Mathematical Modeling_Water Pipe Laying Problem
Topic : During the implementation of the village-to-village tap water project, from the perspective of ensuring the quality of water supply and the convenience of equipment maintenance, a certain area needs to build a central water supply station, 12 primary water supply stations and 168 secondary water supply stations, and water supply stations at all levels The location coordinates of is shown in Table 1, where type A represents a central water supply station, type V represents a primary water supply station, and type P represents a secondary water supply station. Figure 1 is a map of the geographic locations of water supply stations at all levels. Now the tap water at the central water supply station A will be piped to the primary and secondary water supply stations. According to the design requirements, the pipelines laid from the central station A to the primary water supply station are type I pipelines, and the pipelines laid from the primary water supply station to the secondary water supply station are type II pipelines.
Problem 1 : Calculate the Euclidean distance between each point according to the coordinates of each water supply station to obtain the corresponding distance matrix, and then use the Prim algorithm to obtain the minimum spanning tree of type I and type II pipelines, and merge them. In order to remove the loops that may be generated during the merging process, considering that the minimum spanning tree between the central water supply station and the primary water supply station is optimal, it should be kept intact; then the Prim algorithm is used to obtain the overall minimum spanning tree, which is the final Laying scheme. Through MATLAB, the total mileage of type I pipeline is 120.9412 kilometers, the total mileage of type II pipeline is 405.6407 kilometers, and the total mileage of laying is 526.5819 kilometers.
Question 2 : On the basis of question 1, the exhaustive method is used for loop traversal, a total of combinations, compare the mileage of type II pipelines in the minimum spanning tree traversed, and the combination corresponding to the minimum value is the two final upgrades Secondary water supply station. Using MATLAB to solve the upgraded secondary water supply station is He, and the total mileage of laying type II pipelines is at least 394.2375 kilometers, which is 11.4032 kilometers less than the problem one.
Question 3 : On the basis of the minimum spanning tree of question 1, the heuristic method is used to select the optimal upgrade location of the secondary water supply station. First, it is calculated that the secondary water station does not meet the requirements of the problem if it is not upgraded, and then the secondary water supply station is upgraded from The station began to test. Determine the upgraded secondary water supply station by comparing the variance. When continuing to upgrade, continue to upgrade on the basis of the determined first-level water supply station until the restriction that the total pipeline transportation mileage of each first-level water supply station is less than 40 kilometers is met. Using MATLAB to calculate that these 7 secondary water supply stations need to be upgraded, the corresponding total mileage of laying is at least 529.1359 kilometers.
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