1 Introduction

Multi-attribute decision-making (MADM) is a common and important activity in our regular lifestyle. However, the most important step for solving the MADM problems is to choose an adequate environment to do performance assessments of the alternatives. Recently, “fuzzy sets” (Zadeh 1965) and its extensions, “intuitionistic fuzzy set” (IFS) (Atanassov 1986) and “Pythagorean fuzzy set” (PFS) (Yager 2013), have been widely used by the researchers for the alternative assessment. In these environments, various MADM approaches (Chen et al. 2016; Kumar and Chen 2021b, a; Dhankhar and Kumar 2022; Garg and Kumar 2020, 2019; Jiang et al. 2018; Zou et al. 2020; Abdullah et al. 2022; Gupta and Kumar 2022; Joshi and Kumar 2022; Seikh and Mandal 2021a; Garg and Kaur 2020; Ahmad and Sabir 2022; Saad and Rafiq 2022; Chabaane and Kheffache 2022; Ganie 2022; Ashraf et al. 2021; Suresh et al. 2021; Dutta and Doley 2021; Biswas and Deb 2021; Dutta and Saikia 2021; Mishra et al. 2022; Meng and Chen 2021; Pant and Kumar 2022; Ali and Ansari 2022; Ejegwa et al. 2022; Zeb et al. 2022; Chaurasiya and Jain 2022) have been developed by the researchers. For example, Garg and Kumar (2019) defined the improved possibility degree measure (PDM) for intuitionistic fuzzy numbers (IFNs) and a MADM approach based on it. Dhankhar and Kumar (2022) proposed the advanced PDM of IFNs and a MADM approach for the IFNs environment. The power aggregation operator (AO) has the significant advantage of reducing the effects incorrect information given by the biased experts. The power AO allows aggregated values to support each other throughout the aggregation process. Therefore, Garg and Kumar (2020) defined the power AOs for aggregating the connection numbers and a MADM approach for the IFNs environment. Jiang et al. (2018) defined the MADM approach based power AO and entropy measures for the IFNs. Biswas and Deb (2021) presented the Schweizer and Sklar power AOs and MADM for PFSs environment.

Recently, Yager (2017) presented the idea of q-rung orthopair fuzzy set (q-ROFS), where a q-rung orthopair fuzzy value (q-ROFV) \(\chi =\langle \zeta ,\theta \rangle\) with membership grade (MG) \(\zeta\) and non-MG (NMG) \(\theta\) fulfils the constraints \(0\le \zeta \le 1\), \(0\le \theta \le 1\), \(0\le \zeta ^q+\theta ^q\le 1\) and \(q\ge 1\). The q-ROFS is called IFS and PFS when \(q=1\) and \(q=2\), respectively. The q-ROFS provides experts with additional flexibility when evaluating alternatives. In last five years, various MADM approaches (Akram and Shahzadi 2021; Farid and Riaz 2021; Akram et al. 2021; Wei et al. 2018; Yang et al. 2021; Liu and Wang 2018; Liu et al. 2018; Khan et al. 2021; Kumar and Gupta 2023; Feng et al. 2022a; Seikh and Mandal 2021b; Feng et al. 2022b; Verma 2022) have been developed by the researchers in context of q-ROFVs. Akram and Shahzadi (2021) defined the AO based on the Yager’norm for aggregating the q-ROFVs for MADM under the q-ROFVs environment. Farid and Riaz (2021) defined the Einstein interactive geometric AOs for MADM of q-ROFVs. Akram et al. (2021) proposed the q-rung orthopair fuzzy Einstein weighted geometric (q-ROFEWG) AO to develop a MADM approach in the environment of q-ROFVs. Wei et al. (2018) presented the Heronian mean AO to develop a MADM approach in context of q-ROFVs. Yang et al. (2021) defined the interaction Maclaurin symmetric mean AO for MADM approach under the q-ROFVs environment. Liu and Wang (2018) proposed the q-rung orthopair fuzzy weighted geometric (q-ROFWG) AO for MADM in q-ROFVs context. Liu et al. (2018) presented the MADM approach based entropy measure for the q-ROFVs environment.

In this paper, we find that the q-ROFWG AO (Liu and Wang 2018) and the q-ROFEWG AO (Akram et al. 2021) both have the shortcomings that the MG and the NMG of their aggregated q-ROFVs are indeterminate in some instances. Moreover, the q-ROFWG AO (Liu and Wang 2018) and the q-ROFEWG AO (Akram et al. 2021) have the drawbacks that if there is only one q-ROFV with MG 0 among n aggregated q-ROFVs then the MG of the aggregated q-ROFV becomes 0; if there is only one q-ROFV with NMG 1 among n aggregated q-ROFVs then the NMG of the aggregated q-ROFV becomes 1. Therefore, we need a new AO of q-ROFVs to overcome the shortcomings of the q-ROFWG AO (Liu and Wang 2018) and the q-ROFEWG AO (Akram et al. 2021) of q-ROFVs. Moreover, we also find that Akram et al.’s MADM approach (Akram et al. 2021) based on q-ROFEWG AO and Liu and Wang’s MADM approach (Liu and Wang 2018) based on q-ROFWG AO have the shortcomings that they cannot distinguish the ranking ordering (RO) of alternatives in some cases. In order to overcome the shortcomings of the Akram et al.’s MADM approach (Akram et al. 2021) and the Liu and Wang’s MADM approach (Liu and Wang 2018), we need to develop a new MADM approach for the q-ROFVs environment.

In this article, we propose the q-rung orthopair fuzzy Yager power weighted geometric (q-ROFYPWG) AO for aggregating the q-ROFVs by combining the features of PG AO and Yager’s norm. We also present the various properties of the proposed q-ROFYPWG AO. The proposed q-ROFYPWG AO can conquer the shortcomings of the q-ROFWG AO (Liu and Wang 2018) and the q-ROFEWG AO (Akram et al. 2021) of q-ROFVs. Based on the proposed q-ROFYPWG AO, we propose a new MADM approach for q-ROFVs environment. The MADM approach proposed in this paper can conquer the shortcomings of the Akram et al.’s MADM approach (Akram et al. 2021) and the Liu and Wang’s MADM approach (Liu and Wang 2018), where they cannot distinguish the ranking order (RO) of alternatives in some cases. The proposed MADM approach provides us with a very effective method for dealing with MADM challenges in the q-ROFVs environment.

The remaining part of this article is organised as follows: The preliminary materials related to this paper are given in Sect. 2. In Sect. 3, we explore the drawbacks of the q-ROFWG AO (Liu and Wang 2018) and q-ROFEWG AO (Akram et al. 2021) of q-ROFVs. In Sect. 4, we propose the q-ROFYPWG AO of q-ROFVs to overcome the drawbacks of the q-ROFWG AO (Liu and Wang 2018) and q-ROFEWG AO (Akram et al. 2021) of q-ROFVs. Section 5 presents the drawbacks of the Akram et al.’s MADM approach (Akram et al. 2021). In Sect. 6, we develop a new MADM approach based on the proposed q-ROFYPWG AO of q-ROFVs to conquer the drawbacks of the MADM approach given in (Liu and Wang 2018) and (Akram et al. 2021). Finally, Sect. 7 discusses the conclusion of the paper.

2 Preliminaries

In the following, we present some preliminary materials related to this paper.

Definition 1

(Yager 2017) A q-ROFS Q in universal set U is defined as

$$\begin{aligned} Q=\Big \{\langle u, \zeta _Q(u), \theta _Q(u) \rangle \mid u \in U \Big \} \end{aligned}$$
(1)

where \(\zeta (u)\) and \(\theta (u)\) represent the membership grade (MG) and the non-MG (NMG) of element u of q-ROFS Q, respectively, \(0\le \zeta _Q(u), \theta _Q(u) \le 1\) and \(0\le \zeta ^q_Q(u)+ \theta ^q_Q(u) \le 1\). The hesitance degree of u is given by \(\pi _{Q}= (1-\zeta ^q_Q(u)-\theta ^q_Q(u))^{1/q}\) where \(q\ge 1\).

Frequently, Yager (2017) called the pair \(\langle \zeta ,\theta \rangle\) a q-ROFV in the q-ROFS \(Q=\Big \{\langle u, \zeta _Q(u), \theta _Q(u) \rangle \mid u \in U \Big \}\).

Definition 2

(Liu et al. 2019) The Euclidean distance \(d_{E}(\chi _1,\chi _2)\) between two q-ROFVs \(\chi _1=\langle \zeta _1, \theta _1 \rangle\) and \(\chi _2=\langle \zeta _2, \theta _2 \rangle\) is defined as follows:

$$\begin{aligned} d_{E}(\chi _1,\chi _2)=\left( \frac{1}{2}(|\zeta _1^q-\zeta _2^q|^2+|\theta _1^q-\theta _2^q|^2 ) \right) ^{1/2} \end{aligned}$$
(2)

Definition 3

(Yager 2017) Let \(\chi =\langle \zeta , \theta \rangle\) be any q-ROFV. The score value \(S(\chi )\) of \(\chi\) is defined as

$$\begin{aligned} S(\chi )=\zeta ^q-\theta ^q, \end{aligned}$$
(3)

where \(S(\chi )\in [-1,1]\).

Definition 4

(Yager 2017) Let \(\chi =\langle \zeta , \theta \rangle\) be any q-ROFV. The accuracy value \(A_c(\chi )\) of \(\chi\) is defined as

$$\begin{aligned} A_c(\chi )=\zeta ^q+\theta ^q, \end{aligned}$$
(4)

where \(A_c(\chi )\in [0,1]\).

Definition 5

(Yager 2017) Let \(\chi _1\) and \(\chi _2\) be two q-ROFVs. Then

  1. (i)

    If \(S(\chi _1)>S(\chi _2)\), then \(\chi _1\succ \chi _2\).

  2. (ii)

    If \(S(\chi _1)<S(\chi _2)\), then \(\chi _1\prec \chi _2\).

  3. (iii)

    If \(S(\chi _1)=S(\chi _2)\), then

    1. (a)

      If \(A_c(\chi _1)>A_c(\chi _2)\), then \(\chi _1\succ \chi _2\).

    2. (b)

      If \(A_c(\chi _1)<A_c(\chi _2)\), then \(\chi _1\prec \chi _2\).

    3. (c)

      If \(A_c(\chi _1)=A_c(\chi _2)\), then \(\chi _1= \chi _2\).

Definition 6

(Xu and Yager 2010) For aggregating the real numbers \(\chi _{1}\), \(\chi _{2}\),\(\ldots\), \(\chi _{s}\), the PG AO is defined as follows:

$$\begin{aligned} \text {PG}(\chi _{1},\chi _{2},\ldots ,\chi _{s})=\prod \limits _{t=1}^{s} \chi _{t}^{ \dfrac{(1+T(\chi _{t}))}{\sum _{t=1}^{s}(1+T(\chi _{t}))}}, \end{aligned}$$
(5)

where \(T(\chi _{t})=\sum \limits _{\begin{array}{c} l=1\\ l\ne t \end{array}}^{n}Sup(\chi _{t},\chi _{l})\), and \(Sup(\chi _{t},\chi _{l})\) represents the support degree to which \(\chi _{l}\) supports \(\chi _{t}\) and meets the following criteria:

  1. (i)

    \(Sup(\chi _{t},\chi _{l}) \in [0,1]\);

  2. (ii)

    \(Sup(\chi _{t},\chi _{l})= Sup(\chi _{l},\chi _{t})\);

  3. (iii)

    \(Sup(\chi _{t},\chi _{l})\ge Sup(\chi _{m},\chi _{n})\) if \(|\chi _{t}-\chi _{l}| \le |\chi _{m}-\chi _{n}|\).

Definition 7

(Akram and Shahzadi 2021) Let \(\chi _{1}=\langle \zeta _1,\theta _1 \rangle\) and \(\chi _{2}=\langle \zeta _2,\theta _2 \rangle\) be two q-ROFVs, \(\lambda >0\) and \(\delta >0\). Then operation laws of q-ROFVs based on the Yager t-conorm and t-norm are defined as follows:

  1. (a)

    \(\chi _1\oplus \chi _2=\left\langle \root q \of {\min \{1,((\zeta _1^q)^\lambda +(\zeta _2^q)^\lambda )^{1/\lambda }\}},\root q \of {1-\min \{1,((1-\theta _1^q)^\lambda +(1-\theta _2^q)^\lambda )^{1/\lambda }\}} \right\rangle\);

  2. (b)

    \(\chi _1\otimes \chi _2=\left\langle \root q \of {1-\min \{1,((1-\zeta _1^q)^\lambda +(1-\zeta _2^q)^\lambda )^{1/\lambda }\}},\root q \of {\min \{1,((\theta _1^q)^\lambda +(\theta _2^q)^\lambda )^{1/\lambda }\}} \right\rangle\);

  3. (c)

    \(\delta \chi _1=\left\langle \root q \of {\min \{1,(\delta (\zeta _1^q)^\lambda )^{1/\lambda }\}},\root q \of {1-\min \{1,(\delta (1-\theta _1^q)^\lambda )^{1/\lambda }\}} \right\rangle\);

  4. (d)

    \(\chi _1^\delta =\left\langle \root q \of {1-\min \{1,(\delta (1-\zeta _1^q)^\lambda )^{1/\lambda }\}},\root q \of {\min \{1,(\delta (\theta _1^q)^\lambda )^{1/\lambda }\}} \right\rangle\).

3 Analyzing the shortcomings of the existing aggregation operators of q-ROFVs

In this section, we explore the shortcomings of the q-ROFWG AO (Liu and Wang 2018) and q-ROFEWG AO (Akram et al. 2021) of q-ROFVs.

Let \(\chi _{1}=\langle \zeta _1,\theta _1 \rangle\), \(\chi _{2}=\langle \zeta _2,\theta _2 \rangle\),...,\(\chi _{s}=\langle \zeta _s,\theta _s \rangle\) be q-ROFVs. The q-ROFWG AO (Liu and Wang 2018) and q-ROFEWG AO (Akram et al. 2021) of q-ROFVs are reviewed as below:

  1. (1)

    q-ROFWG AO (Liu and Wang 2018):

    $$\begin{aligned} \text {q-ROFWG}\left( \chi _1,\chi _2,\ldots ,\chi _s\right) = \left\langle \prod _{t=1}^s \zeta _t^{w_t}, \left( 1- \prod _{t=1}^s \left( 1-\theta _t^q \right) ^{w_t}\right) ^{1/q} \right\rangle \end{aligned}$$
    (6)

    where \(q\ge 1\), \(w_t \in [0,1]\), \(t=1, 2, \ldots ,s\) and \(\sum \nolimits _{t=1}^{s}w_t=1\).

  2. (2)

    q-ROFEWG AO (Akram et al. 2021):

    $$\begin{aligned}{} & {} q\text {-ROFEWG}\left( \chi _1,\chi _2,\ldots ,\chi _s\right) \nonumber \\= & {} \left\langle \frac{\root q \of {2} \prod _{t=1}^s {\zeta _t}^{w_t} }{\root q \of {\prod _{t=1}^s (2-\zeta _t^q)^{w_t} + \prod _{t=1}^s (\zeta _t^q)^{w_t}}}, \root q \of {\frac{\prod _{t=1}^s (1+\theta _t^q)^{w_t} - \prod _{t=1}^s (1-\theta _t^q)^{w_t}}{\prod _{t=1}^s (1+\theta _t^q)^{w_t} + \prod _{t=1}^s (1-\theta _t^q)^{w_t}}}\right\rangle \end{aligned}$$
    (7)

    where \(q\ge 1\), \(w_t \in [0,1]\), \(t=1, 2, \ldots ,s\) and \(\sum \limits _{t=1}^{s}w_t=1\).

Example 1

Let \(\chi _1 = \langle 0.2, 0.3\rangle\) and \(\chi _2 = \langle 0, 1\rangle\) be two q-ROFVs with weights \(w_1=0.3\) and \(w_2=0.7\), respectively. By using the q-ROFWG AO (Liu and Wang 2018) given in Eq. (6) to aggregate the \(\chi _1\) and \(\chi _2\), it gets \(q-ROFWG(\chi _1,\chi _2)=\langle 0, 1 \rangle\). It can be seen that the MG of the q-ROFV \(\chi _1\) is 0.2, which is not 0 because \(0.2 > 0\), but the MG of the aggregated result \(q-ROFWG(\chi _1,\chi _2)=\langle 0, 1 \rangle\) is 0, which is not satisfactory. Moreover, it can be seen that the NMG of the q-ROFV \(\chi _1\) is 0.3, which is not 1 because \(0.3 < 1\), but the NMG of aggregated result \(q-ROFWG(\chi _1,\chi _2)=\langle 0, 1 \rangle\) is 1, which is not satisfactory. This aggregated value is unreasonable as the q-ROFV \(\chi _1\) loses its effect in the presence of the q-ROFV \(\chi _2\), which is practically infeasible.

Example 2

Let \(\chi _1 = \langle 0.4, 0.6\rangle\) and \(\chi _2 = \langle 0, 1\rangle\) be two q-ROFVs with weights \(w_1=0.4\) and \(w_2=0.6\), respectively. By using the q-ROFEWG AO (Akram et al. 2021) given in Eq. (7) to aggregate the \(\chi _1\) and \(\chi _2\), it gets \(q-ROFEWG(\chi _1,\chi _2)=\langle 0, 1 \rangle\). It can be seen that the MG of the q-ROFV \(\chi _1\) is 0.2, which is not 0 because \(0.4 > 0\), but the MG of aggregated result \(q-ROFEWG(\chi _1,\chi _2)=\langle 0, 1 \rangle\) is 0, which is not satisfactory. Moreover, it can be seen that the NMG of the q-ROFV \(\chi _1\) is 0.6, which is not 1 because \(0.6 < 1\), but the NMG of aggregated result \(q-ROFEWG(\chi _1,\chi _2)=\langle 0, 1 \rangle\) is 1, which is not satisfactory. This aggregated value is unreasonable as the q-ROFV \(\chi _1\) loses its effect in the presence of the q-ROFV \(\chi _2\), which is practically infeasible.

Example 3

Let \(\chi _1=\langle 0.5, 0.3\rangle\) and \(\chi _2=\langle 0, 1 \rangle\) be two q-ROFVs with weights \(w_1=1\) and \(w_2=0\), respectively. By using the q-ROFWG AO (Liu and Wang 2018) given in Eq. (6) to aggregate the \(\chi _1\) and \(\chi _2\), it can be seen that the terms \((\zeta _t)^{w_t}\) and \(\left( 1-\theta _t^q \right) ^{w_t}\) given in Eq. (6) of the q-ROFWG AO (Liu and Wang 2018) are \((\zeta _2)^{w_2}=0^0\) and \(\left( 1-\zeta _2^q \right) ^{w_2}=0^0\), respectively, where \(0^0\) and \(0^0\) are indeterminated values. Hence, the q-ROFWG AO (Liu and Wang 2018) given in Eq. (6) has the above shortcoming in this case.

Example 4

Let \(\chi _1 = \langle 0,1 \rangle\) and \(\chi _2= \langle 0.6,0.2 \rangle\) be two q-ROFVs with weights \(w_1=0\) and \(w_2=1\), respectively. By using the q-ROFEWG AO (Akram et al. 2021) given in Eq. (7) to aggregate the \(\chi _1\) and \(\chi _2\), it can be seen that the terms \((\zeta _t)^{w_t}\) and \(\left( 1-\theta _t^q \right) ^{w_t}\) given in Eq. (7) of the q-ROFEWG AO (Akram et al. 2021) are \((\zeta _1)^{w_1}=0^0\) and \(\left( 1-\zeta _1^q \right) ^{w_1}=0^0\), respectively, where \(0^0\) and \(0^0\) are indeterminated values. Hence, the q-ROFWG AO (Akram et al. 2021) given in Eq. (7) has the above shortcoming in this case.

4 The proposed q-rung orthopair fuzzy Yager power weighted geometric aggregation operator

In this section, based on the Yager’s operations of q-ROFVs given in Definition 7 and the PG AO given in Eq. (5), we propose the q-rung orthopair fuzzy Yager power weighted geometric (q-ROFYPWG) AO to conquer the shortcomings of the q-ROFWG AO (Liu and Wang 2018) and the q-ROFEWG AO (Akram et al. 2021) of the q-ROFVs.

Definition 8

Let \(\chi _{1}=\langle \zeta _1,\theta _1 \rangle\), \(\chi _{2}=\langle \zeta _2,\theta _2 \rangle\), \(\ldots\), \(\chi _{s}=\langle \zeta _s,\theta _s \rangle\) be q-ROFVs. For aggregating the \(\chi _{1}=\langle \zeta _1,\theta _1 \rangle\), \(\chi _{2}=\langle \zeta _2,\theta _2 \rangle\), \(\ldots\), \(\chi _{s}=\langle \zeta _s,\theta _s \rangle\), the proposed q-ROFYPWG AO is as follows:

$$\begin{aligned} q\text {-ROFYPWG}(\chi _{1},\chi _{2},\ldots , \chi _{s})= & {} \bigotimes \limits _{t=1}^{s} (\chi _{t})^{\dfrac{w_t(1+T(\chi _{t}))}{\sum _{t=1}^{s}w_t(1+T(\chi _{t}))}}\nonumber ,\\= & {} \left\langle \root q \of {1-\min \left\{ 1,\left( \sum \limits _{t=1}^s \frac{w_t(1+T(\chi _{t}))}{\sum _{t=1}^{s}w_t(1+T(\chi _{t}))}(1-\zeta _t^q)^\lambda \right) ^{1/\lambda }\right\} },\right. \nonumber \\{} & {} \left. \root q \of {\min \left\{ 1,\left( \sum \limits _{t=1}^s \frac{w_t(1+T(\chi _{t}))}{\sum _{t=1}^{s}w_t(1+T(\chi _{t}))}(\theta _t^q)^\lambda \right) ^{1/\lambda }\right\} } \right\rangle, \end{aligned}$$
(8)

where \(w_{t}\) represents the weight of \(\chi _{t}\), \(w_t \ge 0\), \(\sum \nolimits _{t=1}^s w_t=1\), \(q\ge 1\), \(\lambda >0\), \(T(\chi _{t})=\sum \nolimits _{\begin{array}{c} l=1\\ l\ne t \end{array}}^{s}Sup(\chi _{t},\chi _{l})\), and \(Sup(\chi _{t},\chi _{l}) =1-d_E(\chi _{t},\chi _{l})=1-\left( \frac{1}{2}(|\zeta _t^q-\zeta _l^q|^2+|\theta _t^q-\theta _l^q|^2 ) \right) ^{1/2}\) represents the support degree to which \(\chi _{l}\) supports \(\chi _{t}\) and meets the following criteria:

  1. (i)

    \(Sup(\chi _{t},\chi _{l}) \in [0,1]\);

  2. (ii)

    \(Sup(\chi _{t},\chi _{l})= Sup(\chi _{l},\chi _{t})\);

  3. (iii)

    \(Sup(\chi _{t},\chi _{l})\ge Sup(\chi _{m},\chi _{n})\) if \(|\chi _{t}-\chi _{l}| \le |\chi _{m}-\chi _{n}|\).

Example 5

Let \(\chi _1 = \langle 0.4, 0.5\rangle\), \(\chi _2 = \langle 0.3, 0.6\rangle\) and \(\chi _3 = \langle 0.5, 0.2\rangle\) be three q-ROFVs with weights \(w_1=0.4\), \(w_2=0.3\) and \(w_3=0.3\), respectively. First, we calculate the \(Sup(\chi _{t},\chi _{l})\) between q-ROFVs \(\chi _{t}\) and \(\chi _{l}\), where \(q=3\), \(t,l=1,2,3\), \(t\ne l\),

$$\begin{aligned} Sup(\chi _{1},\chi _{2})= & {} 1-\left( \frac{1}{2}(|\zeta _1^q-\zeta _2^q|^2+|\theta _1^q-\theta _2^q|^2 ) \right) ^{1/2}\\= & {} 1-\left( \frac{1}{2}(|0.4^3-0.3^3|^2+|0.5^3-0.6^3|^2 ) \right) ^{1/2}\\= & {} 0.9305,\\ Sup(\chi _{1},\chi _{3})= & {} 1-\left( \frac{1}{2}(|\zeta _1^q-\zeta _3^q|^2+|\theta _1^q-\theta _3^q|^2 ) \right) ^{1/2}\\= & {} 1-\left( \frac{1}{2}(|0.4^3-0.5^3|^2+|0.5^3-0.2^3|^2 ) \right) ^{1/2}\\= & {} 0.9067,\\ Sup(\chi _{2},\chi _{3})= & {} 1-\left( \frac{1}{2}(|\zeta _2^q-\zeta _3^q|^2+|\theta _2^q-\theta _3^q|^2 ) \right) ^{1/2}\\= & {} 1-\left( \frac{1}{2}(|0.3^3-0.5^3|^2+|0.6^3-0.2^3|^2 ) \right) ^{1/2}\\= & {} 0.8374. \end{aligned}$$

In next step, we calculate the \(T(\chi _{1})\), \(T(\chi _{2})\) and \(T(\chi _{3})\) of the q-ROFVs \(\chi _1\), \(\chi _2\) and \(\chi _3\), respectively, as follows:

$$\begin{aligned} T(\chi _{1})= & {} Sup(\chi _{1},\chi _{2})+Sup(\chi _{1},\chi _{3})\\= & {} 0.9305+0.9067=1.8372,\\ T(\chi _{2})= & {} Sup(\chi _{2},\chi _{1})+Sup(\chi _{2},\chi _{3})\\= & {} 0.9305+0.8374=1.7680,\\ T(\chi _{3})= & {} Sup(\chi _{3},\chi _{2})+Sup(\chi _{3},\chi _{1})\\= & {} 0.8374+0.9067=1.7441. \end{aligned}$$

Then, by using the Eq. (8), we aggregate the q-ROFVs \(\chi _1\), \(\chi _2\) and \(\chi _3\), where \(q=3\), \(\lambda =2\),

$$\begin{aligned}{} & {} q-ROFYPWG(\chi _1,\chi _2,\chi _3)\\= & {} \left\langle \root 3 \of {{1-\min \left\{ 1,\left( \begin{aligned}&\frac{0.4(1+1.8372)}{0.4(1+1.8372)+0.3(1+1.7680)+0.3(1+1.7441)}(1-0.4^3)^2 \\&+\frac{0.3(1+1.7680)}{0.4(1+1.8372)+0.3(1+1.7680)+0.3(1+1.7441)}(1-0.3^3)^2 \\&+\frac{0.3(1+1.7441)}{0.4(1+1.8372)+0.3(1+1.7680)+0.3(1+1.7441)}(1-0.5^3)^2 \\ \end{aligned} \right) ^{1/2}\right\} }}\right. ,\\{} & {} \qquad \left. \root 3 \of {{\min \left\{ 1,\left( \begin{aligned}&\frac{0.4(1+1.8372)}{0.4(1+1.8372)+0.3(1+1.7680)+0.3(1+1.7441)}(0.5^3)^2 \\&+\frac{0.3(1+1.7680)}{0.4(1+1.8372)+0.3(1+1.7680)+0.3(1+1.7441)}(0.6^3)^2 \\&+\frac{0.3(1+1.7441)}{0.4(1+1.8372)+0.3(1+1.7680)+0.3(1+1.7441)}(0.2^3)^2 \\ \end{aligned} \right) ^{1/2}\right\} }}\right\rangle \\= & {} \langle 0.4125,0.5222 \rangle . \end{aligned}$$

In the following, we present a few characteristics of the proposed q-ROFYPWG AO given in Eq. (8).

Property 1

(Idempotency) Let \(\chi _1\), \(\chi _2\), \(\ldots\), \(\chi _s\) be q-ROFVs with weights \(w_1\), \(w_2\), \(\ldots\), \(w_s\), respectively, where \(w_t\ge 0\) and \(\sum _{t=1}^{s}w_t=1\). If \(\chi _1=\chi _2,...=\chi _s=\chi\), then

$$\begin{aligned} q-ROFYPWG(\chi _1,\chi _2, \ldots ,\chi _s)=\chi . \end{aligned}$$

Proof

The weights of q-ROFVs \(\chi _1\), \(\chi _2\), \(\ldots\), \(\chi _s\) are \(w_1\), \(w_2\), \(\ldots\), \(w_s\), respectively, where \(w_t\ge 0\) and \(\sum _{t=1}^{s}w_t=1\). If \(\chi _1=\chi _2,...=\chi _s=\chi\), then by using Eq. (8), we have

$$\begin{aligned} q\text {-ROFYPWG}(\chi _{1},\chi _{2},\ldots , \chi _{s})= & {} \bigotimes \limits _{t=1}^{s} (\chi _{t})^{\dfrac{w_t(1+T(\chi _{t}))}{\sum _{t=1}^{s}w_t(1+T(\chi _{t}))}}\\= & {} \bigotimes \limits _{t=1}^{s} (\chi )^{\dfrac{w_t(1+T(\chi ))}{\sum _{t=1}^{s}w_t(1+T(\chi ))}}\\= & {} \chi ^{\sum _{t=1}^{s}w_t}\\= & {} \chi . \end{aligned}$$

Property 2

Let \(\chi _1\), \(\chi _2\), \(\ldots\), \(\chi _s\) be q-RONs, let \(\chi ^{-}=\min \{\chi _1, \chi _2,..., \chi _s\}\) and let \(\chi ^{+}=\max \{\chi _1, \chi _2,...,\chi _s\}\). Then,

$$\begin{aligned} \chi ^{-}\le \text {{ q}-ROFYPWG}(\chi _{1}, \chi _{2},\ldots , \chi _{s})\le \chi ^{+}. \end{aligned}$$

Proof

Because \(\chi ^{-}=\min \{\chi _1, \chi _2,..., \chi _n\}\) and \(\chi ^{+}=\max \{\chi _1, \chi _2,..., \chi _n\}\), by using Eq. (8), we obtain

$$\begin{aligned} q\text {-ROFYPWG}(\chi _{1}, \chi _{2},\ldots , \chi _{s})= & {} \bigotimes \limits _{t=1}^{s} (\chi _{t})^{\dfrac{w_t(1+T(\chi _{t}))}{\sum _{t=1}^{s}w_t(1+T(\chi _{t}))}} \\\le & {} \bigotimes \limits _{t=1}^{s} (\chi ^{+})^{\dfrac{w_t(1+T(\chi ^{+}))}{\sum _{t=1}^{s}w_t(1+T(\chi ^{+}))}}\\= & {} (\chi ^{+})^{\sum _{t=1}^{s}w_{t}}=\chi ^{+},\\ q\text {-ROFYPWG}(\chi _{1}, \chi _{2},\ldots , \chi _{s})= & {} \bigotimes \limits _{t=1}^{s} (\chi _{t})^{\dfrac{w_t(1+T(\chi _{t}))}{\sum _{t=1}^{s}w_t(1+T(\chi _{t}))}} \\\ge & {} \bigotimes \limits _{t=1}^{s} (\chi ^{-})^{\dfrac{w_t(1+T(\chi ^{-}))}{\sum _{t=1}^{s}w_t(1+T(\chi ^{-}))}}\\= & {} (\chi ^{-})^{\sum _{t=1}^{s}w_{t}}=\chi ^{-}. \end{aligned}$$

Hence, we get \(\chi ^{-}\le\) q-ROFYPWG\((\chi _{1},\chi _{2},\ldots , \chi _{s})\le \chi ^{+}\).

Property 3

(Monotonicity) Let \(\chi _1\), \(\chi _2\), \(\ldots\), \(\chi _s\), \(\chi _1^{*}\), \(\chi _2^{*}\), \(\ldots\), and \(\chi _s^{*}\) be q-ROFVs. If \(\chi _{t} \le \chi _{t}^{*},\) where \(t = 1,2,...,n,\) then

$$\begin{aligned} \textit{q}-ROFYPWG(\chi _1, \chi _2,..., \chi _n) \le \textit{q}-ROFYPWG(\chi _1^{*},\chi _2^{*},...,\chi _n^{*}). \end{aligned}$$

Proof

By utilizing Eq. (8), we obtain

$$\begin{aligned}{} & {} \textit{q}-ROFYPWG(\chi _1, \chi _2,..., \chi _s) = \bigotimes \limits _{t=1}^{s} (\chi _{t})^{\dfrac{w_t(1+T(\chi _{t}))}{\sum _{t=1}^{s}w_t(1+T(\chi _{t}))}},\\{} & {} \textit{q}-ROFYPWG(\chi _1^{*}, \chi _2^{*},..., \chi _s^{*}) =\bigotimes \limits _{t=1}^{s} (\chi _{t}^{*})^{\dfrac{w_t(1+T(\chi _{t}))}{\sum _{t=1}^{s}w_t(1+T(\chi _{t}))}} \end{aligned}$$

Because \(\chi _{t}\le \chi _{t}^{*}\), \(\forall t = 1,2,...,n\), we obtain

$$\begin{aligned} \bigotimes \limits _{t=1}^{s} (\chi _{t})^{\dfrac{w_t(1+T(\chi _{t}))}{\sum _{t=1}^{s}w_t(1+T(\chi _{t}))}} \le \bigotimes \limits _{t=1}^{s} (\chi _{t}^{*})^{\dfrac{w_t(1+T(\chi _{t}))}{\sum _{t=1}^{s}w_t(1+T(\chi _{t}))}}. \end{aligned}$$

Hence, \(\textit{q}-ROFYPWG(\chi _1, \chi _2,..., \chi _s) \le \textit{q}-ROFYPWG(\chi _1^{*},\chi _2^{*},...,\chi _s^{*})\).

In the following, we show how the proposed q-ROFYPWG AO can conquer the shortcomings of the q-ROFWG AO (Liu and Wang 2018) and q-ROFEWG AO (Akram et al. 2021) of the q-ROFVs shown in Sect. 3, as follows:

Example 6

Let us consider same q-ROFVs \(\chi _1 = \langle 0.2, 0.3\rangle\) and \(\chi _2 = \langle 0, 1\rangle\) with weights \(w_1=0.3\) and \(w_2=0.7\), respectively, given in Example 1. By utilizing the q-ROFYPWG AO given in Eq. (8), we obtain

$$\begin{aligned} q-ROFYPWG(\chi _1,\chi _2)=\langle 0.1338, 0.9423 \rangle . \end{aligned}$$

Hence, the q-ROFYPWG AO given in Eq. (8) can conquer the shortcoming of q-ROFWG AO (Liu and Wang 2018), shown in Example 1.

Example 7

Let us consider same q-ROFVs \(\chi _1 = \langle 0.4, 0.6\rangle\) and \(\chi _2 = \langle 0, 1\rangle\) with weights \(w_1=0.4\) and \(w_2=0.6\), respectively, given in Example 2. By utilizing the q-ROFYPWG AO given in Eq. (8), we obtain

$$\begin{aligned} q-ROFYPWG(\chi _1,\chi _2)=\langle 0.2928, 0.9231 \rangle . \end{aligned}$$

Hence, the q-ROFYPWG AO given in Eq. (8) can conquer the shortcoming of q-ROFEWG AO (Akram et al. 2021), shown in Example 2.

Example 8

Let us consider same q-ROFVs \(\chi _1=\langle 0.5, 0.3\rangle\) and \(\chi _2=\langle 0, 1 \rangle\) with weights \(w_1=1\) and \(w_2=0\), respectively, given in Example 3. By utilizing the q-ROFYPWG AO given in Eq. (8), we obtain

$$\begin{aligned} q-ROFYPWG(\chi _1,\chi _2)=\langle 0.5, 0.3 \rangle . \end{aligned}$$

Hence, the proposed q-ROFYPWG AO given in Eq. (8) can conquer the shortcoming of q-ROFWG AO (Liu and Wang 2018), shown in Example 3.

Example 9

Let us consider same q-ROFVs \(\chi _1 = \langle 0,1 \rangle\) and \(\chi _2= \langle 0.6,0.2 \rangle\) with weights \(w_1=0\) and \(w_2=1\), respectively, given in Example 4. By utilizing the q-ROFYPWG AO given in Eq. (8), we obtain

$$\begin{aligned} q-ROFYPWG(\chi _1,\chi _2)=\langle 0.6, 0.2 \rangle . \end{aligned}$$

Hence, the q-ROFYPWG AO given in Eq. (8) can conquer the shortcoming of q-ROFEWG AO (Akram et al. 2021), shown in Example 4.

5 Analyzing the shortcomings of the Akram et al.’s MADM approach

In this section, we explore the shortcomings of Akram et al.’s MADM approach (Akram et al. 2021). Let \(O_1, O_2, \ldots , O_r\) be alternatives and \(C_1, C_2, \ldots , C_s\) be attributes with weights \(w_1\), \(w_2,\) \(\ldots , w_s\) such that \(w_t \in [0,1]\) and \(\sum \nolimits _{t=1}^s w_t=1\). The expert assess the alternative \(O_k\) with respect to the attribute \(C_t\) by utilizing the q-ROFVs \(\widetilde{\chi }_{kt}=\langle \widetilde{\zeta }_{kt},\widetilde{\theta }_{kt}\rangle\) to construct the decision matrix (DMx) \(\widetilde{D}=\left( \widetilde{\chi }_{kt}\right) _{r\times s}\) as follows:

where \(k=1,2,\ldots ,r\) and \(t=1,2,\ldots ,s\). We review the Akram et al.’s MADM approach (Akram et al. 2021) as follows:

Step 1::

By using the q-ROFEWG AO given in Eq. (7), aggregate the q-ROFVs \(\chi _{k1}\), \(\chi _{k2}\),\(\ldots , \chi _{ks}\) appeared in \(k^{th}\) row of NDMx \(D=(\chi _{kt})_{r\times s}\) to obtain the overall aggregated q-ROFV \(\chi _{k}=\langle \zeta _k,\theta _k \rangle\) of alternative \(O_k\), where

$$\begin{aligned} \chi _k= & {} q\text {-ROFEWG}\left( \chi _{k1},\chi _{k2},\ldots ,\chi _{ks}\right) \nonumber \\= & {} \left\langle \frac{\root q \of {2} \prod _{t=1}^s {\zeta _{kt}}^{w_t} }{\root q \of {\prod _{t=1}^s (2-\zeta _{kt}^q)^{w_t} + \prod _{t=1}^s (\zeta _{kt}^q)^{w_t}}}, \root q \of {\frac{\prod _{t=1}^s (1+\theta _{kt}^q)^{w_t} - \prod _{t=1}^s (1-\theta _{kt}^q)^{w_t}}{\prod _{t=1}^s (1+\theta _{kt}^q)^{w_t} + \prod _{t=1}^s (1-\theta _{kt}^q)^{w_t}}}\right\rangle , \end{aligned}$$
(9)

\(k=1,2,\ldots ,r\), \(w_t \in [0,1]\) and \(\sum \limits _{t=1}^s w_t=1\).

Step 2::

Compute the score value \(S(\chi _k)\) of the overall q-ROFV \(\chi _{k}=\langle \zeta _k,\theta _k \rangle\) of alternative \(O_k\) obtained in Step 1, shown as follows:

$$\begin{aligned} S(\chi _k)=(\zeta _k)^q-(\theta _k)^q, \end{aligned}$$
(10)

and compute the accuracy value \(A_c(\chi _k)\) of the overall q-ROFV \(\chi _{k}=\langle \zeta _k,\theta _k \rangle\) obtained in Step 1 as follows:

$$\begin{aligned} A_c(\chi _k)=(\zeta _k)^q+(\theta _k)^q, \end{aligned}$$
(11)

where \(k=1,2,\ldots ,r\).

Step 3::

Find the ranking order (RO) of the alternatives \(O_1\), \(O_2\), \(\ldots , O_r\) based on Definition 5 and select the best alternative.

Example 10

Let \(O_1\), \(O_2\) and \(O_3\) be three alternatives and let \(C_1,C_2\) and \(C_3\) be three beneficiary type attributes with weights \(w_1=0.3\), \(w_2=0.4\) and \(w_3=0.3\). The expert evaluates the alternatives \(O_1, O_2,\) and \(O_3\) with respect to the attribute \(C_1\), \(C_2\) and \(C_3\) by using an q-ROFV \(\widetilde{\chi }_{kt}\) to obtain the DMx \(\widetilde{D}=\left( {\chi }_{kt}\right) _{3\times 3}=\left( \widetilde{\zeta }_{kt},\widetilde{\theta }_{kt}\right) _{3\times 3}\), shown as follows:

Step 1::

By utilizing Eq. (9), Akram et al’s MADM approach (Akram et al. 2021) obtains the overall q-ROFVs \(\chi _1\), \(\chi _2\) and \(\chi _3\) of the alternatives \(O_1, O_2\) and \(O_3\), respectively, where \(q=3\), \(\chi _{1}= \langle 0,1\rangle ,\) \(\chi _{2}= \langle 0,1\rangle\), and \(\chi _{3}= \langle 0,1\rangle\).

Step 2::

By utilizing Eq. (10), Akram et al’s MADM approach (Akram et al. 2021) gets the score values \(S(\chi _1)\), \(S(\chi _2)\) and \(S(\chi _3)\) of the overall q-ROFVs \(\chi _1\), \(\chi _2\) and \(\chi _3\) obtained in Step 1, respectively, where \(q=3\), \(S(\chi _1)=-1\), \(S(\chi _2)=-1\) and \(S(\chi _3)=-1\). Because \(S(\chi _1)=S(\chi _2)=S(\chi _3)\), where \(S(\chi _1)=-1\), \(S(\chi _2)=-1\) and \(S(\chi _3)=-1\), by using Eq. (11), Akram et al’s MADM approach (Akram et al. 2021) gets the accuracy values \(A_c(\chi _1)\), \(A_c(\chi _2)\) and \(A_c(\chi _3)\) of the overall q-ROFVs \(\chi _1\), \(\chi _2\) and \(\chi _3\), respectively, where \(A_c(\chi _1)=1\), \(A_c(\chi _2)=1\) and \(A_c(\chi _3)=1\).

Step 3::

Since \(S(\chi _1)=S(\chi _2)=S(\chi _3)\), where \(S(\chi _1)=-1\), \(S(\chi _2)=-1\) and \(S(\chi _3)=-1\), and because \(A_c(\chi _1)=A_c(\chi _2)=A_c(\chi _3)\), where \(A_c(\chi _1)=1\), \(A_c(\chi _2)=1\) and \(A_c(\chi _3)=1\), based on Definition 5, Akram et al’s MADM approach (Akram et al. 2021) obtains the RO “\(O_1 = O_2 = O_3\)” of the alternatives \(O_1, O_2\) and \(O_3\).

Example 11

Let \(O_1\), \(O_2\) and \(O_3\) be three alternatives and let \(C_1,C_2\) and \(C_3\) be three beneficiary type attributes with weights \(w_1=0.28\), \(w_2=0.35\) and \(w_3=0.37\). The expert evaluates the alternatives \(O_1, O_2,\) and \(O_3\) with respect to the attribute \(C_1\), \(C_2\) and \(C_3\) by using an q-ROFV \(\widetilde{\chi }_{kt}\) to obtain the DMx \(\widetilde{D}=\left( {\chi }_{kt}\right) _{3\times 3}=\left( \widetilde{\zeta }_{kt},\widetilde{\theta }_{kt}\right) _{3\times 3}\), shown as follows:

Step 1::

By utilizing Eq. (9), Akram et al’s MADM approach (Akram et al. 2021) obtains the overall q-ROFVs \(\chi _1\), \(\chi _2\) and \(\chi _3\) of the alternatives \(O_1, O_2\) and \(O_3\), respectively, where \(q=3\), \(\chi _{1}= \langle 0,1\rangle ,\) \(\chi _{2}= \langle 0,1\rangle\), and \(\chi _{3}= \langle 0,1\rangle\).

Step 2::

By utilizing Eq. (10), Akram et al’s MADM approach (Akram et al. 2021) gets the score values \(S(\chi _1)\), \(S(\chi _2)\) and \(S(\chi _3)\) of the overall q-ROFVs \(\chi _1\), \(\chi _2\) and \(\chi _3\) obtained in Step 1, respectively, where \(q=3\), \(S(\chi _1)=-1\), \(S(\chi _2)=-1\) and \(S(\chi _3)=-1\). Because \(S(\chi _1)=S(\chi _2)=S(\chi _3)\), where \(S(\chi _1)=-1\), \(S(\chi _2)=-1\) and \(S(\chi _3)=-1\), by using Eq. (11), Akram et al’s MADM approach (Akram et al. 2021) gets the accuracy values \(A_c(\chi _1)\), \(A_c(\chi _2)\) and \(A_c(\chi _3)\) of the overall q-ROFVs \(\chi _1\), \(\chi _2\) and \(\chi _3\), respectively, where \(A_c(\chi _1)=1\), \(A_c(\chi _2)=1\) and \(A_c(\chi _3)=1\).

Step 3::

Since \(S(\chi _1)=S(\chi _2)=S(\chi _3)\), where \(S(\chi _1)=-1\), \(S(\chi _2)=-1\) and \(S(\chi _3)=-1\), and because \(A_c(\chi _1)=A_c(\chi _2)=A_c(\chi _3)\), where \(A_c(\chi _1)=1\), \(A_c(\chi _2)=1\) and \(A_c(\chi _3)=1\), based on Definition 5, Akram et al’s MADM approach (Akram et al. 2021) obtains the RO “\(O_1 = O_2 = O_3\)” of the alternatives \(O_1, O_2\) and \(O_3\).

6 A new MADM approach based on the proposed q-ROFYWG AO of q-ROFVs

In this section, we develop a new MADM approach by using the proposed q-ROFYWG AO of q-ROFVs. Let \(O_1, O_2, \ldots , O_r\) be alternatives and \(C_1, C_2, \ldots , C_s\) be attributes with weights \(w_1\), \(w_2,\) \(\ldots , w_s\), respectively, where \(w_t \in [0,1]\) and \(\sum \limits _{t=1}^s w_t=1\). By using the q-ROFVs \(\widetilde{\chi }_{kt}=\langle \widetilde{\zeta }_{kt},\widetilde{\theta }_{kt}\rangle\), the expert evaluates the alternative \(O_k\) with respect to the attribute \(C_t\) to construct the DMx \(\widetilde{D}=\left( \widetilde{\chi }_{kt}\right) _{r\times s}\), shown as follows:

where \(k=1,2,\ldots ,r\) and \(t=1,2,\ldots ,s\). The proposed MADM approach is presented as follows:

Step 1::

Transform the DMx \(\widetilde{D}=\left( \widetilde{\chi }_{kt}\right) _{r\times s}=\left( \langle \widetilde{\zeta }_{kt},\widetilde{\theta }_{kt}\rangle \right) _{r\times s}\) into NDMx \(D=(\chi _{kt})_{r\times s}=\langle {\zeta }_{kt},{\theta }_{kt}\rangle\) as follows:

$$\begin{aligned} \chi _{kt}= {\left\{ \begin{array}{ll} \langle \widetilde{\zeta }_{kt},\widetilde{\theta }_{kt}\rangle ~: \text {if } C_{t} \text { is a benefit type attribute} \\ \langle \widetilde{\theta }_{kt},\widetilde{\zeta }_{kt}\rangle ~: \text {if } C_{t} \text { is a cost type attribute}\\ \end{array}\right. } \end{aligned}$$
(12)
Step 2::

Based on Eq. (2), calculate the support measure \(Sup(\chi _{kt},\chi _{kl})\) between q-ROFVs \(\chi _{kt}\) and \(\chi _{kl}\) as follows:

$$\begin{aligned} Sup(\chi _{kt},\chi _{kl})= & {} 1-d_E(\chi _{kt},\mu _{kl})\nonumber \\= & {} 1-\left( \frac{1}{2}(|\zeta _{kt}^q-\zeta _{kl}^q|^2+|\theta _{kt}^q-\theta _{kl}^q|^2 ) \right) ^{1/2} \end{aligned}$$
(13)

where \(k=1,2,\ldots ,r\); \(t,l=1,2,\ldots ,s\); \(l\ne t\), \(d(\chi _{kt},\chi _{kl})\) is Hamming distance between q-ROFVs \(\chi _{kt}\) and \(\chi _{kl}\).

Step 3::

Calculate support \(T_{kt}\) and weight \(\delta _{kt}\) for the q-ROFV \(\chi _{kt}\) as follows:

$$\begin{aligned} T_{kt}=\sum \limits _{\begin{array}{c} l=1\\ l\ne t \end{array}}^{s}Sup(\chi _{kt},\chi _{kl}), \end{aligned}$$
(14)

and

$$\begin{aligned} \delta _{kt}=\frac{w_{t}(1+T(\chi _{kt}))}{\sum _{t=1}^{n}w_{t}(1+T(\chi _{kt}))}, \end{aligned}$$
(15)

where \(\delta _{kt}\in [0,1]\) and \(\sum \nolimits _{t=1}^{s}\delta _{kt}=1.\)

Step 4::

By applying the proposed q-ROFYWG AO given in Eq. (8), aggregate the q-ROFVs \(\chi _{k1}, \chi _{k2},\ldots , \chi _{ks}\) appeared in \(k^{th}\) row of NDMx \(D=(\chi _{kt})_{m\times n}\) to obtain the overall aggregated q-ROFV \(\chi _{k}=\langle \zeta _k,\theta _k \rangle\) of alternative \(O_k\), shown as follows:

$$\begin{aligned} \chi _{k}= & {} \langle \eta _k, \upsilon _k \rangle \nonumber \\= & {} q\text {-ROFYWG}(\chi _{k1},\chi _{k2},\ldots ,\chi _{ks}) \nonumber \\= & {} \left\langle \left( 1-\min \left\{ 1,\left( \sum \limits _{t=1}^{s}\delta _{kt}(1-\zeta _{kt}^q)^\lambda \right) ^{\frac{1}{\lambda }}\right\} \right) ^{\frac{1}{q}},\left( \min \left\{ 1,\left( \sum \limits _{t=1}^{s}\delta _{kt}(\theta _{kt}^q)^\lambda \right) ^{\frac{1}{\lambda }}\right\} \right) ^{\frac{1}{q}} \right\rangle , \end{aligned}$$
(16)

where \(q\ge 1\), \(\lambda >0\), \(\delta _{kt}\ge 0\) and \(\sum \limits _{t=1}^{s}\delta _{kt}=1\).

Step 5::

Calculate the ranking value (RV) \(\psi (\chi _k)\) of the overall q-ROFV \(\chi _{k}=\langle \zeta _k,\theta _k \rangle\) of alternative \(O_k\) obtained in Step 4, shown as follows:

$$\begin{aligned} \psi (\chi _k)=\left( (\zeta _k)^q+\frac{(\pi _k)^q}{2}\right) \left( (\zeta _k)^q+(\theta _k)^q\right) , \end{aligned}$$
(17)

where \((\pi _k)^q=1-(\zeta _k)^q-(\theta _k)^q\) and \(k=1,2,...,r.\)

Step 6::

Arrange the RVs \(\psi (\chi _1)\), \(\psi (\chi _2)\),..., \(\psi (\chi _r)\) of alternatives \(O_1\), \(O_2\),...,\(O_r\) in decreasing sequence and obtain the RO of the alternatives \(O_1\), \(O_2\),...,\(O_r\). If the RVs \(\psi (\chi _\alpha )\) and \(\psi (\chi _\beta )\) of alternatives \(O_\alpha\) and \(O_\beta\), respectively, are equal, i.e., \(\psi (\chi _\alpha )=\psi (\chi _\beta )\), then calculate the accuracy values \(H(\chi _\alpha )=\zeta _\alpha ^q+\theta _\alpha ^q\) and \(H(\chi _\beta )=\zeta _\beta ^q+\theta _\beta ^q\) of the overall q-ROFVs \(\chi _\beta =\langle \zeta _\alpha ,\theta _\alpha \rangle\) and \(\chi _\beta =\langle \zeta _\alpha ,\theta _\alpha \rangle\) respectively. The larger the accuracy value \(H(\chi _k)\), the better the RO of alternative \(O_k\), where \(k=1,2,...,r\). If \(\psi (\chi _\alpha )=\psi (\chi _\beta )\) and \(H(\chi _\alpha )=H(\chi _\beta )\), the alternatives \(O_\alpha\) and \(O_\beta\) have the same RO.

Example 12

(Akram et al. 2021) The government wants to choose the ideal location for a thermal power station (TPS) plant to cover the requirements of electric power. Let \(O_1\), \(O_2\), \(O_3\) and \(O_4\) be the possible locations as alternatives to setup a TPS plant. Let the \(C_1\) (“Availability of coal”), \(C_2\) (“Availability of water”), \(C_3\) (“Transportation facilities”) be the three attributes for the judgement of a best TPS location. The attributes \(C_1\), \(C_2\) and \(C_3\) have the weights \(w_1=0.4\), \(w_2=0.3\) and \(w_3=0.3\), respectively. The expert evaluates the TPS locations \(O_1, O_2,\), \(O_3\) and \(O_4\) with respect to the attribute \(C_1\), \(C_2\) and \(C_3\) by using an q-ROFV \(\widetilde{\chi }_{kt}\) to construct the DMx \(\widetilde{D}=\left( {\chi }_{kt}\right) _{4\times 3}=\left( \widetilde{\zeta }_{kt},\widetilde{\theta }_{kt}\right) _{4\times 3}\), shown as follows:

We utilize the MADM approach presented in this paper to solve this MADM problem as follows:

Step 1::

Since all the attributes \(C_1\), \(C_2\) and \(C_3\) are of benefit type, therefore we get the NDMx \(D=\left( {\zeta }_{kt},{\theta }_{kt}\right) _{4\times 3}=\left( \widetilde{\zeta }_{kt},\widetilde{\theta }_{kt}\right) _{4\times 3}\).

Step 2::

By utilizing Eq. (13), we calculate the support measure \(Sup(\chi _{kt},\chi _{kl})\) between q-ROFVs \(\chi _{kt}\) and \(\chi _{kl}\), where \(q=3\), \(k=1,2,3, 4\); \(t,l=1,2,3\); \(l\ne t\), \(Sup(\chi _{11},\chi _{12})=0.9065\), \(Sup(\chi _{11},\chi _{13})=0.7870\), \(Sup(\chi _{12},\chi _{13})=0.8797\), \(Sup(\chi _{21},\chi _{22})=0.7539\), \(Sup(\chi _{21},\chi _{23})=0.8842\), \(Sup(\chi _{22},\chi _{23})=0.8408\), \(Sup(\chi _{31},\chi _{32})=0.7227\), \(Sup(\chi _{31},\chi _{33})=0.9390\), \(Sup(\chi _{32},\chi _{33})=0.6713\), \(Sup(\chi _{41},\chi _{42})=0.5217\), \(Sup(\chi _{41},\chi _{43})=0.9738\) and \(Sup(\chi _{32},\chi _{33})=0.4959\).

Step 3::

By utilizing Eq. (14), we calculate the support \(T_{kt}\) of the q-ROFVs \(\chi _{kt}\), where

$$\begin{aligned}{} & {} T_{11}= 1.6934, T_{12}= 1.7862, T_{13}= 1.6667,\\{} & {} T_{21}= 1.6381, T_{22}= 1.5948, T_{23}= 1.7250,\\{} & {} T_{31}= 1.6617, T_{32}= 1.3941, T_{33}= 1.6103, \\{} & {} T_{41}= 1.4955, T_{42}= 1.0176, T_{43}= 1.4698, \\ \end{aligned}$$

and, based on Eq. (15), we calculate the weight \(\delta _{kt}\) of the q-ROFV \(\chi _{kt}\), where

$$\begin{aligned}{} & {} \delta _{11}= 0.3971, \delta _{12}= 0.3081, \delta _{13}= 0.2949, \\{} & {} \delta _{21}= 0.3980, \delta _{22}= 0.2936, \delta _{23}= 0.3084, \\{} & {} \delta _{31}= 0.4149, \delta _{32}= 0.2799, \delta _{33}= 0.3052,\\{} & {} \delta _{41}= 0.4258, \delta _{42}= 0.2582, \delta _{43}= 0.3160.\\ \end{aligned}$$
Step 4::

By utilizing Eq. (16), we obtain the overall q-ROFV \(\chi _k\) of the alternative \(O_k\), where \(\lambda =2\), \(q=3\), \(k=1,2,3, 4\), \(\chi _{1}= \langle 0.6919,0.3510 \rangle ,\) \(\chi _{2}= \langle 0.5389, 0.5204\rangle\), \(\chi _{3}= \langle 0.5776,0.4316 \rangle\) and \(\chi _{4}= \langle 0.5525,0.4757 \rangle\).

Step 5::

By utilizing Eq. (17), we obtain the RVs \(\psi (\chi _1)\), \(\psi (\chi _2)\), \(\psi (\chi _3)\) and \(\psi (\chi _4)\) of the overall q-ROFVs \(\chi _1\), \(\chi _2\), \(\chi _3\) and \(\chi _4\) obtained in Step 4, respectively, where \(\psi (\chi _1)=0.2412\), \(\psi (\chi _2)=0.1510\), \(\psi (\chi _3)=0.1519\) and \(\psi (\chi _4)=0.1466\).

Step 6::

Because \(\psi (\chi _1)> \psi (\chi _3)> \psi (\chi _2)>\psi (\chi _4)\), where \(\psi (\chi _1)=0.2412\), \(\psi (\chi _2)=0.1510\), \(\psi (\chi _3)=0.1519\) and \(\psi (\chi _4)=0.1466\), the RO of the alternatives \(O_1\), \(O_2\), \(O_3\) and \(O_4\) is “\(O_1 \succ O_3 \succ O_2\succ O_4\)”, where \(O_1\) is the best alternative among the alternatives \(O_1\), \(O_2\), \(O_3\) and \(O_4\).

Table 1 presents the ROs of the alternatives obtained by using different MADM approaches for Example 12. It shows that Akram et al.’s MADM approach based on the q-ROFEWG AO (Akram et al. 2021), Liu and Wang’s MADM approach based on the q-ROFWG AO (Liu and Wang 2018) and the proposed GDM approach achieve the same RO “\(O_1 \succ O_3 \succ O_2\succ O_4\)” of the alternatives \(O_1\), \(O_2\), \(O_3\) and \(O_4\).

Table 1 The ROs of the alternatives achieved by different MADM approaches for Example 12
Full size table

Example 13

We utilize the MADM approach presented in this paper to solve the Example 10, shown as follows:

Step 1::

Since all the attributes \(C_1\), \(C_2\) and \(C_3\) are of benefit type, therefore we obtain the NDMx \(D=\left( {\zeta }_{kt},{\theta }_{kt}\right) _{3\times 3}=\left( \widetilde{\zeta }_{kt},\widetilde{\theta }_{kt}\right) _{3\times 3}\).

Step 2::

Based on Eq. (13), we calculate the support measure \(Sup(\chi _{kt},\chi _{kl})=S_{kt,kl}\) between q-ROFVs \(\chi _{kt}\) and \(\chi _{kl}\), where \(q=3\), \(k=1,2,3\); \(t,l=1,2,3\); \(l\ne t\), \(Sup(\chi _{11},\chi _{12})=0.8606\), \(Sup(\chi _{11},\chi _{13})=0.3120\), \(Sup(\chi _{12},\chi _{13})=0.4438\), \(Sup(\chi _{21},\chi _{22})=0.5110\), \(Sup(\chi _{21},\chi _{23})=0.7325\), \(Sup(\chi _{22},\chi _{23})=0.3120\), \(Sup(\chi _{31},\chi _{32})=0.4443\), \(Sup(\chi _{31},\chi _{33})=0.3381\) and \(Sup(\chi _{32},\chi _{33})=0.8910\).

Step 3::

By utilizing Eq. (14), we calculate the support \(T_{kt}\) of the q-ROFVs \(\chi _{kt}\), where

$$\begin{aligned}{} & {} T_{11}= 1.1726, T_{12}= 1.3044, T_{13}= 0.7557,\\{} & {} T_{21}= 1.2435, T_{22}= 0.8229, T_{23}= 1.0445,\\{} & {} T_{31}= 0.7834, T_{32}= 1.3363, T_{33}= 1.2291, \\ \end{aligned}$$

and, by utilizing Eq. (15), we calculate the weight \(\delta _{kt}\) of the q-ROFV \(\chi _{kt}\), where

$$\begin{aligned}{} & {} \delta _{11}= 0.3103, \delta _{12}= 0.4389, \delta _{13}= 0.2508, \\{} & {} \delta _{21}= 0.3339, \delta _{22}= 0.3618, \delta _{23}= 0.3043, \\{} & {} \delta _{31}= 0.2502, \delta _{32}= 0.4370, \delta _{33}= 0.3127.\\ \end{aligned}$$
Step 4::

By utilizing Eq. (16), we obtain the overall q-ROFV \(\chi _k\) of the alternative \(O_k\), where \(\lambda =2\), \(q=3\), \(k=1,2,3, 4\), \(\chi _{1}= \langle 0.3111, 0.8047 \rangle ,\) \(\chi _{2}= \langle 0.4105, 0.8588\rangle\) and \(\chi _{3}= \langle 0.2291, 0.8049 \rangle\).

Step 5::

By utilizing the Eq. (17), we obtain the RVs \(\psi (\chi _1)\), \(\psi (\chi _2)\) and \(\psi (\chi _3)\) of the overall q-ROFVs \(\chi _1\), \(\chi _2\) and \(\chi _3\) obtained in Step 4, respectively, where \(\psi (\chi _1)=0.1403\), \(\psi (\chi _2)=0.1531\) and \(\psi (\chi _3)=0.1309\).

Step 6::

Because \(\psi (\chi _2)> \psi (\chi _1)> \psi (\chi _3)\), where \(\psi (\chi _1)=0.1403\), \(\psi (\chi _2)=0.1531\) and \(\psi (\chi _3)=0.1309\), the RO of the alternatives \(O_1\), \(O_2\) and \(O_3\) is “\(O_2 \succ O_1 \succ O_3\)”, where \(O_2\) is the best alternative among the alternatives \(O_1\), \(O_2\) and \(O_3\).

Table 2 represents the ROs of the alternatives obtained by using different MADM approaches for Example 13. It shows that Akram et al.’s MADM approach based on the q-ROFEWG AO (Akram et al. 2021), Liu and Wang’s MADM approach based on the q-ROFWG AO (Liu and Wang 2018) achieve the same RO “\(O_1= O_2 = O_3\)” of the alternatives \(O_1\), \(O_2\) and \(O_3\), where they have the drawbacks that they cannot distinguish the RO of the alternatives \(O_1\), \(O_2\) and \(O_3\) in this situation; the proposed MADM approach gets the RO “\(O_2 \succ O_1 \succ O_3\)” of the alternatives \(O_1\), \(O_2\) and \(O_3\). Hence, the proposed MADM approach can conquer the shortcomings of Akram et al.’s MADM approach based on the q-ROFEWG AO (Akram et al. 2021), Liu and Wang’s MADM approach based on the q-ROFWG AO (Liu and Wang 2018) in this situation.

Table 2 The ROs of the alternatives achieved by different MADM approaches for Example 13
Full size table

Example 14

We utilize the MADM approach presented in this paper to solve the Example 11, shown as follows:

Step 1::

Since all the attributes \(C_1\), \(C_2\) and \(C_3\) are of benefit type, therefore we obtain the NDMx \(D=\left( {\zeta }_{kt},{\theta }_{kt}\right) _{3\times 3}=\left( \widetilde{\zeta }_{kt},\widetilde{\theta }_{kt}\right) _{3\times 3}\).

Step 2::

By utilizing Eq. (13), we calculate the support measure \(Sup(\chi _{kt},\chi _{kl})=S_{kt,kl}\) between q-ROFVs \(\chi _{kt}\) and \(\chi _{kl}\), where \(q=3\), \(k=1,2,3\); \(t,l=1,2,3\); \(l\ne t\), \(Sup(\chi _{11},\chi _{12})=0.3796\), \(Sup(\chi _{11},\chi _{13})=0.5904\), \(Sup(\chi _{12},\chi _{13})=0.4682\), \(Sup(\chi _{21},\chi _{22})=0.7617\), \(Sup(\chi _{21},\chi _{23})=0.2951\), \(Sup(\chi _{22},\chi _{23})=0.3120\), \(Sup(\chi _{31},\chi _{32})=0.3120\), \(Sup(\chi _{31},\chi _{33})=0.7325\) and \(Sup(\chi _{32},\chi _{33})=0.5110\).

Step 3::

By utilizing Eq. (14), we calculate the support \(T_{kt}\) of the q-ROFVs \(\chi _{kt}\), where

$$\begin{aligned}{} & {} T_{11}= 0.9700, T_{12}= 0.8479, T_{13}= 1.0586,\\{} & {} T_{21}= 1.0568, T_{22}= 1.0736, T_{23}= 0.6071,\\{} & {} T_{31}= 1.0445, T_{32}= 0.8229, T_{33}= 1.2435, \\ \end{aligned}$$

and, by utilizing Eq. (15), we calculate the weight \(\delta _{kt}\) of the q-ROFV \(\chi _{kt}\), where

$$\begin{aligned}{} & {} \delta _{11}= 0.2814, \delta _{12}= 0.3300, \delta _{13}= 0.3886, \\{} & {} \delta _{21}= 0.3037, \delta _{22}= 0.3827, \delta _{23}= 0.3136, \\{} & {} \delta _{31}= 0.2805, \delta _{32}= 0.3127, \delta _{33}= 0.4068.\\ \end{aligned}$$
Step 4::

By utilizing Eq. (16), we obtain the overall q-ROFV \(\chi _k\) of the alternative \(O_k\), where \(\lambda =2\), \(q=3\), \(k=1,2,3, 4\), \(\chi _{1}= \langle 0.3111, 0.8047 \rangle ,\) \(\chi _{2}= \langle 0.4105, 0.8588\rangle\) and \(\chi _{3}= \langle 0.2291, 0.8049 \rangle\).

Step 5::

By utilizing Eq. (17), we obtain the RVs \(\psi (\chi _1)\), \(\psi (\chi _2)\) and \(\psi (\chi _3)\) of the overall q-ROFVs \(\chi _1\), \(\chi _2\) and \(\chi _3\) obtained in Step 4, respectively, where \(\psi (\chi _1)=0.2147\), \(\psi (\chi _2)=0.1735\) and \(\psi (\chi _3)=0.1656\).

Step 6::

Because \(\psi (\chi _1)> \psi (\chi _2)> \psi (\chi _3)\), where \(\psi (\chi _1)=0.2147\), \(\psi (\chi _2)=0.1735\) and \(\psi (\chi _3)=0.1656\), the RO of the alternatives \(O_1\), \(O_2\) and \(O_3\) is “\(O_1 \succ O_2 \succ O_3\)” where \(O_1\) is the best alternative among the alternatives \(O_1\), \(O_2\) and \(O_3\).

Table 3 represents the ROs of the alternatives obtained by using different MADM approaches for Example 14. It shows that Akram et al.’s MADM approach MADM approach based on the q-ROFEWG AO (Akram et al. 2021), Liu and Wang’s MADM approach based on the q-ROFWG AO (Liu and Wang 2018) achieve the same RO “\(O_1= O_2 = O_3\)” of the alternatives \(O_1\), \(O_2\) and \(O_3\), where they have the drawbacks that they cannot distinguish the RO of the alternatives \(O_1\), \(O_2\) and \(O_3\) in this situation; the proposed MADM approach gets the RO “\(O_1 \succ O_2 \succ O_3\)” of the alternatives \(O_1\), \(O_2\) and \(O_3\). Hence, the proposed MADM approach can conquer the shortcomings of Akram et al.’s MADM approach based on the q-ROFEWG AO (Akram et al. 2021), Liu and Wang’s MADM approach based on the q-ROFWG AO (Liu and Wang 2018) in this situation.

Table 3 The ROs of the alternatives achieved by different MADM approaches for Example 14
Full size table

7 Conclusion

In this paper, we have proposed the q-rung orthopair fuzzy Yager power weighted geometric (q-ROFYPWG) aggregation operator (AO) based on the PG operator and Yager’s norm for aggregating the q-rung orthopair fuzzy values (q-ROFVs). The proposed q-ROFYPWG AO can reduce the effect of incorrect information given by the biased experts and also allows aggregated values to support each other throughout the aggregation process. The proposed q-ROFYPWG AO can conquer the shortcomings of the q-rung orthopair fuzzy weighted geometric (q-ROFWG) AO (Liu and Wang 2018) and the q-rung orthopair fuzzy Einstein weighted geometric (q-ROFEWG) AO (Akram et al. 2021) of q-ROFVs. Moreover, by utilizing the proposed q-ROFYPWG AO, we have proposed a new multi-attribute decision making (MADM) approach for q-ROFVs environment. The proposed MADM approach can conquer the drawbacks of the Akram et al.’s MADM approach MADM approach based on the q-ROFEWG AO (Akram et al. 2021), Liu and Wang’s MADM approach based on the q-ROFWG AO (Liu and Wang 2018), where they cannot distinguish the ranking orders (ROs) of alternatives in some situations. In future, we can also define the group decision making methods based on the proposed q-ROFYPWG AO under the q-ROFVs environment.