MADM framework based on interval-valued neutrosophic aczel-alsina power average operator and application to translation teaching quality evaluation of college English

Abstract

The huge demand for translation talents has prompted the universities to focus on improving the quality of translation teaching. As a necessary means to improve the quality of teaching, translation teaching quality evaluation plays a role in guiding, regulating and managing translation teaching, which is an important part of translation teaching. The translation teaching quality evaluation of college English is viewed as the multiple attribute decision making (MADM). The interval-valued neutrosophic sets (IVNSs) are used as an effective tool for characterizing uncertain information during the translation teaching quality evaluation of college English. In this paper, interval-valued neutrosophic number (IVNN) Aczel-Alsina power average (IVNNAAPA) operator is proposed based on the classical Aczel-Alsina operator and power average (PA) operator. Then, IVNNAAPA operator is employed to cope with MADM problem. Finally, the numerical decision example for translation teaching quality evaluation of college English is employed to illustrate the produced method and some comparative models are used to prove the effectiveness of the IVNNAAPA method.

Keywords

MADM interval-valued neutrosophic sets (IVNSs)Aczel-Alsina operations translation teaching quality evaluation

1. Introduction

Translation is a highly practical subject, and translation teaching should be closely integrated with extra-curricular practical activities. Extracurricular learning and practice are the extension and expansion of classroom teaching and an important way to cultivate and develop students’ translation ability. They should be carried out purposefully, planned and organized under the guidance of teachers [1, 2, 3, 4, 5]. The extra-curricular learning and practice activities should be based on the contents of classroom teaching and supplement the contents of classroom teaching, stimulate students’ learning interest and cultivate their studying ability, comprehensive language ability, organizational ability, communicative ability, thinking ability and creative ability [6, 7, 8, 9, 10, 11]. Practical teaching mainly includes the following forms: (1) Practice teaching: Students practice oral and written translation or simulation practice in a specific teaching environment or laboratory according to the teacher’s requirements. (2) Professional internship: Students complete teaching activities related to their courses or majors according to the requirements, to get familiar with, experience and understand the meaning of their majors, to master the basic skills of their majors, and to cultivate hands-on ability and team spirit. Professional internship includes cognitive internship and job internship, which can be completed in off-campus internship bases or other internship sites. (3) Academic activities: Students participate in various extracurricular academic activities on their own or under the guidance of teachers, in the form of attending academic lectures, seminars, conferences, research groups, thesis defense, participating in projects, editing publications, participating in disciplinary competitions, etc. (4) Social Practice: Students participate in various social practice activities to gradually understand society, learn about the country, grow in talent, exercise perseverance, and improve social adaptability. College English translation teaching focuses on cultivating students’ pragmatic abilities and mastery of grammar knowledge points, thereby strengthening their core literacy and enabling them to have a deeper understanding of English knowledge and English culture. It should be noted that college English translation is also related to students’ cognitive ability and learning experience [12, 13]. Therefore, while cultivating and strengthening students’ English translation ability, it is also necessary to cultivate students’ self-learning ability in the English subject. Strengthening English translation teaching is an important component that cannot be ignored in college English subject education. In fact, the cultivation of students’ English translation ability should be based on a certain foundation of English, and in the process of strengthening students’ English translation ability, attention should be paid to the synchronous cultivation of basic grammar knowledge, reading ability, and writing ability, promoting the improvement of students’ English comprehensive ability and literacy [14, 15]. Only in this way can students have a deeper understanding of the deep truth contained in English sentences, so that the translated meaning is more relevant and accurate. In college English translation classes, teachers should deeply consider the current situation of uneven English foundation among vocational college students, and provide step-by-step teaching guidance based on the English learning situation of students in this class, such as definitions, summaries, abstracts, bidirectional oral translation, and making written translations [16, 17]. In college English translation teaching, it is necessary to pay attention to cultivating students’ translation awareness, so that students can first understand the characteristics of translation skills, so that translation teaching can be more in-depth. Finally, vocational college students should pay attention to practice. While improving their English translation skills, high-quality and efficient translation teaching can also promote vocational college students’ understanding of the differences between English culture and Chinese culture. This not only has a certain positive effect on cross language communication, but also has a certain positive significance for the dissemination of excellent traditional Chinese culture [18, 19].

Decision-making is a basic human activity. With the help of certain scientific methods, decision-making experts sort out several alternatives and choose the best one to achieve people’s expected goals [20, 21, 22, 23, 24, 25]. In recent years, with the rapid development of science and technology, the decision-making environment has become more complex, and people are full of uncertainty and risk [26, 27, 28, 29, 30, 31, 32]. Aczél and Alsina [33] structured some new operations named as Aczel-Alsina t-norm and t-conorm operations. Yong et al. [34] and Ashraf et al. [35] structured the Aczel-Alsina decision operations to SVNNs and produced the Aczel-Alsina fused operators of SVNNs for MADM. Mahmood, Rehman [36] produced the Aczel-Alsina operators based on bipolar complex fuzzy information in MADM. Ali, Ullah [37] produced the MADM based on intuitionistic fuzzy soft sets and Aczel-Alsina operators. The problems of translation teaching quality evaluation of college English is a classical MADM [38, 39, 40, 41, 42, 43, 44]. The interval-valued neutrosophic sets (IVNSs) [45] are used as an effective tool for characterizing uncertain information during the translation teaching quality evaluation of college English. In this paper, interval-valued neutrosophic number (IVNN) Aczel-Alsina power average (IVNNAAPA) operator is proposed based on the classical Aczel-Alsina operator [46] and power average (PA) operator [47]. Then, IVNNAAPA operator is employed to cope with MADM problem. Finally, the numerical decision example for translation teaching quality evaluation of college English is employed to illustrate the produced method and some comparative models are used to prove the effectiveness of the IVNNAAPA method. The main aim of this defined paper is to expand the Aczel-Alsina operations [33] to cope with MADM under IVNSs. The main study motivations are listed: (1) the Aczel-Alsina operations are extended to IVNSs; (2) the IVNN Aczel-Alsina power average (IVNNAAPA) operator is proposed; (3) the IVNNAAPA operator is designed for MADM; (4) a case study about translation teaching quality decision evaluation of college English is given to show the IVNNAAPA method; (5) some comparative models are used to show the effectiveness of IVNNAAPA method.

The remainder sections of this paper are set out. Section 2 lists the IVNSs. In Sect. 3, the IVNN Aczel-Alsina power average (IVNNAAPA) operator is proposed. In Sect. 4, the IVNNAAPA operator is built for MADM. In Sect. 5, a case study for translation teaching quality decision evaluation of college English is listed and some comparative decision methods are done. The defined decision study ends in Sect. 6.

2. Preliminaries

Wang et al. [45] produced the IVNSs.

Definition 1 [48]. The IVNSs $A$ in $X$ is:

$\displaystyle\tilde{A}=\{{({x,TT_{\tilde{A}}(x),II_{\tilde{A}}(x),FF_{\tilde{A% }}(x)})|{x\in X}}\}$ (1)

with given truth-membership $TT_{\tilde{A}}(x)$ , indeterminacy-membership $II_{\tilde{A}}(x)$ and falsity-membership $FF_{\tilde{A}}(x)$ , $TT_{\tilde{A}}(x)$ , $II_{\tilde{A}}(x)$ , $FF_{\tilde{A}}(x)\in[{0,1}]$ , $0\leqslant\sup TT_{\tilde{A}}(x)+\sup II_{\tilde{A}}(x)+\sup FF_{\tilde{A}}(x)\leqslant 3$ .

The IVNN is expressed as $\tilde{A}=({TT_{\tilde{A}},II_{\tilde{A}},FF_{\tilde{A}}})=({[{TL_{\tilde{A}},% TR_{\tilde{A}}}],[{IL_{\tilde{A}},IR_{\tilde{A}}}],[{FL_{\tilde{A}},FR_{\tilde% {A}}}]})$ , where $TT_{\tilde{A}}\subseteq[{0,1}],II_{\tilde{A}}\subseteq[{0,1}],FF_{\tilde{A}}% \subseteq[{0,1}]$ , $0\leqslant TR_{\tilde{A}}+IR_{\tilde{A}}+FR_{\tilde{A}}\leqslant 3$ .

Definition 2 [49]. Let $\tilde{A}=({[{TL_{\tilde{A}},TR_{\tilde{A}}}],[{IL_{\tilde{A}},IR_{\tilde{A}}}% ],[{FL_{\tilde{A}},FR_{\tilde{A}}}]})$ , the score value is:

$\displaystyle SV(\tilde{A})=\frac{({2+TL_{\tilde{A}}-IL_{\tilde{A}}-FL_{\tilde% {A}}})+({2+TR_{\tilde{A}}-IR_{\tilde{A}}-FR_{\tilde{A}}})}{6},SV(\tilde{A})\in% [{0,1}].$ (2)

Definition 3 [49]. Let $\tilde{A}=({[{TL_{\tilde{A}},TR_{\tilde{A}}}],[{IL_{\tilde{A}},IR_{\tilde{A}}}% ],[{FL_{\tilde{A}},FR_{\tilde{A}}}]})$ , the accuracy value is:

$\displaystyle AV(\tilde{A})=\frac{({TL_{\tilde{A}}+TR_{\tilde{A}}})-({FL_{% \tilde{A}}+FR_{\tilde{A}}})}{2},AV(\tilde{A})\in[{-1,1}].$ (3)

Huang et al. [49] defined the order relation for IVNNs.

Definition 4 [49]. Let $\tilde{A}=({[{TL_{\tilde{A}},TR_{\tilde{A}}}],[{IL_{\tilde{A}},IR_{\tilde{A}}}% ],[{FL_{\tilde{A}},FR_{\tilde{A}}}]})$ and $\tilde{B}=([{TL_{\tilde{B}},TR_{\tilde{B}}}],[IL_{\tilde{B}},\linebreak IR_{% \tilde{B}}],[{FL_{\tilde{B}},FR_{\tilde{B}}}])$ , let

$\displaystyle SV(\tilde{A})=\frac{({2+TL_{\tilde{A}}-IL_{\tilde{A}}-FL_{\tilde% {A}}})+({2+TR_{\tilde{A}}-IR_{\tilde{A}}-FR_{\tilde{A}}})}{6}$

and

$\displaystyle SV(\tilde{B})=\frac{({2+TL_{\tilde{B}}-IL_{\tilde{B}}-FL_{\tilde% {B}}})+({2+TR_{\tilde{B}}-IR_{\tilde{B}}-FR_{\tilde{B}}})}{6},$

and let

$\displaystyle AV(\tilde{A})=\frac{({TL_{\tilde{A}}+TR_{\tilde{A}}})-({FL_{% \tilde{A}}+FR_{\tilde{A}}})}{2}$

and

$\displaystyle AV(\tilde{B})=\frac{({TL_{\tilde{B}}+TR_{\tilde{B}}})-({FL_{% \tilde{B}}+FR_{\tilde{B}}})}{2},$

then if $SV(\tilde{A})<SV(\tilde{B})$ , we have $\tilde{A}<\tilde{B}$ ; if $SV(\tilde{A})=SV(\tilde{B})$ , (1) if $HV(\tilde{A})=HV(\tilde{B})$ , we have $\tilde{A}=\tilde{B}$ ; (2) if $HV(\tilde{A})<HV(\tilde{B})$ , we have $\tilde{A}<\tilde{B}$ .

Aczél and Alsina [33] structured some new operations named as Aczel-Alsina t-norm and t-conorm operations, which have true advantage of changeability through adjusting a decision parameter. Yong et al. [34] and Ashraf et al. [35] structured the Aczel-Alsina operations to SVNNs. Similarly, the Aczel-Alsina operations for IVNNs are produced [50].

Definition 5 [50]. Let $\tilde{A}=({[{TL_{\tilde{A}},TR_{\tilde{A}}}],[{IL_{\tilde{A}},IR_{\tilde{A}}}% ],[{FL_{\tilde{A}},FR_{\tilde{A}}}]})$ and $\tilde{B}=([{TL_{\tilde{B}},TR_{\tilde{B}}}],[IL_{\tilde{B}},\linebreak IR_{% \tilde{B}}],[{FL_{\tilde{B}},FR_{\tilde{B}}}])$ , $\phi\geqslant 1$ , $\lambda>0$ , the Aczel-Alsina operations for IVNNs are produced:

$\displaystyle(1)\tilde{A}\oplus\tilde{B}\left({\begin{array}[]{l}\left[\begin{% array}[]{l}1-e^{-({({-\ln({1-TL_{\tilde{A}}})})^{\phi}+({-\ln({1-TL_{\tilde{B}% }})})^{\phi}})^{1/\phi}},\\ 1-e^{-({({-\ln({1-TR_{\tilde{A}}})})^{\phi}+({-\ln({1-TR_{\tilde{B}}})})^{\phi% }})^{1/\phi}}\end{array}\right],\\ \\ \left[{e^{-({({-\ln({IL_{\tilde{A}}})})^{\phi}+({-\ln({IL_{\tilde{B}}})})^{% \phi}})^{1/\phi}},e^{-({({-\ln({IR_{\tilde{A}}})})^{\theta}+({-\ln({IR_{\tilde% {B}}})})^{\phi}})^{1/\phi}}}\right],\\ \\ \left[{e^{-({({-\ln({FL_{\tilde{A}}})})^{\phi}+({-\ln({FL_{\tilde{B}}})})^{% \phi}})^{1/\phi}},e^{-({({-\ln({FR_{\tilde{A}}})})^{\phi}+({-\ln({FR_{\tilde{B% }}})})^{\phi}})^{1/\phi}}}\right]\\ \end{array}}\right)$ $\displaystyle(2)\tilde{A}\otimes\tilde{B}=\left({\begin{array}[]{l}\left[{e^{-% ({({-\ln({TL_{\tilde{A}}})})^{\phi}+({-\ln({TL_{\tilde{B}}})})^{\phi}})^{1/% \phi}},e^{-({({-\ln({TR_{\tilde{A}}})})^{\phi}+({-\ln({TR_{\tilde{B}}})})^{% \phi}})^{1/\phi}}}\right],\\ \\ \left[{\begin{array}[]{l}1-e^{-({({-\ln({1-IL_{\tilde{A}}})})^{\phi}+({-\ln({1% -IL_{\tilde{B}}})})^{\phi}})^{1/\phi}},\\ 1-e^{-({({-\ln({1-IR_{\tilde{A}}})})^{\phi}+({-\ln({1-IR_{\tilde{B}}})})^{\phi% }})^{1/\phi}}\end{array}}\right],\\ \\ \left[{\begin{array}[]{l}1-e^{-({({-\ln({1-FL_{\tilde{A}}})})^{\phi}+({-\ln({1% -FL_{\tilde{B}}})})^{\phi}})^{1/\phi}},\\ 1-e^{-({({-\ln({1-FR_{\tilde{A}}})})^{\phi}+({-\ln({1-FR_{\tilde{B}}})})^{\phi% }})^{1/\phi}}\end{array}}\right]\\ \end{array}}\right)$ $\displaystyle(3)\lambda A=\left({\begin{array}[]{l}\left[{1-e^{-({\lambda({-% \ln({1-TL_{\tilde{A}}})})^{\phi}})^{1/\phi}},1-e^{-({\lambda({-\ln({1-TR_{% \tilde{A}}})})^{\phi}})^{1/\phi}}}\right],\\ \\ \left[{e^{-({\lambda({-\ln({IL_{\tilde{A}}})})^{\phi}})^{1/\phi}},e^{-({% \lambda({-\ln({IR_{\tilde{A}}})})^{\phi}})^{1/\phi}}}\right],\\ \\ \left[{e^{-({\lambda({-\ln({FL_{\tilde{A}}})})^{\phi}})^{1/\phi}},e^{-({% \lambda({-\ln({FR_{\tilde{A}}})})^{\phi}})^{1/\phi}}}\right]\\ \end{array}}\right)$ $\displaystyle(4)(A)^{\lambda}=\left({\begin{array}[]{l}\left[{e^{-({\lambda({-% \ln({TL_{\tilde{A}}})})^{\phi}})^{1/\phi}},e^{-({\lambda({-\ln({TR_{\tilde{A}}% })})^{\phi}})^{1/\phi}}}\right],\\ \\ \left[{1-e^{-({\lambda({-\ln({1-IL_{\tilde{A}}})})^{\phi}})^{1/\phi}},1-e^{-({% \lambda({-\ln({1-IR_{\tilde{A}}})})^{\phi}})^{1/\phi}}}\right],\\ \\ \left[{1-e^{-({\lambda({-\ln({1-FL_{\tilde{A}}})})^{\phi}})^{1/\phi}},1-e^{-({% \lambda({-\ln({1-FR_{\tilde{A}}})})^{\phi}})^{1/\phi}}}\right]\\ \end{array}}\right)$

3. Some Aczel-Alsina weighted averaging operators with NNs

Karabacak [50] defined the IVNN Aczel-Alsina weighted averaging (IVNNAAWA) operator.

Definition 7 [50]. Let $SA_{i}=({[{TL_{i},TR_{i}}],[{IL_{i},IR_{i}}],[{FL_{i},FR_{i}}]})$ the IVNNs with their weight $sw_{i}=(sw_{1},sw_{2},\ldots,sw_{n})^{T}$ , $\sum\nolimits_{i=1}^{n}{sw_{i}}=1$ , $\phi\geqslant 1$ . If

$\displaystyle\text{IVNNAAWA}_{sw}({SA_{1},SA_{2},\ldots,SA_{n}})=\mathop{% \oplus}\limits_{i=1}^{n}sw_{i}SA_{i}$

(4) $\displaystyle=\left(\@setsize{\scriptsize}{9.5pt}{\viiipt}{\@viiipt}{\begin{% array}[]{l}\left[{1-e^{-\left({\sum\limits_{i=1}^{n}{sw_{i}}({-\ln({1-TL_{j}})% })^{\phi}}\right)^{1/\phi}},1-e^{-\left({\sum\limits_{i=1}^{n}{sw_{i}}({-\ln({% 1-TR_{j}})})^{\phi}}\right)^{1/\phi}}}\right],\\ \\ \left[{e^{-\left({\sum\limits_{i=1}^{n}{sw_{i}}({-\ln({IL_{j}})})^{\phi}}% \right)^{1/\phi}},e^{-\left({\sum\limits_{i=1}^{n}{sw_{i}}({-\ln({IR_{j}})})^{% \phi}}\right)^{1/\phi}}}\right],\\ \\ \left[{e^{-\left({\sum\limits_{i=1}^{n}{sw_{i}}({-\ln({FL_{j}})})^{\phi}}% \right)^{1/\phi}},e^{-\left({\sum\limits_{i=1}^{n}{sw_{i}}({-\ln({FR_{j}})})^{% \phi}}\right)^{1/\phi}}}\right]\\ \end{array}}\right)$

The IVNNAAWA has three properties [50].

Property 1. (idempotency). If $SA_{i}=SA=({[{TL,TR}],[{IL,IR}],[{FL,FR}]})$

$\displaystyle\text{IVNNAAWA}_{sw}({SA_{1},SA_{2},\ldots,SA_{n}})=SA$ (5)

Property 2. (Monotonicity). Let $SA_{i}=({[{TL_{A_{i}},TR_{A_{i}}}],[{IL_{A_{i}},IR_{A_{i}}}],[{FL_{A_{i}},FR_{% A_{i}}}]})$ , $SB_{i}=([TL_{B_{i}},\linebreak TR_{B_{i}}],[{IL_{B_{i}},IR_{B_{i}}}],[{FL_{B_{% i}},FR_{B_{i}}}])$ . If $TL_{A_{i}}\leqslant TL_{B_{i}},IL_{A_{i}}\geqslant IL_{B_{i}},FL_{A_{i}}% \geqslant FL_{B_{i}},TR_{A_{i}}\leqslant TR_{B_{i}},\linebreak IR_{A_{i}}% \geqslant IR_{B_{i}},FR_{A_{i}}\geqslant FR_{B_{i}}$ holds for all i, then

$\displaystyle\text{IVNNAAWA}_{sw}({SA_{1},SA_{2},\ldots,SA_{n}})\leqslant\text% {IVNNAAWA}_{sw}(SB_{1},SB_{2},\ldots,SB_{n})$ (6)

Property 3. (Boundedness). Let $SA_{i}=([{TL_{A_{i}},TR_{A_{i}}}],[{IL_{A_{i}},IR_{A_{i}}}],[{FL_{A_{i}},FR_{A% _{i}}}])$ . If $ZA^{+}=([\max_{i}(TL_{i}),\max_{i}(TR_{i})],[\min_{i}(IL_{i}),\min_{i}(IR_{i})% ],[\min_{i}(FL_{i}),\min_{i}(FR_{i})]),ZA^{+}=([\min_{i}(TL_{i}),\linebreak% \min_{i}(TR_{i})],[\max_{i}(IL_{i}),\max_{i}(IR_{i})],[\max_{i}(FL_{i}),\max_{% i}(FR_{i})])$ then

$\displaystyle ZA^{-}\leqslant\text{IVNNAAWA}_{sw}({SA_{1},SA_{2},\ldots,SA_{n}% })\leqslant ZA^{+}$ (7)

Then, the IVNN Aczel-Alsina power averaging (IVNNAAPA) operator is built on SVNNAAWA [50] operator and PA operator [47].

Definition 7. Let $SA_{i}=({[{TL_{i},TR_{i}}],[{IL_{i},IR_{i}}],[{FL_{i},FR_{i}}]})$ be a set of IVNNs, $\theta\geqslant 1$ . If

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

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

The Theorem 1 is obtained.

Theorem 1. Let $SA_{i}=({[{TL_{i},TR_{i}}],[{IL_{i},IR_{i}}],[{FL_{i},FR_{i}}]})$ the IVNNs with their weight $sw_{i}=(sw_{1},\linebreak sw_{2},\ldots,sw_{n})^{T}$ , $\sum\nolimits_{i=1}^{n}{zw_{i}}=1$ , $\phi\geqslant 1$ . If

$\displaystyle\text{IVNNAAPA}({SA_{1},SA_{2},\ldots,SA_{n}})=\mathop{\oplus}% \limits_{i=1}^{n}\frac{({1+T({SA_{i}})})}{\sum\limits_{i=1}^{n}{({1+T({SA_{i}}% )})}}SA_{i}$ (9) $\displaystyle=\left({\begin{array}[]{l}\left[{1-e^{-\left({\sum\limits_{i=1}^{% n}{\frac{({1+T({SA_{i}})})}{\sum\limits_{i=1}^{n}{({1+T({SA_{i}})})}}}({-\ln({% 1-TL_{j}})})^{\phi}}\right)^{1/\phi}},1-e^{-\left({\sum\limits_{i=1}^{n}{\frac% {({1+T({SA_{i}})})}{\sum\limits_{i=1}^{n}{({1+T({SA_{i}})})}}}({-\ln({1-TR_{j}% })})^{\phi}}\right)^{1/\phi}}}\right],\\ \left[{e^{-\left({\sum\limits_{i=1}^{n}{\frac{({1+T({SA_{i}})})}{\sum\limits_{% i=1}^{n}{({1+T({SA_{i}})})}}}({-\ln({IL_{j}})})^{\phi}}\right)^{1/\phi}},e^{-% \left({\sum\limits_{i=1}^{n}{\frac{({1+T({SA_{i}})})}{\sum\limits_{i=1}^{n}{({% 1+T({SA_{i}})})}}}({-\ln({IR_{j}})})^{\phi}}\right)^{1/\phi}}}\right],\\ \left[{e^{-\left({\sum\limits_{i=1}^{n}{\frac{({1+T({SA_{i}})})}{\sum\limits_{% i=1}^{n}{({1+T({SA_{i}})})}}}({-\ln({FL_{j}})})^{\phi}}\right)^{1/\phi}},e^{-% \left({\sum\limits_{i=1}^{n}{\frac{({1+T({SA_{i}})})}{\sum\limits_{i=1}^{n}{({% 1+T({SA_{i}})})}}}({-\ln({FR_{j}})})^{\phi}}\right)^{1/\phi}}}\right]\\ \end{array}}\right)$

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

(a) Let $i=2$ , then

$\displaystyle\text{IVNNAAPA}({SA_{1},SA_{2}})=\frac{({1+T({SA_{1}})})}{\sum% \limits_{i=1}^{2}{({1+T({SA_{i}})})}}SA_{1}^{\oplus}\frac{({1+T({SA_{2}})})}{% \sum\limits_{i=1}^{2}{({1+T({SA_{i}})})}}SA_{2}$ $\displaystyle\left(\@setsize{\scriptsize}{9.5pt}{\viiipt}{\@viiipt}{\begin{% array}[]{l}\left[{1-e^{-\left({\frac{({1+T({SA_{1}})})}{\sum\limits_{i=1}^{2}{% ({1+T({SA_{i}})})}}({-\ln({1-TL_{1}})})^{\phi}}\right)^{1/\phi}},1-e^{-\left({% \frac{({1+T({SA_{1}})})}{\sum\limits_{i=1}^{2}{({1+T({SA_{i}})})}}({-\ln({1-TR% _{1}})})^{\phi}}\right)^{1/\phi}}}\right],\\ \left[{e^{-\left({\frac{({1+T({SA_{1}})})}{\sum\limits_{i=1}^{2}{({1+T({SA_{i}% })})}}({-\ln({IL_{1}})})^{\phi}}\right)^{1/\phi}},e^{-\left({\frac{({1+T({SA_{% 1}})})}{\sum\limits_{i=1}^{2}{({1+T({SA_{i}})})}}({-\ln({IR_{1}})})^{\phi}}% \right)^{1/\phi}}}\right],\\ \left[{e^{-\left({\frac{({1+T({SA_{1}})})}{\sum\limits_{i=1}^{2}{({1+T({SA_{i}% })})}}({-\ln({FL_{1}})})^{\phi}}\right)^{1/\phi}},e^{-\left({\frac{({1+T({SA_{% 1}})})}{\sum\limits_{i=1}^{2}{({1+T({SA_{i}})})}}({-\ln({FR_{1}})})^{\phi}}% \right)^{1/\phi}}}\right]\\ \end{array}}\right)\oplus$ $\displaystyle\left(\@setsize{\scriptsize}{9.5pt}{\viiipt}{\@viiipt}{\begin{% array}[]{l}\left[{1-e^{-\left({\frac{({1+T({SA_{2}})})}{\sum\limits_{i=1}^{2}{% ({1+T({SA_{i}})})}}({-\ln({1-TL_{2}})})^{\phi}}\right)^{1/\phi}},1-e^{-\left({% \frac{({1+T({SA_{2}})})}{\sum\limits_{i=1}^{2}{({1+T({SA_{i}})})}}({-\ln({1-TR% _{2}})})^{\phi}}\right)^{1/\phi}}}\right],\\ \left[{e^{-\left({\frac{({1+T({SA_{2}})})}{\sum\limits_{i=1}^{2}{({1+T({SA_{i}% })})}}({-\ln({IL_{2}})})^{\phi}}\right)^{1/\phi}},e^{-\left({\frac{({1+T({SA_{% 2}})})}{\sum\limits_{i=1}^{2}{({1+T({SA_{i}})})}}({-\ln({IR_{2}})})^{\phi}}% \right)^{1/\phi}}}\right],\\ \left[{e^{-\left({\frac{({1+T({SA_{2}})})}{\sum\limits_{i=1}^{2}{({1+T({SA_{i}% })})}}({-\ln({FL_{2}})})^{\phi}}\right)^{1/\phi}},e^{-\left({\frac{({1+T({SA_{% 2}})})}{\sum\limits_{i=1}^{2}{({1+T({SA_{i}})})}}({-\ln({FR_{2}})})^{\phi}}% \right)^{1/\phi}}}\right]\\ \end{array}}\right)$ $\displaystyle=\left(\@setsize{\scriptsize}{9.5pt}{\viiipt}{\@viiipt}{\begin{% array}[]{l}\left[{1-e^{-\left({\sum\limits_{j=1}^{2}{\frac{({1+T({SA_{j}})})}{% \sum\limits_{i=1}^{2}{({1+T({SA_{i}})})}}}({-\ln({1-TL_{j}})})^{\phi}}\right)^% {1/\phi}},1-e^{-\left({\sum\limits_{i=1}^{2}{\frac{({1+T({SA_{j}})})}{\sum% \limits_{i=1}^{2}{({1+T({SA_{i}})})}}}({-\ln({1-TR_{j}})})^{\phi}}\right)^{1/% \phi}}}\right],\\ \left[{e^{-\left({\sum\limits_{i=1}^{2}{\frac{({1+T({SA_{j}})})}{\sum\limits_{% i=1}^{2}{({1+T({SA_{i}})})}}}({-\ln({IL_{j}})})^{\phi}}\right)^{1/\phi}},e^{-% \left({\sum\limits_{i=1}^{2}{\frac{({1+T({SA_{j}})})}{\sum\limits_{i=1}^{2}{({% 1+T({SA_{i}})})}}}({-\ln({IR_{j}})})^{\phi}}\right)^{1/\phi}}}\right],\\ \left[{e^{-\left({\sum\limits_{i=1}^{2}{\frac{({1+T({SA_{j}})})}{\sum\limits_{% i=1}^{2}{({1+T({SA_{i}})})}}}({-\ln({FL_{j}})})^{\phi}}\right)^{1/\phi}},e^{-% \left({\sum\limits_{i=1}^{2}{\frac{({1+T({SA_{j}})})}{\sum\limits_{i=1}^{2}{({% 1+T({SA_{i}})})}}}({-\ln({FR_{j}})})^{\phi}}\right)^{1/}\phi}}\right]\\ \end{array}}\right)$

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

$\displaystyle\text{IVNNAAPA}({SA_{1},SA_{2},\ldots,SA_{k}})$ $\displaystyle=\mathop{\oplus}\limits_{i=1}^{k}\frac{({1+T({SA_{i}})})}{\sum% \limits_{i=1}^{k}{({1+T({SA_{i}})})}}SA_{i}$ $\displaystyle=\left({\begin{array}[]{l}\left[{\begin{array}[]{l}1-e^{-\left({% \sum\limits_{i=1}^{k}{\frac{({1+T({SA_{i}})})}{\sum\limits_{i=1}^{k}{({1+T({SA% _{i}})})}}}({-\ln({1-TL_{j}})})^{\phi}}\right)^{1/\phi}},\\ 1-e^{-\left({\sum\limits_{i=1}^{k}{\frac{({1+T({SA_{i}})})}{\sum\limits_{i=1}^% {k}{({1+T({SA_{i}})})}}}({-\ln({1-TR_{j}})})^{\phi}}\right)^{1/\phi}}\end{% array}}\right],\\ \\ \left[{e^{-\left({\sum\limits_{i=1}^{k}{\frac{({1+T({SA_{i}})})}{\sum\limits_{% i=1}^{k}{({1+T({SA_{i}})})}}}({-\ln({IL_{j}})})^{\phi}}\right)^{1/\phi}},e^{-% \left({\sum\limits_{i=1}^{k}{\frac{({1+T({SA_{i}})})}{\sum\limits_{i=1}^{k}{({% 1+T({SA_{i}})})}}}({-\ln({IR_{j}})})^{\phi}}\right)^{1/\phi}}}\right],\\ \\ \left[{e^{-\left({\sum\limits_{i=1}^{k}{\frac{({1+T({SA_{i}})})}{\sum\limits_{% i=1}^{k}{({1+T({SA_{i}})})}}}({-\ln({FL_{j}})})^{\phi}}\right)^{1/\phi}},e^{-% \left({\sum\limits_{i=1}^{k}{\frac{({1+T({SA_{i}})})}{\sum\limits_{i=1}^{k}{({% 1+T({SA_{i}})})}}}({-\ln({FR_{j}})})^{\phi}}\right)^{1/\phi}}}\right]\\ \end{array}}\right)$

(c) Set $i=k+1$ . From Definition 5, we have

$\displaystyle\text{IVNNAAPA}({SA_{1},SA_{2},\ldots,SA_{k},SA_{k+1}})$ $\displaystyle=\mathop{\oplus}\limits_{i=1}^{k}\frac{({1+T({SA_{i}})})}{\sum% \limits_{i=1}^{k+1}{({1+T({SA_{i}})})}}SA_{i}\oplus\frac{({1+T({SA_{k+1}})})}{% \sum\limits_{i=1}^{k+1}{({1+T({SA_{i}})})}}SA_{k+1}$ $\displaystyle=\left({\begin{array}[]{l}\left[{\begin{array}[]{l}1-e^{-\left({% \sum\limits_{i=1}^{k}{\frac{({1+T({SA_{i}})})}{\sum\limits_{i=1}^{k+1}{({1+T({% SA_{i}})})}}}({-\ln({1-TL_{j}})})^{\phi}}\right)^{1/\phi}},\\ 1-e^{-\left({\sum\limits_{i=1}^{k}{\frac{({1+T({SA_{i}})})}{\sum\limits_{i=1}^% {k+1}{({1+T({SA_{i}})})}}}({-\ln({1-TR_{j}})})^{\phi}}\right)^{1/\phi}}\end{% array}}\right],\\ \\ \left[{e^{-\left({\sum\limits_{i=1}^{k}{\frac{({1+T({SA_{i}})})}{\sum\limits_{% i=1}^{k+1}{({1+T({SA_{i}})})}}}({-\ln({IL_{j}})})^{\phi}}\right)^{1/\phi}},e^{% -\left({\sum\limits_{i=1}^{k}{\frac{({1+T({SA_{i}})})}{\sum\limits_{i=1}^{k+1}% {({1+T({SA_{i}})})}}}({-\ln({IR_{j}})})^{\phi}}\right)^{1/\phi}}}\right],\\ \\ \left[{e^{-\left({\sum\limits_{i=1}^{k}{\frac{({1+T({SA_{i}})})}{\sum\limits_{% i=1}^{k+1}{({1+T({SA_{i}})})}}}({-\ln({FL_{j}})})^{\phi}}\right)^{1/\phi}},e^{% -\left({\sum\limits_{i=1}^{k}{\frac{({1+T({SA_{i}})})}{\sum\limits_{i=1}^{k+1}% {({1+T({SA_{i}})})}}}({-\ln({FR_{j}})})^{\phi}}\right)^{1/\phi}}}\right]\\ \end{array}}\right)\oplus$ $\displaystyle\left({\begin{array}[]{l}\left[{1-e^{-\left({\frac{({1+T({SA_{k+1% }})})}{\sum\limits_{i=1}^{k+1}{({1+T({SA_{i}})})}}({-\ln({1-TL_{k+1}})})^{\phi% }}\right)^{1/\phi}},1-e^{-\left({\frac{({1+T({SA_{k+1}})})}{\sum\limits_{i=1}^% {k+1}{({1+T({SA_{i}})})}}({-\ln({1-TR_{k+1}})})^{\phi}}\right)^{1/\phi}}}% \right],\\ \\ \left[{e^{-\left({\frac{({1+T({SA_{k+1}})})}{\sum\limits_{i=1}^{k+1}{({1+T({SA% _{i}})})}}({-\ln({IL_{k+1}})})^{\phi}}\right)^{1/\phi}},e^{-\left({\frac{({1+T% ({SA_{k+1}})})}{\sum\limits_{i=1}^{k+1}{({1+T({SA_{i}})})}}({-\ln({IR_{k+1}})}% )^{\phi}}\right)^{1/\phi}}}\right],\\ \\ \left[{e^{-\left({\frac{({1+T({SA_{k+1}})})}{\sum\limits_{i=1}^{k+1}{({1+T({SA% _{i}})})}}({-\ln({FL_{k+1}})})^{\phi}}\right)^{1/\phi}},e^{-\left({\frac{({1+T% ({SA_{k+1}})})}{\sum\limits_{i=1}^{k+1}{({1+T({SA_{i}})})}}({-\ln({FR_{k+1}})}% )^{\phi}}\right)^{1/\phi}}}\right]\\ \end{array}}\right)$ $\displaystyle=\left({\begin{array}[]{l}\left[{\begin{array}[]{l}1-e^{-\left({% \sum\limits_{i=1}^{k+1}{\frac{({1+T({SA_{i}})})}{\sum\limits_{i=1}^{k+1}{({1+T% ({SA_{i}})})}}}({-\ln({1-TL_{j}})})^{\phi}}\right)^{1/\phi}},\\ 1-e^{-\left({\sum\limits_{i=1}^{k+1}{\frac{({1+T({SA_{i}})})}{\sum\limits_{i=1% }^{k+1}{({1+T({SA_{i}})})}}}({-\ln({1-TR_{j}})})^{\phi}}\right)^{1/\phi}}\end{% array}}\right],\\ \\ \left[{e^{-\left({\sum\limits_{i=1}^{k+1}{\frac{({1+T({SA_{i}})})}{\sum\limits% _{i=1}^{k+1}{({1+T({SA_{i}})})}}}({-\ln({IL_{j}})})^{\phi}}\right)^{1/\phi}},e% ^{-\left({\sum\limits_{i=1}^{k+1}{\frac{({1+T({SA_{i}})})}{\sum\limits_{i=1}^{% k+1}{({1+T({SA_{i}})})}}}({-\ln({IR_{j}})})^{\phi}}\right)^{1/\phi}}}\right],% \\ \\ \left[{e^{-\left({\sum\limits_{i=1}^{k+1}{\frac{({1+T({SA_{i}})})}{\sum\limits% _{i=1}^{k+1}{({1+T({SA_{i}})})}}}({-\ln({FL_{j}})})^{\phi}}\right)^{1/\phi}},e% ^{-\left({\sum\limits_{i=1}^{k+1}{\frac{({1+T({SA_{i}})})}{\sum\limits_{i=1}^{% k+1}{({1+T({SA_{i}})})}}}({-\ln({FR_{j}})})^{\phi}}\right)^{1/\phi}}}\right]\\ \end{array}}\right)\oplus$

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

The IVNNAAPA has three good properties.

Property 1. (idempotency). If $SA_{i}=SA=({[{TL,TR}],[{IL,IR}],[{FL,FR}]})$

$\displaystyle\text{IVNNAAPA}({SA_{1},SA_{2},\ldots,SA_{n}})=SA$ (10)

Property 2. (Monotonicity). Let $SA_{i}=({[{TL_{A_{i}},TR_{A_{i}}}],[{IL_{A_{i}},IR_{A_{i}}}],[{FL_{A_{i}},FR_{% A_{i}}}]})$ , $SB_{i}=([TL_{B_{i}},\linebreak TR_{B_{i}}],[{IL_{B_{i}},IR_{B_{i}}}],[{FL_{B_{% i}},FR_{B_{i}}}])$ . If $TL_{A_{i}}\leqslant TL_{B_{i}},IL_{A_{i}}\geqslant IL_{B_{i}},FL_{A_{i}}% \geqslant FL_{B_{i}},TR_{A_{i}}\leqslant TR_{B_{i}},\linebreak IR_{A_{i}}% \geqslant IR_{B_{i}},FR_{A_{i}}\geqslant FR_{B_{i}}$ holds for all i, then

$\displaystyle\text{IVNNAAPA}({SA_{1},SA_{2},\ldots,SA_{n}})\leqslant\text{% IVNNAAPA}(SB_{1},SB_{2},\ldots,SB_{n})$ (11)

Property 3. (Boundedness). Let $SA_{i}=({[{TL_{A_{i}},TR_{A_{i}}}],[{IL_{A_{i}},IR_{A_{i}}}],[{FL_{A_{i}},FR_{% A_{i}}}]})$ . If $ZA^{+}=([{\max_{i}(TL_{i}),\max_{i}(TR_{i})}],[{\min_{i}(IL_{i}),\min_{i}(IR_{% i})}],[{\min_{i}(FL_{i}),\min_{i}(FR_{i})}]),ZA^{+}=([\min_{i}(TL_{i}),% \linebreak\min_{i}(TR_{i})],[{\max_{i}(IL_{i}),\max_{i}(IR_{i})}],[{\max_{i}(% FL_{i}),\max_{i}(FR_{i})}])$ then

$\displaystyle ZA^{-}\leqslant\text{IVNNAAPA}({SA_{1},SA_{2},\ldots,SA_{n}})% \leqslant ZA^{+}$ (12)
4. Method for MADM based on the IVNNAAPA

The IVNNAAPA is used to build for MADM. Let $CS=\{{CS_{1},CS_{2},\ldots,CS_{m}}\}$ be alternatives, and attributes set $GG=\{{GG_{1},GG_{2},\ldots,GG_{n}}\}$ with weight $z z$ , where $sw_{j}\in[{0,1}]$ , $\sum\limits_{j=1}^{n}{sw_{j}}=1$ . Suppose that values are assessed with IVNNs $QQ=({qq_{ij}})_{m\times n}=({[{TL_{ij},TR_{ij}}],[{IL_{ij},IR_{ij}}],[{FL_{ij}% ,FR_{ij}}]})_{m\times n}$ .

Then, method for MADM is built based on the IVNNAAPA. The given steps are produced.

Step 1. Build the IVNN matrix $QQ=({qq_{ij}})_{m\times n}=({[{TL_{ij},TR_{ij}}],[{IL_{ij},IR_{ij}}],[{FL_{ij}% ,FR_{ij}}]})_{m\times n}$ .

$\displaystyle QQ=[{qq{}_{ij}}]_{m\times n}=\left[{{\begin{array}[]{*{20}c}{qq_% {11}}&{qq_{12}}&\ldots&{qq_{1n}}\\ {qq_{21}}&{qq_{22}}&\ldots&{qq_{2n}}\\ \vdots&\vdots&\vdots&\vdots\\ {qq_{m1}}&{qq_{m2}}&\ldots&{qq_{mn}}\\ \end{array}}}\right]$ (13)

Step 2. Normalize $QQ=({qq_{ij}})_{m\times n}$ to $NQ=[{nq_{ij}}]_{m\times n}$ .

$\displaystyle nq_{ij}=({[{\textit{NTL}_{ij},\textit{NTR}_{ij}}],[{\textit{NIL}% _{ij},\textit{NIR}_{ij}}],[{\textit{NFL}_{ij},\textit{NFR}_{ij}}]})$ (14) $\displaystyle=\left\{{{\begin{array}[]{l}({[{TL_{ij}^{k},TR_{ij}^{k}}],[{IL_{% ij}^{k},IR_{ij}^{k}}],[{FL_{ij}^{k},FR_{ij}^{k}}]}),GG_{j}\textit{ is a % benefit criterion}\\ ({[{FL_{ij}^{k},FR_{ij}^{k}}],[{IL_{ij}^{k},IR_{ij}^{k}}],[{TL_{ij}^{k},TR_{ij% }^{k}}]}),GG_{j}\textit{ is a cost criterion}\\ \end{array}}}\right.$

Step 3. Calculate the supports:

$\displaystyle\textit{Sup}({nq_{ij},nq_{ik}})=1-d({nq_{ij},nq_{ik}}),j,k=1,2,% \ldots,n.$ (15)

Here, without loss of generality, we calculate $d({nq_{ij},nq_{ik}})$ with the normalized Hamming distance:

$\displaystyle d({nq_{ij},nq_{ik}})=\frac{1}{6}\left({\begin{array}[]{l}|{% \textit{NTL}_{ij}-\textit{NTL}_{ik}}|+|{\textit{NTR}_{ij}-\textit{NTR}_{ik}}|% \\ {}+|{\textit{NIL}_{ij}-\textit{NIL}_{ik}}|+|{\textit{NIR}_{ij}-\textit{NIR}_{% ik}}|\\ {}+|{\textit{NFL}_{ij}-\textit{NFL}_{ik}}|+|{\textit{NFR}_{ij}-\textit{NFR}_{% ik}}|\\ \end{array}}\right),j,k=1,2,\ldots,n.$ (16)

Step 4. Calculate the support $T({nq_{ij}})$ of the INN $nq_{ij}$ by the other INN $nq_{ik}({k=1,2,\ldots,n,k\neq j})$ :

$\displaystyle T({nq_{ij}})=\sum\limits_{\begin{subarray}{c}{k=1\atop k\neq j}% \end{subarray}}^{n}{\textit{Sup}({nq_{ij},nq_{ik}})}$ (17)

and calculate the weight $\xi_{ij}({j=1,2,\ldots,n})$ associated with the INNs $nq_{ij}(i=1,2,\ldots,m,j=1,2,\ldots,\linebreak n)$ :

$\displaystyle\xi_{ij}=\frac{({1+T({nq_{ij}})})}{\sum\limits_{j=1}^{n}{({1+T({% nq_{ij}})})}},i=1,2,\ldots,m,j=1,2,\ldots,n.$ (18)

where $\xi_{ij}\geqslant 0,i=1,2,\ldots,m,j=1,2,\ldots,n$ , and $\sum\limits_{j=1}^{n}{\xi_{ij}}=1,i=1,2,\ldots,m$ .

Step 5. According to $NQ=[nq_{ij}]_{m\times n}$ , the overall IVNNs $nq_{i}(i=1,2,\ldots,m)$ are produced through IVNNAAPA operator:

$\displaystyle nq_{i}=\text{IVNNAAPA}(nq_{i1},nq_{i2},\ldots,nq_{in})$ $\displaystyle=\mathop{\oplus}\limits_{j=1}^{n}\frac{({1+T({nq_{ij}})})}{\sum% \limits_{j=1}^{n}{({1+T({nq_{ij}})})}}nq_{ij}$ $\displaystyle=({[{\textit{NTL}_{i},\textit{NTR}_{i}}],[{\textit{NIL}_{i},% \textit{NIR}_{i}}],[{\textit{NFL}_{i},\textit{NFR}_{i}}]})$ $\displaystyle=\left({\begin{array}[]{l}\left[{\begin{array}[]{l}1-e^{-\left({% \sum\limits_{j=1}^{n}{\frac{({1+T({nq_{ij}})})}{\sum\limits_{j=1}^{n}{({1+T({% nq_{ij}})})}}}({-\ln({1-\textit{NTL}_{ij}})})^{\phi}}\right)^{1/\phi}},\\ 1-e^{-\left({\sum\limits_{i=1}^{n}{\frac{({1+T({nq_{ij}})})}{\sum\limits_{j=1}% ^{n}{({1+T({nq_{ij}})})}}}({-\ln({1-\textit{NTR}_{ij}})})^{\phi}}\right)^{1/% \phi}}\end{array}}\right],\\ \left[{e^{-\left({\sum\limits_{i=1}^{n}{\frac{({1+T({nq_{ij}})})}{\sum\limits_% {j=1}^{n}{({1+T({nq_{ij}})})}}}({-\ln({\textit{NIL}_{ij}})})^{\phi}}\right)^{1% /\phi}},e^{-\left({\sum\limits_{i=1}^{n}{\frac{({1+T({nq_{ij}})})}{\sum\limits% _{j=1}^{n}{({1+T({nq_{ij}})})}}}({-\ln({\textit{NIR}_{ij}})})^{\phi}}\right)^{% 1/\phi}}}\right],\\ \left[{e^{-\left({\sum\limits_{i=1}^{n}{\frac{({1+T({nq_{ij}})})}{\sum\limits_% {j=1}^{n}{({1+T({nq_{ij}})})}}}({-\ln({\textit{NFL}_{ij}})})^{\phi}}\right)^{1% /\phi}},e^{-\left({\sum\limits_{i=1}^{n}{\frac{({1+T({nq_{ij}})})}{\sum\limits% _{j=1}^{n}{({1+T({nq_{ij}})})}}}({-\ln({\textit{NFR}_{ij}})})^{\phi}}\right)^{% 1/\phi}}}\right]\\ \end{array}}\right)$ (19)

Step 6. Obtain the $SV(nq_{i}),AV(nq_{i})(i=1,2,\ldots,m)$ .

$\displaystyle SV({nq_{i}})=\frac{({2+\textit{NTL}_{i}-\textit{NIL}_{i}-\textit% {NFL}_{i}})+({2+\textit{NTR}_{i}-\textit{NIR}_{i}-\textit{NFR}_{i}})}{6}$ (20) $\displaystyle AV({({nq_{i}})})=\frac{({\textit{NTL}_{i}+\textit{NTR}_{i}})-({% \textit{NFL}_{i}+\textit{NFR}_{i}})}{2}$

Step 7. Rank the $CS_{i}({i=1,2,\ldots,m})$ through $SV(nq_{i}),AV(nq_{i})$ .

Step 8. End.

5. Numerical example and comparative analysis

5.1 Numerical example for translation teaching quality evaluation of college English

Testing and evaluation are important means to check the implementation of teaching requirements, assess the quality of teaching, understand students’ translation level, and promote teaching reform. Tests should help improve students’ translation ability and translator’s literacy as well as cultivate students’ thinking and analyzing ability. In translation teaching, the test and assessment of students are carried out throughout the whole learning process. The content of the test should include the language ability, translation skills, intercultural communication ability, as well as relevant economic, scientific and technological and cultural knowledge that students must master at each stage of study, as stipulated by the teaching requirements, and at the same time, special attention should be paid to testing students’ ability to analyze and solve the problems. In order to ensure the teaching level of the translation majors and to meet the stipulated requirements, the Teaching Requirements adopt a combination of formative and summative assessment for the evaluation of teaching. In the formative evaluation, various evaluation means and forms are used, including the teachers’ evaluation, college students’ self-evaluation, students’ mutual evaluation, students’ evaluation of teaching, evaluation of students by teaching departments, evaluation of students by internship units, etc., to follow the teaching process and provide feedback on teaching information; the summative evaluation mainly includes course examinations, level examinations and graduation thesis/graduation practice reports, etc. The undergraduate translation majors require students to take the national professional level 4 and 8 exams of the foreign language they are studying, as well as the corresponding oral and interpretation exams. Students are required to take the Level 3 Interpretation and/or Level 3 Translation exams in the China Accreditation Test for Translators and Interpreters (CATTI) sponsored by the Ministry of Human Resources and Social Security. The dissertation/graduation practice report is an important way to examine students’ comprehensive and creative abilities and assess their academic performance. The dissertation/report should be written in a foreign language and should be about 5000 words in length. If a translation practice report is used, students can do a foreign to Chinese translation (at least 2000 words) or a Chinese to foreign translation (at least 2000 Chinese characters), and then do a review of the translation on the basis of that, explaining the reasons for choosing the relevant original, the sources of the text, the problems found in the translation process, and how to solve them. The selected original text should be untranslated and unpublished, and the translation should be faithful and fluent. The paper/report should be reasonable, clear, informative, well-reasoned, well-written, in line with academic writing standards, and have some independent opinions. The translation teaching quality evaluation of college English is looked as MADM. Therefore, it is of great research significance to begin with the translation teaching quality evaluation of college English. There are five possible foreign colleges $CS_{i}(i=1,2,3,4,5)$ to choose. The experts group chooses four attributes to evaluate five possible foreign colleges:

⟀ CG₁ is the quality of multimedia hardware facilities. The quality of multimedia hardware facilities is a factor that directly affects the effectiveness of multimedia teaching in vocational colleges. Therefore, improving the quality of multimedia hardware facilities is very important for improving the effectiveness of multimedia teaching. Therefore, the relevant leaders of vocational colleges should strengthen the investment in multimedia hardware facilities, improve the quality level of hardware facilities implemented, and provide good guarantees for teachers to attend classes.

⟁ CG₂ is the management cost of multimedia teaching. In order to improve the level of multimedia hardware, in addition to increasing capital investment, it is also necessary to strengthen the inspection of existing multimedia equipment, timely identify equipment that needs to be eliminated, and promptly make it worse. In addition, because vocational colleges have a wide range of professional subjects and different requirements for multimedia equipment, enhancing the level of multimedia hardware also requires strengthening the construction of characteristic multimedia classrooms to meet the different needs of different majors.

⟂ CG₃ is the teaching ability for college teachers: The activity that a teacher engages in to achieve teaching objectives. The same content, different teaching methods and strategies lead to varying teaching quality. Teaching ability is the main influencing factor that reflects the quality of teaching and an important element in implementing a multimedia teaching system.

⟃ CG₄ is the learning ability for college students: Learning ability mainly refers to the ability to obtain accurate knowledge and information through fast, simple, and effective multimedia teaching methods, and convert it into one’s own knowledge. Students are always the main factor in participating in the entire multimedia teaching activity, and the content of multimedia teaching can affect students’ learning ability.

The management cost of multimedia teaching (CG₂) is the cost. The IVNNAAPA is built for translation teaching quality evaluation of college English.

Step 1. Build the $QQ=({qq_{ij}})_{5\times 4}$ in Table 1.

Table 1
IVNN information

	CG₁	CG₂
CS₁	([0.63, 0.71], [0.33, 0.42], [0.46, 0.57])	([0.53, 0.62], [0.43, 0.56], [0.43, 0.56])
CS₂	([0.81, 0.92], [0.23, 0.32], [0.75, 0.84])	([0.67, 0.79], [0.36, 0.42], [0.45, 0.63])
CS₃	([0.82, 0.91], [0.28, 0.42], [0.43, 0.56])	([0.92, 1.0], [0.14, 0.24], [0.53, 0.65])
CS₄	([0.65, 0.72], [0.17, 0.28], [0.53, 0.65])	([0.62, 0.75], [0.35, 0.46], [0.43, 0.56])
CS₅	([0.72, 0.91], [0.32, 0.45], [0.46, 0.57])	([0.86, 0.98], [0.56, 0.63], [0.57, 0.67])
	CG₃	CG₄
CS₁	([0.72, 0.83], [0.45, 0.57], [0.43, 0.51])	([0.73, 0.82], [0.36, 0.42], [0.64, 0.72])
CS₂	([0.63, 0.72], [0.56, 0.63], [0.46, 0.54])	([0.71, 0.84], [0.65, 0.73], [0.53, 0.67])
CS₃	([0.72, 0.83], [0.36, 0.42], [0.55, 0.62])	([0.82, 0.95], [0.26, 0.43], [0.62, 0.76])
CS₄	([0.75, 0.82], [0.44, 0.59], [0.62, 0.73])	([0.65, 0.92], [0.17, 0.25], [0.78, 0.89])
CS₅	([0.64, 0.72], [0.76, 0.82], [0.23, 0.34])	([0.73, 0.82], [0.35, 0.52], [0.45, 0.56])

Step 2. Normalize $QQ=[{qq_{ij}}]_{5\times 4}$ to $NQ=[{nq_{ij}}]_{5\times 4}$ (see Table 2).

Table 2

The normalized IVNN matrix

	CG₁	CG₂
CS₁	([0.63, 0.71], [0.33, 0.42], [0.46, 0.57])	([0.43, 0.56], [0.43, 0.56], [0.53, 0.62])
CS₂	([0.81, 0.92], [0.23, 0.32], [0.75, 0.84])	([0.45, 0.63], [0.36, 0.42], [0.67, 0.79])
CS₃	([0.82, 0.91], [0.28, 0.42], [0.43, 0.56])	([0.53, 0.65], [0.14, 0.24], [0.92, 1.0])
CS₄	([0.65, 0.72], [0.17, 0.28], [0.53, 0.65])	([0.43, 0.56], [0.35, 0.46], [0.62, 0.75])
CS₅	([0.72, 0.91], [0.32, 0.45], [0.46, 0.57])	([0.57, 0.67], [0.56, 0.63], [0.86, 0.98])
	CG₃	CG₄
CS₁	([0.72, 0.83], [0.45, 0.57], [0.43, 0.51])	([0.73, 0.82], [0.36, 0.42], [0.64, 0.72])
CS₂	([0.63, 0.72], [0.56, 0.63], [0.46, 0.54])	([0.71, 0.84], [0.65, 0.73], [0.53, 0.67])
CS₃	([0.72, 0.83], [0.36, 0.42], [0.55, 0.62])	([0.82, 0.95], [0.26, 0.43], [0.62, 0.76])
CS₄	([0.75, 0.82], [0.44, 0.59], [0.62, 0.73])	([0.65, 0.92], [0.17, 0.25], [0.78, 0.89])
CS₅	([0.64, 0.72], [0.76, 0.82], [0.23, 0.34])	([0.73, 0.82], [0.35, 0.52], [0.45, 0.56])

Step 3. Utilize Eqs (15)–(18) to calculate the weight information $\xi_{ij}({i=1,2,3,4,5,j=1,2,3,4})$ associated with the INNs $nq_{ij}({i=1,2,3,4,5,j=1,2,3,4})$ , which are contained in the matrix $\xi=({\xi_{ij}})_{5\times 4}$ .

$\displaystyle\xi=\left[{\begin{array}[]{cccc}0.2004&0.0967&0.3035&0.3994\\ 0.2003&0.1009&0.3025&0.3963\\ 0.1995&0.0994&0.3014&0.3998\\ 0.2001&0.1012&0.3064&0.3923\\ 0.1994&0.0987&0.3063&0.3956\\ \end{array}}\right]$

Step 4. Obtain the $nq_{i}(i=1,2,\ldots,5)$ by utilizing IVNNAAPA operator (Table 3).

Table 3

The $nq_{i}(i=1,2,\ldots,5)$ ( $\phi=2$ )

Alternatives	$nq_{i}(i=1,2,\ldots,5)$
CS₁	([0.5467, 0. 7609], [0.6214, 0.7098], [0.5121, 0.6657])
CS₂	([0.8214, 0.9099], [0.2665, 0.4213], [0.6121, 0.7456])
CS₃	([0.6087, 0.7876], [0.3657, 0.5121], [0.4432, 0.5453])
CS₄	([0.6436, 0.7098], [0.1986, 0.2453], [0.7658, 0.8098])
CS₅	([0.7121, 0.8099], [0.3546, 0.4879], [0.6213, 0.7325])

Step 5. Obtain the $SV(nq_{i})(i,1,2,\ldots,5)$ (Table 4).

Table 4

The $SV(nq_{i})(i,1,2,\ldots,5)$

Alternatives	$SV(nq_{i})(i,1,2,\ldots,5)$	Order
CS₁	0.4664	5
CS₂	0.6651	1
CS₃	0.5548	4
CS₄	0.5661	3
CS₅	0.5796	2

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

5.2 Comparative analysis

The IVNNAAPA operator is compared with interval-valued neutrosophic weighted average (IVNNWA) operator [51] and interval-valued neutrosophic weighted geometric (IVNNWG) operator [51] and IVNN-CODAS method [52].

Table 5
The comparative analysis

Models	Order	The best decision choice	The worst decision choice
IVNNWA [51]	$CS_{2}>CS_{4}>CS_{5}>CS_{1}>CS_{3}$	$CS_{2}$	$CS_{3}$
IVNNWG [51]	$CS_{2}>CS_{4}>CS_{5}>CS_{1}>CS_{3}$	$CS_{2}$	$CS_{3}$
IVNN-CODAS [52]	$CS_{2}>CS_{4}>CS_{5}>CS_{1}>CS_{3}$	$CS_{2}$	$CS_{3}$
IVNNAAPA technique	$CS_{2}>CS_{4}>CS_{5}>CS_{1}>CS_{3}$	$CS_{2}$	$CS_{3}$

From Table 5, It can be known that the order of these models is slightly different, however, these models have same best choice. This verifies the rationality and effectiveness of IVNNAAPA. The proposed IVNNAAPWA operator has the following advantages: (1) The built IVNNAAPWA operator considers decision information with relationship between aggregate parameters; (2) The proposed IVNNAAPWA can eliminate the influence of unfairly evaluated information on the decision outcome. The disadvantage of IVNNAAPWA operator is that the calculation of the proposed method is somewhat complicated due to the simultaneous consideration of the PA and HM operators.

6. Conclusion

In formulating the training objectives of undergraduate translation majors, we have fully considered the following pairs of relationships: (1) the relationship between undergraduate translation majors and the training of traditional foreign language talents; (2) the relationship between undergraduate translation majors and the training of translation master’s degree (MTI) talents; (3) the relationship between undergraduate translation majors and the professional qualification of translation; (4) the relationship between undergraduate translation majors and liberal education. (4) the relationship between undergraduate training of translation majors and general education. Taking into account the professional characteristics of translation majors, the following talents cultivation objectives are proposed: “Undergraduate translation majors in higher education aim to cultivate general translation professionals who are well-equipped with both moral and talent and have a broad international perspective. Graduates should be proficient in relevant working languages, have strong logical thinking ability, broad knowledge, high cross-cultural communication quality and good professional ethics, understand Chinese and foreign social culture, be familiar with basic translation theories, better master the professional skills of translation and interpretation, be proficient in using translation tools, understand the operation process of translation and related industries, and have strong independent thinking ability to work, communicate and coordinate. Graduates will have strong independent thinking ability, working ability and coordination ability. Graduates can be engaged in translation, interpretation or other cross-cultural communication with ordinary difficulties in foreign affairs, economy and trade, education, culture, science and technology, military and other fields. It can be seen from the statement of this goal that the training goal of translation major is to train (qualified) professional translators (interpreters or translators). The goal is to train professional interpreters and translators to meet the needs of global economic integration and national international competitiveness, and to meet the needs of national economic, cultural and social construction. The translation teaching quality evaluation of college English is viewed as the MADM. In this paper, interval-valued neutrosophic number (IVNN) Aczel-Alsina power average (IVNNAAPA) operator is proposed based on the classical Aczel-Alsina operator and power average (PA) operator. Then, IVNNAAPA operator is employed to cope with MADM problem. Finally, the numerical decision example for translation teaching quality evaluation of college English is employed to illustrate the produced method and some comparative models are used to prove the effectiveness of the IVNNAAPA method. The main study contributions are listed: (1) the Aczel-Alsina operations are extended to IVNSs; (2) the IVNN Aczel-Alsina power average (IVNNAAPA) operator is proposed; (3) the IVNNAAPA operator is designed for MADM; (4) a case study about translation teaching quality evaluation of college English is given to show the IVNNAAPA method; (5) some comparative models are used to show the effectiveness of the IVNNAAPA method.

This study may have some limitations that can be explored in future studies: (1) The MADM method proposed doesn’t analyze the irrational states of DMs, and applying prospect theory [53, 54, 55, 56] to MADM under IVNSs is a worthwhile research topic for translation teaching quality evaluation of college English; (2) In the future, the evaluation criteria of the translation teaching quality evaluation of college English should be considered from more perspectives.

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MADM framework based on interval-valued neutrosophic aczel-alsina power average operator and application to translation teaching quality evaluation of college English

Abstract

Keywords

1. Introduction

2. Preliminaries

5.1 Numerical example for translation teaching quality evaluation of college English

Table 1 IVNN information

Table 5 The comparative analysis

References

Table 1
IVNN information

Table 5
The comparative analysis