A Mobile Target-Based USV-UAV Collaborative Delivery Method Based on Improved BPC

Abstract

This application relates to a collaborative delivery method for mobile target USV-UAVs based on an improved BPC model. It constructs and enhances a mixed-integer programming model and utilizes an improved branch-pricing-cut plane algorithm to obtain the delivery plan. The improvements include: performing parallel label expansion using a pull paradigm, where nodes actively pull labels from predecessor nodes to eliminate write conflicts; performing one-dimensional mapping and equal-frequency bucket partitioning using a spatiotemporal mapping function weighted by spatial distance, time progress, and time window urgency, triggering dynamic bucket reconstruction when there is load imbalance or significant USV displacement; and introducing a spatiotemporally aware dominating rule, using the inclusion relationship of the set of reachable ships as the basis for dominating pruning. This invention systematically solves the problems of label explosion, parallel inefficiency, and load imbalance in standard solution frameworks in mobile target scenarios, achieving collaborative delivery planning results with minimal total runtime.

Description

Technical Field

[0001] This application relates to the fields of combinatorial optimization and path planning technology, and in particular to a collaborative delivery method for unmanned surface vehicles (USVs) carrying unmanned aerial vehicles (UAVs) based on an improved branch-price-and-cut (BPC) algorithm. Background Technology

[0002] With the continued growth in demand for maritime logistics, the need for replenishing and distributing supplies to ships at sea is becoming increasingly urgent. Traditional ship replenishment methods primarily rely on supply vessels or small transport boats shuttling between ports and target vessels. This approach not only easily causes traffic congestion in port waters and increases the risk of ship collisions, but also suffers from significant problems such as long replenishment cycles, high labor costs, and low safety during operations in adverse weather and sea conditions. Delays caused by factors such as replenishment have long been a persistent problem, severely restricting the overall operational efficiency of maritime transport and operations.

[0003] In recent years, unmanned aerial vehicles (UAVs) have opened up new technological paths for maritime cargo delivery due to their significant advantages such as low cost, rapid response, zero carbon emissions, and flexible deployment. Related research shows that introducing UAVs for anchorage delivery demonstrates clear advantages over traditional methods in terms of cost and speed. However, a single UAV is limited by its own endurance, maximum payload, and operating radius, resulting in limited continuous service capability when facing complex delivery tasks involving multiple targets in the open ocean. To overcome this bottleneck, a system architecture that uses unmanned surface vehicles (USVs) as mobile maritime transport platforms and take-off and landing motherships, carrying multiple UAVs to collaboratively execute delivery tasks, has gradually become a research hotspot. USVs can be deployed flexibly in vast sea areas, greatly extending the effective operating radius of UAVs; after completing delivery, UAVs return to the USV for refueling, thus achieving continuous operation in a cyclical manner. The complementary advantages of both systems form a highly efficient, three-dimensional maritime delivery system.

[0004] At the core decision-making level of path planning and scheduling optimization, existing research has made theoretical and methodological progress in collaborative delivery problems such as "truck-drone" and "ship-drone" scenarios. However, most studies assume that the service objects are stationary targets, lacking systematic modeling and efficient solution methods for dynamic collaborative delivery of moving targets. In real-world maritime applications, target ships continuously move along their predetermined routes, with a strong spatiotemporal coupling between service time and service location. Furthermore, each ship has strict service time windows, and the delivery system must arrive at the corresponding spatial location within the correct time window to complete effective resupply. This fundamental characteristic makes the modeling and solution complexity of collaborative delivery problems for moving targets far exceed that of classic static scenarios.

[0005] The Branch-Price-and-Cut (BPC) algorithm is the mainstream technical framework for accurately solving such strong combinatorial optimization problems. Its core mechanism is to repeatedly call the label setting algorithm in the pricing subproblem of column generation to search for new paths with negative generation costs. However, when this framework is applied to spatiotemporally extended networks in mobile target scenarios, it faces the following structural technical challenges: First, the number of nodes and arcs in spatiotemporally extended networks is much larger than in static problems, causing the number of labels in dynamic programming to explode exponentially with the path length, resulting in a sharp increase in computational overhead. Second, traditional labeling algorithms often use a "push" paradigm to process nodes, which can lead to severe write conflicts in a multi-threaded parallel environment, necessitating the introduction of mutex locks, resulting in low parallel computing efficiency and a large amount of multi-core computing power being idle. Third, standard algorithm frameworks typically divide the time axis into fixed-width buckets to achieve parallel scheduling, but the distribution of ship service time windows in mobile scenarios is highly uneven, leading to a severe imbalance in the computational load between buckets, with some periods becoming computational hotspots while others are almost idle. In addition, existing label domination rules only judge the merits based on limited resource dimensions such as cost, load, power, and the set of visited nodes, completely ignoring the key prior information that the trajectory of the mobile target is known. This results in a large number of labels that are already redundant in terms of reachability being unable to be identified and pruned, causing the label pool to continue to expand and seriously affecting the solution efficiency. These technical problems together constitute the technical barriers that this invention aims to overcome. Summary of the Invention

[0006] Therefore, it is necessary to provide a mobile target USV-UAV collaborative delivery method based on improved BPC to address the above-mentioned technical problems, so as to systematically solve the computational bottlenecks existing in the prior art and obtain a high-quality maritime collaborative delivery planning scheme.

[0007] Firstly, this application provides a mobile target USV-UAV collaborative delivery method based on an improved BPC. The method includes: A mixed-integer programming model is constructed with the objective of minimizing the sum of USV travel time and UAV flight time, and enhanced constraints are introduced to form an enhanced model. The enhanced constraints include energy lower bound constraints, USV moving arc discrete consistency constraints, UAV enable symmetry breaking constraints, long-distance ship flight time lower bound constraints, and dynamic load tracking constraints. The enhanced model is solved by improving the BPC algorithm to obtain the USV travel route and UAV flight scheduling and delivery scheme. The improved BPC algorithm performs the following operations in its column generation pricing subproblem: Parallel label expansion is performed using the pull paradigm, where each node pulls labels from its predecessor node and writes them to its own storage space. Parallel processing is performed when there are no dependencies between buckets. A one-dimensional mapping of nodes is performed using a spatiotemporal mapping function weighted by spatial distance, time progress, and time window urgency. Based on this, equal-frequency buckets are divided. When the tag load ratio between buckets exceeds a preset threshold or the USV displacement exceeds a preset condition, bucket structure reconstruction is triggered. A spatiotemporal awareness domination rule is introduced, using the inclusion relationship of the reachable ship set of the tag as the basis for domination pruning.

[0008] Optionally, in one embodiment of this application, in the pull paradigm, the tag cannot be changed after the dominance check is completed at the time of creation, the node only reads the tag from the predecessor node during processing, and the write operation is limited to the current node's own storage area.

[0009] Optionally, in one embodiment of this application, the dominance check of the parallel tag expansion adopts a single instruction multiple data (SIMD) vectorization method, which loads the same attribute of multiple tags to be compared in batches and compares them simultaneously.

[0010] Optionally, in one embodiment of this application, the spatiotemporal mapping function includes a spatial distance term representing the distance between the ship's position at the corresponding time and the USV's current position, a time progress term representing the relative position at the corresponding time within the planned time domain, and a time window urgency term representing the relative progress at the corresponding time within the ship's service time window.

[0011] Optionally, in one embodiment of this application, the inter-bucket label load ratio exceeding a preset threshold means that the ratio of the number of labels in the largest bucket to the average number of labels in all buckets is greater than a preset value.

[0012] Optionally, in one embodiment of this application, the USV displacement exceeding the preset condition means that the cumulative displacement of the USV since the last bucket reconstruction exceeds the product of a preset sensitivity threshold and the maximum one-way range of the UAV.

[0013] Optionally, in one embodiment of this application, using the inclusion relationship of the reachable ship set of tags as the basis for dominant pruning includes: Based on the existing resource control conditions, a fifth control condition is added: when the reachable vessel set of the first label completely includes the reachable vessel set of the second label, it is determined that the first label controls the second label.

[0014] Optionally, in one embodiment of this application, the method for determining the reachable vessel set of the tag is as follows: for the tag's current node and any unvisited vessel, if at a certain moment within the vessel's service time window, the power required for the UAV to fly from the current node to the vessel's location at that moment does not exceed the tag's remaining power, then the vessel belongs to the tag's reachable vessel set.

[0015] Optionally, in one embodiment of this application, the reachable ship set of each tag is encoded in the form of a bit vector, and the set inclusion relationship is determined by performing a bitwise AND operation between the bit vector of the first tag and the bit vector of the second tag to determine their equality.

[0016] Optionally, in one embodiment of this application, the improved BPC algorithm employs a bidirectional dynamic midpoint search strategy when solving the enhanced model, simultaneously launching a forward search from the path start point and a reverse search from the path end point, with each iteration prioritizing the expansion of the search on the side with the smaller current number of labels.

[0017] Compared with existing technologies, it has the following beneficial effects: First, by adopting a pull-pull paradigm parallel tagging algorithm, the write conflict problem during multi-threaded concurrent processing is fundamentally eliminated, realizing a highly efficient parallel computing mode of read-write separation, which significantly improves the utilization and computing efficiency of multi-core processors.

[0018] Second, a dynamic bucket reconstruction mechanism based on a spatiotemporal mapping function was established to replace the traditional fixed-width time bucket partitioning method. This mechanism can dynamically sense and adapt to the spatial position changes caused by USV movement and the uneven distribution of ship time windows, adaptively adjusting the bucket structure, effectively solving the load imbalance problem and ensuring the continuous high efficiency of parallel computing.

[0019] Third, by introducing a spatiotemporal awareness dominance rule, a future reachability set condition based on ship trajectory prediction is added to the label dominance judgment. This rule can effectively utilize prior information about the target ship's trajectory to identify and remove redundant labels in terms of reachability in advance, significantly reducing the size of the label pool and minimizing invalid computation.

[0020] These three core improvements work together to enable the improved BPC algorithm to efficiently solve large-scale USV-UAV collaborative delivery optimization models for moving targets, and ultimately obtain USV travel paths and UAV flight scheduling plans that minimize total running time. Attached Figure Description

[0021] Figure 1 This is a flowchart illustrating the improved BPC algorithm in one embodiment; Figure 2 This is an internal structural diagram of a computer device in one embodiment. Detailed Implementation

[0022] To make the objectives, technical solutions, and advantages of this application clearer, the following detailed description is provided in conjunction with the accompanying drawings and embodiments. It should be understood that the specific embodiments described herein are merely illustrative and not intended to limit the scope of this application.

[0023] Within the planning time domain, a USV departs from a shore-based station, carrying several UAVs of the same model to perform resupply and delivery missions to a group of ships traveling at a constant speed along a fixed route. The USV and UAVs each travel at a constant speed and have known empty weight, battery capacity, and maximum payload capacity. The resupply requirement of each target ship does not exceed the maximum payload of the UAVs and must be served exactly once within a strict service time window. UAVs complete deliveries via hovering delivery, with negligible service time. After departing from the shore station, the USV proceeds to candidate docking points, and UAVs take off sequentially to perform deliveries. After completing a delivery, the UAVs return to the USV. During this period, the USV dynamically adjusts its docking position based on the geometric median of the positions of all on-the-go UAVs to minimize the average return distance. The optimization objective is to minimize the total runtime for completing all delivery missions while satisfying all constraints.

[0024] In one embodiment, a mobile target USV-UAV collaborative delivery method based on improved BPC is provided, comprising the following steps: Step 1: Establish a mixed integer programming model and introduce enhanced constraints.

[0025] The planning time domain is discretized with a fixed step size, constructing a spatiotemporal network consisting of base nodes, ship spatiotemporal nodes, and USV spatiotemporal nodes. The UAV flight arc includes a takeoff arc connecting a USV node and a ship node, an inter-ship arc connecting two ship nodes, and a return arc connecting a ship node and a USV node. USV movement arcs connect adjacent USV spatiotemporal nodes, and waiting arcs represent USVs remaining at the same location. Based on this spatiotemporal network, a mixed-integer programming model is established with the objective function of minimizing the sum of the total USV travel time and the total flight time of all UAVs. This model characterizes USV path planning, UAV flight scheduling, time window constraints, power consumption, and the cooperative coupling relationship between USVs and UAVs.

[0026] Specifically, since the coordinates of each ship at different times are known, a time-extended network is used to model the aforementioned collaborative delivery path. The planning time domain is divided into steps... Discretization yields a set of time points. Spatiotemporal network node set It consists of three types of nodes: base nodes Ship spatiotemporal nodes and USV spacetime nodes ,in, This represents the collection of ships that require service. This represents the set of candidate USV docking points. Indicates a ship At any moment Location , Indicates that USV is at time Located in UAV flight arc collection It consists of three types of arcs: takeoff arc Connect to USV node With ship nodes The requirement is for the UAV to be located... Fly to The flight time falls within the interval ( [Inner; Intership Arc] Connecting two ship nodes and They meet the same flight time interval conditions; return arc Connecting ship nodes With USV nodes ( ), requiring UAV from Return to location The flight time falls within the interval ( [Inside. USV moving arc] Connect adjacent USV spatiotemporal nodes, waiting arc This indicates that the USV remains in the same location. Additionally, a virtual source node is introduced. With virtual sink node : To the initial USV node of each UAV Connect the arcs of unit capacity. The terminating USV node returned from each UAV is connected to the arc. The complete spatiotemporal network is denoted as... The node size is The arc size is Considering both modeling accuracy and computational efficiency, the following selection was made. Minutes are used as the baseline step size.

[0027] Based on this spatiotemporal network, a mixed integer programming model is established for the USV-UAV collaborative delivery problem. The model aims to minimize the total running time to complete all delivery tasks, while simultaneously characterizing USV path planning, UAV flight scheduling, time window constraints, power consumption, and the collaborative coupling relationship between USV and UAV. The specific model is as follows.

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[0050] in, Represents the UAV set, , Indicates the USV departure arc. Indicates the return arc of the USV. Indicates the arc that the USV is waiting for. Indicates the unloaded weight of the UAV. Indicates the unloaded weight of the USV. For ships The weight of the goods, UAV The weight of the cargo at the node , Indicates the maximum payload capacity of the UAV. Indicates the payload capacity of the USV. Indicates the UAV battery capacity. Indicates the durability of the USV. Indicates the UAV battery consumption coefficient. Indicates the USV energy consumption coefficient. For the great constant, UAV Whether along the arc flight, Indicates whether the USV is along the arc move, UAV Whether it is used, Indicates that USV is at time Is it located in the location? , UAV Reaching the node The remaining battery power at that time Indicates that USV is at time The remaining battery power, Indicates that USV is at a spacetime node Number of UAVs carried UAV Visiting the ship Equation (1) is the objective function, minimizing the sum of the total travel time of the USV and the total flight time of all UAVs. Equation (2) ensures that each vessel awaiting service is visited exactly once by a UAV. Equation (3) stipulates that the USV must be at the initial time... Starting from the base. Equation (4) requires the USV to return to the base after completing the mission, ensuring mission closure. Equation (5) is the flow conservation constraint for the USV at the spatiotemporal nodes, ensuring that the number of arcs to reach a node is equal to the number of arcs to leave that node. Equation (6) is the flow balance constraint for the UAV at each spatiotemporal node, ensuring the continuity of the UAV path. Equation (7) restricts each USV spatiotemporal node to a maximum of only one UAV taking off and landing at any given time. Equation (8) uses large The constraints will use the UAV's flight arc with its enabled state variables. Relatedly, if a UAV is not enabled, it is not allowed to fly. Equation (9) limits the total number of enabled UAVs to no more than the maximum available number. Equation (10) is an auxiliary variable. The definition of UAV Visiting the ship With flight arc variable Explicit association, for direct reference in subsequent time windows and load constraints. Equation (11) ensures that the time of UAV access to the vessel falls within the vessel's service time window. Equation (12) assumes all UAVs start fully charged at the initial moment. Equation (13) assumes the remaining runtime of the USV at the initial moment is... Equation (14) describes the power consumption of the UAV during takeoff and inter-ship arcs. The power consumption varies with flight time and current carried weight (aircraft weight plus cargo weight) by a coefficient. Decreasing. Equation (15) describes the power consumption of the UAV on the return arc. Since the delivery is completed and no longer carrying cargo, the energy consumption is calculated only based on the fuselage weight. Equation (16) describes the remaining range consumption of the USV while traveling along the moving arc. It considers the weight of the USV itself and the total weight of the UAV it carries, calculated using a coefficient. Decreasing. Equations (17) and (18) are non-negative constraints on the remaining endurance of UAVs and USVs, respectively, ensuring that neither will experience power overdraft at any time. Equation (19) limits the number of UAVs carried by a USV at any time and space node to no more than the upper limit allowed by its maximum payload. Equation (20) ensures that the total operating time of a USV does not exceed its endurance limit by accumulating the time of all movement arcs and waiting arcs of the USV. Equation (21) links the take-off and landing behavior of the UAV to the position and state variables of the USV. Relatedly, a UAV can only take off and land at a certain spatiotemporal node if the USV is indeed located at that node. Equation (22) ensures that the USV's position state variables are strictly consistent with the selection of its movement arc through the entry and exit arc constraints, thus guaranteeing the logical integrity of the spatiotemporal position representation.

[0051] To tighten the linear programming relaxation of the model and accelerate the convergence of the subsequent BPC algorithm, enhanced constraints are introduced from the following five dimensions without changing the integer feasible solution set of the original model, forming an enhanced model: An energy lower bound constraint is introduced, requiring that the total energy consumed by the UAV to serve a ship is not less than the theoretical minimum determined by the geometry; a USV moving arc discrete consistency constraint is introduced, limiting the deviation between the nominal arc duration and the actual travel time to a reasonable range; a UAV activation symmetry breaking constraint is introduced, forcing UAVs to be activated in index order, compressing the search space; a long-distance ship flight time lower bound constraint is introduced, requiring that the cumulative flight time of the UAV serving the ship is not less than a preset proportion of its full-charge endurance for ships whose initial distance from the USV exceeds a threshold; and a dynamic load tracking constraint is introduced, explicitly tracking the current load of the UAV at each node to ensure the physical consistency of the unloading logic.

[0052] Specifically, firstly, to eliminate physically infeasible flight paths in advance, the following lower bound constraint on energy is introduced:

[0053] The constraints stipulate that if UAV assigned service vessels Therefore, the total energy consumed during its entire flight mission must not be less than the theoretical minimum energy consumption. ,in:

[0054] in, For UAV flight speed, the first term is the basic energy consumption required for a UAV's empty round-trip flight, and the second term is the additional energy consumption generated during a one-way flight carrying cargo.

[0055] in, express The distance that the USV can reach at any given time from the ship Location The nearest candidate docking points. For ships Within the service window, the minimum distance between the ship's position on the trajectory at each moment and the nearest reachable position of the USV. The energy lower bound constraint is an effective inequality: all physically feasible flight paths must satisfy this constraint because the energy consumed by any actual path is no less than the minimum determined by the problem geometry; at the same time, it tightens the LP relaxation by eliminating fractional solutions with structurally insufficient energy budgets.

[0056] Second, in a spatiotemporal discrete network, each USV moving arc nominal duration yes Integer multiples of the total travel time are not necessarily exactly equal to the continuous travel time. , Let $\frac{ ...

[0057] The constraint limits the deviation between the nominal arc duration and the actual travel time to within Within the range, of which This is the minimum loading and unloading time interval for the UAV. This prevents the model from selecting movement arcs whose discrete time steps are substantially inconsistent with the actual driving capability of the USV. This constraint is in... The minute step acts as a constraint; at this step size, unrestricted arc selection may lead to a significant deviation between nominal and actual travel time.

[0058] Third, since all UAVs have identical physical properties, any feasible solution using a given subset of UAVs can be renumbered to obtain an equivalent solution without changing the objective function value. To eliminate this allocation symmetry, the following UAVs are introduced to enable symmetry breaking constraints:

[0059] Constraints on UAV Must precede UAV Enabled. This breaks the permutation symmetry between UAVs, compressing the effective search space for branch and bound at most. This directly reduces the number of subproblems that the BPC algorithm needs to enumerate by a factor of two.

[0060] Fourth, for vessels far from the USV operating area, any feasible UAV flight mission must consume a significant proportion of electrical power, providing a strong structural lower bound for the constraint. This structure is formalized by defining a set of distant vessels:

[0061] in, This is the initial position of the USV. · This represents the maximum one-way voyage of a UAV. Long-distance vessel convoy. This includes vessels whose closest distance to the USV's initial position exceeds half the UAV's maximum range throughout the entire service window. For the set... Any ship in The UAVs responsible for delivery must dedicate a significant proportion of their endurance to flight, which is explicitly defined by the lower bound constraint on long-distance ship flight time:

[0062] The constraint requires that UAVs assigned to long-distance vessels must have a cumulative flight time of no less than 70% of their full-charge endurance. This threshold is adapted to the step size error in the spatiotemporal discrete network. This constraint, as an effective inequality, tightens the feasible region of long-distance delivery tasks and supports early pruning in the tag algorithm pricing subproblem: when the tag's remaining battery power reaches the vessel... It was already lower than before. In this case, you can skip directly without completing a full resource check.

[0063] Fifth, the basic model implicitly tracks the UAV load through energy constraints. To achieve tighter bounds in pricing subproblems, explicit dynamic load tracking constraints are introduced. The initial carry-on weight upon takeoff from the USV is initialized as follows:

[0064] Constraints will UAV The initial load capacity is set to the sum of the cargo weights of all ships to be delivered in this mission. Each time the UAV arrives at a ship, its load capacity is reduced by the weight of that ship's cargo, enforced by the following pair of inequalities:

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[0066] Enforce precise unloading updates at each ship node: when arc When activated, the two constraints will unite Precisely locked as – Through standard large Linearization enforces equality constraints; this linearization automatically disables when no arc is selected. The UAV must return to the USV without carrying any cargo.

[0067] Ensure UAV The load variable is zero when returning along the return arc, reflecting the physical requirement that all goods have been delivered before recovery. Finally,

[0068] The non-negativity constraint on the load variable at each node, together with the above constraints, ensures the physical consistency of load tracking throughout the entire flight trajectory.

[0069] Step 2: Solve the enhanced model by improving the BPC algorithm.

[0070] Based on the enhanced model described above, an improved BPC algorithm incorporating three core innovations is designed and implemented for accurate solution, to obtain the USV travel path and UAV flight scheduling and delivery plan. For example... Figure 1 As shown, the algorithm performs the following operations in the pricing subproblem of column generation.

[0071] First, it performs parallel label expansion based on the pull paradigm.

[0072] Traditional tagging algorithms employ a push paradigm, actively pushing all tags of a node to its successor nodes during processing. In multi-threaded concurrency, write conflicts necessitate locking for protection, leading to low parallel efficiency. This invention reverses this to a pull paradigm: when processing a node, it actively pulls tags from all predecessor nodes and performs expansion and dominance checks. Each node only writes to its own tag storage space, naturally isolating write operations and fundamentally eliminating write conflicts. Simultaneously, tags undergo dominance checks and are written to fixed locations upon creation, remaining unchanged and never deleted during subsequent processing. This achieves a read-write separation access mode and a structured array storage layout, significantly improving memory management efficiency and CPU cache hit rate, and providing natural support for SIMD vectorization.

[0073] To achieve lock-free bucket-level parallel scheduling, a bucket data structure is introduced to group nodes by time resources. The timeline is then set in steps. Division, Node Belonging Bucket

[0074] A directed dependency graph is constructed between buckets based on possible label expansion directions: if buckets The tags in the bucket can be extended. Then record ,say Depends on Only when After processing, Only then can the algorithm start. The core observation of bucket-level parallelism is that buckets without dependencies can be processed simultaneously. The algorithm maintains a queue of released buckets, initially containing only buckets with a dependency count of zero at the starting point; whenever a bucket is processed, the counts of all its dependent buckets are decremented by one, and those with counts of zero are immediately added to the queue for threads to claim. This achieves automated parallel scheduling.

[0075] In the domination check phase, vectorization optimization is achieved by leveraging the Single Instruction Multiple Data (SIMD) capability of modern CPUs. Domination check is the most frequently performed operation in the tag algorithm; traditional implementations require sequential comparison of each existing tag in the bucket. Taking AVX-256 as an example, it can compare the cost, flight time, remaining battery power, and other attributes of eight tags simultaneously, increasing the throughput of domination check by approximately eight times. To fully utilize the vectorization advantages of SIMD, tag storage adopts a Struct of Arrays (SoA) layout: each attribute forms an independent contiguous array, rather than storing all attributes of a single tag contiguously. The immutability of tags in the pull paradigm provides natural support for the SoA layout—once a tag is written, its position is fixed and it can be directly appended to the end of each attribute array without additional memory cleanup overhead.

[0076] Secondly, a dynamic bucket reconfiguration mechanism (DBR) is constructed. Existing methods often use fixed, equal-width divisions of the time axis. In scenarios where the USV's position is dynamically changing, this can lead to severe load imbalance between buckets due to neglecting spatial heterogeneity and uneven distribution of time windows. This invention constructs a dynamic bucket reconfiguration mechanism based on a spatiotemporal mapping function. This function consists of three weighted normalized components: a spatial distance component, representing the distance between the ship's position at the corresponding moment and the USV's current position; a time progress component, representing the relative position of the current moment within the overall planning time domain; and a time window urgency component, representing the relative progress of the current moment within the ship's service time window. Based on this function, the mapping values ​​of all unprocessed nodes are calculated, and the bucket structure is divided into equal-frequency segments to ensure that each bucket contains approximately an equal number of nodes, achieving load balancing. When the ratio of the maximum bucket label count to the average bucket label count exceeds a preset threshold, or when the cumulative displacement of the USV since the last reconfiguration exceeds the product of a preset sensitivity threshold and the UAV's maximum single-trip distance, dynamic reconfiguration of the bucket structure is triggered, and labels are re-divided and migrated.

[0077] Specifically, let's set For USV at time The actual location is defined by the spatiotemporal mapping function. : Each node is mapped to a one-dimensional value that integrates spatiotemporal features:

[0078] The first term is the normalized spatial distance. For ships At any moment The distance between the current location and the current location of the USV, based on the maximum one-way range of the UAV. Normalization; the second term is time. In the planning time domain The relative position within the timeframe; the third item is the urgency of the time window, i.e., the current position of the ship. Service Window Within the relative progress, nodes with high urgency have values ​​close to 1 and should be processed first. , , For the weight parameters, satisfying . Nodes with similar values ​​have similar spatiotemporal decision-making and should be grouped into the same bucket.

[0079] based on The function uses equal-frequency binning instead of equal-width binning to determine bucket boundaries. The specific steps are: first, calculate the binning rate of all unprocessed nodes. Values ​​are sorted to obtain an ordered sequence. Then, according to the target bucket capacity... The desired number of nodes per bucket is evenly divided so that each bucket contains approximately the same number of nodes:

[0080] In spatiotemporal regions with dense nodes, The values ​​are compactly distributed with small bucket boundary spacing, resulting in fine-grained partitioning. In sparsely populated regions, the bucket boundary spacing is large to avoid over-cutting. Furthermore, the dynamic bucket reconstruction mechanism introduces a time window density function. (defined as time) The number of nearby effective time windows is used to locally adjust the bucket width.

[0081] in For global average density, The standard bin width is used as the baseline. During periods of high density, the bin width is narrowed, while during periods of low density, the bin width is widened to further balance the label load on each bin.

[0082] The dynamic bucket refactoring mechanism not only performs a partitioning operation during algorithm initialization but also dynamically refactors the buckets as needed during runtime. There are two conditions that trigger refactoring: Condition 1 (Load Imbalance): Let... For bucket The current set of tags, when the ratio of the number of tags in the largest bucket to the number of tags in the average bucket exceeds a threshold. ,Right now

[0083] in For bucket Current number of tags, The total number of buckets indicates that the current partitioning has led to a severe hotspot effect, requiring a refactoring.

[0084] Condition 2 (USV location update): Due to The first term in the function depends on the real-time position of the USV. When the USV undergoes significant displacement during the planning process—that is, the displacement exceeds... ( When the sensitivity threshold is reached, the original If a value becomes invalid, the bucket structure needs to be recalculated and reconstructed. The reconstruction operation includes: recalculating the values ​​of all unprocessed nodes. The value is used to determine the new bucket boundary by equal-frequency cutting, the existing labels are migrated to the new bucket, and the bucket dependency graph is updated. Reconstruction itself has computational overhead, and the above two conditions jointly constrain the triggering frequency, achieving a balance between reconstruction benefits and overhead.

[0085] Third, a Spatiotemporally Aware Domination Rule (SADR) is introduced. In the domination check phase of the parallel labeling algorithm, in addition to comparing the four traditional resource states—simplified cost, cumulative load, remaining power, and visited ship set—a fifth domination condition is added: the reachable ship set of a label is defined as the set of all ships that can still be served in the future, starting from the current state, provided that power constraints, time window constraints, and load constraints are met. For two labels, if the reachable ship set of the first label completely contains the reachable ship set of the second label, then, based on the original domination condition, the first label is determined to dominate the second label, and the latter can be safely pruned. The reachable ship sets of each label are efficiently encoded and stored using bit vectors, and the inclusion relationship between sets can be quickly determined through a single bitwise AND operation and equality check.

[0086] Specifically, define tags reachable set For: From Starting from the current state, and under the premise of satisfying the constraints of power supply, time window, and load, the set of ships that can still be served in the future is:

[0087] in, For nodes Corresponding geographic coordinates For tags The collection of visited ships, For the collection of unvisited vessels awaiting service, For position With ships At any moment Location The Euclidean distance between them. Intuitively, If and only if there exists a certain moment landed on the ship Within the time window, the UAV starts from the current node. Depart for The required power should not exceed the remaining power. The reachable set is simultaneously constrained by three factors: spatial distance, time window, and remaining battery power, and is achieved through... The predictability of ship trajectories is explicitly utilized.

[0088] The spatiotemporal awareness dominance rule adds the inclusion of reachable sets as a new judgment dimension to the dominance rule of the pricing subproblem: Domination under the rule of spatiotemporal perception If and only if all the original dominance conditions are satisfied, and additionally the following conditions are also satisfied:

[0089] like The reachable set contains If it is a reachable set, then All ships that can be served Both can provide services, and their resources are comparable. If it is indeed redundant, it can be pruned; conversely, if ,but Able to reach certain Ships that cannot be reached should not be pruned. The spatiotemporal perception domination rule is a strict reinforcement of the original rule: any label pair determined to be dominated by the original rule must also satisfy the spatiotemporal perception domination rule; however, the spatiotemporal perception domination rule can additionally identify redundant labels that the original rule has missed.

[0090] To efficiently store and compare reachable sets, bit vector encoding is used for the ship set. Establish fixed numbering achievable collection Encoded as A bit vector consisting of 64-bit integers , of which The bit is 1 if and only if the ship .

[0091] Set inclusion relationship Equivalent to bitwise operations:

[0092] Right now bit vector and The bit vector, after bitwise AND, is still equal to The judgment can be completed with a single bitwise AND operation and a single comparison. At that time, the entire reachable set occupies only a 64-bit integer, resulting in minimal storage overhead; even It only requires four 64-bit integers, approximately 32 bytes.

[0093] Furthermore, a bidirectional dynamic midpoint search strategy is employed in the solution process. Let the average number of successors per node in the spatiotemporal network be... Average access via UAV path For a single ship, the one-way labeling algorithm must be maintained in the worst case. Each label is unique. The bidirectional search simultaneously performs a forward search from the starting point and a reverse search from the ending point, converging midway through the path to compress the label size. When forward and reverse labels converge at an intermediate node, resource consistency conditions must be met: the sum of their consumed power does not exceed the total battery capacity, and their current loads are equal. The algorithm maintains forward task queues, reverse task queues, and splicing task queues, prioritizing splicing tasks and expanding the direction with fewer current labels between forward and reverse directions to dynamically balance the scale on both sides, thereby compressing the exponential number of labels in unidirectional search into a bidirectional, halved exponential scale.

[0094] To verify the performance of the improved BPC algorithm, numerical experiments were conducted on examples of different sizes, as shown in Table 1. In Table 1, #OPT represents the number of instances that found the optimal solution (gap < 0.1%) within the time limit; #Fea represents the number of instances that found a feasible integer solution within the time limit but whose optimality was not proven; and #Unk represents the number of instances that did not find a feasible solution within the time limit. For instances where feasible solutions are found, the average optimality gap is calculated as (upper bound - lower bound) / upper bound × 100. The maximum optimality gap is defined as the interval between instances where a feasible solution is found; #CPU(s) represents the average computation time (seconds). The naming convention for the examples follows the format "number of ships - number of UAVs," generating 10 random examples for each scale. The experimental environment is a standard computing platform equipped with a multi-core processor, with a time limit of 600 seconds. Table 1 shows that the improved BPC algorithm is robust across all scales of examples. For small-scale examples (10⁻⁵, 20⁻¹⁰), 9 and 8 out of 10 examples respectively reached the optimal solution within the time limit, with average solution times of only 38.8 seconds and 97.3 seconds, respectively, far below the 3600-second time limit, indicating that the algorithm possesses efficient and accurate solution capabilities for small-scale problems. As the problem size increases to medium-to-large scales (30⁻¹⁵ to 50⁻²⁵), #Unk for all examples is 0, indicating that the improved BPC algorithm can stably obtain feasible delivery solutions across all test scales without any solution failures. Regarding the quality of solutions, the average optimality gap for examples of different scales. All do not exceed 1.02%, maximum gap The obtained solutions have a high quality, not exceeding 1.85%. Notably, the average computation time for 50-20 scale examples is 762.7 seconds, but the average gap is only 0.53%, demonstrating the algorithm's ability to obtain high-quality solutions with a reasonable time cost on more difficult examples. These results show that the three core improvements proposed in this invention—pull-parallel label extension, dynamic bucket reconstruction mechanism, and spatiotemporal awareness dominance rule—work synergistically, achieving significant improvements in both computational efficiency and solution quality compared to the standard BPC framework. This verifies the effectiveness and practicality of this invention in solving the USV-UAV collaborative delivery problem.

[0095] Table 1. Performance of the improved BPC algorithm for different scale examples.

[0096] Through the above steps, the improved BPC algorithm completes the accurate solution of the augmentation model, and finally outputs the USV travel path and UAV flight scheduling and delivery scheme that minimizes the total running time.

[0097] It should be understood that although the steps in the flowcharts of the embodiments described above are shown sequentially according to the arrows, these steps are not necessarily executed in the order indicated by the arrows. Unless explicitly stated herein, there is no strict order restriction on the execution of these steps, and they can be executed in other orders. Moreover, at least some steps in the flowcharts of the embodiments described above may include multiple steps or multiple stages. These steps or stages are not necessarily completed at the same time, but can be executed at different times. The execution order of these steps or stages is not necessarily sequential, but can be performed alternately or in turn with other steps or at least some of the steps or stages of other steps.

[0098] In one embodiment, a computer device is provided, which may be a terminal, and its internal structure diagram may be as follows: Figure 2As shown, the computer device includes a processor, memory, communication interface, display screen, and input devices connected via a system bus. The processor provides computing and control capabilities. The memory includes non-volatile storage media and internal memory. The non-volatile storage media stores the operating system and computer programs. The internal memory provides an environment for the operation of the operating system and computer programs stored in the non-volatile storage media. The communication interface is used for wired or wireless communication with external terminals; wireless communication can be achieved through Wi-Fi, mobile cellular networks, NFC (Near Field Communication), or other technologies. When executed by the processor, the computer program implements a mobile target USV-UAV collaborative delivery method based on an improved BPC. The display screen can be an LCD screen or an e-ink display screen. The input devices can be a touch layer covering the display screen, buttons, a trackball, or a touchpad mounted on the computer device casing, or an external keyboard, touchpad, or mouse.

[0099] Those skilled in the art will understand that Figure 2 The structure shown is merely a block diagram of a portion of the structure related to the present application and does not constitute a limitation on the computer device to which the present application is applied. Specific computer devices may include more or fewer components than those shown in the figure, or combine certain components, or have different component arrangements.

[0100] It should be noted that the user information (including but not limited to user device information, user personal information, etc.) and data (including but not limited to data used for analysis, data stored, data displayed, etc.) involved in this application are all information and data authorized by the user or fully authorized by all parties.

[0101] Those skilled in the art will understand that all or part of the processes in the above embodiments can be implemented by a computer program instructing related hardware. The computer program can be stored in a non-volatile computer-readable storage medium. When executed, the computer program can include the processes of the embodiments described above. Any references to memory, databases, or other media used in the embodiments provided in this application can include at least one of non-volatile and volatile memory. Non-volatile memory can include read-only memory (ROM), magnetic tape, floppy disk, flash memory, optical memory, high-density embedded non-volatile memory, resistive random access memory (ReRAM), magnetic random access memory (MRAM), ferroelectric random access memory (FRAM), phase change memory (PCM), graphene memory, etc. Volatile memory can include random access memory (RAM) or external cache memory, etc. By way of illustration and not limitation, RAM can take many forms, such as Static Random Access Memory (SRAM) or Dynamic Random Access Memory (DRAM). The databases involved in the embodiments provided in this application may include at least one type of relational database and non-relational database. Non-relational databases may include, but are not limited to, blockchain-based distributed databases. The processors involved in the embodiments provided in this application may be general-purpose processors, central processing units, graphics processing units, digital signal processors, programmable logic devices, quantum computing-based data processing logic devices, etc., and are not limited to these.

[0102] The technical features of the above embodiments can be combined in any way. For the sake of brevity, not all possible combinations of the technical features in the above embodiments are described. However, as long as there is no contradiction in the combination of these technical features, they should be considered to be within the scope of this specification.

[0103] The embodiments described above are merely illustrative of several implementation methods of this application, and while the descriptions are specific and detailed, they should not be construed as limiting the scope of this patent application. It should be noted that those skilled in the art can make various modifications and improvements without departing from the concept of this application, and these all fall within the protection scope of this application. Therefore, the protection scope of this application should be determined by the appended claims.

Claims

1. An improved BPC-based mobile target USV-UAV collaborative distribution method, characterized in that, The method includes: A mixed-integer programming model is constructed with the objective of minimizing the sum of USV travel time and UAV flight time, and enhanced constraints are introduced to form an enhanced model. The enhanced constraints include energy lower bound constraints, USV moving arc discrete consistency constraints, UAV enable symmetry breaking constraints, long-distance ship flight time lower bound constraints, and dynamic load tracking constraints. The enhanced model is solved by improving the BPC algorithm to obtain the USV travel route and UAV flight scheduling and delivery scheme. The improved BPC algorithm performs the following operations in its column generation pricing subproblem: Parallel label expansion is performed using the pull paradigm, where each node pulls labels from its predecessor node and writes them to its own storage space. Parallel processing is performed when there are no dependencies between buckets. A one-dimensional mapping of nodes is performed using a spatiotemporal mapping function weighted by spatial distance, time progress, and time window urgency. Based on this, equal-frequency buckets are divided. When the tag load ratio between buckets exceeds a preset threshold or the USV displacement exceeds a preset condition, bucket structure reconstruction is triggered. A spatiotemporal awareness domination rule is introduced, using the inclusion relationship of the reachable ship set of the tag as the basis for domination pruning.

2. The improved BPC-based mobile target USV-UAV collaborative distribution method according to claim 1, wherein, In the Pull paradigm, tags cannot be changed after dominance checks are completed during creation. During node processing, tags are only read from the predecessor node, and write operations are limited to the current node's own storage area.

3. The mobile target USV-UAV collaborative delivery method based on improved BPC according to claim 1, characterized in that, The dominance check of the parallel tag expansion adopts the single instruction multiple data (SIMD) vectorization method, which loads the same attribute of multiple tags to be compared in batches and compares them simultaneously.

4. The improved BPC-based mobile target USV-UAV collaborative distribution method according to claim 1, wherein, In the spatiotemporal mapping function, the spatial distance term is the distance between the ship's position at the corresponding time and the current position of the USV, the time progress term is the relative position at the corresponding time within the planned time domain, and the time window urgency term is the relative progress at the corresponding time within the service time window of the ship.

5. The improved BPC-based mobile target USV-UAV collaborative distribution method according to claim 1, wherein, The label load ratio between bins exceeding the preset threshold means that the ratio of the number of labels in the largest bin to the average number of labels in all bins is greater than the preset value.

6. The improved BPC-based mobile target USV-UAV collaborative distribution method according to claim 1, wherein, The USV displacement exceeding the preset condition means that the cumulative displacement of the USV since the last barrel reconfiguration exceeds the product of the preset sensitivity threshold and the maximum single-trip range of the UAV.

7. The improved BPC-based mobile target USV-UAV collaborative distribution method according to claim 1, wherein, The method of using the inclusion relationship of the reachable ship set of tags as the basis for dominant pruning includes: Based on the existing resource control conditions, a fifth control condition is added: when the reachable vessel set of the first label completely includes the reachable vessel set of the second label, it is determined that the first label controls the second label.

8. The mobile target USV-UAV collaborative delivery method based on improved BPC according to claim 7, characterized in that, The method for determining the reachable vessel set of the tag is as follows: for the tag's current node and any unvisited vessel, if at a certain moment within the vessel's service time window, the power required for the UAV to fly from the current node to the vessel's location at that moment does not exceed the tag's remaining power, then the vessel belongs to the tag's reachable vessel set.

9. The mobile target USV-UAV collaborative delivery method based on improved BPC according to claim 7, characterized in that, The reachable ship set of each tag is encoded in bit vector form. The set inclusion relationship is determined by performing a bitwise AND operation between the bit vector of the first tag and the bit vector of the second tag to determine their equality.

10. The improved BPC-based mobile target USV-UAV collaborative distribution method according to claim 1, wherein, The improved BPC algorithm adopts a bidirectional dynamic midpoint search strategy when solving the enhanced model, simultaneously performing forward search from the path starting point and reverse search from the path ending point, and preferentially expanding the search on the side with less current label total number in each iteration.

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