Emergency logistics path optimization method and system based on cooperation of truck and unmanned aerial vehicle

Abstract

The application provides an emergency logistics path optimization method and system based on truck and unmanned aerial vehicle cooperation, belongs to the technical field of logistics path optimization, and comprises the following steps: based on a material center candidate point, a disaster point position and a road network, a main problem is determined, a site of the material center is configured, a number of trucks and unmanned aerial vehicles is configured, a first-stage optimization main problem is optimized to minimize operation cost; meanwhile, truck path reliability of each truck path is calculated according to actual road arc segment damage probability; based on the truck path reliability, the probability that each secondary emergency point needs support is calculated, a sub-problem is constructed, and a second-stage marginal benefit-based tight cut generation strategy is adopted to solve the sub-problem to minimize support scheduling cost; the main problem and the sub-problem constitute a two-stage stochastic programming model, and the total cost minimization of the operation cost and the support scheduling cost is obtained through iterative solving of the programming model to obtain an optimal path. The application improves the solution efficiency of truck and unmanned aerial vehicle cooperative distribution in post-disaster emergency material scheduling.

Description

Technical Field

[0001] This invention belongs to the field of logistics route optimization technology, and in particular relates to an emergency logistics route optimization method and system based on the collaboration of trucks and drones. Background Technology

[0002] The statements in this section are merely background information related to the present invention and do not necessarily constitute prior art.

[0003] Extreme natural disasters pose a serious threat to people's lives and property and social stability. How to quickly and reliably transport limited emergency supplies from storage centers to scattered disaster sites after a disaster has become a core challenge for emergency logistics systems. This process not only demands speed but is also fraught with uncertainty due to the frequent disruption of post-disaster transportation networks, making traditional deterministic route planning methods often inapplicable.

[0004] Against this backdrop, the truck-drone collaborative delivery model has been widely studied due to its flexibility and efficiency, but there are still several shortcomings that need to be addressed: First, although most studies focus on uncertain factors such as demand fluctuations and random transportation times, few systematically incorporate the risk of post-disaster road disruptions into the quantification and constraint system of path reliability; Second, most existing models focus on single-stage pre-planning and lack effective connection with post-event emergency adjustment mechanisms, making it difficult to achieve a balance between robustness and cost control at the system level; Third, the solution efficiency for large-scale two-stage optimization problems is low. Summary of the Invention

[0005] To overcome the shortcomings of the existing technologies, this invention proposes an emergency logistics route optimization method and system based on truck and drone collaboration, in order to solve the problems of systematic lack of road interruption risk, disconnect between single-stage decision-making and post-event adjustment, and low efficiency of large-scale two-stage optimization solutions in the scenario of emergency material dispatching in natural disasters.

[0006] To achieve the above objectives, one or more embodiments of the present invention provide the following technical solutions:

[0007] In a first aspect, this invention discloses an emergency logistics route optimization method based on truck and drone collaboration, comprising:

[0008] Acquire candidate locations of material centers, disaster-stricken areas, and road networks, including emergency points and secondary emergency points;

[0009] Based on the candidate locations of the material centers, the locations of the disaster-stricken areas, and the road network, a main problem is constructed to determine the location of the material centers and the number of trucks and drones to be deployed. The first stage optimizes the main problem to minimize operating costs. At the same time, the reliability of each truck route is calculated based on the actual probability of damage to road arcs.

[0010] Based on the truck route reliability, the probability of needing support for each secondary emergency point is calculated. Sub-problems are constructed according to the probability of needing support. In the second stage, a compact generation strategy based on marginal benefits is used to solve the sub-problems to minimize the support scheduling cost.

[0011] A two-stage stochastic programming model is constructed using the main problem and sub-problems. The optimal path is obtained by iteratively minimizing the total cost of operating costs and support scheduling costs.

[0012] Secondly, this invention discloses an emergency logistics route optimization system based on truck and drone collaboration, comprising:

[0013] The network acquisition module is configured to acquire candidate points of material centers, locations of disaster-stricken points, and road networks, wherein the disaster-stricken points include emergency points and secondary emergency points.

[0014] The first optimization module is configured to: based on the candidate locations of the material center, the location of the disaster-stricken area, and the road network, construct the main problem to determine the location of the material center and the number of trucks and drones to be deployed; optimize the main problem in the first stage to minimize operating costs; at the same time, calculate the reliability of each truck route based on the actual road arc damage probability.

[0015] The second optimization module is configured to: calculate the probability that each secondary emergency point needs support based on the truck route reliability, construct a sub-problem based on the probability that support is needed, and in the second stage, use a compact generation strategy based on marginal benefits to solve the sub-problem to minimize the support scheduling cost.

[0016] The planning and solving module is configured to: construct a two-stage stochastic programming model with the main problem and sub-problems, and obtain the optimal path by iteratively solving the total cost of operating cost and support scheduling cost.

[0017] Thirdly, the present invention discloses an electronic device, including a memory and a processor, and computer instructions stored in the memory and running on the processor, wherein the computer instructions, when run by the processor, complete the steps of the above-mentioned emergency logistics route optimization method based on truck and drone collaboration.

[0018] Fourthly, the present invention discloses a computer-readable storage medium for storing computer instructions, which, when executed by a processor, complete the steps of the above-mentioned emergency logistics route optimization method based on truck and drone collaboration.

[0019] Compared with the prior art, the beneficial effects of the present invention are as follows:

[0020] This invention provides an emergency dispatch decision-making framework for road disruption risk. It constructs a two-stage stochastic optimization model considering path reliability thresholds and disruption scenario responses, and employs an integer L-shaped Benders decomposition algorithm for efficient solution. First, explicitly introducing path reliability constraints and disruption probability thresholds effectively enhances the resilience of the emergency logistics system to road damage risks. By transforming the probability of road segment damage into a path reliability product constraint and setting a minimum reliability threshold, the planned truck routes in the first stage are ensured to be resilient to disruptions, providing a robust initial plan for subsequent emergency support and avoiding global failures due to road disruptions. Furthermore, the two-stage "pre-planning" approach... The "post-event adjustment" decision-making framework achieves coordinated optimization of resource allocation and emergency response. It employs an exact algorithm based on Benders decomposition to solve high-dimensional two-stage stochastic programming problems. Through iterative coordination of the main problem and subproblems, combined with marginal benefit analysis and linear relaxation dual information to generate compact cuts, it significantly improves solution efficiency.

[0021] Advantages of additional aspects of the invention will be set forth in part in the description which follows, and in part will be obvious from the description, or may be learned by practice of the invention. Attached Figure Description

[0022] The accompanying drawings, which form part of this invention, are used to provide a further understanding of the invention. The illustrative embodiments of the invention and their descriptions are used to explain the invention and do not constitute an improper limitation of the invention.

[0023] Figure 1 This is a flowchart of the two-stage stochastic programming model described in Embodiment 1 of the present invention.

[0024] Figure 2 This is a schematic diagram of the first-stage planning process described in Embodiment 1 of the present invention.

[0025] Figure 3 This is a schematic diagram of the second-stage planning process described in Embodiment 1 of the present invention.

[0026] Figure 4 This is a distribution diagram of the 20-node test road network described in Embodiment 1 of the present invention.

[0027] Figure 5 This is a diagram illustrating the convergence process of Benders decomposition as described in Embodiment 1 of the present invention.

[0028] Figure 6 This is a diagram showing the first-stage truck service path and drone allocation relationship as described in Embodiment 1 of the present invention. Detailed Implementation

[0029] It should be noted that the following detailed descriptions are exemplary and intended to provide further illustration of the invention. Unless otherwise specified, all technical and scientific terms used herein have the same meaning as commonly understood by one of ordinary skill in the art to which this invention pertains.

[0030] It should be noted that the terminology used herein is for the purpose of describing particular implementations only and is not intended to limit the exemplary implementations of the present invention.

[0031] Where there is no conflict, the embodiments and features in the embodiments of the present invention can be combined with each other.

[0032] Example 1

[0033] In one or more embodiments, against the backdrop of frequent natural disasters and the severe challenge of the "golden 72 hours" for emergency logistics, the focus is on trucks in the post-disaster emergency material dispatch. This paper addresses the problem of drone-assisted delivery by presenting a method for optimizing emergency logistics routes based on truck-drone collaboration, considering the reliability and uncertainty of delivery paths. The method includes the following steps:

[0034] Step S1: Obtain candidate locations of material centers, disaster-stricken locations, and road networks. The disaster-stricken locations include emergency points and secondary emergency points.

[0035] Following a natural disaster, emergency supplies are transported from the material center I to two types of affected locations: emergency locations E and secondary emergency locations J. Based on the urgency of delivery, the affected locations are categorized: emergency locations require direct delivery via drones, while secondary emergency locations are transported by trucks along the road network. Due to post-disaster road damage, the original transportation route to secondary emergency location J may be disrupted, potentially causing supplies to be delayed. To address this uncertainty, this invention proposes a two-stage stochastic optimization model to guide the pre-positioning and scheduling decisions of emergency supplies.

[0036] In this embodiment, the geographical locations of the candidate material center, the secondary emergency point, and the emergency point, as well as the road network structure (nodes, arcs, and arc lengths), are all known information.

[0037] Whether each road segment is damaged is a random event and they are independent of each other. The probability of damage for each arc segment is known and does not change over time.

[0038] The reliability of a truck route is defined as the probability that all arcs on the route are undamaged, which is the product of the probabilities of each arc on the route being intact. Route reliability must meet a preset minimum threshold.

[0039] Each demand point is served only once by one vehicle, and the tasks of different vehicles are independent of each other.

[0040] Next, a two-stage mixed integer programming model with the goal of minimizing total cost is constructed. The two-stage decision-making logic is embedded through a constraint system, thus laying a solid model foundation for subsequent algorithm design and system analysis.

[0041] Step S2: Based on the candidate locations of the material center, the location of the disaster-stricken area, and the road network, construct the main problem to determine the location of the material center and the number of trucks and drones to be configured. The first stage optimizes the main problem to minimize operating costs. At the same time, in order to ensure the accessibility of the delivery task, calculate the reliability of each truck route according to the actual road arc damage probability, and ensure that it is not lower than the feasible threshold to reduce the risk of interruption.

[0042] The primary optimization problem in the first phase aims to minimize the system's operating cost in the first phase. This cost consists of the cost of setting up the material center, the cost of configuring trucks and drones, the transportation cost of the vehicles, and a lower bound on the expected cost in the second phase, defined as:

[0043] (1)

[0044] in, For operating costs; For the set of candidate points for material centers, index ; To establish a materials center Fixed costs, ; Indicate whether or not The establishment of a materials center is represented by binary data: 1 if it is established, and 0 otherwise. The configuration cost per truck, ; For the material center The number of trucks equipped, a non-negative integer; For a collection of trucks, index ; The number of drones equipped for the materials center i; The configuration cost of a single drone, ; For drone collections, index ; For arc segment Length (distance) ; Let V be a binary number indicating whether truck k passes through arc (i, j), where i,j∈V=I∪J (i.e., the material center and the secondary emergency point), and 1 if yes, 0 otherwise. For the set of less urgent points, index ; This is a set of reachable nodes for all trucks (supply center + secondary emergency point). ; The transportation cost per unit distance traveled by a truck. ; d represents the transportation cost per unit distance traveled by the drone. ; For the material center To the emergency point The straight-line distance; For emergency points Whether by the materials center The drone service d is a binary number, 1 if it is true and 0 otherwise; For the set of urgent points, index ; This is a lower bound approximation of the expected cost in the second stage.

[0045] Equations (2) and (3) are constraints on the consistency of facility opening and vehicle configuration, requiring that trucks and drones can only be configured in established material centers, and that the number of vehicles configured in each material center shall not be less than 1 (if opened), and shall not exceed a sufficiently large upper bound to avoid meaningless resource allocation.

[0046] (2)

[0047] (3)

[0048] in, It is a sufficiently large positive number.

[0049] Equations (4) and (5) are service coverage uniqueness constraints, ensuring that each secondary emergency point is served exactly once by a truck and each emergency point is served exactly once by a drone, thereby ensuring that all demand points are accurately covered.

[0050] (4)

[0051] (5)

[0052] Equations (6) and (7) are constraints on the number of vehicles used, stipulating that the number of trucks and drones departing from any material center shall not exceed the actual number configured in that center, in order to avoid overuse of resources.

[0053] (6)

[0054] (7)

[0055] Equation (8) is a node flow balance constraint, which ensures that each truck must leave after entering any node, thereby maintaining the continuity and integrity of the path.

[0056] (8)

[0057] Equation (9) is a path origin constraint, requiring each truck to start from one and only one material center to ensure that the origin of the delivery path is clear and unique.

[0058] (9)

[0059] Equations (10) and (11) are capacity constraints, which limit the total demand served by each truck and each drone to no more than their rated capacity, thus ensuring the feasibility of the transportation task.

[0060] (10)

[0061] (11)

[0062] in, This refers to the maximum loading capacity of a single truck. ; This represents the maximum payload capacity of a single drone. ; Secondary emergency point average demand ; For emergency points average demand .

[0063] Equation (12) is the flight range constraint for UAVs, which stipulates that the cumulative flight distance of each UAV during the mission shall not exceed its maximum flight range, so as to ensure flight safety and mission completion.

[0064] (12)

[0065] in, d represents the maximum range of a single drone. .

[0066] Equations (13)-(14) eliminate constraints for sub-loops. By introducing node access order variables, we ensure that the delivery path of each truck is an acyclic Hamiltonian path, thereby avoiding unreasonable local loops.

[0067] (13)

[0068] (14)

[0069] in, For trucks Access Node The order.

[0070] Equations (15) and (16) are path reliability constraints. The overall reliability of each truck path is defined as the product of the probability that the arc segment it passes through is not damaged. The reliability is required to be no less than a preset threshold to ensure the robustness of emergency material transportation.

[0071] (15)

[0072] (16)

[0073] in, For arc segment The probability of damage, ; For the set of road network arc segments, index ,in ; For trucks The reliability of the entire path; This is the path reliability threshold.

[0074] Equations (17) to (23) are non-negative and integer constraints, which limit the types of decision variables to ensure the mathematical rigor and feasibility of the model.

[0075] (17)

[0076] (18)

[0077] (19)

[0078] (20)

[0079] (twenty one)

[0080] (twenty two)

[0081] (twenty three)

[0082] In this embodiment, the path reliability of truck k is considered. , which is defined as the product of the probabilities that the traversed arc segment is not damaged, as shown in equation (15). This constraint is nonlinear, because and Taking the natural logarithm of both sides of the equation, we obtain an equivalent linear form:

[0083] (twenty four)

[0084] Further combined with reliability threshold constraints Its equivalent form is:

[0085] (25)

[0086] After the constraint is transformed into a linear form through logarithmic transformation, this invention achieves significant beneficial effects. First, this transformation makes the first-stage model a mixed-integer linear programming problem, which can be directly solved accurately using commercial solvers, avoiding the risk of nonlinear programming failing to solve or getting trapped in local optima during the critical 72 hours after a disaster. Second, linearization significantly improves computational efficiency, accelerating the process by tens of times under large-scale road networks, while nonlinear nonlinear models struggle to converge at the same scale, thus meeting the real-time decision-making needs of emergency logistics. Third, the logarithmic transformation is strictly equivalent, introducing no approximation errors, and accurately ensuring that the actual reliability of each truck route is not lower than the threshold, effectively selecting highly resilient routes in severely damaged disaster areas. Finally, the linearized reliability constraint retains the linear structure of integer variables, making the main problem still a mixed-integer linear programming problem and the subproblems linear programming problems, providing the necessary linear foundation for subsequent integer L-shaped Benders decomposition and marginal benefit cut generation strategies, which is a key prerequisite for achieving accurate and efficient co-optimization in the two stages.

[0087] Step S3: Calculate the probability that each secondary emergency point needs support based on the truck route reliability, construct a sub-problem based on the probability of needing support, and in the second stage, use a compact generation strategy based on marginal benefits to solve the sub-problem to minimize the support scheduling cost.

[0088] The sub-problem of the second-stage optimization aims to minimize the support scheduling cost of the system in the second stage. This cost consists of the support options for the interrupted disaster points. The first option is to dispatch trucks from the material center, including the truck transportation cost and the delayed rescue cost of the interrupted disaster point. The second option is to dispatch trucks from an uninterrupted secondary emergency point to support the interrupted disaster point, including the penalty cost of the interruption of demand at that secondary emergency point, the truck transportation cost, and the delayed rescue cost of the interrupted disaster point. The third option is to dispatch drones from the material center, including the drone transportation cost and the delayed rescue cost of the interrupted disaster point. The fourth option is to dispatch drones from an emergency point to support the interrupted disaster point, including the penalty cost of the interruption of demand at that emergency point, the truck transportation cost, and the delayed rescue cost of the interrupted disaster point. The fifth option is the penalty cost of the interrupted secondary emergency point not receiving support from any of the above four options, resulting in the interruption of demand at that point. Specifically, the sub-problem is constructed as follows:

[0089] (26)

[0090] in, Secondary emergency point Because of trucks The probability of interruption requiring support; Secondary emergency point The support scheme is represented by binary data, with 1 indicating support from trucks and 0 indicating support from drones. This indicates whether to dispatch a truck from supply center i to support emergency point j. It is binary data, with 1 for yes and 0 for no. From point Time The path distance; Secondary emergency point The unit time delay penalty coefficient; Secondary emergency point Emergency weight; k is the average speed of the truck. ; Let d be the average speed of the drone's flight. ; Indicates whether to start from point (Secondary emergency point) Dispatch trucks Support secondary emergency points , which is binary data; it is 1 if it is true and 0 otherwise. Indicate whether to transfer from the currently serving emergency point The drones were dispatched to support the secondary emergency point. , which is binary data; it is 1 if it is true and 0 otherwise. This indicates whether to dispatch a drone from material center i to support emergency point j. It is binary data, with 1 for yes and 0 for no. From point Time The straight-line distance; The penalty cost for service interruption of the demand at the second most urgent point j; For emergency points Fixed penalty for service interruption (when the service is in progress) (Generated during drone deployment). From the secondary emergency point predecessor node (i.e., in the first stage of the truck route) (the previous node) to the supported secondary emergency point The shortest path distance; To support the secondary emergency point To the next emergency point successor node (i.e., in the first stage of the truck route) The shortest path distance to the next node; From the emergency point To the next emergency point The straight-line (Euclidean) distance.

[0091] Equation (27) defines a constraint on the probability of interruption, stipulating that the probability that a secondary emergency point j needs support under the service of truck k is equal to the product of the probability of the truck route being interrupted and the indicator variable of whether the truck is serving the point, thereby linking the reliability of the first-stage route with the emergency needs of the second stage.

[0092] (27)

[0093] Equations (28)-(29) are constraints on the uniqueness of the dispatch source, requiring that each secondary emergency point j requiring support can only dispatch support vehicles from one material center or one secondary emergency point that is performing a mission in truck support mode; and can only dispatch support drones from one material center or one emergency point that is performing a mission in drone support mode, so as to ensure the uniqueness and operability of the support source.

[0094] (28)

[0095] (29)

[0096] Equations (30)-(31) are the original task existence constraints, which stipulate that if a support vehicle is dispatched from a point that is performing a task, the point must actually exist in the path planned in the first phase. That is, the dispatching behavior can only be carried out based on existing service tasks to avoid invalid dispatching.

[0097] (30)

[0098] (31)

[0099] Equations (32)-(33) are resource availability constraints, requiring that the number of support vehicles dispatched from the material center shall not exceed the number of available vehicles remaining in the center after the completion of the first phase of the task, thereby ensuring the actual availability of support resources.

[0100] (32)

[0101] (33)

[0102] Equations (34)-(35) are the vehicle range remaining constraints, stipulating that when a support vehicle is performing an emergency mission, its cumulative range (including the range used in the first phase and the range required for the support mission) shall not exceed its maximum range.

[0103] (34)

[0104] (35)

[0105] in, The first phase of the truck's range has been used; The first phase of the drone's flight range has been used. The maximum driving distance of a single truck.

[0106] Equations (36)-(41) represent non-negative and binary constraints.

[0107] (36)

[0108] (37)

[0109] (38)

[0110] (39)

[0111] (40)

[0112] (41)

[0113] Preferably, in the second stage, the second most urgent point Because of trucks Probability of path interruption requiring support Defined as shown in equation (27), this equation is the product of a continuous variable and a binary variable. Introducing the Big-M linearization method, it is equivalently replaced by the following set of linear constraints:

[0114] (42)

[0115] in, It is a sufficiently large positive number.

[0116] Furthermore, the objective function for the second stage is linearized by piecewise minimization. The objective function for the second stage includes the linearization of each... Minimum value structure:

[0117]

[0118] in, and These are the expected cost expressions for truck support and drone support, respectively. To eliminate this nonlinear term, an auxiliary variable is introduced. Indicates a point The optimal support decision cost is determined, and the following linear constraint is added:

[0119] (43)

[0120] in, It is a sufficiently large positive number; Secondary emergency point Fixed penalty for service interruption (when the service is in progress) (Generated during truck dispatch).

[0121] In this invention, the second-stage sub-problem requires calculating the probability that each secondary emergency point will require support due to road closures, based on the truck route reliability planned in the first stage. The system then determines whether to activate emergency support. To this end, an interruption probability threshold is introduced. The detailed implementation process is as follows:

[0122] threshold Used to distinguish between "high-probability interruption points" and "low-probability interruption points". (Current emergency point) Support probability If a path is deemed unsuitable, a second-stage emergency support plan is initiated; otherwise, the first-stage path is considered sufficiently reliable and does not incur additional rescue costs. The threshold is determined based on historical disaster data: analyzing the historical frequency of road interruptions in similar disasters, and taking the quantile that ensures a rescue success rate no lower than a preset level. This process is dynamically executed in each iteration, ensuring that rescue cost calculations are triggered only at points that genuinely require emergency support.

[0123] Step S4: Construct a two-stage stochastic programming model with the main problem and sub-problems. The optimal path is obtained by iteratively solving the total cost of operating costs and support scheduling costs to minimize the total cost.

[0124] This invention considers a two-stage stochastic programming model. The first stage includes integer decision variables (center location, vehicle configuration, route and allocation), while the second stage includes binary variables (emergency support plan). Direct solution faces significant computational challenges. To address this, an integer L-shaped Benders decomposition method is employed, decomposing the original problem into a master problem (MP) and subproblems (SP). The optimal solution is approximated by iteratively adding Benders cuts.

[0125] The two-stage stochastic programming model is decomposed into the main problem and subproblems mentioned above.

[0126] The main problem (MP) includes all integer variables in the first phase and one auxiliary variable. The latter represents the lower bound of the expected cost in the second stage. The MP form is:

[0127] (44)

[0128] in, For the already generated Benders cut set, For the first The cutting coefficient of the next iteration This represents the number of vehicle configurations in the current MP solution. The number of truck configurations in the current MP solution. This represents the number of drone configurations in the current MP solution. MP is a mixed-integer linear programming problem (MILP) solved using a commercial solver.

[0129] Given an MP solution, the subproblem (SP) evaluates the expected cost of the second stage. Since the SP contains binary variables (such as whether a certain rescue strategy is adopted), standard Benders decomposition cannot directly obtain dual information. Therefore, this invention employs two versions of the subproblem working collaboratively:

[0130] Continuous Version (LP-SP): Relaxes all binary variables into continuous variables to solve linear programming problems. This version is used to extract dual variables of resource constraints (such as the number of trucks and drones) to construct a Benders cut, driving the MP lower bound to increase.

[0131] Integer Version (MIP-SP): Solve mixed integer programming problems while keeping all binary variables unchanged. This version is used to calculate the actual expected cost of the second stage and obtain integer-feasible rescue solutions, which are then used to update the upper bound and the final output.

[0132] The two versions are controlled by the integer parameter, respectively calling the LP and MIP modes of the Gurobi solver, and employing a cut generation strategy, specifically:

[0133] In each iteration, the continuous subproblem (LP-SP) is solved first to obtain the optimal objective value. and the dual variables of resource constraints However, linear relaxation may underestimate the true cost, leading to overly loose cuts. To further improve the tightness of the cuts, this invention introduces marginal benefit analysis: for each center With other decisions remaining unchanged, the number of trucks will be configured accordingly. With the number of drone configurations Increment by 1, resolve the integer subproblem (MIP-SP), obtain the expected cost value for the second stage, and calculate the marginal benefit:

[0134] (45)

[0135] (46)

[0136] in, For the marginal benefits of trucks, For the marginal benefits of drones, Let the objective value be the current integer subproblem. The target value of the sub-problem obtained after adding truck configuration (the expected cost value of the second stage). To increase the objective value of the subproblem (the expected cost value in the second stage) after configuring the drone, if there is a non-zero marginal benefit, the following Benders cut is generated:

[0137] (47)

[0138] in, For the marginal benefits of trucks; To the marginal benefits of drones.

[0139] If all marginal benefits are zero, it degenerates into a cut based on linear relaxation duality:

[0140] (48)

[0141] in, For the first The optimal objective value of the continuous subproblem (LP-SP) in the next iteration.

[0142] like Figure 1 As shown, the specific steps of the algorithm for solving the two-stage stochastic programming model are as follows:

[0143] Step S401: Initialization: Set the number of iterations The lower realm Upper Realm , cut set .

[0144] Step S402: Solve the main problem: Solve the main problem MP, which includes all cuts, to obtain the optimal solution. and the objective function value of the master problem (MP) in the t-th iteration. Update the Nether ;in, For the first The material center location decision vector obtained by solving the main problem in the next iteration, where Indicates whether it is at the candidate point Establish a materials center; : No. The truck configuration vector obtained by solving the main problem in the next iteration, where Indicates the material center Number of trucks equipped; : No. The UAV configuration vector obtained by solving the main problem in the next iteration, where Indicates the material center The number of drones equipped; : No. The set of truck routing decision variables obtained from solving the main problem in the next iteration, among which Indicates truck Does it pass through an arc? ; : No. The set of UAV allocation decision variables obtained by solving the main problem in the next iteration, among which Indicates the material center drones Is the service urgent? .

[0145] Step S403: Solve the subproblems: With the first-stage solution fixed, solve the continuous subproblems (LP-SP) to obtain dual information. ,in, For the first In the next iteration, the continuous subproblems are related to the material center. The truck availability constraint represents the marginal reduction in expected costs for Phase 2 when one more truck is added to the center; For the first In the next iteration, the continuous subproblems are related to the material center. The drone availability constraint represents the marginal reduction in expected cost in the second phase when one drone is added to the center. Solve the integer subproblem (MIP-SP) to obtain the true cost. And integer feasible solutions, update the upper bound. .

[0146] Step S404: Generate a cut: If the marginal benefit is non-zero, generate a cut according to the marginal benefit cut formula; otherwise, generate a cut according to the linear relaxation dual formula; add the new cut. .

[0147] Step S405: Convergence judgment: If If the maximum number of iterations is reached, then stop. This is the convergence threshold; otherwise, the number of iterations is [number missing]. Proceed to step S402.

[0148] Step S406: Output the optimal first-stage solution and the second-stage integer feasible solution obtained by solving the integer subproblems of the optimal solution again.

[0149] The above design uses mathematical transformations to linearize the nonlinear constraints and objective function terms in the model, transforming the original model into a mixed integer linear programming (MILP) algorithm that can be solved efficiently. The algorithm can efficiently solve the established two-stage stochastic programming model, providing theoretical support and practical tools for emergency logistics decision-making.

[0150] This two-stage stochastic programming model and its solution algorithm first employ an integer L-shaped Benders decomposition framework, splitting the original problem into a main problem and subproblems. The main problem is responsible for the first-stage integer decision, while the subproblems evaluate the expected cost in the second stage. Iterative addition of Benders cuts achieves approximation of upper and lower bounds, ensuring global convergence. Second, a collaborative solution mechanism is designed, using continuous version (LP-SP) to extract dual information of resource constraints to generate cuts, and integer version (MIP-SP) to calculate the true expected cost and obtain integer feasible solutions, achieving an organic unity of lower bound-driven and upper bound-verified approaches. Third, marginal benefit analysis is introduced, by adding truck or drone configurations to each center and resolving the integer subproblems, tighter cut coefficients are obtained, effectively compensating for the underestimation of true cost by linear relaxation and significantly improving convergence speed. Finally, logarithmic linearization and reliability constraints are integrated, transforming the path success probability into a linear constraint. Combined with the efficient solution capabilities of the commercial solver (Gurobi), the computational complexity is significantly reduced while maintaining model accuracy.

[0151] In the above model, the locations of candidate material centers, disaster-stricken areas, and road networks are all known information, and the probability of damage to each road segment is independent. The two-stage decision-making process is coupled through vehicle allocation and support plans for secondary emergency points. The main problem is responsible for the first-stage site selection, resource allocation, and service distribution decisions. Sub-problems evaluate the expected cost of emergency support in the second stage based on the main problem's solutions, and minimize the overall system cost by generating and continuously updating the lower bound of the cost in the main problem. The specific flowchart is as follows... Figure 2 and Figure 3 As shown.

[0152] Furthermore, since supporting secondary emergency points disrupted by road damage requires interrupting the execution of current task nodes at disaster-stricken points, a recursive mechanism for disaster-stricken point support is followed, with vehicles from other nodes dispatched to deliver supplies until all disaster-stricken points are fully covered and served only once. This study aims to minimize the total cost of the first and second phases, including the location cost, vehicle configuration cost, and transportation cost of the first phase, and the support scheduling cost of the interrupted points due to road disruptions in the second phase. Key decisions include: the order in which trucks access secondary emergency points; the allocation of drones from service relationships to emergency points; the probability of interruption for secondary emergency points requiring support due to road disruptions; and the support plan for the interrupted disaster-stricken points.

[0153] Furthermore, numerical experiments were conducted on the method proposed in this invention.

[0154] An integer L-shaped Benders decomposition algorithm was implemented on the Python platform. Experimental analysis of the model and algorithm was conducted using a test network with 20 nodes (3 resource centers, 10 secondary emergency points, and 7 emergency points). The experimental environment consisted of an 11th Gen Intel(R) Core(TM) i5-1135G7 @ 2.40GHz processor and 16 GB of memory. Both the main problem and subproblems used the Gurobi solver. The maximum number of iterations for Benders decomposition was set to 10, and the optimality gap threshold was set to 0.1% to verify the effectiveness of the model. The spatial distribution and topology of the nodes are shown below. Figure 4 As shown.

[0155] The network structure comprises 3 material centers, 10 secondary emergency points, and 7 emergency points, with their latitude and longitude coordinates manually determined based on the actual administrative divisions of Jinan City. The road network contains 78 undirected edges, each with an interruption probability randomly generated between 0.10 and 0.65. The path reliability threshold is set to 0.38, and the interruption judgment threshold for secondary emergency points is 0.4. In terms of cost parameters, the configuration costs for trucks and drones are 2000 and 8000 respectively, with unit transportation costs of 10 yuan / km and 15 yuan / km respectively. All parameter values ​​are based on the emergency material reserve standards of a certain province and historical post-disaster dispatch data to ensure the typicality and credibility of the case study.

[0156] To verify the convergence performance of the proposed integer L-shaped Benders decomposition algorithm, the solution results of the main problem, the objective values ​​of the subproblems, the updates of the upper and lower bounds, and the solution time were recorded for each iteration. The iterative process of Benders decomposition, as follows... Figure 5 As shown: In the first iteration: the lower bound of the main problem was 37255.74 yuan, the subproblem objective was 913.01 yuan, the upper bound was 38168.74 yuan, and the initial gap was 2.39%. A feasible solution was obtained through the integer subproblem, and the first Benders cut was generated. In the second iteration: after adding the Benders cut, the lower bound of the main problem increased to 38168.75 yuan, the subproblem objective became 1035.21 yuan, and the upper bound was 38290.96 yuan, but the optimal upper bound remained at 38168.74 yuan. Therefore, the actual gap decreased to 1.68 × 10^6 yuan. -5 The convergence condition is met.

[0157] The algorithm converged in just two iterations, indicating that the initial Benders cut was already very compact and effectively pruned the tree. The first iteration took a relatively long time to solve the main problem (approximately 502 seconds), mainly due to the large number of variables and complex constraints in the initial main problem, and the absence of any effective cuts. In the second iteration, since a compact cut had been added, although the main problem size did not decrease, the feasible region was significantly reduced, and the solution time plummeted to 0.4 seconds, demonstrating the significant effect of Benders decomposition in reducing the search space of the main problem. In the cut generation process, this embodiment incorporates a coefficient scaling mechanism. The cut coefficients generated in the first iteration were all 0, requiring no scaling; although the second iteration did not generate cuts containing resources, the algorithm had reserved scaling logic to handle large coefficient cases. No numerical instability warnings appeared during the overall solution process, indicating that the model's numerical properties are good.

[0158] Furthermore, the optimal solution is analyzed. In the optimal solution, all three candidate material centers are opened, and the specific resource allocation is as follows:

[0159] Center I0 is equipped with one truck and one drone. The truck at the center was responsible for serving three secondary emergency points in subsequent route planning, while the drone was responsible for serving four emergency points. All resources were fully utilized with no idle time.

[0160] Center I1 is equipped with one truck and one drone. The truck serves one secondary emergency point, and the drone serves two emergency points. Both the truck and the drone were deployed for emergency support missions during the second phase, maximizing resource utilization.

[0161] Center I2 is equipped with two trucks and one drone. This center bears the heaviest workload, with the two trucks serving two and four secondary emergency points respectively, and the drone serving one emergency point. Center I2 is centrally located, close to several secondary emergency points, and has relatively good road conditions; therefore, the model allocates more tasks to it, reflecting the optimization logic of on-demand resource allocation.

[0162] In the first phase of route planning, the service routes for trucks and the allocation relationships for drones, such as... Figure 6 As shown. Regarding trucks, all four trucks were put into use in the first phase, with T2 having the highest load rate (86.3%) and T3 the lowest (15.8%). Regarding drones, all three drones were put into use in the first phase, with U2 having the heaviest load (cumulative range of 32.95 km) and U0 the lightest load (7.46 km).

[0163] The specific route sequence of the trucks is shown in Table 1. The truck load rates range from 15.8% to 86.3%, showing a significant difference. This precisely illustrates that the model can flexibly allocate resources according to demand distribution, rather than mechanically pursuing load balancing, reflecting the essential characteristic of "on-demand scheduling" in emergency logistics.

[0164] Table 1. Truck Phase 1 Service Route

[0165]

[0166] Based on the reliability of the first-stage path, the probability of interruption at each emergency point is calculated. Points exceeding a threshold of 0.4 trigger the second-stage support. The support scheme is shown in Table 2.

[0167] Table 2. Phase Two Support Plan

[0168]

[0169] In the second phase of emergency support decision-making, the model selects different support methods for the four secondary emergency points (J4, J6, J7, J9) with interruption probabilities exceeding the threshold, as well as the emergency point E2 that has been interrupted. These schemes are the optimal combinations obtained by the model through exact solution using integer programming after comprehensively considering cost, resource availability, range constraints, interruption probability, and global coordination.

[0170] From a cost minimization perspective, the model selects the lowest-cost support method among the feasible options for each disruption point. Taking J6 as an example, dispatching a drone from emergency point E2 costs only 398.76 yuan, far lower than the 1216.10 yuan for dispatching an idle truck from the center, thus becoming the optimal choice. J7 is supported by the already used drone U7 at a cost of 349.14 yuan, lower than the truck support option; J4 and J9, due to demand exceeding drone capacity, can only be supported by trucks, and the model selects the lowest-cost option of using existing trucks.

[0171] From a resource availability perspective, all support schemes strictly meet range constraints. The model has no idle vehicles available and can only rely on the remaining range of already used vehicles, achieving a "peacetime-wartime integration" and maximizing resource utilization.

[0172] From a global coordination perspective, the model considers support decisions for both secondary and emergency points in a unified manner. E2 was interrupted due to the original drone being reassigned, and was supported by U0 from center I2. This drone only served one point in the first phase, retaining a redundant range of 32.54 kilometers, which just met E2's round-trip requirement of 29.08 kilometers, demonstrating the wisdom of resource reservation and overall optimization.

[0173] From a cost-effectiveness perspective, without any support, the total penalty cost for the four secondary emergency points would reach RMB 2,281,228, while the actual total cost of the second phase was only RMB 913.01, fully demonstrating the necessity and cost-effectiveness of the emergency support mechanism. In this embodiment, the first phase coordinates the selection of the material center location, vehicle configuration, and route planning, while the second phase optimizes the emergency support strategy for road interruption scenarios. Through interruption and delay penalty mechanisms, the system's robustness is ensured while minimizing the total cost. Experiments show that the second phase, with a support cost of only RMB 913, can avoid potential penalty costs exceeding RMB 2.28 million, fully validating the cost-effectiveness and effectiveness of the framework.

[0174] In this invention, the resource on-demand scheduling and dynamic emergency support mechanism achieve an optimal balance in overall system efficiency. In the optimal solution, all three material centers are fully utilized, and all trucks and drones are fully leveraged, with load rates ranging from 15.8% to 86.3%, reflecting the resource allocation logic of "on-demand scheduling and peacetime-wartime integration." In the second phase, the model precisely matches truck or drone support solutions for each interruption point based on the principles of interruption probability, remaining resource range, and cost minimization, achieving efficient resource reuse and global coordination.

[0175] In summary, the proposed integer L-shaped Benders decomposition algorithm converges in just two iterations, with the second iteration solving the main problem in only 0.4 seconds. It also exhibits good numerical stability, verifying the algorithm's efficiency and reliability. The model effectively coordinates the establishment of supply centers, resource allocation, and route planning, and precisely addresses the uncertainties caused by road disruptions in the second phase. In the optimal solution, all three supply centers are operational, all trucks and drones are fully utilized, resources are dispatched on demand with no idle resources, and in the second phase, the model effectively avoids potential penalty costs exceeding 2.28 million yuan with a total support cost of 913 yuan, fully demonstrating the model's optimization capabilities and cost-effectiveness in emergency logistics scenarios.

[0176] In summary, the exact decomposition solution framework for two-stage stochastic programming models proposed in this invention, unlike traditional heuristic methods, achieves rigorous co-optimization of the first-stage integer decision-making and the second-stage emergency support. Furthermore, the algorithm designs a compact cut generation strategy based on marginal benefits. By introducing sensitivity analysis of integer subproblems, it effectively overcomes the overly loose nature of linear relaxation cuts, accelerating the convergence process while ensuring optimality. Crucially, the algorithm integrates a complete convergence recording and analysis module, capable of real-time tracking of upper and lower bound evolution, optimal gap changes, and cut set growth, providing direct evidence for algorithm performance evaluation and parameter tuning. This module also outputs detailed vehicle routing and UAV allocation schemes, deepening insights into the operational efficiency and resource utilization of emergency logistics systems. This integer L-shaped Benders decomposition method not only serves as an exact algorithm to obtain the global optimum within an acceptable timeframe, but its generated cut and dual information can also provide valuable references for other approximation methods (such as Lagrange relaxation and heuristic search), thus providing a rigorous and practical method for solving such complex emergency logistics planning problems.

[0177] Example 2

[0178] In one or more embodiments, an emergency logistics route optimization system based on truck and drone collaboration is disclosed, specifically including:

[0179] The network acquisition module is configured to acquire candidate points of material centers, locations of disaster-stricken points, and road networks, wherein the disaster-stricken points include emergency points and secondary emergency points.

[0180] The first optimization module is configured to: based on the candidate locations of the material center, the location of the disaster-stricken area, and the road network, construct the main problem to determine the location of the material center and the number of trucks and drones to be deployed; optimize the main problem in the first stage to minimize operating costs; at the same time, calculate the reliability of each truck route based on the actual road arc damage probability.

[0181] The second optimization module is configured to: calculate the probability that each secondary emergency point needs support based on the truck route reliability, construct a sub-problem based on the probability that support is needed, and in the second stage, use a compact generation strategy based on marginal benefits to solve the sub-problem to minimize the support scheduling cost.

[0182] The planning and solving module is configured to: construct a two-stage stochastic programming model with the main problem and sub-problems, and obtain the optimal path by iteratively solving the total cost of operating cost and support scheduling cost.

[0183] Example 3

[0184] This embodiment provides an electronic device, including a memory and a processor, as well as computer instructions stored in the memory and running on the processor. When the computer instructions are executed by the processor, they complete the steps of the above-mentioned emergency logistics route optimization method based on truck and drone collaboration.

[0185] Example 4

[0186] This embodiment provides a computer-readable storage medium for storing computer instructions, which, when executed by a processor, complete the steps of the above-described emergency logistics route optimization method based on truck and drone collaboration.

[0187] 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.

[0188] 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.

[0189] These computer program instructions may also be loaded onto a computer or other programmable data processing equipment, whereby a series of operational steps are performed 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.

[0190] The descriptions of each embodiment in the above embodiments have different focuses. For parts not described in detail in a certain embodiment, please refer to the relevant descriptions in other embodiments.

[0191] The above description is merely a preferred embodiment of the present invention and is not intended to limit the invention. Various modifications and variations can be made to the present invention by those skilled in the art. Any modifications, equivalent substitutions, improvements, etc., made within the spirit and principles of the present invention should be included within the scope of protection of the present invention.

Claims

1. An emergency logistics route optimization method based on truck and drone collaboration, characterized in that, include: Acquire candidate locations of material centers, disaster-stricken areas, and road networks, including emergency points and secondary emergency points; Based on the candidate locations of the material centers, the locations of the disaster-stricken areas, and the road network, a main problem is constructed to determine the location of the material centers and the number of trucks and drones to be deployed. The first stage optimizes the main problem to minimize operating costs. At the same time, the reliability of each truck route is calculated based on the actual probability of damage to road arcs. Based on the truck route reliability, the probability of needing support for each secondary emergency point is calculated. Sub-problems are constructed according to the probability of needing support. In the second stage, a compact generation strategy based on marginal benefits is used to solve the sub-problems to minimize the support scheduling cost. A two-stage stochastic programming model is constructed using the main problem and sub-problems. The optimal path is obtained by iteratively minimizing the total cost of operating costs and support scheduling costs. The method employs a compact generation strategy based on marginal benefits to solve subproblems in order to minimize support scheduling costs, and solves subproblems collaboratively using continuous and integer versions: In each iteration, the continuous subproblems are solved first to obtain the optimal objective value and the dual variables of the resource constraints; For each center, keeping other decisions unchanged, increase the number of trucks and drones by 1 respectively, resolve the integer subproblem to obtain the expected cost value of the second stage, and calculate the marginal benefit; If there are non-zero marginal benefits, a Benders cut is generated; if all marginal benefits are zero, it degenerates into a cut based on linear relaxation duality. The solution algorithm for the two-stage stochastic programming model is as follows: Initialization: Set the number of iterations The lower realm Upper Realm , cut set ; Solving the principal problem: Solve the principal problem MP, which includes all cuts, to obtain the optimal solution and the objective function value of the principal problem in the t-th iteration. Update the Nether ; Solving subproblems: With the first-stage solution fixed, solve continuous subproblems to obtain dual information, which includes the solution of the first stage. In the next iteration, the continuous subproblems are related to the material center. Truck availability constraints, the first In the next iteration, the continuous subproblems are related to the material center. UAV availability constraints and the first The optimal objective value of the continuous subproblems in the next iteration; solving the integer subproblems to obtain the true cost. And integer feasible solutions, update the upper bound. ; Generative cut: If the marginal benefit is non-zero, generate a cut according to the marginal benefit cut formula; otherwise, generate a cut according to the linear relaxation dual formula; add the new cut. ; Convergence criterion: If If the maximum number of iterations is reached, then stop. This is the convergence threshold; otherwise, the number of iterations is [number missing]. Then, solve the main problem. Output the optimal first-stage solution and the second-stage integer feasible solution obtained by solving the integer subproblems of the optimal solution again.

2. The emergency logistics route optimization method based on truck and drone collaboration as described in claim 1, characterized in that, The main problem is to minimize operating costs, expressed as: in, For operating costs; For the set of candidate points for material centers, index ; To establish a materials center Fixed costs; Indicate whether or not Establish a materials center; Configuration cost per truck; For the material center Number of trucks equipped; For a collection of trucks, index ; The number of drones equipped for the materials center i; The configuration cost of a single drone; For drone collections, index ; For arc segment Length; Let V be the arc (i, j) to determine whether truck k passes through arc (i, j), where i, j ∈ V = I ∪ J; For the set of less urgent points, index ; For the set of reachable nodes for all trucks, ; The transportation cost per unit distance traveled by a truck. ; d represents the transportation cost per unit distance traveled by the drone. ; For the material center To the emergency point The straight-line distance; For emergency points Whether by the materials center Drone services; For the set of urgent points, index ; This is a lower bound approximation of the expected cost in the second stage.

3. The emergency logistics route optimization method based on truck and drone collaboration as described in claim 1, characterized in that, The reliability of each truck route is calculated based on the actual probability of damage to road arcs. in, For arc segment The probability of damage; For the set of road network arc segments; For trucks The reliability of the entire path; This is the path reliability threshold; Let's determine whether truck k passes through arc (i, j).

4. The emergency logistics route optimization method based on truck and drone collaboration as described in claim 1, characterized in that, Based on the truck route reliability, calculate the probability that support is needed for each secondary emergency point: in, Secondary emergency point Because of trucks The probability of interruption requiring support; Determine whether truck k passes through arc (i, j); For trucks The reliability of the entire path; For the set of less urgent points, index ; For a collection of trucks, index ; This is the set of reachable nodes for all trucks.

5. The emergency logistics route optimization method based on truck and drone collaboration as described in claim 1, characterized in that, The subproblem minimizes the support scheduling cost of the second phase of the system, which consists of the support scheme cost for the interrupted disaster points, specifically: in, Secondary emergency point Because of trucks The probability of interruption requiring support; Secondary emergency point Support solutions; Indicate whether to dispatch trucks from supply center i to support emergency point j; From point Time The path distance; Secondary emergency point The unit time delay penalty coefficient; Secondary emergency point Emergency weight; k is the average speed of the truck. ; Let d be the average speed of the drone's flight. ; Indicates whether to start from point Dispatch trucks Support secondary emergency points ; Indicate whether to transfer from the currently serving emergency point The drones were dispatched to support the secondary emergency point. ; Indicate whether to dispatch drones from the supplies center i to support the secondary emergency point j; From point Time The straight-line distance; The penalty cost for service interruption of the demand at the second most urgent point j; For emergency points Fixed penalties for service interruption; From the secondary emergency point predecessor node To the secondary emergency point being supported The shortest path distance; To support the secondary emergency point To the next emergency point successor node The shortest path distance; From the emergency point To the next emergency point The straight-line distance.

6. An emergency logistics route optimization system based on truck and drone collaboration, characterized in that, include: The network acquisition module is configured to acquire candidate points of material centers, locations of disaster-stricken points, and road networks, wherein the disaster-stricken points include emergency points and secondary emergency points. The first optimization module is configured to: based on the candidate locations of the material center, the location of the disaster-stricken area, and the road network, construct the main problem to determine the location of the material center and the number of trucks and drones to be deployed; optimize the main problem in the first stage to minimize operating costs; at the same time, calculate the reliability of each truck route based on the actual road arc damage probability. The second optimization module is configured to: calculate the probability that each secondary emergency point needs support based on the truck route reliability, construct a sub-problem based on the probability that support is needed, and in the second stage, use a compact generation strategy based on marginal benefits to solve the sub-problem to minimize the support scheduling cost. The planning and solving module is configured to: construct a two-stage stochastic programming model with the main problem and sub-problems, and obtain the optimal path by iteratively solving the total cost of operating cost and support scheduling cost to minimize the optimal path; The method employs a compact generation strategy based on marginal benefits to solve subproblems in order to minimize support scheduling costs, and solves subproblems collaboratively using continuous and integer versions: In each iteration, the continuous subproblems are solved first to obtain the optimal objective value and the dual variables of the resource constraints; For each center, keeping other decisions unchanged, increase the number of trucks and drones by 1 respectively, resolve the integer subproblem to obtain the expected cost value of the second stage, and calculate the marginal benefit; If there are non-zero marginal benefits, a Benders cut is generated; if all marginal benefits are zero, it degenerates into a cut based on linear relaxation duality. The solution algorithm for the two-stage stochastic programming model is as follows: Initialization: Set the number of iterations The lower realm Upper Realm , cut set ; Solving the principal problem: Solve the principal problem MP, which includes all cuts, to obtain the optimal solution and the objective function value of the principal problem in the t-th iteration. Update the Nether ; Solving subproblems: With the first-stage solution fixed, solve continuous subproblems to obtain dual information, which includes the solution of the first stage. In the next iteration, the continuous subproblems are related to the material center. Truck availability constraints, the first In the next iteration, the continuous subproblems are related to the material center. UAV availability constraints and the first The optimal objective value of the continuous subproblems in the next iteration; solving the integer subproblems to obtain the true cost. And integer feasible solutions, update the upper bound. ; Generative cut: If the marginal benefit is non-zero, generate a cut according to the marginal benefit cut formula; otherwise, generate a cut according to the linear relaxation dual formula; add the new cut. ; Convergence criterion: If If the maximum number of iterations is reached, then stop. This is the convergence threshold; otherwise, the number of iterations is [number missing]. Then, solve the main problem. Output the optimal first-stage solution and the second-stage integer feasible solution obtained by solving the integer subproblems of the optimal solution again.

7. An electronic device, characterized in that, It includes a memory and a processor, as well as computer instructions stored in the memory and running on the processor, which, when executed by the processor, perform the emergency logistics route optimization method based on truck and drone collaboration as described in any one of claims 1-5.

8. A computer-readable storage medium, characterized in that, Used to store computer instructions, which, when executed by a processor, complete the emergency logistics route optimization method based on truck and drone collaboration as described in any one of claims 1-5.

Images