Water pipe laying problems
Description of water pipe laying problems
During the implementation of the water supply project for every village, from the perspective of ensuring water supply quality and facilitating equipment maintenance, a certain area needs to build a central water supply station, 12 first-level water supply stations and 168 second-level water supply stations. The locations of water supply stations at all levels The coordinates are shown in Appendix Table 1, where type A represents the central water supply station, type V represents the first-level water supply station, and type P represents the second-level water supply station. Figure 1 in the attachment is a geographical location map of water supply stations at all levels.
Now we need to transport the tap water from central water supply station A to the first-level water supply station and the second-level water supply station through pipelines. According to the design requirements, the pipes laid from central station A to the first-level water supply station are type I pipes, and the pipes laid from the first-level water supply station to the second-level water supply station are type II pipes.
The technical requirements for laying water pipes are as follows:
- 1. The central water supply station can only be connected to the first-level water supply station (laying I-type pipelines), and cannot be directly connected to the second-level water supply station, but the first-level water supply stations can be connected (laying I-type pipelines).
- 2. The first-level water supply station can be connected to the second-level water supply station (laying type II pipelines), and the second-level water supply stations can also be connected (laying type II pipelines).
- 3. The connecting pipes between water supply stations at all levels must start from the position coordinates of the water supply station at the upper level or the same level, and cannot be connected from a point in the middle of any pipe.
- 4. The required pipeline length between two adjacent water supply stations (if there is a pipeline connected) can be simplified to the Euclidean distance.
Please combine the above pipe laying requirements, establish a mathematical model, and complete the following questions
- Question 1: Starting from central water supply station A, how should the tap water pipeline be laid to minimize the total mileage of the pipeline? The laying plan is given graphically, and the total mileage of Type I pipelines and Type II pipelines is given.
- Question 2: Due to the insufficient supply of Type II pipelines in the market, there is an urgent need to reduce the total mileage of Type II pipelines laid from the first-level water supply stations. The initial plan is to upgrade two of the second-level water supply stations to first-level water supply stations. Question: Which two secondary water supply stations should be selected? How should the tap water pipelines be laid to minimize the total mileage of Type II pipelines? Compared with the solution in question 1, how many kilometers has the total mileage of Type II pipeline been reduced?
- Question 3: Based on question 1, if in reality due to the influence of power, the pipeline laid from the first-level water supply station can only supply water for up to 40 kilometers (calculated based on the total mileage of pipeline transportation from the first-level water supply station), but from Pipeline water supply laid from central water supply station A is not subject to this distance restriction. In order to supply water to all water supply stations, several secondary water supply stations need to be upgraded to first-level water supply stations. However, after the upgrade, the pipelines laid from the water supply stations can only supply water for a maximum of 40 kilometers. Question: How many secondary water supply stations can be upgraded at least to achieve water supply to all water supply stations? What is the minimum total number of kilometers of pipeline laid in this configuration?
Modeling ideas and solution design
Symbol Description
Analysis of problem one
For problem one, it is necessary to solve the minimum total mileage of pipelines laid from the central water supply station, and convert it into a minimum spanning tree problem (MST problem) to solve. This article adopts Prim's algorithm, using water stations as nodes and the distance between water stations as edge weights to solve the minimum spanning tree of the total pipeline mileage. The non-hierarchical global optimal solution should not be considered here, that is, the first-level water supply station and the second-level water supply station are regarded as the same node, which will cause some tap water to be transmitted from the second-level water supply station to the first-level water supply station or to the intersection of tap water. This distribution does not form a circulation and does not conform to the distribution of dendritic pipes in real life. Based on the requirements of problem design, the pipeline laying is designed in a dendritic form with fewer pipeline miles. First, the pipelines from the central water supply station to each first-level water supply station are considered, and then the pipelines from the first-level water supply station to each second-level water supply station are considered. , that is, first determine the minimum spanning tree of type I pipeline, and then determine the minimum spanning tree of type II pipeline.
Question one result
Question 2
The results show that. . .
code
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