Regional compulsory education resource layout optimization method considering college house premium

Abstract

The invention discloses a regional compulsory educational resource layout optimization method considering college house premium, and belongs to the technical field of educational science. A bilevel planning model based on a network game is constructed, upper-layer planning is a school layout decision of an education management department, and the goal is to minimize the total cost of the system; the lower layer planning is the school selection behavior of parents, and the maximization of school selection effectiveness is taken as the target. According to the method, factors such as attraction, geographic positions and school area house prices of schools are comprehensively considered, and reasonable allocation of educational resources is realized by using a Stackelberg equilibrium theory. Scientification of school site selection and resource allocation is realized through algorithm optimization and upper-layer planning, and parents are guided to select a school through lower-layer planning, so that the phenomenon of premium of a school area house is effectively relieved, and the problem of uneven distribution of educational resources is improved. According to the method, a scientific decision-making tool is provided for an education management department, the distribution efficiency and fairness of compulsory education resources can be remarkably improved, and high-quality and balanced development of education is promoted.

Description

Technical field:

[0001] The invention proposes a method for optimizing the layout of regional compulsory education resources taking into account the premium of school district housing, and belongs to the field of educational science and technology. Background technology:

[0005] The present invention proposes a method for optimizing the layout of regional compulsory education resources based on the premium of school district housing, aiming to allocate educational resources more reasonably through scientific methods. Specifically, based on network game theory, the present invention constructs a two-level programming model, the upper level of which is the planning decision of the education administration department, and the lower level is the school selection behavior of parents. This two-level programming model can effectively simulate the mutual game relationship between policy makers and parents in the process of educational resource allocation, thereby providing a theoretical basis and practical tools for the rational allocation of educational resources.

[0006] In the optimization of educational resource layout, the upper-level planning is the responsibility of the education administration department, whose goal is to minimize the overall cost of the system by optimizing the school location, including the school construction cost and the student commuting cost; the lower-level planning is centered on the school selection behavior of parents. Parents consider multiple factors when choosing a school, including the school's teaching quality, geographical location, safety, and the price of school district housing, etc. Through this two-level game model, the interests of all parties can be considered more comprehensively, thereby achieving the optimal allocation of educational resources.

[0007] The present invention integrates the decision-making process of parents' school selection and educational resource layout by constructing a mathematical model based on network game, so that the allocation of educational resources not only meets the goals of the education administration department, but also meets the needs of parents. In this model, parents' school selection behavior is regarded as a path selection problem on the education resource network, and the education administration department guides this selection process by planning the location and scale of the school, thereby achieving the optimal balance of the whole system.

[0008] In general, the optimization of the layout of high-quality educational resources involves not only the formulation of educational policies, but also cross-disciplinary research in multiple fields such as economics, game theory, and urban planning. The present invention theoretically fills the research gap in the optimization of educational resource layout under the background of school district housing premium, and in practice provides a scientific decision-making tool for educational management departments, which helps to achieve high-quality and balanced development of compulsory education and promote the realization of educational equity. Summary of the invention: Technical issues:

[0009] The technical problem to be solved by the present invention is how the government education management department can select a site in a two-layer planning model to minimize the total system cost while maximizing the parent utility. Technical solution:

[0010] The traffic distribution formed on the compulsory education resource network is the result of the interaction of two mechanisms. One mechanism is that system users, i.e. parents / students, try to maximize their own utility function by choosing the best path on the network; the other mechanism is that the service level provided by the government education management department to parents / students is closely related to the use of the system. The greater the traffic on the network, the higher the resistance that parents / students encounter in choosing schools, i.e. the corresponding impedance. The interaction of the two mechanisms makes it difficult for people to find the best solution for school selection and the resulting traffic distribution.

[0011] In the problem of optimizing the layout of compulsory education resources under the condition of balanced user allocation, a game relationship is formed between the education administration department (leader) and parents / students (followers) of the school layout plan. The final balance is called Stackelberg equilibrium in game theory and can be represented by a two-level programming mathematical model. In the two-level programming model of the present invention, the upper layer is a 0-1 integer programming of the layout of compulsory education resources, with the sum of the total cost of parents choosing schools and the total cost of the education administration department building schools, that is, the minimization of the total system cost as the goal of layout optimization; the lower layer uses user balance to represent the parents' school district housing selection behavior, with the goal of maximizing the school selection utility. 1. Model Method 1.1 Problem Description

[0012] Assume that the number of school-age students in each starting residential area i in the network is not fixed, and the number of students that each terminal school j can attract is also not fixed. They are related to the housing prices in the starting residential area, the tuition fees and attractiveness of the terminal school, and the commuting costs between the starting and the terminal. However, the total number of school-age students at the starting point, that is, the total number of students in demand, is fixed. The equilibrium allocation model is extended to determine the number of students for each OD pair and the number of students on the road section, that is, the combination of demand distribution and student flow allocation.

[0013] In the context of the multi-school zoning policy, parents will consider a variety of factors when choosing a school. These factors can be divided into two categories: internal factors and external factors. Among them, internal factors include teaching quality, school reputation, teaching staff, school facilities and management, etc., which are uniformly measured by the attraction measurement index M. jTo express. At the same time, the attractiveness of different schools is usually different. For example, the attractiveness of key schools is greater than that of ordinary schools. The external factors that affect parents' school choices include geographical location, safety, community influence, tuition fees, etc. The geographical location and commuting convenience of the school are important factors for parents to consider. Schools close to home addresses are usually more popular, especially in large cities, where commuting time and transportation costs are important considerations. The safety of the school and its surrounding environment is also a major concern for parents. A safe surrounding community contributes to the healthy physical and mental development of children. Therefore, during the compulsory education stage, although the school tuition is usually fixed and does not change with the number of students. But parents will choose to buy school district housing that is close to the school and has high safety, which creates a premium for school district housing. In the process of building a balance in the game of parental school selection, we take into account the cost of parents buying homes for choosing schools, and housing prices will change with changes in traffic.

[0014] It is assumed that parents have complete school choice information, have the same understanding of the net impedance of school choice, and seek to minimize the net impedance of school choice.

[0015] The compulsory education resource network is in a steady-state equilibrium state. The variables and symbols used are defined as follows:

[0016] s a : The number of students on the virtual road section a;

[0017] s b : The number of students on section b;

[0018] s c : The number of students on the terminating section c;

[0019] c a : The impedance of the starting virtual road section a, i.e., the housing price;

[0020] c a (s a ): impedance function of the starting virtual road segment a with flow rate as the independent variable

[0021] c b : Net impedance of section b, i.e. commuting cost

[0022] c c : The net impedance of the virtual road segment c, which is the tuition fee minus the attraction measure

[0023] q j : The capacity of school j (maximum number of students);

[0024] c j : The cost of establishing a school at node j;

[0025] The number of students on the kth path between ODs with departure point i and destination j;

[0026] The net impedance on the kth path between ODs with origin i and destination j;

[0027] y ij : If parents in residential area i choose school j, then y ij =1, otherwise y ij =0;

[0028] Link-path related variables, i.e. 0-1 variables. If virtual link a belongs to the kth path between ODs from departure i to destination j, then otherwise

[0029] Link-path related variables, i.e. 0-1 variables. If link b belongs to the kth path between ODs from departure i to destination j, then otherwise

[0030] Link-path related variables, i.e. 0-1 variables. If the ending virtual link c belongs to the kth path between ODs from departure i to destination j, then otherwise

[0031] x j : Whether node j has established a school, x j =1 means to build a school, x j =0 means not established;

[0032] N: the set of nodes in the network;

[0033] A: the set of starting virtual segments in the network;

[0034] B: the set of road segments in the network;

[0035] C: the set of terminated virtual road segments in the network;

[0036] K: the set of paths in the network;

[0037] I: the set of starting points in the network;

[0038] i: index of residential area;

[0039] J: the set of destinations in the network;

[0040] j: index of the school;

[0041] Kij : The set of all paths between departure point i and destination point j;

[0042] q ij : The demand for OD students between departure point i and destination j;

[0043] c ij : The net impedance between ODs with departure point i and destination j;

[0044] Q max : A large constant used to limit the value of the variable;

[0045] B max : A large constant used to limit the value of a variable. 1.2 Upper-level cost minimization model

[0046] The school capacity of node j cannot exceed a certain predetermined upper limit Q max : In this constraint, q j represents the capacity of school j, x j is a binary variable indicating whether a school is established at the node (1 for establishment, 0 for no establishment). This constraint ensures that the enrollment of each school is within a reasonable range, avoiding overcrowding or waste of resources, thus ensuring that every student can receive high-quality education services.

[0047] The total cost of the school cannot exceed a certain budget B max : Here, c j represents the cost of building a school at the node, and B max is the budget ceiling set by the education administration. This constraint is used to ensure that the total construction cost does not exceed the available budget and to ensure the economic feasibility of the project.

[0048] To ensure that each student can only choose one school, the model sets a unique selection constraint, that is, when each student makes a school choice, he or she can only choose one of the accessible schools as the final destination, which can be expressed as:

[0049] In the upper model, we also need to ensure that the variable x j and ij is a 0-1 integer variable, indicating whether the school is established and the certainty of the relationship between the student's choice, specifically expressed as:

[0050] In order to accurately express the impact of housing costs on parents' school choices, we add a housing price demand function to the model:

[0051] Where p i is the cost of selecting residential node i, p0 is the zero flow cost (the cost required for parents to select a residential area when the node is empty and quiet), o i is the number of students who choose residential node i, c i is the actual accommodation capacity of the residential node i (i.e. the number of households actually accepted by the residential area), α and β are the blocking coefficients (in the present invention, they are taken as 0.7 and 2 respectively, and can also be obtained by regression analysis from actual data).

[0052] The commuting cost between the starting residential area i and the ending school j can be approximated as a constant c b ; Tuition fees for compulsory education are usually fixed and set as a constant f j ·p i +c b +f j is the total cost of school selection for parents, i.e., the net impedance c for parents in residential area i to choose school j ij , then:

[0053] The upper-level planning model is constructed as follows: where u ij ,y ij is the solution to the lower problem. The goal of function (7) is to minimize the total cost of the system, including the commuting cost of students and the construction cost of schools. Based on all constraints, managers can make optimal school location and construction decisions within limited resources and budgets to improve the efficiency of educational resource allocation.

[0054] Through the above mathematical model, the upper-level planning not only focuses on minimizing school construction and commuting costs, but also fully considers the selection behavior of students and parents, the fairness of resource allocation, and the impact of market factors on the demand for educational resources. In this way, the education administration department can reasonably plan the distribution of schools based on the optimization results of the model to achieve an efficient and balanced allocation of educational resources, thereby promoting the fairness and popularization of compulsory education. 1.3 Lower-level utility maximization model

[0055] In the process of balanced allocation of compulsory education resources network, the conservation of the number of students must be ensured, that is, the sum of the number of students on each path between the departure point i and the destination j should be equal to the total demand for OD students, as expressed as follows:

[0056] Number of students on the pathway Student volume with starting virtual section s a , the number of students on the road section b , terminate the number of students in the virtual section c The following conditions should be met, that is, the number of students on a section should be obtained by accumulating the number of students on all (i, j) pairs passing through the section, and the expression is as follows:

[0057] The net impedance of a path should be the sum of the impedances of each section that the path passes through, expressed as follows:

[0058] The number of students in the path should satisfy the non-negative constraint:

[0059] The above problem is that the number of students of the right age at the starting point i is O i , to satisfy the constraints The number of OD students ij}, that is, the problem of parents / students choosing both residential areas and schools must be solved at the same time.

[0060] It is generally assumed that parents / students always try to choose the residential area and school with the greatest attraction and the lowest school selection cost, that is, parents seek M j Maximum and p i +c b +f j Minimum. Finally, a generalized cost function consisting of commuting cost, tuition, housing price and attractiveness is defined to describe the net impedance value of the path. The calculation formula is as follows:

[0061] Assume that each user always chooses the path k that he thinks has the smallest impedance between a certain OD pair (i, j). In this case, the impedance value judged by the user is called “perceived impedance”. to express; to use represents the actual impedance of the path, then: In the formula is a random error term that follows a Gumbel distribution.

[0062] According to the definition of "utility" in utility theory, the utility of the choice can be expressed as the negative value of the path perception impedance: k ,Right now: At this point, the path selection problem is transformed into a utility decision problem. For parents in residential area i, when the perceived impedance of path k is It is the smallest of all possible paths between OD, that is, its negative value is the largest.

[0063] The lower-level planning model is constructed as follows: 1.4 Two-level programming model

[0064] The present invention proposes a two-layer optimization model for the layout of compulsory education resources based on the premium of school district housing. The upper-level planning model is mainly responsible for optimizing the site selection and resource allocation of schools to minimize the overall cost of the system; while the lower-level planning model simulates the school selection behavior of parents, aiming to maximize the utility of parents. The upper and lower-level models achieve dynamic balance through a mutually operating mechanism: the upper-level decisions provide parents with the distribution and resource situation of schools, while the school selection behavior of parents at the lower level in turn affects the school's demand and resource allocation. Through continuous iteration and game-playing, the optimal configuration of educational resources between demand and supply is ultimately achieved, thereby satisfying both the planning goals of the management department and the school selection needs of parents, achieving the global optimality of the system. The structure diagram of the two-layer model is shown in the figure below. Figure 1 As shown. Therefore, the two-level programming model can be obtained: 2. Algorithm Design

[0065] By observing the upper-level planning model, it can be found that the upper-level planning is a simple resource-constrained discrete planning problem. When the upper-level location variable x is given, the lower-level planning can be solved by the MSA algorithm. Therefore, the present invention uses a branch-and-bound algorithm to achieve implicit enumeration of the upper-level location variable.

[0066] Similar to solving the elastic demand balance allocation problem, the present invention adopts the network transformation method to solve the lower-level planning model. This method transforms the lower-level planning model into an equivalent fixed demand balance allocation model by introducing appropriate virtual sections in the basic network. Subsequently, the MSA algorithm is used for solving.

[0067] like Figure 2 As shown, a starting virtual node v1 and a terminating virtual node v2 are set in the basic network, and then a starting virtual segment is connected from the starting virtual node v1 to each starting point i in the basic network, and a terminating virtual segment is connected from each end point j of the basic network to v2. Obviously, the total number of virtual segments is the number of starting nodes plus the number of terminating nodes in the basic network.

[0068] On the starting virtual road segment, from v1 to i, the student flow on it is satisfy The impedance function of the initial virtual section should be The starting point of the transformed network is the virtual node v1, and its end point is the virtual node v2. In the transformed network, the number of OD students from v1 to v2 is fixed. Using the standard user equilibrium allocation method for the transformed network flow distribution, the solution to the parent / student school selection network game equilibrium can be obtained.

[0069] The student school selection demand flow of each section is constantly adjusted to gradually approach a balanced allocation result. In each cycle, an all-or-nothing allocation is performed based on the flow allocated to each section to obtain the additional student volume of each section. Then, the number of students allocated in the next cycle is calculated by taking a weighted average of the number of students allocated to each section in the cycle and the additional student volume obtained. When the number of students allocated in two consecutive cycles is close enough, the calculation is stopped, and the number of students obtained in the last cycle is the final result.

[0070] Step 1: Determine all possible site selection schemes based on the candidate points and perform equivalent network transformation on the schemes.

[0071] Step 2 Initialization. First, use the all-or-nothing method to assign the initial state and obtain the initial solution Set the number of iterations n = 0, the starting virtual section impedance Section impedance End virtual segment impedance

[0072] Step 3 increases the number of iterations, that is, n=n+1. Calculate the net impedance of each virtual road section according to the current student volume of each virtual road section.

[0073] Step 4: Perform an all-or-nothing allocation based on the net impedance value and OD student demand obtained in step 2 to calculate the additional student volume for each starting virtual section. Additional student volume for each route Additional number of students for each terminating virtual segment

[0074] Step 5: Use the MSA method to calculate the current number of students on each road section Current student volume in each section and the current number of students in each virtual section

[0075] Step 6 If and and and If the difference is not large, stop calculating. This is the final allocation result. Otherwise, return to step 2.

[0076] Step 7: Calculate the upper layer objective function value. Go to step 2.

[0077] Step 8 determines the feasible solution with the optimal objective function value, that is, the global optimal solution to the two-level programming problem.

[0078] In step 6, the decision to stop the calculation can be controlled based on the error size or the number of cycles. In step 5, the weight coefficient α is given by the calculator and can be a constant or a variable. In the present invention, α=1 / n, where n is the number of cycles. The algorithm flow chart of the two-layer model is as follows: Figure 3 shown. Beneficial effects:

[0079] The present invention realizes the scientific and reasonable allocation of compulsory education resources in the region by constructing a two-layer planning model, which has significant beneficial effects. First, by optimizing the site selection of schools, the upper-level planning effectively reduces the construction and maintenance costs of schools, and reduces the commuting costs of the overall system, making resource utilization more efficient. Secondly, the model fully considers the school selection behavior of parents, and simulates the selection process of parents through the lower-level planning, so that the configuration of compulsory education resources is more in line with actual needs, thereby improving parental satisfaction. In addition, the present invention also alleviates the phenomenon of premium of school district housing to a certain extent, and reduces the educational unfairness caused by school district housing through a more balanced layout of educational resources, and enhances social fairness. Finally, the present invention provides a scientific decision-making tool for education management departments, which can achieve high-quality and balanced development of compulsory education under limited resources and promote the overall improvement of education quality. Description of the drawings: Figure 1 It is a structural diagram of the double-layer model; Figure 2 For network changes with two starting points and three end points; Figure 3 is the algorithm flow chart; Figure 4 This is the network diagram of the example; Figure 5 Figure 2 is the example road network A and demand distribution after equivalent changes. Specific implementation method:

[0080] like Figure 4As shown in the figure, there are ten location nodes in the area, of which four are residential nodes, namely nodes 1-4. It is now planned to build two schools at nodes 5-8 in the area. The total number of school-age students in the area is known to be 500, and the number of school-age students in each residential area is unknown. The housing cost and capacity of each residential area are shown in Table 1. The basic parameters of the road section are shown in Table 2. The basic parameters of the school are shown in Table 3. Table 1 Basic parameters of residential areas Table 2 Basic parameters of road sections Table 3 Basic parameters of the school

[0081] The branch and bound method is used to solve the upper-level 0-1 integer programming. There are 6 solutions in total, see Table 4. Table 4 School layout plan

[0082] Taking scheme A as an example, the network after equivalent transformation is as follows Figure 4 As shown, nodes 1-4 are residential nodes, nodes 5 and 6 are school nodes, 11 is the starting virtual node, 12 is the ending virtual node, sections 15-18 are the starting virtual sections, and sections 19 and 20 are the ending virtual sections. The basic parameters of the starting virtual section are shown in Table 5. The basic parameters of the ending virtual section are shown in Table 6. Table 5 Basic parameters of the starting virtual road section of the example network Table 6 Basic parameters of the virtual road segment terminated by network A in the example

[0083] Taking scheme A as an example, the network balance distribution results obtained by using R language calculation are as follows: Section impedance in balanced mode Traffic flow of road section in balanced mode Total network utility U A =-547094.7.

[0084] After calculation, the relevant results of each scheme are shown in Table 7: Table 7 Summary of experimental results of each scheme

[0085] According to the calculation results, it can be concluded that the upper-level decision makers in the above example network can choose plan A, plan D or plan E to maximize the total utility of parents / students choosing schools in the network. Whether choosing plan A, that is, building a school at nodes 5 and 6 in the region, or choosing plan D, that is, building a school at nodes 6 and 7 in the region, or choosing plan E, that is, building a school at nodes 6 and 7 in the region, 278 of the 500 school-age students in the entire region will choose to live in the residential area of ​​node 2, and 222 will choose to live in the residential area of ​​node 4; they will all choose the school built at node 6.

Claims

1. The present invention proposes a method for optimizing the layout of regional compulsory education resources taking into account the premium of school district housing, and the method includes the following technical features: (1) It can be represented as a two-level planning model, with the upper level being the system manager and the lower level being the facility selector; (2) The upper model is a 0-1 integer programming model that takes the minimization of the total social cost of the system as the objective function, the limited budget input as the constraint condition, and the location of compulsory education facilities as the decision variable. The user demand allocation of each facility is determined by the lower model. The upper model is expressed as: Where Z is the total system cost, c ij is the net impedance between the OD of residential area i and school j, y ij Indicates whether parents in residential area i choose school j, c j is the cost of building a school at node j, x j Indicates whether node j has established a school, s a is the number of students on the virtual road section a, p i is the cost of selecting residential node i, c b is the commuting cost, c b is the number of students on section b, f j is tuition fee, c is the number of students on the terminal section c, q j is the capacity of school j, Q max is the school capacity limit, B max is the government budget constraint, I is the set of residential areas in the network, and J is the set of schools in the network; (3) The lower model is a user choice equilibrium problem that considers total utility. Assuming that users choose compulsory education facilities with the greatest utility, when parents know the status of the preschool education resource network exactly and try to choose the one with the greatest utility, the result of the game between users will reach a user equilibrium state. The lower model is expressed as: Where U is the school selection utility, M j is the attraction of school j, represents the number of students on the kth path between the ODs of residential area i and school j, q ij represents the demand for OD students between residential area i and school j, O i represents the sum of the number of school-age students in each residential area i, Indicates whether the virtual road segment a belongs to the kth path between the residential area i and the school j. Indicates whether the road segment b belongs to the kth path between the residential area i and the school j. Indicates whether the ending virtual road segment c belongs to the kth path between the ODs of residential area i and school j, where K is the set of paths in the network. is the random error term; (4) Design a heuristic algorithm that uses implicit enumeration for the upper model and network transformation and continuous average algorithm for the lower model to calculate the student flow, total utility and optimal school location plan for each node under the equilibrium mode.

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