Exploring T-spherical fuzzy sets for enhanced evaluation of vocal music classroom teaching

Abstract

Vocal music is a relatively complex skill and skill course, which not only plays an important role in music education in universities, but also cultivates students’ musical level and artistic cultivation. In the process of teaching vocal music in universities, how to improve the quality of teaching has become an important task and goal for teachers. The vocal music classroom teaching quality evaluation is a classical multiple-attribute group decision-making (MAGDM) problem. In this paper, some novel Dice similarity measures (DSM) and some generalized Dice similarity measures (GDSM) are proposed under T-spherical fuzzy sets (T-SFSs). Then, the weighted generalized Dice similarity measures (WGDSM) is used to solve the MAGDM with T-SFSs. Finally, an illustrative example for vocal music classroom teaching quality evaluation is given to demonstrate the efficiency of the GDSMs.

Keywords

Multiple attribute group decision making Dice similarity measures (DSMs)generalized Dice similarity measures (GDSMs)T-spherical fuzzy sets vocal music classroom teaching quality evaluation

1. Introduction

Music education plays an important role and significance in the process of students’ personal development and comprehensive ability development. This requires improving the quality of music teaching in universities [1]. Adopting flexible and varied teaching methods to stimulate students’ interest in learning and encourage them to cooperate with teachers in their subsequent teaching work and activities with a more proactive attitude. At the same time, vocal music teaching is also a very complex skill and skill course [2, 3]. Teachers need to continuously guide students in the process of vocal teaching, so that they can firmly grasp the skills and techniques of vocal music, so that students can learn more efficiently, improve their music level, pronunciation and music literacy, and also improve their music appreciation level [4]. This allows students to develop comprehensively in multiple aspects and cultivate comprehensive and high-quality talents for society. In the process of conducting vocal music teaching in universities, teachers should adopt flexible and varied teaching modes to improve the quality of vocal music teaching and arouse students’ interest in learning. Students should cooperate with the teaching work and activities carried out by teachers with a more positive and upward attitude. This not only allows students to learn vocal knowledge faster and better in a relaxed and happy atmosphere, it can also make students have a deeper impression of vocal knowledge, thereby improving their musical level and vocal literacy [5, 6]. Adopting a flexible and varied teaching mode can not only stimulate students’ interest in learning and maintain their learning motivation, but also enable them to actively engage in vocal learning. Guided by their interests, students can learn vocal knowledge more efficiently, thereby mastering vocal knowledge and basic vocal skills faster and better. Interest is the best teacher [7, 8]. In the process of teaching music in universities, teachers should pay attention to teaching students’ vocal skills and techniques, adopt flexible and varied teaching models, arouse students’ interest in learning, and enable students to be independent [9]. Engage in the learning of vocal skills and techniques to better coordinate with teaching activities organized by teachers. For example, teachers can first explain basic vocal knowledge to students and personally demonstrate the pronunciation skills of vocal music during the vocal teaching process [10]. Then the teacher can select a relatively basic piece of music to help students practice the pronunciation skills and skills of vocal music. During the vocal teaching process, the teacher can also play some music masters’ singing works, allowing students to better feel the pronunciation skills and skills of vocal music in the music atmosphere. This can enable students to learn more efficiently [11]. The impression of vocal skills and techniques is more profound, which can not only arouse students’ interest in learning, but also enhance their musical literacy and vocal level in the music atmosphere, thereby enabling students to develop comprehensively and comprehensively in all aspects [12]. Cultivate students’ core musical literacy. With the continuous development and progress of the times, the internet is becoming increasingly close to people’s lives, and people’s lives are also increasingly inseparable from the internet [13]. Teachers can integrate the teaching mode of information networks into the process of vocal education in universities, thereby improving the teaching quality of vocal education. Integrating online information with vocal teaching to create a modern vocal teaching classroom [14]. This can not only arouse students’ interest in learning and expose them to new vocal teaching models, but also broaden their horizons, inspire their thinking, and make them closer to the Internet in the information and network era, thereby improving their music literacy and level, and strengthening their impression of basic vocal knowledge. Society is increasingly attaching importance to comprehensive and comprehensive training of students in all aspects, while also emphasizing their musical literacy [15]. This requires teachers to focus on cultivating students’ music literacy in the process of music teaching in universities, and the difficulty of cultivating students’ music literacy is to improve the teaching quality of vocal music teaching. Teachers should adopt flexible and varied teaching methods in their teaching to arouse students’ interest in learning and enable them to independently learn vocal knowledge [16, 17].

During the decision making process, most of them need to consider multiple criteria and multiple choices [18, 19, 20, 21, 22, 23], so multi-criteria decision analysis (MCDA) plays an important role within the decision making issues [24, 25, 26, 27]. There are many MCDA methods are used to cope with the decision issues, such as, Characteristic Objects Method (COMET) [28, 29], Stable Preference Ordering Towards Ideal Solution (SPOTIS) method [30] and Data vARIability Assessment?Technique for Order of Preference by Similarity to Ideal Solution?(the DARIA-TOPSIS method) [31]. Multiple attribute group decision making (MAGDM) originates from the Analytic Hierarchy Process (AHP) method, and due to various evaluation requirements in reality, the judgment matrix cannot meet strict consistency requirements, group decision-making methods have emerged [32, 33, 34, 35, 36]. In order to fully express uncertain information [37, 38, 39, 40, 41], Cuong [42] came up with the picture fuzzy sets (PFSs) to deal with the MADM issues. Gundogdu and Kahraman [43] came up with spherical fuzzy sets(T-SFSs). The vocal music classroom teaching quality evaluation is a classical MAGDM problems. The T-SFSs [43] is a useful tool to depict uncertain information during the vocal music classroom teaching quality evaluation. As a powerful tool for solving the MADM, the DSMs [44] has attracted lots of scholar’s attention and applied in many different fields [45, 46, 47, 48, 49, 50]. From above mentioned literatures, we can easily find that the DSMs owns some precious advantages. Furthermore, as far as author knows, the study about DSM, GDSM and WGDSM with T-SFSs are interesting research topics. Thus, it’s meaningful to work for DSM under T-SFSs. To do that, the main purposes of this paper are: (1) two forms of DSM are developed between T-SFSs, (2) the GDSM and WGDSM are developed between T-SFSs, (3) the WGDSM-based MAGDM models are developed between T-SFSs.

In order to do so, the remainder of this paper is organized as follows. In the next section, the T-SFSs is introduced. In Section 3, some GDSM and some weighted GDSM are proposed under T-SFSs. In Section 4, the WGDSM under T-SFSs are applied to MAGDM problems. In Section 5, an illustrative example for vocal music classroom teaching quality evaluation is given to demonstrate the efficiency of WGDSM under T-SFSs. Section 6 concludes the paper with some remarks.

2. Preliminaries

Mahmood et al. [51] formed the TSFSs.

Definition 1 [51]. Assume that $X$ be a fix set. The TSFSs is formed:

$\displaystyle DT=\{{({\theta,d\mu_{DT}(\theta),d\eta_{DT}(\theta),dv_{DT}(% \theta)})|{\theta\in X}}\}$ (1)

where $d\mu_{DT}(\theta),d\eta_{DT}(\theta),dv_{DT}(\theta)$ denotes the positive-membership, neutral-membership and negative-membership. $d\mu_{DT}(\theta),d\eta_{DT}(\theta),dv_{DT}(\theta)\in[{0,1}]$ and meets the condition:

$\displaystyle 0\leqslant({d\mu_{DT}(\theta)})^{q}+({d\eta_{DT}(\theta)})^{q}+(% {dv_{DT}(\theta)})^{q}\leqslant 1$ (2)

The degree of indeterminacy is formed as:

$\displaystyle d\pi_{DT}(\theta)=\sqrt[q]{1-({d\mu_{DT}(\theta)})^{q}-({d\eta_{% DT}(\theta)})^{q}-({dv_{DT}(\theta)})^{q}}$ (3)

For convenience, $DT=({d\mu,d\eta,d\nu})$ is named as a T-spherical fuzzy number (T-SFN).

The concept of the DSM is introduced [44].

Definition 2 [44]. Let $DA=({da_{1},da_{2},\ldots,da_{n}})$ and $DB=({db_{1},db_{2},\ldots,db_{n}})$ be two sets of positive real numbers. Then the DSM is formed:

$\displaystyle\textit{DSM}({DA,DB})=\frac{2DA\cdot DB}{\|{DA}\|_{2}^{2}+\|{DB}% \|_{2}^{2}}=\frac{2\sum\limits_{j=1}^{n}{da_{j}db_{j}}}{\sum\limits_{j=1}^{n}{% ({da_{j}})^{2}+\sum\limits_{j=1}^{n}{({db_{j}})^{2}}}}$ (4)

where $DA\cdot DB=\sum\limits_{j=1}^{n}{da_{j}db_{j}}$ is called the inner product and $\|{DA}\|_{2}=\sqrt{\sum\limits_{j=1}^{n}{({da_{j}})^{2}}}$ and $\|{DB}\|_{2}=\sqrt{\sum\limits_{j=1}^{n}{({db_{j}})^{2}}}$ are the Euclidean norms of $D A$ and $D B$ . Let $\textit{DSM}({DA,DB})=0$ when $da_{j}=db_{j}=0({j=1,2,\ldots,n})$ .

3. Some DSMs for T-SFSs

3.1 DSMs for T-SFSs

In this section, some DSMs and some weighted DSMs between T-SFSs are formed based on the classical DSMs [44].

Definition 3. Let $DT_{x}=({d\mu_{x_{j}},d\eta_{x_{j}},dv_{x_{j}}})$ and $DT_{y}=({d\mu_{y_{j}},d\eta_{y_{j}},d\nu_{y_{j}}})$ , $j=1,2,\ldots,n$ , be two sets of T-SFSs, then the DSM between $DT_{x}$ and $DT_{y}$ is formed:

$\displaystyle\textit{DSM}({DT_{x},DT_{y}})$

(5) $\displaystyle\frac{1}{n}\sum\limits_{j=1}^{n}{\frac{2({d\mu_{x_{j}}^{q}d\mu_{y% _{j}}^{q}+d\eta_{x_{j}}^{q}d\eta_{y_{j}}^{q}+dv_{x_{j}}^{q}dv_{y_{j}}^{q}+d\pi% _{x_{j}}^{q}d\pi_{y_{j}}^{q}})}{\left({\begin{array}[]{l}({({d\mu_{x_{j}}^{q}}% )^{2}+({f\eta_{x_{j}}^{q}})^{2}+({dv_{x_{j}}^{q}})^{2}+({d\pi_{x_{j}}^{q}})^{2% }})\\ {}+({({d\mu_{y_{j}}^{q}})^{2}+({d\eta_{x_{j}}^{q}})^{2}+({dv_{y_{j}}^{q}})^{2}% +({d\pi_{y_{j}}^{q}})^{2}})\\ \end{array}}\right)}}$

The DSM between $DT_{x}=({d\mu_{x_{j}},d\eta_{x_{j}},dv_{x_{j}}})$ and $DT_{y}=({d\mu_{y_{j}},d\eta_{y_{j}},d\nu_{y_{j}}})$ also meets the following properties:

(1)
$0\leqslant\textit{DSM}({DT_{x},DT_{y}})\leqslant 1$ ;
(2)
$\textit{DSM}({DT_{x},DT_{y}})=\textit{DSM}({DT_{y},DT_{x}})$ ;
(3)
$\textit{DSM}({DT_{x},DT_{y}})=1$ , if $DT_{x}=DT_{y}$ , there are $d\mu_{x_{j}}=d\mu_{y_{j}},d\eta_{x_{j}}=d\eta_{y_{j}},dv_{x_{j}}=dv_{y_{j}}$ , for $j=1,2,\ldots,n$ .

If we consider the weights values, the weighted DSM (WDSM) between $DT_{x}=({d\mu_{x_{j}},d\eta_{x_{j}},dv_{x_{j}}})$ and $DT_{y}=({d\mu_{y_{j}},d\eta_{y_{j}},d\nu_{y_{j}}})$ is formed:

$\displaystyle\textit{WDSM}({FT_{x},FT_{y}})=\sum\limits_{j=1}^{n}{d\omega_{j}% \frac{2({d\mu_{x_{j}}^{q}d\mu_{y_{j}}^{q}+d\eta_{x_{j}}^{q}d\eta_{y_{j}}^{q}+% dv_{x_{j}}^{q}dv_{y_{j}}^{q}+d\pi_{x_{j}}^{q}d\pi_{y_{j}}^{q}})}{\left({\begin% {array}[]{l}({({d\mu_{x_{j}}^{q}})^{2}+({f\eta_{x_{j}}^{q}})^{2}+({dv_{x_{j}}^% {q}})^{2}+({d\pi_{x_{j}}^{q}})^{2}})\\ {}+({({d\mu_{y_{j}}^{q}})^{2}+({d\eta_{x_{j}}^{q}})^{2}+({dv_{y_{j}}^{q}})^{2}% +({d\pi_{y_{j}}^{q}})^{2}})\\ \end{array}}\right)}}$ (6)

where $d\omega=({d\omega_{1},d\omega_{2},\ldots,d\omega_{n}})^{T}$ is the weight values of $DT_{x}=({d\mu_{x_{j}},d\eta_{x_{j}},dv_{x_{j}}})$ and $DT_{y}=(d\mu_{y_{j}},\linebreak d\eta_{y_{j}},d\nu_{y_{j}})$ , with $d\omega_{j}\in[{0,1}]$ , $\sum\limits_{j=1}^{n}{d\omega_{j}}=1$ . In particular, if $d\omega=({1\mathord{/{\vphantom{1n}}.\kern-1.2pt}n,1\mathord{/{\vphantom{1n}}.% \kern-1.2pt}n,\ldots,1\mathord{/{\vphantom{1n}}.\kern-1.2pt}n})^{T}$ , then the WDSM reduces to DSM. That’s to say, if we take $d\omega_{j}=\frac{1}{n},j=1,2\ldots,n$ , then there is $\textit{WDSM}({DT_{x},DT_{y}})=\textit{DSM}({DT_{x},DT_{y}})$ .
3.2 The generalized DSM for T-SFSs

In this section, the generalized DSM (GDSM) is developed for T-SFNs. As the generalization of the DSM for T-SFNs, the GDSM for T-SFNs are formed.

Definition 5. Le $DT_{x}=({d\mu_{x_{j}},d\eta_{x_{j}},dv_{x_{j}}})$ and $DT_{y}=({d\mu_{y_{j}},d\eta_{y_{j}},d\nu_{y_{j}}})j=1,2,\ldots,n$ , be two sets of T-SFNs, then the generalized DSM between $DT_{x}$ and $DT_{y}$ is proposed:

$\displaystyle\textit{GDSM}({DT_{x},DT_{y}})$

(7) $\displaystyle\frac{1}{n}\sum\limits_{j=1}^{n}{\frac{2({d\mu_{x_{j}}^{q}d\mu_{y% _{j}}^{q}+d\eta_{x_{j}}^{q}d\eta_{y_{j}}^{q}+dv_{x_{j}}^{q}dv_{y_{j}}^{q}+d\pi% _{x_{j}}^{q}d\pi_{y_{j}}^{q}})}{\left({\begin{array}[]{l}\alpha({({d\mu_{x_{j}% }^{q}})^{2}+({f\eta_{x_{j}}^{q}})^{2}+({dv_{x_{j}}^{q}})^{2}+({d\pi_{x_{j}}^{q% }})^{2}})\\ {}+({1-\alpha})({({d\mu_{y_{j}}^{q}})^{2}+({d\eta_{x_{j}}^{q}})^{2}+({dv_{y_{j% }}^{q}})^{2}+({d\pi_{y_{j}}^{q}})^{2}})\\ \end{array}}\right)}}$

where $\alpha$ is a positive parameter for $0\leqslant\alpha\leqslant 1$ .

Obviously, the GDSM also satisfies the following properties:

(1)
$0\leqslant\textit{GDSM}({DT_{x},DT_{y}})\leqslant 1$ ;
(2)
$\textit{GDSM}({DT_{x},DT_{y}})=\textit{GDSM}({DT_{y},DT_{x}})$ ;
(3)
$\textit{GDSM}({DT_{x},DT_{y}})=1$ , if $DT_{x}=DT_{y},j=1,2,\ldots,n$ .

Then, the GDSM includes some special cases by altering the parameter value $\alpha$ .

If $\alpha=0.5$ , the GDSM reduced to DSM:

$\displaystyle\textit{GDSM}({DT_{x},DT_{y}})=\frac{1}{n}\sum\limits_{j=1}^{n}{% \frac{2({d\mu_{x_{j}}^{q}d\mu_{y_{j}}^{q}+d\eta_{x_{j}}^{q}d\eta_{y_{j}}^{q}+% dv_{x_{j}}^{q}dv_{y_{j}}^{q}+d\pi_{x_{j}}^{q}d\pi_{y_{j}}^{q}})}{\left({\begin% {array}[]{l}\alpha({({d\mu_{x_{j}}^{q}})^{2}+({f\eta_{x_{j}}^{q}})^{2}+({dv_{x% _{j}}^{q}})^{2}+({d\pi_{x_{j}}^{q}})^{2}})\\ {}+({1-\alpha})({({d\mu_{y_{j}}^{q}})^{2}+({d\eta_{x_{j}}^{q}})^{2}+({dv_{y_{j% }}^{q}})^{2}+({d\pi_{y_{j}}^{q}})^{2}})\\ \end{array}}\right)}}=\frac{1}{n}\sum\limits_{j=1}^{n}{\frac{2({d\mu_{x_{j}}^{% q}d\mu_{y_{j}}^{q}+d\eta_{x_{j}}^{q}d\eta_{y_{j}}^{q}+dv_{x_{j}}^{q}dv_{y_{j}}% ^{q}+d\pi_{x_{j}}^{q}d\pi_{y_{j}}^{q}})}{\left({\begin{array}[]{l}O.5({({d\mu_% {x_{j}}^{q}})^{2}+({f\eta_{x_{j}}^{q}})^{2}+({dv_{x_{j}}^{q}})^{2}+({d\pi_{x_{% j}}^{q}})^{2}})\\ {}+({1-0.5})({({d\mu_{y_{j}}^{q}})^{2}+({d\eta_{x_{j}}^{q}})^{2}+({dv_{y_{j}}^% {q}})^{2}+({d\pi_{y_{j}}^{q}})^{2}})\\ \end{array}}\right)}}=\frac{1}{n}\sum\limits_{j=1}^{n}{\frac{2({d\mu_{x_{j}}^{% q}d\mu_{y_{j}}^{q}+d\eta_{x_{j}}^{q}d\eta_{y_{j}}^{q}+dv_{x_{j}}^{q}dv_{y_{j}}% ^{q}+d\pi_{x_{j}}^{q}d\pi_{y_{j}}^{q}})}{\left({\begin{array}[]{l}({({d\mu_{x_% {j}}^{q}})^{2}+({f\eta_{x_{j}}^{q}})^{2}+({dv_{x_{j}}^{q}})^{2}+({d\pi_{x_{j}}% ^{q}})^{2}})\\ {}+({({d\mu_{y_{j}}^{q}})^{2}+({d\eta_{x_{j}}^{q}})^{2}+({dv_{y_{j}}^{q}})^{2}% +({d\pi_{y_{j}}^{q}})^{2}})\\ \end{array}}\right)}}$ (8)

If $\alpha=0,1$ , the GDSM reduced to the following asymmetric similarity measures respectively:

$\displaystyle\textit{GDSM}({DT_{x},DT_{y}})=\frac{1}{n}\sum\limits_{j=1}^{n}{% \frac{2({d\mu_{x_{j}}^{q}d\mu_{y_{j}}^{q}+d\eta_{x_{j}}^{q}d\eta_{y_{j}}^{q}+% dv_{x_{j}}^{q}dv_{y_{j}}^{q}+d\pi_{x_{j}}^{q}d\pi_{y_{j}}^{q}})}{\left({\begin% {array}[]{l}\alpha({({d\mu_{x_{j}}^{q}})^{2}+({f\eta_{x_{j}}^{q}})^{2}+({dv_{x% _{j}}^{q}})^{2}+({d\pi_{x_{j}}^{q}})^{2}})\\ {}+({1-\alpha})({({d\mu_{y_{j}}^{q}})^{2}+({d\eta_{x_{j}}^{q}})^{2}+({dv_{y_{j% }}^{q}})^{2}+({d\pi_{y_{j}}^{q}})^{2}})\\ \end{array}}\right)}}=\frac{1}{n}\sum\limits_{j=1}^{n}{\frac{2({d\mu_{x_{j}}^{% q}d\mu_{y_{j}}^{q}+d\eta_{x_{j}}^{q}d\eta_{y_{j}}^{q}+dv_{x_{j}}^{q}dv_{y_{j}}% ^{q}+d\pi_{x_{j}}^{q}d\pi_{y_{j}}^{q}})}{\left({\begin{array}[]{l}0({({d\mu_{x% _{j}}^{q}})^{2}+({f\eta_{x_{j}}^{q}})^{2}+({dv_{x_{j}}^{q}})^{2}+({d\pi_{x_{j}% }^{q}})^{2}})\\ {}+({1-0})({({d\mu_{y_{j}}^{q}})^{2}+({d\eta_{x_{j}}^{q}})^{2}+({dv_{y_{j}}^{q% }})^{2}+({d\pi_{y_{j}}^{q}})^{2}})\\ \end{array}}\right)}}=\frac{1}{n}\sum\limits_{j=1}^{n}{\frac{2({d\mu_{x_{j}}^{% q}d\mu_{y_{j}}^{q}+d\eta_{x_{j}}^{q}d\eta_{y_{j}}^{q}+dv_{x_{j}}^{q}dv_{y_{j}}% ^{q}+d\pi_{x_{j}}^{q}d\pi_{y_{j}}^{q}})}{({d\mu_{y_{j}}^{q}})^{2}+({d\eta_{x_{% j}}^{q}})^{2}+({dv_{y_{j}}^{q}})^{2}+({d\pi_{y_{j}}^{q}})^{2}}},\quad\text{for% }\alpha=0.\textit{GDSM}({DT_{x},DT_{y}})=\frac{1}{n}\sum\limits_{j=1}^{n}{% \frac{2({d\mu_{x_{j}}^{q}d\mu_{y_{j}}^{q}+d\eta_{x_{j}}^{q}d\eta_{y_{j}}^{q}+% dv_{x_{j}}^{q}dv_{y_{j}}^{q}+d\pi_{x_{j}}^{q}d\pi_{y_{j}}^{q}})}{\left({\begin% {array}[]{l}\alpha({({d\mu_{x_{j}}^{q}})^{2}+({f\eta_{x_{j}}^{q}})^{2}+({dv_{x% _{j}}^{q}})^{2}+({d\pi_{x_{j}}^{q}})^{2}})\\ {}+({1-\alpha})({({d\mu_{y_{j}}^{q}})^{2}+({d\eta_{x_{j}}^{q}})^{2}+({dv_{y_{j% }}^{q}})^{2}+({d\pi_{y_{j}}^{q}})^{2}})\\ \end{array}}\right)}}\textit{GDSM}({DT_{x},DT_{y}})=\frac{1}{n}\sum\limits_{j=% 1}^{n}{\frac{2({d\mu_{x_{j}}^{q}d\mu_{y_{j}}^{q}+d\eta_{x_{j}}^{q}d\eta_{y_{j}% }^{q}+dv_{x_{j}}^{q}dv_{y_{j}}^{q}+d\pi{}_{x_{j}}^{q}d\pi_{y_{j}}^{q}})}{\left% ({\begin{array}[]{l}1({({d\mu_{x_{j}}^{q}})^{2}+({f\eta_{x_{j}}^{q}})^{2}+({dv% _{x_{j}}^{q}})^{2}+({d\pi_{x_{j}}^{q}})^{2}})\\ {}+({1-1})({({d\mu_{y_{j}}^{q}})^{2}+({d\eta_{x_{j}}^{q}})^{2}+({dv_{y_{j}}^{q% }})^{2}+({d\pi_{y_{j}}^{q}})^{2}})\\ \end{array}}\right)}}$ (9) $\displaystyle=\frac{1}{n}\sum\limits_{j=1}^{n}{\frac{2({d\mu_{x_{j}}^{q}d\mu_{% y_{j}}^{q}+d\eta_{x_{j}}^{q}d\eta_{y_{j}}^{q}+dv_{x_{j}}^{q}dv_{y_{j}}^{q}+d% \pi_{x_{j}}^{q}d\pi_{y_{j}}^{q}})}{({d\mu_{x_{j}}^{q}})^{2}+({f\eta_{x_{j}}^{q% }})^{2}+({dv_{x_{j}}^{q}})^{2}+({d\pi_{x_{j}}^{q}})^{2}}},\quad\text{for }% \alpha=1.$ (10)

From above analysis, it can be seen that the above asymmetric similarity measures are the extension of the relative projection measure of the T-SFSs.

In many situations, the weight should be taken into account. Thus, the weighted GDSM(WGDSM) under T-SFSs is proposed:

$\displaystyle\textit{WGDSM}({DT_{x},DT_{y}})=\sum\limits_{j=1}^{n}{d\omega_{j}% \frac{2({d\mu_{x_{j}}^{q}d\mu_{y_{j}}^{q}+d\eta_{x_{j}}^{q}d\eta_{y_{j}}^{q}+% dv_{x_{j}}^{q}dv_{y_{j}}^{q}+d\pi_{x_{j}}^{q}d\pi_{y_{j}}^{q}})}{\left({\begin% {array}[]{l}\alpha({({d\mu_{x_{j}}^{q}})^{2}+({f\eta_{x_{j}}^{q}})^{2}+({dv_{x% _{j}}^{q}})^{2}+({d\pi_{x_{j}}^{q}})^{2}})\\ {}+({1-\alpha})({({d\mu_{y_{j}}^{q}})^{2}+({d\eta_{x_{j}}^{q}})^{2}+({dv_{y_{j% }}^{q}})^{2}+({d\pi_{y_{j}}^{q}})^{2}})\\ \end{array}}\right)}}$ (11)

where $d\omega=({d\omega_{1},d\omega_{2},\ldots,d\omega_{n}})^{T}$ is the weights values of $DT_{x}=({d\mu_{x_{j}},d\eta_{x_{j}},dv_{x_{j}}})$ and $DT_{y}=({d\mu_{y_{j}},d\eta_{y_{j}},d\nu_{y_{j}}})$ , with $d\omega_{j}\in[{0,1}]$ , $\sum\limits_{j=1}^{n}{d\omega_{j}}=1$ . In particular, if $d\omega=({1\mathord{/{\vphantom{1n}}.\kern-1.2pt}n,1\mathord{/{\vphantom{1n}}.% \kern-1.2pt}n,\ldots,1\mathord{/{\vphantom{1n}}.\kern-1.2pt}n})^{T}$ , then the WDSM reduces to DSM. That’s to say, if we take $d\omega_{j}=\frac{1}{n},j=1,2\ldots,n$ , then there is $\textit{WGDSM}({DT_{x},DT_{y}})=\textit{GDSM}({DT_{x},DT_{y}})$ .

Then, the WGDSM includes some special cases by altering the parameter value $\alpha$ .

If $\alpha=0.5$ , the WGDSM reduced to WDSM:

$\displaystyle\textit{WGDSM}({DT_{x},DT_{y}})=\sum\limits_{j=1}^{n}{d\omega_{j}% \frac{2({d\mu_{x_{j}}^{q}d\mu_{y_{j}}^{q}+d\eta_{x_{j}}^{q}d\eta_{y_{j}}^{q}+% dv_{x_{j}}^{q}dv_{y_{j}}^{q}+d\pi_{x_{j}}^{q}d\pi_{y_{j}}^{q}})}{\left({\begin% {array}[]{l}\alpha({({d\mu_{x_{j}}^{q}})^{2}+({f\eta_{x_{j}}^{q}})^{2}+({dv_{x% _{j}}^{q}})^{2}+({d\pi_{x_{j}}^{q}})^{2}})\\ {}+({1-\alpha})({({d\mu_{y_{j}}^{q}})^{2}+({d\eta_{x_{j}}^{q}})^{2}+({dv_{y_{j% }}^{q}})^{2}+({d\pi_{y_{j}}^{q}})^{2}})\\ \end{array}}\right)}}=\sum\limits_{j=1}^{n}{d\omega_{j}\frac{2({d\mu_{x_{j}}^{% q}d\mu_{y_{j}}^{q}+d\eta_{x_{j}}^{q}d\eta_{y_{j}}^{q}+dv_{x_{j}}^{q}dv_{y_{j}}% ^{q}+d\pi_{x_{j}}^{q}d\pi_{y_{j}}^{q}})}{\left({\begin{array}[]{l}0.5({({d\mu_% {x_{j}}^{q}})^{2}+({f\eta_{x_{j}}^{q}})^{2}+({dv_{x_{j}}^{q}})^{2}+({d\pi_{x_{% j}}^{q}})^{2}})\\ {}+({1-0.5})({({d\mu_{y_{j}}^{q}})^{2}+({d\eta_{x_{j}}^{q}})^{2}+({dv_{y_{j}}^% {q}})^{2}+({d\pi_{y_{j}}^{q}})^{2}})\\ \end{array}}\right)}}=\sum\limits_{j=1}^{n}{d\omega_{j}\frac{2({d\mu_{x_{j}}^{% q}d\mu_{y_{j}}^{q}+d\eta_{x_{j}}^{q}d\eta_{y_{j}}^{q}+dv_{x_{j}}^{q}dv_{y_{j}}% ^{q}+d\pi_{x_{j}}^{q}d\pi_{y_{j}}^{q}})}{\left({\begin{array}[]{l}({({d\mu_{x_% {j}}^{q}})^{2}+({f\eta_{x_{j}}^{q}})^{2}+({dv_{x_{j}}^{q}})^{2}+({d\pi_{x_{j}}% ^{q}})^{2}})\\ {}+({({d\mu_{y_{j}}^{q}})^{2}+({d\eta_{x_{j}}^{q}})^{2}+({dv_{y_{j}}^{q}})^{2}% +({d\pi_{y_{j}}^{q}})^{2}})\\ \end{array}}\right)}}$ (12)

If $\alpha=0.1$ , the WGDSM reduced to the following asymmetric weighted similarity measures respectively:

$\displaystyle\textit{WGDSM}({DT_{x},DT_{y}})=\sum\limits_{j=1}^{n}{d\omega_{j}% \frac{2({d\mu_{x_{j}}^{q}d\mu_{y_{j}}^{q}+d\eta_{x_{j}}^{q}d\eta_{y_{j}}^{q}+% dv_{x_{j}}^{q}dv_{y_{j}}^{q}+d\pi_{x_{j}}^{q}d\pi_{y_{j}}^{q}})}{\left({\begin% {array}[]{l}\alpha({({d\mu_{x_{j}}^{q}})^{2}+({f\eta_{x_{j}}^{q}})^{2}+({dv_{x% _{j}}^{q}})^{2}+({d\pi_{x_{j}}^{q}})^{2}})\\ {}+({1-\alpha})({({d\mu_{y_{j}}^{q}})^{2}+({d\eta_{x_{j}}^{q}})^{2}+({dv_{y_{j% }}^{q}})^{2}+({d\pi_{y_{j}}^{q}})^{2}})\\ \end{array}}\right)}}=\sum\limits_{j=1}^{n}{d\omega_{j}\frac{2({d\mu_{x_{j}}^{% q}d\mu_{y_{j}}^{q}+d\eta_{x_{j}}^{q}d\eta_{y_{j}}^{q}+dv_{x_{j}}^{q}dv_{y_{j}}% ^{q}+d\pi_{x_{j}}^{q}d\pi_{y_{j}}^{q}})}{\left({\begin{array}[]{l}0({({d\mu_{x% _{j}}^{q}})^{2}+({f\eta_{x_{j}}^{q}})^{2}+({dv_{x_{j}}^{q}})^{2}+({d\pi_{x_{j}% }^{q}})^{2}})\\ {}+({1-0})({({d\mu_{y_{j}}^{q}})^{2}+({d\eta_{x_{j}}^{q}})^{2}+({dv_{y_{j}}^{q% }})^{2}+({d\pi_{y_{j}}^{q}})^{2}})\\ \end{array}}\right)}}=\sum\limits_{j=1}^{n}{d\omega_{j}\frac{2({d\mu_{x_{j}}^{% q}d\mu{}_{y_{j}}^{q}+d\eta_{x_{j}}^{q}d\eta_{y_{j}}^{q}+dv_{x_{j}}^{q}dv_{y_{j% }}^{q}+d\pi_{x_{j}}^{q}d\pi_{y_{j}}^{q}})}{({d\mu_{y_{j}}^{q}})^{2}+({d\eta_{x% _{j}}^{q}})^{2}+({dv_{y_{j}}^{q}})^{2}+({d\pi_{y_{j}}^{q}})^{2}}},\quad\text{% for }\alpha=0.\textit{WGDSM}({DT_{x},DT_{y}})=\sum\limits_{j=1}^{n}{d\omega_{j% }\frac{2({d\mu_{x_{j}}^{q}d\mu_{y_{j}}^{q}+d\eta_{x_{j}}^{q}d\eta_{y_{j}}^{q}+% dv_{x_{j}}^{q}dv_{y_{j}}^{q}+d\pi_{x_{j}}^{q}d\pi_{y_{j}}^{q}})}{\left({\begin% {array}[]{l}\alpha({({d\mu_{x_{j}}^{q}})^{2}+({f\eta_{x_{j}}^{q}})^{2}+({dv_{x% _{j}}^{q}})^{2}+({d\pi_{x_{j}}^{q}})^{2}})\\ {}+({1-\alpha})({({d\mu_{y_{j}}^{q}})^{2}+({d\eta_{x_{j}}^{q}})^{2}+({dv_{y_{j% }}^{q}})^{2}+({d\pi_{y_{j}}^{q}})^{2}})\\ \end{array}}\right)}}=\sum\limits_{j=1}^{n}{d\omega_{j}\frac{2({d\mu_{x_{j}}^{% q}d\mu_{y_{j}}^{q}+d\eta_{x_{j}}^{q}d\eta_{y_{j}}^{q}+dv_{x_{j}}^{q}dv_{y_{j}}% ^{q}+d\pi_{x_{j}}^{q}d\pi_{y_{j}}^{q}})}{\left({\begin{array}[]{l}1({({d\mu_{x% _{j}}^{q}})^{2}+({f\eta_{x_{j}}^{q}})^{2}+({dv_{x_{j}}^{q}})^{2}+({d\pi_{x_{j}% }^{q}})^{2}})\\ {}+({1-1})({({d\mu_{y_{j}}^{q}})^{2}+({d\eta_{x_{j}}^{q}})^{2}+({dv_{y_{j}}^{q% }})^{2}+({d\pi_{y_{j}}^{q}})^{2}})\\ \end{array}}\right)}}=\sum\limits_{j=1}^{n}{d\omega_{j}\frac{2({d\mu_{x_{j}}^{% q}d\mu_{y_{j}}^{q}+d\eta_{x_{j}}^{q}d\eta_{y_{j}}^{q}+dv_{x_{j}}^{q}dv_{y_{j}}% ^{q}+d\pi_{x_{j}}^{q}d\pi_{y_{j}}^{q}})}{({d\mu_{x_{j}}^{q}})^{2}+({f\eta_{x_{% j}}^{q}})^{2}+({dv_{x_{j}}^{q}})^{2}+({d\pi_{x_{j}}^{q}})^{2}}},\quad\text{for% }\alpha=1.$ (14)

From above analysis, it can be seen that the above asymmetric weighted similarity measures are the extension of the relative weighted projection measure of the T-SFNs.
4. WGDSM for MADM with T-SFNs

In this section, we shall extend the WGDSM for MAGDM with T-SFSs. Let $DY=\{DY_{1},DY_{2},\ldots,\linebreak DY_{m}\}$ be alternatives, and $DX=\{{DX_{1},DX_{2},\ldots,DX_{n}}\}$ be attributes, $d\omega=({d\omega_{1},d\omega_{2},\ldots,d\omega_{n}})^{T}$ is the weight of attribute $DX=\{{DX_{1},DX_{2},\ldots,DX_{n}}\}$ , where $d\omega_{j}>0$ , $\sum\limits_{j=1}^{n}{d\omega_{j}}=1$ . Let $DD=\{{DD_{1},DD_{2},\ldots,DD_{t}}\}$ be decision makers (DMs), $dw=({dw_{1},dw_{2},\ldots,dw_{n}})$ be the weight of DMs, with $dw_{k}>0$ , $\sum\limits_{k=1}^{t}{dw_{k}}=1$ . Suppose that $DR^{(k)}=({DR_{ij}^{(k)}})_{m\times n}=({d\mu_{ij}^{(k)},d\eta_{ij}^{(k)},dv_{% ij}^{(k)}})_{m\times n}$ is the T-spherical fuzzy decision matrix, $d\mu_{ij}^{(k)},d\eta_{ij}^{(k)},dv_{ij}^{(k)}\in[{0,1}]$ , $({d\mu_{ij}^{(k)}})^{q}+({d\mu_{ij}^{(k)}})^{q}+({dv_{ij}^{(k)}})^{q}\leqslant 1% ,q\geqslant 1$ , $i=1,2,\ldots,m$ , $j=1,2,\ldots,n$ , $k=1,2,\ldots,t$ .

Then, The WGDSM are used to solve the MAGDM with T-SFSs.

Step 1. Establish the group decision matrix $DR^{(k)}=({DR_{ij}^{(k)}})_{m\times n}=({d\mu_{ij}^{(k)},d\eta_{ij}^{(k)},dv_{% ij}^{(k)}})_{m\times n}$ , $d\mu_{ij}^{(k)},d\eta_{ij}^{(k)},dv_{ij}^{(k)}\in[{0,1}]$ , $({d\mu_{ij}^{(k)}})^{q}+({d\mu_{ij}^{(k)}})^{q}+({dv_{ij}^{(k)}})^{q}\leqslant 1% ,q\geqslant 1$ .

Step 2. Utilize the decision information given in matrix $DR^{(k)}=({DR_{ij}^{(k)}})_{m\times n}=(d\mu_{ij}^{(k)},d\eta_{ij}^{(k)},% \linebreak dv_{ij}^{(k)})_{m\times n}$ by using the T-SFWG operator [52]

$\displaystyle DR_{ij}=({d\mu_{ij},d\eta_{ij},dv_{ij}})=\text{T-SFWG}({DR_{ij}^% {(1)},DR_{ij}^{(2)},\ldots,DR_{ij}^{(t)}})=\mathop{\otimes}\limits_{k=1}^{t}({% DR_{ij}^{(k)}})^{dw_{k}}=\left({\begin{array}[]{l}\prod\limits_{k=1}^{t}{({d% \mu_{ij}^{(k)}+d\eta_{ij}^{(k)}})^{dw_{k}}-}\prod\limits_{k=1}^{t}{({d\eta_{ij% }^{(k)}})^{dw_{k}}},\\ \prod\limits_{k=1}^{t}{({d\eta_{ij}^{(k)}})^{dw_{k}}},\sqrt[q]{1-\prod\limits_% {k=1}^{t}{({1-({dv_{ij}^{(k)}})^{q}})^{dw_{k}}}}\\ \end{array}}\right)$ (15)

to aggregate all the decision matrices $DR^{(k)}=({DR_{ij}^{(k)}})_{m\times n}=({d\mu_{ij}^{(k)},d\eta_{ij}^{(k)},dv_{% ij}^{(k)}})_{m\times n}$ into a collective decision matrix $DR=({DR_{ij}})_{m\times n}$ .

Step 3. Defining the T-spherical fuzzy positive ideal solution (T-SFPIS) $DY^{+}$ as

$\displaystyle({d\mu^{+},d\eta^{+},d\nu^{+},d\pi^{+}})=({({d\mu_{1}^{+},d\eta_{% 1}^{+},d\nu_{1}^{+},d\pi_{1}^{+}}),({d\mu_{2}^{+},d\eta_{2}^{+},d\nu_{2}^{+},d% \pi_{2}^{+}}),\ldots,({d\mu_{4}^{+},d\eta_{4}^{+},d\nu_{4}^{+},d\pi_{4}^{+}})}% )=({({1,0,0,0}),({1,0,0,0}),\ldots,({1,0,0,0})})$ (16)

Step 4. Calculating the WGDSM between $DY_{i}({i=1,2,\ldots,m})$ and $DY^{+}$ as follows:

$\displaystyle\textit{WGDSM}({DY_{i},DY^{+}})=\sum\limits_{j=1}^{n}{d\omega_{j}% \frac{({d\mu_{ij}})^{q}({d\mu_{j}^{+}})^{q}+({d\eta_{ij}})^{q}({d\eta_{j}^{+}}% )^{q}+({dv_{ij}})^{q}({dv_{j}^{+}})^{q}+({d\pi_{ij}})^{q}({d\pi_{j}^{+}})^{q}}% {\left({\begin{array}[]{l}\alpha({({d\mu_{ij}^{q}})^{2}+({d\eta_{ij}^{q}})^{2}% +({dv_{ij}^{q}})^{2}+({d\pi_{ij}^{q}})^{2}})+({1-\alpha})\\ ({({({d\mu_{j}^{+}})^{q}})^{2}+({({d\eta_{j}^{+}})^{q}})^{2}+({({dv_{j}^{+}})^% {q}})^{2}+({({d\pi_{j}^{+}})^{q}})^{2}})\\ \end{array}}\right)}}$ (17)

Step 5. Rank all the alternatives $DY_{i}({i=1,2,\ldots,m})$ and select the best one(s) in accordance with the $\textit{WGDSM}({DY_{i},DY^{+}})({i=1,2,\ldots,m})$ . If any alternative has the highest $\textit{WGDSM}({DY_{i},DY^{+}})$ value, then, it is the most important alternative.

Step 5. End.

5. Numerical example and compare analysis

5.1 Numerical example

The purpose of vocal teaching is to cultivate professional vocal talents for society, so as to provide higher-level vocal works. Therefore, it is necessary to combine theory and practice to jointly carry out vocal teaching activities and comprehensively improve the level of vocal teaching. In vocal music teaching, vocal skills are the foundation of the entire teaching. If students master vocal skills and possess solid basic skills, they can lay a solid foundation for the learning of subsequent works and emotional expression. At the same time, teachers should also focus on cultivating students’ theoretical knowledge, strengthening their understanding of music theory and various music genres, and helping students better utilize theoretical knowledge for vocal learning. The teaching content of vocal theory is complex, and the learning process is

Table 1
Evaluation information by $DD_{1}$

	DX ${}_{1}$	DX ${}_{2}$	DX ${}_{3}$	DX ${}_{4}$
DY ${}_{1}$	(0.37, 0.95, 0.20)	(0.36, 0.42, 0.06)	(0.69, 0.66, 0.11)	(0.94, 0.42, 0.39)
DY ${}_{2}$	(0.82, 0.52, 0.21)	(0.58, 0.09, 0.44)	(0.08, 0.57, 0.48)	(0.38, 0.42, 0.11)
DY ${}_{3}$	(0.75, 0.62, 0.54)	(0.75, 0.24, 0.10)	(0.41, 0.16, 0.55)	(0.58, 0.10, 0.62)
DY ${}_{4}$	(0.41, 0.75, 0.13)	(0.35, 0.14, 0.04)	(0.93, 0.23, 0.12)	(0.34, 0.66, 0.05)
DY ${}_{5}$	(0.75, 0.66, 0.43)	(0.74, 0.24, 0.38)	(0.59, 0.54, 0.70)	(0.60, 0.35, 0.82)

Table 2

Evaluation information by $DD_{2}$

	DX ${}_{1}$	DX ${}_{2}$	DX ${}_{3}$	DX ${}_{4}$
DY ${}_{1}$	(0.68, 0.80, 0.48)	(0.22, 0.22, 0.14)	(0.57, 0.46, 0.66)	(0.19, 0.61, 0.82)
DY ${}_{2}$	(0.52, 0.38, 0.37)	(0.85, 0.39, 0.18)	(0.82, 0.26, 0.23)	(0.69, 0.31, 0.66)
DY ${}_{3}$	(0.61, 0.27, 0.52)	(0.04, 0.23, 0.52)	(0.20, 0.63, 0.77)	(0.31, 0.74, 0.38)
DY ${}_{4}$	(0.27, 0.35, 0.97)	(0.56, 0.60, 0.11)	(0.24, 0.55, 0.76)	(0.69, 0.14, 0.84)
DY ${}_{5}$	(0.32, 0.04, 0.31)	(0.73, 0.14, 0.55)	(0.43, 0.01, 0.41)	(0.28, 0.11, 0.51)

relatively dull and tedious, which can easily affect students’ interest in learning. In the process of theoretical learning, teachers can introduce a series of new teaching methods and concepts, comprehensively apply various teaching strategies such as multimedia teaching, simulation teaching, and scenario representation teaching, to enhance students’ interest in learning vocal theory knowledge, so that students can fully understand the role and value of theoretical knowledge, and achieve good teaching results. At the same time, increasing the class hours for students’ practical activities allows them to fully understand the charm of vocal music through hands-on practice, clarify the problems and shortcomings in their own learning process, and thus more efficiently improve and optimize. Teachers can encourage students to express themselves more in their daily learning. They can set up special segments in the classroom, requiring students to sing relevant works one by one in front of the podium, and have everyone point out the advantages and disadvantages of singing together, helping students better understand the connotation and significance of the works. In addition, teachers should also encourage students to participate in stage practice, by expressing their works to the audience and collecting feedback from the audience, clarifying the path for further improvement of the works, and creating favorable conditions for the continuous improvement of students’ comprehensive literacy. In summary, the development of the times has promoted the continuous improvement of vocal teaching level and also brought new requirements to vocal teaching activities. Colleges and universities need to continuously optimize their teaching teams, pay attention to the combination of theoretical and practical teaching, leverage the long-term effects of teaching, and optimize teaching content, in order to better adapt to the development of the times, keep up with the pace of the times, meet the needs of society for vocal talents, and promote the continuous improvement of vocal teaching quality. The vocal music classroom teaching quality evaluation is a classical MAGDM problems. In this section, a practical MAGDM problems is used to illustrate the application of the developed approaches. Suppose the university plans to vocal music classroom teaching quality evaluation. There are five potential music colleges $DY_{i}({i=1,2,\ldots,5})$ as candidates. The university employs some external professional experts to aid this decision-making. The project team selects four attributes to evaluate the five potential music colleges: (1) $DX_{1}$ represents the vocal music classroom teaching methods; (2) $DX_{2}$ represents the vocal music classroom teaching contents; (3) $DX_{3}$ represents the peer expert recognition for vocal music classroom teaching; (4) $DX_{4}$ represents the student feedback for vocal music classroom teaching. The five possible five potential music colleges $DY_{i}({i=1,2,\ldots,5})$ are to be evaluated using the T-SFNs by the three decision makers (whose weight $dw=({{0.3},0.4,0.3})^{T}$ ) under the above four attributes (whose weight $d\omega=({0.30,0.20,0.25,0.25})^{T}$ ), and construct, respectively, the decision matrices as listed in the following matrices $DR^{(k)}=({DR_{ij}^{(k)}})_{5\times 4}({k=1,2,3})$ in Tables 1–3.

Table 3

Evaluation information by $DD_{3}$

	DX ${}_{1}$	DX ${}_{2}$	DX ${}_{3}$	DX ${}_{4}$
DY ${}_{1}$	(0.56, 0.31, 0.58)	(0.12, 0.27, 0.69)	(0.76, 0.29, 0.67)	(0.62, 0.26, 0.88)
DY ${}_{2}$	(0.72, 0.13, 0.37)	(0.49, 0.43, 0.25)	(0.60, 0.59, 0.38)	(0.33, 0.29, 0.50)
DY ${}_{3}$	(0.07, 0.89, 0.12)	(0.72, 0.53, 0.20)	(0.31, 0.59, 0.74)	(0.08, 0.85, 0.42)
DY ${}_{4}$	(0.31, 0.30, 0.51)	(0.86, 0.04, 0.66)	(0.24, 0.49, 0.53)	(0.06, 0.32, 0.34)
DY ${}_{5}$	(0.05, 0.03, 0.49)	(0.77, 0.67, 0.24)	(0.34, 0.46, 0.57)	(0.89, 0.57, 0.20)

In the following, we shall utilize the built approach to obtain the most desirable music college:

Step 1. Establish the overall matrix $DR=({DR_{ij}})_{5\times 4}$ by Eq. (15) (Suppose $q=4$ ).

Table 4

The $DR=({DR_{ij}})_{5\times 4}$

	DX ${}_{1}$	DX ${}_{2}$
DY ${}_{1}$	(0.5887, 0.6347, 0.3846)	(0.2204, 0.2850, 0.4235)
DY ${}_{2}$	(0.6922, 0.2999, 0.2386)	(0.6836, 0.2627, 0.2357)
DY ${}_{3}$	(0.5367, 0.4955, 0.3799)	(0.3344, 0.2992, 0.3107)
DY ${}_{4}$	(0.3266, 0.4199, 0.8430)	(0.6510, 0.1772, 0.3941)
DY ${}_{5}$	(0.2651, 0.0834, 0.3148)	(0.7842, 0.2648, 0.3551)
	DX ${}_{3}$	DX ${}_{4}$
DY ${}_{1}$	(0.6754, 0.4448, 0.5195)	(0.5434, 0.4218, 0.7388)
DY ${}_{2}$	(0.5364, 0.4192, 0.2870)	(0.4766, 0.3319, 0.4615)
DY ${}_{3}$	(0.3543, 0.4061, 0.6448)	(0.4690, 0.4187, 0.4007)
DY ${}_{4}$	(0.4573, 0.4061, 0.5510)	(0.4042, 0.2861, 0.6308)
DY ${}_{5}$	(0.6124, 0.0863, 0.4912)	(0.4986, 0.2548, 0.5737)

Step 2. Establish the T-spherical fuzzy positive ideal solution (T-SFPIS) $DY^{+}$ as

$\displaystyle({d\mu^{+},d\eta^{+},d\nu^{+},d\pi^{+}})=({({1,0,0,0}),({1,0,0,0}% ),({1,0,0,0}),({1,0,0,0})})$

Step 3. According to Eq. (17) and different parameter values, the WGDSM between $DY_{i}({i=1,2,\ldots,5})$ and $DY^{+}$ can be obtained, which are shown in Tables 5 and 6.

Table 5

The WGDSM and ranking orders

	$({DY_{1},DY^{+}})$	$({DY_{2},DY^{+}})$	$({DY_{3},DY^{+}})$	$({DY_{4},DY^{+}})$	$({DY_{5},DY^{+}})$
$\alpha=0.0$	0.0732	0.0527	0.1486	0.0604	0.0541
$\alpha=0.1$	0.0811	0.0584	0.1646	0.0669	0.600
$\alpha=0.2$	0.0910	0.0655	0.1846	0.0750	0.0674
$\alpha=0.3$	0.1035	0.0745	0.2100	0.0853	0.0769
$\alpha=0.4$	0.1201	0.0865	0.2435	0.0988	0.0893
$\alpha=0.5$	0.1429	0.1031	0.2899	0.1175	0.1068
$\alpha=0.6$	0.1766	0.1275	0.3578	0.1450	0.1325
$\alpha=0.7$	0.2310	0.1670	0.4676	0.1893	0.1748
$\alpha=0.8$	0.3345	0.2426	0.6749	0.2727	0.2570
$\alpha=0.9$	0.6094	0.4470	1.2149	0.4918	0.4860
$\alpha=1.0$	5.0916	5.6259	11.1552	3.1096	5.9490

Table 6

The ranking orders

	Ranking orders
$\alpha=0.0$	DY ${}_{3}$ $>$ DY ${}_{1}$ $>$ DY ${}_{4}$ $>$ DY ${}_{2}$ $>$ DY ${}_{5}$
$\alpha=0.1$	DY ${}_{3}$ $>$ DY ${}_{1}$ $>$ DY ${}_{4}$ $>$ DY ${}_{2}$ $>$ DY ${}_{5}$
$\alpha=0.2$	DY ${}_{3}$ $>$ DY ${}_{1}$ $>$ DY ${}_{4}$ $>$ DY ${}_{2}$ $>$ DY ${}_{5}$
$\alpha=0.3$	DY ${}_{3}$ $>$ DY ${}_{1}$ $>$ DY ${}_{4}$ $>$ DY ${}_{2}$ $>$ DY ${}_{5}$
$\alpha=0.4$	DY ${}_{3}$ $>$ DY ${}_{1}$ $>$ DY ${}_{4}$ $>$ DY ${}_{2}$ $>$ DY ${}_{5}$
$\alpha=0.5$	DY ${}_{3}$ $>$ DY ${}_{1}$ $>$ DY ${}_{4}$ $>$ DY ${}_{2}$ $>$ DY ${}_{5}$
$\alpha=0.6$	DY ${}_{3}$ $>$ DY ${}_{1}$ $>$ DY ${}_{4}$ $>$ DY ${}_{2}$ $>$ DY ${}_{5}$
$\alpha=0.7$	DY ${}_{3}$ $>$ DY ${}_{1}$ $>$ DY ${}_{4}$ $>$ DY ${}_{2}$ $>$ DY ${}_{5}$
$\alpha=0.8$	DY ${}_{3}$ $>$ DY ${}_{1}$ $>$ DY ${}_{4}$ $>$ DY ${}_{2}$ $>$ DY ${}_{5}$
$\alpha=0.9$	DY ${}_{3}$ $>$ DY ${}_{1}$ $>$ DY ${}_{4}$ $>$ DY ${}_{2}$ $>$ DY ${}_{5}$
$\alpha=1.0$	DY ${}_{1}$ $>$ DY ${}_{3}$ $>$ DY ${}_{4}$ $>$ DY ${}_{5}$ $>$ DY ${}_{2}$

From the Tables 5 and 6, different ranking orders are shown through different $\alpha$ values and different DSM. Then the best music college should belong to $DY_{3}$ or $DY_{1}$ according to the maximum principle of WGDSM between T-SFNs.

5.2 Comparative analyses

The proposed WGDSM under T-SFSs is compared with the T-spherical fuzzy weighted neutral average (TSFWNA) operator [53], T-spherical fuzzy weighted neutral geometric (TSFWNG) operator [53], T-spherical fuzzy power weighted neutral average (TSFPWNA) operator [53], T-spherical fuzzy power weighted neutral geometric (TSFPWNG) operator [53], T-spherical fuzzy Dombi prioritized weighted arithmetic (T-SFDPWA) operator [54] and T-spherical fuzzy Dombi prioritized weighted geometric (T-SFDPWG) operator [54]. The orders by different methods are given in Table 7.

Table 7
The orders by different methods

Methods	Ranking orders
TSFWNA operator [53]	DY ${}_{3}$ $>$ DY ${}_{1}$ $>$ DY ${}_{4}$ $>$ DY ${}_{2}$ $>$ DY ${}_{5}$
TSFWNG operator [53]	DY ${}_{3}$ $>$ DY ${}_{1}$ $>$ DY ${}_{2}$ $>$ DY ${}_{4}$ $>$ DY ${}_{5}$
TSFPWNA operator [53]	DY ${}_{3}$ $>$ DY ${}_{1}$ $>$ DY ${}_{4}$ $>$ DY ${}_{2}$ $>$ DY ${}_{5}$
TSFPWNG operator [53]	DY ${}_{3}$ $>$ DY ${}_{1}$ $>$ DY ${}_{2}$ $>$ DY ${}_{4}$ $>$ DY ${}_{5}$
T-SFDPWA operator [54]	DY ${}_{3}$ $>$ DY ${}_{1}$ $>$ DY ${}_{4}$ $>$ DY ${}_{2}$ $>$ DY ${}_{5}$
T-SFDPWG operator [54]	DY ${}_{3}$ $>$ DY ${}_{1}$ $>$ DY ${}_{2}$ $>$ DY ${}_{4}$ $>$ DY ${}_{5}$

From Table 7, it could be seen that the proposed WGDSM under T-SFSs is compared with the T-spherical fuzzy weighted neutral average (TSFWNA) operator [53], T-spherical fuzzy weighted neutral geometric (TSFWNG) operator [53], T-spherical fuzzy power weighted neutral average (TSFPWNA) operator [53], T-spherical fuzzy power weighted neutral geometric (TSFPWNG) operator [53], T-spherical fuzzy Dombi prioritized weighted arithmetic (T-SFDPWA) operator [54] and T-spherical fuzzy Dombi prioritized weighted geometric (T-SFDPWG) operator [54], the order of these methods is slightly different, however, these decision models have the same optimal music college and worst music college. This verifies the built method is reasonable and effective.

6. Conclusion

Vocal music, as the most prominent and important part of the music system, plays a very important role in expressing emotions and stimulating resonance. With the increasing richness of material life, people have higher aspirations for spiritual life, which has led to rapid development in the vocal industry. In the new historical context, vocal teaching has a broader development space, and more and more schools are carrying out systematic vocal teaching. How to keep up with the development process of the times and promote the continuous improvement of vocal music teaching and education quality is currently a key issue that needs to be focused on and solved. Therefore, it is necessary to conduct in-depth exploration based on the specific problems faced by current vocal teaching, and provide targeted strategies for continuously improving the level of vocal teaching. The vocal music classroom teaching quality evaluation is a classical MAGDM problems. In this work, we shall present some novel DSMs of T-SFSs and the GDSM of T-SFSs and indicates that the DSM and asymmetric measures (projection measures) are the special cases of the GDSMs in some parameter values. Then, we propose the GDSM-based MAGDM models with T-SFSs. Then, we apply the GDSMs between T-SFSs to MAGDM. Finally, an illustrative example is given to demonstrate the efficiency of the GDSMs between T-SFSs. In the future, these decision algorithms and models could be extended to other fuzzy decision environment, like complex T-SFSs, spherical normal FSs etc. On the other hand, many traditional methods such as TOPSIS, TODIM, MARCOS, MULTIMOORA, EDAS, and ELECTRE etc., can be compounded with these decision algorithms and models.

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Exploring T-spherical fuzzy sets for enhanced evaluation of vocal music classroom teaching

Abstract

Keywords

1. Introduction

2. Preliminaries

3.1 DSMs for T-SFSs

5.1 Numerical example

Table 1 Evaluation information by D ⁢ D 1

Table 7 The orders by different methods

References

Table 1
Evaluation information by $DD_{1}$

Table 7
The orders by different methods