Harsh Sea Conditions Research Articles

Analysis and optimization of self-propelled performance of wave-driven vehicle hydrofoil under high sea-level condition

The wave-driven vehicle is a surface vehicle powered by capturing wave energy, which is required to face harsh sea conditions when performing tasks in parts of the ocean. However, wave-driven vehicles are usually small in size, and their seakeeping and speed are generally poor in high sea conditions. Wave driven vehicles are usually equipped with rigid connected hydrofoils to capture wave energy, which can provide power for wave driven vehicles and enhance seakeeping. Aiming at the long-term survival and operation requirements of wave-driven vehicle under high sea conditions, this paper studies the effect of high sea conditions launching wing on the self- propelled performance of wave-driven vehicle. After installing rigid connected hydrofoils on wave-driven vehicles, the structural parameters of the hydrofoils are changed, and the kinematic and dynamic responses of wave-driven vehicles at 0–90 ° wave encounter Angle are numerically simulated based on CFD method. The effects of underwater wing depth, hydrofoil spacing and hydrofoil span length on the self- propelled performance of wave-driven vehicles are analyzed. Based on this, the structural parameters of rigidly connected hydrofoils are optimized, which improves the seakeeping and rapidity of wave-driven vehicles in high sea conditions.

Model experimental studies on active heave compensation control strategy for electric-driven offshore cranes

Ship-mounted heave compensation offshore cranes are indispensable for isolating connected payloads from the support vessel during lifting operations under harsh sea conditions. In this paper, an innovative adaptive robust control strategy is presented for the electric-driven active heave compensation (EDAHC) system, combining an equivalent model predictive control (EMPC) method with a bias proportional integral derivative (BPID) framework to effectively mitigate the adverse effects of wave-induced heave motions from the support vessel on the suspended payload. Building upon the inherent field-oriented control in the permanent magnet synchronous motor (PMSM), the BPID-based control structure is introduced, motivated by its prompt responsiveness and robust resistance against model discrepancies and irregular disturbances. Facilitated by a torque compensation mechanism, the EMPC-based control scheme, synthesized with an autoregressive integrated moving average (ARIMA)-based heave motion prediction algorithm, is subsequently developed to achieve nonlinear friction deadzone correction and adaptive regulation of BPID parameters, thereby ensuring optimal performance of the EDAHC system within specified state and input constraints. Comparative experimental tests conducted on a scaled EDAHC testbed validate the superior capabilities of the proposal in station-keeping, position tracking, constraint satisfaction, and robustness against parametric uncertainties.

Active pendulation control of hoisting systems of ship-mounted cranes under ocean wave excitations: Principle and experimental study

Large-scale ship-mounted cranes are important facilities for marine construction. However, under harsh sea conditions, the hoisting system of a ship-mounted crane is often subjected to wave loadings, leading to frequent occurrences of large-amplitude resonance. This paper proposes an active control method, termed the active displacement compensation (ADC) approach, designed for suppressing the large-amplitude pendulation of the hoisting system of ship-mounted cranes. By controlling the position of the suspension point of the hoisting system using linear-motion actuators, the input excitations of the hoisting system induced by the ship motions can be partially or fully compensated in real-time. Therefore, the proposed ADC approach can significantly suppress the large-amplitude pendulation of the hoisting system. The optimal feedback gain and time delay of the ADC approach are theoretically obtained. The ADC approach has been validated by using a six-degree-of-freedoms shaking table test and numerical simulation on the in-plane pendulation suppression of a small-scale hoisting system. Experimental results indicate that the ADC approach achieves a pendulation reduction ratio between 85 % and 89 % for the hoisting system subjected to scaled ship motion records. This study may help develop efficient active control techniques for hoisting systems of ship-mounted cranes under harsh sea conditions.

Coupling a nonlinear finite element mooring model with an overset CFD solver for dynamic analysis of floating structures in waves

An accurate prediction of motion responses of a moored floating structure requires high-fidelity coupled analysis between the structure and its mooring system in waves, which involves both mooring line dynamics and fluid dynamics. In this paper, we develop a nonlinear finite element dynamic mooring analysis module that allows large axial stretching of mooring lines under the open source CFD framework OpenFOAM, and couple it with an Overset CFD solver to establish a fully coupled finite element mooring-CFD analysis model for floating structures. The coupled model is advantageous over existing models in that it is able to deal with large axial mooring stretching and complex mesh motions induced by significant structure responses under severe operating conditions. The developed dynamic mooring analysis method is firstly validated against publicly available data. In order to validate the coupled model, a series of experimental tests were carried out for a semi-submersible platform moored by eight catenary lines under different operating conditions, including free decay, regular and irregular wave tests. The dynamic responses of the floating structure and mooring tension are analyzed and compared with experimental data. Results show that numerically predicted structure responses and mooring tension are generally in good agreement with experimental measurements. Meanwhile, the strongly nonlinear large-amplitude resonant motions generated by the interaction of irregular waves with the moored floating structure are also investigated. Comparisons reveal that the coupled model can accurately predict the resonant spectral peaks of the structure responses, indicating that the it is capable of simulating large-amplitude motions of moored floating structures. The coupled model developed in this paper can be further applied to study the survivability of moored floating structures under harsh sea conditions.

Unmanned sailboat with self-balancing rotating sail based on elastic rope: Modeling, optimization, and sea trials

Unmanned sailboats use wind energy to assist in propulsion during navigation and are featured by the advantages of clean energy and long endurance, making a hotspot of research on maritime unmanned vehicles. Sails are a core component of unmanned sailboats, and most existing studies on rotating sail have the shortcomings of high energy consumption and load. Therefore, this article proposes a novel method to achieve self-balancing sail rotation based on elastic rope and reduce the energy consumption and load of unmanned sailboats, and establishes a model for the self-balancing rotating sail. We introduced the sum of angular thrust of spherical surface SL as a performance evaluation index for unmanned sailboats, which considers both thrust and navigable range. Based on this index, the mechanical and positional parameters of the model are optimized to obtain the optimal solutions. The performance is increased by 24% after the optimization, and by 76% compared with the sailboat with the fixed sail angle when the same initial sail angle of 38° is adopted. Following the optimal solutions, we develop a prototype of the unmanned sailboat with self-balancing rotating sail. Subsequently, wind turbine experiments are conducted to preliminarily verify the correctness of the model for self-balancing rotating sail and the optimization results. Finally, sea trials are conducted to further verify the performance and seaworthiness of the unmanned sailboat under realistic sea conditions, successfully achieving a 15-day voyage under harsh sea conditions of levels 2 to 4. This study to some extent compensates for the shortcomings of existing studies on rotating sail and provides guidance for subsequent theoretical and engineering research in the field of unmanned sailboats.

Nezha‐IV: A hybrid aerial underwater vehicle in real ocean environments

AbstractHybrid aerial underwater vehicles (HAUVs) that are capable of autonomously operating in both the aerial and underwater environments with repeated rapid transitions have received great interest in recent years. Current HAUV prototypes cannot fully cope with the unstructured, harsh, and hazardous ocean environments considering the high‐pressure and the strong hydrodynamic perturbations. This work reports on the HAUV, Nezha‐IV, with all basic functions validated in a real ocean environment. Nezha‐IV has four aerial propellers and eight underwater thrusters. It is not only capable of maneuvering delicately during flight and underwater but can also perform dynamic transitions under rough sea conditions. We present the design, construction, control, and oceanic testing of Nezha‐IV and demonstrate reliability and robustness, especially during the transition phases. Performance estimation and results gathered from tests in the South China Sea show that Nezha‐IV can operate at various depths ranging from 0 to 50 m at a velocity of 1 kt for 22 min and hover in the air for 15 min or cruise at a velocity of 10 m/s for 7.2 km in 12 min. This project contributes to the field of HAUV with (i) a HAUV capable of dynamic locomotion with high stability in both media and during transitions; (ii) identification of challenges during water–air transition in harsh sea conditions, and a reliable transition strategy that helps the development of fully autonomous HAUVs; (iii) a modular, foldable, and impact‐resistant HAUV structure; (iv) vehicle's capability proved by extensive tests in dynamic ocean environment.

Time-varying structural parameter identification for the offshore gangway

The offshore gangway is a new type of offshore transfer equipment, equivalent to an offshore passage. This device makes it convenient for people to transfer between a vessel and an offshore structure. The control algorithm of the gangway system demands adjusting its control parameters to deal with harsh sea conditions and ensure personnel safety. Thence, it is essential to foreknow its time-varying structural parameters, such as the moment of inertia, the mass, the location of the mass center, and so on. However, due to the highly complex structure, these parameters are difficult to obtain through measurement directly. To adaptively identify these time-varying parameters for better controller design, this paper proposes a novel structural parameter identification approach based on the idea of interval division. During the identification process, the entire period is uniformly divided into several intervals. The time-varying model of the offshore gangway system can be represented in each interval by a quadratic regression equation, the coefficients of which can be derived by the least-squares algorithm. Furthermore, to adapt to the different variation frequencies and amplitudes of the parameters, the divided intervals are of distinct sizes according to the characteristics of each identification parameter. In addition, the particle swarm algorithm is employed to obtain the optimal interval sizes, the cost function of which consists of a series of output error terms. Verified by simulation, the method presented in this paper can effectively estimate the structural parameters of the offshore gangway with high accuracy, with mean square errors reduced by at least 2/3 compared to other methods proposed in previous research.

Performance analysis of the passive heave compensator for hydraulic shipwreck lifting systems in twin-barge salvaging

Passive heave compensation (PHC) is extensively used in marine engineering operations such as offshore hoisting and remote-operated vehicle launching because of its high capacity and no energy requirement. Shanghai Salvage Co. used a new twin-barge hydraulic lifting method with multiple PHC systems to salvage the shipwreck of M.V. Sewol in 2017 and avoid harsh sea conditions. The result opens up a new avenue for rescuing sunken ships with huge tonnage. However, the performance of hydraulic compensators in shipwreck salvaging has rarely been studied. In this paper, we mainly focus on the effect of PHC systems on a twin-barge shipwreck salvaging system. Firstly, the salvage system and the PHC's nonlinear dynamic are mathematically modeled. The frequency characteristics of the salvaging system in five motion states are then analyzed, taking into account compensator stiffness and wave frequencies. Finally, in MATLAB/Simulink, a series of simulations of a salvaging system with multiple PHC systems are carried out. The effects of using PHC systems and changing compensation parameters on load performance, including shipwreck movement and sling tensions, are analyzed. The results suggest that in this salvaging system, PHC systems with proper parameters can compensate for both shipwreck movement and sling tensions with great effectiveness.

Predictive Control of a Heaving Compensation System Based on Machine Learning Prediction Algorithm

Floating structures have become a major part of offshore structure communities as offshore engineering moves from shallow waters to deeper ones. Floating installation ships or platforms are widely used in these engineering operations. Unexpected wave-induced motions affect floating structures, especially in harsh sea conditions. Horizontal motions on the sea surface can be offset by a dynamic positioning system, and heave motions can be controlled by a heave compensation system. Active heave compensation (AHC) systems are applied to control vertical heave motions and improve safety and efficiency. Predictive control based on machine learning prediction algorithms further improves the performance of active heave compensation control systems. This study proposes a predictive control strategy for an active heave compensation system with a machine learning prediction algorithm to minimise the heave motion of crane payload. A predictive active compensation model is presented to verify the proposed predictive control strategy, and proportion–integration–differentiation control with predictive control is adopted. The reliability of back propagation neural network (BPNN) and long short-term memory recurrent neural network (LSTM RNN) prediction algorithms is proven. The influence of the predictive error on compensation performance is analysed by comparing predictive feedforward cases with actual-data feedforward cases. Predictive feedforward control with regular and irregular wave conditions is discussed, and the possible strategies are examined. After implementing the proposed predictive control strategy based on a machine learning algorithm in an active heave compensation system, the heave motion of the payload is reduced considerably. This investigation is expected to contribute to the motion control strategy of floating structures.

Research on the Hydrodynamic Performance of a Pentamaran in Calm Water and Regular Waves

As a typical multi-hull ship, the pentamaran has a wider deck, higher damage stability, and low-speed seaworthiness in harsh sea conditions compared with other multi-hull types such as the catamaran and trimaran, having more potential for becoming a new type of freight merchant ship in line with the development trend of the marine economy. In this paper, on the basis of the finite volume method, the numerical simulations and detailed comparative analysis of the monohull and pentamaran in viscous flow were carried out. The resistance prediction in calm water and regular waves and the changes of each resistance component of the pentamaran were explored. The dynamic fluid body interaction (DFBI) module was used for the calculations in calm water, and the overset mesh was used for the calculations in regular waves. The results show that in calm water, the adverse effect of the side hulls on the resistance was mainly reflected in the low-speed range. In the high-speed range, the influence of the side hulls on the resistance can be ignored. At high speeds, the main hull and side hulls of the pentamaran will produce favorable wave-making interference, thereby improving the wave-making performance of the pentamaran. In regular waves, although the added resistance of the pentamaran increased most of the time, its amplitude was smaller than that of the monohull under the same wave condition. This work provides a technical basis for the research on the design and hydrodynamic performance of the pentamaran.

Semi-analytical study of wave interaction with a submerged permeable sphere applied on a spherical aquaculture cage

Spherical aquaculture cages are a technology of marine fish farming recently developed for installation in offshore areas. In harsh sea conditions, the cage can be temporarily fully immersed and safe from the intense loads. This paper studied semi-analytically linear waves incident to a submerged permeable sphere. The primary purpose of this research is to study the wave excitation forces, hydrodynamic coefficients, and wave strength inside the sphere. The problem's governing equations were established in the context of potential flow theory, considering two inner and outer regions and using multiple methods and Darcy's law for flow passing the porous media. The developed method was applied to the Aquapod spherical cage with a radius of a=9.0 m. The results indicate that the porosity parameter causes wave excitation forces and added mass decreases, but the damping coefficient increases significantly in the particular range of wavenumbers. Moreover, the peak value of the wave excitation force, damping coefficient, and the average of water particle velocities inside the cage occurs at ka=0.5, where k is the wavenumber. Also, at enough immersion, the damping coefficient is independent of depth. The results of this research are helpful in better understanding the behavior of spherical aquaculture cages in submergence conditions.

Analysis of Dynamic Response and Ultimate Strength for Box Girder under Bending Moment

The box girder can be seen as a kind of simplified ship structure that can withstand a vertical bending moment. Dynamic loads play an important role in structural safety analysis, such as ship bow slamming during harsh sea conditions. In this paper, the dynamic elastic–plastic response and ultimate strength of a box girder under a bending moment are analyzed. A box girder with the same cross-section scantlings and span length as the Nishihara experiment is selected as the analysis object. Based on the model experiment results, the non-linear FE analysis method is validated to capture the ultimate strength of a box girder under bending moment. Then, six box girders were designed to study the critical influence factors on the dynamic ultimate moment, including the model length, plate thickness, mass density and load excitation period. On the basis of structural dynamic response, an evaluation criterion of dynamic limit state for the box girder under a bending moment is proposed in this paper. Compared with the static ultimate moment, the change in the dynamic ultimate moment is discussed in detail to obtain the general principal method for dynamic strength analysis. The conclusions in this paper can provide guidance for dynamic ultimate strength evaluation.

A Method for Long-Term Target Anti-Interference Tracking Combining Deep Learning and CKF for LARS Tracking and Capturing

Autonomous underwater vehicles (AUV) recycling in an underwater environment is particularly challenging due to the continuous exploitation of marine resources. AUV recycling via visual technology is the primary method. However, the current visual technology is limited by harsh sea conditions and has problems, such as poor tracking and detection. To solve these problems, we propose a long-term target anti-interference tracking (LTAT) method, which integrates Siamese networks, You Only Look Once (YOLO) networks and online learning ideas. Meanwhile, we propose using the cubature Kalman filter (CKF) for optimization and prediction of the position. We constructed a launch and recovery system (LARS) tracking and capturing the AUV. The system consists of the following parts: First, images are acquired via binocular cameras. Next, the relative position between the AUV and the end of the LARS was estimated based on the pixel positions of the tracking AUV feature points and binocular camera data. Finally, using a discrete proportion integration differentiation (PID) method, the LARS is controlled to capture the moving AUV via a CKF-optimized position. To verify the feasibility of our proposed system, we used the robot operating system (ROS) platform and Gazebo software to simulate the system for experiments and visualization. The experiment demonstrates that in the tracking process when the AUV makes a sinusoidal motion with an amplitude of 0.2 m in the three-dimensional space and the relative distance between the AUV and LARS is no more than 1 m, the estimated position error of the AUV does not exceed 0.03 m. In the capturing process, the final capturing error is about 28 mm. Our results verify that our proposed system has high robustness and accuracy, providing the foundation for future AUV recycling research.

Dynamic Analysis of Active Heave Compensation System for Marine Winch under the Impact of Irregular Waves

The six-degree-of-freedom (6 DOF) motion of the mother ship, especially the heave motion, brings difficulties to offshore operations and even causes safety accidents. To ensure the reliability of the launch and recovery system when the active heave compensation (AHC) function works, its dynamic characteristics are analyzed. Firstly, the dynamic model of the launch and recovery system with a rigid rope is constructed, and the mathematical simulation of the launch and recovery system without the AHC function under the excitation of irregular waves is completed. Secondly, a flexible rope based on the finite element method is established, and the dynamic responses of the launch and recovery system with AHC function under different conditions are compared with those without AHC function. The simulation results demonstrate that the dynamic responses of the system are related to the load, the underwater penetration, and the sea condition. In detail, the increase of the load leads to an increase in the dynamic response of the system. In essence, the underwater penetration affects the dynamic response of the system by affecting the activity of load acceleration. Therefore, a short rope is preferred in the maritime operations. Moreover, the harsh sea condition usually causes an undesirable effect of AHC function. In summary, the overall work is conductive to the modeling of the launch and recovery system, as well as the development of AHC technology so as to improve the quality of offshore operations in the future.

Real-Time Infrared Horizon Detection in Maritime and Land Environments Based on Hyper-Laplace Filter and Convolutional Neural Network

The infrared (IR) horizon is an essential reference for near-ground state awareness and target detection. The traditional horizon detection algorithms based on hand-crafted features claim the advantage of being intuitive and highly targeted, however, their capability of detail extraction and robustness is not strong enough when switching among diverse scenes, meanwhile, the Convolutional Neural Networks (CNN) have the powerful ability in feature acquisition, but has the disadvantages of interpretability and instability. Given this, this paper proposes a novel horizon detection algorithm that combines the traditional method and CNN. Firstly, the Hyper-Laplace filter (HLF) is proposed to eliminate the interferences and enhance the saliency of the target. Then a Positioning Module (PM) based on the traditional idea which fully utilizes the structural characteristics of multi-feature-maps in CNN is present to extract the endpoints of the horizon. Finally, an auxiliary module that consists of a Revise Branch (RB) and Self Attention Module (SAM) is designed to assist the PM with detailed corrections. Compared with the state-of-the-art algorithms, experiments based on three datasets with more than 25000 frames under various scenes demonstrate that the proposed algorithm not only achieves the best accuracy and highest stability but also addresses real-time needs. Notably, the detection deviation is controlled within 2 pixels on all three datasets while achieving a comparable detection speed of 170 frames per second. Besides, our algorithm also shows strong robustness even under harsh sea conditions and urban backgrounds. Code and trained models are available at https://github.com/FJsRepo/InfML-HDD.

Design and Working Performance Evaluation of a Combined Survey Platform under Strong Wave and Deep-Water Conditions

To meet the engineering requirements of long-distance and high-precision elevation transfer under adverse sea conditions, a new structure for a combined survey platform based on the basic principle of vibration reduction and isolation is designed in this paper. The combined survey platform uses the barrier effect of the external support system on the wave-current load to reduce the influence of the wave-current load on the internal support system and the platform, so it can maintain good performance, even in harsh sea conditions, under the premise of no collision between the internal and external support systems. The expected working performance of the structure was verified by numerical simulation, and the influence of the layout, structure size, waves, and water flow on the working performance of the structure was quantitatively analyzed. The results show that: (1) the external support system can better realize the barrier effect of the wave-current load and significantly reduce the influence of the wave-current load on the internal support system and platform; (2) for the independent combined survey platform, when the pile diameters of the internal and external support systems are 0.8 m and 1.2 m and the wall thicknesses are 11.0 mm and 12.0 mm, respectively, the period of the wave is 8.0 s, the water depth is 20.0 m, the speed of water flow is 1.0 m/s, and the wave height is 4.0 m, then the maximum variation of the vertical angle of the platform is only 19.3″; (3) for the attached combined survey platform, the lateral stiffness of the external support system is increased and the displacement of the external support system is significantly reduced because the external support system is connected with the cushion cap through the attachments; therefore, the structure size of the survey platform can be greatly reduced.

A novel motion planning algorithm for a three DoF foldable parallel compensation platform based on prediction and B-spline

Foldable parallel mechanisms are suitable to compensate for the motion of vessels due to their compact structure, multidimensional motion, and large carrying capacity. However, the vessel sways violently under harsh sea conditions, challenging the rational utilization of the mechanism workspace. This work proposes a novel motion planning algorithm based on motion prediction and the B-spline for a three-degrees-of-freedom foldable parallel compensation platform. First, for harsh sea conditions, the motion law of a guardian vessel is studied, and the encoder-decoder model is used to predict the roll, pitch, and heave motions of the vessel. Second, we propose a multidimensional motion compensation strategy under the limits of the mechanism dimensions. Then, a backward kinematics model is developed. Its workspace under the structural and actuator limitations is proposed. Finally, based on the theory of the nonuniform rational B-spline curve and target motion prediction, a dynamic motion planning algorithm is proposed, and experiments are conducted. The motion planning algorithm can provide smooth motion commands under the mechanism workspace, velocity, acceleration constraints. It can be realized using asynchronous calculation, which meets real-time motion control requirements.

Study on Characteristics and Optimal Layout of Components in Shallow Water Mooring System of Floating Wind Turbine

Offshore wind energy resources far exceed those on land. However, the increase in water depth of the continental shelf in China’s sea area slows down as the distance from the shore increases. In view of the characteristics of China’s sea, how to optimize the design of mooring systems to adapt to the water depth conditions in China’s waters and to resist the harsh sea conditions is one of the major problems encountered in the development of floating wind turbines. In this paper, the 5MW-OC4 semi-submersible floating wind turbine is taken as the research object, and the frequency domain and time domain calculation of the floating wind turbine are carried out by using SESAM software under the water depth of 40 m in the Bohai Sea. By comparing the motion response and tension of catenary and tension mooring floating wind turbine. The mooring system is optimized by combining the buoy and clump weight. The results show that the catenary mooring mode is more suitable for the floating wind turbine under shallow water conditions. Mooring accessories used in combination have the same effect on the overall response of the floating wind turbine as changing the same mooring parameters when used alone, but the effect of changing the mooring parameters on the performance optimization of the mooring system is more obvious when used in combination, and a reasonable combination of accessories can significantly change the overall characteristics of the floating system and affect the safety and cost of the system.

Drift simulation of a floating offshore wind turbine with broken mooring lines in a dynamic sea condition

This study investigates the aero-hydro-mooring coupled dynamics of the National Renewable Energy Laboratory (NREL) offshore 5-MW baseline wind turbine supported by the Offshore Code Comparison Collaboration Continuation (OC4) DeepCwind semisubmersible floating platform by considering the effects of suddenly broken mooring behaviour. The wind and wave loads can be selected as the Cummins time-domain equation inputs based on the Norwegian Petroleum Directorate wind and Joint North Sea Wave Project wave spectrums. The nonlinear viscous drag was estimated using the quadratic damping matrix rather than Morison's element. Additionally, the quadratic transfer function matrix was used to calculate the slow-drift force. Compared to the multi-segmented quasi-static mooring model, the lump-mass method is a dynamic mooring model that can solve real-time motion on the platform. Once a single or a pair of mooring lines is intentionally disconnected from the floating platform at a certain time, the transient responses of mooring line tensions and turbine performance will be evaluated. The novelty of this study is that the drift trajectories and power performances of OC4 DeepCwind semisubmersible floating offshore wind turbine under different conditions of environmental loads and broken mooring lines can be approximately estimated in the offshore wind farm. The analysis can be used for the environmental impact assessment in harsh sea conditions.

Vessel Scheduling and Bunker Management With Speed Deviations for Liner Shipping in the Presence of Collaborative Agreements

Vessels face uncertain factors during a voyage, such as bad weather and harsh sea conditions along a route in container liner shipping. As such, the real vessel speed during each leg of a voyage often deviates from the planned one, which may lead to fluctuations in vessel schedule and bunker consumption. This paper investigates the problem of vessel scheduling and bunker management with speed deviations (VSBMSD) for liner shipping in the presence of collaborative agreements. By establishing the worst-case scenario of the maximum bunker consumption function with vessel sailing speed as an independent variable, we develop a mixed-integer nonlinear programming model to minimize the total liner shipping route service cost. A piecewise linear secant approximation method is designed, and then a CPLEX solver is used to solve the problem. The results of the computational experiments conducted for the AEMX route indicate that VSBMSD in the presence of collaborative agreements can enable shipping companies to design vessel schedules reasonably and reduce the total cost of liner shipping route service by at least 2.95% compared to the results from similar studies.