A liner ship schedule robust optimization method and system based on budget uncertainty set and cooperation agreement, and a medium

Abstract

The application provides a liner ship period robust optimization method and system based on a budget uncertainty set and a cooperation agreement, and a medium, and belongs to the field of shipping management. The method first acquires initial plan data and cost parameters of liner route operation; then, under the framework of a port navigation cooperation agreement, a robust optimization model is constructed with the target of minimizing total cost, a budget uncertainty set is used to describe fuel price rising situations, and fuel cost, weekly operation cost, loading and unloading cost, inventory cost and late penalty cost are comprehensively considered; then, a model reconstruction method is designed by combining dynamic tangent approximation and robust counterpart conversion, and the original model with nonlinear relationship and double-layer robust structure is reconstructed into a single-layer mixed integer linear programming model; finally, an optimization solver is used to solve and output a liner ship period robust optimization scheme with the lowest cost. The application fully considers fuel price uncertainty, effectively reduces cost fluctuation and improves ship period scheme stability.

Description

Technical Field

[0001] This invention relates to the field of shipping management technology, and more particularly to a robust optimization method, system and medium for liner schedules based on budget uncertainty sets and cooperation agreements. Background Technology

[0002] Container liner shipping features fixed routes, ports, schedules, and regular departures, providing stable maritime transport services for international trade. As a crucial component of maritime transport, container liner shipping handles a large number of high-frequency, standardized, and networked cargo transport tasks. The rationality of its shipping schedule directly affects the operating costs, service reliability, and market competitiveness of liner companies.

[0003] However, liner shipping costs are influenced by various external factors, with fuel costs being a significant component of the total voyage cost. Fuel prices are highly uncertain due to factors such as international energy market supply and demand and port fuel supply conditions. When fuel prices rise, existing speed schedules, sailing times, and arrival / departure plans based on nominal fuel prices may no longer be economically viable, leading to increased actual operating costs for liner shipping and impacting the stability of schedules. Therefore, considering fuel price uncertainty during schedule design is of practical significance for improving the stability of liner shipping plans.

[0004] With the continuous development of maritime transport organization methods and port service models, the role of port-shipping cooperation agreements in liner schedule design has gradually gained importance. Under these agreements, ports can provide liner companies with multiple optional time windows, multiple optional departure and arrival times, and multiple optional loading and unloading rates, enabling them to coordinate between speed, port operating time, and loading and unloading efficiency. When uncertainties such as fuel price fluctuations occur, liner companies need to re-evaluate factors such as fuel costs, port operating costs, inventory costs, and lateness penalties, adjusting their schedules to reduce total costs and maintain service stability. Therefore, the robust optimization problem of liner schedules, comprehensively considering fuel price uncertainty within the framework of port-shipping cooperation agreements, has significant engineering application value.

[0005] In the field of liner schedule optimization research, although numerous relevant academic journal articles and a small number of patents can be found through publicly available information, such as Dulebenets' article "Minimizing the Total Liner Shipping Route Service Costs via Application of an Efficient Collaborative" published in IEEE Transactions on Intelligent Transportation Systems, 2019, Issue 1. The following papers are cited: “Agreement”, “Joint Optimization of Container Liner Shipping Schedule Design and Refueling Strategy Based on Cooperation Agreement” published by Li Dechang et al. in the Journal of Shanghai Jiaotong University, No. 7, 2022; “Research on Schedule Coordination Considering Horizontal and Vertical Alliance Strategies under Uncertainty” published by Zhao Xu et al. in Transportation Systems Engineering and Information, No. 3, 2023; “Robust Optimization of Liner Shipping Schedule Design Considering Maritime Unexpected Time and Cooperation Agreement” published by Yang Hualong et al. in Transportation Systems Engineering and Information, No. 6, 2021; and the invention patents CN121329259A and CN121329254A, “A Method, System and Medium for Container Liner Shipping Schedule Interference Recovery Based on Improved Spatiotemporal Network”.

[0006] These publicly available studies provide an important foundation for liner shipping schedule optimization. However, existing methods mostly focus on optimizing deterministic cooperation agreements or primarily consider uncertainties such as speed deviations, port hours, and unexpected sailing events. For scenarios involving rising fuel prices, there is still room for improvement in how to construct robust schedule models within the framework of port and shipping cooperation agreements, and further address the nonlinear relationship of fuel consumption and the double-layer robust structure caused by budget uncertainty. Therefore, existing technologies, when facing the problem of optimizing container liner shipping schedules under uncertain fuel prices, still have limitations such as insufficient joint consideration of port and shipping cooperation agreement constraints and fuel price fluctuations, and the high difficulty in solving nonlinear robust optimization models. An effective solution is urgently needed to ensure that container liner shipping can cope with the uncertain impact of fuel price fluctuations while meeting cost control requirements, achieving stable optimization of shipping schedules. Summary of the Invention

[0007] To address the aforementioned technical problems of existing liner schedule optimization methods' insufficient consideration of the combined constraints of port and shipping cooperation agreements and fuel price fluctuations, as well as the high difficulty in solving nonlinear robust optimization models, this invention provides a robust optimization method and system for liner schedules based on budget uncertainty sets and cooperation agreements. This invention primarily utilizes budget uncertainty sets to describe fuel price increases within the framework of port and shipping cooperation agreements. It designs a model reconstruction method that integrates dynamic tangent approximation and robust equivalence transformation to convert the original model containing nonlinear relationships and a two-layer robust structure into a single-layer mixed-integer linear programming model. This mitigates the risk of fuel price fluctuations, reduces the volatility of total voyage costs, and improves the stability of the schedule plan during the planning stage.

[0008] The technical means employed in this invention are as follows: A robust optimization method for liner shipping schedules based on budget uncertainty sets and cooperation agreements includes: S1. Obtain initial planning data and cost parameters for liner operations; S2. Under the framework of the port and shipping cooperation agreement, construct a robust optimization model for liner schedules with the goal of minimizing total cost. The robust optimization model for liner schedules uses a budget uncertainty set to describe the situation of rising fuel prices. S3. Design a model reconstruction method that integrates dynamic tangent approximation and robust equivalence transformation to reconstruct the liner schedule robust optimization model containing nonlinear relationships and a two-layer robust structure into a single-layer mixed integer linear programming model. S4. Use an optimization solver to solve the single-level mixed integer linear programming model and output a robust optimization scheme for liner shipping schedules.

[0009] Further, step S1 includes: S11. Determine the port sets, segment sets, and segment distances of each segment on the liner route according to the order of port calls. S12. Determine the optional service data provided by each port under the framework of the port and shipping cooperation agreement, including the time window set, the start time set, the end time set and the loading and unloading rate set within each time window; S13. Determine the ship's operating parameters, including the upper and lower limits of the ship's speed and the number of ships that can be deployed on the route; S14. Determine fuel price parameters, including nominal fuel price, maximum increase of each port relative to the nominal fuel price, and budget parameters for the number of ports to limit fuel price increases. S15. Determine the operating cost parameters, including the weekly operating cost of ships, the unit loading and unloading cost of each port at each loading and unloading rate, the unit inventory cost of cargo, and the unit lateness penalty cost of each port. S16. Determine the cargo volume parameters, including the total number of containers loaded on each shipping segment and the total number of containers loaded and unloaded at each port.

[0010] Further, step S2 includes: S21. The objective function of the constructed liner schedule robust optimization model includes weekly operating costs, fuel costs under nominal fuel prices, port loading and unloading costs, cargo inventory costs, late arrival penalty costs, and robust costs caused by fuel price uncertainty. The objective function is as follows:

[0011] in, The weekly operating cost for each vessel; The number of vessels allocated to the shipping route; Nominal fuel price; This is a set of indexes of ports and segments on liner routes, numbered according to their call-to-call order. For the segment Fuel consumption; For the port The set of loading and unloading rates provided; For the port loading and unloading rate The unit loading and unloading cost; For the port The loading and unloading volume of containers; It is a 0-1 variable; when the container ship is in port Select the The value is 1 if the loading / unloading rate is specified, and 0 otherwise. Unit inventory cost of goods; For the segment The number of containers shipped; For the segment The sailing time; For the port The unit's cost of penalties for lateness; For the port The time of being late; Describe the port Whether the price of fuel is uncertain is a continuous variable within the interval [0,1]. For the port The maximum increase relative to nominal fuel prices; S22. Construct a budget uncertainty set to describe the scenario of rising fuel prices, assuming a port... The nominal fuel price is The maximum increase in fuel prices was When the port When fuel prices rise, the fuel price changes from... Become ; Introducing variables Indicates port The uncertainty of fuel price increases takes a value that satisfies:

[0012]

[0013] in, This is a budget parameter used to limit the number of ports that can be used to control fuel price increases.

[0014] S23. Construct a port and shipping cooperation agreement framework that includes pre-given constraints on multiple time windows, multiple start and end times, and multiple loading and unloading rates. Specifically, it includes the first constraint, the second constraint, the third constraint, the fourth constraint, the fifth constraint, the sixth constraint, and the seventh constraint, as shown in the following formula:

[0015]

[0016]

[0017]

[0018]

[0019]

[0020]

[0021] Wherein, the first constraint indicates that the vessel can only select one time window at each port; the second, third, fourth, and fifth constraints indicate that the vessel can only select one start time and one end time at each port, and the start time and end time cannot exceed the time window range; the sixth constraint indicates that the vessel must select one loading / unloading rate at each port; the seventh constraint indicates that the loading / unloading time is obtained from the loading / unloading volume and loading / unloading rate; in the above formula, For the port The provided set of time windows; For the port In the time window The set of start times provided; For the port In the time window The provided set of end times; For the port The set of loading and unloading rates provided; For container ships in port Loading and unloading time; The variable is 0-1, and it is used when the container ship selects a time window. Arrival at the port The value is 1 if it is true, and 0 otherwise. The variable is 0-1, representing the time when a container ship chooses to dock at a port. Time window No. The value is 1 at the beginning of the time, otherwise it is 0. The variable is 0-1, representing the time when a container ship chooses to dock at a port. Time window No. The value is 1 if the event ends at a specific time, otherwise it is 0. The variable is 0-1, when the container ship is in port. Select the The value is 1 for loading and unloading rates, and 0 otherwise.

[0022] S24. Construct the continuity of ship schedules and operational constraints (continuity constraints of arrival and departure times, segment sailing time constraints, upper and lower speed limits, lateness calculation constraints, waiting time calculation constraints, total voyage time constraints, and vessel configuration quantity constraints), including the eighth, ninth, tenth, eleventh, twelfth, thirteenth, fourteenth, fifteenth, sixteenth, seventeenth, eighteenth, and nineteenth constraints, as shown in the following formula:

[0023]

[0024]

[0025]

[0026]

[0027]

[0028]

[0029]

[0030]

[0031]

[0032]

[0033]

[0034] Among them, the eighth, ninth, tenth, and eleventh constraints are used to determine whether the vessel is late or waiting; the twelfth, thirteenth, and fourteenth constraints represent the temporal continuity of the vessel's arrival and departure; the fifteenth and sixteenth constraints represent the total time of a round-trip voyage, with the voyage departure frequency being weekly; the seventeenth constraint represents the vessel's sailing time on a segment; the eighteenth constraint represents the upper and lower limits of speed; and the nineteenth constraint represents a 0-1 variable; in the above formula, This represents the total number of ports along the shipping route. For container ships to arrive at the port The moment; For container ships to leave port The moment; For container ships in port Waiting time; For container ships in the voyage The speed of the ship; These are the lower and upper limits of cruising speed; This represents the maximum number of vessels that can be deployed on the route. For the segment The flight distance; For the port In the time window The An ending moment; For the port In the time window The The beginning moment.

[0035] Further, step S3 includes: S31. Establish the nonlinear relationship between fuel consumption and flight time, and obtain the fuel consumption function as follows:

[0036] in, and All are marine fuel consumption coefficients; For the segment The flight distance; For the segment Fuel consumption; For the segment The sailing time; S32. The nonlinear relationship is transformed into a linear constraint by using the dynamic tangent approximation method. S33. Use the robust equivalence transformation method to transform the bi-layer robust structure into a linear dual constraint. S34. Integrating the linear tangent constraints and the linear dual constraints, the robust optimization model for liner schedules is reconstructed into a single-level mixed-integer linear programming model. The objective function of the single-level mixed-integer linear programming model is as follows:

[0037] in, As dual variables, As dual variables, For budget parameters.

[0038] Further, step S32 includes: S321. Set the initial set of tangent points as follows:

[0039] in, and Each segment Lower and upper limits of sailing time; S322, Based on the current set of cut points At the tangent point Constructor First-order tangent approximation constraint:

[0040] in, For the segment No. The point of tangency that the tangent line approaches; For the segment The current set of pointcuts; S323, in the In the next iteration, based on the current set of cut points Construct a linear master problem and solve it to obtain the current solution. and

[0041] S324. Calculate the relative error using the following formula:

[0042] in, ; S325. Determine whether the relative error satisfies the convergence condition:

[0043] If the convergence condition is met, the current tangent approximation is considered to have reached the preset accuracy requirement, and the iteration stops; otherwise, the current solution is... Add to the corresponding flight segment The set of tangent points is defined, and a new tangent approximation constraint is added at that point:

[0044] S326. Repeat steps S323 to S325 until the convergence condition is met or the maximum number of iterations is reached. .

[0045] Further, step S33 includes: S331. Calculate the robust cost term in a two-layer robust structure using the following formula:

[0046] S332. Set the constraints for the robust cost item as follows:

[0047]

[0048] S333, Introducing dual variables for constraints , and The Lagrange function is constructed as follows:

[0049] S334. The objective function of the dual problem is to find the Lagrange function with respect to the worst-case scenario. Let the minimum value of the dual problem be:

[0050] S335, Solving for the Lagrange function with respect to... The maximum value, arranged into about The format is as follows:

[0051] S336, for each port ,like If the dual variable fails to offset the additional fuel cost, it means that the optimal values ​​of the primal and dual problems cannot be guaranteed to be equal; conversely, if the dual variable fails to offset the additional fuel cost, it guarantees that the worst-case scenario is taken into account, and that the optimal values ​​of the primal and dual problems are equal. When the Lagrange function reaches its maximum value, under the constraints set in step S332, the robust cost term is converted into its dual form as follows:

[0052] And satisfy:

[0053]

[0054] in, As the dual variable, it represents the minimum increase in total cost for each additional port that allows for higher fuel prices; As a dual variable, it reflects the marginal contribution of port fuel price fluctuations to total cost.

[0055] Further, step S4 includes: S41. Integrate the objective function of the single-layer mixed integer linear programming model with the constraints of the port and shipping cooperation agreement, the continuity of the shipping schedule, the upper and lower limits of the speed, the dynamic tangent approximation constraint, and the robust dual constraint to construct the single-layer mixed integer linear programming model to be solved. S42. Call the optimization solver to solve the single-level mixed integer linear programming model to be solved, and obtain the optimal decision variable values; S43. Based on the optimal decision variable values, analyze the number of ships, sailing time for each segment, arrival and departure times at each port, port loading and unloading times, time window selection scheme, start and end time selection scheme, and loading and unloading rate selection scheme, and output the liner schedule robust optimization scheme with the lowest cost.

[0056] This invention also provides a robust optimization system for liner schedules based on the aforementioned robust optimization method for liner schedules based on budget uncertainty sets and cooperation agreements. The system includes: a data acquisition module, a robust optimization model construction module for liner schedules, a model reconstruction module, and an optimization scheme generation module, wherein: The data acquisition module is used to acquire initial planning data and cost parameters for liner operations; The liner schedule robust optimization model construction module is used to construct a liner schedule robust optimization model with the objective of minimizing total cost under the framework of the port and shipping cooperation agreement. The liner schedule robust optimization model uses a budget uncertainty set to describe the situation of rising fuel prices. The model reconstruction module is used to design a model reconstruction method that integrates dynamic tangent approximation and robust equivalence transformation, and to reconstruct the liner schedule robust optimization model containing nonlinear relationships and a two-layer robust structure into a single-layer mixed integer linear programming model. The optimization scheme generation module is used to solve a single-layer mixed integer linear programming model using an optimization solver and output a robust optimization scheme for liner shipping schedules.

[0057] Compared with the prior art, the present invention has the following advantages: 1. The present invention provides a robust optimization method for liner schedules based on budget uncertainty set and cooperation agreement. Through collaborative cooperation with ports under multiple time windows, multiple departure and arrival times and multiple loading and unloading rates, it realizes flexible coordination between speed arrangement, port operation time and loading and unloading efficiency, thereby improving the flexibility and applicability of the schedule plan.

[0058] 2. The present invention provides a robust optimization method for liner schedules based on budget uncertainty sets and cooperation agreements. By using budget uncertainty sets to describe the situation of rising fuel prices and incorporating fuel price uncertainty into the liner schedule optimization model, it achieves the pre-emptive protection against the adverse effects of fuel price fluctuations during the planning stage, thereby reducing the fluctuation of total voyage costs and improving the stability of the schedule plan.

[0059] 3. The present invention provides a robust optimization method for liner schedules based on budget uncertainty set and cooperation agreement. By using dynamic tangent approximation method to handle the nonlinear relationship between fuel consumption and sailing time and dynamically adding tangent constraints in an error-driven manner, it achieves precise control of linearization error and avoids the dependence of fixed piecewise linearization method on the number and position of preset discrete points, thereby reducing unnecessary model size expansion.

[0060] 4. The present invention provides a robust optimization method for liner schedules based on budget uncertainty set and cooperation agreement. By using a robust equivalence transformation method to handle the two-layer robust optimization structure caused by budget uncertainty set and transforming the inner-layer maximization problem corresponding to fuel price uncertainty term into linear dual constraints, the original model is effectively reconstructed into a single-layer mixed integer linear programming model, thereby significantly improving the solvability of the model.

[0061] 5. The present invention provides a robust optimization method for liner schedules based on budget uncertainty set and cooperation agreement. By solving the reconstructed single-layer mixed integer linear programming model and outputting the optimization scheme, it realizes reliable schedule design decision support for liner companies under the condition of fuel price uncertainty.

[0062] In summary, by applying the technical solution of this invention, existing technologies, when facing the container liner schedule optimization problem under conditions of fuel price uncertainty, suffer from a lack of robustness in the schedule solutions due to insufficient joint consideration of port and shipping cooperation agreement constraints and fuel price fluctuations, and the nonlinear robust optimization model is difficult to solve. This invention effectively solves the above problems by introducing a budget uncertainty set to describe the fuel price increase scenario within the framework of port and shipping cooperation agreements, and by integrating dynamic tangent approximation and robust equivalence transformation methods to reconstruct the model. Therefore, the technical solution of this invention solves the problems of insufficient joint consideration of port and shipping cooperation agreement constraints and fuel price fluctuations, and the high difficulty in solving the nonlinear robust optimization model in existing technologies.

[0063] Based on the above reasons, this invention can be widely promoted in the fields of shipping management, container liner shipping scheduling, and port and shipping collaborative optimization. Attached Figure Description

[0064] To more clearly illustrate the technical solutions in the embodiments of the present invention or the prior art, the drawings used in the description of the embodiments or the prior art will be briefly introduced below. Obviously, the drawings described below are some embodiments of the present invention. For those skilled in the art, other drawings can be obtained based on these drawings without creative effort.

[0065] Figure 1 This is a flowchart of the method of the present invention.

[0066] Figure 2 This is a schematic diagram of the dynamic tangent approximation linearization process in this invention.

[0067] Figure 3 This is a comparison chart of the model solving performance for the present invention, used in numerical examples. Detailed Implementation

[0068] To enable those skilled in the art to better understand the present invention, the technical solutions of the present invention will be clearly and completely described below with reference to the accompanying drawings of the embodiments of the present invention. Obviously, the described embodiments are only some embodiments of the present invention, and not all embodiments. Based on the embodiments of the present invention, all other embodiments obtained by those skilled in the art without creative effort should fall within the scope of protection of the present invention.

[0069] It should be noted that the terms "first," "second," etc., in the specification, claims, and accompanying drawings of this invention are used to distinguish similar objects and are not necessarily used to describe a specific order or sequence. It should be understood that such data can be interchanged where appropriate so that the embodiments of the invention described herein can be implemented in orders other than those illustrated or described herein. Furthermore, the terms "comprising" and "having," and any variations thereof, are intended to cover a non-exclusive inclusion; for example, a process, method, system, product, or apparatus that comprises a series of steps or units is not necessarily limited to those steps or units explicitly listed, but may include other steps or units not explicitly listed or inherent to such processes, methods, products, or apparatus.

[0070] like Figure 1 As shown, this invention provides a robust optimization method for liner shipping schedules based on budget uncertainty sets and cooperation agreements, including: S1. Obtain initial planning data and cost parameters for liner operations; S2. Under the framework of the port and shipping cooperation agreement, construct a robust optimization model for liner schedules with the goal of minimizing total cost. The robust optimization model for liner schedules uses a budget uncertainty set to describe the situation of rising fuel prices. S3. Design a model reconstruction method that integrates dynamic tangent approximation and robust equivalence transformation to reconstruct the liner schedule robust optimization model containing nonlinear relationships and a two-layer robust structure into a single-layer mixed integer linear programming model. S4. Use an optimization solver to solve the single-level mixed integer linear programming model and output a robust optimization scheme for liner shipping schedules.

[0071] In a specific implementation, as a preferred embodiment of the present invention, step S1 includes: S11. Determine the port sets, segment sets, and segment distances of each segment on the liner route according to the order of port calls. S12. Determine the optional service data provided by each port under the framework of the port and shipping cooperation agreement, including the time window set, the start time set, the end time set and the loading and unloading rate set within each time window; S13. Determine the ship's operating parameters, including the upper and lower limits of the ship's speed and the number of ships that can be deployed on the route; S14. Determine fuel price parameters, including nominal fuel price, maximum increase of each port relative to the nominal fuel price, and budget parameters for the number of ports to limit fuel price increases. S15. Determine the operating cost parameters, including the weekly operating cost of ships, the unit loading and unloading cost of each port at each loading and unloading rate, the unit inventory cost of cargo, and the unit lateness penalty cost of each port. S16. Determine the cargo volume parameters, including the total number of containers loaded on each shipping segment and the total number of containers loaded and unloaded at each port.

[0072] In a specific implementation, as a preferred embodiment of the present invention, step S2 includes: S21. The objective function of the constructed liner schedule robust optimization model includes weekly operating costs, fuel costs under nominal fuel prices, port loading and unloading costs, cargo inventory costs, late arrival penalty costs, and robust costs caused by fuel price uncertainty. The objective function is as follows:

[0073] in, The weekly operating cost for each vessel; The number of vessels allocated to the shipping route; Nominal fuel price; This is a set of indexes of ports and segments on liner routes, numbered according to their call-to-call order. For the segment Fuel consumption; For the port The set of loading and unloading rates provided; For the port loading and unloading rate The unit loading and unloading cost; For the port The loading and unloading volume of containers; It is a 0-1 variable; when the container ship is in port Select the The value is 1 if the loading / unloading rate is specified, and 0 otherwise. Unit inventory cost of goods; For the segment The number of containers shipped; For the segment The sailing time; For the port The unit's cost of penalties for lateness; For the port The time of being late; Describe the port Whether the price of fuel is uncertain is a continuous variable within the interval [0,1]. For the port The objective function is relative to the maximum increase in nominal fuel prices; the first five terms are weekly operating costs, fuel costs under nominal fuel prices, port handling costs, cargo inventory costs, and lateness penalty costs, and the last term is the robust cost caused by fuel price uncertainty.

[0074] S22. Construct a budget uncertainty set to describe the scenario of rising fuel prices, assuming a port... The nominal fuel price is The maximum increase in fuel prices was When the port When fuel prices rise, the fuel price changes from... Become ; Introducing variables Indicates port The uncertainty of fuel price increases takes a value that satisfies:

[0075]

[0076] in, This is a budget parameter used to limit the number of ports affected by rising fuel prices. In this embodiment, the above formula means that, under the worst-case scenario, the number of ports affected by rising fuel prices will not exceed [a certain limit]. value; S23. Construct a port and shipping cooperation agreement framework that includes pre-given constraints on multiple time windows, multiple start and end times, and multiple loading and unloading rates. Specifically, it includes the first constraint, the second constraint, the third constraint, the fourth constraint, the fifth constraint, the sixth constraint, and the seventh constraint, as shown in the following formula:

[0077]

[0078]

[0079]

[0080]

[0081]

[0082]

[0083] Wherein, the first constraint indicates that the vessel can only select one time window at each port; the second, third, fourth, and fifth constraints indicate that the vessel can only select one start time and one end time at each port, and the start time and end time cannot exceed the time window range; the sixth constraint indicates that the vessel must select one loading / unloading rate at each port; the seventh constraint indicates that the loading / unloading time is obtained from the loading / unloading volume and loading / unloading rate; in the above formula, For the port The provided set of time windows; For the port In the time window The set of start times provided; For the port In the time window The provided set of end times; For the port The set of loading and unloading rates provided; For container ships in port Loading and unloading time; The variable is 0-1, and it is used when the container ship selects a time window. Arrival at the port The value is 1 if it is true, and 0 otherwise. The variable is 0-1, representing the time when a container ship chooses to dock at a port. Time window No. The value is 1 at the beginning of the time, otherwise it is 0. The variable is 0-1, representing the time when a container ship chooses to dock at a port. Time window No. The value is 1 if the event ends at a specific time, otherwise it is 0. The variable is 0-1, when the container ship is in port. Select the The value is 1 for loading and unloading rates, and 0 otherwise.

[0084] S24. Construct the continuity of ship schedules and operational constraints (continuity constraints of arrival and departure times, segment sailing time constraints, upper and lower speed limits, lateness calculation constraints, waiting time calculation constraints, total voyage time constraints, and vessel configuration quantity constraints), including the eighth, ninth, tenth, eleventh, twelfth, thirteenth, fourteenth, fifteenth, sixteenth, seventeenth, eighteenth, and nineteenth constraints, as shown in the following formula:

[0085]

[0086]

[0087]

[0088]

[0089]

[0090]

[0091]

[0092]

[0093]

[0094]

[0095]

[0096] Among them, the eighth, ninth, tenth, and eleventh constraints are used to determine whether the vessel is late or waiting; the twelfth, thirteenth, and fourteenth constraints represent the temporal continuity of the vessel's arrival and departure; the fifteenth and sixteenth constraints represent the total time of a round-trip voyage, with the voyage departure frequency being weekly; the seventeenth constraint represents the vessel's sailing time on a segment; the eighteenth constraint represents the upper and lower limits of speed; and the nineteenth constraint represents a 0-1 variable; in the above formula, This represents the total number of ports along the shipping route. For container ships to arrive at the port The moment; For container ships to leave port The moment; For container ships in port Waiting time; For container ships in the voyage The speed of the ship; These are the lower and upper limits of cruising speed; This represents the maximum number of vessels that can be deployed on the route. For the segment The flight distance; For the port In the time window The An ending moment; For the port In the time window The The beginning moment.

[0097] In a specific implementation, as a preferred embodiment of the present invention, step S3 includes: S31. Establish the nonlinear relationship between fuel consumption and flight time, and obtain the fuel consumption function as follows:

[0098] in, and All are marine fuel consumption coefficients; For the segment The flight distance; For the segment Fuel consumption; For the segment The sailing time; S32. The nonlinear relationship is transformed into a linear constraint by using the dynamic tangent approximation method. S33. Use the robust equivalence transformation method to transform the bi-layer robust structure into a linear dual constraint. S34. Integrating the linear tangent constraints and the linear dual constraints, the robust optimization model for liner schedules is reconstructed into a single-level mixed-integer linear programming model. The objective function of the single-level mixed-integer linear programming model is as follows:

[0099] in, As dual variables, As dual variables, For budget parameters.

[0100] In specific implementation, as a preferred embodiment of the present invention, such as Figure 2 As shown, step S32 includes: S321. Set the initial set of tangent points as follows:

[0101] in, and Each segment Lower and upper limits of sailing time; S322, Based on the current set of cut points At the tangent point Constructor First-order tangent approximation constraint:

[0102] in, For the segment No. The point of tangency that the tangent line approaches; For the segment The current set of pointcuts; S323, in the In the next iteration, based on the current set of cut points Construct a linear master problem and solve it to obtain the current solution. and

[0103] S324. Calculate the relative error using the following formula:

[0104] in, ; S325. Determine whether the relative error satisfies the convergence condition:

[0105] If the convergence condition is met, the current tangent approximation is considered to have reached the preset accuracy requirement, and the iteration stops; otherwise, the current solution is... Add to the corresponding flight segment The set of tangent points is defined, and a new tangent approximation constraint is added at that point:

[0106] S326. Repeat steps S323 to S325 until the convergence condition is met or the maximum number of iterations is reached. .

[0107] In a specific implementation, as a preferred embodiment of the present invention, step S33 includes: S331. Calculate the robust cost term in a two-layer robust structure using the following formula:

[0108] S332. Set the constraints for the robust cost item as follows:

[0109]

[0110] S333. According to the strong duality theorem, the extreme values ​​of the Lagrange function are equal to the extreme values ​​of the maximum problem. Therefore, the Lagrange multiplier method is used to extract the dual problem of the maximum problem. Dual variables are introduced to constrain the problem. , and The Lagrange function is constructed as follows:

[0111] S334. The objective function of the dual problem is to ensure the worst-case scenario (the original function with respect to...) Find the Lagrange function with respect to the maximum value. Let the minimum value of the dual problem be:

[0112] S335, Solving for the Lagrange function with respect to... The maximum value, arranged into about The format is as follows:

[0113] S336, for each port ,like If the dual variable fails to offset the additional fuel cost, it means that the optimal values ​​of the primal and dual problems cannot be guaranteed to be equal; conversely, if the dual variable fails to offset the additional fuel cost, it guarantees that the worst-case scenario is taken into account, and that the optimal values ​​of the primal and dual problems are equal. When the Lagrange function reaches its maximum value, under the constraints set in step S332, the robust cost term is converted into its dual form as follows:

[0114] And satisfy:

[0115]

[0116] in, As the dual variable, it represents the minimum increase in total cost for each additional port that allows for higher fuel prices; As a dual variable, it reflects the marginal contribution of port fuel price fluctuations to total cost.

[0117] In a specific implementation, as a preferred embodiment of the present invention, step S4 includes: S41. Integrate the objective function of the single-layer mixed integer linear programming model with the constraints of the port and shipping cooperation agreement, the continuity of the shipping schedule, the upper and lower limits of the speed, the dynamic tangent approximation constraint, and the robust dual constraint to construct the single-layer mixed integer linear programming model to be solved. S42. Call the optimization solver to solve the single-level mixed integer linear programming model to be solved, and obtain the optimal decision variable values; S43. Based on the optimal decision variable values, analyze the number of ships, sailing time for each segment, arrival and departure times at each port, port loading and unloading times, time window selection scheme, start and end time selection scheme, and loading and unloading rate selection scheme, and output the liner schedule robust optimization scheme with the lowest cost.

[0118] This invention also provides a robust liner schedule optimization system based on the above-mentioned robust optimization method for liner schedules based on budget uncertainty sets and cooperation agreements, comprising: a data acquisition module, a liner schedule robust optimization model construction module, a model reconstruction module, and an optimization scheme generation module, wherein: The data acquisition module is used to acquire initial planning data and cost parameters for liner operations; The liner schedule robust optimization model construction module is used to construct a liner schedule robust optimization model with the objective of minimizing total cost under the framework of the port and shipping cooperation agreement. The liner schedule robust optimization model uses a budget uncertainty set to describe the situation of rising fuel prices. The model reconstruction module is used to design a model reconstruction method that integrates dynamic tangent approximation and robust equivalence transformation, and to reconstruct the liner schedule robust optimization model containing nonlinear relationships and a two-layer robust structure into a single-layer mixed integer linear programming model. The optimization scheme generation module is used to solve a single-layer mixed integer linear programming model using an optimization solver and output a robust optimization scheme for liner shipping schedules.

[0119] The embodiments of the present invention are described simply because they correspond to those in the embodiments above. For any similarities, please refer to the descriptions in the embodiments above, which will not be elaborated here.

[0120] In particular, in some preferred embodiments of the present invention, a computer device is also provided, the computer device including a memory, a processor, and a computer program stored in the memory and executable on the processor. When the processor executes the computer program, it implements the steps of a robust optimization method for liner schedules based on budget uncertainty sets and cooperation agreements as described in any of the above embodiments.

[0121] In some other preferred embodiments of the present invention, a computer-readable storage medium is also provided, on which a computer program or computer instructions are stored. When the computer program or computer instructions are executed by a processor, they implement the steps of the robust optimization method for liner schedules based on budget uncertainty sets and cooperation agreements described in any of the above embodiments.

[0122] Those skilled in the art will understand that all or part of the processes in the above embodiments can be implemented by computer program instructions to related hardware. The computer program can be stored in a non-volatile computer-readable storage medium. When the computer program is executed by a processor, it can execute the relevant processes in the above embodiments of the robust optimization method for liner schedules based on budget uncertainty sets and cooperation agreements, which will not be repeated here.

[0123] Computer-readable storage media encompass a variety of types, including persistent and non-persistent, portable and fixed. These media store information using different technologies, and the content can be machine instructions, data structures, program modules, or other types of data. Some typical examples of computer storage media include: phase-change memory (PRAM), static random access memory (SRAM), dynamic random access memory (DRAM), various types of random access memory (RAM), read-only memory (ROM), electrically erasable programmable read-only memory (EEPROM), flash memory and other storage technologies, and optical discs (CDs). Optical storage media such as ROM and digital video optical discs (DVDs), magnetic storage devices such as magnetic tapes and disks, and other non-transferable media are used to store information accessible to computing devices. It should be noted that the computer-readable media described in this invention do not include temporary storage media, such as modulated data signals and carrier waves.

[0124] Those skilled in the art will further recognize that the operation of the module can be achieved using existing technical protocols or programs, without relying on new computer programs themselves. The units and algorithm steps of the various examples described in conjunction with the embodiments disclosed herein can be implemented in electronic hardware, computer software, or a combination of both. To clearly illustrate the interchangeability of hardware and software, the components and steps of the various examples have been generally described in terms of functionality in the foregoing description. Whether these functions are implemented in hardware or software depends on the specific application and design constraints of the technical solution. Those skilled in the art can use different methods to implement the described functions for each specific application, but such implementations should not be considered beyond the scope of this invention.

[0125] The steps of the methods or algorithms described in conjunction with the embodiments disclosed herein can be implemented in hardware, a software module executed by a processor, or a combination of both. The software module can be located in random access memory (RAM), main memory, read-only memory (ROM), electrically programmable ROM, electrically erasable programmable ROM, registers, hard disk, removable disk, CD-ROM, etc. In ROM, or any other form of storage medium known in the art.

[0126] Example To further illustrate the specific implementation process of this invention, a specific shipping route published by a liner company is used as a test example. In this example, the liner calls at 10 ports, and the vessel's container capacity is 10,036 TEU. Assume a unit fuel price of $300 / ton, a maximum fuel price increase of $450 / ton, a budget parameter of 5, and that each port offers 4 loading / unloading rates and 4 time windows, with each time window containing 3 selectable origin and destination times. After running the method described in this invention, the comparison results are as follows: Figure 3 As shown. Under the scenario of fluctuating fuel prices, the actual total cost of the deterministic cooperation agreement model is 17736.07 × The total cost volatility is 14.38% in US dollars; while the actual total cost of the model in this invention is 17081.22 × Compared to the deterministic model, the cost is reduced by approximately 3.69% in US dollars, and because the worst-case scenario was considered during the planning phase, the total cost remained stable. This demonstrates that the present invention can effectively reduce the total cost of liner services under conditions of fuel price uncertainty, mitigate the impact of fuel price fluctuations on shipping schedules, and improve the stability and economy of shipping schedules, proving the effectiveness of the method.

[0127] Finally, it should be noted that the above embodiments are only used to illustrate the technical solutions of the present invention, and not to limit them; although the present invention has been described in detail with reference to the foregoing embodiments, those skilled in the art should understand that modifications can still be made to the technical solutions described in the foregoing embodiments, or equivalent substitutions can be made to some or all of the technical features; and these modifications or substitutions do not cause the essence of the corresponding technical solutions to deviate from the scope of the technical solutions of the embodiments of the present invention.

Claims

1. A robust optimization method for liner shipping schedules based on budget uncertainty sets and cooperation agreements, characterized in that, include: S1. Obtain initial planning data and cost parameters for liner operations; S2. Under the framework of the port and shipping cooperation agreement, construct a robust optimization model for liner schedules with the goal of minimizing total cost. The robust optimization model for liner schedules uses a budget uncertainty set to describe the situation of rising fuel prices. S3. Design a model reconstruction method that integrates dynamic tangent approximation and robust equivalence transformation to reconstruct the liner schedule robust optimization model containing nonlinear relationships and a two-layer robust structure into a single-layer mixed integer linear programming model. S4. Use an optimization solver to solve the single-level mixed integer linear programming model and output a robust optimization scheme for liner shipping schedules.

2. The robust optimization method for liner shipping schedules based on budget uncertainty sets and cooperation agreements as described in claim 1, characterized in that, Step S1 includes: S11. Determine the port sets, segment sets, and segment distances of each segment on the liner route according to the order of port calls. S12. Determine the optional service data provided by each port under the framework of the port and shipping cooperation agreement, including the time window set, the start time set, the end time set and the loading and unloading rate set within each time window; S13. Determine the ship's operating parameters, including the upper and lower limits of the ship's speed and the number of ships that can be deployed on the route; S14. Determine fuel price parameters, including nominal fuel price, maximum increase of each port relative to the nominal fuel price, and budget parameters for the number of ports to limit fuel price increases. S15. Determine the operating cost parameters, including the weekly operating cost of ships, the unit loading and unloading cost of each port at each loading and unloading rate, the unit inventory cost of cargo, and the unit lateness penalty cost of each port. S16. Determine the cargo volume parameters, including the total number of containers loaded on each shipping segment and the total number of containers loaded and unloaded at each port.

3. The robust optimization method for liner shipping schedules based on budget uncertainty sets and cooperation agreements as described in claim 1, characterized in that, Step S2 includes: S21. The objective function of the constructed liner schedule robust optimization model includes weekly operating costs, fuel costs under nominal fuel prices, port loading and unloading costs, cargo inventory costs, late arrival penalty costs, and robust costs caused by fuel price uncertainty. The objective function is as follows: in, The weekly operating cost for each vessel; The number of vessels allocated to the shipping route; Nominal fuel price; This is a set of indexes of ports and segments on liner routes, numbered according to their call-to-call order. For the segment Fuel consumption; For the port The set of loading and unloading rates provided; For the port loading and unloading rate The unit loading and unloading cost; For the port The loading and unloading volume of containers; It is a 0-1 variable; when the container ship is in port Select the The value is 1 if the loading / unloading rate is specified, and 0 otherwise. Unit inventory cost of goods; For the segment The number of containers shipped; For the segment The sailing time; For the port The unit's cost of penalties for lateness; For the port The time of being late; Describe the port Whether the price of fuel is uncertain is a continuous variable within the interval [0,1]. For the port The maximum increase relative to nominal fuel prices; S22. Construct a budget uncertainty set to describe the scenario of rising fuel prices, assuming a port... The nominal fuel price is The maximum increase in fuel prices was When the port When fuel prices rise, the fuel price changes from... Become ; Introducing variables Indicates port The uncertainty of fuel price increases takes a value that satisfies: in, This is a budget parameter used to limit the number of ports that can be used to control fuel price increases. S23. Construct a port and shipping cooperation agreement framework that includes pre-given constraints on multiple time windows, multiple start and end times, and multiple loading and unloading rates. Specifically, it includes the first constraint, the second constraint, the third constraint, the fourth constraint, the fifth constraint, the sixth constraint, and the seventh constraint, as shown in the following formula: Wherein, the first constraint indicates that the vessel can only select one time window at each port; the second, third, fourth, and fifth constraints indicate that the vessel can only select one start time and one end time at each port, and the start time and end time cannot exceed the time window range; the sixth constraint indicates that the vessel must select one loading / unloading rate at each port; the seventh constraint indicates that the loading / unloading time is obtained from the loading / unloading volume and loading / unloading rate; in the above formula, For the port The provided set of time windows; For the port In the time window The set of start times provided; For the port In the time window The provided set of end times; For the port The set of loading and unloading rates provided; For container ships in port Loading and unloading time; The variable is 0-1, and it is used when the container ship selects a time window. Arrival at the port The value is 1 if it is true, and 0 otherwise. The variable is 0-1, representing the time when a container ship chooses to dock at a port. Time window No. The value is 1 at the beginning of the time, otherwise it is 0. The variable is 0-1, representing the time when a container ship chooses to dock at a port. Time window No. The value is 1 if the event ends at a specific time, otherwise it is 0. The variable is 0-1, when the container ship is in port. Select the The value is 1 for loading and unloading rates, and 0 otherwise. S24. Construct the continuity of shipping schedules and operational constraints, including the eighth, ninth, tenth, eleventh, twelfth, thirteenth, fourteenth, fifteenth, sixteenth, seventeenth, eighteenth, and nineteenth constraints, as shown in the following formula: Among them, the eighth, ninth, tenth, and eleventh constraints are used to determine whether the vessel is late or waiting; the twelfth, thirteenth, and fourteenth constraints represent the temporal continuity of the vessel's arrival and departure; the fifteenth and sixteenth constraints represent the total time of a round-trip voyage, with the voyage departure frequency being weekly; the seventeenth constraint represents the vessel's sailing time on a segment; the eighteenth constraint represents the upper and lower limits of speed; and the nineteenth constraint represents a 0-1 variable; in the above formula, This represents the total number of ports along the shipping route. For container ships to arrive at the port The moment; For container ships to leave port The moment; For container ships in port Waiting time; For container ships in the voyage The speed of the ship; These are the lower and upper limits of cruising speed; This represents the maximum number of vessels that can be deployed on the route. For the segment The flight distance; For the port In the time window The An ending moment; For the port In the time window The The beginning moment.

4. The robust optimization method for liner shipping schedules based on budget uncertainty sets and cooperation agreements as described in claim 1, characterized in that, Step S3 includes: S31. Establish the nonlinear relationship between fuel consumption and flight time, and obtain the fuel consumption function as follows: in, and All are marine fuel consumption coefficients; For the segment The flight distance; For the segment Fuel consumption; For the segment The sailing time; S32. The nonlinear relationship is transformed into a linear constraint by using the dynamic tangent approximation method. S33. Use the robust equivalence transformation method to transform the bi-layer robust structure into a linear dual constraint. S34. Integrating the linear tangent constraints and the linear dual constraints, the robust optimization model for liner schedules is reconstructed into a single-level mixed-integer linear programming model. The objective function of the single-level mixed-integer linear programming model is as follows: in, As dual variables, As dual variables, For budget parameters.

5. The robust optimization method for liner shipping schedules based on budget uncertainty sets and cooperation agreements as described in claim 4, characterized in that, Step S32 includes: S321. Set the initial set of tangent points as follows: in, and Each segment Lower and upper limits of sailing time; S322, Based on the current set of cut points At the tangent point Constructor First-order tangent approximation constraint: in, For the segment No. The point of tangency that the tangent line approaches; For the segment The current set of pointcuts; S323, in the In the next iteration, based on the current set of cut points Construct a linear master problem and solve it to obtain the current solution. and S324. Calculate the relative error using the following formula: in, ; S325. Determine whether the relative error satisfies the convergence condition: If the convergence condition is met, the current tangent approximation is considered to have reached the preset accuracy requirement, and the iteration stops; otherwise, the current solution is... Add to the corresponding flight segment The set of tangent points is defined, and a new tangent approximation constraint is added at that point: S326. Repeat steps S323 to S325 until the convergence condition is met or the maximum number of iterations is reached. .

6. The robust optimization method for liner shipping schedules based on budget uncertainty sets and cooperation agreements as described in claim 4, characterized in that, Step S33 includes: S331. Calculate the robust cost term in a two-layer robust structure using the following formula: S332. Set the constraints for the robust cost item as follows: S333, Introducing dual variables for constraints , and The Lagrange function is constructed as follows: S334. The objective function of the dual problem is to find the Lagrange function with respect to the worst-case scenario. Let the minimum value of the dual problem be: S335, Solving for the Lagrange function with respect to... The maximum value, arranged into about The format is as follows: S336, for each port ,like If the dual variable fails to offset the additional fuel cost, it means that the optimal values ​​of the primal and dual problems cannot be guaranteed to be equal; conversely, if the dual variable fails to offset the additional fuel cost, it guarantees that the worst-case scenario is taken into account, and that the optimal values ​​of the primal and dual problems are equal. When the Lagrange function reaches its maximum value, under the constraints set in step S332, the robust cost term is converted into its dual form as follows: And satisfy: in, As the dual variable, it represents the minimum increase in total cost for each additional port that allows for higher fuel prices; As a dual variable, it reflects the marginal contribution of port fuel price fluctuations to total cost.

7. The robust optimization method for liner shipping schedules based on budget uncertainty sets and cooperation agreements as described in claim 1, characterized in that, Step S4 includes: S41. Integrate the objective function of the single-layer mixed integer linear programming model with the constraints of the port and shipping cooperation agreement, the continuity of the shipping schedule, the upper and lower limits of the speed, the dynamic tangent approximation constraint, and the robust dual constraint to construct the single-layer mixed integer linear programming model to be solved. S42. Call the optimization solver to solve the single-level mixed integer linear programming model to be solved, and obtain the optimal decision variable values; S43. Based on the optimal decision variable values, analyze the number of ships, sailing time for each segment, arrival and departure times at each port, port loading and unloading times, time window selection scheme, start and end time selection scheme, and loading and unloading rate selection scheme, and output the liner schedule robust optimization scheme with the lowest cost.

8. A robust optimization system for liner schedules based on budget uncertainty sets and cooperation agreements, implemented using the robust optimization method for liner schedules based on budget uncertainty sets and cooperation agreements as described in any one of claims 1-7, characterized in that, include: The module includes a data acquisition module, a liner schedule robust optimization model construction module, a model reconstruction module, and an optimization scheme generation module, among which: The data acquisition module is used to acquire initial planning data and cost parameters for liner operations; The liner schedule robust optimization model construction module is used to construct a liner schedule robust optimization model with the objective of minimizing total cost under the framework of the port and shipping cooperation agreement. The liner schedule robust optimization model uses a budget uncertainty set to describe the situation of rising fuel prices. The model reconstruction module is used to design a model reconstruction method that integrates dynamic tangent approximation and robust equivalence transformation, and to reconstruct the liner schedule robust optimization model containing nonlinear relationships and a two-layer robust structure into a single-layer mixed integer linear programming model. The optimization scheme generation module is used to solve a single-layer mixed integer linear programming model using an optimization solver and output a robust optimization scheme for liner shipping schedules.

9. A computer-readable storage medium having a computer program / instructions stored thereon, characterized in that: When the computer program / instructions are executed by the processor, they implement the steps of the method according to any one of claims 1 to 7.

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