Multi-Objective Optimization Path of Countermeasure Research on High-Quality and Balanced Development of Compulsory Education in Hebei Province in the Context of Digital Era

Entropy was originally a thermodynamic concept indicating the degree of disorder in a system, labeled with S. It was originally discovered in 1865 while studying the equilibrium state of a thermodynamic system: if the same Carnot heat engine and Carnot cycle are considered, for any closed cycle there is: (1)∮dQT≤0

That is, Clausius’ inequality, currently holds the equal sign only when the reversible cyclic process. This property implies the existence of a state function, and so let: (2)δs=δQrevT

s is the entropy of the system, and δQ_rev represents the heat of the system during a reversible cyclic process, which is equal to the entropy of the heat of the system versus the temperature of the system, a physical quantity that reflects the degree of chaos in a thermodynamic system. Clausius further stated that for an adiabatic system, the entropy never decreases, i.e. the principle of entropy increase.

After that, Boltzmann used the knowledge of statistical mechanics to establish a connection between the entropy, which was originally at the macroscopic level, and the number of microstates of the system, arguing that the greater the number of microstates of a system, the greater the degree of chaos of the system. Boltzmann gave meaning to entropy statistics and concluded: (3)S=klnΩ

Ω is the number of microstates of the system and k is the Boltzmann constant. The entropy at this point is also called the Boltzmann entropy. From the point of view of information theory, information can be categorized into known information and implied information. If you want to predict whether an event occurs or not, then when the more information about the event is known through the source, the more certain the event is whether it occurs or not, the higher the probability of occurrence, and the smaller the information entropy of the system. On the other hand, when the more unknown information or implicit information, then the greater the uncertainty of whether the event occurs, the smaller the probability of occurrence, that is to say, the greater the information entropy of the system. Here we can see a roughly inverse relationship between entropy and the implied event probability and a roughly proportional relationship with the amount of information implied by the event. So what exactly is the relationship between information entropy and event probability? We can be inspired by the Boltzmann expression for entropy: suppose a stable thermal equilibrium system with a Boltzmann expression as in Eq. (4): (4)S=klnΩ

Taking the microstate number Ω as its reciprocal, a sign is added to the right side of the equation in order to equalize it with the original equation, yielding equation (4): (5)S=−kln1Ω

And the inverse of the number of microstates can actually be thought of as the probability that the system is in one of the microstates, so it can be replaced by P to obtain Eq. (6): (6)S=−klnP

Further derivation returns the negative sign to obtain equation (7): (7)S=kln1P

From the equation we can see that the Boltzmann entropy is proportional to the natural logarithm of the inverse of the probability, and the scale factor k is the Boltzmann constant. Due to the assumption of equal probability, all microstates of an isolated system have the same probability of appearing, but extended to the information domain, for a system containing 1, 2…i…n of n individuals, each individual may appear differently under a certain index, and then the probability of the event of each individual should be the probability of the performance of each individual appearing in the performance of the whole group, i.e., P_i. At this time, the entropy of information of an individual is equal to the information entropy of an individual is then equal to: (8)Si=−klnPi

Then the information entropy of the indicator should be equal to the expectation of the information entropy: (9)S=−k∑inPilnPi

1)

Object-element matrix

In our real situation, things are often characterized in more than one way, and their features are multidimensional, if thing N has n feature C₁, C₂, ⋯C_n and its corresponding measure V₁, V₂, ⋯V_n, then thing N can be expressed as: (10)R=( N, C1, V1 C2, V2 ⋮ ⋮ Cn, Vn )=( R1 R2 ⋮ Rn)

where N_{0, ji} denotes the standard object; c denotes the characteristics of N_{0, ji}; v0,ji=〈a0,ji,b0,ji〉 , v_{0, ji} denote N_0j, with respect to c, the range of quantitative values specified, i.e., the range of data taken by the corresponding characteristics of each class - the classical domain. 3)

Determination of the section field

Section Domain: (12)Rp=(Np,C,vp)=( Np, c1, vp1 c2, vp2 ⋮ ⋮ cn, vpn)=( Np, c1, 〈ap1,bp1〉 c2, 〈ap2,bp2〉 ⋮ ⋮ cn, 〈apn,bpn〉)

where N_p denotes the section field object, which is composed of standard things plus things that can be transformed into standards, and vpt=〈apt,bpt〉 , which denotes the range of quantitative values taken by N_p with respect to c_i, called the section field. 4)

Distance and mode

The modulus of the bounded interval X = 〈a, b〉 is: (13)|X|=|b−a|

where the modulus of the point x₀ is specified to be |x0|=0 .

Here, we refer to the open interval (a, b), closed interval [a, b], and half-open, half-closed interval [a, b), (a, b] in classical mathematics as intervals and denote them by X = 〈a, b〉.

Let x, y be is any two points on the real axis, then ρ(x, y) = |x − y| is said to be the distance between x and y. Clearly, ρ(x, y) = ρ(y, x). When x = y, ρ(x, y) = 0. 5)

Calculation of correlation functions

The establishment of the correlation function makes the process of solving the contradiction problem quantitative, and its range of values covers the entire real number axis. In the evaluation of the object element, the correlation function expresses the extent to which the object element meets the required range of values when the value of the measure of the object element is taken to be a point on the real number axis.

Hebei Province is a large province with a large population, and the realization of balanced development of urban and rural compulsory education is the cornerstone of the transformation from a large province with a large population to a province with strong talents, and from a large province with strong human resources to a province with strong human resources, and how to establish a long-term mechanism for the optimal allocation of urban and rural compulsory education resources is an important strategic task that Hebei Province is facing at present.

In recent years, Hebei Province has always insisted on the balanced development of compulsory education in urban and rural areas as the “top priority” of education work, and has taken the purpose of running every school and paying attention to the healthy growth of every student as its goal, and has made the balanced development of compulsory education in urban and rural areas in Hebei Province achieved significant progress by means of clarifying work objectives, innovating management system, exploring development mode, optimizing resource allocation, and perfecting guarantee mechanism. The balanced development of compulsory education in Hebei Province has yielded clear results.

The accounts of the Bureau of Education, Culture and Statistics of Hebei Province are shown in Table 1, and it is found that the investment in compulsory education has continuously increased in the past three years. Education funding has grown from 389,979,700 yuan in 2021 to 529,846,700 yuan in 2023. The provincial standard is 20% higher than the average public funding for compulsory education students. All schools in the region arrange teacher training funds at no less than 5% of the total annual public funding budget of the school. Implementing the standardization of compulsory education schools, 22 schools in urban areas have been upgraded with educational equipment, and newly built and expanded schools have basically reached the provincial education equipment class I standard.

Compulsory education teachers in a region of Hebei Province, as shown in Table 2, through the investigation of the data over the past three years, the comprehensive development of education, to 2023, the region’s primary and secondary schools on the staff of 2,579 people. The number of students enrolled in elementary and middle schools is 47,129. The enrollment rate and the consolidation rate of students enrolled in primary and middle schools are both 100%. Since the introduction of the Compulsory Education Law, the level of development of compulsory education has increased year by year, and in recent years, the enrollment and consolidation rates have been at the forefront of the city. The goal of basic equilibrium in compulsory education has been achieved throughout the country.

The respondents of this study were students enrolled in compulsory education schools, parents of students enrolled in compulsory education schools, and teachers on staff in seven urban areas under the jurisdiction of Hebei Province. The questionnaire in this study was based on a five-level Likert scale, and the respondents answered the questions in the questionnaire according to the degree of conformity of the situation described in the questionnaire.

In this study, based on the importance of China’s education quality evaluation indexes, a total of 41 three-level indexes were selected, and the Delphi method was used to collect experts’ opinions in order to determine the weights of each index in the evaluation system, and eight experts were invited to score to obtain the weights of each sub-indicator. Then the weight entropy value method was used to adjust the weights of the indicators, and finally the weights of each sub-indicator of the evaluation index system for the balanced development of compulsory education in this study were obtained. Table 3 displays the weights of the evaluation indicators adjusted by the entropy value method.

4.2.1

Evaluation of the quality of overall compulsory education

The statistical results of the high-quality balanced development level of compulsory education in Hebei Province are shown in Table 4. Level I to IV: Very unbalanced, more unbalanced, more balanced, very unbalanced. Through the material element measurement method, the data on the level of balanced development is obtained, and the highest level of balanced development is students’ moral character, whose level of balanced development is IV. At the present stage of compulsory education schools in Hebei Province, the overall gap between schools in terms of students’ ideological and moral education is relatively small, reflecting that compulsory education schools in Hebei Province follow the current policies and educational guidelines in education, and carry out students’ ideological and moral quality education in a solid and effective way, with remarkable educational achievements. Due to factors such as the level of local economic development, geographic location, the quality of the student population and teacher resources, the level of balanced development of compulsory education schools in Hebei Province in terms of the quality of students’ academics and students’ physical and mental health needs to be improved. The level of balanced development of students’ work practices and coursework is the lowest, and is “relatively unbalanced”.

Table 4.

Compulsory education high-quality balanced development level statistics

	I level	II level	III level	IV level	max	J0
Student character	-0.1627	-0.1375	-0.1416	0.14934	0.14934	IV
Academic development	-0.1631	-0.1485	0.1175	-0.1093	0.1175	III
Physical and mental health	-0.1522	-0.1329	0.0273	-0.1432	0.0273	III
Labor practice	-0.1608	0.0516	-0.1327	-0.0728	0.0516	II
Class burden	-0.2084	0.1185	-0.2375	-0.1156	0.1185	II
Obligation education quality balanced development level	-0.1816	-0.1693	0.1457	-0.0718	0.1457	III

The radar charts of student character evaluation indicators, academic development index evaluation indicators, physical and mental health evaluation indicators, labor practice and class load evaluation indicators of the interviewed schools are shown in Figure 1. According to the evaluation results of the seven sub-indicators of national identity, the red culture inheritance belongs to a lower level, which indicates that the moral education work of the school needs to strengthen the propaganda of the red culture and the guidance of students’ behavior. According to the evaluation results of the sub-indicators included in the evaluation of academic development, students’ awareness of innovation and application is insufficient, and the degree of progress is small, so in teaching practice, it is necessary to focus on strengthening the cultivation of students’ innovation ability and improving the quality of classroom teaching. According to evaluation indicators of students’ physical and mental health, the development of students’ physical performance and negative emotions is relatively unbalanced. Schools should strengthen their students’ physical exercise and psychological guidance, offer sufficient physical education and mental health classes, and set up special counseling rooms for students’ psychology. In terms of labor practice, the labor participation rate of students is not high, and future educational work needs to focus on increasing the labor participation rate of students. “The integration of five cultivation methods” is the direction of efforts.

Figure 1.

A map of the score of the first level index

4.2.2

Inter-school evaluation of the high-quality balanced development of compulsory education

The distribution of the composite score for compulsory education quality in the sampled schools can be seen in Fig. 2. For ease of expression, the names of each compulsory school participating in the survey are represented by A~V. The black horizontal line in the figure is the average level of the high-quality balanced development of compulsory education. Comprehensive evaluation scores for the education quality of schools participating in the survey. The results of the data analysis show that about 27.3% of the schools in Hebei Province’s compulsory education stage have an average score of their comprehensive evaluation of education quality that is lower than the average level of education quality, and that the quality of education in each school is less balanced as a whole.

Figure 2.

Compulsory education quality comprehensive score distribution

It is worth noting that nearly a third of the comprehensive assessment of the quality of their education is slightly above average, and this category of schools with higher average scores of the quality of education centers on the following two characteristics: first, schools with a high quality of the source of students have higher scores of the quality of education, i.e., the number of students with high quality scores of the quality of education is greater, which makes the average score of the quality of education of this category of schools higher. Secondly, schools with fewer “latecomers” and smaller school sizes have higher education quality scores, i.e., the number of students with low education quality scores is smaller, making the average education quality score of such schools higher. Based on the above analysis, in order to accelerate the quality and balanced development of compulsory education, it is necessary to stabilize the “top students” and reduce the “backward students”.

In order to understand more objectively the impact of the policy on the high-quality balanced development of regional education quality, this study obtained data on the final academic performance of elementary school students in some school districts in Hebei Province in the fall semester of 2023 from educational authorities. Considering that scatter plots have the advantage of presenting the differences in students’ academic performance and the differences between schools in an intuitive manner, this section chooses to present the results of the data analysis using scatter plots. The scatterplot of inter-school differences and academic performance of the surveyed school districts in Hebei Province is shown in Figure 3. 1 to 7 represent the seven school districts in Hebei Province. Overall, the level of balanced development of education quality varies widely among compulsory education school districts in Hebei Province. Most school districts have large inter-school differences, and most of the academic performance is at the middle-low level, which leads to a polarized distribution of academic performance between schools. A small number of districts had small differences but low academic achievement, which suggests that the students in the district are not performing well academically. In short, there is a low level of balanced development of student academic quality between schools.

Figure 3.

Scatter map of intermodulation differences and academic performance

In scientific research and engineering practice, for a complex problem, the need to simultaneously meet multiple demands, but there are often conflicting situations between the various demands, and these multiple problems can be abstracted as multiple nonlinear objective functions, then the process of optimizing these multiple objective problems at the same time is called multi-objective optimization. A problem with multiple objective functions that need to be optimized is called a multi-objective optimization problem, and the goal of the problem is to search for a set of optimal solutions and obtain an objective value acceptable to the decision maker on the solution set. If expressed mathematically, for the multi-objective optimization problem: assuming that there is m objective function, n -dimensional decision variables, see equation (25), and the decision maker needs to choose one of the non-inferior solutions as the final solution from all non-inferior solutions according to the specific needs of the problem, combined with the actual situation of the problem and the decision maker’s preference: (25)minY=F(x)=(f1(x),f2(x),…,fm(x)) s.t gi(x)≤0, i=1,2,…,p hj(x)≤0, i=1,2,…,q x∈D $$\matrix{ {minY = F(x) = ({f_1}(x),{f_2}(x), \ldots ,{f_m}(x))} \cr {s.t\>{g_i}(x) \le 0,\>i = 1,2, \ldots ,p} \cr {{h_j}(x) \le 0,\>i = 1,2, \ldots ,q} \cr {x \in D} \cr } $$

x ∈ D denotes the n -dimensional decision variable, x = (x₁, x₂, …, x_i), x_i denote the ith dimensional decision space. y ∈ S ⊂ R^m is the mth dimensional solution space, and S denotes the mth dimensional solution space. f_m(x) is the mth objective function of x. g_i(x) ≤ 0 and h_j(x) ≤ 0 are the p and qth constraints, respectively, by which the feasible domain of the solution can be determined. minY denotes the minimum value for solving the multi-objective optimization problem (MOP).

Since there is no dominance relationship between the Pareto optimal solutions, improving any objective function for a given set of non-dominated solutions will result in the elimination of at least one of the other objective functions in the set of non-dominated solutions. In addition, if the relative importance of the objectives to be optimized is ambiguous, then the resulting Pareto optimal solutions are equal. For the optimal allocation of educational resources, it is necessary to select the compromise solution among the set of obtained Pareto optimal solutions based on the preference information of the resource allocation decision maker or the demand of the resource allocation system.

Optimal solutions, non-inferior optimal solutions, and optimal solution sets in multi-objective optimization problems can be defined as follows.

Definition 1 Pareto domination:

Assuming that x_a, x_b ∈ Ω is one of the two feasible solutions that exist in the MOP, if f(x_a) is not greater than and less than f(x_b) in at least one of the objectives, it is said to x_a dominate x_b, denoted as x_a ≻ x_b, as shown in equation (26): (26)∀i∈{1,2,…,n},fi(xa)≤fi(xb),∧∃i,fi(xa)<fi(xb)⇒fi(xa)<fi(xb)

Definition 2 Pareto optimal solution:

For an MOP, if x^* ∈ Ω and there exists no x ∈ Ω such that x ≻ x^*, then x^* is called a Pareto-optimal solution of the MOP, or a non-occupied solution.

Definition 3 Set of Pareto optimal solutions:

The set of all Pareto optimal solutions together. Denoted as X^*, it is shown in equation (27): (27)X*={x*|¬∃x∈Ω:x≻x*}

Definition 4 Pareto front:

The vector consisting of all the Pareto optimal solutions in the set of Pareto optimal solutions X^* corresponding to the mapping in the objective space, the final surface is called the Pareto frontier. These nondominated solutions on this surface have fewer conflicts between the individual objectives compared to the other solutions, which provides the decision maker with a better choice space. This is shown in equation (28): (28)PF*={F(x*)=(f1(x*),f2(x*),...,fn(x*))T|x*∈X*}

Among the coordinate points represented by the letters (A, B, C, D and E), point A is the optimal point because the decision variable a associated with point A is better than the other decision variables. For decision variable a, which means that its performance cannot be improved, solutions such as vector a are called Pareto-optimal.