# TOPSIS Method for Teaching Effect Evaluation of College English with Interval-Valued Intuitionistic Fuzzy Information

Authors: Fengling Wang
Journal: Journal of Mathematics
 (2021)

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## Abstract

Teaching effect evaluation of College English is frequently considered as a multiattribute group decision-making (MAGDM) issue. Thus, a novel MAGDM method is needed to tackle it. Depending on the classical TOPSIS method and interval-valued intuitionistic fuzzy sets (IVIFSs), this paper designs a novel intuitive distance-based IVIF-TOPSIS method for teaching effect evaluation of College English. First of all, a related literature review is conducted. Furthermore, some necessary theories related to IVIFSs are briefly reviewed. In addition, the weights of attribute are decided objectively by using the CRITIC method. Afterwards, relying on novel distance measures between IVIFSs, the conventional TOPSIS method is extended to the IVIFSs to calculate closeness degree of each alternative from the interval-valued intuitionistic fuzzy positive ideal solution (IVIF-PIS). Finally, an empirical example about teaching effect evaluation of College English and some comparative analyses have been given. The results show that the designed method is useful for teaching effect evaluation of College English.


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## Body

## 1. Introduction

Since the process of making decision is filled with uncertainty and ambiguity [1–7], in order to cope with the accuracy of decision-making [8–14], Zadeh [15] defined the fuzzy sets (FSs). Atanassov [16] defined the concept of intuitionistic fuzzy sets (IFSs). Liu et al. [17] built some intuitionistic fuzzy BM fused operators with Dombi operations. Gupta et al. [18] extended the fuzzy entropy to IFSs. He et al. [19] integrated the power averaging with IFSs. Garg [20] presented a method related to MAGDM on the basis of intuitionistic fuzzy multiplicative preference and defined several geometric operators. Chen et al. [21] developed TOPSIS method and similarity measures under IFSs. Rouyendegh [22] used the ELECTRE method in IFSs to tackle some MCDM issues. Gan and Luo [23] used the hybrid method with DEMATEL and IFSs. Jin et al. [24] 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. Xiao et al. [25] defined the intuitionistic fuzzy Taxonomy method. Zhao et al. [26] defined TODIM method for IF-MAGDM based on CPT. Cali and Balaman [27] extended ELECTRE I with VIKOR method in IFSs to reflect the decision-makers’ preferences. Hao et al. [28] presented a theory of decision field for IFSs. Gupta et al. [29] modified the SIR method and combined it with IFSs. Li et al. [30] gave a grey target decision-making with IFNs. Gou et al. [31] defined some exponential operational law for IFNs. Khan and Lohani [32] defined similarity measure about IFNs. Bao et al. [33] defined prospect theory and evidential reasoning method under IFSs. Oztaysi et al. [34] solved the research proposals evaluation for grant funding using IVIFSs. Sahu et al. [35] defined the hierarchical clustering of IVIFSs. Xian et al. [36] defined combined weighted averaging operator for GDM under IVIFSs. Zhang et al. [37] defined the programming technique for MAGDM based on Shapley values and incomplete information. Zhang [38] proposed some Frank aggregation operators under IVIFSs. An et al. [39] gave the project delivery system selection with IVIF-MAGDM method. Zeng et al. [40] solved IVIF-MADM based on nonlinear programming methodology and TOPSIS method. Zhao et al. [41] defined the CPT-TODIM method for interval-valued intuitionistic fuzzy MAGDM. Wang and Mendel [42] solved the aggregation methodology for IVIF-MADM with a prioritization of criteria.TOPSIS was initially developed by Hwang and Yoon [43] 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 in various fuzzy settings [44–49]. This paper’s goal is to use TOPSIS method in IVIFSs and build a new decision-making model for actual MADM problems. Thus, the motivation of this study is the following: (1) the weights of attributes are decided objectively by CRITIC method; (2) an empirical example about teaching effect evaluation of College English and some comparative analyses have been given. In order to do so, the reminder of this paper is organized as follows: Some concepts of IVIFSs are reviewed in Section 2. The improved TOPSIS method is defined with IVIFSs and the calculating steps is simply listed in Section 3. An empirical application about teaching effect evaluation of College English 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. IVIFSs

Definition 1.
(see [50]). The interval-valued IFS (IVIFS) on X is(1)I=x,μ˜Ix,ν˜Ix,|x∈X,where μ˜Ix⊂0,1 is named as “membership degree of I” and ν˜Ix⊂0,1 is called “non-membership degree of I,” and μ˜Ix and ν˜Ix meet the following condition: 0≤supμ˜Ix+supν˜Ix≤1, ∀x∈X. For convenience, we call I=μL,μR,νL,νR an IVIFN.Definition 2.
(see [51]). Let I1=μ1L,μ1R,ν1L,ν1R and I2=μ2L,μ2R,ν2L,ν2R be two IVIFNs; the operation formula of them can be defined:(2)I1⊕I2=μ1L+μ2L−μ1Lμ2L,μ1R+μ2R−μ1Rμ2R,ν1Lν2L,ν1Rν2R,I1⊗I2=μ1Lμ2L,μ1Rμ2R,ν1L+ν2L−ν1Lν2L,ν1R+ν2R−ν1Rν2R,λI1=1−1−μ1Lλ,1−1−μ1Rλ,ν1Lλ,ν1Rλ,λ>0,I1λ=μ1Lλ,μ1Rλ,1−1−λ1Lλ,1−1−λ1Rλ,λ>0.Definition 3.
(see [52]). Let I1=μ1L,μ1R,ν1L,ν1R and I2=μ2L,μ2R,ν2L,ν2R be IVIFNs; the score and accuracy values of I1 and I2 can be defined:(3)SI1=μ1L+μ1L1−μ1L−ν1L+μ1R+μ1R1−μ1R−ν1R2,SI2=μ2L+μ2L1−μ2L−ν2L+μ2R+μ2R1−μ2R−ν2R2,HI1=μ1L+ν1L+μ1R+ν1R2,HI2=μ2L+ν2L+μ2R+ν2R2.
For two IVIFNsI1 and I2, according to Definition 3, we have the following:(1)
ifsI1<sI2, then I1<I2(2)
ifsI1=sI2,hI1<hI2, then I1>I2(3)
ifsI1=sI2,hI1=hI2, then I1=I2Definition 4.
(see [53]). Let I1=μ1L,μ1R,ν1L,ν1R and I2=μ2L,μ2R,ν2L,ν2R be IVIFNs; the Euclidean distance between two IVIFNs can be given as follows:(4)IVIFEDI1,I2=14μ1L−μ2L2+μ1R−μ2R2+ν1L−ν2L2+ν1R−ν2R2.


### 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.
(see [54]). Let Ij=μIjL,μIjR,νIjL,νIjRj=1,2,…,n be a set of IVIFNs; the IFWA operator is(5)IVIFWAωI1,I2,…,In=⊕j=1nωjIj=1−∏j=1n1−μIjLωj,1−∏j=1n1−μIjRωj,∏j=1nνIjLωj,∏j=1nνIjRωj,where ω=ω1,ω2,…,ωnT is the weight vector of Ijj=1,2,…,n and ωj>0,∑j=1nωj=1.Definition 6.
(see [51]). Let Ij=μIjL,μIjR,νIjL,νIjRj=1,2,…,n be a set of IVIFNs; the IVIFWG operator is(6)IVIFWGωI1,I2,…,In=⊗j=1nIjωj=∏j=1nμIjLωj,∏j=1nμIjRωj,1−∏j=1n1−νIjLωj,1−∏j=1n1−νIjRωj,where ω=ω1,ω2,…,ωnT is the weight vector of Ijj=1,2,…,n and ωj>0,∑j=1nωj=1.


## 2.1. IVIFSs

Definition 1.
(see [50]). The interval-valued IFS (IVIFS) on X is(1)I=x,μ˜Ix,ν˜Ix,|x∈X,where μ˜Ix⊂0,1 is named as “membership degree of I” and ν˜Ix⊂0,1 is called “non-membership degree of I,” and μ˜Ix and ν˜Ix meet the following condition: 0≤supμ˜Ix+supν˜Ix≤1, ∀x∈X. For convenience, we call I=μL,μR,νL,νR an IVIFN.Definition 2.
(see [51]). Let I1=μ1L,μ1R,ν1L,ν1R and I2=μ2L,μ2R,ν2L,ν2R be two IVIFNs; the operation formula of them can be defined:(2)I1⊕I2=μ1L+μ2L−μ1Lμ2L,μ1R+μ2R−μ1Rμ2R,ν1Lν2L,ν1Rν2R,I1⊗I2=μ1Lμ2L,μ1Rμ2R,ν1L+ν2L−ν1Lν2L,ν1R+ν2R−ν1Rν2R,λI1=1−1−μ1Lλ,1−1−μ1Rλ,ν1Lλ,ν1Rλ,λ>0,I1λ=μ1Lλ,μ1Rλ,1−1−λ1Lλ,1−1−λ1Rλ,λ>0.Definition 3.
(see [52]). Let I1=μ1L,μ1R,ν1L,ν1R and I2=μ2L,μ2R,ν2L,ν2R be IVIFNs; the score and accuracy values of I1 and I2 can be defined:(3)SI1=μ1L+μ1L1−μ1L−ν1L+μ1R+μ1R1−μ1R−ν1R2,SI2=μ2L+μ2L1−μ2L−ν2L+μ2R+μ2R1−μ2R−ν2R2,HI1=μ1L+ν1L+μ1R+ν1R2,HI2=μ2L+ν2L+μ2R+ν2R2.
For two IVIFNsI1 and I2, according to Definition 3, we have the following:(1)
ifsI1<sI2, then I1<I2(2)
ifsI1=sI2,hI1<hI2, then I1>I2(3)
ifsI1=sI2,hI1=hI2, then I1=I2Definition 4.
(see [53]). Let I1=μ1L,μ1R,ν1L,ν1R and I2=μ2L,μ2R,ν2L,ν2R be IVIFNs; the Euclidean distance between two IVIFNs can be given as follows:(4)IVIFEDI1,I2=14μ1L−μ2L2+μ1R−μ2R2+ν1L−ν2L2+ν1R−ν2R2.


## 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.
(see [54]). Let Ij=μIjL,μIjR,νIjL,νIjRj=1,2,…,n be a set of IVIFNs; the IFWA operator is(5)IVIFWAωI1,I2,…,In=⊕j=1nωjIj=1−∏j=1n1−μIjLωj,1−∏j=1n1−μIjRωj,∏j=1nνIjLωj,∏j=1nνIjRωj,where ω=ω1,ω2,…,ωnT is the weight vector of Ijj=1,2,…,n and ωj>0,∑j=1nωj=1.Definition 6.
(see [51]). Let Ij=μIjL,μIjR,νIjL,νIjRj=1,2,…,n be a set of IVIFNs; the IVIFWG operator is(6)IVIFWGωI1,I2,…,In=⊗j=1nIjωj=∏j=1nμIjLωj,∏j=1nμIjRωj,1−∏j=1n1−νIjLωj,1−∏j=1n1−νIjRωj,where ω=ω1,ω2,…,ωnT is the weight vector of Ijj=1,2,…,n and ωj>0,∑j=1nωj=1.


## 3. TOPSIS Method for IVIF-MAGDM with the CRITIC Method

In this section, we build the IVIF-TOPSIS method for MAGDM. The calculating steps of the designed method can be described subsequently. LetR=R1,R2,…,Rn be the group of attributes, and let r=r1,r2,…,rn be the weight of attributes Rj, where rj∈0,1,j=1,2,…,n,∑j=1nrj=1. Assume that H=H1,H2,…,Hl is a set of DMs that have degree of h=h1,h2,…,hl, where hk∈0,1, k=1,2,…,l. ∑k=1lhk=1. Let F=F1,F2,…,Fm be a set of alternatives. Q=qijm×n is the matrix with IVIFNs, where qij means Fi for Rj. Subsequently, the specific calculating steps will be depicted.Step 1. Build each DM’s matrix Qk=qijkm×n with IVIFNs and calculate the overall IVIF decision matrix Q=qijm×n.(7)Qk=qijkm×n=q11kq12k…q1nkq21kq22k…q2nk⋮⋮⋮⋮qm1kqm2k…qmnk,Q=qijm×n=q11q12…q1nq21q22…q2n⋮⋮⋮⋮qm1qm2…qmn,qij=1−∏k=1l1−μqijkLdk,1−∏k=1l1−μqijkRdk,∏k=1lνqijkLdk,∏k=1lνqijkRdk,whereqijk is the assessment value of Fii=1,2,…,m on the basis of the attribute Rjj=1,2,…,n and the DM Hkk=1,2,…,l.Step 2. Normalize the overall matrix Q=qijm×n with IFNs to QN=qijNm×n.(8)qijN=μijL,μijR,νijL,νijR,Zjis a benefit criterion,νijL,νijR,μijL,μijR,Zjis a cost criterion.Step 3. Employ CRITIC method to determine the weighting of attributes.CRiteria Importance Through Intercriteria Correlation (CRITIC) method [55] will be proposed in this part, which is utilized to decide attributes’ weights.(1)
Depending on the normalized overall matrixQN=qijNm×n with IVIFNs, the correlation coefficient between attributes can be defined.(9)IVIFCCjr=∑i=1mSqijN−SqjNSqirN−SqrN∑i=1mSqijN−SqjN2∑i=1mSqirN−SqrN2,j,r=1,2,…,n,whereSqjN=1/m∑i=1mSqijN and SqtN=1/m∑i=1mSqitN.(2)
Obtain attributes’ standard deviation.(10)IVIFSDj=1m−1∑i=1mSqijN−SqjN2,j=1,2,…,n,whereSqjN=1/m∑i=1mSqijN.(3)
Obtain the attributes’ weights.(11)rj=IVIFSDj∑t=1n1−IVIFCCjt∑j=1nIVIFSDj∑t=1n1−IVIFCCjt,j=1,2…,n,whererj∈0,1 and ∑j=1nrj=1.Step 4. Define the interval-valued intuitionistic fuzzy PIS (IVIF-PIS) Aj+ and the interval-valued intuitionistic fuzzy NIS (IVIF-NIS) Aj− as(12)IVIFPISj=μjL+,μjR+,νjL+,νjR+,IVIFNISj=μjL+,μjR+,νjL+,νjR+,whereIVIFPISj=maxjμijL,maxjμijR,minjνijL,minjνijR and IVIFNISj=minjμijL,minjμijR,maxjνijL,maxjνijR.Step 5. Compute the positive distances di+ between each alternative and IVIF-PIS and the negative distances di− between each alternative and IVIF-NIS as(13)di+=∑j=1nrjIVIFEDqijN,Aj+,i=1,2,…,m,di−=∑j=1nrjIVIFEDqijN,Aj−,i=1,2,…,m,whereIVIFEDqijN,Aj+ and IVIFEDqijN,Aj− denote the IVIF Euclidean distances given in Definition 4, and rj is the weight of attributes.Step 6. Compute each alternative’s closeness degree from IVIF-PIS as(14)Ci=di−di−+di+,i=1,2,…,m.Step 7. According to the value, Cii=1,2,…,m. The highest value of Cii=1,2,…,m is the optimal alternative which is designed.


## 4. The Empirical Example and Comparative Analysis

### 4.1. Empirical Example

With the increasing development of economy and more frequent communication between countries, English, as an international language, has more important position and plays a greater role. Accordingly, the requirements for English teaching and learning become higher. Although experts and scholars have been trying to reform the English teaching approaches, the result is not satisfactory. In particular, the recent increasing enrollment has challenged the teaching of College English greatly. The increasing number of students and the lack of faculties lead to the larger number of students in English class. So how to improve the quality of large-class English teaching is the great concern of teachers and students, which is also the ultimate purpose of this research. It is evident that the traditional teacher-centered teaching approach cannot meet the demands of the development. At this moment, the popular cooperative learning approach has gained wide attention. Cooperative learning theories and methods have been researched deeply and are adopted widely in many countries all over the world. The core of the cooperative learning is the group work. It emphasizes the student as center and the teacher as designer, instructor, monitor, etc. By means of such instruments as questionnaires, tests, interviews, and classroom observations, the research on the effect of cooperative learning on the large-class College English teaching is conducted. The result of the research shows that the cooperative learning theories and methods are suitable for the large-class English teaching and are helpful to improve the quality of the teaching. The cooperative learning’s heterogeneous group, positive interdependence, individual accountability and group work, and so forth make the classroom atmosphere relaxed and greatly improve students’ positivity of participation and interest in learning. Students make great progress not only in academic performance but also in communication skills, self-confidence, self-esteem, and so forth. Through this research, some disadvantages of cooperative learning in large-class College English teaching are found, such as students’ inadequate preparations for the group work and unequal opportunities and time for participation of group members. On the basis of these findings in the research, some pedagogical implications are put forward to improve the effect of cooperative learning and the quality of large-class College English teaching. In this chapter, an empirical application about teaching effect evaluation of College English will be provided by making use of IVIF-TOPSIS method. There are five potential College English teaching methodsFii=1,2,3,4,5 preparing to evaluate their investment environment. In order to assess the effect of College English teaching methods 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: ① R1 denotes teaching attitude; ② R2 denotes the teaching methods; ③ R3 denotes student feedback; ④ R4 denotes peer recognition. The decision-making matrices are given in Tables 1–3.Table 1
Decision-making information given byH1.
R1R2R3R4F1([0.16, 0.22], [0.65, 0.78])([0.33, 0.42], [0.50, 0.58])([0.24, 0.30], [0.65, 0.70])([0.47, 0.55], [0.40, 0.45])F2([0.32, 0.40], [0.55, 0.60])([0.17, 0.25], [0.70, 0.75])([0.71, 0.80], [0.14, 0.20])([0.60, 0.70], [0.25, 0.30])F3([0.43, 0.47], [0.50, 0.53])([0.32, 0.40], [0.55,0.60])([0.57, 0.62], [0.30, 0.38])([0.29, 0.36], [0.58, 0.64])F4([0.32, 0.39], [0.41, 0.61])([0.27, 0.36], [0.57, 0.64])([0.34, 0.40], [0.50, 0.60])([0.32, 0.40], [0.55, 0.60])F5([0.25, 0.30], [0.55, 0.70])([0.44, 0.48], [0.50, 0.52])([0.62, 0.70], [0.25, 0.30])([0.60, 0.65], [0.30, 0.35])Table 2
Decision-making information given byH2.
R1R2R3R4F1([0.36, 0.41], [0.56, 0.59])([0.41, 0.45], [0.50, 0.55])([0.74, 0.80], [0.15, 0.20])([0.52, 0.62], [0.32, 0.38])F2([0.70, 0.80], [0.15, 0.20])([0.36, 0.40], [0.57, 0.60])([0.59, 0.65], [0.30, 0.35])([0.66, 0.75], [0.20, 0.25])F3([0.55, 0.62], [0.27, 0.38])([0.29, 0.35], [0.60, 0.65])([0.57, 0.62], [0.32, 0.38])([0.60, 0.65], [0.30, 0.35])F4([0.28, 0.46], [0.50, 0.54])([0.53, 0.60], [0.35, 0.40])([0.68, 0.75], [0.20, 0.25])([0.35, 0.40], [0.55, 0.60])F5([0.52, 0.60], [0.35, 0.40])([0.46, 0.52], [0.40, 0.48])([0.41, 0.52], [0.40, 0.48])([0.58, 0.63], [0.30, 0.37])Table 3
Decision-making information given byH3.
R1R2R3R4F1([0.59, 0.62], [0.26, 0.38])([0.63, 0.70], [0.25, 0.30])([0.37, 0.45], [0.50, 0.55])([0.55, 0.60], [0.32, 0.40])F2([0.65, 0.75], [0.20, 0.25])([0.35, 0.40], [0.55, 0.60])([0.70, 0.80], [0.10, 0.20])([0.52, 0.62], [0.30, 0.38])F3([0.37, 0.40], [0.53, 0.60])([0.42, 0.48], [0.50, 0.52])([0.19, 0.25], [0.70, 0.75])([0.59, 0.65], [0.30, 0.35])F4([0.61, 0.65], [0.30, 0.35])([0.38, 0.42], [0.52, 0.58])([0.62, 0.70], [0.25, 0.30])([0.37, 0.45], [0.55, 0.60])F5([0.35, 0.45], [0.50, 0.55])([0.61, 0.65], [0.30, 0.35])([0.36, 0.40], [0.55, 0.60])([0.55, 0.62], [0.28, 0.38])Then, we shall use the defined TOPSIS method for teaching effect evaluation of College English.:Step 1. Based on the decision-making information Qk=qijkm×ni=1,2,…,m,j=1,2,…,n given in Tables 1–3 and the expert’s weightsh=0.35,0.32,0.33, we can derive the overall matrix Q=qijm×ni=1,2,…,m,j=1,2,…,n according to equation (10), and the computing results are listed in Table 4.Step 2. All the attributes are beneficial attributes; thus, this step is omitted.Step 3. Decide the attribute weights rjj=1,2,…,n by CRITIC method as listed in Table 5.Step 4. Calculate the IVIF-PIS Aj+ and the IVIF-NIS Aj− according to equations (20) and (21).(15)Aj+=0.6862,0.1569,0.4924,0.2844,0.4413,0.1625,0.5054,0.2042,Aj−=0.3169,0.2763,0.1986,0.5885,0.2051,0.3945,0.3078,0.4041.Step 5. Compute the distances di+ and di−; the results are as follows:(16)d1+=0.1823,d2+=0.1978,d2+=0.1043,d2+=0.2123,d2+=0.2213;d1−=0.1246,d2−=0.1623,d2−=0.2509,d2−=0.1366,d2−=0.1735.Step 6. Compute each alternative’s closeness degree Ci from IVIF-PIS by equation (14); the results are as follows:(17)C1=0.3709,C2=0.4982,C3=0.6976,C4=0.3869,C5=0.3916.Step 7. Relying on Ci, all the alternatives can be ordered, and the higher the value of Ci is, the best alternative selected will be. Evidently, the order is F3>F2>F5>F4>F1 and F3 is the optimal College English teaching method.Table 4
The overall matrix with IVIFNs.
R1R2R3R4F1([0.5265, 0.5879], [0.2908, 0.4079])([0.4478, 0.5187], [0.3848, 0.4723])([0.5356, 0.6149], [0.4550, 0.3636])([0.4872, 0.5702], [0.4127, 0.4153])F2([0.5623, 0.6406], [0.3011, 0.3594])([0.3034, 0.3589], [0.5872, 0.6411])([0.6589, 0.7498], [0.1660, 0.2502])([0.5875, 0.6829], [0.2511, 0.3171])F3([0.4125, 0.4685], [0.4625, 0.5315])([0.3638, 0.4299], [0.5243, 0.5701])([0.5273, 0.5805], [0.3471, 0.4195])([0.5144, 0.5866], [0.3420, 0.4134])F4([0.4759, 0.5699], [0.3497, 0.4301])([0.4333, 0.5053], [0.4275, 0.4947])([0.5314, 0.6012], [0.3298, 0.3988])([0.3010, 0.3678], [0.5677, 0.6322])F5([0.3160, 0.4106], [0.5152, 0.5894])([0.4702, 0.5165], [0.4317, 0.4835])([0.5080, 0.5864], [0.3567, 0.4136])([0.4932, 0.5582], [0.3658, 0.4418])Table 5
The attributes weightsrj.
R1R2R3R4rj0.27780.21920.26120.2418


### 4.2. Comparison Analysis

In this section, our defined method is compared with some other methods to show its superiority.First of all, our defined method is compared with IVIFWA and IVIFWG operators [54]. For the IVIFWA operator, the calculating result is SF1=0.0795,SF2=0.1508,SF3=0.3435, SF4=0.0498,SF5=0.0421. Thus, the ranking order is F3>F2>F1>F4>F5. For the IVIFWG operator, the calculating result is SF1=−0.0116,SF2=0.1239,SF3=0.3213, SF4=0.0368,SF5=0.0087. So the ranking order is F3>F2>F4>F5>F1.Furthermore, our defined method is compared with the IVIF-VIKOR method [56]. Then we can obtain the calculating result. The closest ideal score values are the following: CI∗F1=0.9034, CI∗F2=0.6714, CI∗F3=0.0000, CI∗F4=0.9854, and CI∗F5=0.9509; and the farthest worst score values are the following: CI−F1=0.0134, CI−F2=0.3467, CI−F3=1.0000, CI−F4=0.0176, and CI−F5=0.0000. Then the alternatives’ relative closeness is calculated as follows: DRC1=0.9859, DRC2=0.6656, DRC3=0.0000, DRC4=0.9796, and DRC5=1.0000. Hence, the order is F3>F2>F4>F1>F5.In the end, our defined method is also compared with IVIF-CODAS method [57]. Then we can have the calculating result. The total assessment score (AS) of each alternative is calculated as follows: AS1=−0.8023, AS2=0.1650, AS3=1.4827, AS4=−0.3976, and AS5=−0.4436. Therefore, the order is F3>F2>F4>F5>F1.Eventually, the results of these methods are in Table6.Table 6
Evaluation results of these methods.
MethodsRanking orderThe best alternativeThe worst alternativeIVIFWA operator [54]F3>F2>F1>F4>F5F3F5IVIFWG operator [54]F3>F2>F4>F5>F1F3F1IVIF-VIKOR method [56]F3>F2>F4>F1>F5F3F5IVIF-CODAS method [57]F3>F2>F4>F5>F1F3F1The developed methodF3>F2>F5>F4>F1F3F1From Table6, it is evident 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.


## 4.1. Empirical Example

With the increasing development of economy and more frequent communication between countries, English, as an international language, has more important position and plays a greater role. Accordingly, the requirements for English teaching and learning become higher. Although experts and scholars have been trying to reform the English teaching approaches, the result is not satisfactory. In particular, the recent increasing enrollment has challenged the teaching of College English greatly. The increasing number of students and the lack of faculties lead to the larger number of students in English class. So how to improve the quality of large-class English teaching is the great concern of teachers and students, which is also the ultimate purpose of this research. It is evident that the traditional teacher-centered teaching approach cannot meet the demands of the development. At this moment, the popular cooperative learning approach has gained wide attention. Cooperative learning theories and methods have been researched deeply and are adopted widely in many countries all over the world. The core of the cooperative learning is the group work. It emphasizes the student as center and the teacher as designer, instructor, monitor, etc. By means of such instruments as questionnaires, tests, interviews, and classroom observations, the research on the effect of cooperative learning on the large-class College English teaching is conducted. The result of the research shows that the cooperative learning theories and methods are suitable for the large-class English teaching and are helpful to improve the quality of the teaching. The cooperative learning’s heterogeneous group, positive interdependence, individual accountability and group work, and so forth make the classroom atmosphere relaxed and greatly improve students’ positivity of participation and interest in learning. Students make great progress not only in academic performance but also in communication skills, self-confidence, self-esteem, and so forth. Through this research, some disadvantages of cooperative learning in large-class College English teaching are found, such as students’ inadequate preparations for the group work and unequal opportunities and time for participation of group members. On the basis of these findings in the research, some pedagogical implications are put forward to improve the effect of cooperative learning and the quality of large-class College English teaching. In this chapter, an empirical application about teaching effect evaluation of College English will be provided by making use of IVIF-TOPSIS method. There are five potential College English teaching methodsFii=1,2,3,4,5 preparing to evaluate their investment environment. In order to assess the effect of College English teaching methods 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: ① R1 denotes teaching attitude; ② R2 denotes the teaching methods; ③ R3 denotes student feedback; ④ R4 denotes peer recognition. The decision-making matrices are given in Tables 1–3.Table 1
Decision-making information given byH1.
R1R2R3R4F1([0.16, 0.22], [0.65, 0.78])([0.33, 0.42], [0.50, 0.58])([0.24, 0.30], [0.65, 0.70])([0.47, 0.55], [0.40, 0.45])F2([0.32, 0.40], [0.55, 0.60])([0.17, 0.25], [0.70, 0.75])([0.71, 0.80], [0.14, 0.20])([0.60, 0.70], [0.25, 0.30])F3([0.43, 0.47], [0.50, 0.53])([0.32, 0.40], [0.55,0.60])([0.57, 0.62], [0.30, 0.38])([0.29, 0.36], [0.58, 0.64])F4([0.32, 0.39], [0.41, 0.61])([0.27, 0.36], [0.57, 0.64])([0.34, 0.40], [0.50, 0.60])([0.32, 0.40], [0.55, 0.60])F5([0.25, 0.30], [0.55, 0.70])([0.44, 0.48], [0.50, 0.52])([0.62, 0.70], [0.25, 0.30])([0.60, 0.65], [0.30, 0.35])Table 2
Decision-making information given byH2.
R1R2R3R4F1([0.36, 0.41], [0.56, 0.59])([0.41, 0.45], [0.50, 0.55])([0.74, 0.80], [0.15, 0.20])([0.52, 0.62], [0.32, 0.38])F2([0.70, 0.80], [0.15, 0.20])([0.36, 0.40], [0.57, 0.60])([0.59, 0.65], [0.30, 0.35])([0.66, 0.75], [0.20, 0.25])F3([0.55, 0.62], [0.27, 0.38])([0.29, 0.35], [0.60, 0.65])([0.57, 0.62], [0.32, 0.38])([0.60, 0.65], [0.30, 0.35])F4([0.28, 0.46], [0.50, 0.54])([0.53, 0.60], [0.35, 0.40])([0.68, 0.75], [0.20, 0.25])([0.35, 0.40], [0.55, 0.60])F5([0.52, 0.60], [0.35, 0.40])([0.46, 0.52], [0.40, 0.48])([0.41, 0.52], [0.40, 0.48])([0.58, 0.63], [0.30, 0.37])Table 3
Decision-making information given byH3.
R1R2R3R4F1([0.59, 0.62], [0.26, 0.38])([0.63, 0.70], [0.25, 0.30])([0.37, 0.45], [0.50, 0.55])([0.55, 0.60], [0.32, 0.40])F2([0.65, 0.75], [0.20, 0.25])([0.35, 0.40], [0.55, 0.60])([0.70, 0.80], [0.10, 0.20])([0.52, 0.62], [0.30, 0.38])F3([0.37, 0.40], [0.53, 0.60])([0.42, 0.48], [0.50, 0.52])([0.19, 0.25], [0.70, 0.75])([0.59, 0.65], [0.30, 0.35])F4([0.61, 0.65], [0.30, 0.35])([0.38, 0.42], [0.52, 0.58])([0.62, 0.70], [0.25, 0.30])([0.37, 0.45], [0.55, 0.60])F5([0.35, 0.45], [0.50, 0.55])([0.61, 0.65], [0.30, 0.35])([0.36, 0.40], [0.55, 0.60])([0.55, 0.62], [0.28, 0.38])Then, we shall use the defined TOPSIS method for teaching effect evaluation of College English.:Step 1. Based on the decision-making information Qk=qijkm×ni=1,2,…,m,j=1,2,…,n given in Tables 1–3 and the expert’s weightsh=0.35,0.32,0.33, we can derive the overall matrix Q=qijm×ni=1,2,…,m,j=1,2,…,n according to equation (10), and the computing results are listed in Table 4.Step 2. All the attributes are beneficial attributes; thus, this step is omitted.Step 3. Decide the attribute weights rjj=1,2,…,n by CRITIC method as listed in Table 5.Step 4. Calculate the IVIF-PIS Aj+ and the IVIF-NIS Aj− according to equations (20) and (21).(15)Aj+=0.6862,0.1569,0.4924,0.2844,0.4413,0.1625,0.5054,0.2042,Aj−=0.3169,0.2763,0.1986,0.5885,0.2051,0.3945,0.3078,0.4041.Step 5. Compute the distances di+ and di−; the results are as follows:(16)d1+=0.1823,d2+=0.1978,d2+=0.1043,d2+=0.2123,d2+=0.2213;d1−=0.1246,d2−=0.1623,d2−=0.2509,d2−=0.1366,d2−=0.1735.Step 6. Compute each alternative’s closeness degree Ci from IVIF-PIS by equation (14); the results are as follows:(17)C1=0.3709,C2=0.4982,C3=0.6976,C4=0.3869,C5=0.3916.Step 7. Relying on Ci, all the alternatives can be ordered, and the higher the value of Ci is, the best alternative selected will be. Evidently, the order is F3>F2>F5>F4>F1 and F3 is the optimal College English teaching method.Table 4
The overall matrix with IVIFNs.
R1R2R3R4F1([0.5265, 0.5879], [0.2908, 0.4079])([0.4478, 0.5187], [0.3848, 0.4723])([0.5356, 0.6149], [0.4550, 0.3636])([0.4872, 0.5702], [0.4127, 0.4153])F2([0.5623, 0.6406], [0.3011, 0.3594])([0.3034, 0.3589], [0.5872, 0.6411])([0.6589, 0.7498], [0.1660, 0.2502])([0.5875, 0.6829], [0.2511, 0.3171])F3([0.4125, 0.4685], [0.4625, 0.5315])([0.3638, 0.4299], [0.5243, 0.5701])([0.5273, 0.5805], [0.3471, 0.4195])([0.5144, 0.5866], [0.3420, 0.4134])F4([0.4759, 0.5699], [0.3497, 0.4301])([0.4333, 0.5053], [0.4275, 0.4947])([0.5314, 0.6012], [0.3298, 0.3988])([0.3010, 0.3678], [0.5677, 0.6322])F5([0.3160, 0.4106], [0.5152, 0.5894])([0.4702, 0.5165], [0.4317, 0.4835])([0.5080, 0.5864], [0.3567, 0.4136])([0.4932, 0.5582], [0.3658, 0.4418])Table 5
The attributes weightsrj.
R1R2R3R4rj0.27780.21920.26120.2418


## 4.2. Comparison Analysis

In this section, our defined method is compared with some other methods to show its superiority.First of all, our defined method is compared with IVIFWA and IVIFWG operators [54]. For the IVIFWA operator, the calculating result is SF1=0.0795,SF2=0.1508,SF3=0.3435, SF4=0.0498,SF5=0.0421. Thus, the ranking order is F3>F2>F1>F4>F5. For the IVIFWG operator, the calculating result is SF1=−0.0116,SF2=0.1239,SF3=0.3213, SF4=0.0368,SF5=0.0087. So the ranking order is F3>F2>F4>F5>F1.Furthermore, our defined method is compared with the IVIF-VIKOR method [56]. Then we can obtain the calculating result. The closest ideal score values are the following: CI∗F1=0.9034, CI∗F2=0.6714, CI∗F3=0.0000, CI∗F4=0.9854, and CI∗F5=0.9509; and the farthest worst score values are the following: CI−F1=0.0134, CI−F2=0.3467, CI−F3=1.0000, CI−F4=0.0176, and CI−F5=0.0000. Then the alternatives’ relative closeness is calculated as follows: DRC1=0.9859, DRC2=0.6656, DRC3=0.0000, DRC4=0.9796, and DRC5=1.0000. Hence, the order is F3>F2>F4>F1>F5.In the end, our defined method is also compared with IVIF-CODAS method [57]. Then we can have the calculating result. The total assessment score (AS) of each alternative is calculated as follows: AS1=−0.8023, AS2=0.1650, AS3=1.4827, AS4=−0.3976, and AS5=−0.4436. Therefore, the order is F3>F2>F4>F5>F1.Eventually, the results of these methods are in Table6.Table 6
Evaluation results of these methods.
MethodsRanking orderThe best alternativeThe worst alternativeIVIFWA operator [54]F3>F2>F1>F4>F5F3F5IVIFWG operator [54]F3>F2>F4>F5>F1F3F1IVIF-VIKOR method [56]F3>F2>F4>F1>F5F3F5IVIF-CODAS method [57]F3>F2>F4>F5>F1F3F1The developed methodF3>F2>F5>F4>F1F3F1From Table6, it is evident 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.


## 5. Conclusion

With the development of multimedia technology and the wide use of the Internet and computer, College English teaching is becoming more and more multimodal. The rapid development of information technology promotes the change in the ways of communication and the education idea. However, the traditional teaching mode is not adapted to the requirements of the times. This paper offers an effective solution for this issue, since it designs a novel intuitive distance based IVIF-TOPSIS method to build the teaching effect evaluation of College English. Then a numerical example has been given to confirm that this novel method is reasonable. What is 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 numbers 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. Thus, the highlights of this study are the following: (1) the weights of attributes are derived objectively by CRITIC method; (2) an empirical example about teaching effect evaluation of College English and some comparative analyses have been given to show the effectiveness of the designed IVIF-TOPSIS method in MAGDM issues. In our future works, the designed model and algorithm will be needful and meaningful to apply to solve other real MADM or MAGDM problems [58–62], and the designed methods can also be extended to other uncertain settings [63–68].



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