# A Study on the Usefulness of Stochastic Simulation Algorithms for Teaching and Learning in College Physical Education Classrooms

Authors: Ju Xu
Journal: Mathematical Problems in Engineering
 (2022)

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## Abstract

In order to address the problem of the absolute nature of the evaluation of superiority and inferiority in the evaluation of physical education classroom in universities and the problem of inconsistency in the conclusion of multiple evaluations, we develop an “autonomous advantage evaluation method to highlight one's own advantages,” which uses a probabilistic stochastic simulation algorithm to evaluate the advantages of the evaluated objects by calculating the degree of superiority among them. The method is based on an innovative “base to top” approach, with a high degree of independence. The method was validated by means of an algorithm, and the conclusions were obtained with probabilistic information.


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## Body

## 1. Introduction

University students are the future pillars of our country, and it is only when they have a healthy body that they can be most creative and create more value. Therefore, it is very important for universities to provide physical education to students [1]. The health of students is reflected in all aspects of their lives, and physical education is an integral part of it, as learning more about physical education and acquiring skills through practical application can enhance students' physical fitness [2]. The Ministry of Education is paying more and more attention to the health of students, requiring universities to make reasonable physical education programs to improve the physical fitness of students, and according to the regulations, each school is constantly correcting and improving its teaching methods, and gradually tends to diversify and enrich the teaching content [3–5]. Therefore, it is important to evaluate the quality of physical education in order to judge the quality of teaching and learning [6–8]. Data mining is a popular data analysis technology that has received widespread attention. This technology can use known data resources to discover more potential information and the connection between things, firmly grasp this technology and apply it to the evaluation of the quality of physical education in college students, you can find out the factors affecting the quality of physical education and effectively enhance the physical fitness of students [9–12].In China, physical education classroom teaching has been developed for many years, and even though the mechanism of the method varies and the way of solving the problem is different, the conclusion form is mostly determined and consistent, which is expressed as “the absoluteness of superiority and inferiority discrimination” and “the strictness of difference transmission ” [13–15]. The use of different evaluation methods for the same evaluation problem usually results in different evaluation conclusions, resulting in the problem of “nonconsistent multievaluation conclusions” [16]. It is now generally accepted that 'portfolio evaluation' is an effective solution to this problem, but in reality, this is a compromise approach that does not address the essence of the problem at its root [17].In this paper, we develop an “autonomous advantage evaluation method to highlight one's own advantages,” which uses a probabilistic stochastic simulation algorithm to evaluate the advantages of the evaluated objects by calculating the degree of superiority among them [18]. This method produces probabilistic (reliable) evaluation conclusions, which are more interpretable to the actual problem, and proposes an innovative “base to top” comprehensive evaluation method with a high degree of independence, which is added to the evaluation in the form of “components.” The method is based on an innovative “base to top” approach, with a high degree of independence. The validity of the method is verified by means of an example [19].


## 2. Basic Description of Assessment Issues in Physical Education

There is a multi index evaluation systemxij=xjxi, i=1,2,…,n;j=1,2,…,m composed of u1,u2,…,un, m evaluated objects and x1,x2,…,xm indexes, which is about index xj for the evaluated object ui; observed value of the evaluation data matrix (decision matrix) can be expressed as(1)A=xijn×m=x11x12⋯x1mx21x22⋯x2m…⋯⋯⋯xn1xn2⋯xnm,where m, n ≥ 3, and the data in A is the normalized data after preprocessing the physical education evaluation process we describe as a general transformation:(2)yi=fxi1,xi2,…,xin,i∈N,where f is the positive transformation function; yi is the comprehensive evaluation value of the object ui being evaluated, and u1,u2,…,un is ranked according to the u1,u2,…,un value from the largest to the smallest, to complete the u1,u2,…,un comparison of the advantages and disadvantages.


## 3. Description of the Autonomous Strengths Assessment

Hypothesis 1.
is that each of the evaluated subjects has the dual objective of “widening the gap between competitors” and “developing their own strengths,” and in doing so, highlights their own strengths in an integrated manner.
A quantitative description of the idea of autonomous strengths evaluation in hypothesis1:Definition.
αij, βij is the amount of column and row dominance of the evaluated object uii∈N on indicator xii∈N, respectively, and satisfies(3)αij=1n−1∑k≠ixij−xkj,i∈N,j∈M,k∈N,βij=1m−1∑p≠jxij−xip,i∈N,j∈M,p∈M.
If we letλij=μαij+ηβij,i∈N,j∈M, then we say that λij with is the amount of autonomous advantage of the evaluated object uii∈N with respect to indicator xji∈N, where μ is the competitive target coefficient and n is the developmental target coefficient μ,η∈0,1,μ+η=1 .
Column dominanceαiji∈N,j∈M reflects the difference in strength between the jth indicator of the evaluated object μi and the n-1 other evaluated objects as a whole, while row dominance βij reflects the difference in strength between the jth indicator of the evaluated object μi and the m-1 other indicators as a whole.


## 4. Stochastic Simulation Algorithm

### 4.1. Nonlinear Programming Problems Where the Objective Function Is Linear

This paper gives a simulated annealing and evolutionary planning algorithm for nonlinear planning problems with linear objective functions, which transforms problems with constraints into unconstrained ones. Numerical results confirm the high computational accuracy of the method and show good convergence, considering the following optimization problem (wherec is a vector):(4)mincTxs. t.gix≤0i=1,…,rAx≥bIn order to find a feasible solution that satisfies the constraint, we first solve the subproblem:(5)min,f=max0,gixi=1,…,r,s.t.cTx≤cTxk∗−ε,Ax≤b,where xk∗ is the optimal solution at k steps and ε is a small positive number. Let B=CTA, d=cTxk∗−εb, the constraint can be reduced to Bx ≤ d.Bx ≤d can be written as(6)b11x1+b12x2+⋯+b1nxn≤d1,b21x1+b22x2+⋯+b2nxn≤d2,⋯bm+1,1x1+bm+1,2x2+⋯+bm+1,nxn≤dm+1.


### 4.2. Simulated Annealing Algorithms for Nonlinear Programming Global Optimization Problems

Based on the upper and lower bounds of componentxLi, we propose a class of simulated annealing algorithm for solving subproblem (3). The specific steps of the algorithm are as follows: Algorithm 1:Algorithm 1:
Step 0: initialization: the maximum and minimum temperatures areTmax, Tmin, the number of iterations Lmax and the parameters are given respectively ε ＞ 0.Step 1: use the random process to obtain the initial value of the feasible solutionx0=x01,…,x0n, set T=Tmax,t=0,I=0. If f (x0) ≤ 0, then I = 1, y∗=x0. Otherwise, turn to step 2.Step 2: WhileT>Tmin do(a) whilet≤Lmax do(1)
randomly selectlt ∈ {1, 2, ..., n}, and give a uniformly distributed random number λ ∈[−1, 1]. For j = 1,..., n, if λ ＞ 0.zj=xtj+α×blt−xtj×λ,ifj=ltandblt≠∞,xtj+α×λ,ifj=lt,blt=∞,xtj,ifj≠lt.zj=xtj+α×xtj−alt×λ,ifj=ltandalt≠∞,xtj+α×λ,ifj=lt,alt=∞,xtj,ifj≠lt,wherealt and blt are lower and upper bounds, xtlt comes from Algorithm 1, α The initial value of is 1, if 444, then α = 1.(2)
letz = (Z (1),..., Z (n)), if f (z) ≤ 0, then I = 1, y  ∗ = Z. Algorithm 1 stops. Otherwise, turn (3).(3)
Takeη ∈ [0,1], if η≤min1,expfxt−fz/T, set xt+1=z, otherwise xt+1=xt, t=t+1.(b)Lmax=Lmax+d,t=0．(c)
by T =δ × T lower the temperature T. Where parameter D, β And δ is a known constant entered in advance.On the basis of algorithm 1, the initial value of random feasible solutionx0 is given so that k = 0. If I = 1, make xk+1∗=y∗,Tmax=T,Lmax=Lmax,k=k+1.If I = 0, thenxk∗ is the global optimal solution of problem (1).


### 4.3. Evolutionary Planning Algorithms for Nonlinear Programming Global Optimization Problems

This section gives an improved evolutionary planning algorithm for problem (2), where the adaptation value is taken as the objective function value as follows: (Algorithm 2)Algorithm 2:
Step 1: Givenμ initial values, let k = 1 and I = 0. Let the individuals be real-valued vector pairs xi,ηi,∀i∈1,…,μ [20–22].Step 2: Calculate the individual adaptation value. If∃i∈i,…,μ,fxi≤0, then I=1,y∗=xi otherwise turn Step 3.Step 3: For each parentxi,ηi,∀i=1,…,μ, generate a child xi′,ηi′ according to the following steps: Randomly select li from the set { 1, 2, ..., n} to generate a uniformly distributed random parameter λ in the interval [- 1, 1]. For j = 1, ..., n, if λ > 0xi′j=xij+ηibj−xijλ,ifj=liandbli≠∞,xij+ηiλ,iff=li,bli=∞,xij,ifj≠li.xi′j=xij+ηixij−aliλifj=liandali≠∞,xij+ηiλifj=li,ali=∞,xijifj≠li,The initial value ofη is 1 if ηi<10−4, then ηi  = 1.


## 4.1. Nonlinear Programming Problems Where the Objective Function Is Linear

This paper gives a simulated annealing and evolutionary planning algorithm for nonlinear planning problems with linear objective functions, which transforms problems with constraints into unconstrained ones. Numerical results confirm the high computational accuracy of the method and show good convergence, considering the following optimization problem (wherec is a vector):(4)mincTxs. t.gix≤0i=1,…,rAx≥bIn order to find a feasible solution that satisfies the constraint, we first solve the subproblem:(5)min,f=max0,gixi=1,…,r,s.t.cTx≤cTxk∗−ε,Ax≤b,where xk∗ is the optimal solution at k steps and ε is a small positive number. Let B=CTA, d=cTxk∗−εb, the constraint can be reduced to Bx ≤ d.Bx ≤d can be written as(6)b11x1+b12x2+⋯+b1nxn≤d1,b21x1+b22x2+⋯+b2nxn≤d2,⋯bm+1,1x1+bm+1,2x2+⋯+bm+1,nxn≤dm+1.


## 4.2. Simulated Annealing Algorithms for Nonlinear Programming Global Optimization Problems

Based on the upper and lower bounds of componentxLi, we propose a class of simulated annealing algorithm for solving subproblem (3). The specific steps of the algorithm are as follows: Algorithm 1:Algorithm 1:
Step 0: initialization: the maximum and minimum temperatures areTmax, Tmin, the number of iterations Lmax and the parameters are given respectively ε ＞ 0.Step 1: use the random process to obtain the initial value of the feasible solutionx0=x01,…,x0n, set T=Tmax,t=0,I=0. If f (x0) ≤ 0, then I = 1, y∗=x0. Otherwise, turn to step 2.Step 2: WhileT>Tmin do(a) whilet≤Lmax do(1)
randomly selectlt ∈ {1, 2, ..., n}, and give a uniformly distributed random number λ ∈[−1, 1]. For j = 1,..., n, if λ ＞ 0.zj=xtj+α×blt−xtj×λ,ifj=ltandblt≠∞,xtj+α×λ,ifj=lt,blt=∞,xtj,ifj≠lt.zj=xtj+α×xtj−alt×λ,ifj=ltandalt≠∞,xtj+α×λ,ifj=lt,alt=∞,xtj,ifj≠lt,wherealt and blt are lower and upper bounds, xtlt comes from Algorithm 1, α The initial value of is 1, if 444, then α = 1.(2)
letz = (Z (1),..., Z (n)), if f (z) ≤ 0, then I = 1, y  ∗ = Z. Algorithm 1 stops. Otherwise, turn (3).(3)
Takeη ∈ [0,1], if η≤min1,expfxt−fz/T, set xt+1=z, otherwise xt+1=xt, t=t+1.(b)Lmax=Lmax+d,t=0．(c)
by T =δ × T lower the temperature T. Where parameter D, β And δ is a known constant entered in advance.On the basis of algorithm 1, the initial value of random feasible solutionx0 is given so that k = 0. If I = 1, make xk+1∗=y∗,Tmax=T,Lmax=Lmax,k=k+1.If I = 0, thenxk∗ is the global optimal solution of problem (1).


## 4.3. Evolutionary Planning Algorithms for Nonlinear Programming Global Optimization Problems

This section gives an improved evolutionary planning algorithm for problem (2), where the adaptation value is taken as the objective function value as follows: (Algorithm 2)Algorithm 2:
Step 1: Givenμ initial values, let k = 1 and I = 0. Let the individuals be real-valued vector pairs xi,ηi,∀i∈1,…,μ [20–22].Step 2: Calculate the individual adaptation value. If∃i∈i,…,μ,fxi≤0, then I=1,y∗=xi otherwise turn Step 3.Step 3: For each parentxi,ηi,∀i=1,…,μ, generate a child xi′,ηi′ according to the following steps: Randomly select li from the set { 1, 2, ..., n} to generate a uniformly distributed random parameter λ in the interval [- 1, 1]. For j = 1, ..., n, if λ > 0xi′j=xij+ηibj−xijλ,ifj=liandbli≠∞,xij+ηiλ,iff=li,bli=∞,xij,ifj≠li.xi′j=xij+ηixij−aliλifj=liandali≠∞,xij+ηiλifj=li,ali=∞,xijifj≠li,The initial value ofη is 1 if ηi<10−4, then ηi  = 1.


## 5. Application Examples

This paper uses data from the evaluation of the teaching quality of physical education teachers at a university, with the aim of analysing the factors affecting the quality of physical education. Table1 shows that the indicators of teaching quality evaluation are divided into five items based on teaching effectiveness, teaching content, teaching attitudes, and teaching methods [23–25]. It is assumed here that K1: teaching attitude, K2: teaching content, K3: teaching programme, K4: teaching effectiveness, and K5: evaluation result are the data of five training samples, and the evaluation grades are A: excellent (90–100), B: good (80–90), C: moderate (70–80), D: pass (60–70), and E: fail (<60).Table 1
Teaching quality evaluation form.
Teacher numberEvaluating indicatorEvaluation resultsK1K2K3K41BACCGood2BBCBGood3CCAASecondary4CBBCSecondary5AABBGood6BCBCGood7CABCSecondary8BBACExcellent9BCCASecondary10ABBCGoodThe information entropy of each attribute is calculated first. For K1, there are {1, 2, 6, 8, 9} (3 good, 1 moderate and 1 excellent), {3, 4, 7 (3 moderate), and {5, 10} (2 good) for the evaluation of teaching attitude. Then, the information entropy of K1 is calculated as follows:(7)EK1=15×3.53−335+33+1.53−135+13+1.53−135+13=0.3445,EK1=13×3.33−333+33=0,EK1=12×2.23−232+23=0.The information entropy of the teaching attitude K1 is(8)EK1=510.0.3445+310.0+210.0=0.1772.Similarly, we obtain the information entropy of other attributes:(9)EK2=0.2947,EK3=0.2486,EK4=0.2433.Comparing the entropy of each attribute, the ranking isE(K1)<E(K4)<E(K3)<E(K2), so K1 is chosen as the root node and three branches are created, A, B, and C. According to the flow of the algorithm, the test attributes are selected in turn under the branches and nodes are created until the end of the sample division [26].Based on the decision tree created in Figure1, we can see that each branch represents the combined set of attributes tested, and the whole decision tree represents the combined destructions.Figure 1
Evaluation decision tree.It is clear from this analysis that teaching attitude is the most important aspect of teaching. When the teaching attitude is excellent, the result of teaching evaluation is good; when the teaching attitude is medium, the result of teaching evaluation is medium; when the teaching attitude is good, the result of teaching evaluation also depends on the teaching programme, but the teaching attitude is still the dominant factor [27, 28].Evaluation of physical education teaching in universities is ranked. After the calculation of the autonomous evaluation method to obtain the results so thatb = 2, we can get the dominant weight vector ω = 0.4546, 0.2908, 0.1637, 0.0726 to get the ideal order of ranking as(10)u1≻0.9891u2≻0.8971u8≻0.5283u4≻0.0.7272u5≻0.8306u3≻0.5542u6≻0.6636u7≻0.5464u9≻0.7277u10.The results, e.g.u8≻0.5283u4, do not mean that u8 is definitely better than u4, u4 still has a 04717 probability of being better than u8, and this form of conclusion is not suitable for making absolute judgements of superiority between some of the evaluated objects at the intersection of competencies. This form of evaluation gives the most reliable ranking of superiority between objects, but at the same time accommodates a variety of possible rankings (e.g. u8≻0.5283u4 is equivalent to u4≻0.4717u8).This form of evaluation allows multiple absolute evaluation findings to be embedded in a single probabilistic evaluation finding, avoiding the subjective assumptions caused by the “multiple evaluation findings nonconsistency phenomenon” [23–25].Table2 shows the parameter settings for the numerical calculations, while Tables 3 and 4 show the results of the numerical calculations.Table 2
Parameter setting values for the simulated annealing algorithm.
TmaxTminδLmaxdαβ110.0010.9743111.02Table 3
Calculation results based on the simulated annealing algorithm and penalty function method proposed in the paper.
FunctionsOptimal solutionWorst valueAverage best valuePaper method−10.945−10.874−10.912Penalty function method−10.401−10.271−10.397Table 4
Calculation results based on the evolutionary planning algorithm and penalty function method in this paper.
FunctionsEvolutionary algebraPopulation sizeOptimal solutionWorst valueAverage best valuePaper method100010−10.967−10.875−10.926Penalty function method100010−10.567−10.156−10.478


## 6. Conclusions

In this paper, we address the problem of the absolute nature of the evaluation of the merits of traditional physical education classrooms and the inconsistency of the findings of multiple evaluations, and construct a method to evaluate the merits of the evaluated students by highlighting their own strengths. The validity of the method is verified by calculation, and the evaluation conclusion with probability information is obtained.



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