Multi-attribute decision-making methods based on Aczel-Alsina power aggregation operators for English blended teaching quality evaluation under single-valued neutrosophic sets

Abstract

As a new form of teaching, blended teaching can integrate the advantages of classroom teaching and online teaching and improve the overall quality and effectiveness of English teaching under the premise of changing the orientation of teachers and students. The English blended teaching quality evaluation is a multiple attribute decision making (MADM) issue. Aczel-Alsina t-norm and t-conorm operations have the true advantage of changeability by adjusting a decision parameter. power averaging (PA) operator allowed decision attributes to strengthen each other through weight information during fusion, thereby eliminating the influence of subjective weights on the decision results. Motivate through classical Aczel-Alsina operators, PA operator and the single-valued neutrosophic sets (SVNSs), in this paper, the single-valued neutrosophic number Aczel-Alsina power averaging (SVNNAAPA) operator and the single-valued neutrosophic number Aczel-Alsina power weighted averaging (SVNNAAPWA) operator are constructed under SVNSs. The SVNNAAPWA operator is built for MADM. Eventually, an example about English blended teaching quality evaluation and some selected comparative decision analysis is used to depict the SVNNAAPWA method.

Keywords

Multiple attribute decision making (MADM)single-valued neutrosophic sets (SVNSs)power average (PA)English blended teaching quality evaluation

1. Introduction

Blended teaching content is an important factor affecting the quality of English teaching and English teaching objectives [1, 2]. However, in the selection and choice of teaching contents, English teachers generally focus on the contents prescribed by textbooks, often presented to students in the form of courseware and syllabus, which leads to a high degree of abstraction and theoreticality in online teaching sessions, and there are many repetitive and homogeneous problems with classroom teaching sessions, which seriously affects the improvement of English teaching quality [3, 4, 5]. In the classroom teaching, English teachers gradually integrate the objectives of intercultural communication and curriculum thinking into the process of classroom teaching, but the lack of corresponding teaching contents makes it difficult to realize the English teaching objectives of universities effectively [6]. Through theoretical analysis and practical investigation, it can be seen that English teaching in China is still “misaligned” and “disconnected” from social reality, that is, English classroom teaching, online teaching and social development are not connected, and students have difficulty in mastering “As a result, students cannot apply English knowledge well to meet their basic needs in the process of social development and career development [6, 7, 8, 9]. With further deepening of China’s reform and opening up, the more and more exchanges between China and other countries in the world have become more frequent [10, 11]. In such a background, the demand for applied English professionals in China has been rising. As the main training ground for English majors, colleges and universities should actively adapt to the national development needs, rearrange the structure of English education, provide more opportunities for students to practice English language, and improve their English language application ability [10, 11, 12, 13, 14]. The blended teaching mode is a new teaching mode developed by information technology, which effectively integrates the traditional teaching mode with the online learning mode and applies it to the English teaching in colleges and universities, realizing the innovation of English teaching mode and improving the effectiveness of English teaching [15, 16, 17].

The Multicriteria Decision Analysis (MCDA) is an important research domain in modern management science [18, 19, 20, 21, 22]. In many realistic MCDA issues, due to the limitations of the decision maker’s ability to recognize the problem, it is often impossible to supply the accurate information and could only be portrayed through the uncertainty, which constitutes an uncertain MCDA issue [23, 24, 25, 26, 27, 28]. In order to portray uncertain information, Zadeh [29] constructed fuzzy sets (FSs). Atanassov [30] constructed the intuitionistic fuzzy sets (IFSs). To portray inconsistency information, Smarandache [31] constructed the neutrosophic sets (NSs). Then, the NSs has been widely extended and applications in different domains [32, 33, 34, 35, 36]. Pramanik [37] presented an overview of neutrosophic sets. Biswas, Pramanik and Giri [38] constructed entropy based grey relational analysis (GRA) method for MADM about SVNSs. Aczél and Alsina [39] constructed some new operations named as Aczel-Alsina t-norm and t-conorm operations, which have the true advantage of changeability by adjusting a decision parameter. Ashraf et al. [40] constructed the Aczel-Alsina operations to SVNNs and produced the Aczel-Alsina fused operators of SVNNs for MADM method which could reflect the decision flexibility during the practical MADM process. The PA operator [41] allowed decision attributes to strengthen each other through weight information during fusion, thereby eliminating the influence of subjective weights on the decision results. Motivate through classical Aczel-Alsina operators [39], PA operator [41] and the SVNSs, we combine SVNSs with Aczel-Alsina operators and PA operator and construct the SVNN Aczel-Alsina PA (SVNNAAPA) operator and the SVNN Aczel-Alsina power weighted averaging (SVNNAAPWA) operator. The main aim of this paper is propose some power average (PA) operators based on the Aczel-Alsina operations and classical PA operator [41] under SVNNs. Thus, the SVNN Aczel-Alsina PA (SVNNAAPA) operator and the SVNN Aczel-Alsina power weighted averaging (SVNNAAPWA) operator are produced based on the Aczel-Alsina operations and classical PA operator under SNNs. The SVNNAAPWA operator is built for MADM. Finally, an example about English blended teaching quality evaluation and some comparative analysis is constructed. In order to do so, the reminder of this paper proceeds. The SVNSs is reviewed in Section 2. The SVNNAAPA and SVNNAAPWA operators are built in Section 3. The MADM based on the SVNNAAPWA operator is built in in Section 4. An example application for English blended teaching quality evaluation is given in Section 5. The conclusion is listed in Section 6.

2. Preliminaries

Wang et al. [42] constructed the SVNSs.

Definition 1 [42]. The SVNSs $W A$ in $X$ is constructed:

$\displaystyle WA=\{{({x,WT_{A}(x),WI_{A}(x),WF_{A}(x)})|{x\in X}}\}$ (1)

with the truth-membership $WT_{A}(x)$ , indeterminacy-membership $WI_{A}(x)$ and falsity-membership $FF_{A}(x)$ , $WT_{A}(x)$ , $WI_{A}(x)$ , $WF_{A}(x)\in[{0,1}]$ , $0\leqslant WT_{A}(x)+WI_{A}(x)+WF_{A}(x)\leqslant 3$ .

The SVNN is constructed as $WA=({WT_{A},WI_{A},WF_{A}})$ , where $WT_{A}\in[{0,1}]$ , $WI_{A}\in[{0,1}]$ , $WF_{A}\in[{0,1}]$ , and $0\leqslant WT_{A}+WI_{A}+WF_{A}\leqslant 3$ .

Definition 2 [43]. Let $WA=({WT_{A},WI_{A},WF_{A}})$ , the score value is constructed:

$\displaystyle SV({WA})=\frac{({2+WT_{A}-WI_{A}-WF_{A}})}{3},SV({WA})\in[{0,1}].$ (2)

Definition 3 [43]. Let $WA=({WT_{A},WI_{A},WF_{A}})$ , the accuracy value is:

$\displaystyle HV({WA})=WT_{A}-WF_{A},HV({WA})\in[{-1,1}].$ (3)

Peng et al. [43] constructed the order relation.

Definition 4 [43]. Let $WA=({WT_{A},WI_{A},WF_{A}})$ , $WB=({WT_{B},WI_{B},WF_{B}})$ , let

$\displaystyle SV({WA})=\frac{({2+WT_{A}-WI_{A}-WF_{A}})}{3}$

and

$\displaystyle SV({WB})=\frac{({2+WT_{B}-WI_{B}-WF_{B}})}{3},$

and let $HV({WA})=WT_{A}-WF_{A}$ and $HV({WB})=WT_{B}-WF_{B}$ , if $SV({WA})<SV({WB})$ , $WA<WB$ ; if $SV({WA})=SV({WB})$ , (1) if $HV({WA})=HV({WB})$ , $WA=WB$ ; (2) if $HV({WA})<HV({WB})$ , $WA<WB$ .

Definition 5 [40]. Let $WA=({WT_{A},WI_{A},WF_{A}})$ and $WB=({WT_{B},WI_{B},WF_{B}})$ , $\theta\geqslant 1$ , $\lambda>0$ , the constructed Aczel-Alsina operations with SVNNs are:

$\displaystyle(1)WA\oplus WB=\left({\begin{array}[]{l}1-e^{-({({-\ln({1-WT_{A}}% )})^{\theta}+({-\ln({1-WT_{B}})})^{\theta}})^{1/\theta}},\\ e^{-({({-\ln({WI_{A}})})^{\theta}+({-\ln({WI_{B}})})^{\theta}})^{1/\theta}},e^% {-({({-\ln({WF_{A}})})^{\theta}+({-\ln({WF_{B}})})^{\theta}})^{1/\theta}}\\ \end{array}}\right);(2)WA\otimes WB=\left({\begin{array}[]{l}e^{-({({-\ln({WT_% {A}})})^{\theta}+({-\ln({WT_{B}})})^{\theta}})^{1/\theta}},1-e^{-({({-\ln({1-% WI_{A}})})^{\theta}+({-\ln({1-WI_{B}})})^{\theta}})^{1/\theta}},\\ 1-e^{-({({-\ln({1-WF_{A}})})^{\theta}+({-\ln({1-WF_{B}})})^{\theta}})^{1/% \theta}}\\ \end{array}}\right)(3)\lambda WA=\left({1-e^{-({\lambda({-\ln({1-WT_{A}})})^{% \theta}})^{1/\theta}},e^{-({\lambda({-\ln({WI_{A}})})^{\theta}})^{1/\theta}},e% ^{-({\lambda({-\ln({WF_{A}})})^{\theta}})^{1/\theta}}}\right)(4)({WA})^{% \lambda}=\left({e^{-({\lambda({-\ln({WT_{A}})})^{\theta}})^{1/\theta}},1-e^{-(% {\lambda({-\ln({1-WI_{A}})})^{\theta}})^{1/\theta}},1-e^{-({\lambda({-\ln({1-% WF_{A}})})^{\theta}})^{1/\theta}}}\right)$

3. SNNAAPA and SNNAAPWA operator

Yong et al. [44] and Ashraf et al. [40] defined the SVNNAAWA operator.

Definition 6. Let $WA_{i}=(WT_{i},WI_{i},WF_{i})(i=1,2,\ldots,n)$ be the SVNNs with the weight $ww_{i}=(ww_{1},ww_{2},\ldots,ww_{n})^{T}$ , $\sum\nolimits_{i=1}^{n}{ww_{i}}=1$ , $\theta\geqslant 1$ . If

$\displaystyle\text{SVNNAAWA}_{ww}({WA_{1},WA_{2},\ldots,WA_{n}})=\mathop{% \oplus}\limits_{i=1}^{n}ww_{i}WA_{i}$ $\displaystyle=\left({\begin{array}[]{l}1-e^{-\left({\sum\limits_{i=1}^{n}{ww_{% i}}({-\ln({1-WT_{j}})})^{\theta}}\right)^{1/\theta}},\\ e^{-\left({\sum\limits_{i=1}^{n}{ww_{i}}({-\ln({WI_{j}})})^{\theta}}\right)^{1% /\theta}},e^{-\left({\sum\limits_{i=1}^{n}{ww_{i}}({-\ln({WF_{j}})})^{\theta}}% \right)^{1/\theta}}\\ \end{array}}\right)$ (4)

The SVNNAAWA has three decision properties.

Property 1 (idempotency). If $WA_{i}=(WT_{i},WI_{i},WF_{i})$

$\displaystyle\text{SVNNAAWA}_{ww}({WA_{1},WA_{2},\ldots,WA_{n}})=WA$ (5)

Property 2 (Monotonicity). Let $WA_{i}=(WT_{A_{i}},WI_{A_{i}},WF_{A_{i}})$ , $WB_{i}=(WT_{B_{i}},WI_{B_{i}},WF_{B_{i}})$ . If $WT_{A_{i}}\leqslant WT_{B_{i}},WI_{A_{i}}\geqslant WI_{B_{i}},WF_{A_{i}}% \geqslant WF_{B_{i}}$ holds for all i, then

$\displaystyle\text{SVNNAAWA}_{ww}({WA_{1},WA_{2},\ldots,WA_{n}})\leqslant\text% {SVNNAAWA}_{ww}({WB_{1},WB_{2},\ldots,WB_{n}})$ (6)

Property 3 (Boundedness). Let $WA_{i}=(WT_{A_{i}},WI_{A_{i}},WF_{A_{i}})$ . If $WA^{+}=(\max_{i}(WT_{i}),\min_{i}(WI_{i}),\linebreak\min_{i}(WF_{i}))$ , $WA^{-}=(\min_{i}(WT_{i}),\max_{i}(WI_{i}),\max_{i}(WF_{i}))$ , then

$\displaystyle WA^{-}\leqslant\text{SVNNAAWA}_{ww}({WA_{1},WA_{2},\ldots,WA_{n}% })\leqslant WA^{+}$ (7)

Then, the SVNN Aczel-Alsina power averaging (SVNNAAPA) operator is built on SVNNAAWA operator and PA operator [41].

Definition 7. Let $WA_{i}=(WT_{i},WI_{i},WF_{i})$ be the SVNNs, $\theta\geqslant 1$ . If

$\displaystyle\text{SVNNAAPA}({WA_{1},WA_{2},\ldots,WA_{n}})=\mathop{\oplus}% \limits_{i=1}^{n}\frac{({1+T({WA_{i}})})}{\sum\limits_{i=1}^{n}{({1+T({WA_{i}}% )})}}WA_{i}$ (8)

where $T({WA_{a}})=\mathop{\sum}\limits_{\begin{subarray}{c}{j=1\atop a\neq j}% \end{subarray}}^{m}{\textit{Sup}({WA_{a},WA_{j}})}$ , $\textit{Sup}({WA_{a},WA_{j}})$ is the decision support for $WA_{a}$ from $WA_{j}$ , with given conditions: (1) $\textit{Sup}({WA_{a},WA_{b}})\in[{0,1}]$ ; (2) $\textit{Sup}({WA_{b},WA_{a}})=\textit{Sup}({WA_{a},WA_{b}})$ ; (3) $\textit{Sup}({WA_{a},WA_{b}})\geqslant\textit{Sup}({WA_{s},WA_{t}})$ , if $d({WA_{a},WA_{b}})\leqslant d({WA_{s},WA_{t}})$ , where $d$ is a distance.

The Theorem 2 is obtained.

Theorem 2. Let $WA_{i}=(WT_{i},WI_{i},WF_{i})(i=1,2,\ldots,n)$ be the SVNNs, $\theta\geqslant 1$ . If

$\displaystyle\text{SVNNAAPA}({WA_{1},WA_{2},\ldots,WA_{n}})=\mathop{\oplus}% \limits_{i=1}^{n}\frac{({1+T({WA_{i}})})}{\sum\limits_{i=1}^{n}{({1+T({WA_{i}}% )})}}WA_{i}$ (9) $\displaystyle=\left({\begin{array}[]{l}1-e^{-\left({\sum\limits_{i=1}^{n}{% \frac{({1+T({WA_{i}})})}{\sum\limits_{i=1}^{n}{({1+T({WA_{i}})})}}}({-\ln({1-% WT_{j}})})^{\theta}}\right)^{1/\theta}},\\ e^{-\left({\sum\limits_{i=1}^{n}{\frac{({1+T({WA_{i}})})}{\sum\limits_{i=1}^{n% }{({1+T({WA_{i}})})}}}({-\ln({WI_{j}})})^{\theta}}\right)^{1/\theta}},e^{-% \left({\sum\limits_{i=1}^{n}{\frac{({1+T({WA_{i}})})}{\sum\limits_{i=1}^{n}{({% 1+T({WA_{i}})})}}}({-\ln({WF_{j}})})^{\theta}}\right)^{1/\theta}}\\ \end{array}}\right)$

Proof: The proof of Theorem 1 with the mathematical induction is produced as follows:

(a) Let $i=2$ , then

$\displaystyle\text{SVNNAAPA}({WA_{1},WA_{2}})=\frac{({1+T({WA_{1}})})}{\sum% \limits_{i=1}^{2}{({1+T({WA_{1}})})}}WA_{1}\oplus\frac{({1+T({WA_{2}})})}{\sum% \limits_{i=1}^{2}{({1+T({WA_{2}})})}}WA_{2}$ $\displaystyle=\left(\begin{array}[]{l}1-e^{-\left({\frac{({1+T({WA_{1}})})}{% \sum\limits_{i=1}^{2}{({1+T({WA_{1}})})}}({-\ln({1-WT_{1}})})^{\theta}}\right)% ^{1/\theta}},e^{-\left({\frac{({1+T({WA_{1}})})}{\sum\limits_{i=1}^{2}{({1+T({% WA_{1}})})}}({-\ln({WI_{1}})})^{\theta}}\right)^{1/\theta}},\\ e^{-\left({\frac{({1+T({WA_{1}})})}{\sum\limits_{i=1}^{2}{({1+T({WA_{1}})})}}(% {-\ln({WF_{1}})})^{\theta}}\right)^{1/\theta}}\end{array}\right)\oplus$ $\displaystyle\left(\begin{array}[]{l}1-e^{-\left({\frac{({1+T({WA_{2}})})}{% \sum\limits_{i=1}^{2}{({1+T({WA_{2}})})}}({-\ln({1-WT_{2}})})^{\theta}}\right)% ^{1/\theta}},e^{-\left({\frac{({1+T({WA_{2}})})}{\sum\limits_{i=1}^{2}{({1+T({% WA_{2}})})}}({-\ln({WI_{2}})})^{\theta}}\right)^{1/\theta}},\\ e^{-\left({\frac{({1+T({WA_{2}})})}{\sum\limits_{i=1}^{2}{({1+T({WA_{2}})})}}(% {-\ln({WF_{2}})})^{\theta}}\right)^{1/\theta}}\end{array}\right)$ $\displaystyle=\left({\begin{array}[]{l}1-e^{-\left({\sum\limits_{i=1}^{2}{% \frac{({1+T({WA_{i}})})}{\sum\limits_{i=1}^{n}{({1+T({WA_{i}})})}}}({-\ln({1-% WT_{j}})})^{\theta}}\right)^{1/\theta}},\\ e^{-\left({\sum\limits_{i=1}^{2}{\frac{({1+T({WA_{i}})})}{\sum\limits_{i=1}^{n% }{({1+T({WA_{i}})})}}}({-\ln({WI_{j}})})^{\theta}}\right)^{1/\theta}},e^{-% \left({\sum\limits_{i=1}^{2}{\frac{({1+T({WA_{i}})})}{\sum\limits_{i=1}^{n}{({% 1+T({WA_{i}})})}}}({-\ln({WF_{j}})})^{\theta}}\right)^{1/\theta}}\\ \end{array}}\right)$

(b) If Eq. (3) holds for $i=k$ , then

$\displaystyle\text{SVNNAAPA}({WA_{1},WA_{2},\ldots,WA_{k}})=\mathop{\oplus}% \limits_{i=1}^{k}\frac{({1+T({WA_{i}})})}{\sum\limits_{i=1}^{k}{({1+T({WA_{i}}% )})}}WA_{i}$ $\displaystyle=\left({\begin{array}[]{l}1-e^{-\left({\sum\limits_{i=1}^{k}{% \frac{({1+T({WA_{i}})})}{\sum\limits_{i=1}^{k}{({1+T({WA_{i}})})}}}({-\ln({1-% WT_{j}})})^{\theta}}\right)^{1/\theta}},\\ e^{-\left({\sum\limits_{i=1}^{k}{\frac{({1+T({WA_{i}})})}{\sum\limits_{i=1}^{k% }{({1+T({WA_{i}})})}}}({-\ln({WI_{j}})})^{\theta}}\right)^{1/\theta}},e^{-% \left({\sum\limits_{i=1}^{k}{\frac{({1+T({WA_{i}})})}{\sum\limits_{i=1}^{k}{({% 1+T({WA_{i}})})}}}({-\ln({WF_{j}})})^{\theta}}\right)^{1/\theta}}\\ \end{array}}\right)$

$\displaystyle\text{SVNNAAPA}({WA_{1},WA_{2},\ldots,WA_{k}A_{k+1}})$ $\displaystyle=\mathop{\oplus}\limits_{i=1}^{k}\frac{({1+T({WA_{i}})})}{\sum% \limits_{i=1}^{k+1}{({1+T({WA_{i}})})}}WA_{i}^{\oplus}\frac{({1+T({WA_{k+1}})}% )}{\sum\limits_{i=1}^{k+1}{({1+T({WA_{i}})})}}WA_{k+1}$ $\displaystyle=\left({\begin{array}[]{l}1-e^{-\left({\sum\limits_{i=1}^{k}{% \frac{({1+T({WA_{i}})})}{\sum\limits_{i=1}^{k}{({1+T({WA_{i}})})}}}({-\ln({1-% WT_{j}})})^{\theta}}\right)^{1/\theta}},\\ e^{-\left({\sum\limits_{i=1}^{k}{\frac{({1+T({WA_{i}})})}{\sum\limits_{i=1}^{k% }{({1+T({WA_{i}})})}}}({-\ln({WI_{j}})})^{\theta}}\right)^{1/\theta}},e^{-% \left({\sum\limits_{i=1}^{k}{\frac{({1+T({WA_{i}})})}{\sum\limits_{i=1}^{k}{({% 1+T({WA_{i}})})}}}({-\ln({WF_{j}})})^{\theta}}\right)^{1/\theta}}\\ \end{array}}\right)\oplus$ $\displaystyle\left({\begin{array}[]{l}1-e^{-\left({\frac{({1+T({ZW_{k+1}})})}{% \sum\limits_{i=1}^{k+1}{({1+T({WA_{i}})})}}({-\ln({1-WT_{k+1}})})^{\theta}}% \right)^{1/\theta}},\\ e^{-\left({\frac{({1+T({WA_{k+1}})})}{\sum\limits_{i=1}^{k+1}{({1+T({WA_{i}})}% )}}({-\ln({WI_{k+1}})})^{\theta}}\right)^{1/\theta}},e^{-\left({\frac{({1+T({% WA_{k+1}})})}{\sum\limits_{i=1}^{k+1}{({1+T({WA_{i}})})}}({-\ln({WF_{k+1}})})^% {\theta}}\right)^{1/\theta}}\\ \end{array}}\right)$ $\displaystyle=\left({\begin{array}[]{l}1-e^{-\left({\sum\limits_{i=1}^{k+1}{% \frac{({1+T({WA_{i}})})}{\sum\limits_{i=1}^{k+1}{({1+T({WA_{i}})})}}}({-\ln({1% -WT_{j}})})^{\theta}}\right)^{1/\theta}},\\ e^{-\left({\sum\limits_{i=1}^{k+1}{\frac{({1+T({WA_{i}})})}{\sum\limits_{i=1}^% {k+1}{({1+T({WA_{i}})})}}}({-\ln({WI_{j}})})^{\theta}}\right)^{1/\theta}},e^{-% \left({\sum\limits_{i=1}^{k+1}{\frac{({1+T({WA_{i}})})}{\sum\limits_{i=1}^{k+1% }{({1+T({WA_{i}})})}}}({-\ln({WF_{j}})})^{\theta}}\right)^{1/\theta}}\\ \end{array}}\right)$

From the above (a), (b), and (c), it can be seen that Eq. (3) meets any given $i$ .

The SVNNAAPA has three decision properties.

Property 4 (idempotency). If $WA_{i}=WA=(WT,WI,WF)$ ,

$\displaystyle\text{SVNNAAPA}({WA_{1},WA_{2},\ldots,WA_{n}})=WA$ (10)

Property 5 (Monotonicity). Let $WA_{i}=(WT_{A_{i}},WI_{A_{i}},WF_{A_{i}})$ , $WB_{i}=(WT_{B_{i}},WI_{B_{i}},WF_{B_{i}})$ . If $WT_{A_{i}}\leqslant WT_{B_{i}},WI_{A_{i}}\geqslant WI_{B_{i}},WF_{A_{i}}% \geqslant WF_{B_{i}}$ holds for all i, then

$\displaystyle\text{SVNNAAPA}({WA_{1},WA_{2},\ldots,WA_{n}})\leqslant\text{% SVNNAAPA}({WB_{1},WB_{2},\ldots,WB_{n}})$ (11)

Property 6 (Boundedness). Let $WA_{i}=(WT_{A_{i}},WI_{A_{i}},WF_{A_{i}})$ . If $WA^{+}=(\max_{i}(WT_{i}),\min_{i}(WI_{i}),\linebreak\min_{i}(WF_{i}))$ , $WA^{-}=(\min_{i}(WT_{i}),\max_{i}(WI_{i}),\max_{i}(WF_{i}))$ , then

$\displaystyle WA^{-}\leqslant\text{SVNNAAPA}({WA_{1},WA_{2},\ldots,WA_{n}})% \leqslant WA^{+}$ (12)

To pay attention to weights, the SVNN Aczel-Alsina power weight averaging (SVNNAAPWA) operator is built on the SVNNAAWA operator and PA operator [41].

Definition 8. Let $WA_{i}=(WT_{i},WI_{i},WF_{i})$ be the SVNNs with weight $ww=(ww_{1},ww_{2},\ldots,ww_{n})^{T}$ , $\sum\nolimits_{i=1}^{n}{ww_{i}}=1$ , $\theta\geqslant 1$ . If

$\displaystyle\text{SVNNAAPWA}_{ww}({WA_{1},WA_{2},\ldots,WA_{n}})=\mathop{% \oplus}\limits_{i=1}^{n}\frac{ww_{i}({1+T({WA_{i}})})}{\sum\limits_{i=1}^{n}{% ww_{i}({1+T({WA_{i}})})}}WA_{i}$ (13)

where $T({WA_{a}})=\mathop{\sum}\limits_{\begin{subarray}{c}{j=1\atop a\neq j}% \end{subarray}}^{m}{ww_{j}\textit{Sup}({WA_{a},WA_{j}})}$ , $\textit{Sup}({WA_{a},WA_{j}})$ is the decision support for $WA_{a}$ from $WA_{j}$ , with conditions: (1) $\textit{Sup}({WA_{a},WA_{b}})\in[{0,1}]$ ; (2) $\textit{Sup}({WA_{b},WA_{a}})=\textit{Sup}({WA_{a},WA_{b}})$ ; (3) $\textit{Sup}({WA_{a},WA_{b}})\geqslant\textit{Sup}({WA_{s},WA_{t}})$ , if $d({WA_{a},WA_{b}})\leqslant d({WA_{s},WA_{t}})$ , where $d$ is a distance.

The Theorem 2 is obtained.

Theorem 2. Let $WA_{i}=(WT_{i},WI_{i},WF_{i})$ be the SVNNs, $\theta\geqslant 1$ . If

$\displaystyle\text{SVNNAAPWA}_{ww}({WA_{1},WA_{2},\ldots,WA_{n}})=\mathop{% \oplus}\limits_{i=1}^{n}\frac{ww_{i}({1+T({WA_{i}})})}{\sum\limits_{i=1}^{n}{% ww_{i}({1+T({WA_{i}})})}}WA_{i}$ $\displaystyle=\left({\begin{array}[]{l}1-e^{-\left({\sum\limits_{i=1}^{n}{% \frac{ww_{i}({1+T({WA_{i}})})}{\sum\limits_{i=1}^{n}{ww_{i}({1+T({WA_{i}})})}}% }({-\ln({1-WT_{j}})})^{\theta}}\right)^{1/\theta}},\\ e^{-\left({\sum\limits_{i=1}^{n}{\frac{ww_{i}({1+T({WA_{i}})})}{\sum\limits_{i% =1}^{n}{ww_{i}({1+T({WA_{i}})})}}}({-\ln({WI_{j}})})^{\theta}}\right)^{1/% \theta}},e^{-\left({\sum\limits_{i=1}^{n}{\frac{ww_{i}({1+T({WA_{i}})})}{\sum% \limits_{i=1}^{n}{ww_{i}({1+T({WA_{i}})})}}}({-\ln({WF_{j}})})^{\theta}}\right% )^{1/\theta}}\\ \end{array}}\right)$ (14)

The SNNAAPWA has the three decision properties.

Property 4 (idempotency). If $WA_{i}=WA=(WT,WI,WF)$ ,

$\displaystyle\text{SVNNAAPWA}_{ww}({WA_{1},WA_{2},\ldots,WA_{n}})=WA$ (15)

$\displaystyle\text{SVNNAAPWA}_{ww}({WA_{1},WA_{2},\ldots,WA_{n}})\leqslant% \text{SVNNAAPWA}_{ww}({WB_{1},WB_{2},\ldots,WB_{n}})$ (16)

$\displaystyle WA^{-}\leqslant\text{SVNNAAPWA}({WA_{1},WA_{2},\ldots,WA_{n}})% \leqslant WA^{+}$ (17)

4. The MADM method based on SVNNAAPWA with SVNNs

Then, the MADM method is built based on the SVNNAAPWA operator under SVNSs. Let $WZ=\{{WZ_{1},WZ_{2},\ldots,WZ_{n}}\}$ be attributes, $ww=\{{ww_{1},ww_{2},\ldots,ww_{n}}\}$ be weight of $WZ_{j}$ . Let $WP=\{{WP_{1},WP_{2},\ldots,WP_{m}}\}$ be alternatives. $WQ=({WQ_{ij}})_{m\times n}=({WT_{ij},WI_{ij},WF_{ij}})_{m\times n}$ is the SVNNmatrix. The SVNNAAPWA method for MADM will be constructed.

Step 1. Structure SVNN matrix $WQ=({WQ_{ij}})_{m\times n}=({WT_{ij},WI_{ij},WF_{ij}})_{m\times n}$ .

$\displaystyle WQ=[{WQ{}_{ij}}]_{m\times n}=\left[{{\begin{array}[]{*{20}c}{WQ_% {11}}&{WQ_{12}}&\ldots&{WQ_{1n}}\\ {WQ_{21}}&{WQ_{22}}&\ldots&{WQ_{2n}}\\ \vdots&\vdots&\vdots&\vdots\\ {WQ_{m1}}&{WQ_{m2}}&\ldots&{WQ_{mn}}\\ \end{array}}}\right]$ (18) $\displaystyle WQ=[{WQ_{ij}}]_{m\times n}=({WT_{ij},WI_{ij},WF_{ij}})_{m\times n}$ (19) $\displaystyle=\left[{{\begin{array}[]{*{20}c}{({WT_{11},WI_{11},WF_{11}})}&{({% WT_{12},WI_{1},WF_{12}})}&\ldots&{({WT_{1n},WI_{1n},WF_{1n}})}\\ {({WT_{21},WI_{21},WF_{21}})}&{({WT_{22},WI_{22},WF_{22}})}&\ldots&{({WT_{2n},% WI_{2n},WF_{2n}})}\\ \vdots&\vdots&\vdots&\vdots\\ {({WT_{m1},WI_{m1},WF_{m1}})}&{({WT_{m2},WI_{m2},WF_{m2}})}&\ldots&{({WT_{mn},% WI_{mn},WF_{mn}})}\\ \end{array}}}\right]$

Step 2. Normalize the $WQ=[{WQ_{ij}}]_{m\times n}=({WT_{ij},WI_{ij},WF_{ij}})_{m\times n}$ to $\textit{NWQ}=[{\textit{NWQ}_{ij}}]_{m\times n}=({\textit{NWT}_{ij},\textit{NWI% }_{ij},\textit{NWF}_{ij}})_{m\times n}$ .

$\displaystyle\textit{NWQ}_{ij}=({\textit{NWT}_{ij},\textit{NWI}_{ij},\textit{% NWF}_{ij}})$ $\displaystyle=\left\{{{\begin{array}[]{l}({WT_{ij},WI_{ij},WF_{ij}}),WZ_{j}% \textit{ is a benefit criterion}\\ ({WF_{ij},1-WI_{ij},WT_{ij}}),WZ_{j}\textit{ is a cost criterion}\\ \end{array}}}\right.$ (20)

Step 3. According to $\textit{NWQ}=[{\textit{NWQ}_{ij}}]_{m\times n}=({\textit{NWT}_{ij},\textit{NWI% }_{ij},\textit{NWF}_{ij}})_{m\times n}$ , we can fuse all SVNNs $\textit{NWQ}=[{\textit{NWQ}_{ij}}]_{m\times n}=({\textit{NWT}_{ij},\textit{NWI% }_{ij},\textit{NWF}_{ij}})_{m\times n}$ through SVNNAAPWA operator to get the SVNNs $\textit{NWQ}_{i}({i=1,2,\ldots,m})$ :

$\displaystyle\textit{NWQ}_{i}=\text{SVNNAAPWA}_{ww}(\textit{NWQ}_{i1},\textit{% NWQ}_{i2},\ldots,\textit{NWQ}_{in})$ $\displaystyle=\mathop{\oplus}\limits_{j=1}^{n}\frac{ww_{i}({1+T({\textit{NWQ}_% {ij}})})}{\sum\limits_{j=1}^{n}{ww_{i}({1+T({\textit{NWQ}_{ij}})})}}\textit{% NWQ}_{ij}$ (21) $\displaystyle=\left({\begin{array}[]{l}1-e^{-\left({\sum\limits_{i=1}^{n}{% \frac{ww_{i}({1+T({\textit{NWQ}_{ij}})})}{\sum\limits_{j=1}^{n}{ww_{i}({1+T({% \textit{NWQ}_{ij}})})}}}({-\ln({1-\textit{NWT}_{ij}})})^{\theta}}\right)^{1/% \theta}},\\ e^{-\left({\sum\limits_{j=1}^{n}{\frac{ww_{i}({1+T({\textit{NWQ}_{ij}})})}{% \sum\limits_{j=1}^{n}{ww_{i}({1+T({\textit{NWQ}_{ij}})})}}}({-\ln({\textit{NWI% }_{ij}})})^{\theta}}\right)^{1/\theta}},e^{-\left({\sum\limits_{j=1}^{n}{\frac% {ww_{i}({1+T({\textit{NWQ}_{ij}})})}{\sum\limits_{j=1}^{n}{ww_{i}({1+T({% \textit{NWQ}_{ij}})})}}}({-\ln({\textit{NWF}_{ij}})})^{\theta}}\right)^{1/% \theta}}\\ \end{array}}\right)$

Step 4. Obtain the $SV({\textit{NWQ}_{i}})$ , $AV({\textit{NWQ}_{i}})({i=1,2,\ldots,m})$ .

$\displaystyle SV({\textit{NWQ}_{i}})=\frac{({2+\textit{NWT}_{i}-\textit{NWI}_{% i}-\textit{NWF}_{i}})}{3},AV({\textit{NWQ}_{i}})=\textit{NWT}_{i}-\textit{NWF}% _{i}$ (22)

Step 5. Rank the $WP_{i}({i=1,2,\ldots,m})$ through $SV({\textit{NWQ}_{i}})$ , $AV({\textit{NWQ}_{i}})({i=1,2,\ldots,m})$ .

Step 6. End.

5. The decision example and comparative studies

5.1 An decision example

The problem and phenomenon of passive learning and ineffective teaching still exist in English teaching in China’s colleges and universities. Teachers are showing lectures, presenting arguments, teaching knowledge, giving examples and explaining texts, while students are “doing their own things in class”, which seriously affects the effectiveness and efficiency of English teaching in colleges and universities [45]. Although the traditional teaching mode in China can help students have a deep and comprehensive understanding of English grammar, vocabulary and sentence knowledge, the lagging teaching concept and teaching mode are not compatible with the information and modern teaching ecology in which students live, which weakens the subjectivity of students in English teaching and leads to the ineffectiveness of teachers’ teaching behaviors [46, 47]. However, with the support of blended teaching mode, teachers can make full use of modern information technology to innovate teaching forms, stimulate students’ interest, transform the traditional situation of theoretical indoctrination and full classroom duck-filling, and improve the quality of English teaching [48]. However, in order to better play the role and value of blended teaching in English education, we need to have a clear understanding of the definition, classification and practice model of blended teaching. Blended teaching mainly refers to the teaching mode of integrating classroom teaching and online learning, which can cover cognitivism, behaviorism, constructivism and other teaching theories, and apply computer network technology, audio and video recording, slide projection and other means to achieve the goals of streaming video learning, cooperative learning, self-paced learning and virtual classroom teaching [49, 50]. Blended learning was first proposed by foreign scholars and is a form of teaching that combines networked teaching and classroom teaching, which has a pivotal and important function and role in promoting the development of modern education networking and informatization [51]. According to theoretical research and practical analysis, we can find that there are three main types of hybrid teaching mode: network-assisted, hybrid and online, among which network-assisted mainly refers to the teaching form that seldom uses network and only uses network as an auxiliary classroom teaching activity. Hybrid teaching refers to the teaching mode in which network activities replace part of the classroom teaching activities [52, 53]. The online type refers to the teaching mode in which teaching activities are mainly carried out on the Internet. In terms of the degree of implementation, we can classify them into the following four types: (1) the category of using online resources as students’ self-learning grasp; (2) the category of linking classroom teaching with online resources; (3) the category of providing learning materials for students through online platforms; and (4) the category of complementing online learning with classroom teaching. And according to the degree of implementation of blended teaching models, we can also classify them into low intensity, medium intensity and high intensity blended implementation types. In terms of practice mode, blended teaching can incorporate modern advanced teaching mode and teaching methods to form a teaching pattern that adapts to the actual teaching in colleges and universities: (1) flipped classroom. Flipped classroom mainly refers to the process of flipping classroom teaching contents to the end of class and internalizing students’ knowledge, which can focus teachers’ teaching on the level of ability enhancement and knowledge internalization and help teachers better promote students’ overall development [54, 55]. Taking it as the main body of blended teaching can deepen the connotation of online teaching, transform the nature of classroom teaching, and make modern information technology a grip for students’ self-learning. (2) SPOC, short for Small Scale Online Course, can meet the needs of classroom teaching and help students make up for their own shortcomings and problems. And in the process of applying SPOC platform, teachers can provide rich and excellent online teaching resources for blended teaching and improve the efficiency of problem solving [56, 57]. The English blended teaching quality evaluation is a classical MADM issue. In this paper, an empirical application of English blended teaching quality evaluation is given through SNNAAPWA method. There are five English blended teaching colleges are evaluated their grain fermentation process quality. In order to assess five English blended teaching colleges fairly, the experts give the information with four defined attributes: ⟀ WZ₁ is student feedback; ⟁ WZ₂ is blended teaching costs; ⟂ WZ₃ is blended teaching attitude; ⟃ WZ₄ is invited peer expert recognition. Evidently, WZ₂ is the cost, others are the benefit. Then, the SVNNAAPWA method is applied to MADM for English blended teaching quality evaluation with SVNNs. The SVNNAAPWA method involves the decision steps as below ( ${\theta=3}$ ):

Table 1
SVNN matrix

	WZ₁	WZ₂	WZ₃	WZ₄
WP₁	(0.30, 0.59, 0.30)	(0.27, 0.35, 0.48)	(0.38, 0.25, 0.35)	(0.26, 0.25, 0.22)
WP₂	(0.37, 0.32, 0.65)	(0.36, 0.47, 0.35)	(0.32, 0.03, 0.49)	(0.52, 0.22, 0.22)
WP₃	(0.35, 0.27, 0.55)	(0.55, 0.05, 0.55)	(0.55, 0.32, 0.55)	(0.58, 0.22, 0.65)
WP₄	(0.59, 0.43, 0.37)	(0.52, 0.38, 0.63)	(0.59, 0.27, 0.52)	(0.52, 0.35, 0.39)
WP₅	(0.51, 0.23, 0.59)	(0.38, 0.25, 0.51)	(0.50, 0.24, 0.50)	(0.58, 0.25, 0.56)

Table 2

The NQ matrix

	WZ₁	WZ₂	WZ₃	WZ₄
WP₁	(0.30, 0.59, 0.30)	(0.48, 0.65, 0.27)	(0.38, 0.25, 0.35)	(0.26, 0.25, 0.22)
WP₂	(0.37, 0.32, 0.65)	(0.35, 0.53, 0.36)	(0.32, 0.03, 0.49)	(0.52, 0.22, 0.22)
WP₃	(0.35, 0.27, 0.55)	(0.55, 0.95, 0.55)	(0.55, 0.32, 0.55)	(0.58, 0.22, 0.65)
WP₄	(0.59, 0.43, 0.37)	(0.63, 0.62, 0.52)	(0.59, 0.27, 0.52)	(0.52, 0.35, 0.39)
WP₅	(0.51, 0.23, 0.59)	(0.51, 0.75, 0.38)	(0.50, 0.24, 0.50)	(0.58, 0.25, 0.56)

Step 1. Set up the SVNNmatrix $WQ=({WQ_{ij}})_{5\times 4}$ as in Table 1.

Step 2. Normalize $WQ=({WQ_{ij}})_{5\times 4}$ to $\textit{NWQ}=[{\textit{NWQ}_{ij}}]_{5\times 4}$ (see the Table 2).

Step 3. Suppose that the attribute weights are given in Table 3.

Table 3

The weight information

	WZ₁	WZ₂	WZ₃	WZ₄
$ww_{j}$	0.2438	0.1542	0.2804	0.3216

Step 4. Obtain the $\textit{NWQ}_{i}({i=1,2,\ldots,5})$ by utilizing SVNNAAPWA operator (Table 4).

Table 4

The $\textit{NWQ}_{i}({i=1,2,\ldots,5})$ by SVNNAAPWA operator ( $\theta=3$ )

Alternatives	$\textit{NWQ}_{i}({i=1,2,\ldots,5})$
WP₁	(0.5638, 0.6493, 0.3194)
WP₂	(0.6975, 0.2952, 0.3219)
WP₃	(0.5852, 0.5736, 0.3254)
WP₄	(0.5336, 0.4593, 0.3032)
WP₅	(0.5874, 0.6459, 0.1847)

Step 5. Obtain the $\textit{NWQ}_{i}({i=1,2,\ldots,5})$ (Table 5).

Table 5

The $\textit{NWQ}_{i}({i=1,2,\ldots,5})$

Alternatives	$SV({nq_{i}})({i=1,2,\ldots,5})$	Order
WP₁	0.6468	1
WP₂	0.6988	1
WP₃	0.6653	3
WP₄	0.6515	4
WP₅	0.6676	2

Step 6. From Table 5, the order is $WP_{2}>WP_{5}>WP_{3}>WP_{4}>WP_{1}$ , and the best choice is $WP_{2}$ .

5.2 Influence analysis

To show the decision effects on the decision results through different decision parameters of SVNNAAPWA, the obtained results are produced in Tables 6 and 7.

Table 6
Different parameters for SVNNAAPWA

$\theta$	$SV({WP_{1}})$	$SV({WP_{2}})$	$SV({WP_{3}})$	$SV({WP_{4}})$	$SV({WP_{5}})$
1	0.4224	0.5325	0.4456	0.4545	0.4651
2	0.5984	0.6730	0.6207	0.6175	0.6210
3	0.6468	0.6988	0.6653	0.6515	0.6676
4	0.6653	0.7059	0.6822	0.6618	0.6877
5	0.6743	0.7086	0.6909	0.6661	0.6986
6	0.6795	0.7101	0.6964	0.6684	0.7052
7	0.6829	0.7111	0.7003	0.6700	0.7097
8	0.6854	0.7119	0.7033	0.6712	0.7130
9	0.6872	0.7127	0.7057	0.6722	0.7154
10	0.6888	0.7135	0.7078	0.6731	0.7174

Table 7

Different order for SVNNAAPWA

$\theta$	Order
1	WP₂ $>$ WP₅ $>$ WP₄ $>$ WP₃ $>$ WP₁
2	WP₂ $>$ WP₅ $>$ WP₃ $>$ WP₄ $>$ WP₁
3	WP₂ $>$ WP₅ $>$ WP₃ $>$ WP₄ $>$ WP₁
4	WP₂ $>$ WP₅ $>$ WP₃ $>$ WP₁ $>$ WP₄
5	WP₂ $>$ WP₅ $>$ WP₃ $>$ WP₁ $>$ WP₄
6	WP₂ $>$ WP₅ $>$ WP₃ $>$ WP₁ $>$ WP₄
7	WP₂ $>$ WP₅ $>$ WP₃ $>$ WP₁ $>$ WP₄
8	WP₅ $>$ WP₂ $>$ WP₃ $>$ WP₁ $>$ WP₄
9	WP₅ $>$ WP₂ $>$ WP₃ $>$ WP₁ $>$ WP₄
10	WP₅ $>$ WP₂ $>$ WP₃ $>$ WP₁ $>$ WP₄

It can be seen from the decision information in Tables 6 and 7 that when different parameter values are used, the priority of advantages and disadvantages is slightly different. During the MADM process, the choice of parameter values can be changed through the subjective attitude of decision-maker (DM). On the one hand, DMs can get different results through changing the parameter values; On the other hand, the selection of parameter values can also express the DM’s risk preference attitude, and determine which type of decision-maker is risky, conservative or neutral.

5.3 Compare analysis

The SVNNAAPWA operator is compared with SVNN-COPRAS method [58] and SVNN-MARCOS method [59], SVNN-MULTIMOORA method [60] and SVNN-COPRAS method [61]. The results are recorded in Table 8.

Table 8
Results for different methods

Methods	Order	The optimal choice
SVNN-COPRAS method [58]	$WP_{2}>WP_{5}>WP_{3}>WP_{4}>WP_{1}$	$WP_{2}$
SVNN-MARCOS method [59]	$WP_{2}>WP_{5}>WP_{3}>WP_{4}>WP_{1}$	$WP_{2}$
SVNN-MULTIMOORA method [60]	$WP_{2}>WP_{5}>WP_{3}>WP_{4}>WP_{1}$	$WP_{2}$
SVNN-COPRAS method [61]	$WP_{2}>WP_{5}>WP_{3}>WP_{4}>WP_{1}$	$WP_{2}$
SVNNAAPWA operator	$WP_{2}>WP_{5}>WP_{3}>WP_{4}>WP_{1}$	$WP_{2}$

In accordance with WS coefficients [62, 63], the WS coefficient between SVNN-COPRAS method [58] and SVNN-MARCOS method [59], SVNN-MULTIMOORA method [60], SVNN-COPRAS method [61] and the proposed SVNNAAPWA operator is 1.0000, 1.0000, 1.0000, 1.0000, respectively. At the same time, obtained from Table 8, it is obvious that the given optimal decision choice is WP₂, while the worst decision choice is WP₁. In other words, the ordering results of these five models are slightly different. Different models can effectively solve the MADM problem from different angles. The proposed SVNNAAPWA operator has the following advantages: (1) The built SNNAAPWA operator considers decision information with relationship between aggregate parameters; (2) The proposed SVNNAAPWA can effectively capture the intrinsic connection between attributes in MADM problems.

6. Conclusion

In the information age, teachers are no longer the only source of information for students, and the problems of the traditional lecture mode are becoming more and more obvious. Especially in the process of teaching English in colleges and universities, students’ personalized and diversified needs for English learning are becoming more and more obvious, and if traditional theoretical lectures or theoretical indoctrination are continued, students’ needs and goals will be affected. The English blended teaching quality evaluation is a classical MADM issues. In such paper, the Aczel-Alsina operations and PA operator is designed for MADM under SVNSs. The SVNNAAPWA operator is built and then the MADM methods are proposed based on the SVNNAAPWA operator. Finally, an example about English blended teaching quality evaluation and some selected comparative decision analysis is given to produce the SVNNAAPWA method. In the future works, the Aczel-Alsina operations shall be applied to other fuzzy and uncertain decision settings [64, 65, 66, 67, 68, 69, 70].

Footnotes

Acknowledgments

This work was supported by the Eleventh Batch of China Foreign Language Education Fund, National Research Centre for Foreign Language Education, Beijing Foreign Studies University (Project Title: Research on Blended College English Teaching in Private Universities Based on POA, Grant No. ZGWYJYJJ11A167).

References

Thorne

. Blended learning: How to integrate online & traditional learning. London: Kogan Page Publishers; 2003.

Liu

, editor. The construction of college english teaching model based on blended learning. 1st International Workshop on Education Technology and Computer Science; 2009 Mar 07–08; Wuhan, PEOPLES R CHINA. LOS ALAMITOS: Ieee Computer Soc; 2009.

Miao

. Online and offline mixed intelligent teaching assistant mode of english based on mobile information system. Mobile Information Systems. 2021; 2021: 6.

Shao

. Evaluation method of it english blended teaching quality based on the data mining algorithm. Journal of Mathematics. 2021; 2021: 8.

Wang

. Employing blended learning to enhance learners’ english conversation: A preliminary study of teaching with hitutor. Education and Information Technologies. 2021; 26(2): 2407-25.

Cheng

. Research on blended teaching strategies of college english translation based on computer corpus. Wireless Communications & Mobile Computing. 2022; 2022: 11.

Wen

, Ieee, editors. A study on the application of spoc-based blended oral english teaching in higher vocational colleges. 10th International Conference on Educational and Information Technology (ICEIT); 2021 Jan 18–20; Electr Network. NEW YORK: Ieee; 2021.

Gao

. Evaluation and application of college english mixed flipping classroom teaching quality based on the fuzzy judgment model. Security and Communication Networks. 2022; 2022: 9.

. Blended oral english teaching based on core competence training model. Mobile Information Systems. 2022; 2022: 9.

10.

. Psychological adaptability and intervention strategies of college students’ english learning under the mixed foreign language teaching environment monitoring. Journal of Environmental and Public Health. 2022; 2022: 11.

11.

Huang

Wang

. Innovative research on collaborative design of blended english teaching in higher vocational colleges based on digital technology. Scientific Programming. 2022; 2022: 7.

12.

Ning

Ban

. Blended teaching strategies of college english translation under the background of internet. Mobile Information Systems. 2022; 2022: 7.

13.

Pan

. Construction and application of college english blended teaching system based on multidata fusion. Discrete Dynamics in Nature and Society. 2022; 2022: 7.

14.

Qiu

. A study on the application model of blended teaching in english language teaching in colleges and universities under the ecological and internet perspectives. Journal of Environmental and Public Health. 2022; 2022: 10.

15.

Yan

. Effect of blended teaching mode in colleges and universities based on automation technology on college students’ english performance. Mobile Information Systems. 2022; 2022: 10.

16.

Shen

. Analysis of the correlation between academic performance and learning motivation in english course under a corpus-data-driven blended teaching model. Scientific Programming. 2022; 2022: 11.

17.

Zhang

, Ieee, editors. Application of blended teaching into the course of comprehensive english. 11th International Conference on Educational and Information Technology (ICEIT); 2022 Jan 06–08; Sichuan Normal Univ, Chengdu, PEOPLES R CHINA. NEW YORK: Ieee; 2022.

18.

Munier

. A new approach to the rank reversal phenomenon in mcdm with the simus method. Multiple Criteria Decision Making. 2016; 11: 137-52.

19.

Stoilova

. An integrated multi-criteria approach for planning railway passenger transport in the case of uncertainty. Symmetry [Internet]. 2020; 12(6).

20.

Stoilova

Munier

Kendra

Skrúcaný

. Multi-criteria evaluation of railway network performance in countries of the ten-t orient-east med corridor. Sustainability. 2020; 12(4): 1482.

21.

Stoilova

Munier

. A novel fuzzy simus multicriteria decision-making method. An application in railway passenger transport planning. Symmetry [Internet]. 2021; 13(3): 483.

22.

Wiȩckowski

Kizielewicz

Shekhovtsov

Sałabun

. Rancom: A novel approach to identifying criteria relevance based on inaccuracy expert judgments. Engineering Applications of Artificial Intelligence. 2023; 122: 106114.

23.

Linkov

Varghese

Jamil

Seager

Kiker

Bridges

, editors. Multi-criteria decision analysis: A framework for structuring remedial decisions at contaminated sites. NATO Advanced Research Workshop on Comparative Risk Assessment and Environmental Decision Making; 2002 Oct 13–16; Rome, ITALY. DORDRECHT: Springer; 2004.

24.

Pusic

Lusic

. Multi-criteria decision analysis for nautical anchorage selection. Journal of Marine Science and Engineering. 2023; 11(4): 21.

25.

Saha

Villuri

VGK

Bhardwaj

Kumar

. A multi-criteria decision analysis (mcda) approach for landslide susceptibility mapping of a part of darjeeling district in north-east himalaya, india. Applied Sciences-Basel. 2023; 13(8): 23.

26.

Wieckowski

Watrobski

Kizielewicz

Salabun

. Complex sensitivity analysis in multi-criteria decision analysis: An application to the selection of an electric car. Journal of Cleaner Production. 2023; 390: 15.

27.

Bhuyan

Deka

Saikia

. Micro-spatial flood risk assessment in nagaon district, assam (india) using gis-based multi-criteria decision analysis (mcda) and analytical hierarchy process (ahp). Risk Analysis. 2023; 16.

28.

Iweh

Semassou

Ahouansou

. Optimum site selection of hybrid solar photovoltaic (pv)-hydro power plants in off grid locations in cameroon using the multi-criteria decision analysis (mcda). Energy Sources Part a-Recovery Utilization and Environmental Effects. 2023; 45(3): 8076-91.

29.

Zadeh

. Fuzzy sets. Information and Control. 1965; 8(3): 338-53.

30.

Atanassov

. Intuitionistic fuzzy sets. Fuzzy Sets and Systems. 1986; 20(1): 87-96.

31.

Smarandache

. Neutrosophy: Neutrosophic probability, set, and logic: Analytic synthesis & synthetic analysis: American Research Press; 1998.

32.

Mukherjee

Das

. Neutrosophic bipolar vague soft set and its application to decision making problems. Neutrosophic Sets and Systems. 2020; 32: 410-24.

33.

Majumder

Das

Tripathy

. Identification of the most significant risk factor of covid-19 in economy using cosine similarity measure under svpns-environment. Neutrosophic Sets and Systems. 2021; 46: 112-27.

34.

Das

Pramanik

. Single valued bipolar pentapartitioned neutrosophic set and its application in madm strategy. Neutrosophic Sets and Systems. 2022; 49: 145-63.

35.

Das

Mukherjee

Tripathy

. Application of neutrosophic similarity measures in covid-19. Annals of Data Science. 2022; 9(1): 55-70.

36.

Das

Tripathy

. Topology on rough pentapartitioned neutrosophic set. Iraqi Journal of Science. 2022; 63(6): 2630-40.

37.

Pramanik

. Single-valued neutrosophic set: An overview. In: Rezaei N, editor. Transdisciplinarity. Cham: Springer International Publishing; 2022. pp. 563-608.

38.

Biswas

Pramanik

Giri

. Entropy based grey relational analysis method for multi-attribute decision making under single valued neutrosophic assessments. Neutrosophic Sets and Systems. 2014; 2: 102-10.

39.

Aczél

Alsina

. Characterizations of some classes of quasilinear functions with applications to triangular norms and to synthesizing judgements. Aequationes Mathematicae. 1982; 25(1): 313-15.

40.

Ashraf

Ahmad

Naeem

Riaz

Alam

Stevic

. Novel edas methodology based on single-valued neutrosophic aczel-alsina aggregation information and their application in complex decision-making. Complexity. 2022; 2022: 1-18.

41.

Yager

. The power average operator. IEEE Transactions on Systems, Man, and Cybernetics-Part A. 2001; 31(6): 724-31.

42.

Wang

Smarandache

Zhang

Sunderraman

. Single valued neutrosophic sets. Multispace Multistruct. 2010; (4): 410-13.

43.

Peng

Wang

Zhang

Chen

. Simplified neutrosophic sets and their applications in multi-criteria group decision-making problems. International Journal of Systems Science. 2016; 47(10): 2342-58.

44.

Yong

Zhu

Zhang

. Aczel-alsina weighted aggregation operators of simplified neutrosophic numbers and its application in multiple attribute decision making. Cmes-Computer Modeling in Engineering & Sciences. 2022; 132(2): 569-84.

45.

Guo

Liu

Guo

. Research on online and offline blended teaching of higher vocational english under the background of “internet plus”. Basic & Clinical Pharmacology & Toxicology. 2020; 127: 73-73.

46.

, Ieee, editors. Discussion on mixed teaching mode of college english under the background of internet. 5th International Conference on Mechanical, Control and Computer Engineering (ICMCCE); 2020 Dec 25–27; Harbin, PEOPLES R CHINA. NEW YORK: Ieee; 2020.

47.

Yao

. Research on “mixed” english teaching model under mooc platform. Basic & Clinical Pharmacology & Toxicology. 2020; 127: 204-04.

48.

Huang

. Research on improving the teaching effectiveness of crew english course in higher vocational education based on blended teaching. Basic & Clinical Pharmacology & Toxicology. 2021; 128: 192-92.

49.

Zhang

Tsai

. A new online and offline blended teaching system of college english based on computer internet technology. Mathematical Problems in Engineering. 2021; 2021: 12.

50.

Qin

Luo

. Exploration on the construction of online plus offline blended mode of college english teaching. Basic & Clinical Pharmacology & Toxicology. 2021; 128: 97-97.

51.

Wang

. Blended learning-based english teaching mode among new english majors. Agro Food Industry Hi-Tech. 2017; 28(1): 516-19.

52.

Zhang

Man

, editors. A brief study of the application of blended learning in higher vocational college english teaching. International Conference on Advanced Education, Psychology and Sports Science (AEPSS); 2017 Jun 26–28; Guangzhou, PEOPLES R CHINA. LONDON: Francis Acad Press; 2017.

53.

Zhou

. Empirical study on the effectiveness of teaching model of college english writing within blended learning mode. Educational Sciences-Theory & Practice. 2018; 18(5): 1060-76.

54.

, editor. A study on the blended-teaching mode combining moocs and flipping classroom in college english teaching. 7th International Conference on Education, Management, Information and Mechanical Engineering (EMIM); 2017 Apr 28–30; Shenyang, PEOPLES R CHINA. PARIS: Atlantis Press; 2017.

55.

Wang

, editor. The composition of blended learning model and its application in english classroom teaching. 7th International Conference on Mechatronics, Computer and Education Informationization (MCEI); 2017 Nov 03–05; Shenyang, PEOPLES R CHINA. PARIS: Atlantis Press; 2017.

56.

Hui

. Evaluation of blended oral english teaching based on the mixed model of spoc and deep learning. Scientific Programming. 2021; 2021: 9.

57.

Qiu

. Blended teaching mode of higher vocational english based on mooc plus spoc. Wireless Communications & Mobile Computing. 2022; 2022: 9.

58.

Garg

, Algorithms for possibility linguistic single-valued neutrosophic decision-making based on copras and aggregation operators with new information measures. Measurement. 2019; 138: 278-90.

59.

Ul Haq

Saeed

Mateen

Siddiqui

Naqvi

, et al. Sustainable material selection with crisp and ambiguous data using single-valued neutrosophic-merec-marcos framework. Applied Soft Computing. 2022; 128: 109546.

60.

Rani

Mishra

Krishankumar

Ravichandran

Kar

. Multi-criteria food waste treatment method selection using single-valued neutrosophic-critic-multimoora framework. Applied Soft Computing. 2021; 111: 107657.

61.

Hezam

Mishra

Rani

Saha

Smarandache

Pamucar

. An integrated decision support framework using single-valued neutrosophic-maswip-copras for sustainability assessment of bioenergy production technologies. Expert Systems with Applications. 2023; 211: 118674.

62.

Salabun

Urbaniak

, editors. A new coefficient of rankings similarity in decision-making problems. 20th Annual International Conference on Computational Science (ICCS); 2020 Jun 03–05; Amsterdam, NETHERLANDS. CHAM: Springer International Publishing Ag; 2020.

63.

Sałabun

Wa̧tróbski

Shekhovtsov

. Are mcda methods benchmarkable? A comparative study of topsis, vikor, copras, and promethee ii methods. Symmetry. 2020; 12(9): 1549.

64.

Gou

Xiao

Huang

Deng

. Probabilistic double hierarchy linguistic alternative queuing method for real economy development evaluation under the perspective of economic financialization. Economic Research-Ekonomska Istrazivanja. 2021; 34(1): 3225-44.

65.

Gou

Liao

Herrera

. Probabilistic double hierarchy linguistic term set and its use in designing an improved vikor method: The application in smart healthcare. Journal of the Operational Research Society. 2021; 72(12): 2611-30.

66.

Lin

Zhai

Yao

. Multi-attribute group decision-making under probabilistic uncertain linguistic environment. Journal of the Operational Research Society. 2018; 69(2): 157-70.

67.

Xie

Ren

Wang

. The consensus of probabilistic uncertain linguistic preference relations and the application on the virtual reality industry. Knowledge-Based Systems. 2018; 162: 14-28.

68.

Bashir

Ali

Rashid

. Consensus-based robust decision making methods under a novel study of probabilistic uncertain linguistic information and their application in forex investment. Artificial Intelligence Review. 2021; 54(3): 2091-132.

69.

Jana

Muhiuddin

Pal

. Some dombi aggregation of q-rung orthopair fuzzy numbers in multiple-attribute decision making. International Journal of Intelligent Systems. 2019; 34(12): 3220-40.

70.

Krishankumar

Ravichandran

Kar

Cavallaro

Zavadskas

Mardani

. Scientific decision framework for evaluation of renewable energy sources under q-rung orthopair fuzzy set with partially known weight information. Sustainability. 2019; 11(15): 21.

Multi-attribute decision-making methods based on Aczel-Alsina power aggregation operators for English blended teaching quality evaluation under single-valued neutrosophic sets

Abstract

Keywords

1. Introduction

2. Preliminaries

5.1 An decision example

Table 1 SVNN matrix

Table 6 Different parameters for SVNNAAPWA

Table 8 Results for different methods

Footnotes

Acknowledgments

References

Table 1
SVNN matrix

Table 6
Different parameters for SVNNAAPWA

Table 8
Results for different methods