Ship Chain Navigation Co-Scheduling of Three Gorges-Gezhouba Dam under Serial-Lock Scenario

Abstract

Motivated by the operational scenarios of lock scheduling, we propose a serial-lock chain navigation problem (SLCNP) modeled on the Three Gorges-Gezhouba Dam (TGGD) for the first time. Ship grouping, synchronized moving, and grouped waiting operations are integrated into the ship navigation process. A mixed integer programming (MIP) model that incorporates real-world constraints such as ship priority, service fairness, traffic flow equilibrium, and phased ship placement is presented to optimize ship throughput and ship stay time. To solve the SLCNP, a sort-pick strategy-based swarm intelligence algorithm (SPSSIA) framework is developed that integrates the characteristics of SLCNP through a hybrid multi-section encoding method and a two-stage heuristic decoding approach. A swarm intelligence evolution mechanism is used to improve the search ability and robustness of the framework. Several instances are generated based on real data to verify the correctness and effectiveness of the model and algorithm. Computational results demonstrate the applicability and effectiveness of the proposed SPSSIA. Further analysis of the experimental results indicates that the key impact factors significantly influence the navigational performance of the TGGD system. The results of this study will provide practical guidance for the operational processes of inland river hubs with comparable characteristics.

1. Introduction

Inland waterway transportation, as a critical component of the transportation system, holds immeasurable value in promoting regional economic development and ensuring the smooth operation of supply chains. Its advantages—large capacity, low cost, low energy consumption, and minimal environmental impact—make it the preferred mode for bulk cargo transport, particularly in facilitating resource flow and economic integration within river basins. The Three Gorges-Gezhouba Dam (TGGD), a landmark project in the development of inland waterway transport, not only provides significant hydropower, delivering clean energy to the nation, but also plays a decisive role in improving the navigability of the Yangtze River and enhancing its overall capacity. However, with the rapid expansion of inland waterway transport in recent years, the navigational capacity of the TGGD has become insufficient to meet demand. Over the past two years, the average delay time for ships at the Three Gorges complex reached 223 h and 288 h [1], respectively.

The methods to enhance the navigational capacity of TGGD can be classified into two principal categories. (i) The expansion of new navigational facilities can enhance the ship passage capacity of the hub in terms of hardware; and (ii) the optimization of the operation process of navigational control and the maximization of the navigational potential of existing facilities. It is evident that the former option is costly, intricate, and time-consuming in comparison to the latter. In the immediate term, it is challenging to implement substantial alterations to the configuration of the facilities and equipment at the two dams. Consequently, there is an urgent requirement to implement effective measures to enhance the efficiency of navigation, minimize ship delays, and further optimize the management of the waterways to meet the growing transport demand.

The structural features of TGGD are briefly described as follows: The TGGD comprises the double-line five-step locks and ship lift of Three Gorges Dam (TGD), as well as the triple-line single-step locks of Gezhouba Dam (GD), as illustrated from an aerial perspective in Figure 1. The TGD and GD are separated by 38 km, and the lock scheduling plans of the two dams are uniformly formulated and jointly operated. Hence, the TGGD navigation scheduling system is a typical serial-lock scheduling system.

Based on the above requirements and scenario characteristics, this paper proposes a serial-lock chain navigation problem (SLCNP), which aims to smooth the ship navigation process and avoid the waste of resources in the navigation facilities due to ship delays during the scheduling process, so as to maximize the navigation potential of the existing navigation facilities in TGGD. The SLCNP can be decomposed into three core sub-problems: lockage assignment, ship placement, and chain navigation scheduling. Lockage assignment and ship placement are classic problems in lock scheduling. Lockage assignment devises a reasonable lockage schedule based on the origin and destination dams of different ships, while ship placement addresses the location arrangement of ships within a lock chamber for the same lockage. What differentiates this paper is the consideration of practical constraints, such as ship priority and balanced bidirectional traffic flow, making the problems more realistic. The proposed chain navigation scheduling problem ensures seamless coordination of lockage operations within lock groups in each dam, promoting efficient and uninterrupted lock activities.

However, the scheduling problem in a continuous lock scenario implies a series of coupled complex optimization problems [3]. It has been proven that minimizing the total waiting time for ships traveling in the same direction through two identical single-chamber locks is strongly NP-hard, even without ship placement constraints [4]. To the best of our knowledge, there is no mathematical model for the SLCNP described above, and due to its high level of complexity, the development of a solution for this class of problems is still a work in progress. The main contributions of this work are summarized as follows:

(1) This paper introduces a serial-lock chain navigation problem (SLCNP) for inland river hubs for the first time. Ship grouping, synchronized moving, and grouped waiting operations are integrated into the ship navigation process.

(2) A mixed integer programming (MIP) model for SLCNP is developed, optimizing for ship throughput and stay time while considering real-world constraints such as ship priority, service fairness, two-stage ship placement, and the chain navigation process.

(3) A sort-pick strategy-based swarm intelligence algorithm (SPSSIA) framework is developed, which integrates the characteristics of SLCNP through a hybrid multi-section encoding method, a two-stage heuristic decoding approach, and a swarm intelligence evolution mechanism. Experimental case studies are designed to validate the effectiveness of the algorithm and analyze the impact of the key factors.

The rest of the paper is organized as follows: Section 2 reviews the relevant studies; the SLCNP is described in Section 3. The MIP model for SLCNP is detailed in Section 4. Section 5 provides the proposed algorithm for solving SLCNP. The experimental results and analysis are reported in Section 6. Conclusions are drawn, and future research directions are outlined in Section 7.

2. Literature Review

There have been numerous studies on how to improve the navigable capacity of inland river hubs in recent years, which can be categorized into three main areas: performance evaluation, single-lock scheduling, and multi-lock cooperative scheduling.

The evaluation of the navigability of inland waterway hubs involves passage capacity and safety performance. A simulation model and optimization method for lock scheduling is presented in [5] that is capable of simulating the impact of infrastructure changes on the traffic handling at a ship lock with parallel chambers. Zhang et al. [6] constructed a data- and event-driven hybrid simulation model developed based on multi-agent and discrete-event modeling theories to evaluate the passage capacity of the TGGD. Zhang et al. [7] designed an intelligent scheduling system and embedded a hybrid optimization algorithm to refine the lockage schedule, aiming to improve the passing capacity of TGGD. The safety performance of the locks is also critical to the navigability of the hub. The systematic study of the evaluation system of ship lock operation safety was described in [8]. As ships get larger, the mass of the ships and their loads also challenge the conditions of the lock facilities. Farinha et al. [9] presented a crashworthiness case study that evaluated the Douro River lock gates’ strength due to ship collisions. Oni et al. [10] developed a decision-making model to evaluate maintenance decisions for optimizing the conditions of inland navigation locks. Moreover, the evaluation of the performance of lock navigation scheduling rules was studied [11], where they took the TGGD maintenance scenario as an example and achieved accurate identification of the direction, trend, and magnitude of the performance of the scheduling rules under navigation constraints in terms of efficiency, safety, and fairness criteria. Shi et al. [12] developed a decision-making model to assist in the improvement of the passage capacity of TGGD ship locks. The above studies mostly assessed the performance of inland locks from the perspective of status analysis or management decisions. In other words, the priority of these studies is not to optimize lock scheduling to improve the navigational performance of the hub.

In the perspective of lock scheduling optimization, the first issue that comes to mind is the study of the single-lock scheduling problem. Although the single-lock scheduling problem appears to be a small-scale problem, it is nevertheless proved to be NP-hard [13]. Verstichel et al. [14] proposed an exact algorithm (EA) to solve the single-lock scheduling problem, but it took a long computation time. Then they proposed a multi-order best fit heuristic to compare with the exact method, and the experimental results show that the heuristic method is significantly superior to the EA [15]. Passchyn et al. [4] studied a basic lock scheduling problem, namely the lockmaster’s problem, in which the capacity of the lock chamber was ignored. Shortly after that, a new no-wait scheduling problem for a lock with parallel chambers was introduced in [16]. They focused on the relevant case of finding so-called no-wait schedules and obtained algorithmic results for different special cases. A lock-quay co-scheduling problem was studied in [17], where the lock scheduling subproblem incorporates the setup of a single-chamber lock operating in one direction. Meisel and Fagerholt [18] proposed several new optimization models for the problem of scheduling ship traffic in a two-way Kiel Canal, which include variable ship speeds, capacities of siding segments, and limits for waiting times of ships. Yang et al. [19] developed a scheduling optimization model with the objective of minimizing the total fuel cost and delay penalty for ships by considering the TGD as a waterway bottleneck point.

From a global perspective, the distribution of multiple locks along rivers is more prevalent, particularly in critical shipping channels such as the Albertkanaal waterway in Belgium, the Yangtze River in China, the Upper Mississippi River, and the Welland Canal in North America [20]. Ji et al. [21,22] successively proposed various exact approaches and heuristic algorithms to solve the TGGD multi-lock co-scheduling problem. An optimization method for scheduling ship arrivals and their placement at inland waterway locks was proposed based on a MIP model [23]. They solved the optimization problem using a heuristic method based on large neighborhood search (LNS). Segovia et al. [24] presented a scheduling approach for inland ships and locks in which the scheduling strategy was designed in a switching max-plus-linear systems framework. The point of weakness is that the study only considered the single-chamber locks with capacity for one ship. Zhang et al. [25] investigated the ship scheduling problem in water-land transshipment by considering the water discharge constraints, but the possibility that ships can pass through the dam by means of locks was not considered. The approach channel and lock co-scheduling problem of the TGD was examined by [1,26]. However, their analysis did not take into account the ship lift of the TGD and the locks of the GD. With the high profile of environmental issues, some scholars have included fuel consumption [27,28] and energy-efficiency [29] in their research objectives in the study of multi-lock scheduling problems.

Furthermore, in the multi-lock scheduling problem, serial-lock scheduling has long been of interest to some scholars and have been active in the field in recent years. However, the research on serial-lock scheduling is still at a rudimentary level due to its high complexity [3]. Scheduling all the locks on a waterway, or even the network of waterways, as a single system will enable far greater improvements in waiting time and tardiness [20]. For the serial lock scheduling problem, Passchyn et al. [30] developed two models from different perspectives: time-indexed and lockage-indexed. However, their models are relatively simplistic, constraining the origin and destination locks of ships. This means all ships must pass through the locks in a fixed sequence without considering tributaries. Ji et al. [3] adopted similar ideas and considered ships arriving from or departing to tributaries, modeling the general serial lock scheduling problem from a multi-commodity network (MCN) perspective. However, their scenario setting was rather basic, with a conventional lockage process and not considering practical navigation conditions such as ship priority, service fairness, and other real-world constraints.

Comprehensively, we analyze the relevant studies for the multi-lock scheduling problem, as shown in

Table 1. Comparison of literature related to multi-lock scheduling.

Citation	Co-Scheduling	Lockage Assignment	Ship Placement	Serial-Lock Scenario	Tributary Scenario	Ship Priority	Service Fairness	Continuous Operation Process	Solution Method
Ji et al. [3]	✔	✔	✔	✔	✔				MCN-based EA
Verstichel et al. [20]	✔	✔	✔	✔		✔			Heuristic
Ji et al. [21]	✔	✔	✔		✔				Hybrid EA
Ji et al. [22]	✔	✔	✔				✔		ALNS
Guan et al. [23]	✔	✔	✔			✔			LNS-based heuristic
Segovia et al. [24]	✔					✔			EA
Zhang et al. [25]	✔								MPGA
Zhang et al. [26]	✔	✔	✔					✔	MOMA
Golak et al. [27]	✔	✔				✔			LS-based heuristic
Wu et al. [28]	✔	✔	✔	✔	✔	✔			LNS
Zheng et al. [29]	✔	✔	✔		✔				CAMOA
Passchyn et al. [30]	✔	✔		✔			✔		EA
This paper	✔	✔	✔	✔	✔	✔	✔	✔	SPSSIA

ALNS: adaptive large neighborhood search. MPGA: multi-population genetic algorithm. MOMA: multi-objective metaheuristic algorithm. LS: local search. LNS: large neighborhood search. CAMOA: collaborative adaptive multi-objective algorithm.

. As can be found, most of the existing research focuses on multi-lock co-scheduling issues such as lockage allocation and ship placement, and few literatures consider serial lock scenarios and tributary scenarios, especially the continuous operation process, which reflects the operational continuity of the navigational facilities. Therefore, this paper, building on the research of serial lock scheduling, introduces a chain navigation process that ensures the operational efficiency of locks (detailed in Section 3) and innovatively proposes the SLCNP. To closely align the research with real-world issues, we focus on the TGGD serial-lock system, considering practical constraints such as ship priority, service fairness, traffic flow balance, and phased ship placement during navigation. This work aims to contribute modestly to the continued exploration of serial-lock scheduling problems.

3. Serial-Lock Chain Navigation Problem

To facilitate the later elaboration of SLCNP, the following definitions are given first.

Definition 1.

Lockage, the unit of lock transport ships through the hub, refers to a complete process of transporting a group of ships through the lock. Depending on whether there are ships in the lock chamber, a lockage can be categorized as either loaded or unloaded lockage.

Definition 2.

Ship throughput, the number of ships that complete the navigation targets within the hub area over a given cycle.

Definition 3.

The chain navigation process refers to the navigational operation process in which the ship passes through the dam by successively undergoing grouping, synchronized moving, and grouped waiting. This refers specifically to the entire continuous operation process, starting from when a ship begins waiting in the berthing reserve area (such as an anchorage) until it exits the lock. During this process, the ship may undergo one or more of the following operations, possibly multiple times: grouping, synchronized movement, and grouped waiting.

3.1. Overview of the TGGD Navigation Structure

The navigational structure of the TGGD is illustrated in Figure 2. The TGD comprises two five-stage locks (north lock and south lock), a ship lift, and associated guide walls and piers. The GD consists of three single-stage locks, along with their respective guide walls and mooring piers. Generally, the north lock and south lock of TGD perform upward and downward unidirectional operations, respectively, and the ship lift as well as the three locks of GD perform bi-directional operations. Among them, lock 1 of GD is not equipped with a downstream pier due to its geographical location, which results in a long lockage interval when switching from downward operation to upward operation. Therefore, lock 1 of GD usually adopts the mode of regular change of direction to balance the upward and downward ship traffic flow, which is called adjustable lock in this paper.

The navigational structure of the TGGD hub can be mapped as a series service system consisting of two servers, TGD and GD, and anchorages $Z=\left\{z_1^{\prime },z_2^{\prime },z_3^{\prime },z_1^{″},z_2^{″},z_3^{″}\right\}$. Specifically, server 1 comprises ship locks $L_1=\left\{l_{11},l_{12},l_{13}\right\}$ (l₁₁, l₁₂, and l₁₃ represent the north lock, south lock, and the ship lift, respectively) along with their upstream and downstream guide walls $H_1^{\prime }=\left\{h_{11}^{\prime },h_{12}^{\prime },h_{13}^{\prime }\right\}$ and $H_1^{″}=\left\{h_{11}^{″},h_{12}^{″},h_{13}^{″}\right\}$, and upstream and downstream piers $B_1^{\prime }=\left\{b_{11}^{\prime },b_{12}^{\prime },b_{13}^{\prime }\right\}$ and $B_1^{″}=\left\{b_{11}^{″},b_{12}^{″},b_{13}^{″}\right\}$. Server 2 includes ship locks $L_2=\left\{l_{21},l_{22},l_{23}\right\}$ (l₂₁, l₂₂, and l₂₃ represent lock 3, lock 2, and lock 1, respectively) along with their upstream and downstream guide walls $H_2^{\prime }=\left\{h_{21}^{\prime },h_{22}^{\prime },h_{23}^{\prime }\right\}$ and $H_2^{″}=\left\{h_{21}^{″},h_{22}^{″},h_{23}^{″}\right\}$, and upstream and downstream piers $B_2^{\prime }=\left\{b_{21}^{\prime },b_{22}^{\prime },b_{23}^{\prime }\right\}$ and $B_2^{″}=\left\{b_{21}^{″},b_{22}^{″}\right\}$. Anchorage $Z^{\prime }=\left\{z_1^{\prime },z_2^{\prime },z_3^{\prime }\right\}$ is designated for downward ships, while anchorage $Z^{″}=\left\{z_1^{″},z_2^{″},z_3^{″}\right\}$ is for upward ships. Each server executes a chain navigation process to refine lock scheduling, enhancing the overall navigational efficiency of the hub. Based on the TGGD navigational characteristics, this paper proposes the series lock chain navigation problem (SLCNP).

3.2. Serial-Lock Chain Navigation in the TGGD

The SLCNP encompasses three sub-problems: lockage assignment, ship placement, and chain navigation scheduling.

(1)

Lockage assignment

The TGGD management authority directs ships to cross the dam by issuing navigation schedules and real-time scheduling instructions to the ships. There are usually three types of navigation schedules: pre-schedule, security schedule, and lockage schedule. After a ship arrives at the anchor and declares, it is included in the cyclic pre-schedule for lockage schedule development. Pre-schedules for different cycles follow the first-come, first-served principle for ship allocation, while ships within the same cycle are ranked in the pre-schedule according to their priority, as shown in

Table 2. Ship type priority levels description.

Ship Types	General Cargo Ship	Dangerous Cargo Ship	Passenger Ship	Container Ship	Grain/Fresh Cargo Ships	Special Ship
Priority levels	0	1	2	3	4	5

.

Ships in the pre-schedule are incorporated into the cyclic lockage schedule upon passing the security check. The development of the lockage schedule follows two rules of service fairness:

Rule 1: Ships in earlier periods’ lockage schedule have absolute priority over those in subsequent periods.

Rule 2: The order of ships in the same lockage schedule can be partially adjusted to cater for the efficiency of the locks, but it is not allowed that the adjustment leads to the delay of the ships that are supposed to pass the lock in the current cycle to the following cycle; on the contrary, it is allowed and encouraged if the adjustment leads to the advancement of the ships in the following cycle to the current cycle of the schedule.

Additionally, the formulation of the lockage schedule also needs to weigh the balanced distribution of upward and downward ship traffic flow. When the traffic flow ratio of upward and downward ships exceeds the established threshold, adjustable locks should be used to relieve congestion on the busier side.

(2)

Ship placement

The ship placement problem is a variation of the two-dimensional knapsack problem (2D-KP), adhering to the basic constraints of 2D-KP. Furthermore, ships must be moored within the lock chamber. The ship placement process is divided into two stages to mirror actual operational procedures. Stage 1: Ships entering the lock should first moor along the left or right walls of the lock chamber. Stage 2: Ships entering the lock should moor next to ships that are larger and have a similar freeboard height.

(3)

Chain navigation scheduling

In previous research, we conducted a detailed study on the synchronized moving, virtual grouping, and grouped waiting processes of ships between the approach channel and locks of the TGD [26]. This paper extends the study to the entire TGGD navigation system, as shown in Figure 3, which is a diagram of the chain navigation process of TGGD serial locks. The chain navigation process for each lock consists of four stages: (1) ships moving between or out of lock chambers; (2) ships moving from the virtual waiting area into the lock chamber; (3) ships moving from the virtual grouping area to the virtual waiting area; and (4) ships moving from the berthing reserve area to the virtual grouping area. It is noteworthy that the virtual grouping area and virtual waiting area are positioned at guide walls or piers, making their moving environment relatively straightforward. In contrast, the berthing reserve area is generally located at the anchorage or at the entrance of the approach channel, and its mooring environment is more complicated. Therefore, for safety reasons, the first three stages involve synchronized moving, while the movement of ships from the berthing reserve area to the virtual formation area employs a single-ship sequential moving manner.

Taking the five-stage south lock in the downward direction as an example, each of the five lock chambers has dimensions of L × W. The safe moving distance is defined as s₀. The adjacent distances between the lock, the virtual waiting area, the virtual grouping area, and the berthing reserve area are s₁, s₂, and s₃, respectively. The amount of lockages in the virtual waiting area, the virtual grouping area, and the berthing reserve area are n₁, n₂, and n₃, respectively. The location range of the lockage in the virtual waiting area and the virtual grouping area matches the size of the lock chamber, with a position spacing of $\Delta s$. The chain navigation process is as follows:

Stage 1: After lock chambers 1, 3, and 5 descend to their respective water levels, ships in lock chamber 5 begin to exit while ships in lock chambers 1 and 3 simultaneously move to lock chambers 2 and 4. Wherein the moving process adopts the synchronized moving mode, such as ships 7 and 9 of lockage g in lock chamber 1 move towards lock chamber 2 at the same time. When ship 9 reaches a safe distance from ship 8, ship 8 begins moving. This process continues until all ships of the lockage are in lock chamber 2. The moving time for lockage p is defined as $T_p^{mov}=2\left(a_p-1\right)s_0/v_1+L/v_1$, where a_p is the number of ship rows and v₁ is the moving speed between lock chambers.

Stage 2: When lock chamber 1 ascends to match the upstream water level, ships from lockage 1 in the virtual waiting area move synchronously into lock chamber 1. The entry time for the lockage is defined as $T_1^{ent}=2\left(a_1-1\right)s_0/v_2+\left(s_1+L\right)/v_2$, where v₂ is the entry speed. After ships from lockage 1 enter, ships from lockages 2 to n₁ move synchronously to the previous lockage position in sequence. The moving time for lockage q, $1<q\le n_1$, in the virtual waiting area is $T_q^{mov}=2\left(a_q-1\right)s_0/v_3+\left(\Delta s+L\right)/v_3$, where v₃ is the moving speed within the virtual waiting area.

Stage 3: Ships from lockage n₁+1 in the virtual grouping area move to the last lockage n₁ position in the virtual waiting area. The moving time is defined as $T_{n_1+1}^{mov}=2\left(a_{n_1+1}-1\right)s_0/v_4+\left(s_2+L\right)/v_4$, where v₄ is the moving speed within the virtual grouping area. After ships from lockage n₁ + 1 complete their shift, ships from lockage n₁ + 2 to n₁ + n₂ move forward sequentially. The moving time for lockage q, $n_1+1<q\le n_1+n_2$, in the virtual grouping area is $T_q^{mov}=2\left(a_q-1\right)s_0/v_4+\left(\Delta s+L\right)/v_4$.

Stage 4: The first group of ships in the berthing reserve area moves to the virtual grouping area and completes the grouping operation according to the lockage schedule. The moving time of lockage n₁ + n₂ + 1 is identified as $T_{n_1+n_{21}+1}^{mov}=2\left(a_{n_1+n_{21}+1}^{\ast }-1\right)s_0/v_5+s_3/v_5$, where $a_{n_1+n_{21}+1}^{\ast }$ is the number of ships in the lockage, and v₅ is the moving speed of ships from the berthing reserve area to the virtual grouping area. The grouping time is defined as $T_{n_1+n_{21}+1}^{gro}=(a_{n_1+n_{21}+1}^{\ast }-1)T_0^{gro}$, where $T_0^{gro}$ represents the unit grouping time.

The purpose of the SLCNP is to develop a coordinated lockage schedule for the TGGD navigation system, optimizing ship throughput and minimizing ship waiting times. The following sections will detail the modeling and optimization methods for the proposed SLCNP.

4. MIP Model for SLCNP

Building on the work of previous scholars, such as [2,29], this paper further expands on the serial-lock scheduling problem. For the first time, it incorporates a chain navigation process into the model, taking into account practical navigation conditions such as ship priority, bidirectional traffic flow equilibrium, service fairness, and phased ship placement.

4.1. Assumptions

Assumption 1.

The number of ships waiting at both ends of the TGGD is sufficient, with no consideration for sudden incidents such as ship failures.

Assumption 2.

The safety distance is included in the dimensions of the lock chamber and ships when the ships are placed inside the chamber.

Assumption 3.

The water depth of the lock chamber can meet the draft requirements of all ships.

4.2. Notations

The notations used in the SLCNP model are described as follows:

(1)

Set and indices

$D$: set of navigation regulation cycles, indexed by lockage schedule cycle d.

$K$: set of servers, indexed by k.

$L$: set of locks, indexed by l.

$L_k$: subset of L, denoting the set of locks for server k, $L_k\subseteq L$.

$L_{k\ast }$: subset of $L_k$, representing the adjustable locks within server k, $L_{k\ast }\subseteq L_k$.

$L_{k\otimes }$: subset of $L_k$, representing the locks within server k that are authorized to accommodate ships transporting dangerous cargoes, $L_{k\otimes }\subseteq L_k$.

$J$: set of lockages.

$J_d$: subset of J, denoting the set of lockages during lockage schedule cycle d, $J_d\subseteq J$.

$J_{di}$: set of lockages that may transfer ship i during cycle d, and with $J_{di}\subseteq J_d$.

$J_{dk}$: set of lockages for server k in cycle d, and with $J_{dk}\subseteq J_d$.

$J_{d{l}_{km}}$: set of lockages for lock $l_{km}$ in cycle d, and with $J_{d{l}_{km}}\subseteq J_{dk}$.

$J_{d{l}_{km}}^{\prime }$,$J_{d{l}_{km}}^{″}$: subsets of $J_{d{l}_{km}}$, representing downward and upward lockages for lock $l_{km}$ in cycle d, and with $J_{d{l}_{km}}^{\prime }\cup J_{d{l}_{km}}^{″}=J_{d{l}_{km}}$, $J_{d{l}_{km}}^{\prime }\cap J_{d{l}_{km}}^{″}=\varnothing$.

$J_{d{l}_{km}}^{V^{\prime }}$($J_{d{l}_{km}}^{V^{″}}$): set of lockages in the virtual waiting area, virtual grouping area, and berthing reserve area downstream (upstream) of lock $l_{km}$.

$N$: set of ships.

$N_0$: subset of $N$, denoting ships in the pre-schedule, and with $N_0\subseteq N$.

$N_0^{\prime }$,$N_0^{″}$: subset of $N_0$, denoting the remaining downward and upward ships in the pre-schedule, and with $N_0^{\prime }\cap N_0^{″}=\varnothing$.

$N_d$: set of ships passing through TGGD during lockage schedule cycle d, and with $N_d\subseteq N$.

$N_d^{\prime }$,$N_d^{″}$: subsets of $N_d$, denoting downward and upward ships passing through TGGD during lockage schedule cycle d, and with $N_d^{\prime }\cup N_d^{″}=N_d,N_d^{\prime }\cap N_d^{″}=\varnothing$.

$N_{d{l}_{km}}$: set of ships passing through lock $l_{km}$ during cycle d, and with $N_{d{l}_{km}}\subseteq N_d$.

$N_{d{l}_{km}p}$: set of ships transferred via lockage p of lock $l_{km}$ during cycle d, and with $N_{d{l}_{km}p}\subseteq N_{d{l}_{km}}$.

$N_{d{l}_{km}p1}$,$N_{d{l}_{km}p2}$: subsets of $N_{d{l}_{km}p}$, representing ships transferring in the first and second stage at lockage p of lock $l_{km}$ during cycle d, and with $N_{d{l}_{km}p1}\cup N_{d{l}_{km}p2}=N_{d{l}_{km}p},N_{d{l}_{km}p1}\cap N_{d{l}_{km}p2}=\varnothing$.

(2)

Parameters

$l_{km}$: the m-th lock of server k, $l_{km}\in L_k$.

$g_i$: freeboard height of ship i.

$e_g$: limit of difference in freeboard height for ships in the same lockage.

$e_{N_0}$: thresholds for the ratio of the number of ships waiting for service in the upward and downward directions.

$Q_k^{\prime }$,$Q_k^{″}$: the ship throughput in downward and upward direction.

$r_i$: the ship i’s origin node, which is referred to as the source node of the lockage path through which ship i passes.

$u_i$: the ship i’s destination node, which is referred to as the sink node of the lockage path through which ship i passes.

$o_i$: the order of ship i in pre-schedule.

$P_i$: the priority level of ship i.

$s_0$: safe moving distance.

$\Delta s$: the positional interval between lockages in the virtual waiting area, virtual grouping area, and berthing reserve area.

T: average stay time of ships completing the navigational targets.

$T_{pqi}^{tra}$: the traveling time of ship i between the server that operates lockage p and the server that operates lockage q.

$\Delta T_{l_{km}pq}$: the setup time of lock $l_{km}$ between lockage p and lockage q in the same operating direction.

$\Delta T_{l_{km}pq}^{\#}$: the setup time of lock $l_{km}$ between lockage p and lockage q in the opposite operating direction.

$le{n}_i$,$wi{d}_i$: length and width of ship i.

$Le{n}_{l_{km}}$,$Wi{d}_{l_{km}}$: length and width of lock $l_{km}$.

(3)

Variables

$t_p$: start time of lockage p. Particularly, $t_{r_i}$ is defined as the time when ship i arrives at its origin server and $t_{u_i}$ is defined as the time when ship i leaves its destination server.

$T_p^{pro}$: processing time of lockage p.

$T_p^{mov}$: the moving time for lockage p in the virtual waiting area, virtual grouping area, and berthing reserve area.

$T_p^{gro}$: the grouping time for lockage p in the virtual grouping area.

${\beta }_{d{l}_{km}pi}^{\prime }$(${\beta }_{d{l}_{km}pi}^{″}$): binary variable, ${\beta }_{d{l}_{km}pi}^{\prime }=1$ (${\beta }_{d{l}_{km}pi}^{″}=1$) when downward (upward) ship i is transferred via lockage p of lock $l_{km}$ during cycle d; 0, otherwise.

${\chi }_{pqi}$: binary variable, ${\chi }_{pqi}=1$ if and only if ship i transferred sequentially via lockage p and lockage q of two neighboring servers; 0 otherwise.

${\gamma }_{l_{km}p}$: binary variable that indicates whether lockage p of lock $l_{km}$ is loaded (1) or unloaded (0).

${\delta }_i$,${\delta }_p$: binary variable, 1 if ship i (lockage p) runs downwards and 0 if it runs upwards.

${\eta }_{l_{km}pi}$: binary variable that indicates whether ship i is transferred by lockage p of lock $l_{km}$ (1) or not (0).

${\mu }_{l_{km}ij}$: binary variable, ${\mu }_{l_{km}ij}=1$ if and only if ship i is completely to the left of ship j at lock $l_{km}$; 0 otherwise.

${\omega }_{l_{km}ij}$: binary variable, ${\omega }_{l_{km}ij}=1$ if and only if ship i is completely behind ship j at lock $l_{km}$; 0 otherwise.

$X_{l_{km}i}$($Y_{l_{km}i}$): binary variable, $X_{l_{km}i}=1$ ($Y_{l_{km}i}=1$) when ship i is moored to the left (right) wall of lock $l_{km}$; 0 otherwise.

$X_{l_{km}ij}$($Y_{l_{km}ij}$): binary variable, $X_{l_{km}ij}=1$ ($Y_{l_{km}ij}=1$) when ship i is moored to the left (right) of ship j; 0 otherwise.

${\kappa }_{ij}$: binary variable, 1 if ship i and j are transferred via the same lockage; 0 otherwise.

${\alpha }_{l_{km}pq}$: binary variable, 1 if lock $l_{km}$’s lockage p precedes its lockage q and 0 otherwise;

$x_{l_{km}i}$,$y_{l_{km}i}$: x and y coordinate positions of ship i in a lockage being operated at lock $l_{km}$, $x_{l_{km}i},y_{l_{km}i}\in {ℝ}^{+}$.

$x_{l_{km}}$,$y_{l_{km}}$: x and y coordinate positions of the right back vertex of lock chamber in lock $l_{km}$, $x_{l_{km}},y_{l_{km}}\in {ℝ}^{+}$.

4.3. Formulation for SLCNP

Ship waiting time, which is usually the main objective of the optimization of the general lock scheduling problem, refers to the difference between the ship’s arrival time at the anchorage and the start time of the lockage transferring the ship. In the serial lock scheduling problem, ships need to pass through consecutive locks sequentially, and their waiting time can be defined as the sum of the ship’s waiting time in front of each lock. In this paper, considering the complexity of the model solution and the totality of the crossing service system, the stay time of a ship in the system is taken as the objective function, that is, the time interval between the moment when a ship enters the service system and the moment when it leaves the service system. However, this single objective may result in the system wasting resources, such as lock facilities, to achieve the objective. For example, to pursue the shortest stay time of ships in the system, the mutual waiting time of ships in the same lockage is discarded, resulting in the situation of a single ship-single lockage crossing, which is a great waste of facilities such as lock space, water resources, and filling/emptying resources. Therefore, this paper also takes the ship throughput Q as the objective function, which is expressed as the average number of ships leaving the service system per unit schedule cycle. The mathematical expression of the model is as follows:

$$F=\max {\displaystyle \frac{Q}{CT}}$$

(1)

$$T={\displaystyle \frac{1}{Q}}{\displaystyle \sum _{d\in D}{\displaystyle \sum _{i\in N_d}\left(t_{r_i}-t_{u_i}\right)}}$$

(2)

$$Q=\min \left\{Q_k\left|Q_k=Q_k^{\prime }+Q_k^{″}\right.\right\},\forall k\in K$$

(3)

$$Q_k^{\prime }={\displaystyle \frac{1}{\left|D\right|}}{\displaystyle \sum _{d\in D}{\displaystyle \sum _{l_{km}\in L_k}{\displaystyle \sum _{p\in J_{d{l}_{km}}^{\prime }}{\displaystyle \sum _{i\in N_d^{\prime }}{\beta }_{d{l}_{km}pi}^{\prime }\Omega \left(p,i\right)}}}},\forall k\in K$$

(4)

$$Q_k^{″}={\displaystyle \frac{1}{\left|D\right|}}{\displaystyle \sum _{d\in D}{\displaystyle \sum _{l_{km}\in L_k}{\displaystyle \sum _{p\in J_{d{l}_{km}}^{″}}{\displaystyle \sum _{i\in N_d^{″}}{\beta }_{d{l}_{km}pi}^{″}\Omega \left(p,i\right)}}}},\forall k\in K$$

(5)

$$\Omega \left(x,y\right)=\left\{\begin{array}{l}1,{\delta }_x={\delta }_y\\ 0,{\delta }_x\ne {\delta }_y\end{array}\right.,\forall x,y\in J\cup N$$

(6)

Equation (1) is the objective function that pursues the shortest average stay time and the largest average throughput of ships in the TGGD service system, where C denotes the weight adjustment coefficient. Equation (2) is the average ship stay time calculation formula. Equation (3) is the formula for calculating the average ship throughput, and the maximum average ship throughput of the bottleneck server is used as the average ship throughput for the overall serial server system, where the ship throughput contains the throughput in both directions. Equations (4) and (5) represent the average ship throughput in D-cycle in the downward direction and upward direction, respectively. Equation (6) is used to determine the operation direction of lockages and ships.

The constraints contain three parts: lockage assignment constraints, ship placement constraints, and chain navigation constraints.

(1)

Lockage assignment constraints

For each ship i, the lockage schedule requires a complete navigational lockage path for it from $r_i$ to $u_i$. For ease of subsequent presentation, the forward lockage set $I_p$ and the backward lockage set $O_p$ for any lockage $p\in J_{di}$ are defined by constraints (7) and (8). Here, assuming that ship i needs to pass through sever1 and sever2 sequentially.

$$I_p=\left\{\begin{array}{l}{r}_i,\mathrm{if} p\in J_{dk},k=1\\ J_{dk-1},\mathrm{if} p\in J_{dk},k=2\\ J_{dk},\mathrm{if} p=u_i,k=2\end{array}\right.$$

(7)

$$O_p=\left\{\begin{array}{l}{J}_{dk},\mathrm{if} p=r_i,k=1\\ J_{dk+1},\mathrm{if} p\in J_{dk},k=1\\ u_i,\mathrm{if} p\in J_{dk},k=2\end{array}\right.$$

(8)

Constraints (9)–(11) construct a passing path from the origin server to the destination server for each ship passing through the hub. It limits each server on the path to assign only one lockage to transfer the ship. Constraint (12) indicates that ship i can only be transferred via one lockage of one of the parallel locks of server k while passing through the hub. Constraint (13) ensures the reasonableness of the temporal logical relationship between any two lockages of two adjacent servers.

$${\displaystyle \sum _{p\in O_{r_i}}{\chi }_{r_{i}pi}}=1,\forall i\in N_d,d\in D$$

(9)

$${\displaystyle \sum _{p\in I_{u_i}}{\chi }_{p{u}_{i}i}}=1,\forall i\in N_d,d\in D$$

(10)

$${\displaystyle \sum _{p\in I_q}{\chi }_{pqi}}={\displaystyle \sum _{p\in O_q}{\chi }_{qpi}},\forall i\in N_d,q\in J_{di},d\in D$$

(11)

$${\displaystyle \sum _{q\in I_p}{\chi }_{qpi}}={\eta }_{l_{km}pi}\Omega \left(p,i\right),\forall i\in N_d,p\in J_{di},l_{km}\in L_k,k\in K,d\in D$$

(12)

$$t_q-t_p-T_p^{pro}-T_{pqi}^{tra}\ge \left({\chi }_{pqi}-1\right)M,\forall i\in N_d,p\in J_{dk},q\in O_p,k\in K,d\in D$$

(13)

Constraints (14)–(19) restrain lockage operating behavior and ship passing behavior. Constraint (14) describes the relationship between the ship order and priority of any two ships in the pre-schedule. Constraint (15) ensures that a ship passing through a lock must pass through a loaded lockage transfer. Constraint (16) ensures that the loaded lockage is operated with at least one ship in the lock. Constraint (17) ensures orderly operation between different lockages of the same lock. Given a set of dangerous cargo ships, denoted as $N_{d{l}_{km}}^{\otimes }\subseteq N_{d{l}_{km}}$, and a time window [$t_1^{\otimes },t_2^{\otimes }$], Constraint (18) guarantees that dangerous cargo ships can only be transferred during this time window. Constraint (19) regulates the operating direction of the adjustable locks in the server k based on changes in the number of ships waiting for service in the downward and upward directions.

$$\left(o_i-o_j\right)\left(P_i-P_j\right)<0,\forall i,j\in N_0$$

(14)

$${\gamma }_{l_{km}p}-{\chi }_{qpi}\Omega \left(p,i\right)\ge 0,\hspace{0.17em}\forall i\in N_{d{l}_{km}p},p\in J_{d{l}_{km}},q\in I_p,l_{km}\in L_k,k\in K,d\in D$$

(15)

$${\gamma }_{l_{km}p}-{\displaystyle \sum _{i\in N_{d{l}_{km}p}}{\displaystyle \sum _{q\in I_p}{\chi }_{qpi}\Omega \left(p,i\right)}}\le 0,\forall p\in J_{d{l}_{km}},l_{km}\in L_k,k\in K,d\in D$$

(16)

$${\gamma }_{l_{km}p}-{\gamma }_{l_{km}q}\le 0,\forall p>q,p,q\in J_{d{l}_{km}},l_{km}\in L_k,k\in K,d\in D$$

(17)

$$\left\{\begin{array}{l}{\gamma }_{l_{km}p}\Omega \left(p,i\right){\eta }_{l_{km}pi}=1\\ t_1^{\otimes }\le t_p<t_2^{\otimes }\end{array}\right.,\forall i\in N_{d{l}_{km}}^{\otimes },p\in J_{l_{km}},l_{km}\in L_{k\otimes },k\in K,d\in D$$

(18)

$${\delta }_p=\left\{\begin{array}{l}1,\mathrm{if}{\displaystyle \frac{\left|N_0^{″}\right|}{\left|N_0^{\prime }\right|}}<e_{N_0}\\ 0,\mathrm{otherwise}\end{array}\right.,\forall p\in J_{d{l}_{km}},l_{km}\in L_{k\ast },k\in K,d\in D$$

(19)

Ship passing operations are gradual according to the lockage schedule and are strictly constrained by the lockage operation time. Constraints (20)–(22) limit the time sequence relationship between lockages. Specifically, Constraints (20) and (21) ensure the reasonableness of the temporal logic relationship between any two lockages of the same lock. Constraints (22) ensures the service fairness of ships between adjacent schedule cycles.

$$\begin{array}{l}{t}_q-t_p-T_p^{pro}-\Omega \left(p,q\right)\Delta T_{l_{km}pq}-\left(1-\Omega \left(p,q\right)\right)\Delta T_{l_{km}pq}^{\#}\ge \left({\alpha }_{l_{km}pq}+{\gamma }_{l_{km}p}+{\gamma }_{l_{km}q}-3\right)M,\\ \forall p,q\in J_{d{l}_{km}},p<q,l_{km}\in L_k,k\in K,d\in D\end{array}$$

(20)

$$\begin{array}{l}{t}_p-t_q-T_q^{pro}-\Omega \left(q,p\right)\Delta T_{l_{km}qp}-\left(1-\Omega \left(q,p\right)\right)\Delta T_{l_{km}qp}^{\#}\ge \left({\gamma }_{l_{km}p}+{\gamma }_{l_{km}q}-{\alpha }_{l_{km}pq}-2\right)M,\\ \forall p,q\in J_{d{l}_{km}},p<q,l_{km}\in L_k,k\in K,d\in D\end{array}$$

(21)

$$t_{r_i}-t_{r_j}\le 0,\forall i\in N_d,j\in N_{d+1},d\in D$$

(22)

(2)

Ship placement constraints

Constraints (23)–(25) ensure that the two-dimensional vertical projections of any two ships inside the same lock chamber do not overlap each other. Constraints (26)–(29) restrict the two-dimensional vertical projections of ships within the lock chamber to not exceed the lock dimensions.

$$\begin{array}{l}{\mu }_{l_{km}ij}+{\mu }_{l_{km}ji}+{\omega }_{l_{km}ij}+{\omega }_{l_{km}ji}+\left(1-{\eta }_{l_{km}pi}\Omega \left(p,i\right)\right)+\left(1-{\eta }_{l_{km}pj}\Omega \left(p,j\right)\right)\ge 1,\\ \forall i,j\in N_{d{l}_{km}p},i\ne j,p\in J_{d{l}_{km}},l_{km}\in L_k,k\in K,d\in D\end{array}$$

(23)

$$\begin{array}{l}{x}_{l_{km}i}+le{n}_i\le x_{l_{km}j}+\left(3-{\mu }_{l_{km}ij}-{\eta }_{l_{km}pi}-{\eta }_{l_{km}pj}\right)M,\\ \forall i,j\in N_{d{l}_{km}p},i\ne j,p\in J_{d{l}_{km}},l_{km}\in L_k,k\in K,d\in D\end{array}$$

(24)

$$\begin{array}{l}{y}_{l_{km}i}+wi{d}_i\le y_{l_{km}j}+\left(3-{\omega }_{l_{km}ij}-{\eta }_{l_{km}pi}-{\eta }_{l_{km}pj}\right)M,\\ \forall i,j\in N_{d{l}_{km}p},i\ne j,p\in J_{d{l}_{km}},l_{km}\in L_k,k\in K,d\in D\end{array}$$

(25)

$$x_{l_{km}i}\ge x_{l_{km}},\forall i\in N_{d{l}_{km}},l_{km}\in L_k,k\in K,d\in D$$

(26)

$$y_{l_{km}i}\ge y_{l_{km}},\forall i\in N_{d{l}_{km}},l_{km}\in L_k,k\in K,d\in D$$

(27)

$$x_{l_{km}i}+le{n}_i\le x_{l_{km}}+Le{n}_{l_{km}}+\left(1-{\eta }_{l_{km}pi}\right)M,\forall i\in N_{d{l}_{km}},p\in J_{d{l}_{km}},l_{km}\in L_k,k\in K,d\in D$$

(28)

$$y_{l_{km}i}+wi{d}_i\le y_{l_{km}}+Wi{d}_{l_{km}}+\left(1-{\eta }_{l_{km}pi}\right)M,\forall i\in N_{d{l}_{km}},p\in J_{d{l}_{km}},l_{km}\in L_k,k\in K,d\in D$$

(29)

Constraints (30)–(33) describe the constraints for mooring ship i to the left side of ship j ($X_{l_{km}ij}$), mooring ship i to the right side of ship j ($Y_{l_{km}ij}$), mooring ship i to the left wall of lock $l_{km}$ ($X_{l_{km}i}$), and mooring ship i to the right wall of lock $l_{km}$ ($Y_{l_{km}i}$), respectively.

$$\begin{array}{l}{X}_{l_{km}ij}=\left\{\begin{array}{l}1,\mathrm{if} y_{l_{km}i}-y_{l_{km}j}-wi{d}_j=0\wedge x_{l_{km}j}-x_{l_{km}i}\le 0\wedge x_{l_{km}j}+le{n}_j-x_{l_{km}i}-le{n}_i\ge 0\\ 0,\mathrm{otherwise}\end{array}\right.,\\ \forall i\in N_{d{l}_{km}p2},j\in N_{d{l}_{km}p1},l_{km}\in L_k,k\in K,d\in D\end{array}$$

(30)

$$\begin{array}{l}{Y}_{l_{km}ij}=\left\{\begin{array}{l}1,\mathrm{if} y_{l_{km}j}-y_{l_{km}i}-wi{d}_i=0\wedge x_{l_{km}j}-x_{l_{km}i}\le 0\wedge x_{l_{km}j}+le{n}_j-x_{l_{km}i}-le{n}_i\ge 0\\ 0,\mathrm{otherwise}\end{array}\right.,\\ \forall i\in N_{d{l}_{km}p2},j\in N_{d{l}_{km}p1},l_{km}\in L_k,k\in K,d\in D\end{array}$$

(31)

$$X_{l_{km}i}=\left\{\begin{array}{l}1,\mathrm{if} y_{l_{km}}+Le{n}_{l_{km}}-y_{l_{km}i}-le{n}_{l_{km}i}=0\\ 0,\mathrm{otherwise}\end{array}\right.,\forall i\in N_{d{l}_{km}p1},l_{km}\in L_k,k\in K,d\in D$$

(32)

$$Y_{l_{km}i}=\left\{\begin{array}{l}1,\mathrm{if} y_{l_{km}i}-y_{l_{km}}=0\\ 0,\mathrm{otherwise}\end{array}\right.,\forall i\in N_{d{l}_{km}p1},l_{km}\in L_k,k\in K,d\in D$$

(33)

Constraint (34) describes four constraint limitations on the mooring position of ships entering the lock, which means that a ship inside the lock can only be moored to the left or right walls of the lock chamber or to the left or right sides of a ship with a length greater than its own (and can only be moored to one ship).

$$\begin{array}{l}{X}_{l_{km}i}+Y_{l_{km}i}+{\displaystyle \sum _{j\in N_{d{l}_{km}p},j\ne i}{X}_{l_{km}ij}}+{\displaystyle \sum _{j\in N_{d{l}_{km}p},j\ne i}{Y}_{l_{km}ij}}=1,\\ \forall i\in N_{d{l}_{km}p},p\in J_{d{l}_{km}},l_{km}\in L_k,k\in K,d\in D\end{array}$$

(34)

Depending on the ship mooring situation, the ship placement operation is divided into two stages. Stage 1: Ships entering the lock should first moor along the left or right walls of the lock chamber. The ship mooring constraints are as follows:

$$X_{l_{km}i}+Y_{l_{km}i}=1,\forall i\in N_{d{l}_{km}p1},p\in J_{d{l}_{km}},l_{km}\in L_k,k\in K,d\in D$$

(35)

$${\displaystyle \sum _{j\in N_{d{l}_{km}p},j\ne i}{X}_{l_{km}ij}}+{\displaystyle \sum _{j\in N_{d{l}_{km}p},j\ne i}{Y}_{l_{km}ij}}=0,\forall i\in N_{d{l}_{km}p1},p\in J_{d{l}_{km}},l_{km}\in L_k,k\in K,d\in D$$

(36)

Stage 2: Ships entering the lock should moor next to ships that are larger and have a similar freeboard height. The entering ship i is moored alongside the ship j, which has a similar freeboard height and a greater length than its own. The ship i, j mooring constraints are as follows:

$$X_{l_{km}i}+Y_{l_{km}i}=0,\forall i\in N_{d{l}_{km}p2},p\in J_{d{l}_{km}},l_{km}\in L_k,k\in K,d\in D$$

(37)

$${\displaystyle \sum _{j\in N_{d{l}_{km}p1},j\ne i}{X}_{l_{km}ij}}+{\displaystyle \sum _{j\in N_{d{l}_{km}p1},j\ne i}{Y}_{l_{km}ij}}=1,\forall i\in N_{d{l}_{km}p2},p\in J_{d{l}_{km}},l_{km}\in L_k,k\in K,d\in D$$

(38)

$$\begin{array}{l}\left|g_i-g_j\right|\le e_g+\left(1-X_{l_{km}ij}-Y_{l_{km}ij}\right)M,\\ \forall i\in N_{d{l}_{km}p2},j\in N_{d{l}_{km}p1},p\in J_{d{l}_{km}},l_{km}\in L_k,k\in K,d\in D\end{array}$$

(39)

$$\begin{array}{l}{x}_{l_{km}j}-x_{l_{km}i}\le \left(1-X_{l_{km}ij}-Y_{l_{km}ij}\right)M,\\ \forall i\in N_{d{l}_{km}p2},j\in N_{d{l}_{km}p1},p\in J_{d{l}_{km}},l_{km}\in L_k,k\in K,d\in D\end{array}$$

(40)

$$\begin{array}{l}{x}_{l_{km}i}+le{n}_i-x_{l_{km}j}-le{n}_j\le \left(1-X_{l_{km}ij}-Y_{l_{km}ij}\right)M,\\ \forall i\in N_{d{l}_{km}p2},j\in N_{d{l}_{km}p1},p\in J_{d{l}_{km}},l_{km}\in L_k,k\in K,d\in D\end{array}$$

(41)

Constraints (37) and (38) ensure that a ship entering the lock chamber in the second phase can be moored to only one ship that meets the conditions for berthing. Constraint (39) restrains the difference in freeboard height between the two moored ships to be within a certain limit. Constraints (40) and (41) ensure that the entering ship is moored to a ship whose ship length is greater than its own.

Constraint (42) prevents two ships of the same length, neither of which is moored to either wall of the lock chamber, from mooring to each other.

$$X_{l_{km}ij}+Y_{l_{km}ij}+X_{l_{km}ji}+Y_{l_{km}ji}\le 1,\forall i,j\in N_{d{l}_{km}p},p\in J_{d{l}_{km}},l_{km}\in L_k,k\in K,d\in D$$

(42)

In addition, two ships transferring via different lockages in the same lock cannot be moored to each other, which is eliminated by constraints (43) and (44).

$$\begin{array}{l}{\eta }_{l_{km}pi}\Omega \left(p,i\right)-{\eta }_{l_{km}pj}\Omega \left(p,j\right)\le 1-{\kappa }_{ij},\\ \forall i\in N_{d{l}_{km}p},j\in N_{d{l}_{km}}\wedge j\notin N_{d{l}_{km}p},p\in J_{d{l}_{km}},l_{km}\in L_k,k\in K,d\in D\end{array}$$

(43)

$$\begin{array}{l}{X}_{l_{km}ij}+Y_{l_{km}ij}+X_{l_{km}ji}+Y_{l_{km}ji}\le {\kappa }_{ij},\\ \forall i\in N_{d{l}_{km}p},j\in N_{d{l}_{km}}\wedge j\notin N_{d{l}_{km}p},p\in J_{d{l}_{km}},l_{km}\in L_k,k\in K,d\in D\end{array}$$

(44)

(3)

Chain navigation constraints

In the downward direction, for example, the virtual waiting area, the virtual grouping area, and the berthing reserve area in front of the lock $l_{km}$ contain $n_{l_{km}}^1$, $n_{l_{km}}^2$, $n_{l_{km}}^3$ lockages, respectively. The adjacent intervals of lock $l_{km}$, virtual waiting area, virtual grouping area, and berthing reserve area are $s_{l_{km}}^1$, $s_{l_{km}}^2$, $s_{l_{km}}^3$ respectively. The chain navigational constraints are as follows (the same as in the upward direction):

$$t_p+T_p^{mov}-t_{p+1}\le 0,\forall p\in J_{d{l}_{km}}^{V^{\prime }},l_{km}\in L_k,k\in K,d\in D$$

(45)

$$T_p^{pro}=2T_p^{ent}+T_{l_{km}}^{fix}+\left(f_{l_{km}}-1\right)T_p^{mov},\forall p\in J_{d{l}_{km}}^{\prime },l_{km}\in L_k,k\in K,d\in D$$

(46)

$$T_p^{ent}={\displaystyle \frac{2\left(a_p-1\right)s_0+s_{l_{km}}^1+Le{n}_{l_{km}}}{v_2}},\forall p\in J_{d{l}_{km}}^{\prime },l_{km}\in L_k,k\in K,d\in D$$

(47)

$$T_p^{mov}=\left\{\begin{array}{l}{\displaystyle \frac{2\left(a_p-1\right)s_0+Le{n}_{l_{km}}}{v_1}},p\in J_{d{l}_{km}}^{\prime }\\ {\displaystyle \frac{2\left(a_p-1\right)s_0+\Delta s+Le{n}_{l_{km}}}{v_3}},1<p\le n_{l_{km}}^1,p\in J_{d{l}_{km}}^{V^{\prime }}\\ {\displaystyle \frac{2\left(a_p-1\right)s_0+s_{l_{km}}^2+Le{n}_{l_{km}}}{v_4}},p=n_{l_{km}}^1+1,p\in J_{d{l}_{km}}^{V^{\prime }}\\ {\displaystyle \frac{2\left(a_p-1\right)s_0+\Delta s+Le{n}_{l_{km}}}{v_4}},n_{l_{km}}^1+1<p\le n_{l_{km}}^1+n_{l_{km}}^2,p\in J_{d{l}_{km}}^{V^{\prime }}\\ {\displaystyle \frac{2\left(a_p^{\ast }-1\right)s_0+s_{l_{km}}^3}{v_5}}+T_p^{gro},p=n_{l_{km}}^1+n_{l_{km}}^2+1,p\in J_{d{l}_{km}}^{V^{\prime }}\end{array}\right.,\forall l_{km}\in L_k,k\in K,d\in D$$

(48)

$$T_p^{gro}=\left(a_p^{\ast }-1\right)T_0^{gro},\forall p\in J_{d{l}_{km}}^{\prime },l_{km}\in L_k,k\in K,d\in D$$

(49)

Constraint (45) restricts the sequential logic relationships for lockage movements within the virtual waiting area, virtual grouping area, and berthing reserve area. Constraint (46) defines the processing time of lockage p in lock $l_{km}$, where $f_{l_{km}}$ represents the number of chambers in lock $l_{km}$, $T_p^{ent}$ denotes the time for ships of lockage p to enter or exit the lock, and $T_{l_{km}}^{fix}$ is the fixed operation time of the lock $l_{km}$, including gate opening/closing and water filling/emptying. Constraint (47) specifies the time consumed for ships to enter or exit the lockage p, with $a_p$ being the number of rows of ships in lockage p. Constraint (48) defines the moving time of lockage p between adjacent lock chambers, within the virtual waiting area, and within the virtual grouping area, where $a_p^{\ast }$ denotes the number of ships within lockage p. Constraint (49) specifies the grouping time for lockage p, where $T_0^{gro}$ represents the unit grouping time.

5. Proposed SPSSIA for Solving SLCNP

5.1. Overview of the Solution for SLCNP

Passchyn et al. [4] have demonstrated that the problem of solving serial-lock scheduling, even with only two single-chamber locks, is NP-hard. Ji et al. [3] attempted to design exact algorithms to solve the serial-lock scheduling problem, proving that it can only solve single-lock scheduling problems involving 10–20 ships within a reasonable timeframe and simple multi-lock series problems involving fewer than 10 ships. For large-scale or complex serial-lock scheduling problems, it is necessary to design specialized heuristic algorithms to optimize the solution.

For the SLCNP proposed in this paper, the aim is to optimize the lockage schedule. According to the service fairness rules presented in Section 3, during the practical development of lockage schedules, adjustments can be made to the sequence of certain ships within the same schedule cycle to ensure the lock’s navigation efficiency. These adjustments are permitted as long as they do not compromise service fairness among ships across different schedule cycles. The so-called service fairness of ships between schedule cycles implies that the sequence adjustments should not result in ships being delayed from the current schedule cycle to the next one. Conversely, advancing ships from a subsequent schedule cycle into the current one as a result of sequence adjustments is allowed and even encouraged. For instance, as illustrated in Figure 4, the sequence of the first six ships in each schedule cycle can be adjusted while the last four ships are fixed. The original lockage schedule without adjusting the ship sequence is shown in Figure 4a. Figure 4b shows the advancement of ships 11 and 21 from the subsequent schedule cycle as a result of the ship sequence adjustment, which is permitted and encouraged. Within the same schedule cycle, ships in a fixed sequence can be selected by preceding lockages without affecting the lockage sequence of other ships. This is demonstrated in Figure 4a,b, where the advancement of ship 17 does not impact the lockage order of other ships. In Figure 4c, the sequence adjustment of ships 1–6 results in the postponement of ship 10 in schedule cycle 1 and ship 20 in schedule cycle 2 to the next cycle. This is not permitted.

In view of the above lockage schedule development process, we propose a sort-pick strategy executed in the following steps:

Step 1: Define the sort-pick point (sp), which marks the division between the sortable and pickable sections of the ship queue.

Step 2: Optimize the sequence of ships within the sortable section.

Step 3: Assign lockages according to the optimized sequence of the sortable section.

Step 4: For each lockage, pick the most suitable ships from the pickable section based on the remaining capacity of the lock chamber.

Step 5: Assign lockages to the remaining ships in the pickable section, concurrently performing Step 4, until all lockages are allocated.

Based on the aforementioned strategy, this paper proposes a sort-pick strategy-based swarm intelligence algorithm (SPSSIA) framework for the SLCNP. Figure 5 shows the SPSSIA framework, which primarily consists of five components: encoding, population initialization, decoding, fitness evaluation, and swarm intelligence evolution.

5.2. Hybrid Multi-Section Encoding

In optimization algorithms, encoding refers to the process of mapping the solution space of a problem onto a data structure amenable to algorithmic manipulation. A feasible solution of the SLCNP includes two parts, namely, ship sequence (referred to as X) and ship navigational direction sequence (referred to as Nd). To effectively encode the solution, a hybrid three-layer sectional encoding scheme is proposed for the SLCNP. Specifically, X adopts integer encoding to represent the ship number. Nd adopts binary encoding, with zero indicating the downward direction and one indicating the upward direction. Considering the flexibility of element updates in subsequent population operations, a biased random key is used to characterize the ordering of ships. It is worth noting that X consists of three sections: the previous remaining section, the sortable section, and the pickable section. The corresponding Key is also divided into three sections, where the first two sections are sorted within their respective sections. As an example of encoding 13 ships is illustrated in Figure 6, ships 1–4 indicate ships from the previous schedule cycle that have not yet left the system. Ships 5–9 denote ships in the sortable section and ships 10–13 denote ships in the pickable section of the current schedule cycle. The navigational direction of each ship is identified by Nd. The final sequence of ships is formed to arrange the lockages in different directions separately and sequentially. The sequence of ships in the downward direction is [4, 1, 7, 5, 10, 13], and the sequence of ships in the upward direction is [2, 3, 6, 8, 9, 11, 12].

5.3. Population Initialization

The quality of the initial population often affects the speed of convergence for a metaheuristic [31]. Typically, the initialization process should be able to obtain a population that accurately reflects the solution structure of the problem and ensure the solutions of the population are uniformly distributed in the search space. The number of ships remaining from the previous schedule cycle in the SLCNP affects the individual dimensions of the current schedule cycle, which in turn affects the computational complexity of the algorithm. Meanwhile, the number of ships remaining in the current schedule cycle will continue to affect the number of ships remaining in the subsequent schedule cycle due to the navigability of the hub.

Therefore, this paper employs data warming to generate the initial population in order to truly reflect the SPSSIA solution performance. Specifically, we use the actual data of seven schedule cycles to warm up the TGGD system navigational operations. The trend of the remaining ships in the system at the end of each schedule cycle is shown in Figure 7. After seven cycles of warm-up, the number of remaining ships in the system stabilized at around 60. Eventually, the remaining ship data after warm-up is used as the data input for the previous ship section of the initial population.

5.4. Decoding Based on Two-Stage Heuristic Rules

Decoding is the inverse process of encoding, converting the data structure processed by the algorithm back into the solution space of the original problem. The decoding process for the SLCNP is developed as follows:

Step 1: Update the service start time of locks, $T=\left\{\left.t_{l_{km}}\right|l_{km}\in L,k\in K\right\}$.

Step 2: Determine the queue of ships in downward and upward directions based on the encoding structure of the individual.

Step 3: Assigning lockages to ships in different queues based on the ship’s origin and destination servers, combined with the current operation direction of the locks. Determine whether the current lockage is transferring dangerous cargo ships according to constraint (18).

Step 4: When the number of arriving ships in the upward and downward directions is out of balance, the operation mode of the adjustable lock is adjusted by constraint (19).

Step 5: For each lockage, the placement of ships in the lock chamber is necessary and follows the ship placement constraints. Algorithm 1 presents the detailed two-stage ship placement heuristic, which is illustrated in Figure 8.

Step 6: After determining a lockage according to the sequence of ships, suitable ships are picked from the pickable section according to the remaining space in the lock chamber, and the picking process based on the improved best-fit (IBF) heuristic algorithm is shown in Algorithm 2.

Algorithm 1. Two-stage ship placement heuristic
Input: The ship sequence $X_g$ for lockage g. The remaining space sequence Re. Output: A ship placement scheme, Ag.
1:	Set ms = 0
2:	for p in $X_g$ do
3:	r ← Select rectangles that can contain the ship p from Re
4:	if r is not empty, then
5:	r1 ← Select rectangles that met y = 0 or y + w = W from r
6:	if r1 is not empty, then //** Stage 1
7:	r2 ← Select the rectangle [x, y, l, w] with the smallest x from r1
8:	if y = 0, then
9:	Place the ship p at position [x, y, $le{n}_p$, $wi{d}_p$] in Ag
10:	else
11:	Place the ship p at position [x, W-$wi{d}_p$, $le{n}_p$, $wi{d}_p$] in Ag
12:	end if
13:	else //** Stage 2
14:	r3 ← Select rectangles from r whose adjacent placed ships satisfy constraints (37)–(41).
15:	r4 ← Select the rectangle [x, y, l, w] with the smallest x from r3
16:	if ms = 0, then
17:	Place the ship p at position [x, y, $le{n}_p$, $wi{d}_p$] in Ag
18:	Set ms = 1
19:	else
20:	Place the ship p at position [x, y + w-$wi{d}_p$, $le{n}_p$, $wi{d}_p$] in Ag
21:	Set ms = 0
22:	end if
23:	end if
24:	Update Re
25:	else
26:	break
27:	end if
28:	end for

Algorithm 2. Improved best-fit ship picking heuristic
Input: The ship placement scheme, Ag. The pickable ship sequence $X_{pic}$. The remaining space sequence Re. Output: An updated ship placement scheme, Ag.
1:	while $X_{pic}$ is not empty and Re is not empty, do
2:	k ← Select the rectangle [x, y, l, w] with the smallest x from Re
3:	s ← Select ships that can be contained by k from $X_{pic}$
4:	if s is not empty, then
5:	s1 ← Select the ship with maximum $len\times wid$ from s
6:	Place the ship s1 into k with Algorithm 1
7:	Update Re and $X_{pic}$
8:	end if
9:	end while

After decoding operations for each individual in the population, fitness value assessment is needed to determine the strengths or weaknesses of the individual. Fitness evaluation criteria refer to the objective function (1). Therein, the ship throughput is counted as ships passing through their destination servers, and those not completed in the current schedule cycle are included in the next schedule cycle.

5.5. Swarm Intelligence Evolution

Various swarm intelligence algorithms have been applied to address a variety of optimization problems [32]. Swarm intelligence mechanisms can make full use of multiple individuals exploring the solution space in parallel at the same time and guiding the individuals towards the optimal solution using different population information updating strategies.

The incorporation of collective search mechanisms ensures that the algorithm proposed in this paper performs effectively when tackling large-scale problems. With multiple individuals participating in the search process, the swarm intelligence evolution mechanism efficiently explores different regions of the solution space. This distributed search method enhances the probability of finding a global optimum and helps avoid premature convergence to local optima. Additionally, the swarm intelligence mechanism improves the algorithm’s robustness and fault tolerance. Even if some individuals perform poorly or fail, the population can continue to search effectively. Individuals in the swarm can adjust their behavior according to environmental changes and utilize the experiences of other population members through social learning mechanisms, accelerating the search process. Consequently, the algorithm enhanced by the swarm intelligence evolution mechanism is better suited to dynamic or uncertain environmental conditions.

5.6. Computational Complexity

The main computational cost of the SPSSIA comes from the encoding, population initialization, decoding, and swarm intelligence evolution. Assume that the number of ships in schedule cycle d is $\left|N_d\right|$, the number of ships remaining in the schedule cycle d-1 is R, and the population size of the algorithm is PS. The overall worst-case complexity for the encoding is $O\left(\left(\left|N_d\right|+R\right)\ast PS\right)$. During the population initialization, the ships used for data warm-up are selected from ${\Gamma }_0$, thus the overall worst-case complexity for the population initialization is $O\left(N_0\right)$. The lockage schedule obtains the ship placement position by decoding each solution, which requires $O\left(\left|N_d\right|+R\right)$ computations in the worst case. As the swarm intelligence evolution operations are performed for each individual, the computational complexity is $O\left(PS\right)$.

Therefore, the overall worst-case computational complexity of the SPSSIA is $O\left(\left(\left|N_d\right|+R\right)\ast PS\ast G\right)$, where G is the maximum of generations.

6. Experiments and Result Analysis

This section focuses on validating the performance of the proposed model and algorithm for the SLCNP. To establish a comparative standard for validation, we relax the lockage assignment constraints and ship placement constraints of the SLCNP problem to determine its upper bound. The upper bound model (UBM) is shown below:

$$\mathrm{Maximize} \overline{F}$$

Subject to:

Constraints (2)–(8), (12), (13), (15)–(17), (20), (21), (43)–(49)

$$\left\{\begin{array}{l}\overline{F}\ge Q\\ \overline{F}\ge CT\\ Q\ge \overline{F}-\left(1-\omega \right)M\\ CT\ge \overline{F}-\left(1-\upsilon \right)M\\ \omega +\upsilon \ge 1\\ \omega ,\upsilon \in \left\{0,1\right\}\end{array}\right.$$

(50)

$$C=⌊{\displaystyle \frac{Q_{ub}}{T_{ub}}}⌋$$

(51)

$${\displaystyle \sum _{i\in N_{d{l}_{km}p}}le{n}_i\times wi{d}_i\times {\beta }_{d{l}_{km}pi}}\le Le{n}_{l_{km}}\times Wi{d}_{l_{km}},\forall p\in J_{d{l}_{km}},l_{km}\in L_k,k\in K,d\in D$$

(52)

For the convenience of solving the UBM with a solver, the original objective function is linearly transformed using the constraint (50). Constraint (51) defines the weight adjustment coefficient, where $Q_{ub}$ and $T_{ub}$ denote the upper bound value of the ship throughput and the minimum value of the average ship stay time during the solution of the UBM, respectively. Constraint (52) indicates that the two-dimensional area of the ships in each lockage does not exceed the area of the lock chamber. Given that the UBM mainly relaxes the constraints on the placement of the ships inside the lock chamber and restricts the relationship between the ships and the lock chamber through constraint (52). In other words, the solution space of the original model has not been changed; only the structure of the solution has been relaxed, and therefore the results obtained through the UBM are tight upper bounds of the original model.

Striving to make the proposed SPSSIA framework more convincing, in this paper, three classical swarm intelligence optimization algorithms, i.e., the evolutionary strategy (ES), the particle swarm optimization (PSO), and the whale optimization algorithm (WOA), respectively, are embedded into SPSSIA as a swarm intelligence evolutionary module. They are referred to as SPSES, SPSPSO, and SPSWOA, respectively. All of them adopt the elite retention strategy in the process of population renewal in order to perpetuate the optimal individuals and accelerate the population towards the optimal solution.

The UBM is solved using the solver Gurobi 10.0, and all algorithms are coded in Python 3.10. The experimental tests are run on a PC with an AMD R5-3600 CPU of 3.60 Ghz and 16.0 GB RAM (Advanced Micro Devices, Inc., Santa Clara, CA, USA).

6.1. Generation of Instances

Given the scarcity of research on SLCNP, it is difficult to obtain relevant instance sets directly. This study designs and generates test cases based on real-world data to validate the performance of the model and algorithms. Then, we incorporate actual data to compare the optimization results with the actual target values. As shown in

Table 3. The data distribution of ships.

Ship Types	General Cargo Ship	Dangerous Cargo Ship	Passenger Ship	Container Ship	Grain/Fresh Cargo Ships	Special Ship
Length (m)	67.50–130.00	81.80–110.00	87.00–135.20	99.30–110.00	81.80–110.00	10.00–60.00
Width (m)	11.00–19.70	14.03–19.25	15.00–20.20	16.92–17.50	14.00–19.50	4.00–12.00
Freeboard height (m)	8.00–17.80	9.00–16.50	9.00–17.20	10.00–16.50	9.00–16.50	2.50–10.00
Proportions (%)	68.28	11.18	8.46	2.42	9.06	0.60

, the data distribution of ships passing through the locks during a certain period in TGGD is presented.

Table 4. The lock information of the TGGD [2].

Properties	Locks of the TGD			Locks of the GD
	North	South	Ship lift	Lock 1	Lock 2	Lock 3
Length (m)	280	280	120	280	280	120
Width (m)	34	34	18	34	34	18
Number of chambers	5	5	1	1	1	1
Fixed operation time (min)	39	39	12.5	24	24	14
$\mathrm{Setup} \mathrm{time} \Delta T_{l_{km}pq}$ (min)	21	21	13	24	24	12.50
$\mathrm{Setup} \mathrm{time} \Delta T_{l_{km}pq}^{\#}$ (min)	\	\	2	5	5	2

presents the lock information of the TGGD [2].

In reference to the actual planning cycles of the lockage schedule, this paper adopts two basic cycle standards of d = 12 h and d = 24 h. During the development of the lockage schedule, the sortable section and pickable section of the ships are divided by the sort-pick point (sp), and we divide them by the ratio of sp = [0, 0.3, 0.6, 0.9]. Additionally, in practical navigation operations, the management authority promotes the standardization of the general cargo ships to improve the utilization efficiency of the lock chambers. The so-called standardized ship type refers to ships with dimensions of 130 × 16.3 m, where four standardized ships can perfectly occupy the usable area of a lock (except for the ship lift and the lock 3) chamber. Therefore, this paper considers the proportion of standardized ships among general cargo ships cp = [0, 0.3, 0.6, 0.9] as a factor to explore the impact of ship standardization on navigation efficiency. Ultimately, instances are generated by the combination of key impact factors d-sp-cp.

The parameters of the chain navigation process are set as follows with reference to the conditions of the TGGD navigation facilities as well as actual operational data. The lockage numbers of the virtual waiting area, the virtual grouping area, and the berthing reserve area in front of the lock $l_{km}$ are $n_{l_{km}}^1$ = $n_{l_{km}}^2$ = 1, $n_{l_{km}}^3$ = 3, and the positional interval between lockages is $\Delta s$ = 5 m. The adjacent intervals of lock $l_{km}$, virtual waiting area, virtual grouping area, and berthing reserve area are $s_{l_{km}}^1$ = $s_{l_{km}}^2$ = $s_{l_{km}}^3$ = 20 m. The speed of ships in different areas is assigned as follows: $v_1$ = 0.6 m/s, $v_2$ = $v_3$ = $v_4$ = $v_5$ = 1 m/s. The unit grouping time $T_0^{gro}$ = 5 min.

The other parameters used to generate instances are supplemented in

Table 5. Parameters used to generate instances.

Navigation Regulation Cycles D	|D| = 7	Server K	K = {1, 2}
$\mathrm{Freeboard} \mathrm{height} \mathrm{difference} \mathrm{limits} e_g$(m)	0.50	$\mathrm{Traveling} \mathrm{time} T_{pqi}^{tra}$ (min)	U(90, 120), U(150, 180)
$\mathrm{Bidirectional} \mathrm{ship} \mathrm{ratio} \mathrm{thresholds} e_{N_0}$	1.50

. The data in the table is derived from actual investigation and literature [2]. In this case, the traveling time between neighboring servers obeys a uniform distribution of U(90, 120) and U(150, 180) minutes in the downward and upward directions, respectively, which are derived from the statistics of actual data for a certain period at TGGD.

6.2. Parameter Tuning of Relevant Algorithms

The SPSSIA contains some key control parameters. Taking SPSES as an example, it has six key control parameters, namely, the parent population size $\mu$, the offspring population size $\lambda$, the range of variation steps (${\sigma }_{\min }$, ${\sigma }_{\max }$), and the search space ($X_{\min }$, $X_{\max }$). Taguchi analysis [33,34,35] is utilized to obtain the best combination of these parameters based on each instance. We set each parameter to five levels as presented in

Table 6. Level of the parameters for SPSES.

Parameters	Parameter Level
	1	2	3	4	5
$\mu$	20	30	40	50	60
$\lambda$	20	30	40	50	60
${\sigma }_{\max }$	1	0.9	0.8	0.7	0.6
${\sigma }_{\min }$	0.1	0.2	0.3	0.4	0.5
$X_{\max }$	25	20	15	10	5
$X_{\min }$	−25	−20	−15	−10	−5

. To maintain high stability, all combinations are independently run 20 times, and the mean fitness value is used as the response value to choose the parameter combination. A greater mean fitness value represents a better combination. An instance 12–0.6–0.6 is taken as an example to determine the best combination of these parameters. Figure 9 shows the factor level-trend of six parameters. Obviously, the best parameter combination is set as follows: $\mu$ = 30, $\lambda$ = 40, ${\sigma }_{\max }$ = 0.9, ${\sigma }_{\min }$ = 0.2, $X_{\max }$ = 20, and $X_{\min }$ = −5.

The same methodology is used for the determination of the way in which the key control parameters of SPSPSO and SPSWOA are combined. The final selection of key parameters for both algorithms is presented in

Table 7. The parameter values for the two competing algorithms.

Algorithms	Parameter Values
SPSPSO	PS = 50, ${\omega }_{\max }$ = 0.9, ${\omega }_{\min }$ = 0.4, c1 = 2, c2 = 2
SPSWOA	PS = 70, lb = 5, ub = 15, $a_{\max }$ = 2, p = 0.5

PS: population size; (${\omega }_{\min }$,${\omega }_{\max }$): range of inertia weight adjustment; c1: cognitive weight; c2: social weight; (lb, ub): search space; $a_{\max }$: the maximum value of convergence factor; p: the probability to choose between either the shrinking encircling mechanism or the spiral model.

.

6.3. Performance Analysis of Proposed Algorithm

To comprehensively evaluate the performance of SPSSIA, we employ SPSES, SPSPSO, and SPSWOA to solve 32 sets of instances, comparing the results to the upper bound (UB) obtained by the Gurobi solver. Each algorithm is executed independently 20 times per instance, with a maximum of 100 iterations per run, and the average result is recorded. Given the Gurobi solver’s inefficiency in handling large-scale problems, even with relaxed models, it requires substantial time. Consequently, the Gurobi solver is executed only once for the UBM model in this paper. The time limit for all algorithms and the solver is set to 500 s. The final results are presented in

Table 8. The performance of three algorithms based on SPSSIA in comparison with the upper bound of instances.

Instances			UB			SPSES			SPSPSO			SPSWOA
d	sp	cp	F	Q	T(h)	F	Q	T(h)	F	Q	T(h)	F	Q	T(h)
12	0	0	0.9996	85.00	5.6688	0.9168	83.00	6.0353	0.9022	83.00	6.1330	0.9033	83.00	6.1255
		0.3	1.0000	80.00	5.5995	0.9147	78.00	5.9686	0.9056	78.00	6.0286	0.9067	78.00	6.0213
		0.6	1.0000	86.00	5.4576	0.9070	84.00	5.8773	0.8976	84.00	5.9388	0.8971	84.00	5.9421
		0.9	1.0000	86.00	5.5017	0.9098	85.00	5.9768	0.9009	85.00	6.0359	0.9003	85.00	6.0399
	0.3	0	1.0000	85.00	5.2088	0.9105	83.50	5.6199	0.9047	83.60	5.6627	0.9083	83.70	5.6470
		0.3	1.0000	86.00	5.2710	0.9685	84.50	5.3475	0.9621	84.00	5.3512	0.9659	84.10	5.3365
		0.6	1.0000	87.00	5.4576	0.9159	84.00	5.7533	0.8980	84.00	5.8679	0.8983	84.00	5.8660
		0.9	1.0000	88.00	5.5017	0.9094	85.00	5.8436	0.9006	85.00	5.9007	0.9016	85.00	5.8941
	0.6	0	0.9978	88.00	5.2088	0.9193	84.50	5.4287	0.9067	84.10	5.4781	0.9160	85.60	5.5192
		0.3	0.9999	87.00	5.2710	0.9732	85.70	5.3347	0.9606	84.40	5.3227	0.9683	85.10	5.3241
		0.6	1.0000	88.00	5.4610	0.9321	86.30	5.7456	0.9151	86.00	5.8320	0.9096	85.70	5.8468
		0.9	1.0000	88.00	5.5017	0.9193	85.00	5.7806	0.9002	85.00	5.9033	0.9001	85.00	5.9039
	0.9	0	1.0000	84.00	4.9161	0.9697	82.70	4.9913	0.9494	80.70	4.9747	0.9633	82.10	4.9880
		0.3	1.0000	88.00	5.2887	0.9790	86.70	5.3223	0.9776	84.90	5.2193	0.9741	85.30	5.2627
		0.6	0.9975	87.00	5.5826	0.9774	86.60	5.6712	0.9731	86.80	5.7094	0.9708	86.60	5.7098
		0.9	1.0000	90.00	5.5017	0.9448	87.60	5.6679	0.9338	88.00	5.7608	0.9385	88.20	5.7450
24	0	0	0.9972	163.00	5.4487	0.8079	160.00	6.6016	0.7983	160.00	6.6810	0.7985	160.00	6.6793
		0.3	0.9972	168.00	5.4487	0.8131	164.00	6.5233	0.8034	164.00	6.6020	0.8028	164.00	6.6070
		0.6	1.0000	174.00	5.6464	0.8222	169.00	6.6701	0.8130	169.00	6.7456	0.8120	169.00	6.7539
		0.9	0.9983	175.00	5.7431	0.8608	173.00	6.5844	0.8538	173.00	6.6383	0.8507	173.00	6.6625
	0.3	0	1.0000	175.00	5.4399	0.8208	171.00	6.4761	0.8111	167.80	6.4309	0.8103	168.20	6.4526
		0.3	1.0000	176.00	5.4770	0.8292	165.40	6.2074	0.8137	164.60	6.2950	0.8170	165.70	6.3115
		0.6	1.0000	174.00	5.6464	0.8319	169.00	6.5923	0.8187	169.00	6.6986	0.8127	169.00	6.7481
		0.9	0.9983	177.00	5.7431	0.8703	173.00	6.4389	0.8505	173.00	6.5888	0.8506	173.00	6.5880
	0.6	0	1.0000	169.00	5.4627	0.8375	167.50	6.4647	0.8166	167.10	6.6144	0.8170	166.30	6.5795
		0.3	1.0000	171.00	5.4717	0.8322	165.50	6.3635	0.8150	166.30	6.5292	0.8172	166.70	6.5273
		0.6	1.0000	173.00	5.6313	0.8494	171.10	6.5569	0.8321	172.20	6.7363	0.8297	171.30	6.7205
		0.9	0.9983	178.00	5.7431	0.8707	173.00	6.3998	0.8510	173.00	6.5479	0.8509	173.00	6.5487
	0.9	0	1.0000	170.00	5.1134	0.8665	165.10	5.7311	0.8555	163.70	5.7556	0.8551	163.50	5.7512
		0.3	1.0000	171.00	5.3886	0.8654	164.80	6.0010	0.8450	166.30	6.2018	0.8545	169.70	6.2582
		0.6	1.0000	174.00	5.5201	0.8621	173.50	6.3847	0.8639	170.00	6.2429	0.8701	171.70	6.2604
		0.9	0.9984	178.00	5.7423	0.8873	177.10	6.4286	0.8700	176.60	6.5380	0.8684	176.40	6.5426
Ave.			0.9995	129.66	5.4729	0.8911	126.69	6.0223	0.8781	126.32	6.0949	0.8794	126.59	6.1011

, with the best objective values emphasized in bold.

It is evident from the table that the performance of the three algorithms is similar, with the average F-values for the 32 instances differing by no more than 2%. All three algorithms performed well, with the average F-values for the first 16 instances (smaller scale) exceeding 0.9 and for the last 16 instances (larger scale) exceeding 0.82. Among the three algorithms, SPSES performed the best, yielding superior results in the majority of instances.

Figure 10 shows the average computing time distribution of the three algorithms across all instances. The solution environment for all three algorithms was kept consistent for each instance. It is evident from the figure that the computing time for all three algorithms is well below the time limit of 500 s. Specifically, SPSES exhibits the shortest time for the vast majority of instances, while SPSWOA takes the longest. To further confirm the statistical significance of the results, a Kruskal-Wallis variance analysis (ANOVA) is performed on all 32 instances for the three algorithms, with a significance level of 0.05. This non-parametric test is suitable for non-normally distributed data and is used to assess significant differences across multiple algorithms. As seen in Figure 11, the ANOVA significance tests for the F, Q, and T metrics all yield p-values above 0.05. We can conclude that there are no significant differences among the three algorithms, and SPSES is statistically superior to the other two algorithms, which also verifies the conclusions drawn from Table 8. The reason for this is that all three algorithms are based on the SPSSIA framework we proposed, so the performance differences observed are not significant.

In summary, the analysis demonstrates that the three algorithms, built on the SPSSIA framework, performed well on the 32 instances designed using actual data, with only slight variations in performance. These findings validate the superior performance of the proposed model and algorithms.

6.4. Comparative Analysis with Actual Scenario Data

To validate the performance of the proposed model and algorithms in solving practical problems, we directly utilize actual navigation data from seven consecutive lockage schedule cycles of the TGGD over a specific period as the comparison data. Specifically, the average ship passage Q = 73.85 and the average ship stay T = 6.5613 h for the 7 lockage planning cycles with d = 12; the average ship passage Q = 149.29 and the average ship stay T = 6.7650 h for the 7 lockage planning cycles with d = 24. To objectively compare the results obtained from the algorithms with each case, the actual data was converted into the F-value for each case using Equations (1) and (51). In these cases, cp is set according to the actual data, and sp is set to 0, 0.3, 0.6, and 0.9. The settings for three algorithms and solvers are consistent with those in Section 6.3, and the results are presented in

Table 9. The results of the three algorithms solving actual cases in comparison to case data.

Instances		Case Data	UB			SPSES			SPSPSO			SPSWOA
d	sp	F	F	Q	T(h)	F	Q	T(h)	F	Q	T(h)	F	Q	T(h)
12	0	0.7194	1.0000	82.00	5.2411	0.8759	80.00	5.8377	0.8677	79.00	5.8192	0.8675	79.00	5.8206
	0.3	0.6968	1.0000	85.00	5.2621	0.9140	83.40	5.6488	0.9017	83.60	5.7396	0.8995	83.40	5.7399
	0.6	0.6862	1.0000	86.00	5.2430	0.9241	84.70	5.5879	0.9001	83.50	5.6556	0.9040	84.10	5.6716
	0.9	0.6758	0.9994	82.00	4.9265	0.9643	81.90	5.0996	0.9475	79.20	5.0189	0.9511	81.40	5.1388
24	0	0.7493	1.0000	160.00	5.4330	0.8062	156.00	6.5705	0.7961	156.00	6.6539	0.7960	156.00	6.6547
	0.3	0.7087	1.0000	170.00	5.4596	0.8156	168.70	6.6428	0.8062	166.90	6.6485	0.8073	167.70	6.6713
	0.6	0.7173	1.0000	168.00	5.4607	0.8341	164.90	6.4260	0.8082	166.50	6.6963	0.8107	166.50	6.6756
	0.9	0.6850	0.9966	165.00	5.1394	0.8650	161.50	5.7957	0.8511	161.20	5.8794	0.8546	163.40	5.9352
Ave.		0.7048	0.9995	124.75	5.2707	0.8749	122.64	5.9511	0.8598	121.99	6.0139	0.8613	122.69	6.0385

, with the best objective values highlighted in bold.

The results show that all three algorithms performed well in solving the real-world cases, with average objective values exceeding 0.85, significantly higher than the actual case data of 0.7048. The performance differences among the three are minimal, with average objective values differing by less than 2%, and SPSES slightly outperforms the other two by a narrow margin. This observation further corroborates the accuracy and reliability of the experimental conclusions drawn in Section 6.3.

Figure 12 shows the computation times of the three algorithms for the cases in this section. The computation times increase with the size of the cases but remain well below the time limit (500 s), and all three achieve optimal solutions at a relatively fast speed. Among them, SPSES exhibits a slightly faster solving speed overall compared to the other two. This experimental phenomenon indicates that the swarm intelligence optimization mechanism of SPSSIA proposed in this paper has strong convergence and search capabilities.

In conclusion, based on all the experimental results, we can affirm that the model and algorithms proposed in this paper for the TGGD navigation system demonstrate superior performance and efficiency in solving the SLCNP.

6.5. Analysis of Key Impact Factors

This section further analyzes the effect of key impact factors on the TGGD navigational performance from the previous experimental results. Analyzing the data reveals that for both large and small instances in Section 6.3, as the sort-pick ratio (sp) increases, the objective value tends to rise. Figure 13 illustrates the trend of F-values for the three algorithms as sp varies with cp set to 0, 0.3, 0.6, and 0.9. This phenomenon indicates that increasing the sortable portion in the waiting ship sequence of the lockage schedule facilitates more flexible ship combinations, leading to better lockage scheduling and thus improving the ship throughput of the TGGD.

From Table 8 and Figure 13, it is also clear that the variation in the proportion of standardized cargo ships (cp) has little impact on the F-value. To further explore the impact of cp on navigation quality, we analyzed the changes in cargo throughput with varying cp values. Figure 14 shows the trend of cargo throughput as cp varies for the d = 12 and d = 24 instances. It is apparent that as the proportion of standardized cargo ships increases, the cargo throughput also increases linearly. This is because standardized cargo ships have a larger carrying capacity than ordinary cargo ships, so as cp increases, the ship throughput remains relatively stable, but the hub’s cargo throughput rises significantly. This conclusion also validates the correctness of the policy encouraging the standardization of cargo ships.

Figure 15 clearly illustrates the differences between the solutions of the three algorithms and the actual case data, as well as the upper bounds of the case. The trend of the curves also indicates that as the value of sp increases, the F-value shows an upward trend. This suggests that in the actual navigation operation process, the formulation of the lockage schedule should, without violating the principle of service fairness, maximize the flexibility of ship regulation between lockages to enhance the navigation efficiency of TGGD.

7. Conclusions and Future Work

Lock scheduling research has long been a topic of interest, with much of the focus on algorithm optimization for co-scheduling problems and the simulation of scheduling processes. The appearance of the serial lock problem has further enriched this research field. This paper addresses the SLCNP within the context of the TGGD navigation system, presenting the first solution to this issue. A MIP model for SLCNP was developed, optimizing for ship throughput and stay time while considering real-world constraints such as ship priority, service fairness, two-stage ship placement, and the chain navigation process. To solve the SLCNP, we developed the SPSSIA framework, which integrates the characteristics of SLCNP through a hybrid multi-section encoding method, a two-stage heuristic decoding approach, and a swarm intelligence evolution mechanism. The experimental results illustrate the substantial impact of the swarm intelligence evolution mechanism within SPSSIA, highlighting the influence of the sp parameter on lockage schedule development and the cp parameter on hub cargo throughput. The results of this paper could offer valuable insights and practical guidance to the operators and managers of inland river hubs with serial lock scenarios, enabling them to enhance the operational efficiency and resource allocation. The findings can directly inform decision-making processes related to lock management, resource allocation, and service prioritization, enabling hub operators to adapt to changing conditions and optimize their operations dynamically.

While this work represents a significant step forward, several limitations exist. Due to the complexity of the navigation process and data availability constraints, our analysis focused primarily on conventional scenarios encountered at TGGD, limiting the generalizability of the proposed MIP model. The model’s constraints are tailored to the operational characteristics of TGGD facilities and overlook the potential impact of hydrological, meteorological, and ship navigation conditions. These limitations also guide the direction of future research: (i) The generalized SLCNP will be further investigated in the next studies. (ii) We will attempt to use constraint planning to describe the cumbersome navigational constraints and improve the model’s solution capability. (iii) Hydrological, meteorological, and navigational conditions, as well as the influence of human factors on the navigation process, are taken into account more in the modeling for the further study. (iv) Continued exploration of these issues will guide further optimization of solution methods.