School site selection optimization method based on robust adaptive differential evolution algorithm

Abstract

The application discloses a school site selection optimization method based on a robust adaptive differential evolution algorithm, and comprises the following steps: constructing a school site selection optimization model; proposing a dynamic parameter adaptive strategy, recording the speed of updating the optimal solution of each group parameter combination, calculating the selection probability weight, and realizing the collaborative adaptive adjustment of the mutation factor and the crossover probability; designing an acceptance strategy based on the average distance of historical successful individuals; and obtaining a school site selection scheme with the most robust and shortest total commuting distance. The application not only breaks through the limitation that the existing school site selection model can only be planned in a deterministic environment, but also effectively reduces the dependence of the algorithm on artificial parameter adjustment and improves the automation degree and reliability of solving complex real problems through the adaptive parameter mechanism and the stagnation processing strategy considering historical information, so that a quantifiable site selection scheme is provided for coping with population fluctuation and other actual uncertainties.

Description

Technical Field

[0001] This invention relates to the field of urban planning technology, and in particular to a school site selection optimization method based on a robust adaptive differential evolution algorithm. Background Technology

[0002] With the acceleration of urbanization, the school locations determined by traditional site selection problems often have static characteristics. Because they fail to fully consider the uncertainties in real-world scenarios, they are only applicable to the existing environment. For example, fluctuations in student enrollment due to population migration and changes in commuting distances caused by road network modifications can significantly affect the efficiency of school layout.

[0003] Traditional site selection problems primarily focus on determining the optimal school location and student allocation scheme within a given environment, aiming to minimize student commuting distances and balance school capacity. Therefore, developing site selection models that comprehensively consider environmental uncertainties is of crucial practical significance.

[0004] In recent years, research on uncertainties in various application scenarios has increased. For example, studies have focused on the impact of environmental changes in the coming years on the site selection and layout of medical facilities, and on the layout planning of emergency facilities in the context of natural disasters such as earthquakes, floods, and landslides. However, regrettably, there is still a significant gap in current research on site selection under conditions of actual uncertainty. In particular, the changes in birth rates and adjustments in population migration patterns caused by urbanization result in unpredictable fluctuations in student numbers over time. At the same time, road modifications and transportation network updates in urban planning also lead to changes in students' actual commuting distances. Therefore, school site selection and layout decisions must consider both student numbers and commuting distances as uncertainties. However, existing research still lacks systematic and in-depth discussions on school layout optimization under the dual uncertainties of student number fluctuations and commuting distance changes, indicating a significant gap in academic research in this field. Summary of the Invention

[0005] To address the aforementioned technical problems, this invention provides a school site selection optimization method based on a robust adaptive differential evolution algorithm that is simple in algorithm and has good robustness.

[0006] The technical solution of this invention to solve the above-mentioned technical problems is: a school site selection optimization method based on a robust adaptive differential evolution algorithm, comprising the following steps:

[0007] Step 1: Construct a school site selection optimization model, and then use the budget uncertainty set to characterize the fluctuations in student numbers and commuting distances to establish a robust model that can be linearized;

[0008] Step 2: Propose a dynamic parameter adaptive strategy. By recording the speed at which each set of parameter combinations updates the optimal solution, calculate the selection probability weight, and achieve coordinated adaptive adjustment of the mutation factor and crossover probability. At the same time, introduce a parameter scaling mechanism based on the algorithm execution stage to enhance local search in the early stage of iteration and promote global exploration in the later stage of iteration.

[0009] Step 3: Design an acceptance strategy based on the average distance between historically successful individuals. In the selection phase, not only fitness is considered, but also the average distance between individuals who have successfully updated the optimal solution in the past is used as a reference. The probability of accepting the suboptimal solution is set to overcome the premature convergence of the population.

[0010] Step 4: Obtain the most robust school location plan with the shortest total commute distance.

[0011] The specific process of step 1 in the above-mentioned school site selection optimization method based on robust adaptive differential evolution algorithm is as follows:

[0012] Step 11: Obtain data on student locations, number of students, candidate school locations, and school capacity. Under the constraints of school capacity and student allocation, establish a school location optimization model with the goal of minimizing the total commuting distance of all students.

[0013] Step 12: Considering the actual uncertainty of student numbers and commuting distance, introduce a budget uncertainty set to describe parameter fluctuations, and establish a robust optimization model based on this.

[0014] Step 13: Transform the robust optimization model into a solvable linear programming model using linear duality theory.

[0015] In the aforementioned school location optimization method based on robust adaptive differential evolution algorithm, the school location optimization model in step 11 is as follows:

[0016] ;

[0017] In the formula, Indicates from student position Assigned to school The number of students; Indicates the student's location index; Number of student seats; For the number of schools; Indicates a school index; Indicates from student position to school Commute distance; For student positions School The decision variable is the student's location. Students attend school ,but =1, otherwise =0; This is the function to be minimized.

[0018] In the aforementioned school site selection optimization method based on robust adaptive differential evolution algorithm, step 12 includes a budget uncertainty set comprising a student number uncertainty set. and the uncertain set of commuting distance , , ; This represents the deviation between the actual and nominal number of students at each student's location from the school. For student positions to school The deviation between the actual and nominal number of students; This represents the deviation between the actual and nominal distances from a student's location to the school. For student positions to school The deviation between the actual and nominal values ​​of the distance; This represents the upper limit of the total deviation in the number of students in the school, reflecting the degree of conservatism regarding the uncertainty of the number of students in the school. This represents the upper limit of the total deviation of the distance from the student's location to the school, reflecting the degree of conservatism regarding the uncertainty of the distance from the student's location to the school.

[0019] Based on this, a robust optimization model is established:

[0020] ;

[0021] In the formula, Representative student position to school The fluctuation range of the number of students; Representing the position of non-students to school The range of fluctuation in distance; This is the maximization function.

[0022] In the aforementioned school site selection optimization method based on robust adaptive differential evolution algorithm, the linear programming model in step 13 is as follows:

[0023] ;

[0024] In the formula, In order to prevent The marginal cost required to pay in the event of adverse fluctuations; In order to prevent The marginal cost required to pay in the event of adverse fluctuations; The dual variable representing the uncertainty of the corresponding number of people; This represents the dual variable corresponding to the uncertainty of distance.

[0025] The specific process of step 2 in the above-mentioned school site selection optimization method based on robust adaptive differential evolution algorithm is as follows:

[0026] Step 21: Construct the candidate parameter matrix ;

[0027] Step 22: Record the performance of parameter combinations and introduce update speed as an evaluation metric. ; For the first The update speed of parameter combinations; For the first The total number of iterations required to update the current optimal solution using a combination of parameters; For the first The number of times each parameter combination is selected; Index for parameter combinations;

[0028] Step 23: Convert the update rate into selection probability, the expression is:

[0029] ;

[0030] in, For the first The probability of choosing a combination of parameters;

[0031] Normalize the weights of all parameter combinations to obtain the first... The actual probability of a parameter combination being selected The expression is:

[0032] ;

[0033] in, Index for parameter combinations; This represents the total number of parameter combinations.

[0034] Step 24: When a parameter combination is not selected within a set time, it is determined to be a parameter combination that updates slowly, and the parameter combination that updates slowly is subjected to active random perturbation.

[0035] Step 25: Combine the parameter fine-tuning during the algorithm execution phase. When the number of fitness evaluations (NFE) has not reached half of the total number of evaluations, the selected [number]th [number] ... Variation factors in parameter combinations Temporarily reduced to 0.8 Perform a fine-grained local search; when the number of fitness evaluations (NFE) exceeds half of the total number of evaluations, to avoid getting trapped in local optima, the selected [fitness] will be [selected / degraded]. Temporarily magnified to 1.1 .

[0036] In the aforementioned school site selection optimization method based on robust adaptive differential evolution algorithm, step 21 involves the candidate parameter matrix. The mathematical expression is:

[0037] ;

[0038] In the formula, Represents the total number of parameter combinations; Representing the The variation factor in a combination of parameters; Representing the The variation factor in a combination of parameters; Representing the Crossover probability in a combination of parameters; Representing the The crossover probability in a combination of parameters.

[0039] In the aforementioned school site selection optimization method based on robust adaptive differential evolution algorithm, step 22 involves the performance of parameter combinations through... and To reflect, This is the set of times each parameter combination was selected. The set of total iterations required to update the current optimal solution for each parameter combination. , , Indicates the first The number of times each parameter combination is selected; Indicates the first The number of times each parameter combination is selected; Indicates the first The total number of iterations required to update the current optimal solution using a combination of parameters; Indicates the first The total number of iterations required to update the current optimal solution using a combination of parameters.

[0040] In the aforementioned school site selection optimization method based on robust adaptive differential evolution algorithm, the formula for the random perturbation in step 24 is:

[0041] ;

[0042] In the formula, Representing the The variation factor in a combination of parameters; Representing the The variation factor in a combination of parameters; Representing the Crossover probability in a combination of parameters; Represents the total number of parameter combinations; To The result after applying active random perturbation; Representing the Crossover probability in a combination of parameters; To The result after applying active random perturbation; This is a random disturbance term.

[0043] In the aforementioned school site selection optimization method based on robust adaptive differential evolution algorithm, in step 3, after each generation iteration, it is checked whether the current global optimal solution has been updated. If the objective function value of the global optimal solution has not improved for N consecutive generations, the algorithm is determined to have entered a stagnant state. Then, an acceptance probability is dynamically calculated based on the average distance between historically successfully updated individuals and the population center to determine whether to accept a suboptimal solution. The accepted suboptimal solution replaces the worse individuals in the current population, thereby increasing population diversity.

[0044] The beneficial effects of this invention are as follows:

[0045] 1. This invention not only breaks through the limitation of existing school site selection models that can only be planned in deterministic environments, but also effectively reduces the algorithm's dependence on manual parameter tuning and improves the automation and reliability of solving complex real-world problems through its adaptive parameter mechanism and stagnation processing strategy that takes into account historical information. It provides a quantifiable site selection scheme for dealing with real uncertainties such as population fluctuations.

[0046] 2. This invention proposes a dynamic parameter adaptive strategy, avoiding the problems of traditional fixed parameters. This strategy not only dynamically adjusts the selection probability based on the speed of parameter update to the optimal solution, but also regenerates the parameters with slow updates by introducing average deviation and random perturbation, and fine-tunes the mutation factor in conjunction with the algorithm execution phase, thereby achieving a more refined balance between global and local search. Finally, in terms of maintaining population diversity, the stagnation handling mechanism is improved by using the average distance of historically successfully updated individuals instead of the single distance to calculate the probability of accepting the superior individual, thereby helping the algorithm escape local optima. These improvements make the obtained school facility site selection scheme more reasonable and of higher quality. Attached Figure Description

[0047] Figure 1 This is the overall flowchart of the present invention.

[0048] Figure 2 Heatmaps of average fitness values ​​for different population sizes and number of iterations.

[0049] Figure 3 Heatmaps of standard deviations for different population sizes and number of iterations.

[0050] Figure 4 This is a line graph showing the average fitness values ​​of the present invention and other algorithms in six scenarios.

[0051] Figure 5 This is a line graph showing the standard deviation of the present invention and other algorithms in six scenarios. Detailed Implementation

[0052] The present invention will be further described below with reference to the accompanying drawings and embodiments.

[0053] like Figure 1 As shown, the school site selection optimization method based on the robust adaptive differential evolution algorithm includes the following steps:

[0054] Step 1: Construct a school site selection optimization model, and then use the budget uncertainty set to characterize the fluctuations in student numbers and commuting distances to establish a robust model that can be linearized.

[0055] The specific process of step 1 is as follows:

[0056] Step 11: Obtain data on student locations, number of students, candidate school locations, and school capacity. Under the constraints of school capacity and student allocation, establish a school location optimization model with the objective of minimizing the total commuting distance for all students. The school location optimization model is as follows:

[0057] ;

[0058] In the formula, Indicates from student position Assigned to school The number of students; Indicates the student's location index; Number of student seats; For the number of schools; Indicates a school index; Indicates from student position to school Commute distance; For student positions School The decision variable is the student's location. Students attend school ,but =1, otherwise =0; This is the function to be minimized.

[0059] Step 12: Considering the actual uncertainties in the number of students and commuting distance, introduce a budget uncertainty set to describe parameter fluctuations;

[0060] The budget uncertainty set includes the student number uncertainty set. and the uncertain set of commuting distance , , ; This represents the deviation between the actual and nominal number of students at each student's location from the school. For student positions to school The deviation between the actual and nominal number of students; This represents the deviation between the actual and nominal distances from a student's location to the school. For student positions to school The deviation between the actual and nominal values ​​of the distance; This represents the upper limit of the total deviation in the number of students in the school, reflecting the degree of conservatism regarding the uncertainty of the number of students in the school. This represents the upper limit of the total deviation of the distance from the student's location to the school, reflecting the degree of conservatism regarding the uncertainty of the distance from the student's location to the school.

[0061] Based on this, a robust optimization model is established:

[0062] ;

[0063] In the formula, Representative student position to school The fluctuation range of the number of students; Representing the position of non-students to school The range of fluctuation in distance; This is the maximization function.

[0064] Step 13: Transform the robust optimization model into a solvable linear programming model using linear duality theory.

[0065] The linear programming model is as follows:

[0066] ;

[0067] In the formula, In order to prevent The marginal cost required to pay in the event of adverse fluctuations; In order to prevent The marginal cost required to pay in the event of adverse fluctuations; The dual variable representing the uncertainty of the corresponding number of people; This represents the dual variable corresponding to the uncertainty of distance.

[0068] Step 2: Propose a dynamic parameter adaptive strategy. By recording the speed at which each set of parameter combinations updates the optimal solution, calculate the selection probability weight, and achieve coordinated adaptive adjustment of the mutation factor and crossover probability. At the same time, introduce a parameter scaling mechanism based on the algorithm execution stage to enhance local search in the early stage of iteration and promote global exploration in the later stage of iteration.

[0069] The specific process of step 2 is as follows:

[0070] Step 21: Construct the candidate parameter matrix Select parameter matrix The mathematical expression is:

[0071] ;

[0072] In the formula, Represents the total number of parameter combinations; Representing the The variation factor in a combination of parameters; Representing the The variation factor in a combination of parameters; Representing the Crossover probability in a combination of parameters; Representing the The crossover probability in a combination of parameters.

[0073] Step 22: Record the performance of the parameter combinations. The performance of the parameter combinations is achieved through... and To reflect, This is the set of times each parameter combination was selected. The set of total iterations required to update the current optimal solution for each parameter combination. , , Indicates the first The number of times each parameter combination is selected; Indicates the first The number of times each parameter combination is selected; Indicates the first The total number of iterations required to update the current optimal solution using a combination of parameters; Indicates the first The total number of iterations required to update the current optimal solution using a combination of parameters is considered; and the update rate is introduced as an evaluation metric. ; For the first The update speed of parameter combinations; For the first The total number of iterations required to update the current optimal solution using a combination of parameters; For the first The number of times each parameter combination is selected; Index for parameter combinations; The smaller the value, the higher the search efficiency of the parameter combination.

[0074] Step 23: To make the algorithm more inclined to choose efficient parameter combinations, the update speed is converted into selection probability, expressed as:

[0075] ;

[0076] in, For the first The probability of choosing a combination of parameters;

[0077] Normalize the weights of all parameter combinations to obtain the first... The actual probability of a parameter combination being selected The expression is:

[0078] ;

[0079] in, Index for parameter combinations; This represents the total number of parameter combinations.

[0080] This operation ensures that the probability of selecting a parameter combination that performs well will dynamically increase, while the probability of selecting a parameter combination that performs poorly will decrease.

[0081] Step 24: When a parameter combination is not selected within a set time, it is determined to be a slow-updating parameter combination. The slow-updating parameter combination is then subjected to active random perturbation; the formula for random perturbation is:

[0082] ;

[0083] In the formula, Representing the The variation factor in a combination of parameters; Representing the The variation factor in a combination of parameters; Representing the Crossover probability in a combination of parameters; Represents the total number of parameter combinations; To The result after applying active random perturbation; Representing the Crossover probability in a combination of parameters; To The result after applying active random perturbation; This is a random disturbance term.

[0084] Step 25: Combine the parameter fine-tuning during the algorithm execution phase. When the number of fitness evaluations (NFE) has not reached half of the total number of evaluations, the selected [number]th [number] ... Variation factors in parameter combinations Temporarily reduced to 0.8 Perform a fine-grained local search; when the number of fitness evaluations (NFE) exceeds half of the total number of evaluations, to avoid getting trapped in local optima, the selected [fitness] will be [selected / degraded]. Temporarily magnified to 1.1 .

[0085] Step 3: Design an acceptance strategy based on the average distance between historically successful individuals. In the selection phase, in addition to fitness, the average distance between individuals who have successfully updated the optimal solution in the past is used as a reference to set a probability to accept the suboptimal solution, thereby overcoming premature convergence of the population.

[0086] After each iteration, it is checked whether the current global optimum has been updated. If the objective function value of the global optimum has not improved for N consecutive generations, the algorithm is determined to have entered a stagnant state. Then, an acceptance probability is dynamically calculated based on the average distance between the historically successfully updated individuals and the population center to decide whether to accept a suboptimal solution. The accepted suboptimal solution replaces the worse individuals in the current population, thereby increasing population diversity.

[0087] Step 4: Obtain the most robust school location plan with the shortest total commute distance.

[0088] To verify that the method proposed in this invention can solve the school site selection problem under uncertain conditions, real-world case data was used as an example dataset for experiments. This dataset contains... The dataset comprises six student residential areas, each containing three types of information: longitude, latitude, and the number of students residing in that area. The dataset is divided into six experimental scenarios: G1 (…). G2 ), G3 ( ), G4 ), G5 ) and G6 ( ),in This indicates the number of student residential areas. The specific data format is shown in Table 1.

[0089] Table 1 Dataset Format

[0090]

[0091] The data is input into the robust optimization model, and the minimum objective value in the model is used to measure the robustness of the solution. The minimum objective value is represented by RE. The smaller the RE, the stronger the robustness of the solution and the better it performs in uncertain environments. Conversely, the larger the RE, the weaker the robustness.

[0092]

[0093] To verify the effectiveness of robust modeling and decision-making, the robust adaptive differential evolution algorithm (SaDE-RD) was compared with the following benchmark algorithms: Differential Evolution (DE), Reinforcement Learning-Based and Adaptive Parameter Adjustment-Based Differential Evolution (RLAPMDE), Adaptive Parameter Adjustment-Based Differential Evolution (AR-aDE), Mixed Distance-Based Differential Evolution (HDDE), Adaptive Parameter Controlled Differential Evolution (jDE), Adaptive Differential Evolution (JADE), Success History Parameter Adaptive Differential Evolution (SHADE), Algorithm Combining Differential Evolution with Covariance Matrix Adaptive Evolution Strategy (DE / CMA-ES), and Adaptive Weighted Particle Swarm Optimization (AWPSO). Thirty solutions generated by each algorithm were input into the uncertainty model. and Under the given conditions, the RE value of each solution was calculated, and the RE values ​​of each algorithm under different scenarios were obtained, as shown in Table 2.

[0094] Table 2. RE values ​​of various algorithms in different scenarios

[0095]

[0096] As shown in Table 2, under a certain level of uncertainty, the solutions obtained by SaDE-RD exhibit better robustness in most cases. This is because the proposed robust decision-making method further filters solutions based on their robustness, ensuring that the optimal solution has good robustness performance under uncertain environments.

[0097] Since population size and number of iterations significantly affect the final performance of the algorithm, four population sizes (50, 100, 200, 400) and four iteration numbers (100, 200, 400, 800) are predefined. Sixteen parameter configurations are generated by pairing these parameters, and the average fitness, average standard deviation, and running time are calculated. The average fitness represents the overall home-school distance; a smaller value indicates a better solution. The average standard deviation reflects the algorithm's stability. Running time measures computational efficiency. The algorithm performance data for different parameter combinations are shown in Table 3.

[0098] Table 3 Algorithm performance under different parameter combinations

[0099]

[0100] Figure 2 , Figure 3 The heatmaps show the mean fitness and standard deviation for different population sizes and iteration numbers. Darker green in the heatmaps indicates larger mean fitness or standard deviation, while lighter green indicates smaller values. Table 3. Figure 2 and Figure 3 Taking all the results into account, the average fitness was lowest when the population size was 50 and the number of iterations was 800, followed by the standard deviation. Although the stability under this configuration was not the highest, the overall performance was optimal, making it the best parameter combination.

[0101] The optimal parameter combination was used for algorithm evaluation, with evaluation metrics including average fitness (Fit), standard deviation (Std), and the improvement percentage (Imp) of the optimal solution. Average fitness was defined as the total distance students traveled to school; the standard deviation was calculated based on fitness values ​​from 30 repeated trials, reflecting the stability of the algorithm's performance; and the improvement percentage quantified the percentage improvement of the current solution relative to the optimal solution. The performance results of each algorithm on the instance dataset are shown in Table 4. Figure 4 , Figure 5 As shown.

[0102] Table 4 Algorithm Performance Results

[0103]

[0104] Figure 4 , Figure 5 The following are line graphs showing the average fitness and standard deviation of this invention and other algorithms in six scenarios. (See Table 4 for details.) Figure 4 , Figure 5 The results show that SaDE-RD outperforms other algorithms in fitness values ​​in all five scenarios (G1 and G3-G6), and achieves the second-best fitness value in scenario G2. These results fully validate the effectiveness of the proposed SaDE-RD algorithm. Furthermore, AR-aDE achieves the best average fitness in scenario G2. Although SaDE-RD's fitness is slightly worse than AR-aDE in this scenario, AR-aDE has a larger standard deviation, indicating that SaDE-RD's overall performance is still acceptable in this scenario. More importantly, AR-aDE performs worse than SaDE-RD in all other scenarios. Therefore, considering the overall algorithm performance, SaDE-RD still has a significant advantage. Figure 5 Among the ten algorithms, although the mean standard deviation of SaDE-RD is not the lowest and is usually at a moderate level, its stability is still acceptable compared to the significantly higher standard deviations of RLAPMDE and HDDE.

[0105] Table 5 Comparison of School Site Selection Plans Before and After

[0106]

[0107] Referring to Table 5, the proposed solution, applied in a real-world scenario, shows improvements in the number of kindergartens, average number of children enrolled, total school commute distance, average school commute distance per person, and robustness assessment value. The optimized solution reduces the number of kindergartens by 92.42%. The average number of children enrolled is 12.46 times the original number, significantly improving the utilization rate of kindergartens in the study area. The average school commute distance per person decreases by over 50%, and the robustness assessment value decreases by 52.78%. This indicates that the optimized site selection solution exhibits significantly enhanced robustness and better stability when facing uncertainties such as fluctuations in student population and road changes.

Claims

1. A school site selection optimization method based on a robust adaptive differential evolution algorithm, characterized in that, Includes the following steps: Step 1: Construct a school site selection optimization model, and then use the budget uncertainty set to characterize the fluctuations in student numbers and commuting distances to establish a robust model that can be linearized; The specific process of step 1 is as follows: Step 11: Obtain data on student locations, number of students, candidate school locations, and school capacity. Under the constraints of school capacity and student allocation, establish a school location optimization model with the goal of minimizing the total commuting distance of all students. Step 12: Considering the actual uncertainty of student numbers and commuting distance, introduce a budget uncertainty set to describe parameter fluctuations, and establish a robust optimization model based on this. Step 13: Transform the robust optimization model into a solvable linear programming model using linear duality theory; Step 2: Propose a dynamic parameter adaptive strategy. By recording the speed at which each set of parameter combinations updates the optimal solution, calculate the selection probability weight, and achieve coordinated adaptive adjustment of the mutation factor and crossover probability. At the same time, introduce a parameter scaling mechanism based on the algorithm execution stage to enhance local search in the early stage of iteration and promote global exploration in the later stage of iteration. The specific process of step 2 is as follows: Step 21: Construct the candidate parameter matrix ; Step 22: Record the performance of parameter combinations and introduce update speed as an evaluation metric. ; For the first The update speed of parameter combinations; For the first The total number of iterations required to update the current optimal solution using a combination of parameters; For the first The number of times each parameter combination is selected; Index for parameter combinations; Step 23: Convert the update rate into selection probability, the expression is: ； in, For the first The probability of choosing a combination of parameters; Normalize the weights of all parameter combinations to obtain the first... The actual probability of a parameter combination being selected The expression is: ； in, Index for parameter combinations; This represents the total number of parameter combinations. Step 24: When a parameter combination is not selected within a set time, it is determined to be a parameter combination that updates slowly, and the parameter combination that updates slowly is subjected to active random perturbation. Step 25: Combine the parameter fine-tuning during the algorithm execution phase. When the number of fitness evaluations (NFE) has not reached half of the total number of evaluations, the selected [number]th [number] ... Variation factors in parameter combinations Temporarily reduced to 0.8 Perform a fine-grained local search; when the number of fitness evaluations (NFE) exceeds half of the total number of evaluations, to avoid getting trapped in local optima, the selected [fitness] will be [selected / degraded]. Temporarily magnified to 1.1 ; Step 3: Design an acceptance strategy based on the average distance between historically successful individuals. In the selection phase, not only fitness is considered, but also the average distance between individuals who have successfully updated the optimal solution in the past is used as a reference. The probability of accepting the suboptimal solution is set to overcome the premature convergence of the population. In step 3, after each iteration, it is checked whether the current global optimal solution has been updated. If the objective function value of the global optimal solution has not improved for N consecutive generations, it is determined that the algorithm has entered a stagnant state. Then, an acceptance probability is dynamically calculated based on the average distance between the historically successfully updated individuals and the population center to decide whether to accept a suboptimal solution. The accepted suboptimal solution replaces the worse individuals in the current population, thereby increasing population diversity. Step 4: Obtain the most robust school location plan with the shortest total commute distance.

2. The school site selection optimization method based on robust adaptive differential evolution algorithm according to claim 1, characterized in that, In step 11, the school site selection optimization model is as follows: ； In the formula, Indicates from student position Assigned to school The number of students; Indicates the student's location index; Number of student seats; For the number of schools; Indicates a school index; Indicates from student position to school Commute distance; For student positions School The decision variable is the student's location. Students attend school ,but =1, otherwise =0; This is the function to be minimized.

3. The school site selection optimization method based on robust adaptive differential evolution algorithm according to claim 2, characterized in that, In step 12, the budget uncertainty set includes the student number uncertainty set. and the uncertain set of commuting distance , , ; This represents the deviation between the actual and nominal number of students at each student's location from the school. For student positions to school The deviation between the actual and nominal number of students; This represents the deviation between the actual and nominal distances from a student's location to the school. For student positions to school The deviation between the actual and nominal values ​​of the distance; This represents the upper limit of the total deviation in the number of students in the school, reflecting the degree of conservatism regarding the uncertainty of the number of students in the school. This represents the upper limit of the total deviation of the distance from the student's location to the school, reflecting the degree of conservatism regarding the uncertainty of the distance from the student's location to the school. Based on this, a robust optimization model is established: ； In the formula, Representative student position to school The fluctuation range of the number of students; Representing the position of non-students to school The range of fluctuation in distance; This is the maximization function.

4. The school site selection optimization method based on robust adaptive differential evolution algorithm according to claim 3, characterized in that, In step 13, the linear programming model is as follows: ； In the formula, In order to prevent The marginal cost required to pay in the event of adverse fluctuations; In order to prevent The marginal cost required to pay in the event of adverse fluctuations; The dual variable representing the uncertainty of the corresponding number of people; This represents the dual variable corresponding to the uncertainty of distance.

5. The school site selection optimization method based on robust adaptive differential evolution algorithm according to claim 4, characterized in that, In step 21, the candidate parameter matrix The mathematical expression is: ； In the formula, Represents the total number of parameter combinations; Representing the The variation factor in a combination of parameters; Representing the The variation factor in a combination of parameters; Representing the Crossover probability in a combination of parameters; Representing the The crossover probability in a combination of parameters.

6. The school site selection optimization method based on robust adaptive differential evolution algorithm according to claim 5, characterized in that, In step 22, the parameter combination is expressed through... and To reflect, This is the set of times each parameter combination was selected. The set of total iterations required to update the current optimal solution for each parameter combination. , , Indicates the first The number of times each parameter combination is selected; Indicates the first The number of times each parameter combination is selected; Indicates the first The total number of iterations required to update the current optimal solution using a combination of parameters; Indicates the first The total number of iterations required to update the current optimal solution using a combination of parameters.

7. The school site selection optimization method based on robust adaptive differential evolution algorithm according to claim 6, characterized in that, In step 24, the formula for the random perturbation is: ； In the formula, Representing the The variation factor in a combination of parameters; Representing the The variation factor in a combination of parameters; Representing the Crossover probability in a combination of parameters; Represents the total number of parameter combinations; To The result after applying active random perturbation; Representing the Crossover probability in a combination of parameters; To The result after applying active random perturbation; This is a random disturbance term.

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