Vehicle path planning method based on inflection point guided differential evolution algorithm

Abstract

The invention discloses a vehicle path planning method based on an inflection point guided differential evolution algorithm, and the method comprises the steps: building a customer point data model according to a data set of a vehicle path problem or an actual problem; calculating and constructing a distance matrix, and taking the minimum vehicle path distance as a first target function; establishing a time window violation penalty rule, and taking maximized customer satisfaction as a target function 2; constructing a vehicle path problem model with a time window, and determining constraint conditions; and optimizing the vehicle path planning model with the time window based on an inflection point guided differential evolution algorithm, and solving an optimal distribution sequence. The characteristic that the inflection point is used as the optimal solution is utilized, the differential evolution algorithm is used for searching around the inflection point, searching is guided to approach the optimal solution, the convergence rate of the algorithm is increased, the searching capability of the algorithm is enhanced, a decision maker is helped to select a better and feasible scheme from a large number of candidate solutions, and the method is suitable for popularization and application. According to the invention, a better vehicle path solution can be provided within effective time, the logistics distribution cost can be reduced, and the service quality can be improved.

Description

Technical Field

[0001] The invention relates to a vehicle path optimization method, in particular to a multi-objective vehicle path planning problem, and proposes a vehicle path optimization method based on an inflection point guided differential evolution algorithm. Background Art

[0002] The vehicle routing problem is a classic combinatorial optimization problem that is widely used in logistics, distribution, transportation, etc. Its main purpose is to effectively assign a group of vehicles to visit multiple customer points and minimize the travel distance or cost while satisfying constraints.

[0003] Due to its wide range of application backgrounds, the vehicle routing problem has important practical significance and has become a hot topic in recent years. At the same time, in order to adapt to the needs of real life, the vehicle routing problem has also shown new characteristics, among which the time window requirements put forward by customers for logistics distribution is one of the characteristics.

[0004] The time window requirement limits the service time of the vehicle to the customer. When modeling this type of vehicle routing problem, the delivery vehicle is generally subject to time constraints, and the vehicles that arrive early or late are penalized for time costs. Whether the customer's time window requirements can be met is an important evaluation criterion for customer satisfaction in the vehicle routing problem. At the same time, in the absence of decision makers' preferences, among the solutions obtained by various optimization algorithms, only a small number of solutions are suitable for practical applications and can be implemented in reality. Therefore, how to design an effective and feasible solution algorithm to assist decision makers in selecting solutions is of great practical significance for solving vehicle routing problems with time window requirements. Summary of the invention

[0005] Purpose of the invention: The present invention provides a vehicle path planning method based on an inflection point guided differential evolution algorithm. While satisfying the customer's requirements for the service time window as much as possible, the inflection point is used as the characteristic of the optimal solution, and a differential evolution algorithm is used to search around the inflection point to guide the search to the optimal solution, so as to generate higher quality individuals to accelerate the algorithm convergence speed, enhance the algorithm search capability, and help decision makers select better and feasible solutions from a large number of candidate solutions, so as to provide a better vehicle path solution within an effective time.

[0006] Technical solution: A vehicle path planning method based on an inflection point guided differential evolution algorithm, comprising the following steps:

[0007] 1. A vehicle path planning method based on an inflection point guided differential evolution algorithm, characterized in that it comprises the following steps:

[0008] S1. According to the data set or actual problem of vehicle routing planning with time windows, a customer point data model is established, and labels are set for the distribution center and each customer point respectively, and relevant service information is defined;

[0009] S2. Calculate and construct the distance matrix between customer points based on the coordinate information of the distribution center and each customer point, and use minimizing the vehicle travel distance as the first objective function;

[0010] S3, establish time window violation penalty rules according to the actual needs of each customer point, and take maximizing customer satisfaction as the second objective function;

[0011] S4. Construct a vehicle routing problem model with a time window, use objectives 1 and 2 as the objective functions of the model, and determine the constraints of the model according to actual needs;

[0012] S5. Through the inflection point-guided differential evolution algorithm, the next generation of iteration is carried out with the inflection point as the search direction, the vehicle path planning model with time window is optimized to solve the optimal delivery sequence.

[0013] Furthermore, in S1, the customer point data model is established as follows:

[0014] Set label: Set the number of customers to n; set the distribution center label to 0, and the customer point labels to 1, 2, 3, ..., n;

[0015] Define related service information: Set the horizontal coordinates of the customer point and distribution center location to x i , the vertical axis is y i ; Set the demand for goods at customer point i to r i ; Set the customer point time window to [ET i ,LT i ], where ET i and LT i They represent the earliest service time and the latest service time of customer point i respectively; the service duration required by customer point i is set to ST i ; Set the number of transport vehicles in the distribution center to K, and the rated load of each vehicle to Q.

[0016] Furthermore, in S2, the distance matrix between customer points and the vehicle travel distance function are calculated and constructed as follows:

[0017] Calculate the distance between distribution centers and customer points:

[0018]

[0019] The calculation results are stored in the form of an array to obtain a distance matrix, denoted as D, where the array element is d ij, represents the distance the vehicle travels from customer point i to customer point j, where d ii =d jj =0,d ji =d ij , d 0j is the distance from the distribution center to customer j, d i0 is the distance from customer i to the distribution center;

[0020] The first objective function is to minimize the vehicle travel distance, that is:

[0021]

[0022] Among them, x ijk is the decision variable:

[0023]

[0024] Furthermore, in S3, the time window penalty rule and customer satisfaction are as follows:

[0025] Set the penalty factor, that is, set the penalty coefficient epu for early arrival and lpu for late arrival. If the vehicle arrives within the customer time window, the time penalty value is 0; if the vehicle does not arrive within the time window, it will be penalized. The penalty for a single customer point = (vehicle or customer) waiting time * penalty coefficient.

[0026] Time penalty function:

[0027]

[0028] Where K is the total number of vehicles in the distribution center, k is the number of the sub-route and vehicle, i is the customer number, and n k is the number of customers to be served by the vehicle in the kth subpath, ki is the i-th customer served by vehicle k, [ET ki ,LT ki ] is the corresponding time window, T ki is the current time when vehicle k serves customer i.

[0029] If the vehicle arrives within the time window specified by the customer, the time penalty value is 0, and the customer satisfaction is high; conversely, if the vehicle does not arrive within the time window, the customer satisfaction is low. Therefore, there is an inverse relationship between customer satisfaction and time penalty. The customer satisfaction formula is defined as:

[0030]

[0031] The second objective function is to maximize customer satisfaction, which is an inverse relationship, that is, to minimize the time penalty:

[0032] f2=maxS=minT

[0033] Furthermore, in S4, the specific model of the vehicle routing problem with time windows is as follows:

[0034] The objective function is expressed as:

[0035] minF=(f1,f2) T

[0036] The constraints are:

[0037] Constraint 1. Customer point service constraint, that is, each customer point is served only once:

[0038]

[0039] Constraint 2. Start and end constraints:

[0040]

[0041] Constraint 3. Vehicle quantity constraint:

[0042]

[0043] Constraint 4. Capacity constraint:

[0044]

[0045] Constraint 5. Time window constraint:

[0046] It is best for the vehicle to provide delivery service within the time window of the customer point, otherwise it will be penalized according to the time window penalty rule, thereby increasing the time cost.

[0047] Furthermore, in S5, the use of the inflection point-guided differential evolution algorithm to optimize the vehicle routing problem with time windows and solve the optimal delivery sequence includes the following steps:

[0048] S5-1. Related parameter settings: Set the maximum number of iterations G max , the initial iteration number G = 1, set the population size N, set the mutation factor F and crossover operator CR of the differential evolution algorithm;

[0049] S5-2, randomly generate an initial population P according to the customer number, and calculate the individual objective function of the initial population;

[0050] S5-3, non-dominated set selection: select individuals in the current population that dominate other solutions and are not dominated by any other solutions, and save them in the set Q;

[0051] S5-4, inflection point identification: Use the minimum Manhattan distance method to identify inflection points and retain them;

[0052] S5-5. Mutation operation: Randomly select two individuals x from the non-dominated set Q r1 , x r2 , generate variant individuals v through mutation strategy DE / best / 1 i ;

[0053] S5-6, crossover operation: select individual x in the current population i With variant intermediate v i Perform crossover to generate test individuals u i ;

[0054] S5-7, boundary processing;

[0055] S5-8, repeat S5-5 to S5-7 until the population p size is N;

[0056] S5-9, selection operation: calculate the individual objective function value, merge populations P and p, select individuals through non-dominated sorting to generate a new population, and keep the population size unchanged;

[0057] S5-10, repeat S5-3 to S5-9. When the maximum number of iterations or the termination condition is reached, the loop ends and the optimal vehicle path and its solution are obtained.

[0058] Compared with the prior art, the present invention has the following beneficial effects:

[0059] 1. The present invention establishes an actual model based on vehicle routing problem data with time windows, fully considers the time window requirements of customer points, better meets actual service needs, and improves customer satisfaction.

[0060] 2. The present invention proposes a vehicle path planning method based on an inflection point guided differential evolution algorithm, which uses the inflection point as the characteristic of the optimal solution to help decision makers make choices so as to more quickly find a vehicle path plan that meets the objectives and constraints.

[0061] 3. The present invention proposes a vehicle path planning method based on an inflection point guided differential evolution algorithm, which uses a differential evolution algorithm mutation strategy DE / best / 1 with the best individual search capability to search around inflection points to generate population offspring, thereby improving the algorithm convergence capability, being able to more effectively solve vehicle path problems, and providing an effective vehicle path solution. BRIEF DESCRIPTION OF THE DRAWINGS

[0062] Figure 1 is a flow chart of the present invention;

[0063] Figure 2 This is a flow chart of the differential evolution algorithm based on inflection point guidance adopted by the present invention;

[0064] Figure 3Schematic diagram of MMD method for identifying inflection points in two-dimensional space;

[0065] Figure 4 An example of searching directions based on inflection points in two-dimensional space. DETAILED DESCRIPTION

[0066] The specific implementation of the present invention is further described in detail below in conjunction with the accompanying drawings and specific embodiments.

[0067] like Figure 1 As shown, the present invention proposes a vehicle path planning method based on an inflection point guided differential evolution algorithm, comprising the following steps:

[0068] S1. According to the data set or actual problem of vehicle routing planning with time windows, a customer point data model is established, and labels are set for the distribution center and each customer point respectively, and relevant service information is defined:

[0069] Set the labels: the number of customers is n; set the distribution center label to 0, and the customer point labels to 1, 2, 3…, n.

[0070] Define related service information: Set the horizontal coordinates of the customer point and distribution center location to x i , the vertical axis is y i ; Set the demand for goods at customer point i to r i ; Set the customer point time window to [ET i ,LT i ], where ET i and LT i They represent the earliest service time and the latest service time of customer point i respectively; the service duration required by customer point i is set to ST i ; Set the number of transport vehicles in the distribution center to K, and the rated load of each vehicle to Q.

[0071] S2. According to the coordinate information of the distribution center and each customer point, the distance matrix between customer points is calculated and constructed, and the minimization of vehicle travel distance is used as the objective function 1:

[0072] Calculate the distance between distribution centers and customer points:

[0073]

[0074] The calculation results are stored in the form of an array to obtain a distance matrix, denoted as D, where the array element is d ij , represents the distance the vehicle travels from customer point i to customer point j, where d ii =d jj =0,d ji =d ij , d 0jis the distance from the distribution center to customer j, d i0 is the distance from customer i to the distribution center.

[0075] The first objective function is to minimize the vehicle travel distance, that is:

[0076]

[0077] Among them, x ijk is the decision variable:

[0078]

[0079] S3. Establish time window violation penalty rules based on the actual needs of each customer point, and maximize customer satisfaction as objective function 2:

[0080] Set the penalty factor, that is, set the penalty coefficient epu for early arrival and lpu for late arrival. If the vehicle arrives within the customer time window, the time penalty value is 0; if the vehicle does not arrive within the time window, it will be penalized. The penalty for a single customer point = (vehicle or customer) waiting time * penalty coefficient.

[0081] Time penalty function:

[0082]

[0083] Where K is the total number of vehicles in the distribution center, k is the number of the sub-route and vehicle, i is the customer number, and n k is the number of customers to be served by the vehicle in the kth subpath, ki is the i-th customer served by vehicle k, [ET ki ,LT ki ] is the corresponding time window, T ki is the current time when vehicle k serves customer i.

[0084] If the vehicle arrives within the time window specified by the customer, the time penalty value is 0, and the customer satisfaction is high; conversely, if the vehicle does not arrive within the time window, the customer satisfaction is low. Therefore, there is an inverse relationship between customer satisfaction and time penalty.

[0085] Compared with S = 1 / T, S = 1 / e T It is more suitable for complex target trade-off problems, avoiding the problem that S approaches infinity when T approaches 0 in S = 1 / T. At the same time, for S = 1 / e T , when T increases, S decreases exponentially, which better reflects the goal of minimizing T. Therefore, the customer satisfaction formula is defined as:

[0086]

[0087] Objective function 2 is to maximize customer satisfaction, which is inversely proportional to minimize the time penalty:

[0088] f2=maxS=minT

[0089] S4. Construct a vehicle routing problem model with a time window, use objectives 1 and 2 as the objective functions of the model, and determine the constraints of the model according to actual needs:

[0090] The objective function is expressed as:

[0091] minF=(f1,f2) T

[0092] The constraints are:

[0093] Constraint 1. Customer point service constraint, that is, each customer point is served only once:

[0094]

[0095] Constraint 2. Start and end constraints: the vehicle must start from the distribution center and return to the distribution center after completion:

[0096]

[0097] Constraint 3. The number of vehicles in the distribution center is constrained. The number of vehicles used in the routing plan must not exceed the number of vehicles owned by the distribution center:

[0098]

[0099] Constraint 4. Capacity constraint: In the sub-path plan corresponding to each vehicle, the sum of customer demand must not be greater than the vehicle load:

[0100]

[0101] Constraint 5. Time window constraint: It is best for the vehicle to provide delivery service within the time window of the customer point, otherwise it will be penalized according to the time window penalty rule, thereby increasing the time cost.

[0102] S5. Through the inflection point-guided differential evolution algorithm, the inflection point is used as the search direction for the next generation of iterations, and the vehicle path planning model with time windows is optimized to solve the optimal delivery sequence. The overall algorithm process is as follows: Figure 2 As shown, the following steps are included:

[0103] S5-1, related parameter settings. Set the maximum number of iterations G max , the initial number of iterations G = 1, set the population size N, set the mutation factor F and crossover operator CR of the differential evolution algorithm DE.

[0104] S5-2. Randomly generate an initial population P based on the customer number and calculate the individual target values ​​of the initial population.

[0105] Establish the initial population P: P = (x1, x2, ..., x N ), where x1,x2,..,x N There are N individuals in the population, that is, N path plans, and individual x i The weight ratio of the jth gene in Generated according to the following formula:

[0106]

[0107] Among them, rand(0,1) represents a random number within (0,1) generated by a uniform distribution. and Respectively represent the maximum and minimum values ​​of the jth gene weight in the ith path plan. According to the above formula, calculate the gene weight ratio of other individuals in the target population, change the value of rand(0,1) in the formula, generate N initial path plans, and N path plans constitute the initial population.

[0108] S5-3, non-dominated set selection. The offspring of the population should be generated from the better parent individuals, and most of the offspring individuals can be generated near the parent individuals. In order to find an approximate Pareto front with good distribution and good convergence, only non-dominated solutions should be selected to participate in the evolution of the offspring. Non-dominated set selection, that is, select individuals in the current population that dominate other solutions and are not dominated by any other solutions, and save them in the set Q.

[0109] S5-4, inflection point identification. Use the minimum Manhattan distance method to identify inflection points and retain them.

[0110] In the absence of a given decision maker's preference, the inflection point is considered to be the Pareto optimal solution. The minimum Manhattan distance (MMD) method, as one of the inflection point identification methods, combines geometric information analysis and solution performance evaluation. The process of detecting inflection points is as follows:

[0111] Given an M-dimensional Pareto front with K solutions, PF = {x (1) …x (k)}, ideal point and the worst point For the mth target, z * and z nad are respectively the best value and the worst value among all Pareto optimal solutions of the mth objective. The objective function value is standardized as follows:

[0112]

[0113] Among them, fm and f′ m are the original target value and normalized target value of the mth dimension respectively. Thus, MMD can be expressed as:

[0114] f′1(x (i) )+f′2(x (i) )+…+f′ M (x (i) )=v

[0115] Depend on Figure 3 It can be seen that this method approaches the Pareto frontier from the ideal point through a straight line with a slope of 1. The intersection of the straight line and the Pareto frontier is the current global inflection point. At this time, the intercept of the straight line is v, and v is the minimum Manhattan distance of the point. After obtaining the global inflection point, the Pareto frontier can be divided into two, and the local knee search can be recursively performed until all inflection points are found.

[0116] S5-5, mutation operation. Randomly select two individuals x from the non-dominated set Q r1 , x r2 , generate variant individuals v through mutation strategy DE / best / 1 i .

[0117] The vehicle path planning method based on the inflection point guided differential evolution algorithm proposed in this invention aims to find the best inflection point rather than the best entire Pareto frontier. Therefore, the algorithm searches around the inflection point to generate higher quality individuals to improve convergence. Figure 4 As shown in the figure, it shows an example of a search direction based on an inflection point in a two-dimensional space. The differential evolution algorithm has multiple mutation strategies, among which DE / best / 1 relies on the best information search and has the ability to search around the best individual, which can improve the convergence of particles. DE / best / 1 generates a mutant individual v i Here is how:

[0118] v i =x best +F·(x r1 -x r2 )

[0119] where x r1 , x r2 are two different individuals randomly selected from the current population, F is the mutation factor, x best Is the best individual in the current group.

[0120] S5-6, cross operation.

[0121] Select individual x from the current population i With variant intermediate vi Perform crossover to generate test individuals u i , and add it to the population p.

[0122]

[0123] Among them, i (i = 1, 2, ..., N) is the overall index, N is the population size, j (j = 1, 2, ..., D) is the gene index, D is the gene label of the individual, the crossover operator CR takes a value between 0 and 1, and the index j is rand In the range of 1 and D.

[0124] S5-7, boundary processing.

[0125] If during the mutation and crossover operations, individual codes outside the feasible domain are obtained, that is, or Boundary processing is required:

[0126] S5-8. Repeat S5-5 to S5-7 until the population p size reaches N.

[0127] S5-9. Select an operation.

[0128] According to the optimization objective function, calculate the individual objective function value of the new population. Merge populations P and p, perform non-dominated sorting on the mixed population according to the individual objective function value, select better individuals to keep in the next generation, and obtain a new population P, keeping the population size N unchanged.

[0129] S5-10, repeat S5-3 to S5-9. When the maximum number of iterations or the termination condition is reached, the loop ends and the optimal vehicle path and its vehicle distribution plan are obtained.

[0130] In summary, the vehicle path planning method based on the inflection point guided differential evolution algorithm proposed in the present invention combines the inflection point with the differential evolution algorithm, uses the inflection point as the characteristic of the optimal solution, identifies the inflection point through the minimum Manhattan distance method, and combines the mutation strategy of the differential evolution algorithm DE / best / 1 to guide the algorithm search to approach the inflection point and the inflection point area, and solves and optimizes the vehicle path problem model with a time window. Compared with the prior art, the vehicle path planning method based on the inflection point guided differential evolution algorithm proposed in the present invention has improvements in algorithm performance and model fit, has strong search capabilities and stability, and helps decision makers make choices at the same time, and can better solve actual vehicle path planning problems.

[0131] It should be noted that although the embodiments of the present invention described above are illustrative, they are not intended to limit the present invention, and therefore the present invention is not limited to the above specific embodiments. Without departing from the principles of the present invention, any other embodiments obtained by those skilled in the art under the guidance of the present invention are deemed to be within the protection of the present invention.

Claims

1. A vehicle path planning method based on an inflection point guided differential evolution algorithm, characterized in that: The steps include: S1. According to the data set or actual problem of vehicle routing planning with time windows, a customer point data model is established, and labels are set for the distribution center and each customer point respectively, and relevant service information is defined; S2. Calculate and construct the distance matrix between customer points based on the coordinate information of the distribution center and each customer point, and use minimizing the vehicle travel distance as the first objective function; S3, establish time window violation penalty rules according to the actual needs of each customer point, and take maximizing customer satisfaction as the second objective function; S4. Construct a vehicle routing problem model with a time window, use objectives 1 and 2 as the objective functions of the model, and determine the constraints of the model according to actual needs; S5. Through the inflection point-guided differential evolution algorithm, the next generation of iteration is carried out with the inflection point as the search direction, the vehicle path planning model with time window is optimized to solve the optimal delivery sequence.

2. The vehicle path planning method based on inflection point guided differential evolution algorithm according to claim 1 is characterized in that: In S1, the customer point data model is as follows: Set label: Set the number of customers to n; set the distribution center label to 0, and the customer point labels to 1, 2, 3, ..., n; Define related service information: Set the horizontal coordinates of the customer point and distribution center location to x i , the vertical axis is y i ; Set the demand for goods at customer point i to r i ; Set the customer point time window to [ET i ,LT i ], where ET i and LT i They represent the earliest service time and the latest service time of customer point i respectively; Set the service duration required by customer point i to ST i ; Set the number of transport vehicles in the distribution center to K, and the rated load of each vehicle to Q.

3. The vehicle path planning method based on inflection point guided differential evolution algorithm according to claim 1 is characterized in that: In S2, the distance matrix between customer points and the vehicle travel distance function are calculated and constructed as follows: Calculate the distance between distribution centers and customer points: The calculation results are stored in the form of an array to obtain a distance matrix, denoted as D, where the array element is d ij , represents the distance the vehicle travels from customer point i to customer point j, where d ii =d jj =0,d ji =d ij , d 0j is the distance from the distribution center to customer j, d i0 is the distance from customer i to the distribution center; The first objective function is to minimize the vehicle travel distance, that is: Among them, x ijk is the decision variable:

4. The vehicle path planning method based on inflection point guided differential evolution algorithm according to claim 1 is characterized in that: In S3, the time window penalty rule and customer satisfaction are as follows: Set the penalty factor, that is, set the penalty coefficient epu for early arrival and the penalty coefficient lpu for late arrival. If the vehicle arrives within the customer's time window, the time penalty value is 0; if the vehicle does not arrive within the time window, it will be penalized, and the penalty for a single customer point = (vehicle or customer) waiting time * penalty coefficient; Time penalty function: Where K is the total number of vehicles in the distribution center, k is the number of the sub-route and vehicle, i is the customer number, and n k is the number of customers to be served by the vehicle in the kth subpath, ki is the i-th customer served by vehicle k, [ET ki ,LT ki ] is the corresponding time window, T ki is the current time when vehicle k serves customer i; If the vehicle arrives within the time window specified by the customer, the time penalty value is 0, and the customer satisfaction is high; conversely, if the vehicle does not arrive within the time window, the customer satisfaction is low. Therefore, there is an inverse relationship between customer satisfaction and time penalty. The customer satisfaction formula is defined as: Objective function 2 is to maximize customer satisfaction, which is inversely proportional to minimize the time penalty: f2=maxS=minT 5. The vehicle path planning method based on inflection point guided differential evolution algorithm according to claim 1 is characterized in that: In S4, the specific model of the vehicle routing problem with time windows is as follows: The objective function is expressed as: <h2 style=";text-align:left;direction:ltr">minF = (f1,f2)<h2 style=";text-align:left;direction:ltr"> T The constraints are: Constraint 1. Customer point service constraint, that is, each customer point is served only once: Constraint 2. Start and end constraints: Constraint 3. Vehicle quantity constraint: Constraint 4. Capacity constraint: Constraint 5. Time window constraint: It is best for the vehicle to provide delivery service within the time window of the customer point, otherwise it will be penalized according to the time window penalty rule, thereby increasing the time cost.

6. The vehicle path planning method based on inflection point guided differential evolution algorithm according to claim 1 is characterized in that: In S5, the inflection point-guided differential evolution algorithm is used to optimize the vehicle routing problem with time windows to solve the optimal delivery sequence, including the following steps: S5-1. Related parameter settings: Set the maximum number of iterations G max , the initial iteration number G = 1, set the population size N, set the mutation factor F and crossover operator CR of the differential evolution algorithm; S5-2, randomly generate an initial population P according to the customer number, and calculate the individual objective function of the initial population; S5-3, non-dominated set selection. Select individuals in the current population that dominate other solutions and are not dominated by any other solutions, and save them in the set Q; S5-4, inflection point identification: Use the minimum Manhattan distance method to identify inflection points and retain them; S5-5. Mutation operation: Randomly select two individuals x from the non-dominated set Q r1 , x r2 , generate variant individuals v through mutation strategy DE / best / 1 i ; S5-6, crossover operation: select individual x in the current population i With variant intermediate v i Perform crossover to generate test individuals u i ; S5-7, boundary processing; S5-8, repeat S5-5 to S5-7 until the population p size is N; S5-9, selection operation: calculate the individual objective function value, merge populations P and p, select individuals through non-dominated sorting to generate a new population, and keep the population size unchanged; S5-10, repeat S5-3 to S5-9. When the maximum number of iterations or the termination condition is reached, the loop ends and the optimal vehicle path and its solution are obtained.

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