Pickup distribution method based on dynamic modeling of customer pickup behavior and space-time replenishment strategy

Abstract

The application discloses a self-pickup distribution method based on dynamic modeling of customer self-pickup behavior and space-time replenishment strategy, and comprises the following steps: acquiring customer self-pickup behavior data, constructing a model based on a truncated normal distribution, determining a probability density function and a probability function; establishing a self-pickup cabinet available capacity and replenishment model, calculating the residual capacity at each moment, determining the replenishment opportunity and vehicle waiting time and cost; constructing a self-pickup distribution mathematical model with the minimum operation cost as an objective function, and performing self-pickup cabinet site selection, capacity planning, replenishment decision and vehicle path optimization; designing a two-stage heuristic algorithm for solution, generating a self-pickup cabinet site selection, capacity level configuration, replenishment opportunity decision and vehicle path planning scheme, and realizing operation cost optimization. Through dynamic modeling and space-time replenishment strategy, the application improves the space-time utilization rate of the self-pickup cabinet, reduces the operation cost, and provides an efficient solution for express end distribution.

Description

Technical Field

[0001] This invention relates to the field of last-mile delivery technology, specifically to a self-pickup delivery method based on dynamic modeling of customer self-pickup behavior and spatiotemporal replenishment strategies. Background Technology

[0002] With the booming development of online retail, express delivery volume has experienced explosive growth, highlighting the increasingly prominent issue of last-mile delivery. Currently, last-mile delivery faces numerous challenges, including dispersed customer demand, high delivery costs, and low efficiency. Traditional delivery models are no longer adequate to meet the demands of the rapidly developing e-commerce market. While self-pickup services, especially the application of self-service lockers, can reduce delivery costs and improve efficiency to some extent, many problems still need to be addressed in actual operation. Therefore, a new technological solution is urgently needed to optimize the self-pickup delivery system and improve its operational efficiency and economics.

[0003] Traditional self-pickup delivery solutions primarily focus on the static capacity design of self-pickup lockers and a single-delivery model. While this approach can meet basic delivery needs under certain conditions and offers advantages such as relatively simple operation and low planning complexity, its drawbacks are also significant. Because it ignores the dynamic characteristics of customer self-pickup behavior, it cannot accurately grasp the real-time changes in locker capacity, leading to low utilization rates and frequent instances of insufficient or idle lockers. This, in turn, affects the efficiency of the entire delivery system and increases operating costs.

[0004] Existing technologies have explored some aspects of optimizing self-pickup delivery systems. Some studies have considered the location and capacity planning of self-pickup lockers, but most have not integrated customer self-pickup behavior with replenishment strategies. The advantage of existing technologies is that they have conducted some research on the basic layout and capacity of self-pickup lockers, providing a foundation for system optimization. However, their shortcomings lie in the lack of in-depth analysis of the temporal characteristics of customer self-pickup behavior, making it impossible to achieve dynamic management of locker capacity and accurate replenishment. This results in poor flexibility and adaptability of the delivery system, making it difficult to cope with complex real-world operational scenarios. Summary of the Invention

[0005] Based on the above, this application discloses a self-pickup and delivery method based on dynamic modeling of customer self-pickup behavior and spatiotemporal replenishment strategies, which solves the aforementioned technical problems, including:

[0006] S1: Obtain customer self-pickup behavior data, construct a customer self-pickup behavior model based on truncated normal distribution, and determine the probability density function and probability function of customer self-pickup behavior;

[0007] S2: Establish a model for the available capacity and replenishment of self-pickup lockers. Calculate the remaining capacity of the self-pickup lockers at each time point using the probability function of customer self-pickup behavior. Determine the replenishment timing based on the remaining capacity. When the available capacity of the self-pickup lockers meets the replenishment demand, arrange vehicles for replenishment. When the available capacity of the self-pickup lockers cannot meet the replenishment demand, decide whether the vehicle should continue to the next self-pickup locker for delivery or stop nearby and wait. If the vehicle chooses to wait nearby, calculate the vehicle waiting time and waiting cost.

[0008] S3: Construct a self-pickup delivery mathematical model, which uses the minimum operating cost as the objective function to perform site selection, capacity planning, replenishment decision-making, and vehicle route optimization for self-pickup lockers;

[0009] S4: Design a two-stage heuristic algorithm to solve the self-pickup delivery mathematical model. The first stage generates an initial solution through a three-stage progressive construction algorithm, including self-pickup locker location selection, customer allocation, capacity level decision, construction of initial delivery path and construction of replenishment path; the second stage uses variable neighborhood search for post-optimization, including neighborhood structure design, adaptive jitter operation and local search algorithm.

[0010] S5: Based on the solution results of the two-stage heuristic algorithm, generate a self-pickup locker location scheme, a capacity level configuration scheme, a replenishment timing decision scheme, and a vehicle route planning scheme to optimize the operating costs of self-pickup delivery.

[0011] Preferably, in step S1, customer self-pickup behavior data is acquired, and a customer self-pickup behavior model is constructed based on a truncated normal distribution to determine the probability density function and probability function of customer self-pickup behavior. Specifically, through statistical analysis of customer self-pickup behavior data, it is found that the customer's pickup time interval follows a truncated normal distribution. The customer's pickup behavior is modeled as the probability of pickup at time t after receiving the express delivery information. The probability density function formula is: Where 'a' represents the lower cutoff point, i.e., the time when the first successful delivery is completed, 'μ' is the mean of the distribution, 'b' is the upper cutoff point, set as the end time of the cycle, 'σ' is the standard deviation of the distribution, and 'θ' is the normalization coefficient. Meeting the corresponding conditions, the probability of a customer's self-pickup behavior at any time 't' is obtained by integrating the probability density function, as shown in the formula: When t = a, G(t) = 0, and the probability of the customer completing self-pickup approaches 0. When t → b, G(t) = 1, and the probability of the customer completing self-pickup approaches 1.

[0012] Preferably, in step S2, a model for the available capacity and replenishment of the self-pickup locker is established. The remaining capacity of the self-pickup locker at each time point is calculated cumulatively using the customer self-pickup behavior probability function. The replenishment timing is determined based on the remaining capacity. Specifically, the set of customers serving self-pickup locker p is N. p The total demand is The time when vehicle k first arrives at locker p is The time required to complete the express delivery is The formula for calculating the remaining capacity of the self-service locker at any time t is: u is the integral variable; when the available capacity of the self-service locker is sufficient to meet the replenishment demand, a vehicle is arranged to replenish the goods. Each vehicle replenishes each locker at most once, and the replenishment must meet the replenishment demand. When the vehicle arrives at the locker for the second time, if the available capacity of the locker meets the demand of the remaining customers, the vehicle immediately replenishes the goods, and the delivery quantity is... If the demand is not met, the vehicle will park at the nearest available locker and wait until the remaining capacity meets the remaining needs before restocking. The restocking time will be [date / time]. Vehicle waiting time is in, Waiting costs for the vehicle's second arrival at the self-service locker ω represents the vehicle's waiting cost per unit time.

[0013] Preferably, when the vehicle arrives at the self-service locker for the second time, the determination and handling of the locker's available capacity and replenishment demand includes two methods: Replenishment Method 1, the remaining capacity of the locker at the time of the vehicle's second arrival. When the remaining customer demand is met, that is... The vehicle at the time of arrival The self-service lockers were restocked a second time, with a delivery volume of [number missing]. Replenishment successful. The actual replenishment start time for the vehicle is [time]. Replenishment Method 2: Remaining capacity of the self-service locker at the time of the vehicle's second arrival. The vehicle will park nearby and wait until the remaining capacity of the self-service locker meets the remaining demand for restocking. at this time Vehicle waiting time and Waiting costs Calculate, where ω is the vehicle's waiting cost per unit time.

[0014] Preferably, the operating costs in S3 include fixed vehicle usage costs, route costs, self-service locker opening costs, costs for exceeding self-service distance rewards, self-service costs, and vehicle waiting costs, and decision variables and constraints are set accordingly; the decision variables include vehicle arrival time, self-service locker delivery volume, vehicle loading capacity, and self-service locker opening status, and the constraints include vehicle route balance constraints, unique customer allocation constraints, self-service locker capacity constraints, and vehicle load and time constraints.

[0015] Preferably, the mathematical model for self-pickup delivery constructed in S3 is as follows: based on a directed graph G(V, A), where V is a set of nodes including the delivery center's starting point 0 and return point 0', a set of candidate self-pickup lockers P, and a set of virtual nodes directly related to the self-pickup lockers. And a set of self-pickup customers N, where A is an arc set containing the path arc set A. R and self-extracting arc set A C The decision variables include the time it takes for vehicle k to arrive at node i. Delivery volume u to self-service locker p pk Loading amount on arc (i, j) A 0-1 variable indicating whether the vehicle is traveling on arc (i, j) The 0-1 variable of the self-service locker's open level s. Customer i is a 0-1 variable representing the self-service locker p of level s. 0-1 variable for whether to replenish stock and vehicle waiting time The objective function is: Where K represents the vehicle set of the distribution center, F k For the fixed operating cost of vehicle k, c ij Let f(d) be the unit path cost of the vehicle traveling on arc (i,j), S be the set of selectable capacity levels for the self-service lockers, and f(d) be the unit path cost of the vehicle traveling on arc (i,j). ip Let ) be the reward cost function, and c m The cost of self-pickup service for the unit demand, q i This represents the parcel demand of customer i.

[0016] Preferably, in step S4, the first stage generates an initial solution using a three-stage progressive construction algorithm, including locker location selection, customer allocation, capacity level decision, construction of initial delivery routes, and construction of replenishment routes. Specifically, the first stage generates an initial solution using a three-stage progressive construction algorithm. In stage 1, all lockers P are opened, and customers are allocated to lockers according to the principle of minimum pickup cost. The set of lockers P allocated to customers is recorded. open In Phase 2, based on vehicle load and mileage limitations, a set of delivery vehicle routes R0 for single access to self-pickup lockers is constructed using the CW saving algorithm. In Phase 3, the capacity level of self-pickup lockers p in the route is gradually reduced, and a replenishment route is constructed based on the available capacity model and vehicle parking model according to the minimum cost insertion principle.

[0017] Preferably, the second stage in S4 utilizes variable neighborhood search for post-optimization, including neighborhood structure design, adaptive jitter operation, and local search algorithm. Specifically, the second stage utilizes variable neighborhood search for post-optimization, where the neighborhood set H = {N}. L1 NL2 N L3 N V N C1 N C2}, where N L1 N L2 N L3 These represent the locker being closed, open, and its status being reversed, respectively. N V To determine the neighborhood structure for the self-service locker capacity, N C1 N C2 For customer movement and customer swapping, a neighborhood k is selected from H via adaptive jitter operation. s The probability ρ of being selected during the i-th jitter. k，i =w k，i / ∑ k∈H w k,i The weight update formula is: in ψ is the weight decay factor and ψ is the neighborhood contribution coefficient. By designing a local search algorithm that includes path optimization and replenishment optimization, the path optimization adopts a prohibition strategy to avoid inserting different paths for two visits to the self-pickup locker that needs replenishment. The replenishment optimization traverses all possible insertion positions of the self-pickup locker that needs replenishment and selects the replenishment position with the minimum cost.

[0018] Preferably, the fitness function in the two-stage heuristic algorithm is F. fit (X)=F(X)+αP(X)+βQ(X), where F(X) is the objective function value. For vehicle load violation, U k For the maximum load limit of the vehicle, A R For path arc set, Let Q(X) be the load from return point 0' to node j. For violations of driving time, T is the maximum permissible driving time for the vehicle. k The actual driving time of the vehicle is denoted by α and β, which are penalty factors. The heuristic algorithm terminates when the maximum optimal solution is reached without further improvement (ITER). best Alternatively, the maximum number of iterations (ITER) can be used, and an adaptive shanking strategy can be adopted in each main loop during the variable neighborhood search phase. Duplicate solutions are avoided through a hash table, and the optimal solution among all threads is selected for local search. When the local search fails to improve the optimal solution, the neighborhood weight is updated; otherwise, the counter for failure to improve the optimal solution is reset.

[0019] Preferably, in step S5, based on the solution results of the two-stage heuristic algorithm, a self-service locker location scheme, a capacity level configuration scheme, a replenishment timing decision scheme, and a vehicle route planning scheme are generated to optimize the operating cost of the self-service delivery system. Specifically, the values ​​of the decision variables are obtained by solving the two-stage heuristic algorithm, including the self-service locker open state variable. Determine the location and capacity level configuration of the self-pickup lockers based on the delivery volume. pk The available capacity Q of the self-pickup locker is calculated based on the probability function G(t) of customer self-pickup behavior. p (t) Determine the replenishment timing using vehicle driving variables. and arrival time Generate vehicle routing plans and optimize operating costs by reducing the opening costs of self-service lockers, vehicle routing costs, and replenishment waiting costs through various decision-making schemes.

[0020] Compared with the prior art, the technical solution of this application has the following technical effects:

[0021] This invention dynamically models customer self-pickup behavior by constructing a model based on a truncated normal distribution using data from the time interval between receiving delivery confirmation and completing pickup. This model determines the probability density function and the probability function of self-pickup behavior, accurately depicting the probability of customer self-pickup behavior at any given time. This provides an accurate basis for the dynamic calculation of self-pickup locker capacity, enabling the system to monitor changes in locker capacity over time in real time, laying the foundation for the formulation and optimization of subsequent replenishment strategies.

[0022] The self-service locker available capacity and replenishment model established in this invention uses a probability function of customer self-pickup behavior to cumulatively calculate the remaining capacity of the locker at each time. Based on the remaining capacity, the replenishment timing is determined. When the capacity meets the replenishment demand, a vehicle is arranged to replenish the goods. When the capacity does not meet the demand, a decision is made on whether the vehicle continues delivery or waits nearby, and the waiting time and cost are calculated. This dynamic replenishment strategy can effectively balance the self-service locker capacity and customer demand, improve the utilization efficiency of the self-service lockers, reduce delivery problems caused by insufficient capacity, and reduce operating costs.

[0023] The self-pickup delivery mathematical model constructed in this invention takes the minimum operating cost as the objective function, covering costs such as fixed vehicle usage, routes, self-pickup locker opening, distance rewards, self-pickup service, and vehicle waiting. It sets decision variables such as vehicle arrival time and self-pickup locker delivery volume, as well as constraints such as vehicle route balancing and unique customer allocation. It realizes the joint optimization of self-pickup locker site selection, capacity planning, replenishment decision and vehicle route, and provides a scientific mathematical framework for the overall optimization of the self-pickup delivery system.

[0024] The two-stage heuristic algorithm designed in this invention generates an initial solution in the first stage through a three-stage progressive construction algorithm, including locker location selection, customer allocation, capacity level decision, construction of initial delivery path and replenishment path; the second stage uses variable neighborhood search for post-optimization, including neighborhood structure design, adaptive jitter operation and local search algorithm, which can efficiently solve the mathematical model of self-pickup delivery, generate solutions for locker location selection, capacity level configuration, replenishment timing decision and vehicle route planning, etc., realize the optimization of the operating cost of self-pickup delivery system, and provide feasible algorithm support for practical applications.

[0025] The above description is only an overview of the technical solution of this application. In order to better understand the technical means of this application and implement it in accordance with the contents of the specification, and to make the above and other objects, features and advantages of this application more obvious and understandable, the preferred embodiments of this application are described in detail below with reference to the accompanying drawings.

[0026] The above and other objects, advantages and features of this application will become more apparent to those skilled in the art from the following detailed description of specific embodiments in conjunction with the accompanying drawings. Attached Figure Description

[0027] To more clearly illustrate the technical solutions in the embodiments of this application or the prior art, the drawings used in the description of the embodiments or the prior art will be briefly introduced below. Obviously, the drawings described below are some embodiments of this application. For those skilled in the art, other drawings can be obtained based on these drawings without creative effort. In all drawings, similar elements or parts are generally identified by similar reference numerals. In the drawings, the elements or parts are not necessarily drawn to scale.

[0028] Figure 1 Flowchart of a self-pickup and delivery method based on dynamic modeling of customer self-pickup behavior and spatiotemporal replenishment strategy;

[0029] Figure 2 Image showing the restocking process when the vehicle arrives at the self-service locker for the second time;

[0030] Figure 3 : Scenario diagram of prohibition strategy in path optimization;

[0031] Figure 4 : Schematic diagram of self-pickup delivery vehicle route planning;

[0032] Figure 5 : Statistical results of customer pickup time intervals;

[0033] Figure 6 : The impact of replenishment strategies on the delivery system;

[0034] Figure 7 : Time and space capacity change graph of self-service lockers;

[0035] Figure 8 Comparison chart of optional capacity and single capacity strategies;

[0036] Figure 9 : Impact diagram of vehicle waiting strategies on the delivery system;

[0037] Figure 10 Sensitivity analysis chart of customer expected self-pickup distance and unit reward cost; Detailed Implementation

[0038] To make the objectives, technical solutions, and advantages of the embodiments of this application clearer, the technical solutions of the embodiments of this application will be clearly and completely described below with reference to the accompanying drawings. Obviously, the described embodiments are only some embodiments of this application, not all embodiments. In the following description, specific details such as specific configurations and components are provided merely to help fully understand the embodiments of this application. Therefore, those skilled in the art should understand that various changes and modifications can be made to the embodiments described herein without departing from the scope and spirit of this application. In addition, for clarity and brevity, descriptions of known functions and structures are omitted in the embodiments.

[0039] It should be understood that the phrase "an embodiment" or "this embodiment" throughout the specification means that a specific feature, structure, or characteristic related to the embodiment is included in at least one embodiment of this application. Therefore, "an embodiment" or "this embodiment" appearing throughout the specification does not necessarily refer to the same embodiment. Furthermore, these specific features, structures, or characteristics can be combined in any suitable manner in one or more embodiments.

[0040] Furthermore, reference numerals and / or letters may be repeated in different examples within this application. Such repetition is for the purpose of simplification and clarity and does not in itself indicate a relationship between the various embodiments and / or settings discussed.

[0041] In this article, the term "and / or" is merely a description of the relationship between related objects, indicating that three relationships can exist. For example, A and / or B can mean: A exists alone, B exists alone, and A and B exist simultaneously. The term " / and" in this article describes another type of relationship between related objects, indicating that two relationships can exist. For example, A / and B can mean: A exists alone, and A and B exist alone. In addition, the character " / " in this article generally indicates that the related objects before and after it are in an "or" relationship.

[0042] In this article, the term "at least one" is merely a description of the relationship between related objects, indicating that there can be three relationships. For example, "at least one of A and B" can mean: A exists alone, A and B exist simultaneously, or B exists alone.

[0043] It should also be noted that, in this document, relational terms such as "first" and "second" are used merely to distinguish one entity or operation from another, and do not necessarily require or imply any such actual relationship or order between these entities or operations. Furthermore, the terms "comprising," "including," or any other variations thereof are intended to cover non-exclusive inclusion.

[0044] Example 1

[0045] This embodiment mainly describes a self-pickup and delivery method based on dynamic modeling of customer self-pickup behavior and spatiotemporal replenishment strategies, such as... Figure 1 As shown, it includes:

[0046] S1: Obtain customer self-pickup behavior data, construct a customer self-pickup behavior model based on truncated normal distribution, and determine the probability density function and probability function of customer self-pickup behavior;

[0047] S2: Establish a model for the available capacity and replenishment of self-pickup lockers. Calculate the remaining capacity of the self-pickup lockers at each time point using the probability function of customer self-pickup behavior. Determine the replenishment timing based on the remaining capacity. When the available capacity of the self-pickup lockers meets the replenishment demand, arrange vehicles for replenishment. When the available capacity of the self-pickup lockers cannot meet the replenishment demand, decide whether the vehicle should continue to the next self-pickup locker for delivery or stop nearby and wait. If the vehicle chooses to wait nearby, calculate the vehicle waiting time and waiting cost.

[0048] S3: Construct a mathematical model for self-pickup delivery. The mathematical model for self-pickup delivery uses the minimum operating cost as the objective function to carry out site selection, capacity planning, replenishment decision-making, and vehicle route optimization for self-pickup lockers.

[0049] S4: Design a two-stage heuristic algorithm to solve the mathematical model of self-pickup delivery. The first stage generates an initial solution through a three-stage progressive construction algorithm, including self-pickup locker location selection, customer allocation, capacity level decision, construction of initial delivery path and construction of replenishment path; the second stage uses variable neighborhood search for post-optimization, including neighborhood structure design, adaptive jitter operation and local search algorithm.

[0050] S5: Based on the solution results of the two-stage heuristic algorithm, generate a self-pickup locker location scheme, a capacity level configuration scheme, a replenishment timing decision scheme, and a vehicle route planning scheme to optimize the operating costs of self-pickup delivery.

[0051] Furthermore, in S1, customer self-pickup behavior data is obtained, and a customer self-pickup behavior model is constructed based on a truncated normal distribution. The probability density function and probability function of customer self-pickup behavior are determined. Specifically, through statistical analysis of customer self-pickup behavior data, it is found that the customer's pickup time interval follows a truncated normal distribution. The customer's pickup behavior is modeled as the probability of pickup at time t after receiving the delivery information. The probability density function formula is: Where 'a' represents the lower cutoff point, i.e., the time when the first successful delivery is completed, 'μ' is the mean of the distribution, 'b' is the upper cutoff point, set as the end time of the cycle, 'σ' is the standard deviation of the distribution, and 'θ' is the normalization coefficient. Meeting the corresponding conditions, the probability of a customer's self-pickup behavior at any time 't' is obtained by integrating the probability density function, as shown in the formula: When t = a, G(t) = 0, and the probability of the customer completing self-pickup approaches 0. When t → b, G(t) = 1, and the probability of the customer completing self-pickup approaches 1.

[0052] Furthermore, in S2, a model for the available capacity and replenishment of self-pickup lockers is established. The remaining capacity of the lockers at each time point is calculated cumulatively using a probability function of customer self-pickup behavior. The replenishment timing is determined based on the remaining capacity. Specifically, the set of customers serving self-pickup locker p is N. p The total demand is The time when vehicle k first arrives at locker p is The time required to complete the express delivery is The formula for calculating the remaining capacity of the self-service locker at any time t is: u is the integral variable; when the available capacity of the self-service locker is sufficient to meet the replenishment demand, a vehicle is arranged to replenish the goods. Each vehicle replenishes each locker at most once, and the replenishment must meet the replenishment demand. When the vehicle arrives at the locker for the second time, if the available capacity of the locker meets the demand of the remaining customers, the vehicle immediately replenishes the goods, and the delivery quantity is... If the demand is not met, the vehicle will park at the nearest available locker and wait until the remaining capacity meets the remaining needs before restocking. The restocking time will be [date / time]. Vehicle waiting time is in, Waiting costs for the vehicle's second arrival at the self-service locker ω represents the vehicle's waiting cost per unit time.

[0053] Furthermore, when the vehicle arrives at the self-service locker for the second time, the determination and handling of the locker's available capacity and replenishment demand includes two methods: Replenishment Method 1, the remaining capacity of the locker at the time of the vehicle's second arrival. When the remaining customer demand is met, that is... The vehicle at the time of arrival The self-service lockers were restocked a second time, with a delivery volume of [number missing]. Replenishment successful. The actual replenishment start time for the vehicle is [time]. Replenishment Method 2: Remaining capacity of the self-service locker at the time of the vehicle's second arrival. The vehicle will park nearby and wait until the remaining capacity of the self-service locker meets the remaining demand for restocking. at this time Vehicle waiting time and Waiting costs Calculate, where ω is the waiting cost per unit time for the vehicle.

[0054] Furthermore, in S3, operating costs include fixed vehicle usage costs, route costs, self-pickup locker opening costs, costs for exceeding self-pickup distance rewards, self-pickup service costs, and vehicle waiting costs. Decision variables and constraints are set accordingly. Decision variables include vehicle arrival time, self-pickup locker delivery volume, vehicle loading capacity, and self-pickup locker opening status. Constraints include vehicle route balancing constraints, unique customer allocation constraints, self-pickup locker capacity constraints, and vehicle load and time constraints.

[0055] Furthermore, a mathematical model for self-pickup delivery is constructed in S3, specifically based on a directed graph G(V, A), where V is a set of nodes including the delivery center's starting point 0 and return point 0', a set of candidate self-pickup lockers P, and a set of virtual nodes directly related to the self-pickup lockers. And a set of self-pickup customers N, where A is an arc set containing the path arc set A. R and self-extracting arc set A C The decision variables include the time it takes for vehicle k to arrive at node i. Delivery volume u to self-service locker p pk Loading amount on arc (i, j) A 0-1 variable indicating whether the vehicle is traveling on arc (i, j) The 0-1 variable of the self-service locker's open level s. Customer i is a 0-1 variable representing the self-service locker p of level s. 0-1 variable for whether to replenish stock and vehicle waiting time The objective function is: Where K represents the vehicle set of the distribution center, F k For the fixed operating cost of vehicle k, c ij Let f(d) be the unit path cost of the vehicle traveling on arc (i, j), S be the set of selectable capacity levels for the self-service lockers, and f(d) be the unit path cost of the vehicle traveling on arc (i, j). ip Let ) be the reward cost function, and the formula is: Where r is the expected distance for self-pickup, and d ip The distance from customer i to the self-pickup locker p, c mThe cost of self-pickup service for the unit demand, q i This represents the parcel demand of customer i.

[0056] Furthermore, in the first stage of S4, an initial solution is generated through a three-stage progressive construction algorithm, including locker location selection, customer allocation, capacity level decision, construction of initial delivery routes, and construction of replenishment routes. Specifically, in the first stage, an initial solution is generated through a three-stage progressive construction algorithm. In stage 1, all lockers P are opened, and customers are allocated to lockers according to the principle of minimum pickup cost. The set of lockers P to which customers are allocated is recorded. open In Phase 2, based on vehicle load and mileage limitations, a set of delivery vehicle routes R0 for single access to self-pickup lockers is constructed using the CW saving algorithm. In Phase 3, the capacity level of self-pickup lockers p in the route is gradually reduced, and a replenishment route is constructed based on the available capacity model and vehicle parking model according to the minimum cost insertion principle.

[0057] Furthermore, in the second stage of S4, post-optimization is performed using a variable neighborhood search, including neighborhood structure design, adaptive jitter operation, and local search algorithm. Specifically, the second stage uses a variable neighborhood search for post-optimization, where the neighborhood set H = {N}. L1 N L2 N L3 N V N C1 N C2}, where N L1 N L2 N L3 These represent the locker being closed, open, and its status being reversed, respectively. N V To determine the neighborhood structure for the self-service locker capacity, N C1 N C2 For customer movement and customer swapping, a neighborhood is selected from H via adaptive jitter operation, where neighborhood k is the number of neighbors. s The probability ρ of being selected during the i-th jitter. k，i =w k，i / Σ k∈H w k，i The weight update formula is: in ψ is the weight decay factor and ψ is the neighborhood contribution coefficient. By designing a local search algorithm that includes path optimization and replenishment optimization, the path optimization adopts a prohibition strategy to avoid inserting different paths for two visits to the self-pickup locker that needs to be replenished. The replenishment optimization traverses all possible insertion positions for the self-pickup locker that needs to be replenished and selects the replenishment position with the minimum cost.

[0058] Furthermore, the fitness function in the two-stage heuristic algorithm is F. fit(X)=F(X)+αP(X)+βQ(X), where F(X) is the objective function value. For vehicle load violation, U k For the maximum load limit of the vehicle, A R For path arc set, The load from return point 0' to node j. For violations of driving time, T is the maximum permissible driving time for the vehicle. k The actual driving time of the vehicle is denoted by α and β, which are penalty factors. The heuristic algorithm terminates when the maximum optimal solution is reached without further improvement (ITER). best Alternatively, the maximum number of iterations (ITER) can be used, and an adaptive shanking strategy can be adopted in each main loop during the variable neighborhood search phase. Duplicate solutions are avoided through a hash table, and the optimal solution among all threads is selected for local search. When the local search fails to improve the optimal solution, the neighborhood weight is updated; otherwise, the counter for failure to improve the optimal solution is reset.

[0059] Furthermore, based on the solution results of the two-stage heuristic algorithm in S5, a self-service locker location scheme, a capacity level configuration scheme, a replenishment timing decision scheme, and a vehicle route planning scheme are generated to optimize the operating cost of the self-service delivery system. Specifically, the values ​​of decision variables are obtained through the two-stage heuristic algorithm, including the self-service locker open state variable. Determine the location and capacity level configuration of the self-pickup lockers based on the delivery volume. pk The available capacity Q of the self-pickup locker is calculated based on the probability function G(t) of customer self-pickup behavior. p (t) Determine the replenishment timing using vehicle driving variables. and arrival time Generate vehicle routing plans and optimize operating costs by reducing the opening costs of self-service lockers, vehicle routing costs, and replenishment waiting costs through various decision-making schemes.

[0060] This embodiment details a technical solution that differs fundamentally from existing technologies. It is the first to deeply integrate the dynamic characteristics of customer self-pickup behavior with the spatiotemporal replenishment strategy of self-pickup lockers. By dynamically modeling customer self-pickup behavior, the distribution patterns of customer pickup time intervals are accurately grasped, thereby enabling real-time calculation of the remaining capacity of the self-pickup lockers. Based on this, a scientifically sound replenishment strategy is formulated, and a multi-objective joint decision-making model is constructed. Solving this model using a two-stage heuristic algorithm achieves joint optimization of self-pickup locker site selection, capacity planning, replenishment decisions, and vehicle routing. This fundamentally improves the operational efficiency of the self-pickup delivery system, reduces operating costs, and provides a completely new approach and method for solving the last-mile delivery problem.

[0061] Based on Example 1, this example describes in detail the determination and processing of the available capacity and replenishment demand of the self-service locker when the vehicle arrives at the locker for the second time, including two methods:

[0062] Replenishment Method 1: Remaining capacity of the self-service locker at the time of the vehicle's second arrival. When the remaining customer demand is met, that is... The vehicle arrived at the time The self-service lockers were restocked a second time, with a delivery volume of [number missing]. Replenishment successful. The actual replenishment start time for the vehicle is [time]. like Figure 2 As shown in (a), this is the actual time when the vehicle's restocking begins. This refers to the time it takes for the vehicle (k) to reach the self-service locker, and the delivery volume (u) of vehicle (k) at locker (p). pk As shown below;

[0063]

[0064] in, It is a 0-1 decision variable. If customer i is served by locker p with capacity level s, then Otherwise = 0, This represents the maximum available capacity of locker p under capacity class s;

[0065] Replenishment Method 2: Remaining capacity of the self-service locker at the time of the vehicle's second arrival. If the remaining capacity of the self-service lockers cannot meet the remaining customer demand, a decision needs to be made: either the vehicle continues to the next self-service locker for delivery, or the vehicle needs to park nearby and wait (either on-site or in a nearby parking lot) until the remaining capacity of the self-service lockers meets the remaining demand before restocking. If the vehicle ultimately chooses to wait nearby, the restocking time is recorded as [time point missing]. at this time Vehicle waiting time like Figure 2 As shown in (b), and Therefore, the actual replenishment start time of vehicle p Waiting costs Calculate, where ω is the waiting cost per unit time for vehicle k, and the waiting time for vehicle k at locker p. As shown below;

[0066]

[0067] in, It is a 0-1 decision variable. If customer i is served by locker p with capacity level s, then Otherwise = 0, This represents the maximum available capacity of the self-service locker p under capacity class s.

[0068] This embodiment details how, in replenishment scenarios, when the available capacity of the self-service lockers meets the replenishment demand, the system can promptly arrange vehicles for replenishment, ensuring that the locker capacity remains at a reasonable level. This avoids insufficient capacity affecting customer storage needs and guarantees the continuity and stability of the self-service pickup service. When the available capacity of the self-service lockers cannot meet the replenishment demand, the vehicle can decide whether to continue to the next locker or wait at the nearest locker. If waiting is chosen, by calculating waiting time and costs, a balance can be struck between transport capacity and capacity demand, reducing empty runs and ineffective waiting, improving delivery efficiency, reducing cost increases caused by blind delivery, making the replenishment strategy more flexible and adaptable, and optimizing the overall operational effect of the self-service pickup delivery system.

[0069] Based on Embodiment 1, this embodiment describes in detail the two-stage heuristic algorithm of this application, specifically as follows:

[0070] The two-stage heuristic algorithm improves solution efficiency by simultaneously constructing the initial solution and performing post-optimization using the Variable Neighborhood Search (VNS) algorithm through multiple parallel threads. The first stage generates a high-quality initial solution through a problem-feature-based construction algorithm. The post-optimization stage utilizes VNS for global optimization. In the VNS stage, an adaptive shanking strategy is adopted in each main loop to improve the diversity of the search. To address the solution complexity brought about by the replenishment strategy, a two-stage local search algorithm (TSLS) that integrates the prohibition strategy and the replenishment optimization strategy is designed to improve the optimization efficiency of the algorithm. The overall framework of TSHA is as follows.

[0071]

[0072] Based on the solution space of the problem, a three-stage progressive construction algorithm is designed to generate an initial solution, providing high-quality input for the subsequent optimization stage. The specific process is as follows:

[0073] Phase 1: Locker Location Selection - Customer Allocation - Capacity Level Decision. Open all lockers p. Using a greedy strategy, allocate all customers to lockers based on the principle of minimum pickup cost, without violating the maximum locker capacity. Record the set of lockers with customer allocations as P. open And close any lockers without customer assignments; for any open locker p∈P open Set its open capacity to the maximum selectable capacity;

[0074] Phase 2: Constructing the initial delivery routes. Based on vehicle load and mileage constraints, the CW saving algorithm is used to construct delivery vehicle routes for single visits to self-service lockers, resulting in a route set R;

[0075] Phase 3: Constructing the replenishment route. For any vehicle route: r∈R, sequentially add the self-service lockers p∈P along the route. open The capacity levels decrease progressively, and replenishment paths are constructed based on the available capacity model and vehicle parking model, following the principle of minimum cost insertion. This stage optimizes the total cost by balancing the replenishment path cost and the cost of opening self-service lockers.

[0076] Furthermore, based on the analysis of the solution space composition, four types of neighborhood structures are designed for self-service locker location selection, self-service locker capacity decision, customer allocation, and vehicle routing.

[0077] Self-service locker location neighborhood structure (N) L Based on the state space of the self-service locker, three neighborhood structures are designed: self-service locker closed (N...). L1 ): Randomly close one open self-pickup locker and reassign its customers to other open self-pickup lockers according to the principle of minimum self-pickup cost; Open (N) self-pickup lockers L2 ): Randomly open one unopened self-pickup locker and assign it to the customer with the lowest self-pickup cost; the status of the self-pickup lockers is interchanged (N L3 ): Randomly select any open and closed self-pickup lockers, swap the open and closed states of these two lockers, and reassign customers served by the closed lockers according to the principle of proximity;

[0078] Self-service locker capacity decision neighborhood structure (N) V Choose any one locker and randomly replace its capacity class with the largest or smallest class closest to its current capacity class (N). V1 ).

[0079] Customer allocation neighborhood structure (N) C Two neighborhood structures were designed to explore the allocation relationship between customers and self-service lockers: customer movement (N...). C1 Randomly select 1-3 customers and move them to other available self-pickup lockers; customer exchange (N) C2 ): Randomly select 2 open self-pickup lockers, and then swap the service relationships of 1-2 customers whose self-pickup distance exceeds their expected distance.

[0080] Path Neighborhood Structure (N) R It employs three classic neighborhood structures widely used in the VRP algorithm: insert(N) R1 ), swap(N R2 ), inverse(N R3 );

[0081] Furthermore, adaptive shaking is performed. Shaking, as a core component of the VNS algorithm, improves the algorithm's global optimization performance by exploring the solution space in a diversified manner. The neighborhood structures in the designed neighborhood structure that can achieve large-granularity search of the solution space are taken as the shaking neighborhood set, then H = {N} L1 N L2 N L3 N V1 N C1 N C2};

[0082] To improve the efficiency of jittering, a widely used adaptive selection strategy is adopted to select jittering neighborhoods from H. Initially, all neighborhood structures are assigned equal weights. Within the number of algorithm iterations ITER, the weights of each neighborhood are dynamically updated based on the performance of each neighborhood structure in improving the solution quality during the algorithm search process. Roulette wheel selection is used to select neighborhoods.

[0083] Let w be the weight of the neighborhood structure k∈H during the i-th jitter. k，i Then the probability ρ of it being selected in this iteration is... k，i For ρ k，i =w k，i / ∑ k∈H w k，i Then, the weights w are updated based on the new solution X' generated after the jitter. k，i+1 The weight update formula is: Where is the weight adjustment coefficient, and ψ is the scoring function, based on the new solution X', the current solution X, and the optimal solution X. best The relationship between them is calculated as follows:

[0084]

[0085] Among them, F fit (X'), F fit (X best ) and F fit (X) represent X', ... best and X best The fitness function;

[0086] Furthermore, for the local search algorithm, based on the coupling problem of vehicle path optimization and replenishment optimization brought about by replenishment characteristics, a two-stage local search algorithm (TSLS) of "path optimization first and replenishment optimization later" is designed. The specific process is as follows: carry out IterLS sub-path local search (i.e. path optimization), and then carry out replenishment optimization based on the current path after path optimization, so as to achieve dual optimization of vehicle path and replenishment.

[0087] Path optimization, in Iter LS Within the range, from the neighborhood set NR A path operator is randomly selected within the search area for local path search. During the search, a prohibition strategy is employed to avoid inserting two visits to the self-service locker requiring replenishment into different paths, thereby improving path search efficiency by ensuring the integrity of the solution. When using Insert and Swap operations, two typical prohibition scenarios are as follows: Figure 3 (a) and Figure 3 As shown in (b): it can be seen that if the prohibition strategy is not used, the first visit to the self-pickup locker and its replenishment will be on different paths, which will lead to the infeasibility of the solution. This infeasibility will affect the subsequent search process.

[0088] Replenishment optimization is achieved by finding the best replenishment node position in the current path. For self-pickup lockers that need replenishment, after traversing all possible insertion positions, the position with the lowest cost is selected as the replenishment position.

[0089] The pseudocode for the two-phase local search algorithm is shown below:

[0090]

[0091] Furthermore, infeasible solutions during algorithm execution help expand the search space and thus benefit global optimization. Therefore, this paper allows infeasible solutions that violate constraints to participate in the algorithm's search process. For solutions that violate constraints, a penalty coefficient is introduced to control the algorithm's search direction. Violations of the objective function value, vehicle load constraint, and maximum driving time constraint are incorporated into the fitness function, calculated using the formula F. fit (X)=F(X)+αP(X)+βQ(X), where F(X) is the objective function value. For vehicle load violation, U k For the maximum load limit of the vehicle, A R For path arc set, The load from return point 0' to node j. For violations of driving time, T is the maximum permissible driving time for the vehicle. k The actual driving time of the vehicle is denoted by α and β, which are penalty factors. The heuristic algorithm terminates when the maximum optimal solution is reached without further improvement (ITER). best Or the maximum number of iterations (ITER).

[0092] This implementation details a two-stage heuristic algorithm that generates an initial solution through a three-stage progressive construction algorithm, combined with variable neighborhood search for post-optimization, enabling efficient solutions to self-pickup delivery mathematical models. The algorithm dynamically adjusts the neighborhood selection probability through adaptive jittering operations and optimizes path and replenishment decisions using a two-stage local search. This significantly improves solution efficiency in medium-to-large-scale scenarios while maintaining solution quality close to that of commercial solvers. The algorithm's prohibition strategy avoids replenishment path conflicts, and the hash table mechanism prevents duplicate solutions, achieving joint optimization of self-pickup locker location, capacity configuration, replenishment timing, and vehicle routing. This provides an efficient and feasible solution for self-pickup delivery systems, effectively reducing operating costs.

[0093] Based on the above embodiments, this embodiment describes in detail the specific implementation process of this application, specifically as follows:

[0094] The implementation was carried out using the Anaconda platform, with Python programming language, and a CPU configuration of AMD-Ryzen-7-5800H, 3.2GHz, 16GB of memory, and Windows 11 operating system.

[0095] like Figure 4 As shown, during delivery, the vehicle departs from the distribution center, visits the open self-service lockers in sequence, stores the customer's package in the locker, and arranges the replenishment route according to the number of customers served by the self-service locker and the open capacity of the self-service locker, and finally returns to the distribution center.

[0096] A simulation test case set was constructed based on actual operational scenarios. All nodes are located in a 20.0km × 20.0km square area, with the distribution center coordinates (0.0, 0.0). Based on the actual customer distribution, the customer geographical locations are semi-clustered, meaning that 50% of customers are within 0.5km of the nearest self-service locker, while the remaining customers are between 0.5km and 2.0km away. Based on these rules, 10 sets of cases of different sizes were generated. All cases are named according to "Distribution Center - Number of Self-Service Lockers - Number of Customers". The relevant delivery system operating parameters of the model are derived from actual delivery companies and actual survey data. Specifically: r = 0.5km. s m =0.5min, c m =0.5 yuan, c u =2.0 yuan / km, ω =10.0 yuan / hour;

[0097] To determine the truncated normal distribution parameters a, b, μ, and σ in the customer self-pickup behavior model, an experimental design was conducted based on customer satisfaction survey analysis methods, focusing on the customer pickup time interval. Statistical analysis was performed based on the survey results. To ensure the survey results reflect the general characteristics of customer behavior, the sample construction comprehensively considered different age groups (under 18, 18-25, 26-35, 36-50, and over 50), different occupational types (office workers, freelancers, students, and long-term homeowners), and different customer spatial distributions (commercial areas, communities). This survey lasted one month, distributing 900 questionnaires and collecting 502, of which 456 were valid. Categorical statistical results were obtained for customers at different time periods, as shown below. Figure 5 As shown in (a), the overall statistical results are as follows: Figure 5 As shown in (b), it can be seen that both the classification statistics and the overall statistics show a normal distribution, which is consistent with the mathematical model of customer pickup behavior.

[0098] Based on the overall statistical distribution results, the statistical distribution is normalized, and then a = 0 min and b = 360 min are set. Furthermore, according to the formulas for calculating the mean and variance... (n is the number of valid questionnaires, T) i The time interval for customer i to pick up their goods is given by μ = 179.2 min, σ 2 = 4696.75 min. For ease of calculation, the parameters are rounded down, resulting in μ = 180 min, σ 2 =4700min;

[0099] For the parameter Iter LS Iter best The values ​​for ITER are: Iter LS Located at

[3070] (step 10), Iter best Located at [20, 50] (step 5), and ITER located at [3000, 7000] (step 1000), the final values ​​of the three parameters are as follows: Iter LS =50, ITER best =30 and ITER=5000. Other parameters are set as follows: Ω1=50, Ω2=20, Ω3=10, α=50 and β=50;

[0100] By employing time discretization, the probability function of customer self-pickup behavior is transformed into a linear model, which is then solved using Gurobi. The longest vehicle operating time T is then determined. k=360min is divided into 360 time points with a granularity of 1 minute, thus transforming it into a mixed integer linear programming model. Based on the transformed mathematical model, the maximum running time of the Gurobi solver is set to 3600 seconds, and TSHA is run 10 times. The experimental results of TSHA and Gurobi on 10 examples are shown in Table 1. In this table, Obj1 represents the solution obtained by Gurobi (i.e., the upper bound), Obj2 represents the optimal solution obtained by TSHA, Time represents the average running time (unit: seconds), and Gap1 and Gap2 represent the deviations of the current solution from the best solutions obtained by the two algorithms (unit: %).

[0101] Table 1 Comparison of experimental results between TSHA and Gurobi

[0102]

[0103] According to Table 1, in terms of solution quality, Gurobi can only find the optimal solution for five smaller-scale examples within the specified time, while TSHA not only achieves the same results as Gurobi on smaller scales but also obtains better solutions than Gurobi on other larger-scale examples. In terms of solution time, except for examples 1-2-60 and 1-4-120 where Gurobi's solution time is better than TSHA's, Gurobi's solution time increases significantly with the increase in example size. Especially when the number of client nodes exceeds 300, Gurobi can no longer find the optimal solution within the given time, while TSHA can output high-quality solutions within 30 seconds. Therefore, compared to commercial solvers, TSHA has significant advantages in both solution quality and computational efficiency when solving medium-to-large-scale examples, making it particularly suitable for solving medium-to-large-scale problems.

[0104] Meanwhile, the impact of the initial solution construction algorithm (I) and the local search procedure (II) that combines the prohibition strategy and the replenishment optimization strategy on the algorithm's solution performance was verified. The results were compared with those of the algorithm based on random initial solution (TSHA-I) and the algorithm without prohibition strategy and replenishment optimization strategy (TSHA-II), as shown in Table 2.

[0105] Table 2 Algorithm Performance Verification

[0106]

[0107] Table 2 shows that the three-stage progressive construction algorithm designed in this paper, which constructs the initial solution, integrates the prohibition strategy, and the replenishment optimization strategy in its local search procedure, all positively contribute to the algorithm's solution. On average, strategies I and II improve the solution quality by 2.36% and 1.53%, respectively, and improve the solution time by 28.65% and 16.99%, respectively. This indicates that using strategies I and II can significantly improve solution efficiency while enhancing solution quality, with strategy I showing a more significant effect than strategy II.

[0108] To analyze the impact of replenishment strategies on the operating costs of the delivery system, a comparative analysis was conducted on delivery models with and without replenishment strategies based on six sets of case studies (with 270–500 customers) under the same experimental parameters. The results are as follows: Figure 6 As shown

[0109] according to Figure 6 (a) The model employing a "replenishment strategy" outperforms the model without a "replenishment strategy" in all calculations, reducing total delivery costs by 2.22%, 11.43%, 12.54%, 16.10%, 16.85%, and 17.13%, respectively. Furthermore, it is evident that the more customers there are, the more significant the advantage of the replenishment strategy becomes. In addition, according to... Figure 6 (b) It can be seen that the number of self-pickup lockers opened under the "replenishment strategy" model is significantly lower than that under the model without "replenishment strategy"; indicating that "replenishment strategy" helps to improve the economy of self-pickup delivery system by reducing the number of self-pickup lockers opened.

[0110] To visually demonstrate the impact of replenishment strategies on the time and space utilization of self-service lockers, an analysis was conducted based on case study 1-14-500. The algorithm was run to obtain the vehicle route planning for this case study: 0-1-4-6-7-7-4-1-0, 0-3-11-3-5-10-2-14-0, and 0-8-9-8-9-0. It can be seen that the vehicle replenished self-service lockers 1, 3, 4, 7, 8, and 9.

[0111] like Figure 7 (a) and (b) show the capacity changes of lockers 1 and 9 during the delivery cycle (0–360 min), respectively. Figure 7 (a) It can be seen that the vehicle first visits the self-service locker 1 at t=14 and completes the delivery at t=64. Then, the vehicle visits the self-service locker for the second time at t=140. At this time, the available capacity of self-service locker 1 is Q1(140)=6, which meets its replenishment demand D. The vehicle was immediately restocked. Figure 7 In (b), when the vehicle visits the self-service locker 9 for the second time at t=188, its available capacity Q9(218)=5 is insufficient to meet its replenishment demand. Therefore, the vehicle waited for 35 minutes until the replenishment demand was met and the replenishment operation was carried out. Based on the above analysis, the actual number of packages loaded in self-pickup lockers 1 and 9 were 102 and 114, respectively. It can be seen that combining the replenishment strategy with the time and space utilization of self-pickup lockers can improve the utilization rate of self-pickup lockers, and reduce the number of lockers that need to be opened by increasing the utilization rate, thereby helping to reduce the cost of the self-pickup delivery system.

[0112] To analyze the impact of the optional capacity strategy for self-service lockers on the delivery system, a comparative analysis was conducted on a single-capacity model (with locker capacities set to 50, 75, and 100) and an optional capacity model based on a large-scale simulation 1-14-500 under the same experimental parameters. The simulation results are as follows: Figure 8 As shown; by Figure 8 (a) It can be seen that the total delivery cost under the optional capacity strategy is better than that under the arbitrary single capacity strategy. Specifically, compared with the single capacity strategy of 50, 75, and 100, the total delivery cost under the optional capacity strategy is reduced by 8.38%, 5.08%, and 3.59%, respectively. Simulation data shows that when the locker capacity is 50, 14 lockers need to be opened, of which 14 need to be restocked; when the capacity is 75, 13 lockers need to be opened, of which 10 need to be restocked; and when the capacity is 100, 11 lockers need to be opened, of which 6 need to be restocked. It can be observed that as the capacity increases, the number of lockers opened and the number of lockers requiring restocking both decrease. Combined with... Figure 8 Analysis (b) shows that when capacity is selectable, 12 self-service lockers are open, of which 6 require replenishment. This result indicates that the selectable capacity strategy, by rationally configuring self-service lockers of different capacities and employing effective replenishment strategies, reduces the number of open self-service lockers, thereby lowering delivery costs.

[0113] To analyze the impact of vehicle waiting strategies on the operating costs of the delivery system, a comparative test analysis was conducted on six case studies (with 270–500 customers) under the same experimental parameters, comparing whether vehicle waiting was allowed. The results are as follows: Figure 9 As shown, by Figure 9 (a) It can be seen that the vehicle waiting strategy model outperforms the no-vehicle-waiting model in terms of total cost across all cases, especially when the case size is large, this advantage is more pronounced. Through analysis Figure 9(b), (c), and (d) show that when waiting is not allowed, after completing the delivery to the current locker, if there are still lockers to be delivered to, the vehicle will continue to the next locker; if the delivery task is completed, it will return directly to the distribution center. For small-scale cases (270 customers), since there are no additional lockers to deliver to, the vehicle will return directly, which increases the path cost. For medium-to-large-scale cases (300 to 500 customers), the vehicle needs to travel to lockers that are further away, which not only increases the path cost but also leads to an increase in locker opening costs and reward costs, thus increasing the total cost.

[0114] Furthermore, the impact of key parameters in the model on the operating costs of the delivery system is investigated to guide management practices. Based on a 1-14-500 case study, the expected customer pickup distance *r* and unit reward cost *C* are analyzed. u Sensitivity analysis.

[0115] Increase the increment from 0.2 to 0.8 by a step size of 0.1, and take C. u Sensitivity analyses were conducted for samples 1, 2, and 3, and the experimental results are as follows: Figure 10 As shown. According to Figure 10 For any r, the total delivery cost increases with C. u The increases all show an upward trend; in any C u The larger r is, the lower the total delivery cost. This indicates that lower incentive costs and larger customer-expected pickup distances contribute more to the economic efficiency of the delivery system. Specifically, compared to C... u The trend of total delivery cost under different values ​​can be seen as C u The larger the value, the more significant the impact on the total delivery cost; however, as r increases, its impact on the delivery system gradually decreases. This indicates that when the customer's expected self-pickup distance is large, the impact of unit reward cost on the operating cost of the delivery system will decrease.

[0116] This embodiment details how experimental simulation analysis demonstrates the excellent performance of the two-stage heuristic algorithm in solving the optimization model of the self-pickup delivery system. Its solution efficiency significantly outperforms commercial solvers in medium- to large-scale computational examples, while maintaining high solution quality. Analysis of the self-pickup locker replenishment strategy, optional capacity strategy, and vehicle waiting strategy shows that these strategies can effectively reduce the operating costs of the delivery system, improve the utilization rate of self-pickup lockers, and enhance system flexibility. Furthermore, sensitivity analysis of customer-expected self-pickup distance and unit reward cost provides strong evidence for enterprises to optimize cost structure and formulate reasonable operational strategies, verifying the effectiveness and feasibility of this technical solution in practical applications.

[0117] The above are merely preferred embodiments of the present invention and are not intended to limit the scope of protection of the present invention. For those skilled in the art, the present invention can have various modifications and variations. Any changes, modifications, substitutions, integrations, and parameter changes made to these embodiments within the spirit and principles of the present invention, without departing from the principles and spirit of the present invention, through conventional substitutions or to achieve the same function, fall within the scope of protection of the present invention.

Claims

1. A self-pickup and delivery method based on dynamic modeling of customer self-pickup behavior and spatiotemporal replenishment strategies, characterized in that: include: S1: Obtain customer self-pickup behavior data, construct a customer self-pickup behavior model based on truncated normal distribution, and determine the probability density function and probability function of customer self-pickup behavior; S2: Establish a model for the available capacity and replenishment of self-pickup lockers. Calculate the remaining capacity of the self-pickup lockers at each time point using the probability function of customer self-pickup behavior. Determine the replenishment timing based on the remaining capacity. When the available capacity of the self-pickup lockers meets the replenishment demand, arrange vehicles for replenishment. When the available capacity of the self-pickup lockers cannot meet the replenishment demand, decide whether the vehicle should continue to the next self-pickup locker for delivery or stop nearby and wait. If the vehicle chooses to wait nearby, calculate the vehicle waiting time and waiting cost. S3: Construct a self-pickup delivery mathematical model, which uses the minimum operating cost as the objective function to perform site selection, capacity planning, replenishment decision-making, and vehicle route optimization for self-pickup lockers; S4: Design a two-stage heuristic algorithm to solve the self-pickup delivery mathematical model. The first stage generates an initial solution through a three-stage progressive construction algorithm, including self-pickup locker location selection, customer allocation, capacity level decision, construction of initial delivery path and construction of replenishment path. The second stage utilizes variable neighborhood search for post-optimization, including neighborhood structure design, adaptive jitter operation, and local search algorithm. S5: Based on the solution results of the two-stage heuristic algorithm, generate a self-pickup locker location scheme, a capacity level configuration scheme, a replenishment timing decision scheme, and a vehicle route planning scheme to optimize the operating costs of self-pickup delivery.

2. The self-pickup and delivery method based on dynamic modeling of customer self-pickup behavior and spatiotemporal replenishment strategy according to claim 1, characterized in that, In step S1, customer self-pickup behavior data is acquired, and a customer self-pickup behavior model is constructed based on a truncated normal distribution. The probability density function and probability function of customer self-pickup behavior are determined. Specifically, through statistical analysis of customer self-pickup behavior data, it is found that the customer's pickup time interval follows a truncated normal distribution. Therefore, the customer's pickup behavior is modeled as the time interval after receiving the delivery notification. The probability of picking up the item at time t is given by the probability density function formula: ,in, This represents the next cutoff point, i.e., the time when the first successful delivery is completed. Let be the mean of the distribution. Let t be the upper cutoff point, set as the end time of the cycle, σ be the standard deviation of the distribution, and θ be the normalization coefficient. Meeting the corresponding conditions, the probability of a customer picking up their item at any time t is obtained by integrating the probability density function. The formula is: ,when hour, When the probability of a customer completing self-pickup approaches 0, hour, The probability of a customer completing self-pickup approaches 1.

3. The self-pickup and delivery method based on dynamic modeling of customer self-pickup behavior and spatiotemporal replenishment strategy according to claim 1, characterized in that, In step S2, a model for the available capacity and replenishment of the self-pickup lockers is established. The remaining capacity of the lockers at each time point is calculated cumulatively using the probability function of customer self-pickup behavior. The replenishment timing is determined based on the remaining capacity. Specifically, for the self-pickup lockers… The customer base of the service is The total demand is ,vehicle First arrival at the self-service locker The time is The time required to complete the express delivery is Calculate at any time The remaining capacity of the self-service locker is calculated using the following formula: , For integration variables, On behalf of clients The parcel demand is determined; once the available capacity of the self-service lockers is sufficient to meet the replenishment demand, a vehicle will be dispatched to replenish the parcels. Each vehicle will replenish each locker a maximum of once, and the replenishment must meet the replenishment demand. When the vehicle arrives at the locker a second time, if the available capacity meets the remaining customer demand, the vehicle will immediately replenish the parcels. The delivery volume is... If the demand is not met, the vehicle will park nearby and wait until the remaining capacity of the self-service locker meets the remaining demand before restocking. The restocking time will be... The vehicle waiting time is ,in, Waiting costs for the vehicle's second arrival at the self-service locker , The cost of waiting for a vehicle per unit of time.

4. The self-pickup and delivery method based on dynamic modeling of customer self-pickup behavior and spatiotemporal replenishment strategy according to claim 3, characterized in that, When the vehicle arrives at the self-service locker for the second time, the determination and handling of the locker's available capacity and replenishment demand includes two methods: Replenishment Method 1, the remaining capacity of the locker at the time of the vehicle's second arrival. When the remaining customer demand is met, that is... The vehicle at the time of arrival The self-service lockers were restocked a second time, with a delivery volume of [number missing]. Replenishment successful. The actual replenishment start time for the vehicle is [time]. Replenishment method 2: The remaining capacity of the self-service locker at the time of the vehicle's second arrival. The vehicle will park nearby and wait until the remaining capacity of the self-service locker meets the remaining demand for restocking. ,at this time Vehicle waiting time ,and Waiting costs are calculated Calculation, where The cost of waiting for a vehicle per unit of time.

5. The self-pickup and delivery method based on dynamic modeling of customer self-pickup behavior and spatiotemporal replenishment strategy according to claim 1, characterized in that, The operating costs in S3 include fixed vehicle usage costs, route costs, self-pickup locker opening costs, costs for exceeding self-pickup distance rewards, self-pickup service costs, and vehicle waiting costs. Decision variables and constraints are set accordingly. The decision variables include vehicle arrival time at the node, self-pickup locker delivery volume, vehicle loading capacity, and self-pickup locker opening status. The constraints include vehicle route balance constraints, unique customer allocation constraints, self-pickup locker capacity constraints, and vehicle load and time constraints.

6. The self-pickup and delivery method based on dynamic modeling of customer self-pickup behavior and spatiotemporal replenishment strategy according to claim 1 or 5, characterized in that, The self-pickup delivery mathematical model constructed in S3 is specifically: using a directed graph. Based on this, where V is the node set, including the distribution center departure point 0 and return point 0', the candidate self-service locker set P, and the set of virtual nodes directly related to the self-service lockers. And a set of self-pickup customers N, where A is an arc set containing the path arc set. and self-collection arc set ; Decision variables include vehicles Reaching the node time Delivery volume to self-service locker p In the arc Load capacity Is it in the arc? 0-1 variables of the upper driving direction Self-pickup locker open level 0-1 variables ,client The 0-1 variable of the self-service locker p of grade s 0-1 variable for whether to replenish stock and vehicle waiting time The objective function is: ,in For the collection of vehicles at the distribution center, For the distribution center Delivery vehicles, For vehicles Fixed usage costs For self-pickup lockers Open Capacity Level The cost of opening up at that time For vehicles in arc The unit path cost of traveling on the road. This is a set of selectable capacity levels for self-service lockers. For the reward cost function, The cost of self-pickup service for unit needs. On behalf of clients The demand for parcels.

7. The self-pickup and delivery method based on dynamic modeling of customer self-pickup behavior and spatiotemporal replenishment strategy according to claim 1, characterized in that, In the first stage of S4, an initial solution is generated using a three-stage progressive construction algorithm, including locker location selection, customer allocation, capacity level decision, construction of initial delivery routes, and construction of replenishment routes. Specifically, in the first stage, an initial solution is generated using a three-stage progressive construction algorithm. In stage 1, all lockers P are opened, and customers are allocated to lockers according to the principle of minimum pickup cost. The set of lockers allocated to customers is recorded. And close the self-pickup lockers without customer assignments, and set the capacity of open self-pickup lockers to the maximum selectable capacity; Phase 2, based on vehicle load and mileage constraints, utilizes the CW saving algorithm to construct a set of delivery vehicle routes for single access to self-service lockers. ; Phase 3 Self-pickup lockers along the route The capacity levels decrease progressively, and replenishment paths are constructed based on the available capacity model and vehicle parking model, according to the principle of minimum cost insertion.

8. The self-pickup and delivery method based on dynamic modeling of customer self-pickup behavior and spatiotemporal replenishment strategy according to claim 1, characterized in that, The second stage in S4 utilizes variable neighborhood search for post-optimization, including neighborhood structure design, adaptive jitter operation, and local search algorithm. Specifically, the second stage utilizes variable neighborhood search for post-optimization, and the neighborhood set... ,in These represent the locker being closed, open, and its status being switched. The neighborhood structure is determined based on the capacity of the self-pickup lockers. For customer movement and customer swapping, adaptive jitter operation is used from Select neighborhood, neighborhood In the The probability of being selected during the next shake. The weight update formula is: ,in This is the weight decay factor. The neighborhood contribution coefficient is used to design a local search algorithm that includes path optimization and replenishment optimization. In path optimization, a prohibition strategy is used to avoid inserting different paths for two visits to the self-pickup locker that needs replenishment. In replenishment optimization, the algorithm traverses all possible insertion positions of the self-pickup locker that needs replenishment and selects the replenishment position with the lowest cost.

9. The self-pickup and delivery method based on dynamic modeling of customer self-pickup behavior and spatiotemporal replenishment strategy according to claim 8, characterized in that, The fitness function in the two-stage heuristic algorithm is: ,in The objective function value, For vehicle load violations, This refers to the maximum load limit of the vehicle. For path arc set, To return from the point To the node Load capacity per hour For the distribution center Delivery vehicles, The collection of all delivery vehicles in the distribution center; For violations of driving time, The maximum permitted driving time for the vehicle. This refers to the actual driving time of the vehicle. and The penalty factor is used; the termination condition of the heuristic algorithm is reaching the maximum optimal solution without improvement for the specified number of times. or maximum number of iterations Furthermore, in each main loop of the variable neighborhood search phase, an adaptive shanking strategy is adopted to avoid duplicate solutions through a hash table. The optimal solution among all threads is selected for local search. When the local search fails to improve the optimal solution, the neighborhood weight is updated; otherwise, the counter for failure to improve the optimal solution is reset.

10. The self-pickup and delivery method based on dynamic modeling of customer self-pickup behavior and spatiotemporal replenishment strategy according to claim 1, characterized in that, In step S5, based on the solution results of the two-stage heuristic algorithm, a self-pickup locker location scheme, a capacity level configuration scheme, a replenishment timing decision scheme, and a vehicle route planning scheme are generated to optimize the operating costs of the self-pickup delivery system. Specifically, the values ​​of decision variables are obtained by solving the two-stage heuristic algorithm, including the self-pickup locker open state variable. Determine the location and capacity level configuration of the self-pickup lockers based on delivery volume. and based on the probability function of customer self-pickup behavior Calculated available capacity of self-service lockers Determine the timing of replenishment by utilizing vehicle movement variables. and arrival time Generate vehicle routing plans and optimize operating costs by reducing the opening costs of self-service lockers, vehicle routing costs, and replenishment waiting costs through various decision-making schemes.

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