A method for solving a time window constrained coloring traveling salesman problem

Abstract

The application provides a method for solving a time window constrained coloring traveling salesman problem, and the main steps are as follows: reading initial parameters, including the coordinates of each city, time windows, and colors; eliminating edges that do not meet the hard time window constraint; classifying city types and movement operation types to obtain four operation neighborhoods for local search; performing hierarchical jitter operation on the obtained current solution to prevent the algorithm from falling into a local optimum; using a greedy jitter strategy and a preserving order opt operation to perform neighborhood alternating search to improve the solution; when a solution meeting the hard time window constraint cannot be found, relaxing the hard time window constraint into a soft time window constraint, and searching for a solution with minimum comprehensive cost by using an improved ant colony algorithm. The application is used for solving the time window constrained coloring traveling salesman problem model, and the coloring traveling salesman problem can be used for modeling various multi-machine task scheduling problems.

Description

Technical Field

[0001] This invention relates to the field of optimization scheduling, and more specifically, to a method for solving the colored traveling salesman problem with time window constraints. Background Technology

[0002] The Colored Traveling Salesman Problem (CTSP), proposed by Professor Li Jun of Southeast University, cleverly incorporates color to fully describe the homogeneity and heterogeneity of salesmen and the matching relationships between salesmen and cities, breaking the limitations of the Colored Traveling Salesman Problem (MTSP) which is confined to the undifferentiated city visits. The CTSP theoretical method can be used for unified modeling and efficient solution of multiple sequential scheduling problems in various multi-entity task systems, where entities can be machines, humans, or even computational threads. In particular, some new variants of CTSP are also applicable to numerous complex multi-vehicle scheduling problems with vehicle capacity and customer service requirements. Applications of CTSP include: multi-machine task scheduling and path planning in the manufacturing and logistics industries; multi-truss processing and handling systems in prefabrication plants; operation scheduling of various manned cranes and hoisting systems on construction sites; and scheduling of unmanned civilian or military robot swarms and unmanned surface, water, and air vehicles.

[0003] Currently, the Colored Traveling Salesman Problem (TRAS) and its variants only consider constraints such as the relationship between the salesman and the city, priority constraints, and multiple objective functions. In scheduling scenarios such as food delivery, vehicle freight transport, and port container operations, tasks need to be completed within a specified time. Existing algorithms for solving the TRS cannot achieve this. Therefore, algorithms for solving the TRS with time window constraints can help to better apply the TRS in practical engineering. Summary of the Invention

[0004] Purpose of the invention: To address the problem that traditional algorithms for solving the Colored Traveling Salesman Problem (CTSP) cannot consider time window constraints, this invention proposes a method for solving the CTSP with time window constraints. This method is applied to solve the CTSP with time window constraints, generalizing the CTSP model to make it more universal.

[0005] This invention is achieved through the following technical solution:

[0006] A method for solving the Colored Traveling Salesman Problem with Time Window Constraints is presented. This method further considers the time window constraint and can solve the Colored Traveling Salesman Problem with time window constraints. The specific steps are as follows:

[0007] Step S1: Read in the location coordinates, time window, and color of each city, randomly generate an initial solution, and calculate the infeasible edges that violate the time window constraint in the initial solution based on the initial information of each task.

[0008] Step S2: Divide the city type and the type of movement operation into four types of operation neighborhoods, and apply the 1-shift algorithm to the initial solution;

[0009] Step S3: Perform a hierarchical dithering operation on the current solution obtained by the 1-shift algorithm to transform the search neighborhood of the algorithm;

[0010] Step S4: If a feasible solution is found, the feasible solution is improved by using a greedy dithering operation to transform the neighborhood. Then, in the generated neighborhood, a combination of 1-shift and order-preserving opt algorithms is used to perform a variable neighborhood search to improve the feasible solution and find a solution that satisfies the hard time window constraint.

[0011] Step S5: If a feasible solution that satisfies the hard time window constraint cannot be found within the specified time, then execute the soft time window CTSP solution algorithm based on the improved ant colony algorithm. Calculate the timeout based on the access time of the corresponding city in the solution, and apply the order-preserving opt algorithm to the best individual in each round for further optimization.

[0012] Furthermore, the specific operation of calculating the infeasible edges that violate the time window constraint based on the initial information of each task in step S1 is as follows:

[0013] Let t bi To schedule the earliest time for city i to be visited, t ej To schedule the latest time for city j to be visited, w ij This represents the time it takes for a merchant to move from city i to city j; if t bi +w ij >t ej This means that in the most ideal case—that is, after visiting city i at the earliest scheduled time, the merchant directly visits city j. In this case, the merchant will still miss the latest scheduled time for city j, causing the visit to city j to time out. Therefore, city j must be visited before city i. Thus, for any visit sequence, visiting city i first and then city j is necessarily an infeasible solution that violates the time window constraint. During the preprocessing stage, the algorithm will mark these edges that violate the time window constraint so that the algorithm can avoid these infeasible parts during the search process.

[0014] Further, in step S2, the 1-shift algorithm is applied to the solution. The 1-shift algorithm generates four different priority neighborhoods based on the city type and the type of movement operation. Then, the search is performed based on these neighborhoods. The specific neighborhood partitioning steps are as follows:

[0015] Step 3-1: For the classification of city types, cities are divided into non-violation cities based on whether the arrival time exceeds the latest time of the scheduled visit. Violation cities are those whose arrival time exceeds the deadline, and vice versa.

[0016] Step 3-2: For the classification of move operations, the pre-movement operation moves the given city to an earlier position in the sequence, and the city will be visited earlier than its previous position. The post-movement operation moves the given city to a later position in the sequence, and the city will be visited later than its previous position.

[0017] Step 3-3: Based on the city type classification in Step 3-1 and the movement operation type classification in Step 3-2, four operation neighborhoods are divided for the local search stage. The first neighborhood is the solution set generated by the pre-operation of the violating city, the second neighborhood is the solution set generated by the post-operation of the non-violating city, the third neighborhood is the solution set generated by the pre-operation of the non-violating city, and the fourth neighborhood is the solution set generated by the post-operation of the violating city.

[0018] Further, step S3 performs a graded jittering operation on the current solution obtained by the 1-shift algorithm to transform the search neighborhood of the algorithm. Specifically, different levels of jittering are performed according to whether the search neighborhood of the algorithm is trapped in a local optimum. The random 1-shift algorithm is used for jittering, and the solution after jittering is updated by local search. If the solution is better than the solution before jittering, the solution obtained after local search is set as the current solution and the jittering level is reset to 1. If the current solution is not improved, the jittering level is increased by 1. If there is still no improved solution when the jittering level reaches the maximum jittering level, the algorithm will generate a new initial solution and return to step 2 to start searching again.

[0019] Furthermore, in step S4, two local search algorithms, 1-shift and order-preserving opt, are applied to the feasible solution in the generated neighborhood to perform a variable neighborhood alternating search. The variable neighborhood alternating search first performs a 1-shift search on the feasible solution, and then performs an order-preserving opt search on the improved 1-shift solution. By using different local search algorithms to perform neighborhood transformation search, the local optimum solution can be further eliminated.

[0020] Furthermore, in step S5, if no solution satisfying the time window constraint is found in steps S1 to S4 within the specified time, the improved ant colony algorithm under a soft time window is adopted. The specific execution steps are as follows:

[0021] Step 5-1: Randomly generate individual ants in the ant colony. For each individual, assign a single-color city to the corresponding merchant. For each multi-color city, randomly select one from the set of merchants it can access.

[0022] Step 5-2: Traverse the sub-paths of each ant colony individual, calculate the probability transition value of each city based on pheromone and visibility, and then update the city selection solution. The visibility calculation method is as follows:

[0023]

[0024] Where i and j are two different cities, d ij Tout is the distance between the two cities. j Let be the timeout value for city j. This visibility takes into account the impact of the timeout value. The visibility in the unimproved algorithm only considers the impact of the distance between cities. In the improved algorithm, the visibility also includes the timeout penalty when reaching the next city. Including the timeout value in the visibility can increase the probability that individual ants choose cities with smaller timeout values, thereby enhancing the algorithm's sensitivity to the timeout value.

[0025] Step 5-3: After the first generation of ant colony has been traversed, perform order-preserving opt local search operation to optimize the best individual. If it is better than the historical best solution, then update the historical best solution.

[0026] Step 5-4: Update pheromones based on the number of times each individual in this generation visits each path.

[0027] Beneficial effects: This invention proposes a method for solving the Colored Traveling Salesman Problem with time window constraints, and applies it to solving the Colored Traveling Salesman Problem with time window constraints, making the Colored Traveling Salesman Problem more general and applicable to modeling and solving optimization scheduling problems with time window constraints. Attached Figure Description

[0028] Figure 1 This is the main flow of the time window solution algorithm;

[0029] Figure 2 This is the main flow of the hard time window solution algorithm;

[0030] Figure 3 This is the main process of the construction phase of the VNS-based hard time window solution algorithm;

[0031] Figure 4 This is the main process of the improved stage of the VNS-based hard time window solution algorithm;

[0032] Figure 5 The main process is based on the improved soft time window solution algorithm for ant colonies;

[0033] Figure 6 This is a diagram illustrating infeasible edges;

[0034] Figure 7 This is a diagram illustrating the front and rear operation modes;

[0035] Figure 8 This is a schematic diagram of the first neighborhood city cluster;

[0036] Figure 9 This is a schematic diagram of the second neighborhood city cluster;

[0037] Figure 10 This is a schematic diagram of the third neighborhood city cluster;

[0038] Figure 11 This is a schematic diagram of the fourth neighborhood city cluster;

[0039] Figure 12 These are graphs showing the solution results of the time window algorithm in different cases;

[0040] Figure 13 The objective function curve is based on the improved soft time window solution algorithm for ant colonies in the case of a 7-minute time window width.

[0041] Figure 14 The curve of timeout is based on the improved soft time window solution algorithm for ant colonies in the case of a 7-minute time window width.

[0042] Figure 15 This is a comparison chart of the overall costs of the improved ant colony algorithm and the soft time window VNSTW two-stage algorithm;

[0043] Figure 16 This is a comparison chart of the timeout values ​​between the improved ant colony algorithm and the soft time window VNSTW two-phase algorithm; Detailed Implementation

[0044] The present invention will now be described in further detail with reference to the accompanying drawings and embodiments.

[0045] A method for solving the Colored Traveling Salesman Problem with Time Window Constraints is characterized by further considering time window constraints, enabling the solution of the Colored Traveling Salesman Problem with time window constraints. The specific steps are as follows:

[0046] Step 1: Read in the location coordinates, time window, and color of each city, and generate infeasible edges that violate the time window constraint based on the initial information of each task, so as to reduce the amount of computation in the subsequent search process of the algorithm.

[0047] Step 2: Apply the 1-shift algorithm to the current solution for local search. It mainly divides the operation neighborhood into four types based on city type and movement operation type for local search.

[0048] Step 3: Perform a hierarchical dithering operation on the current solution to transform the neighborhood and prevent the algorithm from getting trapped in local optima;

[0049] Step 4: Improve the feasible solution by using a greedy dithering operation to transform the neighborhood, and then perform a combination of 1-shift and order-preserving opt algorithms in the generated neighborhood to perform a variable neighborhood search, thus completing the solution improvement.

[0050] Step 5: If a feasible solution cannot be found within the specified time, the improved ant colony algorithm is executed. The timeout is calculated based on the access time of the corresponding city in the solution, and the order-preserving OPT algorithm is applied to the best individual in each round for further optimization.

[0051] Furthermore, in step S1, infeasible edges are calculated in advance to reduce invalid computations in the subsequent search process. The specific operation is as follows:

[0052] Let t bi To schedule the earliest time for city i to be visited, t ej To schedule the latest time for city j to be visited, w ij This represents the time it takes for a merchant to move from city i to city j. Figure 6 This indicates an infeasible edge. If t bi +w ij >t ej This means that in the ideal scenario—that is, after visiting city i at its earliest scheduled time, the merchant directly visits city j—the merchant will still miss the latest scheduled time for city j, causing the visit to city j to time out. Therefore, city j must be visited before city i. Thus, for any visiting sequence, visiting city i before visiting city j is necessarily an infeasible solution that violates the time window constraint. During the preprocessing stage, the algorithm marks these edges that violate the time window constraint so that the algorithm can avoid these infeasible parts during the search process.

[0053] Further, in step S2, a local 1-shift search operation is performed on the solution. The main idea of ​​1-shift is to generate neighborhoods with different priorities for the solution based on the city type and the type of movement operation, and then perform the search based on these neighborhoods. The specific neighborhood division steps are as follows:

[0054] Step 3-1: For the classification of city types, cities are divided into non-violation cities based on whether the arrival time exceeds the latest time of the scheduled visit. Violation cities are those whose arrival time exceeds the deadline, and vice versa.

[0055] Step 3-2: Classifying move operations, a pre-movement operation moves a given city to an earlier position in the sequence, ensuring the city is visited earlier than before. A post-movement operation moves a given city to a later position in the sequence, ensuring the city is visited later than before. Figure 7 The specific processes of the two operations are shown;

[0056] Step 3-3: Based on the classification of city types in Step 3-1 and the classification of movement operation types in Step 3-2, four types of operation neighborhoods can be divided for the local search stage. Figure 8 The first neighborhood shown is the solution set generated through the pre-operation of the violating city. Figure 9 The second neighborhood shown is the solution set generated by the post-operation on non-violation cities. Figure 10 The third neighborhood shown is the solution set generated through the pre-operation of non-violation cities. Figure 11 The fourth neighborhood shown is the solution set generated by the post-operation of the violating city.

[0057] Furthermore, step S2 performs a graded jittering operation on the algorithm. Different levels of jittering are performed based on whether the search neighborhood of the algorithm is trapped in a local optimum. A random 1-shift operation is used for jittering. The solution after jittering is updated by local search. If the solution is better than the solution before jittering, the solution obtained after local search is set as the current solution and the jittering level is reset to 1. If the current solution is not improved, the jittering level is increased by 1. When the maximum jittering level is reached and there is still no improved solution, the algorithm will generate a new initial solution and return to step 2 to start searching again.

[0058] Furthermore, in step 4, two local search algorithms, 1-shift and order-preserving opt, are used to perform alternating neighborhood searches on feasible solutions. First, a 1-shift local search is performed on the current solution, and then an order-preserving opt search is performed on the improved solution. By using different local search algorithms to perform neighborhood transformation searches, the goal is to further escape local optima.

[0059] Furthermore, the improved ant colony algorithm in step 5. When the hard time window algorithm cannot find a solution within a unit of time, the improved ant colony algorithm under a soft time window is used. Its specific execution steps are as follows:

[0060] Step 5-1: Randomly generate individual ant colonies. For each individual, assign a single-color city to the corresponding merchant. For each multi-color city, randomly select one from the set of merchants it can access.

[0061] Step 5-2: Traverse the sub-paths of each ant colony individual, calculate the probability transition value of each city based on pheromone and visibility, and then update the city selection solution; the visibility calculation method is as follows:

[0062]

[0063] Where i and j are two different cities, d ij Tout is the distance between the two cities. jLet be the timeout value for city j. This visibility takes into account the impact of the timeout value. The visibility in the unimproved algorithm only considers the impact of the distance between cities. In the improved algorithm, the visibility also includes the timeout penalty when reaching the next city. Including the timeout value in the visibility can increase the probability that individual ants choose cities with smaller timeout values, thereby enhancing the algorithm's sensitivity to the timeout value.

[0064] Step 5-3: After the first generation of ant colony has been traversed, perform order-preserving opt local search operation to optimize the best individual. If it is better than the historical best solution, then update the historical best solution.

[0065] Step 5-4: Update pheromones based on the number of times each individual in this generation visits each path;

[0066] Tables 1-3 show the results of the hard time window algorithm VNSTW and the improved ant colony algorithm of the soft time window algorithm on hard time window data. As can be seen from the data in the table, the VNSTW algorithm is better than the improved ant colony algorithm in solving the three hard time window datasets of 20min, 15min and 10min. Figure 12 As shown in the total task time curve, the total task time increases as the time window width decreases.

[0067] Table 1. Comparison of total time consumption for bridge operations between VNSTW and the improved ant colony algorithm under a 20-minute hard time window constraint.

[0068]

[0069] Table 2 Comparison of total time consumption for bridge operations under a 15-minute hard time window constraint between VNSTW and improved ant colony algorithms.

[0070]

[0071]

[0072] Table 3 Comparison of total time consumption for bridge operations under a 10-minute hard time window constraint between VNSTW and improved ant colony algorithms.

[0073]

[0074] Figure 13 Table 4 shows the overall cost curves and solution quality of the improved ant colony algorithm and the unimproved ant colony algorithm within the soft time window of the method involved in this invention. The data in the tables show that the improved ant colony algorithm performs better than the unimproved ant colony algorithm, and in each case, the improved ant colony algorithm obtains a solution with lower cost. Figure 14As shown in the timeout comparison chart, the difference in timeout between the two algorithms increases with the increase in task size. This is because the improved algorithm has visibility that can reflect the timeout, making it more sensitive to the timeout metric during the solution process, while the unimproved algorithm does not consider the timeout factor in its visibility.

[0075] Table 4 shows the improved and unimproved ant colony algorithms. The algorithm performs well in 7... m Comparison of overall cost values ​​under soft time window constraints

[0076]

[0077] Tables 5-7 show the solution results of the improved ant colony algorithm and the VNSTW two-stage (i.e., improved stage) algorithm using the soft time window method of this invention on soft time window data. Figure 15 To compare the performance of the two algorithms, Figure 16 The timeout values ​​for the solutions obtained by the two algorithms are shown in the chart. As can be seen from the data, in the cases comparing the performance of algorithms with soft time window constraints, the overall cost of the improved ant colony algorithm is better than that of the two-stage VNSTW algorithm. This is because the improved ant colony algorithm is more sensitive to the timeout value, while VNSTW cannot effectively adjust the search direction based on the size of the timeout.

[0078] Table 5. Comparison of the overall cost values ​​of VNSTW two-stage algorithm and improved ant colony algorithm under a 5-minute soft time window constraint.

[0079]

[0080] Table 6. Comparison of the overall cost values ​​of VNSTW two-stage algorithm and improved ant colony algorithm under the 3-minute soft time window constraint.

[0081]

[0082] Table 7. Comparison of the overall cost values ​​of VNSTW two-stage algorithm and improved ant colony algorithm under a 1-minute soft time window constraint.

[0083]

[0084]

Claims

1. A method for solving the colored traveling salesman problem with time window constraints, characterized in that, This method considers time window constraints and solves the Colored Traveling Salesman Problem with time window constraints. The time window constraints are refined into two types: hard time windows and soft time windows. The specific steps are as follows: Step S1: Read in the location coordinates, time window, and color of each city, randomly generate an initial solution, and calculate the infeasible edges that violate the time window constraint in the initial solution based on the initial information of each task. Step S2: Divide the city type and the type of movement operation into four types of operation neighborhoods, and apply the 1-shift algorithm to the initial solution; Step S3: Perform a hierarchical dithering operation on the current solution obtained by the 1-shift algorithm to transform the search neighborhood of the algorithm; Step S4: If a feasible solution is found, the feasible solution is improved by using a greedy dithering operation to transform the neighborhood. Then, in the generated neighborhood, a combination of 1-shift and order-preserving opt algorithms is used to perform a variable neighborhood search to improve the feasible solution and find a solution that satisfies the hard time window constraint. Step S5: If a feasible solution that satisfies the hard time window constraint cannot be found within the specified time, then execute the soft time window CTSP solution algorithm based on the improved ant colony algorithm. Calculate the timeout based on the access time of the corresponding city in the solution, and apply the order-preserving opt algorithm to the best individual in each round for further optimization. In step S2, the 1-shift algorithm is applied to the solution. The 1-shift algorithm generates four neighborhoods with different priorities based on the city type and the type of movement operation. Then, the search is performed based on these neighborhoods. The specific neighborhood division steps are as follows: Step 3-1: For the classification of city types, cities are divided into non-violation cities based on whether the arrival time exceeds the latest time of the scheduled visit. Violation cities are those whose arrival time exceeds the deadline, and vice versa. Step 3-2: For the classification of move operations, the pre-movement operation moves the given city to an earlier position in the sequence, and the city will be visited earlier than its previous position. The post-movement operation moves the given city to a later position in the sequence, and the city will be visited later than its previous position. Step 3-3: Based on the city type classification in Step 3-1 and the movement operation type classification in Step 3-2, four operation neighborhoods are divided for the local search stage. The first neighborhood is the solution set generated by the pre-operation of the violating city, the second neighborhood is the solution set generated by the post-operation of the non-violating city, the third neighborhood is the solution set generated by the pre-operation of the non-violating city, and the fourth neighborhood is the solution set generated by the post-operation of the violating city. Step S4 involves performing alternating neighborhood search on feasible solutions in the generated neighborhood using two local search algorithms: 1-shift and order-preserving opt. The alternating neighborhood search first performs a 1-shift search on the feasible solutions, and then performs an order-preserving opt search on the improved 1-shift solutions. By using different local search algorithms to perform neighborhood transformation search, the goal is to further escape local optima.

2. The method for solving the colored traveling salesman problem with time window constraints according to claim 1, characterized in that: The specific operation of calculating infeasible edges that violate the time window constraint based on the initial information of each task in step S1 is as follows: set up Schedule the earliest possible visit time for city i. Schedule the latest time for city j to be visited. This represents the time it takes for a merchant to move from city i to city j; if This means that in the most ideal case—that is, after visiting city i at the earliest scheduled time, the merchant directly visits city j. In this case, the merchant will still miss the latest scheduled time for city j, causing the visit to city j to time out. Therefore, city j must be visited before city i. Thus, for any visit sequence, visiting city i first and then city j is necessarily an infeasible solution that violates the time window constraint. During the preprocessing stage, the algorithm will mark these edges that violate the time window constraint so that the algorithm can avoid these infeasible parts during the search process.

3. The method for solving the colored traveling salesman problem with time window constraints according to claim 1, characterized in that: Step S3 performs a graded jittering operation on the current solution obtained by the 1-shift algorithm to transform the search neighborhood of the algorithm. Specifically, different levels of jittering are performed according to whether the search neighborhood of the algorithm is trapped in a local optimum. The random 1-shift algorithm is used for jittering. The solution after jittering is updated by local search. If the solution is better than the solution before jittering, the solution obtained after local search is set as the current solution and the jittering level is reset to 1. If the current solution is not improved, the jittering level is increased by 1. If there is still no improved solution when the jittering level reaches the maximum jittering level, the algorithm will generate a new initial solution and return to step 2 to start searching again.

4. The method for solving the colored traveling salesman problem with time window constraints according to claim 1, characterized in that: The improved ant colony algorithm described in step S5, when no solution satisfying the time window constraint can be found in steps S1 to S4 within the specified time, adopts an improved ant colony algorithm under a soft time window. The specific execution steps are as follows: Step 5-1: Randomly generate individual ants in the ant colony. For each individual, assign a single-color city to the corresponding merchant. For each multi-color city, randomly select one from the set of merchants it can access. Step 5-2: Traverse the sub-paths of each ant colony individual, calculate the probability transition value of each city based on pheromone and visibility, and then update the city selection solution. The visibility calculation method is as follows: (1) Where i and j are two different cities, d ij Tout is the distance between the two cities. j Let J be the timeout amount for city j. Step 5-3: After the first generation of ant colony has been traversed, perform order-preserving opt local search operation to optimize the best individual. If it is better than the historical best solution, then update the historical best solution. Step 5-4: Update pheromones based on the number of times each individual in this generation visits each path.

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