A two-tier optimization method for last-mile delivery with delivery options and demand guidance.

Abstract

This application provides a two-layer optimization method for last-mile delivery using a distributed bar structure, incorporating delivery options and demand guidance. Belonging to the field of logistics and delivery technology, the method includes: acquiring the delivery network information of the delivery provider, historical parcel demand information, and user utility loss information, including delivery delay utility loss information and delivery mode switching utility loss information; determining the delivery provider's path traversal cost information based on the delivery network information; constructing a distributed bar opportunity constraint on the delivery provider's vehicle capacity based on the historical parcel demand information; and determining vehicle route information, parcel delivery information, and incentive amount information under the distributed bar opportunity constraint, with the objective of minimizing the sum of the total delivery route cost and the total incentive cost, wherein the incentive amount information is not lower than the corresponding utility loss information. This application addresses parcel demand uncertainty through distributed bar opportunity constraints, improving delivery efficiency and reducing delivery costs.

Description

Technical Field

[0001] This application belongs to the technical field of logistics and distribution, and specifically relates to a two-layer optimization method for last-mile delivery that includes delivery options and demand guidance. Background Technology

[0002] The rapid development of e-commerce has driven the continuous growth of urban delivery demand, with consumers (users) constantly raising their expectations for fast delivery. A large number of packages need to be delivered to different addresses within a limited time window, leading to problems such as fragmented delivery orders, route fragmentation, and low vehicle load rates. Statistics show that last-mile delivery costs account for 41% to 50% of total delivery costs, making it the most expensive link in the logistics chain.

[0003] To reduce last-mile delivery costs, existing technologies typically employ demand-driven strategies, incentivizing users to adjust delivery times or methods to aggregate packages. Specifically, demand-driven strategies include two core methods: time integration, which provides delayed delivery options to encourage users to accept later delivery windows to aggregate orders; and spatial integration, which guides users to choose alternative delivery modes such as parcel lockers or self-pickup points to reduce the number of door-to-door deliveries. However, these strategies face the following technical challenges in practical applications: First, existing technologies often treat time and space integration in isolation, lacking an exploration of their synergistic effects. Existing methods either focus solely on delayed delivery or only emphasize delivery mode switching; even when delivery time and delivery methods are involved, they mostly assume that the delivery provider can unilaterally adjust delivery arrangements, without fully considering users' willingness to accept and their corresponding incentive needs.

[0004] Second, existing technologies have shortcomings in incentive design. Most studies assume that users will fully accept incentives or characterize user responses with a fixed probability of acceptance, neglecting the rational decision-making behavior of users as agents maximizing utility. Whether a user accepts an incentive is essentially the result of weighing the gains and losses; that is, they will only accept it if the incentive amount is not less than their psychological loss. This negotiation process between the company and the user is not fully considered in existing technologies, leading to a disconnect between incentive schemes and reality.

[0005] Third, existing technologies are insufficient for handling the uncertainty of parcel demand. In actual delivery, parcel sizes fluctuate randomly, and deterministic optimization methods can easily lead to vehicle overloading and infeasibility, while traditional robust optimization methods are too conservative. How to control costs while ensuring the feasibility of the solution is a pressing technical challenge that needs to be addressed. Summary of the Invention

[0006] This application aims to provide a two-layer optimization method for last-mile delivery with delivery options and demand guidance to solve at least one of the above-mentioned technical problems.

[0007] According to one aspect of this application, a last-mile delivery method incorporating delivery options and demand guidance is provided. The delivery optimization method using a two-tiered approach includes: The system acquires delivery network information, historical parcel demand information, and user utility loss information, including delivery delay utility loss and delivery mode switching utility loss. Based on the delivery network information, it determines the delivery provider's path traversal cost information. Based on the historical parcel demand information, it constructs a distributed bar opportunity constraint on the delivery provider's vehicle capacity. With the objective of minimizing the sum of the delivery provider's total delivery route cost and total incentive cost, and under the distributed bar opportunity constraint, it determines vehicle route information, parcel delivery information, and incentive amount information, where the incentive amount information is not less than the corresponding utility loss information.

[0008] The beneficial effects of the technical solutions provided in this application are: This application obtains delivery network information, historical parcel demand information, and user delivery delay utility loss and delivery mode switching utility loss information from the delivery provider. First, it determines path traversal cost information based on the delivery network information. Then, it constructs a sub-Bruker chance constraint on vehicle capacity based on historical parcel demand information. Finally, with the objective of minimizing the sum of the total delivery route cost and the total incentive cost, it determines vehicle route information, parcel delivery information, and incentive amount information under this sub-Bruker chance constraint, while ensuring that the incentive amount is not lower than the corresponding utility loss information. Thus, this application utilizes the sub-Bruker chance constraint to handle the randomness of parcel size, ensuring vehicle capacity feasibility under the worst-case demand distribution while avoiding excessive conservatism. By linking the incentive amount to the user's utility loss, it makes the incentive scheme conform to the user's rational acceptance conditions. By jointly optimizing the total delivery route cost and the total incentive cost, it achieves coordinated decision-making for vehicle routes, parcel allocation, and incentive amounts, thereby effectively reducing the total cost of last-mile delivery and improving delivery efficiency. Attached Figure Description

[0009] Figure 1 A flowchart illustrating the last-mile delivery split-bar two-layer optimization method including delivery options and demand guidance, according to an embodiment of this application, is shown. Figure 2 A schematic diagram of a two-level last-mile delivery network scenario according to an embodiment of this application is shown; Figure 3 This illustration shows a schematic diagram of the customer decision-making and delivery workflow according to an embodiment of this application; Figure 4 A schematic diagram of the technical solution of an embodiment of this application is shown; Figure 5A schematic diagram of the three-level mathematical heuristic algorithm of this application is shown; Figure 6 A schematic diagram illustrating experimental results of the method of this application and existing methods is shown; Figure 7 A schematic diagram of the total cost in a delivery sensitivity experiment according to an embodiment of this application is shown. Detailed Implementation

[0010] The present application will now be described in detail with reference to the accompanying drawings and embodiments. The various examples are provided by way of explanation and not for limitation of the present application.

[0011] According to the example embodiment, such as Figure 1 As shown, the delivery route and incentive collaborative optimization method includes steps S100-S400. Exemplarily, the method of this application can be executed in a computer system such as a set of executable instructions. Furthermore, although a logical order is shown in the flowchart, under default conditions, the steps shown or described can be executed in a different order than that shown in the flowchart. Exemplarily, the method of this application is applied to solve the last-mile delivery problem for delivery parties. The delivery party is an enterprise or organization undertaking logistics delivery tasks, such as a courier company, e-commerce logistics department, etc. The delivery party is used to deliver packages to users.

[0012] Two-tier last-mile delivery network scenario and customer decision-making and delivery workflow, such as Figure 2 As shown. Packages are first transported by truck from the distribution center to satellite stations or self-pickup points; for packages transported to satellite stations, door-to-door delivery is then completed by freight bicycles, electric bicycles, or small delivery vehicles, such as... Figure 2 In China, AI is used to describe the delivery methods customers specify in advance. However, in actual operation, delivery companies sometimes adjust these preferences, such as transferring door-to-door delivery packages to self-pickup points. While this can consolidate packages that would otherwise be delivered separately to the same delivery station and reduce costs, it can also lead to other issues. Figure 2 While options a-ii are feasible, they may reduce customer satisfaction; conversely, strictly adhering to customer preferences could limit the flexibility of delivery planning and increase operating costs. Therefore, this invention proposes that delivery companies can use economic incentives to guide customers to accept alternative delivery methods or delayed delivery, thereby increasing package aggregation and optimizing delivery routes while ensuring acceptable net customer utility. For example, for packages that would normally require separate door-to-door delivery, some customers can be guided to pick them up at self-pickup points, reducing scattered stops; for packages with similar arrival times but different arrival times, customers can be guided to accept moderately delayed delivery, increasing the integration of deliveries in the same batch and reducing route costs. Figure 2 a-iii; Figure 2 Zhongbihe Figure 2 b-ii represents the delivery route that follows the delivery deadline; Figure 2 b-ii and Figure 2 b-iv represents the delivery route when successfully providing incentives for delayed delivery, and it can be seen that routing costs are reduced in delayed delivery scenarios.

[0013] The relevant parameters are explained below: N represents all nodes (DC, satellite site, pickup point, customer); Represents a set of customers; Indicates the set of pickup points; Represents a set of satellite sites; Represents a set of packages; Represents a set of time periods; Represents a set of delivery modes; This indicates the first-level truck fleet assembly; This indicates a second-tier truck fleet assembly; This represents the set of path edges in the first layer. This represents the set of path edges in the second layer. Indicates the vehicle number; These represent the capacity of Level 1 and Level 2 vehicles, respectively. Representing edges respectively Transportation costs; Indicates vehicle Does it belong to a satellite? ; Indicates customer At the self-pickup point Coverage area; Represents a node Does it support delivery mode? ; Indicates package Is it in time? arrive; Indicates package Is it necessary to Early delivery; Indicates package Belongs to customers ; Indicates package Preferred delivery patterns; Represents a set of delayable times; Indicates the package size (random variable); Indicates the number of time periods of delay; This represents the collection of all incentive amounts provided by the company. This represents the set of all decisions accepted by the customer. This represents the objective function value of a single-layer optimization model; Indicates freight bicycle The base; Indicates the tolerance for capacity constraint violations; This represents the variance of the package dimensions; This represents the average size of the package; p and p' represent two different packages, respectively. In step S100, the delivery network information of the delivery party and the utility loss information of the user are obtained.

[0014] According to the example embodiment, utility loss information includes delivery delay utility loss information and delivery mode switching utility loss information. Delivery delay utility loss information quantifies the psychological cost, time cost, or inconvenience a user experiences due to a package delivery time being later than their expected or agreed-upon deadline. It reflects the user's tolerance for different delay durations (e.g., 1 day, 2 days, etc.) and the corresponding negative feelings. The longer the delay, the greater the utility loss value. Delivery delay utility loss information can be pre-defined through user surveys, historical data statistics, or model back-calculation. Delivery mode switching utility loss information quantifies the inconvenience, time cost, or psychological resistance a user experiences when the package delivery method changes from their preferred mode (e.g., door-to-door delivery) to an alternative mode (e.g., self-pickup point, parcel locker delivery, etc.). Delivery mode switching utility loss information reflects the user's acceptance of different alternative delivery modes. For example, a user who originally wanted door-to-door delivery being directed to a self-pickup point will generate delivery mode switching utility loss information. Information on the loss of utility when switching delivery modes is also obtained through user surveys or historical behavioral data.

[0015] According to the example embodiment, the last-mile delivery problem (LMD - DODS problem) with delivery options and demand-driven delivery is defined on an undirected graph. The last-mile delivery problem can also be called the end-of-logistics delivery problem. The delivery network information of the delivery party is represented by an undirected graph G=(N, E). N is a set of nodes, containing a distribution center DC (index 0), a set of satellite stations S, a set of pickup points L, and a set of users (or user locations) C. Its nodes include: distribution center DC, satellite station s, pickup point l, and user (or user location) c. , , E is the edge set, where each edge represents a path between nodes and its corresponding traversal cost.

[0016] For example, the delivery operation is carried out at two levels, namely a first level and a second level. The first level corresponds to the transportation operation, and the second level corresponds to the delivery operation. N includes the first-level node set N1 and the second-level node set N2, where N1 = {0} ∪ S ∪ L, {0} represents the distribution center DC, and N2 = S ∪ C. Satellite stations S connect the first-level nodes and the second-level nodes. The edge set E is classified by level: the first-level edge set E1 = {ij | i, j ∈ N1, i ≠ j} contains the travel paths (or connection lines) between two different nodes in the distribution center DC, the satellite station set S, and the pickup point set L, as well as the corresponding first-level traversal cost. That is, along the edge The path cost (transportation cost); the second-level edge set E2={mn|m,n∈N2,m≠n} contains the travel path between two different nodes in the satellite station set S and the user set C, and the corresponding second-level traversal cost. That is, along the edge The path cost is calculated as follows. Here, i, m and j, n are node indices, where i and m represent the starting point of the path, and j and n represent the ending point of the path.

[0017] Allocate homogeneous fleets to each tier: This refers to the first-tier trucks stationed at the distribution center (DC). This refers to the second-tier freight bicycles stationed at satellite sites. Bicycle The relationship of stationing, This indicates that the bicycles are stationed at the satellite site. This indicates that the unit is not stationed at this site. Let... and These represent the first-tier car teams. With the second-tier team The vehicle capacity in the system. Consider a planning period divided into several discrete time periods, denoted as [the set of indices of these time periods]. During this planning period, a set of packages needs to be completed. Delivery. There are two delivery methods: Self-pickup point delivery, door-to-door delivery}, each package It has the following characteristics: Arrival indicator variable ,in 1 indicates a package At time t ( Arrive at the distribution center at ) time 0 indicates that the delivery will not arrive at time t; the deadline delivery indicator variable. , Indicates package Must be in time Previously delivered; Indicates a package Does not have to be delivered before a time; the ownership indicator variable , Indicates a package belongs to the customer Indicates a package does not belong to the customer ; the customer-pre-specified preferred delivery mode indicator variable , where 1 indicates that the package prefers the mode Package size 0, which is a non-negative random variable, the random variable characterizes both the package size fluctuation and the arrival uncertainty simultaneously. When it is, it indicates non-arrival, while is the actual size upon arrival. By multiplying the corresponding indicators by , the corresponding indicators are indicator variables corresponding to other random attributes of the package, including: the customer-pre-specified preferred delivery mode, arrival time, deadline, etc., so as to retain the handling of arrival uncertainty and keep the notation concise. When the package does not arrive , the indicator variables of these attributes automatically归零 (should be "become zero"), without generating constraints; when the package arrives , the attributes take effect normally.

[0018] Whether the customer accepts the compensation provided by the delivery enterprise depends on the trade-off between the compensation level and the utility loss caused by the change in the delivery plan. In this plan, these losses are parameterized as follows: For the package , let represent the set of potential delays, that is , where [[ID=四十九]] is the package initial deadline (or the initially required delivery time) indicator variable, indicates that the package P must be delivered before time t, is the number of periods by which the delivery time of the package is delayed, represents the number of periods in the planning horizon. Delaying the package by periods and delivering it to the customer results in a utility loss expressed as . Similarly, let represent the set of alternative delivery modes for the package [[ID=六十六]]; that is, the non-customer-pre-specified preferred delivery mode; switching to [[ID=六十九]] results in a utility loss for the customer expressed as , where When a package is delivered to a self-pickup point, the delivery company must meet the pre-defined service coverage constraints of that self-pickup point. Let... Indicates customer Is it located at a pickup point? Within the service scope. Introducing parameters. To establish the correspondence between primary nodes and delivery models: when and In the self-pickup point delivery mode, or when and When using the door-to-door delivery model, .

[0019] like Figure 3 As shown, based on the incentive plan provided by the delivery provider (including the amount of incentive for delayed delivery and the amount of incentive for switching delivery modes), and considering their own utility loss due to delivery delays and switching delivery modes, the customer calculates the net utility (incentive amount minus utility loss) for each incentive option and selects the incentive option with the highest net utility as the acceptance decision. If the net utility of all incentive options is negative, the customer does not accept any incentive and accepts delivery according to their original preferred delivery method and time.

[0020] In step S200, the path traversal cost information of the delivery party is determined based on the delivery network information.

[0021] According to the example embodiment, the delivery party's path traversal cost information includes the traversal cost of each path at the first level of the delivery network information and the traversal cost of each path at the second level of the delivery network information.

[0022] For example, the first tier consists of a truck fleet stationed at the distribution center, responsible for trunk transportation between the distribution center and satellite stations / pickup points; the second tier consists of a freight bicycle fleet stationed at the satellite stations, responsible for last-mile delivery between the satellite stations and users. Specifically, for each route in the first tier, the computer system determines the corresponding route traversal cost information based on the route distance and the unit transportation cost of the trucks, representing the cost incurred by a truck to travel along that route once. For each route in the second tier, the computer system determines the corresponding route traversal cost information based on the route and the unit transportation cost of the freight bicycles, representing the cost incurred by the freight bicycles to travel along that route once. The delivery party's route traversal cost information is the sum of the path traversal cost information corresponding to the first tier and the path traversal cost information corresponding to the second tier.

[0023] like Figure 4As shown, the technical solution of this application includes three core parts: the first part is the input data, including delivery network information, historical parcel demand information, and user utility loss information; the second part is the core optimization model, including the construction of distributed opportunity constraints and the establishment and transformation of a two-level decision model; the third part is the output results, including vehicle route information, parcel delivery information, and incentive amount information. The entire solution aims to minimize the sum of the total delivery route cost and the total incentive cost, and collaboratively optimizes the delivery route and incentive scheme under the constraint that the incentive amount is not less than the utility loss.

[0024] In step S300, based on historical parcel demand information, a distributed opportunity constraint on the delivery party's vehicle capacity is constructed.

[0025] According to the example embodiment, this step utilizes statistical characteristics from historical parcel demand data to construct a vehicle capacity constraint that can withstand the uncertainty of demand distribution. Specifically, firstly, the mean and variance of each parcel size are statistically derived from the historical parcel demand information; then, a moment fuzzy set is constructed based on these means and variances, which contains all possible probability distributions consistent with the known means and variances; finally, a partially robust chance constraint is established under this fuzzy set, requiring that for any delivery vehicle and any time period, under the worst-case demand distribution, the probability that the vehicle's loading capacity does not exceed its capacity is not lower than a preset confidence level. This constraint replaces the traditional deterministic vehicle capacity constraint, making the optimization scheme more robust to random fluctuations in parcel size.

[0026] In step S400, the objective is to minimize the sum of the total delivery route cost and the total incentive cost of the delivery party. Under the constraint of distributed opportunity, vehicle route information, package delivery information and incentive amount information are determined, wherein the incentive amount information is not less than the corresponding utility loss information.

[0027] According to the example embodiment, this step achieves collaborative optimization between the delivery provider and the user by constructing a two-layer decision model, which includes a delivery provider decision model and a user decision model. In the delivery provider decision model, the delivery provider aims to minimize the sum of the total delivery route cost and the total incentive cost paid to the user, deciding on vehicle routes, package allocation, and incentive amounts. In the user decision model, the user, based on the incentive scheme provided by the delivery provider, aims to maximize their own net utility by choosing which incentives to accept for delayed delivery or delivery mode switching. To solve the model, this step transforms the above two-layer model into a single-layer mixed-integer linear programming model through variable substitution and linearization, and adds individual rationality constraints (i.e., the incentive amount is not less than the corresponding utility loss) and optimal choice constraints (i.e., the user only accepts the incentive option that maximizes their net utility). Finally, the distributed chance constraints constructed in step S300 are applied to this single-layer mixed-integer linear programming model for solving, obtaining vehicle route information, package delivery information, and incentive amount information.

[0028] Through the above embodiments, this application obtains the delivery network information of the delivery party, historical parcel demand information, and the user's delivery delay utility loss information and delivery mode switching utility loss information. First, it determines the path traversal cost information based on the delivery network information. Then, it constructs a sub-Bruker opportunity constraint on vehicle capacity based on the historical parcel demand information. Finally, with the objective of minimizing the sum of the total delivery route cost and the total incentive cost, it determines the vehicle route information, parcel delivery information, and incentive amount information under this sub-Bruker opportunity constraint, while ensuring that the incentive amount is not lower than the corresponding utility loss information. Thus, by utilizing the sub-Bruker opportunity constraint to handle the randomness of parcel size, it ensures the feasibility of vehicle capacity under the worst-case demand distribution while avoiding excessive conservatism. By linking the incentive amount to the user's utility loss, it makes the incentive scheme conform to the user's rational acceptance conditions. By jointly optimizing the total delivery route cost and the total incentive cost, it achieves collaborative decision-making on vehicle routes, parcel allocation, and incentive amounts, thereby effectively reducing the total cost of last-mile delivery and improving delivery efficiency.

[0029] According to the example embodiment, step S400 of this method includes steps S410-S450.

[0030] In step S410, a delivery party decision model is constructed. The delivery party decision model aims to minimize the sum of the total delivery route cost and the total incentive cost, and it is subject to the constraints of the user decision model.

[0031] Delivery companies formulate strategies based on the following decisions: Customer service tasks , Delivery plan, , Delivery route, first floor , Second floor , Incentive scheme, , To minimize total cost (including route cost and incentive cost).

[0032] On the customer's side, they can choose to accept either a delay or a reshipment option. .

[0033] According to the example implementation, the objective function of the delivery party decision model (or upper-level model) is: in, Let be the sum of the total cost of delivery routing and the total cost of incentives; x be the routing decision variable (value 0 or 1); y be the delivery planning decision variable (value 0 or 1); z be the node access decision variable (value 0 or 1); d be the user service allocation decision variable (value 0 or 1); k be the set of all incentive amounts provided by the company; T be the time period set; E (e) For the set of path edges; e is the level (takes a value of 1 or 2, representing the first level and the second level respectively); v is the vehicle number; F (e) Assemble the convoy; Let be the transportation cost of edge ij at level e; Let v be the vehicle at level e and time period t, and let ij be the edge (from i to j). P is the package set; p is the package; a is the set of all user acceptance decisions. For delay time; A set of delay times; To determine whether the user accepts delaying the package delivery. The amount of the incentive; To cause delays in package delivery by the delivery party Incentive amount provided for time-slot delivery; Let O be the set of non-preferred delivery patterns (i.e., alternative patterns) for package p; let O be the set of delivery patterns; and let o be the delivery pattern. This serves as an incentive for the user to switch package p to the non-preferred mode o. This refers to the incentive amount provided by the delivery provider to encourage package p to switch to a non-original preference mode o. .

[0034] The constraints of the delivery party's decision-making model (or the delivery party's decision-making model is subject to the following constraints) are: first-stage routing constraints, second-stage routing constraints, delivery plan constraints, and incentive constraints.

[0035] The first-stage routing constraints are explained below. These constraints correspond to the first level and include: enforcing traffic conservation for first-stage trucks (corresponding to formulas 1-2); connecting routes and access variables (corresponding to formulas 1-3); ensuring that each first-stage node (satellite or pick-up point) is accessed by at most one truck at any given time (corresponding to formulas 1-4); imposing truck capacity limits (corresponding to formulas 1-5); and applying the Danzig-Fulkerson-Johnson subpath elimination constraint (corresponding to formulas 1-6). This embodiment does not limit the specific content of these constraints and may use constraints from existing technologies.

[0036] The relevant expression for this constraint can be: (1-2) (1-3) (1-4) (1-5) (1-6) In the formula, This indicates whether vehicle v has traversed edge ij (from j to i) at level 1 and time period t.

[0037] The second-stage routing constraints (corresponding to formulas 1-7 to 1-11) are explained below. These constraints correspond to the second level. Since the second level corresponds to the first level, the second-stage routing constraints also correspond to the first-stage routing constraints. For an explanation of the second-stage routing constraints, please refer to the above explanation of the first-stage routing constraints; it will not be repeated here. This embodiment does not limit the specific content of these constraints; they can be constraints from the prior art.

[0038] The relevant expression for this constraint can be: (1-7) (1-8) (1-9) (1-10) (1-11) The delivery schedule constraints (or delivery plan constraints) are described, including: Phase 1 package transportation is permitted only when a truck visits the distribution center (Equation 1-12); a specific delivery method is permitted only when a package is assigned to a node that supports that method (Equation 1-13); ensuring that packages are assigned to nodes only when Phase 1 delivery is actually performed (Equation 1-14); allowing package assignment only when the customer to whom the package belongs is served by that node (Equation 1-15); requiring that any assigned node be visited by at least one Phase 1 vehicle (Equation 1-16); aligning delivery with the nodes actually visited and their mode compatibility (Equation 1-17); enforcing pick-up point coverage eligibility (Equation 1-18); and limiting each customer to being served by a maximum of one pick-up point per period (Equation 1-19). The second stage (corresponding to the second shelf level) delivery is only permitted when the delivery bicycle visits the customer to whom the package belongs (corresponding to formula 1-20); access to customers by the delivery bicycle is restricted to the satellite site range assigned to the base station to which the bicycle belongs (corresponding to formula 1-21); and access to satellite sites other than its base station is prohibited (corresponding to formula 1-22); pickup at the distribution center is mandatory (corresponding to formula 1-23); pickup is ensured not to occur before the package arrives (corresponding to formula 1-24); packages picked up from the distribution center must be delivered in the first stage (corresponding to the first level) (corresponding to formula 1-25); and door-to-door delivery packages are ensured to reach their final destination in the second stage (corresponding to formula 1-26). This embodiment does not limit the specific content of this constraint; it can be a logistics matching constraint as described in the prior art.

[0039] The relevant expression for this constraint can be: (1-12) (1-13) (1-14) (1-15) (1-16) (1-17) (1-18) (1-19) (1-20) (1-21) (1-22) (1-23) (1-24) (1-25) (1-26) In the formula, This indicates whether vehicle v in the first layer visits the distribution center during period t. A value of 1 indicates a visit, otherwise 0. This indicates whether customer C is served by a satellite station during time period t.

[0040] The incentive constraints are explained, including: stipulating that the company is only allowed to use a delivery method other than the customer's original preference if the incentive provided by the company for switching delivery methods is accepted by the customer (corresponding to formula 1-27); characterizing the degree of delay relative to the original deadline, and requiring that this degree of delay must be within the range of delay acceptable to the customer, ensuring that the company cannot unilaterally extend the delivery time and must obtain the customer's consent (corresponding to formula 1-28); reflecting the individual rationality condition (corresponding to formulas 1-29~1-30): the customer will only accept the switching of delivery methods and / or delayed delivery if the provided incentive is higher than the corresponding utility loss; specifying the variable domain (corresponding to formula 1-31). The relevant expressions for the incentive constraints are as follows: (1-28) (1-29) (1-30) The upper-level model consists of vehicle dispatching and route planning decisions for the delivery party (or delivery company), as well as financial incentive schemes.

[0041] In step S420, a user decision model is constructed. The user decision model aims to maximize the net utility of the user and has user decision model constraints.

[0042] Customers expect to maximize the utility of their packages, balancing the incentives offered with the inconvenience of delays or changes in delivery methods. The utility between packages can be considered independent between individual customers and within a given customer's package. Under this separability, according to the example embodiment, the objective function of the user decision model (or lower-level model) is: Formula (2) Among them O L (k) represents the user's maximum net utility; To delay package p The utility loss caused to users during the time period, i.e., information on the utility loss due to delivery delays; This refers to the utility loss incurred by the user when switching package p to the non-preferred mode o, i.e., the utility loss information for the delivery mode switching. To delay the incentive effect, This serves as an incentive for mode switching.

[0043] Constraints on the user decision-making model: (1-32) The customer's goal is to maximize the total net benefit of all packages, that is, to choose which incentives to accept under a given incentive scheme k to maximize their total utility. This constraint (corresponding to Equation 1-32) ensures that each package can accept at most one delayed delivery incentive and one change of transportation method.

[0044] The lower-level model represents the customer's order response decision, depicting the two-way interaction between the enterprise and the customer in the delivery system.

[0045] In step S430, the constraints of the delivery party's decision-making model are transformed according to the split-bar chance constraints, so that the constraints of the delivery party's decision-making model contain split-bar chance constraints.

[0046] For example, based on historical information about package demand, the mean and variance of package size are determined; based on the mean and variance, a moment fuzzy set describing the random distribution of package demand is constructed; based on the moment fuzzy set, a split-bar chance constraint is constructed.

[0047] Specifically, let random variable This represents the dimensions (weight or volume) of package p. The average dimensions of package p are calculated from historical package demand information. and variance . For the package random size, For the package The sizes are random. The sizes of different packages are independent of each other.

[0048] Moment fuzzy set Represented as The potential distribution of is specifically represented as: The above formula indicates that the expected value of each package size is equal to the historical estimated mean of package demand; the variance of each package size is equal to the historical estimated variance of package demand, and is greater than 0; The dimensions of the same package are independent of each other (covariance is zero).

[0049] The split-bar chance constraint constructed based on the moment fuzzy set is represented as follows: in, This indicates that if package p is at time 1 / 2... If the delivery is made by vehicle v at level e, the value is 1; otherwise, it is 0. For the capacity of vehicles in Tier E; , indicating the confidence level.

[0050] In other words, to address the uncertainty of package size, this embodiment constructs the upper-level model as a distributed bar optimization model. Let... This represents the size of a random package, which follows a distribution function. Decision-making takes place within a fuzzy set of moments. The worst-case scenario assessment is performed, and the fuzzy set includes all information consistent with the available information. Potential distribution. That is, formula (1-33): This means that the expected value of each package size is equal to the mean of the historical estimates; the variance of each package size is equal to the variance of the historical estimates and is greater than 0; and the sizes of different packages are independent of each other (covariance is zero). Accordingly, constraints (1-5) and (1-10) are reconstructed to confidence levels. The partial bar constraint conditions, where See formula (1-34).

[0051] This represents the worst-case probability of satisfying the capacity constraint among all potential distributions consistent with historical data. This indicates that the probability in the worst-case scenario is at least this value. Therefore, the higher-level problem can be reformulated as equation (1) ( The constraints are (1-2)-(1-4), (1-6)-(1-9), (1-11)-(1-31) and (1-34).

[0052] By implementing the distributionally robust chance constraint transformation, the uncertainty of customer demand and delivery costs can be handled. That is, the upper-level problem is transformed into a distributionally robust chance constraint (DRCC), thereby controlling the probability of constraint violation and improving the robustness of the model without relying on specific distribution assumptions.

[0053] In step S440, the transformed delivery party decision model and user decision model are reconstructed into a single-layer mixed integer linear programming model.

[0054] According to the example embodiment, the goal of this step is to embed the user's rational response logic into the delivery party's decision-making model, thereby eliminating the solution difficulties brought about by the two-level decision-making model. Specifically, firstly, the acceptance decision variable in the user's decision-making model is replaced with the decision variable in the delivery party's decision-making model, achieving unification of the decision variables in the user's decision-making model and the delivery party's decision-making model; then, preset individual rationality constraints and preset optimal choice constraints are added, so that the values ​​of the decision variables are consistent with the user's rational choice aimed at maximizing net utility; finally, the product term of the acceptance decision variable and the incentive amount in the objective function is transformed into a linear form. After the above variable replacement and constraint addition, the two-level decision-making model is equivalently transformed into a single-level mixed integer linear programming model, which can be solved using a standard optimization solver or a heuristic algorithm.

[0055] In application, this can be achieved by reconstructing a framework based on maximum choice, directly embedding the binary responses of followers. The specific construction process is as follows: (i) rewrite the choices among followers that affect upper-level costs or feasibility as binary decision variables at the leader level; (ii) introduce a variable for each package to record its maximum available net utility; (iii) implement an optimistic choice rule by allowing at most one utility-maximizing option for each package; and (iv) linearize the product terms using the Big-M bound derived from the range of incentive values. The final result is a single-level mixed-integer linear programming model.

[0056] A two-layer model was simplified to a single-layer model by converting the two-layer optimization problem into a single-layer optimization model, eliminating nested decision structures and significantly reducing the complexity of problem solving.

[0057] In step S450, a single-layer mixed integer linear programming model is solved to obtain vehicle route information, package delivery information, and incentive amount information.

[0058] According to the example embodiment, this step solves the single-layer mixed-integer linear programming model obtained in step S440. This step employs a three-level heuristic algorithm: the first level constructs a feasible initial solution by decomposing the model and iteratively solving the package allocation and vehicle routing subproblems; the second level adjusts the package allocation and vehicle routing in the initial solution using multiple local search operators to obtain the optimal solution; the third level extracts the actually visited nodes and actually used vehicles from the optimal solution, adds local branch constraints to control the search neighborhood, and obtains a further optimized optimal solution by resolving the problem, thus obtaining vehicle routing information, package delivery information, and incentive amount information. The entire solution process is strictly performed under the bibliometric chance constraints constructed in step S300, ensuring that the final vehicle routing scheme, package delivery assignment, and incentive amount scheme still satisfy the vehicle capacity limit under the worst-case demand distribution.

[0059] Through the above embodiments, this application constructs a delivery party decision model and a user decision model, incorporating the delivery party's decision aimed at minimizing total cost and the user's decision aimed at maximizing net utility into a unified two-level optimization framework. By using variable substitution, adding individual rationality constraints and optimal choice constraints, and linearizing the product term, the two-level model is transformed into a single-level mixed-integer linear programming model that can be solved efficiently, thus solving the difficulty of solving the two-level model. Based on this, a three-stage heuristic algorithm is employed: rapidly constructing initial solutions by decomposing subproblems, fine-tuning using local search operators, and resolving subproblems based on actual nodes and vehicles with added local branch constraints, ensuring high-quality solutions while maintaining efficiency. Furthermore, the entire solution process is conducted under bibliometric chance constraints, ensuring that the final vehicle routing scheme, package delivery scheme, and incentive amount scheme still meet vehicle capacity limitations under the worst-case demand distribution. Therefore, this application achieves collaborative decision-making between delivery parties and users, balancing solution efficiency and quality, and guaranteeing the feasibility and economy of the solution under demand uncertainty.

[0060] According to the example embodiment, step S440 of this method includes steps S441-S443.

[0061] In step S441, the user-side incentive acceptance decision variables in the constraints of the transformed delivery party decision model and user decision model are replaced with delivery party-side incentive acceptance decision variables.

[0062] According to the example embodiment, the user-side incentive decision variables in the constraints of the transformed delivery party decision model and user decision model include: (Whether the user accepts the incentive amount for switching package p to non-preference mode o) and (Does the user accept delaying the package p?) (Incentive amount information), delivery party incentive acceptance decision variables include (Delayed stimulus acceptance variable) and (Delivery mode switching incentive accept variable). [The following is a list of variables:] ... Replace with ,Will Replace with .in, , , and The value can be either 0 or 1, where 1 represents acceptance and 0 represents rejection.

[0063] Within the framework of the two-layer model, for each package The economic incentives of accepting delayed delivery and the net utility of changing the delivery method are respectively expressed as: and The calculation method is as follows: Given the financial incentive policies of courier companies (i.e., delivery companies) and ), customer reactions (represented as and The question is whether to accept or reject these financial incentives. The following equation ensures that the client will always choose the most profitable option.

[0064] Then, the two-layer model is reformulated into a single-layer model in five steps (step 1, step 2, step 3, step 4 and step 5).

[0065] Step 1 (corresponding to step S441). Define two upper-level decision variables. and To replace the lower-level decision variables accordingly Specifically, in constraints (1-27)-(1-30), (1-32), (EC.2.1), and (EC.2.2), They were respectively and replace.

[0066] in, The value is 1 if package p is delivered by the first-level vehicle v at time t, and 0 otherwise.

[0067] in, This indicates that if package p is delivered by the first-level vehicle v at time t, then it is 1; otherwise, it is 0. , indicates whether package p is delivered before time t, 1 for delivery and 0 for non-delivery.

[0068] Among them, constraints (1-27)-(1-30) correspond to (EC.2.4)-(EC.2.7), (1-32) corresponds to (EC.2.8)-(EC.2.9), and (EC.2.1) and (EC.2.2) correspond to (EC.2.10)-(EC.2.12). - The constraints corresponding to (Formulas 3-4) are called the optimal choice constraints below. The incentive decision variables for the delivery party in the optimal choice constraints are also obtained by substituting the incentive decision variables for the user party. (EC.2.4)-(EC.2.7) correspond to individual rationality constraints.

[0069] In step S432, add the optimal selection constraint, the lower bound constraint of the maximum value, and the Big-M constraint.

[0070] According to the example embodiment, the optimal selection constraint is achieved through... This means that linearization (i.e., the lower bound constraint of the maximum value) (corresponding to step 2) is as follows: Formula (3) Formula (4) in, This indicates the maximum net utility of delayed delivery of package p; This represents the maximum net utility of changing the delivery method for package p.

[0071] In other words, the optimal response of the lower layer in equation (EC.2.3) can be obtained by linearization to form the above formulas (3) and (4), formula (3) or... Formula (4) or .

[0072] According to the example embodiment, to ensure that the delivery party can only choose these incentive schemes when the financial incentive schemes for delayed delivery and changes in delivery method are the optimal choice for the customer, a Big-M constraint can be further added (corresponding to step 3), where M is a sufficiently large number greater than 0, for example, greater than... - The maximum possible positive value of the feasible solution is represented as follows: Formula (5) Formula (6) Formula (7) Formula (8) Constraints (5) and (6) introduce binary variables When net utility (for any) When the condition is met, its value is 1; otherwise, its value is 0. Similarly, constraints (7) and (8) introduce binary variables. :when (for any) When ), its value is 1; otherwise, its value is 0.

[0073] Formulas (5), (6), (7), and (8) are also known as , , , .

[0074] According to the example embodiment, the following auxiliary constraints for the optimal selection constraint (corresponding to step 4) are also added: Formula (9) Formula (10) Formula (11) Formula (12) Formula (13) in, This indicates if there is a delay The value is 1 if it is the option with the greatest net utility among all delay options, and 0 otherwise; The value is 1 if changing the delivery method to 'o' is the option with the highest net utility among all alternative delivery methods, and 0 otherwise.

[0075] Formulas (9), (10), (11), (12), and (13) are also known as , , , , .

[0076] The constraints described above are used to link customer responses to the delivery company's financial incentive decisions. These constraints state that the company can only choose these incentive measures if the financial incentives for delayed delivery and changes in delivery methods are optimal for the customer.

[0077] In step S433, the transformed delivery party decision model and user decision model, optimal selection constraints, lower bound constraints of maximum values ​​and Big-M constraints are linearized to obtain a single-level mixed integer linear programming model.

[0078] Using upper-level variables and Replace lower-level variables and Subsequently, the objective function (1) of the upper-level model, as well as the constraints (EC.2.10) and (EC.2.11), include nonlinear terms. and To linearize these terms, we introduce new variables: Formula (14) Formula (15) Formula (16) Formula (17) ∈ ∈ Formula (18) Formula (19) Formula (20) Formula (21) Formula (22) Formula (23) in, This represents the amount actually paid by the company after the user accepts the delayed incentive. This represents the actual amount paid by the company after the user accepts the incentive to switch modes. The above formula transforms the product of the decision variable and the incentive amount information into a linear form. Formulas (14)-(23) are also known as... .

[0079] In other words, the expression for the single-level mixed-integer linear programming model (corresponding to step 5) can be: in, The sum of the driving costs for all vehicles, at all times, and along all sides. The amount of incentives actually paid by the company to its customers. The model is subject to constraints (1-2)-(1-4), (1-6)-(1-9), (1-11)-(1-26), (1-34) and (EC.2.4)-(EC.2.9), (EC.2.12)-(EC.2.33), etc. and These represent the actual amount of incentive the customer receives; the former corresponds to the amount of the package... put off The incentive amount for each delivery cycle, the latter corresponding to the delivery of the package Switch to non-preferred delivery method The amount of incentive at that time.

[0080] The linearized form of equation (1-34) can be expressed as: After linearization, the single-level linear model of LMD-DODS is equation (25)( It is subject to the constraints of (1-2)-(1-4), (1-6)-(1-9), (1-11)-(1-26), (EC.4.11)-(EC.4.16), (EC.2.4)-(EC.2.9), and (EC.2.12)-(EC.2.33).

[0081] Through the above embodiments, this application achieves the unification of decision variables at both the upper and lower levels by replacing the acceptance decision variable in the user decision-making model with the decision variable in the delivery party's decision-making model. By adding individual rationality constraints (incentive amount not less than utility loss) and optimal choice constraints (user chooses the incentive option with the highest net utility), supplemented by auxiliary choice constraints, maximum lower bound constraints, Big-M constraints, and correlation constraints, the value of the acceptance decision variable is made completely consistent with the user's rational choice aimed at maximizing net utility. Finally, by introducing auxiliary variables to transform the product of the acceptance decision variable and the incentive amount into a linear form, the originally difficult-to-solve two-level nonlinear model is transformed into a single-level mixed integer linear programming model. Thus, this application mathematically accurately characterizes the master-slave game relationship between the delivery party and the user, embedding the user's rational response behavior into the delivery party's decision-making model in the form of constraints, avoiding the problem of incentive schemes being out of touch with reality due to neglecting user rationality or simplifying processing in traditional methods, and laying a model foundation for subsequent efficient solution.

[0082] According to the example embodiment, step S450 of this method includes steps S451-S453.

[0083] In step S451, the single-layer mixed integer linear programming model is decomposed into a package allocation subproblem and a vehicle routing subproblem, and the package allocation subproblem and the vehicle routing subproblem are solved by iterative loop to obtain the first solution result.

[0084] In the first stage, the single-layer mixed integer linear programming model is decomposed into a package allocation subproblem and a vehicle routing subproblem. The package allocation subproblem is solved by using the partial boolean chance constraints constructed in step S300 and the linearization constraints obtained in step S440 as constraints to determine which satellite station or self-pickup point each package is allocated to, as well as the incentive amount information for delayed delivery or delivery mode switching for each package.

[0085] In the second stage, based on the parcel allocation results determined in the first stage, the vehicle routing subproblem is solved. Specifically, the driving route and stopping order of each vehicle in each time period are determined to minimize the total cost of delivery routing, and finally the vehicle routing information is obtained.

[0086] The vehicle routing information obtained from the second stage is fed back to the first stage to re-optimize package allocation and incentive amounts; the new allocation results are then passed to the second stage to re-optimize the vehicle routing information. This process is repeated iteratively until the difference between the objective function values ​​of two consecutive iterations is less than a preset threshold or the preset maximum number of iterations is reached, at which point the first solution result is obtained. This first solution result includes vehicle routing information, package delivery information, and incentive amount information.

[0087] In step S452, multiple local search operators are used to locally adjust the package allocation and vehicle path in the first solution result to obtain the second solution result.

[0088] According to the example embodiment, after obtaining the first solution result, a further local search process is employed to improve the solution within its neighborhood. Specifically, a set of local search operators is constructed, including time-related operators, delivery time flexibility operators, and package clustering operators at the package level, as well as path-related operators, minimum loading rate path operators, and random path operators at the path level. In each iteration, a local search operator is randomly selected from the current set of operators. Based on this operator, a corresponding package subset or path subset is selected, and a corresponding subproblem is constructed. The relevant delivery decision variables and incentive decision variables are then re-optimized to obtain candidate solutions and their objective function values, where the objective function value is the sum of the delivery routing cost and the incentive cost.

[0089] If the objective function value of a candidate solution is not higher than that of the current optimal solution, the candidate solution is accepted, and the current optimal solution is updated to the candidate solution; otherwise, the current optimal solution remains unchanged. After completing one operator search, the operator is removed from the current operator set. If the current optimal solution is lower than the objective function value before this round of search, the operator set is restored to the initial operator set, and the neighborhood search restarts; if no improvement is achieved, only operators with random perturbation characteristics are retained or introduced to enhance the search's ability to escape local optima.

[0090] The process of random operator selection, subproblem construction, and candidate solution evaluation is repeated until a preset maximum number of iterations is reached, or until a preset number of iterations fail to improve the current optimal solution. Upon termination, the current optimal solution is output as the second solution. Since updates are only accepted during the local search process when the objective function value of a candidate solution is not higher than the current optimal solution, the objective function value of the second solution is not higher than the objective function value of the first solution.

[0091] In step S453, based on the nodes actually visited and the vehicles actually used in the second solution result, a subproblem is constructed. After adding local branch constraints to the subproblem, it is solved again to obtain vehicle route information, package delivery information and incentive amount information.

[0092] According to the example embodiment, firstly, the nodes actually accessed and the vehicles actually used are determined from the second solution result. Let the node access status of the first-level vehicle v at time t in the current second solution result be... The node access status of the second-level vehicle v is Define the set of nodes in the subgraph at time t. The set of nodes that are actually visited. The collection of vehicles that are actually in use. The threshold for local branch constraints controls the number of allowed node access changes; Let be the edge set of the subgraph, containing all possible connections between nodes.

[0093] The expression for the local branch constraint is as follows: Formula (26) Based on the above subgraph node set Subgraph edge set and A subproblem is constructed for the set of vehicles actually in use. This subproblem has the same objective function and constraint structure as the single-level mixed-integer linear programming model, but its decision variables are limited to the subgraph node set. Kazuko Racing Team Within the scope, only the nodes actually visited and the vehicles actually used in the current second solution result are considered, while unvisited nodes and unused vehicles are excluded.

[0094] Then, the aforementioned local branch constraints are added to this subproblem, and a search is performed within the neighborhood of the current second solution. The objective function is re-solved to obtain a further optimized second solution, yielding vehicle route information, package delivery information, and incentive amount information.

[0095] Because the LMD-DODS (Last-Mile Delivery with Delivery Options and DemandSteering) problem embeds 2E-VRP (Two-Echelon Vehicle Routing Problem), package merging, and incentive design, and the first two are NP-hard problems, the LMD-DODS problem is NP-hard. Therefore, existing algorithms (especially exact algorithms) cannot solve it in a reasonable amount of time, even on smaller instances. For this purpose, see [link to relevant documentation]. Figure 5This application designs a three-level mathematical heuristic algorithm (TLM). The first layer solves three parcel delivery subproblems (PDPs, namely PDP1–PDP3) and multiple traveling salesman problems (TSPs) through a two-stage iterative process, constructing an initial solution (i.e., the first solution result). The second layer improves the initial solution by employing multiple local search operators, such as parcel-level and path-level operators. The third layer further improves the solution obtained in the second layer (i.e., the second solution result) using a path improvement procedure. Specifically, based on the second solution result, a reduced subgraph and sub-vehicle fleet are constructed, the solution is re-solved, and local branch constraints are added, thereby achieving further improvement of the solution while controlling computational costs.

[0096] First layer: Initial solution generation Algorithm 1 demonstrates a two-phase nested loop process. In the PDP subproblem, since the vehicle path is not explicitly solved, the cost of visiting nodes needs to be estimated using approximate costs. Therefore, the estimated access cost is defined. and The inner loop (lines 3–19) repeatedly solves for PDPs and TSPs and updates the estimated access cost. , The process continues until criterion C1 is met (e.g., a preset number of iterations fail to improve the current optimal solution). The outer loop performs diversification (lines 20–26) and stops when C2 or C3 is met (e.g., until a preset maximum number of iterations is reached or multiple consecutive diversification iterations fail to improve the optimal solution). To improve computational efficiency, the Customer Service and Financial Incentive Scheme (CSFIS) is set during the first iteration and each diversification iteration, and remains unchanged until the next diversification. In Algorithm 1, sol* and sol represent the current optimal solution and the current solution, respectively, with objective function values ​​obj* and obj, respectively. This indicates the determined value of the corresponding variable. The variable flag is used to select the variant of PDP: when flag=1, PDP1 is selected; when flag=2, PDP2 is selected; otherwise, PDP3 is selected. Phase 1: Package Delivery Issues PDP coordinates four programs: incentives The subproblem includes customer service allocation (d), package allocation decision (y), and access indicator variable (z). Since this subproblem does not explicitly characterize path planning, a proxy variable for access cost is introduced. and These arrays approximate the values ​​of a first-order vehicle. In time Access Node The cost at that time, and when the second-order vehicle In time Visiting customers The cost at that time. These alternative variables are input as parameters into the PDP.

[0097] This problem needs to satisfy constraints (1-4), (1-9), (1-12)-(1-26), (EC.4.11)-(EC.4.16), (EC.2.4)-(EC.2.9), (EC.2.12)-(EC.2.33) and: The new constraints are designed to ensure operational consistency: constraints (28) and (29) stipulate that trucks are only permitted to access distribution centers and satellite points when loading or delivering packages. Constraints (30) and (31) stipulate that bicycles may only leave satellite points and visit customers when delivering packages and providing services to the corresponding customers.

[0098] Despite the removal of explicit route planning, due to the integrated package-time-space decision-making... The PDP remains difficult to solve when d and y are tightly coupled with the access decision z through constraints (1-13)–(1-17), (1-12), (1-20), and (1-21). To reduce this complexity, we use a custom, multi-stage dimensional simplification scheme that dynamically fixes selected delay and mode switching incentives, customer service allocation, and access indicator variables. This scheme is implemented through three PDP variants.

[0099] PDP1 (Initial Iteration): We filter the search space using package time window slack, mode transition distance difference, and pre-assignment of customers to the nearest service point. Only decision variables with a promising outlook are considered. , and Keep it as a binary variable; all other variables are fixed at zero.

[0100] The specific rules are as follows: We specified the variables that need to be fixed. , and In order to select a fixed delay excitation variable. We base our decisions on each package. Calculate a time window relaxation indicator based on the original arrival time and delivery deadline: A larger Slack(p) indicates a more flexible time window, suggesting that package P is less likely to require delayed excitation. It tends towards 0. Conversely, a smaller Slack(p) indicates a tight time window; in this case, providing a delayed incentive helps to integrate with other packages, making... It is more likely to equal 1.

[0101] To determine which mode-transformation incentive variables to fix We define: in, It is the customer of package P. The distance from package p to the nearest satellite station. The distance from the nearest self-pickup point to the customer of package p. Represents a node With customer c The distance between C. Then calculate the distance difference for mode transition. :

[0102] smaller This means that the distance savings from switching modes are negligible. The smaller, It is very likely to be 0. Larger The more significant the savings, the better. The larger, the more More attractive, closer to 1.

[0103] In calculation and Then, choose The package with the lowest value, and indivual Packages with the highest value are added to the package collection. and For sets The package inside, keep For active variables; for sets The package inside, keep Set as the active variable. For all other packages, set... and They are removed from the set of binary decision variables.

[0104] To determine which customer service variables This process requires fixing the distance between each customer and each satellite (pickup point), and assigning each customer to the nearest satellite (pickup point). This process forms customer clusters for each satellite. , recorded as If a customer does not belong to the cluster of satellite i (pickup point j) Then the customer service variable corresponding to that customer is fixed at zero, that is... Therefore, the PDP1 model is: It complies with constraints (1-4), (1-9), (1-12)-(1-26), (EC.4.11)-(EC.4.16), (EC.2.4)-(EC.2.9), (EC.2.12)-(EC.2.33), (28)-(31), and PDP2 (Non-Diversified Iteration): To enhance the search under the current CSFIS, fixed... , , and And optimize the remaining variables. The specific rules are as follows: Once PDP1 is solved, CSFIS is obtained. In subsequent iterations, unless it is a diversification iteration (indicated by flag=2), the dimensionality reduction strategy will construct the PDP model with CSFIS fixed to the value obtained from the PDP solution, i.e.: With CSFIS fixed at the previously obtained results, the PDP2 model is as follows: It complies with constraints (1-4), (1-9), (1-12)-(1-26), (EC.4.11)-(EC.4.16), (EC.2.4)-(EC.2.9), (EC.2.12)-(EC.2.33), (28)-(31), and: PDP3 (Diversification Iteration): To achieve easily manageable diversification, we apply a cost-based perturbation to fix the z values ​​that are zero in the current optimal solution to zero, and then re-optimize the incentives, customer service, and the remaining access decisions.

[0105] After implementing diversification based on approximate access cost updates (indicated by flag=3), the PDP model is solved to simultaneously update financial incentives. Customer service variable d and access path z. To reduce model size and speed up the solution, a dimensionality reduction strategy fixes specific access path variables to previously established values, i.e.: The basic principle is to use the results obtained in the previous steps to search for the most promising part of the solution space. Based on the previously determined z value, the components in z that need to be fixed are determined as follows: Therefore, the PDP3 model is as follows: It complies with constraints (1-4), (1-9), (1-12)-(1-26), (EC.4.11)-(EC.4.16), (EC.2.4)-(EC.2.9), (EC.2.12)-(EC.2.33), (28)-(31), (EC.5.11) and (EC.5.12).

[0106] This design enables rapid generation of initial solutions, stable improvement in non-diversified iterations, and targeted exploration in diversified processes.

[0107] Phase Two: TSP According to each period Periodic access plan (Truck visits to satellites or pickup points) and (Freight bicycles visiting customers), by solving the maximum Each independent TSP is used to construct the route, with each mode of transport corresponding to one TSP for each period.

[0108] Access cost update After each PDP-TSP cycle, the access cost estimate is updated to approximate the true path cost, capturing improvements in customer clustering, vehicle load, and center distance of vehicle paths.

[0109] in This period Inner vehicle The total delivery volume, and and These are the cost factors related to customer clustering and center distance improvement, respectively. The update steps for these cost factors are described in algorithms EC.6.1 and EC.6.2, respectively. Parameters , and It's the weight. Diversified mechanisms and termination conditions The outer loop alternates between two diverse strategies. First, the path decision x is fixed as the current optimal solution. In this case, re-optimize and Other variables are also considered. A model based on a diversified mechanism for updating customer service and financial incentive plans is presented.

[0110] Satisfying constraints (1-2)-(1-4), (1-6)-(1-9), (1-11)-(1-26), (EC.4.11)-(EC.4.16), and (EC.2.4)-(EC.2.9), (EC.2.12)-(EC.2.33), and If no improved solution is found, reset the access cost using the following method. and : in and It is an approximation of the access cost in the currently known optimal solution; and They are the periods Inner vehicle and The path.

[0111] If continuous If no improvement is made in the next iteration, the inner loop terminates (termination criterion C1). The iteration limit is reached. (Termination criterion C2) or the optimal target value at When a diversification attempt stalls (termination criterion C3), the outer loop stops.

[0112] Second layer: Local search The second layer uses a set of operators. Explore the neighborhood of the solution at level 1. In each iteration, sample an operator, define and solve its subproblems, and then remove the operator. If the solution is improved, reset the set. Otherwise, only the perturbation operator is retained. The search in... Terminate after the nth iteration, or in consecutive iterations The iteration terminates after no further improvements are made. To address the temporal and spatial characteristics of parcel delivery, three parcel-level operators were employed: temporal correlation, delivery time flexibility, and parcel clustering; and three path-level operators: correlation, minimum load, and random path. Each operator selects a subset of parcels or paths and then defines a subproblem to optimize the relevant variables.

[0113] Package-related operators Each package-level operator selects a subset And solve a subproblem. For Other packages, y and z are fixed as the current optimal solution (superscript *); for customers who own these packages, d and z are also fixed.

[0114] Constraints (EC.7.3)-(EC.7.8) ensure: unless the package is selected and recorded in Otherwise, the delivery plan will remain consistent with the current optimal solution. In constraint (EC.7.9), Defined as having a set The set of customers who own the selected package. Therefore, constraint (EC.7.9) requires that for customers who do not own the selected package, their visit plan should remain consistent with the current optimal solution.

[0115] Operators based on parcel time correlation. Given the current optimal first-level delivery plan. Package p and The correlation between them Defined as: in It is a package p and Average time difference between all possible start times: and Measure the package p and in the current optimal solution The difference in delivery time between them. Weighting Two components are used to scale this difference. To construct... Randomly select a seed package p and add an association score. smallest The packages (the lower the score, the stronger the correlation). The target size.

[0116] Parcel Delivery Time Flexibility Operator. In LMD (Location Management Decision), parcels with wider time windows are more likely to find alternative insertion points in the current delivery schedule, thereby reducing transportation costs. This operator calculates the delivery time flexibility for each parcel as follows: The first term reflects the width of the initial time window; the second term captures the deviation between the arrival time and the actual delivery time in the current optimal solution. Weights This operator is used to balance the effects of these two factors. The largest Composed of several packages .

[0117] Package clustering operator. We use the k-means method to divide all packages into clusters based on spatial features (customer location) and temporal features (arrival time, deadline). Groups. Then, packages are drawn from randomly selected clusters until... .

[0118] Path-dependent operators make For vehicles within the current period t The second-level path, and let Each path-dependent operator selects a subset. .make For the set of nodes visited by the selected path, let For service The set of nodes visited by the trucks on the satellite. Similar to the package-level operators, each path-level operator defines and solves a subproblem, where for For all nodes except those specified, the access plan z is fixed as the current optimal solution. The formula is: Constraints (EC.7.10) and (EC.7.11) ensure that: unless a node is included in a set and Otherwise, the access plan must remain consistent with the current optimal solution.

[0119] Path association operator. Randomly select a seed path. Calculate the distance from its centroid to all other paths, and select... Find the nearest path and add all nodes on those paths (including the seed path) to... In the middle. Accordingly defined .

[0120] Lowest load path operator. Calculation. Choose the route with the lowest total delivery volume for each route. A path. Build from these paths. and .

[0121] Random path operator. From Random selection Path, and construct it as described above. and .

[0122] Level 3: Improvement of the current optimal solution The third level is achieved by using the current optimal solution from the second level. The simplified problem is then addressed by applying path improvement methods to optimize the solution. Given access variables... For each time period We define: A subgraph , in , The determination of actual node usage means retaining only those nodes that are actually used in the current optimal solution and removing unused nodes, thereby reducing the problem size.

[0123] and A sub-team for This formula means that only vehicles actually used in the current optimal solution are retained, while idle vehicles are removed.

[0124] Subsequently, for each On the sub-map LMD-DODS, targeting the sub-teams Construct an LMD-DODS model. Correspondingly, in the global collection within the model... and (and its components) and )quilt and What it replaced.

[0125] To speed up the search, local branch constraints are added to ensure that, compared to the current optimal solution, the change in the node access variable for each vehicle in the first and second layers at each time period is no greater than [missing value]. This ensures that the search only occurs within the neighborhood of the current optimal solution, avoiding blind exploration globally. (39) The upper limit is set for the number of times each vehicle accesses nodes in the first and second layers for each time period. According to the above embodiments, this application achieves efficient solution of a single-layer mixed-integer linear programming model through a three-stage heuristic algorithm. The first stage decomposes the model into package allocation and vehicle routing subproblems and solves them iteratively, quickly constructing an initial feasible solution containing vehicle routes, package allocation, and incentive amounts. The second stage employs various local search operators to improve the initial solution through neighborhood search, and introduces a perturbation strategy to escape local optima when trapped, significantly improving the quality of the solution. The third stage constructs a scaled-down subproblem based on the actually visited nodes and actually used vehicles in the current optimal solution, and adds local branch constraints to limit the difference between the new solution and the current optimal solution, further shortening the solution time while ensuring search accuracy. Therefore, this application can obtain high-quality solutions to large-scale delivery route and incentive co-optimization problems within an acceptable computational time, balancing solution efficiency and solution quality, and meeting the application needs of real-world large-scale delivery scenarios.

[0126] According to the example embodiment, in step S400, the vehicle route information is determined while satisfying the constraints of flow conservation, node access, and sub-path elimination.

[0127] According to the example embodiment, the flow conservation constraint is expressed as follows: Formula (40) Formula (41) in, For vehicle v, determine whether it traverses edge ij (from i to j) at level e (e is 1 or 2) and time period t. This is a first-level node; This is a second-level node.

[0128] According to the example embodiment, the node access constraint is represented as follows: Formula (42) Formula (43) Formula (44) Formula (45) in, For vehicle v, determine whether it accesses node j at the first level and during time period t; For vehicle v, whether it accesses node j at the second level and during time period t; Does vehicle v access node i at the first level and during time period t? Does vehicle v access node c at the second level and during time period t?

[0129] According to the example embodiment, the sub-path elimination constraint is represented as follows: Formula (46) Formula (47) Furthermore, capacity constraints can also be included, as shown below: Formula (48) Formula (49) According to the above embodiments, this application constructs a moment fuzzy set containing all potential probability distributions satisfying the mean and variance of package sizes by statistically analyzing the historical package demand information. Based on this, a robust chance constraint is established, requiring that the probability of the vehicle's loading capacity not exceeding its capacity under the worst-case demand distribution is not lower than a preset confidence level, thus replacing the traditional deterministic capacity constraint. Therefore, this application can handle demand uncertainty with only the mean and variance of package sizes known, significantly reducing the risk of vehicle overcapacity compared to deterministic optimization methods, and avoiding excessively conservative costs due to worst-case assumptions compared to traditional robust optimization methods, achieving a balance between feasibility and economy. Furthermore, this application also constrains delivery vehicles through flow conservation constraints, node access constraints, sub-path elimination constraints, and vehicle capacity constraints, ensuring the feasibility of vehicle route information.

[0130] This application proposes: (1) a two-level optimization model for last-mile delivery with delivery options and demand guidance, which innovatively captures the uncertainty of customer-enterprise strategy interaction and demand distribution in a two-level delivery network; (2) a fuzzy set construction method based on moment information is designed to transform the chance constraints of the two-level delivery into linear constraints, thereby achieving efficient and robust optimization under uncertainty; (3) the present invention designs a three-level heuristic algorithm including dimension reduction, neighborhood search, and sub-problem improvement to solve the bottleneck of solving large-scale two-level optimization problems; (4) a collaborative mechanism of "incentive design - demand aggregation - path optimization" is established to improve last-mile delivery efficiency through the superposition effect of time and space integration.

[0131] In the context of insufficient research on time and space synergy integration in existing technologies, lack of dynamic negotiation between companies and customers, uncertainty of package demand, insufficient adaptability of single-level delivery network structure, and insufficient joint optimization of incentives and route planning, this application discloses a last-mile delivery split-bar two-layer optimization method to capture demand uncertainty and stakeholder interaction. The method has the following characteristics: (1) Constructing a two-layer decision-making model between delivery companies and customers. (1) Taking the upper-level delivery companies as the decision-making body, with the goal of minimizing the sum of delivery costs and incentive costs, they jointly decide on the incentives for delayed delivery, the incentives for switching delivery modes, and the delivery plan; taking the lower-level customers as the response body, they combine their own utility loss from delayed delivery and switching delivery modes, and choose the delivery response method that maximizes their net utility under different incentive plans, thereby depicting the hierarchical interaction between delivery companies and customers; (2) To deal with the uncertainty of package demand, a fuzzy set is constructed based on moment information to describe the random distribution of package size and arrival probability, and the vehicle capacity constraint is transformed into a multi-bar chance constraint to ensure the feasibility of the plan under a high confidence level; (3) By transforming the two-level decision model into a solvable single-level mixed integer optimization model, and designing a three-level heuristic solution algorithm, the initial solution is generated by dimension reduction, the local solution is optimized by multi-operator neighborhood search, and the global solution quality is improved by sub-problem improvement, thus solving the mixed integer optimization problem in large-scale scenarios.

[0132] To address the problems of existing last-mile delivery technologies where delayed delivery, delivery mode switching, and route planning are fragmented and fail to simultaneously consider demand uncertainty and customer responsiveness, this application discloses a bilayer optimization method for two-tier last-mile delivery networks. This method jointly optimizes delayed delivery incentives, delivery mode switching incentives, and two-tier delivery routes, while comprehensively considering parcel demand uncertainty and customer responsiveness, thereby achieving cost reduction and efficiency improvement in last-mile delivery through the synergy of time and space integration.

[0133] exist Figure 4 In this process, the partial bob chance constraint is convexly relaxed and equivalently transformed into a second-order cone constraint to obtain a tractable optimization model with a convex structure. On this basis, the second-order cone constraint is linearized and transformed into a linear constraint expression, thereby obtaining a linearized model that is easy to solve using a standard optimization solver, thus improving the computational efficiency and solution stability of the model.

[0134] Technical effects: (1) The proposed method of sub-Bru bar constraint based on moment information fuzzy set has a feasibility rate of 89.3%-100% in tests on five demand distributions: normal distribution, gamma distribution, Poisson distribution, Pareto distribution and bimodal distribution. The average cost is also moderate. It solves the capacity violation risk of deterministic model and avoids excessive cost consumption of conservative robust model, as shown in Table 1. Table 1 Performance of DRO, RO, SO and DM models (2) The three-level heuristic algorithm proposed in this invention can simultaneously optimize economic incentive decisions and delivery route planning decisions, significantly improving computational efficiency while ensuring solution quality. For example Figure 6 As shown, in 72 benchmark instances of last-mile delivery, the algorithm of this invention achieved the best or tied-best objective function values ​​in 66 instances. In instances where all three methods could solve the problem, compared to the existing three-stage heuristic algorithm, the average cost was reduced by 1.78% and the computation time was shortened by 26.2%. Compared to the Gurobi solver, the average cost was reduced by 9.90% and the running time was only 21.8% of the Gurobi solver. For large-scale instances with at least 50 customers, both the existing three-stage heuristic algorithm and Gurobi experienced memory overflow, while the algorithm of this invention could still obtain a feasible solution within a finite time, demonstrating that the algorithm of this invention has strong large-scale solution capability and practical application stability. Figure 6 In the text, Peng et al. (2023) refers to an existing algorithm, specifically Peng, X., L. Zhang, RG Thompson, K. Wang. 2023. A three-phase heuristic for last-mile delivery with spatial-temporalconsolidation and delivery options. International Journal of ProductionEconomics, 266 (12), 109044.

[0135] (3) The three-level heuristic algorithm proposed in this invention has good solution quality and efficiency, and good cross-scenario applicability. In 90 2E-IRP instances, the TLM proposed in this invention obtained the best solution in 73 instances and refreshed 24 new known best solutions (BKS); compared with the existing algorithm, the performance is improved by 4.24%; compared with the known best solution (BKS), it achieves an improvement of 0.02%, and the solution speed of the three-level heuristic algorithm proposed in this invention is 13.4 times faster than the existing algorithm, as shown in Table 2.

[0136] Table 2 (4) The time and space integration strategy proposed in this invention has significant practical benefits. In the 10 selected examples, the integration strategy reduced the average cost by 6.73%, 14.02%, and 36.67% compared with the self-pickup point (DMC), delayed delivery (PD), and first-come, first-served (FIFO) modes, respectively. At the same time, the strategy proposed in this invention reduced the number of truck deliveries from 8 times for DMC and FIFO to 6 times, and the number of freight bicycle deliveries from 2 times for DMC, 6 times for PD, and 8 times for FIFO to 1 time. The truck load efficiency increased from 1.1 for DMC and PD and 0.8 for FIFO to 1.3, and the freight bicycle load efficiency increased from 0.8 for DMC, 0.6 for PD, and 0.5 for FIFO to 0.9, greatly improving vehicle utilization and achieving both economic and environmental benefits, as shown in Table 3. Table 3 (5) The spatiotemporal integration strategy proposed in this invention demonstrates significantly better stability and adaptability than existing strategies in various key parameter fluctuation scenarios. We evaluated how delivery performance is affected by factors such as demand level, package size variance, deadline urgency, vehicle capacity, customer distribution, and incentive costs. In all cases, the integration strategy proposed in this paper consistently achieved the lowest total cost, outperforming other alternatives. It effectively addresses the shortcomings of some strategies that are only effective under specific parameter scenarios and fail when parameters fluctuate, ensuring stability and high feasibility even when parameters change in actual operation, such as... Figure 7 (Partial display).

[0137] As is known from common technical knowledge, this application can be implemented through other embodiments that do not depart from its spirit or essential characteristics. Therefore, the disclosed embodiments are merely illustrative and not the only ones. All modifications within the scope of this application or equivalent to this application are included in this application.

Claims

1. A two-layer optimization method for last-mile delivery with delivery options and demand guidance. Its features are, include: Obtain delivery network information from delivery providers, historical package demand information, and user utility loss information. The utility loss information includes delivery delay utility loss information and delivery mode switching utility loss information; Based on the delivery network information, determine the path traversal cost information of the delivery party; Based on the historical information of package demand, construct a distributed opportunity constraint on the delivery party's vehicle capacity; With the objective of minimizing the sum of the total delivery route cost and the total incentive cost of the delivery party, and under the constraint of the distributed opportunity, vehicle route information, package delivery information, and incentive amount information are determined, wherein the incentive amount information is not less than the corresponding utility loss information.

2. The optimization method according to claim 1, characterized in that, The objective is to minimize the sum of the total delivery route cost and the total incentive cost of the delivery party, and under the distributed opportunity constraint, determine vehicle route information, package delivery information, and incentive amount information, including: Construct a delivery party decision model, the objective of which is to minimize the sum of the total delivery route cost and the total incentive cost, and the delivery party decision model has constraints. Construct a user decision model, which aims to maximize user net utility and has user decision model constraints. The constraints of the delivery party's decision-making model are transformed according to the sub-Bruker chance constraints, so that the constraints of the delivery party's decision-making model contain sub-Bruker chance constraints. The transformed delivery party decision model and the user decision model are reconstructed into a single-layer mixed integer linear programming model; Solve the single-layer mixed integer linear programming model to obtain the vehicle route information, the package delivery information, and the incentive amount information.

3. The optimization method according to claim 2, characterized in that, The step of reconstructing the transformed delivery party decision model and the user decision model into a single-layer mixed integer linear programming model includes: Replace the user-side acceptance decision variables in the constraints of the transformed delivery party decision model and user decision model with delivery party-side incentive acceptance decision variables. Add optimal selection constraints, lower bound constraints for maximum values, and Big-M constraints; The transformed delivery party decision model and user decision model, optimal selection constraints, lower bound constraints of maximum values, and Big-M constraints are linearized to obtain a single-level mixed integer linear programming model.

4. The optimization method according to claim 2, characterized in that, The process involves solving the single-layer mixed-integer linear programming model under the given random chance constraint to obtain the vehicle route information, the package delivery information, and the incentive amount information, including: The single-layer mixed integer linear programming model is decomposed into a package allocation subproblem and a vehicle routing subproblem, and the package allocation subproblem and the vehicle routing subproblem are solved by iterative loops to obtain the first solution result; By employing multiple local search operators, the package allocation and vehicle path in the first solution result are locally adjusted to obtain the second solution result; Based on the nodes actually visited and the vehicles actually used in the second solution result, a subproblem is constructed. After adding local branch constraints to the subproblem, it is solved again to obtain the vehicle route information, the package delivery information, and the incentive amount information.

5. The optimization method according to claim 1, characterized in that, The step of constructing the distribution opportunity constraint of the delivery party's vehicle capacity based on the historical information of package demand includes: Based on the historical information on package demand, determine the mean and variance of package dimensions; Based on the mean and variance, a moment fuzzy set describing the random distribution of package demand is constructed; Based on the given moment fuzzy set, construct the sub-Bruker chance constraint.

6. The optimization method according to claim 1, characterized in that, The vehicle route information is determined in a way that satisfies the flow conservation constraint, node access constraint, and sub-path elimination constraint.

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