Approaches to multiple attribute group decision making under interval-valued pythagorean fuzzy sets and applications to environmental design majors teaching quality evaluation

Abstract

The environmental design major has certain practical characteristics. In the teaching process, it is necessary to effectively combine practical methods and theoretical knowledge in order to improve the overall quality and efficiency of education to a certain extent. The most important training measure currently is project-based teaching method, which has also been recognized by many researchers. The environmental design majors teaching quality evaluation is a classical multiple-attribute group decision-making (MAGDM). Recently, the Taxonomy has been used to cope with MAGDM problems. The interval-valued Pythagorean fuzzy sets (IVPFSs) are useful tools to depict uncertainty in MAGDM. This paper use Taxonomy method to solve MAGDM with IVPFSs. Firstly, the concepts of IVPFSs are presented. Then, the classical Taxonomy method is used to solve MAGDM with IVPFSs. Finally, a numerical example for environmental design majors teaching quality evaluation is proposed and a comparative analysis is also given to show the developed approach. The main study motivations of built paper are outlined: (1) the Taxonomy method is extended to IVPFSs; (2) the score function of IVPFSs and information entropy method is used to derive the attribute weights; (3) the interval-valued Pythagorean fuzzy Taxonomy (IVPF-Taxonomy) method is built to deal with MAGDM; (4) a case study about environmental design majors teaching quality evaluation is given to show the built method; (5) some comparative analysis are given to show the rationality of IVPF-Taxonomy method.

Keywords

Multipleattribute group decisionmaking (MAGDM)interval-valued pythagorean fuzzy sets (IVPFSs)taxonomy method teaching quality evaluation

1. Introduction

Environmental design mainly refers to the practice of using reasonable design methods to integrate and recreate design elements, ultimately achieving the goal of transforming the objective environment [1, 2, 3]. The Environmental Design major belongs to an interdisciplinary field, which involves multiple disciplines such as architecture, urban planning, environment, psychology, society, humanities, etc. It has high requirements for students’ comprehensive professional abilities [4, 5]. The teaching content of environmental design major is relatively complex, requiring students to have high professional theoretical quality and practical operation ability. There are certain differences between the requirements of enterprises for talent abilities and the talent training goals of universities. Enterprise managers usually determine the recruitment conditions for talents based on the needs of market competition and the nature of their business. Usually, the environmental design of enterprises can be divided into landscape design, interior design, exhibition design, and other types. Under larger types, there are several smaller types [6, 7, 8]. For example, interior design can be divided into public space design and residential space design, and public space design can be further divided into commercial space design, office space design, catering space design, entertainment space design, etc [9, 10]. In the teaching process of environmental design, teachers should not only establish a perfect teaching system of professional Curriculum theory, but also pay attention to the systematicness and cohesion between teaching links. Teachers should teach through environmental design to ensure that students’ professional skills meet the requirements of future work. Environmental design is both material and spiritual, and it is not conceptualized or fixed in itself. From the perspective of the design process, designers need to participate in practical activities such as preliminary research, market analysis, sketch creativity, scheme determination, effect performance, design implementation, and post maintenance [11, 12]. Practicality is an important characteristic of environmental design majors, therefore the curriculum and teaching methods of environmental design majors in higher education institutions must have a high degree of practicality. Gropius, a famous German design master, put forward a point in his communication with Chingusai. He believed that “artists must learn how to directly participate in large-scale production, and industrialists must also understand how to accept the value that artists can produce.” Then, how to implement practical teaching in environmental design becomes more practical. The construction of environmental design majors cannot be separated from a highly professional and practical collective. Only in this way can a highly applicable education system be established [13, 14]. Due to the unique nature of the environmental design major, new requirements have been put forward for teaching: (1) the teaching system should have a certain degree of inclusivity and variability, which can be updated and adjusted according to changes in market demand; (2) The curriculum and teaching methods are further updated, allowing for the fastest transfer of practical design experience to students; (3) Encourage environmental design teachers to conduct scientific research and apply the latest achievements to practical teaching. These requirements should be centrally reflected through the talent cultivation plan for the environmental design profession. The talent cultivation plan for the environmental design profession should undergo careful research and rigorous discussion, reflecting professionalism and scientificity [15, 16, 17]. On this basis, it should also have a certain degree of elasticity and not be completely limited. Therefore, the development of talent cultivation plans for environmental design majors in universities must closely combine regional market demand and skill requirements for professional talents, conduct in-depth market research, and determine the overall direction of the plan, such as the structure, types, and class hour distribution of professional courses. There should be a certain degree of flexibility in the specific implementation of the training plan, which can be adjusted appropriately according to changes [18, 19, 20]. When new technologies, materials, and performance methods emerge in the market, they can be introduced and promoted in appropriate courses, so that teaching always closely integrates with market demand and keeps up with the times. Design graduates need to face the market alone and grow rapidly, while our students need to be ‘directly available’. In recent years, through research on some universities, it has been found that students learn professional courses during their school years, with a relatively short teaching cycle [21, 22]. There are also not many courses that can enhance students’ professional skills through practical teaching of professional courses. Usually, students in their freshman and sophomore years study painting courses, professional foundation courses, etc. They only start to truly enter the professional study in their third year, which is relatively late in terms of time, content, and form. As a result, some students’ professional skills during their school years only focused on computer graphics and hand drawing techniques, with vague and ambiguous understanding of professional issues. After entering the workforce, they almost started learning from scratch [23, 24]. Of course, these are not attributed to teachers or students, but we have to admit that design and practice are inseparable. Students’ theoretical knowledge learned from textbooks cannot be effectively integrated with the current situation of actual professional practice, and will inevitably not keep up with the demand for professional talents in the new era. Therefore, investment in project practical teaching urgently needs to be strengthened.

As decision problems and backgrounds become increasingly complex, it is difficult for a single expert to directly provide decision results, so more and more experts need to join in to solve the problem together [25, 26, 27]. During the decision-making process, most of them should pay attention to multiple criteria and multiple choices, thus, multi-criteria decision analysis (MCDA) also plays an important role during the decision-making issues [28, 29, 30, 31]. The environmental design majors teaching quality evaluation is looked as the MAGDM. Compared with COMET method [32, 33], SPOTIS method [34] and DARIA-TOPSIS method [35], Taxonomy method [36] is an important topic in fuzzy set theory and it is an important tool to measure the Euclidean distance to compare the alternatives in pairs to obtain a composite distance matrix, then homogenize the alternatives according to the composite distance matrix, and finally measure the difference between the positive ideal solution and the alternatives [37, 38, 39, 40, 41, 42]. In this case, MAGDM methods have emerged [43, 44, 45, 46]. Atanassov [47] gave the intuitionistic fuzzy sets (IFSs) based on the fuzzy set [48] such that their sum is not greater than one. Pythagorean fuzzy sets (PFSs) [49], an extension of IFSs, has emerged as an effective tool for describing the indeterminacy in uncertain MADM [50, 51, 52, 53, 54]. Zhang [55] extended the PFSs to the interval-valued PFSs (IVPFSs). However, in terms of the information measure theory, a novel accuracy function [56], correlation coefficient [57], improved accuracy function [58], improved score function [59] have been defined under the PFSs and IVPFSs and have been used to solve the MADM. The environmental design majors teaching quality evaluation is always regarded as the MAGDM problem. From the above analysis, there is no scholar to investigate the Taxonomy with IVPFSs. Thus, it is very necessary to keep a watchful eye on this novel and worthy issue. The purpose of our paper is to use the Taxonomy idea to cope with the MAGDM with IVPFSs, toÂ fill upÂ thisÂ vacancy. The main aim of such paper is to extend Taxonomy to solve the MAGDM under IVPFSs. The main study motivations and innovativeness of built paper are outlined: (1) the Taxonomy method is extended to IVPFSs; (2) the score function of IVPFSs and information entropy method is used to derive the attribute weights; (3) the interval-valued Pythagorean fuzzy Taxonomy (IVPF-Taxonomy) method is built to deal with MAGDM; (4) a case study about environmental design majors teaching quality evaluation is given to show the built method; (5) some comparative analysis are given to show the rationality of IVPF-Taxonomy method.

In order to do so, for Sec. 2, the definition of IVPFSs is introduced. The IVPF-Taxonomy is built for MAGDM in Sec. 3. An illustrative example for environmental design majors teaching quality evaluation is pointed out and a comparative analysis is given in Sec. 4. The conclusion is included in Sec. 5.

2. Preliminaries

The PFSs and IVPFSs are introduced.

Definition 1 [49]. Let $U$ be a fix set. A PFS is defined

$\displaystyle T=\{\langle u,(m_{T}(u),n_{T}(u))\rangle|{u\in U}\}$ (1)

where $m_{T}(u)\subseteq[{0,1}],n_{T}(u)\subseteq[{0,1}]$ denotes the membership non-membership of $u\in U$ to $T$ , and, for $u\in U$ , it has the defined condition

$\displaystyle({m_{T}(u)})^{2}+({n_{T}(u)})^{2}\leqslant 1.$ (2)

Definition 2 [49]. Let $t_{1}=({m_{1},n_{1}})$ , $t_{2}=({m_{2},n_{2}})$ , and $t=({m,n})$ be three PFNs, and some operations are given.

$\displaystyle(1)\;t_{1}\oplus t_{2}=({\sqrt{({m_{1}})^{2}+({m_{2}})^{2}-({m_{1% }})^{2}({m_{2}})^{2}},n_{1}n_{2}});$ $\displaystyle(2)\;t_{1}\otimes t_{2}=({m_{1}m_{2},\sqrt{({n_{1}})^{2}+({n_{2}}% )^{2}-({n_{1}})^{2}({n_{2}})^{2}}});$ $\displaystyle(3)\;\lambda t=\left({\sqrt{1-({1-m^{2}})^{\lambda}},n^{\lambda}}% \right),\lambda>0;$ $\displaystyle(4)\;(t)^{\lambda}=\left({m^{\lambda},\sqrt{1-({1-n^{2}})^{% \lambda}}}\right),\lambda>0.$

Zhang [55] developed the interval-valued PFSs (IVPFSs).

Definition 3 [27]. Let $U$ be a fix set. An IVPFS is defined

$\displaystyle\tilde{T}=\{\langle u,(\tilde{m}_{\tilde{T}}(u),\tilde{n}_{\tilde% {T}}(u))\rangle|{u\in U}\}$ (3)

where $\tilde{m}_{\tilde{T}}(u)\subset[0,1]$ , $\tilde{n}_{\tilde{T}}(u)\subset[0,1]$ are interval values, $\tilde{m}_{\tilde{T}}(u)=[{\tilde{m}_{\tilde{T}}^{L}(u),\tilde{m}_{\tilde{T}}^% {R}(u)}],\tilde{n}_{\tilde{T}}(u)=[\tilde{n}_{\tilde{T}}^{L}(u),\linebreak% \tilde{n}_{\tilde{T}}^{R}(u)]$ , $0\leqslant({m_{\tilde{T}}^{R}(u)})^{2}+({n_{\tilde{T}}^{R}(u)})^{2}\leqslant 1$ , $\forall u\in U$ . $\tilde{m}_{\tilde{T}}(u),\tilde{n}_{\tilde{T}}(u)$ denote the membership and no-membership of the $u$ to $\tilde{T}$ .

For convenience, ${\tilde{t}=}({[{m_{\tilde{t}}^{L},m_{\tilde{t}}^{R}}],[{n_{\tilde{t}}^{L},n_{% \tilde{t}}^{R}}]})$ is an interval-valued Pythagorean fuzzy number (IVPFN), and $\tilde{m}_{\tilde{t}}(u)\subset[{0,1}]$ , $\tilde{n}_{\tilde{t}}(u)\subset[{0,1}]$ , $({m_{\tilde{t}}^{R}})^{2}+({n_{\tilde{t}}^{R}})^{2}\leqslant 1$ .

Definition 4 [56]. Let ${\tilde{t}}_{1}=({[{m_{{\tilde{t}}_{1}}^{L},m_{{\tilde{t}}_{1}}^{R}}],[{n_{{% \tilde{t}}_{1}}^{L},n_{{\tilde{t}}_{1}}^{R}}]})$ , ${\tilde{t}}_{2}=({[{m_{{\tilde{t}}_{2}}^{L},m_{{\tilde{t}}_{2}}^{R}}],[{n_{{% \tilde{t}}_{2}}^{L},n_{{\tilde{t}}_{2}}^{R}}]})$ , and ${\tilde{t}=}([{m_{\tilde{t}}^{L},m_{\tilde{t}}^{R}}],[n_{\tilde{t}}^{L},% \linebreak n_{\tilde{t}}^{R}])$ be three IVPFNs, and some operations are given.

$\displaystyle(1)\;\tilde{t}_{1}\oplus\tilde{t}_{2}=\left({\left[{\begin{array}% []{l}\sqrt{({m_{\tilde{t}_{1}}^{L}})^{2}+({m_{\tilde{t}_{2}}^{L}})^{2}-({m_{% \tilde{t}_{1}}^{L}})^{2}({m_{\tilde{t}_{2}}^{L}})^{2}},\\ \sqrt{({m_{\tilde{t}_{1}}^{R}})^{2}+({m_{\tilde{t}_{2}}^{R}})^{2}-({m_{\tilde{% t}_{1}}^{R}})^{2}({m_{\tilde{t}_{2}}^{R}})^{2}}\\ \end{array}}\right],[{n_{\tilde{t}_{1}}^{L}n_{\tilde{t}_{2}}^{L},n_{\tilde{t}_% {1}}^{R}n_{\tilde{t}_{2}}^{R}}]}\right);$ $\displaystyle(2)\;\tilde{t}_{1}\otimes\tilde{t}_{2}=\left({[{m_{\tilde{t}_{1}}% ^{L}m_{\tilde{t}_{2}}^{L},m_{\tilde{t}_{1}}^{R}m_{\tilde{t}_{2}}^{R}}],\left[{% \begin{array}[]{l}\sqrt{({n_{\tilde{t}_{1}}^{L}})^{2}+({n_{\tilde{t}_{2}}^{L}}% )^{2}-({n_{\tilde{t}_{1}}^{L}})^{2}({n_{\tilde{t}_{2}}^{L}})^{2}},\\ \sqrt{({n_{\tilde{t}_{1}}^{R}})^{2}+({n_{\tilde{t}_{2}}^{R}})^{2}-({n_{\tilde{% t}_{1}}^{R}})^{2}({n_{\tilde{t}_{2}}^{R}})^{2}}\\ \end{array}}\right]}\right);$ $\displaystyle(3)\;\lambda\tilde{t}=\left({\left[{\sqrt{1-({1-({m_{\tilde{t}}^{% L}})^{2}})^{\lambda}},\sqrt{1-({1-({m_{\tilde{t}}^{R}})^{2}})^{\lambda}}}% \right],[{({n_{\tilde{t}}^{L}})^{\lambda},({n_{\tilde{t}}^{R}})^{\lambda}}]}% \right),\lambda>0;$ $\displaystyle(4)\;(\tilde{t})^{\lambda}=\left({[{({m_{\tilde{t}}^{L}})^{% \lambda},({m_{\tilde{t}}^{R}})^{\lambda}}],\left[{\sqrt{1-({1-({n_{\tilde{t}}^% {L}})^{2}})^{\lambda}},\sqrt{1-({1-({n_{\tilde{t}}^{R}})^{2}})^{\lambda}}}% \right]}\right),\lambda>0;$ $\displaystyle(5)\;\tilde{t}^{c}=({[{n_{\tilde{t}}^{L},n_{\tilde{t}}^{R}}],[{m_% {\tilde{t}}^{L},m_{\tilde{t}}^{R}}]}).$

Let $\tilde{t}=({[{m_{\tilde{t}}^{L},m_{\tilde{t}}^{R}}],[{n_{\tilde{t}}^{L},n_{% \tilde{t}}^{R}}]})$ be an IVPFN,

$\displaystyle SV(\tilde{t})=\frac{1}{4}[{({1+({m_{\tilde{t}}^{L}})^{2}-({n_{% \tilde{t}}^{L}})^{2}})+({1+({m_{\tilde{t}}^{R}})^{2}-({n_{\tilde{t}}^{R}})^{2}% })}]$

and

$\displaystyle AV(\tilde{t})=\frac{({m_{\tilde{t}}^{L}})^{2}+({m_{\tilde{t}}^{R% }})^{2}+({n_{\tilde{t}}^{L}})^{2}+({n_{\tilde{t}}^{R}})^{2}}{2}$

are score value and accuracy value.

Definition 5 [56]. Let ${\tilde{t}}_{1}=({[{m_{{\tilde{t}}_{1}}^{L},m_{{\tilde{t}}_{1}}^{R}}],[{n_{{% \tilde{t}}_{1}}^{L},n_{{\tilde{t}}_{1}}^{R}}]})$ , ${\tilde{t}}_{2}=({[{m_{{\tilde{t}}_{2}}^{L},m_{{\tilde{t}}_{2}}^{R}}],[{n_{{% \tilde{t}}_{2}}^{L},n_{{\tilde{t}}_{2}}^{R}}]})$ , $SV({\tilde{t}_{1}})$ and $SV({\tilde{t}_{2}})$ is the scores value of $\tilde{t}_{1}$ and $AV({\tilde{t}_{1}})$ and $AV({\tilde{t}_{2}})$ is the accuracy value of $\tilde{t}_{1}$ and $\tilde{t}_{2}$ , then if $SV({\tilde{t}_{1}})<SV({\tilde{t}_{2}})$ , $\tilde{t}_{1}<\tilde{t}_{2}$ ; if $SV({\tilde{t}_{1}})=SV({\tilde{t}_{2}})$ , (1) if $AV({\tilde{t}_{1}})=AV({\tilde{t}_{2}})$ , $\tilde{t}_{1}=\tilde{t}_{2}$ ; (2) if $AV({\tilde{t}_{1}})<AV({\tilde{t}_{2}})$ , $\tilde{t}_{1}<\tilde{t}_{2}$ .

Definition 6 [60]. Let ${\tilde{t}}_{1}=({[{m_{{\tilde{t}}_{1}}^{L},m_{{\tilde{t}}_{1}}^{R}}],[{n_{{% \tilde{t}}_{1}}^{L},n_{{\tilde{t}}_{1}}^{R}}]})$ , ${\tilde{t}}_{2}=({[{m_{{\tilde{t}}_{2}}^{L},m_{{\tilde{t}}_{2}}^{R}}],[{n_{{% \tilde{t}}_{2}}^{L},n_{{\tilde{t}}_{2}}^{R}}]})$ , the normalized Hamming distance is:

$\displaystyle HD({{\tilde{t}}_{1},{\tilde{t}}_{2}})=\frac{1}{4}\left({\begin{% array}[]{l}|{({m_{{\tilde{t}}_{1}}^{L}})^{2}-({m_{{\tilde{t}}_{2}}^{L}})^{2}}|% +|{({m_{{\tilde{t}}_{1}}^{R}})^{2}-({m_{{\tilde{t}}_{2}}^{R}})^{2}}|\\ {}+|{({n_{{\tilde{t}}_{1}}^{L}})^{2}-({n_{{\tilde{t}}_{2}}^{L}})^{2}}|+|{({n_{% {\tilde{t}}_{1}}^{R}})^{2}-({n_{{\tilde{t}}_{2}}^{R}})^{2}}|\\ \end{array}}\right)$ (4)

3. IVPFTaxonomy method for MAGDM problems

Let $\alpha=\{{\alpha_{1},\alpha_{2},\ldots,\alpha_{c}}\}$ be alternatives and $\beta=\{{\beta_{1},\beta_{2},\ldots,\beta_{d}}\}$ be attributes. Let $\gamma=(\gamma_{1},\gamma_{2},\linebreak\ldots,\gamma_{d})$ be the weight of $\beta_{s}$ , $0\leqslant\gamma_{s}\leqslant 1$ , $\sum\limits_{s=1}^{d}{\gamma_{s}}=1$ . Suppose that $\tilde{X}^{(h)}=(\tilde{x}_{gs}^{(h)})_{c\times d}=([m_{gs}^{L(h)},\linebreak m% _{gs}^{R(h)}],[{n_{gs}^{L(h)},n_{gs}^{R(h)}}])_{c\times d}$ is the IVPFNmatrix, with the attribute values $x_{gs}^{(h)}=([{m_{gs}^{L(h)},m_{gs}^{R(h)}}],[n_{gs}^{L(h)},\linebreak n_{gs}% ^{R(h)}])$ , which is given by decision makers $DM_{h}({h=1,2,\ldots,b})$ for alternative $\alpha_{g}$ under attribute $\beta_{s}$ , $[{m_{gs}^{L(h)},m_{gs}^{R(h)}}]\subseteq[{0,1}]$ , $[{n_{gs}^{L(h)},n_{gs}^{R(h)}}]\subseteq[{0,1}]$ .

To settle the MAGDM with IVPFNs, the IVPFTaxonomy method is built. The computing decision procedures are designed.

Step 1. Calculate the IVPFN overall matrix $\tilde{X}=({x_{gs}})_{c\times d}=({[{m_{{gs}}^{L},m_{{gs}}^{R}}],[{n_{{gs}}^{L% },n_{{gs}}^{R}}]})_{c\times d}$ by IVPFWA operator.

$\displaystyle\tilde{x}_{gs}=({[{m_{{gs}}^{L},m_{{gs}}^{R}}],[{n_{{gs}}^{L},n_{% {gs}}^{R}}]})=\text{IVPFWA}_{\alpha}({x_{gs}^{(1)},x_{gs}^{(2)},\ldots,x_{gs}^% {(b)}})=\mathop{\oplus}\limits_{h=1}^{b}({\eta_{h}x_{gs}^{(h)}})=\left({\begin% {array}[]{l}\left[{\sqrt{1-\prod\limits_{h=1}^{b}{({1-({m_{gs}^{L(h)}})^{2}})^% {\eta_{h}}}},\sqrt{1-\prod\limits_{h=1}^{b}{({1-({m_{gs}^{R(h)}})^{2}})^{\eta_% {h}}}}}\right],\\ \left[{\prod\limits_{h=1}^{b}{({n_{gs}^{L(h)}})^{\eta_{h}},\prod\limits_{h=1}^% {b}{({n_{gs}^{R(h)}})^{\eta_{h}}}}}\right]\\ \end{array}}\right)$ (5)

$\eta=({\eta_{1},\eta_{2},\ldots,\eta_{b}})$ be the weight of $DM_{h}({h=1,2,\ldots,b})$ , $0\leqslant\eta_{b}\leqslant 1$ , $\sum\limits_{h=1}^{b}{\eta_{h}}=1$ .

Step 2. Cost attribute is shifted into beneficial one. If the cost value is $({[{m_{{gs}}^{L},m_{{gs}}^{R}}],[{n_{{gs}}^{L},n_{{gs}}^{R}}]})$ , then the corresponding beneficial one is $({[{n_{{gs}}^{L},n_{{gs}}^{R}}],[{m_{{gs}}^{L},m_{{gs}}^{R}}]})$ .

Step 3. Utilize the entropy method to calculate the weight.

Entropy [61] is employed to derive attributes’ weight. Firstly, the normalized IVPFN-matrix $\tilde{x}_{gs}$ is determined:

$\displaystyle\textit{NIVPFN}({S({\tilde{x}_{gs}})})=\frac{SV({\tilde{x}_{gs}})% }{\sum\limits_{g=1}^{c}{SV({\tilde{x}_{gs}})}},s=1,2,\ldots,d,$ (6)

Then, the corresponding Shannon entropy $SE=({SE_{1},SE_{2},\ldots,SE_{d}})$ is derived by Eq. (7):

$\displaystyle SE_{s}=-\frac{1}{\ln c}\sum\limits_{g=1}^{c}{\textit{NIVPFN}({SV% ({x_{gs}})})\ln\textit{NIVPFN}({SV({x_{gs}})})}$ (7)

$\textit{NIVPFN}({SV({\tilde{x}_{gs}})})\ln\textit{NIVPFN}({SV({\tilde{x}_{gs}}% )})=0$ , if $\textit{NIVPFN}({SV({x_{gs}})})=0$ .

The attribute weights $\gamma=({\gamma_{1},\gamma_{2},\ldots,\gamma_{d}})$ is:

$\displaystyle\gamma_{s}=\frac{1-SE_{s}}{\sum\limits_{s=1}^{d}{({1-SE_{s}})}},s% =1,2,\ldots,d.$ (8)

Step 4. Calculate the IVPFN composite distance matrix (IVPFNCDM):

The alternative’s distance from the other alternatives is derived by Eq. (9):

$\displaystyle\textit{IVPFNCDM}_{gk}=\sum\limits_{s=1}^{d}{\gamma_{s}d({({[{m_{% {gs}}^{L},m_{{gs}}^{R}}],[{n_{{gs}}^{L},n_{{gs}}^{R}}]}),({[{m_{{ks}}^{L},m_{{% ks}}^{R}}],[{n_{{ks}}^{L},n_{{ks}}^{R}}]})})}$ (9)

Then, the IVPFNCDM between alternatives is listed as Eq. (11).

$\displaystyle\textit{IVPFNCDM}=\left[{{\begin{array}[]{*{20}c}{\textit{% IVPFNCDM}_{11}}&{\textit{IVPFNCDM}_{12}}&\ldots&{\textit{IVPFNCDM}_{1d}}\\ {\textit{IVPFNCDM}_{21}}&{\textit{PIVPFNCDM}_{22}}&\ldots&{\textit{IVPFNCDM}_{% 2d}}\\ \vdots&\vdots&\vdots&\vdots\\ {\textit{IVPFNCDM}_{c1}}&{\textit{IVPFNCDM}_{c2}}&\ldots&{\textit{IVPFNCDM}_{% cd}}\\ \end{array}}}\right]$ (10)

Step 5. The alternatives are homogenized.

The mean and standard deviation for minimum distance are computed by Eq. (11).

$\displaystyle\textit{IVPFN}\bar{O}=\frac{1}{c}\sum\limits_{g=1}^{c}{\textit{% IVPFNO}_{g}},S_{\textit{IVPFNO}}=\sqrt{\frac{1}{c}\sum\limits_{g=1}^{c}{({% \textit{IVPFNO}_{g}-\textit{IVPFN}\bar{O}})^{2}}}$ (11)

where $\textit{IVPFNO}_{g}$ is the $g t h$ row’s minimum distance in IVPFNCDM. The homogeneity range of IVPFNCDM is derived by Eq. (12).

$\displaystyle\textit{IVPFNO}=\textit{IVPFN}\bar{O}\pm 2S_{\textit{IVPFNO}}$ (12)

Step 6. Define the IVPFN positive ideal solution (IVPFNPIS):

$\displaystyle\textit{IVPFNPIS}=({\textit{IVPFNPIS}_{1},\textit{IVPFNPIS}_{2},% \ldots,\textit{IVPFNPIS}_{d}})$ (13) $\displaystyle\textit{IVPFNPIS}_{s}=({[{m_{s}^{L+},m_{s}^{R+}}],[{n_{s}^{L-}n_{% s}^{R-}}]})$ (14) $\displaystyle SV({({[{m_{s}^{L+},m_{s}^{R+}}],[{n_{s}^{L-}n_{s}^{R-}}]})})=% \mathop{\max}\limits_{g}SV({[{m_{{gs}}^{L},m_{{gs}}^{R}}],[{n_{{gs}}^{L},n_{{% gs}}^{R}}]})$ (15)

Step 7. Calculate the IVPFN development pattern (IVPFNDP) for each defined alternative:

$\displaystyle\textit{IVPFNDP}({a_{g},\textit{IVPFNPIS}})=\sum\limits_{s=1}^{d}% {w_{s}}({d({\tilde{x}_{gs},\textit{IVPFNPIS}_{s}})})$ (16) $\displaystyle\textit{IVPFNDP}({a_{g},\textit{IVPFNPIS}})=\frac{1}{4}\left({% \begin{array}[]{l}|{({m_{{gs}}^{L}})^{2}-({m_{{gs}}^{L+}})^{2}}|+|{({m_{{gs}}^% {R}})^{2}-({m_{{gs}}^{R+}})^{2}}|+\\ |{({n_{{gs}}^{L}})^{2}-({n_{{gs}}^{L+}})^{2}}|+|{({n_{{gs}}^{R}})^{2}-({n_{{gs% }}^{R+}})^{2}}|\\ \end{array}}\right)$ (17)

Step 8. Calculate the IVPFN high limit of development $\textit{IVPFNHLD}_{0}$ and the IVPFN development pattern values $\textit{IVPFNDPV}_{g}({g=1,2,\ldots,c})$ :

The $\textit{IVPFNHLD}_{0}$ is calculated through Eqs (18)–(20)

$\displaystyle\textit{IVPFNHLD}_{0}=\overline{\textit{IVPFNDP}({a_{g},\textit{% IVPFNPIS}})}+2S_{\textit{IVPFNDP}({a_{g},\textit{IVPFNFPIS}})}$ (18) $\displaystyle\overline{\textit{IVPFNDP}({a_{g},\textit{IVPFNPIS}})}=\frac{1}{c% }\sum\limits_{g=1}^{c}{\textit{IVPFNDP}({a_{g},\textit{IVPFNPIS}})}$ (19) $\displaystyle S_{\textit{IVPFNDP}({a_{g},\textit{IVPFNPIS}})}=\sqrt{\frac{1}{c% }\sum\limits_{g=1}^{c}{\left({\begin{array}[]{l}\textit{IVPFNDP}({a_{g},% \textit{IVPFNPIS}})-\\ \overline{\textit{IVPFNDP}({a_{g},\textit{IVPFNPIS}})}\\ \end{array}}\right)^{2}}}$ (20)

The $\textit{IVPFNDPV}_{g}({g=1,2,\ldots,c})$ can be got from Eq. (21).

$\displaystyle\textit{IVPFNDPV}_{g}=\frac{\textit{IVPFNDP}({a_{g},\textit{% IVPFNPIS}})}{\textit{IVPFNHLD}_{0}},g=1,2,\ldots,c.$ (21)

Step 9. Depending on the $\textit{IVPFNDPV}_{g}({g=1,2,\ldots,c})$ , all alternatives’ order can be derived. Therefore, the alternative with the smallest $\textit{IVPFNDPV}_{g}({g=1,2,\ldots,c})$ value is the optimal alternative.

Table 1

The four attributes

Attributes	Attributes interpretation
$\beta_{1}$	Environmental design majors teaching organization
$\beta_{2}$	Environmental design majors teaching resources
$\beta_{3}$	Environmental design majors teaching courses
$\beta_{4}$	Environmental design majors teaching methods

4. Numerical example and comparative analysis

4.1 Numerical example

With the continuous development of society and the improvement of the market economy, the urbanization of construction is also accelerating. The continuous development of the construction industry also drives the improvement of environmental design majors. The Environmental Design major is the most important discipline in the teaching process of art and design. The Environmental Design major includes various disciplines such as architecture, landscaping, landscape, and aesthetics, and has strong practical characteristics. It plays a very important role in establishing a good living environment. The environmental design major has a very prominent practical feature, which corresponds to the practical experience of social production, social processing, and social life. It is determined based on the market demand in practical design. In the overall learning process, there is a lack of opportunities to interact with practical cases, resulting in a lack of certain internship facilities, equipment, corresponding experiments, and venues. Students lack the opportunity to personally design and experience a complete set of practical solutions. They are not familiar with the materials used and the entire process of indoor construction, so they cannot accurately budget the public and material costs involved in the overall construction, And these are common problems in the current teaching process of environmental design majors. In the process of designing practical teaching simulation cases, in most cases, the teacher provides corresponding on-site survey maps. Students use computer software to create renderings or hand draw renderings after design. However, when a specific plan prototype appears, it needs to be completed in the later stage according to the construction site process and reasonable material costs in the budget, without allowing students to consider the actual project firsthand, So students’ design ability can only demonstrate formal beauty or art. In reality, the main medium and venue for communication between students and teachers is the blackboard or drawings during the teaching process. Students cannot fully consider the engineering quotation, market situation, and construction process in classroom teaching. Practice is the only criterion for verifying truth. The prerequisite for implementing project-based teaching methods in environmental design majors is whether actual projects can be interspersed in the classroom teaching process. In the process of revising the plan for cultivating new talents, it is necessary to fully consider the future employment direction of students, as well as the comprehensive qualities and professional abilities of students corresponding to the positions required by the enterprise. With the continuous development of society and the improvement of market economy, the living standards of the people are getting higher and higher. Education in various universities is also undergoing continuous reforms and boldly implementing project-based teaching methods. To a certain extent, it has broken the constraints of traditional teaching models and established an innovative teaching practice system. It is necessary to combine practical cases with course teaching, and meet market demand. Experience should be summarized in practice and professional theory, which is also the most fundamental way to improve the quality of teaching. The environmental design majors teaching quality evaluation is looked as a novel MAGDM. Now, an example for environmental design majors teaching quality evaluation is utilized to reveal built methods with IVPFNs. Five emerging engineering colleges $\alpha_{g}({g=1,2,3,4,5})$ are to be ranked. Four novel attributes are used to evaluate the five possible engineering colleges (See Table 1) [62].

The five engineering colleges are to be evaluated through IVPFNs, as shown in $\tilde{X}_{1},\tilde{X}_{2},\tilde{X}_{3}$ .

$\displaystyle\begin{array}[]{llll}\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;% \beta_{1}&\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\beta_{2}&\;\;% \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\beta_{3}&\;\;\;\;\;\;\;\;\;\;% \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\beta_{4}\end{array}$ $\displaystyle\tilde{X}_{1}={\begin{array}[]{*{20}c}{\alpha_{1}}\\ {\alpha_{2}}\\ {\alpha_{3}}\\ {\alpha_{4}}\\ {\alpha_{5}}\\ \end{array}}\left[{{\begin{array}[]{*{20}c}{({[{0.1,0.6}],[{0.2,0.8}]})}&{({[{% 0.2,0.5}],[{0.3,0.7}]})}&{({[{0.3,0.4}],[{0.6,0.7}]})}&{({[{0.3,0.8}],[{0.2,0.% 6}]})}\\ {({[{0.2,0.3}],[{0.3,0.5}]})}&{({[{0.2,0.4}],[{0.3,0.6}]})}&{({[{0.5,0.6}],[{0% .1,0.5}]})}&{({[{0.3,0.9}],[{0.2,0.4}]})}\\ {({[{0.2,0.6}],[{0.4,0.5}]})}&{({[{0.4,0.7}],[{0.3,0.4}]})}&{({[{0.4,0.9}],[{0% .2,0.3}]})}&{({[{0.1,0.4}],[{0.3,0.5}]})}\\ {({[{0.6,0.7}],[{0.5,0.7}]})}&{({[{0.5,0.8}],[{0.4,0.6}]})}&{({[{0.2,0.6}],[{0% .1,0.4}]})}&{({[{0.2,0.3}],[{0.4,0.8}]})}\\ {({[{0.3,0.6}],[{0.1,0.4}]})}&{({[{0.3,0.9}],[{0.1,0.2}]})}&{({[{0.6,0.8}],[{0% .3,0.4}]})}&{({[{0.3,0.6}],[{0.6,0.7}]})}\\ \end{array}}}\right]\begin{array}[]{llll}\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;% \;\;\;\beta_{1}&\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\beta_{2% }&\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\beta_{3}&\;\;\;\;\;\;\;% \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\beta_{4}\end{array}$ $\displaystyle\tilde{X}_{2}={\begin{array}[]{*{20}c}{\alpha_{1}}\\ {\alpha_{2}}\\ {\alpha_{3}}\\ {\alpha_{4}}\\ {\alpha_{5}}\\ \end{array}}\left[{{\begin{array}[]{*{20}c}{({[{0.3,0.5}],[{0.2,0.7}]})}&{({[{% 0.4,0.5}],[{0.6,0.7}]})}&{({[{0.2,0.5}],[{0.4,0.8}]})}&{({[{0.4,0.5}],[{0.5,0.% 8}]})}\\ {({[{0.1,0.3}],[{0.3,0.4}]})}&{({[{0.2,0.4}],[{0.2,0.6}]})}&{({[{0.3,0.6}],[{0% .3,0.5}]})}&{({[{0.3,0.4}],[{0.4,0.6}]})}\\ {({[{0.2,0.3}],[{0.4,0.9}]})}&{({[{0.4,0.7}],[{0.2,0.4}]})}&{({[{0.4,0.7}],[{0% .2,0.5}]})}&{({[{0.1,0.3}],[{0.2,0.3}]})}\\ {({[{0.5,0.6}],[{0.2,0.8}]})}&{({[{0.5,0.9}],[{0.1,0.2}]})}&{({[{0.2,0.9}],[{0% .3,0.4}]})}&{({[{0.2,0.3}],[{0.2,0.4}]})}\\ {({[{0.3,0.4}],[{0.2,0.4}]})}&{({[{0.3,0.8}],[{0.1,0.3}]})}&{({[{0.6,0.7}],[{0% .3,0.6}]})}&{({[{0.3,0.5}],[{0.5,0.6}]})}\\ \end{array}}}\right]\begin{array}[]{llll}\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;% \;\;\;\beta_{1}&\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\beta_{2% }&\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\beta_{3}&\;\;\;\;\;\;\;% \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\beta_{4}\end{array}$ $\displaystyle\tilde{X}_{3}={\begin{array}[]{*{20}c}{\alpha_{1}}\\ {\alpha_{2}}\\ {\alpha_{3}}\\ {\alpha_{4}}\\ {\alpha_{5}}\\ \end{array}}\left[{{\begin{array}[]{*{20}c}{({[{0.1,0.2}],[{0.2,0.3}]})}&{({[{% 0.2,0.3}],[{0.3,0.4}]})}&{({[{0.2,0.3}],[{0.3,0.4}]})}&{({[{0.3,0.5}],[{0.2,0.% 4}]})}\\ {({[{0.2,0.3}],[{0.3,0.4}]})}&{({[{0.3,0.4}],[{0.3,0.5}]})}&{({[{0.3,0.6}],[{0% .2,0.4}]})}&{({[{0.8,0.9}],[{0.1,0.2}]})}\\ {({[{0.2,0.5}],[{0.5,0.7}]})}&{({[{0.2,0.5}],[{0.3,0.5}]})}&{({[{0.4,0.5}],[{0% .3,0.4}]})}&{({[{0.2,0.3}],[{0.1,0.5}]})}\\ {({[{0.4,0.6}],[{0.3,0.7}]})}&{({[{0.5,0.6}],[{0.4,0.7}]})}&{({[{0.3,0.8}],[{0% .2,0.3}]})}&{({[{0.2,0.7}],[{0.3,0.6}]})}\\ {({[{0.3,0.4}],[{0.2,0.3}]})}&{({[{0.3,0.6}],[{0.1,0.6}]})}&{({[{0.6,0.7}],[{0% .5,0.6}]})}&{({[{0.3,0.4}],[{0.6,0.8}]})}\\ \end{array}}}\right]$

Then, the IVPF-Taxonomy method is utilized for environmental design majors teaching quality evaluation.

Step 1. Calculate the IVPFN overall matrix $\tilde{X}=({x_{gs}})_{5\times 4}$ by IVPFWA operator with equal weight information (See Table 2). Let $\eta=({0.2317,0.4554,3129})$ .

Table 2
IVPFN overall matrix

Alternatives	$\beta_{1}$	$\beta_{2}$
$\alpha_{1}$	([0.1934, 0.4771], [0.2000, 0.5518])	([0.2860, 0.4473], [0.3780, 0.5809])
$\alpha_{2}$	([0.1735, 0.3000], [0.3000, 0.4309])	([0.2387, 0.4000], [0.2621, 0.5646])
$\alpha_{3}$	([0.2000, 0.4912], [0.4309, 0.6804])	([0.3490, 0.6481], [0.2621, 0.4309])
$\alpha_{4}$	([0.5111, 0.6377], [0.3107, 0.7319])	([0.5000, 0.8047], [0.2520, 0.4380])
$\alpha_{5}$	([0.3000, 0.4825], [0.1587, 0.3634])	([0.3000, 0.8047], [0.1000, 0.3302])
Alternatives	$\beta_{3}$	$\beta_{4}$
$\alpha_{1}$	([0.2387, 0.4114], [0.4160, 0.6073])	([0.3376, 0.6425], [0.2714, 0.5769])
$\alpha_{2}$	([0.3832, 0.6000], [0.1817, 0.4642])	([0.5762, 0.8296], [0.2000, 0.3634])
$\alpha_{3}$	([0.4000, 0.7633], [0.2289, 0.3915])	([0.1418, 0.3376], [0.1817, 0.4217])
$\alpha_{4}$	([0.2387, 0.8047], [0.1817, 0.3634])	([0.2000, 0.4997], [0.2884, 0.5769])
$\alpha_{5}$	([0.6000, 0.7389], [0.3557, 0.5241])	([0.3000, 0.5111], [0.5646, 0.6952])

Step 2. Compute the weight values: $\gamma_{1}=$ 0.0891, $\gamma_{2}=$ 0.2711, $\gamma_{3}=$ 0.2158, $\gamma_{4}=$ 0.4240.

Step 3. Calculate the IVPFNCDM (See Table 3).

Table 3

The IVPFNCDM

	$\alpha_{1}$	$\alpha_{2}$	$\alpha_{3}$	$\alpha_{4}$	$\alpha_{5}$
$\alpha_{1}$	–	0.1258	0.1484	0.1446	0.1656
$\alpha_{2}$	0.1258	–	0.1596	0.2034	0.2281
$\alpha_{3}$	0.1484	0.1596	–	0.0866	0.1480
$\alpha_{4}$	0.1446	0.2034	0.0866	–	0.1204
$\alpha_{5}$	0.1656	0.2281	0.1480	0.1204	–

Step 4. Homogenize the alternatives.

The IVPFNO is derived:

$\displaystyle\textit{IVPFNO}_{1}=0.1258,\textit{IVPFNO}_{2}=0.1258,\textit{% IVPFNO}_{3}=0.0866$ $\displaystyle\textit{IVPFNO}_{4}=0.0866,\textit{IVPFNO}_{5}=0.1204$ $\displaystyle\textit{IVPFN}\bar{O}=0.1090,S_{\textit{IVPFNO}}=0.0184$ $\displaystyle\textit{IVPFNO}=\textit{IVPFN}\bar{O}\pm 2S_{\textit{IVPFNO}}=0.1% 090\pm 2{\ast}0.0184=0.1090\pm 0.0368$

Therefore, in IVPFNCDM, all the IVPFN shortest distance values are in this range, and the alternatives are homogeneous.

Step 5. Obtain the IVPFNPIS (See Table 4).

Table 4

The IVPFNPIS

	$\beta_{1}$	$\beta_{2}$
IVPFNPIS	([0.5111, 0.6377], [0.1587, 0.3634])	([0.5000, 0.8047], [0.1000, 0.3302])
	$\beta_{3}$	$\beta_{4}$
IVPFNPIS	([0.6000, 0.8047], [0.1817, 0.3634])	([0.5762, 0.8296], [0.1817, 0.3634])

Step 6. Calculate the IVPFNDP:

$\displaystyle\textit{IVPFNDP}({\alpha_{1},\textit{IVPFNPIS}})=0.1243,\textit{% IVPFNDP}({\alpha_{2},\textit{IVPFNPIS}})={0.0978}$ $\displaystyle\textit{IVPFNDP}({\alpha_{3},\textit{IVPFNPIS}})=0.0889,\textit{% IVPFNDP}({\alpha_{4},\textit{IVPFNPIS}})={0.0479}$ $\displaystyle\textit{IVPFNDP}({\alpha_{5},\textit{IVPFNPIS}})={0.0626}$

Step 8. Calculate the $\textit{IVPFNHLD}_{0}$ and $\textit{IVPFNDPV}_{i}({i=1,2,3,4,5})$ :

$\displaystyle\textit{IVPFNHLD}_{0}=0.2659,\textit{IVPFNDPV}_{1}=0.8262,\textit% {IVPFNDPV}_{2}={0.4189}$ $\displaystyle\textit{IVPFNDPV}_{3}=0.6321,\textit{IVPFNDPV}_{4}=0.5271,\textit% {IVPFNDPV}_{5}={0.6587}$

Step 9. According to $\textit{IVPFNDPV}_{i}({i=1,2,3,4,5})$ , the order is: $\alpha_{2}>\alpha_{4}>\alpha_{3}>\alpha_{5}>\alpha_{1}$ and the optimal college is $\alpha_{2}$ .

4.2 Comparative analysis

Then, IVPF-Taxonomy is compared with IVPFWA operator and IVPFWG operator proposed by Garg [56], IVPF-TOPSIS method [63], IVPF-WASPAS method [64] and IVPF-VIKOR method [65]. The ranking order is shown in Table 5.

Table 5
Order through different methods

Methods	Order
IVPFWA operator [56]	$\alpha_{2}>\alpha_{4}>\alpha_{3}>\alpha_{5}>\alpha_{1}$
IVPFWG operator [56]	$\alpha_{2}>\alpha_{4}>\alpha_{3}>\alpha_{1}>\alpha_{5}$
IVPF-TOPSIS method [63]	$\alpha_{2}>\alpha_{4}>\alpha_{3}>\alpha_{5}>\alpha_{1}$
IVPF-WASPAS method [64]	$\alpha_{2}>\alpha_{4}>\alpha_{3}>\alpha_{5}>\alpha_{1}$
IVPF-VIKOR method [65]	$\alpha_{2}>\alpha_{4}>\alpha_{3}>\alpha_{1}>\alpha_{5}$
IVPF-Taxonomy method	$\alpha_{2}>\alpha_{4}>\alpha_{3}>\alpha_{5}>\alpha_{1}$

Then, the order similarity between the IVPFWA operator and IVPFWG operator proposed by Garg [56], IVPF-TOPSIS method [63], IVPF-WASPAS method [64] and IVPF-VIKOR method [65] and the proposed IVPF-Taxonomy method was analyzed with the WS coefficients [66, 67], the calculating results are shown in Table 6.

Table 6

WS coefficient calculation

	IVPF-Taxonomy method	IVPFWA operator	IVPFWG operator	IVPF-TOPSIS method	IVPF-WASPAS method	IVPF-VIKOR method
$\alpha_{1}$	2	2	2	2	2	2
$\alpha_{2}$	4	4	4	4	4	4
$\alpha_{3}$	3	3	3	3	3	3
$\alpha_{4}$	5	5	1	5	5	1
$\alpha_{5}$	1	1	5	1	1	5
Coefficients	WS	1.00	0.92	1.00	1.00	0.92

Compared the results with IVPFWA operator and IVPFWG operator proposed by Garg [56], IVPF-TOPSIS method [63], IVPF-WASPAS method [64] and IVPF-VIKOR method [65], the order results are slightly different, however, the best alternative is same. Thus, the IVPF-Taxonomy is effective and reasonable. Furthermore, the IVPF-Taxonomy method has the precious characteristics: IVPF-Taxonomy method uses the Euclidean distance to compare the alternatives in pairs to obtain a composite distance matrix, then homogenize the alternatives according to the composite distance matrix, and finally measure the difference between the positive ideal solution and the alternatives.

5. Conclusion

Environmental design is a multi-disciplinary and closely related profession. The market requires environmental design professionals to continuously improve their comprehensive literacy. The reform of practical teaching in environmental design is the foundation for the cultivation and development of applied talents. In terms of teaching, it is necessary to coordinate and adjust the design of professional courses in a timely manner. Starting from the current situation of environmental design majors in higher education institutions, this article proposes effective paths for practical teaching reform in response to existing problems. In order to explore the “teaching and learning” of guiding environmental design through multiple channels, cultivate students’ social activity and practical abilities, lay a solid foundation for them to enter the economy and society, and provide strong support for their future employment. The environmental design majors teaching quality evaluation is often deal with through the MAGDM issue. Recently, the Taxonomy has been used to cope with MAGDM problems. The IVPFSs are useful tools to depict uncertainty in MAGDM. This paper use IVPF-Taxonomy method to solve MAGDM with IVPFSs. Finally, a numerical example for environmental design majors teaching quality evaluation is proposed and a comparative analysis is also given to show the built methods. The main contribution of built paper are outlined: (1) the Taxonomy method is extended to IVPFSs; (2) the score function of IVPFSs and information entropy method is used to derive the attribute weights; (3) the IVPF-Taxonomy method is built to deal with MAGDM; (4) a case study about environmental design majors teaching quality evaluation is given to show the built method; (5) some comparative analysis are given to show the rationality of IVPF-Taxonomy method.

In our future researches, the proposed models and methods of IVPFNs needs to be extended to other MAGDM, risk assessment and some other uncertainty and fuzzy environemnts [45, 68, 69, 70, 71, 72, 73, 74].

Footnotes

Conflict of interest

The authors declare that they have no conflict of interest.

Ethical approval

This article does not contain any studies with human participants or animals performed by any of the authors.

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Approaches to multiple attribute group decision making under interval-valued pythagorean fuzzy sets and applications to environmental design majors teaching quality evaluation

Abstract

Keywords

1. Introduction

2. Preliminaries

4.1 Numerical example

Table 2 IVPFN overall matrix

Table 5 Order through different methods

Footnotes

Conflict of interest

Ethical approval

References

Table 2
IVPFN overall matrix

Table 5
Order through different methods