TODIM method for interval-valued Pythagorean fuzzy multiple attribute decision making

Abstract

Recently, the TODIM method has been used to solve multiple attribute decision making (MADM) problems. The interval-valued Pythagorean fuzzy sets (IVPFSs) are useful tools to depict uncertainty of the MADM. In this paper, we will extend the TODIM method to the MADM with the interval-valued Pythagorean fuzzy numbers (IVPFNs). Firstly, the definition, comparison and distance of IVPFNs are briefly introduced, and the steps of the classical TODIM method for MADM problems are presented. Then, the extended classical TODIM method is proposed to deal with MADM problems with the IVPFNs, and its significant characteristic is that it can fully consider the decision makers’ bounded rationality which is a real action in decision making. Furthermore, we develop the concept of the q-rung interval-valued orthopair fuzzy sets (q-RIVOFSs) and extend the TODIM method to q-RIVOFNs. Finally, a numerical example is proposed and a comparative analysis is given.

Keywords

Multiple attribute decision making (MADM)interval-valued Pythagorean fuzzy numbers (IVPFNs)TODIM method prospect theory q-rung interval-valued orthopair fuzzy sets (q-RIVOFSs)

1. Introduction

Atanassov [1] presented the concept of the intuitionistic fuzzy set (IFS) by considering the pairs of the degree of the membership and non-membership by generalizing the concept of the fuzzy set [2] such that their sum is not greater than one. After their existence, researchers have applied these theories in different disciplines [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17] and found that they are more profitable to handle the uncertainties during the analysis. Since the above defined theories have been successfully defined, but in some case, it is unable to handle the situation by IFS. For instance, if a decision maker (DM) may take the membership degrees of any element is 0.8 and 0.5 then clearly their sum is not less than one. Hence, under such types of cases, IFS have some sort of deficiencies. In order to resolve it, Pythagorean fuzzy set (PFS) [18, 19], an extension of IFSs, has emerged as an effective tool for depicting the uncertainty in the data. In this set, the condition of the sum of the degrees is replaced with their sum of squares is less than one and hence the PFS is more general than the IFS. Further, it is clearly that and hence PFS stand for such cases. After their existence, Zhang and Xu [20] presented the mathematical expression for PFS and developed the Pythagorean fuzzy TOPSIS (Technique for Order Preference by Similarity to Ideal Solution) technique for solving the decision-making problems. Zhang [21] presented Pythagorean fuzzy weighted and ordered weighted aggregation operators and a similarity measure based decision-making approach for solving multi criteria decision-making problems under the Pythagorean fuzzy environment. Peng and Yang [22] developed some fundamental properties of the PFNs. Reformat and Yager [23] applied the PFNs in handling the collaborative-based recommender system. Garg [25, 26] proposed the new generalized Pythagorean fuzzy information aggregation by using Einstein Operations. Zhang [27] extended the PFS to the interval-valued PFSs (IVPFSs). Garg [28] presented the averaging and geometric aggregation operators under the interval-valued PF (IVPF) environment. Also, a novel accuracy function has been presented to rank the IVPF numbers. However, in terms of the information measure theory, a novel accuracy function [28], correlation coefficient [29], improved accuracy function [30], improved score function [31] have been defined under the PFS and IVPFS and applied them to solve the decision-making problems. Recently, an aggregation operation for the PFSs by incorporating the confidence level of the decision makers during the decision-making process named as confidence based Pythagorean fuzzy weighted average and geometric operators have been proposed by Garg [32]. Apart from that, some others researchers are working in the field of the PFS or IVPFS and proposed various types of decision making approaches which are summarized by [24, 27, 30, 31, 33, 34, 35, 36, 37, 38, 39, 40, 41].

In order to depict the increasing complexity in actual world, the DMs’ risk attitudes should be taken into consideration to deal with MADM [42, 43, 44]. Based on the prospect theory, Gomes and Lima [45] established TODIM (an acronym in Portuguese for Interactive Multi-Criteria Decision Making) method to solve the MADM problems which the DMs’ psychological behaviors are considered. Some scholars have paid attention to depict the DMs’ attitudinal characters in the MADM [46, 47, 48]. And some scholars proposed fuzzy TODIM models [49, 50], intuitionistic fuzzy TODIM models [51, 52], Pythagorean fuzzy TODIM approach [42], multi-hesitant fuzzy linguistic TODIM approach [53], Hesitant Fuzzy Linguistic TODIM [54], interval type-2 fuzzy TODIM model [55], intuitionistic linguistic TODIM method [56] and 2-dimension uncertain linguistic TODIM method [57]. But there is no scholar to investigate the TODIM model with IVPFNS. Therefore, it is very necessary to pay abundant attention to this novel and worthy issue. The aim of this paper is to extend the TODIM idea to solve the MADM with the IVPFNS, to fill up this vacancy. In the Section 2, we give the basic concepts of PFSs and IVPFSs and classical TODIM method for MADM problems. In Section 3 we propose the TODIM method for IVPFN MADM problems. In Section 4, we develop the concept of the q-rung interval-valued orthopair fuzzy sets (q-RIVOFSs) and extend the TODIM method to q-RIVOFNs. In Section 5, an illustrative example is pointed out and a comparative analysis is conducted. We give a conclusion in Section 6.

2. Preliminaries

Some basic concepts and definitions of PFSs and IVPFSs are introduced.

2.1 PFSs and IVPFSs

Definition 1 [18, 19]. Let $X$ be a fix set. A PFS is an object having the form

$\displaystyle P=\left\{\left\langle x,\left(\mu_{P}(x),\nu_{P}(x)\right)\right% \rangle\left|x\in X\right.\right\}$ (1)

where the function $\mu_{P}:$ $X\to[0,1]$ defines the degree of membership and the function $\nu_{P}:$ $X\to[0,1]$ defines the degree of non-membership of the element $x\in X$ to $P$ , respectively, and, for every $x\in X$ , it holds that

$\displaystyle\left(\mu_{p}(x)\right)^{2}+\left(\nu_{p}(x)\right)^{2}\leqslant 1$ (2)

Definition 2 [18, 23]. Let $p_{1}=(\mu_{1},\nu_{1})$ , $p_{2}=(\mu_{2}$ , $\nu_{2})$ , and $p=(\mu,\nu)$ be three PFNs, and some basic operations on them are defined as follows:

$\displaystyle(1)p_{1}\oplus p_{2}\!=\!\left(\!\sqrt{(\mu_{1})^{2}\!+\!(\mu_{2}% )^{2}\!-\!(\mu_{1})^{2}(\mu_{2})^{2}},\nu_{1}\nu_{2}\right);$ $\displaystyle(2)p_{1}\otimes p_{2}\!=\!\left(\mu_{1}\mu_{2},\sqrt{(\nu_{1})^{2% }\!+\!(\nu_{2})^{2}\!-\!(\nu_{1})^{2}(\nu_{2})^{2}}\right);$ $\displaystyle\!\!\!\!\!\!\!\!\!\!\!\!(3)\lambda p\!=\!\left(\sqrt{1-(1-\mu^{2}% )^{\lambda},\nu^{\lambda}}\right),\lambda>0;$ $\displaystyle\!\!\!\!\!\!\!\!\!\!\!\!(4)(p)^{\lambda}\!=\!\left(\mu^{\lambda}% \sqrt{1-(1-\nu^{2})^{\lambda}}\right),\lambda>0.$

Furthermore, Zhang [27] developed the concept of the interval-valued Pythagorean fuzzy sets (IVPFSs).

Definition 3 [27]. Let $X$ be a fix set. An IVPFS is an object having the form

$\displaystyle\tilde{P}=\left\{\left\langle x,\left(\tilde{\mu}_{\tilde{P}}(x),% \tilde{\nu}_{\tilde{P}}(x)\right)\right\rangle\left|x\in X\right.\right\}$ (3)

where $\tilde{\mu}_{\tilde{P}}(x)\subset[0,1]$ and $\tilde{\nu}_{\tilde{P}}(x)\subset[0,1]$ are interval numbers $\tilde{\mu}_{\tilde{P}}(x)=\left[\mu_{\tilde{P}}^{L}(x),\mu_{\tilde{P}}^{R}(x)\right]$ , $\tilde{\nu}_{\tilde{P}}(x)=\left[\nu_{\tilde{P}}^{L}(x),\nu_{\tilde{P}}^{R}(x)\right]$ , with the condition 0 $\leqslant\left(\mu_{\tilde{P}}^{R}(x)\right)^{2}+\left(\nu_{\tilde{P}}^{R}(x)% \right)^{2}\leqslant$ 1, $\forall x\in X$ . The numbers $\tilde{\mu}_{\tilde{P}}(x),\tilde{\nu}_{\tilde{P}}(x)$ represent, respectively, the degree of positive membership, degree of negative membership and degree of negative membership of the element $x$ to $\tilde{P}$ . Then for $x\in X$ ,

$\displaystyle\tilde{\pi}_{\tilde{P}}(x)=\left[\pi_{\tilde{P}}^{L}(x),\pi_{% \tilde{P}}^{R}(x)\right]=\left[\sqrt{1-\left(\left(\mu_{\tilde{P}}^{R}(x)% \right)^{2}+\left(\nu_{\tilde{P}}^{R}(x)\right)^{2}\right)}\right.,\left.\sqrt% {1-\left(\left(\mu_{\tilde{P}}^{L}(x)\right)^{2}+\left(\nu_{\tilde{P}}^{L}(x)% \right)^{2}\right)}\right]$

could be called the degree of refusal membership of the element $x$ to $\tilde{P}$ .

For convenience, Zhang [27] called $\tilde{p}=\left(\left[u_{\tilde{p}}^{L},u_{\tilde{p}}^{R}\right]\right.$ , $\left.\left[v_{\tilde{p}}^{L},v_{\tilde{p}}^{R}\right]\right)$ an interval-valued Pythagorean fuzzy number (IVPFN), where $\tilde{u}_{\tilde{p}}\subset[0,1]$ , $\tilde{v}_{\tilde{p}}\subset[0,1]$ , $\left(u_{\tilde{p}}^{R}\right)^{2}+\left(\nu_{\tilde{p}}^{R}\right)^{2}\leqslant$ 1.

Definition 4 [28]. Let $\tilde{p}_{1}\!=\!\left(\left[u_{\tilde{p}_{1}}^{L},u_{\tilde{p}_{1}}^{R}% \right],\left[v_{\tilde{p}_{1}}^{L},v_{\tilde{p}_{1}}^{R}\right]\right)$ , $\tilde{p}_{2}=\left(\left[u_{\tilde{p}_{2}}^{L},u_{\tilde{p}_{2}}^{R}\right],% \left[v_{\tilde{p}_{2}}^{L},v_{\tilde{p}_{2}}^{R}\right]\right)$ , and $\tilde{p}=\left(\left[u_{\tilde{p}}^{L},u_{\tilde{p}}^{R}\right]\right.$ , $\left.\left[v_{\tilde{p}}^{L},v_{\tilde{p}}^{R}\right]\right)$ be three IVPFNs, and some basic operations on them are defined as follows:

$\displaystyle\!\!\!\!\!\!\!\!\!\!\!\!(1)\tilde{p}_{1}\oplus\tilde{p}_{2}=\left% (\left[\begin{array}[]{l}\sqrt{\left(u_{\tilde{p}_{1}}^{L}\right)^{2}+\left(u_% {\tilde{p}_{2}}^{L}\right)^{2}-\left(u_{\tilde{p}_{1}}^{L}\right)^{2}\left(u_{% \tilde{p}_{2}}^{L}\right)^{2}},\\ \sqrt{\left(u_{\tilde{p}_{1}}^{R}\right)^{2}+\left(u_{\tilde{p}_{2}}^{R}\right% )^{2}-\left(u_{\tilde{p}_{1}}^{R}\right)^{2}\left(u_{\tilde{p}_{2}}^{R}\right)% ^{2}}\end{array}\right]\right.,\quad\left.\left[v_{\tilde{p}_{1}}^{L}v_{\tilde% {p}_{2}}^{L},v_{\tilde{p}_{1}}^{R}v_{\tilde{p}_{2}}^{R}\right]\vphantom{\left[% \begin{array}[]{l}\sqrt{\left(u_{\tilde{p}_{1}}^{L}\right)^{2}},\\ \sqrt{\left(u_{\tilde{p}_{1}}^{R}\right)^{2}}\end{array}\right]}\right);$ $\displaystyle\!\!\!\!\!\!\!\!\!\!\!\!(2)\tilde{p}_{1}\otimes\tilde{p}_{2}=% \left(\vphantom{\left[\begin{array}[]{l}\sqrt{\left(v_{\tilde{p}_{1}}^{L}% \right)^{2}},\\ \sqrt{\left(v_{\tilde{p}_{1}}^{R}\right)^{2}}\end{array}\right]}\left[\mu_{% \tilde{p}_{1}}^{L}v_{\tilde{p}_{2}}^{L},\mu_{\tilde{p}_{1}}^{R}v_{\tilde{p}_{2% }}^{R}\right]\right.,\left.\left[\begin{array}[]{l}\sqrt{\left(v_{\tilde{p}_{1% }}^{L}\right)^{2}+\left(v_{\tilde{p}_{2}}^{L}\right)^{2}-\left(v_{\tilde{p}_{1% }}^{L}\right)^{2}\left(v_{\tilde{p}_{2}}^{L}\right)^{2}},\\ \sqrt{\left(v_{\tilde{p}_{1}}^{R}\right)^{2}+\left(v_{\tilde{p}_{2}}^{R}\right% )^{2}-\left(v_{\tilde{p}_{1}}^{R}\right)^{2}\left(v_{\tilde{p}_{2}}^{R}\right)% ^{2}}\end{array}\right]\right);$ $\displaystyle\!\!\!\!\!\!\!\!\!\!\!\!(3)\lambda\tilde{p}=\left(\left[\sqrt{1-% \left(1-\left(u_{\tilde{p}}^{L}\right)^{2}\right)^{\lambda}},\right.\right.% \left.\left.\sqrt{1-\left(1-\left(u_{\tilde{p}}^{R}\right)^{2}\right)^{\lambda% }}\right],\left[\left(v_{\tilde{p}}^{L}\right)^{\lambda},\left(v_{\tilde{p}}^{% R}\right)^{\lambda}\right]\vphantom{\left[\sqrt{1-\left(1-\left(u_{\tilde{p}}^% {L}\right)^{2}\right)^{\lambda}}\right]}\right),\lambda>0;$ $\displaystyle\!\!\!\!\!\!\!\!\!\!\!\!(4)\left(\tilde{p}\right)^{\lambda}=\left% (\left[\left(\mu_{\tilde{p}}^{L}\right)^{\lambda},\left(\mu_{\tilde{p}}^{R}% \right)^{\lambda}\right],\left[\sqrt{1-\left(1-\left(v_{\tilde{p}}^{L}\right)^% {2}\right)^{\lambda}}\right.\right.,\quad\left.\left.\sqrt{1-\left(1-\left(v_{% \tilde{p}}^{R}\right)^{2}\right)^{\lambda}}\right]\right),\lambda>0;$ $\displaystyle\!\!\!\!\!\!\!\!\!\!\!\!(5)\tilde{p}^{c}=\left(\left[v_{\tilde{p}% }^{L},v_{\tilde{p}}^{R}\right],\left[\mu_{\tilde{p}}^{L},\mu_{\tilde{p}}^{R}% \right]\right).$

Based on the Definition 4, Garg [28] derived the following properties easily.

Theorem 1 [28]. Let $\tilde{p}_{1}=\left(\left[u_{\tilde{p}_{1}}^{L},u_{\tilde{p}_{1}}^{R}\right],% \left[v_{\tilde{p}_{1}}^{L},v_{\tilde{p}_{1}}^{R}\right]\right)$ and $\tilde{p}_{2}=\left(\left[u_{\tilde{p}_{2}}^{L},u_{\tilde{p}_{2}}^{R}\right],% \left[v_{\tilde{p}_{2}}^{L},v_{\tilde{p}_{2}}^{R}\right]\right)$ be two IVPFNs, $\lambda,\lambda_{1},\lambda_{2}>$ 0, then

$\displaystyle(1)\tilde{p}_{1}\oplus\tilde{p}_{2}=\tilde{p}_{2}\oplus\tilde{p}_% {1};$ $\displaystyle(2)\tilde{p}_{1}\otimes\tilde{p}_{2}=\tilde{p}_{2}\otimes\tilde{p% }_{1}$ $\displaystyle(3)\lambda\left(\tilde{p}_{1}\oplus\tilde{p}_{2}\right)=\lambda% \tilde{p}_{1}\oplus\lambda\tilde{p}_{2};$ $\displaystyle(4)\left(\tilde{p}_{1}\otimes\tilde{p}_{2}\right)^{\lambda}=\left% (\tilde{p}_{1}\right)^{\lambda}\otimes\left(\tilde{p}_{2}\right)^{\lambda};$ $\displaystyle(5)\lambda_{1}\tilde{p}_{1}\oplus\lambda_{2}\tilde{p}_{1}=\left(% \lambda_{1}+\lambda_{2}\right)\tilde{p}_{1};$ $\displaystyle(6)\left(\tilde{p}_{1}\right)^{\lambda_{1}}\otimes\left(\tilde{p}% _{1}\right)^{\lambda_{2}}=\left(\tilde{p}_{1}\right)^{(\lambda_{1}+\lambda_{2}% )};$ $\displaystyle(7)\left(\left(\tilde{p}_{1}\right)^{\lambda_{1}}\right)^{\lambda% _{2}}=\left(\tilde{p}_{1}\right)^{\lambda_{1}\lambda_{2}}.$

Let $\tilde{p}=\left(\left[u_{p}^{L},u_{p}^{R}\right],\left[v_{p}^{L},v_{p}^{R}% \right]\right)$ be an IVPFN, $S(\tilde{p})$ $=\frac{1}{4}\left[\left(1\!+\!\left(u_{p}^{L}\right)^{2}\!-\!\left(v_{p}^{L}% \right)^{2}\right)+\left(1\!+\!\left(u_{p}^{R}\right)^{2}\!-\!\left(v_{p}^{R}% \right)^{2}\right)\right]$ and $H(\tilde{p})=\frac{\left(u_{p}^{L}\right)^{2}+\left(u_{p}^{R}\right)^{2}+\left% (v_{p}^{L}\right)^{2}+\left(v_{p}^{R}\right)^{2}}{2}$ are the score function and accuracy function of an IVPFN $\tilde{p}$ .

Definition 5. Let $\tilde{p}_{1}=\left(\left[u_{\tilde{p}_{1}}^{L},u_{\tilde{p}_{1}}^{R}\right],% \left[v_{\tilde{p}_{1}}^{L},v_{\tilde{p}_{1}}^{R}\right]\right)$ and $\tilde{p}_{2}=\left(\left[u_{p_{2}}^{L},u_{p_{2}}^{R}\right],\left[v_{p_{2}}^{% L},v_{p_{2}}^{R}\right]\right)$ be two IVPFNs, $S\left(\tilde{p}_{1}\right)$ and $S\left(\tilde{p}_{2}\right)$ be the scores of $\tilde{p}_{1}$ and $\tilde{p}_{2}$ , respectively, and let $H\left(\tilde{p}_{1}\right)$ and $H\left(\tilde{p}_{2}\right)$ be the accuracy degrees of $\tilde{p}_{1}$ and $\tilde{p}_{2}$ , respectively, then if $S\left(\tilde{p}_{1}\right)<S\left(\tilde{p}_{2}\right)$ , then $\tilde{p}_{1}<\tilde{p}_{2}$ ; if $S\left(\tilde{p}_{1}\right)=S\left(\tilde{p}_{2}\right)$ , then (1) if $H\left(\tilde{p}_{1}\right)=H\left(\tilde{p}_{2}\right)$ , then $\tilde{p}_{1}=\tilde{p}_{2}$ ; (2) if $H\left(\tilde{p}_{1}\right)<H\left(\tilde{p}_{2}\right)$ , then $\tilde{p}_{1}<\tilde{p}_{2}$ .

Definition 6 [58]. Let $\tilde{p}_{1}=\left(\left[u_{\tilde{p}_{1}}^{L},u_{\tilde{p}_{1}}^{R}\right],% \left[v_{\tilde{p}_{1}}^{L},v_{\tilde{p}_{1}}^{R}\right]\right)$ and $\tilde{p}_{2}=\left(\left[u_{p_{2}}^{L},u_{p_{2}}^{R}\right],\left[v_{p_{2}}^{% L},v_{p_{2}}^{R}\right]\right)$ be two IVPFN, then the normalized Hamming distance between $\tilde{p}_{1}$ and $\tilde{p}_{2}$ is:

$\displaystyle d\left(\tilde{p}_{1},\tilde{p}_{2}\right)=$ $\displaystyle\frac{1}{4}\left(\begin{array}[]{l}\left|\left(u_{\tilde{p}_{1}}^% {L}\right)^{2}\!-\!\left(u_{p_{2}}^{L}\right)^{2}\right|+\left|\left(u_{\tilde% {p}_{1}}^{R}\right)^{2}\!-\!\left(u_{p_{2}}^{R}\right)^{2}\right|+\\ \left|\left(v_{\tilde{p}_{1}}^{L}\right)^{2}\!-\!\left(v_{p_{2}}^{L}\right)^{2% }\right|+\left|\left(v_{\tilde{p}_{1}}^{R}\right)^{2}\!-\!\left(v_{p_{2}}^{R}% \right)^{2}\right|\end{array}\right)$ (4)

2.2 The TODIM approach

Gomes and Lima [45] proposed the TODIM approach to solve the MADM which considers the DM’s psychological behavior.

Let $G=\{G_{1},G_{2},\ldots,G_{n}\}$ be the attributes, $w=(w_{1},w_{2},\ldots,w_{n})$ be the weight of $G_{j}$ , 0 $\leqslant w_{j}\leqslant$ 1, $\sum_{j=1}^{n}w_{j}=$ 1. $A=\{A_{1},A_{2},\ldots,A_{m}\}$ are alternatives. Let $A=(a_{ij})_{m\times n}$ be a decision matrix, where $a_{ij}$ is given for the alternative $A_{i}$ under the $G_{j}$ , $i=$ 1, 2, …, $m$ , $j=$ 1, 2, …, $n$ . We set $w_{jr}=w_{j}/w_{r}$ ( $j,r=$ 1, 2, …, $n$ ) are relative weight of $G_{j}$ to $G_{r}$ , and $w_{r}=\max\{w_{j}|j=1,2,\ldots,n\}$ , and 0 $\leqslant w_{jr}\leqslant$ 1.

Then the traditional TODIM model concludes the following computing steps:

Step 1. Normalizing $A=(a_{ij})_{m\times n}$ into $B=(b_{ij})_{m\times n}$ .

Step 2. Computing the dominance degree $A_{i}$ over every alternative $A_{t}$ under attribute $G_{j}$ :

$\displaystyle\!\!\!\!\!\!\!\delta(A_{i},A_{t})=\sum_{j=1}^{n}\phi_{j}(A_{i},A_% {t}),(i,t=1,2,\ldots,m)$ (5)

where

$\displaystyle\phi_{j}(A_{i},A_{t})=$ $\displaystyle\left\{\begin{array}[]{ll}\sqrt{w_{jr}(b_{ij}-b_{tj})}\Big{/}\sum% \limits_{j=1}^{n}w_{jr},&\text{if }b_{ij}-b_{tj}>0\\ 0,&\text{if }b_{ij}-b_{tj}=0\\ \!-\frac{1}{\theta}\sqrt{\!\left(\sum\limits_{j=1}^{n}w_{jr}\!\right)\!(b_{tj}% \!-\!b_{ij})/w_{jr}},&\text{if }b_{ij}-b_{tj}<0\end{array}\right.$ (6)

and the parameter $\theta$ shows the attenuation factor of the losses. If $b_{ij}-b_{tj}>$ 0, then $\phi_{j}(A_{i},A_{t})$ represents a gain; if $b_{ij}-b_{tj}<$ 0, then $\phi_{j}(A_{i},A_{t})$ signifies a loss.

Step 3. Deriving the overall dominance value of $A_{i}$ by the Eq. (2.2):

$\displaystyle\phi(A_{i})=$ $\displaystyle\frac{\sum\limits_{t=1}^{m}\delta(A_{i},A_{t})-\mathop{\min}% \limits_{i}\left\{\sum\limits_{t=1}^{m}\delta(A_{i},A_{t})\right\}}{\mathop{% \max}\limits_{i}\left\{\sum\limits_{t=1}^{m}\delta(A_{i},A_{t})\right\}-% \mathop{\min}\limits_{i}\left\{\sum\limits_{t=1}^{m}\delta(A_{i},A_{t})\right% \}},$ (7) $\displaystyle i=1,2,\ldots,m.$

Step 4. Ranking all alternatives and selecting the most desirable alternative in accordance with $\phi(A_{i})$ . The alternative with minimum value is the worst. Inversely, the maximum value is the best one.

3. TODIM method for MADM problems with IVPFNs

Let $A=\{A_{1},A_{2},\ldots,A_{m}\}$ be alternatives, and $G=$ $\{G_{1}$ , $G_{2}$ , …, $G_{n}\}$ be attributes. Let $w=(w_{1}$ , $w_{2}$ , …, $w_{n})$ be the weight of attributes, where $w_{j}\in[0,1]$ , $\sum_{j=1}^{n}w_{j}=$ 1. Suppose that $\tilde{R}=\left(\tilde{r}_{ij}\right)_{m\times n}=\left(\left[\mu_{ij}^{L},\mu% _{ij}^{R}\right],\left[\nu_{ij}^{L},\nu_{ij}^{R}\right]\right)_{m\times n}$ be a IVPFN matrix, where $\tilde{r}_{ij}=\left(\left[\mu_{ij}^{L},\mu_{ij}^{R}\right],\left[\nu_{ij}^{L}% ,\nu_{ij}^{R}\right]\right)$ , which is an attribute value, given by an expert, for the alternative under $A_{i}$ under $G_{j}$ , $\left[\mu_{ij}^{L},\mu_{ij}^{R}\right]\subset[0,1]$ , $\left[\nu_{ij}^{L},\nu_{ij}^{R}\right]\subset[0,1]$ , $\left(\mu_{ij}^{R}\right)^{2}+\left(\nu_{ij}^{R}\right)^{2}\leqslant$ 1, $i=$ 1, 2, …, $m$ , $j=$ 1, 2, …, $n$ .

To solve the MADM problem with IVPFNs, we try to present an IVPF TODIM model based on the prospect theory and can depict the DMs’ behaviors under risk.

Firstly, we calculate the relative weight of each attribute $G_{j}$ as:

$\displaystyle w_{jr}=w_{j}/w_{r},j,r=1,2,\ldots,n.$ (8)

where $w_{j}$ is the weight of the attribute of $G_{j}$ , $w_{r}=\max\{w_{j}|j=1,2,\ldots,n\}$ , and 0 $\leqslant w_{jr}\leqslant$ 1.

Based on the Eq. (9), we can derive the dominance degree of $A_{i}$ over each alternative $A_{t}$ with respect to the attribute $G_{j}$ :

$\displaystyle\phi_{j}(A_{i},A_{t})=$ $\displaystyle\left\{\begin{array}[]{ll}\sqrt{w_{jr}d\left(\tilde{r}_{ij},% \tilde{r}_{tj}\right)\Big{/}\sum\limits_{j=1}^{n}w_{jr}},&\text{if }\tilde{r}_% {ij}>\tilde{r}_{tj}\\ 0,&\text{if }\tilde{r}_{ij}=\tilde{r}_{tj}\\ -\frac{1}{\theta}\sqrt{\!\left(\sum\limits_{j=1}^{n}w_{jr}\!\right)d\left(% \tilde{r}_{ij},\tilde{r}_{tj}\right)/w_{jr}},&\text{if }\tilde{r}_{ij}<\tilde{% r}_{tj}\end{array}\right.$ (9) $\displaystyle d\left(\tilde{r}_{ij},\tilde{r}_{tj}\right)=$ $\displaystyle\frac{1}{4}\left(\begin{array}[]{l}\left|\left(u_{ij}^{L}\right)^% {2}\!-\!\left(u_{tj}^{L}\right)^{2}\right|+\left|\left(u_{ij}^{R}\right)^{2}\!% -\!\left(u_{tj}^{R}\right)^{2}\right|+\\ \left|\left(v_{ij}^{L}\right)^{2}\!-\!\left(v_{tj}^{L}\right)^{2}\right|+\left% |\left(v_{ij}^{R}\right)^{2}\!-\!\left(v_{tj}^{R}\right)^{2}\right|\end{array}\right)$ (10)

where the parameter $\theta$ shows the attenuation factor of the losses, and $d\left(\tilde{r}_{ij},\tilde{r}_{tj}\right)$ is to measure the distances between the IVPFNs $\tilde{r}_{ij}$ and $\tilde{r}_{tj}$ by Definition 6. If $\tilde{r}_{ij}>\tilde{r}_{tj}$ , then $\phi_{j}(A_{i},A_{t})$ represents a gain; if $\tilde{r}_{ij}<\tilde{r}_{tj}$ , then $\phi_{j}(A_{i},A_{t})$ signifies a loss.

For indicating functions $\phi_{j}(A_{i},A_{t})$ clearly, a dominance degree matrix $\phi_{j}=[\phi_{j}(A_{i},A_{t})]_{m\times m}$ under $G_{j}$ is expressed as:

$\displaystyle\phi_{j}=[\phi_{j}(A_{i},A_{t})]_{m\times m}=$ $\displaystyle\bordermatrix{&A_{1}&A_{2}&\ldots&A_{m}\cr A_{1}&0&\phi_{j}(A_{1}% ,A_{2})&\ldots&\phi_{j}(A_{1},A_{m})\cr A_{2}&\phi_{j}(A_{2},A_{1})&0&\ldots&% \phi_{j}(A_{2},A_{m})\cr\vdots&\vdots&\ldots&\vdots\cr A_{m}&\phi_{j}(A_{m},A_% {1})&\phi_{j}(A_{m},A_{2})&\ldots&0}$ $\displaystyle\quad j=1,2,\ldots,n$ (11)

On the basis of Eq. (3), the overall dominance degree $\delta(A_{i},A_{t})$ of the $A_{i}$ over each $A_{t}$ can be calculated:

$\displaystyle\!\!\!\!\!\!\!\delta(A_{i},A_{t})\!=\!\sum_{j=1}^{n}\phi_{j}(A_{i% },A_{t}),(i,t=1,2,\ldots,m)$ (12)

Thus, the overall dominance degree matrix $\delta=[\delta(A_{i},A_{t})]_{m\times m}$ can be derived by Eq. (12):

$\displaystyle\delta_{j}=[\delta_{j}(A_{i},A_{t})]_{m\times m}=$ $\displaystyle\bordermatrix{&A_{1}&A_{2}&\ldots&A_{m}\cr A_{1}&0&\delta_{j}(A_{% 1},A_{2})&\ldots&\delta_{j}(A_{1},A_{m})\cr A_{2}&\delta_{j}(A_{2},A_{1})&0&% \ldots&\delta_{j}(A_{2},A_{m})\cr&\vdots&\vdots&\ldots&\vdots\cr A_{m}&\delta_% {j}(A_{m},A_{1})&\delta_{j}(A_{m},A_{2})&\ldots&0}$ (13)

Then, the overall value of each $A_{i}$ can be calculated Eq. (3):

$\displaystyle\delta(A_{i})=$ $\displaystyle\frac{\sum\limits_{t=1}^{m}\delta(A_{i},A_{t})-\mathop{\min}% \limits_{i}\left\{\sum\limits_{t=1}^{m}\delta(A_{i},A_{t})\right\}}{\mathop{% \max}\limits_{i}\left\{\sum\limits_{t=1}^{m}\delta(A_{i},A_{t})\right\}-% \mathop{\min}\limits_{i}\left\{\sum\limits_{t=1}^{m}\delta(A_{i},A_{t})\right% \}},$ (14) $\displaystyle i=1,2,\ldots,m.$

And the greater the overall value $\delta(A_{i})$ , the better the alternative $A_{i}$ .

In general, IVPFN TODIM model includes the computing steps:

Step 1. Identify the IVPFN matrix $\tilde{R}=\left(\tilde{r}_{ij}\right)_{m\times n}=\left(\left[\mu_{ij}^{L},\mu% _{ij}^{R}\right],\left[\nu_{ij}^{L},\nu_{ij}^{R}\right]\right)_{m\times n}$ in the MADM, where $\tilde{r}_{ij}$ is an IVPFN.

Step 2. Calculate the relative weight of $G_{j}$ by using Eq. (8).

Step 3. Calculate the dominance degree $\phi_{j}(A_{i},A_{t})$ of $A_{i}$ over each alternative $A_{t}$ under attribute $G_{j}$ by Eq. (9).

Step 4. Calculate the overall dominance degree $\delta(A_{i},A_{t})$ of $A_{i}$ over each alternative $A_{t}$ by using Eq. (12).

Step 5. Derive the overall value $\delta(A_{i})$ of each alternative $A_{i}$ using Eq. (3).

Step 6. Determine the order of the alternatives in accordance with $\delta(A_{i})$ ( $i=$ 1, 2, …, $m$ ).

4. TODIM method for MADM problems with q-RIVOFNs

Based on the q-rung orthopair fuzzy sets (q-ROFSs) [59, 60] and interval-valued Pythagorean fuzzy sets (IVPFSs) [27]. Furthermore, we develop the concept of the q-rung interval-valued orthopair fuzzy sets (q-RIVOFSs).

Definition 7. Let $X$ be a fix set. A q-RIVOFSs is an object having the form

$\displaystyle\tilde{Q}=\left\{\left\langle x,\left(\tilde{\mu}_{\tilde{Q}}(x),% \tilde{\nu}_{\tilde{Q}}(x)\right)\right\rangle\left|x\in X\right.\right\}$ (15)

where $\tilde{\mu}_{\tilde{Q}}(x)\subset[0,1]$ and $\tilde{\nu}_{\tilde{Q}}(x)\subset[0,1]$ are interval numbers $\tilde{\mu}_{\tilde{Q}}(x)=\left[\mu_{\tilde{Q}}^{L}(x),\mu_{\tilde{Q}}^{R}(x)% \right],\tilde{\nu}_{\tilde{Q}}(x)=\left[\nu_{\tilde{Q}}^{L}(x),\nu_{\tilde{Q}% }^{R}(x)\right]$ , with the condition 0 $\leqslant\left(\mu_{\tilde{Q}}^{R}(x)\right)^{q}+\left(\nu_{\tilde{Q}}^{R}(x)% \right)^{q}\leqslant$ 1, $\forall x\in X$ , $q\geqslant$ 1. The numbers $\tilde{\mu}_{\tilde{Q}}(x),{\tilde{\nu}_{\tilde{Q}}}(x)$ represent, respectively, the degree of positive membership, degree of negative membership and degree of negative membership of the element $x$ to $\tilde{Q}$ . Then for $x\in X$ ,

$\displaystyle\tilde{\pi}_{\tilde{Q}}(x)=\left[\pi_{\tilde{Q}}^{L}(x),\pi_{% \tilde{Q}}^{R}(x)\right]=\left[\sqrt[q]{1-\left(\left(\mu_{\tilde{Q}}^{R}(x)% \right)^{2}+\left(\nu_{\tilde{Q}}^{R}(x)\right)^{2}\right)}\right.,\left.\sqrt% [q]{1-\left(\left(\mu_{\tilde{Q}}^{L}(x)\right)^{2}+\left(\nu_{\tilde{Q}}^{L}(% x)\right)^{2}\right)}\right]$

could be called the degree of refusal membership of the element $x$ to $\tilde{Q}$ .

For convenience, we called $\tilde{q}=\left(\left[u_{\tilde{q}}^{L},u_{\tilde{q}}^{R}\right],\left[v_{% \tilde{q}}^{L}\right.\right.$ , $\left.\left.v_{\tilde{q}}^{R}\right]\right)$ a q-rung interval-valued orthopair fuzzy number (q-RIVOFN).

Let $\tilde{q}=\left(\left[u_{\tilde{q}}^{L},u_{\tilde{q}}^{R}\right],\left[v_{% \tilde{q}}^{L},v_{\tilde{q}}^{R}\right]\right)$ be an q-RIVOFN, $S\left({\tilde{q}}\right)=\frac{1}{4}\left[\left(1+\left(u_{\tilde{q}}^{L}% \right)^{q}-\left(v_{\tilde{q}}^{L}\right)^{q}\right)+\left(1+\left(u_{\tilde{% q}}^{R}\right)^{q}\right.\right.$ $-$ $\left.\left.\left(v_{\tilde{q}}^{R}\right)^{q}\right)\right]$ and $H\left({\tilde{q}}\right)=\frac{\left(u_{\tilde{q}}^{L}\right)^{q}+\left(u_{% \tilde{q}}^{R}\right)^{q}+\left(v_{\tilde{q}}^{L}\right)^{q}+\left(v_{\tilde{q% }}^{R}\right)^{q}}{2}$ are the score function and accuracy function of an q-RIVOFN $\tilde{q}$ .

Definition 8.Let ${\tilde{q}}_{1}=\left(\left[u_{{\tilde{q}}_{1}}^{L},u_{{\tilde{q}}_{1}}^{R}% \right],\left[v_{{\tilde{q}}_{1}}^{L},v_{{\tilde{q}}_{1}}^{R}\right]\right)$ and ${\tilde{q}}_{2}=\left(\left[u_{{\tilde{q}}_{2}}^{L},u_{{\tilde{q}}_{2}}^{R}% \right],\left[v_{{\tilde{q}}_{2}}^{L},v_{{\tilde{q}}_{2}}^{R}\right]\right)$ be two q-RIVOFNs, $S\left({\tilde{q}}_{1}\right)$ and $S\left({\tilde{q}}_{2}\right)$ be the scores of ${\tilde{q}}_{1}$ and ${\tilde{q}}_{2}$ , respectively, and let $H\left({\tilde{q}}_{1}\right)$ and $H\left({\tilde{q}}_{2}\right)$ be the accuracy degrees of ${\tilde{q}}_{1}$ and ${\tilde{q}}_{2}$ , respectively, then if $S\left({\tilde{q}}_{1}\right)<S\left({\tilde{q}}_{2}\right)$ , then ${\tilde{q}}_{1}<{\tilde{q}}_{2}$ ; if $S\left({\tilde{q}}_{1}\right)=S\left({\tilde{q}}_{2}\right)$ , then (1) if $H\left({\tilde{q}}_{1}\right)=H\left({\tilde{q}}_{2}\right)$ , then ${\tilde{q}}_{1}={\tilde{q}}_{2}$ ; (2) if $H\left({\tilde{q}}_{1}\right)<H\left({\tilde{q}}_{2}\right)$ , then ${\tilde{q}}_{1}<{\tilde{q}}_{2}$ .

Definition 9. Let ${\tilde{q}}_{1}=\left(\left[u_{{\tilde{q}}_{1}}^{L},u_{{\tilde{q}}_{1}}^{R}% \right],\left[v_{{\tilde{q}}_{1}}^{L},v_{{\tilde{q}}_{1}}^{R}\right]\right)$ , ${\tilde{q}}_{2}=\left(\left[u_{{\tilde{q}}_{2}}^{L},u_{{\tilde{q}}_{2}}^{R}% \right],\left[v_{{\tilde{q}}_{2}}^{L},v_{{\tilde{q}}_{2}}^{R}\right]\right)$ , and ${\tilde{q}}=\!\left(\left[u_{\tilde{q}}^{L},u_{\tilde{q}}^{R}\right]\right.$ , $\left.\left[v_{\tilde{q}}^{L},v_{\tilde{q}}^{R}\right]\right)$ be three q-RIVOFNs, and some basic operations on them are defined as follows:

$\displaystyle\!\!\!\!\!\!\!\!\!\!\!\!(1)\tilde{q}_{1}\oplus\tilde{q}_{2}=\left% (\left[\begin{array}[]{l}\sqrt[q]{\left(u_{\tilde{q}_{1}}^{L}\right)^{2}+\left% (u_{\tilde{q}_{2}}^{L}\right)^{2}-\left(u_{\tilde{q}_{1}}^{L}\right)^{2}\left(% u_{\tilde{q}_{2}}^{L}\right)^{2}},\\ \sqrt[q]{\left(u_{\tilde{q}_{1}}^{R}\right)^{2}+\left(u_{\tilde{q}_{2}}^{R}% \right)^{2}-\left(u_{\tilde{q}_{1}}^{R}\right)^{2}\left(u_{\tilde{q}_{2}}^{R}% \right)^{2}}\end{array}\right]\right.,\quad\left.\left[v_{\tilde{q}_{1}}^{L}v_% {\tilde{q}_{2}}^{L},v_{\tilde{q}_{1}}^{R}v_{\tilde{q}_{2}}^{R}\right]\vphantom% {\left[\begin{array}[]{l}\sqrt[q]{\left(u_{\tilde{q}_{1}}^{L}\right)^{2}},\\ \sqrt[q]{\left(u_{\tilde{q}_{1}}^{R}\right)^{2}}\end{array}\right]}\right);$ $\displaystyle\!\!\!\!\!\!\!\!\!\!\!\!(2)\tilde{q}_{1}\otimes\tilde{q}_{2}=% \left(\vphantom{\left[\begin{array}[]{l}\sqrt[q]{\left(v_{\tilde{q}_{1}}^{L}% \right)^{2}},\\ \sqrt[q]{\left(v_{\tilde{q}_{1}}^{R}\right)^{2}}\end{array}\right]}\left[\mu_{% \tilde{q}_{1}}^{L}v_{\tilde{q}_{2}}^{L},\mu_{\tilde{q}_{1}}^{R}v_{\tilde{q}_{2% }}^{R}\right]\right.,\left.\left[\begin{array}[]{l}\sqrt[q]{\left(v_{\tilde{q}% _{1}}^{L}\right)^{2}+\left(v_{\tilde{q}_{2}}^{L}\right)^{2}-\left(v_{\tilde{q}% _{1}}^{L}\right)^{2}\left(v_{\tilde{q}_{2}}^{L}\right)^{2}},\\ \sqrt[q]{\left(v_{\tilde{q}_{1}}^{R}\right)^{2}+\left(v_{\tilde{q}_{2}}^{R}% \right)^{2}-\left(v_{\tilde{q}_{1}}^{R}\right)^{2}\left(v_{\tilde{q}_{2}}^{R}% \right)^{2}}\end{array}\right]\right);$ $\displaystyle\!\!\!\!\!\!\!\!\!\!\!\!(3)\lambda\tilde{q}=\left(\left[\sqrt[q]{% 1-\left(1-\left(u_{\tilde{q}}^{L}\right)^{2}\right)^{\lambda}},\right.\right.% \left.\left.\sqrt[q]{1-\left(1-\left(u_{\tilde{q}}^{R}\right)^{2}\right)^{% \lambda}}\right],\left[\left(v_{\tilde{q}}^{L}\right)^{\lambda},\left(v_{% \tilde{q}}^{R}\right)^{\lambda}\right]\vphantom{\left[\sqrt[q]{1-\left(1-\left% (u_{\tilde{q}}^{L}\right)^{2}\right)^{\lambda}}\right]}\right),\lambda>0;$ $\displaystyle\!\!\!\!\!\!\!\!\!\!\!\!(4)\left(\tilde{q}\right)^{\lambda}=\left% (\left[\left(\mu_{\tilde{q}}^{L}\right)^{\lambda},\left(\mu_{\tilde{q}}^{R}% \right)^{\lambda}\right],\left[\sqrt[q]{1-\left(1-\left(v_{\tilde{q}}^{L}% \right)^{2}\right)^{\lambda}}\right.\right.,\quad\left.\left.\sqrt[q]{1-\left(% 1-\left(v_{\tilde{q}}^{R}\right)^{2}\right)^{\lambda}}\right]\right),\lambda>0;$ $\displaystyle\!\!\!\!\!\!\!\!\!\!\!\!(5)\tilde{q}^{c}=\left(\left[v_{\tilde{q}% }^{L},v_{\tilde{q}}^{R}\right],\left[\mu_{\tilde{q}}^{L},\mu_{\tilde{q}}^{R}% \right]\right).$

Let $A=\{A_{1},A_{2},\ldots,A_{m}\}$ be alternatives, and $G=\{G_{1}$ , $G_{2}$ , …, $G_{n}\}$ be attributes. Let $w=(w_{1}$ , $w_{2}$ , …, $w_{n})$ be the weight of attributes, where $w_{j}\in[0,1]$ , $\sum_{j=1}^{n}w_{j}=$ 1. Suppose that $\tilde{Q}=\left({\tilde{q}}_{ij}\right)_{m\times n}=\left(\left[\mu_{ij}^{L},% \mu_{ij}^{R}\right],\left[\nu_{ij}^{L},\nu_{ij}^{R}\right]\right)_{m\times n}$ be a q-rung interval-valued orthopair fuzzy matrix, where ${\tilde{q}}_{ij}=\left(\left[\mu_{ij}^{L},\mu_{ij}^{R}\right],\left[\nu_{ij}^{% L},\nu_{ij}^{R}\right]\right)$ , which is an attribute value, given by an expert, for the alternative $A_{i}$ under $G_{j}$ , $\left[\mu_{ij}^{L},\mu_{ij}^{R}\right]\subset[0,1]$ , $\left[\nu_{ij}^{L},\nu_{ij}^{R}\right]\subset[0,1]$ , $\left(\mu_{ij}^{R}\right)^{q}+\left(\nu_{ij}^{R}\right)^{q}\leqslant$ 1, $i=$ 1, 2, …, $m$ , $j=$ 1, 2, …, $n$ .

To solve the MADM problem with q-RIVOFNs, we try to present a q-RIVOFN TODIM model based on the prospect theory and can depict the DMs’ behaviors under risk.

Firstly, we calculate the relative weight of each attribute $G_{j}$ as:

$\displaystyle w_{jr}=w_{j}/w_{r},j,r=1,2,\ldots,n.$ (16)

where $w_{j}$ is the weight of the attribute of $G_{j}$ , $w_{r}=\max\{w_{j}|j=1,2,\ldots,n\}$ , and 0 $\leqslant w_{jr}\leqslant$ 1.

Based on the Eq. (17), we can derive the dominance degree of $A_{i}$ over each alternative $A_{t}$ with respect to the attribute $G_{j}$ :

$\displaystyle\phi_{j}(A_{i},A_{t})=$ $\displaystyle\left\{\begin{array}[]{ll}\sqrt{w_{jr}d\left(\tilde{r}_{ij},% \tilde{r}_{tj}\right)\Big{/}\sum\limits_{j=1}^{n}w_{jr}},&\text{if }\tilde{r}_% {ij}>\tilde{r}_{tj}\\ 0,&\text{if }\tilde{r}_{ij}=\tilde{r}_{tj}\\ -\frac{1}{\theta}\sqrt{\!\left(\sum\limits_{j=1}^{n}w_{jr}\!\right)d\left(% \tilde{r}_{ij},\tilde{r}_{tj}\right)/w_{jr}},&\text{if }\tilde{r}_{ij}<\tilde{% r}_{tj}\end{array}\right.$ (17) $\displaystyle d\left(\tilde{r}_{ij},\tilde{r}_{tj}\right)=$ $\displaystyle\frac{1}{4}\left(\begin{array}[]{l}\left|\left(u_{ij}^{L}\right)^% {q}\!-\!\left(u_{tj}^{L}\right)^{q}\right|+\left|\left(u_{ij}^{R}\right)^{q}\!% -\!\left(u_{tj}^{R}\right)^{q}\right|+\\ \left|\left(v_{ij}^{L}\right)^{q}\!-\!\left(v_{tj}^{L}\right)^{q}\right|+\left% |\left(v_{ij}^{R}\right)^{q}\!-\!\left(v_{tj}^{R}\right)^{q}\right|\end{array}\right)$ (18)

where the parameter $\theta$ shows the attenuation factor of the losses, and $d\left(\tilde{r}_{ij},\tilde{r}_{tj}\right)$ is to measure the distances between the q-RIVOFNs $\tilde{r}_{ij}$ and $\tilde{r}_{tj}$ . If $\tilde{r}_{ij}>\tilde{r}_{tj}$ , then $\phi_{j}(A_{i},A_{t})$ represents a gain; if $\tilde{r}_{ij}<\tilde{r}_{tj}$ , then $\phi_{j}(A_{i},A_{t})$ signifies a loss.

For indicating functions $\phi_{j}(A_{i},A_{t})$ clearly, a dominance degree matrix $\phi_{j}=\left[\phi_{j}(A_{i},A_{t})\right]_{m\times m}$ under $G_{j}$ is expressed as:

$\displaystyle\phi_{j}=[\phi_{j}(A_{i},A_{t})]_{m\times m}=$ $\displaystyle\bordermatrix{&A_{1}&A_{2}&\ldots&A_{m}\cr A_{1}&0&\phi_{j}(A_{1}% ,A_{2})&\ldots&\phi_{j}(A_{1},A_{m})\cr A_{2}&\phi_{j}(A_{2},A_{1})&0&\ldots&% \phi_{j}(A_{2},A_{m})\cr&\vdots&\vdots&\ldots&\vdots\cr A_{m}&\phi_{j}(A_{m},A% _{1})&\phi_{j}(A_{m},A_{2})&\ldots&0}$ (19) $\displaystyle\quad j=1,2,\ldots,n$

On the basis of Eq. (4), the overall dominance degree $\delta(A_{i},A_{t})$ of the $A_{i}$ over each $A_{t}$ can be calculated:

$\displaystyle\!\!\!\!\!\!\!\delta(A_{i},A_{t})\!=\!\sum_{j=1}^{n}\phi_{j}(A_{i% },A_{t}),(i,t=1,2,\ldots,m)$ (20)

Thus, the overall dominance degree matrix $\delta=\left[\delta(A_{i},A_{t})\right]_{m\times m}$ can be derived by Eq. (20):

Then, the overall value of each $A_{i}$ can be calculated Eq. (4):

And the greater the overall value $\delta(A_{i})$ , the better the alternative $A_{i}$ .

5. Numerical example and comparative analysis

5.1 Numerical example

In this part, a numerical example is given to show potential evaluation of emerging technology commercialization with IVPFNs. Five possible emerging technology enterprises (ETEs) $A_{i}$ ( $i=$ 1, 2, 3, 4, 5) are to be evaluated and selected. Four attributes are selected to evaluate the five possible ETEs: (1) $G_{1}$ is the employment creation; (2) $G_{2}$ is the development of science and technology; (3) $G_{3}$ is the technical advancement; (4) $G_{4}$ is the industrialization infrastructure. The five ETEs $A_{i}$ ( $i=$ 1, 2, 3, 4, 5) are to be evaluated by using the IVPFNs under the above four attributes (whose weighting vector $\omega=(0.2,0.1,0.3,0.4)^{T}$ ), as listed in the following matrix.

$\displaystyle\!\!\!\!\!\!\!\!\!\tilde{R}=\left[\begin{array}[]{ll}([0.4,0.5],[% 0.3,0.4])&([0.5,0.6],[0.3,0.4])\\ ([0.6,0.7],[0.4,0.5])&([0.6,0.7],[0.1,0.2])\\ ([0.5,0.6],[0.3,0.4])&([0.4,0.5],[0.6,0.7])\\ ([0.7,0.8],[0.1,0.2])&([0.5,0.6],[0.2,0.3])\\ ([0.5,0.6],[0.3,0.4])&([0.3,0.4],[0.7,0.8])\end{array}\right.\left.\begin{% array}[]{ll}([0.2,0.3],[0.5,0.6])&([0.4,0.5],[0.6,0.7])\\ ([0.6,0.7],[0.1,0.2])&([0.3,0.4],[0.4,0.5])\\ ([0.4,0.5],[0.2,0.3])&([0.2,0.4],[0.5,0.6])\\ ([0.2,0.3],[0.3,0.4])&([0.4,0.5],[0.5,0.6])\\ ([0.6,0.7],[0.5,0.6])&([0.4,0.5],[0.7,0.8])\end{array}\right]$

Then, we use the proposed model to select the best ETE.

Firstly, since $w_{4}=\max\{w_{1},w_{2},w_{3},w_{4}\}$ , then $G_{4}$ is the reference attribute and the reference attribute’s weight is $w_{r}=$ 0.4. Then, we can calculate the relative weights of the attributes $G_{j}$ ( $j=$ 1, 2, 3, 4) as: $w_{1r}=$ 0.50, $w_{2r}=$ 0.25, $w_{3r}=$ 0.75 and $w_{4r}=$ 1.00. Let $\theta=$ 2.5, then the dominance degree matrix $\phi_{j}(A_{i},A_{t})$ ( $i=$ 1, 2, 3, 4, 5) with respect to $G_{j}$ can be calculated:

$\displaystyle\phi_{1}=$ $\displaystyle\bordermatrix{&A_{1}&A_{2}&A_{3}&A_{4}&A_{5}\cr A_{1}&0.0000&-0.3% 464&-0.2000&-0.4290&-0.2000\cr A_{2}&0.1732&0.0000&0.1414&-0.3578&0.1414\cr A_% {3}&0.1000&-0.2828&0.0000&-0.3795&0.0000\cr A_{4}&0.2145&0.1789&0.1897&0.0000&% 0.1897\cr A_{5}&0.1000&-0.2828&0.0000&-0.3795&0.0000}$ $\displaystyle\phi_{2}=$ $\displaystyle\bordermatrix{&A_{1}&A_{2}&A_{3}&A_{4}&A_{5}\cr A_{1}&0.0000&-0.4% 195&0.1414&-0.2191&0.1761\cr A_{2}&0.1049&0.0000&0.1761&0.0894&0.2049\cr A_{3}% &-0.5657&-0.7043&0.0000&-0.6066&0.1049\cr A_{4}&0.0548&-0.3578&0.1517&0.0000&0% .1844\cr A_{5}&-0.7043&-0.8198&-0.4195&-0.7376&0.0000}$ $\displaystyle\phi_{3}=$ $\displaystyle\bordermatrix{&A_{1}&A_{2}&A_{3}&A_{4}&A_{5}\cr A_{1}&0.0000&-0.4% 131&-0.3183&-0.2191&-0.3098\cr A_{2}&0.3098&0.0000&0.1975&0.2627&0.2049\cr A_{% 3}&0.2387&-0.2633&0.0000&0.1732&0.2627\cr A_{4}&0.1643&-0.3502&-0.2309&0.0000&% -0.3795\cr A_{5}&0.2324&-0.2733&-0.3502&0.2846&0.0000}$ $\displaystyle\phi_{4}=$ $\displaystyle\bordermatrix{A_{1}&A_{2}&A_{3}&A_{4}&A_{5}\cr A_{1}&0.0000&-0.24% 49&-0.2121&-0.1549&0.1673\cr A_{2}&0.2449&0.0000&0.1581&0.1897&0.2966\cr A_{3}% &0.2121&-0.1581&0.0000&-0.1449&0.2702\cr A_{4}&0.1549&-0.1897&0.1449&0.0000&0.% 2280\cr A_{5}&-0.1673&-0.2966&-0.2702&-0.2280&0.0000}$

The overall dominance degree $\delta(A_{i},A_{t})$ of the candidate $A_{i}$ over each candidate $A_{t}$ can be derived by Eq. (12):

$\displaystyle\delta=$ $\displaystyle\bordermatrix{&A_{1}&A_{2}&A_{3}&A_{4}&A_{5}\cr A_{1}&0.0000&-1.4% 240&-0.5890&-1.0220&-0.1664\cr A_{2}&0.8329&0.0000&0.6731&0.1841&0.8479\cr A_{% 3}&-0.0148&-1.4085&0.0000&-0.9578&0.6377\cr A_{4}&0.5885&-0.7189&0.2554&0.0000% &0.2227\cr A_{5}&-0.5392&-1.6725&-1.0399&-1.0605&0.0000\cr}$

Table 1
The aggregating values of the emerging technology enterprises by the IVPFWA (IVPFWG) operators

	IVPFWA	IVPFWG
$A_{1}$	([0.3681,0.4666], [0.4614,0.5651])	([0.3322,0.4369], [0.5040,0.6049])
$A_{2}$	([0.5131,0.6143], [0.2297,0.3466])	([0.4547,0.5596], [0.3208,0.4149])
$A_{3}$	([0.3657,0.4896], [03492,0.4564])	([0.3170,0.4743], [0.4168,0.5156])
$A_{4}$	([0.4665,0.5697], [0.2837,0.3979])	([0.3716,0.4799], [0.3726,0.4690])
$A_{5}$	([0.4869,0.5877], [0.5342,0.6389])	([0.4590,0.5610], [0.5976,0.7023])

Then, we get the overall value $\delta(A_{i})$ ( $i=$ 1, 2, 3, 4, 5) by using Eq. (3):

$\displaystyle\!\!\!\!\!\!\delta(A_{1})=0.1621,\delta(A_{2})=1.0000,\delta(A_{3% })=0.3750,$ $\displaystyle\!\!\!\!\!\!\delta(A_{4})=0.6803,\delta(A_{5})=0.0000.$

Finally, we get order of ETEs by $\delta(A_{i})$ ( $i=$ 1, 2, 3, 4, 5): $A_{2}\succ A_{4}\succ A_{3}\succ A_{1}\succ A_{5}$ , and thus the most desirable ETE is $A_{2}$ .

5.2 Comparative analysis

In what follows, we compare our proposed method with other existing methods including the IVPFWA operator and IVPFWG operator proposed by Garg [28] as follows:

Definition 7 [28]. Suppose that $\tilde{R}=\left(\tilde{r}_{ij}\right)_{m\times n}=\left(\left[\mu_{ij}^{L},\mu% _{ij}^{R}\right],\left[\nu_{ij}^{L},\nu_{ij}^{R}\right]\right)_{m\times n}$ be a IVPFN matrix, $w=\left(w_{1},w_{2},\ldots,w_{n}\right)^{T}$ be the weight of $\tilde{r}_{ij}$ ( $j=$ 1, 2, …, $n$ ), and $w_{j}>$ 0, $\sum_{j=1}^{n}w_{j}=$ 1. Then

$\displaystyle\tilde{r}_{i}=\text{IVPFWA}_{w}\left(\tilde{r}_{i1},\tilde{r}_{i2% },\ldots,\tilde{r}_{in}\right)=\mathop{\oplus}_{j=1}^{n}\left(w_{j}\tilde{r}_{% ij}\right)=\left(\left[\sqrt{1-\prod\limits_{j=1}^{n}\left(1-\left(\mu_{ij}^{L% }\right)^{2}\right)^{w_{j}}},\right.\right.\quad\left.\sqrt{1-\prod\limits_{j=% 1}^{n}\left(1-\left(\mu_{ij}^{R}\right)^{2}\right)^{w_{j}}}\,\right],\quad% \left.\left[\prod\limits_{j=1}^{n}\left(\nu_{ij}^{L}\right)^{w_{j}},\prod% \limits_{j=1}^{n}\left(\nu_{ij}^{R}\right)^{w_{j}}\right]\right)i=1,2,\ldots,m$ (23) $\displaystyle\tilde{r}_{i}=\text{IVPFWG}_{w}\left(\tilde{r}_{i1},\tilde{r}_{i2% },\ldots,\tilde{r}_{in}\right)=\mathop{\otimes}\limits_{j=1}^{n}\left(\tilde{r% }_{ij}\right)^{w_{j}}=\left(\left[\prod\limits_{j=1}^{n}\left(\mu_{ij}^{L}% \right)^{w_{j}},\prod\limits_{j=1}^{n}\left(\mu_{ij}^{R}\right)^{w_{j}}\right]% ,\right.\quad\left[\sqrt{1-\prod\limits_{j=1}^{n}\left(1-\left(\nu_{ij}^{L}% \right)^{2}\right)^{w_{j}}},\right.\quad\left.\left.\sqrt{1-\prod\limits_{j=1}% ^{n}\left(1-\left(\nu_{ij}^{R}\right)^{2}\right)^{w_{j}}}\,\right]\right)i=1,2% ,\ldots,m$ (24)

Table 2

The score functions of the emerging technology enterprises

	IVPFWA	IVPFWG
$A_{1}$	0.4553	0.4203
$A_{2}$	0.6169	0.5612
$A_{3}$	0.5108	0.4715
$A_{4}$	0.5758	0.5024
$A_{5}$	0.4723	0.4188

Table 3

Order of the emerging technology enterprises

	Order
IVPFWA	$A_{2}>A_{4}>A_{3}>A_{5}>A_{1}$
IVPFWG	$A_{2}>A_{4}>A_{3}>A_{1}>A_{5}$

By utilizing $\tilde{R}$ and the IVPFWA and IVPFWG operators, the aggregating values are derived in Table 1.

According to the aggregating results in Table 1 and the score functions are in Table 2.

According to the score functions shown in Table 2, the order of the emerging technology enterprises is in Table 3.

From the above analysis, it can be seen that two operators have the same best emerging technology enterprise $A_{2}$ and two methods’ ranking results are slightly different. The IVPFN TODIM approach can reasonably depict the DMs’ psychological behaviors under risk, and thus, it may deal with the above issue effectively. But the IVPFWA (IVPFWG) operators can’t reasonably depict the DMs’ psychological behaviors under risk. This verifies the method we proposed is reasonable and effective in this paper.

6. Conclusion

In this paper, we will extend TODIM method to the MADM with the IVPFNs. Firstly, the definition, comparison and distance of IVPFNs are briefly presented, and the steps of the classical TODIM method for MADM problems are introduced. Then, the extended classical TODIM method is proposed to deal with MADM problems with the IVPFNs, and its significant characteristic is that it can fully consider the decision makers’ bounded rationality which is a real action in decision making. Furthermore, we develop the concept of the q-rung interval-valued orthopair fuzzy sets (q-RIVOFSs) and extend the TODIM method to q-RIVOFNs. Finally, a numerical example is proposed to verify the developed approach and a comparative analysis is also given.

In the future, the application of the proposed models and methods of IVPFNs needs to be explored in decision making [61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74], risk analysis and many other uncertain and fuzzy environments [75, 76, 77, 78, 79, 80, 81, 82, 83].

Footnotes

Acknowledgments

The work was supported by the National Natural Science Foundation of China (grant no. 71571128), the Humanities and Social Sciences Foundation of Ministry of Education of the People’s Republic of China (17XJA630003), and the Construction Plan of Scientific Research Innovation Team for Colleges and Universities in Sichuan Province (15TD0004).

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TODIM method for interval-valued Pythagorean fuzzy multiple attribute decision making

Abstract

Keywords

1. Introduction

2. Preliminaries

2.1 PFSs and IVPFSs

5.1 Numerical example

Table 1 The aggregating values of the emerging technology enterprises by the IVPFWA (IVPFWG) operators

Footnotes

Acknowledgments

References

Table 1
The aggregating values of the emerging technology enterprises by the IVPFWA (IVPFWG) operators