Research on teaching quality evaluation of college english based on the CODAS method under interval-valued intuitionistic fuzzy information

Abstract

With the rapid development of China’s economic globalization in the new era, the demand for English majors is obviously on the rise, which puts forward new and higher requirements for application-oriented undergraduate colleges to train compound English majors. However, from the perspective of teaching quality evaluation of English majors in application-oriented undergraduate colleges, the results are not optimistic. Therefore, it is an important task for higher education research in China to explore the problems existing in the process of teaching quality evaluation for English majors in application-oriented undergraduate colleges and how to better train qualified and versatile talents for English majors to adapt to the economic and social development in the new era. The teaching quality evaluation of college English is frequently viewed as a multi-attribute group decision-making (MAGDM). Thus, a novel MAGDM method is used to tackle it. Depending on the conventional CODAS method and interval-valued intuitionistic fuzzy sets (IVIFSs), this paper designs a novel distance based IVIF-CODAS method to assess the teaching quality evaluation of college English. First of all, a related literature review is conducted. What’s more, some necessary theories related to IVIFSs are briefly reviewed. In addition, since subjective randomness frequently exists in determining criteria weights, the weights of criteria is decided objectively by utilizing CRITIC method. Afterwards, relying on novel distance measures between IVIFSs, the conventional CODAS method is extended to the IVIFSs to calculate assessment score of every alternative. Therefore, all alternatives can be ranked and the one with the best teaching quality. Eventually, an application about teaching quality evaluation of college English and some comparative methods have been employed to show the superiority of the developed method. The results illustrate that the defined framework is very useful for assessing the teaching quality of college English.

Keywords

MAGDM issues interval-valued intuitionistic fuzzy sets (IVIFSs)CODAS method CRITIC method teaching quality college English

1 Introduction

Since the process of making decision is filled with uncertainty and ambiguity [1 –5], thus, in order to cope with the accuracy of decision-making, Zadeh [6] proposed the fuzzy sets (FSs). Atanassov [7] proposed the intuitionistic fuzzy sets (IFSs). Garg [8] presented a method related to MAGDM on the basis of intuitionistic fuzzy multiplicative preference and defined several geometric operators. Chen, Cheng and Lan [9] developed TOPSIS method and similarity measures under IFSs. Rouyendegh [10] used the ELECTRE method in IFSs to tackle some MCDM issues. Gupta, Arora and Tiwari [11] extended the fuzzy entropy to IFSs. Gan and Luo [12] used the hybrid method with DEMATEL and IFSs. Xiao, Zhang, Wei, Wu, Wei, Guo and Wei [13] defined the intuitionistic fuzzy Taxonomy method. He, He and Huang [14] integrated the power averaging with IFSs. Li and Wu [15] presented the intuitionistic fuzzy cross entropy distance. Zhao, Wei, Wei and Wu [16] defined TODIM method for IF-MAGDM based on CPT. Cali and Balaman [17] extended ELECTRE I with VIKOR method in IFSs to reflect the decision makers’ preferences. Hao, Xu, Zhao and Zhang [18] presented a theory of decision field for IFSs. Jin, Ni, Chen and Li [19] 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. Gou, Xu and Lei [20] defined some exponential operational law for IFNs. Khan, Lohani and Ieee [21] defined similarity measure about IFNs. Bao, Xie, Long and Wei [22] defined prospect theory and evidential reasoning method under IFSs. Gupta, Mehlawat, Grover and Chen [23] modified the SIR method and combined it with IFSs. Krishankumar, Arvinda, Amrutha, Premaladha, Ravichandran and Ieee [24] integrated AHP with IFSs to design a GDM method for effective cloud vendor selection. Li, Liu, Liu, Su and Wu [25] gave a grey target decision making with IFNs. Liu, Liu and Chen [26] built some intuitionistic fuzzy BM fused operators with Dombi operations. Oztaysi, Onar, Goztepe and Kahraman [27] solved the research proposals evaluation for grant funding using IVIFSs. An, Wang, Li and Ding [28] gave the project delivery system selection with IVIF-MAGDM method. Zhang, Ju and Liu [29] defined the programming technique for MAGDM based on Shapley values and incomplete information. Zhang [30] proposed some Frank aggregation operators under IVIFSs. Wang and Mendel [31] solved the aggregation methodology for IVIF-MADM with a prioritization of criteria. Xian, Dong and Yin [32] defined combined weighted averaging operator for GDM under IVIFSs. Sahu, Gupta and Mehra [33] defined the hierarchical clustering of IVIFSs. Zhao, Wei, Wei, Wu and Wei [34] extended CPT-TODIM method for interval-valued intuitionistic fuzzy MAGDM.

CODAS method was initially developed with Ghorabaee, Zavadskas, Turskis and Antucheviciene [35] to solve MAGDM issues. Compare with other MAGDM models, CODAS method improves the precision of ranking results by integrating Euclidean distance and Hamming distance. Wang, Wang, Wei, Wu, Wei and Wei [36] defined the CODAS procedures for MAGDM under 2-tuple linguistic neutrosophic information. Pamucar, Badi, Sanja and Obradovic [37] presented an original Pairwise-CODAS method for MCDM. He, Zhang, Wei, Wang, Wu and Wei [38] defined the CODAS procedures for 2-tuple linguistic pythagorean fuzzy MAGDM. Roy, Das, Kar and Pamucar [39] established the CODAS method for MCDM issues with IVIFNs. Zhang, Wei, Wang, Wu, Wei, Guo and Wei [40] defined the CODAS procedures for a green supplier selection. Ghorabaee, Amiri, Zavadskas, Hooshmand and Antuchevičienė [41] designed the CODAS method by using the trapezoidal fuzzy numbers. Lan, Wu, Guo, Wei, Wei and Gao [42] defined the CODAS methods for MAGDM with interval-valued bipolar uncertain linguistic information. Wei, Wu, Guo and Wei [43] defined the CODAS method in probabilistic uncertain linguistic environment.

The reminder of such paper proceeds as follows. A literature review is shown in section 2. Some knowledge of IVIFSs is listed in section 3. The CODAS method with IVIFNs and the calculating steps is defined in section 4. An empirical example about teaching quality evaluation of college English is given and some comparative analysis are also given Section 5. Then, an overall conclusion is given in Section 6.

2 Preliminaries

2.1 IVIFSs

Definition 1 [44]. The interval-valued IFSs (IVIFSs) on the Xis: $I = {〈 x, {\tilde{μ}}_{I} (x), {\tilde{ν}}_{I} (x) 〉 | x \in X}$ (1) where ${\tilde{μ}}_{I} (x) \subset [0, 1]$ is named as “membership degree of I” and ${\tilde{ν}}_{I} (x) \subset [0, 1]$ is called as “non-membership degree of I”, and ${\tilde{μ}}_{I} (x)$ , ${\tilde{ν}}_{I} (x)$ meet condition: $0 ⩽ sup {\tilde{μ}}_{I} (x) + sup {\tilde{ν}}_{I} (x) ⩽ 1$ , ∀ x ∈ X. For convenience, we call $I = ([μ^{L}, μ^{R}], [ν^{L}, ν^{R}])$ is an IVIFN.

Definition 2 [45]. Let $I_{1} = ([μ_{1}^{L}, μ_{1}^{R}], [ν_{1}^{L}, ν_{1}^{R}])$ and $I_{2} = ([μ_{2}^{L}, μ_{2}^{R}], [ν_{2}^{L}, ν_{2}^{R}])$ be two IVIFNs, the operation formula of them can be defined: $I_{1} \oplus I_{2} = ([μ_{1}^{L} + μ_{2}^{L} - μ_{1}^{L} μ_{2}^{L}, μ_{1}^{R} + μ_{2}^{R} - μ_{1}^{R} μ_{2}^{R}], [ν_{1}^{L} ν_{2}^{L}, ν_{1}^{R} ν_{2}^{R}])$ (2) $I_{1} \otimes I_{2} = ([μ_{1}^{L} μ_{2}^{L}, μ_{1}^{R} μ_{2}^{R}], [ν_{1}^{L} + ν_{2}^{L} - ν_{1}^{L} ν_{2}^{L}, ν_{1}^{R} + ν_{2}^{R} - ν_{1}^{R} ν_{2}^{R}])$ (3) $λ I_{1} = ([1 - {(1 - μ_{1}^{L})}^{λ}, 1 - {(1 - μ_{1}^{R})}^{λ}], [{(ν_{1}^{L})}^{λ}, {(ν_{1}^{R})}^{λ}]), λ > 0$ (4) $I_{1}^{λ} = ([{(μ_{1}^{L})}^{λ}, {(μ_{1}^{R})}^{λ}], [1 - {(1 - λ_{1}^{L})}^{λ}, 1 - {(1 - λ_{1}^{R})}^{λ}]), λ > 0$ (5)

Definition 3 [46]. Let $I_{1} = ([μ_{1}^{L}, μ_{1}^{R}], [ν_{1}^{L}, ν_{1}^{R}])$ and $I_{2} = ([μ_{2}^{L}, μ_{2}^{R}], [ν_{2}^{L}, ν_{2}^{R}])$ be IVIFNs, the score and accuracy values of I₁ and I₂ can be defined: $\begin{matrix} S (I_{1}) = \frac{μ_{1}^{L} + μ_{1}^{L} (1 - μ_{1}^{L} - ν_{1}^{L}) + μ_{1}^{R} + μ_{1}^{R} (1 - μ_{1}^{R} - ν_{1}^{R})}{2} \\ S (I_{2}) = \frac{μ_{2}^{L} + μ_{2}^{L} (1 - μ_{2}^{L} - ν_{2}^{L}) + μ_{2}^{R} + μ_{2}^{R} (1 - μ_{2}^{R} - ν_{2}^{R})}{2} \end{matrix}$ (6) $\begin{matrix} H (I_{1}) = \frac{μ_{1}^{L} + ν_{1}^{L} + μ_{1}^{R} + ν_{1}^{R}}{2}, \\ H (I_{2}) = \frac{μ_{2}^{L} + ν_{2}^{L} + μ_{2}^{R} + ν_{2}^{R}}{2} \end{matrix}$ (7)

For two IVIFNs I₁ and I₂, according to Definition 3, then $\begin{matrix} (1) if s (I_{1}) < s (I_{2}), then I_{1} < I_{2}; \\ (2) if s (I_{1}) > s (I_{2}), then I_{1} > I_{2}; \\ (3) if s (I_{1}) = s (I_{2}), h (I_{1}) < h (I_{2}), then I_{1} < I_{2}; \\ (4) if s (I_{1}) = s (I_{2}), h (I_{1}) = h (I_{2}), then I_{1} = I_{2} . \end{matrix}$

Definition 4 [47]. Let $I_{1} = ([μ_{1}^{L}, μ_{1}^{R}], [ν_{1}^{L}, ν_{1}^{R}])$ and $I_{2} = ([μ_{2}^{L}, μ_{2}^{R}], [ν_{2}^{L}, ν_{2}^{R}])$ be IVIFNs, the Euclidean distance between two IVIFNs can be given as follows: $IVIFED (I_{1}, I_{2}) = \sqrt{\frac{1}{4} [{(μ_{1}^{L} - μ_{2}^{L})}^{2} + {(μ_{1}^{R} - μ_{2}^{R})}^{2} + {(ν_{1}^{L} - ν_{2}^{L})}^{2} + {(ν_{1}^{R} - ν_{2}^{R})}^{2}]}$ (8)

2.2 Two aggregation operators under IVIFSs

Under the IVIFSs, some fused operators will be introduced in this section, including IVIFWA fused operator and IVIFWG fused operator.

Definition 5 [48]. Let $I_{j} = ([μ_{I_{j}}^{L}, μ_{I_{j}}^{R}], [ν_{I_{j}}^{L}, ν_{I_{j}}^{R}]) (j = 1, 2, \dots, n)$ be a set of IVIFNs, the IFWA operator is: $\begin{matrix} {IVIFWA}_{ω} (I_{1}, I_{2}, \dots, I_{n}) = \oplus_{j = 1}^{n} (ω_{j} I_{j}) \\ = ([1 - \prod_{j = 1}^{n} {(1 - μ_{I_{j}}^{L})}^{ω_{j}}, 1 - \prod_{j = 1}^{n} {(1 - μ_{I_{j}}^{R})}^{ω_{j}}], [\prod_{j = 1}^{n} {(ν_{I_{j}}^{L})}^{ω_{j}}, \prod_{j = 1}^{n} {(ν_{I_{j}}^{R})}^{ω_{j}}]) \end{matrix}$ (9) where ω = (ω₁, ω₂, . . . , ω_n) ^T be the weight vector of I_j (j = 1, 2, . . . , n) and $ω_{j} > 0, \sum_{j = 1}^{n} ω_{j} = 1 .$

Definition 6 [45]. Let I_j = $([μ_{I_{j}}^{L}, μ_{I_{j}}^{R}], [ν_{I_{j}}^{L}, ν_{I_{j}}^{R}]) (j = 1, 2, \dots, n)$ be a set of IVIFNs, the IVIFWG operator is: $\begin{matrix} {IVIFWG}_{ω} (I_{1}, I_{2}, \dots, I_{n}) = \otimes_{j = 1}^{n} {(I_{j})}^{ω_{j}} \\ = ([\prod_{j = 1}^{n} {(μ_{I_{j}}^{L})}^{ω_{j}}, \prod_{j = 1}^{n} {(μ_{I_{j}}^{R})}^{ω_{j}}], \\ [1 - \prod_{j = 1}^{n} {(1 - ν_{I_{j}}^{L})}^{ω_{j}}, 1 - \prod_{j = 1}^{n} {(1 - ν_{I_{j}}^{R})}^{ω_{j}}]) \end{matrix}$ (10) where ω = (ω₁, ω₂, . . . , ω_n) ^T be the weight vector of I_j (j = 1, 2, . . . , n) and $ω_{j} > 0, \sum_{j = 1}^{n} ω_{j} = 1 .$

3 CODAS method for MAGDM with IVIFNs

Integrating CODAS model with IVIFNs, we build the IVIF-CODAS method with IVIFNs. The calculating procedures of the defined method can be described subsequently. Let R ={ R₁, R₂, … R_n } be the set 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 significant degree of h = { h₁, h₂, … h_l }, where $h_{k} \in [0, 1], k = 1, 2, \dots, l . \sum_{k = 1}^{l} h_{k} = 1$ . Let F = { F₁, F₂, … F_m }be a discrete group of alternatives. And $Q = {(q_{ij})}_{m \times n}$ is the overall matrix, $q_{ij}$ with IVIFNs. Subsequently, the calculating procedures are listed.

Step 1. Set up decision matrix $Q^{(k)} = {(q_{ij}^{k})}_{m \times n}$ under IFSs and calculate the overall matrix Q = (q_ij) _m×n with IFNs. $Q^{(k)} = {[q_{ij}^{k}]}_{m \times n} = [\begin{matrix} q_{11}^{k} & q_{12}^{k} & \dots & q_{1 n}^{k} \\ q_{21}^{k} & q_{22}^{k} & \dots & q_{2 n}^{k} \\ ⋮ & ⋮ & ⋮ & ⋮ \\ q_{m 1}^{k} & q_{m 2}^{k} & \dots & q_{mn}^{k} \end{matrix}]$ (11) $Q = {[q_{ij}]}_{m \times n} = [\begin{matrix} q_{11} & q_{12} & \dots & q_{1 n} \\ q_{21} & q_{22} & \dots & q_{2 n} \\ ⋮ & ⋮ & ⋮ & ⋮ \\ q_{m 1} & q_{m 2} & \dots & q_{mn} \end{matrix}]$ (12)

$\begin{matrix} q_{ij} = ([1 - \prod_{k = 1}^{l} {(1 - μ_{q_{ij}^{k}}^{L})}^{d_{k}}, 1 - \prod_{k = 1}^{l} {(1 - μ_{q_{ij}^{k}}^{R})}^{d_{k}}], \\ [\prod_{k = 1}^{l} {(ν_{q_{ij}^{k}}^{L})}^{d_{k}}, \prod_{k = 1}^{l} {(ν_{q_{ij}^{k}}^{R})}^{d_{k}}]) \end{matrix}$ (13) where $q_{ij}^{k}$ is the assessment value of F_i (i = 1, 2, …, m) on the basis of R_j (j = 1, 2, …, n) and the DM H_k (k = 1, 2, …, l).

Step 2. Normalize the overall matrix Q = (q_ij) _m×n to $Q^{N} = {[q_{ij}^{N}]}_{m \times n}$ . $q_{ij}^{N} = {\begin{matrix} ([μ_{ij}^{L}, μ_{ij}^{R}], [ν_{ij}^{L}, ν_{ij}^{R}]) Z_{j} is a benefit criterion \\ ([ν_{ij}^{L}, ν_{ij}^{R}], [μ_{ij}^{L}, μ_{ij}^{R}]) Z_{j} is a cost criterion \end{matrix}$ (14)

Step 3. Utilize CRITIC model to derive the weight of attributes.

CRITIC method will be introduced to decide attributes’ weights [49]. Then, the calculating steps of such method is presented.

(1) Depending on the normalized matrix $Q^{N} = {(q_{ij}^{N})}_{m \times n}$ , the correlation coefficient between attributes is given. $\begin{matrix} {IVIFCC}_{jr} = \\ \frac{\sum_{i = 1}^{m} (S (q_{ij}^{N}) - S (q_{j}^{N})) (S (q_{ir}^{N}) - S (q_{r}^{N}))}{\sqrt{\sum_{i = 1}^{m} {(S (q_{ij}^{N}) - S (q_{j}^{N}))}^{2}} \sqrt{\sum_{i = 1}^{m} {(S (q_{ir}^{N}) - S (q_{r}^{N}))}^{2}}}, \\ j, r = 1, 2, \dots, n \end{matrix}$ (15) 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) Calculate attributes’ standard deviation. ${IVIFSD}_{j} = \sqrt{\frac{1}{m - 1} \sum_{i = 1}^{m} {(S (q_{ij}^{N}) - S (q_{j}^{N}))}^{2}}, j = 1, 2, \dots, n$ (16) where $S (q_{j}^{N}) = \frac{1}{m} \sum_{i = 1}^{m} S (q_{ij}^{N})$ .

(3) Calculate the attributes’ weights. $\begin{matrix} r_{j} = \frac{{IVIFSD}_{j} \sum_{t = 1}^{n} (1 - {IVIFCC}_{jt})}{\sum_{j = 1}^{n} ({IVIFSD}_{j} \sum_{t = 1}^{n} (1 - {IVIFCC}_{jt}))}, \\ j = 1, 2 \dots, n . \end{matrix}$ (17) where r_j ∈ [0, 1] and $\sum_{j = 1}^{n} r_{j} = 1$ .

Step 4. Compute weighted matrix with IVIFNs. The weighted normalized p_ij are computed as in Equations (21) and (22): $P = {[p_{ij}]}_{m \times n}$ (18) $p_{ij} = r_{j} \otimes q_{ij}^{N}$ (19) where r_jdenotes the weights of jth criterion.

Step 5. Decide interval-valued intuitionistic fuzzy NIS (IVIFNIS) as in Equations (23) and (24): $IVIFNIS = {[{IVIFNIS}_{j}]}_{1 \times m}$ (20) $\begin{matrix} {IVIFNIS}_{j} = ([min_{j} (μ_{ij}^{L}), min_{j} (μ_{ij}^{R})], \\ [max_{j} (ν_{ij}^{L}), max_{j} (ν_{ij}^{R})]) \end{matrix}$ (21)

Step 6. Compute interval-valued intuitionistic fuzzy Euclidean distances and interval-valued intuitionistic fuzzy Hamming distances from IVIFNIS as in Equations (25) and (26): ${IVIFED}_{i} = \sum_{j = 1}^{n} IVIFED (p_{ij,} {IVIFNIS}_{j})$ (22) ${IVIFHD}_{i} = \sum_{j = 1}^{n} IVIFHD (p_{ij,} {IVIFNIS}_{j})$ (23)

Step 7. Derive the interval-valued intuitionistic fuzzy relative assessment (IVIFRA) matrix as in the Equations (27-28): $IVIFRA = {[g_{ik}]}_{m \times m}$ (24) $\begin{matrix} g_{ik} = ({IVIFED}_{i} - {IVIFED}_{k}) + (t ({IVIFED}_{i} - \\ {IVIFED}_{k}) \times ({IVIFHD}_{i} - {IVIFHD}_{k})) (28) \end{matrix}$ where k∈ { 1, 2, … ∞ , m } and t is a threshold function that is given in Equation (29): $t (x) = {\begin{matrix} 1 if | x | ⩾ θ \\ 0 f | x | < θ \end{matrix}$ (25)

In this research, θ = 0.02 is taken for calculations.

Step 8. Obtain the alternative’s assessment scoreAS_i (i = 1, 2, …, m) in Equation (30): ${AS}_{i} = \sum_{k = 1}^{m} g_{ik}$ (26)

Step 9. In the light of theAS_i (i = 1, 2, …, m). The highest value of AS_i (i = 1, 2, …, m) is, the optimal alternative is designed.

4 Empirical example and comparative analysis

4.1 An empirical example

With the acceleration of the development of science and technology and the globalization of world economy, English has become a necessary means of political, economic, cultural exchange and tools. Chinese’s rapid economic development, all-round, wide-ranging, multi-level opening-up pattern is advancing, the status of English in running the global economy becomes more and more important, that is playing an increasingly important role in various international affairs. Chinese people are increasing emphasis on foreign language learning, especially after China joining the WTO and successful bid to host the 2008 Olympic Games, people for learning English enthusiasm unprecedented upsurge. Therefore, University English teaching as teaching English knowledge and the basic course of excellent western culture, how to enhance and improve teaching methods, to get rid of the traditional teaching concept, how to reflect the university English teaching characteristics, and how to implement quality education has been the university English teaching and research problems. Because this is not only the need of development of era, but it also the need of our training cross century talents with the world. In such chapter, an empirical application of evaluating teaching quality of college English is provided based on the IVIF-CODAS method. Since the school wants to select one best teacher with the best teaching quality of college English, there are five potential teachersF_i (i = 1, 2, 3, 4, 5) preparing to evaluate their teaching quality of college English. In order to assess these teachers fairly, five experts H = { H₁, H₂, H₃, H₄, H₅ }(expert’s weight h = (0 . 20, 0 . 20, 0 . 20, 0 . 20, 0 . 20) are invited. All experts express their assessment information according to four subsequently attributes: ding1 R₁ is teaching attitude; ding2 R₂ is teaching methods; ding3 R₃ is the student feedback; ding4 R₄ is peer recognition. All attributes are benefit attributes. The decision-making matrix are given in Table 1 –3.

Table 1
Decision making information given by H₁

R₁ R₂

P₁ ([0.60,0.66],[0.30,0.34]) ([0.35,0.40],[0.50,0.60])

P₂ ([0.50,0.50],[0.50,0.50]) ([0.28,0.32],[0.60,0.68])

P₃ ([0.43,0.46],[0.51,0.54]) ([0.49,0.55],[0.40,0.45])

P₄ ([0.70,0.78],[0.17,0.22]) ([0.33,0.43],[0.51,0.57])

P₅ ([0.22,0.34],[0.58,0.66]) ([0.41,0.45],[0.50,0.55])

T₃ T₄

P₁ ([0.72,0.80],[0.15,0.20]) ([0.28,0.35],[0.58,0.65])

P₂ ([0.65,0.72],[0.20,0.28]) ([0.41,0.55],[0.37,0.45])

P₃ ([0.65,0.70],[0.25,0.30]) ([0.62,0.72],[0.20,0.28])

P₄ ([0.56,0.62],[0.30,0.38]) ([0.19,0.25],[0.19,0.25])

P₅ ([0.51,0.60],[0.35,0.40]) ([0.33,0.42],[0.50,0.58])

	R₁	R₂
P₁	([0.60,0.66],[0.30,0.34])	([0.35,0.40],[0.50,0.60])
P₂	([0.50,0.50],[0.50,0.50])	([0.28,0.32],[0.60,0.68])
P₃	([0.43,0.46],[0.51,0.54])	([0.49,0.55],[0.40,0.45])
P₄	([0.70,0.78],[0.17,0.22])	([0.33,0.43],[0.51,0.57])
P₅	([0.22,0.34],[0.58,0.66])	([0.41,0.45],[0.50,0.55])
	T₃	T₄
P₁	([0.72,0.80],[0.15,0.20])	([0.28,0.35],[0.58,0.65])
P₂	([0.65,0.72],[0.20,0.28])	([0.41,0.55],[0.37,0.45])
P₃	([0.65,0.70],[0.25,0.30])	([0.62,0.72],[0.20,0.28])
P₄	([0.56,0.62],[0.30,0.38])	([0.19,0.25],[0.19,0.25])
P₅	([0.51,0.60],[0.35,0.40])	([0.33,0.42],[0.50,0.58])

Table 2

Decision making information given by H₂

	R₁	R₂
P₁	([0.73,0.80],[0.10,0.20])	([0.46,0.55],[0.30,0.45])
P₂	([0.55,0.60],[0.30,0.40])	([0.34,0.41],[0.53,0.59])
P₃	([0.24,0.35],[0.58,0.65])	([0.27,0.34],[0.60,0.66])
P₄	([0.31,0.42],[0.50,0.58])	([0.59,0.65],[0.27,0.35])
P₅	([0.18,0.30],[0.65,0.70])	([0.40,0.45],[0.50,0.55])
	T₃	T₄
P₁	([0.38,0.45],[0.50,0.55])	([0.57,0.67],[0.26,0.33])
P₂	([0.63,0.75],[0.15,0.25])	([0.69,0.75],[0.18,0.25])
P₃	([0.55,0.60],[0.30,0.40])	([0.39,0.45],[0.48,0.55])
P₄	([0.36,0.41],[0.52,0.59])	([0.26,0.32],[0.60,0.68])
P₅	([0.59,0.65],[0.30,0.35])	([0.34,0.41],[0.52,0.59])

Table 3

Decision making information given by H₃

	R₁	R₂
F₁	([0.41,0.45],[0.50,0.55])	([0.74,0.80],[0.15,0.20])
F₂	([0.36,0.40],[0.57,0.60])	([0.59,0.65],[0.30,0.35])
F₃	([0.29,0.35],[0.60,0.65])	([0.57,0.62],[0.32,0.38])
F₄	([0.53,0.60],[0.35,0.40])	([0.68,0.75],[0.20,0.25])
F₅	([0.46,0.52],[0.40,0.48])	([0.41,0.52],[0.40,0.48])
	R₃	R₄
F₁	([0.16,0.22],[0.65,0.78])	([0.24,0.30],[0.65,0.70])
F₂	([0.32,0.40],[0.55,0.60])	([0.71,0.80],[0.14,0.20])
F₃	([0.43,0.47],[0.50,0.53])	([0.57,0.62],[0.30,0.38])
F₄	([0.32,0.39],[0.41,0.61])	([0.34,0.40],[0.50,0.60])
F₅	([0.25,0.30],[0.55,0.70])	([0.62,0.70],[0.25,0.30])

Then, we shall use the defined IVIF-CODAS method for teaching quality evaluation of College English.

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 Table 1–3 and the expert’s weightsh = (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 (13), the computing results are list in Table 4.

Table 4

Overall matrix with IVIFNs

	R₁	R₂
F₁	([0.4872,0.5702],[0.4127,0.4153])	([0.5265,0.5879],[0.2908,0.4079])
F₂	([0.5875,0.6829],[0.2511,0.3171])	([0.5623,0.6406],[0.3011,0.3594])
F₃	([0.5144,0.5866],[0.3420,0.4134])	([0.4125,0.4685],[0.4625,0.5315])
F₄	([0.3010,0.3678],[0.5677,0.6322])	([0.4759,0.5699],[0.3497,0.4301])
F₅	([0.4932,0.5582],[0.3658,0.4418])	([0.3160,0.4106],[0.5152,0.5894])
	R₃	R₄
F₁	([0.4478,0.5187],[0.3848,0.4723])	([0.5356,0.6149],[0.4550,0.3636])
F₂	([0.3034,0.3589],[0.5872,0.6411])	([0.6589,0.7498],[0.1660,0.2502])
F₃	([0.3638,0.4299],[0.5243,0.5701])	([0.5273,0.5805],[0.3471,0.4195])
F₄	([0.4333,0.5053],[0.4275,0.4947])	([0.5314,0.6012],[0.3298,0.3988])
F₅	([0.4702,0.5165],[0.4317,0.4835])	([0.5080,0.5864],[0.3567,0.4136])

Step 2. All the attributes are benefit attributes, thus this step is omitted.

Step 3. Decide the attribute weights r_j (j = 1, 2, …, n) by CRITIC method as listed in Table 5.

Table 5

IVIF weighted normalized performance values of alternatives

	R₁	R₂
F₁	([0.5356,0.6149],[0.4550,0.3636])	([0.4478,0.5187],[0.3848,0.4723])
F₂	([0.6589,0.7498],[0.1660,0.2502])	([0.3034,0.3589],[0.5872,0.6411])
F₃	([0.5273,0.5805],[0.3471,0.4195])	([0.3638,0.4299],[0.5243,0.5701])
F₄	([0.5314,0.6012],[0.3298,0.3988])	([0.4333,0.5053],[0.4275,0.4947])
F₅	([0.5080,0.5864],[0.3567,0.4136])	([0.4702,0.5165],[0.4317,0.4835])
	R₃	R₄
F₁	([0.5265,0.5879],[0.2908,0.4079])	([0.4872,0.5702],[0.4127,0.4153])
F₂	([0.5623,0.6406],[0.3011,0.3594])	([0.5875,0.6829],[0.2511,0.3171])
F₃	([0.4125,0.4685],[0.4625,0.5315])	([0.5144,0.5866],[0.3420,0.4134])
F₄	([0.4759,0.5699],[0.3497,0.4301])	([0.3010,0.3678],[0.5677,0.6322])
F₅	([0.3160,0.4106],[0.5152,0.5894])	([0.4932,0.5582],[0.3658,0.4418])

Step 4. Obtain IVIF weighted normalized assessing matrix (Table 6).

Table 6

The attributes weights r_j

	R₁	R₂	R₃	R₄
r _j	0.2766	0.2204	0.2624	0.2406

Step 5. Decide IVIFNIS by utilizing Equations (20–21) in terms of Table 6 and give it in Table 7.

Table 7

IVIFNIS

	IFNIS
R₁	(0 . 3078, 0 . 4041)
R₂	(0 . 2051, 0 . 3945)
R₃	(0 . 1986, 0 . 5885)
R₄	(0 . 3169, 0 . 2763)

Step 6. Derive the alternatives’ weighted ED and weighted HD by utilizing Equations (22) and (23). The results are recorded in Table 8.

Table 8

Alternatives’ Euclidean & Hamming distances

	ED	HD
F₁	0.0656	0.1194
F₂	0.0799	0.1465
F₃	_0.0147	_0.0269
F₄	0.0488	0.0872
F₅	0.2556	0.4657

Step 7. Obtain the RA matrix by utilizing Equations (24–26) as in Table 9.

Table 9

Relative assessment matrix

	F₁	F₂	F₃	F₄	F₅
F₁	0.0000	–0.5856	–0.0188	–0.0934	0.0946
F₂	0.5856	0.0000	0.5367	0.4929	0.6786
F₃	0.0188	–0.5367	0.0000	–0.0158	0.1449
F₄	0.0934	–0.4929	0.0158	0.0000	0.1868
F₅	–0.0946	–0.6786	–0.1449	–0.1868	0.0000

Step 8. Calculate each alternative’s total assessment score (AS) as given in Equation (27) and the corresponding results are recorded in Table 10.

Table 10

Assessment Score

Alternative	Assessment Score
F₁	–0.5966
F₂	2.2997
F₃	–0.3898
F₄	–0.2043
F₅	–1.1069

Step 9. Relying on the calculating values of AS, all the alternatives could be ordered, the higher value of AS, the optimal alternative is. Evidently, the order of these five alternatives is F₂ > F₄ > F₃ > F₁ > F₅ and F₂ is the optimal enterprise.

4.2 Compare analysis

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

Besides, our defined method is compared with IVIF-GRA method [50]. Then we can have the calculating result. The grey relational grades of each alternative are: γ₁ = 0 . 8102,γ₂ = 0 . 8879,γ₃ = 0 . 9849,γ₄ = 0 . 8348,γ₅ = 0 . 8216. Therefore, the order is:F₃ > F₂ > F₄ > F₅ > F₁.

What’s more, our defined method is compared with the IVIF-VIKOR method [51]. Then we can obtain the calculating result. The closest ideal score values are:CI^* (F₁) = 0 . 9034, CI^* (F₂) = 0 . 6714, CI^* (F₃) = 0 . 0000, CI^* (F₄) = 0 . 9854, CI^* (F₅) = 0 . 9509 . And the farthest worst score values are: CI^- (F₁) = 0 . 0134, CI^- (F₂) = 0 . 3467, CI^- (F₃) = 1 . 0000, CI^- (F₄) = 0 . 0176, CI^- (F₅) = 0 . 0000 . Then each alternatives’ relative closeness are calculated as: DRC₁ = 0 . 9859,DRC₂ = 0 . 6656,DRC₃ = 0 . 0000,DRC₄ = 0 . 9796,DRC₅ = 1 . 0000. Hence, the order isF₃ > F₂ > F₄ > F₁ > F₅.

First of all, our defined method is compared with IVIFWA and IVIFWG operators [48]. For the IVIFWA operator, the calculating result is: S (F₁) = 0 . 0795, S (F₂) = 0 . 1508, S (F₃) = 0 . 3435, S (F₄) = 0 . 0498, S (F₅) = 0 . 0421 . Thus, the ranking order is F₃ > F₂ > F₁ > F₄ > F₅. For the IVIFWG operator, the calculating result is S (F₁) = - 0 . 0116, S (F₂) = 0 . 1239, S (F₃) = 0 . 3213, S (F₄) = 0 . 0368, S (F₅) = 0 . 0087 .So the ranking order isF₃ > F₂ > F₄ > F₅ > F₁.

Eventually, the results of these methods are in Table 11.

Table 11
Evaluation results of these methods

Methods Ranking order The best alternative The worst alternative

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

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

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

IVIF- GRA 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
IVIFWA operator	F₃ > F₂ > F₁ > F₄ > F₅	F ₃	F ₅
IVIFWG operator	F₃ > F₂ > F₄ > F₅ > F₁	F ₃	F ₁
IVIF-VIKOR method	F₃ > F₂ > F₄ > F₁ > F₅	F ₃	F ₅
IVIF- GRA method	F₃ > F₂ > F₄ > F₅ > F₁	F ₃	F ₁
The developed method	F₃ > F₂ > F₅ > F₄ > F₁	F ₃	F ₁

From Table 11, it is evidently that the best alternative is F₃, while the worst alternative is F₁ in most situations. In other words, these methods’ order is slightly different. Different methods can tackle MAGDM from different angles.

6 Conclusion

With the deepening of the reform of our higher education, the conflicts between quantity and quality are becoming increasingly serious. The development of social economy made request to the change of cultivation of university talents and its ideology. To comply with the requirement, the ideology of university course is adjusting now. College English course, as a basic course of cultivating university talents, its quality has a direct influence over the cultivation of interdisciplinary talents and it has become a worthy subject for deeper research. This paper offers an effective solution idea for this kind of issue, since it designs a novel intuitive distance based IVIF-CODAS method to build the evaluation system of teaching quality evaluation of college English. And then a numerical example is given to confirm that the IVIF-CODAS method is reasonable. What’s more, to show the validity and feasibility of the defined method, some comparative analysis is also given. However, the main drawback of this paper is that the number of DMs and attributes are very small and interdependency of these four attributes is not considered, which may limit the application scope of the developed method to some extent. Future research could tackle the interdependency of attributes by utilizing some other methods including ANP, AHP, information entropy. Furthermore, the developed method can be utilized to tackle many other MAGDM issues like risk evaluation [52 –54] and project selection [55 –59]. And it can also be applied to many other diverse uncertain and ambiguous environments [60 –65].

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