contribute theoretical methods to energy conservation and the emission reduction efforts of the waterway transportation industry. Specifically, the contributions of this paper are as follows:

Firstly, under the influence of streamflow, the selection of multiple bottleneck passing ways and the waiting time of vessels at the bottleneck are considered, and a systematic optimal scheduling model for a mixed fleet passing through a river bottleneck is established with the goal of minimizing the total fuel cost and potential opportunity cost. To efficiently solve this model, the nonlinear model is linearized through constraint linearization and the outer approximation of the nonlinear functions while ensuring that the approximation error remains in an absolute objective value tolerance. In view of the influence of an excessive number of approximate line segments on the solving speed, the optimal value range of the variable of sailing time in the model is proven to avoid the additional increase in the solving time of the approximate model. Finally, a mixed fleet with green and traditional vessels on the Yangtze River is used as a case study to analyze the model’s ability to deal with the practical problem. Furthermore, the impact of fuel price, streamflow velocity, and proportion of LNG-fuelled vessels on the results is explored. This study offers a scheduling method for a mixed fleet passing through a river bottleneck, providing a series of important management insights and theoretical foundations for authorities while outlining future research directions in light of this study’s limitations.

The remainder of this paper is organized as follows. Section 2 reviews related studies. Section 3 provides a problem description. Section 4 presents the modelling of the problem. Section 5 contains numerical experiments based on the Yangtze River. Section 6 presents implications. Section 7 concludes this paper.

2. Literature Review

This paper studies the vessel scheduling problem, focusing on optimizing vessel sailing speed and the arrangement of a fleet passing through a river bottleneck. Therefore, the literature is reviewed from these two aspects.

In past research, significant progress has been made in sailing speed optimization in marine shipping. Most of the early studies on marine shipping service optimization assumed that vessels sailed at a given speed [6,7,8,9]. Subsequently, some scholars investigated the optimal sailing speed based on the power relationship between daily fuel consumption and sailing speed [10]. Corbett et al. [11] and Ronen [4] focused on the optimal speed for single vessel routes. Further, scholars conducted a more comprehensive study on speed optimization considering fuel bunkering [12,13], fuel price fluctuations [14,15,16], emission control areas [17,18,19,20,21], sea conditions [22,23], and emerging statistical methods and data mining technologies [24,25,26]. It can be seen that relevant studies extended from single-vessel to multi-vessel studies, from single fuel cost to multi-cost consideration, and from single optimization to multi-problem collaboration. Although reducing speed can reduce fuel cost, vessels generally arrive within port time windows or before their scheduled arrival time. If a vessel is delayed in arriving at a destination port, it incurs opportunity cost losses. Dulebenets [27,28] studied the problem of speed optimization under multiple time windows and multiple handling rates. Considering port disruptions, Abioye et al. [29], Brouer et al. [30], Elmi et al. [31], Wen et al. [32], and Zhao et al. [33] studied the problem of vessel schedule recovery. Given the uncertainty of port service time, Wang and Meng [34] optimized vessel speed, and considering vessel lateness, Wang and Meng [35] designed robust schedules for shipping companies under the influence of late penalties. Aydin et al. [36] studied the vessel speed dynamic optimization method to determine the sailing speed of the next leg according to the time spent in port during the previous stage. There have been many achievements in marine shipping, but it is worth mentioning that the speed optimization problem in inland waterway transportation differs from that in marine shipping, and these methods may not be applicable.

To address this inapplicability, scholars began developing some specific models for river vessel scheduling. The container liner shipping service was optimized by taking into account the influence of streamflow velocity during navigation [2,37]. Deng et al. [38] and Tan et al. [39] considered the passage time of locks to optimize vessel speed in inland waterway transportation. And facing the green transformation of the transportation sector, the inland river shipping industry also encourages companies to shift towards more environmentally friendly practices [40], and relevant policies are gradually being implemented, such as accelerating the green transition of traditional oil-fuelled river vessels [41,42,43]. In this context, Li et al. [1] conducted an economic feasibility analysis and emission assessment for LNG-fuelled river vessels. These achievements deepen our understanding of speed optimization in inland waterway transportation and the green development direction of river vessels.

Mitigating congestion at bottlenecks has been a major concern. Earlier, Ramanathan and Schonfeld [44] used a continuous flow theory model to estimate delays caused by service interruptions. To address the delays, Griffiths [45] studied strategies for reducing queuing delays in the Suez Canal, combining linear programming, queuing theory, and simulation techniques. Smith et al. [46] constructed a discrete event simulation model to compare strategies such as prioritization, the use of auxiliary vessels, and lock expansion to mitigate lock congestion in the upper reaches of the Mississippi River. Smith et al. [47] studied the method of improving the passing performance of locks in the upper reaches of the Mississippi River by using a mathematical programming method. Laih and Sun [48] studied the method of eliminating waiting in the Panama Canal from the perspective of queuing and charging. Recently, Lübbecke et al. [49] proposed a mathematical formulation to minimize total sailing time for two-way vessel traffic on the liner shipping network of the Kiel Canal. Passchyn et al. [50] studied the single lock scheduling problem to reduce the waiting time and proposed a polynomial-time algorithm. Passchyn et al. [51] studied the scheduling problem of bidirectional vessels passing through a lock with multiple chambers and designed an exact algorithm. Although the above study deals with vessel queuing and congestion delays at bottlenecks, the proposed method does not involve vessel speed optimization. For this reason, Yang et al. [52] studied the vessel scheduling problem in which vessels pass through the bottleneck in the same way under the background of no speed range constraint or bottleneck service time window. Under the same research background, Buchem et al. [53] studied dynamic speed adjustment for vessels passing through bottlenecks, considering the uncertainty of bottleneck passing time. Golak et al. [54] studied speed optimization across multiple river sections and lock groups to minimize fuel consumption.

Existing research on speed optimization and vessel scheduling has significantly advanced our understanding of marine shipping, but several key issues and research gaps remain in the field of inland waterway transportation. Firstly, river vessels often face the minimum speed constraints required for navigation safety and vessel control, which have frequently been overlooked in previous studies. Secondly, the diversity of vessel types results in differences in operating cost and expected arrival time, further complicating the scheduling process, with existing models typically failing to adequately account for these variations. Thirdly, the multiple passing ways at river bottlenecks, each with unique tonnage limits and service times, add to scheduling challenges. These factors are critical for meeting maintenance needs and managing traffic flow, yet previous studies often fail to fully integrate these complexities. In terms of methodology, many scheduling approaches rely on discretization methods [27] and static linear approximation methods [28,31], which are based on fixed parameters. This leads to greater deviation from real-world conditions in the optimization results and affects the computational time of the models. To address these challenges, this paper aims to fill the aforementioned gaps by proposing a scheduling method that incorporates the minimum speed constraints, vessel diversity, and the complexities of river bottlenecks. Moreover, a dynamic linear outer approximation algorithm plays a crucial role in this model, as it reduces the number of binary variables and avoids the complexity explosion associated with discretization methods. By continuously adjusting to parameter changes, this method improves the model’s accuracy, minimizes deviations from real-world conditions, and outperforms static methods that rely on fixed parameters. Thus, this study not only identifies key issues in river vessel scheduling but also offers practical solutions.

3. Problem Description

3.1. Problem Definition

To alleviate potential congestion at river bottlenecks, authorities have continuously built or expanded multiple locks and developed various bottleneck passing ways. Common ways to pass through river bottlenecks mainly include three types: locks, ship lifts, and trans-shipment facilities to bypass bottlenecks. It is worth noting that each passing way has own characteristics, as follows: For locks, they can pass multiple vessels at a time within a fixed operating time. Ship lifts, on the other hand, have limited carrying capacity and generally impose restrictions on the tonnage of vessels that can use them. As for using a trans-shipment facility to bypass bottlenecks, which involves unloading cargo, transporting it passing through the river bottleneck, and then reloading it onto another vessel, shipowners typically select this way based on their cargo and operational organization. It can be seen that for locks and ship lifts, authorities can assign vessels to specific bottleneck passing modes based on the vessels’ tonnage and location, while a shipowner can only choose a trans-shipment facility to bypass bottlenecks by weighing the potential opportunity cost of waiting in place and the cost of trans-shipment.

Due to the safety inspection and maintenance activities conducted by authorities on bottleneck passing ways, as well as changes in the expected navigation conditions, the service time $\left\{T_{d1}^l,\dots ,T_{d{a}^n}^l\right\}$ of the different passing ways in the scheduling plan may vary. This means that vessels constrained by the schedule must arrive at the bottleneck before the selected service time [38]. However, arriving too early can result in waiting times, leading to potential opportunity costs for shipowners. It is important to note that this opportunity cost arises from waiting at the bottleneck or facing delays at the destination port. The costs stem from three main factors: firstly, the ongoing expense of paying crew wages during idle time; secondly, the reduced return on capital due to vessel inactivity, resulting in the loss of effective asset utilization; thirdly, delayed arrival at the port postpones the start of the next voyage, resulting in missed potential revenue. Advantageously, the additional time resulting from delays allows the vessel to sail at a lower speed, rather than maintaining a higher speed to arrive at the destination. This reduced speed can significantly decrease fuel consumption, thereby lowering operating cost. Therefore, scheduling should account for the opportunity costs incurred from waiting at the bottleneck and port delays to improve the economic efficiency of the fleet. Figure 1 displays a scheduling diagram of the fleet at the river bottleneck.

In Figure 1, it is assumed that vessels report their plans to pass through the bottleneck at time $t^r$ (including location information and shipping schedule, i.e., $\mathrm{Origin} \left(T_s^o,L_{1s}\right)$ and $\mathrm{Destination} \left(T_s^m,L_{2s}\right)$, $s\in S$), so the vessel’s scheduling period is denoted as $\max \left\{T_{d{a}^n}^l-t^r\right\}$. $T_{d{a}^n}^l$ is the last arrival time allowed by bottleneck passing way $d$. Generally speaking, $2\le \max \left\{T_{d{a}^n}^l-t^r\right\}\le 4$, because in most cases, shipowners will have already finalized their cargo transportation plans and made schedule arrangements for their vessels within 2 to 4 days. Authorities determine which vessels need to be scheduled during this period based on the order in which each vessel’s plan is reported or the priority of the vessel for passing through the bottleneck. It is important to note that to avoid excessive vessels arriving at the bottleneck, which could cause danger and congestion on the inland waterway, vessels not scheduled during this period are required to wait at nearby anchorages for subsequent scheduling. These assumptions are very realistic in river vessel scheduling. For example, for shipping in the Yangtze River, vessels passing through the bottleneck need to report their lock passing plans in advance via the Yangtze e+ app, after which all vessels are subjected to unified scheduling by the Three Gorges Navigation Authority. The authority determines the scheduling priority based on the order of the vessels’ reports and their priority levels, creating yellow, blue, and white schedules for the vessels passing through the lock. Vessels listed in the yellow, blue, and white schedules are scheduled in sequence, with each schedule typically covering a period of 3 to 4 days. According to the Three Gorges Navigation Authority’s scheduling plan, vessels on the yellow schedule must proceed to the bottleneck within the specified period, while those on the blue and white schedules must wait at nearby anchorages.

A mixed fleet mainly consists of tramp vessels, including bulk carriers and dangerous goods vessels, which are engaged in shipping between two points. This is significant because tramp vessels account for more than 90% of inland waterway transportation. In addition, passenger vessels, ro-ro vessels, and container vessels enjoy priority scheduling to accommodate their specific schedules. Figure 2 shows the typical process of scheduling a mixed fleet passing through a river bottleneck in multiple ways.

As can be seen from Figure 2, for each vessel $s$, $s\in S$ that is within the scheduling plan, it reports its place of origin at time $T_s^o$ and the expected arrival at destination $T_s^m$ so that the distance between its origin and the bottleneck $L_{1s}$ (i.e., the first leg of the voyage) and the distance from the bottleneck to the destination $L_{2s}$ (i.e., the second leg of the voyage) can be obtained. There are generally multiple ways to pass bottlenecks in inland waterway transportation, and a single lock can also handle multiple vessels at the same time, and the handling time of vessels through each way is also different. A time ${\varpi }_d$ is set for the vessel to pass through the bottleneck by way $d$, and the service time for each passing way is determined. Vessels are affected by streamflow velocity and direction during navigation in inland waterways [39,55], which significantly affects fuel consumption costs. Moreover, vessels have minimum speed $v_s^{\min }$ and maximum speed $v_s^{\max }$ restrictions, as sailing below the minimum speed compromises navigation safety and vessel control, while excessive speed leads to engine inefficiency and a sharp increase in fuel consumption. During fleet scheduling at a river bottleneck, due to the influence of vessel location and fuel consumption characteristics, some vessels may arrive earlier than scheduled even when sailing at the minimum speed $v_s^{\min }$, so the vessels need to wait appropriately at the bottleneck. In the unified scheduling process, to achieve overall system optimization, some vessels may arrive at their destination later than expected, increasing their potential opportunity costs [38]. Therefore, in the scheduling of a mixed fleet, unified scheduling arrangements were carried out for all vessels with the goal of minimizing the total fuel consumption cost and potential opportunity cost of the fleet so as to determine the sailing speed of the vessels on each leg, the way of passing through the river bottleneck, and the time of arriving at each selected way.

3.2. Assumptions

According to the actual situation of inland waterway transportation and referring to previous research methods, the following assumptions or clarifications are made:

(I). All vessels that need to pass through a river bottleneck are subject to the unified schedule of the authority [56].

(II). During the scheduling period, the streamflow velocity of the river is expected to remain constant [57].

(III). Vessels not included in the current scheduling plan can seek the nearest anchorage and wait until the next scheduling period [56].

(IV). Considering the trend in standardization in river vessels, it is assumed that under the “multi-vessel per lock” way, the number of vessels passing through a single lock at one time is fixed, and the energy consumption of the operation of the bottleneck passing way is negligible and not considered in the model [56].

4. Problem Modelling

4.1. Notations

(I). Sets

$S\in \left\{1,\dots ,a^s\right\}$ is the set of vessels scheduled to pass through a river bottleneck within the scheduling period;

$D_s\in \left\{1,\dots ,a^d\right\}$, $s\in S$ is the set of the bottleneck passing way that vessel $s$ can select;

$N_d\in \left\{1,\dots ,a^n\right\}$, $d\in D_s$ is the set of the selectable arrival time of bottleneck passing way $d$.

(II). Parameters

$L_{1s}$, $s\in S$ is the distance from the vessel’s origin to the bottleneck (km);

$L_{2s}$, $s\in S$ is the distance from the bottleneck to the vessel’s destination (km);

${\widehat{v}}_{1s}$, $s\in S$ is the streamflow velocity from the vessel’s origin to the bottleneck (km/h);

${\widehat{v}}_{2s}$, $s\in S$ is the streamflow velocity from the bottleneck to the vessel’s destination (km/h);

$T_s^o$, $s\in S$ is the departure time of vessel $s$ from its origin (h);

$T_s^m$, $s\in S$ is the expected arrival time of vessel $s$ at its destination (h);

$C_s^p$, $s\in S$ is the potential opportunity cost per hour for vessel $s$ (CNY/h);

${\alpha }_s,{\alpha }_s>0$, $s\in S$ is the linear coefficient of the fuel consumption function for vessel $s$;

${\beta }_s,{\beta }_s>2$, $s\in S$ is the power coefficient of the fuel consumption function for vessel $s$;

$v_s^{\min }$, $s\in S$ is the minimum sailing speed of vessel $s$ (km/h);

$v_s^{\max }$, $s\in S$ is the maximum sailing speed of vessel $s$ (km/h);

${\varpi }_d$, $d\in D_s$ is the time required for each vessel to pass through the bottleneck when selecting bottleneck passing way $d$ (h);

$G_d$, $d\in D_s$ is the vessel capacity of the bottleneck passing way $d$ at a single pass during the scheduling period (vessels);

$T_{dn}^l$, $d\in D_s;n\in N_d$ is the specific time of the nth arrival time for bottleneck passing way d (h);

$C_s^f$, $s\in S$ is the price of fuel used by vessel $s$ (CNY/ton).

(III). Variables

$t_s^e$, $s\in S$ is the arrival time of vessel $s$ at its destination (h);

$t_s^w$, $s\in S$ is the waiting time of vessel $s$ at the bottleneck (h);

$r_{sdn}$, $s\in S;d\in D_s;n\in N_d$ is a binary variable. If vessel $s$ selects the nth arrival time for bottleneck passing way $d$, then $r_{sdr}=1$; otherwise, $r_{sdr}=0$.

4.2. Model Formulation

Before presenting the model, the following explanation is necessary:

(I). The fuel consumption of vessel $s$ is related to its sailing speed $v$ by a power relationship. Referring to [2], this study defines the hourly fuel consumption function of vessel $s$ as follows:

$$f_s\left(v\right)={\alpha }_s{v}^{{\beta }_s},s\in S$$

(1)

where $f_s\left(v\right)$ is the fuel consumption of vessel $s$ per hour (tons/h).

(II). Assuming that the potential opportunity cost of vessel $s$ is a linear function of the time delay in arriving at the destination and the waiting time at the river bottleneck [2], it is specified as follows:

$$C_s^{pz}\left(t_s^e,t_s^w\right)=C_s^p\left(\max \left\{0,t_s^e-T_s^m\right\}+t_s^w\right),s\in S$$

(2)

where $C_s^{pz}\left(t_s^e,t_s^w\right)$ is the total potential opportunity cost incurred by vessel $s$ (CNY).

To obtain the optimal scheduling plan for a mixed fleet passing through a river bottleneck, the model [M1] is constructed. The specific modelling is as follows:

Objective function:

$$\begin{array}{l}\min {\displaystyle \sum _{s\in S}{C}_s^f\left(\left({\displaystyle \sum _{d\in D_s}{\displaystyle \sum _{n\in N_d}{r}_{sdn}{T}_{dn}^l}}-T_s^o-t_s^w\right){\alpha }_s{\left(\frac{L_{1s}}{{\displaystyle \sum _{d\in D_s}{\displaystyle \sum _{n\in N_d}{r}_{sdn}{T}_{dn}^l}}-T_s^o-t_s^w}-{\widehat{v}}_{1s}\right)}^{{\beta }_s}\right.}\\ \left.+\left(t_s^e-{\displaystyle \sum _{d\in D_s}{\displaystyle \sum _{n\in N_d}{r}_{sdn}{T}_{dn}^l}}-{\displaystyle \sum _{d\in D_s}{\displaystyle \sum _{n\in N_d}{\varpi }_d\cdot r_{sdn}}}\right){\alpha }_s{\left({\displaystyle \frac{L_{2s}}{t_s^e-{\displaystyle \sum _{d\in D_s}{\displaystyle \sum _{n\in N_d}{r}_{sdn}{T}_{dn}^l}}-{\displaystyle \sum _{d\in D_s}{\displaystyle \sum _{n\in N_d}{\varpi }_d\cdot r_{sdn}}}}}-{\widehat{v}}_{2s}\right)}^{{\beta }_s}\right)\\ +{\displaystyle \sum _{s\in S}{C}_s^p\left(\max \left\{0,t_s^e-T_s^m\right\}+t_s^w\right)}\end{array}$$

(3)

Constraints:

$$T_s^o+{\displaystyle \frac{L_{1s}}{v_s^{\max }+{\widehat{v}}_{1s}}}\le {\displaystyle \sum _{d\in D_s}{\displaystyle \sum _{n\in N_d}{r}_{sdn}{T}_{dn}^l}}-t_s^w\le T_s^o+{\displaystyle \frac{L_{1s}}{v_s^{\min }+{\widehat{v}}_{1s}}},s\in S$$

(4)

$${\displaystyle \sum _{d\in D_s}{\displaystyle \sum _{n\in N_d}{r}_{sdn}{T}_{dn}^l}}+{\displaystyle \sum _{d\in D_s}{\displaystyle \sum _{n\in N_d}{\varpi }_d\cdot r_{sdn}}}+{\displaystyle \frac{L_{2s}}{v_s^{\max }+{\widehat{v}}_{2s}}}\le t_s^e\le {\displaystyle \sum _{d\in D_s}{\displaystyle \sum _{n\in N_d}{r}_{sdn}{T}_{dn}^l}}+{\displaystyle \sum _{d\in D_s}{\displaystyle \sum _{n\in N_d}{\varpi }_d\cdot r_{sdn}}}+{\displaystyle \frac{L_{2s}}{v_s^{\min }+{\widehat{v}}_{2s}}},s\in S$$

(5)

$${\displaystyle \sum _{d\in D_s}{\displaystyle \sum _{n\in N_d}{r}_{sdn}}}=1,s\in S$$

(6)

$${\displaystyle \sum _{s\in S}{r}_{sdn}}\le G_d,d\in D_s;n\in N_d$$

(7)

$$r_{sdn}\in \left\{0,1\right\};t_s^e\ge 0;t_s^w\ge 0,s\in S;d\in D_s;n\in N_d$$

(8)

Objective function (3) aims to minimize the total fuel cost and potential opportunity cost of a mixed fleet passing through a river bottleneck during the scheduling period. In this function, the first term represents the fuel consumption cost for all vessels, and the second term represents the potential opportunity cost for all vessels. Constraint (4) ensures that the time of the vessel’s arrival at the bottleneck cannot exceed the arrival time of the vessel at the maximum speed. If the vessel arrives earlier than the selected arrival time even when sailing at the minimum speed, then the vessel must wait at the bottleneck. Constraint (5) ensures that the arrival time of the vessel at the destination cannot exceed the time range permitted by the vessel speed. Constraint (6) ensures that a vessel can only select one of the multiple available arrival times that this vessel can select. Constraint (7) ensures that the number of vessels using bottleneck passing way $d$ cannot exceed the total number of vessels that can be handled by this way in a single instance. Constraint (8) defines the value range of the variables.

4.3. Properties of Model

Through the analysis of the model, the following propositions and corollaries can be derived for a deeper understanding of the research problem.

Proposition 1.

For the same sailing distance and time, the fuel consumption of vessel s at a fixed speed is the lowest.

Proof. 

Proposition 1 can be proven by analyzing the strict convexity of the fuel consumption per hour and the vessel’s sailing time, which corresponds to the fuel consumption term in the objective function (3) of the model. Before proving this, the sailing distance and time of vessel $s$ are denoted by $L$ and $t$, respectively. Then, a scenario is considered where a vessel sails at two different speeds during the sailing time $t$: at a higher speed $v_1$ for $\delta t$, $0<\delta <1$ and at a lower speed $v_2$, $v_2>v_1$ for $\left(1-\delta \right)t$, satisfying $v_1\delta t+v_2\left(1-\delta \right)t=L$. In this case, the total fuel consumption for the vessel is $f_s\left(v_1\right)\delta t+f_s\left(v_2\right)\left(1-\delta \right)t$. If the vessel sails at an average speed $v_1\delta +v_2\left(1-\delta \right)$, the fuel consumption sailing the same distance $L$ over the same time interval $t$ is $f_s\left(v_1\delta +v_2\left(1-\delta \right)\right)t$. Due to the strict convexity of the fuel consumption function, $f_s\left(v_1\delta +v_2\left(1-\delta \right)\right)t<f_s\left(v_1\right)\delta t+f_s\left(v_2\right)\left(1-\delta \right)t$ can be obtained. Thus, sailing at a fixed speed results in lower fuel consumption. □

Corollary 1.

Under unified scheduling, if the speed range of vessels allows, it is not optimal for any vessel s to arrive earlier than the expected arrival time.

Proof. 

Corollary 1 is valid because if a vessel queues at a river bottleneck, the authority can coordinate the fleet to sail at a lower speed while still adhering to the speed range limit. This coordination eliminates queuing at the bottleneck and ensures that every vessel arrives at the bottleneck at the selected time without delays. This reduces the fuel cost of a mixed fleet without increasing the potential opportunity cost for all vessels, achieving optimal unified scheduling. In addition, proposition 1 indicates that it is optimal for each vessel to sail at a fixed speed before and after the bottleneck, with speed ${\displaystyle \frac{L_{1s}}{{\displaystyle \sum _{d\in D_s}{\displaystyle \sum _{n\in N_d}{r}_{sdn}{T}_{dn}^l}}-T_s^o-t_s^w}}-{\widehat{v}}_{1s}$ before the bottleneck and ${\displaystyle \frac{L_{2s}}{t_s^e-{\displaystyle \sum _{d\in D_s}{\displaystyle \sum _{n\in N_d}{r}_{sdn}{T}_{dn}^l}}-{\displaystyle \sum _{d\in D_s}{\displaystyle \sum _{n\in N_d}{\varpi }_d\cdot r_{sdn}}}}}-{\widehat{v}}_{2s}$ after the bottleneck. Therefore, the problem is equivalent to finding the optimal $r_{sdn}$, $t_s^e$ and $t_s^w$ to minimize the total fuel consumption and potential opportunity cost of the fleet. □

Corollary 2.

When a vessel s sails at the minimum speed, if it still cannot arrive at the bottleneck at the selected arrival time $T_{dn}^l$, the vessel must wait at the bottleneck.

Proof. 

Corollary 2 is valid because if a vessel cannot arrive at the bottleneck at the selected arrival time $T_{dn}^l$, even when sailing at the minimum speed, it will arrive early at the bottleneck. In this case, the vessel cannot use a reduced speed to extend the sailing time and reduce fuel consumption. Therefore, to meet the scheduling plan, a constraint $T_s^o+{\displaystyle \frac{L_{1s}}{v_s^{\max }+{\widehat{v}}_{1s}}}\le {\displaystyle \sum _{d\in D_s}{\displaystyle \sum _{n\in N_d}{r}_{sdn}{T}_{dn}^l}}-t_s^w\le T_s^o+{\displaystyle \frac{L_{1s}}{v_s^{\min }+{\widehat{v}}_{1s}}},s\in S$ is introduced to ensure that the vessel’s sailing time adheres to the speed range requirements and to model the waiting time of vessels at the bottleneck. □

This section provides theoretical support for the assumption of the optimal sailing speed in Proposition 1 and offers mathematical explanations for speed adjustment and waiting times at bottlenecks in the subsequent analysis of the results.

4.4. Linearization Methods

Model [M1] is clearly a nonlinear model. To linearize the potential opportunity cost function and nonlinear fuel cost function, variable substitution methods and piecewise linear approximation can be used [58], respectively.

4.4.1. Variable Substitution

Firstly, a non-negative auxiliary variable $t_s^p$ can be introduced to linearize the potential opportunity time due to delay time. Then, ${\displaystyle \sum _{s\in S}{C}_s^p\left(\max \left\{0,t_s^e-T_s^m\right\}+t_s^w\right)}$ can be replaced in the objective function (3) with ${\displaystyle \sum _{s\in S}{C}_s^p\left(t_s^p+t_s^w\right)}$ while adding the following constraints:

$$t_s^p\ge 0,s\in S$$

(9)

$$t_s^p\ge t_s^e-T_s^m,s\in S$$

(10)

To linearize the fuel consumption cost in the objective function (3), constraints involving decision variable $t_s^e$ can first be replaced with sailing times $t_s^y$ and $t_s^c$ for the first and second legs of the voyage, respectively, where $t_s^y$ is the sailing time before the bottleneck, and $t_s^c$ is the sailing time after the bottleneck. The definitions of $t_s^y$ and $t_s^c$ are as follows:

$$t_s^y={\displaystyle \sum _{d\in D_s}{\displaystyle \sum _{n\in N_d}{r}_{sdn}{T}_{dn}^l}}-T_s^o-t_s^w,s\in S$$

(11)

$$t_s^c=t_s^e-{\displaystyle \sum _{d\in D_s}{\displaystyle \sum _{n\in N_d}{r}_{sdn}{T}_{dn}^l}}-{\displaystyle \sum _{d\in D_s}{\displaystyle \sum _{n\in N_d}{\varpi }_d\cdot r_{sdn}}},s\in S$$

(12)

Therefore, constraints (4), (5), and (10) are rewritten as follows:

$$t_s^y\ge {\displaystyle \frac{L_{1s}}{v_s^{\max }+{\widehat{v}}_{1s}}},s\in S$$

(13)

$$t_s^y\le {\displaystyle \frac{L_{1s}}{v_s^{\min }+{\widehat{v}}_{1s}}},s\in S$$

(14)

$$t_s^c\ge {\displaystyle \frac{L_{2s}}{v_s^{\max }+{\widehat{v}}_{2s}}},s\in S$$

(15)

$$t_s^c\le {\displaystyle \frac{L_{2s}}{v_s^{\min }+{\widehat{v}}_{2s}}},s\in S$$

(16)

$$t_s^p\ge T_s^o+t_s^y+t_s^w+{\displaystyle \sum _{d\in D_s}{\displaystyle \sum _{n\in N_d}{\varpi }_d\cdot r_{sdn}}}+t_s^c-T_s^m,s\in S$$

(17)

Now, the only formula left in the model to calculate the fuel consumption cost is nonlinear, so the piecewise linearization method for the nonlinear fuel consumption function is introduced next. Given the characteristics of this approximation method, the relationship between the approximation error and the computational speed with respect to the number of tangent lines is first discussed. Based on this relationship, the outer approximation method is introduced to balance these factors.

4.4.2. Discussion of Approximate Error

To realize the approximation of the nonlinear function, this model adopts the outer approximation method. The core idea of this method is to approximate nonlinear curves with multiple tangent lines by local linearization. Specifically, we compute the derivatives of the objective function at each key point and construct linear tangents from these derivative values to locally approximate the curve. This process effectively reduces the computational complexity caused by nonlinear functions and ensures the accuracy within a certain error range while preserving the main characteristics of the original problem. Figure 3 demonstrates the relationship between the number of tangent lines and the approximation error, showing errors when using 1, 2, 4, and 9 tangent lines to approximate the nonlinear function $Q_s\left(t_s\right)$. As illustrated in Figure 3, there is a trend of diminishing error ${\epsilon }_1>{\epsilon }_2>{\epsilon }_3>{\epsilon }_4$ as the tangent line continues to increase, indicating that the approximation error decreases with an increasing number of tangent lines. Therefore, as long as the number of tangent lines is sufficiently high, the linear approximation functions will become infinitely close to the original nonlinear function, resulting in the linearized model’s outcomes aligning closely with those of the original model. However, research indicates that the number of tangent lines affects the computational speed of the model [28], highlighting the need for a method to balance computational accuracy and the number of tangent lines.

When using linear tangent approximations, the approximation error depends on both the number and distribution of tangent lines. The more tangent lines used, the higher the approximation accuracy, but this also increases computational time. The outer approximation method provided in this paper ensures that, while the approximation error is adequately controlled, no additional tangent lines are introduced, thus achieving a balance between accuracy and computational efficiency. To better understand this method and apply it more effectively to the unified scheduling problem of mixed fleets, the process of the outer approximation method is further introduced in the following section. This process involves the allocation scheme of absolute objective value tolerance for each vessel and the method for generating the optimal set of tangent lines.

4.4.3. Outer Approximation Method for Linearized Nonlinear Fuel Consumption Function

The fuel consumption function can be rewritten as $t_s^y{\alpha }_s{\left({\displaystyle \frac{L_{1s}}{t_s^y}}-{\widehat{v}}_{1s}\right)}^{{\beta }_s}$ and $t_s^c{\alpha }_s{\left({\displaystyle \frac{L_{2s}}{t_s^c}}-{\widehat{v}}_{2s}\right)}^{{\beta }_s}$. To linearize this, $Q_{1s}\left(t_s^y\right)$ and $Q_{2s}\left(t_s^c\right)$ are defined as the fuel consumption per kilometer of the vessel, as follows:

$$Q_{1s}\left(t_s^y\right)={\displaystyle \frac{t_s^y}{L_{1s}}}{\alpha }_s{\left({\displaystyle \frac{L_{1s}}{t_s^y}}-{\widehat{v}}_{1s}\right)}^{{\beta }_s},s\in S$$

(18)

$$Q_{2s}\left(t_s^c\right)={\displaystyle \frac{t_s^c}{L_{2s}}}{\alpha }_s{\left({\displaystyle \frac{L_{2s}}{t_s^c}}-{\widehat{v}}_{2s}\right)}^{{\beta }_s},s\in S$$

(19)

Considering the consistency in the functional expressions of $Q_{1s}\left(t_s^y\right)$ and $Q_{2s}\left(t_s^c\right)$, $Q_s\left(t_s\right)={\displaystyle \frac{t_s}{L_s}}{\alpha }_s{\left({\displaystyle \frac{L_s}{t_s}}-{\widehat{v}}_s\right)}^{{\beta }_s},s\in S$ is introduced to extend the piecewise linear approximation method when presenting it. This method uses multiple piecewise linear functions to approximate them, ensuring that the target error does not exceed absolute objective value tolerance $\epsilon$ (CNY). The algorithm aims to approximate the nonlinear function with as few linear segments as possible while maintaining the required level of accuracy. To achieve this $\epsilon$ control, the cost error per kilometer $\overline{\epsilon }$ (tons/km) is allocated for the vessels based on the fuel price $C_s^f$ and the total distance sailed by all vessels ${\displaystyle \sum _{s\in S}{C}_s^f\left(L_{1s}+L_{2s}\right)}$, calculating the tolerance $\overline{\epsilon }$ using Equation (20). As long as the approximation error of $Q_s\left(t_s\right)$ remains within the defined limit $\overline{\epsilon }$, the corresponding error in the objective value will not exceed $\epsilon$.

$$\overline{\epsilon }={\displaystyle \frac{\epsilon }{{\displaystyle \sum _{s\in S}{C}_s^f\left(L_{1s}+L_{2s}\right)}}}$$

(20)

To ensure that the function approximation remains within $\overline{\epsilon }$ for any $t_s$, $t_s\in \left[T_s^{\min },T_s^{\max }\right]$, while also minimizing the number of linear segments, the following method is introduced, as shown in Figure 4. Let the derivative of $Q_s\left(t_s\right)$ at $t_s$ be denoted as $Q_s^{\prime }\left(t_s\right)$, specifically:

$$Q_s^{\prime }\left(t_s\right)={\displaystyle \frac{{\alpha }_s}{L_s}}{\left({\displaystyle \frac{L_s}{t_s}}-{\widehat{v}}_s\right)}^{{\beta }_s}-{\displaystyle \frac{{\alpha }_s{\beta }_s}{t_s}}{\left({\displaystyle \frac{L_s}{t_s}}-{\widehat{v}}_s\right)}^{{\beta }_s-1}$$

(21)

Piecewise linear approximation function ${\overline{Q}}_s\left(t_s\right)$ of $Q_s\left(t_s\right)$ can be generated according to Algorithm 1, as follows:

Algorithm 1. The generation of a piecewise linear approximation function.

Step 0. Let ${\Lambda }_s$ be a set of lines, $T_s^{\min }$, $T_s^{\max }$. Set ${\Lambda }_s:=\varnothing$, $k:=0$, $t_1=T_s^{\min }$, $Q_{s1}=Q_s\left(t_{s1}\right)-\overline{\epsilon }$. Step 1. Set $k:=k+1$. If the following inequality holds $${\displaystyle \frac{Q_s\left(T_s^{\max }\right)-Q_{sk}}{T_s^{\max }-t_{sk}}}\ge Q_s^{\prime }\left(T_s^{\max }\right)$$ (22) specifically, point $\left(t_{sk},Q_s\left(t_{sk}\right)\right)$ is on or below the tangent line of $Q_s\left(t_s\right)$ at $T_s^{\max }$; refer to Figure 4a, and add to ${\Lambda }_g$ line $k$ as follows $$Q_s-Q_s\left(T_s^{\max }\right)=Q_s^{\prime }\left(T_s^{\max }\right)\left(t_s-T_s^{\max }\right)$$ (23) and go to Step 3. Else, add to ${\Lambda }_s$ line $k$ that passes the point $\left(t_{sk},Q_{sk}\right)$ and supports the epigraph of $Q_s\left(t_s\right)$. Suppose that line $k$ supports the epigraph of $Q_s\left(t_s\right)$ at the point $\left({\widehat{t}}_{sk},{\widehat{Q}}_{sk}\right)$; then, the point can be calculated using Formula (24). $${\widehat{Q}}_{sk}={\displaystyle \frac{{\widehat{t}}_{sk}}{L_s}}{\alpha }_s{\left({\displaystyle \frac{L_s}{{\widehat{t}}_{sk}}}-{\widehat{v}}_s\right)}^{{\beta }_s}$$ (24) By definition, $${\displaystyle \frac{{\widehat{Q}}_{sk}-Q_{sk}}{{\widehat{t}}_{sk}-t_{sk}}}=Q_s^{\prime }\left({\widehat{t}}_{sk}\right)={\displaystyle \frac{{\alpha }_s}{L_s}}{\left({\displaystyle \frac{L_s}{{\widehat{t}}_{sk}}}-{\widehat{v}}_s\right)}^{{\beta }_s}-{\displaystyle \frac{{\alpha }_s{\beta }_s}{{\widehat{t}}_{sk}}}{\left({\displaystyle \frac{L_s}{{\widehat{t}}_{sk}}}-{\widehat{v}}_s\right)}^{{\beta }_s-1}$$ (25) By combining Formulas (24) and (25), ${\widehat{t}}_{sk}$ can be numerically estimated using the bisection search method. Consequently, line $k$ can be derived according to Formula (26). $$Q_s-Q_{sk}={\displaystyle \frac{{\widehat{Q}}_{sk}-Q_{sk}}{{\widehat{t}}_{sk}-t_{sk}}}\left(t_s-t_{sk}\right)$$ (26) Go to Step 2. Step 2. For line $k$, determine if $Q_s$ satisfies the following inequality when $t_s$ takes the value $T_s^{\max }$. $$Q_{sk}+{\displaystyle \frac{{\widehat{Q}}_{sk}-Q_{sk}}{{\widehat{t}}_{sk}-t_{sk}}}\left(T_s^{\max }-t_{sk}\right)\ge Q_s\left(T_s^{\max }\right)-\overline{\epsilon }$$ (27) If the inequality above is satisfied, the gap between line $k$ and $Q_s\left(t_s\right)$ will not exceed $\overline{\epsilon }$, even when $t_s$ takes the value $T_s^{\max }$. This conclusion can be extended to any value of $t_s$ within the interval $\left[t_{sk},T_s^{\max }\right]$, as illustrated in Figure 4b; go to Step 3. If this condition is not met, there will be exactly one point $\left(t_{s\left(k+1\right)},Q_{s\left(k+1\right)}\right)$ on line $k$ such that $t_{sk}<t_{s\left(k+1\right)}<T_s^{\max }$ and $Q_{s\left(k+1\right)}=Q_s\left(t_{s\left(k+1\right)}\right)-\overline{\epsilon }$, as shown in Figure 4c. $t_{s\left(k+1\right)}$ can be estimated numerically using the bisection search method, and then return to Step 1. Step 3. Let $K_s$ (the number of lines in ${\Lambda }_s$) be the current value of $k$, and $k$, $k=1,2,\dots ,K_s$ in ${\Lambda }_s$ is represented using the generic form. $$Q_s={\theta }_{sk}\cdot t_s+{\gamma }_{sk}$$ (28) The piecewise linear approximation function ${\overline{Q}}_s\left(t_s\right)$ (see Figure 4d) can be written as $${\overline{Q}}_s\left(t_s\right)=\max \left\{0,{\theta }_{sk}\cdot t_s+{\gamma }_{sk}\right\},s\in S;k=1,2,\dots ,K_s$$ (29)

At this point, the introduction to the outer approximation method is complete. According to the algorithm described, $Q_{1s}\left(t_s^y\right)$ and $Q_{2s}\left(t_s^c\right)$ can be represented using the piecewise linear approximation functions ${\overline{Q}}_{1s}\left(t_s^y\right)$ and ${\overline{Q}}_{2s}\left(t_s^c\right)$, respectively.

$${\overline{Q}}_{1s}\left(t_s^y\right)=\max \left\{0,{\theta }_{1sk}{t}_s^y+{\gamma }_{1sk}\right\},s\in S;k=1,2,\dots ,K_{1s}$$

(30)

$${\overline{Q}}_{2s}\left(t_s^c\right)=\max \left\{0,{\theta }_{2sk}{t}_s^c+{\gamma }_{2sk}\right\},s\in S;k=1,2,\dots ,K_{2s}$$

(31)

where ${\theta }_{1sk},{\gamma }_{1sk},{\theta }_{2sk}$, and ${\gamma }_{2sk}$ are the coefficients of linear approximation functions, and $K_{1s}$ and $K_{2s}$ are the number of linear segments. The piecewise linear approximation functions ${\overline{Q}}_{1s}\left(t_s^y\right)$ and ${\overline{Q}}_{2s}\left(t_s^c\right)$ for $Q_{1s}\left(t_s^y\right)$ and $Q_{2s}\left(t_s^c\right)$, respectively, satisfy the conditions $Q_{1s}\left(t_s^y\right)-{\overline{Q}}_{1s}\left(t_s^y\right)\le \overline{\epsilon }$ and $Q_{2s}\left(t_s^c\right)-{\overline{Q}}_{2s}\left(t_s^c\right)\le \overline{\epsilon }$. Therefore, this approximation method ensures that the error in the approximation of the total fuel cost for a mixed fleet does not exceed the predefined tolerance $\epsilon$, i.e., ${\displaystyle \sum _{s\in S}{C}_s^f\left(\left(L_{1s}{Q}_{1s}\left(t_s^y\right)+L_{2s}{Q}_{2s}\left(t_s^c\right)\right)-\left(L_{1s}{\overline{Q}}_{1s}\left(t_s^y\right)+L_{2s}{\overline{Q}}_{2s}\left(t_s^c\right)\right)\right)}\le \epsilon$.

4.4.4. Range of Linearized Variables

According to the above method, the model [M1] was linearized. However, the number of $K_{1s}$ and $K_{2s}$ will affect computational time [28]. To improve solution efficiency, the optimal range for $t_s^y$ and $t_s^c$ was determined. These bounds can effectively narrow the search range for the optimal solution of variables $t_s^y$ and $t_s^c$ and reduce the number of linear segments used in Formulas (30) and (31).

Specifically, the upper bounds $t_s^{y\max }$ and $t_s^{c\max }$ of $t_s^y$ and $t_s^c$ can be obtained from Formulas (32) and (33), respectively.

$$t_s^{y\max }=\min \left\{\max \left\{T_s^m-T_s^o-\min \left\{{\varpi }_d\right\}-{\displaystyle \frac{L_{2s}}{v_s^{\max }+{\widehat{v}}_{2s}}},{\overline{t}}_s^y\right\}+\max \left\{T_{d{a}^n}^l-T_{d1}^l\right\},{\displaystyle \frac{L_{1s}}{v_s^{\min }+{\widehat{v}}_{1s}}}\right\},s\in S;d\in D_s$$

(32)

$$t_s^{c\max }=\min \left\{\max \left\{T_s^m-T_s^o-\min \left\{{\varpi }_d\right\}-{\displaystyle \frac{L_{1s}}{v_s^{\max }+{\widehat{v}}_{1s}}},{\overline{t}}_s^c\right\},{\displaystyle \frac{L_{2s}}{v_s^{\min }+{\widehat{v}}_{2s}}}\right\},s\in S;d\in D_s$$

(33)

And the lower bounds $t_s^{y\min }$ and $t_s^{c\min }$ of $t_s^y$ and $t_s^c$ can be obtained from Formulas (34) and (35), respectively.

$$t_s^{y\min }=\max \left\{T_{d1}^l-T_s^o,{\displaystyle \frac{L_{1s}}{v_s^{\max }+{\widehat{v}}_{1s}}}\right\},s\in S;d\in D_s$$

(34)

$$t_s^{c\min }=\max \left\{T_s^m-T_s^o-\max \left\{{\varpi }_d\right\}-{\displaystyle \frac{L_{1s}}{v_s^{\min }+{\widehat{v}}_{1s}}},{\displaystyle \frac{L_{2s}}{v_s^{\max }+{\widehat{v}}_{2s}}}\right\},s\in S;d\in D_s$$

(35)

Next, proofs for the upper and lower bounds of $t_s^y$ and $t_s^c$ will be provided.

(I). Proof of upper bounds $t_s^{y\max }$ and $t_s^{c\max }$.

Proof. 

It is proven that Formula (32) provides the optimal upper bound for $t_s^y$. Let $C_s^T\left(t_s^y,t_s^c\right)=C_s^f\left[L_{1s}{Q}_{1s}\left(t_s^y\right)+L_{2s}{Q}_{2s}\left(t_s^c\right)\right]+C_s^p\left(t_s^p+t_s^w\right)$ represent the total fuel consumption and potential opportunity cost for vessel $s$. The value of $t_s^y$ that minimizes $C_s^T\left(t_s^y,t_s^c\right)$ will be found. To conduct this, an assumption is made that $t_s^c$ is fixed, and only a single vessel passes through the bottleneck.

Firstly, when $t_s^y\le T_s^m-T_s^o-\min \left\{{\varpi }_d\right\}-t_s^w-t_s^c$, $t_s^p+t_s^w=t_s^w$ is obtained, i.e., $t_s^p=0$. In this scenario, $C_s^T\left(t_s^y,t_s^c\right)$ decreases as $t_s^y$ decreases because $Q_{1s}\left(t_s^y\right)$ is a monotonically decreasing function of $t_s^y$.

Secondly, when $t_s^y>T_s^m-T_s^o-\min \left\{{\varpi }_d\right\}-t_s^w-t_s^c$, $t_s^p+t_s^w=T_s^o+t_s^y+2t_s^w+{\displaystyle \sum _{d\in D_s}{\displaystyle \sum _{n\in N_d}{\varpi }_d\cdot r_{sdn}}}+t_s^c-T_s^m$ will show a linear increase with respect to $t_s^y$. Thus, if $t_s^y>T_s^m-T_s^o-\min \left\{{\varpi }_d\right\}-t_s^w-t_s^c$, Formula (36) is obtained:

$$C_s^T\left(t_s^y,t_s^c\right)=C_s^f\left[L_{1s}{Q}_{1s}\left(t_s^y\right)+L_{2s}{Q}_{2s}\left(t_s^c\right)\right]+C_s^p\left(T_s^o+t_s^y+2t_s^w+{\displaystyle \sum _{d\in D_s}{\displaystyle \sum _{n\in N_d}{\varpi }_d\cdot r_{sdn}}}+t_s^c-T_s^m\right)$$

(36)

The first derivative of $C_s^T\left(t_s^y,t_s^c\right)$ with respect to $t_s^y$ is as follows:

$${\displaystyle \frac{\partial C_s^T}{\partial t_s^y}}=C_s^f{\alpha }_s{\left({\displaystyle \frac{L_{1s}}{t_s^y}}-{\widehat{v}}_s\right)}^{{\beta }_s}-{\displaystyle \frac{C_s^f{L}_{1s}{\alpha }_s{\beta }_s}{t_s^y}}{\left({\displaystyle \frac{L_{1s}}{t_s^y}}-{\widehat{v}}_s\right)}^{{\beta }_s-1}+C_s^p$$

(37)

In addition, the second derivative of $C_s^T\left(t_s^y,t_s^c\right)$ with respect to $t_s^y$ is as follows:

$${\displaystyle \frac{{\partial }^2{C}_s^T}{{\partial }^2{t}_s^y}}={\displaystyle \frac{C_s^f{L}_{1s}^2{\alpha }_s{\beta }_s\left({\beta }_s-1\right)}{t_s^{y3}}}{\left({\displaystyle \frac{L_{1s}}{t_s^y}}-{\widehat{v}}_s\right)}^{{\beta }_s-2}>0$$

(38)

Therefore, when $t_s^y>T_s^m-T_s^o-\min \left\{{\varpi }_d\right\}-t_s^w-t_s^c$ and $t_s^c$ is fixed, $C_s^T\left(t_s^y,t_s^c\right)$ is convex for $t_s^y$, and the minimum value of $C_s^T\left(t_s^y,t_s^c\right)$ is obtained when ${\displaystyle \frac{\partial C_s^T}{\partial t_s^y}}=0$. $t_s^y$ at ${\displaystyle \frac{\partial C_s^T}{\partial t_s^y}}=0$ is denoted as ${\overline{t}}_s^y$.

When $t_s^y\le T_s^m-T_s^o-\min \left\{{\varpi }_d\right\}-t_s^w-t_s^c$, $Q_{1s}\left(t_s^y\right)$ is a decreasing function about $t_s^y$, and $C_s^T\left(t_s^y,t_s^c\right)=C_s^f\left[L_{1s}{Q}_{1s}\left(t_s^y\right)+L_{2s}{Q}_{2s}\left(t_s^c\right)\right]+C_s^p{t}_s^w$ shows a decreasing trend with the increase in $t_s^y$. $C_s^T\left(t_s^y,t_s^c\right)$ is convex for $t_s^y$ when $t_s^y>T_s^m-T_s^o-\min \left\{{\varpi }_d\right\}-t_s^w-t_s^c$ is satisfied. Then, with $t_s^c$ fixed, the optimal sailing time ${\overline{\overline{t}}}_s^y$ of a single vessel $s$ passing through the bottleneck to minimize the fuel cost before the bottleneck is as follows:

$${\overline{\overline{t}}}_s^y=\left\{T_s^m-T_s^o-\min \left\{{\varpi }_d\right\}-t_s^w-t_s^c,{\overline{t}}_s^y\right\}$$

(39)

Since $t_s^c\ge {\displaystyle \frac{L_{2s}}{v_s^{\max }+{\widehat{v}}_{2s}}}$, for any single vessel passing through the bottleneck, the sailing time $t_s^y$ is less than the bound ${\overline{\overline{t}}}_s^y$ and will only increase the cost of vessel $s$, where ${\overline{\overline{t}}}_s^{y\max }$ is as follows:

$${\overline{\overline{t}}}_s^{y\max }=\left\{T_s^m-T_s^o-\min \left\{{\varpi }_d\right\}-{\displaystyle \frac{L_{2s}}{v_s^{\max }+{\widehat{v}}_{2s}}},{\overline{t}}_s^y\right\}$$

(40)

An extreme scenario is assumed where the number of vessels is equal to the bottleneck’s handling capacity. Taking into account the interactions between vessels in the fleet, a vessel $s$ may be queued up after others to arrive at the bottleneck, resulting in it possibly not being able to pass through the bottleneck in a time ${\overline{\overline{t}}}_s^y$ that minimizes cost (upper bound of ${\overline{\overline{t}}}_s^y$ is ${\overline{\overline{t}}}_s^{y\max }$). Therefore, the worst-case scenario is considered, in which the current vessel $s$ is positioned after all other vessels $\left|S_d\right|-1$. Assume the arrival times of the other vessels $\left|S_d\right|-1$ to the bottleneck are $T_{d1}^l$, $T_{d2}^l$, $T_{d3}^l$, …, $T_{d{a}^n-1}^l$. In this case, the last vessel cannot pass through the bottleneck within the time frame $\left[T_{d1}^l,T_{d{a}^n}^l-\widehat{\epsilon }\right]$ without delaying other vessels, where $\widehat{\epsilon }$ is a very small positive value. That is, the last vessel must pass through the bottleneck at time $T_{d{a}^n}^l$ or later. Since ${\overline{\overline{t}}}_s^y+\max \left\{T_{d{a}^n}^l-\widehat{\epsilon }-T_{d1}^l\right\}<{\overline{\overline{t}}}_s^y+\max \left\{T_{d{a}^n}^l-T_{d1}^l\right\}\le {\overline{\overline{t}}}_s^{y\max }+\max \left\{T_{d{a}^n}^l-T_{d1}^l\right\}$, the upper bound on $t_s^y$ is set to ${\overline{\overline{t}}}_s^{y\max }+\max \left\{T_{d{a}^n}^l-T_{d1}^l\right\}$ without considering the vessel speed limit.

In summary, the bottleneck service times have constraints on $T_{d1}^l$, $T_{d2}^l$, $T_{d3}^l$, …, $T_{d{a}^n-1}^l$, and the value of $t_s^{y\max }$ is directly related to ${\overline{\overline{t}}}_s^{y\max }$ and $T_{dn}^l$. Therefore, for generality, the upper bound of $t_s^y$ without considering the vessel speed limit is set to ${\overline{\overline{t}}}_s^{y\max }+\max \left\{T_{d{a}^n}^l-T_{d1}^l\right\}$.

Finally, the sailing time of a vessel is influenced by its speed range $\left[v_s^{\min },v_s^{\max }\right]$ and streamflow velocity ${\widehat{v}}_{1s}$, and the maximum sailing time for a vessel $s$ over a distance $L_{1s}$ satisfies $t_s^{y\max }\le {\displaystyle \frac{L_{1s}}{v_s^{\min }+{\widehat{v}}_{1s}}}$. Therefore, the optimal upper bound $t_s^{y\max }$ about the piecewise linear function ${\overline{Q}}_{1s}\left(t_s^y\right)$ is determined as provided in Formula (32).

The upper bound $t_s^{c\max }$ can also be proven similarly. The only difference is that vessels do not need to be distinguished by their sequence after passing through the bottleneck. Therefore, the optimal upper bound $t_s^{c\max }$ of $t_s^c$ does not include $\max \left\{T_{d{a}^n}^l-T_{d1}^l\right\}$, as shown in Formula (33). □

(II). Proof of lower bounds $t_s^{y\min }$ and $t_s^{c\min }$.

Proof. 

The optimal lower bound of $t_s^y$ is first proven to be $t_s^{y\min }$. This is because the bottleneck exists at the service time $T_{d1}^l$, $T_{d2}^l$, $T_{d3}^l$, …, $T_{d{a}^n-1}^l$. A special case is considered where, for any vessel $s$, it is always scheduled to pass through the bottleneck at $T_{d1}^l$. In this situation, the speed of the vessel during the first leg $L_{1s}$ of the voyage is always maximized, and under the condition of the maximum allowed speed $v_s^{\max }$, $T_{d1}^l-T_s^o$ is the minimum value of $t_s^{y\min }$.

In addition, the minimum sailing time of a vessel $s$ is constrained by its maximum speed $v_s^{\max }$ and the streamflow velocity ${\widehat{v}}_{1s}$, meaning that the minimum sailing time $t_s^{y\min }$ over a distance $L_{1s}$ satisfies $t_s^{y\min }\ge {\displaystyle \frac{L_{1s}}{v_s^{\max }+{\widehat{v}}_{1s}}}$. Therefore, Formula (34) is proven.

Then, the optimal lower bound $t_s^{c\min }$ for $t_s^c$ is proven. According to ${\overline{\overline{t}}}_s^c$, in the case where $t_s^y$ is fixed, it can be analogously determine that the optimal sailing time ${\tilde{t}}_s^c$ for all vessels minimizing fuel cost after passing through the bottleneck is given by the following:

$${\tilde{t}}_s^c=T_s^m-T_s^o-\max \left\{{\varpi }_d\right\}-t_s^w-t_s^y$$

(41)

Given $t_s^y\le {\displaystyle \frac{L_{1s}}{v_s^{\min }+{\widehat{v}}_{1s}}}$, under the constraint of vessel speed limit, the sailing time $t_s^c$ for any vessel passing through the bottleneck will be greater than this lower bound ${\tilde{t}}_s^{c\min }$, where ${\tilde{t}}_s^{c\min }$ is as follows:

$${\tilde{t}}_s^{c\min }=T_s^m-T_s^o-\max \left\{{\varpi }_d\right\}-{\displaystyle \frac{L_{1s}}{v_s^{\min }+{\widehat{v}}_{1s}}}$$

(42)

Similarly, due to the scheduling requirements within the scheduling period and the speed limit of the vessels, $t_s^{c\min }$ must satisfy $t_s^{c\min }\ge {\displaystyle \frac{L_{2s}}{v_s^{\max }+{\widehat{v}}_{2s}}}$, which confirms that Formula (35) holds true. □

4.5. Linearized Model

Based on the above description, the linearized model can be obtained. Before constructing the linearized model, it is necessary to introduce some new parameters and variables, as detailed below:

(I). Parameters

${\theta }_{1sk}$, $s\in S;k\in \left\{1,2,\dots ,K_{1s}\right\}$ is the slope of the piecewise linear approximation function ${\overline{Q}}_{1s}\left(t_s^y\right)$ used to construct the fuel consumption rate for the first leg of the voyage, where $K_{1s}$ is the total number of linear segments used in ${\overline{Q}}_{1s}\left(t_s^y\right)$;

${\gamma }_{1sk}$, $s\in S;k\in \left\{1,2,\dots ,K_{1s}\right\}$ is the intercept of the piecewise linear approximation function ${\overline{Q}}_{1s}\left(t_s^y\right)$ used to construct the fuel consumption rate for the first leg of the voyage;

${\theta }_{2sk}$, $s\in S;k\in \left\{1,2,\dots ,K_{2s}\right\}$ is the slope of the piecewise linear approximation function ${\overline{Q}}_{2s}\left(t_s^c\right)$ used to construct the fuel consumption rate for the second leg of the voyage, where $K_{2s}$ is the total number of linear segments used in ${\overline{Q}}_{2s}\left(t_s^c\right)$;

${\gamma }_{2sk}$, $s\in S;k\in \left\{1,2,\dots ,K_{2s}\right\}$ is the intercept of the piecewise linear approximation function ${\overline{Q}}_{2s}\left(t_s^c\right)$ used to construct the fuel consumption rate for the second leg of the voyage.

(II). Variables

$q_{1s}$, $s\in S$ is the fuel consumption rate of vessel $s$ in the first leg of the voyage before the bottleneck (tons/km);

$q_{2s}$, $s\in S$ is the fuel consumption rate of vessel $s$ in the second leg of the voyage after the bottleneck (tons/km);

$t_s^y$, $s\in S$ is the sailing time of vessel $s$ for the first leg of the voyage before the bottleneck (h);

$t_s^c$, $s\in S$ is the sailing time of vessel $s$ for the second leg of the voyage after the bottleneck (h);

$t_s^p$, $s\in S$ is the delay time in the arrival of vessel $s$ at its destination (h).

The model [M1] can be transformed into the following mixed-integer linear programming model [M2]:

Objective function:

$$\min {\displaystyle \sum _{s\in S}\left(C_s^f\left(L_{1s}{q}_{1s}+L_{2s}{q}_{2s}\right)+C_s^p\left(t_s^p+t_s^w\right)\right)}$$

(43)

Constraints: Constraints (6)–(9), constraints (11)–(17)

$$q_{1s}\ge {\theta }_{1sk}{t}_s^y+{\gamma }_{1sk},s\in S;k=1,2,\dots ,K_{1s}$$

(44)

$$q_{1s}\ge 0$$

(45)

$$q_{2s}\ge {\theta }_{2sk}{t}_s^c+{\gamma }_{2sk},s\in S;k=1,2,\dots ,K_{2s}$$

(46)

$$q_{2s}\ge 0$$

(47)

The model [M2] can be solved directly using a commercial solver such as CPLEX.

5. Numerical Case Study

Section 5.1 introduces the case background and parameter settings of the Yangtze River. In Section 5.2, the optimal fleet scheduling plan is studied. Section 5.3 analyzes the impact of changes in fuel prices, streamflow velocities, and proportion of LNG-fuelled vessels on the scheduling results.

5.1. Case Description

The Three Gorges Project consists of one of the busiest river bottlenecks in the world. The ways for passing through this bottleneck include the Three Gorges five-class ship lock, ship lift, trans-shipment facility to bypass bottlenecks, and a new planned ship lock that may be put into use in the future. The following is a detailed description of these ways:

(I). Three Gorges five-class ship lock: The design’s daily operating time for the Three Gorges and Gezhouba locks is 22 h, but in practice, they operate 24 h a day without scheduled maintenance or repairs. The two lines of the Three Gorges lock adopt the operation mode of one-way continuous passing through the lock, one lock chamber for upstream and one for downstream traffic, each allowing for four vessels per lockage, and the operation interval between two adjacent lockages is about 1.5 h. Recently, the average number of vessels passing through the Three Gorges lock and Gezhouba locks No. 1 and No. 2 per lockage has been about four. The navigation management authority of the Three Gorges ensures unified scheduling for vessels passing through the lock, and the handling capacity of the Three Gorges and Gezhouba locks is generally consistent. Studies show that with 4 vessels per lockage, each line of the Three Gorges lock operates 16 lockages per day, totalling 32 lockages per day for both lines. For the Gezhouba lock, during one-way operation, Lock No. 2 operates 17.4 lockages per day, and Lock No. 1 operates 15–16 lockages per day, making the total daily lockages exceed 32, which matches the capacity of the Three Gorges lock. Therefore, Gezhouba locks No. 1 and No. 2 mainly adopt the one-way operation mode to match the operation of the two-line mode of the Three Gorges lock. The Three Gorges lock is located 38 km from the Gezhouba lock, and vessels take about 10 h to pass through this section. In addition, the average passing times of the Three Gorges and Gezhouba locks are 3 h and 1.5 h, respectively.

(II). Ship lift: The ship lift takes about 1 h to lift a single vessel. After lifting, the vessel passes through the waterway connecting the Three Gorges and Gezhouba locks and then through the Gezhouba lock. Due to the need for regular maintenance, the ship lift operates intermittently.

(III). Trans-shipment facility to bypass bottlenecks: As for using trans-shipment facilities to bypass bottlenecks, most of them are often selected by the shipowner, and the authority schedules the vessel according to the selection of shipowners. The Zigui and Yichang ports provide transfer services. Currently, there are four berths available for simultaneous operations, with each loading or unloading operation taking about 5 h per berth. The time taken by trans-shipment facilities to bypass bottlenecks includes loading, unloading, and road transport time. Among these, the average speed of road transport is about 50 km/h, and the transport distance is 100 km.

(IV). New planned ship lock: Since the new lock of the Yangtze River is still in the planning stage, it is assumed that the new planned ship lock, which may be put into use in the future, will have a similar daily operation and vessel handling capacity to those of the existing five-class ship lock.

In summary, the basic parameters of each bottleneck passing way in the Three Gorges Project are provided in

Table 1. Basic parameters of each bottleneck passing way in Three Gorges Project.

No	Bottleneck Passing Way	${\mathit{\varpi }}_{\mathit{d}}$ (h)	${\mathit{T}}_{\mathit{d}\mathit{n}}^{\mathit{l}}$ (h)	${\mathit{G}}_{\mathit{d}}$ (Vessels)	Tonnage Limit (DWT)
1	Three Gorges five-class ship lock	14.5	72.0/73.5/75.0/76.5/78.0/79.5/81.0/82.5/ 84.0/85.5/87.0/88.5/90.0/91.5/93.0/94.5	4	-
2	Ship lift	12.5	72.0/74.0/75.5/76.5/78.0/79.0/80.0/81.5/ 83.0/84.0/85.5/87.0/89.5/90.0/91.5/93.0	1	3000
3	Trans-shipment facility	12.0	72.0/77.0/82.0/87.0	4	-
4	New planned ship lock	14.5	72.5/74.0/75.5/77.0/78.5/80.0/81.5/83.0/ 84.5/86.0/87.5/89.0/90.5/92.0/93.5/95.0	4	-

“-” means that there is no tonnage limit on the bottleneck passing way.

.

In recent years, inland waterway transportation has begun to develop in a greener direction. Among green-powered river vessels, LNG-fuelled vessels are the most mature and widely promoted. These LNG-fuelled vessels are also expected to be the mainstay of green-powered river shipping in China in the medium to short term. Compared to oil-fuelled vessels, LNG-fuelled vessels have advantages in terms of fuel price, but they have higher vessel capital and operating cost. Therefore, in the unified scheduling process of a mixed fleet, the two main vessel types in the future are comprehensively considered to adapt to the arrangement of the vessel scheduling plan. Statistics indicate that the annual average price of No. 0 diesel is about 6500 CNY/ton, while the price of LNG fuel is generally set at around 80% of the price of No. 0 diesel [1]. In river shipping, the actual sailing speed of vessels and the operating cost are influenced by streamflow velocity. For the sake of generality, the streamflow velocity is set to 3 km/h.

According to the actual river shipping scenario, a total of 160 vessels are selected to pass through the site of the Three Gorges Project. With reference to previous studies on river shipping [2], fuel consumption parameters, speed range, and other parameters of vessels are set. $C_s^p$ depends on the attributes of the vessel, and it can be estimated according to vessel capital and operating cost [1,55,59]. The origin ports are selected from major ports on the Yangtze River main line, including the following: Badong, Wushan, Fengjie, Wanzhou, Xinsheng, Fengdu, Fuling, Guoyuan, Chongqing, Jiangjin, Luzhou, and Yibin. When the vessel reports its bottleneck passing plan, it may be at the port or on passage. $T_s^o$ and $T_s^m$ are related to sailing distance, vessel economic speed, and the cargo owner’s required arrival time. The destination ports are selected from Zhicheng, Jingzhou, Yueyang, Honghu, Jiayu, Wuhan, Huangzhou, Ezhou, Huangshi, and Jiujiang. Based on these settings, a typical river section including the Three Gorges Project will be constructed. The locations of these ports and the Three Gorges Project are shown in Figure 5, and the distances between ports and the basic parameters of the vessels passing through the bottleneck during the scheduling period are listed in

Table 2. Distances between river ports.

No	Leg	Distance (km)	No	Leg	Distance (km)
1	Yibin → Luzhou	130	13	Zigui (Three Gorges lock) → Yichang (Gezhouba lock)	38
2	Luzhou → Jiangjin	168	14	Yichang (Gezhouba lock) → Zhicheng	60
3	Jiangjin → Chongqing	86	15	Zhicheng → Jingzhou	232
4	Chongqing → Guoyuan	31	16	Jingzhou → Yueyang	244
5	Guoyuan → Fuling	89	17	Yueyang → Honghu	51
6	Fuling → Fengdu	55	18	Honghu →Jiayu	49
7	Fengdu → Xinsheng	63	19	Jiayu → Wuhan	131
8	Xinsheng → Wanzhou	83	20	Wuhan → Huangzhou	74
9	Wanzhou → Fengjie	126	21	Huangzhou → Ezhou	18
10	Fengjie → Wushan	39	22	Ezhou → Huangshi	32
11	Wushan → Badong	49	23	Huangshi → Jiujiang	127
12	Badong → Zigui (Three Gorges lock)	75

and

Table A1. The basic parameters of vessels through a river bottleneck on the Yangtze River main line.

Vessel	Origin	Destination	${\mathit{L}}_{1\mathit{s}}$ (km)	${\mathit{L}}_{2\mathit{s}}$ (km)	${\mathit{T}}_{\mathit{s}}^{\mathit{o}}$ (h)	${\mathit{T}}_{\mathit{s}}^{\mathit{m}}$ (h)	${\mathit{C}}_{\mathit{s}}^{\mathit{p}}$ (CNY/h)	${\mathit{v}}_{\mathit{s}}^{\mathbf{min}}$ (km/h)	${\mathit{v}}_{\mathit{s}}^{\mathbf{max}}$ (km/h)	${\mathit{\alpha }}_{\mathit{s}}$	${\mathit{\beta }}_{\mathit{s}}$	${\mathit{D}}_{\mathit{s}}$
1-O	Badong	Zhicheng	75	60	65.9	91.9	400	7	28	0.000124	2.234	1/4
2-O	Badong	Jingzhou	75	292	67.9	106.0	1000	8	29	0.000307	2.186	1/4
3-L	Badong	Yueyang	75	536	68.2	114.0	1600	5	25	0.000287	2.148	1/4
4-L	Badong	Honghu	75	587	66.7	128.6	1400	8	28	0.000271	2.389	3
5-O	Badong	Jiayu	75	636	67.5	138.7	1000	6	27	0.000273	2.390	1/4
6-O	Badong	Wuhan	75	767	67.7	138.5	600	8	25	0.000125	2.342	1/4
7-O	Badong	Huangzhou	75	841	67.0	149.4	500	6	25	0.000299	2.118	1/4
8-O	Badong	Ezhou	75	859	67.2	151.1	500	5	25	0.000175	2.254	1/4
9-O	Badong	Huangshi	75	891	67.2	144.4	400	7	27	0.000131	2.213	1/2/4
10-O	Badong	Jiujiang	75	1018	65.9	169.5	400	7	30	0.000131	2.126	1/2/4
11-O	Wushan	Zhicheng	124	60	63.3	95.3	500	5	28	0.000268	2.229	1/4
12-O	Wushan	Jingzhou	124	292	64.5	111.8	1000	8	30	0.000326	2.328	1/4
13-L	Wushan	Yueyang	124	536	64.5	119.4	2000	7	25	0.000274	2.360	1/4
14-L	Wushan	Honghu	124	587	65.3	118.9	1200	5	29	0.000209	2.209	3
15-O	Wushan	Jiayu	124	636	60.8	149.1	400	6	26	0.000318	2.157	1/4
16-O	Wushan	Wuhan	124	767	61.8	151.7	400	7	29	0.000155	2.276	1/4
17-O	Wushan	Huangzhou	124	841	63.5	144.9	400	7	27	0.000121	2.105	1/2/4
18-O	Wushan	Ezhou	124	859	62.3	160.3	400	7	25	0.000171	2.241	1/4
19-O	Wushan	Huangshi	124	891	63.0	164.3	400	8	27	0.000150	2.313	1/4
20-O	Wushan	Jiujiang	124	1018	61.7	171.5	400	6	30	0.000138	2.274	1/4
21-O	Fengjie	Zhicheng	163	60	63.5	92.6	800	7	25	0.000180	2.255	1/4
22-O	Fengjie	Jingzhou	163	292	62.2	109.7	600	8	28	0.000157	2.329	3
23-L	Fengjie	Yueyang	163	536	60.3	125.4	800	6	25	0.000119	2.425	1/4
24-L	Fengjie	Honghu	163	587	61.6	124.3	800	5	29	0.000136	2.219	1/4
25-O	Fengjie	Jiayu	163	636	62.6	131.5	800	5	29	0.000123	2.418	1/4
26-O	Fengjie	Wuhan	163	767	59.8	146.7	400	6	29	0.000159	2.209	1/4
27-O	Fengjie	Huangzhou	163	841	62.1	149.3	400	8	26	0.000205	2.133	3
28-O	Fengjie	Ezhou	163	859	61.8	154.7	600	7	30	0.000164	2.340	1/4
29-O	Fengjie	Huangshi	163	891	62.4	150.6	1000	7	26	0.000214	2.352	1/4
30-O	Fengjie	Jiujiang	163	1018	63.7	181.7	1000	6	30	0.000341	2.409	1/4
31-O	Wanzhou	Zhicheng	289	60	51.1	97.8	900	6	26	0.000299	2.393	1/4
32-O	Wanzhou	Jingzhou	289	292	55.9	108.8	900	6	26	0.000206	2.274	1/4
33-L	Wanzhou	Yueyang	289	536	51.7	124.6	800	7	25	0.000155	2.153	1/2/4
34-L	Wanzhou	Honghu	289	587	53.7	124.2	1000	5	30	0.000187	2.326	1/4
35-O	Wanzhou	Jiayu	289	636	56.6	140.6	1000	8	28	0.000269	2.355	1/4
36-O	Wanzhou	Wuhan	289	767	56.6	147.5	1000	8	27	0.000278	2.318	1/4
37-O	Wanzhou	Huangzhou	289	841	51.7	158.2	500	8	27	0.000295	2.145	3
38-O	Wanzhou	Ezhou	289	859	55.2	151.8	1000	5	27	0.000317	2.255	1/4
39-O	Wanzhou	Huangshi	289	891	53.3	159.1	1000	5	26	0.000320	2.298	1/4
40-O	Wanzhou	Jiujiang	289	1018	55.6	160.2	900	5	28	0.000195	2.333	1/4
41-O	Xinsheng	Zhicheng	372	60	44.4	98.7	400	7	25	0.000243	2.156	1/4
42-O	Xinsheng	Jingzhou	372	292	46.5	115.8	500	8	25	0.000274	2.152	1/4
43-L	Xinsheng	Yueyang	372	536	33.3	142.8	800	5	30	0.000204	2.316	1/4
44-L	Xinsheng	Honghu	372	587	45.2	129.2	800	5	26	0.000170	2.384	1/4
45-O	Xinsheng	Jiayu	372	636	47.3	137.2	400	6	28	0.000127	2.230	1/4
46-O	Xinsheng	Wuhan	372	767	52.4	137.7	1000	7	30	0.000310	2.177	1/4
47-O	Xinsheng	Huangzhou	372	841	53.2	137.9	1000	6	30	0.000177	2.278	1/4
48-O	Xinsheng	Ezhou	372	859	50.9	162.0	1000	7	26	0.000308	2.382	3
49-O	Xinsheng	Huangshi	372	891	35.4	178.5	400	5	25	0.000295	2.196	1/4
50-O	Xinsheng	Jiujiang	372	1018	52.6	172.9	1000	7	27	0.000282	2.350	1/4
51-O	Fengdu	Zhicheng	435	60	49.0	96.4	1000	7	30	0.000261	2.242	1/4
52-O	Fengdu	Jingzhou	435	292	42.2	118.2	400	7	26	0.000165	2.263	1/4
53-L	Fengdu	Yueyang	435	536	39.0	127.7	800	6	27	0.000238	2.214	1/4
54-L	Fengdu	Honghu	435	587	39.0	131.6	800	6	25	0.000271	2.298	1/4
55-O	Fengdu	Jiayu	435	636	46.7	137.1	500	7	25	0.000136	2.278	1/4
56-O	Fengdu	Wuhan	435	767	49.1	132.3	600	8	29	0.000154	2.181	3
57-O	Fengdu	Huangzhou	435	841	47.1	144.7	700	5	28	0.000238	2.168	1/4
58-O	Fengdu	Ezhou	435	859	37.1	184.0	400	6	26	0.000195	2.373	1/4
59-O	Fengdu	Huangshi	435	891	43.0	168.7	400	8	25	0.000203	2.219	1/4
60-O	Fengdu	Jiujiang	435	1018	37.0	197.9	500	5	30	0.000297	2.301	1/4
61-O	Fuling	Zhicheng	490	60	42.1	98.1	800	8	29	0.000144	2.429	1/4
62-O	Fuling	Jingzhou	490	292	45.5	110.6	800	7	29	0.000307	2.121	1/4
63-L	Fuling	Yueyang	490	536	35.4	127.1	1600	8	29	0.000319	2.402	3
64-L	Fuling	Honghu	490	587	45.1	119.2	800	7	25	0.000163	2.168	1/2/4
65-O	Fuling	Jiayu	490	636	35.0	135.4	400	8	25	0.000169	2.126	1/2/4
66-O	Fuling	Wuhan	490	767	36.5	163.8	700	6	25	0.000301	2.327	1/4
67-O	Fuling	Huangzhou	490	841	42.4	170.7	1000	7	25	0.000326	2.393	1/4
68-O	Fuling	Ezhou	490	859	39.0	163.1	500	7	26	0.000295	2.157	1/4
69-O	Fuling	Huangshi	490	891	42.0	155.1	1000	6	26	0.000300	2.266	1/4
70-O	Fuling	Jiujiang	490	1018	46.3	152.0	900	8	25	0.000323	2.119	1/4
71-O	Guoyuan	Zhicheng	579	60	30.9	103.9	500	7	26	0.000173	2.388	1/4
72-O	Guoyuan	Jingzhou	579	292	43.0	111.4	1000	6	28	0.000344	2.150	1/4
73-L	Guoyuan	Yueyang	579	536	36.2	120.1	800	5	28	0.000121	2.199	1/2/4
74-L	Guoyuan	Honghu	579	587	34.2	125.4	1000	8	28	0.000152	2.403	1/4
75-O	Guoyuan	Jiayu	579	636	38.2	137.7	500	7	26	0.000211	2.145	1/4
76-O	Guoyuan	Wuhan	579	767	29.0	145.5	400	5	25	0.000132	2.218	1/4
77-O	Guoyuan	Huangzhou	579	841	35.2	167.1	1000	8	26	0.000288	2.393	1/4
78-O	Guoyuan	Ezhou	579	859	38.1	151.2	800	6	26	0.000193	2.319	3
79-O	Guoyuan	Huangshi	579	891	36.2	158.3	800	6	29	0.000290	2.231	1/4
80-O	Guoyuan	Jiujiang	579	1018	16.4	210.4	400	6	29	0.000331	2.265	1/4
81-O	Chongqing	Zhicheng	610	60	17.1	92.4	400	7	30	0.000136	2.176	1/2/4
82-O	Chongqing	Jingzhou	610	292	26.0	127.3	700	6	30	0.000272	2.395	1/4
83-L	Chongqing	Yueyang	610	536	40.8	114.4	1800	7	27	0.000251	2.224	1/4
84-L	Chongqing	Honghu	610	587	16.8	140.1	800	7	30	0.000338	2.156	1/4
85-O	Chongqing	Jiayu	610	636	36.3	137.2	500	8	25	0.000169	2.212	1/4
86-O	Chongqing	Wuhan	610	767	38.3	159.2	1000	7	30	0.000342	2.310	1/4
87-O	Chongqing	Huangzhou	610	841	35.7	157.4	1000	8	27	0.000250	2.355	1/4
88-O	Chongqing	Ezhou	610	859	31.8	162.7	700	5	25	0.000292	2.248	1/4
89-O	Chongqing	Huangshi	610	891	40.3	143.9	1000	6	27	0.000312	2.145	3
90-O	Chongqing	Jiujiang	610	1018	24.1	196.7	800	7	30	0.000305	2.407	1/4
91-O	Jiangjin	Zhicheng	696	60	30.8	100.3	500	8	28	0.000191	2.192	1/4
92-O	Jiangjin	Jingzhou	696	292	37.1	110.0	800	7	26	0.000196	2.177	1/4
93-L	Jiangjin	Yueyang	696	536	15.7	130.3	800	7	25	0.000257	2.108	1/4
94-L	Jiangjin	Honghu	696	587	37.0	116.5	1000	8	28	0.000166	2.167	3
95-O	Jiangjin	Jiayu	696	636	36.3	138.1	1000	7	25	0.000273	2.252	1/4
96-O	Jiangjin	Wuhan	696	767	6.3	202.8	400	5	26	0.000331	2.173	1/4
97-O	Jiangjin	Huangzhou	696	841	32.8	151.5	700	6	27	0.000117	2.398	1/4
98-O	Jiangjin	Ezhou	696	859	24.9	168.3	700	6	30	0.000258	2.316	1/4
99-O	Jiangjin	Huangshi	696	891	32.4	155.3	1000	7	29	0.000206	2.351	1/4
100-O	Jiangjin	Jiujiang	696	1018	23.8	178.6	400	5	27	0.000264	2.135	1/4
101-O	Luzhou	Zhicheng	864	60	44.9	114.6	700	6	25	0.000170	2.318	1/4
102-O	Luzhou	Jingzhou	864	292	2.6	146.2	400	5	25	0.000301	2.393	1/4
103-L	Luzhou	Yueyang	864	536	22.9	156.3	800	5	29	0.000220	2.332	1/4
104-L	Luzhou	Honghu	864	587	38.2	150.3	2000	5	26	0.000329	2.420	3
105-O	Luzhou	Jiayu	864	636	34.9	161.2	700	8	25	0.000270	2.288	1/4
106-O	Luzhou	Wuhan	864	767	46.4	178.0	1000	7	29	0.000339	2.429	1/4
107-O	Luzhou	Huangzhou	864	841	39.1	174.3	800	7	26	0.000220	2.358	1/4
108-O	Luzhou	Ezhou	864	859	32.5	179.0	400	6	28	0.000243	2.173	1/4
109-O	Luzhou	Huangshi	864	891	28.5	188.4	1000	8	26	0.000331	2.391	1/4
110-O	Luzhou	Jiujiang	864	1018	12.1	218.1	400	5	30	0.000193	2.418	1/4
111-O	Yibin	Zhicheng	994	60	23.6	115.4	400	7	30	0.000145	2.115	1/2/4
112-O	Yibin	Jingzhou	994	292	29.6	130.5	400	7	27	0.000128	2.111	1/2/4
113-L	Yibin	Yueyang	994	536	20.8	151.5	800	8	25	0.000128	2.154	1/2/4
114-L	Yibin	Honghu	994	587	25.0	152.9	800	7	29	0.000149	2.154	1/2/4
115-O	Yibin	Jiayu	994	636	17.2	161.4	400	8	27	0.000144	2.137	1/2/4
116-O	Yibin	Wuhan	994	767	30.2	161.8	400	7	25	0.000134	2.106	1/2/4
117-O	Yibin	Huangzhou	994	841	37.0	160.9	400	8	30	0.000120	2.177	1/2/4
118-O	Yibin	Ezhou	994	859	36.7	162.2	400	8	27	0.000117	2.144	1/2/4
119-O	Yibin	Huangshi	994	891	43.0	163.5	500	7	29	0.000163	2.170	3
120-O	Yibin	Jiujiang	994	1018	18.3	190.6	400	8	29	0.000139	2.148	1/2/4
121-O	On passage	Zhicheng	862	60	0.0	104.9	600	8	29	0.000293	2.313	1/4
122-O	On passage	Jingzhou	839	292	0.0	141.1	400	6	29	0.000164	2.223	1/4
123-L	On passage	Yueyang	894	536	0.0	154.2	800	5	25	0.000133	2.121	1/2/4
124-L	On passage	Honghu	732	587	0.0	148.9	800	5	27	0.000220	2.370	1/4
125-O	On passage	Jiayu	924	636	0.0	148.2	400	5	26	0.000160	2.114	1/2/4
126-O	On passage	Wuhan	852	767	0.0	165.7	400	5	28	0.000227	2.193	1/4
127-O	On passage	Huangzhou	942	841	0.0	176.3	400	8	26	0.000129	2.147	1/2/4
128-O	On passage	Ezhou	919	859	0.0	185.8	500	8	26	0.000278	2.275	1/4
129-O	On passage	Huangshi	937	891	0.0	163.9	400	6	26	0.000126	2.164	1/2/4
130-O	On passage	Jiujiang	978	1018	0.0	200.2	400	5	26	0.000174	2.347	1/4
131-O	On passage	Zhicheng	964	60	0.0	114.8	400	6	27	0.000284	2.337	1/4
132-O	On passage	Jingzhou	828	292	0.0	128.3	400	5	27	0.000157	2.205	1/4
133-L	On passage	Yueyang	737	536	0.0	147.6	800	8	25	0.000295	2.252	1/4
134-L	On passage	Honghu	866	587	0.0	162.4	800	6	27	0.000130	2.275	1/4
135-O	On passage	Jiayu	764	636	0.0	174.8	400	7	27	0.000215	2.397	1/4
136-O	On passage	Wuhan	721	767	0.0	190.3	400	6	26	0.000134	2.355	1/4
137-O	On passage	Huangzhou	984	841	0.0	174.3	800	7	27	0.000327	2.340	3
138-O	On passage	Ezhou	905	859	0.0	190.6	400	5	29	0.000227	2.300	1/4
139-O	On passage	Huangshi	855	891	0.0	180.0	400	6	27	0.000260	2.180	1/4
140-O	On passage	Jiujiang	870	1018	0.0	190.0	400	5	26	0.000299	2.167	1/4
141-O	On passage	Zhicheng	865	60	0.0	111.4	400	7	26	0.000156	2.260	1/4
142-O	On passage	Jingzhou	872	292	0.0	124.7	400	7	30	0.000181	2.283	1/4
143-L	On passage	Yueyang	949	536	0.0	153.8	800	6	25	0.000127	2.237	1/2/4
144-L	On passage	Honghu	962	587	0.0	162.9	800	7	27	0.000205	2.328	1/4
145-O	On passage	Jiayu	972	636	0.0	152.4	400	6	29	0.000192	2.216	1/4
146-O	On passage	Wuhan	923	767	0.0	178.2	400	7	30	0.000190	2.121	3
147-O	On passage	Huangzhou	883	841	0.0	191.3	400	5	29	0.000269	2.110	1/4
148-O	On passage	Ezhou	931	859	0.0	191.3	400	6	27	0.000300	2.233	1/4
149-O	On passage	Huangshi	903	891	0.0	202.5	400	5	28	0.000335	2.235	1/4
150-O	On passage	Jiujiang	864	1018	0.0	215.3	400	8	27	0.000141	2.323	1/4
151-O	On passage	Zhicheng	930	60	0.0	108.0	900	8	26	0.000330	2.365	1/4
152-O	On passage	Jingzhou	963	292	0.0	134.9	500	6	30	0.000237	2.359	1/4
153-L	On passage	Yueyang	977	536	0.0	145.0	1000	8	28	0.000300	2.259	1/4
154-L	On passage	Honghu	962	587	0.0	144.7	800	8	26	0.000244	2.355	1/4
155-O	On passage	Jiayu	868	636	0.0	178.0	400	8	25	0.000178	2.126	1/2/4
156-O	On passage	Wuhan	943	767	0.0	166.4	400	7	26	0.000210	2.258	3
157-O	On passage	Huangzhou	932	841	0.0	182.9	400	7	29	0.000314	2.172	1/4
158-O	On passage	Ezhou	918	859	0.0	186.6	400	6	27	0.000145	2.132	1/2/4
159-O	On passage	Huangshi	930	891	0.0	194.5	400	8	26	0.000276	2.254	1/4
160-O	On passage	Jiujiang	947	1018	0.0	186.4	400	8	29	0.000130	2.131	1/2/4

“*-O” means that the vessel * is an oil-fuelled vessel, and “*-L” means that the vessel * is an LNG-fuelled vessel.

, respectively.

For the approximation model [M2], considering the impact of the sailing distance on the objective value, the absolute objective value tolerance $\epsilon$ is set to CNY 5000. Model [M2] is solved using the IBM ILOG CPLEX 12.6.2 MILP solver, with the software parameters set to their default values. All tests were conducted on a PC with an AMD Ryzen 5-4600H with Radeon graphics at 3.00 GHz and 16.0 GB RAM memory, under Windows 10 64-bit. And all scenarios for this model can be solved precisely within 60 s.

5.2. Optimization Results

Using the scheduling model in this paper, vessels that need to pass through the river bottleneck under different passing ways and service times are organized, as shown in

Table 3. Vessel scheduling arrangement for different bottleneck passing ways.

Bottleneck Passing Way	Service Time	Vessel
Three Gorges five-class ship lock	72.0	3-L/13-L/23-L/133-L
	73.5	5-O/6-O/7-O/34-L
	75.0	12-O/21-O/29-O/74-L
	76.5	24-L/32-O/47-O/135-O
	78.0	38-O/40-O/46-O/53-L
	79.5	30-O/36-O/43-L/51-O
	81.0	57-O/61-O/70-O/80-O
	82.5	59-O/72-O/79-O/92-O
	84.0	76-O/87-O/124-L/128-O
	85.5	52-O/58-O/66-O/95-O
	87.0	67-O/68-O/141-O/142-O
	88.5	71-O/86-O/91-O/153-L
	90.0	123-L/132-O/140-O/157-O
	91.5	130-O/138-O/148-O/152-O
	93.0	114-L/115-O/116-O/122-O
	94.5	105-O/108-O/109-O/111-O
Ship lift	72.0	10-O
	74.0	9-O
	75.5	17-O
	76.5	64-L
	78.0	81-O
	79.0	65-O
	80.0	33-L
	81.5	73-L
	83.0	155-O
	84.0	160-O
	85.5	127-O
	87.0	129-O
	89.5	125-O
	90.0	143-L
	91.5	118-O
	93.0	117-O
Trans-shipment facility	72.0	4-L/14-L/63-L/94-L
	77.0	22-O/27-O/37-O/56-O
	82.0	48-O/78-O/89-O/156-O
	87.0	104-L/119-O/137-O/146-O
New planned ship lock	72.5	1-O/2-O/8-O/83-L
	74.0	15-O/16-O/18-O/19-O
	75.5	20-O/25-O/44-L/54-L
	77.0	11-O/26-O/28-O/84-L
	78.5	31-O/39-O/121-O/150-O
	80.0	35-O/42-O/69-O/136-O
	81.5	41-O/49-O/62-O/93-L
	83.0	45-O/50-O/55-O/99-O
	84.5	90-O/97-O/151-O/159-O
	86.0	75-O/77-O/88-O/154-L
	87.5	60-O/82-O/85-O/98-O
	89.0	100-O/126-O/139-O/145-O
	90.5	96-O/134-L/147-O/158-O
	92.0	113-L/120-O/144-L/149-O
	93.5	103-L/110-O/112-O/131-O
	95.0	101-O/102-O/106-O/107-O

. This scheduling plan can be published in advance on an official website to provide a basis for the scheduling of a mixed fleet and bottleneck passing ways.

From Table 3, it can be observed that LNG-fuelled vessels are likely to be scheduled for an earlier service time to pass through the bottleneck at the same fuel consumption. This is because LNG-fuelled vessels benefit from lower fuel costs, allowing them to increase speed to better coordinate with the scheduling of oil-fuelled vessels. In addition, since LNG-fuelled vessels have higher potential opportunity costs, the costs associated with waiting or delays are greater. The subsequent sensitivity analysis revealed that with a rising fuel price, increased streamflow velocity, and a decrease in the proportion of LNG-fuelled vessels, the fleet’s delay time at the destination port increased, as an inevitable result of balancing fuel consumption cost and opportunity cost. These findings also indirectly indicate that LNG-fuelled vessels tend to be prioritized in scheduling. Therefore, under similar conditions, prioritizing the passage of LNG-fuelled vessels through the bottleneck can help reduce the total cost of a mixed fleet.

Figure 6 shows the optimal sailing, delay, and waiting times for the fleet passing through the bottleneck. Through a joint analysis of Figure 6a,b, it can be seen that under unified scheduling, vessels sailing longer distances will reduce their sailing speed according to their shipping plans. At the same time, vessels with relatively higher fuel consumption rates and lower potential opportunity costs will select to pass through the bottleneck later under unified scheduling and extend the planned total voyage time to reduce fuel consumption. In this way, each vessel can arrive at the bottleneck in an orderly manner and adjust its sailing plans, thereby reducing the total cost of the mixed fleet.

5.3. Sensitivity Analysis

5.3.1. Fuel Price

Due to fluctuations in fuel price influenced by international oil price, which historically range between 4000 CNY/ton and 9000 CNY/ton, our numerical case study uses six representative prices, 4000 CNY/ton, 5000 CNY/ton, 6000 CNY/ton, 7000 CNY/ton, 8000 CNY/ton, and 9000 CNY/ton, to account for changes in the fuel market. Comparing solutions at different fuel prices can reveal how fuel price affects the optimal scheduling plan and associated costs.

Figure 7 shows how the delay time changes with fluctuations in fuel prices. When the fleet is operating downstream, an increase in fuel price leads vessels to extend their planned sailing time, exchanging higher potential opportunity costs for reduced fuel consumption cost. And with the increase in fuel price, this phenomenon becomes more obvious.

Figure 8 illustrates that waiting time occurs only when a vessel arrives at the selected bottleneck arrival time earlier than its minimum sailing speed. All vessels must pass through the bottleneck within the scheduling period. Therefore, as fuel price increases, vessels extend their planned sailing time to exchange higher potential opportunity costs for reduced fuel consumption cost. This also leads to the optimization of each vessel’s bottleneck passing way and timing to avoid increased waiting time. Excessive waiting time at the bottleneck only adds extra costs without contributing to a reduced fuel consumption cost.

Figure 9 illustrates the changes in bottleneck passing time for the scheduled fleet as fuel price changes. This figure shows that as fuel price increases, vessels with higher fuel consumption rates adjust to pass through the bottleneck later based on their planned arrival time and fuel consumption. This adjustment helps to optimize sailing time and achieve more significant fuel saving.

Figure 10 reveals a positive correlation between the total cost of the fleet passing through a river bottleneck and the potential opportunity cost incurred due to vessel delays with fluctuations in fuel prices. As fuel price increases, the opportunity cost resulting from delays in vessels rise correspondingly. However, the opportunity cost arising from vessel waiting time remains relatively unchanged.

5.3.2. Streamflow Velocity

Streamflow velocity can vary due to seasonal changes and tidal effects, and it has a significant impact on fuel cost of vessel. For this reason, to study the effect of streamflow velocity on the optimization results, six values for streamflow velocity are set: +4 km/h, +3 km/h, +2 km/h, −2 km/h, −3 km/h, and −4 km/h.

Figure 11 shows that under unified scheduling, vessels experience a longer delay time when sailing upstream compared to downstream. This is because the fuel consumption cost is higher when sailing against the streamflow, leading to a longer optimal delay time for vessels. However, given that the fuel consumption function is related to both time and the streamflow velocity, the total delay time of the fleet is influenced by the vessel opportunity cost, fuel price, and fuel consumption rate.

Figure 12 explains how vessel waiting time changes with variations in streamflow velocity. It can be seen from this figure that when vessels are sailing downstream with higher streamflow velocities, the actual minimum sailing speed is greater, resulting in a longer total waiting time for the fleet. On the contrary, when vessels are sailing upstream, the actual minimum sailing speed is lower, and there is a greater range for adjusting sailing speed, allowing vessels to reduce their speed to avoid potential waiting time at the bottleneck.

Figure 13 presents the arrival time of fleets at the bottleneck under different streamflow velocities. When vessels are sailing downstream and waiting time is likely, the system adjusts the scheduling plan to prioritize vessels with lower fuel consumption for waiting. In contrast, when the streamflow velocity is lower or vessels are sailing upstream, the bottleneck passing plan is adjusted based on vessel parameters and related costs.

Figure 14 shows that the total cost increases as the streamflow velocity decreases. When vessels are sailing downstream, the delay time slightly decreases as streamflow velocity decreases. On the contrary, when vessels are sailing upstream, the delay time initially increases and then slightly decreases with increasing streamflow velocity. The causes of these phenomena are similar to the motivation of vessel delay and waiting times and scheduling plan adjustment.

5.3.3. Proportion of LNG-Fuelled Vessels

In recent years, the application of green vessels has been gradually increasing, so the impact of the proportion of LNG-fuelled vessels on the scheduling results needs to be discussed. Considering the development rate of LNG-fuelled vessels, seven scenarios with proportions ranging from 0% to 60% are designed. The scheduling results for these different scenarios are shown in Figure 15.

Figure 15a shows the phenomenon of a reduced vessel delay time as the proportion of LNG-fuelled vessels increases. When the proportion of LNG-fuelled vessels increases from 0% to 60%, the total scheduling cost decreases by 11.7%, and the fleet’s delay time decreases by 84.0%. This phenomenon aligns with the rule observed in reduced fuel prices. However, unlike fuel price effects alone, the proportion of LNG-fuelled vessels also changes the potential opportunity cost for vessels. Since LNG fuel is less expensive but the opportunity cost for LNG-fuelled vessels is higher, the proportion of LNG-fuelled vessels changes the composition of vessel scheduling costs. As the proportion of LNG-fuelled vessels increases, the number of vessels in the fleet with lower potential fuel costs and higher opportunity costs will increase during scheduling, and these vessels tend to increase their sailing speed to ensure shorter arrival delays. Correspondingly, the schedule for vessels arriving at the bottleneck is adjusted to balance sailing time, thereby optimizing both fuel cost and opportunity cost. This phenomenon also indirectly reflects the rationale for prioritizing green vessels over traditional ones under the condition of the same fuel consumption level. Based on the above reasons, when the proportion of LNG-fuelled vessels changes, various costs show the rule as shown in Figure 15b.

6. Implications

Through the above case analysis, a series of management insights are derived:

(I). For a mixed fleet passing through a river bottleneck in multiple passing ways, authorities can ensure the efficient implementation of operations at the bottleneck and realize the optimization of the total cost of the fleet by arranging the vessels of each way and their passing bottleneck sequence in unified scheduling.

(II). Compared to oil-fuelled vessels, LNG-fuelled vessels have lower fuel costs and higher potential opportunity costs. At the same fuel consumption level, authorities should schedule LNG-fuelled vessels to pass through the bottleneck earlier to reduce potential opportunity costs and total costs of mixed fleets.

(III). The expectation of fuel costs and potential opportunity costs has an important impact on fleet scheduling. Shipowners should timely adjust the potential opportunity cost of vessels based on shipping market forecasts and fuel price. For some vessels that require a lower probability of delay in arriving at the destination port, shipowners need to adjust the expected arrival time based on fuel price. In addition, under varying fuel prices, reasonable expectations of opportunity cost can help save unified scheduling costs.

(IV). Streamflow velocity has a significant impact on the fuel consumption of vessels. Authorities should adjust the maintenance schedules for locks, lifts, and other equipment based on expected streamflow velocities and change the length of the scheduling period appropriately. Furthermore, shipowners should develop appropriate planned arrival times and opportunity costs based on expected future streamflow velocities. Specifically, when vessels are sailing downstream with high streamflow velocities, authorities should avoid all kinds of equipment maintenance during this period to ensure the availability of multiple lock passing times and reduce vessel waiting times. Conversely, when vessels are sailing upstream, they should extend the arrival deadline to ensure a higher probability of a timely arrival while managing the shipping cost.

(V). The proportion of LNG-fuelled vessels will impact the fleet’s delay rate and scheduling cost. Authorities should adjust the proportion of green vessels in the mixed fleet based on the lock-passing vessel density to ensure the optimal scheduling plan.

(VI). When certain exogenous conditions change, such as unpredictable weather conditions during navigation, tidal shifts, and the captain’s operational proficiency, these variations in navigation conditions and vessel operations will affect fuel consumption [13], which in turn influences the scheduling results. Therefore, authorities should fully consider these conditions and establish a more realistic fuel consumption function to ensure the robustness of the scheduling plan. Moreover, even with a reduced number of scheduled vessels, the model and method remain applicable and effective. It is worth noting that in such scenarios, the flexibility of vessel scheduling increases, significantly reducing the time of port arrival delays.

7. Conclusions

This paper investigates the unified scheduling problem of a mixed fleet passing through a river bottleneck in multiple passing ways. In the unique context of inland waterway transportation, it comprehensively considers the types and number of vessels that can be handled by different passing ways, as well as the selectable service time, vessel delays, and waiting time at the bottleneck, to explore the unified scheduling of the mixed fleet. The outer approximation method is applied, and the optimal value range of nonlinear variables is derived to ensure both the solving speed and accuracy of the scheduling model. Taking the Yangtze River as a case study, the model is validated, and the impact of fuel price, streamflow velocity, and the proportion of LNG-fuelled vessels on the scheduling results is analyzed.

The unified scheduling of mixed fleets passing through a river bottleneck can optimize the total cost of the fleet by effectively coordinating the sequence of vessels. LNG-fuelled vessels have lower fuel costs but higher opportunity costs, so they should be prioritized for passing through locks under the same fuel consumption conditions as traditional oil-fuelled vessels, to minimize total costs. Authorities could also adjust scheduling based on expectations for fuel price, opportunity cost, streamflow velocity, and weather conditions. Properly optimizing the proportion of green vessels passing through the bottleneck according to vessel density, adjusting maintenance schedules, and allowing shipowners to change their expected arrival times based on market conditions can improve scheduling efficiency and reduce delays. Additionally, changes in exogenous conditions, such as weather and tides, will affect scheduling, necessitating the establishment of a robust fuel consumption model to ensure both flexibility and effectiveness in scheduling.

Our study has several limitations that could be addressed and refined in future research. Firstly, this study ignores the impact of a few non-standard vessels on the number of vessels passing through the bottleneck in a single instance. Secondly, some vessels are willing to incur additional costs to gain priority through the bottleneck to ensure their shipping schedules are adhered to, whereas our model does not address vessel priority levels in the scheduling process. Therefore, future research could explore the unified scheduling of a mixed fleet based on vessel priority classification and the pricing of priority service. Finally, the development of a robust fuel consumption function forecasting technique and river vessel scheduling methods based on unpredictable weather conditions, as well as methods for optimizing the proportion of green vessels passing through a river bottleneck under different traffic densities, is also an interesting research direction. Despite these limitations, there is no denying that our research has contributed to inland waterway transportation in both theoretical and managerial aspects.