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

Abstract

Under the background of the national fitness craze, the demand space for social sports professionals is constantly expanding. However, according to the author’s investigation, the overall situation shows that the number of high-quality social sports professionals in Chinese colleges and universities is relatively small. Among them, the unsound teaching quality evaluation system of social sports major is one of the important reasons affecting the cultivation of high-quality talents, so it is imperative to construct a sound teaching quality evaluation system of social sports major. At the same time, the perfect social physical education teaching quality evaluation system is an important basis for teachers’ teaching job evaluation and strengthening teachers’ management. And it is frequently considered as a multi-attribute group decision-making (MAGDM) issue. Thus, a novel MAGDM method is needed to tackle it. Depending on the conventional TOPSIS method and intuitionistic fuzzy sets (IFSs), this essay designs a novel intuitive distance based IF-TOPSIS method for teaching quality evaluation of physical education. First of all, a related literature review is conducted. What’s more, some necessary theories related to IFSs are briefly reviewed. In addition, since subjective randomness frequently exists in determining criteria weights, the weights of criteria are decided objectively by utilizing CRITIC method. Afterwards, relying on novel distance measures between IFNs, the conventional TOPSIS method is extended to the intuitionistic fuzzy environment to calculate assessment score of each alternative. Eventually, an application about teaching quality evaluation of physical education and some comparative analysis have been given. The results think that the designed method is useful for teaching quality evaluation of physical education.

Keywords

Multi-attribute group decision-making (MAGDM)intuitionistic fuzzy sets (IFSs)TOPSIS method CRITIC method teaching quality evaluation physical education

1 Introduction

Zadeh [1] initially presented the theory of fuzzy sets (FSs). Atanassov [2] defined the concept of intuitionistic fuzzy sets (IFSs). Garg [3]presented a method related to MAGDM on the basis of intuitionistic fuzzy multiplicative preference and defined several geometric operators. Gou, Xu and Lei [4] defined some exponential operational law for IFNs. He, He and Huang [5]integrated the power averaging with IFSs. Liu, Liu and Chen [6] built some intuitionistic fuzzy BM fused operators with Dombi operations. Gupta, Arora and Tiwari [7]extended the fuzzy entropy to IFSs. Li and Wu [8]presented the intuitionistic fuzzy cross entropy distance. Khan, Lohani and Ieee [9]defined similarity measure about IFNs. Li, Liu, Liu, Su and Wu [10]gave a grey target decision making with IFNs. Bao, Xie, Long and Wei [11]defined prospect theory and evidential reasoning method under IFSs. Jin, Ni, Chen and Li [12]defined two GDM methods which can obtain the normalized intuitionistic fuzzy priority weights from IFPRs on the basis of the order consistency and the multiplicative consistency. Chen, Cheng and Lan [13] developed TOPSIS method and similarity measures under IFSs. Zhao, Wei, Wei and Wu [14] improved TODIM method for IF-MAGDM based on cumulative prospect theory. Gupta, Mehlawat, Grover and Chen [15]modified the SIR method and combined it with IFSs. Hao, Xu, Zhao and Zhang [16]presented a theory of decision field for IFSs. Krishankumar, Arvinda, Amrutha, Premaladha, Ravichandran and Ieee [17]integrated AHP with IFSs to design a GDM method for effective cloud vendor selection. Rouyendegh [18]used the ELECTRE method in IFSs to tackle some MCDM issues. Cali and Balaman [19]extended ELECTRE I with VIKOR method in IFSs to reflect the decision makers’ preferences. Xiao, Zhang, Wei, Wu, Wei, Guo and Wei [20] gave the green supplier selection in steel industry with intuitionistic fuzzy Taxonomy method. Liang, He, Wang, Chen and Li [21]extended MABAC method to IFSs through distance measures.

With the comprehensive development of the quality education, more and more colleges begin to pay attention to the physical education for the college students. Evaluating the teaching quality for the physical education can enhance the physical teaching management of the colleges. Huang and Feng [22] connected the AHP method with the TOPSIS method and proposed the AHP-TOPSIS method in order to evaluate accurately the teaching quality for the college physical education. Wu [23] drawn some merits from previous research and developed the teaching quality evaluation system and model of physical education in colleges and universities. Jing [24] introduced the “Johari window” model theory and discussed how to put forward the connotation, mode, methods of the “Johari window” mode methods with application in assessment of instruction in physical education. Ma [25] analyzed the fuzzy comprehensive evaluation of the teaching quality of public physical education in colleges and universities. Chen and Huang [26] dealt with the AHP and put forward a new FAHP according to the characteristics of PE teaching and by referring to the relevant knowledge of fuzzy mathematics. Huang and Chen [27] defined a new comprehensive evaluation method of teaching quality for public physical education in colleges and universities. Han [28] introduced AHP method to the evaluation of PE teaching quality in colleges. Zeng [29] analyzed the development status of the teaching quality evaluation system of PE curriculum in colleges and proposed a mathematical model for evaluating the quality of PE teaching in colleges.

TOPSIS was initially developed by Hwang and Yoon [30] to solve MAGDM issues. Compared with other MAGDM, TOPSIS method can consider the distances degree of every alternative from PIS and NIS. This method has been used to various fuzzy setting [31 –38]. This paper’s goal is to use TOPSIS method to IFSs and build a new decision-making model for actual MADM problems. Thus, the motivation of this study is: (1) the IF-TOPSIS method is designed based on the novel Euclidean distances; (2) the weights of criteria are decided objectively by utilizing CRITIC method; (3) an application about teaching quality evaluation of physical education and some comparative analysis have been given. In order to do so, the reminder of this paper proceeds. Some concept of IFSs is reviewed in section 2. The improved TOPSIS method is defined with IFNs and the calculating steps is simply listed in section 3. An empirical application about fire safety assessment based on the high building is given to show the superiority of this designed approach and some comparative analyses are given to prove the merits of such method in section 4. At last, we make an overall conclusion of such work in section 5.

2 Preliminaries

2.1 IFSs

Definition 1 [2.] An IFS on the universe X is an object: $I = {〈 x, μ_{I} (x), ν_{I} (x) 〉 | x \in X}$ (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: $I_{1} \oplus I_{2} = (μ_{1} + μ_{2} - μ_{1} μ_{2}, ν_{1} ν_{2})$ (2) $I_{1} \otimes I_{2} = (μ_{1} μ_{2}, ν_{1} + ν_{2} - ν_{1} ν_{2})$ (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};

λ (I₁ ⊕ I₂) = λI₁ ⊕ λI₂, (I₁ ⊗ I₂) ^λ = (I₁) ^λ ⊗ (I₂) ^λ;

λ₁I₁ ⊕ λ₂I₁ = (λ₁ + λ₂) I₁, (I₁) ^{λ
₁} ⊗ (I₁) ^{λ
₂} = (I₁) ^{(λ₁+λ₂)} .

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

$\begin{matrix} S (I_{1}) & = μ_{1} + μ_{1} (1 - μ_{1} - ν_{1}), \\ S (I_{2}) & = μ_{2} + μ_{2} (1 - μ_{2} - ν_{2}) \end{matrix}$ (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;

if s (I₁) > s (I₂) , then I₁ > I₂;

if s (I₁) = s (I₂) , h (I₁) < h (I₂) , then I₁ < I₂;

if s (I₁) = s (I₂) , h (I₁) > h (I₂) , then I₁ > I₂;

if s (I₁) = s (I₂) , h (I₁) = h (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: $IFED (I_{1}, I_{2}) = \sqrt{\frac{1}{6} ({(ℓ_{1})}^{2} + {(ℓ_{2})}^{2} + {(ℓ_{3})}^{2})}$ (8) where $ℓ_{1} = \frac{| μ_{1} - μ_{2} | + | ν_{1} - ν_{2} | + | (μ_{1} + 1 - ν_{1}) - (μ_{2} + 1 - ν_{2}) |}{2}$

$ℓ_{2} = \frac{π_{1} + π_{2}}{2}$

$ℓ_{3} = max (| μ_{1} - μ_{2} |, | ν_{1} - ν_{2} |, \frac{| π_{1} - π_{2} |}{2})$

2.2 Intuitionistic fuzzy aggregation operators

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: ${IFWA}_{ω} (I_{1}, I_{2}, \dots, I_{n}) = \oplus_{j = 1}^{n} (ω_{j} I_{j})$ (9) where ω = (ω₁, ω₂, . . . , ω_n) ^T be the weight vector of I_j (j = 1, 2, . . . , n) and $ω_{j} > 0, \sum_{j = 1}^{n} ω_{j} = 1$ .

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

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

$\begin{matrix} {IFWA}_{ω} (I_{1}, I_{2}, \dots, I_{n}) = \oplus_{j = 1}^{n} \\ (ω_{j} I_{j}) = (1 - \prod_{j = 1}^{n} {(1 - μ_{I_{j}})}^{ω_{j}}, \prod_{j = 1}^{n} {(ν_{I_{j}})}^{ω_{j}}) \end{matrix}$ (10) where ω = (ω₁, ω₂, . . . , ω_n) ^T be the weight vector of I_j (j = 1, 2, . . . , n) and $ω_{j} > 0, \sum_{j = 1}^{n} ω_{j} = 1$ .

Definition 6 [39.] Let I_j (j = 1, 2, ⋯ , n) be a set of IFNs, the IFWG operator is: ${IFWG}_{ω} (I_{1}, I_{2}, \dots, I_{n}) = \otimes_{j = 1}^{n} {(I_{j})}^{ω_{j}}$ (11) where ω = (ω₁, ω₂, . . . , ω_n) ^T be the weight vector of I_j (j = 1, 2, . . . , n) and $ω_{j} > 0, \sum_{j = 1}^{n} ω_{j} = 1$

From the Definition 6, the result can be obtained:

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

$\begin{matrix} {IFWG}_{ω} (I_{1}, I_{2}, \dots, I_{n}) = \otimes_{j = 1}^{n} \\ {(I_{j})}^{ω_{j}} = (\prod_{j = 1}^{n} {(μ_{I_{j}})}^{ω_{j}}, 1 - \prod_{j = 1}^{n} {(1 - ν_{I_{j}})}^{ω_{j}}) \end{matrix}$ (12) where ω = (ω₁, ω₂, . . . , ω_n) ^T be the weight vector of I_j (j = 1, 2, . . . , n) and $ω_{j} > 0, \sum_{j = 1}^{n} ω_{j} = 1$ .

3 The TOPSIS method with Intuitionistic fuzzy information

In this section, we build the IF-TOPSIS method for MAGDM. The calculating steps of the designed method can be described subsequently.

Let R ={ R₁, R₂, … R_n } be the group of attributes, r = { r₁, r₂, … r_n } be the weight of attributes R_j, where $r_{j} \in [0, 1], j = 1, 2, \dots, n, \sum_{j = 1}^{n} r_{j} = 1$ . Assume H = { H₁, H₂, … H_l } be a set of DMs that have degree of h = { h₁, h₂, … h_l }, where h_k ∈ [0, 1], k = 1, 2, …, l . $\sum_{k = 1}^{l} h_{k} = 1$ . Let F = { F₁, F₂, … F_m } be a set of alternatives. And $Q = {(q_{ij})}_{m \times n}$ is the matrix with IFNs, $q_{ij}$ means the F_i for R_j. Subsequently, the specific calculating steps will be depicted.

Step 1. Build each DM’s matrix $Q^{(k)} = {(q_{ij}^{k})}_{m \times n}$ with IFNs and calculate the overall intuitionistic fuzzy decision matrix Q = (q_ij) _m×n. $Q^{(k)} = {[q_{ij}^{k}]}_{m \times n} = [\begin{matrix} q_{11}^{k} & q_{12}^{k} & \dots & q_{1 n}^{k} \\ q_{21}^{k} & q_{22}^{k} & \dots & q_{2 n}^{k} \\ ⋮ & ⋮ & ⋮ & ⋮ \\ q_{m 1}^{k} & q_{m 2}^{k} & \dots & q_{mn}^{k} \end{matrix}]$ (13) $Q = {[q_{ij}]}_{m \times n} = [\begin{matrix} q_{11} & q_{12} & \dots & q_{1 n} \\ q_{21} & q_{22} & \dots & q_{2 n} \\ ⋮ & ⋮ & ⋮ & ⋮ \\ q_{m 1} & q_{m 2} & \dots & q_{mn} \end{matrix}]$ (14) $q_{ij} = (1 - \prod_{k = 1}^{l} {(1 - μ_{q_{ij}^{k}})}^{h_{k}}, \prod_{k = 1}^{l} {(ν_{q_{ij}^{k}})}^{h_{k}})$ (15) where $q_{ij}^{k}$ is the assessment value of F_i (i = 1, 2, … , m) on the basis of the attribute R_j (j = 1, 2, …, n) and the DM H_k (k = 1, 2, …, l).

Step 2. Normalize the overall matrixQ = (q_ij) _m×n with IFNs to $Q^{N} = {[q_{ij}^{N}]}_{m \times n}$ . $q_{ij}^{N} = {\begin{matrix} (μ_{ij}, ν_{ij}) R_{j} is a benefit criterion \\ (ν_{ij}, μ_{ij}), R_{j} is a cost criterion \end{matrix}$ (16)

Step 3. Employ CRITIC method to determine the weighting of attributes.

CRiteria Importance Through Intercriteria Correlation (CRITIC) method will be proposed in this part which is utilized to decide attributes’ weights. This method was initially defined by Diakoulaki, Mavrotas and Papayannakis [42]. Subsequently, the calculating steps of such method will be presented.

(1) Depending on the normalized overall matrix $Q^{N} = {(q_{ij}^{N})}_{m \times n}$ with IFNs, the correlation coefficient between attributes can be defined.

$\begin{matrix} {IFCC}_{jt} \\ = \frac{\sum_{i = 1}^{m} (S (q_{ij}^{N}) - S (q_{j}^{N})) (S (q_{it}^{N}) - S (q_{t}^{N}))}{\sqrt{\sum_{i = 1}^{m} {(S (q_{ij}^{N}) - S (q_{j}^{N}))}^{2}} \sqrt{\sum_{i = 1}^{m} {(S (q_{it}^{N}) - S (q_{t}^{N}))}^{2}}}, \\ j, t = 1, 2, \dots, n \end{matrix}$ (17)

where $S (q_{j}^{N}) = \frac{1}{m} \sum_{i = 1}^{m} S (q_{ij}^{N})$ and $S (q_{t}^{N}) = \frac{1}{m} \sum_{i = 1}^{m} S (q_{it}^{N})$ .

(2) Obtain attributes’ standard deviation.

$\begin{matrix} {IFSD}_{j} & = \sqrt{\frac{1}{m - 1} \sum_{i = 1}^{m} {(S (q_{ij}^{N}) - S (q_{j}^{N}))}^{2}}, \\ j = 1, 2, \dots, n \end{matrix}$ (18) where $S (q_{j}^{N}) = \frac{1}{m} \sum_{i = 1}^{m} S (q_{ij}^{N})$ .

(3) Obtain the attributes’ weights.

$\begin{matrix} r_{j} = & \frac{{IFSD}_{j} \sum_{t = 1}^{n} (1 - {IFCC}_{jt})}{\sum_{j = 1}^{n} ({IFSD}_{j} \sum_{t = 1}^{n} (1 - {IFCC}_{jt}))}, \\ j = 1, 2 \dots, n \end{matrix}$ (19) where r_j ∈ [0, 1] and $\sum_{j = 1}^{n} r_{j} = 1$ .

Step 4. Define the intuitionistic fuzzy PIS (IF-PIS) $A_{j}^{+}$ and the intuitionistic fuzzy NIS (IF-NIS) $A_{j}^{-}$ as: $A_{j}^{+} = max_{i} (q_{ij}^{N}) = (max_{i} (μ_{ij}^{N}), min_{i} (ν_{ij}^{N}))$ (20) $A_{j}^{-} = min_{i} (q_{ij}^{N}) = (min_{i} (μ_{ij}^{N}), max_{i} (ν_{ij}^{N}))$ (21) where $q_{ij}^{N}$ denotes the normalized intuitionistic fuzzy nummbers.

Step 5. Compute the positive distances $d_{i}^{+}$ between each alternative and IF-PIS and the negative distances $d_{i}^{-}$ between each aternative and IF-PIS as: $d_{i}^{+} = \sum_{j = 1}^{n} r_{j} IFED (q_{ij}^{N}, A_{j}^{+}), i = 1, 2, \dots, m$ (22) $d_{i}^{-} = \sum_{j = 1}^{n} r_{j} IFED (q_{ij}^{N}, A_{j}^{-}), i = 1, 2, \dots, m$ (23) where $IFED (q_{ij}^{N}, A_{j}^{+})$ and $IFED (q_{ij}^{N}, A_{j}^{-})$ denote the intuitionistic fuzzy Euclidean distances given in Definition 4, and r_j is the weight of attributes.

Step 6. Compute each alternative’s correlation coefficient as: $C_{i} = \frac{d_{i}^{-}}{d_{i}^{-} + d_{i}^{+}}, i = 1, 2, \dots, m$ (24)

Step 7. According to the C_i (i = 1, 2, …, m). The highest value of C_i (i = 1, 2, …, m) is, the optimal alternative is designed.

4 The empirical example and comparative analysis

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.

Table 1
Decision making information given by H₁

R₁ R₂ R₃ R₄

F₁ (0.6,0.3) (0.5,0.2) (0.3,0.6) (0.4,0.2)

F₂ (0.5,0.4) (0.2,0.4) (0.1,0.3) (0.7,0.1)

F₃ (0.8,0.1) (0.4,0.5) (0.5,0.4) (0.4,0.3)

F₄ (0.5,0.4) (0.3,0.3) (0.1,0.3) (0.5,0.2)

F₅ (0.4,0.5) (0.2,0.4) (0.2,0.5) (0.4,0.1)

	R₁	R₂	R₃	R₄
F₁	(0.6,0.3)	(0.5,0.2)	(0.3,0.6)	(0.4,0.2)
F₂	(0.5,0.4)	(0.2,0.4)	(0.1,0.3)	(0.7,0.1)
F₃	(0.8,0.1)	(0.4,0.5)	(0.5,0.4)	(0.4,0.3)
F₄	(0.5,0.4)	(0.3,0.3)	(0.1,0.3)	(0.5,0.2)
F₅	(0.4,0.5)	(0.2,0.4)	(0.2,0.5)	(0.4,0.1)

Table 2

Decision making information given by H₂

	R₁	R₂	R₃	R₄
F₁	(0.4,0.2)	(0.3,0.4)	(0.5,0.1)	(0.2,0.7)
F₂	(0.3,0.5)	(0.5,0.1)	(0.4,0.2)	(0.3,0.3)
F₃	(0.6,0.2)	(0.3,0.6)	(0.5,0.1)	(0.6,0.2)
F₄	(0.1,0.1)	(0.4,0.5)	(0.2,0.5)	(0.4,0.3)
F₅	(0.3,0.2)	(0.5,0.2)	(0.1,0.3)	(0.4,0.5)

Table 3

Decision making information given by H₃

	R₁	R₂	R₃	R₄
F₁	(0.7,0.2)	(0.8,0.1)	(0.4,0.2)	(0.3,0.5)
F₂	(0.4,0.1)	(0.3,0.6)	(0.5,0.2)	(0.4,0.3)
F₃	(0.6,0.2)	(0.2,0.4)	(0.3,0.1)	(0.5,0.4)
F₄	(0.4,0.5)	(0.2,0.2)	(0.4,0.1)	(0.3,0.4)
F₅	(0.5,0.2)	(0.1,0.2)	(0.3,0.4)	(0.5,0.3)

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

Table 4

Overall matrix with IFNS

	R₁	R₂	R₃	R₄
F₁	(0.5858,0.2305)	(0.5885,0.1986)	(0.4026,0.2353)	(0.3078,0.4041)
F₂	(0.4086,0.2719)	(0.3414,0.2934)	(0.3489,0.2305)	(0.5054,0.2042)
F₃	(0.6862,0.1569)	(0.3069,0.4924)	(0.4413,0.1625)	(0.5038,0.2897)
F₄	(0.3169,0.2763)	(0.3037,0.3090)	(0.2418,0.2458)	(0.4077,0.2862)
F₅	(0.4065,0.2756)	(0.2844,0.2549)	(0.2051,0.3945)	(0.4350,0.2405)

Step 2. Normalize the intuitionistic fuzzy evaluation matrix $Q = {[q_{ij}]}_{m \times n}$ to $Q^{N} = {[q_{ij}^{N}]}_{m \times n}$ .

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

Table 5

The normalized matrix with IFNS

	R₁	R₂	R₃	R₄
F₁	(0.5858,0.2305)	(0.1986, 0.5885)	(0.4026,0.2353)	(0.3078,0.4041)
F₂	(0.4086,0.2719)	(0.2934,0.3414)	(0.3489,0.2305)	(0.5054,0.2042)
F₃	(0.6862,0.1569)	(0.4924, 0.3069)	(0.4413,0.1625)	(0.5038,0.2897)
F₄	(0.3169,0.2763)	(0.3090, 0.3037)	(0.2418,0.2458)	(0.4077,0.2862)
F₅	(0.4065,0.2756)	(0.2549, 0.2844)	(0.2051,0.3945)	(0.4350,0.2405)

Table 6

The attributes weights r_j

	R₁	R₂	R₃	R₄
r _j	0.2799	0.2171	0.2606	0.2423

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

Step 5. Compute the distances $d_{i}^{+}$ and $d_{i}^{-}$ , the results are list as follows. $\begin{matrix} d_{1}^{+} = 0.1879, d_{2}^{+} = 0.1765, d_{2}^{+} = 0.1065, \\ d_{2}^{+} = 0.2099, d_{2}^{+} = 0.2074; \\ d_{1}^{-} = 0.1102, d_{2}^{-} = 0.1618, d_{2}^{-} = 0.2424, \\ d_{2}^{-} = 0.1300, d_{2}^{-} = 0.1331 . \end{matrix}$

Step 6. Compute each alternative’s correlation coefficient C_i by Equation (24), the results are list as: $\begin{matrix} C_{1} = 0.3697, C_{2} = 0.4781, C_{3} = 0.6948, \\ C_{4} = 0.3824, C_{5} = 0.3908 \end{matrix}$

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.

4.2 Compare analysis

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.

Table 7
Evaluation results of these methods

Methods Ranking order The best alternative The worst alternative

IFWA operator F₃ > F₂ > F₁ > F₄ > F₅ F ₃ F ₅

IFWG operator F₃ > F₂ > F₄ > F₅ > F₁ F ₃ F ₁

IF-VIKOR method F₃ > F₂ > F₄ > F₁ > F₅ F ₃ F ₅

IF-GRA method F₃ > F₂ > F₄ > F₅ > F₁ F ₃ F ₁

IF-CODAS method F₃ > F₂ > F₄ > F₅ > F₁ F ₃ F ₁

The developed method F₃ > F₂ > F₅ > F₄ > F₁ F ₃ F ₁

Methods	Ranking order	The best alternative	The worst alternative
IFWA operator	F₃ > F₂ > F₁ > F₄ > F₅	F ₃	F ₅
IFWG operator	F₃ > F₂ > F₄ > F₅ > F₁	F ₃	F ₁
IF-VIKOR method	F₃ > F₂ > F₄ > F₁ > F₅	F ₃	F ₅
IF-GRA method	F₃ > F₂ > F₄ > F₅ > F₁	F ₃	F ₁
IF-CODAS method	F₃ > F₂ > F₄ > F₅ > F₁	F ₃	F ₁
The developed method	F₃ > F₂ > F₅ > F₄ > F₁	F ₃	F ₁

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.

5 Conclusion

Since the beginning of the 21st century, with the expansion of college enrollment scale and the scale of higher education, China’s colleges and universities are facing a severe competition situation, and the good guarantee of teaching quality is the goal pursued by many researchers. The Higher Education Department of the Ministry of Education also emphasized in the No.2 document that “the quality of teaching in China’s colleges and universities still has a lot of room for improvement”. The quality of teaching in the secondary colleges of physical education colleges is for the quality management of schools and the first-line teaching of various colleges. The connection of activities and the improvement of the quality of school teaching play an important role. This paper offers an effective solution for this issue, since it designs a novel intuitive distance based IF-TOPSIS method to build the evaluation system of fire safety assessment based on the high building. And then a numerical example has been given to confirm that this novel method is reasonable. What’s more, to verify the validity and feasibility of the developed method, some comparative analysis is also given. However, the main drawback of this paper is that the number of DMs and attributes are small and interdependency of attributes is not taken into consideration, which may limit the application scope of the developed method to some extent. In our future works, the designed model and algorithm will be needful and meaningful to apply to solve other real MADM or MAGDM problems [46 –52], and the designed methods can also be extended to other uncertain setting [53 –59].

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

Abstract

Keywords

1 Introduction

2 Preliminaries

2.1 IFSs

4.1 An empirical example

Table 1 Decision making information given by H1 R1 R2 R3 R4 F1 (0.6,0.3) (0.5,0.2) (0.3,0.6) (0.4,0.2) F2 (0.5,0.4) (0.2,0.4) (0.1,0.3) (0.7,0.1) F3 (0.8,0.1) (0.4,0.5) (0.5,0.4) (0.4,0.3) F4 (0.5,0.4) (0.3,0.3) (0.1,0.3) (0.5,0.2) F5 (0.4,0.5) (0.2,0.4) (0.2,0.5) (0.4,0.1)

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