Aversive facility site selection method based on power graph and improved genetic algorithm

Abstract

The invention provides an aversive facility site selection method based on a power graph and an improved genetic algorithm, and belongs to the technical field of facility planning and site selection. By combining the Maximin site selection model, the Power graph, the improved genetic algorithm and the local search algorithm, the decision fairness and acceptability of aversive facility site selection are improved, the aversive facility site selection method has more accurate global search capability, and the negative influence on nearby residents or environments is reduced to the greatest extent. Specifically, double targets of maximizing public benefits and minimizing adverse effects are considered through a Maximin site selection model; the Power graph can accurately describe the proximity and influence range of each potential position; the application of the improved genetic algorithm more effectively finds a globally optimal solution in a complex search space, and effective position optimization is carried out by simulating an evolutionary process; the local search algorithm further improves the understanding precision, and ensures that a better solution is achieved in a local area.

Description

Technical Field

[0001] The present invention belongs to the technical field of facility planning and site selection, and particularly relates to a method for locating aversive facilities based on power diagrams and an improved genetic algorithm. Background Technique

[0002] With the continuous deepening of global industrialization, environmental problems have become increasingly prominent; in modern urban planning, the problem of locating aversive facilities has become a key challenge; aversive facilities refer to those public facilities that are necessary for the whole society but have a negative impact on the sanitary environment of the neighboring areas and are therefore not welcomed by local residents, such as landfills, power plants, wastewater treatment plants, etc.; their location not only concerns environmental protection but also directly affects the quality of life of residents and the sustainable development of communities.

[0003] In the context of urban renewal operations, the tightening of land resources and the issue of sustainable development make the location of aversive facilities particularly crucial; traditional planning of aversive facilities often relies on overall demand analysis at the macro level, that is, determining the location of facilities based on simple geometric centers or equalization principles; for example, if the location of landfills and wastewater treatment plants is only based on geometric centers, factors such as terrain, hydrological conditions, and resident distribution may be ignored, resulting in environmental pollution and a decline in the quality of life of residents.

[0004] It can be seen that traditional methods for locating aversive facilities are inefficient in resource allocation when dealing with large-scale and complex urban environments, often ignoring the spatial distribution characteristics of resident needs and the impact on the environment, leading to waste of public resources and inaccurate location decisions. Summary of the Invention

[0005] Aiming at the problems existing in the prior art, the present invention proposes a method for locating aversive facilities based on power diagrams and an improved genetic algorithm, aiming to combine the Maximin location model, power diagrams, improved genetic algorithm, and local search algorithm to effectively improve the efficiency of locating aversive facilities while considering residents' wishes and environmental pollution; it solves the problems of waste of public resources, inaccurate decision-making, and various limitations of traditional location methods.

[0006] To this end, the specific technical solution adopted by the method for locating aversive facilities based on power diagrams and an improved genetic algorithm of the present invention is as follows:

[0007] Step 1: Quantify basic data: Quantify the degree of aversion of residents to facilities, randomly assign values according to the levels of "like", "slightly like", "average", "slightly dislike", "dislike", and "very dislike" as the weight values of community locations, and perform data cleaning to remove invalid or incorrect data;

[0008] Step 2. Construct a power diagram:

[0009] (1) In the overall area, regard the community points as the sites of the power diagram, and use their corresponding population data values as the site weight values;

[0010] (2) Adopt an embedding method to elevate the two-dimensional problem to three-dimensional space to obtain the patches of the three-dimensional convex hull, and then map them back to two-dimensional space to obtain the power diagram boundary;

[0011] (3) For the case with a boundary, generate symmetric points through symmetric transformation of the boundary points, merge the symmetric points with the original point set, and recalculate the power diagram to ensure that all patches are within the boundary;

[0012] (4) Calculate the angles of each patch and sort the vertices to obtain the power diagram vertices as the candidate locations for the noxious facilities;

[0013] Step 3. Calculate the minimum weighted distance: For each vertex of the power diagram, calculate its minimum weighted distance to the nearest facility;

[0014] Step 4. Use an improved genetic algorithm to search for the optimal location of the facilities;

[0015] Step 5. After the improved genetic algorithm selects the optimal individual, use a local search algorithm to perform local optimization based on the current solution by closing one facility and then opening another facility to find a better solution within the local area;

[0016] Step 6. Draw an image: Plot the community points, the existing facility locations, the power diagram, and the selected facility locations in an image to obtain the final site selection result diagram.

[0017] Preferably, each region of the power diagram in Step 2 corresponds to a site x i , and the set of candidate noxious facility locations in this region is And define the weighted distance d between point V and the community as i :

[0018]

[0019] For each point i and its weight w i , the cells of the power diagram are regarded as the regions where all points j satisfy the following conditions:

[0020]

[0021] where (x i , x j ) are the coordinates of points i and j, and w i and w j are their weights.

[0022] Preferably, the specific process of dimension elevation to three-dimensional space in the second step is as follows:

[0023] ① Expand the coordinate matrix: Given a two-dimensional point S = {(x i , y i )} and a weight w i , expand each point (x i , y i ) to three-dimensional space:

[0024]

[0025] is the third-dimensional coordinate of the point in three-dimensional space, and the calculation method is based on the definition of the Power diagram, with the weight introduced into the calculation;

[0026] ② Calculate the convex hull: Calculate the three-dimensional convex hull of the expanded point set; the convex hull is the smallest convex polyhedron that contains all points; in three-dimensional space, each face of the convex hull represents the boundary of the Power diagram generated in the original two-dimensional space;

[0027] The specific process of mapping the three-dimensional convex hull faces back to two-dimensional space is as follows:

[0028] ③ Classify the faces: The three-dimensional convex hull generates some faces, and these faces are divided into two categories:

[0029] Lower-half faces: Located in the lower half of the convex hull, representing the valid Power diagram area;

[0030] Upper-half faces: Located in the upper half of the convex hull, not belonging to the valid Power diagram area and need to be removed;

[0031] ④ Map the faces back to two-dimensional space: For each valid face, map it back to two-dimensional space through polar coordinate transformation to obtain the boundary of the Power diagram.

[0032] Preferably, the formula for calculating the minimum weighted distance in the third step is:

[0033] Preferably, the process of improving the genetic algorithm to search for the optimal position in the fourth step is as follows:

[0034] (1) Gene encoding and decoding: Adopt binary encoding method to implement the encoding function encode and the decoding function decode, and convert the selected intersection position and the binary encoding to each other;

[0035]

[0036] (2) Initialize the population using the dynamic programming method: Use the dynamic programming table dp[i][j] to record all combinations of selecting j positions among the first i positions, randomly select a combination as the initial population, and convert it into an individual in binary coding form;

[0037] (3) Evaluate the fitness: Calculate the fitness function of each individual, and measure the quality of the individual solution with the goal of maximizing the minimum weighted distance;

[0038] (4) Selection operation: Arrange in non-decreasing order according to the magnitude of the fitness function, and then use the greedy selection method to select and determine whether an individual is selected as a parent in turn;

[0039] (5) Crossover operation: Use a hybrid crossover method to combine the genes of two parent individuals to generate new offspring individuals. The hybrid crossover method includes single-point crossover, multi-point crossover in the first half, and simulated binary crossover in the second half;

[0040] (6) Mutation operation: Randomly change some gene positions of an individual with a certain probability, and use diversity-enhanced differential mutation to introduce the diversity of the population, which helps to avoid premature convergence to a local optimal solution;

[0041] (7) Reward mechanism: Use the fitness ranking to assign rewards to high-quality individuals, thereby motivating high-quality individuals and improving the quality of the population.

[0042] Preferably, the specific process of initializing the population using the dynamic programming method is as follows:

[0043] ① Initialize the DP table: dp[0][0] is [[]], indicating that the combination of selecting 0 positions among the first 0 positions is an empty list;

[0044] ② State transition process: For each position i and the number of selections j, dp[i][j] is obtained by inheriting from dp[i - 1][j] (excluding the i-th position) and dp[i - 1][j - 1] (including the i-th position):

[0045] dp[i][j] = dp[i - 1][j] ∪ {comb + [i - 1] | comb ∈ dp[i - 1][j - 1]}

[0046] ③ Sample from the DP table: Randomly select a specified number of combinations from dp[K][p] as the initial population, and convert the selected combinations into individuals in binary coding form of length K, indicating the selected positions.

[0047] Preferably, the Maximin location model for evaluating the fitness is: max{min{d(p i , x j )}}.

[0048] Preferably, the specific process of the mutation operation is as follows:

[0049] ① Select three different individuals a, b, and c, where a, b, and c are all individuals encoded in binary; the calculation formula for the difference vector is:

[0050] d non-linear = b - a

[0051] ② Perform a non-linear transformation on the difference vector d:

[0052] d = sign(d) * (|d|)^ n

[0053] where sign(d) is the sign (+1 or -1) of each element in the difference vector d, |d| is the absolute value of the difference vector d, and n is the power of the non-linear transformation;

[0054] ③ Generate a mutant individual by adding the ind of the current individual to the scaled difference vector F × d non-linear :

[0055] mutant = ind + F × d non-linear

[0056] ④ To further enhance the diversity of the population, apply an additional random site inversion operation on the generated mutant individual; with a certain probability p flip perform the inversion operation; select several random positions in the mutant individual and invert these positions: 0 becomes 1, and 1 becomes 0; the formula for the inversion operation is:

[0057] mutant[i] = 1 - mutant[i]

[0058] where i is the index of the randomly selected site, and mutant[i] is the value at the i-th position.

[0059] Preferably, the specific process of the reward mechanism is as follows:

[0060] First, sort all individuals according to fitness. The sorted individual indices are R, where R[1] is the optimal individual, R[2] is the second-best individual, and so on; the rank r[i] represents the position of individual i, starting from 1;

[0061] Secondly, assign rewards R[i] to individuals based on the rank r[i]. The higher the rank, the greater the reward for the individual; the reward is calculated by the following formula:

[0062]

[0063] Finally, during the population update process, a reward mechanism is used to guide the generation of new individuals; and after a new individual is generated, it is checked whether the new individual already exists in the reward mechanism; if it exists, the update is skipped; if it does not exist, the fitness of the individual is calculated and a reward is assigned to it.

[0064] Preferably, the specific process of the local search algorithm in step five is as follows:

[0065] (1) Initialization: Randomly select an initial solution from the solution space of the problem;

[0066] (2) Local search: For each opened facility, try to close it and open a facility at another location, and calculate the fitness of the new solution; check the facility number limit and the distance requirement between facilities. If the requirements are met and the new solution is better, update the optimal solution;

[0067] (3) Loop iteration: Perform multiple local searches on the current optimal solution until a better solution cannot be found, and finally return the optimized optimal solution.

[0068] Compared with the prior art, the beneficial effects of the present invention are as follows:

[0069] By combining the Maximin location model, the Power diagram, the improved genetic algorithm, and the local search algorithm, the present invention improves the decision fairness and acceptability of the siting of aversive facilities, has a more accurate global search ability, and minimizes the negative impact on the nearby residents or the environment.

[0070] 1. The Maximin location model of the present invention aims to maximize the minimum distance of all individuals, thereby ensuring that the negative impact on any one individual is minimized; this means that the negative impact on the nearby residents or the environment can be minimized to the greatest extent when selecting the location of an aversive facility. At the same time, the Maximin location model reflects the fairness and acceptability of the decision-making. It can ensure that the interests of all stakeholders are fairly treated and avoid any party being overly negatively affected by the construction of the facility; this design effectively optimizes the location model and helps to improve the overall performance.

[0071] 2. The power diagram of the present invention has obvious visualization advantages and can graphically express the influence range of the location; in the problem of siting aversive facilities, the power diagram can help decision-makers intuitively understand the influence degree of different sitings on the surrounding environment and population, so as to better evaluate and compare the advantages and disadvantages of candidate locations. And through the power diagram, the influence of each location can be more intuitively understood, which helps to make a more objective and comprehensive decision.

[0072] 3. The improved genetic algorithm of the present invention has stronger global search ability. It is a heuristic search algorithm, suitable for global search in a large range. In the facility location problem, the improved genetic algorithm can explore a large number of possible solutions, and with the help of crossover and mutation operations, gradually optimize the quality of the solutions. The improved genetic algorithm can adapt to diverse problem spaces, is not restricted by the complexity of the search space, and can handle complex constraint conditions and non-linear objective functions. For the aversive facility location problem, it can handle various types of constraint conditions and changing geographical environments.

[0073] Furthermore, through diversity-enhanced differential mutation and reward mechanism, the improved genetic algorithm can effectively avoid premature convergence to local optimal solutions, maintain the diversity of the population, ensure wider exploration in the solution space, and finally obtain the optimal solution.

[0074] 4. The local search algorithm of the present invention can be further refined and optimized in the facility location problem. Based on the solutions obtained by the improved genetic algorithm, it can finely adjust the quality of the solutions. By conducting a small-range search and adjustment near the current solution, it can effectively optimize the local quality of the existing solutions. The local search algorithm can quickly converge to local optimal solutions. Compared with global search, local search is faster, which helps to reduce the computational cost and time while ensuring the quality of the solutions.

[0075] In addition, during the process of using the local search algorithm to optimize the solutions, by calculating the distances between the positions of the new solutions and the existing facilities, it is ensured that there will be no greater impact between the facilities, meeting various constraint conditions in practical applications. BRIEF DESCRIPTION OF THE DRAWINGS

[0076] Figure 1 is a flowchart of the present invention;

[0077] Figure 2 is a schematic diagram of the relationship between the power diagram and weights of the present invention;

[0078] Figure 3 is a schematic diagram of the residential points and weights of Embodiment 1 of the present invention;

[0079] Figure 4 is a schematic diagram of the residential points and power diagram of Embodiment 1 of the present invention;

[0080] Figure 5 is a schematic diagram of the location result obtained in Embodiment 1 of the present invention;

[0081] Figure 6 is a schematic diagram of the community points and weights of Embodiment 2 of the present invention;

[0082] Figure 7 is a schematic diagram of the community points and power diagram of Embodiment 2 of the present invention;

[0083] Figure 8 Schematic diagram of the site selection result obtained in Embodiment 2 of the present invention;

[0084] Figure 9 Schematic diagram of community points and weights in Embodiment 3 of the present invention;

[0085] Figure 10 Schematic diagram of community points and Power diagram in Embodiment 3 of the present invention;

[0086] Figure 11 Schematic diagram of the site selection result obtained in Embodiment 3 of the present invention. Detailed implementation manner

[0087] To better understand the content of the present invention, the present invention will be further elaborated below in conjunction with specific embodiments. The following embodiments are based on the technology of the present invention and give detailed implementation manners and operation steps. However, the protection scope of the present invention is not limited to the following embodiments, that is: all other embodiments obtained by those of ordinary skill in the art without creative efforts based on the embodiments of the present invention belong to the protection scope of the present invention.

[0088] Please refer to Figure 1 , a method for locating aversive facilities based on a power diagram and an improved genetic algorithm, comprising the following steps:

[0089] Step 1: Quantify the basic data;

[0090] First, quantify the degree of aversion of residents to the facility, randomly assign values according to the levels of "like", "slightly like", "average", "slightly averse", "averse", and "very averse" as the weight values of the community location, and then perform data cleaning to remove invalid or incorrect data to ensure the accuracy and consistency of the data.

[0091] Step 2: Construct a power diagram;

[0092] 1. First, in the overall area, regard the community points as the sites of the power diagram, and the corresponding population data values as the weight values of the sites.

[0093] Let represent a continuous and closed convex geometric domain in d-dimensional space, be a given set of sites, and define the corresponding weight value w i for site x i , and denote the weight set as The power diagram divides the entire geometric domain into n non-overlapping cells according to the positions of the sites and their weight values and As Figure 2 shown, site x iThe corresponding power unit is defined as

[0094]

[0095] where ||·|| is the Euclidean distance.

[0096] Each region of the Power diagram corresponds to a site x i , and the set of candidate locations of the aversive facilities in this region (points to be calculated) is And the weighted distance (also known as the power distance) d between point V and the community is defined as i :

[0097]

[0098] 2. Construct the power diagram using the embedding method. First, lift the two-dimensional problem to three-dimensional space to obtain the facets of the three-dimensional convex hull, and then map it back to two-dimensional space to obtain the boundary of the power diagram. The steps are as follows:

[0099] (1) Define the power diagram

[0100] The power diagram (or Laguerre diagram) is a weighted variant of the Voronoi diagram, which takes into account the weights of each point; for each point i and its weight w i , the cell of the power diagram can be regarded as the region where all points j satisfy the following conditions:

[0101]

[0102] where (x i , x j ) are the coordinates of points i and j, and w i and w j are their weights.

[0103] (2) Lift to three-dimensional space

[0104] To calculate the power diagram, first, the two-dimensional problem needs to be lifted to three-dimensional space; this process includes the following steps:

[0105] ① Expand the coordinate matrix: Given the two-dimensional points S = {(x i , y i )} and the weights w i , each point (x i , y i ) is expanded to three-dimensional space:

[0106]

[0107] Here is the third - dimensional coordinate of a point in three - dimensional space. The calculation method is based on the definition of the Power diagram, and weights are introduced into the calculation.

[0108] ② Calculate the convex hull: Calculate the three - dimensional convex hull of the expanded point set; the convex hull is the smallest convex polyhedron that contains all points. In three - dimensional space, each face (or called a patch) of the convex hull represents the boundary of the Power diagram generated in the original two - dimensional space.

[0109] (3) Process the convex hull faces

[0110] The patches of the calculated three - dimensional convex hull are mapped back to two - dimensional space in the following way:

[0111] ① Classify the patches: The three - dimensional convex hull generates some patches, and these patches are divided into two categories:

[0112] Lower - half patches: These patches are located in the lower half of the convex hull and represent the valid Power diagram regions.

[0113] Upper - half patches: These patches are located in the upper half of the convex hull and usually need to be removed because they do not belong to the valid Power diagram regions (e.g., regions at infinity).

[0114] ② Map the patches back to two - dimensional space: For each valid patch, map it back to two - dimensional space through polar coordinate transformation (or other mapping methods) to obtain the boundary of the Power diagram.

[0115] 3. Apply boundary handling. For the case where the patches exceed the boundary, generate symmetric points and recalculate the Power diagram to ensure that the patches are within the boundary;

[0116] ① Generate symmetric points: Generate symmetric points, which are obtained through symmetric transformation of the boundary points; for example, if a point exceeds the boundary, a symmetric point can be generated by reflection or other means.

[0117] ② Recalculate the Power diagram: Combine these symmetric points with the original point set and then recalculate the Power diagram. This will ensure that all patches are within the boundary.

[0118] 4. To ensure that the patches of the Power diagram are arranged in order, calculate the angle of each patch and sort the vertices to obtain the vertices of the power diagram as candidate positions for the aversive facilities;

[0119] ① Calculate the angle of each patch: Convert the vertex coordinates of each patch to polar coordinates (or other appropriate coordinate systems) to determine its sorting order.

[0120] ② Sort vertices: Sort the vertices according to the angles, so as to obtain the Power graph with the vertices of the patches arranged in order, and obtain the vertices of the power graph, which are used as the candidate positions for the aversive facilities.

[0121] Step 3: Calculate the minimum weighted distance;

[0122] For each vertex of the power graph Calculate the minimum weighted distance from the vertex to the nearest facility x (i.e., the site), that is

[0123] Step 4: Use the improved genetic algorithm to search for the optimal positions of the facilities;

[0124] (1) Gene encoding and decoding

[0125] Adopt the binary encoding method to implement the encoding function encode and the decoding function decode, and convert the selected intersection positions into binary encodings, or restore the selected intersections from the binary encodings, as follows:

[0126]

[0127] (2) Initialize the population using the dynamic programming method

[0128] Initialize the population, that is, the initial set of individuals, and each individual represents a potential solution to the problem. Use the dynamic programming method to initialize the population, and use the dynamic programming table dp[i][j] to record all combinations of selecting j positions among the first i positions. The process of initializing the population is as follows:

[0129] ① Initialize the DP table: dp[0][0] is [[]], indicating that the combination of selecting 0 positions among the first 0 positions is an empty list.

[0130] ② State transition process: For each position i and the number of selections j, dp[i][j] can be obtained by inheriting from dp[i - 1][j] (excluding the i-th position) and dp[i - 1][j - 1] (including the i-th position):

[0131] dp[i][j] = dp[i - 1][j] ∪ {comb + [i - 1] | comb ∈ dp[i - 1][j - 1]}

[0132] ③ Sample from the DP table: Randomly select a specified number of combinations from dp[K][p] as the initial population, and convert the selected combinations into individuals in the form of binary encodings of length K, representing the selected positions.

[0133] (3) Evaluate the fitness

[0134] Calculate the fitness function for each individual, which measures the quality of the individual solution. The fitness function is to maximize the minimum weighted distance, that is, the Maximin location model:

[0135] max{min{d(p i ,x j )}}

[0136] (4) Selection operation

[0137] Arrange in non - decreasing order according to the size of the fitness function of each individual, and then use the greedy selection method to select and determine whether an individual is selected as a parent in turn.

[0138] (5) Crossover operation

[0139] Generate new offspring individuals by combining the genes of two parent individuals in a certain way. The present invention uses a hybrid crossover method, performing crossover operations on the parents with single - point crossover, multi - point crossover in the first half, and simulated binary crossover in the second half. First, use single - point crossover to randomly select a crossover point in the parent individuals and exchange part of the genes of the parents; then use multi - point crossover in the first half to exchange genes before the crossover point to increase gene diversity; in the second half, use simulated binary crossover operation to generate new individuals. This method can effectively explore the solution space and improve the global search ability of the algorithm. This operation aims to retain excellent genes and introduce new changes to increase the diversity of the population.

[0140] (6) Mutation operation

[0141] Randomly change some gene positions of an individual with a certain probability, and use diversity - enhanced differential mutation to introduce the diversity of the population, which helps to avoid premature convergence to the local optimal solution. The specific operations are as follows:

[0142] ① Select three different individuals a, b, and c, where a, b, and c are all individuals with binary encoding; the calculation formula for the difference vector is:

[0143] d non-linear =b - a

[0144] ② Perform a non - linear transformation on the difference vector d:

[0145] d = sign(d)*(|d|) n

[0146] where sign(d) is the sign (+1 or - 1) of each element in the difference vector d, |d| is the absolute value of the difference vector d, and n is the power of the non - linear transformation.

[0147] ③ By adding the current individual's ind to the scaled difference vector F×d non-linearGenerate mutant individuals by addition:

[0148] mutant = ind + F × d non-linear

[0149] ④ To further enhance the diversity of the population, an additional random site inversion operation is applied to the generated mutant individuals. With a certain probability p flip perform the inversion operation. Select several random positions in the mutant individual and invert these positions (0 becomes 1, 1 becomes 0). The formula for the inversion operation is:

[0150] mutant[i] = 1 - mutant[i]

[0151] where i is the index of the randomly selected site, and mutant[i] is the value at the i-th position.

[0152] Through the crossover and mutation operations, the diversity of the population can be maintained, which helps to explore a wider area in the solution space, avoid falling into local optimal solutions, and finally obtain the optimal solution. The goal of the present invention is to maximize the minimum weighted distance to ensure the optimal facility coverage effect.

[0153] (7) Reward mechanism

[0154] Assign rewards to high-quality individuals using fitness ranking to motivate high-quality individuals and improve the quality of the population.

[0155] First, sort all individuals according to fitness. The sorted individual index is R, where R[1] is the optimal individual, R[2] is the second-best individual, and so on. The ranking r[i] represents the position of individual i, starting from 1. Secondly, assign a reward R[i] to the individual based on the ranking r[i]. The reward can be calculated by the following formula:

[0156]

[0157] That is, the higher the ranking, the greater the reward for the individual. Finally, during the population update process, use the reward mechanism to guide the generation of new individuals; and after generating a new individual, check whether the new individual already exists in the reward mechanism. If it exists, skip the update; if not, calculate the fitness of the individual and assign a reward to it. This can ensure that new individuals can also obtain certain rewards and promote the diversity of the population.

[0158] Step Five: Optimize the solution using a local search algorithm;

[0159] After using the improved genetic algorithm to select the optimal individuals, a local search algorithm is used to perform local optimization on the basis of the current solution by closing one facility and opening another. This strategy helps to find a better solution in the local area to improve the quality of the solution. The specific steps are as follows:

[0160] (1) Initialization: Randomly select an initial solution from the solution space of the problem.

[0161] (2) Local search: Generate new solutions in the neighborhood of the current solution; First, for each opened facility, try to close it and try to open a facility at other locations, and then calculate the fitness of the new solution. Secondly, check whether the newly added facilities meet the facility number limit. And check whether the distance between the selected facility locations meets certain distance requirements. If the distance requirements are met and the new solution is better than the current best solution, update the optimal solution.

[0162] (3) Loop iteration: Perform multiple local searches on the current best solution until no better solution can be found, and finally return the optimal solution optimized by local search.

[0163] In the process of using the local search algorithm to optimize the solution, the Maximin location model is used to determine the optimal solution, and in the process of determining the optimal solution, the distance between the position of the new solution and the existing facilities needs to be calculated and controlled not to be less than the minimum safety distance to ensure that there will be no greater impact between the facilities. If the position of the newly selected facility is less than the safety distance from the position of the existing facility, skip this position and select the next one until the required number of facility positions is selected.

[0164] Step 6: Draw an image;

[0165] Plot the community points, the positions of the existing facilities, the power diagram, and the selected facility positions in the image for display, making the results clearer and more understandable.

[0166] In the problem of locating aversive facilities, the present invention uses the Maximin location model in combination with the power diagram, the improved genetic algorithm, and the local search algorithm to search for the optimal location of the facilities; among them, the best facility location is determined by the Maximin location model, considering the dual objectives of maximizing the public interest and minimizing the adverse effects; the Power diagram, as a tool for calculating convex polyhedra, can accurately describe the proximity and influence range of each potential location, providing key data support for subsequent decision-making; the application of the improved genetic algorithm more effectively finds the global optimal solution in the complex search space and performs effective location optimization by simulating the evolution process; the local search algorithm further improves the accuracy of the solution and ensures a better solution in the local area.

[0167] Through visualization tools and scientific algorithms, the present invention improves the transparency and fairness of the decision-making process, enhances the public's acceptance of the siting decision; makes the location of the facility not only meet the technical and economic feasibility, but also minimizes the negative impacts on the environment and the community; this includes but is not limited to avoiding ecosystem damage, reducing noise and pollution, respecting local culture and community interests, etc. By comprehensively considering these factors, a siting method for aversive facilities based on the power diagram and improved genetic algorithm provided by the present invention ensures the fairness and sustainability of the decision-making through various optimization means, and has significant practical value and social benefits.

[0168] Relying on the technical solution of the present invention, the following further illustrates with specific embodiments:

[0169] Embodiment 1

[0170] In a small village with an area of about 25 square kilometers, at present, the village plans to build a garbage station to centrally process the domestic garbage generated in the village. There are 20 residential points distributed throughout the village, and these residential points constitute the main gathering areas of the village population. Since the area of the village is relatively small, once the garbage station is built, no matter where it is located, it will inevitably have an impact on these 20 residential points. Considering that the distance from each residential point to the garbage station is relatively close, and the possibility and degree of being affected are not very different, the weight value assigned to each residential point is between 10 and 20. The setting of this weight range is based on the following factors: on the one hand, the odor, noise and possible environmental pollution generated by the garbage station will cause a certain degree of negative impact on the daily life quality of residents, so the weight cannot be too low; on the other hand, since the degree of influence on each residential point is not particularly significant, the weight will not have a very large span.

[0171] Step 1: Quantify the basic data: First, quantify the degree of aversion of residents to the facility, randomly assign values between 10 and 20 as the weight values of the residential points, and perform data cleaning to remove invalid or incorrect data. The location distribution and weights of the residents in the small village are as Figure 3 shown;

[0172] Step 2: Construct the power diagram:

[0173] (1) In the overall area, regard the 20 residential points as the sites of the power diagram, and their corresponding weights as the site weight values;

[0174] (2) Adopt the embedded method to elevate the two-dimensional problem to the three-dimensional space to obtain the patches of the three-dimensional convex hull, and then map it back to the two-dimensional space to obtain the power diagram boundary. In the scenario of this small village, use mathematical algorithms to construct a convex hull of the residential points in the three-dimensional space, and then project it back to the two-dimensional plane to determine the preliminary boundary of the power diagram;

[0175] (3) Since the village area has boundaries, symmetric transformation is performed on the boundary points to generate symmetric points. The symmetric points are combined with the original point set, and the Power diagram is recalculated to ensure that all patches are within the boundaries. For example, for the residential points near the edge of the village, symmetric points are generated outside the boundary through symmetric transformation, and then these symmetric points are added to the original point set. The Power diagram is recalculated again to make the obtained Power diagram completely within the village area;

[0176] (4) Calculate the angle of each patch and sort the vertices, and obtain the vertices of the power diagram as the candidate positions for the noxious facilities. Through the mathematical analysis of the Power diagram, a series of vertices are obtained, and these vertices are the possible construction positions of the garbage station, as Figure 4 shown;

[0177] Step 3: Calculate the minimum weighted distance: For each vertex of the power diagram, calculate its minimum weighted distance to the nearest residential point;

[0178] Step 4: Use the improved genetic algorithm to search for the optimal location of the facility;

[0179] (1) Encoding: Encode the vertices of the power diagram, and each encoding represents a possible garbage station location selection scheme.

[0180] (2) Fitness function: Use the minimum weighted distance as the fitness function. The smaller the distance, the higher the fitness.

[0181] (3) Improved genetic operations: Through genetic operations such as selection, crossover, and mutation, continuously iterate and optimize to find the location selection scheme corresponding to the minimum weighted distance.

[0182] Step 5: After selecting the optimal individual by the improved genetic algorithm, use the local search algorithm to perform local optimization on the basis of the current solution by closing one facility and then opening another facility, and find a better solution within the local area. In this small village, conduct a detailed search in the area near the optimal solution obtained by the genetic algorithm, try different location selection combinations, and find a better solution within the local area;

[0183] Step 6: Draw the image: Plot the selected facility locations and residential points in the image, and visually display the village map, residential points, and the finally determined garbage station location selection to obtain the final location selection result map, providing a clear visual basis for decision-making. The finally selected facility location in this village is (35.9, 4.4), as Figure 5 .

[0184] Please refer to Figure 5, which is the schematic diagram of the final site selection result of this embodiment. One facility location is selected in a village with 20 residential points. The black points are residential points, the red points are the vertices of the power diagram, the black lines are the unit boundaries of the power diagram, and the blue X is the finally selected location of the disliked facility.

[0185] Embodiment 2

[0186] In a medium-sized town with an area of about 100 square kilometers, there are 50 community points distributed throughout the town, and these communities form the main gathering areas of the town's population. In order to achieve centralized and effective treatment of garbage and alleviate the environmental pollution and health hazards caused by random garbage stacking, the town plans to build two garbage stations. At the same time, considering introducing a chemical plant to promote the diversified development of the industry and drive economic growth. However, during the operation of the garbage stations and the chemical plant, problems such as odor, noise, and sewage discharge may occur, which will have a negative impact on the quality of life of the surrounding community residents. Therefore, the weight values assigned to each community point are between 1 and 30.

[0187] Step 1: Quantify the basic data: First, quantify the degree of disgust of residents towards facilities. Randomly assign values from 1 to 30 for each community as the weight value of the community point, and perform data cleaning to remove invalid or incorrect data. The community location distribution and weights are as Figure 6 shown;

[0188] Step 2: Construct the power diagram:

[0189] (1) In the overall area, the 50 community points are regarded as the sites of the power diagram, and their corresponding weights are used as the site weight values;

[0190] (2) Adopt the embedding method to lift the two-dimensional problem to three-dimensional space to obtain the patches of the three-dimensional convex hull, and then map it back to two-dimensional space to obtain the power diagram boundary. In the scenario of this medium-sized town, use a mathematical algorithm to construct a convex hull of the community points in three-dimensional space, and then project it back to the two-dimensional plane to determine the preliminary boundary of the power diagram;

[0191] (3) Since the town area has boundaries, perform symmetric transformation on the boundary points to generate symmetric points, combine the symmetric points with the original point set, and recalculate the Power diagram to ensure that all patches are within the boundary. For example, for the community points near the town edge, generate symmetric points outside the boundary through symmetric transformation, and then add these symmetric points to the original point set and recalculate the Power diagram to make the obtained Power diagram completely within the town area;

[0192] (4) Calculate the angle of each patch and sort the vertices to obtain the vertices of the power diagram as candidate locations for the noxious facilities. Through the mathematical analysis of the power diagram, a series of vertices are obtained, which are the possible construction locations of the waste station, as Figure 7 shown;

[0193] Step 3: Calculate the minimum weighted distance: For each vertex of the power diagram, calculate its minimum weighted distance to the nearest community point;

[0194] Step 4: Use the improved genetic algorithm to search for the optimal location of the facility;

[0195] (1) Encoding: Encode the vertices of the power diagram, and each encoding represents a possible waste station location selection scheme.

[0196] (2) Fitness function: Use the minimum weighted distance as the fitness function. The smaller the distance, the higher the fitness.

[0197] (3) Improved genetic operations: Through genetic operations such as selection, crossover, and mutation, continuously iterate and optimize to find the location selection scheme corresponding to the minimum weighted distance.

[0198] Step 5: After selecting the optimal individual using the improved genetic algorithm, use the local search algorithm to perform local optimization on the basis of the current solution by closing one candidate facility and then opening another candidate facility, and find a better solution within the local area. In this medium-sized town, conduct a detailed search in the area near the optimal solution obtained by the genetic algorithm, try different location selection combinations, and find a better solution within the local area;

[0199] Step 6: Draw an image: Plot the selected facility locations and community points in the image, visually display the town community points and the finally determined waste station location selection, obtain the final location selection result map, and provide a clear visual basis for decision-making. The hazards of factory pollutants are large and easy to accumulate in the long term, and the threat to health far exceeds that of the landfill. Therefore, the finally selected factory location is farther from the community than the landfill. The location is (92.1, 94.5), and the landfill locations are (71.8, 39.4) and (29.2, 94.5), as Figure 8 .

[0200] Please refer to Figure 8 , for the final location selection result schematic diagram of this embodiment. Three facility locations are selected in a medium-sized town with 50 communities. The black dots are community points, the red dots are the vertices of the power diagram, the black lines are the unit boundaries of the power diagram, and the blue Xs are the finally selected noxious facility locations.

[0201] Embodiment 3

[0202] In a large town with an area of approximately 150 square kilometers, there are 100 community points distributed throughout the town, and these communities constitute the main gathering areas of the town's population. Currently, the town plans to introduce a total of 3 waste landfills, power plants, factories and other disliked facilities, and the weight values assigned to each community point range from 1 to 30.

[0203] Step 1: Quantify basic data: First, quantify the degree of dislike of residents for facilities. Randomly assign values from 1 to 30 for each community as the weight value of the community point. The community location distribution and weights are as Figure 9 shown;

[0204] Step 2: Construct a power diagram:

[0205] (1) In the overall area, the 100 community points are regarded as the sites of the power diagram, and their corresponding weights are used as the site weight values;

[0206] (2) Adopt an embedding method to elevate the two-dimensional problem to three-dimensional space to obtain the faces of the three-dimensional convex hull, and then map back to two-dimensional space to obtain the power diagram boundary. In the scenario of this large town, use a mathematical algorithm to construct a convex hull for the community points in three-dimensional space, and then project back to the two-dimensional plane to determine the preliminary boundary of the power diagram;

[0207] (3) Since the town area has boundaries, perform a symmetry transformation on the boundary points to generate symmetric points, combine the symmetric points with the original point set, and recalculate the power diagram to ensure that all faces are within the boundary. For example, for community points near the town edge, generate symmetric points outside the boundary through symmetry transformation, then add these symmetric points to the original point set, and recalculate the power diagram to make the obtained power diagram completely within the town area;

[0208] (4) Calculate the angle of each face and sort the vertices to obtain the vertices of the power diagram as candidate locations for disliked facilities. Through mathematical analysis of the power diagram, a series of vertices are obtained, and these vertices are the possible construction locations of the facilities, as Figure 10 shown;

[0209] Step 3: Calculate the minimum weighted distance: For each vertex of the power diagram, calculate its minimum weighted distance to the nearest community point;

[0210] Step 4: Use an improved genetic algorithm to search for the optimal location of the facilities;

[0211] (1) Encoding: Encode the vertices of the power diagram, and each encoding represents a possible waste landfill siting scheme.

[0212] (2) Fitness function: Use the minimum weighted distance as the fitness function. The smaller the distance, the higher the fitness.

[0213] (3) Improve genetic operations: Through genetic operations such as selection, crossover, and mutation, continuously iterate and optimize to find the site selection scheme corresponding to the minimum weighted distance.

[0214] Step Five: After the improved genetic algorithm selects the optimal individual, use the local search algorithm to perform local optimization on the basis of the current solution by closing one candidate facility and then opening another candidate facility, and find a better solution in the local area. In this large town, conduct a detailed search in the area near the optimal solution obtained by the genetic algorithm, try different site selection combinations, and find a better solution in the local area;

[0215] Step Six: Draw an image: Plot the selected facility locations and community points in the image to visually display the town community points and the finally determined waste station site selection locations, and obtain the final site selection result map, providing a clear visual basis for decision-making, such as Figure 11 。

[0216] Please refer to Figure 11 , which is the schematic diagram of the final site selection result of this embodiment. Three facility locations are selected in a large town with 100 communities. The black dots are community points, the red dots are the vertices of the power diagram, the black lines are the cell boundaries of the power diagram, and the blue Xs are the finally selected locations of the aversion facilities.

[0217] The present invention may have other forms of embodiments according to the above method, which will not be listed one by one. Therefore, any person skilled in the art, without departing from the scope of the technical solution of the present invention, any simple modification, equivalent change and modification made to the above embodiments based on the technical essence of the present invention shall still fall within the scope of the technical solution of the present invention.

Claims

1. A method for locating aversive facilities based on the power diagram and improved genetic algorithm, characterized in that It includes the following steps: Step 1. Quantify the basic data: Quantify the degree of residents' aversion to facilities, randomly assign values according to the levels of "like", "slightly like", "average", "slightly dislike", "dislike", and "strongly dislike" as the weight values of community locations, and perform data cleaning to remove invalid or incorrect data; Step 2. Construct a power diagram: (1) In the overall area, regard the community points as the stations of the power diagram, and the corresponding population data values as the station weight values; (2) Adopt an embedding method to elevate the two-dimensional problem to three-dimensional space to obtain the facets of the three-dimensional convex hull, and then map it back to two-dimensional space to obtain the power diagram boundary; (3) For the case with a boundary, generate symmetric points through symmetric transformation of the boundary points, combine the symmetric points with the original point set, and recalculate the power diagram to ensure that all facets are within the boundary; (4) Calculate the angles of each facet and sort the vertices to obtain the power diagram vertices as the candidate locations for the aversion-type facilities; Step 3. Calculate the minimum weighted distance: For each vertex of the power diagram, calculate its minimum weighted distance to the nearest facility; Step 4. Use an improved genetic algorithm to search for the optimal location of the facility; Step 5. After selecting the optimal individual using the improved genetic algorithm, use a local search algorithm to perform local optimization on the basis of the current solution by closing one facility and opening another facility to find a better solution within the local area; Step 6. Draw an image: Draw the community points, the existing facility locations, the power diagram, and the selected facility locations in the image to obtain the final site selection result diagram.

2. The aversive facility location method based on power diagram and improved genetic algorithm according to claim 1, characterized in that, In step two, each area of the Power diagram corresponds to a site x i , and the set of candidate locations of aversive facilities in this area is and define the weighted distance d between point V and the community i as: For each point i and its weight w i , the cell of the Power diagram is considered as the region of all points j that satisfy the following conditions: where (x i , x j ) are the coordinates of point i, j, w i and w j are their weights.

3. The aversive facility location method based on the power diagram and the improved genetic algorithm according to claim 2, wherein The specific process of elevating to three-dimensional space in Step 2 is as follows: ① Extended Coordinate Matrix: Given a two-dimensional point S = {(x i , y i ),} and a weight w i , each point (x i , y i ) is extended to three-dimensional space: is the third-dimensional coordinate of a point in three-dimensional space, and the calculation method is based on the definition of the Power diagram, with weights introduced into the calculation; ② Calculate the convex hull: Calculate the three-dimensional convex hull of the extended point set; The convex hull is the smallest convex polyhedron that contains all points; In three-dimensional space, each facet of the convex hull represents the boundary of the power diagram generated in the original two-dimensional space; The specific process of mapping the three-dimensional convex hull facets back to two-dimensional space is as follows: ③ Classify the facets: The three-dimensional convex hull generates some facets, and these facets are divided into two categories: Lower-half facets: Located in the lower half of the convex hull, representing the effective power diagram area; Upper-half facets: Located in the upper half of the convex hull, not belonging to the effective power diagram area, and need to be removed; ④ Map the facets back to two-dimensional space: For each effective facet, map it back to two-dimensional space through polar coordinate transformation to obtain the power diagram boundary.

4. The aversive facility location method based on the power diagram and the improved genetic algorithm as claimed in claim 3, wherein The formula for calculating the minimum weighted distance in Step 3 is as follows:

5. The aversive facility location method based on the power diagram and the improved genetic algorithm according to claim 4, wherein The process of using the improved genetic algorithm to search for the optimal location in Step 4 is as follows: (1) Gene encoding and decoding: Adopt binary encoding, implement the encoding function encode and the decoding function decode, and convert the selected intersection positions and binary encoding to each other; (2) Initialize the population using dynamic programming: Use the dynamic programming table dp[i][j] to record all combinations of selecting j positions from the first i positions, randomly select a combination as the initial population and convert it into an individual in binary encoding form; (3) Evaluate the fitness: Calculate the fitness function of each individual, and measure the quality of the individual solution with the goal of maximizing the minimum weighted distance; (4) Selection operation: Arrange in non-decreasing order according to the magnitude of the fitness function, and then use the greedy selection method to sequentially select and determine whether an individual is selected as a parent; (5) Crossover operation: Combine the genes of two parent individuals using a hybrid crossover method to generate new offspring individuals. The hybrid crossover method includes single-point crossover, multi-point crossover in the first half, and simulated binary crossover in the second half; (6) Mutation operation: Randomly change some gene positions of an individual with a certain probability, and use diversity-enhanced differential mutation to introduce the diversity of the population, which helps to avoid premature convergence to a local optimal solution; (7) Reward mechanism: Allocate rewards to high-quality individuals using fitness ranking, thereby motivating high-quality individuals and improving the quality of the population.

6. The aversive facility location method based on the power diagram and the improved genetic algorithm as claimed in claim 5, wherein, The specific process of initializing the population using the dynamic programming method is as follows: ① Initialize the DP table: dp[0][0] is [[]], indicating that the combination of selecting 0 positions among the first 0 positions is an empty list; ② State transition process: For each position i and the number of selections j, dp[i][j] is obtained by inheriting from dp[i - 1][j] (excluding the i-th position) and dp[i - 1][j - 1] (including the i-th position): dp[i][j] = dp[i - 1][j] ∪ {comb + [i - 1] | comb ∈ dp[i - 1][j - 1]} ③ Sample from the DP table: Randomly select a specified number of combinations from dp[K][p] as the initial population, and convert the selected combinations into individuals in the form of binary encoding with a length of K, indicating the selected positions.

7. The aversive facility location method based on power diagram and improved genetic algorithm according to claim 6, characterized in that, The Maximin site selection model for evaluating fitness is: max{min{d(p i ,x j )}}。 8. The aversive facility location method based on the power diagram and the improved genetic algorithm according to claim 7, characterized in that, The specific process of the mutation operation is as follows: ① Select three different individuals a, b, and c, where a, b, and c are all individuals in binary encoding; the calculation formula for the difference vector is: d non-linear = b - a ② Perform a non-linear transformation on the difference vector d: d = sign(d) * (|d|) n where sign(d) is the sign (+1 or -1) of each element in the difference vector d, |d| is the absolute value of the difference vector d, and n is the power of the non-linear transformation; ③ By adding the ind of the current individual to the scaled difference vector F×d non-linear to generate a mutant individual: mutant = ind + F×d non-linear ④ To further enhance the diversity of the population, an additional random site inversion operation is applied to the generated mutant individuals; with a certain probability p flip perform the inversion operation; select several random positions in the mutant individuals and invert these positions: 0 becomes 1, 1 becomes 0; the formula for the inversion operation is: mutant[i] = 1 - mutant[i] where i is the index of the randomly selected locus, and mutant[i] is the value at the i-th position.

9. The aversive facility location method based on the power diagram and the improved genetic algorithm as described in claim 8, wherein The specific process of the reward mechanism is as follows: First, sort all individuals according to fitness. The sorted individual indices are R, where R[1] is the optimal individual, R[2] is the second-best individual, and so on; the ranking r[i] represents the position of individual i, with the value starting from 1; Secondly, allocate rewards R[i] to individuals based on the ranking r[i]. The higher the ranking, the greater the reward; the reward is calculated by the following formula: Finally, during the population update process, use the reward mechanism to guide the generation of new individuals; and after generating a new individual, check whether the new individual already exists in the reward mechanism; if it exists, skip the update; if it does not exist, calculate the fitness of the individual and allocate a reward to it.

10. A method for locating aversion facilities based on power diagram and improved genetic algorithm as claimed in claim 1, characterized in that, The specific process of the local search algorithm in step five is as follows: (1) Initialization: Randomly select an initial solution from the solution space of the problem; (2) Local search: For each opened facility, try to close it and open a facility at another location, and calculate the fitness of the new solution; check the facility number limit and the distance requirement between facilities. If the requirements are met and the new solution is better, then update the optimal solution. Iterative loop: Conduct multiple local searches on the current optimal solution until no better solution can be found, and finally return the optimized optimal solution.

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