Multi-modal multi-target travel salesman problem optimization method

Abstract

The invention discloses a multi-modal multi-target travel salesman problem optimization method, which is used for searching more equivalent optimal solutions. An edge crossing strategy adopting non-dominated sorting is introduced to generate excellent individuals; the Spearman distance is combined with a diversity measurement standard based on the same edge in a decision space, and the similarity between different loop solutions is allowed to be evaluated more accurately; in addition, a concept of a special crowding distance is introduced in an environment selection process so as to ensure that a population keeps diversity in a balanced manner in a decision space and a target space; and after the evolution is finished, selecting the best individual to evaluate on the test set. According to the method, the multi-target traveling salesman problem is solved through MMTSPDS, and the algorithm provides various different solutions for the traveling salesman problem by storing a plurality of high-quality multi-modal solutions, so that the scientificity and flexibility of decision making are improved.

Description

Technical Field

[0001] This invention belongs to the technical field of the Traveling Salesman Problem (TSP), and specifically relates to a multimodal, multi-objective optimization method for the TSP; it aims to solve the multi-objective TSP by utilizing multimodal solutions to improve the flexibility and adaptability of route planning. Background Art

[0002] The Traveling Salesman Problem (TSP), a classic problem in combinatorial optimization, has broad and profound applications. In the transportation sector, the TSP provides important theoretical support for vehicle routing, logistics distribution, and flight scheduling. In the electronics manufacturing industry, it helps optimize printed circuit board (PCB) vias and improve production efficiency. In communications engineering, cable and optical fiber wiring design has also benefited from TSP research. Therefore, in-depth research on the TSP is not only of theoretical value but also has a positive impact on practical applications across multiple industries. However, most current approaches to solving the TSP often focus on finding a single, high-quality Pareto solution, ignoring the multimodal nature of the TSP problem. This multimodal nature implies that the solution space of the problem contains multiple near-optimal solutions, each of which may possess unique advantages in different scenarios. Therefore, pursuing only a single optimal solution may limit decision makers' options and fail to fully meet the diverse needs of practical applications.

[0003] Given this situation, exploring the multimodal properties of the TSP is particularly important. By comprehensively exploring the multimodal solution set, we can provide decision makers with more high-quality alternatives. This not only helps improve the scientific nature and flexibility of decision-making, but also helps maximize economic benefits in practical applications while effectively mitigating potential risks. For example, in the transportation sector, multimodal solutions can provide multiple route options to address uncertainties such as traffic congestion and weather changes, ensuring smooth logistics and transportation. Therefore, in-depth research on the multimodal properties of the TSP has important practical significance and application value. Summary of the Invention

[0004] This paper studies the multi-objective optimization in the traveling salesman problem. Based on the classic multimodal multi-objective traveling salesman test set example, the paper conducts basic problem modeling of the multi-objective traveling salesman problem and proposes a multimodal multi-objective optimization algorithm (MMTSP_DS) with a diversity judgment mechanism to solve the problem. The algorithm can distinguish the differences between individuals and thus find more equivalent optimal solutions, providing decision makers with more decision-making options.

[0005] The technical solution adopted in this invention is as follows:

[0006] A multi-modal and multi-objective traveling salesman problem optimization method comprises the following steps:

[0007] S1, reads the cost matrix of the problem and uses it to calculate the two target values ​​of the route individuals;

[0008] S2, sets the maximum number of evaluations of the algorithm and initializes the current number of fitness evaluations;

[0009] S3, initialize the population;

[0010] S4, determine whether the number of evaluations is greater than the maximum number of evaluations. If it is less, go to S5; if it is greater, go to S9;

[0011] S5, evaluate the fitness of the population and perform non-dominated sorting to obtain the individuals on each non-dominated frontier;

[0012] S6 uses a crossover strategy to generate new individuals;

[0013] S7, evaluate the fitness of the generated new individuals and merge them with the parent individuals;

[0014] S8, screening individuals through environmental selection and retaining excellent individuals;

[0015] S9, after the evolution is completed, the final population is tested on the test set.

[0016] Furthermore, the specific steps of S1 are:

[0017] S11, using a test problem generator to generate corresponding test problems and corresponding data such as optimal solutions;

[0018] S12, Initialize the cost matrix to store different objective values;

[0019] S13, read the data of different target values ​​into the corresponding cost matrix.

[0020] Furthermore, the specific steps of S3 are as follows:

[0021] S31, create a matrix to store the individuals of the current population;

[0022] S32, create a matrix to store the non-dominant front index of each individual in the population;

[0023] S33, create a matrix for storing two fitness values ​​of each individual in the population.

[0024] Furthermore, the specific steps of S31 are:

[0025] S311, the population size is set to 100, and the size of the individual, i.e. the number of cities, is defined according to the problem;

[0026] S312, each individual is displayed in a loop, such as 1-2-3-4-5-1, where each number represents the city number, and each city is appended with the number information of the two adjacent cities on its left and right.

[0027] Furthermore, the specific steps of S5 are:

[0028] S51, traverse the travel path of each individual and calculate its two fitness values ​​based on the given cost matrix;

[0029] S52, Initialize the non-dominated front sequence number and dominance count;

[0030] S53, Traverse the population and assess dominance relationships;

[0031] S54, constructing a non-dominated frontier;

[0032] S55, output the final sorting result.

[0033] Furthermore, the specific steps of S53 are as follows:

[0034] S531, For each individual i, compare it with other individuals j in the population;

[0035] S532, if individual i is superior to individual j in all objectives, then individual i dominates individual j, j is added to the dominance set of i, and the dominance count of individual j is incremented by 1; if individual j dominates individual i, then the dominance count of individual i is incremented.

[0036] S533, if the dominance count of individual i is still zero, then individual i is marked as the first layer of non-dominated frontier.

[0037] Furthermore, the specific steps of S6 are:

[0038] S61, randomly select two individuals A and B as parent individuals to form G AB And find the AB cycles;

[0039] S62, constructing a set E, i.e., the union of AB cycles, using the specified strategy for the identified AB cycles;

[0040] S63, remove the edge of individual A in set E and add the edge of individual B to obtain an intermediate solution;

[0041] S64, Non-dominated sorting determines the quality of all candidate solutions by considering the dominance of each candidate connection scheme, thereby selecting the most suitable scheme to generate the final complete solution from the intermediate solutions.

[0042] Furthermore, the specific steps of S61 are:

[0043] S611 transforms the representation of individuals A and B into vertices plus all edges, forming a set G. AB ;

[0044] S612, randomly select a vertex as the starting point, in G AB Search for edge E in sequence A and edge E B ;

[0045] S613, when two candidate edges are found, randomly select one edge and start from G. AB Delete the selected edges until you reach the starting point again;

[0046] S614, in G AB If a new AB cycle is found, the edge of that cycle is deleted; continue searching G. AB The search continues until all remaining edges are deleted, marking the end of the search process.

[0047] Furthermore, the specific steps of S64 are:

[0048] S641, the intermediate solution consists of multiple sub-cycles;

[0049] S642, remove some edges from the intermediate solution and add some edges to generate the final complete solution;

[0050] S643 performs a non-dominated sort of all connection schemes and selects the optimal connection scheme to obtain the final solution.

[0051] Furthermore, the specific steps of S8 are as follows:

[0052] S81, after merging the offspring and parent populations, performs non-dominated sorting;

[0053] S82, obtains multiple subpopulations with non-dominant levels from 1 to n;

[0054] S83, If the number of individuals in the first frontier is less than the population size N, calculate the difference K and proceed to step S85;

[0055] S84, If the number of individuals in the first frontier is greater than the population size N, then proceed to step S87;

[0056] S85, continue to select individuals from the second frontier subpopulation if the number of individuals in the second frontier is greater than the difference K;

[0057] S86 selects the top K best individuals in the second frontier subpopulation by adopting a special crowding distance (SCD) method, thereby making up the final required number N;

[0058] S87, If the number of individuals in the second frontier is less than the difference K, then select individuals in the third frontier in sequence until the number exceeds the difference K, and then select according to the SCD method;

[0059] S88 sorts the individuals in the first frontier by a special crowding distance and selects the top N individuals.

[0060] Furthermore, the specific steps of S86 are as follows:

[0061] S861, the effect of the special crowding distance is to give individuals with above-average diversity more opportunities in at least one space and give individuals with poor diversity less opportunities in both spaces. The expression of SCD is as follows;

[0062]

[0063] in This represents the SCD value of the i-th individual. and Let represent the crowding distance of individual i in the decision space and the goal space, respectively. and Let represent the average crowding distance in the decision and goal spaces, respectively;

[0064] S862, calculates the crowding distance of each individual in the decision space;

[0065] S863, calculate the crowding distance of each individual in the target space;

[0066] S864, calculate the average crowding distance in the decision space and the target space respectively;

[0067] S865: If the two crowding distances of an individual are greater than the corresponding average value in at least one space, the larger value in the two spaces is taken as the special crowding distance; if the two crowding distances of an individual are less than the corresponding average value in both spaces, the smaller value in the two spaces is taken as the special crowding distance.

[0068] Furthermore, the specific steps of S862 are as follows:

[0069] S8621, calculate the number of identical edges between individuals, as shown in the following formula:

[0070]

[0071] in Let the set of edges of individual p be denoted as , Let the set of edges for an individual q be denoted as Individual p is represented as Individual q is represented as ;

[0072] S8622, calculate the Spearman distance between individuals, as shown in the following formula:

[0073]

[0074] in This represents the number of vertices of an individual. and Let represent the ranks of individuals p and q at the i-th vertex, respectively. The rank of individual p at the i-th vertex is calculated as follows:

[0075]

[0076] In this context, Iverson brackets “[ ]” are used to represent conditional statements; for Iverson brackets [P], if P is true, its result is 1, otherwise it is 0. and These represent the indices of the i-th and j-th vertices of an individual, respectively.

[0077] S8623 normalizes the number of common edges and Spearman distance between individuals, converting them to the same scale for summation. A larger number of common edges V indicates higher similarity between individuals, while a smaller Spearman distance S also indicates higher similarity. Therefore, the final S is obtained by subtracting the normalized S value from 1, as shown in the following formula:

[0078]

[0079]

[0080]

[0081] S8624, with the weights of both key variables set to 0.5, is used to obtain a similarity measure between individuals in the decision space, as shown in the following formula:

[0082]

[0083] S8625, to determine the diversity measure for each individual in the decision space, this invention uses the sum of the distances between an individual and its nearest neighbor and the distances to all other individuals; first, the distance between individual p and its nearest neighbor q (i.e., their similarity) is denoted as... And the sum of the distances between individual p and all other individuals, denoted as Then, multiply these two parts to obtain the crowding distance of individual p in the decision space, as shown in the following formula:

[0084]

[0085] In the decision space, individuals are represented as loops formed by connecting vertices, and these loops are high-dimensional decision vectors. Therefore, the number of shared edges between individuals is a clear measure of similarity. However, even if two individuals share edges, if other edges differ significantly, the number of shared edges is not comprehensive enough. Therefore, the Spearman distance is introduced, which allows us to view the TSP individuals as an irregular permutation, where the difference in rank represents the difference between loop individuals. Thus, combining the number of shared edges and the Spearman distance provides a comprehensive reflection of the diversity of individuals in the decision space.

[0086] Furthermore, the specific steps of S863 are as follows:

[0087] The diversity of individual p is calculated using the k-nearest neighbor method based on Euclidean distance, as shown in the following formula:

[0088]

[0089] in, Let K represent the k-th nearest neighbor of individual i, and let K be 3. This reflects the crowding around the current individual, where a larger value indicates better diversity.

[0090] Furthermore, the specific steps of S865 are as follows:

[0091]

[0092] in, This represents the specific crowding distance for individual p. and represents the average diversity index in the decision and goal spaces, respectively; the calculation of SCD aims to more accurately assess the crowding level of individuals in different spaces. In the non-dominated ordination process, this method prioritizes solutions exhibiting superior diversity in at least one space, thus achieving a balance in population distribution across the decision and goal spaces.

[0093] The beneficial effects of this invention are:

[0094] 1. The introduction of common edge count and Spearman distance among individuals makes the similarity judgment between individuals in the decision space more accurate;

[0095] 2. Calculate the diversity index of individuals in the decision-making space using the minimum distance and total distance to reflect the level of individual diversity;

[0096] 3. The concept of special crowding distance is introduced, which allows for consideration of both decision-making and population diversity in the target space during the environmental selection process; Attached Figure Description

[0097] Figure 1 This is a flowchart of the method of the present invention. DETAILED DESCRIPTION

[0098] The present invention will be further described below with reference to the accompanying drawings.

[0099] like Figure 1 As shown, this invention is an optimization method for the multimodal, multi-objective traveling salesman problem, comprising the following steps:

[0100] S1, reads the cost matrix of the problem and uses it to calculate the two target values ​​of the route individuals;

[0101] S2, sets the maximum number of evaluations of the algorithm and initializes the current number of fitness evaluations;

[0102] S3, initialize the population;

[0103] S31, create a matrix to store the individuals of the current population;

[0104] S311, the population size is set to 100, and the size of the individual, i.e. the number of cities, is defined according to the problem;

[0105] S312, each individual is displayed in a loop, such as 1-2-3-4-5-1, where each number represents the city number, and each city is appended with the number information of the two adjacent cities on its left and right.

[0106] S32, create a matrix to store the non-dominant front index of each individual in the population;

[0107] S33, create a matrix for storing two fitness values ​​of each individual in the population.

[0108] S4. Determine if the number of evaluations is greater than the maximum number of evaluations. If it is less, proceed to S5; if it is greater, proceed to S9.

[0109] S5, evaluate the fitness of the population and perform non-dominated sorting to obtain the individuals on each non-dominated frontier;

[0110] S51, Traverse the travel path of each individual and calculate its two fitness values ​​based on the given cost matrix;

[0111] S52, Initialize the non-dominated front sequence number and dominance count;

[0112] S53, Traverse the population and assess dominance relationships;

[0113] S531, For each individual i, compare it with other individuals j in the population;

[0114] S532, if individual i is superior to individual j in all objectives, then individual i dominates individual j, j is added to the dominance set of i, and the dominance count of individual j is incremented by 1; if individual j dominates individual i, then the dominance count of individual i is incremented.

[0115] S533, if the dominance count of individual i is still zero, then individual i is marked as the first layer of non-dominated frontier.

[0116] S54, constructing a non-dominated frontier;

[0117] S55, output the final sorting result.

[0118] S6 uses a crossover strategy to generate new individuals;

[0119] S61, randomly select two individuals A and B as parent individuals to form G AB And find the AB cycles;

[0120] S611 transforms the representation of individuals A and B into vertices plus all edges, forming a set G. AB ;

[0121] S612, randomly select a vertex as the starting point, in G AB Search for edge E in sequence A and edge E B ;

[0122] S613, when two candidate edges are found, randomly select one edge and start from G. AB Delete the selected edges until you reach the starting point again;

[0123] S614, in G AB If a new AB cycle is found, the edge of that cycle is deleted; continue searching G. AB The search continues until all remaining edges are deleted, marking the end of the search process.

[0124] S62, constructing a set E, i.e., the union of AB cycles, using the specified strategy for the identified AB cycles;

[0125] S63, remove the edge of individual A in set E and add the edge of individual B to obtain an intermediate solution;

[0126] S64, Non-dominated sorting determines the quality of all candidate solutions by considering the dominance of each candidate connection scheme, thereby selecting the most suitable scheme to generate the final complete solution from the intermediate solutions.

[0127] S641, the intermediate solution consists of multiple sub-cycles;

[0128] S642, remove some edges from the intermediate solution and add some edges to generate the final complete solution;

[0129] S643 performs a non-dominated sort of all connection schemes and selects the optimal connection scheme to obtain the final solution.

[0130] S7, evaluate the fitness of the generated new individuals and merge them with the parent individuals;

[0131] S8, screening individuals through environmental selection and retaining excellent individuals;

[0132] S81, after merging the offspring and parent populations, performs non-dominated sorting;

[0133] S82, obtains multiple subpopulations with non-dominant levels from 1 to n;

[0134] S83, If the number of individuals in the first frontier is less than the population size N, calculate the difference K and proceed to step S84;

[0135] S84, If the number of individuals in the first frontier is greater than the population size N, then proceed to step S87;

[0136] S85, continue to select individuals from the second frontier subpopulation, if the number of individuals in the second frontier is greater than the difference K;

[0137] S86 selects the top K best individuals in the second frontier subpopulation by adopting a special crowding distance (SCD) method, thereby making up the final required number N;

[0138] S861, the effect of the special crowding distance is to give individuals with above-average diversity more opportunities in at least one space and give individuals with poor diversity less opportunities in both spaces. The expression of SCD is as follows;

[0139]

[0140] in This represents the SCD value of the i-th individual. and Let represent the crowding distance of individual i in the decision space and the goal space, respectively. and Let represent the average crowding distance in the decision and goal spaces, respectively;

[0141] S862, calculates the crowding distance of each individual in the decision space;

[0142] S8621, calculate the number of identical edges between individuals, as shown in the following formula:

[0143]

[0144] in, Let the set of edges of individual p be denoted as , Let the set of edges for an individual q be denoted as Individual p is represented as Individual q is represented as ;

[0145] S8622, calculate the Spearman distance between individuals, as shown in the following formula:

[0146]

[0147] in This represents the number of vertices of an individual. and Let represent the ranks of individuals p and q at the i-th vertex, respectively. The rank of individual p at the i-th vertex is calculated as follows:

[0148]

[0149] In this context, Iverson brackets “[ ]” are used to represent conditional statements; for example, for Iverson brackets [P], if P is true, its evaluation result is 1, otherwise it is 0; and These represent the indices of the i-th and j-th vertices of an individual, respectively.

[0150] S8623, normalize the number of common edges and Spearman distance between individuals, convert them to the same scale and sum them. The larger the number of common edges V, the higher the similarity between individuals, and the smaller the Spearman distance S, the higher the similarity between individuals. Therefore, take 1 minus the normalized S value as the final S, the formula is as follows:

[0151]

[0152]

[0153]

[0154] S8624, with the weights of both key variables set to 0.5, is used to obtain a similarity measure between individuals in the decision space, as shown in the following formula:

[0155]

[0156] S8625, to determine the diversity measure for each individual in the decision space, this invention uses the sum of the distances between an individual and its nearest neighbor and the distances to all other individuals; first, the distance between individual p and its nearest neighbor q (i.e., their similarity) is denoted as... And the sum of the distances between individual p and all other individuals, denoted as Then, multiply these two parts to obtain the crowding distance of individual p in the decision space, as shown in the following formula:

[0157]

[0158] In the decision space, individuals are represented as loops formed by connecting vertices, and these loops are high-dimensional decision vectors. Therefore, the number of shared edges between individuals is a clear measure of similarity. However, even if two individuals share edges, if other edges differ significantly, the number of shared edges is not comprehensive enough. Therefore, the Spearman distance is introduced, which allows us to view the TSP individuals as an irregular permutation, where the difference in rank represents the difference between loop individuals. Thus, combining the number of shared edges and the Spearman distance provides a comprehensive reflection of the diversity of individuals in the decision space.

[0159] S863, calculate the crowding distance of each individual in the target space;

[0160] The diversity of individual p is calculated using the k-nearest neighbor method based on Euclidean distance, as shown in the following formula:

[0161]

[0162] in, Let K represent the k-th nearest neighbor of individual i, and let K be 3. This reflects the crowding around the current individual, where a larger value indicates better diversity.

[0163] S864, calculate the average crowding distance in the decision space and the target space respectively;

[0164] S865: If the two crowding distances of an individual are greater than the corresponding average value in at least one space, the larger value in the two spaces is taken as the special crowding distance; if the two crowding distances of an individual are less than the corresponding average value in both spaces, the smaller value in the two spaces is taken as the special crowding distance.

[0165]

[0166] in, This represents the specific crowding distance for individual p. and represents the average diversity index in the decision and goal spaces, respectively; the calculation of SCD aims to more accurately assess the crowding level of individuals in different spaces. In the non-dominated ordination process, this method prioritizes solutions exhibiting superior diversity in at least one space, thus achieving a balance in population distribution across the decision and goal spaces.

[0167] S87, If the number of individuals in the second frontier is less than the difference K, then select individuals in the third frontier in sequence until the number exceeds the difference K, and then select according to the SCD method;

[0168] S88 sorts the individuals in the first frontier by a special crowding distance and selects the top N individuals.

[0169] S9, after the evolution is completed, the final population is tested on the test set.

[0170] To further illustrate the superiority of the present invention in solving the traveling salesman problem, Table 1 shows the experimental results of the present invention and some typical multimodal multi-objective traveling salesman algorithms TSPXEA, DNEA-TSP, EAG-MOEAD and NSGA-II on the processing test set.

[0171] These test problems consist of a C-type matrix composed of single-objective Traveling Salesman Problems (TSPs) such as “uni1”, “bays29”, and “att48”, with 1, 29, and 48 cities respectively. Additionally, there are 10 sets of D-type vertices, from D1 to D10, each containing 10 cities. Finally, the D-type vertices are combined with the C-type vertices to form three test sets for the Multimodal Multi-Objective Traveling Salesman Problem (MMTSP), with 11, 39, and 58 cities respectively. Furthermore, all these problems contain two objectives to be optimized. The population size for all comparison algorithms is set to 100, and the maximum number of fitness evaluations is set to 50,000. All algorithms are run independently 30 times on each test problem, and the experimental results are given based on the Wilcoxon rank-sum test after 30 runs. The symbols “+” (better), “=" (equivalent), and “-” (worse) are used to indicate whether the proposed algorithm is superior, a match, or inferior to the comparison algorithms.

[0172] Table 1 shows the IGDM indices obtained by the algorithm, which reflect a comprehensive measure of the diversity and convergence of the population in the decision space and objective space. The comparison shows that the proposed method exhibits superior performance in the multimodal multi-objective traveling salesman problem. Specifically, the proposed method outperforms other methods in both the proximity of the obtained route to the true optimal route and the proximity of the obtained Pareto front to the true Pareto front. These results also demonstrate that the proposed method can find better multimodal solutions to the multi-objective traveling salesman problem.

[0173] In summary, the multimodal, multi-objective travel salesman problem optimization method proposed in this invention can effectively handle the multimodal, multi-objective travel salesman problem and find a good multimodal optimal solution.

[0174] Table 1 Comparison Results of IGDM Indicators

[0175] Multimodal multi-objective optimization strategies can automatically balance and optimize multiple conflicting objectives, exploring more diverse potential solutions to find the optimal comprehensive configuration. Evolutionary computation, a widely validated technique with superior global search capabilities, can efficiently search for globally optimal or near-optimal solution sets. Combining multimodal multi-objective optimization with evolutionary computation can more quickly and accurately identify the best comprehensive solution, significantly improving the efficiency and effectiveness of the optimization process.

[0176] This invention introduces an individual diversity discrimination mechanism to accurately assess the diversity of individuals in the decision space, truly reflecting the degree of crowding and preventing the elimination of superior individuals during environmental selection. The diversity equilibrium mechanism maintains the diversity of the two spatial populations during environmental selection, avoiding a situation where one spatial population has high diversity while the other has low diversity, while ensuring convergence. In summary, in the field of the Traveling Salesman Problem (TSP), this invention provides a novel multimodal multi-objective optimization algorithm with a diversity discrimination mechanism to solve the problem. This method can distinguish the differences between individuals, find the optimal travel route, and has significant advantages over other multimodal multi-objective TSP algorithms.

[0177] Although the above describes the specific embodiments of the present invention in conjunction with the accompanying drawings, it is not intended to limit the scope of protection of the present invention. Those skilled in the art should understand that various modifications or variations that can be made by those skilled in the art on the basis of the technical solution of the present invention without any creative work are still within the scope of protection of the present invention.

Claims

1. A multi-modal multi-objective traveling salesman problem optimization method, characterized in that: The following steps are involved: S1, reads the cost matrix of the problem and is used to calculate the two objective values ​​of the route individuals; S2, set the maximum number of evaluations of the algorithm and initialize the current number of fitness evaluations; S3, initialize the population; S4, determine whether the number of evaluations is greater than the maximum number of evaluations, if less, go to S5, if greater, go to S9; S5, evaluate the fitness of the population and perform non-dominated sorting to obtain the individuals on each non-dominated frontier; S6, using the crossover strategy to generate new individuals; S7, evaluate the fitness of the generated new individuals and merge them with the parent individuals; S8, screening individuals through environmental selection and retaining excellent individuals; S9, after the evolution is completed, the final population is tested on the test set.

2. The multi-modal multi-objective traveling salesman problem optimization method according to claim 1, characterized in that: The specific steps of S1 are: S11, using a test problem generator to generate corresponding test problems and corresponding data such as optimal solutions; S12, initializing the cost matrix storing different target values; S13, reading the data of different target values ​​into the corresponding cost matrix.

3. The multi-modal multi-objective traveling salesman problem optimization method according to claim 1, characterized in that: The specific steps of S3 are: S31, create a matrix to store the individuals of the current population; S32, create a matrix to store the non-dominated frontier sequence number of each individual in the population; S33, create a matrix for storing two fitness values ​​for each individual in the population.

4. The multi-modal multi-objective traveling salesman problem optimization method according to claim 3, characterized in that: The specific steps of S31 are: S311, the population size is set to 100, and the size of the individual, i.e., the number of cities, is defined according to the problem; S312, each individual is displayed in a loop form, each number in which represents the number of a city, and each city is attached with the number information of its adjacent cities on the left and right.

5. The multi-modal multi-objective traveling salesman problem optimization method according to claim 1, characterized in that: The specific steps of S5 are: S51, traverse the travel path of each individual and calculate its two fitness values ​​according to the given cost matrix; S52, initializing the non-dominated front number and the dominance count; S53, traverse the population and evaluate dominance relationships; S54, constructing nondominated frontiers; S55, output the final sorting result.

6. The multi-modal multi-objective traveling salesman problem optimization method according to claim 5, characterized in that: The specific steps of S53 are: S531, for each individual i, compare it with other individuals j in the population; S532, if individual i is superior to individual j in all objectives, then individual i dominates individual j, adds j to the domination set of i, and increases the domination count of individual j by 1; if individual j dominates individual i, increases the domination count of individual i; S533: If the domination count of individual i is still zero, individual i is marked as the first-layer non-dominated frontier.

7. The multi-modal multi-objective traveling salesman problem optimization method according to claim 1, characterized in that: The specific steps of S6 are: S61, randomly select two individuals A and B as parent individuals to form G AB And find out the AB cycles; S62, constructing a set E, i.e., a union of the AB cycles, using a specified strategy for the identified AB cycles; S63, remove the edge of individual A in set E, add the edge of individual B, and get an intermediate solution; S64, non-dominated sorting determines the quality of all candidate solutions by considering the dominance relationship of each candidate connection solution, so as to select the most suitable solution to generate the final complete solution from the intermediate solution.

8. The multi-modal multi-objective traveling salesman problem optimization method according to claim 7, characterized in that: The specific steps of S61 are: S611, transform the expression of individuals A and B into vertices plus all edges, and form a set G AB ; S612, randomly select a vertex as the starting point, in G AB Search for edge E in turn A and edge E B ; S613, when two candidate edges are found, randomly select an edge and select it from G AB Delete the selected edges until you reach the starting point again; S614, in G AB If a new AB cycle is found in G, delete the edge of the cycle; continue searching G AB The remaining edges in the search process are deleted until all edges are deleted, marking the end of the search process.

9. The multi-modal multi-objective traveling salesman problem optimization method according to claim 7, characterized in that: The specific steps of S64 are: S641, the intermediate solution consists of multiple sub-cycles; S642, deleting some edges and adding some edges to the intermediate solution to generate the final complete solution; S643, perform non-dominated sorting on all connection schemes and select the best connection scheme to obtain the final solution.

10. The multi-modal multi-objective traveling salesman problem optimization method according to claim 1, characterized in that: The specific steps of S8 are: S81, merge the offspring population with the parent population and perform non-dominated sorting; S82, obtaining a plurality of subpopulations with non-dominated levels ranging from 1 to n; S83, if the number of individuals in the first frontier is less than the population size N, calculate the difference K and go to step S85; S84, if the number of individuals in the first frontier is greater than the population size N, go to step S87; S85, continue to select individuals from the second frontier subpopulation, if the number of individuals in the second frontier is greater than the difference K; S86, by adopting the SCD method to select the top K excellent individuals in the second frontier sub-population, so as to make up the final required quantity N; S87, if the number of individuals in the second front is less than the difference K, select individuals in the third front in sequence until the number exceeds the difference K, and then select according to the SCD method; S88, sort the individuals in the first frontier by special crowding distance and select the first N individuals.

11. The multi-modal multi-objective traveling salesman problem optimization method according to claim 10, characterized in that: The specific steps of S86 are: S861, the effect of the special crowding distance is to give more opportunities to individuals with above-average diversity in at least one space, and give less opportunities to individuals with less diversity in both spaces. The expression of SCD is as follows; in, represents the SCD value of the i-th individual, and denote the crowding distance of individual i in the decision and target space, respectively. and denote the average crowding distance in decision and target space respectively; S862, calculate the crowding distance of each individual in the decision space; S863, calculate the crowding distance of each individual in the target space; The k-nearest neighbor method based on Euclidean distance is used to calculate the diversity of individual p. The formula is as follows: in, represents the kth nearest neighbor individual of individual i, and K is set to 3; S864, calculating the average crowding distance in the decision space and the target space respectively; S865, if the two crowding distances of an individual are greater than the corresponding average value in at least one space, the larger value in the two spaces is taken as the special crowding distance; if the two crowding distances of an individual are less than the corresponding average value in both spaces, the smaller value in the two spaces is taken as the special crowding distance, the formula is as follows: in, represents the special crowding distance of individual p, and denote the average diversity metrics in the decision and target spaces, respectively.

12. The multi-modal multi-objective traveling salesman problem optimization method according to claim 11, characterized in that: The specific steps of S862 are: S8621, calculate the number of identical edges between individuals, as shown below: in, Represents the edge set of individual p, expressed as , The edge set of individual q is expressed as , individual p is represented by , individual q is expressed as ; S8622, calculate the Spearman distance between individuals, as shown below: in, represents the number of vertices of an individual, and They represent the ranks of individuals p and q at the i-th vertex respectively; the rank value of the i-th vertex of individual p is calculated as follows: The Iverson brackets "[ ]" are used to represent conditional statements. For an Iverson bracket [P], if P is true, its calculation result is 1, otherwise it is 0; and Respectively represent the serial numbers of the i-th and j-th vertices of the individual; S8623, normalize the number of common edges and Spearman distance between individuals, convert them to the same scale and sum them; the larger the number of common edges V, the higher the similarity between individuals, and the smaller the Spearman distance S, the higher the similarity between individuals; therefore, take 1 minus the normalized S value as the final S, the formula is as follows: S8624, the weights of the two key variables are both set to 0.5 to obtain the similarity measure between individuals in the decision space, the formula is as follows: S8625, in order to determine the diversity measure for each individual in the decision space, the distance between the individual and its nearest neighbor and the sum of the distances to all other individuals is defined; first, the distance between individual p and its nearest neighbor q is denoted as , and the sum of the distances between individual p and all other individuals, denoted as , and then multiply these two parts to get the crowding distance of individual p in the decision space, the formula is as follows: 。

Images