# Probabilistic Double Hierarchy Linguistic TOPSIS Method for MAGDM and Its Application to Teaching Quality Evaluation of College English

Authors: Fengling Wang
Journal: Discrete Dynamics in Nature and Society
 (2022)

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## Abstract

College English is a compulsory course for college students. The teaching level of teachers in this course has naturally been evaluated by various schools. The teaching quality evaluation of college English is frequently viewed as a multiple attribute group decision-making (MAGDM) issue. The probabilistic double hierarchy linguistic not only conforms to people’s language expression habit of “adverb + adjective,” but can also accurately express its importance. Therefore, this paper comes up with the PDHL-TOPSIS mean for MAGDM based on the PDHLTS environment and applies it to the teaching quality evaluation of college English. On the one hand, we bring information entropy into the determination of target weight. The PDHL-TOPSIS technique is constructed and utilized for teaching level evaluation. Eventually, the PDHL-TOPSIS technique is compared with other decision-making means, and the novel mean turns out to be significant and valid.


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## Body

## 1. Introduction

To better integrate decision information, multiattribute group decision-making technology came into being [1, 2]. After MAGDM theory came into being, it has been widely used in finance [3, 4], engineering [5, 6], uncertain decision-making [7, 8], and many other aspects [9, 10]. In view of the intricateness and fuzzification of decision circumstances [11, 12], in many MCDM issues, expert opinions are often stated as fuzzy data [13, 14]. For this reason, Zadeh [15] raised that an expert’s perspective could be delivered by language term set (LTS). In many situations, LTS cannot exactly formulate a proficient perspective. Hence, hesitant fuzzy LTS (HFLTS) entered Rodriguez, Martinez, and Herrera [16]. Nevertheless, the probability distribution of LTS failed to be obtained in HFLTS. An idea about probabilistic linguistic term sets (PLTSs) came to Pang, Wang, and Xu [17]. Soon afterwards, critical malfunction matters were finished off by the PLAMM and PLWAMM formulas derived by Liu and Teng [18]. Liang, Dai, and Wang [19] assessed the customer satisfaction of O2O takeaway based on online reviews by integrating fuzzy comprehensive evaluation with AHP and PLTSs. Wei, Wei, and Guo [20] proposed the EDAS method for probabilistic linguistic MAGDM. The performance estimation system of college teachers was finished off by the PLPA and PLPWG formulas derived by Kobina, Liang, and He [21]. Wang, Wei, Lu, Wu, Wei, and Chen [22] defined the GRP and CRITIC methods for probabilistic uncertain linguistic MAGDM. Reference [23]defined the generalized dice similarity measures for probabilistic uncertain linguistic MAGDM. Su, Zhao, Wei, and Chen [24] built the PT-TODIM method for probabilistic linguistic MAGDM. Su, Zhao, Wei, Wei, and Chen [25] built the probabilistic uncertain linguistic EDAS method based on prospect theory for MAGDM. Gou, Liao, Xu, and Herrera [26] made a conceptual layout of the double hierarchy linguistic term set (DHLTS) and the double hierarchy hesitant fuzzy linguistic term set (DHHFLTS). Many research results have emerged one after another [27–35]. Soon afterwards, Gou, Xu, Liao, and Herrera [36] made a project about the probabilistic double hierarchy linguistic term set (PDHLTS). Lei, Wei, and Chen [37] defined a sequence of probabilistic double hierarchy linguistic polymerization formulas. Lei, Wei, and Chen [38]built the PDHL-CODAS model to rank Internet shopping terraces. Lei, Wei, Shen, and Guo [39] defined the PDHL-EDAS method for MAGDM with 3D printer selection.TOPSIS was initially built by Hwang and Yoon [40] to tackle MAGDM. Zhang, Zhan, and Yao [41] built the TOPSIS models for CVPIFRS under IFSs. Wu, Lu, Huang, Lan, Wu, and Huang [42] built the TOPSIS method to evaluate urban subway tunnel construction. Porro, Pardo-Bosch, Agell, and Sanchez [43] connected AHP with fuzzy linguistic TOPSIS. Prabhu, Abdullah, Ahmed, Nambirajan, and Pandiyan [44] designed triangular interval-valued fuzzy TOPSIS to cope with MADM. Sarwar [45] developed a rough D-TOPSIS with rough information. Peng, Peng, and Wang [46] built the IVHF-TOPSIS for MADM. Karaaslan and Hunu [47] defined TOPSIS for MAGDM under a type-2 single-valued neutrosophic setting. Gundogdu and Kahraman [48] defined interval-valued spherical fuzzy TOPSIS. Sen, Yang, Caihong, and Chengliang [49] assessed the Beijing-Tianjin-Hebei Region with entropy-TOPSIS.The critical aim of the thesis is to utilize the TOPSIS algorithm to finish off a few MAGDM matters on the strength of PDHLTSs. The research work of this paper is arranged as follows: (1) the TOPSIS is constructed on account of PDHLTSs; (2) the PDHL-TOPSIS mean is applied for finish off the MAGDM matter under PDHLTSs; (3) a numerical study for teaching level appraisal of university English is designed for developed method; and (4) a few contrasts and analysis are employed to state the rationality of PDHL-TOPSIS model. To this end, the definition of the PDHLTSs and elementary calculation principles are given. Subsequently, the defined PDHL-TOPSIS method is given. Then, the stated PDHL-TOPSIS mean is proved by teaching quality evaluation of college English and compared with PDHL-CODAS technology. The verdict is drawn at the end of the essay.


## 2. Preliminaries

First, let us learn some basics about PDHLTS.Definition 1.
(see [26]). Let us say DHL=ΓϑΙΩ|ϑ=−Α,…,−1,0,1,…,Α;Ω=−Β,…,−1,0,1,…,Β; is a DHLTS, the definition of the DHLTS is invented as(1)DHLTS=ΓϑΙΩΔ|ϑ=−A,…,−1,0,1,…,A;Ω=−B,…,−1,0,1,…,B;,here Δ=1,2,…,ΞDHL, the Δ−th double hierarchy linguistic element (DHLE) is narrated as ΓϑΔΙΩ, the quantities of all DHLEs is ΞDHL, and all DHLEs are sorted in ascending sequence.Definition 2.
(see [26]). Let us say DHL=ΓϑΙΩ|ϑ=−Α,…,−1,0,1,…Α;Ω=−Β,…,−1,0,1,…Β; is a DHLTS, the PDHLTS is created as(2)PDHLƛ=ΓϑΙΩΔƛΔ|ΓϑΙΩΔ∈DHL,ƛΔ≥0,∑Δ=1ΞPDHLƛƛΔ≤1,where Δ=1,2,…,ΞPDHLƛ, the Δ−th probabilistic double hierarchy linguistic element (PDHLE) is narrated as ΓϑΙΩΔƛΔ, the quantities of all PDHLEs are shown as ΞPDHLƛ.According to ϒΓϑΙΩΔƛΔ, all PDHLE are sorted in ascending order, the function ϒ is determined by the formula (3).Definition 3.
(see [36]). Let us assume DHL=ΓϑΙΩ|ϑ=−Α,…,−1,0,1,…Α;Ω=−Β,…,−1,0,1,…,Β; be a DHLTS, and PDHLƛ=ΓϑΙΩΔƛΔ|ΓϑΙΩΔ∈DHL,ƛΔ≥0,∑Δ=1ΞPDHLƛƛΔ≤1 be a PDHLTS. The above conversion function ϒ for PDHLE ΓϑΙΩΔƛΔ is designed as follows:(3)ϒ:−A,A×−B,B⟶0,1,ϒϑ,Ω=Ω+A+ϑB2AB=ϖ,(4)ϒ−1:0,1⟶−A,A×−B,B,ϒ−1ϖ=2Aϖ−AΙB2Aϖ−A−2Aϖ−Aor2Aϖ−A+1ΙB2Aϖ−A−2Aϖ−A−B.
Due to the probability sum of all PDHLEs in PDHLTS may be less than 1, we had to standardize PDHLTS and the specific measures as follows:(5)PDHLƛ=ΓϑΙΩΔƛΔ|ΓϑΙΩΔ∈DHL,ƛΔ≥0,∑Δ=1ΞPDHLƛƛΔ≤1,where ƛ˜Δ=ƛΔ/∑Δ=1ΞPDHLƛƛΔ;ϑ∈−A,A;Ω∈−B,B . Here A,B are all integers.Definition 4.
(see [38]). Let DHL=ΓϑΙΩ|ϑ=−Α,…,−1,0,1,…,Α;Ω=−Β,…,−1,0,1,…,Β; be a DHLTS. PDHL˜1ƛ˜=ΓϑΙΩΔƛ˜1Δ|ΓϑΙΩΔ∈DHL;Δ=1,2,…,ΞPDHL˜1ƛ˜ and PDHL˜2ƛ˜=ΓϑΙΩΔƛ˜2Δ|ΓϑΙΩΔ∈DHL;Δ=1,2,…,ΞPDHL˜2ƛ˜ be two different PDHLTSs, where #PDHL˜1ƛ˜ and #PDHL˜2ƛ˜ are the lengths of all PDHLEs in PDHL˜1ƛ˜ and PDHL˜2ƛ˜, respectively. Especially, if ΞPDHL˜1ƛ˜>ΞPDHL˜2ƛ˜, then the lengths of ΞPDHL˜1ƛ˜−ΞPDHL˜2ƛ˜ DHLEs are raised to PDHL˜2ƛ˜. The added PDHLEs should not be greater than any of the elements in the PDHL˜2ƛ˜, and the probability should be set to 0.Definition 5.
(see [38]). Let PDHL˜ƛ˜=ΓϑΙΩΔƛ˜Δ|ΓϑΙΩΔ∈DHL;Δ=1,2,…,ΞPDHL˜ƛ˜ be a PDHLTS. The expected values χPDHL˜ƛ˜ and deviation degree γPDHL˜ƛ˜ of PDHL˜ƛ˜ are built as χPDHL˜1ƛ˜=χPDHL˜2ƛ˜(6)χPDHL˜ƛ˜=∑Δ=1ΞPDHL˜ƛ˜ϒPDHL˜ƛ˜ƛ˜Δ∑Δ=1ΞPDHL˜ƛ˜ƛ˜Δ,(7)γPDHL˜ƛ˜=∑Δ=1ΞPDHL˜ƛ˜ϒPDHL˜ƛ˜ƛ˜Δ−χPDHL˜ƛ˜2∑Δ=1ΞPDHL˜ƛ˜ƛ˜Δ.
According to formulas (6) and (7), we can compare the size between any two PDHLTS as follows:(1)
ifχPDHL˜1ƛ˜>χPDHL˜2ƛ˜, then PDHL˜1ƛ˜>PDHL˜2ƛ˜;(2)
ifχPDHL˜1ƛ˜=χPDHL˜2ƛ˜, then if γPDHL˜1ƛ˜=γPDHL˜2ƛ˜, then PDHL˜1ƛ˜=PDHL˜2ƛ˜. In addition, if γPDHL˜1ƛ˜<γPDHL˜2ƛ˜, then, PDHL˜1ƛ˜>PDHL˜2ƛ˜.Definition 6.
(see [38]). Let DHL=ΓϑΙΩ|ϑ=−Α,…,−1,0,1,…,Α;Ω=−Β,…,−1,0,1,…,Β; be a DHLTS. PDHL˜1ƛ˜=ΓϑΙΩΔƛ˜1Δ|ΓϑΙΩΔ∈DHL;Δ=1,2,…,ΞPDHL˜1ƛ˜ and PDHL˜2ƛ˜=ΓϑΙΩΔƛ˜2Δ|ΓϑΙΩΔ∈DHL;Δ=1,2,…,ΞPDHL˜2ƛ˜ are two PDHLTSs, where ΞPDHL˜1ƛ˜=ΞPDHL˜2ƛ˜=ΞPDHL˜ƛ˜, then the Hamming distance HDPDHL˜1ƛ˜,PDHL˜2ƛ˜ between PDHL˜1ƛ˜ and PDHL˜2ƛ˜ are determined as follows:(8)HDPDHL˜1ƛ˜,PDHL˜2ƛ˜=∑Δ=1ΞPDHL˜ƛ˜ϒΓ2ϑΙΩΔƛ˜1Δ−ϒΓ2ϑΙΩΔƛ˜2ΔΞPDHL˜ƛ˜.


## 3. PDHL-TOPSIS Method for MAGDM with Entropy Weight

Now, TOPSIS mean in the context of PDHLTSs is proposed to deal with MAGDM matters and a complete MAGDM issue is narrated as follows: whole alternatives are shown asC=C1,C2,…,Ca, the D=D1,D2,…,Db is denoted a sequence of attributes, and the weight vector is ℑ=ℑ1,ℑ2,…,ℑb, where ℑσ∈0,1, σ=1,2,…,b, ∑σ=1bℑb=1, and JK=JK1,JK2,…,JKΤ are Τ experts, ℜ=ℜ1,ℜ2,…,ℜΤ is weight vector of all experts. Suppose that q-th expert JKq is evaluated τ−th alternative Cτ under σ−th attribute Dσ as PDHLτσqƛ=ΓτσϑΙΩΔqƛτσΔq|ΓτσϑΙΩΔq∈DHL,ƛτσΔq≥0,∑Δ=1ΞPDHLƛƛτσΔq≤1τ=1,2,…,a,σ=1,2,…,b,q=1,2,…,Τ.Furthermore, PDHL-TOPSIS mean is created to dispose of MAGDM issue with entropy weight.Step 1.
Established all decision-makers’ decision matrixes PDHLTSQq=PDHLqƛa×b.(9)PDHLTS=⊕q=1ΤwqPDHL˜qƛ˜=ϒ−1∪1−∏q=1Τ1−ϒΓϑΙΩΔℜq∑q=1Tƛ˜qΔq.Step 2.
Converted cost index into benefit index. LetPDHL˜ƛ˜=ΓϑΙΩΔƛ˜Δ|ΓϑΙΩΔ∈DHL;Δ=1,2,…,ΞPDHL˜ƛ˜ be a PDHLTS, if ΓϑΙΩΔƛ˜Δ is an evaluation on cost, we need to translate it into the benefit evaluation ΓϑΙΩΔƛ˜Δ.Step 3.
Computed the normalized decision matrixQ˜q=PDHL˜τσqƛ˜a×b.Step 4.
The proportion of each attribute is calculated depending on the entropy formula.
Entropy [50] is one of the important tools to ascertain the proportion of each attribute.
The first thing to do is to ascertain the normalized decision matrixNLijp as(10)PDHL˜τσqƛ˜=∑Δ=1ΞPDHL˜ƛ˜ϒΓτσϑΙΩΔqƛ˜τσΔq∑τ=1a∑Δ=1ΞPDHL˜ƛ˜ϒΓτσϑΙΩΔqƛ˜τσΔq,σ=1,2,…,b,secondly, the information of Shannon entropy E=E1,E2,…,Eb is got by the following formula:(11)Eσ=−1lna∑τ=1aPDHL˜τσqƛ˜lnPDHL˜τσqƛ˜,and PDHL˜τσqƛ˜lnPDHL˜τσqƛ˜ is defined as 0, if PDHL˜τσqƛ˜=0.
Finally, the attribute weightsℑ=ℑ1,ℑ2,⋯,ℑb are computed as(12)ℑσ=1−Eσ∑σ=1b1−Eσ,σ=1,2,⋯,b.Step 5.
We confirm the probabilistic double hierarchy linguistic ideal scheme more than zero (PDHLPIS) and the probabilistic double hierarchy linguistic ideal scheme less than zero (PDHLNIS) as follows:(13)PDHLPIS=PDHLPIS1,PDHLPIS2,…,PDHLPISb,PDHLNIS=PDHLNIS1,PDHLNIS2,…,PDHLNISb,where(14)PDHLPISσ=ΓϑΙΩΔƛ˜Δ|Δ=1,2,…,ΞPDHL˜ƛ˜=maxτϒΓϑΙΩΔƛ˜Δ,PDHLNISσ=ΓϑΙΩΔƛ˜Δ|Δ=1,2,…,ΞPDHL˜ƛ˜=minτϒΓϑΙΩΔƛ˜Δ.Step 6.
A calculation is made about the distances of each scheme from PDHLPIS and PDHLNIS, respectively, as follows:(15)dPDHLAτ,PDHLPIS=∑τ=1aℑτdPDHLAτσ,PDHLPISσ,dPDHLAτ,PDHLNIS=∑τ=1aℑσdPDHLAτσ,PDHLNISσHDPDHLAτσ,PDHLPISσ=∑Δ=1ΞPDHLτσƛ˜ϒΓσϑΙΩΔƛ˜Δ−ϒΓσϑΙΩΔƛ˜ΔΞPDHLτσƛ˜,HDPDHLAτσ,PDHLNISσ=∑Δ=1ΞPDHLτσƛ˜ϒΓσϑΙΩΔƛ˜Δ−ϒΓσϑΙΩΔƛ˜ΔΞPDHLτσƛ˜.Step 7.
A computation is made for the probabilistic double hierarchy linguistic relative closeness degree (PDHLRCD) of each scheme from PDHLPIS.(16)PDHLRCDPDHLAτ,PDHLPIS=dPDHLAτ,PDHLPISdPDHLAτ,PDHLPIS+dPDHLAτ,PDHLNIS,τ=1,2,…,a.Step 8.
The final sorting outcomes of each scheme are ascertained according to thePDHLRCDPDHLAτ,PDHLPIS. The optimal scheme is the one nearest to PDHLPIS and outermost from PDHLNIS. Hence, if one scheme is the minimum PDHLRCDPDHLAτ,PDHLPIS data, it is the first-best scheme.


## 4. Numerical Example and Comparative Analysis

### 4.1. Numerical Example

Since 2003, the state conducted a comprehensive evaluation of the teaching work of all ordinary colleges and universities in China over a five-year period and form a five-year round teaching evaluation system. The evaluation of undergraduate teaching level reflects the shift in the focus of higher education. That is, while the scale is developing continuously, the improvement of quality is put in a more prominent position. The evaluation of undergraduate teaching level is an important step for the ministry of education to adhere to the scientific outlook on development, attach great importance to teaching work, pay close attention to teaching quality, build a quality monitoring, and guarantee system of higher education with Chinese characteristics. The evaluation includes many aspects, and the important point of these treaties is the teacher’s classroom teaching quality evaluation by undergraduates and training school students in literature and history classes in our country’s common colleges and universities. The evaluation is to perfect classroom teaching in an institution of higher learning, to extend the teacher’s managerial theory, and to improve classroom teaching quality. The higher school class teaching quality mainly includes study of the evaluation’s features, classes, functions, major parts, and contents. It clearly defines classroom teaching, classroom teaching quality, classroom teaching quality evaluation, teaching quality evaluation by students, and the evaluation target system. The study’s theory summaries analyze the evaluation’s aims, functions, processes, patterns, the result’s presentation, and the target system. It also sums up the gratifying achievements of our country’s teaching evaluation. Our principal problems in evaluating are lack of formation in guiding ideology, lack of science in the target system, lack of standards in operating, and lack of effectiveness for a given period of time in the result feedback. We mainly analyze two most important effect factors: teacher and student. There are four primary aspects in the theory proposition of perfecting a teacher’s classroom teaching evaluation in colleges and universities. They are establishing a modern teaching evaluation view; establishing a scientific, reasonable, and well operating evaluation target system; establishing a perfect online evaluation system by using modern network technology; and bringing every positive factor into play, perfecting the whole evaluating process. However, these problems can be attributed to the multiattribute group decision-making problem. This paper analyzes teaching quality evaluation problems of college English based on the proposed PDHL-TOPSIS method. There are five given college English teaching waysC=C1,C2,C3,C4,C5 , which maybe the best. For the sake of assessing the teaching effect of college English teaching patterns fairly, three experts JK=JK1,JK2,JK3 (expert’s weight ℜ=0.40,0.33,0.27 are invited. All experts depict their assessment information through four subsequent attributes: ① D1 is the teaching attitude; ② D2 is the teaching method; ③ D3 is student feedback; and ④ D4 is the peer recognition. Evidently, all attributes are benefits, and ℑ=ℑ1,ℑ2,ℑ3,ℑ4 is the weight of four attributes, where ℑσ∈0,1, ℑ=1,2,3,4, ∑σ=14ℑσ=1. We suppose that q−th expert JKq evaluated τ−th alternative Cτ under σ−th attribute Dσ as PDHLτσqƛ=ΓτσϑΙΩΔqƛτσΔq|ΓτσϑΙΩΔq∈DHL,ƛτσΔq≥0,∑Δ=1ΞPDHLƛƛτσΔq≤1τ=1,2,…,5,σ=1,2,…,4,q=1,2,3, where, the double linguistic hierarchy evaluation information tables are given as follows:(17)Γ=Γ−3=extremely poor,Γ−2=very poor,Γ−1=poor,Γ0=medium,Γ1=good,Γ2=very good,Γ3=extremely good,Ι=Ι−3=far form,Ι−2=only a little,Ι−1=a little,Ι0=just right,Ι1=much,Ι2=very much,Ι2=very much,Ι3=extirely.Then, the decision matrixes of each invited expert are expressed in Tables1–3.Table 1
The PDHLTS evaluation of all alternatives is provided by JK1.
D1D2D3D4C1Γ−1Ι−30.3,Γ−1Ι10.3,Γ2Ι10.4Γ−1Ι−10.3,Γ0Ι−20.5,Γ1Ι−20.2Γ−2Ι01.0Γ−2Ι−10.2,Γ−1Ι10.1,Γ−1Ι20.7C2Γ−3Ι−10.3,Γ2Ι−10.2,Γ3Ι00.5Γ−1Ι10.4,Γ0Ι30.6Γ1Ι−20.2,Γ3Ι−20.5,Γ2Ι10.3Γ1Ι−20.2,Γ2Ι00.7,Γ−1Ι10.1C3Γ1Ι−10.4,Γ2Ι10.1,Γ2Ι20.5Γ−3Ι00.2,Γ−2Ι00.3,Γ−1Ι00.5Γ−3Ι10.6,Γ1Ι−10.1,Γ3Ι00.3Γ−2Ι10.8,Γ0Ι00.1,Γ0Ι10.1C4Γ−2Ι10.2,Γ−1Ι20.4,Γ0Ι20.4Γ−1Ι10.4,Γ0Ι−20.5,Γ3Ι00.1Γ−3Ι10.1,Γ−1Ι20.5,Γ2Ι30.4Γ−1Ι20.3,Γ−1Ι10.7,C5Γ−3Ι00.6,Γ−2Ι20.1,Γ2Ι30.3Γ−1Ι10.3,Γ0Ι−10.4,Γ1Ι20.3Γ0Ι−10.1,Γ2Ι00.9Γ3Ι01.0Table 2
The PDHLTS evaluation of all alternatives is provided by JK2.
D1D2D3D4C1Γ−2Ι10.8,Γ−1Ι−10.2Γ0Ι−11.0Γ−1Ι30.2,Γ2Ι−20.2,Γ2Ι00.6Γ−1Ι−30.4,Γ−1Ι00.5,Γ2Ι−10.1C2Γ2Ι10.2,Γ2Ι00.5,Γ3Ι00.3Γ−3Ι00.2,Γ−3Ι10.6,Γ−3Ι20.2,Γ3Ι01.0Γ−2Ι10.2,Γ3Ι00.4,Γ−2Ι−20.4C3Γ−1Ι−10.4,Γ1Ι20.1,Γ1Ι30.5Γ−2Ι20.5,Γ2Ι00.5Γ2Ι00.2,Γ2Ι20.8Γ1Ι10.9,Γ2Ι10.1C4Γ−1Ι00.3,Γ0Ι00.1,Γ0Ι30.6Γ−3Ι−10.1,Γ−3Ι−10.3,Γ3Ι00.6Γ−2Ι00.4,Γ−2Ι20.6Γ−2Ι−10.3,Γ−1Ι00.2,Γ−1Ι30.5C5Γ2Ι−10.1,Γ2Ι00.5,Γ2Ι20.4Γ−2Ι−30.5,Γ1Ι30.3,Γ2Ι30.2Γ2Ι−31.0Γ2Ι−30.2,Γ2Ι−10.7,Γ3Ι00.1Table 3
The PDHLTS evaluation of all alternatives is provided by JK3.
D1D2D3D4C1Γ−1Ι20.4,Γ2Ι10.2,Γ2Ι30.4Γ−2Ι00.3,Γ−3Ι20.6,Γ1Ι20.1Γ−1Ι21.0Γ−2Ι10.4,Γ0Ι10.6C2Γ0Ι01.0Γ−2Ι00.6,Γ0Ι10.4Γ3Ι−20.2,Γ3Ι−10.4,Γ3Ι00.4Γ1Ι−30.2,Γ1Ι−10.3,Γ2Ι00.5C3Γ3Ι−20.1,Γ2Ι00.8,Γ2Ι20.1Γ−2Ι−30.3,Γ−2Ι00.4,Γ−2Ι30.3Γ3Ι−20.7,Γ−1Ι10.3Γ0Ι−20.4,Γ2Ι00.1,Γ1Ι10.5C4Γ−2Ι00.1,Γ−2Ι10.3,Γ−0Ι10.6Γ3Ι−20.1,Γ3Ι00.4,Γ1Ι10.5Γ−3Ι20.1,Γ−2Ι30.3,Γ−1Ι30.6Γ−1Ι10.3,Γ0Ι10.2,Γ1Ι20.5C5Γ−3Ι00.2,Γ−2Ι10.4,Γ2Ι20.4Γ0Ι−30.3,Γ0Ι−10.1,Γ0Ι30.6Γ−3Ι−10.2,Γ−1Ι00.2,Γ1Ι20.6Γ1Ι20.1,Γ2Ι30.6,Γ−3Ι20.3Now, the built PDHL-TOPSIS method is used to select the optimal college English teaching method.(i)
Step 1. The evaluation matrix of the three experts is standardized (Tables4–6).(ii)
Step 2. According to the weighted average operator, the evaluation of three experts is aggregated into a total decision matrix, which has been converted to real data (see Table7).(iii)
Step 3. The weight of the decision attribute is given.(18)ℑ1=0.1077,ℑ2=0.4564,ℑ3=0.1323,ℑ4=0.3036.(iv)
Step 4. The positive ideal point and the negative ideal point are determined according to the global decision matrix, which has been converted to real data (see Tables8 and 9).(v)
Step 5. The distance between each scheme and the positive and negative ideal points is calculated, respectively (see Tables10 and 11).(vi)
Step 6. The probabilistic double hierarchy linguistic relative closeness degree (PDHLRCD) of each alternative from PDHLPIS is calculated (see Table12).(vii)
Step 7. We can get the optimal college English teaching methodsC5.Table 4
The standardized decision matrix is provided by JK1.
D1D2D3D4C1Γ−1Ι−30.3,Γ−1Ι10.3,Γ2Ι10.4Γ−1Ι−10.3,Γ0Ι−20.5,Γ1Ι−20.2Γ−2Ι00,Γ−2Ι00,Γ−2Ι01.0Γ−2Ι−10.2,Γ−1Ι00.1,Γ−1Ι20.7C2Γ−3Ι−10.3,Γ2Ι−10.2,Γ3Ι00.5Γ−1Ι10,Γ−1Ι10.4,Γ0Ι30.6Γ1Ι−20.2,Γ3Ι−20.5,Γ2Ι10.3Γ1Ι−20.2,Γ2Ι00.7,Γ−1Ι10.1C3Γ1Ι−10.4,Γ2Ι10.1,Γ2Ι20.5Γ−3Ι00.2,Γ−2Ι00.3,Γ−1Ι00.5Γ−3Ι10.6,Γ1Ι−10.1,Γ3Ι00.3Γ−2Ι10.8,Γ0Ι00.1,Γ0Ι10.1C4Γ−2Ι10.2,Γ−1Ι20.4,Γ0Ι20.4Γ−1Ι10.4,Γ0Ι−20.5,Γ3Ι00.1Γ−3Ι10.1,Γ−1Ι20.5,Γ2Ι30.4Γ−1Ι20,Γ−1Ι20.3,Γ−1Ι10.7,C5Γ−3Ι00.6,Γ−2Ι20.1,Γ2Ι30.3Γ−1Ι10.3,Γ0Ι−10.4,Γ1Ι20.3Γ0Ι−10,Γ0Ι−10.1,Γ2Ι00.9Γ3Ι01.0,Γ3Ι01.0,Γ3Ι01.0Table 5
The standardized decision matrix is provided by JK2.
D1D2D3D4C1Γ−2Ι10,Γ−2Ι10.8,Γ−1Ι−10.2Γ0Ι−10,Γ0Ι−10,Γ0Ι−11.0Γ−1Ι30.2,Γ2Ι−20.2,Γ2Ι00.6Γ−1Ι−30.4,Γ−1Ι00.5,Γ2Ι−10.1C2Γ2Ι10.2,Γ2Ι00.5,Γ3Ι00.3Γ−3Ι00.2,Γ−3Ι10.6,Γ−3Ι20.2,Γ3Ι00,Γ3Ι00,Γ3Ι01.0Γ−2Ι10.2,Γ3Ι00.4,Γ−2Ι−20.4C3Γ−1Ι−10.4,Γ1Ι20.1,Γ1Ι30.5Γ−2Ι20,Γ−2Ι20.5,Γ2Ι00.5Γ2Ι00,Γ2Ι00.2,Γ2Ι20.8Γ1Ι10,Γ1Ι10.9,Γ2Ι10.1C4Γ−1Ι00.3,Γ0Ι00.1,Γ0Ι30.6Γ−3Ι−10.1,Γ−3<Ι−10.3,Γ3Ι00.6Γ−2Ι00,Γ−2Ι00.4,Γ−2Ι20.6Γ−2Ι−10.3,Γ−1Ι00.2,Γ−1Ι30.5C5Γ2Ι−10.1,Γ2Ι00.5,Γ2Ι20.4Γ−2Ι−30.5,Γ1Ι30.3,Γ2Ι30.2Γ2Ι−30,Γ2Ι−30,Γ2Ι−31.0Γ2Ι−30.2,Γ2Ι−10.7,Γ3Ι00.1Table 6
The standardized decision matrix is provided by JK3.
D1D2D3D4C1Γ−1Ι20.4,Γ2Ι10.2,Γ2Ι30.4Γ−2Ι00.3,Γ−3Ι20.6,Γ1Ι20.1Γ−1Ι20,Γ−1Ι20,Γ−1Ι21.0Γ−2Ι10,Γ−2Ι10.4,Γ0Ι10.6C2Γ0Ι00,Γ0Ι00,Γ0Ι01.0Γ−2Ι00,Γ−2Ι00.6,Γ0Ι10.4Γ3Ι−20.2,Γ3Ι−10.4,Γ3Ι00.4Γ1Ι−30.2,Γ1Ι−10.3,Γ2Ι00.5C3Γ3Ι−20.1,Γ2Ι00.8,Γ2Ι20.1Γ−2Ι−30.3,Γ−2Ι00.4,Γ−2Ι30.3Γ3Ι−20,Γ3Ι−20.7,Γ−1Ι10.3Γ0Ι−20.4,Γ2Ι00.1,Γ1Ι10.5C4Γ−2Ι00.1,Γ−2Ι10.3,Γ−0Ι10.6Γ3Ι−20.1,Γ3Ι00.4,Γ1Ι10.5Γ−3Ι20.1,Γ−2Ι30.3,Γ−1Ι30.6Γ−1Ι10.3,Γ0Ι10.2,Γ1Ι20.5C5Γ−3Ι00.2,Γ−2Ι10.4,Γ2Ι20.4Γ0Ι−30.3,Γ0Ι−10.1,Γ0Ι30.6Γ−3Ι−10.2,Γ−1Ι00.2,Γ1Ι20.6Γ1Ι20.1,Γ2Ι30.6,Γ−3Ι20.3Table 7
The overall evaluation matrix.
D1D2D3D4C1Γ1Ι1.14190.2333,Γ−1Ι1.51720.4333,Γ−3Ι0.00000.3333Γ1Ι0.39140.2000,Γ0Ι2.79420.3667,Γ−1Ι1.14150.4333Γ0Ι2.35890.0667,Γ0Ι0.35610.0667,Γ−1Ι1.90470.8667Γ2Ι0.10820.2000,Γ1Ι0.51000.3333,Γ−1Ι0.95850.4667C2Γ−1Ι2.44490.2667,Γ−2Ι1.52810.1333,Γ−3Ι0.00000.6000Γ1Ι2.07150.0667,Γ1Ι1.80860.5333,Γ−1Ι2.96300.4000Γ−3Ι0.00000.1333,Γ−3Ι0.00000.3000,Γ−3Ι0.00000.5667Γ0Ι0.93350.2000,Γ−3Ι0.00000.4667,Γ−1Ι2.94190.3333C3Γ−1Ι0.12240.3000,Γ−3Ι2.80490.3333,Γ−3Ι1.43700.3667Γ2Ι1.16720.1667,Γ1Ι2.30810.4000,Γ−1Ι1.59450.4333Γ−2Ι2.38160.2000,Γ−2Ι0.77370.3333,Γ−3Ι0.00000.4667Γ0Ι0.3387<>0.4000,Γ−2Ι2.51040.5000,Γ−2Ι1.45960.1000C4Γ1Ι1.55580.2000,Γ0Ι2.80490.2667,Γ−1Ι0.89710.5333Γ−1Ι2.31430.2000,Γ−3Ι0.00000.4000,Γ−3Ι0.00000.4000Γ2Ι1.04730.0667,Γ1Ι0.00850.4000,Γ−3Ι0.00000.5333Γ1Ι0.44780.2000,Γ0Ι1.38770.3667,Γ−1Ι1.54150.4333C5Γ0Ι1.95750.3000,Γ−3Ι2.80490.3333,Γ−3Ι1.43700.3667Γ1Ι1.24880.3667,Γ−1Ι0.72120.2667,Γ−3Ι0.00000.3667Γ0Ι1.04740.0667,Γ−1Ι2.87500.1000,Γ−2Ι1.07560.8333Γ−3Ι0.00000.1000,Γ−3Ι0.00000.4333,Γ−3Ι0.00000.4667Table 8
The positive ideal point.
D1D2D3D4Γ−1Ι2.44490.2667,Γ−2Ι1.52810.1333,Γ−3Ι0.00000.6000Γ−1Ι2.31430.2000,Γ−3Ι0.00000.4000,Γ−3Ι0.00000.4000Γ−3Ι0.00000.1333,Γ−3Ι0.00000.3000,Γ−3Ι0.00000.5667Γ−3Ι0.00000.1000,Γ−3Ι0.00000.4333,Γ−3Ι0.00000.4667Table 9
The Negative ideal point.
D1D2D3D4Γ1Ι1.55580.2000,Γ0Ι2.80490.2667,Γ−1Ι0.89710.5333Γ1Ι2.07150.0667,Γ1Ι1.80860.5333,Γ−1Ι2.96300.4000Γ0Ι2.35890.0667,Γ0Ι0.35610.0667,Γ−1Ι1.90470.8667Γ2Ι0.10820.2000,Γ1Ι0.51000.3333,Γ−1Ι0.95850.4667Table 10
dPDHLAτ,PDHLPIS.
C1C2C3C4C5dPDHLAτ,PDHLPIS0.16630.12110.16690.08160.0734Table 11
dPDHLAτ,PDHLNIS.
C1C2C3C4C5dPDHLAi,PDHLNIS0.02260.08830.10470.10560.1174Table 12
PDHLRCD.
C1C2C3C4C5PDHLRCD0.88040.57820.61450.43580.3846


### 4.2. Comparative Analysis

Finally, we compared it with the PDHL-VIKOR method [36], the PDHL-CODAS method [38], the PDHLWA operator, the PDHLWG operator, the PDHLPWA operator, and the PDHLPWG operator. The results and analysis are as follows (See Table 13).Table 13
The numerical results and rank were derived by the PDHL-CODAS.
PDHL-TOPSISRankPDHL-CODASRankC10.88045−0.50405C20.578230.07353C30.614540.16722C40.435820.07344C50.384610.28471The expected values of the PDHLWA operatorRankThe expected values of the PDHLWA operatorRankC10.440950.44335C20.444440.65993C30.648830.76112C40.733320.56094C50.884110.88231The expected values of the PDHLPWA operatorRankThe expected values of the PDHLPWG operatorRankC10.443150.44555C20.646630.66213C30.551040.76332C40.735520.56314C50.886310.88451It can be seen from Table13 that although the six methods are different, the optimal scheme obtained is the same. Only schemes 3 and 4 have slight differences between the PDHLWA operator and other methods. Therefore, the PDHL-TOPSIS method proposed by us can scientifically and effectively solve the investment decision problem.


## 4.1. Numerical Example

Since 2003, the state conducted a comprehensive evaluation of the teaching work of all ordinary colleges and universities in China over a five-year period and form a five-year round teaching evaluation system. The evaluation of undergraduate teaching level reflects the shift in the focus of higher education. That is, while the scale is developing continuously, the improvement of quality is put in a more prominent position. The evaluation of undergraduate teaching level is an important step for the ministry of education to adhere to the scientific outlook on development, attach great importance to teaching work, pay close attention to teaching quality, build a quality monitoring, and guarantee system of higher education with Chinese characteristics. The evaluation includes many aspects, and the important point of these treaties is the teacher’s classroom teaching quality evaluation by undergraduates and training school students in literature and history classes in our country’s common colleges and universities. The evaluation is to perfect classroom teaching in an institution of higher learning, to extend the teacher’s managerial theory, and to improve classroom teaching quality. The higher school class teaching quality mainly includes study of the evaluation’s features, classes, functions, major parts, and contents. It clearly defines classroom teaching, classroom teaching quality, classroom teaching quality evaluation, teaching quality evaluation by students, and the evaluation target system. The study’s theory summaries analyze the evaluation’s aims, functions, processes, patterns, the result’s presentation, and the target system. It also sums up the gratifying achievements of our country’s teaching evaluation. Our principal problems in evaluating are lack of formation in guiding ideology, lack of science in the target system, lack of standards in operating, and lack of effectiveness for a given period of time in the result feedback. We mainly analyze two most important effect factors: teacher and student. There are four primary aspects in the theory proposition of perfecting a teacher’s classroom teaching evaluation in colleges and universities. They are establishing a modern teaching evaluation view; establishing a scientific, reasonable, and well operating evaluation target system; establishing a perfect online evaluation system by using modern network technology; and bringing every positive factor into play, perfecting the whole evaluating process. However, these problems can be attributed to the multiattribute group decision-making problem. This paper analyzes teaching quality evaluation problems of college English based on the proposed PDHL-TOPSIS method. There are five given college English teaching waysC=C1,C2,C3,C4,C5 , which maybe the best. For the sake of assessing the teaching effect of college English teaching patterns fairly, three experts JK=JK1,JK2,JK3 (expert’s weight ℜ=0.40,0.33,0.27 are invited. All experts depict their assessment information through four subsequent attributes: ① D1 is the teaching attitude; ② D2 is the teaching method; ③ D3 is student feedback; and ④ D4 is the peer recognition. Evidently, all attributes are benefits, and ℑ=ℑ1,ℑ2,ℑ3,ℑ4 is the weight of four attributes, where ℑσ∈0,1, ℑ=1,2,3,4, ∑σ=14ℑσ=1. We suppose that q−th expert JKq evaluated τ−th alternative Cτ under σ−th attribute Dσ as PDHLτσqƛ=ΓτσϑΙΩΔqƛτσΔq|ΓτσϑΙΩΔq∈DHL,ƛτσΔq≥0,∑Δ=1ΞPDHLƛƛτσΔq≤1τ=1,2,…,5,σ=1,2,…,4,q=1,2,3, where, the double linguistic hierarchy evaluation information tables are given as follows:(17)Γ=Γ−3=extremely poor,Γ−2=very poor,Γ−1=poor,Γ0=medium,Γ1=good,Γ2=very good,Γ3=extremely good,Ι=Ι−3=far form,Ι−2=only a little,Ι−1=a little,Ι0=just right,Ι1=much,Ι2=very much,Ι2=very much,Ι3=extirely.Then, the decision matrixes of each invited expert are expressed in Tables1–3.Table 1
The PDHLTS evaluation of all alternatives is provided by JK1.
D1D2D3D4C1Γ−1Ι−30.3,Γ−1Ι10.3,Γ2Ι10.4Γ−1Ι−10.3,Γ0Ι−20.5,Γ1Ι−20.2Γ−2Ι01.0Γ−2Ι−10.2,Γ−1Ι10.1,Γ−1Ι20.7C2Γ−3Ι−10.3,Γ2Ι−10.2,Γ3Ι00.5Γ−1Ι10.4,Γ0Ι30.6Γ1Ι−20.2,Γ3Ι−20.5,Γ2Ι10.3Γ1Ι−20.2,Γ2Ι00.7,Γ−1Ι10.1C3Γ1Ι−10.4,Γ2Ι10.1,Γ2Ι20.5Γ−3Ι00.2,Γ−2Ι00.3,Γ−1Ι00.5Γ−3Ι10.6,Γ1Ι−10.1,Γ3Ι00.3Γ−2Ι10.8,Γ0Ι00.1,Γ0Ι10.1C4Γ−2Ι10.2,Γ−1Ι20.4,Γ0Ι20.4Γ−1Ι10.4,Γ0Ι−20.5,Γ3Ι00.1Γ−3Ι10.1,Γ−1Ι20.5,Γ2Ι30.4Γ−1Ι20.3,Γ−1Ι10.7,C5Γ−3Ι00.6,Γ−2Ι20.1,Γ2Ι30.3Γ−1Ι10.3,Γ0Ι−10.4,Γ1Ι20.3Γ0Ι−10.1,Γ2Ι00.9Γ3Ι01.0Table 2
The PDHLTS evaluation of all alternatives is provided by JK2.
D1D2D3D4C1Γ−2Ι10.8,Γ−1Ι−10.2Γ0Ι−11.0Γ−1Ι30.2,Γ2Ι−20.2,Γ2Ι00.6Γ−1Ι−30.4,Γ−1Ι00.5,Γ2Ι−10.1C2Γ2Ι10.2,Γ2Ι00.5,Γ3Ι00.3Γ−3Ι00.2,Γ−3Ι10.6,Γ−3Ι20.2,Γ3Ι01.0Γ−2Ι10.2,Γ3Ι00.4,Γ−2Ι−20.4C3Γ−1Ι−10.4,Γ1Ι20.1,Γ1Ι30.5Γ−2Ι20.5,Γ2Ι00.5Γ2Ι00.2,Γ2Ι20.8Γ1Ι10.9,Γ2Ι10.1C4Γ−1Ι00.3,Γ0Ι00.1,Γ0Ι30.6Γ−3Ι−10.1,Γ−3Ι−10.3,Γ3Ι00.6Γ−2Ι00.4,Γ−2Ι20.6Γ−2Ι−10.3,Γ−1Ι00.2,Γ−1Ι30.5C5Γ2Ι−10.1,Γ2Ι00.5,Γ2Ι20.4Γ−2Ι−30.5,Γ1Ι30.3,Γ2Ι30.2Γ2Ι−31.0Γ2Ι−30.2,Γ2Ι−10.7,Γ3Ι00.1Table 3
The PDHLTS evaluation of all alternatives is provided by JK3.
D1D2D3D4C1Γ−1Ι20.4,Γ2Ι10.2,Γ2Ι30.4Γ−2Ι00.3,Γ−3Ι20.6,Γ1Ι20.1Γ−1Ι21.0Γ−2Ι10.4,Γ0Ι10.6C2Γ0Ι01.0Γ−2Ι00.6,Γ0Ι10.4Γ3Ι−20.2,Γ3Ι−10.4,Γ3Ι00.4Γ1Ι−30.2,Γ1Ι−10.3,Γ2Ι00.5C3Γ3Ι−20.1,Γ2Ι00.8,Γ2Ι20.1Γ−2Ι−30.3,Γ−2Ι00.4,Γ−2Ι30.3Γ3Ι−20.7,Γ−1Ι10.3Γ0Ι−20.4,Γ2Ι00.1,Γ1Ι10.5C4Γ−2Ι00.1,Γ−2Ι10.3,Γ−0Ι10.6Γ3Ι−20.1,Γ3Ι00.4,Γ1Ι10.5Γ−3Ι20.1,Γ−2Ι30.3,Γ−1Ι30.6Γ−1Ι10.3,Γ0Ι10.2,Γ1Ι20.5C5Γ−3Ι00.2,Γ−2Ι10.4,Γ2Ι20.4Γ0Ι−30.3,Γ0Ι−10.1,Γ0Ι30.6Γ−3Ι−10.2,Γ−1Ι00.2,Γ1Ι20.6Γ1Ι20.1,Γ2Ι30.6,Γ−3Ι20.3Now, the built PDHL-TOPSIS method is used to select the optimal college English teaching method.(i)
Step 1. The evaluation matrix of the three experts is standardized (Tables4–6).(ii)
Step 2. According to the weighted average operator, the evaluation of three experts is aggregated into a total decision matrix, which has been converted to real data (see Table7).(iii)
Step 3. The weight of the decision attribute is given.(18)ℑ1=0.1077,ℑ2=0.4564,ℑ3=0.1323,ℑ4=0.3036.(iv)
Step 4. The positive ideal point and the negative ideal point are determined according to the global decision matrix, which has been converted to real data (see Tables8 and 9).(v)
Step 5. The distance between each scheme and the positive and negative ideal points is calculated, respectively (see Tables10 and 11).(vi)
Step 6. The probabilistic double hierarchy linguistic relative closeness degree (PDHLRCD) of each alternative from PDHLPIS is calculated (see Table12).(vii)
Step 7. We can get the optimal college English teaching methodsC5.Table 4
The standardized decision matrix is provided by JK1.
D1D2D3D4C1Γ−1Ι−30.3,Γ−1Ι10.3,Γ2Ι10.4Γ−1Ι−10.3,Γ0Ι−20.5,Γ1Ι−20.2Γ−2Ι00,Γ−2Ι00,Γ−2Ι01.0Γ−2Ι−10.2,Γ−1Ι00.1,Γ−1Ι20.7C2Γ−3Ι−10.3,Γ2Ι−10.2,Γ3Ι00.5Γ−1Ι10,Γ−1Ι10.4,Γ0Ι30.6Γ1Ι−20.2,Γ3Ι−20.5,Γ2Ι10.3Γ1Ι−20.2,Γ2Ι00.7,Γ−1Ι10.1C3Γ1Ι−10.4,Γ2Ι10.1,Γ2Ι20.5Γ−3Ι00.2,Γ−2Ι00.3,Γ−1Ι00.5Γ−3Ι10.6,Γ1Ι−10.1,Γ3Ι00.3Γ−2Ι10.8,Γ0Ι00.1,Γ0Ι10.1C4Γ−2Ι10.2,Γ−1Ι20.4,Γ0Ι20.4Γ−1Ι10.4,Γ0Ι−20.5,Γ3Ι00.1Γ−3Ι10.1,Γ−1Ι20.5,Γ2Ι30.4Γ−1Ι20,Γ−1Ι20.3,Γ−1Ι10.7,C5Γ−3Ι00.6,Γ−2Ι20.1,Γ2Ι30.3Γ−1Ι10.3,Γ0Ι−10.4,Γ1Ι20.3Γ0Ι−10,Γ0Ι−10.1,Γ2Ι00.9Γ3Ι01.0,Γ3Ι01.0,Γ3Ι01.0Table 5
The standardized decision matrix is provided by JK2.
D1D2D3D4C1Γ−2Ι10,Γ−2Ι10.8,Γ−1Ι−10.2Γ0Ι−10,Γ0Ι−10,Γ0Ι−11.0Γ−1Ι30.2,Γ2Ι−20.2,Γ2Ι00.6Γ−1Ι−30.4,Γ−1Ι00.5,Γ2Ι−10.1C2Γ2Ι10.2,Γ2Ι00.5,Γ3Ι00.3Γ−3Ι00.2,Γ−3Ι10.6,Γ−3Ι20.2,Γ3Ι00,Γ3Ι00,Γ3Ι01.0Γ−2Ι10.2,Γ3Ι00.4,Γ−2Ι−20.4C3Γ−1Ι−10.4,Γ1Ι20.1,Γ1Ι30.5Γ−2Ι20,Γ−2Ι20.5,Γ2Ι00.5Γ2Ι00,Γ2Ι00.2,Γ2Ι20.8Γ1Ι10,Γ1Ι10.9,Γ2Ι10.1C4Γ−1Ι00.3,Γ0Ι00.1,Γ0Ι30.6Γ−3Ι−10.1,Γ−3<Ι−10.3,Γ3Ι00.6Γ−2Ι00,Γ−2Ι00.4,Γ−2Ι20.6Γ−2Ι−10.3,Γ−1Ι00.2,Γ−1Ι30.5C5Γ2Ι−10.1,Γ2Ι00.5,Γ2Ι20.4Γ−2Ι−30.5,Γ1Ι30.3,Γ2Ι30.2Γ2Ι−30,Γ2Ι−30,Γ2Ι−31.0Γ2Ι−30.2,Γ2Ι−10.7,Γ3Ι00.1Table 6
The standardized decision matrix is provided by JK3.
D1D2D3D4C1Γ−1Ι20.4,Γ2Ι10.2,Γ2Ι30.4Γ−2Ι00.3,Γ−3Ι20.6,Γ1Ι20.1Γ−1Ι20,Γ−1Ι20,Γ−1Ι21.0Γ−2Ι10,Γ−2Ι10.4,Γ0Ι10.6C2Γ0Ι00,Γ0Ι00,Γ0Ι01.0Γ−2Ι00,Γ−2Ι00.6,Γ0Ι10.4Γ3Ι−20.2,Γ3Ι−10.4,Γ3Ι00.4Γ1Ι−30.2,Γ1Ι−10.3,Γ2Ι00.5C3Γ3Ι−20.1,Γ2Ι00.8,Γ2Ι20.1Γ−2Ι−30.3,Γ−2Ι00.4,Γ−2Ι30.3Γ3Ι−20,Γ3Ι−20.7,Γ−1Ι10.3Γ0Ι−20.4,Γ2Ι00.1,Γ1Ι10.5C4Γ−2Ι00.1,Γ−2Ι10.3,Γ−0Ι10.6Γ3Ι−20.1,Γ3Ι00.4,Γ1Ι10.5Γ−3Ι20.1,Γ−2Ι30.3,Γ−1Ι30.6Γ−1Ι10.3,Γ0Ι10.2,Γ1Ι20.5C5Γ−3Ι00.2,Γ−2Ι10.4,Γ2Ι20.4Γ0Ι−30.3,Γ0Ι−10.1,Γ0Ι30.6Γ−3Ι−10.2,Γ−1Ι00.2,Γ1Ι20.6Γ1Ι20.1,Γ2Ι30.6,Γ−3Ι20.3Table 7
The overall evaluation matrix.
D1D2D3D4C1Γ1Ι1.14190.2333,Γ−1Ι1.51720.4333,Γ−3Ι0.00000.3333Γ1Ι0.39140.2000,Γ0Ι2.79420.3667,Γ−1Ι1.14150.4333Γ0Ι2.35890.0667,Γ0Ι0.35610.0667,Γ−1Ι1.90470.8667Γ2Ι0.10820.2000,Γ1Ι0.51000.3333,Γ−1Ι0.95850.4667C2Γ−1Ι2.44490.2667,Γ−2Ι1.52810.1333,Γ−3Ι0.00000.6000Γ1Ι2.07150.0667,Γ1Ι1.80860.5333,Γ−1Ι2.96300.4000Γ−3Ι0.00000.1333,Γ−3Ι0.00000.3000,Γ−3Ι0.00000.5667Γ0Ι0.93350.2000,Γ−3Ι0.00000.4667,Γ−1Ι2.94190.3333C3Γ−1Ι0.12240.3000,Γ−3Ι2.80490.3333,Γ−3Ι1.43700.3667Γ2Ι1.16720.1667,Γ1Ι2.30810.4000,Γ−1Ι1.59450.4333Γ−2Ι2.38160.2000,Γ−2Ι0.77370.3333,Γ−3Ι0.00000.4667Γ0Ι0.3387<>0.4000,Γ−2Ι2.51040.5000,Γ−2Ι1.45960.1000C4Γ1Ι1.55580.2000,Γ0Ι2.80490.2667,Γ−1Ι0.89710.5333Γ−1Ι2.31430.2000,Γ−3Ι0.00000.4000,Γ−3Ι0.00000.4000Γ2Ι1.04730.0667,Γ1Ι0.00850.4000,Γ−3Ι0.00000.5333Γ1Ι0.44780.2000,Γ0Ι1.38770.3667,Γ−1Ι1.54150.4333C5Γ0Ι1.95750.3000,Γ−3Ι2.80490.3333,Γ−3Ι1.43700.3667Γ1Ι1.24880.3667,Γ−1Ι0.72120.2667,Γ−3Ι0.00000.3667Γ0Ι1.04740.0667,Γ−1Ι2.87500.1000,Γ−2Ι1.07560.8333Γ−3Ι0.00000.1000,Γ−3Ι0.00000.4333,Γ−3Ι0.00000.4667Table 8
The positive ideal point.
D1D2D3D4Γ−1Ι2.44490.2667,Γ−2Ι1.52810.1333,Γ−3Ι0.00000.6000Γ−1Ι2.31430.2000,Γ−3Ι0.00000.4000,Γ−3Ι0.00000.4000Γ−3Ι0.00000.1333,Γ−3Ι0.00000.3000,Γ−3Ι0.00000.5667Γ−3Ι0.00000.1000,Γ−3Ι0.00000.4333,Γ−3Ι0.00000.4667Table 9
The Negative ideal point.
D1D2D3D4Γ1Ι1.55580.2000,Γ0Ι2.80490.2667,Γ−1Ι0.89710.5333Γ1Ι2.07150.0667,Γ1Ι1.80860.5333,Γ−1Ι2.96300.4000Γ0Ι2.35890.0667,Γ0Ι0.35610.0667,Γ−1Ι1.90470.8667Γ2Ι0.10820.2000,Γ1Ι0.51000.3333,Γ−1Ι0.95850.4667Table 10
dPDHLAτ,PDHLPIS.
C1C2C3C4C5dPDHLAτ,PDHLPIS0.16630.12110.16690.08160.0734Table 11
dPDHLAτ,PDHLNIS.
C1C2C3C4C5dPDHLAi,PDHLNIS0.02260.08830.10470.10560.1174Table 12
PDHLRCD.
C1C2C3C4C5PDHLRCD0.88040.57820.61450.43580.3846


## 4.2. Comparative Analysis

Finally, we compared it with the PDHL-VIKOR method [36], the PDHL-CODAS method [38], the PDHLWA operator, the PDHLWG operator, the PDHLPWA operator, and the PDHLPWG operator. The results and analysis are as follows (See Table 13).Table 13
The numerical results and rank were derived by the PDHL-CODAS.
PDHL-TOPSISRankPDHL-CODASRankC10.88045−0.50405C20.578230.07353C30.614540.16722C40.435820.07344C50.384610.28471The expected values of the PDHLWA operatorRankThe expected values of the PDHLWA operatorRankC10.440950.44335C20.444440.65993C30.648830.76112C40.733320.56094C50.884110.88231The expected values of the PDHLPWA operatorRankThe expected values of the PDHLPWG operatorRankC10.443150.44555C20.646630.66213C30.551040.76332C40.735520.56314C50.886310.88451It can be seen from Table13 that although the six methods are different, the optimal scheme obtained is the same. Only schemes 3 and 4 have slight differences between the PDHLWA operator and other methods. Therefore, the PDHL-TOPSIS method proposed by us can scientifically and effectively solve the investment decision problem.


## 5. Conclusion

This essay assesses the level of college English teaching from the perspective of MAGDM, based on the probabilistic double hierarchy linguistics. Because it has advantages that other language environments do not have, it has become another important tool to solve the problem of multiattribute group decision-making. Then, PDHL-TOPSIS technology is proposed to solve the evaluation of college English teaching level. Then, compared with PDHL-CODAS technology, which verifies that our proposed method can finish off a sequence of enriching evaluation problems in a more meaningful and effective way. In our future research, the proposed model and algorithm will be needful and meaningful to apply to working out other real decision-making problems [51–53] and the developed approaches can also be extended to other fuzzy settings [54–56]and uncertain settings [57–59].



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*Source: 6079906-2022-03-29.xml*