Research on the teaching quality evaluation of physical education with intuitionistic fuzzy TOPSIS method

2.1 IFSs

Definition 1 [2.] An IFS on the universe X is an object: I = { 〈 x , μ I ( x ) , ν I ( x ) 〉 | x ∈ X }(1) where μ _I  (x) ∈ [0, 1] is named as the “degree of membership of I” and ν _I  (x) ∈ [0, 1] is named as the “degree of non-membership of I”, and μ _I  (x), ν _I  (x) meet the condition: 0 ⩽ μ _I  (x) + ν _I  (x) ⩽ 1, ∀ x ∈ X. Then for x ∈ X, π _I  (x) = 1 - (μ _I  (x) + ν _I  (x)) is called the degree of hesitancy of x in I.

Definition 2 [39.] Let I₁ =  (μ₁, ν₁) and I₂ =  (μ₂, ν₂) be two IFNs, the operation formula is: I 1 ⊕ I 2 = ( μ 1 + μ 2 - μ 1 μ 2 , ν 1 ν 2 )(2) I 1 ⊗ I 2 = ( μ 1 μ 2 , ν 1 + ν 2 - ν 1 ν 2 )(3) λ I 1 = ( 1 - ( 1 - μ 1 ) λ , ν 1 λ ) , λ > 0(4) I 1 λ = ( μ 1 λ , 1 - ( 1 - ν 1 ) λ ) , λ > 0(5)

Derived from the Definition 2, the following properties of the operation laws can be obtained.

I₁ ⊕ I₂ = I₂ ⊕ I₁, I₁ ⊗ I₂ = I₂ ⊗ I₁, ((I₁)  ^{λ ₁} )  ^{λ ₂}  = (I₁)  ^{λ ₁ λ ₂} ;

Definition 3 [40.] Let I₁ =  (μ₁, ν₁) and I₂ =  (μ₂, ν₂) be IFNs, the score and accuracy functions of I₁ and I₂ is:

S ( I 1 ) = μ 1 + μ 1 ( 1 - μ 1 - ν 1 ) , S ( I 2 ) = μ 2 + μ 2 ( 1 - μ 2 - ν 2 )(6) H ( I 1 ) = μ 1 + ν 1 , H ( I 2 ) = μ 2 + ν 2(7)

For two IFNs I₁ and I₂, regarding on the Definition 3, then

if s (I₁) < s (I₂) ,  then I₁ < I₂;

Definition 4 [41.] Let I₁ =  (μ₁, ν₁) and I₂ =  (μ₂, ν₂) be IFNs, the novel Euclidean distances between two IFNs can be defined as follows: IFED ( I 1 , I 2 ) = 1 6 ( ( ℓ 1 ) 2 + ( ℓ 2 ) 2 + ( ℓ 3 ) 2 )(8) where ℓ 1 = | μ 1 - μ 2 | + | ν 1 - ν 2 | + | ( μ 1 + 1 - ν 1 ) - ( μ 2 + 1 - ν 2 ) | 2

ℓ 2 = π 1 + π 2 2

ℓ 3 = max ( | μ 1 - μ 2 | , | ν 1 - ν 2 | , | π 1 - π 2 | 2 )

Under the IFSs, some fused operators will be introduced in this section, including intuitionistic fuzzy weighted averaging (IFWA) fused operator and intuitionistic fuzzy weighted geometric (IFWG) fused operator.

Definition 5 [39.] Let I _j  =  (μ _{I j} , ν _{I j} ) (j = 1, 2,  ⋯ , n) be a set of IFNs, the IFWA operator is: IFWA ω ( I 1 , I 2 , … , I n ) = ⊕ j = 1 n ( ω j I j )(9) where ω =  (ω₁, ω₂, . . . , ω _n )  ^T be the weight vector of I _j  (j = 1, 2, . . . , n) and ω j > 0 , ∑ j = 1 n ω j = 1 .

Derived from the Definition 5, the result could be obtained:

Theorem 1 [39.] The fused value by IFWA operator is also a IFN, where

IFWA ω ( I 1 , I 2 , … , I n ) = ⊕ j = 1 n ( ω j I j ) = ( 1 - ∏ j = 1 n ( 1 - μ I j ) ω j , ∏ j = 1 n ( ν I j ) ω j )(10) where ω =  (ω₁, ω₂, . . . , ω _n )  ^T be the weight vector of I _j  (j = 1, 2, . . . , n) and ω j > 0 , ∑ j = 1 n ω j = 1 .

Definition 6 [39.] Let I _j  (j = 1, 2, ⋯ , n) be a set of IFNs, the IFWG operator is: IFWG ω ( I 1 , I 2 , … , I n ) = ⊗ j = 1 n ( I j ) ω j(11) where ω =  (ω₁, ω₂, . . . , ω _n )  ^T be the weight vector of I _j  (j = 1, 2, . . . , n) and ω j > 0 , ∑ j = 1 n ω j = 1

From the Definition 6, the result can be obtained:

Theorem 2 [39.] The fused value by IFWG operator is a IFN,

IFWG ω ( I 1 , I 2 , … , I n ) = ⊗ j = 1 n ( I j ) ω j = ( ∏ j = 1 n ( μ I j ) ω j , 1 - ∏ j = 1 n ( 1 - ν I j ) ω j )(12) where ω =  (ω₁, ω₂, . . . , ω _n )  ^T be the weight vector of I _j  (j = 1, 2, . . . , n) and ω j > 0 , ∑ j = 1 n ω j = 1 .

4.1 An empirical example

Science and technology is the first productive force of social development, international competition will eventually be the talent competition, in the strategy of rejuvenating the country through science and policy requirements and new era demand, universities are responsible for the important task of cultivating talents, cultivate outstanding talents of high quality and highly sophisticated science and technology workers. This is not only keep the fighting capacity of the premise for the future of the country, is everlasting higher education principles. China as a traditional great power of education, school education continuous stretches for thousands of years, in the long river of history continue to accumulate valuable educational experience, rid themselves of some does not conform to the era of education concepts and methods. Since the founding of the people’s Republic of China, PE in Colleges and universities of our country from theory to practice has been a rapid development, and achieved certain results, this and the national culture policy of talent are inseparable colleges and universities, but also will continue to strengthen itself ahead of the results. In the national promulgation and implementation of “sustainable development strategy” and of cultural and educational undertakings of physical culture and sports to develop in an all-round way is great and decision-making in pointed out that the next phase of the main task is the integration of the diversified development of the education, economy and culture. As colleges and universities to develop the cradle of high-tech talent, in all respects to the country and society carrying a large number of outstanding young talent, and a lot of achievements in scientific research also from colleges and universities emerged, in a certain extent also confirmed the importance of higher education and social acceptance. The higher sports colleges and universities each year for the country trained a large number of outstanding teachers, social sports workers and sports organizer, plays an indispensable role in the school education, in the excavation and cultivation of sports talents play an important role, as well as a comprehensive fitness program implementation, provides a lot of scientific material. Public sports are one of the important courses in Institutions of higher learning stage, the course to carry out in a certain sense to mobilize the enthusiasm and the necessity of students’ participation in sports, but also enrich the students’ learning content, the same to enhance the physical fitness of students also played a key role. In this chapter, an empirical application about teaching quality evaluation of Physical Education will be provided by making use of IF-TOPSIS method. There are five potential physical education schools F _i  (i = 1, 2, 3, 4, 5) preparing to evaluate their teaching quality. In order to assess these physical education schools fairly, three experts H = { H₁, H₂, H₃ } (expert’s weight h = (0 . 35, 0 . 32, 0.33) are invited. All experts depict their assessment information through four subsequent attributes: ①R₁ is teaching content; ②R₂ is the teachers’ specialization degree; ③R₃ is teachers’ artistic accomplishment; ④R₄ is teachers’ appreciation ability. Evidently, R₂ is the cost attribute, while R₁ R₃ and R₄ are the benefit attributes, and the decision making matrix are given in Tables 1–3.

Step 1. Based on the decision making information Q ( k ) = ( q ij k ) m × n ( i = 1 , 2 , … , m , j = 1 , 2 , … , n ) given in Tables 1–3 and the expert’s weights h = (0 . 35, 0 . 32, 0.33), we can derive the overall matrix Q = ( q ij ) m × n ( i = 1 , 2 , … , m , j = 1 , 2 , … , n ) according to Equation (16), the computing results are list in Table 4.

Step 2. Normalize the intuitionistic fuzzy evaluation matrix Q = [ q ij ] m × n to Q N = [ q ij N ] m × n .

Step 3. Decide the attribute weights r _j  (j = 1, 2,  … , n) by CRITIC method as listed in Table 6.

Step 4. Calculate the IF-PIS A j + and the IF-NIS A j - according to Equations (20) and (21); A j + = { ( 0.6862 , 0 . 1569 ) , ( 0.4924 , 0 . 2844 ) , ( 0.4413 , 0 . 1625 ) , ( 0.5054 , 0 . 2042 ) } A j - = { ( 0.3169 , 0 . 2763 ) , ( 0.1986 , 0 . 5885 ) , ( 0.2051 , 0 . 3945 ) , ( 0.3078 , 0 . 4041 ) }

Step 5. Compute the distances d i + and d i - , the results are list as follows. d 1 + = 0.1879 , d 2 + = 0.1765 , d 2 + = 0.1065 , d 2 + = 0.2099 , d 2 + = 0.2074 ; d 1 - = 0.1102 , d 2 - = 0.1618 , d 2 - = 0.2424 , d 2 - = 0.1300 , d 2 - = 0.1331 .

Step 6. Compute each alternative’s correlation coefficient C _i by Equation (24), the results are list as: C 1 = 0.3697 , C 2 = 0.4781 , C 3 = 0.6948 , C 4 = 0.3824 , C 5 = 0.3908

Step 9. Relying on the C _i , all the alternatives can be ordered, the higher value of C _i is, the best alternative selected will be. Evidently, the order is F₃ > F₂ > F₅ > F₄ > F₁ and F₃ is the optimal physical education school.

In this section, our defined method is made comparison with some other methods to show its superiority.

First of all, our defined method is compared with IFWA and IFWG operators [39]. For the IFWA operator, the calculating result is: S (F₁) = 0.0790, S (F₂) = 0.1411, S (F₃) = 0.3351, S (F₄) = 0.0443, S (F₅) = 0.0403. Thus, the ranking order isF₃ > F₂ > F₁ > F₄ > F₅. For the IFWG operator, the calculating result is S (F₁) =  - 0 . 0102, S (F₂) = 0.1227, S (F₃) = 0.3018, S (F₄) = 0.0352, S (F₅) = 0.0099. So the ranking order isF₃ > F₂ > F₄ > F₅ > F₁.

What’s more, our defined method is compared with the IF-VIKOR method [43]. Then we can obtain the calculating result. The closest ideal score values are: CI^* (F₁) = 0.9045, CI^* (F₂) = 0.6782, CI^* (F₃) = 0.0000, CI^* (F₄) = 0.9832, CI^* (F₅) = 0.9576. And the farthest worst score values are: CI^- (F₁) = 0.0128, CI^- (F₂) = 0.3459, CI^- (F₃) = 1.0000, CI^- (F₄) = 0.0168, CI^- (F₅) = 0.0000. Then each alternatives’ relative closeness are calculated as: DRC₁ = 0.9860, DRC₂ = 0.6622, DRC₃ = 0.0000, DRC₄ = 0.9832, DRC₅ = 1.0000. Hence, the order is F₃ > F₂ > F₄ > F₁ > F₅.

Besides, our defined method is compared with IF-GRA method [44]. Then we can have the calculating result. The grey relational grades of each alternative are: γ₁ = 0.8099, γ₂ = 0.8775, γ₃ = 0.9827, γ₄ = 0.8240, γ₅ = 0.8172. Therefore, the order is: F₃ > F₂ > F₄ > F₅ > F₁.

In the end, our defined method is also compared with IF- CODAS method [45]. Then we can have the calculating result. The total assessment score (AS) of each alternative is calculated as: AS₁ = -0.8019, AS₂ = 0.1404, AS₃ = 1.4818, AS₄ = -0.3860, AS₅ = -0.4343. Therefore, the order is F₃ > F₂ > F₄ > F₅ > F₁.

Eventually, the results of these methods are in Table 7.

From Table 7, it is evidently that the best alternative is F₃, while the worst alternative is F₁ in most situations. In other words, these methods’ order are slightly different. Different methods can tackle MAGDM from different angles. IFWA and IFWG operator emphasis to fuse evaluation information. The modified IF-VIKOR method emphasis the closest to the PIS and the farthest to the NIS. The IF-GRA method emphasis the degree of similarity between two sequences. However, compared with the above methods, our designed method is more precision, since it considers the distances of each alternative from PIS and NIS. What’s more, compared with the IF-CODAS method, our designed method utilizes novel distance measures and CRITIC method. The novel distance measures can not only reflect intuitionistic fuzzy information more comprehensive but also take waver in IFSs into consideration and do not generate counterintuitive situations. The CRITIC method can minimize subjective randomness while the criteria weights are determined.