A scheduling method for tramp shipping considering uncertain scenarios

Abstract

The application discloses a scheduling method for irregular ship transportation considering uncertain scenarios, and relates to shipping scheduling and operation optimization. The method comprises the following steps: a scheduling optimization model is constructed, including a first stage model and a second stage model; wherein the first stage model is used to generate a benchmark scheduling scheme according to a preset optimization criterion before the uncertain scenario is realized; the second stage model is used to select and generate an emergency scheduling scheme from a plurality of preset recovery strategies by taking the benchmark scheduling scheme as input after the uncertain scenario is realized; a decomposition method based on a cut plane is used to solve the scheduling optimization model, and the scheduling optimization model is divided into a main problem and a plurality of scenario sub-problems corresponding to each uncertain scenario; the main problem and the plurality of scenario sub-problems are iteratively fed back with cut plane information until a convergence condition is met, and an optimized benchmark scheduling scheme and a corresponding recovery strategy are output.

Description

Technical Field

[0001] This invention relates to a scheduling method for irregular shipping that takes into account uncertain scenarios, belonging to the field of shipping scheduling and operations optimization technology. Background Technology

[0002] Turbo-freight shipping is the main form of organization for international dry bulk and liquid bulk cargo transportation. Its operating model is similar to that of a sea taxi, where ships flexibly arrange routes according to the needs of cargo owners, rather than operating on a fixed schedule. This flexibility allows turbo-freight ships to adapt to changing freight demands, but it also makes them highly exposed to uncertainties such as port congestion, severe weather, traffic control, and port equipment failures.

[0003] Currently, traditional scheduling methods in the industry mainly employ deterministic optimization models, treating sailing time, port operation time, and waterway traffic conditions as known fixed parameters to generate the most cost-effective scheduling scheme under static conditions. However, once unexpected disturbances such as port congestion, severe weather, or waterway interruptions occur in actual operations, the original scheduling scheme often results in problems such as arrival delays, unbalanced loading and unloading, and order delays, which can further trigger a chain reaction of delays, leading to high default penalties, port surcharges, and rental losses.

[0004] Regarding recovery strategies, existing research typically considers only a single recovery method, such as speed adjustment, waiting strategies, or local route modification, making it difficult to simultaneously characterize port inefficiency, route disruptions, order fulfillment risks, and the coupling relationships between multiple recovery strategies. Furthermore, when a large number of discrete decision variables, such as the activation of backup ports, route reconfiguration, and the selection of expedited processing modes, are introduced into the recovery model, the traditional Benders decomposition method based on continuous dual information becomes difficult to apply directly, easily leading to weak cutting planes, slow convergence, or even solutions lacking physical feasibility.

[0005] Therefore, existing technologies have significant shortcomings in the following aspects:

[0006] (1) It is impossible to internalize multiple potential disturbance risks in the initial planning stage;

[0007] (2) There is a lack of a unified scheduling model that simultaneously covers multiple flexible recovery mechanisms such as speed adjustment, port rush work, activation of backup ports, route reconstruction and land-based cargo transshipment. Summary of the Invention

[0008] The purpose of this invention is to provide a scheduling method for irregular ship transportation considering uncertain scenarios. By constructing a two-stage scheduling optimization model and introducing five recovery strategies, and by using a cutting plane to achieve convergence, the two-stage scheduling and recovery optimization method can achieve the effect of having the lowest expected cost and being both economical and robust under various uncertain scenarios.

[0009] To achieve the above objectives, the present invention is implemented using the following technical solution.

[0010] On one hand, the present invention provides a scheduling method for irregular ship transportation considering uncertain scenarios, comprising:

[0011] A pre-built scheduling optimization model is invoked, which includes a first-stage model and a second-stage model. The first-stage model is used to generate a baseline scheduling scheme before the uncertain scenario is realized. The second-stage model is used to select and combine various preset recovery strategies from the baseline scheduling scheme as input after the uncertain scenario is realized to generate an emergency scheduling scheme. The recovery strategies include speed adjustment, port expediting, activation of backup ports, route reconstruction, and land-based cargo transshipment.

[0012] The scheduling optimization model is solved using a cutting plane-based decomposition method, which divides the scheduling optimization model into a main problem and multiple scenario sub-problems corresponding to each uncertainty scenario. The solution is obtained by iteratively feeding back cutting plane information between the main problem and the multiple scenario sub-problems until the convergence condition is met, and the baseline scheduling scheme and the corresponding emergency scheduling scheme under each uncertainty scenario are output.

[0013] Furthermore, the first-stage model generates the baseline scheduling scheme with the objective of minimizing the overall expected cost, and satisfies constraints on ship activation logic, network flow balancing, loading and unloading operations, port time windows, and order delivery time; wherein, the overall expected cost includes fixed activation cost, fuel cost, rental cost, and baseline delay penalty.

[0014] Furthermore, the second-stage model aims to minimize the actual expected recovery cost under each of the aforementioned uncertain scenarios, and generates the corresponding emergency dispatch plan for each of the aforementioned uncertain scenarios; the actual expected recovery cost includes incremental fuel cost, incremental rental cost, expedited work surcharge, backup port activation cost, multimodal transport cost, and actual delay penalty.

[0015] Furthermore, the decomposition method based on the cutting plane is specifically a master-slave decomposition method based on the combined cutting plane; the master problem is used to generate the baseline scheduling scheme and the lower bound of the true expected recovery cost, the lower bound being the lowest estimate of the true expected recovery cost in the second stage in the master problem; the scenario sub-problem is used to solve for the variables related to the recovery strategy for each uncertainty scenario, given the baseline scheduling scheme, and to calculate the true recovery cost under each uncertainty scenario based on the solution results of the variables.

[0016] Furthermore, under the baseline scheduling scheme, if there exists any scenario subproblem corresponding to an uncertain scenario that has no feasible solution, then the combination of values ​​of the variables related to the recovery strategy corresponding to the baseline scheduling scheme is the currently infeasible discrete decision combination.

[0017] Furthermore, the cutting plane information fed back between the main problem and the scenario subproblems includes combinatorial optimality cut and combinatorial feasibility cut; the combinatorial optimality cut is used to correct the lower bound of the true expected recovery cost to avoid the main problem underestimating the true expected recovery cost; the combinatorial feasibility cut is used to exclude currently infeasible discrete decision combinations.

[0018] Furthermore, when all the subproblems of the scenario are feasible, and the lower bound of the true expected recovery cost is less than the true expected recovery cost, the combined optimality cut is generated and fed back; when a certain subproblem of the scenario is infeasible, the combined feasibility cut is generated and fed back; wherein, the true expected recovery cost is the weighted average of the true recovery costs of all the subproblems of the scenario.

[0019] Furthermore, one or more of the following physical feasibility enhancement constraints are introduced into the sub-problem of the scenario:

[0020] Path reconfiguration is only permitted if the baseline path contains blocked segments;

[0021] In the second phase, cargo inventory balance constraints are established independently to ensure that the cargo load of the ship does not exceed its capacity and follows the order of loading before unloading.

[0022] The expedited or standby port measures may only be activated when the port is undertaking actual loading and unloading tasks and is affected by reduced efficiency.

[0023] Furthermore, the method also includes introducing a minimum infeasible subsystem (MIS) identification mechanism in the scenario subproblem, extracting a set of core conflict variables that cause infeasibility from the infeasible scenario subproblem, and generating an enhanced feasibility cut based on the set of core conflict variables to exclude invalid solution spaces containing the conflict structure.

[0024] Furthermore, the best baseline scheduling scheme obtained during the iteration process is taken as the optimal scheduling scheme, and the corresponding total cost is recorded as the optimal value. When the relative difference between the upper and lower bounds of the optimal value is less than a preset tolerance threshold, or when the number of iterations reaches the preset maximum number of iterations, the convergence condition is met. The lower bound is provided by the continuous variables in the main problem, and the upper bound is determined by the total cost of the baseline scheduling scheme that has been found so that all scenario subproblems have feasible solutions.

[0025] Compared with the prior art, the beneficial effects achieved by the present invention are as follows:

[0026] This invention constructs a scheduling optimization model comprising a first-stage model and a second-stage model. The first-stage model generates a baseline scheduling scheme before uncertain scenarios are realized. It does not rely solely on a specific uncertain scenario to generate a specific baseline scheduling scheme, but rather uses preset optimization criteria as the objective, ensuring that the generated baseline scheduling scheme has good robustness. The second-stage model, based on the baseline scheduling scheme, incorporates five preset recovery strategies—speed adjustment, port expediting, activation of backup ports, route reconstruction, and land-based cargo transshipment—as constraints. This allows the system to find economically feasible solutions when facing various disturbances ranging from minor delays to complete failure, thereby avoiding systemic paralysis caused by the failure of a single recovery method. Attached Figure Description

[0027] Figure 1 The diagram shown is a flowchart illustrating a scheduling method for irregular ship transportation considering uncertain scenarios provided in this embodiment.

[0028] Figure 2 The diagram shown is a path diagram of the baseline scheduling scheme of a scheduling method for irregular ship transportation considering uncertain scenarios provided in this embodiment.

[0029] Figure 3 The figure shown is a schematic diagram of the ship's spatiotemporal trajectory time axis of the baseline scheduling scheme of an irregular ship transportation scheduling method considering uncertain scenarios provided in this embodiment.

[0030] Figure 4 The diagram shown is a schematic of the recovery scheduling scheme after being affected by scenario S1 in a scheduling method for irregular ship transportation considering uncertain scenarios provided in this embodiment.

[0031] Figure 5 The diagram shown is a time axis representation of the ship's spatiotemporal trajectory after being affected by scenario S1, representing a recovery scheduling scheme for an irregular ship transportation scheduling method considering uncertain scenarios provided in this embodiment.

[0032] Figure 6 The diagram shows a recovery scheduling scheme after being affected by scenario S2, which is a scheduling method for irregular ship transportation considering uncertain scenarios provided in this embodiment.

[0033] Figure 7 The diagram shown is a schematic representation of ship scheduling in an unscheduled shipping scenario with an oil price of 5, considering uncertain scenarios provided in this embodiment.

[0034] Figure 8 The diagram shown is a schematic representation of ship scheduling in an unscheduled shipping scenario with an oil price of 50, considering uncertain situations provided in this embodiment.

[0035] Figure 9 The diagram shown is a schematic representation of ship scheduling in an unscheduled shipping scenario with an oil price of 200, considering uncertain situations provided in this embodiment.

[0036] Figure 10 The diagram shown is a schematic diagram illustrating the changes in fleet size and average speed across the entire network under different delay penalty coefficients for irregular shipping in uncertain scenarios, as provided in this embodiment.

[0037] Figure 11 The diagram shown illustrates the change in the actual expected recovery cost in the second stage under different delay penalty coefficients for irregular shipping in uncertain scenarios, as provided in this embodiment.

[0038] Figure 12 The diagram shown is a schematic diagram of the ship scheduling situation when the delay penalty coefficient is low in an irregular ship transportation considering uncertain scenarios provided in this embodiment;

[0039] Figure 13 The diagram shown is a schematic representation of ship scheduling in an irregular shipping scenario considering uncertain situations, with the delay penalty coefficient as the baseline, according to this embodiment.

[0040] Figure 14 The diagram shown is a schematic representation of ship scheduling in an irregular shipping scenario considering uncertain conditions, where the delay penalty coefficient is extremely high, according to this embodiment. Detailed Implementation

[0041] The technical solution of the present invention will be described in detail below with reference to the accompanying drawings and specific embodiments. It should be understood that the embodiments of the present invention and the specific features in the embodiments are detailed descriptions of the technical solution of the present invention, rather than limitations thereof. In the absence of conflict, the embodiments of the present invention and the technical features in the embodiments can be combined with each other.

[0042] The term "and / or" simply describes the relationship between related objects, indicating that three relationships can exist. For example, A and / or B can represent: A alone, A and B simultaneously, or B alone. Additionally, the character " / " generally indicates that the preceding and following related objects have an "or" relationship.

[0043] Example 1

[0044] This embodiment provides a scheduling method for irregular ship transportation considering uncertain scenarios, the method including:

[0045] The pre-built scheduling optimization model is invoked. The scheduling optimization model includes a first-stage model and a second-stage model. The first-stage model is used to generate a baseline scheduling scheme according to preset optimization criteria before uncertain scenarios are realized.

[0046] Specifically, uncertain scenarios refer to various random disturbances or environmental changes that may occur in the future. Therefore, the first-stage model is mainly used to generate a baseline scheduling scheme in an environment without specific disturbance information. The baseline scheduling scheme includes ship activation decisions, baseline route planning, speed selection, and order allocation schemes. In the first stage, the optimization criterion preset in this embodiment aims to minimize the overall expected cost and satisfies constraints such as ship activation logic, network flow balancing, loading and unloading operations, port time windows, and order delivery times. The overall expected cost includes fixed activation costs. Fuel costs Rental costs and benchmark delay penalty Specifically, as shown in the following expression:

[0047] (1),

[0048] In the formula, the fixed start-up cost K represents the ship collection; This represents the fixed commissioning cost of the k-th ship. ; Let be a 0-1 decision variable, representing whether the k-th ship is activated. This indicates that the k-th ship is activated, when This indicates that the k-th ship has not been activated.

[0049] fuel costs (i,j) represents a voyage segment, where i represents the index of the starting port of the voyage segment in the physical port set N, j represents the index of the ending port of the voyage segment in the physical port set N, and A represents the set of feasible voyage segments. Let represent the discrete set of speeds of the k-th ship; This represents the fuel cost of the k-th ship at the v-th discrete speed; This represents the distance of flight segment (i,j); Let v be a 0-1 decision variable, representing whether the k-th ship travels at the v-th discrete speed on segment (i,j). This indicates that the k-th ship is traveling at the v-th discrete speed on segment (i,j), when... This indicates that the k-th ship does not sail at the v-th discrete speed in segment (i,j) or is not sailing in segment (i,j).

[0050] Rental costs , This represents the daily rental cost of the k-th vessel. This represents the total time occupied by the k-th ship. , This represents the reference time for the k-th ship to arrive at its destination port, End. This represents the reference time when the k-th ship departs from the starting port, Start.

[0051] Baseline Delay Penalty R represents the set of freight orders; This represents the order delay level parameter, where p is the index; This represents the unit penalty for delays at level p; This represents the delay duration of the r-th freight order in the Z-level delay interval.

[0052] As a specific implementation method, this embodiment makes the following assumptions for ease of modeling:

[0053] 1. In the first phase, sailing time, port operation efficiency, and waterway passage status are all considered as fixed parameters:

[0054] Using fixed parameters can avoid the randomness caused by the introduction of random parameters in the first stage;

[0055] 2. The first phase does not consider recovery measures such as backup ports, route reconfiguration, and port expediting. These measures will only be activated after the second phase scenario is implemented:

[0056] The first phase mainly targets potential future disturbances, while recovery measures such as backup ports, route reconfiguration, and port expediting are mainly recovery measures implemented for disturbances that have already occurred.

[0057] 3. Each order must be transported independently by one vessel; splitting the shipment is not permitted.

[0058] The baseline scheduling scheme includes an order allocation scheme. If orders are split for delivery, additional complex constraints such as order splitting and cargo transfer will be introduced, causing the complexity of the first-stage model to increase dramatically. By holisticizing the orders, the problem can be simplified and the model complexity reduced.

[0059] 4. Vessels can flexibly choose the order of ports of call based on the origin and destination requirements of the order, and are allowed to visit the same physical port multiple times on a single route:

[0060] The baseline scheduling scheme includes baseline route planning, which allows for repeated visits to the same ports, enabling more efficient use of capacity and generating the most cost-effective routes.

[0061] 5. The ship's speed is selected from a discrete set of options, and the choice of speed affects both sailing time and fuel cost.

[0062] The constraints specifically include:

[0063] 1. Ship activation logic constraints

[0064] The specific logic for activating the ship is shown in the following expression:

[0065] (2),

[0066] In the formula, This represents the set of ports that includes a virtual start point (Start) and a virtual end point (End).

[0067] When the k-th ship is activated, then at this time Currently activated vessels must depart from a virtual starting point and can only have one departure edge; Currently activated vessels must reach the virtual endpoint, and can only have one reach edge.

[0068] If the kth ship is not activated, then at this time Currently, vessels that are not in use have a flow of 0 on all segments, meaning that the vessels do not have any activity from the starting point nor will they appear at the destination.

[0069] 2. Network flow balance constraints

[0070] For any intermediate port node that does not contain a start and end point, the number of inbound segments of a vessel equals the number of outbound segments, as shown in the following expression:

[0071] (3),

[0072] For any physical port n, the left side of the equation is the sum of all segments into n for the k-th ship, and the right side is the sum of all segments out of n.

[0073] If a ship visits a port, it must have both inflows and outflows, meaning the inflows must equal the outflows, to ensure that the ship does not terminate its journey or become stuck in that port.

[0074] If the ship does not visit the port, both sides are 0, and the equation is naturally true.

[0075] 3. Order allocation restrictions and constraints

[0076] Each order must be assigned to one and only one vessel; that is, each order cannot be split for delivery, and all orders must be served, with no orders allowed to remain unassigned.

[0077] 4. Full load and unload constraints

[0078] If an order is assigned to the k-th vessel, the k-th vessel must load all cargo under the order at the port of origin and unload all cargo at the port of destination. Partial loading or unloading is not permitted; the integrity of the order must be guaranteed.

[0079] 5. Cargo flow balance constraints

[0080] When the kth vessel departs from the port, the remaining cargo quantity for a certain order on board is equal to the cargo quantity upon arrival plus the cargo quantity loaded in port minus the cargo quantity unloaded in port, ensuring that the cargo does not appear out of thin air or disappear.

[0081] 6. Vessel capacity restrictions

[0082] The total amount of cargo allocated to a vessel in an order must not exceed the vessel's current maximum cargo capacity to prevent overloading and ensure transportation safety.

[0083] 7. Job entry constraints

[0084] Ensure that any loading or unloading operations at a port can only begin after the vessel has actually arrived at that port.

[0085] 8. Time propagation constraints

[0086] The time propagation constraint is specifically shown in the following expression:

[0087] (4),

[0088] In the formula, The time when the k-th ship arrives at port j This indicates the time when the k-th ship leaves port i; This represents the operating time at port i. This represents the quantity of cargo loaded by the k-th ship at port i for the r-th order. This represents the amount of cargo unloaded from the r-th order by the k-th ship at port i. Z represents the operational efficiency of port i; Z represents a pre-set, sufficiently large positive number.

[0089] Expression (4) means that when k ships choose to sail through segment (i,j) at speed v, the time of arrival at port j must not be earlier than the time of departure from port i + the working time at port i + the sailing time in segment (i,j).

[0090] When this route / speed is not selected, i.e. Z makes the right side of the inequality extremely negative, and the constraint is automatically relaxed.

[0091] 9. Order Time Window Constraints

[0092] This includes two constraints: the earliest available delivery time and the latest deadline for the order. The specific expression is as follows:

[0093] (5),

[0094] In the formula, This represents the moment when the k-th ship departs from the port of origin Origin(r) of the r-th order. This represents the ready time of the r-th order, i.e., the earliest time it can be loaded onto the ship. Let be a 0-1 decision variable, representing whether the r-th order is assigned to the k-th ship. This indicates allocation, when This indicates that the allocation is unassigned;

[0095] This represents the time when the k-th ship arrives at the destination port Dest(r) of the r-th order. This represents the expected delivery time for the r-th order.

[0096] Specifically, the earliest deliverable time for an order, i.e., the time when the vessel departs from the port of origin of the order, cannot be earlier than the ready time of the goods in that order; if the r-th order is assigned to the k-th vessel, i.e. When the departure time of a vessel is greater than or equal to its readiness time; if the r-th order is not assigned to the k-th vessel, i.e. hour, The order time window constraint becomes extremely easy to satisfy, requiring only... The constraint is then valid.

[0097] The latest deadline for an order, i.e., the time when the vessel arrives at the destination port, cannot exceed the expected delivery time of the order plus the total delay; if the r-th order is assigned to the k-th vessel, i.e. When the arrival time is less than or equal to the expected delivery time plus the total delay; if the r-th order is not assigned to the k-th vessel, i.e. When using the preset Z, When the time window becomes extremely large, the order time window constraint also becomes very easy to satisfy, requiring only... The constraint is then valid.

[0098] 10. Port Opening Hours Restrictions

[0099] Vessels must arrive no earlier than the port’s opening time and depart no later than the port’s closing time to ensure that the scheduling plan meets the operational requirements of each port in terms of time.

[0100] 11. Delay Calculation Constraints

[0101] Delay calculation constraints are driven by the cost of the objective function, automatically ensuring that the delay variable is exactly equal to the actual time exceeded, without the need for additional enforced equality constraints.

[0102] There are three levels of order delays. The first two levels have maximum delay limits, while the third level has no maximum delay limit. Specifically:

[0103] When p=1 When the delay time exceeds The remaining delay time enters the second-level delay region. When p=2, When the delay time exceeds The remaining delay time will enter the third-level delay zone, which has no upper limit.

[0104] The specific expression for the delay calculation constraint is as follows:

[0105] (6),

[0106] In the formula, This represents the sum of delay times for the r-th order across the three levels, and the expression on the right represents the actual arrival time of the ship minus the expected delivery time.

[0107] When the actual arrival time is no later than the expected time, the right-hand side of the constraint is less than or equal to zero, and the left-hand delay variable is allowed to be zero; based on the principle of cost minimization, the optimization process will automatically set it to zero. When the actual arrival time is later than the expected time, the right-hand side is positive, and the left-hand delay variable should be at least equal to the excess value.

[0108] 12. Variable type constraints

[0109] Variable types are specifically divided into two types, one of which is... , , Both are binary variables of 0-1, which can only take the value 0 or 1, representing do / don't do, choose / don't choose; the other is a continuous non-negative variable, which can take any non-negative real number and is used to represent quantities that can change continuously, such as time, quantity, and cost.

[0110] The second-stage model is used to generate an emergency scheduling plan after an uncertain scenario is realized, taking the baseline scheduling plan generated by the first-stage model as input, and selecting from a variety of preset recovery strategies. The recovery strategies include at least speed adjustment, port expediting, activation of backup ports, route reconstruction, and land-based cargo transshipment. Specifically:

[0111] 1. Speed ​​Adjustment: This mainly addresses disturbances such as port congestion, increased or decreased transit time, and increased pressure on time windows. It resolves disturbances by reselecting a speed from a discrete set of speeds.

[0112] 2. Port expedited processing: This mainly addresses disruptions such as port congestion and decreased loading and unloading efficiency. By selecting different expedited processing modes and paying additional operating costs, port loading and unloading efficiency can be improved, thus avoiding order delays.

[0113] 3. Activation of backup ports: This is mainly to address disruptions such as severe congestion or malfunction of the originally planned port of call. Loading and unloading operations are completed by transferring to a nearby backup port to avoid order delays.

[0114] 4. Path Reconstruction: This mainly addresses disturbances such as segment blockages in the baseline path, adjusting subsequent flight paths while keeping the already visited nodes unchanged.

[0115] 5. Cargo transshipment by land: This mainly targets situations where core maritime routes are completely blocked or disrupted, and some orders are unloaded in advance at safe transit ports, with subsequent transportation completed through land transshipment arcs.

[0116] The second-stage model aims to minimize the true expected recovery cost under various uncertainty scenarios. The target is to determine the actual expected recovery costs, which include incremental fuel costs, incremental rental costs, expedited work surcharges, costs of activating alternative ports, actual delay penalties, and multimodal transport costs. The specific expression is as follows:

[0117] (7),

[0118] In the formula, Represents a set of uncertain scenarios. Indicates an index; Indicating uncertain scenarios The probability of occurrence; in uncertain scenarios Down, This represents the incremental fuel cost of the k-th ship. This represents the incremental cost of the rental fee for the k-th ship. This represents the surcharge for the k-th ship. This represents the cost of activating the backup port for the k-th ship. This represents the actual delay penalty for the k-th ship. This represents the multimodal transport cost for the k-th vessel.

[0119] Among these factors, accelerating to save time will increase fuel consumption; however, slowing down due to traffic congestion will save fuel but may lead to increased charter fees. Therefore, the incremental fuel cost for the k-th vessel is... Specifically, as shown in the following expression:

[0120] (8),

[0121] In the formula, Indicates the unit price of fuel; This represents the fuel consumption per unit distance for the k-th ship at a speed of v; Indicates an uncertain scenario Next, determine whether the k-th ship is selected to sail at speed v on segment (i,j); This represents the baseline fuel cost of the k-th vessel on segment (i,j) in the baseline scheduling scheme of the first phase. This indicates whether the k-th ship in the first phase has passed through segment (i,j).

[0122] The incremental cost of charter rate for the k-th vessel is due to the additional time spent in port due to slower sailing or port congestion. Specifically, as shown in the following expression:

[0123] (9),

[0124] In the formula, This represents the unit-time rental rate for the k-th ship; Indicates an uncertain scenario The time when the k-th ship leaves the virtual endpoint End is the final time when the ship completes all transportation tasks and leaves the network. This represents the start time of the k-th ship's departure from the virtual starting point; This represents the total planned time occupied by the k-th ship in the baseline scheduling scheme of the first phase, that is, the time from departure to completion of the task.

[0125] Expedited work surcharge The additional costs incurred to activate the rush work mode to cope with congestion are shown in the following expression:

[0126] (10)

[0127] In the formula, This represents the additional operating cost per unit of goods under the rush mode m. M={0,1,2,3} represents the set of rush modes, with each level increasing efficiency by 50%. m is the index of M={0,1,2,3}. Indicates an uncertain scenario Next, port i selects the rush mode m. When m=0, the rush mode is not selected and the speed remains unchanged; when m=1, the rush mode is selected and the speed increases.

[0128] The total cost of activating a backup port when the base port becomes completely unavailable is the backup port activation cost. Specifically, as shown in the following expression:

[0129] (11),

[0130] In the formula, This indicates the fixed activation cost of the backup port; This represents the unit cargo handling cost at the standby port (RMB / ton). Indicate whether to activate the backup port. Indicates that it is enabled. This indicates that it is not enabled.

[0131] Actual delay penalty This refers to the penalty for delays incurred due to failure to deliver on time despite adjustments, as shown in the following expression:

[0132] (12)

[0133] In the formula, P1, P2, and P3 represent the unit time penalty amount for delays of levels 1, 2, and 3, respectively; and represent the penalties under uncertain scenarios. Down, , , These represent the duration for order r to fall within the first, second, and third level delay intervals, respectively.

[0134] Multimodal transport costs Specifically, this refers to all direct costs related to land transportation incurred when sea routes are completely blocked and a strategy for restoring cargo transshipment by land is implemented, as shown in the following expression:

[0135] (13)

[0136] In the formula, Represents the set of feasible land transport arcs; Indicates an uncertain scenario If the k-th ship terminates its physical sea transport at node i, and the cargo it carries is transported to node j via a land route, then... Indicates approval, when This indicates that the application was rejected. This represents the unit operation and transportation cost of land transport arc (i,j).

[0137] While retaining the physical constraints of the baseline model, the second phase introduces parameters for uncertainty. and Adaptive logic constraints:

[0138] 1. Segment inheritance and speed reselection constraints

[0139] The second phase ensures that vessels must sail along the baseline route of the first phase, without arbitrarily adding or deleting segments. However, different speeds can be selected, such as increasing speed to save time or decreasing speed to save fuel. This is the basis of the speed adjustment and recovery strategy, enabling the model to cope with time delays or save fuel by changing speed while maintaining the original path structure.

[0140] 2. Time constraints for each leg of the journey

[0141] The time a vessel arrives at a downstream port cannot be earlier than "the time it departs from the upstream port + the actual sailing time between the upstream and downstream ports". Specifically, as shown in the following expression:

[0142] (14)

[0143] In the formula, Indicates an uncertain scenario The time when the k-th ship arrives at the downstream port j; Indicates an uncertain scenario The moment when the k-th ship leaves the upstream port i.

[0144] The time constraint for each leg of the voyage introduces an extreme maritime disruption factor, i.e., an uncertain scenario. The following is the growth factor of the flight time The actual sailing time is calculated as: (distance ÷ selected discrete speed) × sailing time growth factor. Specifically:

[0145] when At that time, flight segment (i,j) is in an uncertain scenario The sailing time was consistent with normal conditions and was not affected by any disturbances;

[0146] when At that time, flight segment (i,j) is in an uncertain scenario The sailing time becomes twice as long as normal, which can simulate the deceleration of ships caused by bad weather, worsening sea conditions, temporary traffic control, etc.

[0147] when At that time, flight segment (i,j) is in an uncertain scenario As the travel time approaches infinity, the segment (i,j) becomes impassable, forcing the model to choose other recovery methods.

[0148] When a shipping segment is completely blocked and impassable, land transportation becomes the only feasible path. Therefore, this embodiment proposes a time propagation constraint for a land transportation arc. This constraint is independent of extreme maritime blocking factors and provides an alternative path unaffected by weather or congestion, ensuring that orders can still be delivered even in extreme circumstances. The specific expression is as follows:

[0149] (15)

[0150] In the formula, This represents the fixed travel time of the land transport arc (i,j).

[0151] From expression (15), it can be seen that when land transportation is selected, The time is directly determined by land transport time, and is completely unaffected by extreme disruptions at sea. The impact.

[0152] When the k-th ship terminates its sea transport at port i and chooses land transport, and the goods are transported to the destination port j via the land transport arc (i,j), at this time... The time of arrival at port j must not be earlier than the time of departure from port i. + Fixed transit time by land ".

[0153] Land transport time is completely unrelated to maritime disruptions; even in the event of a complete blockage at sea, the land transport time remains constant. This is still a normal value, which provides the model with an alternative path that is not affected by maritime risks.

[0154] 3. Port Operation Mode Selection

[0155] If a vessel is assigned to visit port i (i.e., inflow is 1), an operating mode must be selected or a backup port must be activated, and this must be done when port i is in an uncertain scenario. Fixed operation time growth factor hour, Specifically, as shown in the following expression:

[0156] (16)

[0157] The left side of expression (16) is the sum of the operating modes selected by the kth vessel at port i, and the right side is whether the vessel plans to call at port i in the first stage baseline scheduling scheme.

[0158] when At that time, i.e., in the first phase of the baseline scheduling scheme, the planned port of call for the ship is i, left-hand side. This means that a work mode or backup port must be selected. Work modes include normal operation and expedited operation.

[0159] when When, i.e., in the first phase of the baseline scheduling plan, the ship does not plan to call at port i, left-hand side This means that no work can be performed.

[0160] and and Both conditions cannot be true simultaneously, meaning that a ship in the same port can only choose one option: either continue working at the original port or be transferred to an alternative port.

[0161] This embodiment proposes a logical constraint to force the activation of a backup port when a port fails, the specific expression of which is as follows:

[0162] (17)

[0163] When port i is normal, i.e. Regardless of whether the ship visits port i, the maximum value of the left side of expression (17) is 998, and the maximum value of the right side is... At this point, expression (17) naturally holds true;

[0164] When port i is severely ineffective, i.e. Furthermore, when the ship actually visits port i, the left side of expression (17) is greater than or equal to 999. At this time, if The right side of expression (17) is only 998, so expression (17) is not true.

[0165] Therefore, when a ship is required to visit a port according to a baseline plan, but that port is in an uncertain scenario... If the model fails completely, a backup port must be enabled; otherwise, the solution is not feasible.

[0166] 4. Port operation time constraints

[0167] The minimum dwell time for a vessel in a port is specified as follows: Departure time ≥ Arrival time + Actual operation time. The actual operation time can be the normal operation time remaining unchanged, the time shortened for expedited work, or the fixed operation time increased by using a backup port. The specific expression is as follows:

[0168] (18)

[0169] In the formula, This indicates the port's baseline operating time, including fixed time plus variable loading and unloading time; Indicates the rate of increase in work efficiency; This indicates the additional fixed operation time required when activating a backup port.

[0170] Actual operating time in Hong Kong = (Fixed operating time + Floating time (Total loading / unloading volume ÷ Operating efficiency)) ÷ Acceleration rate :

[0171] In normal mode, m=0, At this time, the operation time remains unchanged;

[0172] In the rush mode, m=1, At this point, the work time is reduced to 2 / 3 of the original time;

[0173] when That is, when the ship does not visit the port, the Z-term constraint is relaxed.

[0174] 5. Delay calculation constraint

[0175] The delay time for an order must not be less than the difference between the actual delivery time and the expected delivery time.

[0176] If an order arrives on time or ahead of schedule, the delay time is set to 0 (0 is taken when the objective function is optimal).

[0177] If an order arrives late, delay calculation constraints must be met, resulting in delay costs.

[0178] 6. Port operation rate indicator variable constraints

[0179] Expedited work is only permitted when the port is affected, as specified in the following expression:

[0180] (19)

[0181] In the formula, This represents a binary indicator variable, based on the situation of uncertainty. The operational efficiency of port i determines whether to expedite the work; It represents a very small positive number.

[0182] when When, the right side of expression (19) is (Slightly less than 1), the left-hand side can only be less than or equal to a value slightly less than 1, that is, only when The expression (19) is true only when the time is right;

[0183] when When, it is necessary to use To enlarge the right side of expression (19);

[0184] when When, the right side of expression (19) is (Slightly greater than 1), forced implementation .

[0185] The constraint on the port operation rate indicator variable prevents the model from choosing to accelerate operations when the port is operating at full capacity, which would be ineffective and wasteful. Accelerated operation mode is only allowed when port efficiency declines. This ensures the economic rationality of recovery measures.

[0186] 7. Access order constraints for segment interruptions and route reconstruction

[0187] These two constraints work together to strictly lock subsequent routes when a port has been visited or has not triggered a path reconfiguration before the disturbance, and subsequent segments must be completely consistent with the first phase; only nodes that have triggered path reconfiguration and have not yet been visited are allowed to change course.

[0188] 8. Constraints on the coexistence of maritime and land transport in flow balance

[0189] The total flow into the port (maritime arc + land arc) equals the total flow out of the port. The maritime arc and the land arc are equal in status and together satisfy the flow balance.

[0190] 9. Even after the adjustment, you still need to visit the order's origin / destination port.

[0191] Even if a route rescheduling occurs, the vessel must still visit the port of origin and port of destination of the order; otherwise, the order cannot be fulfilled. This is the bottom line for performance, consistent with the allocation results of the first phase.

[0192] 10. Cargo Sequence Constraints

[0193] For each order handled by the vessel, unloading must be completed after loading, meaning the departure time from the port of origin must be no later than the arrival time at the port of destination. This avoids the model generating erroneous conclusions about simultaneous arrivals or arrivals earlier than departures.

[0194] 11. Destination Port Arrival Mode Constraints

[0195] The arrival methods at the destination port are relaxed: not limited to sea transport, arrival via land transshipment can also be considered as completing the order. This provides flexibility for extreme disruption scenarios and avoids the model directly determining infeasibility due to the lack of a sea route.

[0196] 12. Restriction on prohibiting re-sea transport after land transport.

[0197] Physically, once goods are unloaded from a ship at port j and transferred to land transport, the ship no longer carries those goods. If the model allowed goods to depart again by sea from port j, it would be equivalent to the goods instantly moving to another ship or continuing their voyage, violating continuity. This constraint prevents such illogical jumps and ensures the physical consistency of multimodal transport routes.

[0198] 13. The revised plan must ensure that orders are still delivered.

[0199] This means that after the plan is adjusted, the shipper's order will not be discarded due to the recovery strategy. Instead, it will ensure that the goods of each allocated order are eventually delivered to the port of destination by requiring the ship to visit both the port of origin and the port of destination and that the unloading volume equals the order volume.

[0200] A cutting plane-based decomposition method is used to solve the scheduling optimization model, which is divided into a main problem and multiple sub-problems corresponding to various uncertainty scenarios.

[0201] See Figure 1As a specific implementation method, the decomposition method based on the cutting plane used in this embodiment is specifically a master-slave decomposition method based on combined cutting planes. Traditional cutting plane decomposition (such as classic Benders decomposition) requires that the subproblems be continuous linear programming problems in order to construct the cutting plane through dual multipliers. Once integer variables appear in the subproblems, the dual information becomes meaningless, and the traditional method fails. However, this embodiment involves a mixed integer programming problem with a large number of discrete recovery decisions. Therefore, the master-slave decomposition method based on combined cutting planes used in this embodiment no longer relies on the dual information of the subproblems, but instead:

[0202] 1. Directly utilize the specific combinations of integer variables in the main problem, such as which orders are assigned to which ships and which routes;

[0203] 2. Construct combinatorial optimality cuts and combinatorial feasibility cuts; these cut planes operate directly on the space of integer variables.

[0204] 3. When a subproblem is infeasible, a more compact combinatorial cut is generated through the Minimal Infeasible Subsystem (MIS) identification mechanism, resulting in a stronger pruning effect.

[0205] The combinatorial optimality cut refers to the cut plane generated when all subproblems in the scenario are feasible, but the estimated cost of the second-stage recovery in the main problem is lower than the actual expected recovery cost. The combinatorial optimality cut plane is used to correct the lower bound estimate in the main problem, ensuring that the lower bound is not underestimated when the same discrete decision combination is selected again in subsequent iterations.

[0206] A combinatorial feasibility cut is a cutting plane generated when a subproblem in a given scenario becomes infeasible. The combinatorial feasibility cut plane is used to eliminate discrete decision combinations that lead to infeasibility, requiring the main problem to change at least one decision variable in that combination in subsequent iterations.

[0207] A discrete decision combination refers to a set of specific values ​​for all 0-1 decision variables in the main problem, corresponding to a complete baseline scheduling scheme. 0-1 decision variables include ship activation variables, order allocation variables, and baseline route selection variables. When a discrete decision combination results in no feasible solution for any subproblem in the scenario, the discrete decision combination is called a currently infeasible discrete decision combination.

[0208] This invention decomposes a two-stage scheduling model into a main problem and several independent sub-problems corresponding to various uncertain scenarios. The main problem generates a baseline scheduling scheme and a lower bound for the actual expected recovery cost. The sub-problems, given the baseline scheduling scheme, independently optimize the recovery decision for each uncertain scenario and calculate the actual recovery cost. The solution is obtained by iteratively feeding back cut plane information between the main problem and the multiple sub-problems until the convergence condition is met, outputting the optimized baseline scheduling scheme and the corresponding recovery strategy.

[0209] The main problem primarily includes the first-stage decision variables, including the baseline route variable. Order allocation variables Ship activation variables and a continuous variable used to represent the lower bound of the true expected recovery cost in the second stage. The objective function of the main problem is to minimize the sum of the baseline operating cost in the first phase and the estimated actual expected recovery cost in the second phase. The specific expression is as follows:

[0210] (20)

[0211] The main problem constraints include physical constraints such as network flow balancing, order allocation, capacity limits, loading and unloading operations, and time window constraints in the first-stage baseline scheduling model, as well as a set of combinatorial cut planes dynamically added during the iteration process.

[0212] The constraints include all deterministic physical constraints in the first phase, as well as a set of cutting planes that are dynamically added during the iteration process, with the initial set of cutting planes set to empty.

[0213] Given an integer feasible solution to the main problem, let its baseline path and order allocation result be denoted as follows: For each uncertain scenario Construct an independent sub-problem of the scenario Sub-problems Optimize uncertain scenarios for input. Recovery decisions under these circumstances include variables such as speed adjustments, port expedited operations, activation of backup ports, route restructuring, and land-based cargo transshipment.

[0214] For a given uncertain scenario The goal of the subproblem is to minimize the actual recovery cost in this scenario, as shown in the following expression:

[0215] (twenty one),

[0216] The true expected recovery cost under all uncertain scenarios is the weighted average of the true recovery costs of all sub-problems in all scenarios. .

[0217] In the combinatorial cutting plane framework, the algorithm generates cutting planes based on the combination of activated discrete variables in the integer solutions to the current master problem. Let S be the set of key binary variables with a value of 1 in the solution to the current master problem, including the set of baseline path variables. and order allocation variables Set, that is , This represents the number of variables in set S.

[0218] When all subproblems in the scenario are feasible, and the lower bound of the actual expected recovery cost is less than the actual expected recovery cost, i.e. , This represents a preset threshold, which is used to generate and feedback a combined optimality cut to correct the second-stage cost estimate corresponding to the current discrete scheme. Let L be the effective lower bound of the true expected recovery cost in the second stage; then the combined optimality cut can be constructed, with the following specific expression:

[0219] (twenty two),

[0220] When the main problem chooses the exact same discrete decision combination again, all variables in set S take the value 1. At this time, the value inside the parentheses is 1, and expression (22) degenerates into This ensures that the cost of the second phase of the plan is not underestimated.

[0221] If the main problem changes any key variable in the current combination, the value of the summation term will be less than or equal to... The restrictive effect of the cutting plane on the new scheme is weakened, and expression (22) is transformed into Constraint failure. Unrestricted, the main problem can freely explore new solutions, and can continue to explore other discrete combinations.

[0222] When a subproblem in a scenario becomes infeasible, it means that the current baseline scheduling scheme cannot achieve an executable solution in that scenario through recovery measures. In this case, the algorithm adds a combinatorial feasibility cut to the main problem to eliminate currently infeasible discrete decision combinations. The specific expression is as follows:

[0223] (twenty three),

[0224] The cutting plane requires that the main problem change at least one key decision variable in the current combination in subsequent iterations, thereby avoiding the repeated generation of the same infeasible baseline scheduling scheme.

[0225] Furthermore, since the second-stage sub-problem involves complex recovery decisions such as path reconstruction, activation of backup ports, and land-based cargo transshipment, relying solely on general network flow constraints may result in solutions inconsistent with actual transportation logic. For example, the model might over-trigger path reconstruction when there are no obstacles, or cause problems such as discontinuous cargo flow, ship overloading, and port downtime despite no operations after path adjustments. To improve the executability of the sub-problem solution, this invention introduces the following physical feasibility enhancement constraints into the scenario sub-problem:

[0226] 1. Path reconstruction triggering constraints and second-stage flow balancing constraints

[0227] Path Reconstruction Variables Based on the definition of a ship, it is used to represent the k-th ship in an uncertain scenario. Whether deviation from the first-stage baseline path is permitted. To avoid unnecessary path reconstruction in undisturbed scenarios, this invention stipulates that deviation is only permitted if the baseline path of the k-th vessel contains a blocked segment. Otherwise, it is mandatory. This allows ships to continue operating along the baseline path. Meanwhile, after path reconfiguration is triggered, the second phase must still satisfy complete node flow balance constraints to ensure the adjusted route remains continuously feasible within the network structure.

[0228] 2. Second-stage inventory flow constraints

[0229] Segment disruptions can alter the subsequent port call sequence of vessels. Without re-characterizing changes in cargo inventory, the model may generate path schemes that violate vessel capacity or loading / unloading order. Therefore, this invention introduces an independent cargo inventory variable in the second stage to characterize the order cargo volume carried by the vessel at each departure node. Through inventory balance constraints, capacity constraints, and load-before-unload constraints, it ensures that the vessel's cargo volume at any given time does not exceed its capacity. Furthermore, orders must be loaded at the port of origin before they can be unloaded at the port of destination or land transport transit point.

[0230] 3. State-dependent port operation constraints

[0231] During path reconstruction, ships may pass through intermediate nodes that do not undertake loading or unloading tasks. For ports that only serve as transit nodes, forcibly imposing constraints on expedited loading, backup ports, or loading / unloading operations may cause unnecessary logical conflicts. This invention determines the port status based on the loading / unloading volume at a node: when the loading / unloading volume of the k-th ship at port i for all orders is 0, the node is considered a transit node, and expedited loading and backup port activation decisions are not triggered, with the corresponding dwell time set to 0; only when the node undertakes actual loading / unloading tasks and is affected by port inefficiency are expedited loading or backup port recovery measures allowed. This mechanism helps reduce ineffective recovery decisions and improves the stability of subproblem solutions.

[0232] In combinatorial Benders decomposition, when a subproblem in a scenario becomes infeasible, the most direct approach is to add a feasibility cut to the set S of all variables activated by the current main problem. This cut plane can eliminate currently infeasible solutions, but its scope is narrow because it only requires subsequent solutions to change any one variable in the current combination, without identifying the core conflict that leads to infeasibility. In fact, the infeasibility of a subproblem is often caused by only a few key decisions. For example, if two orders with tight time windows are assigned to the same ship, and their loading and unloading sequence cannot meet the time requirements under all recovery strategies, then what truly causes infeasibility is the order allocation combination, not all paths and allocation variables in the current baseline scheduling scheme.

[0233] To enhance the binding force of feasibility cuts, this invention introduces a MIS identification mechanism to extract the set of core conflicting variables that lead to infeasibility from the sub-problems of infeasible scenarios, and generate reinforced feasibility cuts to more efficiently remove invalid solution space.

[0234] Let C be the set of core conflicting variables that make the current subproblem infeasible. and By conducting conflict analysis on the reasons for the infeasibility of sub-problems, the set of critical path variables is extracted. and the set of order allocation variables And generate enhanced feasibility cuts, as shown in the following expression:

[0235] (twenty four),

[0236] Compared to feasibility cuts based on the entire set S, MIS cuts only exclude combinations of core variables that cause infeasibility. Therefore, it can simultaneously remove a class of infeasible solutions containing the conflict structure, thereby improving the search efficiency of the main problem and reducing the number of times invalid subproblems are solved repeatedly.

[0237] After initializing the upper and lower bounds and the set of cutting planes, the main problem is solved, the scene subproblems are evaluated, the cutting planes are generated and the convergence is judged in a loop until the tolerance requirements are met or the search space is fully explored, and the optimal baseline scheduling scheme and the corresponding scene recovery strategy are output.

[0238] As a specific implementation method, this embodiment employs a main tree search mechanism to solve the main problem. Specifically, during the main problem solving process, a branch-and-bound main tree search mechanism of a mixed integer programming solver is used to search the discrete decision space of the first stage. The main tree nodes correspond to partially or completely fixed states of binary decisions such as ship activation variables, baseline route selection variables, and order allocation variables. The solver continuously generates integer feasible candidate solutions in the main tree and calculates the current objective value of the main problem, which serves as the lower bound of the overall objective function of the two-stage problem.

[0239] When the main tree search yields an integer feasible solution that satisfies the physical constraints of the first stage, its corresponding baseline scheduling scheme is extracted. This involves fixing the discrete variable values ​​of the current solution to known parameters, denoted as... Simultaneously, the objective value of the current main problem is calculated as the lower bound LB of the overall objective function of the two-stage problem, as shown in the following expression:

[0240] (25)

[0241] In the formula, This indicates the fixed activation cost for the first phase. This indicates the fuel cost in the first phase. This represents the rental cost for the first phase. This indicates the baseline delay penalty for the first phase.

[0242] When the main tree search yields an integer feasible solution that satisfies the physical constraints of the first stage, its corresponding baseline scheduling scheme is extracted, including ship activation status, order allocation relationships, and baseline route paths. This baseline scheduling scheme is not immediately accepted as the final feasible solution, but is further passed to each scenario subproblem for the second stage of recovery feasibility and actual recovery cost assessment.

[0243] The total cost of the candidate solution is then calculated based on this cost, and used to update the current upper bound. During the iterative solution process, if all sub-problems corresponding to the candidate solution are feasible, then the weighted average true recovery cost, i.e., the true expected recovery cost, is calculated based on the recovery cost and probability of occurrence of each scenario. This is achieved by adding the baseline cost of the first stage to this true expected recovery cost to obtain the global total cost G of the current solution. If this total cost is less than the upper bound of the current record... Then update the upper bound UB to the smaller value. It records the current baseline scheduling scheme as the current optimal scheduling scheme, and the corresponding total cost as the optimal value.

[0244] The upper bound is determined by the total cost of the benchmark scheduling scheme that has been found so far, which can make all sub-problems in the scenario feasible. The total cost is the sum of the expected cost of the first stage and the actual expected recovery cost of each scenario, and is the minimum value among all feasible schemes.

[0245] The lower bound is determined by the continuous variables in the main problem. The provided value is the minimum estimate of the expected recovery cost in the second phase for the main problem. The objective value of the main problem (cost of the first phase + ...) This constitutes the lower bound of the total cost of the two stages.

[0246] The specific expression for updating the upper bound is shown below:

[0247] (26)

[0248] In the formula, This represents the baseline cost for the first phase; This represents the actual expected recovery cost for each scenario in the second phase.

[0249] When the relative difference between the upper bound LB and the lower bound UB of the optimal value is less than the preset tolerance threshold, that is... Convergence is achieved when the number of iterations reaches the preset maximum number of iterations.

[0250] If any subproblem in the scenario is infeasible, a cutting plane is generated to exclude that candidate solution.

[0251] By combining main tree search and sub-problem evaluation, the gap between the upper and lower bounds is continuously narrowed until convergence. Specifically, main tree search is responsible for systematically enumerating and screening candidate solutions in the first stage, while scenario sub-problems are responsible for verifying the feasibility and economy of the candidate solution under uncertain scenarios.

[0252] If all sub-problems corresponding to the candidate solution are feasible, then calculate the actual expected recovery cost based on the recovery cost of each scenario and update the upper bound of the current two-stage problem; if the candidate solution is not feasible in a certain scenario, it means that the baseline scheduling combination represented by the main tree node cannot meet the recovery requirements under uncertain disturbances, and it needs to be excluded or restricted by cutting plane.

[0253] To avoid adding a large number of combined cut planes at once in the initial stage of the main problem, this embodiment uses a lazy constraint callback mechanism to dynamically generate cut planes. Specifically, during the main tree branch bound search of the main problem, when the solver discovers a new integer feasible candidate solution, the lazy constraint callback process is triggered. The callback function first reads the set of key binary variables in the current candidate solution, including the baseline path variable and the order allocation variable; then, it fixes the baseline scheduling scheme and solves the scenario subproblems under each uncertain scenario.

[0254] If all subproblems in the scenario are feasible, then calculate the true expected recovery cost corresponding to the candidate solution. If the continuous variable in the main problem used to represent the lower bound of the true expected recovery cost in the second stage... If the recovery cost is less than the actual expected cost, then a combinatorial optimality cut is added to the callback, forcing the main problem to choose the same key discrete combination again. The cost must not be lower than the actual recovery cost of the plan.

[0255] If at least one subproblem in a scenario is infeasible, a combinatorial feasibility cut is added to the callback to exclude currently infeasible discrete decision combinations. Furthermore, if a minimal infeasible subsystem can be identified through conflict analysis, an enhanced feasibility cut (MIS) is generated only for the core path variables and order allocation variables that cause infeasibility, thereby eliminating a class of infeasible solutions containing the same conflict structure.

[0256] By employing a lazy constraint callback mechanism, the main problem does not need to pre-enumerate all scenario cut planes. Instead, optimal cuts and feasible cuts are generated on demand during the main tree search process. This method can significantly reduce the initial size of the main problem, improve solution efficiency, and ensure that the final baseline scheduling scheme has been verified for recovery feasibility under all considered uncertain scenarios.

[0257] As order and port scale expand, the discrete decision variables in the first-stage main problem, such as ship activation, order allocation, and route selection, form a huge combinatorial search space. Exact solution methods (such as Benders+CPLEX) face bottlenecks in medium- to large-scale computational examples, with solution time increasing dramatically and even failing to obtain feasible solutions. Meanwhile, the scenario subproblems in this invention still require accurate solutions to ensure the accuracy of recovery cost estimation and the effectiveness of the combinatorial cut plane. Therefore, while maintaining overall optimization quality, an Adaptive Large Neighborhood Search (ALNS) is introduced as a heuristic solver for the main problem, replacing the exact main problem. This algorithm rapidly generates high-quality baseline scheduling candidates, delegating only the accurate evaluation and cut generation to the subproblem modules. This loosely coupled design leverages ALNS's efficient search capabilities for large-scale combinatorial optimization while preserving the reliability of accurate subproblem evaluation, thereby significantly reducing solution time within an acceptable error range and meeting the rapid decision-making needs of shipping companies for practical engineering applications.

[0258] Example 2

[0259] This embodiment, based on Embodiment 1, constructs an irregular shipping network comprising 5 physical ports and 10 freight orders. The 5 physical ports are designated as Port A, Port B, Port C, Port D, and Port E.

[0260] Experimental parameter settings:

[0261] 1. Ship parameters:

[0262] This embodiment configures a total of 8 available, irregularly operated vessels, with specific parameters shown in Table 1. Fuel consumption is directly proportional to the cube of the speed, and the fuel cost per unit distance is calculated as: unit fuel price × fuel consumption coefficient a × speed to the power of b.

[0263] Table 1: Ship Parameter Table

[0264]

[0265] 2. Port coefficient:

[0266] A total of 5 physical ports were set up, and their operational efficiency, opening time windows and backup port costs are shown in Table 2.

[0267] Table 2: Port Parameter Table

[0268]

[0269] 3. Port distance parameters:

[0270] The sea voyage distances between the ports are shown in Table 3. These distances represent the shortest navigable routes and take into account channel curvature and navigation restrictions.

[0271] Table 3: Port Distance Parameter Table

[0272]

[0273] 4. Cost parameters: as shown in Table 4.

[0274] Table 4: Cost Parameter Table

[0275]

[0276] 5. Convergence Conditions

[0277] The algorithm converges when the upper bound (UB) and lower bound (LB) of the global optimum are less than the preset tolerance threshold ε, i.e., (UB - LB) / |UB| < 1e-4 (0.01%), or the number of iterations reaches a maximum of 300. Iteration stops when either condition is met.

[0278] 6. Computing Environment

[0279] All comparative experiments were conducted in a unified hardware and software environment: the operating system was Windows 11 (64-bit); the CPU was an Intel Core i7-12700K @ 3.60GHz; the memory was 32 GB DDR4 3600MHz; the main solver was IBM ILOGCPLEX 12.10; the ALNS algorithm was implemented in Python 3.10 and called CPLEX as the exact solver for subproblems.

[0280] The port of origin, port of destination, cargo volume, cargo readiness time, and expected delivery time for each order are shown in Table 5.

[0281] Table 5: Order OD Table

[0282]

[0283] Three uncertainty scenarios are set up, and the parameters are shown in Table 6:

[0284] Table 6: Parameters for Uncertain Scenarios

[0285]

[0286] Scenario S0 is a normal scenario with an impact factor of 1 and an occurrence probability of 0.4; Scenario S1 mainly describes port congestion and increased transit time on local shipping segments, including the failure of port B (…). Port E operation time growth coefficient 2, navigation time growth coefficient for segment 2-3 2, probability 0.3; Scenario S2 describes the superposition of segment blockage and port failure, including segment 3-1 blockage ( ), voyage time growth coefficient for segment 4-2, port D failure ( ), with a probability of 0.3. This indicates that the port is experiencing severe congestion or malfunction. This indicates a flight segment is blocked.

[0287] The solution is obtained by using the method proposed in Example 1. The first stage generates a baseline scheduling scheme: three ships (S1, S2, and S4) are activated, and the routes and order allocations are shown in Table 7.

[0288] Table 7: Baseline Scheduling Scheme

[0289]

[0290] This solution distributes orders across multiple independent routes, effectively reducing the risk of widespread delays caused by disruptions to a single route. The baseline operating plan is as follows: Figure 2 and Figure 3 As shown; where, Figure 2 This is a path diagram. Figure 3 This is the timeline diagram of the corresponding ship's spatiotemporal trajectory.

[0291] After scenario S1 was implemented, the second-phase recovery model addressed port inefficiency and segment delays by adjusting ship speeds (increasing speeds to 26kt on some segments) and activating backup ports to correct the baseline scheduling plan, ensuring the delivery bottom line for core orders. The post-recovery scheduling plan is as follows: Figure 4 and Figure 5 As shown; where, Figure 4 This is a path diagram. Figure 5 This is the timeline diagram of the corresponding ship's spatiotemporal trajectory.

[0292] After implementing scenario S2, facing the combined effects of route disruption and port failure, the model triggers a joint recovery strategy involving path reconstruction, port expediting, and the activation of backup ports, incurring a small delay cost (e.g., a 3.6-hour delay) to avoid systemic default and collapse. The post-recovery scheduling scheme is as follows: Figure 6 and Figure 7 As shown, where, Figure 6 This is a path diagram. Figure 7 This is the timeline diagram of the corresponding ship's spatiotemporal trajectory.

[0293] Furthermore, sensitivity analysis was conducted by changing fuel prices (5, 50, 200) and delay penalty coefficients (low, medium, and high groups), and the results are as follows. Figures 8 to 10 As shown.

[0294] Specifically Figure 8 , 9 10 shows the ship scheduling situation under different oil prices. Figure 8 The table below shows the ship scheduling when the fuel price is 5. Figure 9 The table below shows the ship scheduling when the fuel price is 50. Figure 10 The table below shows the ship scheduling when the fuel price is 200. The scheduling scheme is shown in Table 10.

[0295] Figure 11 The diagram illustrates the different penalty levels for different time periods. When the delay penalty coefficient is low, the delay penalty levels are 5 / 10 / 20; when the delay penalty coefficient is the baseline, the delay penalty levels are 50 / 100 / 200; and when the delay penalty coefficient is extremely high, the delay penalty levels are 500 / 1000 / 2000. Figure 11 (a) shows the changes in the size of the activated fleet and the average speed across the network when the delay penalty coefficients are low, baseline, and extremely high. Figure 11 Figure (b) shows the changes in the actual expected recovery cost in the second phase for low, baseline, and extremely high delay penalty coefficients.

[0296] Figure 12 , 13 Section 14 illustrates the ship scheduling situation under different time penalties. Figure 12 The table below shows the ship scheduling when the delay penalty coefficient is low. Figure 13 The table below shows the ship scheduling when the delay penalty coefficient is the baseline. Figure 14 The table below shows the ship scheduling when the delay penalty coefficient is low. The scheduling scheme is shown in Table 13.

[0297] Experiments show that rising fuel prices prompt the model to increase the number of ships and shorten the route of a single ship; increased delay penalties drive the model to shift from a cost-saving strategy to a performance guarantee strategy, verifying the ability of the method of this invention to generate flexible scheduling schemes that take into account transportation efficiency, performance reliability and operating costs under different economic parameters.

[0298] 1. Impact of unit fuel cost on dispatching scheme: Tables 8 to 10 show the dispatching schemes when the fuel price is 5 / 50 / 200 respectively.

[0299] Table 8: Dispatch Plan When Fuel Price is 5

[0300]

[0301] Table 9: Dispatch Plan When Fuel Price is 50

[0302]

[0303] Table 10: Dispatch Plan When Fuel Price is 200

[0304]

[0305] Experimental results show that changes in fuel prices significantly affect the number of vessels deployed, route length, and speed selection. Under low fuel price conditions, the model tends to reduce the number of vessels deployed and serve more orders through longer, sequential routes. Simultaneously, due to the lower fuel cost pressure from high-speed navigation, the model is more inclined to increase speed to cope with time window constraints and scenario disturbances. Under high fuel price conditions, the model tends to increase the number of vessels deployed, distributing orders across multiple shorter routes to reduce fuel costs associated with long-distance and high-speed navigation per vessel.

[0306] This result demonstrates that fuel prices not only influence tactical decisions such as speed, but also further affect the number of ships deployed and the order allocation structure. The model can balance fixed deployment costs, fuel costs, and delay penalties to generate scheduling schemes that adapt to different fuel price levels.

[0307] 2. Impact of Delay Penalty Coefficient on Scheduling Scheme: Tables 11 to 13 show the scheduling schemes when the delay penalty coefficient is low / baseline / extremely high, respectively: when the delay penalty coefficient is low, the delay penalty level is 5 / 10 / 20; when the delay penalty coefficient is baseline, the delay penalty level is 50 / 100 / 200; when the delay penalty coefficient is extremely high, the delay penalty level is 500 / 1000 / 2000.

[0308] Table 11: Scheduling schemes with low penalty coefficients

[0309]

[0310] Table 12: Scheduling schemes based on penalty coefficients

[0311]

[0312] Table 13: Scheduling schemes with extremely high penalty coefficients

[0313]

[0314] Experimental results show that as the delay penalty coefficient increases, the model places greater emphasis on on-time order delivery. Under low delay penalties, the model tends to reduce the number of vessels deployed and accept a certain degree of delay to reduce fixed deployment costs and fuel costs. As the delay penalty increases, the weight of delay costs in the total cost rises, and the model gradually reduces delay risk by increasing the number of vessels, shortening the service chain per vessel, increasing speed, or implementing expedited measures.

[0315] Under conditions of high delay penalties, the model may significantly increase recovery costs in exchange for a reduction in delay penalties. This indicates that when contract performance constraints are strong, the scheduling system will shift from a cost-saving strategy to a performance-guarantee strategy. This result demonstrates the adaptability of the two-stage scheduling model under different contractual pressures and shows that the introduction of recovery measures can provide enterprises with a quantitative basis for weighing transportation costs against performance reliability.

[0316] To further verify the effectiveness and applicable scenarios of the combined cutting plane exact solution framework and ALNS acceleration strategy in the method of this invention, this embodiment uses the Benders+CPLEX exact algorithm as the benchmark method and conducts a comparative experiment with the ALNS acceleration algorithm under the same computational scale.

[0317] Seven sets of computational examples with different scales were selected, each with a fixed number of ports (5) and order numbers (3, 5, 7, 10, 15, 20, and 30). Under the same computing environment, parameter settings, and convergence criteria, the solution time, target value, and relative error of the two algorithms were recorded. The comparison results are shown in Table 14.

[0318] Table 14: Comparison of Solution Results between Exact Algorithm and Accelerated Algorithm

[0319]

[0320] Experimental results show that in small-scale examples (such as 5+3 and 5+5), both algorithms can obtain the same optimal solution in a short time with limited differences in computation time. This verifies that the proposed precise solution framework for scheduling irregular ship transportation considering uncertain scenarios has good solvability and optimality guarantee capabilities in small-scale scenarios. As the order size increases to 5+10 or more, the solution time of the precise algorithm rises rapidly, while the ALNS acceleration algorithm can significantly shorten the computation time (up to approximately 7.8 times the speedup) while maintaining a small deviation in the target value (maximum deviation not exceeding 4.35%). This comparison demonstrates that the combined cutting plane precise method in this invention is suitable for small-scale scenarios or scenarios requiring strict optimality proof. Furthermore, by introducing the ALNS acceleration strategy, the method can be effectively extended to fast near-optimal solutions for medium-to-large-scale instances, balancing solution quality and computational efficiency.

[0321] Those skilled in the art will understand that embodiments of the present invention can be provided as methods, systems, or computer program products. Therefore, the present invention can take the form of a completely hardware embodiment, a completely software embodiment, or an embodiment combining software and hardware aspects. Furthermore, the present invention can take the form of a computer program product embodied on one or more computer-usable storage media (including, but not limited to, disk storage, CD-ROM, optical storage, etc.) containing computer-usable program code.

[0322] This invention is described with reference to flowchart illustrations and / or block diagrams of methods, apparatus (systems), and computer program products according to embodiments of the invention. It will be understood that each block of the flowchart illustrations and / or block diagrams, and combinations of blocks in the flowchart illustrations and / or block diagrams, can be implemented by computer program instructions. These computer program instructions can be provided to a processor of a general-purpose computer, special-purpose computer, embedded processor, or other programmable data processing apparatus to produce a machine, such that the instructions, which execute via the processor of the computer or other programmable data processing apparatus, generate instructions for implementing the flowchart illustrations and / or block diagrams. Figure 1 One or more processes and / or boxes Figure 1 A device that provides the functions specified in one or more boxes.

[0323] These computer program instructions may also be stored in a computer-readable storage medium that can direct a computer or other programmable data processing device to function in a particular manner, such that the instructions stored in the computer-readable storage medium produce an article of manufacture including instruction means, which are implemented in a process Figure 1 One or more processes and / or boxes Figure 1 The function specified in one or more boxes.

[0324] These computer program instructions may also be loaded onto a computer or other programmable data processing equipment to cause a series of operational steps to be performed on the computer or other programmable equipment to produce a computer-implemented process, thereby providing instructions that execute on the computer or other programmable equipment for implementing the process. Figure 1 One or more processes and / or boxes Figure 1 The steps of the function specified in one or more boxes.

[0325] The embodiments of the present invention have been described above with reference to the accompanying drawings. However, the present invention is not limited to the specific embodiments described above. The specific embodiments described above are merely illustrative and not restrictive. Those skilled in the art can make many other forms under the guidance of the present invention without departing from the spirit and scope of the claims. All of these forms are within the protection scope of the present invention.

Claims

1. A scheduling method for irregular ship transportation considering uncertain scenarios, characterized in that, include: A pre-built scheduling optimization model is invoked, which includes a first-stage model and a second-stage model. The first-stage model is used to generate a baseline scheduling scheme before the uncertain scenario is realized. The second-stage model is used to select and combine various preset recovery strategies from the baseline scheduling scheme as input after the uncertain scenario is realized to generate an emergency scheduling scheme. The recovery strategies include speed adjustment, port expediting, activation of backup ports, route reconstruction, and land-based cargo transshipment. The scheduling optimization model is solved using a cutting plane-based decomposition method, which divides the scheduling optimization model into a main problem and multiple scenario sub-problems corresponding to each of the uncertain scenarios. The solution is obtained by iteratively feeding back cutting plane information between the main problem and the multiple scenario sub-problems until the convergence condition is met, and the baseline scheduling scheme and the corresponding emergency scheduling scheme under each of the uncertain scenarios are output. The decomposition method based on the cutting plane is specifically a master-slave decomposition method based on the combined cutting plane; the master problem is used to generate the baseline scheduling scheme and the lower bound of the actual expected recovery cost, the lower bound being the lowest estimate of the actual expected recovery cost in the second stage in the master problem; the scenario subproblem is used to solve for the variables related to the recovery strategy for each uncertain scenario, given the baseline scheduling scheme, and to calculate the actual recovery cost under each uncertain scenario based on the solution results of the variables; Under the baseline scheduling scheme, if there is any scenario subproblem corresponding to any uncertain scenario that has no feasible solution, then the combination of values ​​of the variables related to the recovery strategy corresponding to the baseline scheduling scheme is the currently infeasible discrete decision combination. The cutting plane information fed back between the main problem and the sub-problems of the scenario includes combinatorial optimality cut and combinatorial feasibility cut; the combinatorial optimality cut is used to correct the lower bound of the true expected recovery cost to avoid the main problem underestimating the true expected recovery cost; the combinatorial feasibility cut is used to exclude currently infeasible discrete decision combinations. When all the subproblems in the scenario are feasible, and the lower bound of the true expected recovery cost is less than the true expected recovery cost, the combined optimality cut is generated and fed back; when a certain subproblem in the scenario is infeasible, the combined feasibility cut is generated and fed back; wherein, the true expected recovery cost is the weighted average of the true recovery costs of all the subproblems in the scenario.

2. The scheduling method for irregular ship transportation considering uncertain scenarios according to claim 1, characterized in that, The first-stage model generates the baseline scheduling scheme with the goal of minimizing the overall expected cost, and satisfies constraints on ship activation logic, network flow balancing, loading and unloading operations, port time windows, and order delivery time; wherein, the overall expected cost includes fixed activation cost, fuel cost, rental cost, and baseline delay penalty.

3. The scheduling method for irregular ship transportation considering uncertain scenarios according to claim 1, characterized in that, The second-stage model aims to minimize the actual expected recovery cost under each of the aforementioned uncertain scenarios, and generates the corresponding emergency dispatch plan for each of the aforementioned uncertain scenarios; the actual expected recovery cost includes incremental fuel cost, incremental rental cost, expedited work surcharge, backup port activation cost, multimodal transport cost, and actual delay penalty.

4. The scheduling method for irregular ship transportation considering uncertain scenarios according to claim 1, characterized in that, The scenario subproblem introduces one or more of the following physical feasibility enhancement constraints: Path reconfiguration is only permitted if the baseline path contains blocked segments; In the second phase, cargo inventory balance constraints are established independently to ensure that the cargo load of the ship does not exceed its capacity and follows the order of loading before unloading. The expedited or standby port measures are only permitted when the port is undertaking actual loading and unloading tasks and is affected by reduced efficiency.

5. The scheduling method for irregular ship transportation considering uncertain scenarios according to claim 1, characterized in that, The method further includes introducing a minimum infeasible subsystem (MIS) identification mechanism in the subproblem of the scenario, extracting a set of core conflict variables that cause infeasibility from the infeasible subproblem of the scenario, and generating an enhanced feasibility cut based on the set of core conflict variables to exclude invalid solution space containing conflict structures.

6. The scheduling method for irregular ship transportation considering uncertain scenarios according to claim 1, characterized in that, The best baseline scheduling scheme obtained during the iteration process is taken as the optimal scheduling scheme, and the corresponding total cost is recorded as the optimal value. When the relative difference between the upper and lower bounds of the optimal value is less than a preset tolerance threshold, or when the number of iterations reaches the preset maximum number of iterations, the convergence condition is met. The lower bound is provided by the continuous variables in the main problem, and the upper bound is determined by the total cost of the baseline scheduling scheme that has been found so that all sub-problems in the scenario have feasible solutions.

Images