# Grey Relational Analysis Method for Probabilistic Double Hierarchy Linguistic Multiple Attribute Group Decision Making and Its Application to College Tennis Classroom Teaching Effect Evaluation

Authors: Lihua Wang; Huiming Li; Jianpeng Zhang; Jin Yang
Journal: Mathematical Problems in Engineering
 (2022)

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## Abstract

The college tennis classroom teaching effect evaluation is viewed as the multiattribute group decision making (MAGDM). The probabilistic double hierarchy linguistic term set (PDHLTS) not only conforms to people’s language expression habit of “adverb + adjective” but also can accurately depict its importance in real MAGDM. Therefore, this paper comes up with the probabilistic double hierarchy linguistic grey relational analysis (PDHL-GRA) method based on the grey relational analysis (GRA) process for MAGDM based on PDHLTS environment and applies it to the college tennis classroom teaching effect evaluation. Finally, a practical case for college tennis classroom teaching effect evaluation is presented to demonstrate the steps of our method, and a comparison analysis illustrates its feasibility and effectiveness.


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## Body

## 1. Introduction

To better fuse decision information, MAGDM technology came into being [1–5]. After MAGDM theory came into being, it has been widely used in finance, engineering, corporate decision making, and many other aspects [6–10]. In view of the intricateness and fuzzification of decision circumstances [11–15], in many MAGDM issues, expert opinions are often stated as fuzzy data [16–18]. For this reason, Zadeh [19] raised concept of a linguistic variable for approximate reasoning. In many environments, the linguistic variable cannot exactly formulate proficient’s perspective. Hence, hesitant fuzzy LTS (HFLTS) was proposed by Rodriguez, Martinez and Herrera [20]. An idea about probabilistic linguistic term sets (PLTSs) was proposed by Pang et al. [21]. Soon afterwards, critical malfunction matters were finished off by the PLAMM and PLWAMM formulas derived by Liu and Teng [22]. The performance estimation system of college teachers was finished off by the PLPA and PLPWG formulas derived by Kobina et al. [23]. Wei et al. [24] built the EDAS method for PL-MAGDM. The extensive similarity measure based on probabilistic language circumstances was derived by Wei et al. [25]. Su et al. [26] defined the PT-TODIM method for PL-MAGDM. Lin et al. [27] defined the probabilistic uncertain linguistic term sets (PULTs). Wang et al. [28] developed the GRP and CRITIC methods for PUL-MAGDM. Wei et al. [29] built the generalized Dice similarity measures for PUL-MAGDM. Zhao et al. [30] built the PUL-TODIM method based on prospect theory. He et al. [31] built the taxonomy-based MAGDM method with probabilistic uncertain linguistic assessment information. He et al. [32] built the bidirectional projection method for PUL-MAGDM. Nevertheless, a few sophisticated proficient estimation perspectives cannot be remarked in existing language terms such as “only a tiny bit poor” or “only a tiny bit good.” Hence, Gou et al. [33] made a conceptual layout about double hierarchy linguistic term set (DHLTS) and double hierarchy hesitant fuzzy linguistic term set (DHHFLTS). Many research results have emerged one after another [34–41]. Soon afterwards, Gou et al. [42] made a project about probabilistic double hierarchy linguistic term set (PDHLTS). Lei et al. [43] built the PDHL-CODAS model to rank online shopping platform. Lei et al. [44] defined a sequence of probabilistic double hierarchy linguistic polymerization formulas. Lei et al. [45] defined the PDHL-EDAS method for MAGDM.GRA was initially defined by Deng [46] to cope with real MAGDM. Compared with other real MAGDM methods [47–51], the GRA method could consider the shape similarity of every given alternative from PIS as well as NIS. Javanmardi et al. [52] explored grey system theory-based methods and applications in sustainability studies. Javanmardi and Liu [53] explored the human cognitive capacity in understanding systems: a grey system theory perspective. Zhang et al. [54] used the GRA method based on cumulative prospect theory for IF-MAGDM. Javanmardi et al. [55] explored the philosophical paradigm of grey system theory as a postmodern theory. With the purpose of discerning the carbon market, Zhu et al. [56] took advantage of the GRA process as well as EMD. Malek et al. [57] built a revised hybrid GRA for green supply. Kung and Wen [58] used the GRA process to solve grey MADM. Javanmardi and Liu [59] explored grey system theory-based methods and applications in analyzing socioeconomic systems. Javanmardi et al. [60] explored the philosophical foundations of grey system theory. Alptekin et al. [61] solved the low carbon development based on the GRA process. Zhang et al. [62] defined the SF-GRA method based on cumulative prospect theory for MAGDM.The main contributions of this paper are to utilize the GRA algorithm to build the MAGDM matters on the strength of PDHLTSs. The main research work of this paper is arranged as follows: (1) the GRA is constructed on account of PDHLTSs; (2) the PDHL-GRA method is applied to finish off the MAGDM issue under PDHLTSs; (3) a practical case for college tennis classroom teaching effect evaluation is presented to demonstrate the steps of our method; and (4) a comparison analysis illustrates its feasibility and effectiveness. The framework of this article is as follows. Section2 reviews some concepts of PDHLTSs. Section 3 designs a PDHL-GRA method for MAGDM with entropy weight. Section 4 provides a practical example to illustrate the method and a comparison analysis illustrates its effectiveness. Finally, Section 5 summarizes this study.


## 2. Preliminaries

First, let us learn some basics about PDHLTS.Definition 1.
(see [33]). Let us say DHL=ΓϑΙΩ|ϑ=−Α,⋯,−1,0,1,⋯Α;Ω=−Β,⋯,−1,0,1,⋯Β; is a DHLTS, and the definition of the DHLTS is(1)DHLTS=ΓϑΙΩΔ|ϑ=−A,⋯,−1,0,1,⋯A;Ω=−B,⋯,−1,0,1,⋯B;,where Δ=1,2,...,ΞDHL, the Δ−th double hierarchy linguistic element (DHLE) is narrated as ΓϑΙΩΔ, the quantity of all DHLEs is ΞDHL, and all DHLEs are sorted in ascending sequence.Definition 2 (see [42]).
Let us sayDHL=ΓϑΙΩ|ϑ=−Α,⋯,−1,0,1,⋯Α;Ω=−Β,⋯,−1,0,1,⋯Β; is a DHLTS, and 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 denoted as ΞPDHLƛ, and according to ϒΓϑΙΩΔƛΔ, PDHLE is sorted in ascending order; the function is determined by formula (3).Definition 3 (see [42]).
LetDHL=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,ϒ−1:0,1⟶−A,A×−B,B,ϒ−1ϖ=2Aϖ−AΙB2Aϖ−A−2Aϖ−Aor2Aϖ−A+1ΙB2Aϖ−A−2Aϖ−A−Bor.
Because the probability sum of all PDHLEs in PDHLTS may be less than 1, we had to standardize PDHLTS, and the specific measures are as follows:(4)PDHLƛ=ΓϑΙΩΔƛΔ|ΓϑΙΩΔ∈DHL,ƛΔ≥0,∑Δ=1ΞPDHLƛƛΔ≤1,where ƛ˜Δ=ƛΔ/∑Δ=1ΞPDHLƛƛΔ;ϑ∈−A,A;Ω∈−B,B;A,B are all integers.Definition 4.
(see [43]). Let DHL=ΓϑΙΩ|ϑ=−Α,⋯,−1,0,1,⋯Α;Ω=−Β,⋯,−1,0,1,⋯Β; be a DHLTS and PDHL˜1ƛ˜=Γ1ϑΙΩΔƛ˜1Δ|Γ1ϑΙΩΔ∈DHL;Δ=1,2,...,ΞPDHL˜1ƛ˜ and PDHL˜2ƛ˜=Γ2ϑΙΩΔƛ˜2Δ|Γ2ϑΙΩΔ∈DHL;Δ=1,2,...,ΞPDHL˜2ƛ˜ be two different PDHLTSs, where #PDHL˜1ƛ˜, #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 [42]). Let PDHL˜ƛ˜=ΓϑΙΩΔƛ˜Δ|ΓϑΙΩΔ∈DHL;Δ=1,2,...,ΞPDHL˜ƛ˜ be a PDHLTS, and the expected values χPDHL˜ƛ˜ and deviation degree γPDHL˜ƛ˜ of PDHL˜ƛ˜ are built as χPDHL˜1ƛ˜=χPDHL˜2ƛ˜:(5)χPDHL˜ƛ˜=∑Δ=1ΞPDHL˜ƛ˜ϒPDHL˜ƛ˜ƛ˜Δ∑Δ=1PDHL˜ƛ˜ƛ˜Δ,γPDHL˜ƛ˜=∑Δ=1ΞPDHL˜ƛ˜ϒPDHL˜ƛ˜ƛ˜Δ−χPDHL˜ƛ˜2∑Δ=1ΞPDHL˜ƛ˜ƛ˜Δ.Definition 6.
(see [43]). Let DHL=ΓϑΙΩ|ϑ=−Α,⋯,−1,0,1,⋯Α;Ω=−Β,⋯,−1,0,1,⋯Β; be a DHLTS, and PDHL˜1ƛ˜=Γ1ϑΙΩΔƛ˜1Δ|Γ1ϑΙΩΔ∈DHL;Δ=1,2,...,ΞPDHL˜1ƛ˜ and PDHL˜2ƛ˜=Γ2ϑΙΩΔƛ˜2Δ|Γ2ϑΙΩΔ∈DHL;Δ=1,2,...,ΞPDHL˜2ƛ˜ are two PDHLTSs, where ΞPDHL˜1ƛ˜=ΞPDHL˜2ƛ˜=ΞPDHL˜ƛ˜; then, Hamming distance HDPDHL˜1ƛ˜,PDHL˜2ƛ˜ is determined.(6)HDPDHL˜1ƛ˜,PDHL˜2ƛ˜=∑Δ=1ΞPDHL˜ƛ˜ϒΓ1ϑΙΩΔƛ˜1Δ−Γ2ϑΙΩΔƛ˜2ΔΞPDHL˜ƛ˜.


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

Now, GRA mean in the context of PDHLTSs is proposed to deal with MAGDM matters. Also, a complete MAGDM issue is narrated as follows. Whole alternatives is shown asC=C1,C2,⋯,Ca, 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, and 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-GRA mean is created to dispose of MAGDM issue with entropy weight.Step 1. Establish all decision makers’ decision matrixes PDHLTS Qq=PDHLqƛa×b.(7)PDHLTS=⊕q=1ΤwqPDHL˜qƛ˜,=ϒ−1∪1−∏q=1Τ1−ϒΓϑΙΩΔℜq∑q=1Tƛ˜qΔq.Step 2. Convert cost index into benefit index. Let PDHL˜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 Γ−ϑΙΩΔƛ˜Δ.Step 3. Compute the normalized decision matrix Q˜q=PDHL˜τσqƛ˜a×b.Step 4. The proportion of each attribute is calculated depending on the entropy formula.Entropy [63] is one of the important tools to ascertain the proportion of each attribute.The first thing to do is ascertaining the normalized decision matrixNLijp:(8)PDHL˜τσqƛ˜=∑Δ=1ΞPDHL˜ƛ˜ϒΓτσϑΙΩΔqƛ˜τσΔq∑τ=1a∑Δ=1ΞPDHL˜ƛ˜ϒΓτσϑΙΩΔqƛ˜τσΔq,σ=1,2,⋯,b.Secondly, the Shannon entropyE=E1,E2,⋯,Eb is obtained by the following formula:(9)Eσ=−1lna∑τ=1aPDHL˜τσqƛ˜lnPDHL˜τσqƛ˜,andPDHL˜τσqƛ˜lnPDHL˜τσqƛ˜ is defined as 0, if PDHL˜τσqƛ˜=0.Finally, the attribute weightsℑ=ℑ1,ℑ2,⋯,ℑb are computed:(10)ℑσ=1−Eσ∑σ=1b1−Eσ,σ=1,2,⋯,b.Step 5. Confirm the probabilistic double hierarchy linguistic positive ideal scheme more than zero (PDHLPIS) and probabilistic double hierarchy linguistic negative ideal scheme less than zero (PDHLNIS):(11)PDHLPIS=PDHLPIS1,PDHLPIS2,⋯,PDHLPISb,PDHLPIS=PDHLPIS1,PDHLPIS2,⋯,PDHLPISb,PDHLNIS=PDHLNIS1,PDHLNIS2,⋯,PDHLNISb,where(12)PDHLPISσ=ΓϑΙΩΔƛ˜Δ|Δ=1,2,...,ΞPDHL˜ƛ˜,=maxτϒΓϑΙΩΔƛ˜Δ,PDHLNISσ=ΓϑΙΩΔƛ˜Δ|Δ=1,2,...,ΞPDHL˜ƛ˜,=minτϒΓϑΙΩΔƛ˜Δ.Step 6. Compute the grey rational coefficients of every given attribute of every given alternative from the PDHLPIS and PDHLNIS.(13)PDHLPISξτσ=min1≤i≤mmin1≤j≤ndPDHLAτσ,PDHLPISσ+ρmax1≤i≤mmax1≤j≤ndPDHLAτσ,PDHLPISσdPDHLAτσ,PDHLPISσ+ρmax1≤i≤mmax1≤j≤ndPDHLAτσ,PDHLPISσ,PDHLPISξτσ=min1≤i≤mmin1≤j≤ndPDHLAτσ,PDHLNISσ+ρmax1≤i≤mmax1≤j≤ndPDHLAτσ,PDHLNISσdPDHLAτσ,PDHLNISσ+ρmax1≤i≤mmax1≤j≤ndPDHLAτσ,PDHLNISσ,τ=1,2,⋯,a,σ=1,2,⋯,b.Step 7. Figure out the degree of GRC of all given alternatives from PDHLPIS as well as PDHLNIS:(14)PDHLPISξτ=∑τ=1aℑτPDHLPISξτσ,τ=1,2,⋯,a,PDHLPISξτ=∑τ=1aℑτPDHLNISξτσ,τ=1,2,⋯,a.Step 8. Compute each alternative’s PDHL relative relational degree (PDHLRRD) of all given alternatives from PDHLPIS:(15)PDHLRRDτ=PDHLPISξτPDHLNISξτ+PDHLPISξτ,τ=1,2,⋯,a.Step 9. According to PDHLRRDττ=1,2,⋯,a. The highest value of PDHLRRDττ=1,2,⋯,a, the optimal choice is.


## 4. Numerical Example and Comparative Analysis

### 4.1. Numerical Example

Based on the research on the development of tennis teachers in colleges and universities and the evaluation requirements of the new round of basic education curriculum reform, it is of great significance to measure whether tennis teaching meets the expected goals. The core courses in the curriculum reform were implemented, and the fundamental way is to implement curriculum classroom. Curriculum reform embodies an important issue that every school and teacher is thinking about. Classroom evaluation reform to carry out scientific and effective evaluation of classroom teaching and establish an effective evaluation system mechanism should be the core of the curriculum reform. According to the current and future period of teaching reform and development, classroom evaluation should be “developmental classroom evaluation.” Classroom evaluation helps to overcome the limitations and deficiencies of current evaluation. Classroom evaluation reflects the latest trend of current teacher evaluation, evaluation of advanced ideas, and evaluation functions. Classroom evaluation conducts reflection and analysis on teachers, evaluates teachers' development potential, teachers' classroom status and the process of value judgment. However, the evaluation of teaching in the field of teaching is a worldwide problem, but also the key to promoting quality education process. There is a clear gap between the current evaluation theories, methods and systems, and quality education. Similar problems exist in teacher teaching evaluation. These serious constraints restricted the promotion of quality education. Therefore, the establishment of the quality of classroom education development of the concept of evaluation system is the full implementation of the objective of quality education, and at the same time, it also pushes the design and implementation of teaching activities to a new stage. However, these problems can be attributed to the MAGDM problem. This paper analyzes college tennis classroom teaching effect evaluation problems based on the proposed PDHL-GRA method. There are five given latent college tennis teachersC=C1,C2,C3,C4,C5, who may be the best. For the sake of assessing the college tennis classroom teaching effect 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 teaching attitude; ② D2 represents the teaching methods; ③ D3 is student feedback; and ④ D4 is teaching quality. Obviously, all attributes are benefit, and ℑ=ℑ1,ℑ2,ℑ3,ℑ4 is the weight of four attributes where ℑσ∈0,1, ℑ=1,2,3,4, ∑σ=14ℑσ=1. 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:(16)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 formΙ−2=onlyalittle,Ι−1=alittle,Ι0=just right,Ι1=much,Ι2=very much,Ι3=extirely much.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−2Ι−10.1,Γ−1Ι10.3,Γ0Ι10.6Γ−3Ι10.2,Γ0Ι−20.1,Γ1Ι10.7Γ−3Ι01.0Γ2Ι−10.5,Γ−1Ι−20.3,Γ−1Ι−10.2C2Γ−3Ι11Γ−3Ι00.3,Γ0Ι20.7Γ−3Ι−20.6,Γ−3Ι−10.2,Γ2Ι−10.2Γ1Ι20.3,Γ−2Ι00.4,Γ−1Ι10.3C3Γ0Ι−10.1,Γ1Ι−10.1,Γ2Ι20.8Γ3Ι00.2,Γ2Ι00.4,Γ−1Ι10.4Γ−3Ι00.8,Γ1Ι−20.1,Γ3Ι00.1Γ−3Ι10.1,Γ1Ι00.8,Γ2Ι10.1C4Γ2Ι−10.3,Γ1Ι20.3,Γ2Ι20.4Γ−2Ι10.4,Γ0Ι−30.4,Γ3Ι−20.2Γ−3Ι20.4,Γ−1Ι20.5,Γ1Ι30.1Γ−1Ι20.5,Γ−1Ι30.5,C5Γ−2Ι00.5,Γ−1Ι20.2,Γ0Ι30.3Γ−1Ι−10.4,Γ2Ι−10.4,Γ1Ι00.2Γ0Ι00.4,Γ2Ι10.6Γ2Ι11.0Table 2
The PDHLTS evaluation of all alternatives is provided by JK2.
D1D2D3D4C1Γ−3Ι20.7,Γ−1Ι10.3Γ1Ι−11.0Γ1Ι−30.2,Γ1Ι−20.6,Γ1Ι00.2Γ−1Ι−20.5,Γ−1Ι10.1,Γ2Ι20.4C2Γ2Ι−10.6,Γ2Ι00.1,Γ2Ι10.3Γ−3Ι10.7,Γ−3Ι20.1,Γ−3Ι30.2,Γ0Ι01.0Γ−2Ι−30.2,Γ1Ι10.5,Γ2Ι10.3C3Γ1Ι−10.1,Γ1Ι00.3,Γ2Ι10.6Γ−3Ι20.8,Γ2Ι10.2Γ−2Ι10.1,Γ−2Ι20.9Γ1Ι−10.5,Γ2Ι−10.5C4Γ1Ι00.6,Γ2Ι00.2,Γ1Ι30.2Γ−2Ι00.3,Γ−2Ι10.6,Γ−1Ι10.1Γ−2Ι10.8,Γ−2Ι30.2Γ−2Ι10.3,Γ−1Ι10.1,Γ−1Ι20.6C5Γ1Ι10.2,Γ1Ι20.5,Γ2Ι20.3Γ−2Ι−20.5,Γ−1Ι−10.4,Γ0Ι20.1Γ3Ι−21.0Γ1Ι−30.1,Γ1Ι−10.2,Γ2Ι10.7Table 3
The PDHLTS evaluation of all alternatives is provided by JK3.
D1D2D3D4C1Γ−1Ι10.4,Γ1Ι10.2,Γ0Ι30.4Γ−3Ι−30.3,Γ−3Ι−20.3,Γ1Ι−20.4Γ1Ι01.0Γ−2Ι−30.6,Γ1Ι10.4C2Γ0Ι31.0Γ−1Ι00.3,Γ1Ι10.7Γ2Ι−20.4,Γ2Ι−10.4,Γ2Ι00.2Γ0Ι−30.1,Γ2Ι−10.2,Γ2Ι00.7C3Γ1Ι−20.7,Γ2Ι−10.2,Γ2Ι00.1Γ−1Ι30.2,Γ1Ι10.5,Γ2Ι10.3Γ2Ι−10.4,Γ−1Ι00.6Γ0Ι−30.3,Γ1Ι00.1,Γ2Ι10.6C4Γ−1Ι−30.3,Γ−2Ι10.6,Γ1Ι10.1Γ−2Ι−20.4,Γ0Ι00.4,Γ1Ι00.5Γ−3Ι−20.4,Γ−1Ι−30.1,Γ1Ι−30.5Γ1Ι−10.3,Γ1Ι10.4,Γ2Ι20.3C5Γ−3Ι−10.5,Γ−2Ι00.4,Γ2Ι10.1Γ1Ι−30.1,Γ1Ι−10.3,Γ1Ι30.6Γ−3Ι−20.6,Γ−1Ι10.2,Γ1Ι30.2Γ−1Ι−20.5,Γ2Ι20.1,Γ3Ι00.4Now, the built PDHL-GRA method is used to select the optimal latent college tennis teacher.Step 1. Standardize the evaluation matrix of the three experts (Tables 4–6).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 the PDHLTSs (see Table 7).Step 3. Calculate the weight of the decision attribute.(17)ℑ1=0.1432ℑ2=0.3496ℑ3=0.3217ℑ4=0.1855.Step 4. The PDHLPIS and the PDHLNIS are determined according to the global decision matrix, which has been converted to the PDHLTSs (see Tables 8 and 9).Step 5. Figure out the GRC of every alternative from PDHLPIS as well as PDHLNIS (Tables 10 and 11).Step 6. Figure out the degree of GRC of all alternatives from PDHLPIS as well as PDHLNIS (Table 12).Step 7. Calculate the PDHLRRDτ of each given alternative from PDHLPIS (Table 13).Step 8. According to the PDHLRRDτ, all given alternatives are ranked, the higher the PDHLRRDτ, the better the alternative selected. Evidently, the order is C5>C4>C2>C3>C1 and C5 is the best one.Table 4
The standardized decision matrix is provided by JK1.
D1D2D3D4C1Γ−2Ι−10.1,Γ−1Ι10.3,Γ0Ι10.6Γ−3Ι10.2,Γ0Ι−20.1,Γ1Ι10.7Γ−3Ι00,Γ−3Ι00,Γ−3Ι01.0Γ2Ι−10.5,Γ−1Ι−20.3,Γ−1Ι−10.2C2Γ−3Ι10,Γ−3Ι10,Γ−3Ι11Γ−3Ι00,Γ−3Ι00.3,Γ0Ι20.7Γ−3Ι−20.6,Γ−3Ι−10.2,Γ2Ι−10.2Γ1Ι20.3,Γ−2Ι00.4,Γ−1Ι10.3C3Γ0Ι−10.1,Γ1Ι−10.1,Γ2Ι20.8Γ3Ι00.2,Γ2Ι00.4,Γ−1Ι10.4Γ−3Ι00.8,Γ1Ι−20.1,Γ3Ι00.1Γ−3Ι10.1,Γ1Ι00.8,Γ2Ι10.1C4Γ2Ι−10.3,Γ1Ι20.3,Γ2Ι20.4Γ−2Ι10.4,Γ0Ι−30.4,Γ3Ι−20.2Γ−3Ι20.4,Γ−1Ι20.5,Γ1Ι30.1Γ−1Ι20,Γ−1Ι20.5,Γ−1Ι30.5,C5Γ−2Ι00.5,Γ−1Ι20.2,Γ0Ι30.3Γ−1Ι−10.4,Γ2Ι−10.4,Γ1Ι00.2Γ0Ι00,Γ0Ι00.4,Γ2Ι10.6Γ2Ι10,Γ2Ι10,Γ2Ι11.0Table 5
The standardized decision matrix is provided by JK2.
D1D2D3D4C1Γ−3Ι20,Γ−3Ι20.7,Γ−1Ι10.3Γ1Ι−10,Γ1Ι−10,Γ1Ι−11.0Γ1Ι−30.2,Γ1Ι−20.6,Γ1Ι00.2Γ−1Ι−20.5,Γ−1Ι10.1,Γ2Ι20.4C2Γ2Ι−10.6,Γ2Ι00.1,Γ2Ι10.3Γ−3Ι10.7,Γ−3Ι20.1,Γ−3Ι30.2,Γ0Ι00,Γ0Ι00,Γ0Ι01.0Γ−2Ι−30.2,Γ1Ι10.5,Γ2Ι10.3C3Γ1Ι−10.1,Γ1Ι00.3,Γ2Ι10.6Γ−3Ι20,Γ−3Ι20.8,Γ2Ι10.2Γ−2Ι10,Γ−2Ι10.1,Γ−2Ι20.9Γ1Ι−10,Γ1Ι−10.5,Γ2Ι−10.5C4Γ1Ι00.6,Γ2Ι00.2,Γ1Ι30.2Γ−2Ι00.3,Γ−2Ι10.6,Γ−1Ι10.1Γ−2Ι10,Γ−2Ι10.8,Γ−2Ι30.2Γ−2Ι10.3,Γ−1Ι10.1,Γ−1Ι20.6C5Γ1Ι10.2,Γ1Ι20.5,Γ2Ι20.3Γ−2Ι−20.5,Γ−1Ι−10.4,Γ0Ι20.1Γ3Ι−20,Γ3Ι−20,Γ3Ι−21.0Γ1Ι−30.1,Γ1Ι−10.2,Γ2Ι10.7Table 6
The standardized decision matrix is provided by JK3.
D1D2D3D4C1Γ−1Ι10.4,Γ1Ι10.2,Γ0Ι30.4Γ−3Ι−30.3,Γ−3Ι−20.3,Γ1Ι−20.4Γ1Ι00,Γ1Ι00,Γ1Ι01.0Γ−2Ι−30,Γ−2Ι−30.6,Γ1Ι10.4C2Γ0Ι30,Γ0Ι30,Γ0Ι31.0Γ−1Ι00,Γ−1Ι00.3,Γ1Ι10.7Γ2Ι−20.4,Γ2Ι−10.4,Γ2Ι00.2Γ0Ι−30.1,Γ2Ι−10.2,Γ2Ι00.7C3Γ1Ι−20.7,Γ2<Ι−10.2,Γ2Ι00.1Γ−1Ι30.2,Γ1Ι10.5,Γ2Ι10.3Γ2Ι−10,Γ2Ι−10.4,Γ−1Ι00.6Γ0Ι−30.3,Γ1Ι00.1,Γ2Ι10.6C4Γ−1Ι−30.3,Γ−2Ι10.6,Γ1Ι10.1Γ−2Ι−20.4,Γ0Ι00.4,Γ1Ι00.5Γ−3Ι−20.4,Γ−1Ι−30.1,Γ1Ι−30.5Γ1Ι−10.3,Γ1Ι10.4,Γ2Ι20.3C5Γ−3Ι−10.5,Γ−2Ι00.4,Γ2Ι10.1Γ1Ι−30.1,Γ1Ι−10.3,Γ1Ι30.6Γ−3Ι−20.6,Γ−1Ι10.2,Γ1Ι30.2Γ−1Ι−20.5,Γ2Ι20.1,Γ3Ι00.4Table 7
The overall evaluation matrix.
D1D2D3D4C1Γ3Ι0.58990.1667,Γ−2Ι0.17560.4000,Γ−2Ι1.23700.4333Γ−3Ι0.76170.1667,Γ−3Ι0.91520.1333,Γ−1Ι2.15980.7000Γ−3Ι0.49040.0667,Γ−3Ι1.55230.2000,Γ−1Ι0.37190.7333Γ−3Ι2.97410.3333,Γ−3Ι1.38770.3333,Γ−2Ι1.56390.3333C2Γ−3Ι2.00780.2000,Γ−3Ι0.35780.0333,Γ−1Ι2.94460.7667Γ−3Ι0.58940.2333,Γ−3Ι0.66090.2333,Γ−2Ι2.21600.5333Γ−3Ι2.47700.3333,Γ−3Ι1.65020.2000,Γ−1Ι0.14290.4667Γ−3Ι1.83200.2000,Γ−2Ι0.92130.3667,Γ−2Ι2.88780.4333C3Γ−3Ι2.88910.3000,Γ−3Ι2.45600.2000,Γ−1Ι2.15450.5000Γ−3Ι2.40000.1333,Γ−1Ι0.80950.5667,Γ−2Ι1.21340.3000Γ−3Ι0.31710.2667,Γ−3Ι2.00400.2000,Γ−0Ι0.60000.5333Γ−3Ι0.86050.1333,Γ−2Ι2.45390.4667,Γ−1Ι0.19440.4000C4Γ−2Ι1.58650.4000,Γ−2Ι1.72930.3667,Γ−2Ι0.68220.2333Γ−3Ι1.06580.3667,Γ−3Ι2.94810.4667,Γ−3Ι2.21290.1667Γ−3Ι0.46370.2667,Γ−3Ι2.58200.4667,Γ−2Ι0.09940.2667Γ−3Ι1.57030.2000,Γ−2Ι0.14730.3333,Γ−2Ι2.99730.4667C5Γ−3Ι2.74940.4000,Γ−2Ι0.57660.3667,Γ−2Ι0.62390.2333Γ−3Ι1.71300.3333,Γ−2Ι1.08310.3667,Γ−2Ι0.82930.3000Γ−3Ι2.24060.2000,Γ−3Ι2.44320.2000,Γ0Ι0.46120.6000Γ−3Ι2.48880.2000,Γ−3Ι1.54920.1000,Γ1Ι0.60000.7000Table 8
The PDHLPIS.
D1D2Γ−3Ι2.88910.3000,Γ−3Ι2.45600.2000,Γ−1Ι2.15450.5000Γ−3Ι2.40000.1333,Γ−1Ι0.80950.5667,Γ−2Ι1.21340.3000D3D4Γ−3Ι2.24060.2000,Γ−3Ι2.44320.2000,Γ0Ι0.46120.6000Γ−3Ι2.48880.2000,Γ−3Ι1.54920.1000,Γ1Ι0.60000.7000Table 9
The PDHLNIS.
D1D2Γ3Ι0.58990.1667,Γ−2Ι0.17560.4000,Γ−2Ι1.23700.4333Γ−3Ι1.06580.3667,Γ−3Ι2.94810.4667,Γ−3Ι2.21290.1667D3D4Γ−3Ι0.46370.2667,Γ−3Ι2.58200.4667,Γ−2Ι0.09940.2667Γ−3Ι2.97410.3333,Γ−3Ι1.38770.3333,Γ−2Ι1.56390.3333Table 10
GRC of each alternative from PDHLPIS.
AlternativesD1D2D3D4C10.57251.00000.22330.5195C20.43430.54430.30750.6043C30.59490.79251.00001.0000C40.40810.60430.32430.5931C51.00000.54430.30750.6281Table 11
GRC of each alternative from PDHLNIS.
AlternativesD1D2D3D4C10.40721.00000.30670.5645C20.67951.11000.89000.9433C30.33120.47591.00000.5047C40.89000.62720.60431.0000C51.00000.56450.37390.5869Table 12
PDHLPISξτ and PDHLNISξτ of every alternative.
AlternativesIVIFPISξiIVIFNISξiC10.69530.5446C20.60891.0698C30.98240.5046C40.61560.8749C50.75750.6166Table 13
PDHLRRD of each alternative from PDHLPIS.
AlternativesC1C2C3C4C5PDHLRRDτ0.17480.40490.23860.42330.5373


### 4.2. Comparative Analysis

Finally, we compared it with the PDHL-VIKOR method [64], PDHL-CODAS method [43], PDHLWA operator, PDHLWG operator, PDHLPWA operator, and PDHLPWG operator. The results and analysis are as follows (see Table 14). It can be seen from Table 14 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-GRA method proposed by us can scientifically and effectively solve the investment decision problem.Table 14
The numerical results and rank derived by the PDHL-CODAS.
PDHL-TOPSISRankPDHL-CODASRankC10.88045−0.50405C20.578230.07353C30.614540.16722C40.435820.07344C50.384610.28471The expected values of PDHLWA operatorRankThe expected values of PDHLWA operatorRankC10.440950.44335C20.444440.65993C30.648830.76112C40.733320.56094C50.884110.88231The expected values of PDHLPWA operatorRankThe expected values of PDHLPWG operatorRankC10.443150.44555C20.646630.66213C30.551040.76332C40.735520.56314C50.886310.88451


## 4.1. Numerical Example

Based on the research on the development of tennis teachers in colleges and universities and the evaluation requirements of the new round of basic education curriculum reform, it is of great significance to measure whether tennis teaching meets the expected goals. The core courses in the curriculum reform were implemented, and the fundamental way is to implement curriculum classroom. Curriculum reform embodies an important issue that every school and teacher is thinking about. Classroom evaluation reform to carry out scientific and effective evaluation of classroom teaching and establish an effective evaluation system mechanism should be the core of the curriculum reform. According to the current and future period of teaching reform and development, classroom evaluation should be “developmental classroom evaluation.” Classroom evaluation helps to overcome the limitations and deficiencies of current evaluation. Classroom evaluation reflects the latest trend of current teacher evaluation, evaluation of advanced ideas, and evaluation functions. Classroom evaluation conducts reflection and analysis on teachers, evaluates teachers' development potential, teachers' classroom status and the process of value judgment. However, the evaluation of teaching in the field of teaching is a worldwide problem, but also the key to promoting quality education process. There is a clear gap between the current evaluation theories, methods and systems, and quality education. Similar problems exist in teacher teaching evaluation. These serious constraints restricted the promotion of quality education. Therefore, the establishment of the quality of classroom education development of the concept of evaluation system is the full implementation of the objective of quality education, and at the same time, it also pushes the design and implementation of teaching activities to a new stage. However, these problems can be attributed to the MAGDM problem. This paper analyzes college tennis classroom teaching effect evaluation problems based on the proposed PDHL-GRA method. There are five given latent college tennis teachersC=C1,C2,C3,C4,C5, who may be the best. For the sake of assessing the college tennis classroom teaching effect 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 teaching attitude; ② D2 represents the teaching methods; ③ D3 is student feedback; and ④ D4 is teaching quality. Obviously, all attributes are benefit, and ℑ=ℑ1,ℑ2,ℑ3,ℑ4 is the weight of four attributes where ℑσ∈0,1, ℑ=1,2,3,4, ∑σ=14ℑσ=1. 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:(16)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 formΙ−2=onlyalittle,Ι−1=alittle,Ι0=just right,Ι1=much,Ι2=very much,Ι3=extirely much.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−2Ι−10.1,Γ−1Ι10.3,Γ0Ι10.6Γ−3Ι10.2,Γ0Ι−20.1,Γ1Ι10.7Γ−3Ι01.0Γ2Ι−10.5,Γ−1Ι−20.3,Γ−1Ι−10.2C2Γ−3Ι11Γ−3Ι00.3,Γ0Ι20.7Γ−3Ι−20.6,Γ−3Ι−10.2,Γ2Ι−10.2Γ1Ι20.3,Γ−2Ι00.4,Γ−1Ι10.3C3Γ0Ι−10.1,Γ1Ι−10.1,Γ2Ι20.8Γ3Ι00.2,Γ2Ι00.4,Γ−1Ι10.4Γ−3Ι00.8,Γ1Ι−20.1,Γ3Ι00.1Γ−3Ι10.1,Γ1Ι00.8,Γ2Ι10.1C4Γ2Ι−10.3,Γ1Ι20.3,Γ2Ι20.4Γ−2Ι10.4,Γ0Ι−30.4,Γ3Ι−20.2Γ−3Ι20.4,Γ−1Ι20.5,Γ1Ι30.1Γ−1Ι20.5,Γ−1Ι30.5,C5Γ−2Ι00.5,Γ−1Ι20.2,Γ0Ι30.3Γ−1Ι−10.4,Γ2Ι−10.4,Γ1Ι00.2Γ0Ι00.4,Γ2Ι10.6Γ2Ι11.0Table 2
The PDHLTS evaluation of all alternatives is provided by JK2.
D1D2D3D4C1Γ−3Ι20.7,Γ−1Ι10.3Γ1Ι−11.0Γ1Ι−30.2,Γ1Ι−20.6,Γ1Ι00.2Γ−1Ι−20.5,Γ−1Ι10.1,Γ2Ι20.4C2Γ2Ι−10.6,Γ2Ι00.1,Γ2Ι10.3Γ−3Ι10.7,Γ−3Ι20.1,Γ−3Ι30.2,Γ0Ι01.0Γ−2Ι−30.2,Γ1Ι10.5,Γ2Ι10.3C3Γ1Ι−10.1,Γ1Ι00.3,Γ2Ι10.6Γ−3Ι20.8,Γ2Ι10.2Γ−2Ι10.1,Γ−2Ι20.9Γ1Ι−10.5,Γ2Ι−10.5C4Γ1Ι00.6,Γ2Ι00.2,Γ1Ι30.2Γ−2Ι00.3,Γ−2Ι10.6,Γ−1Ι10.1Γ−2Ι10.8,Γ−2Ι30.2Γ−2Ι10.3,Γ−1Ι10.1,Γ−1Ι20.6C5Γ1Ι10.2,Γ1Ι20.5,Γ2Ι20.3Γ−2Ι−20.5,Γ−1Ι−10.4,Γ0Ι20.1Γ3Ι−21.0Γ1Ι−30.1,Γ1Ι−10.2,Γ2Ι10.7Table 3
The PDHLTS evaluation of all alternatives is provided by JK3.
D1D2D3D4C1Γ−1Ι10.4,Γ1Ι10.2,Γ0Ι30.4Γ−3Ι−30.3,Γ−3Ι−20.3,Γ1Ι−20.4Γ1Ι01.0Γ−2Ι−30.6,Γ1Ι10.4C2Γ0Ι31.0Γ−1Ι00.3,Γ1Ι10.7Γ2Ι−20.4,Γ2Ι−10.4,Γ2Ι00.2Γ0Ι−30.1,Γ2Ι−10.2,Γ2Ι00.7C3Γ1Ι−20.7,Γ2Ι−10.2,Γ2Ι00.1Γ−1Ι30.2,Γ1Ι10.5,Γ2Ι10.3Γ2Ι−10.4,Γ−1Ι00.6Γ0Ι−30.3,Γ1Ι00.1,Γ2Ι10.6C4Γ−1Ι−30.3,Γ−2Ι10.6,Γ1Ι10.1Γ−2Ι−20.4,Γ0Ι00.4,Γ1Ι00.5Γ−3Ι−20.4,Γ−1Ι−30.1,Γ1Ι−30.5Γ1Ι−10.3,Γ1Ι10.4,Γ2Ι20.3C5Γ−3Ι−10.5,Γ−2Ι00.4,Γ2Ι10.1Γ1Ι−30.1,Γ1Ι−10.3,Γ1Ι30.6Γ−3Ι−20.6,Γ−1Ι10.2,Γ1Ι30.2Γ−1Ι−20.5,Γ2Ι20.1,Γ3Ι00.4Now, the built PDHL-GRA method is used to select the optimal latent college tennis teacher.Step 1. Standardize the evaluation matrix of the three experts (Tables 4–6).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 the PDHLTSs (see Table 7).Step 3. Calculate the weight of the decision attribute.(17)ℑ1=0.1432ℑ2=0.3496ℑ3=0.3217ℑ4=0.1855.Step 4. The PDHLPIS and the PDHLNIS are determined according to the global decision matrix, which has been converted to the PDHLTSs (see Tables 8 and 9).Step 5. Figure out the GRC of every alternative from PDHLPIS as well as PDHLNIS (Tables 10 and 11).Step 6. Figure out the degree of GRC of all alternatives from PDHLPIS as well as PDHLNIS (Table 12).Step 7. Calculate the PDHLRRDτ of each given alternative from PDHLPIS (Table 13).Step 8. According to the PDHLRRDτ, all given alternatives are ranked, the higher the PDHLRRDτ, the better the alternative selected. Evidently, the order is C5>C4>C2>C3>C1 and C5 is the best one.Table 4
The standardized decision matrix is provided by JK1.
D1D2D3D4C1Γ−2Ι−10.1,Γ−1Ι10.3,Γ0Ι10.6Γ−3Ι10.2,Γ0Ι−20.1,Γ1Ι10.7Γ−3Ι00,Γ−3Ι00,Γ−3Ι01.0Γ2Ι−10.5,Γ−1Ι−20.3,Γ−1Ι−10.2C2Γ−3Ι10,Γ−3Ι10,Γ−3Ι11Γ−3Ι00,Γ−3Ι00.3,Γ0Ι20.7Γ−3Ι−20.6,Γ−3Ι−10.2,Γ2Ι−10.2Γ1Ι20.3,Γ−2Ι00.4,Γ−1Ι10.3C3Γ0Ι−10.1,Γ1Ι−10.1,Γ2Ι20.8Γ3Ι00.2,Γ2Ι00.4,Γ−1Ι10.4Γ−3Ι00.8,Γ1Ι−20.1,Γ3Ι00.1Γ−3Ι10.1,Γ1Ι00.8,Γ2Ι10.1C4Γ2Ι−10.3,Γ1Ι20.3,Γ2Ι20.4Γ−2Ι10.4,Γ0Ι−30.4,Γ3Ι−20.2Γ−3Ι20.4,Γ−1Ι20.5,Γ1Ι30.1Γ−1Ι20,Γ−1Ι20.5,Γ−1Ι30.5,C5Γ−2Ι00.5,Γ−1Ι20.2,Γ0Ι30.3Γ−1Ι−10.4,Γ2Ι−10.4,Γ1Ι00.2Γ0Ι00,Γ0Ι00.4,Γ2Ι10.6Γ2Ι10,Γ2Ι10,Γ2Ι11.0Table 5
The standardized decision matrix is provided by JK2.
D1D2D3D4C1Γ−3Ι20,Γ−3Ι20.7,Γ−1Ι10.3Γ1Ι−10,Γ1Ι−10,Γ1Ι−11.0Γ1Ι−30.2,Γ1Ι−20.6,Γ1Ι00.2Γ−1Ι−20.5,Γ−1Ι10.1,Γ2Ι20.4C2Γ2Ι−10.6,Γ2Ι00.1,Γ2Ι10.3Γ−3Ι10.7,Γ−3Ι20.1,Γ−3Ι30.2,Γ0Ι00,Γ0Ι00,Γ0Ι01.0Γ−2Ι−30.2,Γ1Ι10.5,Γ2Ι10.3C3Γ1Ι−10.1,Γ1Ι00.3,Γ2Ι10.6Γ−3Ι20,Γ−3Ι20.8,Γ2Ι10.2Γ−2Ι10,Γ−2Ι10.1,Γ−2Ι20.9Γ1Ι−10,Γ1Ι−10.5,Γ2Ι−10.5C4Γ1Ι00.6,Γ2Ι00.2,Γ1Ι30.2Γ−2Ι00.3,Γ−2Ι10.6,Γ−1Ι10.1Γ−2Ι10,Γ−2Ι10.8,Γ−2Ι30.2Γ−2Ι10.3,Γ−1Ι10.1,Γ−1Ι20.6C5Γ1Ι10.2,Γ1Ι20.5,Γ2Ι20.3Γ−2Ι−20.5,Γ−1Ι−10.4,Γ0Ι20.1Γ3Ι−20,Γ3Ι−20,Γ3Ι−21.0Γ1Ι−30.1,Γ1Ι−10.2,Γ2Ι10.7Table 6
The standardized decision matrix is provided by JK3.
D1D2D3D4C1Γ−1Ι10.4,Γ1Ι10.2,Γ0Ι30.4Γ−3Ι−30.3,Γ−3Ι−20.3,Γ1Ι−20.4Γ1Ι00,Γ1Ι00,Γ1Ι01.0Γ−2Ι−30,Γ−2Ι−30.6,Γ1Ι10.4C2Γ0Ι30,Γ0Ι30,Γ0Ι31.0Γ−1Ι00,Γ−1Ι00.3,Γ1Ι10.7Γ2Ι−20.4,Γ2Ι−10.4,Γ2Ι00.2Γ0Ι−30.1,Γ2Ι−10.2,Γ2Ι00.7C3Γ1Ι−20.7,Γ2<Ι−10.2,Γ2Ι00.1Γ−1Ι30.2,Γ1Ι10.5,Γ2Ι10.3Γ2Ι−10,Γ2Ι−10.4,Γ−1Ι00.6Γ0Ι−30.3,Γ1Ι00.1,Γ2Ι10.6C4Γ−1Ι−30.3,Γ−2Ι10.6,Γ1Ι10.1Γ−2Ι−20.4,Γ0Ι00.4,Γ1Ι00.5Γ−3Ι−20.4,Γ−1Ι−30.1,Γ1Ι−30.5Γ1Ι−10.3,Γ1Ι10.4,Γ2Ι20.3C5Γ−3Ι−10.5,Γ−2Ι00.4,Γ2Ι10.1Γ1Ι−30.1,Γ1Ι−10.3,Γ1Ι30.6Γ−3Ι−20.6,Γ−1Ι10.2,Γ1Ι30.2Γ−1Ι−20.5,Γ2Ι20.1,Γ3Ι00.4Table 7
The overall evaluation matrix.
D1D2D3D4C1Γ3Ι0.58990.1667,Γ−2Ι0.17560.4000,Γ−2Ι1.23700.4333Γ−3Ι0.76170.1667,Γ−3Ι0.91520.1333,Γ−1Ι2.15980.7000Γ−3Ι0.49040.0667,Γ−3Ι1.55230.2000,Γ−1Ι0.37190.7333Γ−3Ι2.97410.3333,Γ−3Ι1.38770.3333,Γ−2Ι1.56390.3333C2Γ−3Ι2.00780.2000,Γ−3Ι0.35780.0333,Γ−1Ι2.94460.7667Γ−3Ι0.58940.2333,Γ−3Ι0.66090.2333,Γ−2Ι2.21600.5333Γ−3Ι2.47700.3333,Γ−3Ι1.65020.2000,Γ−1Ι0.14290.4667Γ−3Ι1.83200.2000,Γ−2Ι0.92130.3667,Γ−2Ι2.88780.4333C3Γ−3Ι2.88910.3000,Γ−3Ι2.45600.2000,Γ−1Ι2.15450.5000Γ−3Ι2.40000.1333,Γ−1Ι0.80950.5667,Γ−2Ι1.21340.3000Γ−3Ι0.31710.2667,Γ−3Ι2.00400.2000,Γ−0Ι0.60000.5333Γ−3Ι0.86050.1333,Γ−2Ι2.45390.4667,Γ−1Ι0.19440.4000C4Γ−2Ι1.58650.4000,Γ−2Ι1.72930.3667,Γ−2Ι0.68220.2333Γ−3Ι1.06580.3667,Γ−3Ι2.94810.4667,Γ−3Ι2.21290.1667Γ−3Ι0.46370.2667,Γ−3Ι2.58200.4667,Γ−2Ι0.09940.2667Γ−3Ι1.57030.2000,Γ−2Ι0.14730.3333,Γ−2Ι2.99730.4667C5Γ−3Ι2.74940.4000,Γ−2Ι0.57660.3667,Γ−2Ι0.62390.2333Γ−3Ι1.71300.3333,Γ−2Ι1.08310.3667,Γ−2Ι0.82930.3000Γ−3Ι2.24060.2000,Γ−3Ι2.44320.2000,Γ0Ι0.46120.6000Γ−3Ι2.48880.2000,Γ−3Ι1.54920.1000,Γ1Ι0.60000.7000Table 8
The PDHLPIS.
D1D2Γ−3Ι2.88910.3000,Γ−3Ι2.45600.2000,Γ−1Ι2.15450.5000Γ−3Ι2.40000.1333,Γ−1Ι0.80950.5667,Γ−2Ι1.21340.3000D3D4Γ−3Ι2.24060.2000,Γ−3Ι2.44320.2000,Γ0Ι0.46120.6000Γ−3Ι2.48880.2000,Γ−3Ι1.54920.1000,Γ1Ι0.60000.7000Table 9
The PDHLNIS.
D1D2Γ3Ι0.58990.1667,Γ−2Ι0.17560.4000,Γ−2Ι1.23700.4333Γ−3Ι1.06580.3667,Γ−3Ι2.94810.4667,Γ−3Ι2.21290.1667D3D4Γ−3Ι0.46370.2667,Γ−3Ι2.58200.4667,Γ−2Ι0.09940.2667Γ−3Ι2.97410.3333,Γ−3Ι1.38770.3333,Γ−2Ι1.56390.3333Table 10
GRC of each alternative from PDHLPIS.
AlternativesD1D2D3D4C10.57251.00000.22330.5195C20.43430.54430.30750.6043C30.59490.79251.00001.0000C40.40810.60430.32430.5931C51.00000.54430.30750.6281Table 11
GRC of each alternative from PDHLNIS.
AlternativesD1D2D3D4C10.40721.00000.30670.5645C20.67951.11000.89000.9433C30.33120.47591.00000.5047C40.89000.62720.60431.0000C51.00000.56450.37390.5869Table 12
PDHLPISξτ and PDHLNISξτ of every alternative.
AlternativesIVIFPISξiIVIFNISξiC10.69530.5446C20.60891.0698C30.98240.5046C40.61560.8749C50.75750.6166Table 13
PDHLRRD of each alternative from PDHLPIS.
AlternativesC1C2C3C4C5PDHLRRDτ0.17480.40490.23860.42330.5373


## 4.2. Comparative Analysis

Finally, we compared it with the PDHL-VIKOR method [64], PDHL-CODAS method [43], PDHLWA operator, PDHLWG operator, PDHLPWA operator, and PDHLPWG operator. The results and analysis are as follows (see Table 14). It can be seen from Table 14 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-GRA method proposed by us can scientifically and effectively solve the investment decision problem.Table 14
The numerical results and rank derived by the PDHL-CODAS.
PDHL-TOPSISRankPDHL-CODASRankC10.88045−0.50405C20.578230.07353C30.614540.16722C40.435820.07344C50.384610.28471The expected values of PDHLWA operatorRankThe expected values of PDHLWA operatorRankC10.440950.44335C20.444440.65993C30.648830.76112C40.733320.56094C50.884110.88231The expected values of PDHLPWA operatorRankThe expected values of PDHLPWG operatorRankC10.443150.44555C20.646630.66213C30.551040.76332C40.735520.56314C50.886310.88451


## 5. Conclusion

Life changes and people’s ideas and educational expectation have brought great challenges to contemporary school education, especially to college tennis education. With the gradual development of social needs, schools seem hard to meet the more and more advanced and complex education needs of the society. In order to promote whole-person education to students, family-school cooperation has become one of the effective ways to collect common effort and establish collaboration for education. Family and school cooperation not only provides an opportunity for in-depth development by prioritizing education environment and exploring potentiality of education resources but also is a booster for the development of students’ physical and mental health. However, while there are achievements in family-school cooperative management, there are still difficulties and problems. Also, the theoretical basis and teaching practices need further exploration. Affordance theory proposed by Gibson [65] claims that there is an interaction between humans (individuals) and the environment (the nature). There is potentiality of potential act in the affordance environment. Its existence is closely related to actors’ capability and understanding of the environment. That is to say, affordance is characterized not only by the environment but also by the individuals and emerges only when the two factors interact. Generally, we may put our focus on the affordance of language, the affordance of social culture, and the affordance of situations. Although focal difference exists between these types of affordances, there are similarities. Classroom management can be considered as an environment created together by the child, the teacher, and the parents, as compared with the traditional classroom management, which put emphasis on the interactive rule of the teacher and the student and the environment managed by the teacher. However, parents’ participation in college tennis class management provides a possible route for affordable learning environment. This paper defines an useful method for this kind of issue, since it builds the PDHL-GRA method for college tennis classroom teaching effect evaluation. And then a numerical example is used to evaluate the College tennis classroom teaching effect. Furthermore, to check on the feasibility as well as availability of the new proposed method, useful comparative analysis is also designed. In the near future, we shall pay attention to the consensus reaching process [66–71], influence of DMs’ psychological factors [72–77], and how to deal with the situations when criteria weights are incompletely known [78–83].



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*Source: 7419496-2022-06-10.xml*